Landauer-Datta-Lundstrom Generalized Transport Model for Nanoelectronics

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


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


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)


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)


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)


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)


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)


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


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


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


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


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)



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)


2119863] =

1 S Similarly for other specific coefficients 119904119879= 119878119879119871119882 120581

0=


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)


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)


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)]


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


[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 ofNanomaterials


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)


D1D

D2D

D3D

M1D

M2D

M3D

EC

EEC

E

EC

E

EC

E

EC

EC

E

E

1


where

119863 (119864) = ⟨]2119909120591⟩ = ]2120591 (119864) 1

1

21

3 (lowast)


120590 (119864) =1199022

ℎ119872 (119864) 120582 (119864) 1

1

1198821

119860 (19b)






120590 (119864) =1199022120591 (119864)

119898 (119864)119873 (119864)

119871119873 (119864)

119871119882119873 (119864)

119871119860 (19c)


119898(119864) =119901 (119864)

V (119864) (20)


119898lowast=119864119865

V2119865

(21)


119889119881)119881rarr0



1198910(119864) asymp 119891

1(119864) gt 119891

2(119864) (22)


bias

119881 =Δ119864119865

119902=(1198641198651minus 1198641198652)

119902(23)


1198912= 1198911+1205971198911

120597119864119865

Δ119864119865= 1198911+1205971198911

120597119864119865

119902119881 (24)


+120597119891

120597119864119865

= minus120597119891

120597119864 (25)

one finds

1198911minus 1198912= (minus

1205971198910

120597119864) 119902119881 (26)


119865119879(119864 119864119865) = 119896119879(minus

120597119891

120597119864) (27)



1010

1

5

00

0 0minus10minus10

minus5

05 0201 03


120597E)

E minus EF

kT

uarr



int

+infin

minusinfin

119889119864119865119879(119864 119864119865) = 119896119879 (28)









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)



119869119909=120590

119902

119889 (119864119865)

119889119909 (31)



120590119878=21199022

ℎint1198722119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (32)


120590119878= int120590

1015840

119878(119864) 119889119864 (33a)


1205901015840

119878(119864) =

21199022

ℎ1198722119863

(119864) 120582 (119864) (minus1205971198910

120597119864) (33b)



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)


119866 = int1198661015840(119864) 119889119864 [S] (35a)

1198661015840(119864) =

21199022

ℎ119872 (119864) 119879 (119864) (minus

1205971198910

120597119864) (35b)







thermocurrent

119868119876=2

ℎint (119864 minus 119864

119865) 119879 (119864)119872 (119864) (119891

1minus 1198912) 119889119864 (36)



temperature and get


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



119868 = int 1198681015840(119864) 119889119864 (38a)


1198681015840(119864) =

2119902

ℎ119879 (119864)119872 (119864) (119891

1minus 1198912) (38b)


1198681015840(119864) = 119866

1015840(119864) Δ119881 + 119878

1015840

119879(119864) Δ119879 (39a)

where

1198661015840(119864) =

21199022

ℎ119879 (119864)119872 (119864) (minus

1205971198910

120597119864) (39b)


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)




below 119864119865(119901-resistors)


119868 = 119866Δ119881 + 119878119879Δ119879 [A] (40a)

119868119876= minus 119879119878

119879Δ119881 minus 119870

0Δ119879 [W] (40b)


119878119879= int 1198781015840

119879(119864) 119889119864 = minus

119896

119902int(

119864 minus 119864119865

119896119879)1198661015840(119864) 119889119864 [AK]

(40c)


1198700= 119879(

119896

119902)

2

int(119864 minus 119864

119865

119896119879)

2

1198661015840(119864) 119889119864 [WK] (40d)





119869119909= 120590

119889 (119864119865119902)

119889119909minus 119904119879

119889119879

119889119909 [0m2] (41a)

119869119876119909

= 119879119904119879

119889 (119864119865119902)

119889119909minus 1205810

119889119879

119889119909[Wm2] (41b)


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)



Δ119881 = 119877119868 minus 119878Δ119879 (42a)

119868119876= minus Π119868 minus 119870Δ119879 (42b)

where

119878 =119878119879

119866 (42c)

Π = 119879119878 (42d)

119870 = 1198700minus Π119878119866 (42e)



119889 (119864119865119902)

119889119909= 120588119869119909+ 119878

119889119879

119889119909 (43a)

119869119876119909

= 119879119878119869119909minus 120581

119889119879

119889119909(43b)


120588 =1

120590 (43c)

119878 =119904119879

120590 (43d)

120581 = 1205810minus 1198782120590119879 (43e)



1198723119863

(119864) = 119892V119898lowast

2120587ℏ2(119864 minus 119864

119862) (119864 ge 119864

119862) (44a)



119872(V)3119863

(119864) = 119892V

119898lowast

119901

2120587ℏ2(119864119881minus 119864) (119864 le 119864

119881) (44b)


120590 =1199022

ℎint

infin

119864119862

1198723119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (45a)


120590119901=1199022

ℎint

119864119881

minusinfin

119872V3119863

(119864) 120582119901(119864) (minus

1205971198910

120597119864)119889119864 (45b)


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)


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)



120590tot

equiv 120590 + 120590119901=1199022

ℎint

1198642

1198641

119872tot3119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (48a)

119872tot3119863

(119864) = 1198723119863

(119864) + 119872V3119863

(119864) (48b)


3119863(119864) and 120582(119864)



0(119864) with 1 minus 119891

0(119864)


119904tot119879

= minus119896

119902int

+infin

minusinfin

(119864 minus 119864

119865

119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878

119901120590119901 (49a)


119878tot

=

119878120590 + 119878119901120590119901

120590 + 120590119901

(49b)






119869ph119876119909

= minus120581119871

119889119879

119889119909[Wm2] (50)








119868 =2119902

ℎint119879el (119864)119872el (119864) (1198911 minus 119891

2) 119889119864 (51)


1198990(ℏ120596) =

1

119890ℏ120596119896119879 minus 1 (52)







119876 =1


2) 119889 (ℏ120596) [W]

(53)



1205971198990

120597119879Δ119879 (54)


1205971198990

120597119879=ℏ120596

119879(minus

1205971198990

120597 (ℏ120596)) (55)

with

1205971198990

120597 (ℏ120596)= (minus

1

119896119879)

119890ℏ120596119896119879

(119890ℏ120596119896119879 minus 1)2 (56)


119876 = minus119870119871Δ119879 (57)


119870119871=1198962119879

ℎint119879ph (ℏ120596)119872ph (ℏ120596)

times [(ℏ120596

119896119879)

2

(minus1205971198990

120597 (ℏ120596))] 119889 (ℏ120596) [WK]

(58)


119866 =21199022

ℎint119879el (119864)119872el (119864) (minus

1205971198910

120597119864)119889119864 (59)

The derivative

119882el (119864) equiv (minus1205971198910

120597119864) (60)


int

+infin

minusinfin

(minus1205971198910

120597119864)119889119864 = 1 (61)


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


minus13WK2) 119879 (64)




119865119879(119909) equiv

119890119909

(119890119909 + 1)

2(65)



119865ph119879(119909) equiv

3

1205872

1199092119890119909

(119890119909 minus 1)

2(66)




119879ph (ℏ120596) =120582ph (ℏ120596)

120582ph (ℏ120596) + 119871

100381610038161003816100381610038161003816100381610038161003816119871≫120582ph

997888rarr

120582ph (ℏ120596)

119871 (67)


119872ph (ℏ120596) prop 119860 (68)


10

5

0

minus5

minus100 01 02 03

FT(x)

FT(x)

FT(x)

phx



119876

119860equiv 119869

ph119876119909

= minus120581119871

119889119879

119889119909(69)


120581119871= 119870119871

119871

119860 (70)


120581119871=12058721198962119879

3ℎint

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)


⟨

119872ph

119860⟩ equiv int

119872ph (ℏ120596)

119860119882ph (ℏ120596) 119889 (ℏ120596) (72)

Then

120581119871=12058721198962119879

3ℎ⟨

119872ph

119860⟩⟨⟨120582ph⟩⟩ (73)


⟨⟨120582ph⟩⟩ =

int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)

(74)


119869119909=120590

119902

119889 (119864119865)

119889119909 (75)

120590 =21199022

ℎ⟨119872el119860

⟩⟨⟨120582el⟩⟩ (76)



