Investigation of Transport Phenomena of Thermal Acoustic Excitations in Semi-Crystalline and Amorphous Materials Using Transient

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Investigation of Transport Phenomena of Thermal Acoustic Excitations in Semi-Crystalline and Amorphous Materials Using Transient Grating Spectroscopy
Citation
Kim, Taeyong
(2021)
Investigation of Transport Phenomena of Thermal Acoustic Excitations in Semi-Crystalline and Amorphous Materials Using Transient Grating Spectroscopy.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/k364-ga14.
Abstract
The physics of transport of heat-carrying atomic vibrations in amorphous and semi-crystalline solids is a topic of fundamental interest. Diverse tools have been employed to study thermal transport in these materials, including cryogenic thermal conductivity measurements and various inelastic scattering tools. However, unambiguously identifying the damping mechanisms of few THz and smaller frequency excitations remains difficult owing to the lack of the experimental probes in the frequency band. As a result, debate has remained regarding the microscopic origin of weak acoustic damping in amorphous silicon (Si), the unusually high thermal conductivity of ultra-drawn polyethylene, and other topics.
In this thesis, we investigate the transport properties of heat-carrying acoustic excitations in semi-crystalline and amorphous solids using transient grating spectroscopy. This optical method permits the creation of thermal gradients over sub-micron length scales which may be comparable to the attenuation lengths of the excitations. We show how these measurements can be used to constrain the damping mechanisms in the sub-THz range that has been historically inaccessible by typical methods such as inelastic scattering.
First, we report measurements of the bulk thermal conductivity and elastic properties of MoS₂ thin films. Specifically, we use TG to measure the in-plane longitudinal sound velocity and thermal conductivity. We do not observe any size effects of thermal conductivity with grating period, indicating that the propagating distance of heat-carrying acoustic phonons are smaller than the thermal length scale accessible in the experiment. This result is consistent with the mean free paths predicted from ab-initio numerical methods.
Second, we utilize the capability of TG to resolve the microscopic heat transport properties of phonons in highly oriented semi-crystalline polyethylene (PE). Earlier experimental studies have reported thermal conductivities of up to ~ 100 Wm⁻¹ K⁻¹ crystalline polyethylene, orders of magnitude larger than the bulk value of ~ 0.4 Wm⁻¹ K⁻¹. However, the microscopic origin of the high thermal conductivity remains unclear. We address this question by applying TG to highly oriented polyethylene to show that mean free paths on micron length scales are the dominant heat carriers. Using a low-energy anisotropic Debye model to interpret these data, we find evidence of one-dimensional phonon density of states for excitations of frequency less than ~ 2 THz. This transition frequency is consistent with the unique features of ultradrawn PE, in particular the stiff longitudinal branch leading to wavelengths of 8 nm at 2 THz frequency; and fiber diameters < 10 nm observed in prior structural studies of ultradrawn polymers; so that the wavelength does indeed exceed the fiber diameter at the relevant frequencies.
Finally, we report the measurements of the frequency-resolved mean free path of heat-carrying acoustic excitation in amorphous silicon (aSi), for the first time. The heat-carrying acoustic excitations of amorphous silicon are of interest because their mean free paths approach the micron scale at room temperature. Despite extensive investigation, the origin of the weak acoustic damping in the heat-carrying frequencies remains a topic of debate for decades. A prior study suggested a framework of classifying the vibrations into propagons, diffusons, and locons. Propagons were considered phonon-like, delocalized, propagating vibrations; locons as localized vibrations, and diffusons as delocalized yet non-propagating vibrations. Following the framework, numerous works have predicted mechanism of acoustic damping in aSi, but the predictions have contradicted to observations in experiments. In this work, we obtained measurements of the frequency-dependent mean free path in amorphous silicon thin films from ~0.1-3 THz and over temperatures from 60 - 315 K using picosecond acoustics (PSA) and transient grating spectroscopy. We first describe our PSA experiments to resolve the attenuation of 0.1 THz acoustic excitations in aSi. We then present our table-top approach to resolve MFP of heat-carrying acoustic excitation between ~ 0.1-3 using TG spectroscopy. The mean free paths are independent of temperature and exhibit a Rayleigh scattering trend over most of this frequency range. The observed trend is inconsistent with the predictions of numerical studies based on normal mode analysis, but agrees with diverse measurements on other glasses. The micron-scale MFPs in amorphous Si arise from the absence of Akhiezer and two-level system damping in the sub-THz frequencies, leading to heat-carrying acoustic excitations with room-temperature damping comparable to that of other glasses at cryogenic temperatures. Our results allow us to establish a clear picture for the origin of micron-scale damping in aSi by understanding vibrations as acoustic excitation rather than propagons, diffusons, and locons.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Thermal conductivity; nanoscale heat transport; low frequency phonon; TMDC; Molybdenum disulfide; polymers; polyethylene; amorphous materials; amorphous silicon; transient grating; heterodyne; picosecond sound attenuation; thin films; anisotropic; quasi-ballistic; mean free path spectroscopy
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Minnich, Austin J.
Thesis Committee:
Fultz, Brent T. (chair)
Faraon, Andrei
Vahala, Kerry J.
Minnich, Austin J.
Defense Date:
27 August 2020
Funders:
Funding Agency
Grant Number
2017 GIST-Caltech Research Collaboration
UNSPECIFIED
Office of Naval Research (ONR)
N00014-18-1-2101
2018 GIST-Caltech Research Collaboration
UNSPECIFIED
Record Number:
CaltechTHESIS:10162020-115452109
Persistent URL:
DOI:
10.7907/k364-ga14
Related URLs:
URL
URL Type
Description
DOI
Article adapted for Chapter 3.
arXiv
Article adapted for Chapter 5.
ORCID:
Author
ORCID
Kim, Taeyong
0000-0003-2452-1065
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
13978
Collection:
CaltechTHESIS
Deposited By:
Taeyong Kim
Deposited On:
22 Oct 2020 15:41
Last Modified:
08 Nov 2023 00:34
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Investigation of transport phenomena of thermal acoustic
excitations in semi-crystalline and amorphous materials
using transient grating spectroscopy

