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A microscopic formulation of the phonon transmission at the nanoscaleY. Chalopin and S. Volz Citation: Appl. Phys. Lett. 103, 051602 (2013); doi: 10.1063/1.4816738 View online: http://dx.doi.org/10.1063/1.4816738 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v103/i5 Published by the AIP Publishing LLC. Additional information on Appl. Phys. Lett.Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors
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A microscopic formulation of the phonon transmission at the nanoscale
Y. Chalopina) and S. VolzLaboratoire d’Energ�etique Mol�eculaire et Macroscopique, CNRS UPR 288, Ecole Centrale Paris,F-92295 Chatenay-Malabry, France
(Received 20 March 2013; accepted 1 July 2013; published online 31 July 2013)
We present a microscopic approach for estimating the frequency vs. wave-vector dependent
phonon transmission across a solid-solid interface. We show that the spectral properties of the
heat flux can be generally deduced from the equilibrium displacements fluctuations of the
contact atoms. We have applied and demonstrated our formalism with molecular dynamics
simulations to predict the angular and mode dependent phonon transport in silicon and
germanium thin films. This notably unveils the existence of confined interface mode at the
thermal contacts. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4816738]
Understanding the transport of phonons at an interface
between two solids is of primary importance in the study of
thermal properties of condensed matter. Interfaces often play
a significant role in the overall thermal properties of the mate-
rials, for instance when grain boundaries,1 heterojunctions,2 or
aggregates3 are involved. Thermal boundary resistance corre-
sponds to heat carrier scattering due to discontinuities within
the crystal lattice. It can be enhanced in some situations by
atomic defects4 such as vacancies, dislocations, isotopic
impurities, atom mixing, or roughness. However, an atomi-
cally perfect interface between dissimilar lattices still gener-
ates the so called Kapitza resistance, which is commonly
interpreted as a mismatch between the vibrational density of
states of the two materials.5,6
The actual theoretical modeling of the Kapitza transmis-
sion mainly consists in either the acoustic mismatch model
(AMM) or the diffuse mismatch model (DMM)7 or the atom-
istic Green’s functions theory (AGF).8 The AMM is known
to be valid for long-wavelength carriers, which restricts its
application to low temperature problems. This method is
based on the extrapolation of the low frequency impedances
of the bulk materials and thus neglect, for instance, the
description of inelastic processes. The DMM assumes a dif-
fusive interface scattering, which also limits its application
to disordered or rough interfaces. These models do not usu-
ally agree with measurement data,9–13 often by several orders
of magnitude. Note that both descriptions rely on the bulk
phonon properties. Alternatively, AGFs model the carrier
interactions within the harmonic approximation14 and do not
track the inelastic processes.15,16 On the other hand, with a
direct non-equilibrium method.17 molecular dynamics simu-
lations have been used to artificially simulate the reflection
and transmission of a phonon wave packet at an inter-
face,18,19 as well as to estimate the contact conductance.20
As a matter of fact, providing a relevant microscopic
model to understand and to accurately estimate the Kapitza
phonon transmission is one of the few remaining challenges
of the last decades in solid state physics.21 The cornerstone
of such a model lies in the recovery of the selection rules
that govern the phonon transmission at a solid/solid
interface. This task is even more critical for nanoscale sys-
tems in which interfacial scattering are predominant over the
internal scattering. For instance, a low thermal resistance at
interfaces is of particular concern to the development of
microelectronic semiconductor devices.22
The manuscript first presents a microscopic formula-
tion of the interface mode-to-mode conductance. Owing to
Molecular Dynamics (MD) simulations, our approach is
validated by estimating the thermal resistance between
semiconductor thin films (Si:Ge). We derive the directional
transmission that can be obtained from the sampling of
the frequency dependent phonon scattering in the two-
dimensional interfacial Brillouin zone. We finally discuss
the existence of interface modes and demonstrates that they
play an important role in transferring heat.
