Post on 20-Apr-2023
Review ArticleLandauer-Datta-Lundstrom Generalized TransportModel for Nanoelectronics
Yuriy Kruglyak
Department of Information Technologies Odessa State Environmental University Odessa 65016 Ukraine
Correspondence should be addressed to Yuriy Kruglyak quantumnetyandexru
Received 20 June 2014 Accepted 14 August 2014 Published 17 September 2014
Academic Editor Xizhang Wang
Copyright copy 2014 Yuriy Kruglyak This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The Landauer-Datta-Lundstrom electron transport model is briefly summarized If a band structure is given the number ofconduction modes can be evaluated and if a model for a mean-free-path for backscattering can be established then the near-equilibrium thermoelectric transport coefficients can be calculated using the final expressions listed below for 1D 2D and 3Dresistors in ballistic quasiballistic and diffusive linear response regimes when there are differences in both voltage and temperatureacross the device The final expressions of thermoelectric transport coefficients through the Fermi-Dirac integrals are collected for1D 2D and 3D semiconductors with parabolic band structure and for 2D graphene linear dispersion in ballistic and diffusiveregimes with the power law scattering
1 Introduction
Theobjectives of this short review is to give a condensed sum-mary of Landauer-Datta-Lundstrom (LDL) electron trans-port model [1ndash5] which works at the nanoscale as well asat the macroscale for 1D 2D and 3D resistors in ballisticquasiballistic and diffusive linear response regimes whenthere are differences in both voltage and temperature acrossthe device
Appendices list final expressions of thermoelectric trans-port coefficients through the Fermi-Dirac integrals for 1D2D and 3D semiconductors with parabolic band structureand for 2D graphene linear dispersion in ballistic and diffu-sive regimes with the power law scattering
2 Generalized Model for Current
The generalized model for current can be written in twoequivalent forms
119868 =2119902
ℎint 120574 (119864) 120587
119863 (119864)
2(1198911minus 1198912) 119889119864 (1a)
119868 =2119902
ℎint119879 (119864)119872 (119864) (119891
1minus 1198912) 119889119864 (1b)
where ldquobroadeningrdquo 120574(119864) relates to transit time for electronsto cross the resistor channel
120574 (119864) equivℏ
120591 (119864) (2)
density of states 119863(119864) with the spin degeneracy factor 119892119904=
2 is included 119872(119864) is the integer number of modes ofconductivity at energy 119864 the transmission
119879 (119864) =120582 (119864)
120582 (119864) + 119871 (3)
where 120582(119864) is the mean-free-path for backscattering and 119871 isthe length of the conductor Fermi function
119891 (119864) =1
119890(119864minus119864119865)119896119879 + 1(4)
is indexed with the resistor contact numbers 1 and 2 119864119865is the
Fermi energy which as well as temperature119879may be differentat both contacts
Hindawi Publishing CorporationJournal of NanoscienceVolume 2014 Article ID 725420 15 pageshttpdxdoiorg1011552014725420
2 Journal of Nanoscience
Equation (3) can be derived with relatively few assump-tions and it is valid not only in the ballistic and diffusionlimits but in between as well
Diffusive 119871 ≫ 120582 119879 =120582
119871≪ 1
Ballistic 119871 ≪ 120582 119879 997888rarr 1
Quasi-ballistic 119871 asymp 120582 119879 lt 1
(5)
The LDL transport model can be used to describe all threeregimes
It is now clearly established that the resistance of a ballisticconductor can be written in the form
119877ball
=ℎ
1199022
1
119872 (119864) (6)
where ℎ1199022 is fundamental Klitzing constant and number
of modes 119872(119864) represents the number of effective parallelchannels available for conduction
This result is now fairly well known but the commonbelief is that it applies only to short resistors and belongsto a course on special topics like mesoscopic physics ornanoelectronicsWhat is not well known is that the resistancefor both long and short conductors can be written in the form
119877 (119864) =ℎ
1199022
1
119872 (119864)(1 +
119871
120582 (119864)) (7)
Ballistic and diffusive conductors are not two differentworlds but rather a continuum as the length 119871 is increasingBallistic limit is obvious for 119871 ≪ 120582 while for 119871 ≫ 120582 it reducesinto standard Ohmrsquos law
119877 equiv119881
119868= 120588
119871
119860 (8)
Indeed we could rewrite 119877(119864) above as
119877 (119864) =120588 (119864)
119860[119871 + 120582 (119864)] (9)
with a new expression for specific resistivity
120588 (119864) =ℎ
1199022(
1
119872 (119864)119860)
1
120582 (119864) (10)
which provides a different view of resistivity in terms of thenumber of modes per unit area and the mean-free-path
Number of modes
119872(119864) = 1198721119863
(119864) =ℎ
4⟨V+119909(119864)⟩119863
1119863(119864) (11a)
119872(119864) = 1198821198722119863
(119864) = 119882ℎ
4⟨V+119909(119864)⟩119863
2119863(119864) (11b)
119872(119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (11c)
is proportional to the width119882 of the resistor in 2D and to thecross-sectional area119860 in 3D ⟨V+
119909(119864)⟩ is the average velocity in
the +119909 direction from contact 1 to contact 2
For parabolic energy bands
119864 (119896) = 119864119862+ℏ21198962
2119898lowast(12)
the 1D 2D and 3D densities of states are given by
119863 (119864) = 1198631119863
(119864) 119871 =119871
120587ℏradic
2119898lowast
(119864 minus 119864119862)119867 (119864 minus 119864
119862) (13a)
119863 (119864) = 1198632119863
(119864)119860 = 119860119898lowast
120587ℏ2119867(119864 minus 119864
119862) (13b)
119863 (119864) = 1198633119863
(119864)Ω = Ω
119898lowastradic2119898lowast (119864 minus 119864
119862)
1205872ℏ3119867(119864 minus 119864
119862)
(13c)
where 119860 is the area of the 2D resistor Ω is the volume of the3D resistor and 119867(119864 minus 119864
119862) is the Heaviside step function
Then number of modes is
119872(119864) = 1198721119863
(119864) = 119867 (119864 minus 119864119862) (14a)
119872(119864) = 1198821198722119863
(119864) = 119882119892V
radic2119898lowast (119864 minus 119864119862)
120587ℏ119867 (119864 minus 119864
119862)
(14b)
119872(119864) = 1198601198723119863
(119864) = 119860119892V119898lowast(119864 minus 119864
119862)
2120587ℏ2119867(119864 minus 119864
119862) (14c)
where 119892V is the valley degeneracyFigure 1 shows qualitative behavior of the density of states
and number of modes for resistors with parabolic bandstructure
For linear dispersion in graphene
119864 (119896) = plusmnℏV119865119896 (15)
where +sign corresponds to conductivity band with 119864119865gt 0
(119899-type graphene) and minussign corresponds to valence bandwith 119864
119865lt 0 (119901-type graphene)
V (119896) =1
ℏ
120597119864
120597119896equiv V119865asymp 1 times 10
8 cms (16)
Density of states in graphene is
119863 (119864) =2 |119864|
120587ℏ2V2119865
(17)
and number of modes is
119872(119864) = 1198822 |119864|
120587ℏV119865
(18)
Two equivalent expressions for specific conductivitydeserve attention one as a product of119863(119864) and the diffusioncoefficient119863(119864)
120590 (119864) = 1199022119863 (119864)
119863 (119864)
1198711
1
1198821
119860 (19a)
Journal of Nanoscience 3
D1D
D2D
D3D
M1D
M2D
M3D
EC
EEC
E
EC
E
EC
E
EC
EC
E
E
1
Figure 1 Comparison of the density of states 119863(119864) and number ofmodes119872(119864) for 1D 2D and 3D resistors with parabolic dispersion
where
119863 (119864) = ⟨]2119909120591⟩ = ]2120591 (119864) 1
1
21
3 (lowast)
with 120591(119864) being the mean free time after which an electrongets scattered and the other as a product of119872(119864) and 120582(119864)
120590 (119864) =1199022
ℎ119872 (119864) 120582 (119864) 1
1
1198821
119860 (19b)
where the three items in parenthesis correspond to 1D 2Dand 3D resistors
Although (19b) is not well known the equivalent versionin (19a) is a standard result that is derived in textbooks Both(19a) and (19b) are far more generally applicable comparedwith traditional Drude model For example these equationsgive sensible answers even for materials like graphene whosenonparabolic bands make the meaning of electron masssomewhat unclear causing considerable confusion whenusing Drude model In general we must really use (19a) and(19b) and not Drude model to shape our thinking aboutconductivity
These conceptual equations are generally applicable evento amorphous materials and molecular resistors Irrespectiveof the specific119864(119901) relation at any energy the density of states119863(119864) velocity ](119864) and momentum 119901(119864) are related to thetotal number of states 119873(119864) with energy less than 119864 by thefundamental relation
119863 (119864) ] (119864) 119901 (119864) = 119873 (119864) sdot 119889 (lowastlowast)
where 119889 is the number of dimensions Being combinedwith (19a) it gives one more fundamental equation forconductivity
120590 (119864) =1199022120591 (119864)
119898 (119864)119873 (119864)
119871119873 (119864)
119871119882119873 (119864)
119871119860 (19c)
where electron mass is defined as
119898(119864) =119901 (119864)
V (119864) (20)
For parabolic 119864(119901) relations the mass is independent ofenergy but in general it could be energy-dependent as forexample in graphene the effective mass
119898lowast=119864119865
V2119865
(21)
21 Linear Response Regime Near-equilibrium transport orlow field linear response regime corresponds to lim(119889119868
119889119881)119881rarr0
There are several reasons to develop low fieldtransport model First near-equilibrium transport is thefoundation for understanding transport in general Conceptsintroduced in the study of near-equilibrium regime areoften extended to treat more complicated situations andnear-equilibrium regime provides a reference point whenwe analyze transport in more complex conditions Secondnear-equilibrium transport measurements are widely usedto characterize electronic materials and to understand theproperties of new materials And finally near-equilibriumtransport strongly influences and controls the performanceof most electronic devices
Under the low field condition let
1198910(119864) asymp 119891
1(119864) gt 119891
2(119864) (22)
where1198910(119864) is the equilibriumFermi function and an applied
bias
119881 =Δ119864119865
119902=(1198641198651minus 1198641198652)
119902(23)
is small enough Using Taylor expansion under constanttemperature condition
1198912= 1198911+1205971198911
120597119864119865
Δ119864119865= 1198911+1205971198911
120597119864119865
119902119881 (24)
and property of the Fermi function
+120597119891
120597119864119865
= minus120597119891
120597119864 (25)
one finds
1198911minus 1198912= (minus
1205971198910
120597119864) 119902119881 (26)
The derivative of the Fermi function multiplied by 119896119879 tomake it dimensionless
119865119879(119864 119864119865) = 119896119879(minus
120597119891
120597119864) (27)
is known as thermal broadening function and shown inFigure 2
4 Journal of Nanoscience
1010
1
5
00
0 0minus10minus10
minus5
05 0201 03
rarr f(E) rarr kT (minus 120597f
120597E)
E minus EF
kT
uarr
Figure 2 Fermi function and the dimensionless normalized ther-mal broadening function
If one integrates 119865119879over all energy range the total area
int
+infin
minusinfin
119889119864119865119879(119864 119864119865) = 119896119879 (28)
so that we can approximately visualize 119865119879as a rectangular
pulse centered around 119864 = 119864119865with a peak value of 14 and a
width of sim4119896119879The derivative (minus120597119891
0120597119864) is known as the Fermi conduc-
tion window functionWhether a conductor is good or bad isdetermined by the availability of the conductor energy statesin an energy window simplusmn2119896119879 around the electrochemicalpotential 119864
1198650 which can vary widely from one material
to another Current is driven by the difference 1198911minus 1198912
in the ldquoagendardquo of the two contacts which for low biasis proportional to the derivative of the equilibrium Fermifunction (26) With this near-equilibrium assumption forcurrent (1b) we have
119868 = [21199022
ℎint119879 (119864)119872 (119864) (minus
1205971198910
120597119864)119889119864]119881 = 119866119881 (29)
with conductivity
119866 =21199022
ℎint119879 (119864)119872 (119864) (minus
1205971198910
120597119864)119889119864 (30)
known as the Landauer expression which is valid in 1D 2Dand 3D resistors if we use the appropriate expressions for119872(119864)
For ballistic limit 119879(119864) = 1 For diffusive transport 119879(119864)is given by (3) For a conductor much longer than a mean-free-path the current density equation for diffusive transportis
119869119909=120590
119902
119889 (119864119865)
119889119909 (31)
where the electrochemical potential 119864119865is also known as the
quasi-Fermi levelFor a 2D conductor the surface specific conductivity is
120590119878=21199022
ℎint1198722119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (32)
or in a different form
120590119878= int120590
1015840
119878(119864) 119889119864 (33a)
where differential specific conductivity
1205901015840
119878(119864) =
21199022
ℎ1198722119863
(119864) 120582 (119864) (minus1205971198910
120597119864) (33b)
Similar expressions can be written for 1D and 3D resistorsAnother way to write the conductance is the product of
the quantum of conductance times the average transmissiontimes the number of modes in the Fermi windows
119866 =21199022
ℎ⟨⟨119879⟩⟩ ⟨119872⟩ (34a)
⟨119872⟩ = int119872(119864) (minus1205971198910
120597119864)119889119864 (34b)
⟨⟨119879⟩⟩ =
int119879 (119864)119872 (119864) (minus1205971198910120597119864) 119889119864
int119872(119864) (minus1205971198910120597119864) 119889119864
=⟨119872119879⟩
⟨119872⟩ (34c)
Yet another way to write the conductance is in terms ofthe differential conductance 1198661015840(119864) as
119866 = int1198661015840(119864) 119889119864 [S] (35a)
1198661015840(119864) =
21199022
ℎ119872 (119864) 119879 (119864) (minus
1205971198910
120597119864) (35b)
22 Thermocurrent and Thermoelectric Coefficients Elec-trons carry both charge and heat The charge current is givenby (1a) and (1b) To get the equation for the heat currentone notes that electrons in the contacts flow at an energy119864 asymp 119864
119865 To enter a mode119872(119864) in the resistor electrons must
absorb (if 119864 gt 119864119865) or emit (if 119864 lt 119864
119865) a thermal energy
119864 minus 119864119865 We conclude that to get the heat current equation
we should insert (119864 minus 119864119865)119902 inside the integral The resulting
thermocurrent
119868119876=2
ℎint (119864 minus 119864
119865) 119879 (119864)119872 (119864) (119891
1minus 1198912) 119889119864 (36)
It is important from practical point of view that bothexpressionsmdashfor the electric current (1a) and (1b) and ther-mocurrent (36)mdashare suitable for analysis of conductivity ofany materials from metals to semiconductors up to modernnanocomposites
When there are differences in both voltage and temper-ature across the resistor then we must the Fermi difference(1198911minus 1198912) expands to Taylor series in both voltage and
temperature and get
1198911minus 1198912asymp (minus
1205971198910
120597119864) 119902Δ119881 minus (minus
1205971198910
120597119864)119864 minus 119864
119865
119879Δ119879 (37)
where Δ119881 = 1198812minus 1198811 Δ119879 = 119879
2minus 1198791 and 119879 = (119879
1+ 1198792)2
Journal of Nanoscience 5
Deriving a general near-equilibrium current equation isnow straightforward The total current is the sum of thecontributions from each energy mode
119868 = int 1198681015840(119864) 119889119864 (38a)
where the differential current is
1198681015840(119864) =
2119902
ℎ119879 (119864)119872 (119864) (119891
1minus 1198912) (38b)
Using (37) we obtain
1198681015840(119864) = 119866
1015840(119864) Δ119881 + 119878
1015840
119879(119864) Δ119879 (39a)
where
1198661015840(119864) =
21199022
ℎ119879 (119864)119872 (119864) (minus
1205971198910
120597119864) (39b)
is the differential conductance and
1198781015840
119879(119864) = minus
21199022
ℎ119879 (119864)119872 (119864) (
119864 minus 119864119865
119902119879)(minus
1205971198910
120597119864)
= minus119896
119902(119864 minus 119864
119865
119896119879)1198661015840(119864)
(39c)
is the Soret coefficient for electrothermal diffusion in differ-ential formNote that 1198781015840
119879(119864) is negative formodeswith energy
above 119864119865(119899-resistors) and positive for modes with energy
below 119864119865(119901-resistors)
Now we integrate (39a) over all energy modes and find
119868 = 119866Δ119881 + 119878119879Δ119879 [A] (40a)
119868119876= minus 119879119878
119879Δ119881 minus 119870
0Δ119879 [W] (40b)
with three transport coefficients namely conductivity givenby (35a) and (35b) the Soret electrothermal diffusion coeffi-cient
119878119879= int 1198781015840
119879(119864) 119889119864 = minus
119896
119902int(
119864 minus 119864119865
119896119879)1198661015840(119864) 119889119864 [AK]
(40c)
and the electronic heat conductance under the short circuitconditions (Δ119881 = 0)
1198700= 119879(
119896
119902)
2
int(119864 minus 119864
119865
119896119879)
2
1198661015840(119864) 119889119864 [WK] (40d)
where current 119868 is defined to be positive when it flows inconductor from contact 2 to contact 1 with electrons flowingin opposite direction The heat current 119868
119876is positive when it
flows in the +119909 direction out of contact 2Equations (40a) (40b) (40c) and (40d) for long diffusive
resistors can be written in the common form used to describebulk transport as
119869119909= 120590
119889 (119864119865119902)
119889119909minus 119904119879
119889119879
119889119909 [0m2] (41a)
119869119876119909
= 119879119904119879
119889 (119864119865119902)
119889119909minus 1205810
119889119879
119889119909[Wm2] (41b)
with three specific transport coefficients
120590 = int1205901015840(119864) 119889119864
1205901015840(119864) =
21199022
ℎ1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864) [1Ω sdotm sdot J]
(41c)
119904119879= minus
119896
119902int(
119864 minus 119864119865
119896119879)1205901015840(119864) 119889119864 [0m sdot K] (41d)
1205810= (
119896
119902)
2
int(119864 minus 119864
119865
119896119879)
2
1205901015840(119864) 119889119864 [Wm sdot K]
(41e)
These equations have the same form for 1D and 2D resistorsbut the units of the various terms differ
The inverted form of (40a) (40b) (40c) and (40d) isoften preferred in practice namely
Δ119881 = 119877119868 minus 119878Δ119879 (42a)
119868119876= minus Π119868 minus 119870Δ119879 (42b)
where
119878 =119878119879
119866 (42c)
Π = 119879119878 (42d)
119870 = 1198700minus Π119878119866 (42e)
In this form of the equations the contributions from eachenergy mode are not added for example 119877 = int119877(119864)119889119864
Similarly the inverted form of the bulk transport equa-tions (41a) (41b) (41c) (41d) and (41e) becomes
119889 (119864119865119902)
119889119909= 120588119869119909+ 119878
119889119879
119889119909 (43a)
119869119876119909
= 119879119878119869119909minus 120581
119889119879
119889119909(43b)
with transport coefficients
120588 =1
120590 (43c)
119878 =119904119879
120590 (43d)
120581 = 1205810minus 1198782120590119879 (43e)
In summary when a band structure is given number ofmodes can be evaluated from (11a) (11b) and (11c) and if amodel for the mean-free-path for backscattering 120582(119864) can bechosen then the near-equilibrium transport coefficients canbe evaluated using the expressions listed above
23 Bipolar Conduction Let us consider a 3D semiconductorwith parabolic dispersion For the conduction band
1198723119863
(119864) = 119892V119898lowast
2120587ℏ2(119864 minus 119864
119862) (119864 ge 119864
119862) (44a)
6 Journal of Nanoscience
and for the valence band
119872(V)3119863
(119864) = 119892V
119898lowast
119901
2120587ℏ2(119864119881minus 119864) (119864 le 119864
119881) (44b)
The conductivity is provided with two contributions forthe conduction band
120590 =1199022
ℎint
infin
119864119862
1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (45a)
and for the valence band
120590119901=1199022
ℎint
119864119881
minusinfin
119872V3119863
(119864) 120582119901(119864) (minus
1205971198910
120597119864)119889119864 (45b)
The Seebeck coefficient for electrons in the conductionband follows from (41a) (41b) (41c) (41d) and (41e)
120590 = int
infin
119864119862
1205901015840(119864) 119889119864 (46a)
1205901015840(119864) =
21199022
ℎ1198723119863
(119864 minus 119864119862) 120582 (119864) (minus
1205971198910
120597119864) (46b)
119904119879= minus
119896
119902int
infin
119864119862
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 (46c)
119878 =119904119879
120590 (46d)
Similarly for electrons in the valence band we have
120590119901= int
119864119881
minusinfin
1205901015840
119901(119864) 119889119864 (47a)
1205901015840
119901(119864) =
21199022
ℎ119872
V3119863
(119864119881minus 119864) 120582
119901(119864) (minus
1205971198910
120597119864) (47b)
119904(V)119879
= minus119896
119902int
119864119881
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840
119901(119864) 119889119864 (47c)
119878119901=119904(V)119879
120590119901
(47d)
but the sign of 119878119901will be positive
What is going on when both the conduction and valencebands contribute to conductionThis can happen for narrowbandgap conductors or at high temperatures In such a casewe have to simply integrate over all the modes and will find
120590tot
equiv 120590 + 120590119901=1199022
ℎint
1198642
1198641
119872tot3119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (48a)
119872tot3119863
(119864) = 1198723119863
(119864) + 119872V3119863
(119864) (48b)
moreover we do not have to be worried about integratingto the top of the conduction band or from the bottom ofthe valence band because the Fermi function ensures thatthe integrand falls exponentially to zero away from the bandedge What is important is that in both cases we integratethe same expression with the appropriate 119872
3119863(119864) and 120582(119864)
over the relevant energy difference 1198642minus 1198641 Electrons carry
current in both bands Our general expression is the same forthe conduction and valence bandsThere is no need to changesigns for the valence band or to replace 119891
0(119864) with 1 minus 119891
0(119864)
To calculate the Seebeck coefficient when both bandscontribute let us be reminded that in the first direct form ofthe transport coefficients (41a) (41b) (41c) (41d) and (41e)the contributions from eachmode are added in parallel so thetotal specific Soret coefficient
119904tot119879
= minus119896
119902int
+infin
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878
119901120590119901 (49a)
then the Seebeck coefficient for bipolar conduction
119878tot
=
119878120590 + 119878119901120590119901
120590 + 120590119901
(49b)
Since the Seebeck coefficients for the conduction and valencebands have opposite signs the total Seebeck coefficientjust drops for high temperatures and the performance of athermoelectrical device falls down
In summary given a band structure dispersion thenumber of modes can be evaluated and if a model for amean-free-path for backscattering can be established thenthe near-equilibrium transport coefficients can be calculatedusing final expressions listed above
3 Heat Transfer by Phonons
Electrons transfer both charge and heat Electrons carry mostof the heat in metals In semiconductors electrons carry onlya part of the heat but most of the heat is carried by phonons
The phonon heat flux is proportional to the temperaturegradient
119869ph119876119909
= minus120581119871
119889119879
119889119909[Wm2] (50)
with coefficient 120581119871known as the specific lattice thermal
conductivity Such an exceptional thermal conductor likediamond has 120581
119871asymp 2 sdot 10
3Wm sdotK while such a poor thermalconductor like glass has 120581
119871asymp 1Wm sdot K Note that electrical
conductivities of solids vary over more than 20 orders ofmagnitude but thermal conductivities of solids vary over arange of only 3-4 orders of magnitude We will see that thesame methodology used to describe electron transport canbe also used for phonon transport We will also discuss thedifferences between electron and phonon transport For athorough introduction to phonons use classical books [6ndash9]
To describe the phonon current we need an expressionlike for the electron current (1b) written now as
119868 =2119902
ℎint119879el (119864)119872el (119864) (1198911 minus 119891
2) 119889119864 (51)
For electrons the states in the contacts were filled accord-ing to the equilibrium Fermi functions but phonons obeyBose statistics thus the phonon states in the contacts arefilled according to the equilibriumBose-Einstein distribution
1198990(ℏ120596) =
1
119890ℏ120596119896119879 minus 1 (52)
Journal of Nanoscience 7
Let temperature for the left and the right contacts be1198791and 119879
2 As for the electrons both contacts are assumed
ideal Thus the phonons that enter a contact are not able toreflect back and transmission coefficient119879ph(119864) describes thephonon transmission across the entire channel
It is easy now to rewrite (51) to the phonon heat currentElectron energy 119864 we replace by the phonon energy ℏ120596 Inthe electron current we have charge 119902moving in the channelin case of the phonon current the quantum of energy ℏ120596
is moving instead thus we replace 119902 in (51) with ℏ120596 andmove it inside the integral The coefficient 2 in (51) reflectsthe spin degeneracy of an electron In case of the phononswe remove this coefficient and instead the number of thephonon polarization states that contribute to the heat flowlets us include to the number of the phononmodes119872ph(ℏ120596)Finally the heat current due to phonons is
119876 =1
ℎint (ℏ120596) 119879ph (ℏ120596)119872ph (ℏ120596) (1198991 minus 119899
2) 119889 (ℏ120596) [W]
(53)
In the linear response regime by analogy with (26)
1198991minus 1198992asymp minus
1205971198990
120597119879Δ119879 (54)
where the derivative according to (52) is
1205971198990
120597119879=ℏ120596
119879(minus
1205971198990
120597 (ℏ120596)) (55)
with
1205971198990
120597 (ℏ120596)= (minus
1
119896119879)
119890ℏ120596119896119879
(119890ℏ120596119896119879 minus 1)2 (56)
Now (53) for small differences in temperature becomes
119876 = minus119870119871Δ119879 (57)
where the thermal conductance
119870119871=1198962119879
ℎint119879ph (ℏ120596)119872ph (ℏ120596)
times [(ℏ120596
119896119879)
2
(minus1205971198990
120597 (ℏ120596))] 119889 (ℏ120596) [WK]
(58)
Equation (57) is simply Fourierrsquos law stating that heatflows down to a temperature gradient It is also useful tonote that the thermal conductance (58) displays certainsimilarities with the electrical conductance
119866 =21199022
ℎint119879el (119864)119872el (119864) (minus
1205971198910
120597119864)119889119864 (59)
The derivative
119882el (119864) equiv (minus1205971198910
120597119864) (60)
known as the Fermi window function that picks out thoseconduction modes which only contribute to the electriccurrent The electron windows function is normalized
int
+infin
minusinfin
(minus1205971198910
120597119864)119889119864 = 1 (61)
In case of phonons the term in square brackets of (58) actsas a window function to specify which modes carry the heatcurrent After normalization
119882ph (ℏ120596) =3
1205872(ℏ120596
119896119879)(
1205971198990
120597 (ℏ120596)) (62)
thus finally
119870119871=12058721198962119879
3ℎint119879ph (ℏ120596)119872ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (63)
with
1198920equiv12058721198962119879
3ℎasymp (9456 times 10
minus13WK2) 119879 (64)
known as the quantum of thermal conductance experimen-tally observed first in 2000 [10]
Comparing (59) and (63) one can see that the electricaland thermal conductances are similar in structure bothare proportional to corresponding quantum of conductancetimes an integral over the transmission times the number ofmodes times a window function
The thermal broadening functions for electrons andphonons have similar shapes and each has a width of a few119896119879 In case of electrons this function is given by (27) or
119865119879(119909) equiv
119890119909
(119890119909 + 1)
2(65)
with 119909 equiv (119864minus119864119865)119896119879 and is shown on Figure 2This function
for phonons is given by (62) or
119865ph119879(119909) equiv
3
1205872
1199092119890119909
(119890119909 minus 1)
2(66)
with 119909 equiv ℏ120596119896119879 Both functions are normalized to a unityand shown together on Figure 3
Along with the number of modes determined by thedispersion relation these two window functions play a keyrole in determining the electrical and thermal conductances
31 Thermal Conductivity of the Bulk Conductors The ther-mal conductivity of a large diffusive resistor is a key materialproperty that controls performance of any electronic devicesBy analogy with the transmission coefficient (3) for electrontransport the phonon transmission
119879ph (ℏ120596) =120582ph (ℏ120596)
120582ph (ℏ120596) + 119871
100381610038161003816100381610038161003816100381610038161003816119871≫120582ph
997888rarr
120582ph (ℏ120596)
119871 (67)
It is also obvious that for large 3D conductors the numberof phonon modes is proportional to the cross-sectional areaof the sample
119872ph (ℏ120596) prop 119860 (68)
8 Journal of Nanoscience
10
5
0
minus5
minus100 01 02 03
FT(x)
FT(x)
FT(x)
phx
Figure 3 Broadening function for phonons compared to that ofelectrons
Now let us return to (57) dividing and multiplying it by119860119871 which immediately gives (50) for the phonon heat fluxpostulated above
119876
119860equiv 119869
ph119876119909
= minus120581119871
119889119879
119889119909(69)
with specific lattice thermal conductivity
120581119871= 119870119871
119871
119860 (70)
or substituting (67) into (63) one for the lattice thermalconductivity finally obtains
120581119871=12058721198962119879
3ℎint
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)
It is useful now to define the average number of phononmodes per cross-sectional area of the conductor that partici-pate in the heat transport
⟨
119872ph
119860⟩ equiv int
119872ph (ℏ120596)
119860119882ph (ℏ120596) 119889 (ℏ120596) (72)
Then
120581119871=12058721198962119879
3ℎ⟨
119872ph
119860⟩⟨⟨120582ph⟩⟩ (73)
where the average mean-free-path is defined now as
⟨⟨120582ph⟩⟩ =
int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)
(74)
Thus the couple of the phonon transport equations (69)and (73) corresponds to similar electron transport equations
119869119909=120590
119902
119889 (119864119865)
119889119909 (75)
120590 =21199022
ℎ⟨119872el119860
⟩⟨⟨120582el⟩⟩ (76)
The thermal conductivity (73) and the electrical conduc-tivity (76) have the same structure It is always a product of thecorresponding quantum of conductance times the numberof modes that participate in transport times the averagemean-free-path These three quantities for phonons will bediscussed later
32 Specific Heat versus Thermal Conductivity The connec-tion between the lattice specific thermal conductivity andthe lattice specific heat at constant volume is well known[6ndash9] We will show now that corresponding proportionalitycoefficient is a product of an appropriately-defined mean-free-path ⟨⟨Λ ph⟩⟩ and an average phonon velocity ⟨Vph⟩namely
120581119871=1
3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)
The total phonon energy per unit volume
119864ph = int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)
where119863ph(ℏ120596) is the phonon density of states By definition
119862119881equiv
120597119864ph
120597119879=
120597
120597119879int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596)
= int
infin
0
(ℏ120596)119863ph (ℏ120596) (1205971198990(ℏ120596)
120597119879)119889 (ℏ120596)
=12058721198962119879
3int
infin
0
119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(79)
