# Evaluation of diffuse mismatch model for phonon scattering at disordered
interfaces

Qichen Song<EMAIL_ADDRESS>Department of Mechanical Engineering,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Gang Chen<EMAIL_ADDRESS>Department of Mechanical Engineering, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA

###### Abstract

Diffuse phonon scattering strongly affects the phonon transport through a
disordered interface. The often-used diffuse mismatch model assumes that
phonons lose memory of their origin after being scattered by the interface.
Using mode-resolved atomistic Green’s function simulation, we demonstrate that
diffuse phonon scattering by a single disordered interface cannot make a
phonon lose its memory and thus the applicability of diffusive mismatch model
is limited. An analytical expression for diffuse scattering probability based
on the continuum approximation is also derived and shown to work reasonably
well at low frequencies.

Diffuse phonon scattering, interfacial phonon transport, thermal boundary
resistance, disordered interfaces

##  I Introduction

The interface between two dissimilar materials, or even same materials but
different crystal orientations, can scatter phonons and gives rise to the
thermal boundary resistance for heat flow across the interface[1, 2, 3]. One
model for the thermal boundary resistance is based on assuming that phonons
are specularly scattered at the interface and computing the phonon
transmittance and reflectance based on acoustic wave equations, i.e., the
acoustic mismatch model (AMM)[4, 2]. However, it was found that the AMM only
works at very low temperatures at which the phonon wavelengths are long. At
elevated temperatures, phonons of short wavelengths carry most of the heat,
and they do not experience specular transmission/reflection due to interface
imperfections, such as atomic mixing. The diffuse mismatch model (DMM) is
proposed as an extreme to describe phonon transport across such rough
interfaces[1]. Two major assumptions are made in the DMM. Firstly, the
transmittance is isotropic, i.e., transmittance is angle-independent.
Secondly, phonons lose memory of their origin after being scattered by the
interface such that one cannot distinguish if a phonon has just been through a
transmission or reflection process. Although the DMM has improved the
agreement with experimental measurements of thermal boundary resistance at
high temperatures[5, 6], the assumptions behind DMM have never been examined
in detail.

The thermal boundary resistance has been studied using equilibrium molecular
dynamics[7, 8, 9] and nonequilibrium molecular dynamics[10, 11, 12]. In
particular, phonon-mode-resolved transmittance had been formulated, which
builds upon the atomic trajectories at steady state from molecular dynamics
(MD) simulations[13], where the anharmonicity of phonons is intrinsically
included. It has been applied to study mode-resolved transmittance through
perfect interfaces, yet it has not been used to examine details of diffuse
phonon transmittance across a disordered interface. Phonon wave-packet
dynamics technique has been applied to compute the transmittance and
reflectance of each phonon mode, and the Kapiza resistance can be calculated
using Landauer formalism[14, 15]. The MD simulations and, especially the wave-
packet dynamics simulations, however, require large structures in real space,
including two bulk regions and the interface region, and simulating phonon
transport through a rough interface with a large lateral dimension becomes
computationally extensive.

The atomistic Green’s function (AGF) has been shown as an effective method to
study phonon interfacial transport[16, 17, 18]. The method is formulated in
reciprocal space such that one does not have to deal with large-scale
simulations of atomic displacements in real space. Recent advances in
calculation of interfacial thermal resistance using the AGF have provided more
insights in understanding interfacial thermal resistance with detailed
information including mode-resolved transmission coefficients[19, 20, 21, 16].
Specifically, Ong et al studied the phonon specularity and coherence for
phonon transport through a disordered grain boundary in two-dimensional
graphene, and showed that incoherent phonon scatterings at interface are
almost perfectly diffusive[22]. Ong also demonstrated that the specularity
parameters are different for transmittance and reflectance for graphene grain-
boundaries[23]. Using AGF combined with ab initio inter-atomic force
constants, Tian et al found that the intermixing of atoms for Si/Ge interface
can enhance interfacial thermal conductance[18]. Sadasivam et al demonstrated
using phonon-eigenspectrum-based formulation of AGF that the enhanced
interfacial thermal conductance of a Si/Ge interface with atom intermixing
comes from diffuse transmission channels where the in-plane momentum is not
conserved[20]. The scattering boundary method (SBM), a mathematically
equivalent method to AGF has been proposed by Young et al[24] and generalized
by Zhao et al[25] to study mode-resolved phonon scattering at the interface.
Simon et al[26] has applied SBM to study phonon scattering at the interface
between two-dimensional materials. Recently, Latour et al have demonstrated
the transmission spectra across a perfect interface as a function of incident
angle of phonons, using mode-resolved AGF[21]. However, in order to study
diffuse phonon scattering, a supercell of a rough interface with a large
lateral dimension is required. In addition, the folded lateral wavevector in
the supercell must be carefully mapped back to the wavevector defined in the
original unitcell. We realize that, despite these studies, none of them had
critically examined the validity of DMM for diffuse phonon scattering.

In this work, we conduct mode-resolved AGF calculation of transmittance and
reflectance, and revisit the assumptions of DMM. Our study reveals that most
phonons do not lose their memory of origin. We also derive an analytical
expression for the diffuse transmittance and reflectance based on a continuum
model, and show that it works reasonably well at low frequencies.

##  II Methodology

###  II.1 Revisiting DMM

To derive the DMM, Swartz and Pohl[1] have made two major assumptions. The
first assumption is that phonons are diffusely scattered by the interface and
the transmittance is isotropic,

$T_{L\to R}(\omega,\mathbf{q}\nu)=T_{L\to R}(\omega)$ (1)

where $\mathbf{q}$ is the wavevector of the phonon with frequency $\omega$ and
$\nu$ is the phonon branch index on the left side. The second assumption is
that the transmittance from one side must equal the reflectance from the other
side, i.e., complete loss of memory,

$R_{R\to R}(\omega,\mathbf{q}\nu)=T_{L\to R}(\omega)$ (2)

Consequently, the transmittance from right side writes,

$T_{R\to L}(\omega,\mathbf{q}\nu)=1-T_{L\to R}(\omega)$ (3)

At a given frequency, by invoking the principle of detailed balance, the
transmittance in the elastic scattering limit writes,

$\begin{split}&T_{L\to R}(\omega)\\\
&=\frac{\sum^{+}_{\mathbf{q}\nu}\frac{v_{Rz,\mathbf{q}\nu}}{V_{\mathrm{uc},R}}\delta(\omega-\omega_{R,\mathbf{q}\nu})}{\sum^{+}_{\mathbf{q}\nu}\frac{v_{Lz,\mathbf{q}\nu}}{V_{\mathrm{uc},L}}\delta(\omega-\omega_{L,\mathbf{q}\nu})+\sum^{+}_{\mathbf{q}\nu}\frac{v_{Rz,\mathbf{q}\nu}}{V_{\mathrm{uc},R}}\delta(\omega-\omega_{R,\mathbf{q}\nu})}\\\
&=\frac{\Theta_{\mathrm{bulk},R}(\omega)}{\Theta_{\mathrm{bulk},L}(\omega)+\Theta_{\mathrm{bulk},R}(\omega)}\end{split}$
(4)

where $v_{\alpha z,\mathbf{q}\nu}$ with $\alpha=L,R$ is the group velocity
normal to the interface for phonons from the left or the right side,
$\omega_{\alpha,\mathbf{q}\nu}$ is the phonon frequency and $V_{uc,\alpha}$ is
the volume of unitcell of the left and the right side. The superscript + means
that only forward-moving states with $v_{\alpha z,\mathbf{q}\nu}>0$ are
included in the summation. The transmittance can also be written in terms of
the ratio of transmission functions, as expressed in the second line of the
equation. $\Theta_{\mathrm{bulk},L/R}(\omega)$ is the bulk transmission
function for the left/right side, which is a measure of the number of heat
conduction channels.

To assess the validity of DMM, we examine if the transmittance and reflectance
are indeed isotropic and if the transmittance from one side and reflectance
from the other side are the same.

###  II.2 Mode-resolved atomistic Green’s function formalism

The essential physical quantities to study diffuse phonon scattering by a
rough interface are the transmission probability matrix $T_{mn}(\omega)$ and
reflection probability matrix $R_{ln}(\omega)$ at a given phonon frequency
$\omega$, which describe the transition probability from the initial state $n$
to the final state $l$ or $m$ via interface scattering processes. These
matrices are computed from mode-resolved atomistic Green’s function formalism
as outlined in Ref. [19] and Ref. [22], with details provided in the
supplementary material. Specifically, we divide the system of interest into
three parts, the left lead, the right lead and the device. A propagating state
coming out of one lead can be transmitted through (or be reflected by) the
device region and travel to the other lead (or the same lead). We then compute
the ratio of the heat flux along the z direction of the outgoing state $m$ (or
state $l$), to the heat flux along the z direction of the initial state $n$,
which is the element of transmission probability matrix $T_{mn}(\omega)$ (or
reflection probability matrix $R_{ln}(\omega)$).

For a rough interface created by atomic mixing at the interface, the
transverse translational symmetry is broken by the interfacial disorders. It
is impractical to compute scatterings of an infinitely large rough interface.
Instead, we construct a supercell of two materials and a rough interface
between them with periodic boundary conditions along the transverse directions
(x-direction and y-direction). Because of the transverse periodicity of the
supercell, the phonon state of the lead region defined at a given transverse
wavevector $\mathbf{q}_{\mathrm{sc},\parallel}$ can only be scattered into
phonon states of the lead region (either the left or the right lead) with the
same wavevector $\mathbf{q}_{\mathrm{sc},\parallel}$.

The lead part of the supercell contains $N_{x}\times N_{y}$ repeated
unitcells, as depicted in Fig. 1 (a). The period lengths of the lead along the
direction normal to the interface are $a_{z,L}$ for the left lead and
$a_{z,R}$ for the right lead. The phonon wavevectors parallel to the interface
in the supercell and in the unitcell representations are related via
$\mathbf{q}_{\mathrm{uc},\parallel}=\mathbf{q}_{\mathrm{sc},\parallel}+a\mathbf{G}_{\mathrm{sc},x}+b\mathbf{G}_{\mathrm{sc},y}$,
where $a$ and $b$ are integers. $\mathbf{G}_{\mathrm{sc},x}$ and
$\mathbf{G}_{\mathrm{sc},y}$ are transverse reciprocal lattice vectors of the
supercell. The phonon states at the corresponding equivalent wavevectors (with
same $q_{z}$ and same branch index) in the two representations are
equivalent[27]. The phonon state in the unitcell representation is preferred
as it is much easier to interpret than the supercell representation (we will
hide subscript uc in the following for visual clarity). However, for a given
supercell state $\mathbf{q}_{\mathrm{sc},\parallel}$, there are multiple
possible choices of $a$ and $b$. To find out the correct pair of $a$ and $b$
for wavevector $\mathbf{q}_{\parallel}$ is known as an unfolding problem. We
have adopted the unfolding scheme by Popescu et al[28] and the details can be
found in the supplementary material.

For an interface scattering event, the transverse wavevectors for initial
state $n$ and final state $m$ in the unitcell representation are constrained
by,

$\mathbf{q}_{n}=\mathbf{q}_{m}+a\mathbf{G}_{\mathrm{sc},x}+b\mathbf{G}_{\mathrm{sc},y},$
(5)

where $a$ and $b$ are unknown integers, as wavevectors of initial and final
states in the supercell representation can be unfolded differently.

Figure 1: (a) In AGF, the system is partitioned into three parts: two semi-
infinite leads and a rough interface as the device region. The lead region in
the supercell, contains $N_{\mathrm{uc}}$ unitcells and transverse lattice
vector $\mathbf{R}_{\mathrm{sc}}=N_{\mathrm{uc}}\mathbf{R}_{\mathrm{uc}}$. The
numbers 0, 1, … denote the index of repeated cells for left and right leads.
The period lengths along the direction normal to the interface are $a_{z,L}$
and $a_{z,R}$ for the left and right lead, respectively. (b) The in-plane
wavevector for incident, transmitted and reflected phonons,
$\mathbf{q}_{\parallel,n}$, $\mathbf{q}_{\parallel,m}$ and
$\mathbf{q}_{\parallel,l}$. $a=0$ corresponds to specular transmission, while
$a\neq 0$ corresponds to diffuse transmission. Similarly, $b=0$ corresponds to
specular reflection, while $b\neq 0$ corresponds to diffuse reflection. Note
the schematic is drawn for two-dimensional system for visual clarity and for
three-dimensional system the partitioning and wavevector conservation laws can
be analogously defined.