120581119871=1

3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)


119864ph = int

infin

0

(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)


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)


= [

[

(1ℎ) intinfin

0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881



1205821119863

(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)


120582 (119864) = 2

⟨V2119909120591⟩

⟨1003816100381610038161003816V119909




1205822119863

(119864) =120587

2V (119864) 120591 (119864) (81b)

1205823119863

(119864) =4

3V (119864) 120591 (119864) (81c)


120591 (119864) = 1205910(119864 minus 119864

119862

119896119879)

119904

(82)



120582 (119864) = 1205820(119864 minus 119864

119862

119896119879)

119903

(83)




120582ph (ℏ120596) =4

3Vph (ℏ120596) 120591ph (ℏ120596) (84)


Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)

and finally

120582ph (ℏ120596) =4

3Λ ph (ℏ120596) (86)


119872el (119864) = 1198601198723119863

(119864) = 119860ℎ

4⟨V+119909(119864)⟩119863

3119863(119864) (87)



of states

1198633119863

(119864) = 21198631015840

3119863(119864) (88)


⟨V+119909(119864)⟩ =

Vel (119864)2

(89)


119872ph (ℏ120596) = 119860ℎ

2(

Vph (ℏ120596)2

) 2119863ph (ℏ120596)

= 119860ℎ

4Vph (ℏ120596)119863ph (ℏ120596)

(90)


120581119871

= [

[

(13) intinfin


intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881

(91)


int

infin

0

Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)



⟨⟨Λ ph⟩⟩

equiv

intinfin


intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(93)


⟨Vph⟩ equiv

intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(94)






ℏ120596 = ℏV119863119902 (95)



119904prop 119898minus12







119863ph (ℏ120596) =3(ℏ120596)

2



119872ph (ℏ120596) =3(ℏ120596)

2

4120587(ℏV119863)2 (97)



int

ℏ120596119863

0

119863ph (ℏ120596) 119889 (ℏ120596) (98)


ℏ120596119863= ℏV119863(61205872119873

Ω)

13

equiv 119896119879119863 (99)



perature

119879119863=ℏ120596

119896 (100)




120581119871=12058721198962119879

3ℎint

ℏ120596119863

0

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)




ℏ 1199021+ ℏ 1199022= ℏ 1199023

ℏ1+ ℏ2= ℏ3

(102)



1

120591ph (ℏ120596)=

1

120591119880(ℏ120596)

+1

120591119863(ℏ120596)

+1

120591119861(ℏ120596)

(103)


1

120582ph (ℏ120596)=

1

120582119880(ℏ120596)

+1

120582119863(ℏ120596)

+1

120582119861(ℏ120596)

(104)


1

120591119863(ℏ120596)

prop 1205964 (105)



1

120591119861(ℏ120596)

prop

Vph (ℏ120596)119871

(106)


1

120591119880(ℏ120596)

prop 11987931205962119890minus119879119863119887119879 (107)





119871(119879)



100 101101

102

102

103

103

104

120581ph

(Wm

minus1K

minus1)

T (K)



⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)


119863 all of







119866ball

=21199022

ℎ119872(119864119865) (109)


119870119871=12058721198962119879

3ℎ119879ph (0)119872ph (0) (110)


119870119871=12058721198962119879

3ℎ119872ph (111)



1198920equiv12058721198962119879

3ℎ

(112)


4 Conclusions







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




D1D

D2D

D3D

M1D

M2D

M3D

EC

EEC

E

EC

E

EC

E

EC

EC

E

E

1


where

119863 (119864) = ⟨]2119909120591⟩ = ]2120591 (119864) 1

1

21

3 (lowast)


120590 (119864) =1199022

ℎ119872 (119864) 120582 (119864) 1

1

1198821

119860 (19b)






120590 (119864) =1199022120591 (119864)

119898 (119864)119873 (119864)

119871119873 (119864)

119871119882119873 (119864)

119871119860 (19c)


119898(119864) =119901 (119864)

V (119864) (20)


119898lowast=119864119865

V2119865

(21)


119889119881)119881rarr0



1198910(119864) asymp 119891

1(119864) gt 119891

2(119864) (22)


bias

119881 =Δ119864119865

119902=(1198641198651minus 1198641198652)

119902(23)


1198912= 1198911+1205971198911

120597119864119865

Δ119864119865= 1198911+1205971198911

120597119864119865

119902119881 (24)


+120597119891

120597119864119865

= minus120597119891

120597119864 (25)

one finds

1198911minus 1198912= (minus

1205971198910

120597119864) 119902119881 (26)


119865119879(119864 119864119865) = 119896119879(minus

120597119891

120597119864) (27)



1010

1

5

00

0 0minus10minus10

minus5

05 0201 03


120597E)

E minus EF

kT

uarr



int

+infin

minusinfin

119889119864119865119879(119864 119864119865) = 119896119879 (28)









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)



119869119909=120590

119902

119889 (119864119865)

119889119909 (31)



120590119878=21199022

ℎint1198722119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (32)


120590119878= int120590

1015840

119878(119864) 119889119864 (33a)


1205901015840

119878(119864) =

21199022

ℎ1198722119863

(119864) 120582 (119864) (minus1205971198910

120597119864) (33b)



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)


119866 = int1198661015840(119864) 119889119864 [S] (35a)

1198661015840(119864) =

21199022

ℎ119872 (119864) 119879 (119864) (minus

1205971198910

120597119864) (35b)







thermocurrent

119868119876=2

ℎint (119864 minus 119864

119865) 119879 (119864)119872 (119864) (119891

1minus 1198912) 119889119864 (36)



temperature and get


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



119868 = int 1198681015840(119864) 119889119864 (38a)


1198681015840(119864) =

2119902

ℎ119879 (119864)119872 (119864) (119891

1minus 1198912) (38b)


1198681015840(119864) = 119866

1015840(119864) Δ119881 + 119878

1015840

119879(119864) Δ119879 (39a)

where

1198661015840(119864) =

21199022

ℎ119879 (119864)119872 (119864) (minus

1205971198910

120597119864) (39b)


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)




below 119864119865(119901-resistors)


119868 = 119866Δ119881 + 119878119879Δ119879 [A] (40a)

119868119876= minus 119879119878

119879Δ119881 minus 119870

0Δ119879 [W] (40b)


119878119879= int 1198781015840

119879(119864) 119889119864 = minus

119896

119902int(

119864 minus 119864119865

119896119879)1198661015840(119864) 119889119864 [AK]

(40c)


1198700= 119879(

119896

119902)

2

int(119864 minus 119864

119865

119896119879)

2

1198661015840(119864) 119889119864 [WK] (40d)





119869119909= 120590

119889 (119864119865119902)

119889119909minus 119904119879

119889119879

119889119909 [0m2] (41a)

119869119876119909

= 119879119904119879

119889 (119864119865119902)

119889119909minus 1205810

119889119879

119889119909[Wm2] (41b)


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)



Δ119881 = 119877119868 minus 119878Δ119879 (42a)

119868119876= minus Π119868 minus 119870Δ119879 (42b)

where

119878 =119878119879

119866 (42c)

Π = 119879119878 (42d)

119870 = 1198700minus Π119878119866 (42e)



119889 (119864119865119902)

119889119909= 120588119869119909+ 119878

119889119879

119889119909 (43a)

119869119876119909

= 119879119878119869119909minus 120581

119889119879

119889119909(43b)


120588 =1

120590 (43c)

119878 =119904119879

120590 (43d)

120581 = 1205810minus 1198782120590119879 (43e)



1198723119863

(119864) = 119892V119898lowast

2120587ℏ2(119864 minus 119864

119862) (119864 ge 119864

119862) (44a)



119872(V)3119863

(119864) = 119892V

119898lowast

119901

2120587ℏ2(119864119881minus 119864) (119864 le 119864

119881) (44b)


120590 =1199022

ℎint

infin

119864119862

1198723119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (45a)


120590119901=1199022

ℎint

119864119881

minusinfin

119872V3119863

(119864) 120582119901(119864) (minus

1205971198910

120597119864)119889119864 (45b)


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)


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)



120590tot

equiv 120590 + 120590119901=1199022

ℎint

1198642

1198641

119872tot3119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (48a)

119872tot3119863

(119864) = 1198723119863

(119864) + 119872V3119863

(119864) (48b)


