PHYSICAL REVIEW B
97
, 024201 (2018)
Propagating elastic vibrations dominate thermal conduction in amorphous silicon
Jaeyun Moon, Benoit Latour, and Austin J. Minnich
*
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
(Received 26 April 2017; revised manuscript received 15 December 2017; published 10 January 2018)
The thermal atomic vibrations of amorphous solids can be distinguished by whether they propagate as elastic
waves or do not propagate due to lack of atomic periodicity. In
a
-Si, prior works concluded that nonpropagating
waves are the dominant contributors to heat transport, with propagating waves being restricted to frequencies
less than a few THz and scattered by anharmonicity. Here, we present a lattice and molecular dynamics analysis
of vibrations in
a
-Si that supports a qualitatively different picture in which propagating elastic waves dominate
the thermal conduction and are scattered by local fluctuations of elastic modulus rather than anharmonicity. We
explicitly demonstrate the propagating nature of waves up to around 10 THz, and further show that pseudoperiodic
structures with homogeneous elastic properties exhibit a marked temperature dependence characteristic of
anharmonic interactions. Our work suggests that most heat is carried by propagating elastic waves in
a
-Si and
demonstratesthatmanipulatinglocalelasticmodulusvariationsisapromisingroutetorealizeamorphousmaterials
with extreme thermal properties.
DOI:
10.1103/PhysRevB.97.024201
I. INTRODUCTION
Amorphous materials are of interest for a wide range of
applications due to their low thermal conductivity [
1
,
2
]. While
in crystals heat is carried by propagating lattice waves, or
phonons, in amorphous solids heat carriers are classified as
propagons, diffusons, and locons depending on the degree of
delocalization of the atomic vibrations and their mean free
paths [
3
,
4
].
This classification has been widely used to analyze the
vibrations responsible for thermal transport in amorphous
materials, especially for pure
a
-Si. Numerical studies using
equilibrium molecular dynamics (EMD) and lattice dynamics
(LD) have attempted to determine the fraction of heat carried
by each category of vibration [
5
,
6
]. While the general con-
sensus is that diffusons carry the majority of the heat, prior
works have reported that propagons may carry 20%–50% of
thermal conductivity in
a
-Si due to their long mean free paths
[
3
,
5
]. Using normal-mode analysis, Larkin and McGaughey
reported that propagons have a lifetime scaling of
ω
−
2
which
suggests plane-wave-like propagation that is not affected by
atomic disorder [
6
]. The normal mode lifetime analysis of
Lv and Henry concluded that the phonon gas model is not
applicable to amorphous materials [
7
]. Experimental works
have qualitatively confirmed some of these predictions, par-
ticularly regarding the important contribution of propagons
[
8
–
13
]. For instance, Kwon
et al.
observed size effects in
a
-Si
nanostructures, indicating the presence of propagons [
12
].
Despite these efforts, numerous puzzles remain. One dis-
crepancy concerns the conclusion that the lifetimes of few THz
vibrationsaregovernedbyanharmonicity[
6
].Ifthatisthecase,
explaining the low thermal conductivity of
a
-Si is challenging
because the same vibrations contribute 75 W m
−
1
K
−
1
to
*
aminnich@caltech.edu
thermal conductivity in
c
-Si. Accounting for the low thermal
conductivity of
a
-Si only by changes in anharmonicity requires
either large increases in anharmonic force constants or in the
scattering phase space. These changes would in turn affect
other properties like the heat capacity of
a
-Si that have not
been observed [
10
]. Along similar lines, if lifetimes of few
THz vibrations are governed by anharmonicity the reported
thermal conductivities of films of the same thickness should be
reasonably uniform, yet the data vary widely [
8
,
10
,
13
]. Over-
all, an unambiguous classification of the propagating nature
and scattering mechanisms of vibrational modes transporting
heat in amorphous solids is poorly developed, impeding efforts
to synthesize, for example, novel materials with exceptionally
low thermal conductivity.
In this work, we address these questions using lattice and
molecular dynamics to calculate dynamic structure factors and
thermal transport properties of
a
-Si. Our analysis supports
a qualitatively different picture of atomic vibrations in
a
-Si
from the conventional one in which propagating elastic waves
dominate the thermal conduction and are scattered by local
variations in elastic modulus rather than anharmonicity. Our
work provides strong evidence that, unintuitively, elastic waves
with frequencies up to around 10 THz carry substantial heat
in disordered media and demonstrates that manipulating local
elastic modulus variations is a promising route to realize
amorphous materials with extreme thermal properties.