This expression indicates that the interface scattering can either be a
momentum conserved (specular), when $a=b=0$, or momentum non-conserved
(diffuse) process for other $a$ and $b$ values. Depending on the transverse
wavevector of state $n$ and state $m$, the transmission probability matrix can
be categorized into specular and diffuse transmission parts
$T_{mn}(\omega)=T_{\mathrm{s},mn}(\omega)+T_{\mathrm{d},mn}(\omega)$, where

$\begin{cases}T_{\mathrm{s},mn}(\omega)=T_{mn}(\omega),\>\mathrm{when}\>\mathbf{q}_{\parallel,n}=\mathbf{q}_{\parallel,m}\\\
T_{\mathrm{d},mn}(\omega)=T_{mn}(\omega),\>\mathrm{when}\>\mathbf{q}_{\parallel,n}\neq\mathbf{q}_{\parallel,m}\\\
\end{cases}$ (6)

The reflection probability matrix can be analogously expressed by,
$R_{ln}(\omega)=R_{\mathrm{s},ln}(\omega)+R_{\mathrm{d},ln}(\omega)$, where

$\begin{cases}R_{\mathrm{s},ln}(\omega)=R_{ln}(\omega),\>\mathrm{when}\>\mathbf{q}_{\parallel,n}=\mathbf{q}_{\parallel,l}\\\
R_{\mathrm{d},ln}(\omega)=R_{ln}(\omega),\>\mathrm{when}\>\mathbf{q}_{\parallel,n}\neq\mathbf{q}_{\parallel,l}\\\
\end{cases}$ (7)

The diffuse transmittance for a given incident phonon $n$ from the left side
is defined by summing over the scattering probabilities of all possible
outgoing states,

$\begin{split}T_{\mathrm{d},L\to R}(\omega,\Omega_{L})=\sum_{m}T^{L\to
R}_{\mathrm{d},mn}(\omega)\end{split}$ (8)

where $\Omega_{L}=(\theta,\phi)$ indicates transmittance is a directional
quantity. The polar and azimuthal angles are defined in a coordinate system
where the interface normal lies along the z-axis,
$\theta=\mathrm{arccos}\frac{v_{z}}{|\mathbf{v}|}$,
$\phi=\mathrm{arctan}\frac{v_{y}}{v_{x}}$. $\mathbf{v}=(v_{x},v_{y},v_{z})$ is
the group velocity for incident phonon $n$ from the left side and we use $L\to
R$ to denote the trajectory of the phonon. The reason for using the angle of
the group velocity rather than the phase velocity (or wavevector) is that
group velocity is a uniquely defined quantity irrelevant to the choice of in-
plane Brillouin zone while not for the phase velocity. Likewise, the diffuse
reflectance for a given initial state $n$ in the left side and in the right
side read,

$\begin{split}R_{\mathrm{d},\alpha\to\alpha}(\omega,\Omega_{\alpha})=\sum_{m}R^{\alpha\to\alpha}_{\mathrm{d},mn}(\omega)\end{split}$
(9)

where $\alpha=L,R$. The specular transmittance and reflectance can be
similarly defined.

Figure 2: The angle-resolved (a) diffuse transmittance
$T_{d,\mathrm{Si}\to\mathrm{Ge}}(\Omega_{\mathrm{Si}})$ from the Si side and
(b) diffuse reflectance
$R_{d,\mathrm{Ge}\to\mathrm{Ge}}(\Omega_{\mathrm{Ge}})$ from the Ge side at
$\omega=5$ THz. The angle-resolved (c) diffuse transmittance
$T_{d,\mathrm{Ge}\to\mathrm{Si}}(\Omega_{\mathrm{Ge}})$ from the Ge side and
(d) diffuse reflectance
$R_{d,\mathrm{Si}\to\mathrm{Si}}(\Omega_{\mathrm{Si}})$ from the Si side at
$\omega=5$ THz. $\Omega_{\mathrm{\alpha}}=(\theta_{\alpha},\phi_{\alpha})$,
$\alpha=\mathrm{Si,Ge}$ is the direction of incident group velocity. The
radial coordinate corresponds to the polar angle $\theta_{\alpha}$ (angle of
incidence) and the polar axis corresponds to the azimuthal angle
$\phi_{\alpha}$. (e), (f) The difference between maximum and minimum diffuse
scattering probability as a measure of the anisotropy of diffuse scattering
probability. The diffuse scattering probability is obtained by taking the
ensemble average of calculations for 21 structures of 8 ml disordered
configurations with a 20$\times$20$\mathbf{q}_{\mathrm{sc},\parallel}$-point
mesh.

Furthermore, to study the impact of diffuse phonon scattering on interfacial
transport, we compute the transmission function $\Theta(\omega)$, which
accounts for the total phonon conduction transmission at a given frequency
$\omega$, defined by,

$\Theta(\omega)=\Theta_{\mathrm{s}}(\omega)+\Theta_{\mathrm{d}}(\omega)$ (10)

$\Theta_{\mathrm{s}}(\omega)$ and $\Theta_{\mathrm{d}}(\omega)$ are specular
and diffuse transmission function obtained by summing over all possible
incoming and outgoing states at a given frequency,

$\begin{split}\Theta_{\mathrm{s}}(\omega)&=\sum_{mn}T^{L\to
R}_{\mathrm{s},mn}(\omega)\\\ \Theta_{\mathrm{d}}(\omega)&=\sum_{mn}T^{L\to
R}_{\mathrm{d},mn}(\omega)\\\ \end{split}$ (11)

The specular and diffuse reflection function is defined by,

$\begin{split}\Xi_{\mathrm{s},\alpha}(\omega)&=\sum_{mn}R^{\alpha\to\alpha}_{\mathrm{s},mn}(\omega)\\\
\Xi_{\mathrm{d},\alpha}(\omega)&=\sum_{mn}R^{\alpha\to\alpha}_{\mathrm{d},mn}(\omega)\\\
\end{split}$ (12)

with $\alpha=L,R$. Note that the transmission function for two sides are the
same due to the time-reversal symmetry of the transmission probability matrix
but not for the reflection function.

###  II.3 Continuum modeling

In addition to the AGF simulation, we also derived analytical formulas for
diffuse the transmittance and reflectance from continuum modeling with details
provided in the Appendix A. The model assumes scalar acoustic waves and random
mass disorders distributed at the interface, and hence neglects mode
conversion at the interface. The model is derived based on perturbation theory
and it only takes the density $\rho_{L}$, $\rho_{R}$, bulk modulus $\mu_{L}$,
$\mu_{R}$ and number of pairs of swapped atoms per unit area $n$ as
parameters. The model captures the specific contributions to the total
transmittance/reflectance of specular and diffuse scattering processes and
allows one to calculate diffuse and specular transmittance and reflectance
components analytically.

We assumes a linear dispersion $\omega=c|\mathbf{q}|$, where the sound
velocities for the left and right side are $c_{L}=\sqrt{\mu_{L}/\rho_{L}}$ and
$c_{R}=\sqrt{\mu_{R}/\rho_{R}}$, respectively. Due to the simple dispersion
relation, the transverse $\mathbf{q}_{\parallel}$ can uniquely define a
forward-moving phonon state. Thus, we use $\mathbf{q}_{\parallel}$ and
$\mathbf{q}^{\prime}_{\parallel}$ to denote the initial and final states,
instead of using $m$ and $n$. The momentum for a phonon state in the left side
is
$(\mathbf{q}_{\parallel},q_{L})=\frac{\omega}{c_{L}}(\mathrm{sin}\theta_{L}\mathrm{cos}\phi,\mathrm{sin}\theta_{L}\mathrm{sin}\phi,\mathrm{cos}\theta_{L})$,
where $q_{L}$ is the perpendicular momentum. For a specular transmission
process from the left side to the right side, the transverse momentum is
conserved. Thus, the corresponding transmitted phonon state on the right side
is
$(\mathbf{q}_{\parallel},q_{R})=\frac{\omega}{c_{R}}(\mathrm{sin}\theta_{R}\mathrm{cos}\phi,\mathrm{sin}\theta_{R}\mathrm{sin}\phi,\mathrm{cos}\theta_{R})$.
It follows that the perpendicular velocities for the initial and final state
are
$v_{L}=c_{L}\mathrm{cos}\theta_{L}=c_{L}\sqrt{1-c^{2}_{L}|\mathbf{q}_{\parallel}|^{2}/\omega^{2}}$,
$v_{R}=c_{R}\mathrm{cos}\theta_{R}=c_{R}\sqrt{1-c^{2}_{R}|\mathbf{q}_{\parallel}|^{2}/\omega^{2}}$,
respectively.

Figure 3: (a), (b) The diffuse transmittance $T_{d,\alpha}$ from one side and
the diffuse reflectance $R_{d,\beta}$ from the other side as a function of the
polar angle of the incident phonon at $\omega=3.3$ THz. The markers are
obtained by integrating $T_{d}(\Omega)$ and $R_{d}(\Omega)$ from AGF
calculation for 8 ml structures over the azimuthal angle $\phi$ divided by
$2\pi$. The solid lines are predictions from continuum modeling with the
number of pairs of swapped atoms per unit area $n=2.78/a^{2}$ and $a=5.527$ Å.
(c), (d) The average diffuse transmittance $T_{d,\alpha}(\omega)$ from one
side and the average diffuse reflectance $R_{d,\beta}(\omega)$ from the other
side as a function of frequency from AGF calculation in solid lines, compared
with DMM in dash-dot lines. (e)-(h) are the specular transmittance and
specular reflectance corresponding to (a)-(d). Figure 4: The frequency-
resolved (a) diffuse and (b) total transmittance/reflectance from Si and Ge
side with different $m_{\mathrm{Ge}}/m_{\mathrm{Si}}$ from AGF calculation for
8 ml structures and DMM. $\omega_{\mathrm{max}}$ is the maximum allowed
frequency for non-zero transmission function $\Theta(\omega)$.
$\omega_{\mathrm{max}}$ = 17.2 THz when
$m_{\mathrm{\mathrm{Ge}}}/m_{\mathrm{\mathrm{Si}}}=1.1$, and
$\omega_{\mathrm{max}}$ = 9.1 THz when
$m_{\mathrm{\mathrm{Ge}}}/m_{\mathrm{\mathrm{Si}}}=4.0$.

From our continuum model, the diffuse transmittance for a given incident state
from the left side writes,

$\begin{split}T_{\mathrm{d},L\to
R}(\omega,\Omega_{L})=\int\frac{d^{2}\mathbf{q}^{\prime}_{\parallel}}{(2\pi)^{2}}T^{L\to
R}_{\mathrm{d}}(\omega,\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel})\end{split}$
(13)

where

$\begin{split}&T^{L\to
R}_{\mathrm{d}}(\omega,\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel})=\\\
&4\omega^{-2}V_{2}\frac{\rho_{R}v^{\prime}_{R}}{\left|\rho_{L}v^{\prime}_{L}+\rho_{R}v^{\prime}_{R}\right|^{2}}\frac{\rho_{L}v_{L}}{{\left|\rho_{L}v_{L}+\rho_{R}v_{R}\right|^{2}}}\end{split}$
(14)

Here, $V_{2}$ is a quantity related to density of mixed atoms at interface
(see Appendix A). Essentially, we have integrated over all possible final
states $\mathbf{q}_{\parallel}^{\prime}$ on the right-hand side. Note that the
final transverse momentum is bounded by
$|\mathbf{q}^{\prime}_{\parallel}|\leq\frac{\omega}{c_{R}}$.