Thesis by

Taeyong Kim

In Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2020
Defended August 27, 2020

ii

Taeyong Kim
ORCID: 0000-0003-2452-1065

iii

ACKNOWLEDGEMENTS
Throughout my six-year journey for my graduate study at Caltech, I have been
fortunate for being helped by many great people. Without their support, I would not
have been able to complete my thesis work. I cannot list all the names here, but I
would like to express my sincere gratitude to all of them.
First and foremost, I would like to present my sincere thank to my academic advisor,
Prof. Austin Minnich for his unconditional support, passion and patience. When I
just joined his group, he helped me by walking me through thermal science, which
I would never forget. He always dedicated his time and efforts to advise me, which,
for me, became a source for overcoming any risky challenges that I had, especially
when I got lost. I am grateful to him for the discussions that I had with him, from
which I was able to improve my scientific insights. Not only as a researcher, but
also as a mentor, Austin was my role model. I sincerely thank him for constantly
encouraging me during my years at Caltech, and for providing me advice on my
career paths when I became a senior graduate student.
I would like to thank Prof. Brent Fultz, Prof. Kerry Vahala, and Prof. Andrei
Faraon, who graciously serve on my dissertation committee, for providing useful
comments and discussions for my works over many years.
I wish to acknowledge the financial support from the Jeongsong graduate fellowship,
Light-Material Interactions in Energy Conversion Energy Frontier Research Center
(LMI-EFRC) at Caltech, GIST-Caltech Research Collaboration, and Office of Naval
Research.
I benefited from the support of many people at Caltech. I thank the Joint-Center
for Artificial Photosynthesis (JCAP) for allowing me to build a time-resolved setup.
Thank you to the post-docs and students at JCAP, especially to Dr. Ke Sun, and
Dr. Stefan Omelchenko. I also wish to acknowledge Kavli Nanoscience Institute
(KNI) for letting me use their great facilities. Thank you to the staff members at
KNI, especially Matthew Hunt and Nathan Sangkook Lee. I also want to thank Dr.
Tiziana Luccio in Prof. Julie Kornfield’s group for fruitful discussions. I am also
grateful to Prof. Harry Atwater and his research group for allowing me to use their
research facilities. Thank you to the staff members at Caltech for their support:
Sonya Lincoln, Jennifer Blankenship, and Tiffany Kimoto.
I had the good fortune of collaborating with many researchers from various institutes.