3119863(119864) and 120582(119864)



0(119864) with 1 minus 119891

0(119864)


119904tot119879

= minus119896

119902int

+infin

minusinfin

(119864 minus 119864

119865

119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878

119901120590119901 (49a)


119878tot

=

119878120590 + 119878119901120590119901

120590 + 120590119901

(49b)






119869ph119876119909

= minus120581119871

119889119879

119889119909[Wm2] (50)








119868 =2119902

ℎint119879el (119864)119872el (119864) (1198911 minus 119891

2) 119889119864 (51)


1198990(ℏ120596) =

1

119890ℏ120596119896119879 minus 1 (52)







119876 =1


2) 119889 (ℏ120596) [W]

(53)



1205971198990

120597119879Δ119879 (54)


1205971198990

120597119879=ℏ120596

119879(minus

1205971198990

120597 (ℏ120596)) (55)

with

1205971198990

120597 (ℏ120596)= (minus

1

119896119879)

119890ℏ120596119896119879

(119890ℏ120596119896119879 minus 1)2 (56)


119876 = minus119870119871Δ119879 (57)


119870119871=1198962119879

ℎint119879ph (ℏ120596)119872ph (ℏ120596)

times [(ℏ120596

119896119879)

2

(minus1205971198990

120597 (ℏ120596))] 119889 (ℏ120596) [WK]

(58)


119866 =21199022

ℎint119879el (119864)119872el (119864) (minus

1205971198910

120597119864)119889119864 (59)

The derivative

119882el (119864) equiv (minus1205971198910

120597119864) (60)


int

+infin

minusinfin

(minus1205971198910

120597119864)119889119864 = 1 (61)


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


minus13WK2) 119879 (64)




119865119879(119909) equiv

119890119909

(119890119909 + 1)

2(65)



119865ph119879(119909) equiv

3

1205872

1199092119890119909

(119890119909 minus 1)

2(66)




119879ph (ℏ120596) =120582ph (ℏ120596)

120582ph (ℏ120596) + 119871

100381610038161003816100381610038161003816100381610038161003816119871≫120582ph

997888rarr

120582ph (ℏ120596)

119871 (67)


119872ph (ℏ120596) prop 119860 (68)


10

5

0

minus5

minus100 01 02 03

FT(x)

FT(x)

FT(x)

phx



119876

119860equiv 119869

ph119876119909

= minus120581119871

119889119879

119889119909(69)


120581119871= 119870119871

119871

119860 (70)


120581119871=12058721198962119879

3ℎint

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)


⟨

119872ph

119860⟩ equiv int

119872ph (ℏ120596)

119860119882ph (ℏ120596) 119889 (ℏ120596) (72)

Then

120581119871=12058721198962119879

3ℎ⟨

119872ph

119860⟩⟨⟨120582ph⟩⟩ (73)


⟨⟨120582ph⟩⟩ =

int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)

(74)


119869119909=120590

119902

119889 (119864119865)

119889119909 (75)

120590 =21199022

ℎ⟨119872el119860

⟩⟨⟨120582el⟩⟩ (76)



120581119871=1

3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)


119864ph = int

infin

0

(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)


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)


= [

[

(1ℎ) intinfin

0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881



1205821119863

(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)


120582 (119864) = 2

⟨V2119909120591⟩

⟨1003816100381610038161003816V119909




1205822119863

(119864) =120587

2V (119864) 120591 (119864) (81b)

1205823119863

(119864) =4

3V (119864) 120591 (119864) (81c)


120591 (119864) = 1205910(119864 minus 119864

119862

119896119879)

119904

(82)



120582 (119864) = 1205820(119864 minus 119864

119862

119896119879)

119903

(83)




120582ph (ℏ120596) =4

3Vph (ℏ120596) 120591ph (ℏ120596) (84)


Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)

and finally

120582ph (ℏ120596) =4

3Λ ph (ℏ120596) (86)


119872el (119864) = 1198601198723119863

(119864) = 119860ℎ

4⟨V+119909(119864)⟩119863

3119863(119864) (87)



of states

1198633119863

(119864) = 21198631015840

3119863(119864) (88)


⟨V+119909(119864)⟩ =

Vel (119864)2

(89)


119872ph (ℏ120596) = 119860ℎ

2(

Vph (ℏ120596)2

) 2119863ph (ℏ120596)

= 119860ℎ

4Vph (ℏ120596)119863ph (ℏ120596)

(90)


120581119871

= [

[

(13) intinfin


intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881

(91)


int

infin

0

Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)



⟨⟨Λ ph⟩⟩

equiv

intinfin


intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(93)


⟨Vph⟩ equiv

intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(94)






ℏ120596 = ℏV119863119902 (95)



119904prop 119898minus12







119863ph (ℏ120596) =3(ℏ120596)

2



119872ph (ℏ120596) =3(ℏ120596)

2

4120587(ℏV119863)2 (97)



int

ℏ120596119863

0

119863ph (ℏ120596) 119889 (ℏ120596) (98)


ℏ120596119863= ℏV119863(61205872119873

Ω)

13

equiv 119896119879119863 (99)



perature

119879119863=ℏ120596

119896 (100)




120581119871=12058721198962119879

3ℎint

ℏ120596119863

0

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)




ℏ 1199021+ ℏ 1199022= ℏ 1199023

ℏ1+ ℏ2= ℏ3

(102)



1

120591ph (ℏ120596)=

1

120591119880(ℏ120596)

+1

120591119863(ℏ120596)

+1

120591119861(ℏ120596)

(103)


1

120582ph (ℏ120596)=

1

120582119880(ℏ120596)

+1

120582119863(ℏ120596)

+1

120582119861(ℏ120596)

(104)


1

120591119863(ℏ120596)

prop 1205964 (105)



1

120591119861(ℏ120596)

prop

Vph (ℏ120596)119871

(106)


1

120591119880(ℏ120596)

prop 11987931205962119890minus119879119863119887119879 (107)





119871(119879)



100 101101

102

102

103

103

104

120581ph

(Wm

minus1K

minus1)

T (K)



⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)


119863 all of







119866ball

=21199022

ℎ119872(119864119865) (109)


119870119871=12058721198962119879

3ℎ119879ph (0)119872ph (0) (110)


119870119871=12058721198962119879

3ℎ119872ph (111)



1198920equiv12058721198962119879

3ℎ

(112)


4 Conclusions







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




1010

1

5

00

0 0minus10minus10

minus5

05 0201 03


120597E)

E minus EF

kT

uarr



int

+infin

minusinfin

119889119864119865119879(119864 119864119865) = 119896119879 (28)









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)



119869119909=120590

119902

119889 (119864119865)

119889119909 (31)



120590119878=21199022

ℎint1198722119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (32)


120590119878= int120590

1015840

119878(119864) 119889119864 (33a)


1205901015840

119878(119864) =

21199022

ℎ1198722119863

(119864) 120582 (119864) (minus1205971198910

120597119864) (33b)



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)


119866 = int1198661015840(119864) 119889119864 [S] (35a)

1198661015840(119864) =

21199022

ℎ119872 (119864) 119879 (119864) (minus

1205971198910

120597119864) (35b)







thermocurrent

119868119876=2

ℎint (119864 minus 119864

119865) 119879 (119864)119872 (119864) (119891

1minus 1198912) 119889119864 (36)



temperature and get


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



119868 = int 1198681015840(119864) 119889119864 (38a)


1198681015840(119864) =

2119902

ℎ119879 (119864)119872 (119864) (119891

1minus 1198912) (38b)


1198681015840(119864) = 119866

1015840(119864) Δ119881 + 119878

1015840

119879(119864) Δ119879 (39a)

where

1198661015840(119864) =

21199022

ℎ119879 (119864)119872 (119864) (minus

1205971198910

120597119864) (39b)


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)




below 119864119865(119901-resistors)


119868 = 119866Δ119881 + 119878119879Δ119879 [A] (40a)

119868119876= minus 119879119878

119879Δ119881 minus 119870

0Δ119879 [W] (40b)


119878119879= int 1198781015840

119879(119864) 119889119864 = minus

119896

119902int(

119864 minus 119864119865

119896119879)1198661015840(119864) 119889119864 [AK]

(40c)


1198700= 119879(

119896

119902)

2

int(119864 minus 119864

119865

119896119879)