The diffuse reflectance for a phonon state from the left-hand side is,

$\begin{split}R_{\mathrm{d},L\to
L}(\omega,\Omega_{L})=\int\frac{d^{2}\mathbf{q}^{\prime}_{\parallel}}{(2\pi)^{2}}R^{L\to
L}_{\mathrm{d}}(\omega,\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel})\end{split}$
(15)

where

$\begin{split}&R^{L\to
L}_{\mathrm{d}}(\omega,\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel})=\\\
&4\omega^{-2}V_{2}\frac{\rho_{L}v^{\prime}_{L}}{\left|\rho_{L}v^{\prime}_{L}+\rho_{R}v^{\prime}_{R}\right|^{2}}\frac{\rho_{L}v_{L}}{\left|\rho_{L}v_{L}+\rho_{R}v_{R}\right|^{2}}\end{split}$
(16)

and the transverse momentum of the final state is bounded by
$|\mathbf{q}^{\prime}_{\parallel}|\leq\frac{\omega}{c_{L}}$.

If we denote
$F(\omega)=\int\frac{d^{2}\mathbf{q}^{\prime}_{\parallel}}{(2\pi)^{2}}4\omega^{-2}V_{2}\frac{\rho_{L}v^{\prime}_{L}}{\left|\rho_{L}v^{\prime}_{L}+\rho_{R}v^{\prime}_{R}\right|^{2}}$,
the diffuse transmittance from one side and the diffuse reflectance from the
other side can be respectively expressed by,

$T_{\mathrm{d},R\to
L}(\omega,\Omega^{\prime}_{R})=\frac{\rho_{R}v^{\prime}_{R}}{\left|\rho_{L}v^{\prime}_{L}+\rho_{R}v^{\prime}_{R}\right|^{2}}F(\omega)$
(17) $R_{\mathrm{d},L\to
L}(\omega,\Omega_{L})=\frac{\rho_{L}v_{L}}{\left|\rho_{L}v_{L}+\rho_{R}v_{R}\right|^{2}}F(\omega)$
(18)

It is evident that they are both anisotropic since they depend on
perpendicular incident velocity. Their ratio writes,

$\frac{T_{\mathrm{d},R\to L}(\omega,\Omega^{\prime}_{R})}{R_{\mathrm{d},L\to
L}(\omega,\Omega_{L})}=\frac{\rho_{R}v^{\prime}_{R}}{\rho_{L}v_{L}}\frac{\left|\rho_{L}v_{L}+\rho_{R}v_{R}\right|^{2}}{\left|\rho_{L}v^{\prime}_{L}+\rho_{R}v^{\prime}_{R}\right|^{2}}$
(19)

which is not a constant. Thereby, our continuum model suggests that the
diffuse transmittance from one side and the diffuse reflectance from the other
side are generally not equal.

##  III Results and discussions

We study phonon transport through a disordered [001] Si/Ge interface by
creating a 3$\times$3 supercell (along x and y direction). We use the average
of Si’s and Ge’s lattice constants, $a=5.527$ Å, as the lattice constant in
generating the supercell structures and the Stillinger-Weber inter-atomic
potential to compute the dynamical matrix and Green’s function[29]. The
interface is constructed by randomly swapping Si and Ge atoms with the same
distances to the interface and the further away from the interface the fewer
atoms are swapped. At even further distances from the interface, no Si and Ge
atoms are swapped. For instance, when we have 2 layers of Si and 2 layers of
Ge atoms are mixed, 2 pairs of Si and Ge atoms will be swapped in the Si and
the Ge layer closest to the interface and 1 pair of Si and Ge atoms will be
swapped in the Si and the Ge layer secondly closest to interface. In this
case, we have a 1|2|2|1 configuration, with each number denoting the number of
swapped atoms within the same layer. We label such interface structure by 4
ml, in short for 4 mixing layers in total. Apparently, a larger ml number
means a larger degree of disorder. We generate 21 configurations for each
given total mixing layers and compute the ensemble average of the transmission
and reflection probability matrix $T^{L\to R}_{mn}(\omega)$, $T^{R\to
L}_{mn}(\omega)$, $R^{L\to L}_{mn}(\omega)$ and $R^{R\to R}_{mn}(\omega)$.

We first examine the angular dependence of ensemble-averaged diffuse
transmittance and reflectance at a given frequency to ascertain whether or not
they are isotropic. In Fig. 2, we find strong angle dependence of diffuse
transmittance and diffuse reflectance from the Si side as well as the Ge side,
contradicting the assumption of isotropic reflectance and transmittance
underlying the DMM. The diffuse reflectance from Ge to Ge is found to be
overall higher than the diffuse transmittance from Si to Ge. The diffuse
reflectance from Si to Si is in a similar range compared with the diffuse
transmittance from Ge to Si, although their explicit angle dependences are
drastically different.

If the DMM is valid, the diffuse transmittance from one side and the diffuse
reflectance from the other side should be isotropic. We compute the difference
between the maximum and minimum scattering probability at different
frequencies, as a measure of anisotropy, shown in Fig. 2 (e) and (f). We
observe that the diffuse scattering probability generally varies in a wide
range. For example, the diffuse reflectance from the Si side at high
frequencies ranges almost from zero to one. Hence, we conclude the diffuse
transmittance and reflectance from both sides are highly anisotropic.

Additionally, we find out that the patterns of diffuse transmittance and
diffuse reflectance do not have two perfect diagonal reflection axes as the
perfect Si/Ge [001] interface structure does. This is the consequence of
ensemble average of disordered structures, where each structure might have
broken the reflection symmetry, and it is not guaranteed to recover the
original symmetry after ensemble average. Still, we can clearly observe a
clover-like pattern for the diffuse transmittance and reflectance from the Si
side, which originates from the pmm symmetry of the phonon bands for a perfect
[001] Si/Ge interface.

In particular, to quantitatively study how the diffuse transmittance and
diffuse reflectance depend on the polar angle, we integrate the angle-
dependent scattering probability over the azimuthal angle. As shown in Fig. 3
(a), at $\omega=3.3$ THz, the diffuse transmittance from Si is lower than the
reflectance from Ge, while in Fig. 3 (b), the diffuse transmittance from Ge is
higher than the diffuse reflectance from Si. It is interesting to note that
the continuum model agrees well with the results from AGF when incident states
are from Si, suggesting the model captures the physics of diffuse phonon
scattering. On the contrary, the continuum model predicts a different diffuse
scattering probability profile compared with the calculation from AGF when
incident phonons are from Ge. From the analytical expression in the continuum
model for diffuse scattering probability (Eq. 13 and Eq. 15), we learn that
the peak in diffuse transmittance and reflectance profile corresponds to the
critical angle for total internal reflection (37 degrees). In comparison, the
scattering probability profile from AGF has two peaks (9 degrees and around 50
degrees). In Fig. 3 (e) and (f), we find that the specular transmittance and
reflectance have a much stronger dependence on the polar angle compared with
their diffuse counterparts. Our continuum model for specular scattering
probabilities again shows good agreement with the AGF calculation except
incapable of capturing the multiple peaks arising from total internal
reflection. Note that the continuum modeling is based on a scalar field, where
the polarization vectors are not included. The difference between continuum
modeling and AGF calculation suggests that without considering mode conversion
among different polarizations, the scalar continuum model cannot accurately
describe the actual number of available diffuse transmission and reflection
pathways.

To study the diffuse transmittance at different frequencies, we define an
average diffuse transmittance for phonon modes with the same frequency by,

$T_{\mathrm{d},\alpha}(\omega)=\frac{\sum^{+}_{n}T^{\alpha\to\beta}_{\mathrm{d},n}(\omega)v_{z,n}}{\sum^{+}_{n}1\cdot
v_{z,n}}=\frac{\Theta_{\mathrm{d}}(\omega)}{\Theta_{\mathrm{bulk},\alpha}(\omega)}$
(20)

where $v_{z,n}$ is the group velocity normal to the interface.
$\Theta_{\mathrm{bulk},\alpha}(\omega)$ is the transmission function for bulk
material $\alpha$ (we use two leads and device all consisting of $\alpha$
atoms in AGF calculation). The average total transmittance including specular
transmittance and diffuse transmittance is defined by,
$T_{\alpha}(\omega)=\frac{\Theta(\omega)}{\Theta_{\mathrm{bulk},\alpha}(\omega)}$.
The average diffuse and total reflectance are
$R_{\mathrm{d},\alpha}(\omega)=\frac{\Xi_{\mathrm{d},\alpha}(\omega)}{\Theta_{\mathrm{bulk},\alpha}(\omega)}$
and
$R_{\alpha}(\omega)=\frac{\Xi_{\alpha}(\omega)}{\Theta_{\mathrm{bulk},\alpha}(\omega)}$,
respectively. Note that the sum of transmittance and reflectance is one, yet
the sum of diffuse transmittance and diffuse reflectance is less than one, as
not all phonons are diffusely scattered. On the other hand, the sum of
transmittance and reflectance given by DMM is always unity. Therefore, when we
compare the diffuse transmittance/reflectance with transmittance/reflectance
by DMM, it is entirely possible their values do not match. However, what we
are more interested in answering is whether or not the phonon loses its
memory, i.e., whether the diffuse transmittance from one side equals the
diffuse reflectance from the other side[30].

In Fig. 3 (c), we find that the diffuse transmittance from Si is lower than
the diffuse reflectance from Ge for all frequencies. They both deviate from
DMM given by Eq. 4 at low frequencies. At high frequencies, the DMM’s
prediction is close to the reflectance from Ge. In Fig. 3 (d), the diffuse
transmittance from Ge is lower than reflectance from Si except for low
frequencies. And they are both different from DMM. The crossing point at 4.4
THz for transmittance and reflectance suggest that at this frequency, the
average transmittance from Ge side is the same with average diffuse
reflectance from Si side, although they individually have strong angle
dependence. The gap between diffuse transmittance from one side and diffuse
reflectance from the other side suggests that diffuse phonon scattering
depends on the initial state such that phonons actually do not lose their
memory of origin. In Fig. 3 (g) and (h), we see that the specular scattering
probability is generally much higher than the diffuse scattering probability
at low frequencies, suggesting that at low frequencies, the interface
scattering is almost all specular. This trend is also consistent with previous
findings[20, 31].

An important factor that is relevant to the diffuse phonon scattering is the
amount of dissimilarity between two materials. According to Pohl and
Swartz[1], DMM predicts that diffuse scattering increases thermal boundary
resistance of the interface between similar solids, suggesting that diffuse
scattering plays a significant role when the mass ratio between the two
materials is close to one. In Fig. 4, we present the frequency-resolved
transmittance and reflectance with different mass ratios
$m_{\mathrm{Ge}}/m_{\mathrm{Si}}$ (we fix the mass $m_{\mathrm{Si}}$ and vary
the mass $m_{\mathrm{Ge}}$). We find that when the two sides are similar, the
diffuse transmittance from one side is similar to the diffuse reflectance from
the other side. This partially aligns with the assumptions of DMM, although
the diffuse transmittance and diffuse reflectance still have strong anisotropy
in directions (see Fig. 6 in the supplementary material). When the mass ratio
is large, the transmittance from one side is no longer similar to the
reflectance from the other side, indicating that the scattering probability
strongly depends on where the initial states are from. As for the total
transmittance and reflectance, due to the inclusion of the specular scattering
probability, the difference between transmittance from one side and
reflectance from the other side is enlarged. It is interesting to note that
when two sides are similar, although DMM cannot correctly describe either
scattering probability, it is close to the average of the total transmittance
from one side and the total reflectance from the other side. When the two
sides are dissimilar, the DMM’s prediction becomes similar to the total
transmittance in Ge side at low frequencies and the total transmittance in Si
side at high frequencies. This suggests that for certain cases, DMM is able to
roughly describe the total transmittance from one side in a certain frequency
range but not for all frequencies.