II. COMPUTATIONAL APPROACH
We used lattice and molecular dynamics to examine the
atomic vibrations of various amorphous domains. The molecu-
lar dynamics calculations were performed using the large-scale
atomic/molecular massively parallel simulator (LAMMPS)
with a time step of 0.5 fs [
14
]. Periodic boundary conditions
were imposed and the Stillinger-Weber interatomic potential
was used [
15
]. The initial structure we considered contained
2469-9950/2018/97(2)/024201(6)
024201-1
©2018 American Physical Society
JAEYUN MOON, BENOIT LATOUR, AND AUSTIN J. MINNICH
PHYSICAL REVIEW B
97
, 024201 (2018)
FIG. 1. Dynamical structure factor for (a) longitudinal waves and (b) transverse waves for 4096-atom pure
a
-Si domain. Bright yellow
indicates a high intensity of vibrations with the given frequency and wave vector. A clear phonon band is observed up to around 10 THz despite
the atomic disorder. (c) Constant wave vector slice of the dynamical structure factor at
q
=
6
.
0nm
−
1
in the longitudinal direction. Anharmonic
broadening is negligible at 300 K.
4096 atoms and was created by first melting crystalline silicon
at 3500 K for 500 ps in an
NVT
ensemble. Next, the liquid
silicon was quenched to 1000 K with the quench rate of
100 K
/
ps. The structures were annealed at 1000 K for 25 ns to
reduce metastabilities [
16
]. Finally, the domain was quenched
atarateof100K
/
psto300Kandequilibratedat300Kfor10ns
in an
NVT
ensemble using a Nose-Hoover thermostat. The
structure was then equilibrated at 300 K for 500 ps in an
NVT
ensemble. After an additional equilibration in an
NVE
ensem-
ble for 500 ps, the heat fluxes were computed for 1.6 ns in the
same
NVE
ensemble. We use Green-Kubo theory to compute
the thermal conductivity of the structure to be 1.5 W m
−
1
K
−
1
,
a value that is consistent with prior works [
6
,
16
,
17
].
We begin our analysis to gain more insight into the vibra-
tions carrying heat by characterizing the propagating nature
of the normal modes of vibration of the amorphous domain.
A convenient metric for this characterization is the dynamic
structure factor, given by
S
L,T
(
q
,ω
)
=
∑
ν
E
L,T
(
q
,ν
)
δ
(
ω
−
ω
(
q
=
0
,ν
)
)
,
(1)
where the
q
is phonon wave vector,
ω
is frequency, and
the summation is over all the modes
ν
at gamma point.
E
L
and
E
T
refer to the longitudinal polarization and transverse
polarization and are defined as
E
L,T
(
q
,ν
)
=
∣
∣
∣
∣
∣
∑
i
u
L,T
i
e
i
q
·
r
i
∣
∣
∣
∣
∣
2
,
(2)
where the summation is over all atoms indexed by
i
in the do-
main and
r
i
are the equilibrium positions. Here
u
L
i
=
ˆ
q
·
e
(
ν,i
)
and
u
T
i
=
ˆ
q
×
e
(
ν,i
), where
ˆ
q
is a unit vector and
e
(
ν,i
)isthe
eigenvector. The dynamic structure factor is precisely what
is measured in scattering experiments to measure dispersion
relations in crystals and can be applied to search for plane
waves in disordered media.
We calculated the eigenvectors of the 4096 atom structure
using the general utility lattice program (GULP) with equili-
brated structures from MD [
18
]. As amorphous Si is isotropic,
we average the dynamic structure factor over all wave vectors
of the same magnitude. If propagating waves exist despite the
atomic disorder, the dynamic structure factor will exhibit a
clear phonon band with a dispersion; if propagating waves
are not supported, the vibrational modes will appear diffuse
without an apparent dispersion.
III. RESULTS
A. Dynamic structure factor
The dynamic structure factor for longitudinal waves is
presented in Fig.