2

1198661015840(119864) 119889119864 [WK] (40d)





119869119909= 120590

119889 (119864119865119902)

119889119909minus 119904119879

119889119879

119889119909 [0m2] (41a)

119869119876119909

= 119879119904119879

119889 (119864119865119902)

119889119909minus 1205810

119889119879

119889119909[Wm2] (41b)


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)



Δ119881 = 119877119868 minus 119878Δ119879 (42a)

119868119876= minus Π119868 minus 119870Δ119879 (42b)

where

119878 =119878119879

119866 (42c)

Π = 119879119878 (42d)

119870 = 1198700minus Π119878119866 (42e)



119889 (119864119865119902)

119889119909= 120588119869119909+ 119878

119889119879

119889119909 (43a)

119869119876119909

= 119879119878119869119909minus 120581

119889119879

119889119909(43b)


120588 =1

120590 (43c)

119878 =119904119879

120590 (43d)

120581 = 1205810minus 1198782120590119879 (43e)



1198723119863

(119864) = 119892V119898lowast

2120587ℏ2(119864 minus 119864

119862) (119864 ge 119864

119862) (44a)



119872(V)3119863

(119864) = 119892V

119898lowast

119901

2120587ℏ2(119864119881minus 119864) (119864 le 119864

119881) (44b)


120590 =1199022

ℎint

infin

119864119862

1198723119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (45a)


120590119901=1199022

ℎint

119864119881

minusinfin

119872V3119863

(119864) 120582119901(119864) (minus

1205971198910

120597119864)119889119864 (45b)


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)


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)



120590tot

equiv 120590 + 120590119901=1199022

ℎint

1198642

1198641

119872tot3119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (48a)

119872tot3119863

(119864) = 1198723119863

(119864) + 119872V3119863

(119864) (48b)


3119863(119864) and 120582(119864)



0(119864) with 1 minus 119891

0(119864)


119904tot119879

= minus119896

119902int

+infin

minusinfin

(119864 minus 119864

119865

119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878

119901120590119901 (49a)


119878tot

=

119878120590 + 119878119901120590119901

120590 + 120590119901

(49b)






119869ph119876119909

= minus120581119871

119889119879

119889119909[Wm2] (50)








119868 =2119902

ℎint119879el (119864)119872el (119864) (1198911 minus 119891

2) 119889119864 (51)


1198990(ℏ120596) =

1

119890ℏ120596119896119879 minus 1 (52)







119876 =1


2) 119889 (ℏ120596) [W]

(53)



1205971198990

120597119879Δ119879 (54)


1205971198990

120597119879=ℏ120596

119879(minus

1205971198990

120597 (ℏ120596)) (55)

with

1205971198990

120597 (ℏ120596)= (minus

1

119896119879)

119890ℏ120596119896119879

(119890ℏ120596119896119879 minus 1)2 (56)


119876 = minus119870119871Δ119879 (57)


119870119871=1198962119879

ℎint119879ph (ℏ120596)119872ph (ℏ120596)

times [(ℏ120596

119896119879)

2

(minus1205971198990

120597 (ℏ120596))] 119889 (ℏ120596) [WK]

(58)


119866 =21199022

ℎint119879el (119864)119872el (119864) (minus

1205971198910

120597119864)119889119864 (59)

The derivative

119882el (119864) equiv (minus1205971198910

120597119864) (60)


int

+infin

minusinfin

(minus1205971198910

120597119864)119889119864 = 1 (61)


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


minus13WK2) 119879 (64)




119865119879(119909) equiv

119890119909

(119890119909 + 1)

2(65)



119865ph119879(119909) equiv

3

1205872

1199092119890119909

(119890119909 minus 1)

2(66)




119879ph (ℏ120596) =120582ph (ℏ120596)

120582ph (ℏ120596) + 119871

100381610038161003816100381610038161003816100381610038161003816119871≫120582ph

997888rarr

120582ph (ℏ120596)

119871 (67)


119872ph (ℏ120596) prop 119860 (68)


10

5

0

minus5

minus100 01 02 03

FT(x)

FT(x)

FT(x)

phx



119876

119860equiv 119869

ph119876119909

= minus120581119871

119889119879

119889119909(69)


120581119871= 119870119871

119871

119860 (70)


120581119871=12058721198962119879

3ℎint

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)


⟨

119872ph

119860⟩ equiv int

119872ph (ℏ120596)

119860119882ph (ℏ120596) 119889 (ℏ120596) (72)

Then

120581119871=12058721198962119879

3ℎ⟨

119872ph

119860⟩⟨⟨120582ph⟩⟩ (73)


⟨⟨120582ph⟩⟩ =

int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)

(74)


119869119909=120590

119902

119889 (119864119865)

119889119909 (75)

120590 =21199022

ℎ⟨119872el119860

⟩⟨⟨120582el⟩⟩ (76)



120581119871=1

3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)


119864ph = int

infin

0

(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)


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)


= [

[

(1ℎ) intinfin

0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881



1205821119863

(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)


120582 (119864) = 2

⟨V2119909120591⟩

⟨1003816100381610038161003816V119909




1205822119863

(119864) =120587

2V (119864) 120591 (119864) (81b)

1205823119863

(119864) =4

3V (119864) 120591 (119864) (81c)


120591 (119864) = 1205910(119864 minus 119864

119862

119896119879)

119904

(82)



120582 (119864) = 1205820(119864 minus 119864

119862

119896119879)

119903

(83)




120582ph (ℏ120596) =4

3Vph (ℏ120596) 120591ph (ℏ120596) (84)


Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)

and finally

120582ph (ℏ120596) =4

3Λ ph (ℏ120596) (86)


119872el (119864) = 1198601198723119863

(119864) = 119860ℎ

4⟨V+119909(119864)⟩119863

3119863(119864) (87)



of states

1198633119863

(119864) = 21198631015840

3119863(119864) (88)


⟨V+119909(119864)⟩ =

Vel (119864)2

(89)


119872ph (ℏ120596) = 119860ℎ

2(

Vph (ℏ120596)2

) 2119863ph (ℏ120596)

= 119860ℎ

4Vph (ℏ120596)119863ph (ℏ120596)

(90)


120581119871

= [

[

(13) intinfin


intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881

(91)


int

infin

0

Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)



⟨⟨Λ ph⟩⟩

equiv

intinfin


intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(93)


⟨Vph⟩ equiv

intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(94)






ℏ120596 = ℏV119863119902 (95)



119904prop 119898minus12







119863ph (ℏ120596) =3(ℏ120596)

2



119872ph (ℏ120596) =3(ℏ120596)

2

4120587(ℏV119863)2 (97)



int

ℏ120596119863

0

119863ph (ℏ120596) 119889 (ℏ120596) (98)


ℏ120596119863= ℏV119863(61205872119873

Ω)

13

equiv 119896119879119863 (99)



perature

119879119863=ℏ120596

119896 (100)




120581119871=12058721198962119879

3ℎint

ℏ120596119863

0

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)




ℏ 1199021+ ℏ 1199022= ℏ 1199023

ℏ1+ ℏ2= ℏ3

(102)



1

120591ph (ℏ120596)=

1

120591119880(ℏ120596)

+1

120591119863(ℏ120596)

+1

120591119861(ℏ120596)

(103)


1

120582ph (ℏ120596)=

1

120582119880(ℏ120596)

+1

120582119863(ℏ120596)

+1

120582119861(ℏ120596)

(104)


1

120591119863(ℏ120596)

prop 1205964 (105)



1

120591119861(ℏ120596)

prop

Vph (ℏ120596)119871

(106)


1

120591119880(ℏ120596)

prop 11987931205962119890minus119879119863119887119879 (107)





119871(119879)



100 101101

102

102

103

103

104

120581ph

(Wm

minus1K

minus1)

T (K)



⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)


119863 all of







119866ball

=21199022

ℎ119872(119864119865) (109)


119870119871=12058721198962119879

3ℎ119879ph (0)119872ph (0) (110)


119870119871=12058721198962119879

3ℎ119872ph (111)



1198920equiv12058721198962119879

3ℎ

(112)


4 Conclusions







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





119868 = int 1198681015840(119864) 119889119864 (38a)


1198681015840(119864) =

2119902

ℎ119879 (119864)119872 (119864) (119891

1minus 1198912) (38b)