Figure 5: The similarity between the diffuse transmittance from one side and
the diffuse reflectance from the other side. The blue lines correspond to the
cases when the final states are on the Ge side and the orange lines correspond
to the cases where the final states are on the Si side. The diffuse
transmittance and reflectance for evaluating the similarity are from the data
presented in Fig. 4 (a).  Figure 6: (a) The total transmission function, (b)
the specular transmission function and (c) the diffuse transmission function
versus phonon frequency for [001] Si/Ge interfaces from AGF. Inset: the
ensemble-averaged mass profile as a function of atom layer number at the
interface region obtained by averaging the mass of atoms within the same
layer. The right axis is the Ge fraction in each atom layer. The distance
between adjacent atom layers is $a/4=1.382$ Å. (d)-(f) The total, specular and
diffuse reflection function from the Si side. (g)-(i) The total, specular and
diffuse reflection function from the Ge side.

To study how much memory the phonon loses regarding its origin in a
quantitative manner, we define a similarity measure by,

$S_{\beta}(\omega)=\mathrm{exp}\left(-\frac{\left|T_{\mathrm{d},\alpha}(\omega)-R_{\mathrm{d},\beta(\omega)}\right|}{R_{\mathrm{d},\beta}(\omega)}\right)$
(21)

where $T_{\alpha}(\omega)$ is the diffuse transmittance from one side and
$R_{\beta}(\omega)$ is the diffuse reflectance from the other side, and the
final states for these scattering processes are on the $\beta$ side.
$S_{\beta}\to 1$ means high similarity between the diffuse transmittance and
reflectance when the final state is on the $\beta$ side, i.e. phonon
completely loses its memory of origin. $S_{\beta}\to 0$ means low similarity
between the diffuse transmittance from one side and diffuse reflectance from
the other side, and phonon does not lose its memory. A large mass ratio
generally lead to a smaller similarity measure. We also find that the
similarity measure depends which side the final state resides at. For example,
in Fig. 5 (b), we find that at $\omega=0.39\omega_{\mathrm{max}}$, when phonon
is scattered into Si side, it loses its memory. However, this is not true when
phonon is scattered into Ge side. In comparison, for DMM, the similarity is
always one when the final states are on either side of the interface.

Lastly, we want to discuss the diffuse phonon scattering’s role in interfacial
transport. In Fig. 6 (a), we plot the total transmission function as a
function of frequency with different numbers of mixing layers. And we find
that the total transmission can either be enhanced or reduced compared with
the perfect Si/Ge interface as a result of the competition between the
specular transmission versus diffuse transmission. From Fig. 6 (b) and (c), we
find as the degree of disorder increases, the specular transmission decreases
while the diffuse transmission increases. In other words, the disorders remove
specular channels while creating new diffuse channels. As a result, we do not
observe significant changes in the transmission function due to atomic mixing.
The opposite trends for specular and diffuse transmission versus the degree of
disorder eventually leads to the maximum thermal conductance for 4 ml
structures. It is interesting to note that similar enhancements for
transmission enabled by disorders have been discovered in electron transport
in heterostructures[32]. From Fig. 6 (d)-(i), we learn that the increasing
amount of disorders always reduce the specular reflection function and
increases the diffuse reflection function. Thus, the disorders can both
increase the diffuse transmission (enhancing the interface conductance) and
the diffuse reflection (worsening the interface conductance). This competition
is another reason why we cannot observe significant enhancement of thermal
conductance by disorders. Our current analysis is based on elastic scattering,
yet phonon anharmonicity can contribute to interfacial transport by enabling
“vertical coupling” between conduction channels of different frequencies.
There are several works on understanding the anharmonicity’s role in thermal
interface conductance for abrupt interfaces[33, 34, 35, 36], but how diffuse
scattering and anharmonicity affect heat conduction channels remains unclear.

##  IV Conclusion

Through AGF calculation, we have demonstrated that the diffuse phonon
scattering by a single disordered interface depends both on initial incoming
states as well as the final outgoing states. The transmittance and reflectance
strongly depend on the polar angle of group velocity. Also, the transmittance
from one side and reflectance from the other side are generally different.
That is to say, phonons do not lose their memory after diffuse scattering by a
single interface. When two materials are similar, the diffuse transmittance
from one side and the diffuse reflectance from the other side become similar
to each other. However, the total transmittance from one side and total
reflectance from the other side are still different.

The number of specular transmission channels for interfacial transport is
always reduced by interfacial disorders while new transmission channels are
created by diffuse phonon scatterings. The competing roles of specular and
diffuse transmission can lead to either enhanced or reduced transmission
function and interfacial thermal conductance.

We also derived the expressions for transmittance and reflectance for diffuse
scattering processes based on the continuum approximation, which works
reasonably well in the low-frequency range. The model leads to different
analytical expressions for diffuse transmittance from one side and diffuse
reflectance from the other side. Our model also shows that the diffuse
transmission opens up new transmission channels even for those states above
the critical angle for total internal reflection.

## Acknowledgements

We want to thank Dr. Jiawei Zhou and Dr. Jonathan Mendoza for helpful
discussions. The funding for this work is supported by the MRSEC Program of
the National Science Foundation under award number DMR-1419807.

##  Appendix A The continuum modeling of diffuse phonon scattering

###  A.1 The transmission and reflection matrix

We consider interfacial transport in the case of scalar phonon model in the
continuum limit, where the equation of motion (EOM) for displacements
writes[37],

$\rho(\mathbf{r})\frac{\partial^{2}u}{\partial
t^{2}}-\nabla\cdot\left(\mu(z)\nabla u\right)=0$ (22)

where $\mu(z)$ is the bulk modulus. For an interface between two dissimilar
solids, we have $\mu(z)=\mu_{L}$ for $z<0$ and $\mu(z)=\mu_{R}$ for $z\geq 0$.
$\rho(\mathbf{r})=\rho_{0}(z)+\Delta\rho(\mathbf{r})$ is the density, where
$\rho_{0}(z)$ is the density without mass disorder at the interface, expressed
by $\rho_{0}(z<0)=\rho_{L}$ and $\rho_{0}(z>0)=\rho_{R}$.
$\Delta\rho(\mathbf{r})=\sum_{i}\Delta
m_{i}\delta(\mathbf{r}_{\parallel}-\mathbf{r}_{i,\parallel})\delta(z)$ is the
density fluctuations due to atomic mixing at the interface, where $\Delta
m_{i}$ is the change of mass at atom site $i$ and $\mathbf{r}_{i,\parallel}$
is the in-plane position of atom site $i$. The choice of the delta function
form for density fluctuations suggests that the atomic mixing only exists
exactly at the interface, thus our model does not apply to the cases where the
atomic mixing exists even far away from the interface. When atomic mixing is
realized by randomly swapping pairs of atoms on two sides of the interface,
the average of mass fluctuations is zero $\sum_{i}\Delta m_{i}=0$. The mass
disorders are distributed randomly in the x-y plane. The ensemble average of
any physical quantity $P$ over many configurations of mass disorders is
obtained by integrating over all possible positions of mass disorders,

$\begin{split}\langle
P\rangle=\int\prod_{j}\frac{d^{2}\mathbf{r}_{\parallel,j}}{A}P\end{split}$
(23)

where $A$ is the cross-section area. After ensemble average, the average of
mass fluctuations is still zero $\langle\sum_{i}\Delta m_{i}=0\rangle$. We
further assume an independent distribution of mass fluctuations such that,

$\langle\sum_{i,j}\Delta m_{i}\Delta m_{j}\rangle=\sum_{i}\langle\left(\Delta
m_{i}\right)^{2}\rangle$ (24)

The time-harmonic solution of Eq. 22 reads,

$u=\sum_{\mathbf{q}_{\parallel}}u_{\mathbf{q}_{\parallel}}(z)\frac{e^{i\left(-\omega
t+\mathbf{q}_{\parallel}\cdot\mathbf{r}_{\parallel}\right)}}{\sqrt{A}}$ (25)

where $\mathbf{q}_{\parallel}=(q_{x},q_{y})$ is the transverse wavevector,
$\omega$ is the phonon frequency, $\mathbf{r}_{\parallel}=(x,y)$ is the
transverse position and $u_{\mathbf{q}_{\parallel}}(z)$ is the z-dependent
component of the solution. The perpendicular wavevectors
$q_{\perp}(z)=q_{z,L}$, when $z<0$, and $q_{\perp}(z)=q_{z,R}$, when $z\geq
0$, are determined by the dispersion relation
$\omega^{2}=c^{2}_{L/R}(q^{2}_{\parallel}+q_{\perp,L/R}^{2})$. Here, $c_{L/R}$
is the sound velocity defined by $c_{L/R}=\sqrt{\mu_{L/R}/\rho_{L/R}}$.
Plugging in the time-harmonic solution to EOM, we have the following equation
for $u_{\mathbf{q}_{\parallel}}(z)$,

$\sum_{\mathbf{q}_{\parallel}}\left[\left(\rho_{0}(z)+\Delta\rho(\mathbf{r})\right)\omega^{2}+\nabla\cdot\mu(z)\nabla\right]u_{\mathbf{q}_{\parallel}}(z)\frac{e^{i\mathbf{q}_{\parallel}\cdot\mathbf{r}_{\parallel}}}{\sqrt{A}}=0$
(26)

Multiply both sides of Eq. 26 by $\int
d^{2}\mathbf{r}_{\parallel}e^{-i\mathbf{q}_{\parallel}^{\prime}\cdot\mathbf{r}_{\parallel}}/\sqrt{A}$.
The orthogonality relations for plane waves leads to,

$\left[\mu(z)q_{\perp}^{2}(z)+\frac{\partial}{\partial
z}\mu(z)\frac{\partial}{\partial
z}\right]u_{\mathbf{q}_{\parallel}^{\prime}}=\sum_{\mathbf{q}_{\parallel}}M_{\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel}}\delta(z)u_{\mathbf{q}_{\parallel}}$
(27)

The scattering matrix
$M_{\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel}}$ is defined by,

$M_{\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel}}=-\sum_{i}\Delta
m_{i}\omega^{2}{A}^{-1}e^{i\left(\mathbf{q}_{\parallel}-\mathbf{q}^{\prime}_{\parallel}\right)\cdot\mathbf{r}_{\parallel,i}}.$
(28)

The solution to Eq. 27 is expressed by,

$\begin{split}u_{\mathbf{q}_{\parallel}}(z)&=\delta_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}e^{iq_{L}z}+r_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\frac{\sqrt{v^{\prime}_{L}}}{\sqrt{v_{L}}}e^{-iq_{L}z},\;z\leq
0\\\
u_{\mathbf{q}_{\parallel}(z)}&=t_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\frac{\sqrt{\rho_{L}v^{\prime}_{L}}}{\sqrt{\rho_{R}v_{R}}}e^{iq_{R}z},\;z\geq
0\\\ \end{split}$ (29)

where $q_{L/R}=q_{z,L/R}$ is the phonon wavevector normal to the interface.
$t_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}$ is the
transmission matrix and
$r_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}$ is the reflection
matrix. $v_{L/R}=c_{L/R}\cos\theta_{L/R}$ is the group velocity perpendicular
to the interface, where $\theta_{L/R}$ is the angle between the direction of
phonon velocity and the axis normal to the interface.