1(a)
. The figure demonstrates that despite the
atomicdisorderacleardispersionexistsuptofrequencyashigh
as 10 THz for longitudinal waves, corresponding to a wave-
length of 6.5
̊
A. In the transverse direction, Fig.
1(b)
, a clear
dispersion with broadening is also observed up to
∼
5THz,
with a similar transition wavelength of 6.6
̊
A . For sufficiently
high frequency vibrations with wavelengths comparable to
interatomic distances, the structure factor is very broad and
identifying plane waves with definite frequency and wave
vector is not possible. However, the figure clearly shows that
propagating elastic waves comprise a substantial portion of the
vibrational spectrum. Specifically, by calculating the density
of states of the propagating vibrations with a Debye model,
we estimate that about 24% of all modes are propagating
waves. Our observation is consistent with prior calculations
of dynamical structure factor [
6
,
19
] but is inconsistent with
prior conclusions that propagons have frequencies less than
2–3 THz in amorphous silicon [
3
,
5
–
7
,
17
,
20
].
We also observe that the lines are not narrow but have a clear
broadening indicating the presence of a scattering mechanism.
In crystals, this broadening is typically due to anharmonic
interactions. In the harmonic lattice dynamics calculations of
amorphous silicon, anharmonic interactions cannot play any
role. Instead, the broadening must be due to fluctuations of the
local elastic modulus. To assess how broadening due to elastic
fluctuations compares to that from anharmonic interactions, we
also calculate dynamic structure factors using velocity outputs
from MD at 300 K [
21
]. The longitudinal dynamic structure
factors at
q
=
6
.
0nm
−
1
with harmonic and anharmonic forces
are depicted in Fig.
1(c)
, demonstrating that the two are nearly
identical. Anharmonic broadening has essentially no effect
on the lifetimes and the broadening is solely due to elastic
modulus fluctuations. Therefore, the picture that emerges from
our calculation of dynamical structural factor of
a
-Si is a
vibrational spectrum that is dominated by elastic waves that
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FIG. 2. (a) Spectral mean free path and (b) lifetime multiplied by frequency vs frequency for longitudinal waves with harmonic and
anharmonic forces for the 4096-atom pure
a
-Si domain. We observe an excellent agreement between mean free paths obtained from dynamic
structure factors and tuning fork calculations that explicitly simulate the damping of propagating waves. The Ioffe-Regel criterion occurs when
lifetime multiplied by frequency equals 1. Propagons are observed up to around 10 THz for longitudinal waves as predicted from the dispersion.
are scattered by elastic modulus variations in the disordered
solid.
We next aim to extract quantitative information from the
observed broadened lines. Prior works used normal mode anal-
ysis to extract lifetimes from molecular dynamics simulations
[
5
,
6
]. Here, we use the standard scattering theory approach to
obtain lifetimes by fitting a constant wave vector slice of the
dynamic structure factor with a damped harmonic oscillator
(DHO) model [
19
,
21
–
25
]. The lifetime
τ
at a certain frequency
is related to the full width at half maximum
by
τ
=
1
/π
[
26
]. By multiplying the lifetimes by the group velocity given
by the slope of the dispersion, we also obtain mean free paths.
The results are shown in Fig.
2(a)
. We see that the mean free
paths span from 0.5 nm to 10 nm. At still lower frequencies that
cannot be included in the present simulations mean free paths
are likely even longer, as suggested by experiment [
12
]. In
addition, Fig.
2(b)
plots the product of lifetime and vibrational
frequency. In this plot, the Ioffe-Regel (IR) crossover from
propagons to diffusons, defined as when the lifetime is equal
to the period of a wave, can be indicated as a horizontal line
[
23
]. For longitudinal waves, the IR crossover is observed
at
∼
10 THz and
∼
5 THz for transverse waves (not shown);
both of these values are in good agreement with the qualitative
estimate of the transition frequency from the structure factor.
Having established that propagons comprise a substantial
fraction of the vibrational spectrum, we next estimate the
propagon contribution to thermal conductivity given knowl-
edge of the linear, isotropic dispersion, the group veloc-
ity, and the mean free paths from Figs.