1198681015840(119864) = 119866

1015840(119864) Δ119881 + 119878

1015840

119879(119864) Δ119879 (39a)

where

1198661015840(119864) =

21199022

ℎ119879 (119864)119872 (119864) (minus

1205971198910

120597119864) (39b)


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)




below 119864119865(119901-resistors)


119868 = 119866Δ119881 + 119878119879Δ119879 [A] (40a)

119868119876= minus 119879119878

119879Δ119881 minus 119870

0Δ119879 [W] (40b)


119878119879= int 1198781015840

119879(119864) 119889119864 = minus

119896

119902int(

119864 minus 119864119865

119896119879)1198661015840(119864) 119889119864 [AK]

(40c)


1198700= 119879(

119896

119902)

2

int(119864 minus 119864

119865

119896119879)

2

1198661015840(119864) 119889119864 [WK] (40d)





119869119909= 120590

119889 (119864119865119902)

119889119909minus 119904119879

119889119879

119889119909 [0m2] (41a)

119869119876119909

= 119879119904119879

119889 (119864119865119902)

119889119909minus 1205810

119889119879

119889119909[Wm2] (41b)


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)



Δ119881 = 119877119868 minus 119878Δ119879 (42a)

119868119876= minus Π119868 minus 119870Δ119879 (42b)

where

119878 =119878119879

119866 (42c)

Π = 119879119878 (42d)

119870 = 1198700minus Π119878119866 (42e)



119889 (119864119865119902)

119889119909= 120588119869119909+ 119878

119889119879

119889119909 (43a)

119869119876119909

= 119879119878119869119909minus 120581

119889119879

119889119909(43b)


120588 =1

120590 (43c)

119878 =119904119879

120590 (43d)

120581 = 1205810minus 1198782120590119879 (43e)



1198723119863

(119864) = 119892V119898lowast

2120587ℏ2(119864 minus 119864

119862) (119864 ge 119864

119862) (44a)



119872(V)3119863

(119864) = 119892V

119898lowast

119901

2120587ℏ2(119864119881minus 119864) (119864 le 119864

119881) (44b)


120590 =1199022

ℎint

infin

119864119862

1198723119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (45a)


120590119901=1199022

ℎint

119864119881

minusinfin

119872V3119863

(119864) 120582119901(119864) (minus

1205971198910

120597119864)119889119864 (45b)


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)


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)



120590tot

equiv 120590 + 120590119901=1199022

ℎint

1198642

1198641

119872tot3119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (48a)

119872tot3119863

(119864) = 1198723119863

(119864) + 119872V3119863

(119864) (48b)


3119863(119864) and 120582(119864)



0(119864) with 1 minus 119891

0(119864)


119904tot119879

= minus119896

119902int

+infin

minusinfin

(119864 minus 119864

119865

119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878

119901120590119901 (49a)


119878tot

=

119878120590 + 119878119901120590119901

120590 + 120590119901

(49b)






119869ph119876119909

= minus120581119871

119889119879

119889119909[Wm2] (50)








119868 =2119902

ℎint119879el (119864)119872el (119864) (1198911 minus 119891

2) 119889119864 (51)


1198990(ℏ120596) =

1

119890ℏ120596119896119879 minus 1 (52)







119876 =1


2) 119889 (ℏ120596) [W]

(53)



1205971198990

120597119879Δ119879 (54)


1205971198990

120597119879=ℏ120596

119879(minus

1205971198990

120597 (ℏ120596)) (55)

with

1205971198990

120597 (ℏ120596)= (minus

1

119896119879)

119890ℏ120596119896119879

(119890ℏ120596119896119879 minus 1)2 (56)


119876 = minus119870119871Δ119879 (57)


119870119871=1198962119879

ℎint119879ph (ℏ120596)119872ph (ℏ120596)

times [(ℏ120596

119896119879)

2

(minus1205971198990

120597 (ℏ120596))] 119889 (ℏ120596) [WK]

(58)


119866 =21199022

ℎint119879el (119864)119872el (119864) (minus

1205971198910

120597119864)119889119864 (59)

The derivative

119882el (119864) equiv (minus1205971198910

120597119864) (60)


int

+infin

minusinfin

(minus1205971198910

120597119864)119889119864 = 1 (61)


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


minus13WK2) 119879 (64)




119865119879(119909) equiv

119890119909

(119890119909 + 1)

2(65)



119865ph119879(119909) equiv

3

1205872

1199092119890119909

(119890119909 minus 1)

2(66)




119879ph (ℏ120596) =120582ph (ℏ120596)

120582ph (ℏ120596) + 119871

100381610038161003816100381610038161003816100381610038161003816119871≫120582ph

997888rarr

120582ph (ℏ120596)

119871 (67)


119872ph (ℏ120596) prop 119860 (68)


10

5

0

minus5

minus100 01 02 03

FT(x)

FT(x)

FT(x)

phx



119876

119860equiv 119869

ph119876119909

= minus120581119871

119889119879

119889119909(69)


120581119871= 119870119871

119871

119860 (70)


120581119871=12058721198962119879

3ℎint

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)


⟨

119872ph

119860⟩ equiv int

119872ph (ℏ120596)

119860119882ph (ℏ120596) 119889 (ℏ120596) (72)

Then

120581119871=12058721198962119879

3ℎ⟨

119872ph

119860⟩⟨⟨120582ph⟩⟩ (73)


⟨⟨120582ph⟩⟩ =

int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)

(74)


119869119909=120590

119902

119889 (119864119865)

119889119909 (75)

120590 =21199022

ℎ⟨119872el119860

⟩⟨⟨120582el⟩⟩ (76)



120581119871=1

3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)


119864ph = int

infin

0

(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)


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)


= [

[

(1ℎ) intinfin

0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881



1205821119863

(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)


120582 (119864) = 2

⟨V2119909120591⟩

⟨1003816100381610038161003816V119909




1205822119863

(119864) =120587

2V (119864) 120591 (119864) (81b)

1205823119863

(119864) =4

3V (119864) 120591 (119864) (81c)


120591 (119864) = 1205910(119864 minus 119864

119862

119896119879)

119904

(82)



120582 (119864) = 1205820(119864 minus 119864

119862

119896119879)

119903

(83)




120582ph (ℏ120596) =4

3Vph (ℏ120596) 120591ph (ℏ120596) (84)


Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)

and finally

120582ph (ℏ120596) =4

3Λ ph (ℏ120596) (86)


119872el (119864) = 1198601198723119863

(119864) = 119860ℎ

4⟨V+119909(119864)⟩119863

3119863(119864) (87)



of states

1198633119863

(119864) = 21198631015840

3119863(119864) (88)


⟨V+119909(119864)⟩ =

Vel (119864)2

(89)


119872ph (ℏ120596) = 119860ℎ

2(

Vph (ℏ120596)2

) 2119863ph (ℏ120596)

= 119860ℎ

4Vph (ℏ120596)119863ph (ℏ120596)

(90)


120581119871

= [

[

(13) intinfin


intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881

(91)


int

infin

0

Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)



⟨⟨Λ ph⟩⟩

equiv

intinfin


intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(93)


⟨Vph⟩ equiv

intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(94)






ℏ120596 = ℏV119863119902 (95)



119904prop 119898minus12







119863ph (ℏ120596) =3(ℏ120596)

2



119872ph (ℏ120596) =3(ℏ120596)

2

4120587(ℏV119863)2 (97)



int

ℏ120596119863

0

119863ph (ℏ120596) 119889 (ℏ120596) (98)


ℏ120596119863= ℏV119863(61205872119873

Ω)

13

equiv 119896119879119863 (99)



perature

119879119863=ℏ120596

119896 (100)




120581119871=12058721198962119879

3ℎint

ℏ120596119863

0

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)




ℏ 1199021+ ℏ 1199022= ℏ 1199023

ℏ1+ ℏ2= ℏ3

(102)



1

120591ph (ℏ120596)=

1

120591119880(ℏ120596)

+1

120591119863(ℏ120596)

+1

120591119861(ℏ120596)

(103)


1

120582ph (ℏ120596)=

1

120582119880(ℏ120596)

+1

120582119863(ℏ120596)

+1

120582119861(ℏ120596)

(104)


1

120591119863(ℏ120596)

prop 1205964 (105)



1

120591119861(ℏ120596)

prop

Vph (ℏ120596)119871

(106)