iv
Thanks to my collaborators for providing me a lot of a great source for my research.
I am grateful to Prof. Young-dahl Jho, and Dr. Jong-Hyuck Yim at Gwangju
Institute of Science and Technology (GIST) in South Korea. I am also thankful to
Prof. Sara Ronca, Prof. Ignacio Martin-Faviany, and Dr. Stavros Drakopoulos at
Loughborough University in the UK. I am grateful to Prof. Clivia Sotomayor-Torres
and her group at Catalan Institute of Nanoscience and Nanotechnology (ICN2) in
Spain. I thank Prof. Brian Daly at Vassar College for useful discussions on sound
attenuation measurements.
For the past six years, I have been able to work with past and contemporary Minnich
lab members. Thank you to Andrew Robbins, Ding Ding, Nate Thomas, Nick
Dou, Chengyun hua, Navaneetha Ravichandran, Junlong Kou, Xiangwen Cheng,
and Jaeyun Moon. Particularly, I would like to express my sincere thanks to Andrew
Robbins and Ding Ding, without whom many parts of my thesis would not have
been made. I also would like to acknowledge current Minnich group members: Alex
Choi, Peishi Cheng, Adrian Tan, Erika Ye, Shi-Ning Sun, Zoila Jurado, and other
members.
I thank friends at Caltech for their friendships. Thanks to many Korean friends
at Caltech, especially Jaeyun Moon, Myung-gyun Suh, Jeesoon Choi, Yonil Jung,
Chang Ho Sohn, Youngjoon Choi, Dennis Sungtae Kim, Areum Kim, Gunho Kim,
Hyun-Gi Yoon, Kyuhyun Lee, Yonghwi Kim, Dr. Sonjong Hwang, and Dr. Byungkuk Yoo. Thank you all to other friends at Caltech, especially to John Pang, Jinglin
Huang, Hengjiang Ren, and Ke Ke. A special thanks to Daniel Naftalovich for being
my terrific roommate in my first year at Caltech, and for a life-long friendship.
Last, but not the least, I sincerely thank my family for their love, support, and the
sacrifices that they had to make for me. I would like to thank my mom, dad, and my
sister without whom I would not have been able to be a grown up. Thanks to you, I
was able to endure many ups and downs during my doctoral years.

ABSTRACT
The physics of transport of heat-carrying atomic vibrations in amorphous and semicrystalline solids is a topic of fundamental interest. Diverse tools have been employed to study thermal transport in these materials, including cryogenic thermal
conductivity measurements and various inelastic scattering tools. However, unambiguously identifying the damping mechanisms of few THz and smaller frequency
excitations remains difficult owing to the lack of the experimental probes in the
frequency band. As a result, debate has remained regarding the microscopic origin
of weak acoustic damping in amorphous silicon (Si), the unusually high thermal
conductivity of ultra-drawn polyethylene, and other topics.
In this thesis, we investigate the transport properties of heat-carrying acoustic excitations in semi-crystalline and amorphous solids using transient grating spectroscopy.
This optical method permits the creation of thermal gradients over sub-micron length
scales which may be comparable to the attenuation lengths of the excitations. We
show how these measurements can be used to constrain the damping mechanisms
in the sub-THz range that has been historically inaccessible by typical methods such
as inelastic scattering.
First, we report measurements of the bulk thermal conductivity and elastic properties
of MoS2 thin films. Specifically, we use TG to measure the in-plane longitudinal
sound velocity and thermal conductivity. We do not observe any size effects of
thermal conductivity with grating period, indicating that the propagating distance of
heat-carrying acoustic phonons are smaller than the thermal length scale accessible
in the experiment. This result is consistent with the mean free paths predicted from
ab-initio numerical methods.
Second, we utilize the capability of TG to resolve the microscopic heat transport properties of phonons in highly oriented semi-crystalline polyethylene (PE).
Earlier experimental studies have reported thermal conductivities of up to ∼ 100
Wm−1 K−1 in crystalline polyethylene, orders of magnitude larger than the bulk
value of ∼ 0.4 Wm−1 K−1 . However, the microscopic origin of the high thermal
conductivity remains unclear. We address this question by applying TG to highly
oriented polyethylene to show that mean free paths on micron length scales are the
dominant heat carriers. Using a low-energy anisotropic Debye model to interpret
these data, we find evidence of one-dimensional phonon density of states for ex-