where (55) and (62) were used Next multiply and divide (71)by (79) and obtain the proportionality we are looking for120581119871
= [
[
(1ℎ) intinfin
0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(80)To obtain final expression (77) and correct interpretation
of the proportionality coefficient we need to return to (67)This expression can be easily derived for 1D conductorwith several simplifying assumptions Nevertheless it worksvery well in practice for a conductor of any dimensionDerivation of (67) is based on the interpretation of themean-free-path 120582(119864) or 120582(ℏ120596) as that its inverse value is theprobability per unit length that a positive flux is convertedto a negative flux This is why 120582 is often called a mean-free-path for backscattering Let us relate it to the scattering time120591 The distinction between mean-free-path and mean-free-path for backscattering is easiest to see for 1D conductorLet an electron undergo a scattering event For isotropicscattering the electron can forward scatter or backscatterOnly backscattering is relevant for the mean-free-path forscattering so the time between backscattering events is 2120591Thus the mean-free-path for backscattering is twice themean-free-path for scattering
1205821119863
(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)
It was shown that the proper definition of the mean-free-path for backscattering for a conductor of any dimension [11]is
120582 (119864) = 2
⟨V2119909120591⟩
⟨1003816100381610038161003816V119909
1003816100381610038161003816⟩ (lowast lowast lowast)
Journal of Nanoscience 9
where averaging is performed over angles For isotropicbands
1205822119863
(119864) =120587
2V (119864) 120591 (119864) (81b)
1205823119863
(119864) =4
3V (119864) 120591 (119864) (81c)
The scattering time is often approximately written as thepower law scattering
120591 (119864) = 1205910(119864 minus 119864
119862
119896119879)
119904
(82)
where exponent 119904 describes the specific scattering mecha-nism for acoustic phonon scattering in 3D conductor withparabolic dispersion 119904 = minus12 and for ionized impurityscattering 119904 = +32 [12]
Analogous power law is often used for mean-free-path
120582 (119864) = 1205820(119864 minus 119864
119862
119896119879)
119903
(83)
For parabolic zone structure V(119864) prop 11986412 thus 119903 = 119904 + 12
with 119903 = 0 for acoustic phonon scattering and 119903 = 2 forionized impurity scattering
Coming back to our initial task to derive (77) from (80)for 3D conductor according to (81c) we have
120582ph (ℏ120596) =4
3Vph (ℏ120596) 120591ph (ℏ120596) (84)
where according to (81a)
Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)
and finally
120582ph (ℏ120596) =4
3Λ ph (ℏ120596) (86)
It was stated above in (11c) that the density of states andnumber of modes for electrons in 3D are
119872el (119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (87)
Let us rewrite this formula for phonons Note that the spindegeneracy for electrons 119892
119878= 2 is included with the density
of states
1198633119863
(119864) = 21198631015840
3119863(119864) (88)
and for spherical bands in 3D conductor
⟨V+119909(119864)⟩ =
Vel (119864)2
(89)
Collecting (87) up to (89) all together in case of phonons wehave
119872ph (ℏ120596) = 119860ℎ
2(
Vph (ℏ120596)2
) 2119863ph (ℏ120596)
= 119860ℎ
4Vph (ℏ120596)119863ph (ℏ120596)
(90)
Substituting (86) and (90) into (80) we obtain
120581119871
= [
[
(13) intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(91)
Multiplying and dividing (91) by
int
infin
0
Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)
we finally get (77) with proportionality coefficient between 120581119871
and 119862119881as the product of an average mean-free-path as
⟨⟨Λ ph⟩⟩
equiv
intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(93)
and an average velocity as
⟨Vph⟩ equiv
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(94)
with the appropriate averagingEquation (77) is often used to estimate the average mean-
free-path from the measured 120581119871and 119862
119881 if we know the
average velocity which is frequently assumed to be the lon-gitudinal sound velocity The derivation above has identifiedthe precise definitions of the ⟨⟨Λ ph⟩⟩ and ⟨Vph⟩ If a phonondispersion is chosen one can always compute the averagevelocity according to (94) and it is typically very differentfrom the longitudinal sound velocity Thus estimates of theaverage mean-free-path can be quite wrong if one assumesthe longitudinal sound velocity [13]
33 Debye Model For 3D conductors there are three polar-ization states for lattice vibrations one for atoms displacedin the direction of propagation (longitudinal119871) and two foratoms displaced orthogonally to the direction of propagation(transverse119879) The low energy modes are called acousticmodes119860 one 119871119860 mode analogous to sound waves propa-gating in air and two 119879119860 modes Near 119902 rarr 0 dispersion ofacoustic modes is linear
ℏ120596 = ℏV119863119902 (95)
and is known as Debye approximation The Debye velocityV119863is an average velocity of the 119871 and 119879 acoustic modes In
case of 119871119860mode V119863is simply the sound velocity V
119904prop 119898minus12
with 119898 being an effective mass of vibrating atom TypicallyV119904asymp 5 times 10
3ms about 20 times slower that the velocity of atypical electron
The bandwidth of the electronic dispersion is typicallyBW ≫ 119896119879 so only states near the bottom of the conductionband where the effective mass model is reasonably accurate
10 Journal of Nanoscience
are occupied For phonons the situation ismuch different thebandwidth BW asymp 119896119879 so states across the entire Brillouinzone are occupied The widely used Debye approximation(95) fits the acoustic branches as long as 119902 is not too far fromthe center of the Brillouin zone
With the Debye approximation (95) it is easy to find thedensity of the phonon states
119863ph (ℏ120596) =3(ℏ120596)
2
21205872(ℏV119863)3 [Jminus1 sdotmminus3] (96)
where the factor of three is for the three polarizations Thenone can obtain the number of phonon modes per cross-sectional area from (90)
119872ph (ℏ120596) =3(ℏ120596)
2
4120587(ℏV119863)2 (97)
Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures we are to be sure that we accountfor the correct number of states For a crystal there are 3119873Ω
states per unit volume To find the total number of states wehave to integrate the density of states
int
ℏ120596119863
0
119863ph (ℏ120596) 119889 (ℏ120596) (98)
with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states namely
ℏ120596119863= ℏV119863(61205872119873
Ω)
13
equiv 119896119879119863 (99)
The Debye frequency defines a cutoff frequency abovewhich no states are accounted for This restriction can alsobe expressed via a cutoff wave vector 119902
119863or as a Debye tem-
perature
119879119863=ℏ120596
119896 (100)
For 119879 ≪ 119879119863 only states with 119902 rarr 0 for which the Debye
approximation is accurate are occupiedNow we can calculate the lattice thermal conductivity by
integrating (71) to the Debye cutoff energy
120581119871=12058721198962119879
3ℎint
ℏ120596119863
0
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)
and estimate119872ph(ℏ120596) according to (71) The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used This is how the lattice thermalconductivities were first calculated [14 15] The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11 13 16]
34 Phonon Scattering Phonons can scatter from defectsimpurity atoms isotopes surfaces and boundaries and elec-trons and from other phonons Phonon-phonon scattering
occurs because the potential energy of the bonds in the crystalis not exactly harmonic All higher order terms are treatedas a scattering potential Two types of phonon scattering areconsidered In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved
ℏ 1199021+ ℏ 1199022= ℏ 1199023
ℏ1+ ℏ2= ℏ3
(102)
The total momentum of the phonon ensemble is conservedthus this type of scattering has little effect on the heat flux
In a second type of scattering umklapp119880-scatteringthe two initial phonons have larger momentum thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions The 119880-scatterings are thebasic processes in the heat transport especially at hightemperatures Scattering on defects119863 and on boundaries119861is also important Scattering rates are additive thus the totalphonon scattering rate is
1
120591ph (ℏ120596)=
1
120591119880(ℏ120596)
+1
120591119863(ℏ120596)
+1
120591119861(ℏ120596)
(103)
or alternatively in terms of the mean-free-path (84)
1
120582ph (ℏ120596)=
1
120582119880(ℏ120596)
+1
120582119863(ℏ120596)
+1
120582119861(ℏ120596)
(104)
Expressions for each of the scattering rates are developed[17] For scattering from point defects
1
120591119863(ℏ120596)
prop 1205964 (105)
known as the Rayleigh scattering which is like the scatteringof light from the dust
For boundaries and surfaces
1
120591119861(ℏ120596)
prop
Vph (ℏ120596)119871
(106)
where 119871 is the shortest dimension of the sampleA commonly used expression for 119880-scattering is
1
120591119880(ℏ120596)
prop 11987931205962119890minus119879119863119887119879 (107)
With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity
35 Lattice Thermal Conductivity versus Temperature Thetemperature dependence of the lattice thermal conductivity120581119871is illustrated for bulk Si on Figure 4According to (73) 120581
119871is proportional to the number of the
phononmodes that are occupied ⟨119872ph119860⟩ and to the averagevalue of the phononmean-free-path ⟨⟨120582ph⟩⟩The curve 120581
119871(119879)
can be explained by understanding how ⟨119872ph⟩ and ⟨⟨120582ph⟩⟩vary with temperature
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
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Nano
materials
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Journal ofNanomaterials
2 Journal of Nanoscience
Equation (3) can be derived with relatively few assump-tions and it is valid not only in the ballistic and diffusionlimits but in between as well
Diffusive 119871 ≫ 120582 119879 =120582
119871≪ 1
Ballistic 119871 ≪ 120582 119879 997888rarr 1
Quasi-ballistic 119871 asymp 120582 119879 lt 1
(5)
The LDL transport model can be used to describe all threeregimes
It is now clearly established that the resistance of a ballisticconductor can be written in the form
119877ball
=ℎ
1199022
1
119872 (119864) (6)
where ℎ1199022 is fundamental Klitzing constant and number
of modes 119872(119864) represents the number of effective parallelchannels available for conduction
This result is now fairly well known but the commonbelief is that it applies only to short resistors and belongsto a course on special topics like mesoscopic physics ornanoelectronicsWhat is not well known is that the resistancefor both long and short conductors can be written in the form
119877 (119864) =ℎ
1199022
1
119872 (119864)(1 +
119871
120582 (119864)) (7)
Ballistic and diffusive conductors are not two differentworlds but rather a continuum as the length 119871 is increasingBallistic limit is obvious for 119871 ≪ 120582 while for 119871 ≫ 120582 it reducesinto standard Ohmrsquos law
119877 equiv119881
119868= 120588
119871
119860 (8)
Indeed we could rewrite 119877(119864) above as
119877 (119864) =120588 (119864)
119860[119871 + 120582 (119864)] (9)
with a new expression for specific resistivity
120588 (119864) =ℎ
1199022(
1
119872 (119864)119860)
1
120582 (119864) (10)
which provides a different view of resistivity in terms of thenumber of modes per unit area and the mean-free-path
Number of modes
119872(119864) = 1198721119863
(119864) =ℎ
4⟨V+119909(119864)⟩119863
1119863(119864) (11a)
119872(119864) = 1198821198722119863
(119864) = 119882ℎ
4⟨V+119909(119864)⟩119863
2119863(119864) (11b)
119872(119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (11c)
is proportional to the width119882 of the resistor in 2D and to thecross-sectional area119860 in 3D ⟨V+
119909(119864)⟩ is the average velocity in
the +119909 direction from contact 1 to contact 2
For parabolic energy bands
119864 (119896) = 119864119862+ℏ21198962
2119898lowast(12)
the 1D 2D and 3D densities of states are given by
119863 (119864) = 1198631119863
(119864) 119871 =119871
120587ℏradic
2119898lowast
(119864 minus 119864119862)119867 (119864 minus 119864
119862) (13a)
119863 (119864) = 1198632119863
(119864)119860 = 119860119898lowast
120587ℏ2119867(119864 minus 119864
119862) (13b)
119863 (119864) = 1198633119863
(119864)Ω = Ω
119898lowastradic2119898lowast (119864 minus 119864
119862)
1205872ℏ3119867(119864 minus 119864
119862)
(13c)
where 119860 is the area of the 2D resistor Ω is the volume of the3D resistor and 119867(119864 minus 119864
119862) is the Heaviside step function
Then number of modes is
119872(119864) = 1198721119863
(119864) = 119867 (119864 minus 119864119862) (14a)
119872(119864) = 1198821198722119863
(119864) = 119882119892V
radic2119898lowast (119864 minus 119864119862)
120587ℏ119867 (119864 minus 119864
119862)
(14b)
119872(119864) = 1198601198723119863
(119864) = 119860119892V119898lowast(119864 minus 119864
119862)
2120587ℏ2119867(119864 minus 119864
119862) (14c)
where 119892V is the valley degeneracyFigure 1 shows qualitative behavior of the density of states
and number of modes for resistors with parabolic bandstructure
For linear dispersion in graphene
119864 (119896) = plusmnℏV119865119896 (15)
where +sign corresponds to conductivity band with 119864119865gt 0
(119899-type graphene) and minussign corresponds to valence bandwith 119864
119865lt 0 (119901-type graphene)
V (119896) =1
ℏ
120597119864
120597119896equiv V119865asymp 1 times 10
8 cms (16)
Density of states in graphene is
119863 (119864) =2 |119864|
120587ℏ2V2119865
(17)
and number of modes is
119872(119864) = 1198822 |119864|
120587ℏV119865
(18)
Two equivalent expressions for specific conductivitydeserve attention one as a product of119863(119864) and the diffusioncoefficient119863(119864)
120590 (119864) = 1199022119863 (119864)
119863 (119864)
1198711
1
1198821
119860 (19a)
Journal of Nanoscience 3
D1D
D2D
D3D
M1D
M2D
M3D
EC
EEC
E
EC
E
EC
E
EC
EC
E
E
1
Figure 1 Comparison of the density of states 119863(119864) and number ofmodes119872(119864) for 1D 2D and 3D resistors with parabolic dispersion
where
119863 (119864) = ⟨]2119909120591⟩ = ]2120591 (119864) 1
1
21
3 (lowast)
with 120591(119864) being the mean free time after which an electrongets scattered and the other as a product of119872(119864) and 120582(119864)
120590 (119864) =1199022
ℎ119872 (119864) 120582 (119864) 1
1
1198821
119860 (19b)
where the three items in parenthesis correspond to 1D 2Dand 3D resistors
Although (19b) is not well known the equivalent versionin (19a) is a standard result that is derived in textbooks Both(19a) and (19b) are far more generally applicable comparedwith traditional Drude model For example these equationsgive sensible answers even for materials like graphene whosenonparabolic bands make the meaning of electron masssomewhat unclear causing considerable confusion whenusing Drude model In general we must really use (19a) and(19b) and not Drude model to shape our thinking aboutconductivity
These conceptual equations are generally applicable evento amorphous materials and molecular resistors Irrespectiveof the specific119864(119901) relation at any energy the density of states119863(119864) velocity ](119864) and momentum 119901(119864) are related to thetotal number of states 119873(119864) with energy less than 119864 by thefundamental relation
119863 (119864) ] (119864) 119901 (119864) = 119873 (119864) sdot 119889 (lowastlowast)
where 119889 is the number of dimensions Being combinedwith (19a) it gives one more fundamental equation forconductivity
120590 (119864) =1199022120591 (119864)
119898 (119864)119873 (119864)
119871119873 (119864)
119871119882119873 (119864)
119871119860 (19c)
where electron mass is defined as
119898(119864) =119901 (119864)
V (119864) (20)
For parabolic 119864(119901) relations the mass is independent ofenergy but in general it could be energy-dependent as forexample in graphene the effective mass
119898lowast=119864119865
V2119865
(21)
21 Linear Response Regime Near-equilibrium transport orlow field linear response regime corresponds to lim(119889119868
119889119881)119881rarr0
There are several reasons to develop low fieldtransport model First near-equilibrium transport is thefoundation for understanding transport in general Conceptsintroduced in the study of near-equilibrium regime areoften extended to treat more complicated situations andnear-equilibrium regime provides a reference point whenwe analyze transport in more complex conditions Secondnear-equilibrium transport measurements are widely usedto characterize electronic materials and to understand theproperties of new materials And finally near-equilibriumtransport strongly influences and controls the performanceof most electronic devices
Under the low field condition let
1198910(119864) asymp 119891
1(119864) gt 119891
2(119864) (22)
where1198910(119864) is the equilibriumFermi function and an applied
bias
119881 =Δ119864119865
119902=(1198641198651minus 1198641198652)
119902(23)
is small enough Using Taylor expansion under constanttemperature condition
1198912= 1198911+1205971198911
120597119864119865
Δ119864119865= 1198911+1205971198911
120597119864119865
119902119881 (24)
and property of the Fermi function
+120597119891
120597119864119865
= minus120597119891
120597119864 (25)
one finds
1198911minus 1198912= (minus
1205971198910
120597119864) 119902119881 (26)
The derivative of the Fermi function multiplied by 119896119879 tomake it dimensionless
119865119879(119864 119864119865) = 119896119879(minus
120597119891
120597119864) (27)
is known as thermal broadening function and shown inFigure 2
4 Journal of Nanoscience
1010
1
5
00
0 0minus10minus10
minus5
05 0201 03
rarr f(E) rarr kT (minus 120597f
120597E)
E minus EF
kT
uarr
Figure 2 Fermi function and the dimensionless normalized ther-mal broadening function
If one integrates 119865119879over all energy range the total area
int
+infin
minusinfin
119889119864119865119879(119864 119864119865) = 119896119879 (28)
so that we can approximately visualize 119865119879as a rectangular
pulse centered around 119864 = 119864119865with a peak value of 14 and a
width of sim4119896119879The derivative (minus120597119891
0120597119864) is known as the Fermi conduc-
tion window functionWhether a conductor is good or bad isdetermined by the availability of the conductor energy statesin an energy window simplusmn2119896119879 around the electrochemicalpotential 119864
1198650 which can vary widely from one material
to another Current is driven by the difference 1198911minus 1198912
in the ldquoagendardquo of the two contacts which for low biasis proportional to the derivative of the equilibrium Fermifunction (26) With this near-equilibrium assumption forcurrent (1b) we have
119868 = [21199022
ℎint119879 (119864)119872 (119864) (minus
1205971198910
120597119864)119889119864]119881 = 119866119881 (29)
with conductivity
119866 =21199022
ℎint119879 (119864)119872 (119864) (minus
1205971198910
120597119864)119889119864 (30)
known as the Landauer expression which is valid in 1D 2Dand 3D resistors if we use the appropriate expressions for119872(119864)
For ballistic limit 119879(119864) = 1 For diffusive transport 119879(119864)is given by (3) For a conductor much longer than a mean-free-path the current density equation for diffusive transportis
119869119909=120590
119902
119889 (119864119865)
119889119909 (31)
where the electrochemical potential 119864119865is also known as the
quasi-Fermi levelFor a 2D conductor the surface specific conductivity is
120590119878=21199022
ℎint1198722119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (32)
or in a different form
120590119878= int120590
1015840
119878(119864) 119889119864 (33a)
where differential specific conductivity
1205901015840
119878(119864) =
21199022
ℎ1198722119863
(119864) 120582 (119864) (minus1205971198910
120597119864) (33b)
Similar expressions can be written for 1D and 3D resistorsAnother way to write the conductance is the product of
the quantum of conductance times the average transmissiontimes the number of modes in the Fermi windows
119866 =21199022
ℎ⟨⟨119879⟩⟩ ⟨119872⟩ (34a)
⟨119872⟩ = int119872(119864) (minus1205971198910
120597119864)119889119864 (34b)
⟨⟨119879⟩⟩ =
int119879 (119864)119872 (119864) (minus1205971198910120597119864) 119889119864
int119872(119864) (minus1205971198910120597119864) 119889119864
=⟨119872119879⟩
⟨119872⟩ (34c)
Yet another way to write the conductance is in terms ofthe differential conductance 1198661015840(119864) as
119866 = int1198661015840(119864) 119889119864 [S] (35a)
1198661015840(119864) =
21199022
ℎ119872 (119864) 119879 (119864) (minus
1205971198910
120597119864) (35b)
22 Thermocurrent and Thermoelectric Coefficients Elec-trons carry both charge and heat The charge current is givenby (1a) and (1b) To get the equation for the heat currentone notes that electrons in the contacts flow at an energy119864 asymp 119864
119865 To enter a mode119872(119864) in the resistor electrons must
absorb (if 119864 gt 119864119865) or emit (if 119864 lt 119864
119865) a thermal energy
119864 minus 119864119865 We conclude that to get the heat current equation
we should insert (119864 minus 119864119865)119902 inside the integral The resulting
thermocurrent
119868119876=2
ℎint (119864 minus 119864
119865) 119879 (119864)119872 (119864) (119891
1minus 1198912) 119889119864 (36)
It is important from practical point of view that bothexpressionsmdashfor the electric current (1a) and (1b) and ther-mocurrent (36)mdashare suitable for analysis of conductivity ofany materials from metals to semiconductors up to modernnanocomposites
When there are differences in both voltage and temper-ature across the resistor then we must the Fermi difference(1198911minus 1198912) expands to Taylor series in both voltage and
temperature and get
1198911minus 1198912asymp (minus
1205971198910
120597119864) 119902Δ119881 minus (minus
1205971198910
120597119864)119864 minus 119864
119865
119879Δ119879 (37)
where Δ119881 = 1198812minus 1198811 Δ119879 = 119879
2minus 1198791 and 119879 = (119879
1+ 1198792)2
Journal of Nanoscience 5
Deriving a general near-equilibrium current equation isnow straightforward The total current is the sum of thecontributions from each energy mode
119868 = int 1198681015840(119864) 119889119864 (38a)
where the differential current is
1198681015840(119864) =
2119902
ℎ119879 (119864)119872 (119864) (119891
1minus 1198912) (38b)
Using (37) we obtain
1198681015840(119864) = 119866
1015840(119864) Δ119881 + 119878
1015840
119879(119864) Δ119879 (39a)
where
1198661015840(119864) =
21199022
ℎ119879 (119864)119872 (119864) (minus
1205971198910
120597119864) (39b)
is the differential conductance and
1198781015840
119879(119864) = minus
21199022
ℎ119879 (119864)119872 (119864) (
119864 minus 119864119865
119902119879)(minus
1205971198910
120597119864)
= minus119896
119902(119864 minus 119864
119865
119896119879)1198661015840(119864)
(39c)
is the Soret coefficient for electrothermal diffusion in differ-ential formNote that 1198781015840
119879(119864) is negative formodeswith energy
above 119864119865(119899-resistors) and positive for modes with energy
below 119864119865(119901-resistors)
Now we integrate (39a) over all energy modes and find
119868 = 119866Δ119881 + 119878119879Δ119879 [A] (40a)
119868119876= minus 119879119878
119879Δ119881 minus 119870
0Δ119879 [W] (40b)
with three transport coefficients namely conductivity givenby (35a) and (35b) the Soret electrothermal diffusion coeffi-cient
119878119879= int 1198781015840
119879(119864) 119889119864 = minus
119896
119902int(
119864 minus 119864119865
119896119879)1198661015840(119864) 119889119864 [AK]
(40c)
and the electronic heat conductance under the short circuitconditions (Δ119881 = 0)
1198700= 119879(
119896
119902)
2
int(119864 minus 119864
119865
119896119879)
2
1198661015840(119864) 119889119864 [WK] (40d)
where current 119868 is defined to be positive when it flows inconductor from contact 2 to contact 1 with electrons flowingin opposite direction The heat current 119868
119876is positive when it
flows in the +119909 direction out of contact 2Equations (40a) (40b) (40c) and (40d) for long diffusive
resistors can be written in the common form used to describebulk transport as
119869119909= 120590
119889 (119864119865119902)
119889119909minus 119904119879
119889119879
119889119909 [0m2] (41a)
119869119876119909
= 119879119904119879
119889 (119864119865119902)
119889119909minus 1205810
119889119879
119889119909[Wm2] (41b)
with three specific transport coefficients
120590 = int1205901015840(119864) 119889119864
1205901015840(119864) =
21199022
ℎ1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864) [1Ω sdotm sdot J]
(41c)
119904119879= minus
119896
119902int(
119864 minus 119864119865
119896119879)1205901015840(119864) 119889119864 [0m sdot K] (41d)
1205810= (
119896
119902)
2
int(119864 minus 119864
119865
119896119879)
2
1205901015840(119864) 119889119864 [Wm sdot K]
(41e)
These equations have the same form for 1D and 2D resistorsbut the units of the various terms differ
The inverted form of (40a) (40b) (40c) and (40d) isoften preferred in practice namely
Δ119881 = 119877119868 minus 119878Δ119879 (42a)
119868119876= minus Π119868 minus 119870Δ119879 (42b)
where
119878 =119878119879
119866 (42c)
Π = 119879119878 (42d)
119870 = 1198700minus Π119878119866 (42e)
In this form of the equations the contributions from eachenergy mode are not added for example 119877 = int119877(119864)119889119864
Similarly the inverted form of the bulk transport equa-tions (41a) (41b) (41c) (41d) and (41e) becomes
119889 (119864119865119902)
119889119909= 120588119869119909+ 119878
119889119879
119889119909 (43a)
119869119876119909
= 119879119878119869119909minus 120581
119889119879
119889119909(43b)
with transport coefficients
120588 =1
120590 (43c)
119878 =119904119879
120590 (43d)
120581 = 1205810minus 1198782120590119879 (43e)
In summary when a band structure is given number ofmodes can be evaluated from (11a) (11b) and (11c) and if amodel for the mean-free-path for backscattering 120582(119864) can bechosen then the near-equilibrium transport coefficients canbe evaluated using the expressions listed above
23 Bipolar Conduction Let us consider a 3D semiconductorwith parabolic dispersion For the conduction band
1198723119863
(119864) = 119892V119898lowast
2120587ℏ2(119864 minus 119864
119862) (119864 ge 119864
119862) (44a)
6 Journal of Nanoscience
and for the valence band
119872(V)3119863
(119864) = 119892V
119898lowast
119901
2120587ℏ2(119864119881minus 119864) (119864 le 119864
119881) (44b)
The conductivity is provided with two contributions forthe conduction band
120590 =1199022
ℎint
infin
119864119862
1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (45a)
and for the valence band
120590119901=1199022
ℎint
119864119881
minusinfin
119872V3119863
(119864) 120582119901(119864) (minus
1205971198910
120597119864)119889119864 (45b)
The Seebeck coefficient for electrons in the conductionband follows from (41a) (41b) (41c) (41d) and (41e)
120590 = int
infin
119864119862
1205901015840(119864) 119889119864 (46a)
1205901015840(119864) =
21199022
ℎ1198723119863
(119864 minus 119864119862) 120582 (119864) (minus
1205971198910
120597119864) (46b)
119904119879= minus
119896
119902int
infin
119864119862
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 (46c)
119878 =119904119879
120590 (46d)
Similarly for electrons in the valence band we have
120590119901= int
119864119881
minusinfin
1205901015840
119901(119864) 119889119864 (47a)
1205901015840
119901(119864) =
21199022
ℎ119872
V3119863
(119864119881minus 119864) 120582
119901(119864) (minus
1205971198910
120597119864) (47b)
119904(V)119879
= minus119896
119902int
119864119881
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840
119901(119864) 119889119864 (47c)
119878119901=119904(V)119879
120590119901
(47d)
but the sign of 119878119901will be positive
What is going on when both the conduction and valencebands contribute to conductionThis can happen for narrowbandgap conductors or at high temperatures In such a casewe have to simply integrate over all the modes and will find
120590tot
equiv 120590 + 120590119901=1199022
ℎint
1198642
1198641
119872tot3119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (48a)
119872tot3119863
(119864) = 1198723119863
(119864) + 119872V3119863
(119864) (48b)
moreover we do not have to be worried about integratingto the top of the conduction band or from the bottom ofthe valence band because the Fermi function ensures thatthe integrand falls exponentially to zero away from the bandedge What is important is that in both cases we integratethe same expression with the appropriate 119872
3119863(119864) and 120582(119864)
over the relevant energy difference 1198642minus 1198641 Electrons carry
current in both bands Our general expression is the same forthe conduction and valence bandsThere is no need to changesigns for the valence band or to replace 119891
0(119864) with 1 minus 119891
0(119864)
To calculate the Seebeck coefficient when both bandscontribute let us be reminded that in the first direct form ofthe transport coefficients (41a) (41b) (41c) (41d) and (41e)the contributions from eachmode are added in parallel so thetotal specific Soret coefficient
119904tot119879
= minus119896
119902int
+infin
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878
119901120590119901 (49a)
then the Seebeck coefficient for bipolar conduction
119878tot
=
119878120590 + 119878119901120590119901
120590 + 120590119901
(49b)
Since the Seebeck coefficients for the conduction and valencebands have opposite signs the total Seebeck coefficientjust drops for high temperatures and the performance of athermoelectrical device falls down
In summary given a band structure dispersion thenumber of modes can be evaluated and if a model for amean-free-path for backscattering can be established thenthe near-equilibrium transport coefficients can be calculatedusing final expressions listed above
3 Heat Transfer by Phonons
Electrons transfer both charge and heat Electrons carry mostof the heat in metals In semiconductors electrons carry onlya part of the heat but most of the heat is carried by phonons
The phonon heat flux is proportional to the temperaturegradient
119869ph119876119909
= minus120581119871
119889119879
119889119909[Wm2] (50)
with coefficient 120581119871known as the specific lattice thermal
conductivity Such an exceptional thermal conductor likediamond has 120581
119871asymp 2 sdot 10
3Wm sdotK while such a poor thermalconductor like glass has 120581
119871asymp 1Wm sdot K Note that electrical
conductivities of solids vary over more than 20 orders ofmagnitude but thermal conductivities of solids vary over arange of only 3-4 orders of magnitude We will see that thesame methodology used to describe electron transport canbe also used for phonon transport We will also discuss thedifferences between electron and phonon transport For athorough introduction to phonons use classical books [6ndash9]
To describe the phonon current we need an expressionlike for the electron current (1b) written now as
119868 =2119902
ℎint119879el (119864)119872el (119864) (1198911 minus 119891
2) 119889119864 (51)
For electrons the states in the contacts were filled accord-ing to the equilibrium Fermi functions but phonons obeyBose statistics thus the phonon states in the contacts arefilled according to the equilibriumBose-Einstein distribution
1198990(ℏ120596) =
1
119890ℏ120596119896119879 minus 1 (52)
Journal of Nanoscience 7
Let temperature for the left and the right contacts be1198791and 119879
2 As for the electrons both contacts are assumed
ideal Thus the phonons that enter a contact are not able toreflect back and transmission coefficient119879ph(119864) describes thephonon transmission across the entire channel