1
and
2
using
a Debye model. In this model, we separate the propagon
contribution into longitudinal and transverse modes with
group velocities obtained from the dispersion as 8000 and
3610 m s
−
1
, respectively. Recalling the bulk thermal conduc-
tivity of 1.5 W m
−
1
K
−
1
from the Green-Kubo calculation, we
estimate that propagons contribute about 1.35 W m
−
1
K
−
1
,or
90% of the thermal conductivity. The primary uncertainty in
this estimate is the role of vibrations of frequency less than 2
THz that are challenging to include in both the Green-Kubo
and structure factor calculations; however, our conclusion still
holds even in the absence of these additional propagating
vibrations in our analysis. This contribution is much larger
than the values reported previously and suggests that, counter-
intuitively, heat transport in
a
-Si is dominated by propagating
waves despite the atomic disorder.
FIG. 3. (a) Dispersion for longitudinal waves and (b) for transverse waves for 4096-atom pure
a
-Si domain using Tersoff potential from
dynamic structure factor calculations. Bright yellow indicates a high intensity of vibrations with the given frequency and wave vector. A clear
phonon band is observed up to around 10 THz for longitudinal and 5 THz for transverse directions.
024201-3
JAEYUN MOON, BENOIT LATOUR, AND AUSTIN J. MINNICH
PHYSICAL REVIEW B
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TABLE I. Thermal conductivity comparison for
a
-Si at 300 K
using Tersoff potential without the quantum correction in the specific
heat.
Source
Thermal conductivity (W m
−
1
K
−
1
)
This work (GK)
2
.
32
±
0
.
30
This work (SF+AF)
2.61
He
et al.
(GK) [
5
]2
.
4
±
0
.
35
Lv and Henry (GK) [
17
]2
±
0
.
32
Lv and Henry (GKMA) [
17
]
1.75
B. Calculations using Tersoff potential
In addition to the SW potential to describe the interatomic
interactions in
a
-Si, we have calculated thermal conductivity
values from the GK formalism and dynamic structure factors
using the Tersoff potential. Figure
3
shows the dispersion
of longitudinal and transverse directions using the Tersoff
potential [
27
] for 4096 atoms from dynamic structure factor
calculations. As observed in the dispersions for the Stillinger-
Weber potential, we see clear and well-defined phonon bands
up to around 10 THz and 5 THz for longitudinal and transverse
directions, respectively. The Ioffe-Regel crossover frequencies
using Tersoff potential and the lifetimes from the structure
factor agree well with those from Stillinger-Weber potential
calculations.
The thermal conductivity of amorphous silicon was then
calculated using the Debye model with longitudinal sound ve-
locity of 8179 m
/
s and transverse sound velocity of 4198 m
/
s
for the propagons from the structure factors (SF) and Allen
and Feldman (AF) diffusivities for diffusons and is tabulated
among the works by He
et al.
and Lv and Henry as shown in
Table
I
[
5
,
17
]. For the frequency range between longitudinal
and transverse Ioffe-Regel frequencies, we multiplied a factor
of 2
/
3 to the Allen and Feldman diffusivities to account for
transverse vibrations only. We see that our thermal conduc-
tivity prediction from structure factor and Allen and Feldman
diffusivities agree well with the available data. About 90% of
thermal conductivity is calculated to be from propagons and
the rest from diffusons. Calculations using the Tersoff potential
also confirm that propagons dominate the thermal conduction
in amorphous materials as predicted by our calculations using
the Stillinger-Weber potential.
C. Triggered wave analysis
We provide further support for our conclusions with two ad-
ditional calculations. First, we explicitly demonstrate the prop-
agating nature of vibrational modes by conducting a “tuning
fork experiment” in which imposed oscillatory atomic motions
at one edge of the atomic domain trigger a longitudinal wave
through the
a
-Si. To perform this calculation, we first create
a domain by repeating the 4096-atom cell 10 times along one
direction, resulting in a supercell of size 4
.
3
×
4
.
3
×
43 nm.
In the long dimension, the domain is divided into 80 slabs
of width 5.431
̊
A. Periodic boundary conditions are applied
and the temperature is set at 0.1 K to avoid additional thermal
displacements. The calculation begins by rigidly displacing the
first slab in the longitudinal direction for 2 ps with a sinusoidal
wave with amplitude 0.01
̊
A and a specified frequency ranging
from 2 to 10 THz. We computed the longitudinal displacements
of every atom for time durations less than 2 ps to prevent edge
effects, and subsequently averaged the atomic displacements
within each slab.
The wave propagation in
a
-Si at different frequencies is
shown in Fig.