The transmission and reflection probability matrix are defined by the ratio of
the transmitted flux normal to the interface of phonon
$\mathbf{q}_{\parallel}$ and the reflected flux normal to the interface of
phonon $\mathbf{q}_{\parallel}$ to the incident flux normal to the interface
of phonon state $\mathbf{q}^{\prime}_{\parallel}$,
$T_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}=\frac{J_{\mathrm{t},\mathbf{q}_{\parallel}}}{J_{\mathrm{inc},\mathbf{q}^{\prime}_{\parallel}}}$
and
$R_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}=\frac{J_{\mathrm{r},\mathbf{q}_{\parallel}}}{J_{\mathrm{inc},\mathbf{q}^{\prime}_{\parallel}}}$,
respectively, where the time-averaged energy flux for a phonon mode reads[38],

$J=\frac{\mu(z)}{-i\omega A}\int\left(u^{*}\frac{\partial u}{\partial
z}-u\frac{\partial u^{*}}{\partial z}\right)d^{2}\mathbf{r}_{\parallel}$ (30)

The resultant expressions for transmission probability matrix and reflection
probability matrix are,

$T_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}=\left|t_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\right|^{2}$
(31a)
$R_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}=\left|r_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\right|^{2}$
(31b)

The boundary conditions for displacement $u_{\mathbf{q}_{\parallel}}(z)$ for
solving the transmission and reflection matrix write,

$u_{\mathbf{q}_{\parallel}}(0^{-})=u_{\mathbf{q}_{\parallel}}(0^{+})$ (32a)
$\mu(z)\frac{\partial}{\partial
z}u_{\mathbf{q}_{\parallel}}(z)\Big{|}^{0^{+}}_{0^{-}}=\sum_{\mathbf{q}^{\prime\prime}_{\parallel}}M_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime\prime}_{\parallel}}u_{\mathbf{q}^{\prime\prime}_{\parallel}}(0),$
(32b)

where the second boundary condition is obtained by integrating Eq. 27 from
$-\eta$ to $\eta$, with $\eta\to 0^{+}$. Plug in the expression in Eq. 29 into
the boundary condition. We obtain the following expressions,

$\delta_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}+r_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}=t_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}$
(33a)
$\sum_{\mathbf{q}^{\prime\prime}_{\parallel}}(\delta_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime\prime}_{\parallel}}+i\Gamma_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime\prime}_{\parallel}})t_{\mathbf{q}^{\prime\prime}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}=\Lambda_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}$
(33b)

where

$\Gamma_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime\prime}_{\parallel}}=\frac{M_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime\prime}_{\parallel}}}{2\omega\bar{\rho}\bar{v}}\sqrt{\frac{v_{R}}{v_{R}^{\prime\prime}}}$
(34a)
$\Lambda_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}=\delta_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\frac{\sqrt{v_{L}v_{R}}}{\bar{v}}$
(34b) $\bar{v}=\frac{\rho_{L}v_{L}+\rho_{R}v_{R}}{2\bar{\rho}}$ (34c)
$\bar{\rho}=\sqrt{\rho_{L}\rho_{R}}$ (34d)

In particular, we can reorganize Eq. 33b and identify that the transmission
matrix can be expanded in series as,

$t=\sum_{N=0}^{\infty}(-i\Gamma)^{N}\Lambda.$ (35)

which is a summation of terms arising from multiple scatterings of different
orders. We can discard high-order terms to obtain the approximate expression
for the transmission matrix.

###  A.2 The Green’s function in the continuum limit

The transmission matrix can be computed from the Green’s function of the whole
system. We choose to compute the Green’s function because of the mathematical
convenience in perturbation expansions using Dyson’s equation. In the
following, we will illustrate the exact relationship between the transmission
matrix and the Green’s function.

To start with, we evaluate the unperturbed Green’s function for a disorder-
free interface. The unperturbed EOM writes,

$\left[\rho_{0}(z)\omega^{2}+\frac{\partial}{\partial
z}\mu(z)\frac{\partial}{\partial z}\right]u(z)=0$ (36)

which can be identified as a Sturm-Liouville equation. Two sets of solutions
are given by,

$u_{<}(z)=\begin{cases}t_{L}e^{-iq_{L}z},&z<0\\\
e^{-iq_{R}z}+r_{L}e^{iq_{R}z},&z>0\\\ \end{cases}$ (37)

and

$u_{>}(z)=\begin{cases}t_{R}e^{iq_{R}z},&z>0\\\
e^{iq_{L}z}+r_{R}e^{-iq_{L}z},&z<0\\\ \end{cases}$ (38)

The continuity condition at interface gives,

$t_{L}=1+r_{L}=\frac{2\mu_{R}q_{R}}{\mu_{L}q_{L}+\mu_{R}q_{R}}$ (39)

and

$t_{R}=1+r_{R}=\frac{2\mu_{L}q_{L}}{\mu_{L}q_{L}+\mu_{R}q_{R}}$ (40)

For a Sturm-Liouville equation, the Wronskian writes[39],

$\begin{split}W&=u_{<}(z)\frac{du_{>}(z)}{z}-u_{>}(z)\frac{du_{<}(z)}{z}\\\
&=\begin{cases}\frac{4i\mu_{L}q_{L}q_{R}}{\mu_{L}q_{L}+\mu_{R}q_{R}},&z>0\\\
\frac{4i\mu_{R}q_{L}q_{R}}{\mu_{L}q_{L}+\mu_{R}q_{R}},&z<0\end{cases}\end{split}$
(41)

and the Green’s function is defined by,

$\begin{split}G_{0}(z,z^{\prime})=&\begin{cases}\frac{u_{<}(z)u_{>}(z^{\prime})}{\mu(z^{\prime})W(z^{\prime})},&-\infty<z<z^{\prime}\\\
\frac{u_{<}(z^{\prime})u_{>}(z)}{\mu(z^{\prime})W(z^{\prime})},&z^{\prime}<z<\infty\end{cases}\\\
=&\begin{cases}-\frac{i}{2\mu_{L}}\frac{t_{R}}{q_{L}}e^{-iq_{L}z}e^{iq_{R}z^{\prime}},&z<0,z^{\prime}>0\\\
-\frac{i}{2\mu_{R}}\frac{t_{L}}{q_{R}}e^{-iq_{L}z^{\prime}}e^{iq_{R}z},&z>0,z^{\prime}<0\\\
-\frac{i}{2\mu_{L}}\frac{e^{-iq_{L}|z^{\prime}-z|}+r_{R}e^{-iq_{L}(z+z^{\prime})}}{q_{L}},&z<0,z^{\prime}<0\\\
-\frac{i}{2\mu_{R}}\frac{e^{iq_{R}|z^{\prime}-z|}+r_{L}e^{iq_{R}(z+z^{\prime})}}{q_{R}},&z>0,z^{\prime}>0\end{cases}\end{split}$
(42)

When $z$ and $z^{\prime}$ both approaches zero, the unperturbed Green’s
function at interface is

$G_{0}^{+}=-\frac{i}{\mu_{L}q_{L}+\mu_{R}q_{R}}=-\frac{i}{2\omega\bar{\rho}\bar{v}}$
(43)

where the superscript + is added to represents the retarded Green’s function.

Then, we study the Green’s function for the scenario where atomic mixing is
present at the interface. It is convenient to define the Green’s function
operator,

$\hat{G}^{\pm}=\left[\rho(\mathbf{r})\omega^{2}-\hat{K}\pm i\eta\right]^{-1}$
(44)

where the operator $\hat{K}=-\nabla\cdot\mu(z)\nabla$ and $\eta$ is an
infinitesimal positive real number. The Green’s function in the real space
representation can then be expressed by
$G^{\pm}(\mathbf{r},\mathbf{r}^{\prime})=\bra{\mathbf{r}}\hat{G}^{\pm}\ket{\mathbf{r}^{\prime}}$.
The Green’s function that describes the scattering channel between mode $a$ of
left side and mode $b$ of right side is,

$\begin{split}G_{b,a}^{\pm}(z,z^{\prime})=&A^{-1}\iint
d\mathbf{r}_{\parallel}d\mathbf{r}^{\prime}_{\parallel}e^{-i\left(\mathbf{q}_{\parallel}\cdot\mathbf{r}_{\parallel}-\mathbf{q}^{\prime}_{\parallel}\cdot\mathbf{r}^{\prime}_{\parallel}\right)}\\\
&\times\bra{u_{\mathbf{q}_{\parallel}}}\hat{G}^{\pm}\ket{u^{\prime}_{\mathbf{q}^{\prime}_{\parallel}}}\\\
\end{split}$ (45)

where the transverse wavevector for mode $a$ and mode $b$ are
$\mathbf{q}^{\prime}_{\parallel}$ and $\mathbf{q}_{\parallel}$, respectively.

When disorders are introduced, the perturbed eigenvector is related to the
unperturbed eigenvector via[40],

$u=u^{\prime}+\int
G^{+}(\mathbf{r},\mathbf{r}^{\prime})V(\mathbf{r}^{\prime})u^{\prime}(\mathbf{r}^{\prime})d\mathbf{r}^{\prime},$
(46)

where $u^{\prime}$ is the eigenstate for the disorder-free case and the
perturbation $V(\mathbf{r})=-\Delta\rho(\mathbf{r})\omega^{2}$. Specifically,
the second term on the right-hand side of Eq. 46 equals,

$\begin{split}&\int
d\mathbf{r}^{\prime}G^{+}(\mathbf{r},\mathbf{r}^{\prime})V(\mathbf{r}^{\prime})u^{\prime}(\mathbf{r}^{\prime})\\\
=&\int
d\mathbf{r}^{\prime}\bigg{\\{}G^{+}(\mathbf{r},\mathbf{r}^{\prime})\hat{K}^{\prime}-\delta(\mathbf{r}-\mathbf{r}^{\prime})-\left[\hat{K}^{\prime}G^{+}(\mathbf{r},\mathbf{r}^{\prime})\right]\bigg{\\}}u^{\prime}(\mathbf{r}^{\prime})\end{split}$
(47)

Thus, Eq. 46 is equivalent to,

$\begin{split}u&=\int\mu(\mathbf{r}^{\prime})\left(u^{\prime}\frac{\partial
G^{+}}{\partial z^{\prime}}-G^{+}\frac{\partial u^{\prime}}{\partial
z^{\prime}}\right)\hat{\mathbf{e}}_{z}\cdot d\mathbf{S}^{\prime}\end{split}$
(48)

It is easy to show that,

$\begin{split}\bra{u^{\prime}}V\ket{u}_{b,a}&=\bra{\hat{K}u^{\prime}}\ket{u}_{b,a}-\bra{u^{\prime}}\hat{K}\ket{u}_{b,a}\\\
&=\int\left[u_{a}\hat{K}u^{\prime*}_{b}-u^{\prime*}_{b}\hat{K}u_{a}\right]d^{3}\mathbf{r}\\\
&=\int\mu(\mathbf{r})\left[u^{\prime*}_{b}\frac{\partial u_{a}}{\partial
z}-\frac{\partial u^{\prime*}_{b}}{\partial
z}u_{a}\right]\hat{\mathbf{e}}_{z}\cdot d\mathbf{S}\\\ \end{split}$ (49)

where we applied integration by part and the divergence theorem.

Plugging in the expression of $u$ given by Eq. 48 into Eq. 49, we have,

$\begin{split}&\bra{u^{\prime}}V\ket{u}_{b,a}=\int\left[\hat{K}u^{\prime*}_{b}(\mathbf{r})\right]u_{a}(\mathbf{r})d\mathbf{r}\\\
&-\iint u^{\prime*}_{b}(\mathbf{r})\mu(\mathbf{r}^{\prime})\\\
&\times\left[u^{\prime}_{a}(\mathbf{r}^{\prime})\frac{\partial\hat{K}G^{+}}{\partial
z^{\prime}}-\frac{\partial u^{\prime}_{a}(\mathbf{r}^{\prime})}{\partial
z^{\prime}}\hat{K}G^{+}\right]dS^{\prime}d\mathbf{r}\\\
=&\iint\mu(\mathbf{r})\mu(\mathbf{r}^{\prime})\big{\\{}\frac{\partial
u^{\prime*}_{b}(\mathbf{r})}{\partial z}\\\
&\times\big{[}-u^{\prime}_{a}(\mathbf{r}^{\prime})\frac{\partial
G^{+}}{\partial z^{\prime}}+G^{+}\frac{\partial
u^{\prime}_{a}(\mathbf{r}^{\prime})}{\partial z^{\prime}}\big{]}\\\
&+u^{\prime*}_{b}(\mathbf{r})\big{[}u^{\prime}_{a}(\mathbf{r}^{\prime})\frac{\partial^{2}G^{+}}{\partial
z\partial z^{\prime}}-\frac{\partial
u^{\prime}_{a}(\mathbf{r}^{\prime})}{\partial z^{\prime}}\frac{\partial
G^{+}}{\partial z}\big{]}\big{\\}}dSdS^{\prime}\\\
=&-4\mu_{L}\mu_{R}t_{b,R}^{*}e^{-ik_{b,R}z_{2}+ik_{a,L}z_{1}^{\prime}}G^{+}(z_{2},z_{1}^{\prime})q_{b,R}q_{a,L}\\\
&-4\mu^{2}_{L}r_{b,R}^{*}e^{ik_{b,L}z_{1}+ik_{a,L}z^{\prime}_{1}}G^{+}_{1}(z_{1},z_{1}^{\prime})q_{b,L}q_{a,L}\\\
&+4\mu_{L}^{2}r^{*}_{b,R}e^{ik_{b,L}z_{1}-ik_{a,L}z_{1}^{\prime}}G^{+}_{2}(z_{1},z_{1}^{\prime})q_{b,L}q_{a,L}\end{split}$
(50)

where
$G^{+}(z_{1},z_{1}^{\prime})=G^{+}_{1}(z_{1},z_{1}^{\prime})+G^{+}_{2}(z_{1},z_{1}^{\prime})$
and we do not need to know the exact expression of $G^{+}_{1}$ and
$G^{+}_{2}$. Note that we have used the form of Green’s function in its
asymptotic limit in deriving the above expression. Denote $L$ the length of
the domain containing disorders. Then, $z_{1},z_{1}^{\prime}<0$ and
$z_{2},z_{2}^{\prime}>L$ are the boundary for integration. Since the random
mass disorders are localized at the interface at $z=0$, we have $L\to 0$, such
that we can set $z_{1}=z_{1}^{\prime}=0^{-}$ and $z_{2}=z_{2}^{\prime}=0^{+}$.