It is easy now to rewrite (51) to the phonon heat currentElectron energy 119864 we replace by the phonon energy ℏ120596 Inthe electron current we have charge 119902moving in the channelin case of the phonon current the quantum of energy ℏ120596
is moving instead thus we replace 119902 in (51) with ℏ120596 andmove it inside the integral The coefficient 2 in (51) reflectsthe spin degeneracy of an electron In case of the phononswe remove this coefficient and instead the number of thephonon polarization states that contribute to the heat flowlets us include to the number of the phononmodes119872ph(ℏ120596)Finally the heat current due to phonons is
119876 =1
ℎint (ℏ120596) 119879ph (ℏ120596)119872ph (ℏ120596) (1198991 minus 119899
2) 119889 (ℏ120596) [W]
(53)
In the linear response regime by analogy with (26)
1198991minus 1198992asymp minus
1205971198990
120597119879Δ119879 (54)
where the derivative according to (52) is
1205971198990
120597119879=ℏ120596
119879(minus
1205971198990
120597 (ℏ120596)) (55)
with
1205971198990
120597 (ℏ120596)= (minus
1
119896119879)
119890ℏ120596119896119879
(119890ℏ120596119896119879 minus 1)2 (56)
Now (53) for small differences in temperature becomes
119876 = minus119870119871Δ119879 (57)
where the thermal conductance
119870119871=1198962119879
ℎint119879ph (ℏ120596)119872ph (ℏ120596)
times [(ℏ120596
119896119879)
2
(minus1205971198990
120597 (ℏ120596))] 119889 (ℏ120596) [WK]
(58)
Equation (57) is simply Fourierrsquos law stating that heatflows down to a temperature gradient It is also useful tonote that the thermal conductance (58) displays certainsimilarities with the electrical conductance
119866 =21199022
ℎint119879el (119864)119872el (119864) (minus
1205971198910
120597119864)119889119864 (59)
The derivative
119882el (119864) equiv (minus1205971198910
120597119864) (60)
known as the Fermi window function that picks out thoseconduction modes which only contribute to the electriccurrent The electron windows function is normalized
int
+infin
minusinfin
(minus1205971198910
120597119864)119889119864 = 1 (61)
In case of phonons the term in square brackets of (58) actsas a window function to specify which modes carry the heatcurrent After normalization
119882ph (ℏ120596) =3
1205872(ℏ120596
119896119879)(
1205971198990
120597 (ℏ120596)) (62)
thus finally
119870119871=12058721198962119879
3ℎint119879ph (ℏ120596)119872ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (63)
with
1198920equiv12058721198962119879
3ℎasymp (9456 times 10
minus13WK2) 119879 (64)
known as the quantum of thermal conductance experimen-tally observed first in 2000 [10]
Comparing (59) and (63) one can see that the electricaland thermal conductances are similar in structure bothare proportional to corresponding quantum of conductancetimes an integral over the transmission times the number ofmodes times a window function
The thermal broadening functions for electrons andphonons have similar shapes and each has a width of a few119896119879 In case of electrons this function is given by (27) or
119865119879(119909) equiv
119890119909
(119890119909 + 1)
2(65)
with 119909 equiv (119864minus119864119865)119896119879 and is shown on Figure 2This function
for phonons is given by (62) or
119865ph119879(119909) equiv
3
1205872
1199092119890119909
(119890119909 minus 1)
2(66)
with 119909 equiv ℏ120596119896119879 Both functions are normalized to a unityand shown together on Figure 3
Along with the number of modes determined by thedispersion relation these two window functions play a keyrole in determining the electrical and thermal conductances
31 Thermal Conductivity of the Bulk Conductors The ther-mal conductivity of a large diffusive resistor is a key materialproperty that controls performance of any electronic devicesBy analogy with the transmission coefficient (3) for electrontransport the phonon transmission
119879ph (ℏ120596) =120582ph (ℏ120596)
120582ph (ℏ120596) + 119871
100381610038161003816100381610038161003816100381610038161003816119871≫120582ph
997888rarr
120582ph (ℏ120596)
119871 (67)
It is also obvious that for large 3D conductors the numberof phonon modes is proportional to the cross-sectional areaof the sample
119872ph (ℏ120596) prop 119860 (68)
8 Journal of Nanoscience
10
5
0
minus5
minus100 01 02 03
FT(x)
FT(x)
FT(x)
phx
Figure 3 Broadening function for phonons compared to that ofelectrons
Now let us return to (57) dividing and multiplying it by119860119871 which immediately gives (50) for the phonon heat fluxpostulated above
119876
119860equiv 119869
ph119876119909
= minus120581119871
119889119879
119889119909(69)
with specific lattice thermal conductivity
120581119871= 119870119871
119871
119860 (70)
or substituting (67) into (63) one for the lattice thermalconductivity finally obtains
120581119871=12058721198962119879
3ℎint
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)
It is useful now to define the average number of phononmodes per cross-sectional area of the conductor that partici-pate in the heat transport
⟨
119872ph
119860⟩ equiv int
119872ph (ℏ120596)
119860119882ph (ℏ120596) 119889 (ℏ120596) (72)
Then
120581119871=12058721198962119879
3ℎ⟨
119872ph
119860⟩⟨⟨120582ph⟩⟩ (73)
where the average mean-free-path is defined now as
⟨⟨120582ph⟩⟩ =
int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)
(74)
Thus the couple of the phonon transport equations (69)and (73) corresponds to similar electron transport equations
119869119909=120590
119902
119889 (119864119865)
119889119909 (75)
120590 =21199022
ℎ⟨119872el119860
⟩⟨⟨120582el⟩⟩ (76)
The thermal conductivity (73) and the electrical conduc-tivity (76) have the same structure It is always a product of thecorresponding quantum of conductance times the numberof modes that participate in transport times the averagemean-free-path These three quantities for phonons will bediscussed later
32 Specific Heat versus Thermal Conductivity The connec-tion between the lattice specific thermal conductivity andthe lattice specific heat at constant volume is well known[6ndash9] We will show now that corresponding proportionalitycoefficient is a product of an appropriately-defined mean-free-path ⟨⟨Λ ph⟩⟩ and an average phonon velocity ⟨Vph⟩namely
120581119871=1
3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)
The total phonon energy per unit volume
119864ph = int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)
where119863ph(ℏ120596) is the phonon density of states By definition
119862119881equiv
120597119864ph
120597119879=
120597
120597119879int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596)
= int
infin
0
(ℏ120596)119863ph (ℏ120596) (1205971198990(ℏ120596)
120597119879)119889 (ℏ120596)
=12058721198962119879
3int
infin
0
119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(79)
where (55) and (62) were used Next multiply and divide (71)by (79) and obtain the proportionality we are looking for120581119871
= [
[
(1ℎ) intinfin
0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(80)To obtain final expression (77) and correct interpretation
of the proportionality coefficient we need to return to (67)This expression can be easily derived for 1D conductorwith several simplifying assumptions Nevertheless it worksvery well in practice for a conductor of any dimensionDerivation of (67) is based on the interpretation of themean-free-path 120582(119864) or 120582(ℏ120596) as that its inverse value is theprobability per unit length that a positive flux is convertedto a negative flux This is why 120582 is often called a mean-free-path for backscattering Let us relate it to the scattering time120591 The distinction between mean-free-path and mean-free-path for backscattering is easiest to see for 1D conductorLet an electron undergo a scattering event For isotropicscattering the electron can forward scatter or backscatterOnly backscattering is relevant for the mean-free-path forscattering so the time between backscattering events is 2120591Thus the mean-free-path for backscattering is twice themean-free-path for scattering
1205821119863
(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)
It was shown that the proper definition of the mean-free-path for backscattering for a conductor of any dimension [11]is
120582 (119864) = 2
⟨V2119909120591⟩
⟨1003816100381610038161003816V119909
1003816100381610038161003816⟩ (lowast lowast lowast)
Journal of Nanoscience 9
where averaging is performed over angles For isotropicbands
1205822119863
(119864) =120587
2V (119864) 120591 (119864) (81b)
1205823119863
(119864) =4
3V (119864) 120591 (119864) (81c)
The scattering time is often approximately written as thepower law scattering
120591 (119864) = 1205910(119864 minus 119864
119862
119896119879)
119904
(82)
where exponent 119904 describes the specific scattering mecha-nism for acoustic phonon scattering in 3D conductor withparabolic dispersion 119904 = minus12 and for ionized impurityscattering 119904 = +32 [12]
Analogous power law is often used for mean-free-path
120582 (119864) = 1205820(119864 minus 119864
119862
119896119879)
119903
(83)
For parabolic zone structure V(119864) prop 11986412 thus 119903 = 119904 + 12
with 119903 = 0 for acoustic phonon scattering and 119903 = 2 forionized impurity scattering
Coming back to our initial task to derive (77) from (80)for 3D conductor according to (81c) we have
120582ph (ℏ120596) =4
3Vph (ℏ120596) 120591ph (ℏ120596) (84)
where according to (81a)
Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)
and finally
120582ph (ℏ120596) =4
3Λ ph (ℏ120596) (86)
It was stated above in (11c) that the density of states andnumber of modes for electrons in 3D are
119872el (119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (87)
Let us rewrite this formula for phonons Note that the spindegeneracy for electrons 119892
119878= 2 is included with the density
of states
1198633119863
(119864) = 21198631015840
3119863(119864) (88)
and for spherical bands in 3D conductor
⟨V+119909(119864)⟩ =
Vel (119864)2
(89)
Collecting (87) up to (89) all together in case of phonons wehave
119872ph (ℏ120596) = 119860ℎ
2(
Vph (ℏ120596)2
) 2119863ph (ℏ120596)
= 119860ℎ
4Vph (ℏ120596)119863ph (ℏ120596)
(90)
Substituting (86) and (90) into (80) we obtain
120581119871
= [
[
(13) intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(91)
Multiplying and dividing (91) by
int
infin
0
Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)
we finally get (77) with proportionality coefficient between 120581119871
and 119862119881as the product of an average mean-free-path as
⟨⟨Λ ph⟩⟩
equiv
intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(93)
and an average velocity as
⟨Vph⟩ equiv
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(94)
with the appropriate averagingEquation (77) is often used to estimate the average mean-
free-path from the measured 120581119871and 119862
119881 if we know the
average velocity which is frequently assumed to be the lon-gitudinal sound velocity The derivation above has identifiedthe precise definitions of the ⟨⟨Λ ph⟩⟩ and ⟨Vph⟩ If a phonondispersion is chosen one can always compute the averagevelocity according to (94) and it is typically very differentfrom the longitudinal sound velocity Thus estimates of theaverage mean-free-path can be quite wrong if one assumesthe longitudinal sound velocity [13]
33 Debye Model For 3D conductors there are three polar-ization states for lattice vibrations one for atoms displacedin the direction of propagation (longitudinal119871) and two foratoms displaced orthogonally to the direction of propagation(transverse119879) The low energy modes are called acousticmodes119860 one 119871119860 mode analogous to sound waves propa-gating in air and two 119879119860 modes Near 119902 rarr 0 dispersion ofacoustic modes is linear
ℏ120596 = ℏV119863119902 (95)
and is known as Debye approximation The Debye velocityV119863is an average velocity of the 119871 and 119879 acoustic modes In
case of 119871119860mode V119863is simply the sound velocity V
119904prop 119898minus12
with 119898 being an effective mass of vibrating atom TypicallyV119904asymp 5 times 10
3ms about 20 times slower that the velocity of atypical electron
The bandwidth of the electronic dispersion is typicallyBW ≫ 119896119879 so only states near the bottom of the conductionband where the effective mass model is reasonably accurate
10 Journal of Nanoscience
are occupied For phonons the situation ismuch different thebandwidth BW asymp 119896119879 so states across the entire Brillouinzone are occupied The widely used Debye approximation(95) fits the acoustic branches as long as 119902 is not too far fromthe center of the Brillouin zone
With the Debye approximation (95) it is easy to find thedensity of the phonon states
119863ph (ℏ120596) =3(ℏ120596)
2
21205872(ℏV119863)3 [Jminus1 sdotmminus3] (96)
where the factor of three is for the three polarizations Thenone can obtain the number of phonon modes per cross-sectional area from (90)
119872ph (ℏ120596) =3(ℏ120596)
2
4120587(ℏV119863)2 (97)
Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures we are to be sure that we accountfor the correct number of states For a crystal there are 3119873Ω
states per unit volume To find the total number of states wehave to integrate the density of states
int
ℏ120596119863
0
119863ph (ℏ120596) 119889 (ℏ120596) (98)
with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states namely
ℏ120596119863= ℏV119863(61205872119873
Ω)
13
equiv 119896119879119863 (99)
The Debye frequency defines a cutoff frequency abovewhich no states are accounted for This restriction can alsobe expressed via a cutoff wave vector 119902
119863or as a Debye tem-
perature
119879119863=ℏ120596
119896 (100)
For 119879 ≪ 119879119863 only states with 119902 rarr 0 for which the Debye
approximation is accurate are occupiedNow we can calculate the lattice thermal conductivity by
integrating (71) to the Debye cutoff energy
120581119871=12058721198962119879
3ℎint
ℏ120596119863
0
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)
and estimate119872ph(ℏ120596) according to (71) The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used This is how the lattice thermalconductivities were first calculated [14 15] The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11 13 16]
34 Phonon Scattering Phonons can scatter from defectsimpurity atoms isotopes surfaces and boundaries and elec-trons and from other phonons Phonon-phonon scattering
occurs because the potential energy of the bonds in the crystalis not exactly harmonic All higher order terms are treatedas a scattering potential Two types of phonon scattering areconsidered In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved
ℏ 1199021+ ℏ 1199022= ℏ 1199023
ℏ1+ ℏ2= ℏ3
(102)
The total momentum of the phonon ensemble is conservedthus this type of scattering has little effect on the heat flux
In a second type of scattering umklapp119880-scatteringthe two initial phonons have larger momentum thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions The 119880-scatterings are thebasic processes in the heat transport especially at hightemperatures Scattering on defects119863 and on boundaries119861is also important Scattering rates are additive thus the totalphonon scattering rate is
1
120591ph (ℏ120596)=
1
120591119880(ℏ120596)
+1
120591119863(ℏ120596)
+1
120591119861(ℏ120596)
(103)
or alternatively in terms of the mean-free-path (84)
1
120582ph (ℏ120596)=
1
120582119880(ℏ120596)
+1
120582119863(ℏ120596)
+1
120582119861(ℏ120596)
(104)
Expressions for each of the scattering rates are developed[17] For scattering from point defects
1
120591119863(ℏ120596)
prop 1205964 (105)
known as the Rayleigh scattering which is like the scatteringof light from the dust
For boundaries and surfaces
1
120591119861(ℏ120596)
prop
Vph (ℏ120596)119871
(106)
where 119871 is the shortest dimension of the sampleA commonly used expression for 119880-scattering is
1
120591119880(ℏ120596)
prop 11987931205962119890minus119879119863119887119879 (107)
With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity
35 Lattice Thermal Conductivity versus Temperature Thetemperature dependence of the lattice thermal conductivity120581119871is illustrated for bulk Si on Figure 4According to (73) 120581
119871is proportional to the number of the
phononmodes that are occupied ⟨119872ph119860⟩ and to the averagevalue of the phononmean-free-path ⟨⟨120582ph⟩⟩The curve 120581
119871(119879)
can be explained by understanding how ⟨119872ph⟩ and ⟨⟨120582ph⟩⟩vary with temperature
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Biomaterials
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Advances in
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BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 3
D1D
D2D
D3D
M1D
M2D
M3D
EC
EEC
E
EC
E
EC
E
EC
EC
E
E
1
Figure 1 Comparison of the density of states 119863(119864) and number ofmodes119872(119864) for 1D 2D and 3D resistors with parabolic dispersion
where
119863 (119864) = ⟨]2119909120591⟩ = ]2120591 (119864) 1
1
21
3 (lowast)
with 120591(119864) being the mean free time after which an electrongets scattered and the other as a product of119872(119864) and 120582(119864)
120590 (119864) =1199022
ℎ119872 (119864) 120582 (119864) 1
1
1198821
119860 (19b)
where the three items in parenthesis correspond to 1D 2Dand 3D resistors
Although (19b) is not well known the equivalent versionin (19a) is a standard result that is derived in textbooks Both(19a) and (19b) are far more generally applicable comparedwith traditional Drude model For example these equationsgive sensible answers even for materials like graphene whosenonparabolic bands make the meaning of electron masssomewhat unclear causing considerable confusion whenusing Drude model In general we must really use (19a) and(19b) and not Drude model to shape our thinking aboutconductivity
These conceptual equations are generally applicable evento amorphous materials and molecular resistors Irrespectiveof the specific119864(119901) relation at any energy the density of states119863(119864) velocity ](119864) and momentum 119901(119864) are related to thetotal number of states 119873(119864) with energy less than 119864 by thefundamental relation
119863 (119864) ] (119864) 119901 (119864) = 119873 (119864) sdot 119889 (lowastlowast)
where 119889 is the number of dimensions Being combinedwith (19a) it gives one more fundamental equation forconductivity
120590 (119864) =1199022120591 (119864)
119898 (119864)119873 (119864)
119871119873 (119864)
119871119882119873 (119864)
119871119860 (19c)
where electron mass is defined as
119898(119864) =119901 (119864)
V (119864) (20)
For parabolic 119864(119901) relations the mass is independent ofenergy but in general it could be energy-dependent as forexample in graphene the effective mass
119898lowast=119864119865
V2119865
(21)
21 Linear Response Regime Near-equilibrium transport orlow field linear response regime corresponds to lim(119889119868
119889119881)119881rarr0
There are several reasons to develop low fieldtransport model First near-equilibrium transport is thefoundation for understanding transport in general Conceptsintroduced in the study of near-equilibrium regime areoften extended to treat more complicated situations andnear-equilibrium regime provides a reference point whenwe analyze transport in more complex conditions Secondnear-equilibrium transport measurements are widely usedto characterize electronic materials and to understand theproperties of new materials And finally near-equilibriumtransport strongly influences and controls the performanceof most electronic devices
Under the low field condition let
1198910(119864) asymp 119891
1(119864) gt 119891
2(119864) (22)
where1198910(119864) is the equilibriumFermi function and an applied
bias
119881 =Δ119864119865
119902=(1198641198651minus 1198641198652)
119902(23)
is small enough Using Taylor expansion under constanttemperature condition
1198912= 1198911+1205971198911
120597119864119865
Δ119864119865= 1198911+1205971198911
120597119864119865
119902119881 (24)
and property of the Fermi function
+120597119891
120597119864119865
= minus120597119891
120597119864 (25)
one finds
1198911minus 1198912= (minus
1205971198910
120597119864) 119902119881 (26)
The derivative of the Fermi function multiplied by 119896119879 tomake it dimensionless
119865119879(119864 119864119865) = 119896119879(minus
120597119891
120597119864) (27)
is known as thermal broadening function and shown inFigure 2
4 Journal of Nanoscience
1010
1
5
00
0 0minus10minus10
minus5
05 0201 03
rarr f(E) rarr kT (minus 120597f
120597E)
E minus EF
kT
uarr
Figure 2 Fermi function and the dimensionless normalized ther-mal broadening function
If one integrates 119865119879over all energy range the total area
int
+infin
minusinfin
119889119864119865119879(119864 119864119865) = 119896119879 (28)
so that we can approximately visualize 119865119879as a rectangular
pulse centered around 119864 = 119864119865with a peak value of 14 and a
width of sim4119896119879The derivative (minus120597119891
0120597119864) is known as the Fermi conduc-
tion window functionWhether a conductor is good or bad isdetermined by the availability of the conductor energy statesin an energy window simplusmn2119896119879 around the electrochemicalpotential 119864
1198650 which can vary widely from one material
to another Current is driven by the difference 1198911minus 1198912
in the ldquoagendardquo of the two contacts which for low biasis proportional to the derivative of the equilibrium Fermifunction (26) With this near-equilibrium assumption forcurrent (1b) we have
119868 = [21199022
ℎint119879 (119864)119872 (119864) (minus
1205971198910
120597119864)119889119864]119881 = 119866119881 (29)
with conductivity
119866 =21199022
ℎint119879 (119864)119872 (119864) (minus
1205971198910
120597119864)119889119864 (30)
known as the Landauer expression which is valid in 1D 2Dand 3D resistors if we use the appropriate expressions for119872(119864)
For ballistic limit 119879(119864) = 1 For diffusive transport 119879(119864)is given by (3) For a conductor much longer than a mean-free-path the current density equation for diffusive transportis
119869119909=120590
119902
119889 (119864119865)
119889119909 (31)
where the electrochemical potential 119864119865is also known as the
quasi-Fermi levelFor a 2D conductor the surface specific conductivity is
120590119878=21199022
ℎint1198722119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (32)
or in a different form
120590119878= int120590
1015840
119878(119864) 119889119864 (33a)
where differential specific conductivity
1205901015840
119878(119864) =
21199022
ℎ1198722119863
(119864) 120582 (119864) (minus1205971198910
120597119864) (33b)
Similar expressions can be written for 1D and 3D resistorsAnother way to write the conductance is the product of
the quantum of conductance times the average transmissiontimes the number of modes in the Fermi windows
119866 =21199022
ℎ⟨⟨119879⟩⟩ ⟨119872⟩ (34a)
⟨119872⟩ = int119872(119864) (minus1205971198910
120597119864)119889119864 (34b)
⟨⟨119879⟩⟩ =
int119879 (119864)119872 (119864) (minus1205971198910120597119864) 119889119864
int119872(119864) (minus1205971198910120597119864) 119889119864
=⟨119872119879⟩
⟨119872⟩ (34c)
Yet another way to write the conductance is in terms ofthe differential conductance 1198661015840(119864) as
119866 = int1198661015840(119864) 119889119864 [S] (35a)
1198661015840(119864) =
21199022
ℎ119872 (119864) 119879 (119864) (minus
1205971198910
120597119864) (35b)
22 Thermocurrent and Thermoelectric Coefficients Elec-trons carry both charge and heat The charge current is givenby (1a) and (1b) To get the equation for the heat currentone notes that electrons in the contacts flow at an energy119864 asymp 119864
119865 To enter a mode119872(119864) in the resistor electrons must
absorb (if 119864 gt 119864119865) or emit (if 119864 lt 119864
119865) a thermal energy
119864 minus 119864119865 We conclude that to get the heat current equation
we should insert (119864 minus 119864119865)119902 inside the integral The resulting
thermocurrent
119868119876=2
ℎint (119864 minus 119864
119865) 119879 (119864)119872 (119864) (119891
1minus 1198912) 119889119864 (36)
It is important from practical point of view that bothexpressionsmdashfor the electric current (1a) and (1b) and ther-mocurrent (36)mdashare suitable for analysis of conductivity ofany materials from metals to semiconductors up to modernnanocomposites
When there are differences in both voltage and temper-ature across the resistor then we must the Fermi difference(1198911minus 1198912) expands to Taylor series in both voltage and
temperature and get
1198911minus 1198912asymp (minus
1205971198910
120597119864) 119902Δ119881 minus (minus
1205971198910
120597119864)119864 minus 119864
119865
119879Δ119879 (37)
where Δ119881 = 1198812minus 1198811 Δ119879 = 119879
2minus 1198791 and 119879 = (119879
1+ 1198792)2
Journal of Nanoscience 5
Deriving a general near-equilibrium current equation isnow straightforward The total current is the sum of thecontributions from each energy mode
119868 = int 1198681015840(119864) 119889119864 (38a)
where the differential current is
1198681015840(119864) =
2119902
ℎ119879 (119864)119872 (119864) (119891
1minus 1198912) (38b)
Using (37) we obtain
1198681015840(119864) = 119866
1015840(119864) Δ119881 + 119878
1015840
119879(119864) Δ119879 (39a)
where
1198661015840(119864) =
21199022
ℎ119879 (119864)119872 (119864) (minus
1205971198910
120597119864) (39b)
is the differential conductance and
1198781015840
119879(119864) = minus
21199022
ℎ119879 (119864)119872 (119864) (
119864 minus 119864119865
119902119879)(minus
1205971198910
120597119864)
= minus119896
119902(119864 minus 119864
119865
119896119879)1198661015840(119864)
(39c)
is the Soret coefficient for electrothermal diffusion in differ-ential formNote that 1198781015840
119879(119864) is negative formodeswith energy
above 119864119865(119899-resistors) and positive for modes with energy
below 119864119865(119901-resistors)
Now we integrate (39a) over all energy modes and find
119868 = 119866Δ119881 + 119878119879Δ119879 [A] (40a)
119868119876= minus 119879119878
119879Δ119881 minus 119870
0Δ119879 [W] (40b)
with three transport coefficients namely conductivity givenby (35a) and (35b) the Soret electrothermal diffusion coeffi-cient
119878119879= int 1198781015840
119879(119864) 119889119864 = minus
119896
119902int(
119864 minus 119864119865
119896119879)1198661015840(119864) 119889119864 [AK]
(40c)
and the electronic heat conductance under the short circuitconditions (Δ119881 = 0)
1198700= 119879(
119896
119902)
2
int(119864 minus 119864
119865
119896119879)
2
1198661015840(119864) 119889119864 [WK] (40d)
where current 119868 is defined to be positive when it flows inconductor from contact 2 to contact 1 with electrons flowingin opposite direction The heat current 119868
119876is positive when it
flows in the +119909 direction out of contact 2Equations (40a) (40b) (40c) and (40d) for long diffusive
resistors can be written in the common form used to describebulk transport as
119869119909= 120590
119889 (119864119865119902)
119889119909minus 119904119879
119889119879
119889119909 [0m2] (41a)
119869119876119909
= 119879119904119879
119889 (119864119865119902)
119889119909minus 1205810
119889119879
119889119909[Wm2] (41b)
with three specific transport coefficients
120590 = int1205901015840(119864) 119889119864
1205901015840(119864) =
21199022
ℎ1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864) [1Ω sdotm sdot J]
(41c)
119904119879= minus
119896
119902int(
119864 minus 119864119865
119896119879)1205901015840(119864) 119889119864 [0m sdot K] (41d)
1205810= (
119896
119902)
2
int(119864 minus 119864
119865
119896119879)
2
1205901015840(119864) 119889119864 [Wm sdot K]
(41e)
These equations have the same form for 1D and 2D resistorsbut the units of the various terms differ
The inverted form of (40a) (40b) (40c) and (40d) isoften preferred in practice namely
Δ119881 = 119877119868 minus 119878Δ119879 (42a)
119868119876= minus Π119868 minus 119870Δ119879 (42b)
where
119878 =119878119879
119866 (42c)
Π = 119879119878 (42d)
119870 = 1198700minus Π119878119866 (42e)
In this form of the equations the contributions from eachenergy mode are not added for example 119877 = int119877(119864)119889119864
Similarly the inverted form of the bulk transport equa-tions (41a) (41b) (41c) (41d) and (41e) becomes
119889 (119864119865119902)
119889119909= 120588119869119909+ 119878
119889119879
119889119909 (43a)
119869119876119909
= 119879119878119869119909minus 120581
119889119879
119889119909(43b)
with transport coefficients
120588 =1
120590 (43c)
119878 =119904119879
120590 (43d)
120581 = 1205810minus 1198782120590119879 (43e)
In summary when a band structure is given number ofmodes can be evaluated from (11a) (11b) and (11c) and if amodel for the mean-free-path for backscattering 120582(119864) can bechosen then the near-equilibrium transport coefficients canbe evaluated using the expressions listed above
23 Bipolar Conduction Let us consider a 3D semiconductorwith parabolic dispersion For the conduction band
1198723119863
(119864) = 119892V119898lowast
2120587ℏ2(119864 minus 119864
119862) (119864 ge 119864
119862) (44a)
6 Journal of Nanoscience
and for the valence band
119872(V)3119863
(119864) = 119892V
119898lowast
119901
2120587ℏ2(119864119881minus 119864) (119864 le 119864
119881) (44b)
The conductivity is provided with two contributions forthe conduction band
120590 =1199022
ℎint
infin
119864119862
1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (45a)
and for the valence band
120590119901=1199022
ℎint
119864119881
minusinfin
119872V3119863
(119864) 120582119901(119864) (minus
1205971198910
120597119864)119889119864 (45b)
The Seebeck coefficient for electrons in the conductionband follows from (41a) (41b) (41c) (41d) and (41e)
120590 = int
infin
119864119862
1205901015840(119864) 119889119864 (46a)
1205901015840(119864) =
21199022
ℎ1198723119863
(119864 minus 119864119862) 120582 (119864) (minus
1205971198910
120597119864) (46b)
119904119879= minus
119896
119902int
infin
119864119862
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 (46c)
119878 =119904119879
120590 (46d)
Similarly for electrons in the valence band we have
120590119901= int
119864119881
minusinfin
1205901015840
119901(119864) 119889119864 (47a)
1205901015840
119901(119864) =
21199022
ℎ119872
V3119863
(119864119881minus 119864) 120582
119901(119864) (minus
1205971198910
120597119864) (47b)
119904(V)119879
= minus119896
119902int
119864119881
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840
119901(119864) 119889119864 (47c)
119878119901=119904(V)119879
120590119901
(47d)
but the sign of 119878119901will be positive
What is going on when both the conduction and valencebands contribute to conductionThis can happen for narrowbandgap conductors or at high temperatures In such a casewe have to simply integrate over all the modes and will find
120590tot
equiv 120590 + 120590119901=1199022
ℎint
1198642
1198641
119872tot3119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (48a)
119872tot3119863
(119864) = 1198723119863
(119864) + 119872V3119863
(119864) (48b)
moreover we do not have to be worried about integratingto the top of the conduction band or from the bottom ofthe valence band because the Fermi function ensures thatthe integrand falls exponentially to zero away from the bandedge What is important is that in both cases we integratethe same expression with the appropriate 119872
3119863(119864) and 120582(119864)
over the relevant energy difference 1198642minus 1198641 Electrons carry
current in both bands Our general expression is the same forthe conduction and valence bandsThere is no need to changesigns for the valence band or to replace 119891
0(119864) with 1 minus 119891
0(119864)
To calculate the Seebeck coefficient when both bandscontribute let us be reminded that in the first direct form ofthe transport coefficients (41a) (41b) (41c) (41d) and (41e)the contributions from eachmode are added in parallel so thetotal specific Soret coefficient