4
. It is apparent that waves do indeed propagate
through
a
-Si at 3 THz and 8 THz as predicted by the dynamic
structure factor calculations. We obtain mean free paths from
these simulations by identifying the location at which the wave
amplitude has decreased to 1
/e
of its original value. These
mean free paths are in quantitative agreement with those from
dynamic structure factor calculations as shown in Fig.
2(a)
.
On the other hand, the excited wave at 16 THz is damped
very quickly, and by the second slab the amplitude is already
less than 1
/e
of the original value. This observation indicates
that at 16 THz the vibration is nonpropagating. Therefore, the
“tuning fork experiment” explicitly confirms that longitudinal
propagating waves exist up to a high frequency of around
10 THz in
a
-Si.
D. Role of elastic fluctuations
Second, we examine how the thermal conductivity is af-
fected by the partial elimination of elastic modulus fluctu-
ations. If our assertion regarding scattering by elastic fluc-
tuations is true, we should observe a marked increase in
thermal conductivity when these fluctuations are partially
FIG. 4. Temporal displacement of atoms in each slab with triggering frequencies (a) 3 THz, (b) 8 THz, and (c) 16 THz in the longitudinal
direction. Each sinusoidal wave represents the averaged displacements of the atoms in a slab. By observing where the amplitudes of the
displacement decrease by 1
/e
, we estimate that the mean free paths are around 9 and 2 nm for 3 THz and 8 THz waves, respectively. The mean
free path of the 16 THz wave is comparable to the interatomic spacing and hence the vibration is nonpropagating.
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FIG. 5. (a) Dynamic structure factor for longitudinal vibrations
for the 64 AUC tiled structure. (b) Constant wave vector slice of
the dynamical structure factor at
q
=
6
.
0nm
−
1
with harmonic and
anharmonic forces for the 64 AUC structure. While the harmonic
calculation indicates the presence of closely spaced, discrete modes
as occurs in crystals, the anharmonic case exhibits a single broadened
mode due to phonon-phonon interactions. (c) Thermal conductivity
vs temperature for three amorphous structures. No temperature
dependence is observed for 512- and 4096-atom AUC structures,
while a noticeable dependence in temperature for the 64-atom AUC
tiled structure is evident.
eliminated along with a temperature dependence of thermal
conductivity that reflects the renewed dominance of phonon-
phonon interactions. To test this hypothesis, we generated
two additional domains designed to possess reduced elastic
fluctuations consisting of 512- and 64-atom amorphous unit
cell (AUC) tiled to create 4096-atom structures. The 512 and
64 AUC domains were created using the same melt-quench
procedure described earlier. Elastic fluctuations over a length
scale equal to the AUC domain size should be eliminated
because the same unit cell is tiled repeatedly in space to form
the 4096 atom final structure.
We followed the same procedure as described earlier
to obtain dynamic structure factors for the tiled structures.
The structure factor for the 512-atom AUC tiled structure
appears almost identical to that of the original calculation
(not shown). That for the 64-atom AUC tiled structure are
shown in Fig.
5(a)
. We observe discrete points rather than a
continuous broadening, indicative of the dynamic structure
factor having delta-function-like peaks as occurs in
c
-Si.
From a constant wave vector slice of the dynamic structure
factor for the 64-atom AUC tiled structure in Fig.
5(b)
,we
observe that anharmonicity broadens those individual peaks
of the modes from the harmonic calculations, indicating that
anharmonicity plays a role in scattering these modes. Overall,
these calculations indicate that the 64-atom AUC structure
possesses vibrations that are characteristic of a semicrystalline
solid while the 512-atom AUC remains effectively amorphous.
We now compute the thermal conductivity of the three
structures using Green-Kubo theory. The resulting thermal
conductivity calculations of these structures are shown in
Fig.
5(c)
. The figure shows that the pure
a
-Si and the 512-atom
AUC tiled structure have identical thermal conductivity with
little temperature dependence. This result confirms that the
512-atom AUC structure is effectively amorphous. However,
we observe a significant increase in thermal conductivity of the
64-atom AUC tiled structure, by more than a factor of 2 at room
temperature, along with a marked temperature dependence. At
100 K, the thermal conductivity of the 64-atom AUC tiled
structure is
∼
10 W m
−
1
K
−
1
, more than six times that of 4096
AUC structure. Therefore, the 64-atom AUC tiled structure
exhibits characteristics of crystals, and the key disorder length
scale that sets the transition of thermal vibrations from crys-
talline to amorphous character lies between 10 and 20
̊
A.