Directly plugging in the general expression of $u^{\prime}$ for unperturbed
system given by Eq. 38 and $u$ for perturbed system given by Eq. 29, we can
obtain another expression for matrix element $\bra{u^{\prime}}V\ket{u}_{b,a}$,

$\begin{split}\bra{u^{\prime}}V\ket{u}_{b,a}=&2i\omega
t^{*}_{b,R}t_{b,a}\sqrt{\rho_{L}v_{a,L}\rho_{R}v_{b,R}}\\\
+&2i\omega\left(r^{*}_{b,R}r_{b,a}\frac{\sqrt{v_{a,L}}}{\sqrt{v_{b,L}}}-\delta_{b,a}\right)\rho_{L}v_{b,L}\end{split}$
(51)

By equating Eq. 50 to Eq. 51, we have found the relationship between
transmission matrix and Green’s function,

$\begin{split}t_{b,a}&=2i\omega
G^{+}_{b,a}(0^{+},0^{-})\sqrt{\rho_{L}\rho_{R}v_{a,L}v_{b,R}}\\\
&=2i\omega\bar{\rho}\sqrt{v_{R}v^{\prime}_{L}}G^{+}_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\end{split}$
(52)

Furthermore, using the boundary condition given by Eq. 33a, we find that the
reflection matrix is related to Green’s function through,

$\begin{split}r_{b,a}&=2i\omega
G^{+}_{b,a}(0^{-},0^{-})\rho_{L}\sqrt{v_{a,L}v_{b,L}}-\delta_{ba}.\\\
&=2i\omega\rho_{L}\sqrt{v_{L}v^{\prime}_{L}}G^{+}_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}-\delta_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\end{split}$
(53)

###  A.3 The ensemble averaged Green’s function

From the series expansion of the transmission matrix in Eq. 35 and
relationship between transmission matrix and Green’s function given by Eq. 52,
we can obtain the following series for the ensemble averaged Green’s function,

$\begin{split}\langle
G^{+}_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\rangle&=-i\frac{1}{2\omega\bar{\rho}\bar{v}^{\prime}}\frac{\sqrt{v_{R}^{\prime}}}{\sqrt{v_{R}}}\sum_{N}^{\infty}\bigg{\langle}(-i\Gamma)^{N}\bigg{\rangle}_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\end{split}$
(54)

According to Eq. 23, the ensemble average of matrix
$\left(-i\Gamma\right)^{N}$ is obtained by integrating over all possible
impurity positions,

$\begin{split}\langle\left(-i\Gamma\right)^{N}\rangle=\int\prod_{j}\frac{d^{2}\mathbf{r}_{\parallel,j}}{A}\left(-i\Gamma\right)^{N}\end{split}$
(55)

In the weak perturbation limit, using Eq. 24, Eq. 28 and Eq. 34a, we write
down the Green’s function in Eq. 54 up to the second order,

$\langle
G^{+}_{{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}}\rangle=\left(G^{+}_{0}(\mathbf{q}^{\prime}_{\parallel})+G^{+2}_{0}(\mathbf{q}^{\prime}_{\parallel})\frac{V_{2}}{A}\sum_{\mathbf{q}_{\parallel}^{\prime\prime}}G^{+}_{0}(\mathbf{q}^{\prime\prime}_{\parallel})\right)\delta_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}$
(56)

where $V_{2}=\langle\sum_{i}\frac{m_{i}^{2}}{A}\omega^{4}\rangle$. The first-
order term vanishes due to ensemble average
$\langle\sum_{i}m_{i}\omega^{2}\rangle=0$. The diagonal form of Eq. 56 implies
that the ensemble average recovers the in-plane translational symmetry of the
unperturbed Green’s function.

###  A.4 The transmission and reflection probability matrix

From Eq. 31a, we find that the transmission probability matrix is related to
the product of retarded and advanced Green’s fuction,

$\langle
T_{{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}}\rangle=4\omega^{2}\bar{\rho}^{2}v^{\prime}_{L}v_{R}\langle
G^{+}_{{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}}G^{-}_{{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}}\rangle.$
(57)

The ensemble averaged
$G^{+}_{{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}}G^{-}_{{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}}$
can be expressed by,

$\begin{split}\langle&G^{+}_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}G^{-}_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\rangle=|\langle
G^{+}_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\rangle|^{2}\\\
&+\sum_{\mathbf{q}^{\prime\prime}_{\parallel},\mathbf{q}^{\prime\prime\prime}_{\parallel}}|\langle
G^{+}(\mathbf{q}_{\parallel},\mathbf{q}^{\prime\prime}_{\parallel})\rangle|^{2}W_{\mathbf{q}^{\prime\prime}_{\parallel},\mathbf{q}^{\prime\prime\prime}_{\parallel}}|\langle
G^{+}(\mathbf{q}^{\prime\prime\prime}_{\parallel},\mathbf{q}^{\prime}_{\parallel})\rangle|^{2}\end{split}$
(58)

where the term
$W_{\mathbf{q}^{\prime\prime}_{\parallel},\mathbf{q}^{\prime\prime\prime}_{\parallel}}$
is called the reducible vertex function. To the lowest order, the vertex
function reads[41],

$W_{\mathbf{q}^{\prime\prime}_{\parallel},\mathbf{q}^{\prime\prime\prime}_{\parallel}}=\frac{V_{2}}{A}$
(59)

From Eq. 56 to Eq. 59, we obtain the expression for the transmission
probability matrix,

$\begin{split}\langle&T_{{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}}\rangle=\delta_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\frac{4\rho_{L}\rho_{R}v_{L}v_{R}}{\left|\rho_{L}v_{L}+\rho_{R}v_{R}\right|^{2}}\\\
&\times\left[1-2\mathrm{Re}\,\mathcal{G}^{+}\omega^{-1}V_{2}\left(\frac{1}{\rho_{L}v_{L}+\rho_{R}v_{R}}\right)\right]\\\
&+\frac{4\omega^{-2}V_{2}}{A}\frac{\rho_{R}v_{R}}{\left|\rho_{L}v_{L}+\rho_{R}v_{R}\right|^{2}}\frac{\rho_{L}v^{\prime}_{L}}{{\left|\rho_{L}v^{\prime}_{L}+\rho_{R}v^{\prime}_{R}\right|^{2}}}\end{split}$
(60)

where
$\mathcal{G}^{+}=\left(\sum_{\mathbf{q}_{\parallel}^{\prime\prime}}iG^{+}_{0}(\mathbf{q}^{\prime\prime}_{\parallel})\right)/A$
and the analytical expression for $\mathcal{G}^{+}$ can be found in the next
session. We identify that the diagonal term is responsible for specular
transmission, while the off-diagonal term is responsible for diffuse
transmission. Similarly, from Eq. 31b, Eq. 53 and Eq. 56 , we derive that the
reflection probability matrix writes,

$\begin{split}&\langle
R_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\rangle=\delta_{\mathbf{q}_{\parallel},\mathbf{q}^{\prime}_{\parallel}}\frac{\left|\rho_{L}v_{L}-\rho_{R}v_{R}\right|^{2}}{\left|\rho_{L}v_{L}+\rho_{R}v_{R}\right|^{2}}\\\
&\times\left[1-4\mathrm{Re}\,\mathcal{G}^{+}\omega^{-1}V_{2}\mathrm{Re}\left(\frac{\rho_{L}v_{L}}{\rho^{2}_{L}v^{2}_{L}-\rho^{2}_{R}v^{2}_{R}}\right)\right]\\\
&+\frac{4\omega^{-2}V_{2}}{A}\frac{\rho_{L}v_{L}}{\left|\rho_{L}v_{L}+\rho_{R}v_{R}\right|^{2}}\frac{\rho_{L}v^{\prime}_{L}}{{\left|\rho_{L}v^{\prime}_{L}+\rho_{R}v^{\prime}_{R}\right|^{2}}}.\end{split}$
(61)

###  A.5 The analytical expression of $\mathcal{G}^{+}$

The term $\mathcal{G}^{+}$ in Eq. 60 and Eq. 61 can be written in terms of an
integral over all transverse wavevectors,

$\begin{split}&\mathcal{G}^{+}\\\
&=\int\frac{d^{2}\mathbf{q}_{\parallel}}{(2\pi)^{2}}\frac{1}{\sqrt{\mu_{L}(\rho_{L}\omega^{2}-\mu_{L}q^{2}_{\parallel})}+\sqrt{\mu_{R}(\rho_{R}\omega^{2}-\mu_{R}q^{2}_{\parallel})}}\end{split}$
(62)

Introduce the ratio of bulk moduli as $a=\rho_{R}/\rho_{L}$ and
$b=\mu_{R}/\mu_{L}$. Depending on the bulk moduli and densities of two sides,
the expression for the real part of $\mathcal{G}^{+}$ is as follows. When
$(b-a)(1-b^{2})>0$,

$\begin{split}\mathrm{Re}\,\mathcal{G}^{+}&=\frac{\omega\mathcal{G}^{+}_{0}}{1-b^{2}}\Big{[}1-\sqrt{ab}+\sqrt{\frac{b(b-a)}{1-b^{2}}}\\\
&\times\left(\mathrm{atan}\sqrt{\frac{a(1-b^{2})}{b-a}}-\mathrm{atan}\sqrt{\frac{1-b^{2}}{b(b-a)}}\right)\Big{]}\end{split}$
(63)

where
$\mathcal{G}^{+}_{0}=\frac{1}{2\pi\mu_{L}}\sqrt{\frac{\rho_{L}}{\mu_{L}}}$ and
for Si, $\mathcal{G}^{+}_{0}=2.62\times 10^{-16}\mathrm{s^{3}/kg}$. When
$(b-a)(1-b^{2})<0$,

$\begin{split}\mathrm{Re}\,\mathcal{G}^{+}&=\frac{\omega\mathcal{G}^{+}_{0}}{1-b^{2}}\Big{[}1-\sqrt{ab}+\frac{1}{2}\sqrt{\frac{b(a-b)}{1-b^{2}}}\\\
&\times\left(\mathrm{ln}\frac{|1-\sqrt{\frac{b(a-b)}{1-b^{2}}}|}{1+\sqrt{\frac{b(a-b)}{1-b^{2}}}}-\mathrm{ln}\frac{|1-\sqrt{\frac{a-b}{a(1-b^{2})}}|}{1+\sqrt{\frac{a-b}{a(1-b^{2})}}}\right)\Big{]}\end{split}$
(64)

When $b=1$ and $a\neq 1$,

$\mathrm{Re}\,\mathcal{G}^{+}=\frac{\omega\mathcal{G}^{+}_{0}}{3}\frac{1-a^{\frac{3}{2}}}{1-a}$
(65)

When $a=b$,

$\mathrm{Re}\,\mathcal{G}^{+}=\frac{\omega\mathcal{G}^{+}_{0}}{1+b}$ (66)

###  A.6 The specular and diffuse transmittance/reflectance

The transmittance for a given initial state $\mathbf{q}_{\parallel}$ is
defined by summing transition probabilities to different final states
$\mathbf{q}^{\prime}_{\parallel}$,

$T_{L\to
R}(\mathbf{q}_{\parallel})=\sum_{\mathbf{q}_{\parallel}^{\prime}}\langle
T_{\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel}}\rangle$ (67)

where $\langle
T_{\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel}}\rangle$ is the
transmission probability matrix defined in Eq. 60.