119904tot119879
= minus119896
119902int
+infin
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878
119901120590119901 (49a)
then the Seebeck coefficient for bipolar conduction
119878tot
=
119878120590 + 119878119901120590119901
120590 + 120590119901
(49b)
Since the Seebeck coefficients for the conduction and valencebands have opposite signs the total Seebeck coefficientjust drops for high temperatures and the performance of athermoelectrical device falls down
In summary given a band structure dispersion thenumber of modes can be evaluated and if a model for amean-free-path for backscattering can be established thenthe near-equilibrium transport coefficients can be calculatedusing final expressions listed above
3 Heat Transfer by Phonons
Electrons transfer both charge and heat Electrons carry mostof the heat in metals In semiconductors electrons carry onlya part of the heat but most of the heat is carried by phonons
The phonon heat flux is proportional to the temperaturegradient
119869ph119876119909
= minus120581119871
119889119879
119889119909[Wm2] (50)
with coefficient 120581119871known as the specific lattice thermal
conductivity Such an exceptional thermal conductor likediamond has 120581
119871asymp 2 sdot 10
3Wm sdotK while such a poor thermalconductor like glass has 120581
119871asymp 1Wm sdot K Note that electrical
conductivities of solids vary over more than 20 orders ofmagnitude but thermal conductivities of solids vary over arange of only 3-4 orders of magnitude We will see that thesame methodology used to describe electron transport canbe also used for phonon transport We will also discuss thedifferences between electron and phonon transport For athorough introduction to phonons use classical books [6ndash9]
To describe the phonon current we need an expressionlike for the electron current (1b) written now as
119868 =2119902
ℎint119879el (119864)119872el (119864) (1198911 minus 119891
2) 119889119864 (51)
For electrons the states in the contacts were filled accord-ing to the equilibrium Fermi functions but phonons obeyBose statistics thus the phonon states in the contacts arefilled according to the equilibriumBose-Einstein distribution
1198990(ℏ120596) =
1
119890ℏ120596119896119879 minus 1 (52)
Journal of Nanoscience 7
Let temperature for the left and the right contacts be1198791and 119879
2 As for the electrons both contacts are assumed
ideal Thus the phonons that enter a contact are not able toreflect back and transmission coefficient119879ph(119864) describes thephonon transmission across the entire channel
It is easy now to rewrite (51) to the phonon heat currentElectron energy 119864 we replace by the phonon energy ℏ120596 Inthe electron current we have charge 119902moving in the channelin case of the phonon current the quantum of energy ℏ120596
is moving instead thus we replace 119902 in (51) with ℏ120596 andmove it inside the integral The coefficient 2 in (51) reflectsthe spin degeneracy of an electron In case of the phononswe remove this coefficient and instead the number of thephonon polarization states that contribute to the heat flowlets us include to the number of the phononmodes119872ph(ℏ120596)Finally the heat current due to phonons is
119876 =1
ℎint (ℏ120596) 119879ph (ℏ120596)119872ph (ℏ120596) (1198991 minus 119899
2) 119889 (ℏ120596) [W]
(53)
In the linear response regime by analogy with (26)
1198991minus 1198992asymp minus
1205971198990
120597119879Δ119879 (54)
where the derivative according to (52) is
1205971198990
120597119879=ℏ120596
119879(minus
1205971198990
120597 (ℏ120596)) (55)
with
1205971198990
120597 (ℏ120596)= (minus
1
119896119879)
119890ℏ120596119896119879
(119890ℏ120596119896119879 minus 1)2 (56)
Now (53) for small differences in temperature becomes
119876 = minus119870119871Δ119879 (57)
where the thermal conductance
119870119871=1198962119879
ℎint119879ph (ℏ120596)119872ph (ℏ120596)
times [(ℏ120596
119896119879)
2
(minus1205971198990
120597 (ℏ120596))] 119889 (ℏ120596) [WK]
(58)
Equation (57) is simply Fourierrsquos law stating that heatflows down to a temperature gradient It is also useful tonote that the thermal conductance (58) displays certainsimilarities with the electrical conductance
119866 =21199022
ℎint119879el (119864)119872el (119864) (minus
1205971198910
120597119864)119889119864 (59)
The derivative
119882el (119864) equiv (minus1205971198910
120597119864) (60)
known as the Fermi window function that picks out thoseconduction modes which only contribute to the electriccurrent The electron windows function is normalized
int
+infin
minusinfin
(minus1205971198910
120597119864)119889119864 = 1 (61)
In case of phonons the term in square brackets of (58) actsas a window function to specify which modes carry the heatcurrent After normalization
119882ph (ℏ120596) =3
1205872(ℏ120596
119896119879)(
1205971198990
120597 (ℏ120596)) (62)
thus finally
119870119871=12058721198962119879
3ℎint119879ph (ℏ120596)119872ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (63)
with
1198920equiv12058721198962119879
3ℎasymp (9456 times 10
minus13WK2) 119879 (64)
known as the quantum of thermal conductance experimen-tally observed first in 2000 [10]
Comparing (59) and (63) one can see that the electricaland thermal conductances are similar in structure bothare proportional to corresponding quantum of conductancetimes an integral over the transmission times the number ofmodes times a window function
The thermal broadening functions for electrons andphonons have similar shapes and each has a width of a few119896119879 In case of electrons this function is given by (27) or
119865119879(119909) equiv
119890119909
(119890119909 + 1)
2(65)
with 119909 equiv (119864minus119864119865)119896119879 and is shown on Figure 2This function
for phonons is given by (62) or
119865ph119879(119909) equiv
3
1205872
1199092119890119909
(119890119909 minus 1)
2(66)
with 119909 equiv ℏ120596119896119879 Both functions are normalized to a unityand shown together on Figure 3
Along with the number of modes determined by thedispersion relation these two window functions play a keyrole in determining the electrical and thermal conductances
31 Thermal Conductivity of the Bulk Conductors The ther-mal conductivity of a large diffusive resistor is a key materialproperty that controls performance of any electronic devicesBy analogy with the transmission coefficient (3) for electrontransport the phonon transmission
119879ph (ℏ120596) =120582ph (ℏ120596)
120582ph (ℏ120596) + 119871
100381610038161003816100381610038161003816100381610038161003816119871≫120582ph
997888rarr
120582ph (ℏ120596)
119871 (67)
It is also obvious that for large 3D conductors the numberof phonon modes is proportional to the cross-sectional areaof the sample
119872ph (ℏ120596) prop 119860 (68)
8 Journal of Nanoscience
10
5
0
minus5
minus100 01 02 03
FT(x)
FT(x)
FT(x)
phx
Figure 3 Broadening function for phonons compared to that ofelectrons
Now let us return to (57) dividing and multiplying it by119860119871 which immediately gives (50) for the phonon heat fluxpostulated above
119876
119860equiv 119869
ph119876119909
= minus120581119871
119889119879
119889119909(69)
with specific lattice thermal conductivity
120581119871= 119870119871
119871
119860 (70)
or substituting (67) into (63) one for the lattice thermalconductivity finally obtains
120581119871=12058721198962119879
3ℎint
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)
It is useful now to define the average number of phononmodes per cross-sectional area of the conductor that partici-pate in the heat transport
⟨
119872ph
119860⟩ equiv int
119872ph (ℏ120596)
119860119882ph (ℏ120596) 119889 (ℏ120596) (72)
Then
120581119871=12058721198962119879
3ℎ⟨
119872ph
119860⟩⟨⟨120582ph⟩⟩ (73)
where the average mean-free-path is defined now as
⟨⟨120582ph⟩⟩ =
int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)
(74)
Thus the couple of the phonon transport equations (69)and (73) corresponds to similar electron transport equations
119869119909=120590
119902
119889 (119864119865)
119889119909 (75)
120590 =21199022
ℎ⟨119872el119860
⟩⟨⟨120582el⟩⟩ (76)
The thermal conductivity (73) and the electrical conduc-tivity (76) have the same structure It is always a product of thecorresponding quantum of conductance times the numberof modes that participate in transport times the averagemean-free-path These three quantities for phonons will bediscussed later
32 Specific Heat versus Thermal Conductivity The connec-tion between the lattice specific thermal conductivity andthe lattice specific heat at constant volume is well known[6ndash9] We will show now that corresponding proportionalitycoefficient is a product of an appropriately-defined mean-free-path ⟨⟨Λ ph⟩⟩ and an average phonon velocity ⟨Vph⟩namely
120581119871=1
3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)
The total phonon energy per unit volume
119864ph = int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)
where119863ph(ℏ120596) is the phonon density of states By definition
119862119881equiv
120597119864ph
120597119879=
120597
120597119879int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596)
= int
infin
0
(ℏ120596)119863ph (ℏ120596) (1205971198990(ℏ120596)
120597119879)119889 (ℏ120596)
=12058721198962119879
3int
infin
0
119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(79)
where (55) and (62) were used Next multiply and divide (71)by (79) and obtain the proportionality we are looking for120581119871
= [
[
(1ℎ) intinfin
0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(80)To obtain final expression (77) and correct interpretation
of the proportionality coefficient we need to return to (67)This expression can be easily derived for 1D conductorwith several simplifying assumptions Nevertheless it worksvery well in practice for a conductor of any dimensionDerivation of (67) is based on the interpretation of themean-free-path 120582(119864) or 120582(ℏ120596) as that its inverse value is theprobability per unit length that a positive flux is convertedto a negative flux This is why 120582 is often called a mean-free-path for backscattering Let us relate it to the scattering time120591 The distinction between mean-free-path and mean-free-path for backscattering is easiest to see for 1D conductorLet an electron undergo a scattering event For isotropicscattering the electron can forward scatter or backscatterOnly backscattering is relevant for the mean-free-path forscattering so the time between backscattering events is 2120591Thus the mean-free-path for backscattering is twice themean-free-path for scattering
1205821119863
(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)
It was shown that the proper definition of the mean-free-path for backscattering for a conductor of any dimension [11]is
120582 (119864) = 2
⟨V2119909120591⟩
⟨1003816100381610038161003816V119909
1003816100381610038161003816⟩ (lowast lowast lowast)
Journal of Nanoscience 9
where averaging is performed over angles For isotropicbands
1205822119863
(119864) =120587
2V (119864) 120591 (119864) (81b)
1205823119863
(119864) =4
3V (119864) 120591 (119864) (81c)
The scattering time is often approximately written as thepower law scattering
120591 (119864) = 1205910(119864 minus 119864
119862
119896119879)
119904
(82)
where exponent 119904 describes the specific scattering mecha-nism for acoustic phonon scattering in 3D conductor withparabolic dispersion 119904 = minus12 and for ionized impurityscattering 119904 = +32 [12]
Analogous power law is often used for mean-free-path
120582 (119864) = 1205820(119864 minus 119864
119862
119896119879)
119903
(83)
For parabolic zone structure V(119864) prop 11986412 thus 119903 = 119904 + 12
with 119903 = 0 for acoustic phonon scattering and 119903 = 2 forionized impurity scattering
Coming back to our initial task to derive (77) from (80)for 3D conductor according to (81c) we have
120582ph (ℏ120596) =4
3Vph (ℏ120596) 120591ph (ℏ120596) (84)
where according to (81a)
Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)
and finally
120582ph (ℏ120596) =4
3Λ ph (ℏ120596) (86)
It was stated above in (11c) that the density of states andnumber of modes for electrons in 3D are
119872el (119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (87)
Let us rewrite this formula for phonons Note that the spindegeneracy for electrons 119892
119878= 2 is included with the density
of states
1198633119863
(119864) = 21198631015840
3119863(119864) (88)
and for spherical bands in 3D conductor
⟨V+119909(119864)⟩ =
Vel (119864)2
(89)
Collecting (87) up to (89) all together in case of phonons wehave
119872ph (ℏ120596) = 119860ℎ
2(
Vph (ℏ120596)2
) 2119863ph (ℏ120596)
= 119860ℎ
4Vph (ℏ120596)119863ph (ℏ120596)
(90)
Substituting (86) and (90) into (80) we obtain
120581119871
= [
[
(13) intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(91)
Multiplying and dividing (91) by
int
infin
0
Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)
we finally get (77) with proportionality coefficient between 120581119871
and 119862119881as the product of an average mean-free-path as
⟨⟨Λ ph⟩⟩
equiv
intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(93)
and an average velocity as
⟨Vph⟩ equiv
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(94)
with the appropriate averagingEquation (77) is often used to estimate the average mean-
free-path from the measured 120581119871and 119862
119881 if we know the
average velocity which is frequently assumed to be the lon-gitudinal sound velocity The derivation above has identifiedthe precise definitions of the ⟨⟨Λ ph⟩⟩ and ⟨Vph⟩ If a phonondispersion is chosen one can always compute the averagevelocity according to (94) and it is typically very differentfrom the longitudinal sound velocity Thus estimates of theaverage mean-free-path can be quite wrong if one assumesthe longitudinal sound velocity [13]
33 Debye Model For 3D conductors there are three polar-ization states for lattice vibrations one for atoms displacedin the direction of propagation (longitudinal119871) and two foratoms displaced orthogonally to the direction of propagation(transverse119879) The low energy modes are called acousticmodes119860 one 119871119860 mode analogous to sound waves propa-gating in air and two 119879119860 modes Near 119902 rarr 0 dispersion ofacoustic modes is linear
ℏ120596 = ℏV119863119902 (95)
and is known as Debye approximation The Debye velocityV119863is an average velocity of the 119871 and 119879 acoustic modes In
case of 119871119860mode V119863is simply the sound velocity V
119904prop 119898minus12
with 119898 being an effective mass of vibrating atom TypicallyV119904asymp 5 times 10
3ms about 20 times slower that the velocity of atypical electron
The bandwidth of the electronic dispersion is typicallyBW ≫ 119896119879 so only states near the bottom of the conductionband where the effective mass model is reasonably accurate
10 Journal of Nanoscience
are occupied For phonons the situation ismuch different thebandwidth BW asymp 119896119879 so states across the entire Brillouinzone are occupied The widely used Debye approximation(95) fits the acoustic branches as long as 119902 is not too far fromthe center of the Brillouin zone
With the Debye approximation (95) it is easy to find thedensity of the phonon states
119863ph (ℏ120596) =3(ℏ120596)
2
21205872(ℏV119863)3 [Jminus1 sdotmminus3] (96)
where the factor of three is for the three polarizations Thenone can obtain the number of phonon modes per cross-sectional area from (90)
119872ph (ℏ120596) =3(ℏ120596)
2
4120587(ℏV119863)2 (97)
Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures we are to be sure that we accountfor the correct number of states For a crystal there are 3119873Ω
states per unit volume To find the total number of states wehave to integrate the density of states
int
ℏ120596119863
0
119863ph (ℏ120596) 119889 (ℏ120596) (98)
with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states namely
ℏ120596119863= ℏV119863(61205872119873
Ω)
13
equiv 119896119879119863 (99)
The Debye frequency defines a cutoff frequency abovewhich no states are accounted for This restriction can alsobe expressed via a cutoff wave vector 119902
119863or as a Debye tem-
perature
119879119863=ℏ120596
119896 (100)
For 119879 ≪ 119879119863 only states with 119902 rarr 0 for which the Debye
approximation is accurate are occupiedNow we can calculate the lattice thermal conductivity by
integrating (71) to the Debye cutoff energy
120581119871=12058721198962119879
3ℎint
ℏ120596119863
0
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)
and estimate119872ph(ℏ120596) according to (71) The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used This is how the lattice thermalconductivities were first calculated [14 15] The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11 13 16]
34 Phonon Scattering Phonons can scatter from defectsimpurity atoms isotopes surfaces and boundaries and elec-trons and from other phonons Phonon-phonon scattering
occurs because the potential energy of the bonds in the crystalis not exactly harmonic All higher order terms are treatedas a scattering potential Two types of phonon scattering areconsidered In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved
ℏ 1199021+ ℏ 1199022= ℏ 1199023
ℏ1+ ℏ2= ℏ3
(102)
The total momentum of the phonon ensemble is conservedthus this type of scattering has little effect on the heat flux
In a second type of scattering umklapp119880-scatteringthe two initial phonons have larger momentum thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions The 119880-scatterings are thebasic processes in the heat transport especially at hightemperatures Scattering on defects119863 and on boundaries119861is also important Scattering rates are additive thus the totalphonon scattering rate is
1
120591ph (ℏ120596)=
1
120591119880(ℏ120596)
+1
120591119863(ℏ120596)
+1
120591119861(ℏ120596)
(103)
or alternatively in terms of the mean-free-path (84)
1
120582ph (ℏ120596)=
1
120582119880(ℏ120596)
+1
120582119863(ℏ120596)
+1
120582119861(ℏ120596)
(104)
Expressions for each of the scattering rates are developed[17] For scattering from point defects
1
120591119863(ℏ120596)
prop 1205964 (105)
known as the Rayleigh scattering which is like the scatteringof light from the dust
For boundaries and surfaces
1
120591119861(ℏ120596)
prop
Vph (ℏ120596)119871
(106)
where 119871 is the shortest dimension of the sampleA commonly used expression for 119880-scattering is
1
120591119880(ℏ120596)
prop 11987931205962119890minus119879119863119887119879 (107)
With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity
35 Lattice Thermal Conductivity versus Temperature Thetemperature dependence of the lattice thermal conductivity120581119871is illustrated for bulk Si on Figure 4According to (73) 120581
119871is proportional to the number of the
phononmodes that are occupied ⟨119872ph119860⟩ and to the averagevalue of the phononmean-free-path ⟨⟨120582ph⟩⟩The curve 120581
119871(119879)
can be explained by understanding how ⟨119872ph⟩ and ⟨⟨120582ph⟩⟩vary with temperature
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
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Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
4 Journal of Nanoscience
1010
1
5
00
0 0minus10minus10
minus5
05 0201 03
rarr f(E) rarr kT (minus 120597f
120597E)
E minus EF
kT
uarr
Figure 2 Fermi function and the dimensionless normalized ther-mal broadening function
If one integrates 119865119879over all energy range the total area
int
+infin
minusinfin
119889119864119865119879(119864 119864119865) = 119896119879 (28)
so that we can approximately visualize 119865119879as a rectangular
pulse centered around 119864 = 119864119865with a peak value of 14 and a
width of sim4119896119879The derivative (minus120597119891
0120597119864) is known as the Fermi conduc-
tion window functionWhether a conductor is good or bad isdetermined by the availability of the conductor energy statesin an energy window simplusmn2119896119879 around the electrochemicalpotential 119864
1198650 which can vary widely from one material
to another Current is driven by the difference 1198911minus 1198912
in the ldquoagendardquo of the two contacts which for low biasis proportional to the derivative of the equilibrium Fermifunction (26) With this near-equilibrium assumption forcurrent (1b) we have
119868 = [21199022
ℎint119879 (119864)119872 (119864) (minus
1205971198910
120597119864)119889119864]119881 = 119866119881 (29)
with conductivity
119866 =21199022
ℎint119879 (119864)119872 (119864) (minus
1205971198910
120597119864)119889119864 (30)
known as the Landauer expression which is valid in 1D 2Dand 3D resistors if we use the appropriate expressions for119872(119864)
For ballistic limit 119879(119864) = 1 For diffusive transport 119879(119864)is given by (3) For a conductor much longer than a mean-free-path the current density equation for diffusive transportis
119869119909=120590
119902
119889 (119864119865)
119889119909 (31)
where the electrochemical potential 119864119865is also known as the
quasi-Fermi levelFor a 2D conductor the surface specific conductivity is
120590119878=21199022
ℎint1198722119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (32)
or in a different form
120590119878= int120590
1015840
119878(119864) 119889119864 (33a)
where differential specific conductivity
1205901015840
119878(119864) =
21199022
ℎ1198722119863
(119864) 120582 (119864) (minus1205971198910
120597119864) (33b)
Similar expressions can be written for 1D and 3D resistorsAnother way to write the conductance is the product of
the quantum of conductance times the average transmissiontimes the number of modes in the Fermi windows
119866 =21199022
ℎ⟨⟨119879⟩⟩ ⟨119872⟩ (34a)
⟨119872⟩ = int119872(119864) (minus1205971198910
120597119864)119889119864 (34b)
⟨⟨119879⟩⟩ =
int119879 (119864)119872 (119864) (minus1205971198910120597119864) 119889119864
int119872(119864) (minus1205971198910120597119864) 119889119864
=⟨119872119879⟩
⟨119872⟩ (34c)
Yet another way to write the conductance is in terms ofthe differential conductance 1198661015840(119864) as
119866 = int1198661015840(119864) 119889119864 [S] (35a)
1198661015840(119864) =
21199022
ℎ119872 (119864) 119879 (119864) (minus
1205971198910
120597119864) (35b)
22 Thermocurrent and Thermoelectric Coefficients Elec-trons carry both charge and heat The charge current is givenby (1a) and (1b) To get the equation for the heat currentone notes that electrons in the contacts flow at an energy119864 asymp 119864
119865 To enter a mode119872(119864) in the resistor electrons must
absorb (if 119864 gt 119864119865) or emit (if 119864 lt 119864
119865) a thermal energy
119864 minus 119864119865 We conclude that to get the heat current equation
we should insert (119864 minus 119864119865)119902 inside the integral The resulting
thermocurrent
119868119876=2
ℎint (119864 minus 119864
119865) 119879 (119864)119872 (119864) (119891
1minus 1198912) 119889119864 (36)
It is important from practical point of view that bothexpressionsmdashfor the electric current (1a) and (1b) and ther-mocurrent (36)mdashare suitable for analysis of conductivity ofany materials from metals to semiconductors up to modernnanocomposites
When there are differences in both voltage and temper-ature across the resistor then we must the Fermi difference(1198911minus 1198912) expands to Taylor series in both voltage and
temperature and get
1198911minus 1198912asymp (minus
1205971198910
120597119864) 119902Δ119881 minus (minus
1205971198910
120597119864)119864 minus 119864
119865
119879Δ119879 (37)
where Δ119881 = 1198812minus 1198811 Δ119879 = 119879
2minus 1198791 and 119879 = (119879
1+ 1198792)2
Journal of Nanoscience 5
Deriving a general near-equilibrium current equation isnow straightforward The total current is the sum of thecontributions from each energy mode
119868 = int 1198681015840(119864) 119889119864 (38a)
where the differential current is
1198681015840(119864) =
2119902
ℎ119879 (119864)119872 (119864) (119891
1minus 1198912) (38b)
Using (37) we obtain
1198681015840(119864) = 119866
1015840(119864) Δ119881 + 119878
1015840
119879(119864) Δ119879 (39a)
where
1198661015840(119864) =
21199022
ℎ119879 (119864)119872 (119864) (minus
1205971198910
120597119864) (39b)
is the differential conductance and
1198781015840
119879(119864) = minus
21199022
ℎ119879 (119864)119872 (119864) (
119864 minus 119864119865
119902119879)(minus
1205971198910
120597119864)
= minus119896
119902(119864 minus 119864
119865
119896119879)1198661015840(119864)
(39c)
is the Soret coefficient for electrothermal diffusion in differ-ential formNote that 1198781015840
119879(119864) is negative formodeswith energy
above 119864119865(119899-resistors) and positive for modes with energy
below 119864119865(119901-resistors)
Now we integrate (39a) over all energy modes and find
119868 = 119866Δ119881 + 119878119879Δ119879 [A] (40a)
119868119876= minus 119879119878
119879Δ119881 minus 119870
0Δ119879 [W] (40b)
with three transport coefficients namely conductivity givenby (35a) and (35b) the Soret electrothermal diffusion coeffi-cient
119878119879= int 1198781015840
119879(119864) 119889119864 = minus
119896
119902int(
119864 minus 119864119865
119896119879)1198661015840(119864) 119889119864 [AK]
(40c)
and the electronic heat conductance under the short circuitconditions (Δ119881 = 0)
1198700= 119879(
119896
119902)
2
int(119864 minus 119864
119865
119896119879)
2
1198661015840(119864) 119889119864 [WK] (40d)
where current 119868 is defined to be positive when it flows inconductor from contact 2 to contact 1 with electrons flowingin opposite direction The heat current 119868
119876is positive when it
flows in the +119909 direction out of contact 2Equations (40a) (40b) (40c) and (40d) for long diffusive
resistors can be written in the common form used to describebulk transport as
119869119909= 120590
119889 (119864119865119902)
119889119909minus 119904119879
119889119879
119889119909 [0m2] (41a)
119869119876119909
= 119879119904119879
119889 (119864119865119902)
119889119909minus 1205810
119889119879
119889119909[Wm2] (41b)
with three specific transport coefficients
120590 = int1205901015840(119864) 119889119864
1205901015840(119864) =
21199022
ℎ1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864) [1Ω sdotm sdot J]
(41c)
119904119879= minus
119896
119902int(
119864 minus 119864119865
119896119879)1205901015840(119864) 119889119864 [0m sdot K] (41d)
1205810= (
119896
119902)
2
int(119864 minus 119864
119865
119896119879)
2
1205901015840(119864) 119889119864 [Wm sdot K]
(41e)
These equations have the same form for 1D and 2D resistorsbut the units of the various terms differ
The inverted form of (40a) (40b) (40c) and (40d) isoften preferred in practice namely
Δ119881 = 119877119868 minus 119878Δ119879 (42a)
119868119876= minus Π119868 minus 119870Δ119879 (42b)
where
119878 =119878119879
119866 (42c)
Π = 119879119878 (42d)
119870 = 1198700minus Π119878119866 (42e)
In this form of the equations the contributions from eachenergy mode are not added for example 119877 = int119877(119864)119889119864
Similarly the inverted form of the bulk transport equa-tions (41a) (41b) (41c) (41d) and (41e) becomes
119889 (119864119865119902)
119889119909= 120588119869119909+ 119878
119889119879
119889119909 (43a)
119869119876119909
= 119879119878119869119909minus 120581
119889119879
119889119909(43b)
with transport coefficients
120588 =1
120590 (43c)
119878 =119904119879
120590 (43d)
120581 = 1205810minus 1198782120590119879 (43e)
In summary when a band structure is given number ofmodes can be evaluated from (11a) (11b) and (11c) and if amodel for the mean-free-path for backscattering 120582(119864) can bechosen then the near-equilibrium transport coefficients canbe evaluated using the expressions listed above
23 Bipolar Conduction Let us consider a 3D semiconductorwith parabolic dispersion For the conduction band
1198723119863
(119864) = 119892V119898lowast
2120587ℏ2(119864 minus 119864
119862) (119864 ge 119864
119862) (44a)
6 Journal of Nanoscience
and for the valence band
119872(V)3119863
(119864) = 119892V
119898lowast
119901
2120587ℏ2(119864119881minus 119864) (119864 le 119864
119881) (44b)
The conductivity is provided with two contributions forthe conduction band
120590 =1199022
ℎint
infin
119864119862
1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (45a)
and for the valence band
120590119901=1199022
ℎint
119864119881
minusinfin
119872V3119863
(119864) 120582119901(119864) (minus
1205971198910
120597119864)119889119864 (45b)
The Seebeck coefficient for electrons in the conductionband follows from (41a) (41b) (41c) (41d) and (41e)
120590 = int
infin
119864119862
1205901015840(119864) 119889119864 (46a)
1205901015840(119864) =
21199022
ℎ1198723119863
(119864 minus 119864119862) 120582 (119864) (minus
1205971198910
120597119864) (46b)
119904119879= minus
119896
119902int
infin
119864119862
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 (46c)
119878 =119904119879
120590 (46d)
Similarly for electrons in the valence band we have
120590119901= int
119864119881
minusinfin
1205901015840
119901(119864) 119889119864 (47a)
1205901015840
119901(119864) =
21199022
ℎ119872
V3119863
(119864119881minus 119864) 120582
119901(119864) (minus
1205971198910
120597119864) (47b)
119904(V)119879
= minus119896
119902int
119864119881
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840
119901(119864) 119889119864 (47c)
119878119901=119904(V)119879
120590119901
(47d)
but the sign of 119878119901will be positive
What is going on when both the conduction and valencebands contribute to conductionThis can happen for narrowbandgap conductors or at high temperatures In such a casewe have to simply integrate over all the modes and will find
120590tot
equiv 120590 + 120590119901=1199022
ℎint
1198642
1198641
119872tot3119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (48a)
119872tot3119863
(119864) = 1198723119863
(119864) + 119872V3119863
(119864) (48b)
moreover we do not have to be worried about integratingto the top of the conduction band or from the bottom ofthe valence band because the Fermi function ensures thatthe integrand falls exponentially to zero away from the bandedge What is important is that in both cases we integratethe same expression with the appropriate 119872
3119863(119864) and 120582(119864)
over the relevant energy difference 1198642minus 1198641 Electrons carry
current in both bands Our general expression is the same forthe conduction and valence bandsThere is no need to changesigns for the valence band or to replace 119891
0(119864) with 1 minus 119891
0(119864)
To calculate the Seebeck coefficient when both bandscontribute let us be reminded that in the first direct form ofthe transport coefficients (41a) (41b) (41c) (41d) and (41e)the contributions from eachmode are added in parallel so thetotal specific Soret coefficient
119904tot119879
= minus119896
119902int
+infin
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878
119901120590119901 (49a)
then the Seebeck coefficient for bipolar conduction
119878tot
=
119878120590 + 119878119901120590119901
120590 + 120590119901
(49b)
Since the Seebeck coefficients for the conduction and valencebands have opposite signs the total Seebeck coefficientjust drops for high temperatures and the performance of athermoelectrical device falls down
In summary given a band structure dispersion thenumber of modes can be evaluated and if a model for amean-free-path for backscattering can be established thenthe near-equilibrium transport coefficients can be calculatedusing final expressions listed above
3 Heat Transfer by Phonons
Electrons transfer both charge and heat Electrons carry mostof the heat in metals In semiconductors electrons carry onlya part of the heat but most of the heat is carried by phonons
The phonon heat flux is proportional to the temperaturegradient
119869ph119876119909
= minus120581119871
119889119879
119889119909[Wm2] (50)