IV. DISCUSSION
Our analysis differs from prior works in two key respects.
First, we use a standard definition of lifetime from scattering
theory as the broadening of the inelastic peak of the structure
factor rather than the typical normal-mode lifetime. Although
the normal-mode and structure factor definitions coincide at
frequencies below around 2 THz [
6
], they do not agree at higher
frequency. Our tuning fork analysis explicitly shows that the
physical lifetime corresponding to the damping of a wave is
given by the lifetime from the structure factor. Those given
by normal-mode analysis give lifetimes that are 3 to 10 times
higher than those observed in the tuning fork calculation.
Second, we determine the propagon-diffuson transition
frequency using the standard Ioffe-Regel criterion with the life-
timesfromthestructurefactor,leadingtotransitionfrequencies
of around
∼
5 THz and
∼
10 THz for transverse and longitudi-
nal vibrations, respectively. Prior works used different criteria
to identify propagon-diffuson transition frequency, such as
the frequency at which diffusivities calculated by different
methods are equal [
6
], leading to the commonly cited transition
frequency of 2–3 THz. Our tuning fork analysis in Figs.
2
and
4
explicitly shows that modes that were previously considered
to be diffusons by the latter criterion are in fact propagating.
Our conclusion that propagons dominate thermal conduction
in amorphous Si is a direct and unavoidable consequence of
these observations.
The picture of a gas of delocalized elastic vibrations
transporting heat in amorphous solids suggests follow-on
experiments as well as new strategies to realize exceptional
thermal materials. First, our prediction of propagons existing
up to around 10 THz can be verified with additional thermal
measurements on amorphous nanostructures with character-
istic dimensions of less than 10 nm as well as with scattering
methods such as inelastic x-ray scattering. Second, our analysis
suggests that fully dense solids with exceptionally low thermal
conductivity can be achieved by disrupting the propagating
024201-5
JAEYUN MOON, BENOIT LATOUR, AND AUSTIN J. MINNICH
PHYSICAL REVIEW B
97
, 024201 (2018)
modes via enhancement of variations of local elastic modu-
lus, expanding the physical range of thermal conductivity of
solids.
V. CONCLUSIONS
In summary, we have examined the atomic vibrations in
a
-Si using lattice and molecular dynamics calculations. Our
study reveals a qualitatively different view of atomic vibrations
in
a
-Si from a conventional one in which propagating elastic
waves dominate the thermal conduction and are scattered
by elastic fluctuations instead of anharmonicity. Our work
provides important insights into the long-standing problem of
thermal transport in disordered solids.
ACKNOWLEDGMENTS
This work was supported by the Samsung Scholar-
ship, NSF CAREER Award No. CBET 1254213, and the
Resnick Fellowship from the Resnick Sustainability Institute at
Caltech.
[1] M. C. Foote, M. Kenyon, T. R. Krueger, T. A. McCann, R.
Chacon, E. W. Jones, M. R. Dickie, J. T. Schofield, D. J.
McCleese, and S. Gaalema (unpublished).
[2] M. C. Wingert, J. Zheng, S. Kwon, and R. Chen, Thermal
transport in amorphous materials: A review,
Semicond. Sci.
Technol.
31
,
113003
(
2016
).
[3] P. B. Allen, J. L. Feldman, J. Fabian, and F. Wooten, Diffusons,
locons and propagons: Character of atomic vibrations in amor-
phous Si,
Philos. Mag. B
79
,
1715
(
1999
).
[4] J. L. Feldman, M. D. Kluge, P. B. Allen, and F. Wooten, Thermal
conductivity and localization in glasses: Numerical study of a
model of amorphous silicon,
Phys.Rev.B
48
,
12589
(
1993
).
[5] Y. He, D. Donadio, and G. Galli, Heat transport in amorphous
silicon: Interplay between morphology and disorder,
Appl. Phys.
Lett.
98
,
144101
(
2011
).
[6] J. M. Larkin and A. J. H. McGaughey, Thermal conductivity
accumulation in amorphous silica and amorphous silicon,
Phys.
Rev. B
89
,
144303
(
2014
).
[7] W. Lv and A. Henry, Examining the validity of the phonon gas
model in amorphous materials,
Sci. Rep.
