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)

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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)

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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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