vi
citations of frequency less than ∼ 2 THz. This transition frequency is consistent
with the unique features of ultradrawn PE, in particular the stiff longitudinal branch
leading to wavelengths of 8 nm at 2 THz frequency; and fiber diameters < 10 nm
observed in prior structural studies of ultradrawn polymers; so that the wavelength
does indeed exceed the fiber diameter at the relevant frequencies.
Finally, we report the measurements of the frequency-resolved mean free path of
heat-carrying acoustic excitation in amorphous silicon (aSi), for the first time. The
heat-carrying acoustic excitations of amorphous silicon are of interest because their
mean free paths approach the micron scale at room temperature. Despite extensive
investigation, the origin of the weak acoustic damping in the heat-carrying frequencies remains a topic of debate for decades. A prior study suggested a framework
of classifying the vibrations into propagons, diffusons, and locons. Propagons were
considered phonon-like, delocalized, propagating vibrations; locons as localized
vibrations, and diffusons as delocalized yet non-propagating vibrations. Following
the framework, numerous works have predicted mechanism of acoustic damping
in aSi, but the predictions have contradicted to observations in experiments. In
this work, we obtained measurements of the frequency-dependent mean free path
in amorphous silicon thin films from ∼ 0.1 − 3 THz and over temperatures from
60 - 315 K using picosecond acoustics (PSA) and transient grating spectroscopy.
We first describe our PSA experiments to resolve the attenuation of 0.1 THz acoustic excitations in aSi. We then present our table-top approach to resolve MFP of
heat-carrying acoustic excitation between ∼ 0.1 − 3 using TG spectroscopy. The
mean free paths are independent of temperature and exhibit a Rayleigh scattering
trend over most of this frequency range. The observed trend is inconsistent with the
predictions of numerical studies based on normal mode analysis, but agrees with
diverse measurements on other glasses. The micron-scale MFPs in amorphous Si
arise from the absence of Akhiezer and two-level system damping in the sub-THz
frequencies, leading to heat-carrying acoustic excitations with room-temperature
damping comparable to that of other glasses at cryogenic temperatures. Our results
allow us to establish a clear picture for the origin of micron-scale damping in aSi
by understanding vibrations as acoustic excitation rather than propagons, diffusons,
and locons.

vii

PUBLISHED CONTENT AND CONTRIBUTIONS

[1] Ding Ding, Taeyong Kim, and Austin J. Minnich. Active thermal extraction
of near-field thermal radiation. Phys. Rev. B, 93:081402, 2 2016. doi: 10.
1103/PhysRevB.93.081402. URL https://link.aps.org/doi/10.1103/
PhysRevB.93.081402.
T.K. conducted the calculations, and analyzed the results.
[2] Ding Ding, Taeyong Kim, and Austin J. Minnich. Active thermal extraction and
temperature sensing of near-field thermal radiation. Scientific Reports, 6(1):
32744, 2016. doi: 10.1038/srep32744. URL https://doi.org/10.1038/
srep32744.
T.K. conducted the calculations, and analyzed the results.
[3] Taeyong Kim, Ding Ding, Jong-Hyuck Yim, Young-Dahl Jho, and Austin J.
Minnich. Elastic and thermal properties of free-standing molybdenum disulfide membranes measured using ultrafast transient grating spectroscopy. APL
Materials, 5(8):086105, 2017. doi: 10.1063/1.4999225. URL https:
//doi.org/10.1063/1.4999225.
T.K. prepared the experimental data, analyzed the results, and participated in
the writing of the manuscript.
[4] Taeyong Kim, Jaeyun Moon, and Austin J. Minnich. Origin of micron-scale
propagation lengths of heat-carrying acoustic excitations in amorphous silicon.
July 2020. URL https://arxiv.org/abs/2007.15777v1.
T.K. participated in the conception of the project, prepared the experimental
data, analyzed the results, and participated in the writing of the manuscript.
[5] Taeyong Kim, Andrew B. Robbins, Stavros X. Drakopoulos, Ignacio MartinFabiani, Sara Ronca, and Austin J. Minnich. Heat-carrying phonons with
micron-scale mean free paths in highly oriented polyethylene (in preparation).
T.K. participated in the conception of the project, prepared the experimental
data, analyzed the results, and participated in the writing of the manuscript,
2020.
[6] Stavros X. Drakopoulos, Georgios C. Psarras, Aurora Nogales, Tiberio A.
Ezquerra, Taeyong Kim, Andrew B Robbins, Austin J. Minnich, Georgia C. Manika, Gianfranco Claudio, Ignacio Martin-Fabiani, and Sara Ronca.
Gold/ultra high molecular weight polyethylene nanocomposites for electrical
energy storage: Enhanced recovery efficiency and thermal conductivity upon
uniaxial deformation (in review).
T.K. prepared the thermal characterization data, analyzed the thermal characterization results, and participated in the writing of the manuscript, 2020.