with coefficient 120581119871known as the specific lattice thermal
conductivity Such an exceptional thermal conductor likediamond has 120581
119871asymp 2 sdot 10
3Wm sdotK while such a poor thermalconductor like glass has 120581
119871asymp 1Wm sdot K Note that electrical
conductivities of solids vary over more than 20 orders ofmagnitude but thermal conductivities of solids vary over arange of only 3-4 orders of magnitude We will see that thesame methodology used to describe electron transport canbe also used for phonon transport We will also discuss thedifferences between electron and phonon transport For athorough introduction to phonons use classical books [6ndash9]
To describe the phonon current we need an expressionlike for the electron current (1b) written now as
119868 =2119902
ℎint119879el (119864)119872el (119864) (1198911 minus 119891
2) 119889119864 (51)
For electrons the states in the contacts were filled accord-ing to the equilibrium Fermi functions but phonons obeyBose statistics thus the phonon states in the contacts arefilled according to the equilibriumBose-Einstein distribution
1198990(ℏ120596) =
1
119890ℏ120596119896119879 minus 1 (52)
Journal of Nanoscience 7
Let temperature for the left and the right contacts be1198791and 119879
2 As for the electrons both contacts are assumed
ideal Thus the phonons that enter a contact are not able toreflect back and transmission coefficient119879ph(119864) describes thephonon transmission across the entire channel
It is easy now to rewrite (51) to the phonon heat currentElectron energy 119864 we replace by the phonon energy ℏ120596 Inthe electron current we have charge 119902moving in the channelin case of the phonon current the quantum of energy ℏ120596
is moving instead thus we replace 119902 in (51) with ℏ120596 andmove it inside the integral The coefficient 2 in (51) reflectsthe spin degeneracy of an electron In case of the phononswe remove this coefficient and instead the number of thephonon polarization states that contribute to the heat flowlets us include to the number of the phononmodes119872ph(ℏ120596)Finally the heat current due to phonons is
119876 =1
ℎint (ℏ120596) 119879ph (ℏ120596)119872ph (ℏ120596) (1198991 minus 119899
2) 119889 (ℏ120596) [W]
(53)
In the linear response regime by analogy with (26)
1198991minus 1198992asymp minus
1205971198990
120597119879Δ119879 (54)
where the derivative according to (52) is
1205971198990
120597119879=ℏ120596
119879(minus
1205971198990
120597 (ℏ120596)) (55)
with
1205971198990
120597 (ℏ120596)= (minus
1
119896119879)
119890ℏ120596119896119879
(119890ℏ120596119896119879 minus 1)2 (56)
Now (53) for small differences in temperature becomes
119876 = minus119870119871Δ119879 (57)
where the thermal conductance
119870119871=1198962119879
ℎint119879ph (ℏ120596)119872ph (ℏ120596)
times [(ℏ120596
119896119879)
2
(minus1205971198990
120597 (ℏ120596))] 119889 (ℏ120596) [WK]
(58)
Equation (57) is simply Fourierrsquos law stating that heatflows down to a temperature gradient It is also useful tonote that the thermal conductance (58) displays certainsimilarities with the electrical conductance
119866 =21199022
ℎint119879el (119864)119872el (119864) (minus
1205971198910
120597119864)119889119864 (59)
The derivative
119882el (119864) equiv (minus1205971198910
120597119864) (60)
known as the Fermi window function that picks out thoseconduction modes which only contribute to the electriccurrent The electron windows function is normalized
int
+infin
minusinfin
(minus1205971198910
120597119864)119889119864 = 1 (61)
In case of phonons the term in square brackets of (58) actsas a window function to specify which modes carry the heatcurrent After normalization
119882ph (ℏ120596) =3
1205872(ℏ120596
119896119879)(
1205971198990
120597 (ℏ120596)) (62)
thus finally
119870119871=12058721198962119879
3ℎint119879ph (ℏ120596)119872ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (63)
with
1198920equiv12058721198962119879
3ℎasymp (9456 times 10
minus13WK2) 119879 (64)
known as the quantum of thermal conductance experimen-tally observed first in 2000 [10]
Comparing (59) and (63) one can see that the electricaland thermal conductances are similar in structure bothare proportional to corresponding quantum of conductancetimes an integral over the transmission times the number ofmodes times a window function
The thermal broadening functions for electrons andphonons have similar shapes and each has a width of a few119896119879 In case of electrons this function is given by (27) or
119865119879(119909) equiv
119890119909
(119890119909 + 1)
2(65)
with 119909 equiv (119864minus119864119865)119896119879 and is shown on Figure 2This function
for phonons is given by (62) or
119865ph119879(119909) equiv
3
1205872
1199092119890119909
(119890119909 minus 1)
2(66)
with 119909 equiv ℏ120596119896119879 Both functions are normalized to a unityand shown together on Figure 3
Along with the number of modes determined by thedispersion relation these two window functions play a keyrole in determining the electrical and thermal conductances
31 Thermal Conductivity of the Bulk Conductors The ther-mal conductivity of a large diffusive resistor is a key materialproperty that controls performance of any electronic devicesBy analogy with the transmission coefficient (3) for electrontransport the phonon transmission
119879ph (ℏ120596) =120582ph (ℏ120596)
120582ph (ℏ120596) + 119871
100381610038161003816100381610038161003816100381610038161003816119871≫120582ph
997888rarr
120582ph (ℏ120596)
119871 (67)
It is also obvious that for large 3D conductors the numberof phonon modes is proportional to the cross-sectional areaof the sample
119872ph (ℏ120596) prop 119860 (68)
8 Journal of Nanoscience
10
5
0
minus5
minus100 01 02 03
FT(x)
FT(x)
FT(x)
phx
Figure 3 Broadening function for phonons compared to that ofelectrons
Now let us return to (57) dividing and multiplying it by119860119871 which immediately gives (50) for the phonon heat fluxpostulated above
119876
119860equiv 119869
ph119876119909
= minus120581119871
119889119879
119889119909(69)
with specific lattice thermal conductivity
120581119871= 119870119871
119871
119860 (70)
or substituting (67) into (63) one for the lattice thermalconductivity finally obtains
120581119871=12058721198962119879
3ℎint
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)
It is useful now to define the average number of phononmodes per cross-sectional area of the conductor that partici-pate in the heat transport
⟨
119872ph
119860⟩ equiv int
119872ph (ℏ120596)
119860119882ph (ℏ120596) 119889 (ℏ120596) (72)
Then
120581119871=12058721198962119879
3ℎ⟨
119872ph
119860⟩⟨⟨120582ph⟩⟩ (73)
where the average mean-free-path is defined now as
⟨⟨120582ph⟩⟩ =
int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)
(74)
Thus the couple of the phonon transport equations (69)and (73) corresponds to similar electron transport equations
119869119909=120590
119902
119889 (119864119865)
119889119909 (75)
120590 =21199022
ℎ⟨119872el119860
⟩⟨⟨120582el⟩⟩ (76)
The thermal conductivity (73) and the electrical conduc-tivity (76) have the same structure It is always a product of thecorresponding quantum of conductance times the numberof modes that participate in transport times the averagemean-free-path These three quantities for phonons will bediscussed later
32 Specific Heat versus Thermal Conductivity The connec-tion between the lattice specific thermal conductivity andthe lattice specific heat at constant volume is well known[6ndash9] We will show now that corresponding proportionalitycoefficient is a product of an appropriately-defined mean-free-path ⟨⟨Λ ph⟩⟩ and an average phonon velocity ⟨Vph⟩namely
120581119871=1
3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)
The total phonon energy per unit volume
119864ph = int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)
where119863ph(ℏ120596) is the phonon density of states By definition
119862119881equiv
120597119864ph
120597119879=
120597
120597119879int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596)
= int
infin
0
(ℏ120596)119863ph (ℏ120596) (1205971198990(ℏ120596)
120597119879)119889 (ℏ120596)
=12058721198962119879
3int
infin
0
119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(79)
where (55) and (62) were used Next multiply and divide (71)by (79) and obtain the proportionality we are looking for120581119871
= [
[
(1ℎ) intinfin
0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(80)To obtain final expression (77) and correct interpretation
of the proportionality coefficient we need to return to (67)This expression can be easily derived for 1D conductorwith several simplifying assumptions Nevertheless it worksvery well in practice for a conductor of any dimensionDerivation of (67) is based on the interpretation of themean-free-path 120582(119864) or 120582(ℏ120596) as that its inverse value is theprobability per unit length that a positive flux is convertedto a negative flux This is why 120582 is often called a mean-free-path for backscattering Let us relate it to the scattering time120591 The distinction between mean-free-path and mean-free-path for backscattering is easiest to see for 1D conductorLet an electron undergo a scattering event For isotropicscattering the electron can forward scatter or backscatterOnly backscattering is relevant for the mean-free-path forscattering so the time between backscattering events is 2120591Thus the mean-free-path for backscattering is twice themean-free-path for scattering
1205821119863
(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)
It was shown that the proper definition of the mean-free-path for backscattering for a conductor of any dimension [11]is
120582 (119864) = 2
⟨V2119909120591⟩
⟨1003816100381610038161003816V119909
1003816100381610038161003816⟩ (lowast lowast lowast)
Journal of Nanoscience 9
where averaging is performed over angles For isotropicbands
1205822119863
(119864) =120587
2V (119864) 120591 (119864) (81b)
1205823119863
(119864) =4
3V (119864) 120591 (119864) (81c)
The scattering time is often approximately written as thepower law scattering
120591 (119864) = 1205910(119864 minus 119864
119862
119896119879)
119904
(82)
where exponent 119904 describes the specific scattering mecha-nism for acoustic phonon scattering in 3D conductor withparabolic dispersion 119904 = minus12 and for ionized impurityscattering 119904 = +32 [12]
Analogous power law is often used for mean-free-path
120582 (119864) = 1205820(119864 minus 119864
119862
119896119879)
119903
(83)
For parabolic zone structure V(119864) prop 11986412 thus 119903 = 119904 + 12
with 119903 = 0 for acoustic phonon scattering and 119903 = 2 forionized impurity scattering
Coming back to our initial task to derive (77) from (80)for 3D conductor according to (81c) we have
120582ph (ℏ120596) =4
3Vph (ℏ120596) 120591ph (ℏ120596) (84)
where according to (81a)
Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)
and finally
120582ph (ℏ120596) =4
3Λ ph (ℏ120596) (86)
It was stated above in (11c) that the density of states andnumber of modes for electrons in 3D are
119872el (119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (87)
Let us rewrite this formula for phonons Note that the spindegeneracy for electrons 119892
119878= 2 is included with the density
of states
1198633119863
(119864) = 21198631015840
3119863(119864) (88)
and for spherical bands in 3D conductor
⟨V+119909(119864)⟩ =
Vel (119864)2
(89)
Collecting (87) up to (89) all together in case of phonons wehave
119872ph (ℏ120596) = 119860ℎ
2(
Vph (ℏ120596)2
) 2119863ph (ℏ120596)
= 119860ℎ
4Vph (ℏ120596)119863ph (ℏ120596)
(90)
Substituting (86) and (90) into (80) we obtain
120581119871
= [
[
(13) intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(91)
Multiplying and dividing (91) by
int
infin
0
Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)
we finally get (77) with proportionality coefficient between 120581119871
and 119862119881as the product of an average mean-free-path as
⟨⟨Λ ph⟩⟩
equiv
intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(93)
and an average velocity as
⟨Vph⟩ equiv
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(94)
with the appropriate averagingEquation (77) is often used to estimate the average mean-
free-path from the measured 120581119871and 119862
119881 if we know the
average velocity which is frequently assumed to be the lon-gitudinal sound velocity The derivation above has identifiedthe precise definitions of the ⟨⟨Λ ph⟩⟩ and ⟨Vph⟩ If a phonondispersion is chosen one can always compute the averagevelocity according to (94) and it is typically very differentfrom the longitudinal sound velocity Thus estimates of theaverage mean-free-path can be quite wrong if one assumesthe longitudinal sound velocity [13]
33 Debye Model For 3D conductors there are three polar-ization states for lattice vibrations one for atoms displacedin the direction of propagation (longitudinal119871) and two foratoms displaced orthogonally to the direction of propagation(transverse119879) The low energy modes are called acousticmodes119860 one 119871119860 mode analogous to sound waves propa-gating in air and two 119879119860 modes Near 119902 rarr 0 dispersion ofacoustic modes is linear
ℏ120596 = ℏV119863119902 (95)
and is known as Debye approximation The Debye velocityV119863is an average velocity of the 119871 and 119879 acoustic modes In
case of 119871119860mode V119863is simply the sound velocity V
119904prop 119898minus12
with 119898 being an effective mass of vibrating atom TypicallyV119904asymp 5 times 10
3ms about 20 times slower that the velocity of atypical electron
The bandwidth of the electronic dispersion is typicallyBW ≫ 119896119879 so only states near the bottom of the conductionband where the effective mass model is reasonably accurate
10 Journal of Nanoscience
are occupied For phonons the situation ismuch different thebandwidth BW asymp 119896119879 so states across the entire Brillouinzone are occupied The widely used Debye approximation(95) fits the acoustic branches as long as 119902 is not too far fromthe center of the Brillouin zone
With the Debye approximation (95) it is easy to find thedensity of the phonon states
119863ph (ℏ120596) =3(ℏ120596)
2
21205872(ℏV119863)3 [Jminus1 sdotmminus3] (96)
where the factor of three is for the three polarizations Thenone can obtain the number of phonon modes per cross-sectional area from (90)
119872ph (ℏ120596) =3(ℏ120596)
2
4120587(ℏV119863)2 (97)
Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures we are to be sure that we accountfor the correct number of states For a crystal there are 3119873Ω
states per unit volume To find the total number of states wehave to integrate the density of states
int
ℏ120596119863
0
119863ph (ℏ120596) 119889 (ℏ120596) (98)
with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states namely
ℏ120596119863= ℏV119863(61205872119873
Ω)
13
equiv 119896119879119863 (99)
The Debye frequency defines a cutoff frequency abovewhich no states are accounted for This restriction can alsobe expressed via a cutoff wave vector 119902
119863or as a Debye tem-
perature
119879119863=ℏ120596
119896 (100)
For 119879 ≪ 119879119863 only states with 119902 rarr 0 for which the Debye
approximation is accurate are occupiedNow we can calculate the lattice thermal conductivity by
integrating (71) to the Debye cutoff energy
120581119871=12058721198962119879
3ℎint
ℏ120596119863
0
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)
and estimate119872ph(ℏ120596) according to (71) The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used This is how the lattice thermalconductivities were first calculated [14 15] The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11 13 16]
34 Phonon Scattering Phonons can scatter from defectsimpurity atoms isotopes surfaces and boundaries and elec-trons and from other phonons Phonon-phonon scattering
occurs because the potential energy of the bonds in the crystalis not exactly harmonic All higher order terms are treatedas a scattering potential Two types of phonon scattering areconsidered In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved
ℏ 1199021+ ℏ 1199022= ℏ 1199023
ℏ1+ ℏ2= ℏ3
(102)
The total momentum of the phonon ensemble is conservedthus this type of scattering has little effect on the heat flux
In a second type of scattering umklapp119880-scatteringthe two initial phonons have larger momentum thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions The 119880-scatterings are thebasic processes in the heat transport especially at hightemperatures Scattering on defects119863 and on boundaries119861is also important Scattering rates are additive thus the totalphonon scattering rate is
1
120591ph (ℏ120596)=
1
120591119880(ℏ120596)
+1
120591119863(ℏ120596)
+1
120591119861(ℏ120596)
(103)
or alternatively in terms of the mean-free-path (84)
1
120582ph (ℏ120596)=
1
120582119880(ℏ120596)
+1
120582119863(ℏ120596)
+1
120582119861(ℏ120596)
(104)
Expressions for each of the scattering rates are developed[17] For scattering from point defects
1
120591119863(ℏ120596)
prop 1205964 (105)
known as the Rayleigh scattering which is like the scatteringof light from the dust
For boundaries and surfaces
1
120591119861(ℏ120596)
prop
Vph (ℏ120596)119871
(106)
where 119871 is the shortest dimension of the sampleA commonly used expression for 119880-scattering is
1
120591119880(ℏ120596)
prop 11987931205962119890minus119879119863119887119879 (107)
With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity
35 Lattice Thermal Conductivity versus Temperature Thetemperature dependence of the lattice thermal conductivity120581119871is illustrated for bulk Si on Figure 4According to (73) 120581
119871is proportional to the number of the
phononmodes that are occupied ⟨119872ph119860⟩ and to the averagevalue of the phononmean-free-path ⟨⟨120582ph⟩⟩The curve 120581
119871(119879)
can be explained by understanding how ⟨119872ph⟩ and ⟨⟨120582ph⟩⟩vary with temperature
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
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Journal of Nanoscience 5
Deriving a general near-equilibrium current equation isnow straightforward The total current is the sum of thecontributions from each energy mode
119868 = int 1198681015840(119864) 119889119864 (38a)
where the differential current is
1198681015840(119864) =
2119902
ℎ119879 (119864)119872 (119864) (119891
1minus 1198912) (38b)
Using (37) we obtain
1198681015840(119864) = 119866
1015840(119864) Δ119881 + 119878
1015840
119879(119864) Δ119879 (39a)
where
1198661015840(119864) =
21199022
ℎ119879 (119864)119872 (119864) (minus
1205971198910
120597119864) (39b)
is the differential conductance and
1198781015840
119879(119864) = minus
21199022
ℎ119879 (119864)119872 (119864) (
119864 minus 119864119865
119902119879)(minus
1205971198910
120597119864)
= minus119896
119902(119864 minus 119864
119865
119896119879)1198661015840(119864)
(39c)
is the Soret coefficient for electrothermal diffusion in differ-ential formNote that 1198781015840
119879(119864) is negative formodeswith energy
above 119864119865(119899-resistors) and positive for modes with energy
below 119864119865(119901-resistors)
Now we integrate (39a) over all energy modes and find
119868 = 119866Δ119881 + 119878119879Δ119879 [A] (40a)
119868119876= minus 119879119878
119879Δ119881 minus 119870
0Δ119879 [W] (40b)
with three transport coefficients namely conductivity givenby (35a) and (35b) the Soret electrothermal diffusion coeffi-cient
119878119879= int 1198781015840
119879(119864) 119889119864 = minus
119896
119902int(
119864 minus 119864119865
119896119879)1198661015840(119864) 119889119864 [AK]
(40c)
and the electronic heat conductance under the short circuitconditions (Δ119881 = 0)
1198700= 119879(
119896
119902)
2
int(119864 minus 119864
119865
119896119879)
2
1198661015840(119864) 119889119864 [WK] (40d)
where current 119868 is defined to be positive when it flows inconductor from contact 2 to contact 1 with electrons flowingin opposite direction The heat current 119868
119876is positive when it
flows in the +119909 direction out of contact 2Equations (40a) (40b) (40c) and (40d) for long diffusive
resistors can be written in the common form used to describebulk transport as
119869119909= 120590
119889 (119864119865119902)
119889119909minus 119904119879
119889119879
119889119909 [0m2] (41a)
119869119876119909
= 119879119904119879
119889 (119864119865119902)
119889119909minus 1205810
119889119879
119889119909[Wm2] (41b)
with three specific transport coefficients
120590 = int1205901015840(119864) 119889119864
1205901015840(119864) =
21199022
ℎ1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864) [1Ω sdotm sdot J]
(41c)
119904119879= minus
119896
119902int(
119864 minus 119864119865
119896119879)1205901015840(119864) 119889119864 [0m sdot K] (41d)
1205810= (
119896
119902)
2
int(119864 minus 119864
119865
119896119879)
2
1205901015840(119864) 119889119864 [Wm sdot K]
(41e)
These equations have the same form for 1D and 2D resistorsbut the units of the various terms differ
The inverted form of (40a) (40b) (40c) and (40d) isoften preferred in practice namely
Δ119881 = 119877119868 minus 119878Δ119879 (42a)
119868119876= minus Π119868 minus 119870Δ119879 (42b)
where
119878 =119878119879
119866 (42c)
Π = 119879119878 (42d)
119870 = 1198700minus Π119878119866 (42e)
In this form of the equations the contributions from eachenergy mode are not added for example 119877 = int119877(119864)119889119864
Similarly the inverted form of the bulk transport equa-tions (41a) (41b) (41c) (41d) and (41e) becomes
119889 (119864119865119902)
119889119909= 120588119869119909+ 119878
119889119879
119889119909 (43a)
119869119876119909
= 119879119878119869119909minus 120581
119889119879
119889119909(43b)
with transport coefficients
120588 =1
120590 (43c)
119878 =119904119879
120590 (43d)
120581 = 1205810minus 1198782120590119879 (43e)
In summary when a band structure is given number ofmodes can be evaluated from (11a) (11b) and (11c) and if amodel for the mean-free-path for backscattering 120582(119864) can bechosen then the near-equilibrium transport coefficients canbe evaluated using the expressions listed above
23 Bipolar Conduction Let us consider a 3D semiconductorwith parabolic dispersion For the conduction band
1198723119863
(119864) = 119892V119898lowast
2120587ℏ2(119864 minus 119864
119862) (119864 ge 119864
119862) (44a)
6 Journal of Nanoscience
and for the valence band
119872(V)3119863
(119864) = 119892V
119898lowast
119901
2120587ℏ2(119864119881minus 119864) (119864 le 119864
119881) (44b)
The conductivity is provided with two contributions forthe conduction band
120590 =1199022
ℎint
infin
119864119862
1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (45a)
and for the valence band
120590119901=1199022
ℎint
119864119881
minusinfin
119872V3119863
(119864) 120582119901(119864) (minus
1205971198910
120597119864)119889119864 (45b)
The Seebeck coefficient for electrons in the conductionband follows from (41a) (41b) (41c) (41d) and (41e)
120590 = int
infin
119864119862
1205901015840(119864) 119889119864 (46a)
1205901015840(119864) =
21199022
ℎ1198723119863
(119864 minus 119864119862) 120582 (119864) (minus
1205971198910
120597119864) (46b)
119904119879= minus
119896
119902int
infin
119864119862
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 (46c)
119878 =119904119879
120590 (46d)
Similarly for electrons in the valence band we have
120590119901= int
119864119881
minusinfin
1205901015840
119901(119864) 119889119864 (47a)
1205901015840
119901(119864) =
21199022
ℎ119872
V3119863
(119864119881minus 119864) 120582
119901(119864) (minus
1205971198910
120597119864) (47b)
119904(V)119879
= minus119896
119902int
119864119881
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840
119901(119864) 119889119864 (47c)
119878119901=119904(V)119879
120590119901
(47d)
but the sign of 119878119901will be positive
What is going on when both the conduction and valencebands contribute to conductionThis can happen for narrowbandgap conductors or at high temperatures In such a casewe have to simply integrate over all the modes and will find
120590tot
equiv 120590 + 120590119901=1199022
ℎint
1198642
1198641
119872tot3119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (48a)
119872tot3119863
(119864) = 1198723119863
(119864) + 119872V3119863
(119864) (48b)
moreover we do not have to be worried about integratingto the top of the conduction band or from the bottom ofthe valence band because the Fermi function ensures thatthe integrand falls exponentially to zero away from the bandedge What is important is that in both cases we integratethe same expression with the appropriate 119872
3119863(119864) and 120582(119864)
over the relevant energy difference 1198642minus 1198641 Electrons carry
current in both bands Our general expression is the same forthe conduction and valence bandsThere is no need to changesigns for the valence band or to replace 119891
0(119864) with 1 minus 119891
0(119864)
To calculate the Seebeck coefficient when both bandscontribute let us be reminded that in the first direct form ofthe transport coefficients (41a) (41b) (41c) (41d) and (41e)the contributions from eachmode are added in parallel so thetotal specific Soret coefficient
119904tot119879
= minus119896
119902int
+infin
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878
119901120590119901 (49a)
then the Seebeck coefficient for bipolar conduction
119878tot
=
119878120590 + 119878119901120590119901
120590 + 120590119901
(49b)
Since the Seebeck coefficients for the conduction and valencebands have opposite signs the total Seebeck coefficientjust drops for high temperatures and the performance of athermoelectrical device falls down
In summary given a band structure dispersion thenumber of modes can be evaluated and if a model for amean-free-path for backscattering can be established thenthe near-equilibrium transport coefficients can be calculatedusing final expressions listed above
3 Heat Transfer by Phonons
Electrons transfer both charge and heat Electrons carry mostof the heat in metals In semiconductors electrons carry onlya part of the heat but most of the heat is carried by phonons
The phonon heat flux is proportional to the temperaturegradient
119869ph119876119909
= minus120581119871
119889119879
119889119909[Wm2] (50)
with coefficient 120581119871known as the specific lattice thermal
conductivity Such an exceptional thermal conductor likediamond has 120581
119871asymp 2 sdot 10
3Wm sdotK while such a poor thermalconductor like glass has 120581
119871asymp 1Wm sdot K Note that electrical
conductivities of solids vary over more than 20 orders ofmagnitude but thermal conductivities of solids vary over arange of only 3-4 orders of magnitude We will see that thesame methodology used to describe electron transport canbe also used for phonon transport We will also discuss thedifferences between electron and phonon transport For athorough introduction to phonons use classical books [6ndash9]
To describe the phonon current we need an expressionlike for the electron current (1b) written now as
119868 =2119902
ℎint119879el (119864)119872el (119864) (1198911 minus 119891
2) 119889119864 (51)
For electrons the states in the contacts were filled accord-ing to the equilibrium Fermi functions but phonons obeyBose statistics thus the phonon states in the contacts arefilled according to the equilibriumBose-Einstein distribution
1198990(ℏ120596) =
1
119890ℏ120596119896119879 minus 1 (52)
Journal of Nanoscience 7
Let temperature for the left and the right contacts be1198791and 119879
2 As for the electrons both contacts are assumed
ideal Thus the phonons that enter a contact are not able toreflect back and transmission coefficient119879ph(119864) describes thephonon transmission across the entire channel
It is easy now to rewrite (51) to the phonon heat currentElectron energy 119864 we replace by the phonon energy ℏ120596 Inthe electron current we have charge 119902moving in the channelin case of the phonon current the quantum of energy ℏ120596
is moving instead thus we replace 119902 in (51) with ℏ120596 andmove it inside the integral The coefficient 2 in (51) reflectsthe spin degeneracy of an electron In case of the phononswe remove this coefficient and instead the number of thephonon polarization states that contribute to the heat flowlets us include to the number of the phononmodes119872ph(ℏ120596)Finally the heat current due to phonons is
119876 =1
ℎint (ℏ120596) 119879ph (ℏ120596)119872ph (ℏ120596) (1198991 minus 119899
2) 119889 (ℏ120596) [W]
(53)
In the linear response regime by analogy with (26)
1198991minus 1198992asymp minus
1205971198990
120597119879Δ119879 (54)
where the derivative according to (52) is
1205971198990
120597119879=ℏ120596
119879(minus
1205971198990
120597 (ℏ120596)) (55)
with
1205971198990
120597 (ℏ120596)= (minus
1
119896119879)
119890ℏ120596119896119879
(119890ℏ120596119896119879 minus 1)2 (56)
Now (53) for small differences in temperature becomes
119876 = minus119870119871Δ119879 (57)
where the thermal conductance
119870119871=1198962119879
ℎint119879ph (ℏ120596)119872ph (ℏ120596)
times [(ℏ120596
119896119879)
2
(minus1205971198990
120597 (ℏ120596))] 119889 (ℏ120596) [WK]
(58)
Equation (57) is simply Fourierrsquos law stating that heatflows down to a temperature gradient It is also useful tonote that the thermal conductance (58) displays certainsimilarities with the electrical conductance
119866 =21199022
ℎint119879el (119864)119872el (119864) (minus
1205971198910
120597119864)119889119864 (59)
The derivative
119882el (119864) equiv (minus1205971198910
120597119864) (60)
known as the Fermi window function that picks out thoseconduction modes which only contribute to the electriccurrent The electron windows function is normalized
int
+infin
minusinfin
(minus1205971198910
120597119864)119889119864 = 1 (61)
In case of phonons the term in square brackets of (58) actsas a window function to specify which modes carry the heatcurrent After normalization
119882ph (ℏ120596) =3
1205872(ℏ120596
119896119879)(
1205971198990
120597 (ℏ120596)) (62)
thus finally
119870119871=12058721198962119879
3ℎint119879ph (ℏ120596)119872ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (63)
with
1198920equiv12058721198962119879
3ℎasymp (9456 times 10
minus13WK2) 119879 (64)
known as the quantum of thermal conductance experimen-tally observed first in 2000 [10]
Comparing (59) and (63) one can see that the electricaland thermal conductances are similar in structure bothare proportional to corresponding quantum of conductancetimes an integral over the transmission times the number ofmodes times a window function
The thermal broadening functions for electrons andphonons have similar shapes and each has a width of a few119896119879 In case of electrons this function is given by (27) or
119865119879(119909) equiv
119890119909
(119890119909 + 1)
2(65)
with 119909 equiv (119864minus119864119865)119896119879 and is shown on Figure 2This function
for phonons is given by (62) or
119865ph119879(119909) equiv
3
1205872
1199092119890119909
(119890119909 minus 1)
2(66)
with 119909 equiv ℏ120596119896119879 Both functions are normalized to a unityand shown together on Figure 3
Along with the number of modes determined by thedispersion relation these two window functions play a keyrole in determining the electrical and thermal conductances
31 Thermal Conductivity of the Bulk Conductors The ther-mal conductivity of a large diffusive resistor is a key materialproperty that controls performance of any electronic devicesBy analogy with the transmission coefficient (3) for electrontransport the phonon transmission
119879ph (ℏ120596) =120582ph (ℏ120596)
120582ph (ℏ120596) + 119871
100381610038161003816100381610038161003816100381610038161003816119871≫120582ph
997888rarr
120582ph (ℏ120596)
119871 (67)
It is also obvious that for large 3D conductors the numberof phonon modes is proportional to the cross-sectional areaof the sample