In the following, we will use direction $\Omega_{L}=(\theta_{L},\phi_{L})$ to
denote a phonon state $\mathbf{q}_{\parallel}$, where
$(\mathbf{q}_{\parallel},q_{L})=\frac{\omega}{c_{L}}(\mathrm{sin}\theta_{L}\mathrm{cos}\phi,\mathrm{sin}\theta_{L}\mathrm{sin}\phi,\mathrm{cos}\theta_{L})$.
Note that the group velocity is parallel to the wavevector thus the angles for
the group velocity and the wavevector are the same. After integration, the
transmittance in Eq. 67 is given by,

$\begin{split}T_{L\to R}(\Omega_{L})&=T_{\mathrm{s},L\to
R}(\Omega_{L})+T_{\mathrm{d},L\to R}(\Omega_{L})\\\
&=T_{\mathrm{AMM}}(\Omega_{L})p_{T}(\Omega_{L})+T_{\mathrm{d},L\to
R}(\Omega_{L})\\\ \end{split}$ (68)

The first term is the specular transmittance, which is the product of
transmittance from AMM[4, 2],

$T_{\mathrm{AMM}}(\Omega_{L})=\frac{4\rho_{L}\rho_{R}v_{L}v_{R}}{\left|\rho_{L}v_{L}+\rho_{R}v_{R}\right|^{2}}$
(69)

and the specularity parameter for transmittance,

$p_{T}(\Omega_{L})=1-2\mathrm{Re}\,\mathcal{G}^{+}\omega^{-1}V_{2}\left(\frac{1}{\rho_{L}v_{L}+\rho_{R}v_{R}}\right)$
(70)

The second term, diffuse transmittance, is given by Eq. 13 in Sec. II.

Similarly, the reflectance for a given incident state from the left side is
given by,

$\begin{split}&R_{L\to
L}(\Omega_{L})=\sum_{\mathbf{q}^{\prime}_{\parallel}}\langle
R_{\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel}}\rangle\\\
&=R_{\mathrm{s},L\to L}(\Omega_{L})+R_{\mathrm{d},L\to L}(\Omega_{L})\\\
&=R_{\mathrm{AMM}}(\Omega_{L})p_{R}(\Omega_{L})+R_{\mathrm{d},L\to
L}(\Omega_{L})\\\ \end{split}$ (71)

where $\langle
R_{\mathbf{q}^{\prime}_{\parallel},\mathbf{q}_{\parallel}}\rangle$ is the
transmission probability matrix defined in Eq. 61. The reflectance by AMM
writes,

$R_{\mathrm{AMM}}(\Omega_{L})=\frac{\left|\rho_{L}v_{L}-\rho_{R}v_{R}\right|^{2}}{\left|\rho_{L}v_{L}+\rho_{R}v_{R}\right|^{2}}$
(72)

The specularity parameter for reflectance is given by,

$p_{R}(\Omega_{L})=1-4\mathrm{Re}\,\mathcal{G}^{+}\omega^{-1}V_{2}\mathrm{Re}\left(\frac{\rho_{L}v_{L}}{\rho^{2}_{L}v^{2}_{L}-\rho^{2}_{R}v^{2}_{R}}\right)$
(73)

And the diffuse reflectance $R_{d,L\to L}(\Omega_{L})$ is defined by Eq. 15.

Figure 7: The transmittance and reflectance for a rough Si/Ge interface
predicted from the continuum model. (a) and (b): The specular and diffuse
transmittance of acoustic phonons at 4 THz in Si and Ge compared with AMM. (c)
and (d): The specular and diffuse reflectance of acoustic phonons at 4 THz in
Si and Ge compared with AMM. (e) and (f): The diffuse transmittance from one
side and reflectance from the other side of acoustic phonons at 4 THz.
$\theta$ is the velocity angle of the incident state.

We want to stress that the expressions in Eq. 68 and Eq. 71 add up to one in
the current lowest-order perturbation theory, which means our continuum model
is a self-consistent theory. However, this is not a guaranteed property at
higher orders.

Figure 8: (a)-(d) The frequency-resolved transmittance and reflectance from Si
side and Ge side from continuum modeling. (e) The transmission function
$\Theta(\omega)$ for Si/Ge interface as a function of frequency. A
multiplicity factor of three is multiplied in the transmission function as
there are three acoustic phonon branches. When the frequency is much higher
than 5 Thz, the lowest perturbation theory is no longer valid, as the
perturbed part becomes large.

In the previous study of partially specular and partially specular interface
scattering by a disordered interface[42], the transmittance and reflectance
are often phenomenologically written as,

$T(\Omega)=p(\Omega)T_{\mathrm{AMM}}(\Omega)+(1-p(\Omega))T_{\mathrm{DMM}}(\Omega)$
(74a)
$R(\Omega)=p(\Omega)R_{\mathrm{AMM}}(\Omega)+(1-p(\Omega))R_{\mathrm{DMM}}(\Omega)$
(74b)

where $p$ is the specularity parameter calculated by Ziman’s equation[43].
However, in our continuum modeling, there are two specularity parameters, one
for transmittance (Eq. 70), one for reflectance (eq. 73) and they are
generally not equal. The necessity of two specularity parameters has been
hypothesized by Li el al[44] and our analytical model gives direct support for
the hypothesis of two different specularity parameters. What’s more, it is
entirely possible to have $p_{R}$ in our model larger than one (this is also
observed in AGF calculation presented in the supplementary material), while
the specularity parameter $p$ given by Ziman’s equation is bounded by one.
Thus, the specularity parameter is merely a correction factor and cannot be
interpreted as probability of being specularly scattered.

Similar to Eq. 20, we further compute the frequency-resolved average
transmittance by integrating over solid angle,

$T_{L\to
R}(\omega)=2\int_{0}^{\pi/2}d\theta\mathrm{sin}\theta\mathrm{cos}\theta
T_{L\to R}(\Omega_{L})$ (75)

and the reflectance can be similarly computed. The energy-resolved
transmission function, which measures the number of conduction channels for
interfacial thermal transport, is obtained by,

$\begin{split}\Theta(\omega)&=A\int\frac{d^{2}\mathbf{q}_{\parallel}}{(2\pi)^{2}}T_{L\to
R}(\omega,\mathbf{q}_{\parallel})\\\ &=2\pi
A\int\frac{d^{2}\mathbf{q}_{\parallel}dq_{L}}{(2\pi)^{3}}T_{L\to
R}(\Omega_{L})v_{L}\delta(\omega-c_{L}\sqrt{q^{2}_{\parallel}+q^{2}_{L}})\\\
&=\pi
AD_{L}(\omega)c_{L}\int_{0}^{\pi/2}d\theta\mathrm{sin}\theta\mathrm{cos}\theta
T_{L\to R}(\Omega_{L})\end{split}$ (76)

where $D_{L}(\omega)=\frac{\omega^{2}}{2\pi^{2}c_{L}^{3}}$ is the density of
states of left side and
$\mathrm{sin}\theta=|\mathbf{q}_{\parallel}|c_{L}/\omega$. The two-probe
interfacial thermal conductance per unit area is determined by the
transmission function,

$G=\frac{1}{2\pi A}\int_{0}^{\infty}\hbar\omega\Theta(\omega)\frac{\partial
f(\omega,T)}{\partial T}d\omega$ (77)

where $f(\omega,T)$ is the Bose-Einstein distribution function.

###  A.7 The interface scattering transition probability for a rough Si/Ge
interface

We apply the derived equations for transmittance and reflectance for a rough
Si/Ge interface along [001] direction. If we assume the atomic mixing is
realized by swapping Si and Ge atoms on two sides of interface, the variance
of mass fluctuations is estimated to be $\langle
m^{2}_{i}\rangle\approx\left(m_{\mathrm{Si}}-m_{\mathrm{Ge}}\right)^{2}=1.985\times
10^{3}\mathrm{u}^{2}$. Thus, the parameter $V_{2}=n\langle
m^{2}_{i}\rangle\omega^{4}$, where $n$ is the number of pairs of swapped Si
and Ge atoms per unit area. We choose $n=2/a^{2}$ in the following
calculation, and $a=5.527$ Å is the lattice constant, obtained by taking the
average of Si’s and Ge’s lattice constants. The bulk moduli of Si and Ge are
$\mu_{L}=95$ GPa and $\mu_{R}=77.2$ GPa. The densities of Si and Ge are
$\rho_{L}=2.329\times 10^{3}\,\mathrm{kg/m^{3}}$ and $\rho_{R}=5.323\times
10^{3}\,\mathrm{kg/m^{3}}$.

As shown in Fig. 7, we find that the phonon transmittance of Si is smaller
compared with AMM. Although there are more transmission channels due to
diffuse scattering, the reduction of transmittance is mainly due to fewer
specular transmission channels, which are removed by interfacial disorders. In
addition, we note that the diffuse transmission from the Ge side opens new
transmission channels above the critical angle for total reflection.
Furthermore, we find that the reflectance from the Si side increases with the
angle, similar to the trend of AMM. For reflectance from the Ge side, the
specular part is smaller than AMM. Due to large diffuse reflectance shown in
Fig. 7 (e), the total reflection probability is eventually higher than
predictions of AMM below the critical angle and lower above the critical
angle. In fact, from the expression of Eq. 60 and Eq. 61, we find that the
specular transmittance is always reduced by disorders while the specular
reflectance can either be enhanced or reduced depending on the sign of
$\rho_{L}v_{L}-\rho_{R}v_{R}$.

From Fig. 7 (e) and (f), we observe that generally the diffuse transmittance
from one side is different from the diffuse reflectance from the other
side.Furthermore, we find that as frequency increases, the specular scattering
probability decreases while the diffuse scattering probability increases, as
shown in Fig. 8 (a)-(d). For the transmittance from both sides, the reduction
in the specular part is always larger than the increment in the diffuse part,
hence a reduced total transmittance. In contrast, for reflectance, the
increment in the diffuse part prevails over the reduction in the specular
part, causing a greater total reflectance. When we compare the phonon
transmission function for Si/Ge interface in Fig. 8 (e), the interface
disorders lead to a smaller total transmission, thus a smaller thermal
conductance. Note that when $\omega>5$ THz, the specular reflectance from Ge
side will become negative, because the perturbation is no longer a small
quantity. From Eq. 73, we see that the reduction in the specularity parameter
for reflectance $p_{R}$ varies drastically with frequency with $\omega^{4}$
scaling. When $p_{R}\sim 0$, we have a critical frequency
$\omega\sim\left(\frac{\mu^{2}A}{\langle m^{2}_{i}\rangle}\right)^{1/4}$, and
our model only works below this critical frequency.

To summarize, continuum modeling using perturbation theory to the lowest order
suggests that the diffuse scattering cannot make a phonon forget its origin,
opposing the picture of DMM. However, we want to point out limitations of the
continuum model of scalar phonons. First of all, the phonon mode conversion is
not considered. Secondly, the model is valid for low-frequency acoustic
phonons thus at lifted temperatures, where high-frequency phonons are playing
an important role in interfacial phonon transport, the model is no longer
valid.