viii

CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter II: Introduction to transient grating spectroscopy . . . . . . . . . . .
2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Signal monitoring via heterodyne detection . . . . . . . . . . . . . .
2.3 Thermal signal traces and fitting . . . . . . . . . . . . . . . . . . . .
2.4 TG arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter III: In-plane elastic and thermal properties of Molybdenum disulfide
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Ultrafast transient grating . . . . . . . . . . . . . . . . . . . . . . .
3.2 Material: Molybdenum disulfide . . . . . . . . . . . . . . . . . . .
3.3 TG experiments on MoS2 . . . . . . . . . . . . . . . . . . . . . . .
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter IV: Heat-carrying phonons with micron-scale mean free paths in
highly oriented polyethylene . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Bulk thermal property of PE . . . . . . . . . . . . . . . . . . . . . .
4.3 TG application on highly oriented PE . . . . . . . . . . . . . . . . .
4.4 Grating period dependent thermal conductivity . . . . . . . . . . . .
4.5 Ballistic transport of the heat-carrying phonons in DR196 . . . . . .
4.6 Low energy Debye model . . . . . . . . . . . . . . . . . . . . . . .
4.7 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter V: Origin of acoustic excitations in amorphous silicon . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Anomalous thermal properties observed in amorphous materials . . .
5.3 Prior empirical models . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Picosecond sound acoustics (PSA) on aSi . . . . . . . . . . . . . . .
5.5 Thermal characterization of aSi . . . . . . . . . . . . . . . . . . . .
5.6 Unusually weak damping of acoustic excitation in aSi . . . . . . . .
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii
vii
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12
15
16
19
21
21
21
28
31
37
39
39
39
42
44
51
53
58
58
62
64
65
65
65
67
71
76
83
94
97

ix
Chapter VI: Conclusion and future works . . . . . . . . . . . . . . . . . . . 98
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2 Outlook and future insight . . . . . . . . . . . . . . . . . . . . . . . 99
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Chapter 1

INTRODUCTION
Thermal conductivity of polymers and amorphous materials
Bulk amorphous dielectric materials and polymers are of importance for modern
device applications due to unusual properties compared to their crystalline counterparts. For instance, owing to cost-effectiveness and chemical stability combined
with excellent electronic and optical properties, these substances have been used in
various applications such as photovoltaic solar cells, gate dielectrics for transistors,
channel materials for thin-film transistors, and mini-channel heat exchangers [1–3].
Concerning the thermal perspective, polymers were traditional thermal insulator materials. Due to the lack of long-range structural order, thermal conductivity on the
order of ∼ 0.1 Wm−1 K−1 was reported [4]. Contrary to the traditional understanding,
some of earlier studies have reported orders of magnitude increase. For instance, a
factor 6 of increase in thermal conductivity was observed in stretched polyacetylene
[5]. Several other studies have also reported observed thermal conductivity enhancement in polymethylmethacrylate (PMMA), polyethylene (PE), polypropelene,
and polyethylene [6–8].
Despite great strides that have been made, its microscopic origin that leads to these
dramatic changes is poorly understood. For instance, some of the prior works
have argued that overall thermal conductivity increase originates from a substantial
increase in the amorphous fraction, while the others demonstrated the opposite
reports. For instance, Xu and Chen have shown that the thermal conductivity of
110 times stretched PE is as high as 60 Wm−1 K−1 with amorphous contribution 25%
[9]. Similar conclusion was drawn for oriented amorphous polythiophene [10]. On
the other hand, Ronca and Rastogi measured thermal conductivity of 220 times
stretched PE to be around 50 Wm−1 K−1 , but with substantial contribution from
crystalline domain 94%.
Mechanism of the damping of the low energy excitations in amorphous silicon (aSi)
is another topic under debate for more than 20 years due to unusual properties compared to other glasses. In most glasses, earlier works have shown that the thermal
vibrational mean free path (MFP), an average travel distance before the scattering,
is of order of nanometers due to the lack of long-range order. Due to the similar-

ity in the bulk properties of the glassy substances, corresponding mechanism of the
damping of the vibration was considered to be universal. However, the experimental
observations in aSi indicate that the aSi is an unusual glass due to the following reasons: First, some experiments have measured the thermal conductivity of aSi as high
as 4 Wm−1 K−1 , which is anomalously high compared to the values in typical glassy
substances. Second, recent experiments reported an evidence that heat-carrying
acoustic excitations in aSi can propagate micron-length distances [11]. Third, there
is some experimental evidence that the excess heat capacity at low temperature, a
characteristic of the glassy materials, is not present in amorphous silicon [12]. A
classification of the vibrations was formulated to explain the damping of vibrations:
propagons, diffusons, and locons [13, 14]. Various numerical approaches have
demonstrated diverse arguments under this classification, but the predictions have
yielded discrepancies with the experimental results. For instance, measured MFP of
100 GHz acoustic excitation in aSi is of order of 10 𝜇m [15], an order of magnitude
higher than the numerical prediction [16].
Prior experiments have strived to identify damping mechanisms of acoustic excitations as an origin of the bulk thermal properties. Experimentally resolving the
damping of the acoustic vibrations is of significant challenging due to the lack of experimental probes. To be specific, amorphous silicon and semi-crystalline polymers
are typically fabricated as a thin-film, therefore difficult to be studied by in-elastic
neutron scattering and in-elastic x-ray scattering. Although a considerable amount
of the volumes of the samples would be provided, the instrumental resolution of
in-elastic x-ray scattering is not sufficient to probe excitations with low frequencies.
More precisely, kinematic constraints complicate the measurement of the dispersion
and broadening of low energy excitations. Overall, these experimental challenges
highlight a need for an experimental probe that can probe the dynamics of the low
energy excitations.
Traditional approach to infer mean free path
Traditionally, the transport properties of the low energy excitations were estimated
based on thermal transport measurements. The theory of the heat conduction is
based on phonons as the dominant microscopic heat carriers, collective vibrational
excitations arisen by periodic arrangements of consisting atoms. The thermal conductivity of a bulk material is well-described by the kinetic theory, given by
𝜅(𝑇) = 𝐶𝑠 (𝜔)𝑣 𝑠 (𝜔)Λ𝑠 (𝜔, 𝑇)𝑑𝜔
(1.1)