119872ph (ℏ120596) prop 119860 (68)
8 Journal of Nanoscience
10
5
0
minus5
minus100 01 02 03
FT(x)
FT(x)
FT(x)
phx
Figure 3 Broadening function for phonons compared to that ofelectrons
Now let us return to (57) dividing and multiplying it by119860119871 which immediately gives (50) for the phonon heat fluxpostulated above
119876
119860equiv 119869
ph119876119909
= minus120581119871
119889119879
119889119909(69)
with specific lattice thermal conductivity
120581119871= 119870119871
119871
119860 (70)
or substituting (67) into (63) one for the lattice thermalconductivity finally obtains
120581119871=12058721198962119879
3ℎint
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)
It is useful now to define the average number of phononmodes per cross-sectional area of the conductor that partici-pate in the heat transport
⟨
119872ph
119860⟩ equiv int
119872ph (ℏ120596)
119860119882ph (ℏ120596) 119889 (ℏ120596) (72)
Then
120581119871=12058721198962119879
3ℎ⟨
119872ph
119860⟩⟨⟨120582ph⟩⟩ (73)
where the average mean-free-path is defined now as
⟨⟨120582ph⟩⟩ =
int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)
(74)
Thus the couple of the phonon transport equations (69)and (73) corresponds to similar electron transport equations
119869119909=120590
119902
119889 (119864119865)
119889119909 (75)
120590 =21199022
ℎ⟨119872el119860
⟩⟨⟨120582el⟩⟩ (76)
The thermal conductivity (73) and the electrical conduc-tivity (76) have the same structure It is always a product of thecorresponding quantum of conductance times the numberof modes that participate in transport times the averagemean-free-path These three quantities for phonons will bediscussed later
32 Specific Heat versus Thermal Conductivity The connec-tion between the lattice specific thermal conductivity andthe lattice specific heat at constant volume is well known[6ndash9] We will show now that corresponding proportionalitycoefficient is a product of an appropriately-defined mean-free-path ⟨⟨Λ ph⟩⟩ and an average phonon velocity ⟨Vph⟩namely
120581119871=1
3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)
The total phonon energy per unit volume
119864ph = int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)
where119863ph(ℏ120596) is the phonon density of states By definition
119862119881equiv
120597119864ph
120597119879=
120597
120597119879int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596)
= int
infin
0
(ℏ120596)119863ph (ℏ120596) (1205971198990(ℏ120596)
120597119879)119889 (ℏ120596)
=12058721198962119879
3int
infin
0
119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(79)
where (55) and (62) were used Next multiply and divide (71)by (79) and obtain the proportionality we are looking for120581119871
= [
[
(1ℎ) intinfin
0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(80)To obtain final expression (77) and correct interpretation
of the proportionality coefficient we need to return to (67)This expression can be easily derived for 1D conductorwith several simplifying assumptions Nevertheless it worksvery well in practice for a conductor of any dimensionDerivation of (67) is based on the interpretation of themean-free-path 120582(119864) or 120582(ℏ120596) as that its inverse value is theprobability per unit length that a positive flux is convertedto a negative flux This is why 120582 is often called a mean-free-path for backscattering Let us relate it to the scattering time120591 The distinction between mean-free-path and mean-free-path for backscattering is easiest to see for 1D conductorLet an electron undergo a scattering event For isotropicscattering the electron can forward scatter or backscatterOnly backscattering is relevant for the mean-free-path forscattering so the time between backscattering events is 2120591Thus the mean-free-path for backscattering is twice themean-free-path for scattering
1205821119863
(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)
It was shown that the proper definition of the mean-free-path for backscattering for a conductor of any dimension [11]is
120582 (119864) = 2
⟨V2119909120591⟩
⟨1003816100381610038161003816V119909
1003816100381610038161003816⟩ (lowast lowast lowast)
Journal of Nanoscience 9
where averaging is performed over angles For isotropicbands
1205822119863
(119864) =120587
2V (119864) 120591 (119864) (81b)
1205823119863
(119864) =4
3V (119864) 120591 (119864) (81c)
The scattering time is often approximately written as thepower law scattering
120591 (119864) = 1205910(119864 minus 119864
119862
119896119879)
119904
(82)
where exponent 119904 describes the specific scattering mecha-nism for acoustic phonon scattering in 3D conductor withparabolic dispersion 119904 = minus12 and for ionized impurityscattering 119904 = +32 [12]
Analogous power law is often used for mean-free-path
120582 (119864) = 1205820(119864 minus 119864
119862
119896119879)
119903
(83)
For parabolic zone structure V(119864) prop 11986412 thus 119903 = 119904 + 12
with 119903 = 0 for acoustic phonon scattering and 119903 = 2 forionized impurity scattering
Coming back to our initial task to derive (77) from (80)for 3D conductor according to (81c) we have
120582ph (ℏ120596) =4
3Vph (ℏ120596) 120591ph (ℏ120596) (84)
where according to (81a)
Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)
and finally
120582ph (ℏ120596) =4
3Λ ph (ℏ120596) (86)
It was stated above in (11c) that the density of states andnumber of modes for electrons in 3D are
119872el (119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (87)
Let us rewrite this formula for phonons Note that the spindegeneracy for electrons 119892
119878= 2 is included with the density
of states
1198633119863
(119864) = 21198631015840
3119863(119864) (88)
and for spherical bands in 3D conductor
⟨V+119909(119864)⟩ =
Vel (119864)2
(89)
Collecting (87) up to (89) all together in case of phonons wehave
119872ph (ℏ120596) = 119860ℎ
2(
Vph (ℏ120596)2
) 2119863ph (ℏ120596)
= 119860ℎ
4Vph (ℏ120596)119863ph (ℏ120596)
(90)
Substituting (86) and (90) into (80) we obtain
120581119871
= [
[
(13) intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(91)
Multiplying and dividing (91) by
int
infin
0
Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)
we finally get (77) with proportionality coefficient between 120581119871
and 119862119881as the product of an average mean-free-path as
⟨⟨Λ ph⟩⟩
equiv
intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(93)
and an average velocity as
⟨Vph⟩ equiv
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(94)
with the appropriate averagingEquation (77) is often used to estimate the average mean-
free-path from the measured 120581119871and 119862
119881 if we know the
average velocity which is frequently assumed to be the lon-gitudinal sound velocity The derivation above has identifiedthe precise definitions of the ⟨⟨Λ ph⟩⟩ and ⟨Vph⟩ If a phonondispersion is chosen one can always compute the averagevelocity according to (94) and it is typically very differentfrom the longitudinal sound velocity Thus estimates of theaverage mean-free-path can be quite wrong if one assumesthe longitudinal sound velocity [13]
33 Debye Model For 3D conductors there are three polar-ization states for lattice vibrations one for atoms displacedin the direction of propagation (longitudinal119871) and two foratoms displaced orthogonally to the direction of propagation(transverse119879) The low energy modes are called acousticmodes119860 one 119871119860 mode analogous to sound waves propa-gating in air and two 119879119860 modes Near 119902 rarr 0 dispersion ofacoustic modes is linear
ℏ120596 = ℏV119863119902 (95)
and is known as Debye approximation The Debye velocityV119863is an average velocity of the 119871 and 119879 acoustic modes In
case of 119871119860mode V119863is simply the sound velocity V
119904prop 119898minus12
with 119898 being an effective mass of vibrating atom TypicallyV119904asymp 5 times 10
3ms about 20 times slower that the velocity of atypical electron
The bandwidth of the electronic dispersion is typicallyBW ≫ 119896119879 so only states near the bottom of the conductionband where the effective mass model is reasonably accurate
10 Journal of Nanoscience
are occupied For phonons the situation ismuch different thebandwidth BW asymp 119896119879 so states across the entire Brillouinzone are occupied The widely used Debye approximation(95) fits the acoustic branches as long as 119902 is not too far fromthe center of the Brillouin zone
With the Debye approximation (95) it is easy to find thedensity of the phonon states
119863ph (ℏ120596) =3(ℏ120596)
2
21205872(ℏV119863)3 [Jminus1 sdotmminus3] (96)
where the factor of three is for the three polarizations Thenone can obtain the number of phonon modes per cross-sectional area from (90)
119872ph (ℏ120596) =3(ℏ120596)
2
4120587(ℏV119863)2 (97)
Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures we are to be sure that we accountfor the correct number of states For a crystal there are 3119873Ω
states per unit volume To find the total number of states wehave to integrate the density of states
int
ℏ120596119863
0
119863ph (ℏ120596) 119889 (ℏ120596) (98)
with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states namely
ℏ120596119863= ℏV119863(61205872119873
Ω)
13
equiv 119896119879119863 (99)
The Debye frequency defines a cutoff frequency abovewhich no states are accounted for This restriction can alsobe expressed via a cutoff wave vector 119902
119863or as a Debye tem-
perature
119879119863=ℏ120596
119896 (100)
For 119879 ≪ 119879119863 only states with 119902 rarr 0 for which the Debye
approximation is accurate are occupiedNow we can calculate the lattice thermal conductivity by
integrating (71) to the Debye cutoff energy
120581119871=12058721198962119879
3ℎint
ℏ120596119863
0
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)
and estimate119872ph(ℏ120596) according to (71) The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used This is how the lattice thermalconductivities were first calculated [14 15] The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11 13 16]
34 Phonon Scattering Phonons can scatter from defectsimpurity atoms isotopes surfaces and boundaries and elec-trons and from other phonons Phonon-phonon scattering
occurs because the potential energy of the bonds in the crystalis not exactly harmonic All higher order terms are treatedas a scattering potential Two types of phonon scattering areconsidered In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved
ℏ 1199021+ ℏ 1199022= ℏ 1199023
ℏ1+ ℏ2= ℏ3
(102)
The total momentum of the phonon ensemble is conservedthus this type of scattering has little effect on the heat flux
In a second type of scattering umklapp119880-scatteringthe two initial phonons have larger momentum thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions The 119880-scatterings are thebasic processes in the heat transport especially at hightemperatures Scattering on defects119863 and on boundaries119861is also important Scattering rates are additive thus the totalphonon scattering rate is
1
120591ph (ℏ120596)=
1
120591119880(ℏ120596)
+1
120591119863(ℏ120596)
+1
120591119861(ℏ120596)
(103)
or alternatively in terms of the mean-free-path (84)
1
120582ph (ℏ120596)=
1
120582119880(ℏ120596)
+1
120582119863(ℏ120596)
+1
120582119861(ℏ120596)
(104)
Expressions for each of the scattering rates are developed[17] For scattering from point defects
1
120591119863(ℏ120596)
prop 1205964 (105)
known as the Rayleigh scattering which is like the scatteringof light from the dust
For boundaries and surfaces
1
120591119861(ℏ120596)
prop
Vph (ℏ120596)119871
(106)
where 119871 is the shortest dimension of the sampleA commonly used expression for 119880-scattering is
1
120591119880(ℏ120596)
prop 11987931205962119890minus119879119863119887119879 (107)
With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity
35 Lattice Thermal Conductivity versus Temperature Thetemperature dependence of the lattice thermal conductivity120581119871is illustrated for bulk Si on Figure 4According to (73) 120581
119871is proportional to the number of the
phononmodes that are occupied ⟨119872ph119860⟩ and to the averagevalue of the phononmean-free-path ⟨⟨120582ph⟩⟩The curve 120581
119871(119879)
can be explained by understanding how ⟨119872ph⟩ and ⟨⟨120582ph⟩⟩vary with temperature
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
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Journal ofNanomaterials
6 Journal of Nanoscience
and for the valence band
119872(V)3119863
(119864) = 119892V
119898lowast
119901
2120587ℏ2(119864119881minus 119864) (119864 le 119864
119881) (44b)
The conductivity is provided with two contributions forthe conduction band
120590 =1199022
ℎint
infin
119864119862
1198723119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (45a)
and for the valence band
120590119901=1199022
ℎint
119864119881
minusinfin
119872V3119863
(119864) 120582119901(119864) (minus
1205971198910
120597119864)119889119864 (45b)
The Seebeck coefficient for electrons in the conductionband follows from (41a) (41b) (41c) (41d) and (41e)
120590 = int
infin
119864119862
1205901015840(119864) 119889119864 (46a)
1205901015840(119864) =
21199022
ℎ1198723119863
(119864 minus 119864119862) 120582 (119864) (minus
1205971198910
120597119864) (46b)
119904119879= minus
119896
119902int
infin
119864119862
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 (46c)
119878 =119904119879
120590 (46d)
Similarly for electrons in the valence band we have
120590119901= int
119864119881
minusinfin
1205901015840
119901(119864) 119889119864 (47a)
1205901015840
119901(119864) =
21199022
ℎ119872
V3119863
(119864119881minus 119864) 120582
119901(119864) (minus
1205971198910
120597119864) (47b)
119904(V)119879
= minus119896
119902int
119864119881
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840
119901(119864) 119889119864 (47c)
119878119901=119904(V)119879
120590119901
(47d)
but the sign of 119878119901will be positive
What is going on when both the conduction and valencebands contribute to conductionThis can happen for narrowbandgap conductors or at high temperatures In such a casewe have to simply integrate over all the modes and will find
120590tot
equiv 120590 + 120590119901=1199022
ℎint
1198642
1198641
119872tot3119863
(119864) 120582 (119864) (minus1205971198910
120597119864)119889119864 (48a)
119872tot3119863
(119864) = 1198723119863
(119864) + 119872V3119863
(119864) (48b)
moreover we do not have to be worried about integratingto the top of the conduction band or from the bottom ofthe valence band because the Fermi function ensures thatthe integrand falls exponentially to zero away from the bandedge What is important is that in both cases we integratethe same expression with the appropriate 119872
3119863(119864) and 120582(119864)
over the relevant energy difference 1198642minus 1198641 Electrons carry
current in both bands Our general expression is the same forthe conduction and valence bandsThere is no need to changesigns for the valence band or to replace 119891
0(119864) with 1 minus 119891
0(119864)
To calculate the Seebeck coefficient when both bandscontribute let us be reminded that in the first direct form ofthe transport coefficients (41a) (41b) (41c) (41d) and (41e)the contributions from eachmode are added in parallel so thetotal specific Soret coefficient
119904tot119879
= minus119896
119902int
+infin
minusinfin
(119864 minus 119864
119865
119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878
119901120590119901 (49a)
then the Seebeck coefficient for bipolar conduction
119878tot
=
119878120590 + 119878119901120590119901
120590 + 120590119901
(49b)
Since the Seebeck coefficients for the conduction and valencebands have opposite signs the total Seebeck coefficientjust drops for high temperatures and the performance of athermoelectrical device falls down
In summary given a band structure dispersion thenumber of modes can be evaluated and if a model for amean-free-path for backscattering can be established thenthe near-equilibrium transport coefficients can be calculatedusing final expressions listed above
3 Heat Transfer by Phonons
Electrons transfer both charge and heat Electrons carry mostof the heat in metals In semiconductors electrons carry onlya part of the heat but most of the heat is carried by phonons
The phonon heat flux is proportional to the temperaturegradient
119869ph119876119909
= minus120581119871
119889119879
119889119909[Wm2] (50)
with coefficient 120581119871known as the specific lattice thermal
conductivity Such an exceptional thermal conductor likediamond has 120581
119871asymp 2 sdot 10
3Wm sdotK while such a poor thermalconductor like glass has 120581
119871asymp 1Wm sdot K Note that electrical
conductivities of solids vary over more than 20 orders ofmagnitude but thermal conductivities of solids vary over arange of only 3-4 orders of magnitude We will see that thesame methodology used to describe electron transport canbe also used for phonon transport We will also discuss thedifferences between electron and phonon transport For athorough introduction to phonons use classical books [6ndash9]
To describe the phonon current we need an expressionlike for the electron current (1b) written now as
119868 =2119902
ℎint119879el (119864)119872el (119864) (1198911 minus 119891
2) 119889119864 (51)
For electrons the states in the contacts were filled accord-ing to the equilibrium Fermi functions but phonons obeyBose statistics thus the phonon states in the contacts arefilled according to the equilibriumBose-Einstein distribution
1198990(ℏ120596) =
1
119890ℏ120596119896119879 minus 1 (52)
Journal of Nanoscience 7
Let temperature for the left and the right contacts be1198791and 119879
2 As for the electrons both contacts are assumed
ideal Thus the phonons that enter a contact are not able toreflect back and transmission coefficient119879ph(119864) describes thephonon transmission across the entire channel
It is easy now to rewrite (51) to the phonon heat currentElectron energy 119864 we replace by the phonon energy ℏ120596 Inthe electron current we have charge 119902moving in the channelin case of the phonon current the quantum of energy ℏ120596
is moving instead thus we replace 119902 in (51) with ℏ120596 andmove it inside the integral The coefficient 2 in (51) reflectsthe spin degeneracy of an electron In case of the phononswe remove this coefficient and instead the number of thephonon polarization states that contribute to the heat flowlets us include to the number of the phononmodes119872ph(ℏ120596)Finally the heat current due to phonons is
119876 =1
ℎint (ℏ120596) 119879ph (ℏ120596)119872ph (ℏ120596) (1198991 minus 119899
2) 119889 (ℏ120596) [W]
(53)
In the linear response regime by analogy with (26)
1198991minus 1198992asymp minus
1205971198990
120597119879Δ119879 (54)
where the derivative according to (52) is
1205971198990
120597119879=ℏ120596
119879(minus
1205971198990
120597 (ℏ120596)) (55)
with
1205971198990
120597 (ℏ120596)= (minus
1
119896119879)
119890ℏ120596119896119879
(119890ℏ120596119896119879 minus 1)2 (56)
Now (53) for small differences in temperature becomes
119876 = minus119870119871Δ119879 (57)
where the thermal conductance
119870119871=1198962119879
ℎint119879ph (ℏ120596)119872ph (ℏ120596)
times [(ℏ120596
119896119879)
2
(minus1205971198990
120597 (ℏ120596))] 119889 (ℏ120596) [WK]
(58)
Equation (57) is simply Fourierrsquos law stating that heatflows down to a temperature gradient It is also useful tonote that the thermal conductance (58) displays certainsimilarities with the electrical conductance
119866 =21199022
ℎint119879el (119864)119872el (119864) (minus
1205971198910
120597119864)119889119864 (59)
The derivative
119882el (119864) equiv (minus1205971198910
120597119864) (60)
known as the Fermi window function that picks out thoseconduction modes which only contribute to the electriccurrent The electron windows function is normalized
int
+infin
minusinfin
(minus1205971198910
120597119864)119889119864 = 1 (61)
In case of phonons the term in square brackets of (58) actsas a window function to specify which modes carry the heatcurrent After normalization
119882ph (ℏ120596) =3
1205872(ℏ120596
119896119879)(
1205971198990
120597 (ℏ120596)) (62)
thus finally
119870119871=12058721198962119879
3ℎint119879ph (ℏ120596)119872ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (63)
with
1198920equiv12058721198962119879
3ℎasymp (9456 times 10
minus13WK2) 119879 (64)
known as the quantum of thermal conductance experimen-tally observed first in 2000 [10]
Comparing (59) and (63) one can see that the electricaland thermal conductances are similar in structure bothare proportional to corresponding quantum of conductancetimes an integral over the transmission times the number ofmodes times a window function
The thermal broadening functions for electrons andphonons have similar shapes and each has a width of a few119896119879 In case of electrons this function is given by (27) or
119865119879(119909) equiv
119890119909
(119890119909 + 1)
2(65)
with 119909 equiv (119864minus119864119865)119896119879 and is shown on Figure 2This function
for phonons is given by (62) or
119865ph119879(119909) equiv
3
1205872
1199092119890119909
(119890119909 minus 1)
2(66)
with 119909 equiv ℏ120596119896119879 Both functions are normalized to a unityand shown together on Figure 3
Along with the number of modes determined by thedispersion relation these two window functions play a keyrole in determining the electrical and thermal conductances
31 Thermal Conductivity of the Bulk Conductors The ther-mal conductivity of a large diffusive resistor is a key materialproperty that controls performance of any electronic devicesBy analogy with the transmission coefficient (3) for electrontransport the phonon transmission
119879ph (ℏ120596) =120582ph (ℏ120596)
120582ph (ℏ120596) + 119871
100381610038161003816100381610038161003816100381610038161003816119871≫120582ph
997888rarr
120582ph (ℏ120596)
119871 (67)
It is also obvious that for large 3D conductors the numberof phonon modes is proportional to the cross-sectional areaof the sample
119872ph (ℏ120596) prop 119860 (68)
8 Journal of Nanoscience
10
5
0
minus5
minus100 01 02 03
FT(x)
FT(x)
FT(x)
phx
Figure 3 Broadening function for phonons compared to that ofelectrons
Now let us return to (57) dividing and multiplying it by119860119871 which immediately gives (50) for the phonon heat fluxpostulated above
119876
119860equiv 119869
ph119876119909
= minus120581119871
119889119879
119889119909(69)
with specific lattice thermal conductivity
120581119871= 119870119871
119871
119860 (70)
or substituting (67) into (63) one for the lattice thermalconductivity finally obtains
120581119871=12058721198962119879
3ℎint
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)
It is useful now to define the average number of phononmodes per cross-sectional area of the conductor that partici-pate in the heat transport
⟨
119872ph
119860⟩ equiv int
119872ph (ℏ120596)
119860119882ph (ℏ120596) 119889 (ℏ120596) (72)
Then
120581119871=12058721198962119879
3ℎ⟨
119872ph
119860⟩⟨⟨120582ph⟩⟩ (73)
where the average mean-free-path is defined now as
⟨⟨120582ph⟩⟩ =
int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)
(74)
Thus the couple of the phonon transport equations (69)and (73) corresponds to similar electron transport equations
119869119909=120590
119902
119889 (119864119865)
119889119909 (75)
120590 =21199022
ℎ⟨119872el119860
⟩⟨⟨120582el⟩⟩ (76)
The thermal conductivity (73) and the electrical conduc-tivity (76) have the same structure It is always a product of thecorresponding quantum of conductance times the numberof modes that participate in transport times the averagemean-free-path These three quantities for phonons will bediscussed later
32 Specific Heat versus Thermal Conductivity The connec-tion between the lattice specific thermal conductivity andthe lattice specific heat at constant volume is well known[6ndash9] We will show now that corresponding proportionalitycoefficient is a product of an appropriately-defined mean-free-path ⟨⟨Λ ph⟩⟩ and an average phonon velocity ⟨Vph⟩namely
120581119871=1
3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)
The total phonon energy per unit volume
119864ph = int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)
where119863ph(ℏ120596) is the phonon density of states By definition
119862119881equiv
120597119864ph
120597119879=
120597
120597119879int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596)
= int
infin
0
(ℏ120596)119863ph (ℏ120596) (1205971198990(ℏ120596)
120597119879)119889 (ℏ120596)
=12058721198962119879
3int
infin
0
119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(79)
where (55) and (62) were used Next multiply and divide (71)by (79) and obtain the proportionality we are looking for120581119871
= [
[
(1ℎ) intinfin
0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(80)To obtain final expression (77) and correct interpretation
of the proportionality coefficient we need to return to (67)This expression can be easily derived for 1D conductorwith several simplifying assumptions Nevertheless it worksvery well in practice for a conductor of any dimensionDerivation of (67) is based on the interpretation of themean-free-path 120582(119864) or 120582(ℏ120596) as that its inverse value is theprobability per unit length that a positive flux is convertedto a negative flux This is why 120582 is often called a mean-free-path for backscattering Let us relate it to the scattering time120591 The distinction between mean-free-path and mean-free-path for backscattering is easiest to see for 1D conductorLet an electron undergo a scattering event For isotropicscattering the electron can forward scatter or backscatterOnly backscattering is relevant for the mean-free-path forscattering so the time between backscattering events is 2120591Thus the mean-free-path for backscattering is twice themean-free-path for scattering
1205821119863
(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)
It was shown that the proper definition of the mean-free-path for backscattering for a conductor of any dimension [11]is
120582 (119864) = 2
⟨V2119909120591⟩
⟨1003816100381610038161003816V119909
1003816100381610038161003816⟩ (lowast lowast lowast)
Journal of Nanoscience 9
where averaging is performed over angles For isotropicbands
1205822119863
(119864) =120587
2V (119864) 120591 (119864) (81b)
1205823119863
(119864) =4
3V (119864) 120591 (119864) (81c)
The scattering time is often approximately written as thepower law scattering
120591 (119864) = 1205910(119864 minus 119864
119862
119896119879)
119904
(82)
where exponent 119904 describes the specific scattering mecha-nism for acoustic phonon scattering in 3D conductor withparabolic dispersion 119904 = minus12 and for ionized impurityscattering 119904 = +32 [12]
Analogous power law is often used for mean-free-path
120582 (119864) = 1205820(119864 minus 119864
119862
119896119879)
119903
(83)
For parabolic zone structure V(119864) prop 11986412 thus 119903 = 119904 + 12
with 119903 = 0 for acoustic phonon scattering and 119903 = 2 forionized impurity scattering
Coming back to our initial task to derive (77) from (80)for 3D conductor according to (81c) we have
120582ph (ℏ120596) =4
3Vph (ℏ120596) 120591ph (ℏ120596) (84)
where according to (81a)
Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)
and finally
120582ph (ℏ120596) =4
3Λ ph (ℏ120596) (86)
It was stated above in (11c) that the density of states andnumber of modes for electrons in 3D are
119872el (119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (87)
Let us rewrite this formula for phonons Note that the spindegeneracy for electrons 119892
119878= 2 is included with the density
of states
1198633119863
(119864) = 21198631015840
3119863(119864) (88)
and for spherical bands in 3D conductor
⟨V+119909(119864)⟩ =
Vel (119864)2
(89)
Collecting (87) up to (89) all together in case of phonons wehave
119872ph (ℏ120596) = 119860ℎ
2(
Vph (ℏ120596)2
) 2119863ph (ℏ120596)
= 119860ℎ
4Vph (ℏ120596)119863ph (ℏ120596)
(90)
Substituting (86) and (90) into (80) we obtain
120581119871
= [
[
(13) intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(91)
Multiplying and dividing (91) by
int
infin
0
Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)
we finally get (77) with proportionality coefficient between 120581119871
and 119862119881as the product of an average mean-free-path as
⟨⟨Λ ph⟩⟩
equiv
intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(93)
and an average velocity as
⟨Vph⟩ equiv
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(94)
with the appropriate averagingEquation (77) is often used to estimate the average mean-
free-path from the measured 120581119871and 119862
119881 if we know the
average velocity which is frequently assumed to be the lon-gitudinal sound velocity The derivation above has identifiedthe precise definitions of the ⟨⟨Λ ph⟩⟩ and ⟨Vph⟩ If a phonondispersion is chosen one can always compute the averagevelocity according to (94) and it is typically very differentfrom the longitudinal sound velocity Thus estimates of theaverage mean-free-path can be quite wrong if one assumesthe longitudinal sound velocity [13]
33 Debye Model For 3D conductors there are three polar-ization states for lattice vibrations one for atoms displacedin the direction of propagation (longitudinal119871) and two foratoms displaced orthogonally to the direction of propagation(transverse119879) The low energy modes are called acousticmodes119860 one 119871119860 mode analogous to sound waves propa-gating in air and two 119879119860 modes Near 119902 rarr 0 dispersion ofacoustic modes is linear
ℏ120596 = ℏV119863119902 (95)
and is known as Debye approximation The Debye velocityV119863is an average velocity of the 119871 and 119879 acoustic modes In
case of 119871119860mode V119863is simply the sound velocity V
119904prop 119898minus12
with 119898 being an effective mass of vibrating atom TypicallyV119904asymp 5 times 10
3ms about 20 times slower that the velocity of atypical electron
The bandwidth of the electronic dispersion is typicallyBW ≫ 119896119879 so only states near the bottom of the conductionband where the effective mass model is reasonably accurate
10 Journal of Nanoscience
are occupied For phonons the situation ismuch different thebandwidth BW asymp 119896119879 so states across the entire Brillouinzone are occupied The widely used Debye approximation(95) fits the acoustic branches as long as 119902 is not too far fromthe center of the Brillouin zone
With the Debye approximation (95) it is easy to find thedensity of the phonon states
119863ph (ℏ120596) =3(ℏ120596)
2
21205872(ℏV119863)3 [Jminus1 sdotmminus3] (96)
where the factor of three is for the three polarizations Thenone can obtain the number of phonon modes per cross-sectional area from (90)
119872ph (ℏ120596) =3(ℏ120596)
2
4120587(ℏV119863)2 (97)
Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures we are to be sure that we accountfor the correct number of states For a crystal there are 3119873Ω
states per unit volume To find the total number of states wehave to integrate the density of states
int
ℏ120596119863
0
119863ph (ℏ120596) 119889 (ℏ120596) (98)
with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states namely
ℏ120596119863= ℏV119863(61205872119873
Ω)
13
equiv 119896119879119863 (99)
The Debye frequency defines a cutoff frequency abovewhich no states are accounted for This restriction can alsobe expressed via a cutoff wave vector 119902
119863or as a Debye tem-
perature
119879119863=ℏ120596
119896 (100)
For 119879 ≪ 119879119863 only states with 119902 rarr 0 for which the Debye
approximation is accurate are occupiedNow we can calculate the lattice thermal conductivity by
integrating (71) to the Debye cutoff energy
120581119871=12058721198962119879
3ℎint
ℏ120596119863
0
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)
and estimate119872ph(ℏ120596) according to (71) The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used This is how the lattice thermalconductivities were first calculated [14 15] The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11 13 16]