To express the frequency and wave-vector dependent
phonon transmission, we start from the expression of the
heat flux Q obtained from the hamiltonian equation of
motion23 of two interacting phonon systems whose hamilto-
nians are H ¼ HA þ HB with Q ¼ �h _HAi ¼ h½H; HA�i;
QðtÞ ¼ 1=2X
i2 A; j2 Ba;d2 fx;y; zg
ka;di;j ðh _ua
i ðt0Þudj ðtÞi� hua
i ðtÞ _udj ðt0ÞiÞ; (1)
ui refers to the atomic displacement and _ui the instantaneous
velocity of atom i. The a and d exponents refer to the x, y, or
z component. ka;di;j is the interatomic force constant between
atoms i and j. Rewriting the correlation h _uai ðt0Þud
j ðtÞi as
h _uai ðt0Þud
j ðtÞi ¼ limt0!t
d
dt0hua
i ðt0Þudj ðtÞi ¼ lim
s!0
d
dshua
i ðsÞudj ð0Þi;
introduces a characteristic time s ¼ t� t0, which is related to
the vibrational dynamics. Applying the Fourier transform on
the correlations yields
QðxÞ¼ 1=2X
i2A; j2Ba;d2fx;y;zg
ka;di;j ½ _ua
i ðxÞu�dj ðxÞ�u�ai ðxÞ _udj ðxÞ�: (2)
Here, u� is the complex conjugate of u. Eq. (2) can be com-
pared to the following Landauer expression:24
QLDðxÞ ¼ �hxðn0aðxÞ � n0
bðxÞÞTðxÞ; (3)a)Author to whom correspondence should be addressed. Electronic mail:
yann.chalopin@ecp.fr
0003-6951/2013/103(5)/051602/5/$30.00 VC 2013 AIP Publishing LLC103, 051602-1
APPLIED PHYSICS LETTERS 103, 051602 (2013)
Downloaded 05 Aug 2013 to 140.112.103.175. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions
where �hx stands for the energy quantum and T for the phonon
transmission. na;b corresponds to the Bose Einstein occupation
numbers of systems A and B, respectively. Identifying the first
term of the difference of Eq. (2) to the one in Eq. (3) and con-
sidering the classical limit (�hx� kBT) allow us to extract the
following frequency dependent transmission:
TðxÞ ¼ b=2X
i 2 A; j 2 Ba; d 2 fx; y; zg
ka;di;j ½ _ua
i ðxÞu�dj ðxÞ � u�ai ðxÞ _udj ðxÞ�:
(4)
With
TðxÞ ¼ bRe
(�ix
Xi 2 A; j 2 B
a; d 2 fx; y; zg
ka;di;j ua
i ðxÞu�dj ðxÞ); (5)
where b ¼ 1=kBT and kB denotes the Boltzmann constant.
Eq. (4) highlights that the information on the transmission is
provided by the fluctuations of the atoms in interaction
across the interface. The phonon transmission thus corre-
sponds to the cross spectral density of these contact atoms
weighted by their interatomic force constant. From this con-
sideration, an additional characteristic length lc arises for an
interface, it corresponds to the thickness that includes the set
of atoms in interaction across the contact (note that In MD
simulations, lc is artificially prescribed by the cutoff radius
of the interatomic potentials).
A further step can be taken by considering the atomic
velocities in the wave-vector reciprocal space defined by
_uakðxÞ ¼
Xi
_uai ðxÞeik�r0
i ;
where r0i refers to the equilibrium position of atom i.