where 𝜅 is the thermal conductivity of a solid, index 𝑠 is the polarization, 𝜔 is
frequency, 𝐶 is the heat capacity, 𝑣 is the group velocity, and Λ is the mean free
path (MFP). The kinetic theory (Eq. 1.1) provides two important insights. First,
the higher speed of the phonons as well as the longer travel distance results in the
higher thermal conductivity. Second, the microscopic properties have a spectrum,
therefore the information on the spectrum is a key to understanding corresponding
bulk properties.
Earlier studies have used the above relation to infer the MFP. First approach is
known as gray approximation which does not consider frequency dependence given
in Eq. 1.1. According to this approximation, in SiO2 for example, the mean free path
of the phonons are an order of ∼ 50 nm (∼ 1 nm) for crystal (amorphous) at room
temperature. [17, 18]. Similarly, dominant phonon approximation was employed
to analyze the thermal conductivity data. In this approximation, it was assumed
that heat at a temperature is predominantly carried by a single phonon mode with a
frequency [19]
𝑘 𝐵𝑇
(1.2)
𝑓𝑑𝑜𝑚 = 4.25
where 𝑘 𝐵 is the Boltzmann constant. Corresponding average phonon mean free path
for an isotropic material was estimated to be

Λ( 𝑓 = 𝑓𝑑𝑜𝑚 ) = 𝜅 𝐶𝑣 𝑣

 −1
(1.3)

where 𝜅 is the thermal conductivity of a solid, 𝐶𝑣 is the specific heat, and 𝑣 is the
sound velocity. The approximation was primarily used to explain the transport of
low frequency excitation in amorphous materials such as vitreous silica, vitreous
germanium, and glycerol, as well as amorphous polymers such as polystyrene and
polymethylmethacrylate (PMMA) [18–20].
However, as is often discussed in many prior literature, the above estimations are
known to typically underestimate the MFP, yielding discrepancies with actual values
[21, 22]. In fact, the vibrations of the solids are dispersive; the group velocity
decreases as the frequency increases, meaning that the velocity that is used for the
above estimation is generally overestimated. Therefore corresponding predictions
for 𝐶𝑣 and 𝑣 get underestimated. Second, the lifetimes for the atomic vibrations
in solids are highly frequency dependent; high frequency vibrations are typically
scattered stronger than low frequency vibrations, which is not accurately captured
in the description above. Therefore, the use of the approximations above had been

only limited for analyzing low temperature thermal conductivity data to characterize
the damping of the acoustic vibrations.
Another approach to estimate the MFP was from the reduced thermal conductivity in
tailored structures. According to prior literature [22, 23], the thermal conductivity
of a thin-film with a thickness 𝑑 is
3Λ
𝜅 ∼ 𝜅𝑏 1 −
(1.4)
8𝑑
where 𝜅 𝑏 , and 𝜅 represent thermal conductivity of bulk, thin-film, respectively.
This phenomenon is known to be classical size effects of diffuse transport of the
phonon at the boundary of the thin-film. Phonon heat conduction based on the
observation of the classical size effects was discussed in various literature. As an
example, an earlier work changed the thickness of the crystalline silicon down to
∼ 100 nm, and reported the presence of phonons with MFP as high as 300 nm, far
exceeding the gray approximation [24]. While the approach mentioned above can
provide a useful insight on the MFP distribution, direct measurements regarding
frequency dependence were challenging, therefore details on microscopic transport
were missing.
Experimental methods to obtain mean free path
Conventionally, inelastic x-ray scattering (IXS) and inelastic neutron scattering
(INS) has been used for studying acoustic dispersion, as demonstrated in many
prior studies [25–27]. Furthermore, INS and IXS showed their potential to probe
transport property of the vibrations. More precisely, the technique measures the
broadening from dynamic structure factor as a function of the frequency, therefore
suggesting an avenue for a better understanding of the microscopic phenomena,
[28, 29]. For instance, the INS experiments were applied to a single-crystalline
PbTe, an exceptionally low thermal conductivity material with its physical origin
previously unknown. The experiments have observed anharmonic coupling among
LA, TO, and LO, leading to a strong damping behavior as an origin of such low
thermal conductivity [29]. However, as earlier mentioned, the instrumental energy
resolution as well as kinematic constraints have impeded the application of IXS and
INS toward studying low energy excitations in glassy substances.
Therefore as an alternative, mean free path spectroscopy has emerged as a tool
to understand the mean free spectrum in solids [25]. This technique is based on
the observation of the ballistic transport of the heat-carrying acoustic vibrations.