1

120591119880(ℏ120596)

prop 11987931205962119890minus119879119863119887119879 (107)





119871(119879)



100 101101

102

102

103

103

104

120581ph

(Wm

minus1K

minus1)

T (K)



⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)


119863 all of







119866ball

=21199022

ℎ119872(119864119865) (109)


119870119871=12058721198962119879

3ℎ119879ph (0)119872ph (0) (110)


119870119871=12058721198962119879

3ℎ119872ph (111)



1198920equiv12058721198962119879

3ℎ

(112)


4 Conclusions







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





119872(V)3119863

(119864) = 119892V

119898lowast

119901

2120587ℏ2(119864119881minus 119864) (119864 le 119864

119881) (44b)


120590 =1199022

ℎint

infin

119864119862

1198723119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (45a)


120590119901=1199022

ℎint

119864119881

minusinfin

119872V3119863

(119864) 120582119901(119864) (minus

1205971198910

120597119864)119889119864 (45b)


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)


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)



120590tot

equiv 120590 + 120590119901=1199022

ℎint

1198642

1198641

119872tot3119863

(119864) 120582 (119864) (minus1205971198910

120597119864)119889119864 (48a)

119872tot3119863

(119864) = 1198723119863

(119864) + 119872V3119863

(119864) (48b)


3119863(119864) and 120582(119864)



0(119864) with 1 minus 119891

0(119864)


119904tot119879

= minus119896

119902int

+infin

minusinfin

(119864 minus 119864

119865

119896119879)1205901015840(119864) 119889119864 = 119878120590 + 119878

119901120590119901 (49a)


119878tot

=

119878120590 + 119878119901120590119901

120590 + 120590119901

(49b)






119869ph119876119909

= minus120581119871

119889119879

119889119909[Wm2] (50)








119868 =2119902

ℎint119879el (119864)119872el (119864) (1198911 minus 119891

2) 119889119864 (51)


1198990(ℏ120596) =

1

119890ℏ120596119896119879 minus 1 (52)







119876 =1


2) 119889 (ℏ120596) [W]

(53)



1205971198990

120597119879Δ119879 (54)


1205971198990

120597119879=ℏ120596

119879(minus

1205971198990

120597 (ℏ120596)) (55)

with

1205971198990

120597 (ℏ120596)= (minus

1

119896119879)

119890ℏ120596119896119879

(119890ℏ120596119896119879 minus 1)2 (56)


119876 = minus119870119871Δ119879 (57)


119870119871=1198962119879

ℎint119879ph (ℏ120596)119872ph (ℏ120596)

times [(ℏ120596

119896119879)

2

(minus1205971198990

120597 (ℏ120596))] 119889 (ℏ120596) [WK]

(58)


119866 =21199022

ℎint119879el (119864)119872el (119864) (minus

1205971198910

120597119864)119889119864 (59)

The derivative

119882el (119864) equiv (minus1205971198910

120597119864) (60)


int

+infin

minusinfin

(minus1205971198910

120597119864)119889119864 = 1 (61)


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


minus13WK2) 119879 (64)




119865119879(119909) equiv

119890119909

(119890119909 + 1)

2(65)



119865ph119879(119909) equiv

3

1205872

1199092119890119909

(119890119909 minus 1)

2(66)




119879ph (ℏ120596) =120582ph (ℏ120596)

120582ph (ℏ120596) + 119871

100381610038161003816100381610038161003816100381610038161003816119871≫120582ph

997888rarr

120582ph (ℏ120596)

119871 (67)


119872ph (ℏ120596) prop 119860 (68)


10

5

0

minus5

minus100 01 02 03

FT(x)

FT(x)

FT(x)

phx



119876

119860equiv 119869

ph119876119909

= minus120581119871

119889119879

119889119909(69)


120581119871= 119870119871

119871

119860 (70)


120581119871=12058721198962119879

3ℎint

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)


⟨

119872ph

119860⟩ equiv int

119872ph (ℏ120596)

119860119882ph (ℏ120596) 119889 (ℏ120596) (72)

Then

120581119871=12058721198962119879

3ℎ⟨

119872ph

119860⟩⟨⟨120582ph⟩⟩ (73)


⟨⟨120582ph⟩⟩ =

int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)

(74)


119869119909=120590

119902

119889 (119864119865)

119889119909 (75)

120590 =21199022

ℎ⟨119872el119860

⟩⟨⟨120582el⟩⟩ (76)



120581119871=1

3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)


119864ph = int

infin

0

(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)


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)


= [

[

(1ℎ) intinfin

0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881



1205821119863

(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)


120582 (119864) = 2

⟨V2119909120591⟩

⟨1003816100381610038161003816V119909




1205822119863

(119864) =120587

2V (119864) 120591 (119864) (81b)

1205823119863

(119864) =4

3V (119864) 120591 (119864) (81c)


120591 (119864) = 1205910(119864 minus 119864

119862

119896119879)

119904

(82)



120582 (119864) = 1205820(119864 minus 119864

119862

119896119879)

119903

(83)




120582ph (ℏ120596) =4

3Vph (ℏ120596) 120591ph (ℏ120596) (84)


Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)

and finally

120582ph (ℏ120596) =4

3Λ ph (ℏ120596) (86)


119872el (119864) = 1198601198723119863

(119864) = 119860ℎ

4⟨V+119909(119864)⟩119863

3119863(119864) (87)



of states

1198633119863

(119864) = 21198631015840

3119863(119864) (88)


⟨V+119909(119864)⟩ =

Vel (119864)2

(89)


119872ph (ℏ120596) = 119860ℎ

2(

Vph (ℏ120596)2

) 2119863ph (ℏ120596)

= 119860ℎ

4Vph (ℏ120596)119863ph (ℏ120596)

(90)


120581119871

= [

[

(13) intinfin


intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881

(91)


int

infin

0

Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)



⟨⟨Λ ph⟩⟩

equiv

intinfin


intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(93)


⟨Vph⟩ equiv

intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(94)






ℏ120596 = ℏV119863119902 (95)



119904prop 119898minus12







119863ph (ℏ120596) =3(ℏ120596)

2



119872ph (ℏ120596) =3(ℏ120596)

2

4120587(ℏV119863)2 (97)



int

ℏ120596119863

0

119863ph (ℏ120596) 119889 (ℏ120596) (98)


ℏ120596119863= ℏV119863(61205872119873

Ω)

13

equiv 119896119879119863 (99)



perature

119879119863=ℏ120596

119896 (100)




120581119871=12058721198962119879

3ℎint

ℏ120596119863

0

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)




ℏ 1199021+ ℏ 1199022= ℏ 1199023

ℏ1+ ℏ2= ℏ3

(102)



1

120591ph (ℏ120596)=

1

120591119880(ℏ120596)

+1

120591119863(ℏ120596)

+1

120591119861(ℏ120596)

(103)


1

120582ph (ℏ120596)=

1

120582119880(ℏ120596)

+1

120582119863(ℏ120596)

+1

120582119861(ℏ120596)

(104)


1

120591119863(ℏ120596)

prop 1205964 (105)



1

120591119861(ℏ120596)

prop

Vph (ℏ120596)119871

(106)


1

120591119880(ℏ120596)

prop 11987931205962119890minus119879119863119887119879 (107)





119871(119879)



100 101101

102

102

103

103

104

120581ph

(Wm

minus1K

minus1)

T (K)



⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)


119863 all of







119866ball

=21199022

ℎ119872(119864119865) (109)


119870119871=12058721198962119879

3ℎ119879ph (0)119872ph (0) (110)


119870119871=12058721198962119879

3ℎ119872ph (111)



1198920equiv12058721198962119879

3ℎ

(112)


4 Conclusions







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials









119876 =1


2) 119889 (ℏ120596) [W]

(53)



1205971198990

120597119879Δ119879 (54)


1205971198990

120597119879=ℏ120596

119879(minus

1205971198990

120597 (ℏ120596)) (55)

with

1205971198990

120597 (ℏ120596)= (minus

1

119896119879)

119890ℏ120596119896119879

(119890ℏ120596119896119879 minus 1)2 (56)


119876 = minus119870119871Δ119879 (57)


119870119871=1198962119879

ℎint119879ph (ℏ120596)119872ph (ℏ120596)

times [(ℏ120596

119896119879)