34 Phonon Scattering Phonons can scatter from defectsimpurity atoms isotopes surfaces and boundaries and elec-trons and from other phonons Phonon-phonon scattering
occurs because the potential energy of the bonds in the crystalis not exactly harmonic All higher order terms are treatedas a scattering potential Two types of phonon scattering areconsidered In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved
ℏ 1199021+ ℏ 1199022= ℏ 1199023
ℏ1+ ℏ2= ℏ3
(102)
The total momentum of the phonon ensemble is conservedthus this type of scattering has little effect on the heat flux
In a second type of scattering umklapp119880-scatteringthe two initial phonons have larger momentum thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions The 119880-scatterings are thebasic processes in the heat transport especially at hightemperatures Scattering on defects119863 and on boundaries119861is also important Scattering rates are additive thus the totalphonon scattering rate is
1
120591ph (ℏ120596)=
1
120591119880(ℏ120596)
+1
120591119863(ℏ120596)
+1
120591119861(ℏ120596)
(103)
or alternatively in terms of the mean-free-path (84)
1
120582ph (ℏ120596)=
1
120582119880(ℏ120596)
+1
120582119863(ℏ120596)
+1
120582119861(ℏ120596)
(104)
Expressions for each of the scattering rates are developed[17] For scattering from point defects
1
120591119863(ℏ120596)
prop 1205964 (105)
known as the Rayleigh scattering which is like the scatteringof light from the dust
For boundaries and surfaces
1
120591119861(ℏ120596)
prop
Vph (ℏ120596)119871
(106)
where 119871 is the shortest dimension of the sampleA commonly used expression for 119880-scattering is
1
120591119880(ℏ120596)
prop 11987931205962119890minus119879119863119887119879 (107)
With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity
35 Lattice Thermal Conductivity versus Temperature Thetemperature dependence of the lattice thermal conductivity120581119871is illustrated for bulk Si on Figure 4According to (73) 120581
119871is proportional to the number of the
phononmodes that are occupied ⟨119872ph119860⟩ and to the averagevalue of the phononmean-free-path ⟨⟨120582ph⟩⟩The curve 120581
119871(119879)
can be explained by understanding how ⟨119872ph⟩ and ⟨⟨120582ph⟩⟩vary with temperature
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
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Journal of Nanoscience 7
Let temperature for the left and the right contacts be1198791and 119879
2 As for the electrons both contacts are assumed
ideal Thus the phonons that enter a contact are not able toreflect back and transmission coefficient119879ph(119864) describes thephonon transmission across the entire channel
It is easy now to rewrite (51) to the phonon heat currentElectron energy 119864 we replace by the phonon energy ℏ120596 Inthe electron current we have charge 119902moving in the channelin case of the phonon current the quantum of energy ℏ120596
is moving instead thus we replace 119902 in (51) with ℏ120596 andmove it inside the integral The coefficient 2 in (51) reflectsthe spin degeneracy of an electron In case of the phononswe remove this coefficient and instead the number of thephonon polarization states that contribute to the heat flowlets us include to the number of the phononmodes119872ph(ℏ120596)Finally the heat current due to phonons is
119876 =1
ℎint (ℏ120596) 119879ph (ℏ120596)119872ph (ℏ120596) (1198991 minus 119899
2) 119889 (ℏ120596) [W]
(53)
In the linear response regime by analogy with (26)
1198991minus 1198992asymp minus
1205971198990
120597119879Δ119879 (54)
where the derivative according to (52) is
1205971198990
120597119879=ℏ120596
119879(minus
1205971198990
120597 (ℏ120596)) (55)
with
1205971198990
120597 (ℏ120596)= (minus
1
119896119879)
119890ℏ120596119896119879
(119890ℏ120596119896119879 minus 1)2 (56)
Now (53) for small differences in temperature becomes
119876 = minus119870119871Δ119879 (57)
where the thermal conductance
119870119871=1198962119879
ℎint119879ph (ℏ120596)119872ph (ℏ120596)
times [(ℏ120596
119896119879)
2
(minus1205971198990
120597 (ℏ120596))] 119889 (ℏ120596) [WK]
(58)
Equation (57) is simply Fourierrsquos law stating that heatflows down to a temperature gradient It is also useful tonote that the thermal conductance (58) displays certainsimilarities with the electrical conductance
119866 =21199022
ℎint119879el (119864)119872el (119864) (minus
1205971198910
120597119864)119889119864 (59)
The derivative
119882el (119864) equiv (minus1205971198910
120597119864) (60)
known as the Fermi window function that picks out thoseconduction modes which only contribute to the electriccurrent The electron windows function is normalized
int
+infin
minusinfin
(minus1205971198910
120597119864)119889119864 = 1 (61)
In case of phonons the term in square brackets of (58) actsas a window function to specify which modes carry the heatcurrent After normalization
119882ph (ℏ120596) =3
1205872(ℏ120596
119896119879)(
1205971198990
120597 (ℏ120596)) (62)
thus finally
119870119871=12058721198962119879
3ℎint119879ph (ℏ120596)119872ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (63)
with
1198920equiv12058721198962119879
3ℎasymp (9456 times 10
minus13WK2) 119879 (64)
known as the quantum of thermal conductance experimen-tally observed first in 2000 [10]
Comparing (59) and (63) one can see that the electricaland thermal conductances are similar in structure bothare proportional to corresponding quantum of conductancetimes an integral over the transmission times the number ofmodes times a window function
The thermal broadening functions for electrons andphonons have similar shapes and each has a width of a few119896119879 In case of electrons this function is given by (27) or
119865119879(119909) equiv
119890119909
(119890119909 + 1)
2(65)
with 119909 equiv (119864minus119864119865)119896119879 and is shown on Figure 2This function
for phonons is given by (62) or
119865ph119879(119909) equiv
3
1205872
1199092119890119909
(119890119909 minus 1)
2(66)
with 119909 equiv ℏ120596119896119879 Both functions are normalized to a unityand shown together on Figure 3
Along with the number of modes determined by thedispersion relation these two window functions play a keyrole in determining the electrical and thermal conductances
31 Thermal Conductivity of the Bulk Conductors The ther-mal conductivity of a large diffusive resistor is a key materialproperty that controls performance of any electronic devicesBy analogy with the transmission coefficient (3) for electrontransport the phonon transmission
119879ph (ℏ120596) =120582ph (ℏ120596)
120582ph (ℏ120596) + 119871
100381610038161003816100381610038161003816100381610038161003816119871≫120582ph
997888rarr
120582ph (ℏ120596)
119871 (67)
It is also obvious that for large 3D conductors the numberof phonon modes is proportional to the cross-sectional areaof the sample
119872ph (ℏ120596) prop 119860 (68)
8 Journal of Nanoscience
10
5
0
minus5
minus100 01 02 03
FT(x)
FT(x)
FT(x)
phx
Figure 3 Broadening function for phonons compared to that ofelectrons
Now let us return to (57) dividing and multiplying it by119860119871 which immediately gives (50) for the phonon heat fluxpostulated above
119876
119860equiv 119869
ph119876119909
= minus120581119871
119889119879
119889119909(69)
with specific lattice thermal conductivity
120581119871= 119870119871
119871
119860 (70)
or substituting (67) into (63) one for the lattice thermalconductivity finally obtains
120581119871=12058721198962119879
3ℎint
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)
It is useful now to define the average number of phononmodes per cross-sectional area of the conductor that partici-pate in the heat transport
⟨
119872ph
119860⟩ equiv int
119872ph (ℏ120596)
119860119882ph (ℏ120596) 119889 (ℏ120596) (72)
Then
120581119871=12058721198962119879
3ℎ⟨
119872ph
119860⟩⟨⟨120582ph⟩⟩ (73)
where the average mean-free-path is defined now as
⟨⟨120582ph⟩⟩ =
int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)
(74)
Thus the couple of the phonon transport equations (69)and (73) corresponds to similar electron transport equations
119869119909=120590
119902
119889 (119864119865)
119889119909 (75)
120590 =21199022
ℎ⟨119872el119860
⟩⟨⟨120582el⟩⟩ (76)
The thermal conductivity (73) and the electrical conduc-tivity (76) have the same structure It is always a product of thecorresponding quantum of conductance times the numberof modes that participate in transport times the averagemean-free-path These three quantities for phonons will bediscussed later
32 Specific Heat versus Thermal Conductivity The connec-tion between the lattice specific thermal conductivity andthe lattice specific heat at constant volume is well known[6ndash9] We will show now that corresponding proportionalitycoefficient is a product of an appropriately-defined mean-free-path ⟨⟨Λ ph⟩⟩ and an average phonon velocity ⟨Vph⟩namely
120581119871=1
3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)
The total phonon energy per unit volume
119864ph = int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)
where119863ph(ℏ120596) is the phonon density of states By definition
119862119881equiv
120597119864ph
120597119879=
120597
120597119879int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596)
= int
infin
0
(ℏ120596)119863ph (ℏ120596) (1205971198990(ℏ120596)
120597119879)119889 (ℏ120596)
=12058721198962119879
3int
infin
0
119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(79)
where (55) and (62) were used Next multiply and divide (71)by (79) and obtain the proportionality we are looking for120581119871
= [
[
(1ℎ) intinfin
0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(80)To obtain final expression (77) and correct interpretation
of the proportionality coefficient we need to return to (67)This expression can be easily derived for 1D conductorwith several simplifying assumptions Nevertheless it worksvery well in practice for a conductor of any dimensionDerivation of (67) is based on the interpretation of themean-free-path 120582(119864) or 120582(ℏ120596) as that its inverse value is theprobability per unit length that a positive flux is convertedto a negative flux This is why 120582 is often called a mean-free-path for backscattering Let us relate it to the scattering time120591 The distinction between mean-free-path and mean-free-path for backscattering is easiest to see for 1D conductorLet an electron undergo a scattering event For isotropicscattering the electron can forward scatter or backscatterOnly backscattering is relevant for the mean-free-path forscattering so the time between backscattering events is 2120591Thus the mean-free-path for backscattering is twice themean-free-path for scattering
1205821119863
(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)
It was shown that the proper definition of the mean-free-path for backscattering for a conductor of any dimension [11]is
120582 (119864) = 2
⟨V2119909120591⟩
⟨1003816100381610038161003816V119909
1003816100381610038161003816⟩ (lowast lowast lowast)
Journal of Nanoscience 9
where averaging is performed over angles For isotropicbands
1205822119863
(119864) =120587
2V (119864) 120591 (119864) (81b)
1205823119863
(119864) =4
3V (119864) 120591 (119864) (81c)
The scattering time is often approximately written as thepower law scattering
120591 (119864) = 1205910(119864 minus 119864
119862
119896119879)
119904
(82)
where exponent 119904 describes the specific scattering mecha-nism for acoustic phonon scattering in 3D conductor withparabolic dispersion 119904 = minus12 and for ionized impurityscattering 119904 = +32 [12]
Analogous power law is often used for mean-free-path
120582 (119864) = 1205820(119864 minus 119864
119862
119896119879)
119903
(83)
For parabolic zone structure V(119864) prop 11986412 thus 119903 = 119904 + 12
with 119903 = 0 for acoustic phonon scattering and 119903 = 2 forionized impurity scattering
Coming back to our initial task to derive (77) from (80)for 3D conductor according to (81c) we have
120582ph (ℏ120596) =4
3Vph (ℏ120596) 120591ph (ℏ120596) (84)
where according to (81a)
Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)
and finally
120582ph (ℏ120596) =4
3Λ ph (ℏ120596) (86)
It was stated above in (11c) that the density of states andnumber of modes for electrons in 3D are
119872el (119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (87)
Let us rewrite this formula for phonons Note that the spindegeneracy for electrons 119892
119878= 2 is included with the density
of states
1198633119863
(119864) = 21198631015840
3119863(119864) (88)
and for spherical bands in 3D conductor
⟨V+119909(119864)⟩ =
Vel (119864)2
(89)
Collecting (87) up to (89) all together in case of phonons wehave
119872ph (ℏ120596) = 119860ℎ
2(
Vph (ℏ120596)2
) 2119863ph (ℏ120596)
= 119860ℎ
4Vph (ℏ120596)119863ph (ℏ120596)
(90)
Substituting (86) and (90) into (80) we obtain
120581119871
= [
[
(13) intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(91)
Multiplying and dividing (91) by
int
infin
0
Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)
we finally get (77) with proportionality coefficient between 120581119871
and 119862119881as the product of an average mean-free-path as
⟨⟨Λ ph⟩⟩
equiv
intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(93)
and an average velocity as
⟨Vph⟩ equiv
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(94)
with the appropriate averagingEquation (77) is often used to estimate the average mean-
free-path from the measured 120581119871and 119862
119881 if we know the
average velocity which is frequently assumed to be the lon-gitudinal sound velocity The derivation above has identifiedthe precise definitions of the ⟨⟨Λ ph⟩⟩ and ⟨Vph⟩ If a phonondispersion is chosen one can always compute the averagevelocity according to (94) and it is typically very differentfrom the longitudinal sound velocity Thus estimates of theaverage mean-free-path can be quite wrong if one assumesthe longitudinal sound velocity [13]
33 Debye Model For 3D conductors there are three polar-ization states for lattice vibrations one for atoms displacedin the direction of propagation (longitudinal119871) and two foratoms displaced orthogonally to the direction of propagation(transverse119879) The low energy modes are called acousticmodes119860 one 119871119860 mode analogous to sound waves propa-gating in air and two 119879119860 modes Near 119902 rarr 0 dispersion ofacoustic modes is linear
ℏ120596 = ℏV119863119902 (95)
and is known as Debye approximation The Debye velocityV119863is an average velocity of the 119871 and 119879 acoustic modes In
case of 119871119860mode V119863is simply the sound velocity V
119904prop 119898minus12
with 119898 being an effective mass of vibrating atom TypicallyV119904asymp 5 times 10
3ms about 20 times slower that the velocity of atypical electron
The bandwidth of the electronic dispersion is typicallyBW ≫ 119896119879 so only states near the bottom of the conductionband where the effective mass model is reasonably accurate
10 Journal of Nanoscience
are occupied For phonons the situation ismuch different thebandwidth BW asymp 119896119879 so states across the entire Brillouinzone are occupied The widely used Debye approximation(95) fits the acoustic branches as long as 119902 is not too far fromthe center of the Brillouin zone
With the Debye approximation (95) it is easy to find thedensity of the phonon states
119863ph (ℏ120596) =3(ℏ120596)
2
21205872(ℏV119863)3 [Jminus1 sdotmminus3] (96)
where the factor of three is for the three polarizations Thenone can obtain the number of phonon modes per cross-sectional area from (90)
119872ph (ℏ120596) =3(ℏ120596)
2
4120587(ℏV119863)2 (97)
Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures we are to be sure that we accountfor the correct number of states For a crystal there are 3119873Ω
states per unit volume To find the total number of states wehave to integrate the density of states
int
ℏ120596119863
0
119863ph (ℏ120596) 119889 (ℏ120596) (98)
with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states namely
ℏ120596119863= ℏV119863(61205872119873
Ω)
13
equiv 119896119879119863 (99)
The Debye frequency defines a cutoff frequency abovewhich no states are accounted for This restriction can alsobe expressed via a cutoff wave vector 119902
119863or as a Debye tem-
perature
119879119863=ℏ120596
119896 (100)
For 119879 ≪ 119879119863 only states with 119902 rarr 0 for which the Debye
approximation is accurate are occupiedNow we can calculate the lattice thermal conductivity by
integrating (71) to the Debye cutoff energy
120581119871=12058721198962119879
3ℎint
ℏ120596119863
0
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)
and estimate119872ph(ℏ120596) according to (71) The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used This is how the lattice thermalconductivities were first calculated [14 15] The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11 13 16]
34 Phonon Scattering Phonons can scatter from defectsimpurity atoms isotopes surfaces and boundaries and elec-trons and from other phonons Phonon-phonon scattering
occurs because the potential energy of the bonds in the crystalis not exactly harmonic All higher order terms are treatedas a scattering potential Two types of phonon scattering areconsidered In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved
ℏ 1199021+ ℏ 1199022= ℏ 1199023
ℏ1+ ℏ2= ℏ3
(102)
The total momentum of the phonon ensemble is conservedthus this type of scattering has little effect on the heat flux
In a second type of scattering umklapp119880-scatteringthe two initial phonons have larger momentum thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions The 119880-scatterings are thebasic processes in the heat transport especially at hightemperatures Scattering on defects119863 and on boundaries119861is also important Scattering rates are additive thus the totalphonon scattering rate is
1
120591ph (ℏ120596)=
1
120591119880(ℏ120596)
+1
120591119863(ℏ120596)
+1
120591119861(ℏ120596)
(103)
or alternatively in terms of the mean-free-path (84)
1
120582ph (ℏ120596)=
1
120582119880(ℏ120596)
+1
120582119863(ℏ120596)
+1
120582119861(ℏ120596)
(104)
Expressions for each of the scattering rates are developed[17] For scattering from point defects
1
120591119863(ℏ120596)
prop 1205964 (105)
known as the Rayleigh scattering which is like the scatteringof light from the dust
For boundaries and surfaces
1
120591119861(ℏ120596)
prop
Vph (ℏ120596)119871
(106)
where 119871 is the shortest dimension of the sampleA commonly used expression for 119880-scattering is
1
120591119880(ℏ120596)
prop 11987931205962119890minus119879119863119887119879 (107)
With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity
35 Lattice Thermal Conductivity versus Temperature Thetemperature dependence of the lattice thermal conductivity120581119871is illustrated for bulk Si on Figure 4According to (73) 120581
119871is proportional to the number of the
phononmodes that are occupied ⟨119872ph119860⟩ and to the averagevalue of the phononmean-free-path ⟨⟨120582ph⟩⟩The curve 120581
119871(119879)
can be explained by understanding how ⟨119872ph⟩ and ⟨⟨120582ph⟩⟩vary with temperature
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
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Nano
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Journal ofNanomaterials
8 Journal of Nanoscience
10
5
0
minus5
minus100 01 02 03
FT(x)
FT(x)
FT(x)
phx
Figure 3 Broadening function for phonons compared to that ofelectrons
Now let us return to (57) dividing and multiplying it by119860119871 which immediately gives (50) for the phonon heat fluxpostulated above
119876
119860equiv 119869
ph119876119909
= minus120581119871
119889119879
119889119909(69)
with specific lattice thermal conductivity
120581119871= 119870119871
119871
119860 (70)
or substituting (67) into (63) one for the lattice thermalconductivity finally obtains
120581119871=12058721198962119879
3ℎint
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)
It is useful now to define the average number of phononmodes per cross-sectional area of the conductor that partici-pate in the heat transport
⟨
119872ph
119860⟩ equiv int
119872ph (ℏ120596)
119860119882ph (ℏ120596) 119889 (ℏ120596) (72)
Then
120581119871=12058721198962119879
3ℎ⟨
119872ph
119860⟩⟨⟨120582ph⟩⟩ (73)
where the average mean-free-path is defined now as
⟨⟨120582ph⟩⟩ =
int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)
(74)
Thus the couple of the phonon transport equations (69)and (73) corresponds to similar electron transport equations
119869119909=120590
119902
119889 (119864119865)
119889119909 (75)
120590 =21199022
ℎ⟨119872el119860
⟩⟨⟨120582el⟩⟩ (76)
The thermal conductivity (73) and the electrical conduc-tivity (76) have the same structure It is always a product of thecorresponding quantum of conductance times the numberof modes that participate in transport times the averagemean-free-path These three quantities for phonons will bediscussed later
32 Specific Heat versus Thermal Conductivity The connec-tion between the lattice specific thermal conductivity andthe lattice specific heat at constant volume is well known[6ndash9] We will show now that corresponding proportionalitycoefficient is a product of an appropriately-defined mean-free-path ⟨⟨Λ ph⟩⟩ and an average phonon velocity ⟨Vph⟩namely
120581119871=1
3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)
The total phonon energy per unit volume
119864ph = int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)
where119863ph(ℏ120596) is the phonon density of states By definition
119862119881equiv
120597119864ph
120597119879=
120597
120597119879int
infin
0
(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596)
= int
infin
0
(ℏ120596)119863ph (ℏ120596) (1205971198990(ℏ120596)
120597119879)119889 (ℏ120596)
=12058721198962119879
3int
infin
0
119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(79)
where (55) and (62) were used Next multiply and divide (71)by (79) and obtain the proportionality we are looking for120581119871
= [
[
(1ℎ) intinfin
0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(80)To obtain final expression (77) and correct interpretation
of the proportionality coefficient we need to return to (67)This expression can be easily derived for 1D conductorwith several simplifying assumptions Nevertheless it worksvery well in practice for a conductor of any dimensionDerivation of (67) is based on the interpretation of themean-free-path 120582(119864) or 120582(ℏ120596) as that its inverse value is theprobability per unit length that a positive flux is convertedto a negative flux This is why 120582 is often called a mean-free-path for backscattering Let us relate it to the scattering time120591 The distinction between mean-free-path and mean-free-path for backscattering is easiest to see for 1D conductorLet an electron undergo a scattering event For isotropicscattering the electron can forward scatter or backscatterOnly backscattering is relevant for the mean-free-path forscattering so the time between backscattering events is 2120591Thus the mean-free-path for backscattering is twice themean-free-path for scattering
1205821119863
(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)
It was shown that the proper definition of the mean-free-path for backscattering for a conductor of any dimension [11]is
120582 (119864) = 2
⟨V2119909120591⟩
⟨1003816100381610038161003816V119909
1003816100381610038161003816⟩ (lowast lowast lowast)
Journal of Nanoscience 9
where averaging is performed over angles For isotropicbands
1205822119863
(119864) =120587
2V (119864) 120591 (119864) (81b)
1205823119863
(119864) =4
3V (119864) 120591 (119864) (81c)
The scattering time is often approximately written as thepower law scattering
120591 (119864) = 1205910(119864 minus 119864
119862
119896119879)
119904
(82)
where exponent 119904 describes the specific scattering mecha-nism for acoustic phonon scattering in 3D conductor withparabolic dispersion 119904 = minus12 and for ionized impurityscattering 119904 = +32 [12]
Analogous power law is often used for mean-free-path
120582 (119864) = 1205820(119864 minus 119864
119862
119896119879)
119903
(83)
For parabolic zone structure V(119864) prop 11986412 thus 119903 = 119904 + 12
with 119903 = 0 for acoustic phonon scattering and 119903 = 2 forionized impurity scattering
Coming back to our initial task to derive (77) from (80)for 3D conductor according to (81c) we have
120582ph (ℏ120596) =4
3Vph (ℏ120596) 120591ph (ℏ120596) (84)
where according to (81a)
Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)
and finally
120582ph (ℏ120596) =4
3Λ ph (ℏ120596) (86)
It was stated above in (11c) that the density of states andnumber of modes for electrons in 3D are
119872el (119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (87)
Let us rewrite this formula for phonons Note that the spindegeneracy for electrons 119892
119878= 2 is included with the density
of states
1198633119863
(119864) = 21198631015840
3119863(119864) (88)
and for spherical bands in 3D conductor
⟨V+119909(119864)⟩ =
Vel (119864)2
(89)
Collecting (87) up to (89) all together in case of phonons wehave
119872ph (ℏ120596) = 119860ℎ
2(
Vph (ℏ120596)2
) 2119863ph (ℏ120596)
= 119860ℎ
4Vph (ℏ120596)119863ph (ℏ120596)
(90)
Substituting (86) and (90) into (80) we obtain
120581119871
= [
[
(13) intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(91)
Multiplying and dividing (91) by
int
infin
0
Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)
we finally get (77) with proportionality coefficient between 120581119871
and 119862119881as the product of an average mean-free-path as
⟨⟨Λ ph⟩⟩
equiv
intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(93)
and an average velocity as
⟨Vph⟩ equiv
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(94)
with the appropriate averagingEquation (77) is often used to estimate the average mean-
free-path from the measured 120581119871and 119862
119881 if we know the
average velocity which is frequently assumed to be the lon-gitudinal sound velocity The derivation above has identifiedthe precise definitions of the ⟨⟨Λ ph⟩⟩ and ⟨Vph⟩ If a phonondispersion is chosen one can always compute the averagevelocity according to (94) and it is typically very differentfrom the longitudinal sound velocity Thus estimates of theaverage mean-free-path can be quite wrong if one assumesthe longitudinal sound velocity [13]
33 Debye Model For 3D conductors there are three polar-ization states for lattice vibrations one for atoms displacedin the direction of propagation (longitudinal119871) and two foratoms displaced orthogonally to the direction of propagation(transverse119879) The low energy modes are called acousticmodes119860 one 119871119860 mode analogous to sound waves propa-gating in air and two 119879119860 modes Near 119902 rarr 0 dispersion ofacoustic modes is linear
ℏ120596 = ℏV119863119902 (95)
and is known as Debye approximation The Debye velocityV119863is an average velocity of the 119871 and 119879 acoustic modes In
case of 119871119860mode V119863is simply the sound velocity V
119904prop 119898minus12
with 119898 being an effective mass of vibrating atom TypicallyV119904asymp 5 times 10
3ms about 20 times slower that the velocity of atypical electron
The bandwidth of the electronic dispersion is typicallyBW ≫ 119896119879 so only states near the bottom of the conductionband where the effective mass model is reasonably accurate
10 Journal of Nanoscience
are occupied For phonons the situation ismuch different thebandwidth BW asymp 119896119879 so states across the entire Brillouinzone are occupied The widely used Debye approximation(95) fits the acoustic branches as long as 119902 is not too far fromthe center of the Brillouin zone
With the Debye approximation (95) it is easy to find thedensity of the phonon states
119863ph (ℏ120596) =3(ℏ120596)
2
21205872(ℏV119863)3 [Jminus1 sdotmminus3] (96)
where the factor of three is for the three polarizations Thenone can obtain the number of phonon modes per cross-sectional area from (90)
119872ph (ℏ120596) =3(ℏ120596)
2
4120587(ℏV119863)2 (97)
Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures we are to be sure that we accountfor the correct number of states For a crystal there are 3119873Ω
states per unit volume To find the total number of states wehave to integrate the density of states
int
ℏ120596119863
0
119863ph (ℏ120596) 119889 (ℏ120596) (98)
with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states namely
ℏ120596119863= ℏV119863(61205872119873
Ω)
13
equiv 119896119879119863 (99)
The Debye frequency defines a cutoff frequency abovewhich no states are accounted for This restriction can alsobe expressed via a cutoff wave vector 119902
119863or as a Debye tem-
perature
119879119863=ℏ120596
119896 (100)
For 119879 ≪ 119879119863 only states with 119902 rarr 0 for which the Debye
approximation is accurate are occupiedNow we can calculate the lattice thermal conductivity by
integrating (71) to the Debye cutoff energy
120581119871=12058721198962119879
3ℎint
ℏ120596119863
0
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)
and estimate119872ph(ℏ120596) according to (71) The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used This is how the lattice thermalconductivities were first calculated [14 15] The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11 13 16]
34 Phonon Scattering Phonons can scatter from defectsimpurity atoms isotopes surfaces and boundaries and elec-trons and from other phonons Phonon-phonon scattering
occurs because the potential energy of the bonds in the crystalis not exactly harmonic All higher order terms are treatedas a scattering potential Two types of phonon scattering areconsidered In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved
ℏ 1199021+ ℏ 1199022= ℏ 1199023
ℏ1+ ℏ2= ℏ3
(102)
The total momentum of the phonon ensemble is conservedthus this type of scattering has little effect on the heat flux
In a second type of scattering umklapp119880-scatteringthe two initial phonons have larger momentum thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions The 119880-scatterings are thebasic processes in the heat transport especially at hightemperatures Scattering on defects119863 and on boundaries119861is also important Scattering rates are additive thus the totalphonon scattering rate is
1
120591ph (ℏ120596)=
1
120591119880(ℏ120596)
+1
120591119863(ℏ120596)
+1
120591119861(ℏ120596)
(103)
or alternatively in terms of the mean-free-path (84)
1
120582ph (ℏ120596)=
1
120582119880(ℏ120596)
+1
120582119863(ℏ120596)
+1
120582119861(ℏ120596)
(104)
Expressions for each of the scattering rates are developed[17] For scattering from point defects
1
120591119863(ℏ120596)
prop 1205964 (105)
known as the Rayleigh scattering which is like the scatteringof light from the dust
For boundaries and surfaces
1
120591119861(ℏ120596)
prop
Vph (ℏ120596)119871
(106)
where 119871 is the shortest dimension of the sampleA commonly used expression for 119880-scattering is
1
120591119880(ℏ120596)
prop 11987931205962119890minus119879119863119887119879 (107)
With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity
35 Lattice Thermal Conductivity versus Temperature Thetemperature dependence of the lattice thermal conductivity120581119871is illustrated for bulk Si on Figure 4According to (73) 120581
119871is proportional to the number of the
phononmodes that are occupied ⟨119872ph119860⟩ and to the averagevalue of the phononmean-free-path ⟨⟨120582ph⟩⟩The curve 120581
119871(119879)
can be explained by understanding how ⟨119872ph⟩ and ⟨⟨120582ph⟩⟩vary with temperature
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 9
where averaging is performed over angles For isotropicbands
1205822119863
(119864) =120587
2V (119864) 120591 (119864) (81b)
1205823119863
(119864) =4
3V (119864) 120591 (119864) (81c)
The scattering time is often approximately written as thepower law scattering