A key to this approach is tunable length scale of a temperature gradient that is
comparable to the MFPs. If the thermal length scale is far longer than many of
the MFPs in the material, the observed thermal conductivity is close to the bulk
value. As the thermal length approaches a value close to the MFP, corresponding
thermal conductivity becomes suppressed, and deviates from the heat diffusion
theory, indicating ballistic transport of the heat-carrying phonons. The physical
mechanism for this is ballistic travel of the phonons external to the thermal gradient,
therefore resulting in the actual scattering being less likely to occur within the
thermal gradient. Corresponding heat flux from a source becomes less than the
diffusion theory, resulting in the reduction of the thermal conductivity that depends
on the length scale of the heat source [30].
The observation of the ballistic transport improved the understanding of the phonon
mean free path spectrum. Earlier experimental studies have varied the thermal
length scale in different ways. For example, Koh and Cahill have reported that
optical modulation frequency changes the thermal diffusion length across the cross
plane, and that the contribution of phonons with MFP longer than the diffusion
length to the thermal conductivity is less than predicted [31]. Another approach
was reducing the in-plane thermal gradient length. Minnich and Chen varied the
diameter of the laser spot size and measured the accumulated thermal conductivity
versus the mean free path distribution in crystalline silicon [32].
The heat-source size dependent thermal conductivity further advanced to develop a
technique to probe the MFP of the heat-carrying vibrations. As a result of this effort,
thermal transient grating spectroscopy (TG) has provided a new insight into understanding the MFPs of the heat carrying phonons in crystals. First measurements of
the evidence of the ballistic transport using TG was reported in early 2010’s [33].
Johnson and Nelson have measured the MFPs in 400 nm thick silicon membranes
[33], and found an evidence of micron length scale MFP in crystalline silicon. Corresponding quantitative analysis based on TG measurements was further provided
[34]. In brief, thermal conductivity at a given grating period 𝐿 𝑖 can be expressed as,
𝑆(𝑞𝑖 , Λ𝑠 )𝐶𝑠 (𝜔)𝑣 𝑠 (𝜔)Λ𝑠 (𝜔, 𝑇)𝑑𝜔

𝜅𝑖 =

where index 𝑞𝑖 = 2𝜋/𝐿 𝑖 is the grating wave vector inversely proportional to the
thermal length, index 𝑠 is polarization for phonons, and 𝑆(𝑞, Λ) is the phonon
thermal suppression function accounting for the thermal conductivity reduction

due to the ballistic transport, first introduced by Ref. [34]. Using Eq. 1, Maznev
and Nelson have shown that Akhiezer scattering mechanism of the phonons is the
dominant process for heat conduction in crystalline silicon [34].
The TG technique was further applied to discover other microscopic heat properties:
accumulated thermal conductivity and phonon specularity parameter. Minnich
proposed a method to reconstruct accumulated thermal conductivity over mean
free path without prior knowledge of the microscopic properties [35]. Minnich in
Ref. [35] has shown that Eq. 1 can be re-written as
𝜅𝑖 = 𝑞 𝑖