2

(minus1205971198990

120597 (ℏ120596))] 119889 (ℏ120596) [WK]

(58)


119866 =21199022

ℎint119879el (119864)119872el (119864) (minus

1205971198910

120597119864)119889119864 (59)

The derivative

119882el (119864) equiv (minus1205971198910

120597119864) (60)


int

+infin

minusinfin

(minus1205971198910

120597119864)119889119864 = 1 (61)


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


minus13WK2) 119879 (64)




119865119879(119909) equiv

119890119909

(119890119909 + 1)

2(65)



119865ph119879(119909) equiv

3

1205872

1199092119890119909

(119890119909 minus 1)

2(66)




119879ph (ℏ120596) =120582ph (ℏ120596)

120582ph (ℏ120596) + 119871

100381610038161003816100381610038161003816100381610038161003816119871≫120582ph

997888rarr

120582ph (ℏ120596)

119871 (67)


119872ph (ℏ120596) prop 119860 (68)


10

5

0

minus5

minus100 01 02 03

FT(x)

FT(x)

FT(x)

phx



119876

119860equiv 119869

ph119876119909

= minus120581119871

119889119879

119889119909(69)


120581119871= 119870119871

119871

119860 (70)


120581119871=12058721198962119879

3ℎint

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)


⟨

119872ph

119860⟩ equiv int

119872ph (ℏ120596)

119860119882ph (ℏ120596) 119889 (ℏ120596) (72)

Then

120581119871=12058721198962119879

3ℎ⟨

119872ph

119860⟩⟨⟨120582ph⟩⟩ (73)


⟨⟨120582ph⟩⟩ =

int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)

(74)


119869119909=120590

119902

119889 (119864119865)

119889119909 (75)

120590 =21199022

ℎ⟨119872el119860

⟩⟨⟨120582el⟩⟩ (76)



120581119871=1

3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)


119864ph = int

infin

0

(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)


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)


= [

[

(1ℎ) intinfin

0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881



1205821119863

(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)


120582 (119864) = 2

⟨V2119909120591⟩

⟨1003816100381610038161003816V119909




1205822119863

(119864) =120587

2V (119864) 120591 (119864) (81b)

1205823119863

(119864) =4

3V (119864) 120591 (119864) (81c)


120591 (119864) = 1205910(119864 minus 119864

119862

119896119879)

119904

(82)



120582 (119864) = 1205820(119864 minus 119864

119862

119896119879)

119903

(83)




120582ph (ℏ120596) =4

3Vph (ℏ120596) 120591ph (ℏ120596) (84)


Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)

and finally

120582ph (ℏ120596) =4

3Λ ph (ℏ120596) (86)


119872el (119864) = 1198601198723119863

(119864) = 119860ℎ

4⟨V+119909(119864)⟩119863

3119863(119864) (87)



of states

1198633119863

(119864) = 21198631015840

3119863(119864) (88)


⟨V+119909(119864)⟩ =

Vel (119864)2

(89)


119872ph (ℏ120596) = 119860ℎ

2(

Vph (ℏ120596)2

) 2119863ph (ℏ120596)

= 119860ℎ

4Vph (ℏ120596)119863ph (ℏ120596)

(90)


120581119871

= [

[

(13) intinfin


intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881

(91)


int

infin

0

Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)



⟨⟨Λ ph⟩⟩

equiv

intinfin


intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(93)


⟨Vph⟩ equiv

intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(94)






ℏ120596 = ℏV119863119902 (95)



119904prop 119898minus12







119863ph (ℏ120596) =3(ℏ120596)

2



119872ph (ℏ120596) =3(ℏ120596)

2

4120587(ℏV119863)2 (97)



int

ℏ120596119863

0

119863ph (ℏ120596) 119889 (ℏ120596) (98)


ℏ120596119863= ℏV119863(61205872119873

Ω)

13

equiv 119896119879119863 (99)



perature

119879119863=ℏ120596

119896 (100)




120581119871=12058721198962119879

3ℎint

ℏ120596119863

0

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)




ℏ 1199021+ ℏ 1199022= ℏ 1199023

ℏ1+ ℏ2= ℏ3

(102)



1

120591ph (ℏ120596)=

1

120591119880(ℏ120596)

+1

120591119863(ℏ120596)

+1

120591119861(ℏ120596)

(103)


1

120582ph (ℏ120596)=

1

120582119880(ℏ120596)

+1

120582119863(ℏ120596)

+1

120582119861(ℏ120596)

(104)


1

120591119863(ℏ120596)

prop 1205964 (105)



1

120591119861(ℏ120596)

prop

Vph (ℏ120596)119871

(106)


1

120591119880(ℏ120596)

prop 11987931205962119890minus119879119863119887119879 (107)





119871(119879)



100 101101

102

102

103

103

104

120581ph

(Wm

minus1K

minus1)

T (K)



⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)


119863 all of







119866ball

=21199022

ℎ119872(119864119865) (109)


119870119871=12058721198962119879

3ℎ119879ph (0)119872ph (0) (110)


119870119871=12058721198962119879

3ℎ119872ph (111)



1198920equiv12058721198962119879

3ℎ

(112)


4 Conclusions







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




10

5

0

minus5

minus100 01 02 03

FT(x)

FT(x)

FT(x)

phx



119876

119860equiv 119869

ph119876119909

= minus120581119871

119889119879

119889119909(69)


120581119871= 119870119871

119871

119860 (70)


120581119871=12058721198962119879

3ℎint

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (71)


⟨

119872ph

119860⟩ equiv int

119872ph (ℏ120596)

119860119882ph (ℏ120596) 119889 (ℏ120596) (72)

Then

120581119871=12058721198962119879

3ℎ⟨

119872ph

119860⟩⟨⟨120582ph⟩⟩ (73)


⟨⟨120582ph⟩⟩ =

int (119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

int (119872ph (ℏ120596) 119860)119882ph (ℏ120596) 119889 (ℏ120596)

(74)


119869119909=120590

119902

119889 (119864119865)

119889119909 (75)

120590 =21199022

ℎ⟨119872el119860

⟩⟨⟨120582el⟩⟩ (76)



120581119871=1

3⟨⟨Λ ph⟩⟩ ⟨Vph⟩119862119881 (77)


119864ph = int

infin

0

(ℏ120596)119863ph (ℏ120596) 1198990 (ℏ120596) 119889 (ℏ120596) (78)


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)


= [

[

(1ℎ) intinfin

0(119872ph (ℏ120596) 119860) 120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881



1205821119863

(119864) = 2Λ (119864) = 2V (119864) 120591 (119864) (81a)


120582 (119864) = 2

⟨V2119909120591⟩

⟨1003816100381610038161003816V119909




1205822119863

(119864) =120587

2V (119864) 120591 (119864) (81b)

1205823119863

(119864) =4

3V (119864) 120591 (119864) (81c)


120591 (119864) = 1205910(119864 minus 119864

119862

119896119879)

119904

(82)



120582 (119864) = 1205820(119864 minus 119864

119862

119896119879)

119903

(83)




120582ph (ℏ120596) =4

3Vph (ℏ120596) 120591ph (ℏ120596) (84)


Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)

and finally

120582ph (ℏ120596) =4

3Λ ph (ℏ120596) (86)


119872el (119864) = 1198601198723119863

(119864) = 119860ℎ

4⟨V+119909(119864)⟩119863

3119863(119864) (87)



of states

1198633119863

(119864) = 21198631015840

3119863(119864) (88)


⟨V+119909(119864)⟩ =

Vel (119864)2

(89)


119872ph (ℏ120596) = 119860ℎ

2(

Vph (ℏ120596)2

) 2119863ph (ℏ120596)

= 119860ℎ

4Vph (ℏ120596)119863ph (ℏ120596)

(90)


120581119871

= [

[

(13) intinfin


intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881

(91)


int

infin

0

Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)



⟨⟨Λ ph⟩⟩

equiv

intinfin


intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(93)


⟨Vph⟩ equiv

intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(94)






ℏ120596 = ℏV119863119902 (95)



119904prop 119898minus12







119863ph (ℏ120596) =3(ℏ120596)

2



119872ph (ℏ120596) =3(ℏ120596)

2

4120587(ℏV119863)2 (97)



int

ℏ120596119863

0

119863ph (ℏ120596) 119889 (ℏ120596) (98)


ℏ120596119863= ℏV119863(61205872119873

Ω)