6
,
37675
(
2016
).
[8] D. G. Cahill, M. Katiyar, and J. R. Abelson, Thermal conduc-
tivity of
a
-Si: H thin films,
Phys.Rev.B
50
,
6077
(
1994
).
[9] R.Sultan,A.D.Avery,J.M.Underwood,S.J.Mason,D.Bassett,
and B. L. Zink, Heat transport by long mean free path vibrations
in amorphous silicon nitride near room temperature,
Phys. Rev.
B
87
,
214305
(
2013
).
[10] B. L. Zink, R. Pietri, and F. Hellman, Thermal Conductivity and
Specific Heat of Thin-Film Amorphous Silicon,
Phys. Rev. Lett.
96
,
055902
(
2006
).
[11] J. L. Braun, C. H. Baker, A. Giri, M. Elahi, K. Artyushkova,
T. E. Beechem, P. M. Norris, Z. C. Leseman, J. T. Gaskins,
and P. E. Hopkins, Size effects on the thermal conductivity of
amorphous silicon thin films,
Phys.Rev.B
93
,
140201
(
2016
).
[12] S. Kwon, J. Zheng, M. C. Wingert, S. Cui, and R. Chen, Un-
usually high and anisotropic thermal conductivity in amorphous
silicon nanostructures,
ACS Nano
11
,
2470
(
2017
).
[13] X. Liu, J. L. Feldman, D. G. Cahill, R. S. Crandall, N. Bernstein,
D. M. Photiadis, M. J. Mehl, and D. A. Papaconstantopoulos,
High Thermal Conductivity of a Hydrogenated Amorphous
Silicon Film,
Phys.Rev.Lett.
102
,
035901
(
2009
).
[14] S. Plimpton, Fast parallel algorithms for short-range molecular
dynamics,
J. Comput. Phys.
117
,
1
(
1995
).
[15] F. H. Stillinger and T. A. Weber, Computer simulation of local
order in condensed phases of silicon,
Phys. Rev. B
31
,
5262
(
1985
).
[16] J.MoonandA.J.Minnich,Sub-amorphous thermalconductivity
in amorphous heterogeneous nanocomposites,
RSC Adv.
6
,
105154
(
2016
).
[17] W. Lv and A. Henry, Direct calculation of modal contributions
to thermal conductivity via Green-Kubo modal analysis,
New J.
Phys.
18
,
013028
(
2016
).
[18] J. D. Gale, GULP: A computer program for the symmetry-
adapted simulation of solids,
J. Chem. Soc., Faraday Trans.
93
,
629
(
1997
).
[19] Y. M. Beltukov, C. Fusco, D. A. Parshin, and A. Tanguy,
Boson peak and Ioffe-Regel criterion in amorphous siliconlike
materials: The effect of bond directionality,
Phys. Rev. E
93
,
023006
(
2016
).
[20] H.RezaSeyfandA.Henry,Amethodfordistinguishingbetween
propagons, diffusions, and locons,
J. Appl. Phys.
120
,
025101
(
2016
).
[21] H. Shintani and H. Tanaka, Universal link between the boson
peak and transverse phonons in glass,
Nat. Mater.
7
,
870
(
2008
).
[22] S. M. Shapiro, J. D. Axe, G. Shirane, and T. Riste, Critical
Neutron Scattering in SrTiO
3
and KMnF
3
,
Phys. Rev. B
6
,
4332
(
1972
).
[23] S. N. Taraskin and S. R. Elliott, Low-frequency vibrational
excitations in vitreous silica: The Ioffe-Regel limit,
J. Phys.:
Condens. Matter
11
,
A219
(
1999
).
[24] G. T. Barkema and N. Mousseau, High-quality continuous
random networks,
Phys. Rev. B
62
,
4985
(
2000
).
[25] T. Damart, V. M. Giordano, and A. Tanguy, Nanocrystalline
inclusions as a low-pass filter for thermal transport in
a
-Si,
Phys.
Rev. B
92
,
094201
(
2015
).
[26] G. Nilsson and G. Nelin, Phonon dispersion relations in Ge at
80 K,
Phys.Rev.B
3
,
364
(
1971
).
[27] J. Tersoff, Empirical interatomic potential for silicon
with improved elastic properties,
Phys. Rev. B
38
,
9902
(
1988
).
024201-6