Replacing the velocities by their expressions in Eq. (4), the
transmission from wave-vector k to k0 at frequency x can be
written as
Tðx; k; k0Þ ¼ Reibx
Xa;d2fx;y;zg
_uakðxÞ _u�dk0 ðxÞDa;dðk; k0Þ
( );
(6)
where Da;d stands for the interfacial phonon dynamical ma-
trix as
Da;dðk; k0Þ ¼X
i2A;j2B
ka;di;j eiðk�r0
i�k0�r0j Þ:
We now turn to apply this formalism to the computation of
the phonon transmission in a semiconductor stacked Si:Ge:Si
heterostructure made of germanium layers with different
thicknesses L. A detailed atomic representation of the system
is depicted in Fig. 1. In MD simulations, the covalent Si:Si/
Ge:Ge/Si:Ge interactions are modeled by using a Stillinger-
Weber (SW) interatomic potential.25 When computing the
transmission, the three-body SW term can be neglected.23
Periodic boundary conditions have been applied along the
[100] and [010] directions to form an infinite contact plan
with interfacial in-plane wave-vectors k ¼ ðkx; kyÞ defined in
the intervals, kx 2 ½0� p=a; 0; 0� and ky 2 ½0; 0� p=a; 0�.
From Eq. (4), we have hence calculated the Si:Ge trans-
mission for L ranging between 0.5 nm and 10 nm as well as
that of the perfect Si:Si transmission. The equilibrium tem-
perature was set to 400 K to keep the classical approximation
relevant. Results are reported in Fig. 2 inset (a), (b), and (c)
highlighting two important features. Inset (a) shows that the
optical modes coming from the Si substrate above the fre-
quency � > 15 THz are poorly transmitted while a signifi-
cant contribution to transmission (noted S on inset (b)) is
observed at � � 14 THz. This later peak does not appear in
perfect bulk silicon (inset (c)) because, as demonstrated later,
FIG. 1. Configuration of the Si:Ge:Si heterostructure. Periodic boundary
conditions are imposed in the directions X ([100]) and Y ([010]). L denotes
the Ge layer thickness.
FIG. 2. Phonon transmission at T¼ 400 K for several Ge layer thicknesses.
Inset (a) compares the perfect Si:Si transmission (hatching curve) to that of
Si-Ge for L¼ 0.54 nm (red) and L¼ 10.9 nm (black). Inset (b) details the
structure of the Si:Ge transmissions for various Ge thicknesses. Inset (c) dis-
plays the transmission of the bulk Si substrate.
051602-2 Y. Chalopin and S. Volz Appl. Phys. Lett. 103, 051602 (2013)
Downloaded 05 Aug 2013 to 140.112.103.175. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions
it corresponds to a specific interface mode. We also observe
that additional modes are transmitted above the Ge bulk cut-
off frequency �Ge (Fig. 2 inset (a) represented by the dotted
vertical line). This indicates that inelastic processes occur at
the contact layers with additional transport channels transfer-
ring thermal energy above �Ge.
At this stage, it is important to validate and benchmark
the degree of accuracy of this microscopic description by
comparing the total thermal resistance of the junction to that
retrieved by more conventional approaches reported in the
literature. We have compared the thermal resistance R
obtained from
R�1 ¼ kB
ðx
TðxÞdx; (7)
to Non Equilibrium Molecular Dynamics (NEMD) simula-
tions20 and AGFs26 calculations. Both calculations rely on
the same interatomic potential.25
As seen on Fig. 3, all calculations indicate that the ther-
mal resistance increases when the layer thickness increases.
Note that the NEMD data trend (red crosses) is recovered by
our theoretical model (blue triangles) within a satisfying
range of accuracy. AGFs overestimate the thermal resistance
by less than thirty percents when compared to other predic-
tions. The harmonic feature of the theory can be considered
at the origin of this mismatch.