13

equiv 119896119879119863 (99)



perature

119879119863=ℏ120596

119896 (100)




120581119871=12058721198962119879

3ℎint

ℏ120596119863

0

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)




ℏ 1199021+ ℏ 1199022= ℏ 1199023

ℏ1+ ℏ2= ℏ3

(102)



1

120591ph (ℏ120596)=

1

120591119880(ℏ120596)

+1

120591119863(ℏ120596)

+1

120591119861(ℏ120596)

(103)


1

120582ph (ℏ120596)=

1

120582119880(ℏ120596)

+1

120582119863(ℏ120596)

+1

120582119861(ℏ120596)

(104)


1

120591119863(ℏ120596)

prop 1205964 (105)



1

120591119861(ℏ120596)

prop

Vph (ℏ120596)119871

(106)


1

120591119880(ℏ120596)

prop 11987931205962119890minus119879119863119887119879 (107)





119871(119879)



100 101101

102

102

103

103

104

120581ph

(Wm

minus1K

minus1)

T (K)



⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)


119863 all of







119866ball

=21199022

ℎ119872(119864119865) (109)


119870119871=12058721198962119879

3ℎ119879ph (0)119872ph (0) (110)


119870119871=12058721198962119879

3ℎ119872ph (111)



1198920equiv12058721198962119879

3ℎ

(112)


4 Conclusions







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





1205822119863

(119864) =120587

2V (119864) 120591 (119864) (81b)

1205823119863

(119864) =4

3V (119864) 120591 (119864) (81c)


120591 (119864) = 1205910(119864 minus 119864

119862

119896119879)

119904

(82)



120582 (119864) = 1205820(119864 minus 119864

119862

119896119879)

119903

(83)




120582ph (ℏ120596) =4

3Vph (ℏ120596) 120591ph (ℏ120596) (84)


Vph (ℏ120596) 120591ph (ℏ120596) = Λ ph (ℏ120596) (85)

and finally

120582ph (ℏ120596) =4

3Λ ph (ℏ120596) (86)


119872el (119864) = 1198601198723119863

(119864) = 119860ℎ

4⟨V+119909(119864)⟩119863

3119863(119864) (87)



of states

1198633119863

(119864) = 21198631015840

3119863(119864) (88)


⟨V+119909(119864)⟩ =

Vel (119864)2

(89)


119872ph (ℏ120596) = 119860ℎ

2(

Vph (ℏ120596)2

) 2119863ph (ℏ120596)

= 119860ℎ

4Vph (ℏ120596)119863ph (ℏ120596)

(90)


120581119871

= [

[

(13) intinfin


intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

]

]

119862119881

(91)


int

infin

0

Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (92)



⟨⟨Λ ph⟩⟩

equiv

intinfin


intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(93)


⟨Vph⟩ equiv

intinfin

0Vph (ℏ120596)119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

intinfin

0119863ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596)

(94)






ℏ120596 = ℏV119863119902 (95)



119904prop 119898minus12







119863ph (ℏ120596) =3(ℏ120596)

2



119872ph (ℏ120596) =3(ℏ120596)

2

4120587(ℏV119863)2 (97)



int

ℏ120596119863

0

119863ph (ℏ120596) 119889 (ℏ120596) (98)


ℏ120596119863= ℏV119863(61205872119873

Ω)

13

equiv 119896119879119863 (99)



perature

119879119863=ℏ120596

119896 (100)




120581119871=12058721198962119879

3ℎint

ℏ120596119863

0

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)




ℏ 1199021+ ℏ 1199022= ℏ 1199023

ℏ1+ ℏ2= ℏ3

(102)



1

120591ph (ℏ120596)=

1

120591119880(ℏ120596)

+1

120591119863(ℏ120596)

+1

120591119861(ℏ120596)

(103)


1

120582ph (ℏ120596)=

1

120582119880(ℏ120596)

+1

120582119863(ℏ120596)

+1

120582119861(ℏ120596)

(104)


1

120591119863(ℏ120596)

prop 1205964 (105)



1

120591119861(ℏ120596)

prop

Vph (ℏ120596)119871

(106)


1

120591119880(ℏ120596)

prop 11987931205962119890minus119879119863119887119879 (107)





119871(119879)



100 101101

102

102

103

103

104

120581ph

(Wm

minus1K

minus1)

T (K)



⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)


119863 all of







119866ball

=21199022

ℎ119872(119864119865) (109)


119870119871=12058721198962119879

3ℎ119879ph (0)119872ph (0) (110)


119870119871=12058721198962119879

3ℎ119872ph (111)



1198920equiv12058721198962119879

3ℎ

(112)


4 Conclusions







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






119863ph (ℏ120596) =3(ℏ120596)

2



119872ph (ℏ120596) =3(ℏ120596)

2

4120587(ℏV119863)2 (97)



int

ℏ120596119863

0

119863ph (ℏ120596) 119889 (ℏ120596) (98)


ℏ120596119863= ℏV119863(61205872119873

Ω)

13

equiv 119896119879119863 (99)



perature

119879119863=ℏ120596

119896 (100)




120581119871=12058721198962119879

3ℎint

ℏ120596119863

0

119872ph (ℏ120596)

119860120582ph (ℏ120596)119882ph (ℏ120596) 119889 (ℏ120596) (101)




ℏ 1199021+ ℏ 1199022= ℏ 1199023

ℏ1+ ℏ2= ℏ3

(102)



1

120591ph (ℏ120596)=

1

120591119880(ℏ120596)

+1

120591119863(ℏ120596)

+1

120591119861(ℏ120596)

(103)


1

120582ph (ℏ120596)=

1

120582119880(ℏ120596)

+1

120582119863(ℏ120596)

+1

120582119861(ℏ120596)

(104)


1

120591119863(ℏ120596)

prop 1205964 (105)



1

120591119861(ℏ120596)

prop

Vph (ℏ120596)119871

(106)


1

120591119880(ℏ120596)

prop 11987931205962119890minus119879119863119887119879 (107)





119871(119879)



100 101101

102

102

103

103

104

120581ph

(Wm

minus1K

minus1)

T (K)



⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)


119863 all of







119866ball

=21199022

ℎ119872(119864119865) (109)


119870119871=12058721198962119879

3ℎ119879ph (0)119872ph (0) (110)


119870119871=12058721198962119879

3ℎ119872ph (111)



1198920equiv12058721198962119879

3ℎ

(112)


4 Conclusions







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




100 101101

102

102

103

103

104

120581ph

(Wm

minus1K

minus1)

T (K)



⟨119872ph⟩ prop 1198793 (119879 997888rarr 0) (108)


119863 all of







119866ball

=21199022

ℎ119872(119864119865) (109)


119870119871=12058721198962119879

3ℎ119879ph (0)119872ph (0) (110)


119870119871=12058721198962119879

3ℎ119872ph (111)



1198920equiv12058721198962119879

3ℎ

(112)


4 Conclusions







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







Appendices



I119895(120578119865) =

1

Γ (119895 + 1)int

infin

0

120578119895

exp (120578 minus 120578119865) + 1

119889120578 (A1)




120578119865=119864119865minus 119864119862

119896119879 (A2)


120582 (119864) = 1205820(119864

119896119879)

119903

(A3)


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)


119866 =21199022

ℎ(1205820

119871) Γ (119903 + 1)I

119903minus1(120578119865)

119878119879= minus

119896

119902

21199022

ℎ(1205820

119871) Γ (119903 + 1)


(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)


1119863] = 1 S sdot



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)



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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





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)


2119863] =


0=


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)


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)


3119863] =


1205810= 1198700119871119860 120581 = 119870119871119860




119879= 119878119879119871119882 120581

0=


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)


minus 120578119865[I0(120578119865) +I0(minus120578119865)]


119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




119870ball

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


I0(120578119865) +I0(minus120578119865)

119870ball0

= 11988211987921199022

ℎ(2119896119879

120587ℏV119865

)(119896

119902)

2

times 6 [I2(120578119865) +I2(minus120578119865)]


+ 1205782

119865[I0(120578119865) +I0(minus120578119865)]

(B1)


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)



Acknowledgments


References



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Landauer-Datta-Lundstrom Generalized Transport Model for Nanoelectronics

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Transcript of Landauer-Datta-Lundstrom Generalized Transport Model for Nanoelectronics