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# A note on multicolor Ramsey number of small odd cycles versus a large clique

Zixiang Xu${}^{\text{a,}}$ and Gennian Ge${}^{\text{a,}}$
${}^{\text{a}}$ School of Mathematical Sciences, Capital Normal University,
Beijing 100048, China.  e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Research supported by the National Key Research and Development Program of
China under Grant No. 2020YFA0712100, National Natural Science Foundation of
China under Grant No. 11971325, and Beijing Scholars Program.

###### Abstract

Let $R_{k}(H;K_{m})$ be the smallest number $N$ such that every coloring of
the edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color
$i$ for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color
$k+1$. In this short note, we study the lower bound for $R_{k}(H;K_{m})$ when
$H$ is $C_{5}$ or $C_{7}$, respectively. We show that

$R_{k}(C_{5};K_{m})=\Omega(m^{\frac{3k}{8}+1}/(\log{m})^{\frac{3k}{8}+1}),$

and

$R_{k}(C_{7};K_{m})=\Omega(m^{\frac{2k}{9}+1}/(\log{m})^{\frac{2k}{9}+1}),$

for fixed positive integer $k$ and $m\rightarrow\infty$. These slightly
improve the previously known lower bound
$R_{k}(C_{2\ell+1};K_{m})=\Omega(m^{\frac{k}{2\ell-1}+1}/(\log
m)^{k+\frac{2k}{2\ell-1}})$ obtained by Alon and Rödl. The proof is based on
random block constructions and random blowups argument.

Key words and phrases: Multicolor Ramsey problems, odd cycles

AMS subject classifications: 05C55.

##  1 Introduction

For a given graph $H$ and a positive integer $k$, the multicolor Ramsey number
$R_{k}(H;K_{m})$ is the smallest number $N$ such that every edge coloring of
$K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$ for some
$1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$.
Obtaining good upper and lower bounds on $R_{k}(H;K_{m})$ is a central
question in Ramsey theory, and despite more than $80$ years of effort, still
not much is known. For example, the most well-studied case is that of $k=1$
and $H=K_{m}$, and the theoretical bounds $2^{\frac{m}{2}}\leqslant
R_{1}(K_{m};K_{m})\leqslant 2^{2m}$ were obtained by Erdős and Szekeres [9,
11] in $1940s$. However, only lower term improvements [5, 15, 17] have been
made to these bounds. For larger values of $k\geqslant 2,$ even less has been
known. When $H=K_{m}$, a recent breakthrough of Conlon and Ferber [6] yielded
an exponential improvement on the lower bounds for multicolor diagonal Ramsey
numbers $R_{k}(K_{m};K_{m})$ using mixed random algebraic approaches. More
recently, the lower bounds for $R_{k}(K_{m};K_{m})$ were improved by Wigderson
[20] and Sawin [16] via different methods.

In this paper, we mainly study the lower bound for $R_{k}(H;K_{m})$ when $H$
is an odd cycle $C_{2\ell+1}$. For $k=1$, Erdős, Faudree, Rousseau and Schelp
[10] proved that $R_{1}(C_{\ell};K_{m})=O(m^{1+\frac{1}{t}})$, where
$t=\lceil\frac{\ell}{2}\rceil-1$. These were later improved by Sudakov [18],
who proved that
$R_{1}(C_{2\ell+1};K_{m})=O(\frac{m^{1+1/\ell}}{\log^{1/\ell}m})$. When
$\ell=1$, the matched lower bound for
$R_{1}(C_{3},K_{m})=\Theta(\frac{m^{2}}{\log{m}})$ was shown in [1, 13].
Moreover, by analyzing the $C_{\ell}$-free process, Bohman and Keevash [3]
proved that
$R_{1}(C_{\ell};K_{m})=\Omega(\frac{m^{(\ell-1)/(\ell-2)}}{\log{m}})$. For
$\ell\in\\{5,6,7,10\\}$, the best known lower bounds for
$R_{1}(C_{\ell};K_{m})$ were improved by Mubayi and Verstraëte [8] via
different approaches. When $k\geqslant 2$, Alon and Rödl [2] proved that
$R_{k}(C_{3};K_{m})=\Theta(m^{k+1}\text{poly}\log{m})$ and
$R_{k}(C_{2\ell+1};K_{m})=\Omega(m^{1+\frac{k}{2\ell-1}}/(\log
m)^{k+\frac{2k}{2\ell-1}})$. For more literature on the Ramsey problems, we
refer the readers to the dynamic survey [14] and the references therein.
Inspired by the random homomorphisms method in the recent work [20] on
multicolor diagonal Ramsey number, we prove the lower bound for
$R_{k}(H;K_{m})$ when $H$ is $C_{5}$ or $C_{7}$, respectively. The appropriate
base graphs come from the random block construction in [8].

######  Theorem 1.1.

For fixed positive integer $k$ and $m\rightarrow\infty$, we have

$R_{k}(C_{5};K_{m})=\Omega\big{(}(\frac{m}{\log{m}})^{\frac{3k}{8}+1}\big{)},$

and

$R_{k}(C_{7};K_{m})=\Omega\big{(}(\frac{m}{\log{m}})^{\frac{2k}{9}+1}\big{)}.$

These slightly improve the previously known lower bounds
$R_{k}(C_{5};K_{m})=\Omega(m^{\frac{k}{3}+1}\text{poly}\log{m})$ and
$R_{k}(C_{7};K_{m})=\Omega(m^{\frac{k}{5}+1}\text{poly}\log{m})$ respectively.

##  2 Proof of Theorem 1.1

Our first step is to construct a base graph which is $C_{5}$-free and has no
large independent set. The construction is based on a recent work of Mubayi
and Verstraëte [8]. By the existence of generalized hexagons of order
$(q,q^{3})$ (see [7, 12, 19]), there exists a graph $G$ with bipartition
$G=A\cup B$ of sizes $|A|=(q+1)(q^{8}+q^{4}+1)$ and
$|B|=(q^{3}+1)(q^{8}+q^{4}+1)$ such that every vertex in $A$ has degree
$q^{3}+1$ and every vertex in $B$ has degree $q+1$. Moreover, the girth of $G$
is at least $12$, that is, there is no cycle of length less than $12$ in $G$.
For a vertex $a\in A,$ we can randomly partition the neighbors $N_{G}(a)$ into
two parts $S_{a}$ and $T_{a}$ uniformly, and then we form a complete bipartite
graph between $S_{a}$ and $T_{a}$. Furthermore, we repeat these operations for
all vertices in $A$ and finally obtain a new graph $F$ on vertex set $V(F)=B.$
Note that there is no multiple edge in the new graph $F$ since $G$ is
$C_{4}$-free. Moreover, $F$ is $C_{5}$-free, otherwise by definition of graph
$F$, there will be a copy of $C_{10}$ in $G$, a contraction. We also claim
that there is no independent set of size $t=(1+o(1))q^{8}$ with positive
probability. To see this, let $I$ be a subset of $F$ with $|I|=t$. Set
$t_{a}=|I\cap N_{G}(a)|$ for $a\in A,$ it is easy to see the probability that
$I\cap N_{G}(a)$ is an independent set in $F$ is $2^{1-t_{a}}$. Since the
partitions are independently at random for different $a\in A$, the probability
that $I$ forms an independent set in $F$ is $\prod\limits_{a\in A}2^{1-t_{a}}$
and the expected number of independent sets of size $t$ in $F$ is
$\binom{|F|}{t}\prod\limits_{a\in A}2^{1-t_{a}}<1.$

Now we fix a graph $F$ which is $C_{5}$-free and has no independent set of
size $t=(1+o(1))q^{8}.$ We say $F^{\prime}$ is an $r$-blowup of the graph $F$
means the graph is obtained by replacing each vertex of $F$ by an independent
set of size $r$ and each edge of $F$ by a complete bipartite graph. Set
$r=c_{k}q^{3(k-1)}$ and $m=d_{k}q^{8}\log{q}$ for some suitable constants
$c_{k},d_{k}>0$. We claim that the number of independent sets of size $m$ in
$F^{\prime}$ is at most $\binom{|F|}{t}\binom{tr}{m}$. To see this, since
there is no independent set of size $t$ in $F$, there are at most
$\binom{|F|}{t}$ blocks which contain the independent sets in $F^{\prime}$.
Moreover, each vertex of the independent set in $F^{\prime}$ of size $m$ is
one of the $tr$ vertices in these blocks since each block contains exactly $r$
vertices.

We then show that there exists a sequence of $k+1$ graphs
$F_{1},F_{2},\ldots,F_{k},F_{k+1}$ on vertex set $V(F^{\prime})$, such that
$F_{i}$ is isomorphic to $F^{\prime}$ with $1\leqslant i\leqslant k$ and
$F_{k+1}$ is the graph whose edges are all pairs of vertices of
$V(F^{\prime})$ that do not lie in any $F_{i}$ with $1\leqslant i\leqslant k$.
Moreover, $F_{k+1}$ is $K_{m}$-free. To show this, we pick $k$ random copies
$F_{1},F_{2},\ldots,F_{k}$ of $F^{\prime}$ on $V(F^{\prime})$ independently,
then the graph $F_{k+1}$ is determined by the above rules. For a fixed subset
$M$ in $V(F^{\prime})$ with $|M|=m$, the probability that $M$ forms an
independent set in any $F_{i}$ is
$\frac{\binom{|F|}{t}\binom{tr}{m}}{\binom{|F^{\prime}|}{m}}$, thus the
probability that $M$ forms a clique in $F_{k+1}$ is at most
$\big{(}\frac{\binom{|F|}{t}\binom{tr}{m}}{\binom{|F^{\prime}|}{m}}\big{)}^{k}$,
and the expected number of $K_{m}$ in $F_{k+1}$ is at most
$\binom{|F^{\prime}|}{m}\big{(}\frac{\binom{|F|}{t}\binom{tr}{m}}{\binom{|F^{\prime}|}{m}}\big{)}^{k}<1$
by choices of the parameters $r$ and $m$. Hence, the required graph $F_{k+1}$
exists, with positive probability.

Finally, we fix a required sequence of graphs $F_{1},F_{2},\ldots,F_{k+1}$ as
above. Let $N:=r|F|$ and color each edge of $K_{N}$ by minimum index
$i\in\\{1,2,\ldots,k\\}$ such that the edge belongs to $F_{i}$. Moreover, if
no such $i$ exists, we color the edge by color $k+1$. Through the above
analysis, we can see there is no $C_{5}$ in color $i$ with $1\leqslant
i\leqslant k$ and no $K_{m}$ in color $k+1$. This gives
$R_{k}(C_{5};K_{m})=\Omega\big{(}(\frac{m}{\log{m}})^{\frac{3k}{8}+1}\big{)}$.

We can also obtain the lower bound for $R_{k}(C_{7};K_{m})$ through the
similar analysis. The required base graphs of girth at least $16$ can be
constructed from Ree-Tits octagons (see [7, 12, 19]), we omit the details.

##  3 Remarks

In [20], Wigderson pointed out that the random induced subgraphs and random
blowups are closely related and are both part of a more general framework of
random homomorphisms. We do agree that the framework of random homomorphisms
is great, and in some sense we can regard the random induced subgraphs and the
random blowups as the same thing. However, in this paper, we prefer to use the
random blowups method, since our $C_{5}$-free base graph from the random block
construction has too many independent sets of small order. In diagonal Ramsey
case [6, 20], the base graph is constructed algebraically, which provides a
good estimation of the number of small independent sets. The random blowups
method has certain limitations, for example, it cannot work when the graph $H$
is bipartite, unless one chooses other ways to define the “blowups”.

It seems to be very hard to obtain the tight asymptotic bound for
$R_{k}(C_{2\ell+1};K_{m})$, even for the simplest case $R_{1}(C_{5};K_{m})$.
The best known upper bound
$R_{1}(C_{5};K_{m})=O(\frac{m^{3/2}}{\sqrt{\log{m}}})$ was proven in [4]. For
multicolor case, we conjecture that
$R_{k}(C_{5};K_{m})=O(m^{\frac{k}{2}+1}\text{poly}\log{m})$ holds.

## Acknowledgements

The authors thank Xin Wei for carefully reading an early draft of this paper.

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