To corroborate the local features of the Kapitza trans-
mission mechanisms, we have reported the phonon disper-
sion relation of the Si (inset (a)) and the Ge (inset (b))
interacting atoms when L ¼ 2 nm in Fig. 5. These local dis-
persion relations largely differ from those of bulk materials
(Fig. 4) by (i) the existence of an interface mode (S), which
is weakly dispersive between 13 and 14 THz and by (ii) the
opening of additional transport channels. These new modes
appear due to the boundary condition imposed by the
z-confined cavity, which are projected in the directions of
the interfacial plan. The presence of (S) is explained by the
fact that the interfaces formed with bonded atoms of differ-
ent masses produces a vibration that differs from that of the
bulk modes.27,28 Consequently, an additional mode appear in
a gap bounded by the cutoff frequencies of the bulk materials
(here Si and Ge). Such a vibration remains confined at the
interface as illustrated on Fig. 5 inset (c) where the local
Density of States (LDOS) obtained from atomic planes
parallel to the interface confirms the local feature of this
interface mode (red lines). The amplitude of vibration of the
(S) mode decays over �5 nm suggesting that this interface
mode is evanescent. This aspect is not captured by harmonic
approaches.26
We now extend our analysis by considering Eq. (6) to
compute the k-dependent transmission governing the Si:Ge
interfacial phonon transport when k0 ¼ k. This latter condition
selects the specular processes among all other transmission
FIG. 3. Thermal resistance obtained from the transmission of Fig. 2 and
compared to the NEMD direct calculation20 (red circle) and AGFs26 (dark
square) for various layer thicknesses.
FIG. 4. Bulk density of states of Si and Ge and phonon dispersion along the
[100] and [110] directions calculated from MD simulations.
FIG. 5. Phonon dispersion curves along the [110] and [100] for the atomic
planes at the interface for Si (a) and Ge (b). Inset (c) details the LDOS of
atomic planes parallel to the interface revealing the confinement of the S
mode in the vicinity of 14 THz.
051602-3 Y. Chalopin and S. Volz Appl. Phys. Lett. 103, 051602 (2013)
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mechanisms. Figure 6 reports the wave-vector k and fre-
quency dependent transmission Tðx; kÞ when Ge layers have
two different thicknesses. k is constructed in the plane
defined by the [100] and [110] directions. The right hand
side graphs represent the total transmissions previously
calculated.
As reported, a significant part of the transmitted energy
arises in the low frequency range, i.e., below the Ge optical
modes frequencies, where phonon modes exist in both
materials. The localized interface mode (S) is found to be
weakly dispersive and, as deduced from the spectral distri-
bution, it has important contributions to the interfacial heat
transfer.
From this established wave-vector description of the
transmission, it is possible to investigate the angular depend-
ent interface phonon conductance. This is done by sampling
each point within the two-dimensional Brillouin zone of the
interface. Note that the wave-vector conservation is consid-
ered in Eq. (6), which restricts the scope of this calculation
to the specular processes. We have qualitatively checked that
the diffuse processes are not important. We report that Eq.
(6) allows to extract the specular energy transferred as this
information is relevant for Boltzmann phonon transport cal-
culations.29 Fig. 7 illustrates the distribution of the energy
transmission in a two dimensional reciprocal space defined
by the wave-vectors ½6p=a; 0; 0� and ½0;6p=a; 0�, at the fre-
quency of the interface mode (S) at � ¼ 14 THz.
Fig. 7 reveals a preferential direction observed for
jkxj ¼ jkyj.We think that this approach provides a convenient tool
to study interface effects on phonon focusing30,31 as well as
to predict the angular dependence of the interface conduct-
ance for Boltzmann simulations.
In conclusion, we have formulated a microscopic theory
of the Kapitza resistance which captures the frequency and
wave-vector dependence of the transmission from equilib-
rium atomic fluctuations. We have shown that interface
transmission can be viewed as the cross power spectral den-
sity of the contact atoms. A wave-vector decomposition and
a detailed analysis of the modes that govern the phonon
transmission for Si:Ge interfaces were presented.
We have demonstrated that interfacial transport cannot
be accurately estimated from bulk properties. We have for
instance reported the existence of a localized, non-
dispersive, and directional interface mode, which signifi-
cantly contributes to the heat transfer between silicon and
germanium. The present formulation of the interfacial
transport has the advantage of remaining general and rather
simple to implement.
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051602-5 Y. Chalopin and S. Volz Appl. Phys. Lett. 103, 051602 (2013)
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