120591 (119864) = 1205910(119864 minus 119864
119862
119896119879)
119904
(82)
where exponent 119904 describes the specific scattering mecha-nism for acoustic phonon scattering in 3D conductor withparabolic dispersion 119904 = minus12 and for ionized impurityscattering 119904 = +32 [12]
Analogous power law is often used for mean-free-path
120582 (119864) = 1205820(119864 minus 119864
119862
119896119879)
119903
(83)
For parabolic zone structure V(119864) prop 11986412 thus 119903 = 119904 + 12
with 119903 = 0 for acoustic phonon scattering and 119903 = 2 forionized impurity scattering
Coming back to our initial task to derive (77) from (80)for 3D conductor according to (81c) we have
120582ph (ℏ120596) =4
3Vph (ℏ120596) 120591ph (ℏ120596) (84)
where according to (81a)
Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)
and finally
120582ph (ℏ120596) =4
3Λ ph (ℏ120596) (86)
It was stated above in (11c) that the density of states andnumber of modes for electrons in 3D are
119872el (119864) = 1198601198723119863
(119864) = 119860ℎ
4⟨V+119909(119864)⟩119863
3119863(119864) (87)
Let us rewrite this formula for phonons Note that the spindegeneracy for electrons 119892
119878= 2 is included with the density
of states
1198633119863
(119864) = 21198631015840
3119863(119864) (88)
and for spherical bands in 3D conductor
⟨V+119909(119864)⟩ =
Vel (119864)2
(89)
Collecting (87) up to (89) all together in case of phonons wehave
119872ph (ℏ120596) = 119860ℎ
2(
Vph (ℏ120596)2
) 2119863ph (ℏ120596)
= 119860ℎ
4Vph (ℏ120596)119863ph (ℏ120596)
(90)
Substituting (86) and (90) into (80) we obtain
120581119871
= [
[
(13) intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
]
]
119862119881
(91)
Multiplying and dividing (91) by
int
infin
0
Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)
we finally get (77) with proportionality coefficient between 120581119871
and 119862119881as the product of an average mean-free-path as
⟨⟨Λ ph⟩⟩
equiv
intinfin
0Λ ph (ℏ120596) Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(93)
and an average velocity as
⟨Vph⟩ equiv
intinfin
0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
intinfin
0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)
(94)
with the appropriate averagingEquation (77) is often used to estimate the average mean-
free-path from the measured 120581119871and 119862
119881 if we know the
average velocity which is frequently assumed to be the lon-gitudinal sound velocity The derivation above has identifiedthe precise definitions of the ⟨⟨Λ ph⟩⟩ and ⟨Vph⟩ If a phonondispersion is chosen one can always compute the averagevelocity according to (94) and it is typically very differentfrom the longitudinal sound velocity Thus estimates of theaverage mean-free-path can be quite wrong if one assumesthe longitudinal sound velocity [13]
33 Debye Model For 3D conductors there are three polar-ization states for lattice vibrations one for atoms displacedin the direction of propagation (longitudinal119871) and two foratoms displaced orthogonally to the direction of propagation(transverse119879) The low energy modes are called acousticmodes119860 one 119871119860 mode analogous to sound waves propa-gating in air and two 119879119860 modes Near 119902 rarr 0 dispersion ofacoustic modes is linear
ℏ120596 = ℏV119863119902 (95)
and is known as Debye approximation The Debye velocityV119863is an average velocity of the 119871 and 119879 acoustic modes In
case of 119871119860mode V119863is simply the sound velocity V
119904prop 119898minus12
with 119898 being an effective mass of vibrating atom TypicallyV119904asymp 5 times 10
3ms about 20 times slower that the velocity of atypical electron
The bandwidth of the electronic dispersion is typicallyBW ≫ 119896119879 so only states near the bottom of the conductionband where the effective mass model is reasonably accurate
10 Journal of Nanoscience
are occupied For phonons the situation ismuch different thebandwidth BW asymp 119896119879 so states across the entire Brillouinzone are occupied The widely used Debye approximation(95) fits the acoustic branches as long as 119902 is not too far fromthe center of the Brillouin zone
With the Debye approximation (95) it is easy to find thedensity of the phonon states
119863ph (ℏ120596) =3(ℏ120596)
2
21205872(ℏV119863)3 [Jminus1 sdotmminus3] (96)
where the factor of three is for the three polarizations Thenone can obtain the number of phonon modes per cross-sectional area from (90)
119872ph (ℏ120596) =3(ℏ120596)
2
4120587(ℏV119863)2 (97)
Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures we are to be sure that we accountfor the correct number of states For a crystal there are 3119873Ω
states per unit volume To find the total number of states wehave to integrate the density of states
int
ℏ120596119863
0
119863ph (ℏ120596) 119889 (ℏ120596) (98)
with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states namely
ℏ120596119863= ℏV119863(61205872119873
Ω)
13
equiv 119896119879119863 (99)
The Debye frequency defines a cutoff frequency abovewhich no states are accounted for This restriction can alsobe expressed via a cutoff wave vector 119902
119863or as a Debye tem-
perature
119879119863=ℏ120596
119896 (100)
For 119879 ≪ 119879119863 only states with 119902 rarr 0 for which the Debye
approximation is accurate are occupiedNow we can calculate the lattice thermal conductivity by
integrating (71) to the Debye cutoff energy
120581119871=12058721198962119879
3ℎint
ℏ120596119863
0
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)
and estimate119872ph(ℏ120596) according to (71) The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used This is how the lattice thermalconductivities were first calculated [14 15] The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11 13 16]
34 Phonon Scattering Phonons can scatter from defectsimpurity atoms isotopes surfaces and boundaries and elec-trons and from other phonons Phonon-phonon scattering
occurs because the potential energy of the bonds in the crystalis not exactly harmonic All higher order terms are treatedas a scattering potential Two types of phonon scattering areconsidered In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved
ℏ 1199021+ ℏ 1199022= ℏ 1199023
ℏ1+ ℏ2= ℏ3
(102)
The total momentum of the phonon ensemble is conservedthus this type of scattering has little effect on the heat flux
In a second type of scattering umklapp119880-scatteringthe two initial phonons have larger momentum thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions The 119880-scatterings are thebasic processes in the heat transport especially at hightemperatures Scattering on defects119863 and on boundaries119861is also important Scattering rates are additive thus the totalphonon scattering rate is
1
120591ph (ℏ120596)=
1
120591119880(ℏ120596)
+1
120591119863(ℏ120596)
+1
120591119861(ℏ120596)
(103)
or alternatively in terms of the mean-free-path (84)
1
120582ph (ℏ120596)=
1
120582119880(ℏ120596)
+1
120582119863(ℏ120596)
+1
120582119861(ℏ120596)
(104)
Expressions for each of the scattering rates are developed[17] For scattering from point defects
1
120591119863(ℏ120596)
prop 1205964 (105)
known as the Rayleigh scattering which is like the scatteringof light from the dust
For boundaries and surfaces
1
120591119861(ℏ120596)
prop
Vph (ℏ120596)119871
(106)
where 119871 is the shortest dimension of the sampleA commonly used expression for 119880-scattering is
1
120591119880(ℏ120596)
prop 11987931205962119890minus119879119863119887119879 (107)
With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity
35 Lattice Thermal Conductivity versus Temperature Thetemperature dependence of the lattice thermal conductivity120581119871is illustrated for bulk Si on Figure 4According to (73) 120581
119871is proportional to the number of the
phononmodes that are occupied ⟨119872ph119860⟩ and to the averagevalue of the phononmean-free-path ⟨⟨120582ph⟩⟩The curve 120581
119871(119879)
can be explained by understanding how ⟨119872ph⟩ and ⟨⟨120582ph⟩⟩vary with temperature
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
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MaterialsJournal of
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Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
10 Journal of Nanoscience
are occupied For phonons the situation ismuch different thebandwidth BW asymp 119896119879 so states across the entire Brillouinzone are occupied The widely used Debye approximation(95) fits the acoustic branches as long as 119902 is not too far fromthe center of the Brillouin zone
With the Debye approximation (95) it is easy to find thedensity of the phonon states
119863ph (ℏ120596) =3(ℏ120596)
2
21205872(ℏV119863)3 [Jminus1 sdotmminus3] (96)
where the factor of three is for the three polarizations Thenone can obtain the number of phonon modes per cross-sectional area from (90)
119872ph (ℏ120596) =3(ℏ120596)
2
4120587(ℏV119863)2 (97)
Since all the states in the Brillouin zone tend to be occupiedat moderate temperatures we are to be sure that we accountfor the correct number of states For a crystal there are 3119873Ω
states per unit volume To find the total number of states wehave to integrate the density of states
int
ℏ120596119863
0
119863ph (ℏ120596) 119889 (ℏ120596) (98)
with the upper limit as ℏ times the so-called Debye frequencyto produce the correct number of states namely
ℏ120596119863= ℏV119863(61205872119873
Ω)
13
equiv 119896119879119863 (99)
The Debye frequency defines a cutoff frequency abovewhich no states are accounted for This restriction can alsobe expressed via a cutoff wave vector 119902
119863or as a Debye tem-
perature
119879119863=ℏ120596
119896 (100)
For 119879 ≪ 119879119863 only states with 119902 rarr 0 for which the Debye
approximation is accurate are occupiedNow we can calculate the lattice thermal conductivity by
integrating (71) to the Debye cutoff energy
120581119871=12058721198962119879
3ℎint
ℏ120596119863
0
119872ph (ℏ120596)
119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)
and estimate119872ph(ℏ120596) according to (71) The integral can betaken numerically or analytically if appropriate expression forthe mean-free-path is used This is how the lattice thermalconductivities were first calculated [14 15] The theory andcomputational procedures for the thermoelectric transportcoefficients were developed further in [11 13 16]
34 Phonon Scattering Phonons can scatter from defectsimpurity atoms isotopes surfaces and boundaries and elec-trons and from other phonons Phonon-phonon scattering
occurs because the potential energy of the bonds in the crystalis not exactly harmonic All higher order terms are treatedas a scattering potential Two types of phonon scattering areconsidered In the normal process two phonons interact andcreate a third phonon with energy and momentum beingconserved
ℏ 1199021+ ℏ 1199022= ℏ 1199023
ℏ1+ ℏ2= ℏ3
(102)
The total momentum of the phonon ensemble is conservedthus this type of scattering has little effect on the heat flux
In a second type of scattering umklapp119880-scatteringthe two initial phonons have larger momentum thus theresulting phonon would have a momentum outside theBrillouin zone due to unharmonic phonon-phonon as wellas electron-phonon interactions The 119880-scatterings are thebasic processes in the heat transport especially at hightemperatures Scattering on defects119863 and on boundaries119861is also important Scattering rates are additive thus the totalphonon scattering rate is
1
120591ph (ℏ120596)=
1
120591119880(ℏ120596)
+1
120591119863(ℏ120596)
+1
120591119861(ℏ120596)
(103)
or alternatively in terms of the mean-free-path (84)
1
120582ph (ℏ120596)=
1
120582119880(ℏ120596)
+1
120582119863(ℏ120596)
+1
120582119861(ℏ120596)
(104)
Expressions for each of the scattering rates are developed[17] For scattering from point defects
1
120591119863(ℏ120596)
prop 1205964 (105)
known as the Rayleigh scattering which is like the scatteringof light from the dust
For boundaries and surfaces
1
120591119861(ℏ120596)
prop
Vph (ℏ120596)119871
(106)
where 119871 is the shortest dimension of the sampleA commonly used expression for 119880-scattering is
1
120591119880(ℏ120596)
prop 11987931205962119890minus119879119863119887119879 (107)
With this background we are now able to understand thetemperature dependence of the lattice thermal conductivity
35 Lattice Thermal Conductivity versus Temperature Thetemperature dependence of the lattice thermal conductivity120581119871is illustrated for bulk Si on Figure 4According to (73) 120581
119871is proportional to the number of the
phononmodes that are occupied ⟨119872ph119860⟩ and to the averagevalue of the phononmean-free-path ⟨⟨120582ph⟩⟩The curve 120581
119871(119879)
can be explained by understanding how ⟨119872ph⟩ and ⟨⟨120582ph⟩⟩vary with temperature
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Advances in
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BioMed Research International
MaterialsJournal of
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Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 11
100 101101
102
102
103
103
104
120581ph
(Wm
minus1K
minus1)
T (K)
Figure 4 The experimental [18] and calculated [13] thermal con-ductivity of bulk Si as a function of temperature
It can be shown using (72) that at low temperatures
⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)
so the initial rise in thermal conductivity is due to thefact that the number of populated modes rises quickly withtemperature At low temperatures boundary scattering isimportant As the temperature increases more short-wave-length phonons are produced These phonons scatter frompoint defects so defect scattering becomes more and moreimportant As the temperature approaches the 119879
119863 all of
the phonon modes are populated and further increases intemperature do not change ⟨119872ph⟩ Instead the higher tem-peratures increase the phonon scattering by119880-processes andthe thermal conductivity drops with increasing temperatures
36 Difference between Lattice Thermal and Electrical Con-ductivities We have already noted the similarity betweenphonon transport equations (69) and (73) and electrontransport equations (75) and (76) The average electron andphonon mean-free-paths are of the same order of magni-tude Why then does the electrical conductance vary overmany more orders of magnitude while the lattice thermalconductance varies only over a few The answer lies in thecorresponding windows functions (60) and (62) For bothelectrons and phonons higher temperatures broaden thewindow function and increase the population of states Forelectrons however the position of the Fermi level has adramatic effect on the magnitude of the window functionBy controlling the position of the Fermi level the electricalconductivity can be varied over many orders of magnitudeFor the phonons the width of the window function isdetermined by temperature only
Another key difference between electrons and phononsrelates to how the states are populated For a thermoelectricdevice 119864
119865asymp 119864119862 and the electron and phonon window
functions are quite similar However for electrons the BWof the dispersion is very large so only a few states near thebottom of the conduction band are populated the effectivemass approximation works well for these states and it is easyto obtain analytical solutions For phonons the BW of thedispersion is small Atmoderate temperatures states all acrossthe entire Brillouin zone are occupied simple analyticalapproximations do not work and it is hard to get analyticalsolutions for the lattice thermal conductivity
37 Lattice Thermal Conductivity Quantization By analogywith the electronic conduction quantization
119866ball
=21199022
ℎ119872(119864119865) (109)
over 30 years ago Pendry [19] stated the existence of thequantum limits to the heat flow In fact if119879 rarr 0 in (63) thenthe phonon window119882ph(ℏ120596) is sharply peaked near ℏ120596 = 0
119870119871=12058721198962119879
3ℎ119879ph (0)119872ph (0) (110)
For a bulk conductor 119872ph(ℏ120596) rarr 0 as ℏ120596 rarr 0but for nanoresistors like a nanowire or nanoribbon one canhave a finite number of phonon modes For ballistic phonontransport 119879ph = 1 and one can expect that
119870119871=12058721198962119879
3ℎ119872ph (111)
Exactly this result was proved experimentally using 4-mode resistor at 119879 lt 08K [10] thermoconductivity mea-surements agree with the predictions for 1D ballistic resistors[20ndash22]
The quantum of thermal conductance
1198920equiv12058721198962119879
3ℎ
(112)
represents themaximumpossible value of energy transportedper phonon mode Surprisingly it does not depend onparticle statistics the quantum of thermal conductance isuniversal for fermions bosons and anyons [23ndash25]
4 Conclusions
In summary we see that the LDL concept used to describeelectron transport can be generalized for phonons In bothcases the Landauer approach generalized and extended byDatta and Lundstom gives correct quantitative descriptionof transport processes for resistors of any nature and anydimension and size in ballistic quasiballistic and diffusivelinear response regimes when there are differences in bothvoltage and temperature across the device We saw that thelattice thermal conductivity can be written in a form that isvery similar to the electrical conductivity but there are twoimportant differences
The first difference between electrons and phonons is thedifference in bandwidths of their dispersions For electrons
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
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MaterialsJournal of
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Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
12 Journal of Nanoscience
the dispersion BW ≫ 119896119879 at room temperature so onlylow energy states are occupied For phonons BW asymp 119896119879so at room temperature all of the acoustic modes across theentire Brillouin zone are occupied As a result the simpleDebye approximation to the acoustic phonon dispersiondoes not work nearly as well as the simple effective massapproximation to the electron dispersion
The second difference between electrons and phonons isthat for electrons the mode populations are controlled by thewindow function which depends on the position of the Fermilevel and the temperature For phonons the window functiondepends only on the temperature The result is that electricalconductivities vary over many orders of magnitude as theposition of the Fermi level varies while lattice conductivitiesvary over only a few orders of magnitude
Finally we also collect below the thermoelectric coeffi-cients for parabolic band semiconductors and for graphene[5 26]
Appendices
A Thermoelectric Coefficients for1D 2D and 3D Semiconductors withParabolic Dispersion for Ballistic andDiffusive Regimes
Thermoelectric coefficients are expressed through the Fermi-Dirac integral of order 119895 defined as
I119895(120578119865) =
1
Γ (119895 + 1)int
infin
0
120578119895
exp (120578 minus 120578119865) + 1
119889120578 (A1)
where the location of the Fermi level 119864119865relative to the
conduction band edge 119864119862
is given by the dimensionlessparameter
120578119865=119864119865minus 119864119862
119896119879 (A2)
In expressions below thermoelectric coefficients (40a)(40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)for diffusive regime were calculated with the power lawscattering
120582 (119864) = 1205820(119864
119896119879)
119903
(A3)
Thermoelectric coefficients for 1D ballistic resistors are
119866 =21199022
ℎIminus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ[I0(120578119865) minus 120578119865Iminus1(120578119865)]
119878 = minus119896
119902[I0(120578119865)
Iminus1(120578119865)minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ[2I1(120578119865) minus 2120578
119865I0(120578119865) + 1205782
119865Iminus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ[2I1(120578119865) minus
I20(120578119865)
Iminus1(120578119865)]
(A4)
Thermoelectric coefficients for 1D diffusive resistors are
119866 =21199022
ℎ(1205820
119871) Γ (119903 + 1)I
119903minus1(120578119865)
119878119879= minus
119896
119902
21199022
ℎ(1205820
119871) Γ (119903 + 1)
times [(119903 + 1)I119903(120578119865) minus 120578119865I119903minus1
(120578119865)]
119878 = minus119896
119902[(119903 + 1)I
119903(120578119865)
I119903minus1
(120578119865)
minus 120578119865]
1198700= 119879(
119896
119902)
221199022
ℎ(1205820
119871)
times [Γ (119903 + 3)I119903+1
(120578119865) minus 2120578
119865Γ (119903 + 2)
timesI119903(120578119865) + 1205782
119865Γ (119903 + 1)I
119903minus1(120578119865)]
119870 = 119879(119896
119902)
221199022
ℎ(1205820
119871) Γ (119903 + 2)
times [(119903 + 2)I119903+1
(120578119865) minus
(119903 + 1)I2119903(120578119865)
I119903minus1
(120578119865)
]
(A5)
Conductivity 119866 = 1205901119863119871 is given in Siemens [120590
1119863] = 1 S sdot
m Similarly for other specific coefficients 119904119879= 119878119879119871 1205810=
1198700119871 120581 = 119870119871Thermoelectric coefficients for 2D ballistic resistors are
119866 = 11988221199022
ℎ
radic2120587119898lowast119896119879
ℎIminus12
(120578119865)
119878119879= minus119882
119896
119902
21199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [3
2I12
(120578119865) minus 120578119865Iminus12
(120578119865)]
119878 = minus119896
119902[3I12
(120578119865)
2Iminus12
(120578119865)minus 120578119865]
1198700= 119882119879(
119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus 3120578
119865I12
(120578119865) + 1205782
119865Iminus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ
radic2120587119898lowast119896119879
ℎ
times [15
4I32
(120578119865) minus
9I212
(120578119865)
4Iminus12
(120578119865)]
(A6)
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 13
Thermoelectric coefficients for 2D diffusive resistors are
119866 = 11988221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
3
2)I119903minus12
(120578119865)
119878 = minus119896
119902[(119903 + 32)I
119903+12(120578119865)
I119903minus12
(120578119865)
minus 120578119865]
119878119879= minus119882
119896
119902
21199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +5
2)I119903+12
(120578119865) minus 120578119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
1198700= 119882119879(
119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎ
times [Γ (119903 +7
2)I119903+32
(120578119865) minus 2120578
119865Γ (119903 +
5
2)
times I119903+12
(120578119865) + 1205782
119865Γ (119903 +
3
2)I119903minus12
(120578119865)]
119870 = 119882119879(119896
119902)
221199022
ℎ(1205820
119871)radic2119898lowast119896119879
120587ℎΓ (119903 +
5
2)
times [(119903 +5
2)I119903+32
(120578119865) minus
(119903 + 32)I2119903+12
(120578119865)
I119903minus12
(120578119865)
]
(A7)
Conductivity 119866 = 1205902119863119882119871 is given in Siemens [120590
2119863] =
1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Thermoelectric coefficients for 3D ballistic resistors are
119866 = 11986021199022
ℎ
119898lowast119896119879
2120587ℏ2I0(120578119865)
119878119879= minus119860
119896
119902
21199022
ℎ
119898lowast119896119879
2120587ℏ2[2I1(120578119865) minus 120578119865I0(120578119865)]
119878 = minus119896
119902[2I1(120578119865)
I0(120578119865)minus 120578119865]
1198700= 119860119879(
119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2
times [6I2(120578119865) minus 4120578
119865I1(120578119865) + 1205782
119865I0(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ
119898lowast119896119879
2120587ℏ2[6I2(120578119865) minus
4I21(120578119865)
I0(120578119865)]
(A8)
Thermoelectric coefficients for 3D diffusive resistors are
119866 = 11986021199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 2)I
119903(120578119865)
119878 = minus119896
119902[(119903 + 2)I
119903+1(120578119865)
I119903(120578119865)
minus 120578119865]
119878119879= minus119860
119896
119902
21199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 3)I119903+1
(120578119865) minus 120578119865Γ (119903 + 2)I
119903(120578119865)]
1198700= 119860119879(
119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2
times [Γ (119903 + 4)I119903+2
(120578119865) minus 2120578
119865Γ (119903 + 3)I
119903+1(120578119865)
+ 1205782
119865Γ (119903 + 2)I
119903(120578119865)]
119870 = 119860119879(119896
119902)
221199022
ℎ(1205820
119871)119898lowast119896119879
2120587ℏ2Γ (119903 + 3)
times [(119903 + 3)I119903+2
(120578119865) minus
(119903 + 2)I2119903+1
(120578119865)
I119903(120578119865)
]
(A9)
Conductivity 119866 = 1205903119863119860119871 is given in Siemens [120590
3119863] =
1 Sm Similarly for other specific coefficients 119904119879= 119878119879119871119860
1205810= 1198700119871119860 120581 = 119870119871119860
B Thermoelectric Coefficients forGraphene with Linear Dispersion forBallistic and Diffusive Regimes
Graphene is a 2D conductor with a unique linear bandstructure (15) Its transport coefficients are calculated from(40a) (40b) (40c) (40d) (42a) (42b) (42c) (42d) and (42e)with the number of modes given by (18) The power lawscattering for diffusive regime (A3) is used
Conductivity 119866 = 120590119882119871 is given in Siemens [120590] = 1 SSimilarly for other specific coefficients 119904
119879= 119878119879119871119882 120581
0=
1198700119871119882 120581 = 119870119871119882Ballistic regime are
119866ball
= 11988221199022
ℎ(2119896119879
120587ℏV119865
) [I0(120578119865) +I0(minus120578119865)]
119878ball
= minus119896
1199022 [I1(120578119865) minusI1(minus120578119865)]
I0(120578119865) +I0(minus120578119865)
minus 120578119865
119878ball119879
= minus11988221199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
times 2 [I1(120578119865) minusI1(minus120578119865)]
minus 120578119865[I0(120578119865) +I0(minus120578119865)]
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
14 Journal of Nanoscience
119870ball
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus4[I1(120578119865) minusI1(minus120578119865)]2
I0(120578119865) +I0(minus120578119865)
119870ball0
= 11988211987921199022
ℎ(2119896119879
120587ℏV119865
)(119896
119902)
2
times 6 [I2(120578119865) +I2(minus120578119865)]
minus 4120578119865[I1(120578119865) minusI1(minus120578119865)]
+ 1205782
119865[I0(120578119865) +I0(minus120578119865)]
(B1)
Diffusive regime are
119866diff
= 11988221199022
ℎ(2119896119879
120587ℏV119865
)(1205820
119871) Γ (119903 + 2)
times [I119903(120578119865) +I119903(minus120578119865)]
119878diff
= minus119896
119902(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]
I119903(120578119865) +I119903(minus120578119865)
minus 120578119865
119878diff119879
= minus11988221199022
ℎ
119896
119902(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3) [I119903+1
(120578119865) minusI119903+1
(minus120578119865)]
minus 120578119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
119870 = 11988211987921199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 3)
times (119903 + 3) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus(119903 + 2) [I
119903+1(120578119865) minusI119903+1
(minus120578119865)]2
I119903(120578119865) +I119903(minus120578119865)
1198700= 119882119879
21199022
ℎ(119896
119902)
2
(2119896119879
120587ℏV119865
)(1205820
119871)
times Γ (119903 + 4) [I119903+2
(120578119865) +I119903+2
(minus120578119865)]
minus 2120578119865Γ (119903 + 3) [I
119903+1(120578119865) +I119903+1
(minus120578119865)]
+ 1205782
119865Γ (119903 + 2) [I
119903(120578119865) +I119903(minus120578119865)]
(B2)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thanks are due to Professor Supriyo Datta and ProfessorMark Lundstrom for giving modern lectures ldquoFundamentalsof Nanoelectronics Part I Basic Conceptsrdquo and ldquoNear-Equilibrium Transport Fundamentals and Applicationsrdquoonline in 2011 and 2012 under initiative of Purdue Universi-tynanoHUB-U (httpsnanohuborggroupsu)This reviewis closely based on these lectures and books [4 5]
References
[1] R Landauer ldquoSpatial variation of currents and fields dueto localized scatterers in metallic conductionrdquo InternationalBusiness Machines Corporation Journal of Research and Devel-opment vol 1 no 3 pp 223ndash231 1957
[2] R Landauer ldquoSpatial variation of currents and fields due tolocalized scatterers in metallic conductionrdquo Journal of Mathe-matical Physics vol 37 no 10 pp 5259ndash5268 1996
[3] R Landauer ldquoElectrical resistance of disordered one dimen-sional latticesrdquo Philosophical Magazine vol 21 pp 863ndash8671970
[4] S Datta Lessons from Nanoelectronics A New Perspective onTransport World Scientific Publishing Company New JerseyNJ USA 2012
[5] M Lundstrom and J Changwook Near-Equilibrium TransportFundamentals and Applications World Scientific New JerseyNJ USA 2013
[6] J M Ziman Electrons and Phonons The Theory of TransportPhenomena in Solids Clarendon Press Oxford UK 1960
[7] J M Ziman Principles of the Theory of Solids CambridgeUniversity Press Cambridge UK 1964
[8] C Kittel Introduction to Solid State Physics JohnWiley amp SonsNew York NY USA 1971
[9] N W Ashcroft and N D Mermin Solid State Physics SaundersCollege Philadelphia Pa USA 1976
[10] K Schwab E A Henriksen J M Worlock and M LRoukes ldquoMeasurement of the quantum of thermal conduc-tancerdquo Nature vol 404 no 6781 pp 974ndash977 2000
[11] C Jeong R Kim M Luisier S Datta and M Lundstrom ldquoOnLandauer versus Boltzmann and full band versus effective massevaluation of thermoelectric transport coefficientsrdquo Journal ofApplied Physics vol 107 no 2 Article ID 023707 2010
[12] M Lundstrom Fundamentals of Carrier Transport CambridgeUniversity Press 2012
[13] C Jeong S Datta and M Lundstrom ldquoFull dispersion versusDebye model evaluation of lattice thermal conductivity with aLandauer approachrdquo Journal of Applied Physics vol 109 no 7Article ID 073718 2011
[14] J Callaway ldquoModel for lattice thermal conductivity at lowtemperaturesrdquo Physical Review vol 113 no 4 pp 1046ndash10511959
[15] M G Holland ldquoAnalysis of lattice thermal conductivityrdquo Phys-ical Review vol 132 no 6 pp 2461ndash2471 1963
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 15
[16] C Jeong S Datta and M Lundstrom ldquoThermal conductivityof bulk and thin-film silicon a Landauer approachrdquo Journal ofApplied Physics vol 111 no 9 Article ID 093708 2012
[17] C Gang Nanoscale Energy Transport and Conversion A Par-allel Treatment of Electrons Molecules Phonons and PhotonsOxford University Press New York NY USA 2005
[18] C J Glassbrenner and G A Slack ldquoThermal conductivity ofsilicon and germanium from 3∘K to the melting pointrdquo PhysicalReview vol 134 no 4A pp A1058ndashA1069 1964
[19] J B Pendry ldquoQuantum limits to the flow of information andentropyrdquo Journal of Physics A vol 16 no 10 pp 2161ndash2171 1983
[20] D E AngelescuM C Cross andM L Roukes ldquoHeat transportin mesoscopic systemsrdquo Superlattices and Microstructures vol23 no 3-4 pp 673ndash689 1998
[21] L G C Rego and G Kirczenow ldquoQuantized thermal conduc-tance of dielectric quantum wiresrdquo Physical Review Letters vol81 no 1 pp 232ndash235 1998
[22] M P Blencowe ldquoQuantum energy flow inmesoscopic dielectricstructuresrdquo Physical Review BmdashCondensed Matter and Materi-als Physics vol 59 no 7 pp 4992ndash4998 1999
[23] L G C Rego and G Kirczenow ldquoFractional exclusion statisticsand the universal quantum of thermal conductance A unifyingapproachrdquo Physical Review B Condensed Matter and MaterialsPhysics vol 59 no 20 pp 13080ndash13086 1999
[24] I V Krive and E R Mucciolo ldquoTransport properties of quasi-particles with fractional exclusion statisticsrdquo Physical ReviewBmdashCondensed Matter and Materials Physics vol 60 no 3 pp1429ndash1432 1999
[25] CM Caves and P D Drummond ldquoQuantum limits on bosoniccommunication ratesrdquo Reviews ofModern Physics vol 66 no 2pp 481ndash537 1994
[26] R S Kim Physics and Simulation of Nanoscale Electronic andThermoelectric Devices Purdue University West Lafayette IndUSA 2011
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials