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PHYSICAL REVIEW MATERIALS
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Thermal acoustic excitations with atomic-scale wavelengths in amorphous silicon
Jaeyun Moon
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
Raphaël P. Hermann and Michael E. Manley
Material Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
Ahmet Alatas and Ayman H. Said
Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 64039, USA
Austin J. Minnich
*
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
(Received 4 February 2019; revised manuscript received 6 May 2019; published 3 June 2019)
The vibrational properties of glasses remain a topic of intense interest due to several unresolved puzzles,
including the origin of the Boson peak and the mechanisms of thermal transport. Inelastic scattering measure-
ments have revealed that amorphous solids support collective acoustic excitations with low THz frequencies
despite the atomic disorder, but these frequencies are well below most of the thermal vibrational spectrum.
Here, we report the observation of acoustic excitations with frequencies up to 10 THz in amorphous silicon.
The excitations have atomic-scale wavelengths as short as 6 Å and exist well into the thermal vibrational
frequencies. Simulations indicate that these high-frequency waves are supported due to the high group velocity
and monatomic composition of a-Si, suggesting that other glasses with these characteristics may also exhibit
such excitations. Our findings demonstrate that a substantial portion of thermal vibrational modes in amorphous
materials can still be described as a phonon gas despite the lack of atomic order.
DOI:
10.1103/PhysRevMaterials.3.065601
I. INTRODUCTION
Amorphous materials possess a number of peculiar vi-
brational properties compared to those of their crystalline
counterparts [
1
7
]. For vibrations of frequency less than a few
THz, the wavelengths are sufficiently long that the vibrational
properties would be expected to be unaffected by the atomic
disorder [
8
]. However, the observed density of states and
heat capacity of glass differ from the Debye predictions at
low temperatures below 10 K [
1
7
]. Various explanations
have been proposed for these observations, including the
soft potential model [
9
,
10
], elastic heterogeneities [
11
], and
lower mass density of amorphous solids compared to their
crystalline counterparts [
6
,
7
].
The introduction of high-brilliance inelastic x-ray scat-
tering (IXS) experiments has enabled acoustic excitations
to be resolved by wave vector and energy outside of the
dynamic range accessible by neutron scattering [
12
16
]. The
measurements have shown that many glasses [
17
22
] and
liquids [
23
26
] support isotropic acoustic excitations with
frequencies up to
1 THz despite the atomic disorder. Fur-
ther, the observed inelastic peaks exhibit a broadening,

(
q
),
that increases with wave vector
q
following a power law
[
17
22
,
25
30
]. The acoustic excitation ceases to be well-
defined when the broadening is on the order of the frequency
*
aminnich@caltech.edu
of the excitation, typically around a few THz. These general
trends have been reported in a variety of glasses including
polymers [
18
,
28
,
30
], metallic glasses [
21
], and monatomic
glasses such as selenium [
20
], and others [
17
,
19
,
22
,
27
,
29
].
Numerical methods such as molecular dynamics simula-
tions are often utilized to gain microscopic understanding of
vibrational properties in amorphous materials and to guide
experimental measurements. Amorphous silicon has been
used for decades as a representative material to understand
the vibrational properties of amorphous materials due to its
monoatomic composition and widely used available potentials
such as Stillinger-Weber [
31
,
32
] and Tersoff [
33
,
34
]. Prior
computational works based on normal mode decomposition
have concluded that propagating acoustic waves exist up to
2 THz to 3 THz [
35
40
] and that nonpropagating vibrations
dominate the thermal conductivity [
35
,
37
40
]. On the other
hand, recent calculations predicted that acousticlike vibra-
tions are present well into the thermal frequencies [
41
,
42
].
The conflict between these reports remains unresolved due
to the lack of inelastic scattering measurements that pro-
vide direct information on the acoustic excitations supported
in a-Si.
Here, we report the observation of acoustic excitations with
atomic-scale wavelengths as small as 6 Å, corresponding to
frequencies up to 10 THz using IXS. Molecular dynamics
simulations show that these high-frequency acoustic waves
are supported due to the high group velocity and monatomic
composition of a-Si, and that these results hold for a
2475-9953/2019/3(6)/065601(8)
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©2019 American Physical Society
JAEYUN MOON
et al.
PHYSICAL REVIEW MATERIALS
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FIG. 1. Structural characterization of a-Si powders. (a) Radial distribution function (blue circles) of the sample A1 compared to a
calculation using an amorphous structure from molecular dynamics (line) [
42
]. (b) X-ray diffraction pattern measured at 300 K. Each tick
mark represents 2
θ
=
2
.
5
at x-ray energy of 21.657 keV. Broadened features in both the RDF and XRD measurements indicate an amorphous
atomic structure.
variety of atomic configurations that match the observed
radial distribution function (RDF). Such excitations ex-
tending in the thermal frequencies might also be found
in other elastically stiff monatomic glasses such as tetra-
hedral amorphous carbon. Our findings indicate that the
phonon gas model describes thermal atomic vibrations
accurately in amorphous Si despite the lack of atomic
order.
II. SAMPLE PREPARATION
We prepared two samples (A1 and A2) by depositing
amorphous Si onto sapphire substrates by plasma-enhanced
chemical vapor deposition (PECVD) at separate times with
silane gas diluted (5%) in argon gas at a deposition table
temperature of 473 K. To make the samples suitable for
the x-ray beam with absorption length of 2 mm, the thin
films (3
μ
m thick) were powderized in a glovebox under
either nitrogen or argon and placed in quartz capillary tubes
with 10
μ
m wall thickness. The structural characterization of
sample A1 is shown in Fig.
1
. The RDF of this sample was
measured by neutron scattering at Nanoscale-Ordered Mate-
rials Diffractometer (NOMAD), Spallation Neutron Source
(SNS) at Oak Ridge National Laboratory. The neutron total
scattering structure function was produced by normalizing the
sample scattering intensity to the scattering intensity from
a solid vanadium rod and subtracting the background of an
empty 2-mm quartz capillary. The radial distribution func-
tion was obtained through the Fourier transform of the total
scattering function with momentum transfer between 0.1 and
31
.
1
. The RDF and x-ray diffraction pattern show broad-
ened features, indicating that the samples are disordered. The
RDF indicates that residual hydrogen (
20 at. %) is present in
the sample as indicated by the negative peak at the Si-H 1.4 Å
distance, but we expect little influence on our measurements,
considering that prior work observed no systematic change in
thermal conductivity of PECVD amorphous silicon films with
hydrogen content varying from 1 at. % to 20 at. % [
43
].
III. RESULTS
A. Inelastic x-ray scattering measurements
Dynamic structure factors from both samples (A1 and
A2) were independently measured using spectrometers at
sector HERIX-3 and HERIX-30 with energies of 21.657
and 23.71 keV at the Advanced Photon Source [
44
47
],
respectively. The measurements for the longitudinal branch at
different momentum transfers at room temperature are shown
in Figs.
2(a)
and
2(b)
, respectively. For both samples, distinct
inelastic peaks are clearly visible at thermal frequencies up to
around 10 THz, indicating the presence of collective acoustic
excitations with well-defined frequencies and wave vectors.
The wavelengths of these excitations are as small as 6 Å, com-
parable to the interatomic spacing in a-Si. A sudden increase
in the broadening of the inelastic peaks is observed between
7
.
79 nm
1
and 11
.
12 nm
1
for A1 and between 10
.
0nm
1
and 12
.
0nm
1
for A2, respectively. This visual observation
indicates that collective excitations with well-defined wave
vectors and frequencies are not supported beyond these wave
vectors and frequencies.
To determine the center frequencies and the broadening
of the collective excitations at each momentum transfer, the
dynamic structure factor spectra were modeled with a function
S
(
q
) consisting of a Lorentzian for the central elastic peak
and a damped harmonic oscillator for the inelastic peaks,
S
(
q
)
=
I
0
(
q
)

0
(
q
)
2

0
(
q
)
2
+
ν
2
+
[
n
(
ν
)
+
1]
I
(
q
)
×
ν
(
q
)
2

(
q
)
[

(
q
)
2
ν
2
]
2
+

(
q
)
2
ν
2
,
(1)
where
q
is the wave vector,
ν
is the frequency,
I
0
(
q
) and

0
(
q
)
are the intensity and width of the central peak,
I
(
q
) and

(
q
)
are the intensity and full width half maximum of the inelastic
peak with the peak frequency

(
q
), and
n
(
ν
) is the Bose factor
[
23
]. The function
S
(
q
) is then convolved with a pseudo-
Voigt function representing the resolution function to fit to the
experimental data [
44
].
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FIG. 2. Inelastic x-ray scattering spectra of (a) sample A1 (black
circles) and (b) sample A2 (blue circles) along with the fit (red lines).
For both samples, sharp inelastic peaks are observed at frequencies
approaching 10 THz, above which a significant increase in the
broadening is seen.
The dispersion relations of the measured acoustic excita-
tions are plotted in Fig.
3
along with the calculated dynamic
structure factor from Ref. [
42
]. The vertical bars represent the
full width at half maximum of the inelastic peaks. Excellent
agreement between the simulation and the IXS measurements
is observed, with the sound velocities agreeing to within 4%.
We attempted to include the 20 at. % hydrogen present in
the sample in MD simulations using available interatomic
potentials [
48
]; however, the thermal conductivity of the struc-
ture was inconsistent with experiments [
49
]. This discrepancy
may arise because the interatomic potential was optimized for
structural rather than dynamic properties. The good agreement
between simulations with pure a-Si and experiments suggests
that the presence of hydrogen does not affect the dispersion of
the acoustic excitations. In both simulations and experiments,
vibrations with frequencies less than
10 THz have a well-
defined frequency and wave vector. As shown in Fig. S1 in the
Supplemental Material [
50
], the calculated dynamic structure
factors of Refs. [
38
,
41
] are also in good agreement with
the measurements. These works also utilized Stillinger-Weber
potential [
31
]. Comparison with other theoretical studies that
focus on quantities of relevance to Allen-Feldman theory
[
35
,
51
] is difficult because such quantities are not directly
measured in an IXS experiment.
Next, we performed additional temperature-dependent IXS
measurements on Sample A1 at 35 K and 500 K. The
extracted inelastic peak frequencies and broadenings are
FIG. 3. Dispersion relation from IXS measurements (diamonds
from sample A1 using HERIX sector 3 and filled circles from
sample A2 using HERIX sector 30) and calculated dynamic structure
factor from Moon
et al.
[
42
]. Vertical bars represent the FWHM of
the inelastic peaks. The sound velocity from the measurements is
7850 m s
1
, within 4% of the predicted sound velocity of 8179 m s
1
.
shown in Fig.
4
. We observe a slight softening with tem-
perature for the peak frequency,

(
q
), but no temperature
dependence of the broadening is seen. This observation in-
dicates that the origin of the broadening of the inelastic
peaks is structural rather than anharmonic. The temperature
independence of the broadening has also been reported in
other amorphous solids such as glycerol and silica [
19
,
27
,
52
].
IV. DISCUSSION
A. Dynamic structure factor of various atomic configurations
We note that a-Si does not have a uniquely determined
structure. Prior calculations [
53
,
54
] have shown that paracrys-
talline amorphous silicon structures yield an RDF that is indis-
tinguishable with that of experiments. We therefore generated
three different amorphous silicon configurations (continuous
random network, melt-quench, and crystal seed nucleation)
that closely match with our RDF from neutron scattering. All
three structures contain 4096 atoms and use the same SW
potential with a time constant of 0.5 fs. The continuous ran-
dom network model was provided by N. Mousseau and was
generated from the modified Wooten-Winer-Weaire algorithm
[
55
]. The description of the melt-quench method is provided
in detail in Moon
et al.
[
42
]. For the crystal seed nucleation
method, the crystalline silicon was first melted at 3500 K at
constant volume for 50 ps while the spherical crystal seed
atoms (1 at. %) are kept fixed at their positions. We then
quenched the structure to 1000 K at 100 K ps
1
. The entire
structure was then annealed at 1000 K and 0 bar for 2.75 ns
in NPT followed by quenching to room temperature at the
same rate as before. The resulting structure from the crystal
seed nucleation, therefore, has a crystalline region as shown
in Fig.
5(a)
. Only the crystallite atoms are displayed in the
otherwise amorphous domain.
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FIG. 4. Temperature-dependent IXS measurements for sample A1. (a) Inelastic peak frequency and (b) broadening at wave vectors of 7.79
(blue circles), 11.12 (red diamonds), and 14
.
44 nm
1
(black squares) at various temperatures. A slight softening of the peak frequencies with
temperature is observed but no clear temperature dependence is found for the broadenings.
FIG. 5. (a) Paracrystalline amorphous silicon structure with only the crystalline region displayed. (b) Zoomed-in view of the crystallite.
The red atoms denote the initial crystal seed and the black atoms represent the crystal growth from annealing. The crystallite is estimated to
compose 8 at. % of the structure. Crystalline order is clearly observed. For scale, the average interatomic distance is around 2.35 Å. (c) Radial
distribution function of continuous random network (black line), melt-quench (blue line), and paracrystalline (red line) amorphous silicon
structures compared to that of experimental data (green circles).
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FIG. 6. Dispersion relations for longitudinal waves for (a) continuous random network, (b) melt-quench, and (c) crystal seed nucleation
amorphous silicon structures. All of them show a crisp phonon dispersion up to around 10 THz, above which a significant broadening is
observed.
A zoomed-in view of the crystallite is depicted in Fig.
5(b)
.
The red atoms denote the initial crystal seed (1 at. %) and
the black atoms represent the crystal growth from the seed
during the annealing process. The crystalline region with
well-defined tetrahedral local structures is easily observed
and is estimated to be 8 at. % by dividing the number of
atoms in the crystallite by the total number of atoms. The
RDFs of the above-mentioned structures are plotted against
the experimental data from neutron scattering in Fig.
5(c)
and show good agreement. A larger crystal seed with 3 at. %
was also used to create a paracrystalline silicon structure with
the same procedure above, but additional distinctive peaks
were clearly seen in the RDF, suggesting that the structural
heterogeneity is large enough to affect the sample average of
atomic density fluctuations.
The calculated dynamic structure factors of these three
amorphous silicon structures are depicted in Fig.
6
. We ob-
serve that all structures exhibit crisp phonon dispersion lines
up to around 10 THz above which significant broadening
is clearly observed, consistent with our IXS measurements.
Comparisons of the RDFs and dynamic structure factor calcu-
lations between the PECVD a-Si and three amorphous silicon
models suggest that acoustic excitations with frequencies up
to 10 THz with atomic scale wavelengths exist in several
possible atomic configurations of a-Si.
B. IXS measurement comparison among
several amorphous materials
The broadening of the inelastic peaks,

(
q
), of a-Si at
300 K versus wave vector are shown in Fig.
7(a)
along with
those of various types of amorphous materials from metallic
glasses to polymers studied by IXS [
18
,
21
,
27
,
30
] and the
computational work of Ref. [
42
]. A

(
q
)
α
q
2
scaling is
also shown as a guide to the eye. Previous works in other
materials show a clear power-law dependence. In contrast, the
FIG. 7. Inelastic peak broadening for a-Si (A1 and A2) and other amorphous materials. (a) Broadening

versus wave vector
q
of various
amorphous materials from IXS: Present measurements at 300 K (black filled triangles for A1 and black filled circles for A2), simulations
from Ref. [
42
] (solid black line), silica at 1050 K (blue crosses) [
27
], polybutadiene at 140 K (PB, orange squares) [
18
], Ni
33
Zr
67
metallic
glass at room temperature (MG, yellow diamonds) [
21
], and amorphous drugs of Indomethacin (IMC, purple circles) and Celecoxib (CXB,
green crosses) at room temperature [
30
]. A temperature dependence of the broadenings were not observed in these materials; therefore, direct
comparison of our measurements at 300 K is possible. The
q
2
dependence of broadening on wave vectors for these materials is not observed
in amorphous silicon. (b) Broadening versus frequency for the same materials as in (a). The dotted line is the Ioffe-Regel crossover defined by

=
π
. The Ioffe-Regel crossover occurs at around
10 THz for a-Si, well into the thermal frequencies. The vertical bars in the measured
data are the uncertainties of fitting the damped harmonic oscillator model to the measurements. The simulated broadening lies within the
vertical bars.
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FIG. 8. Calculated dynamic structure factor for longitudinal vibrations in (a) a-SiO
2
and (b) a-SiC. Noticeable broadening is observed for
frequencies below 10 THz. The Ioffe-Regel crossover frequency is
1
.
5 THz and
8 THz for (a) and (b), respectively.
broadening of a-Si for both the IXS and dynamic structure
factor calculations [
41
,
42
] demonstrate a sudden increase
rather than a power-law dependence. This sudden increase
with frequency in the broadening can also be visually ob-
served in the raw data shown in Fig.
2
. The origin of this
increase is at present not clear and will be the subject of future
work.
The definition of the frequency at which an acoustic ex-
citation is no longer well-defined is conventionally taken
to be when the broadening

(
q
)
=

(
q
)
, known as the
Ioffe-Regel crossover [
2
,
56
]. Using this criterion, we find a
strikingly high crossover frequency of around 10 THz, as
shown in Fig.
7(b)
. This crossover frequency (corresponding
to ̄
h
ω/
k
B
=
480 K) is well within the portion of the vibra-
tional spectrum thermally occupied at 295 K and implies that
acoustic excitations are supported for wavelengths as small as
6 Å, only a few times larger than the interatomic distance of
amorphous silicon (
2
.
4Å).
Our simulations show that this unusually high crossover
frequency can be explained by two features of a-Si. First,
acoustic excitations cease to possess a well-defined wave vec-
tor and frequency if the disorder is sufficiently strong. a-Si is
monatomic with only minor isotopic mass disorder and hence
lacks the degree of disorder present in polyatomic glasses such
as mass or force constant disorder. Using Tersoff potentials
[
34
,
57
], dynamic structure factors of a-SiO
2
and a-SiC were
calculated. The longitudinal sound velocities of a-SiO
2
and
a-SiC from dynamic structure factors are calculated to be 5567
and 9844 m s
1
, which are within 5% of the experimental
results of 5800 and 9462 m s
1
, respectively [
27
,
58
]. Clear
additional broadening in the dispersions is observed compared
to a-Si as shown in Fig.
8
.
Second, a-Si has a high group velocity due to the low
atomic mass of Si and stiff covalent bonds. Thus, for a
given wave vector, a-Si supports higher frequency excita-
tions than for a heavier and weaker bonded amorphous
material like glassy selenium. Inelastic neutron and x-ray
scattering studies on glassy selenium reported a longitudinal
sound velocity of around 2000 m s
1
, leading to a Ioffe-
Regel crossover frequency of around 1–2 THz [
20
,
59
]. For
a given wave vector, a-Si supports a vibrational frequency
around four times larger than that of a-Se, owing to its
higher group velocity. These factors explain the presence of
acoustic excitations at frequencies up to 10 THz in a-Si.
Other glasses with similar characteristics, such as tetrahedral
amorphous carbon, may exhibit such excitations as well, a
prediction that can be verified with further inelastic scattering
experiments.
V. CONCLUSION
In summary, we report the observation of thermal acoustic
excitations with atomic-scale wavelengths in a-Si despite the
atomic disorder. The excitations possess wavelengths as small
as 6 Å and are supported due to the monatomic composition
and high group velocity of a-Si. Our findings demonstrate
that the description of thermal vibrations in a-Si as a gas of
acoustic excitations is unexpectedly accurate despite the lack
of crystalline order, suggesting that other monoatomic glasses
with high sound velocity may also support acoustic waves in
the thermal spectrum.
ACKNOWLEDGMENTS
The authors thank Nathan Sangkook Lee for helpful dis-
cussions in sample preparations, Dr. Jörg Neuefeind and
Michelle Everett for assistance in data collection at NOMAD,
and Dr. Bianca Haberl for helpful discussions. The authors
thank Dr. John Budai for assistance in data collection at
HERIX-30. This work was supported by a Samsung Scholar-
ship and a Resnick Fellowship from the Resnick Sustainabil-
ity Institute at Caltech, and the U.S. Department of Energy,
Office of Science, Basic Energy Sciences, Materials Sciences
and Engineering Division. This research used resources of
the Advanced Photon Source, a U.S. Department of Energy
(DOE) Office of Science User Facility operated for the DOE
Office of Science by Argonne National Laboratory under Con-
tract No. DE-AC02-06CH11357. A portion of this research
used resources at the Spallation Neutron Source, a DOE Office
of Science User Facility operated by the Oak Ridge National
Laboratory.
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This paper has been co-authored by employees of UT-
Battelle, LLC, under Contract No. DE AC0500OR22725
with the U.S. Department of Energy. The United States
Government retains and the publisher, by accepting the article
for publication, acknowledges that the United States Govern-
ment retains a nonexclusive, paid-up, irrevocable, worldwide
license to publish or reproduce the published form of this
paper, or allow others to do so, for the United States Gov-
ernment purposes. The Department of Energy will provide
public access to these results of federally sponsored research
in accordance with the DOE Public Access Plan.
J.M. and A.J.M. conceived the project. J.M. synthesized
the a-Si samples. J.M., A.A., A.H.S, R.P.H., and M.E.M con-
ducted the IXS experiments and analyzed the results. R.P.H.
performed the RDF measurements. J.M. performed molecular
dynamics calculations. J.M. and A.J.M. wrote the paper with
input from all authors. A.J.M. supervised the project.
The authors declare no competing interests.
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