𝐾 (𝑥𝑖 )𝐹 (Λ𝜔 )𝑑Λ𝜔

∫Λ
where 𝑥 = (2𝜋Λ𝜔 )/𝐿 𝑖 , 𝐾 = −𝑑𝑆/𝑑𝑥, and 𝐹 (Λ𝜔 ) = 0 𝜔 𝑓 (Λ0𝜔 )𝑑Λ0𝜔 is the accumulated thermal conductivity. As a method to experimentally reconstruct 𝐹 (Λ𝜔 )
using Eq. 1, convex optimization was used, and showed an excellent agreement with
computational predictions [35]. Other recent work reported that the application
of the TG along with the Bayesian statistics can reveal frequency-resolved phonon
specularity parameters [36].
Despite a short history of TG, the TG technique showed potential to probe heatcarrying acoustic vibrations in partially oriented semicrystalline polymers [37],
suggesting its potential to allow us to study low energy excitations.
In this thesis, we exploit TG’s capability as an experimental probe to address
challenges for characterizing semi-crystalline and amorphous materials, therefore
provide a comprehensive picture of the thermal transport in them.
Outline and scope of the thesis
As discussed, a main challenge for bulk thermal properties in semi-crystalline and
amorphous materials is the lack of the experimental probe that enables to access
the transport of the low energy excitations. The purpose of this thesis is a better
understanding of the damping of the acoustic vibrations using an experimental probe
with an appropriate length and time scales required for accessing the dynamics of
the excitations. The approach presented in this thesis will provide key insights into
revealing the origin of the damping in glassy substances, therefore substantially
advance the field of the thermal science.
In chapter 2, we introduce the thermal transient grating (TG). We start from discussing working principles and requirements of TG. We then describe the typical

arrangements of TG and associated advantages in TG. Finally, we apply the TG to
measure thermal properties of crystalline silicon thin-film.
In chapter 3, we present measurements of bulk vibrational properties of crystal thinfilm. For this purpose, we describe our development of the ultrafast TG spectroscopy
to enable to simultaneously measure bulk elastic and thermal properties of crystals.
The adaptiveness of the TG to enable ultrafast temporal resolution for detection
will be discussed. We then demonstrate measurements of in-plane bulk elastic
and thermal properties of MoS2 using ultrafast TG. First, we begin with our sound
velocity measurements, which agrees with ab-initio calculations of longitudinal
acoustic phonon group velocity along Γ to 𝐾 Brillouine zone. The sound velocity
will be further compared with prior INS data to demonstrate the frequency range
that TG can measure. Next, we report measurements of the thermal conductivity
of MoS2 thin film in which we do not observe grating period dependence. The
reason will be further examined based on most recent ab-initio calculation and other
experimental results.
In chapter 4, we expand the experimental capability of TG to establish the microscopic picture of the heat transport in highly oriented semi-crystalline polyethylene
(PE). We first present results of the bulk thermal conductivity measurements and
infer the associated microscopic heat transport properties of partially oriented PE.
Next, we move to study highly oriented PE to measure mean free path of heatcarrying vibrations. We demonstrate the evidence of the micron-length scale mean
free path using our measurements along with a rigorous analysis using a Bayesian
inference. Further, we provide an evidence of one-dimensional propagation of
the heat-carrying vibrations using a low-energy anisotropic Debye model. Physical origin of our observation will be discussed under a careful consideration of
structure-transport relationships in highly oriented PE.
In chapter 5, we report first measurements of the damping of the heat-carrying
acoustic excitation in amorphous silicon (aSi) using picosecond acoustics (PSA)
and TG. In this chapter, we seek to elucidate the primary damping mechanism of the
acoustic excitation, therefore resolving the discrepancy between prior experimental
implications and numerical predictions in aSi. For this, we describe the measurements of the MFP of 100 GHz acoustic excitations. Then, we apply a novel approach
to obtain a frequency resolved MFP for frequency 0.1 − 3 THz in aSi using TG.
Finally, we discuss the presence and the origin of the weak damping in aSi, which
is in contradiction with the prior numerical predictions. The results in this chapter

provide a clear picture for understanding atomic vibrations in glasses quantitatively,
a long standing challenge for bulk amorphous materials.
Finally, in chapter 6, we summarize the results of this thesis and discuss an outlook
for future works.

Chapter 2

INTRODUCTION TO TRANSIENT GRATING SPECTROSCOPY
In the following discussion, we introduce a thermal characterization tool, transient
grating (TG) spectroscopy. The transient grating is based on optical holography
and was originally employed in other fields, for example for the study of ultrasonic
waves, among others [38]. Concerning the thermal transport, the TG was applied to
measure the bulk thermal diffusivity of polyamide thin-film [39]. The improvements
in the spatial and temporal resolution further expanded its capability toward MFP
spectroscopy [33, 37]. We start from the theoretical background and the principles.
2.1

Fundamentals

Thermal grating formation
Typically in TG, two different laser sources are employed: pulsed laser as a pump
and continuous wave laser as a probe, respectively. We start by describing pump
induced material response.
Figure 2.1 shows a schematic of the TG experiment. At phase mask, pump diffracts
from the sample, and a pair of first order diffracted beams are created. Following
Bragg’s law of the diffraction, the separation between two beams are given by
sin 𝜃 1 =

(2.1)

𝑑 𝑃𝑀

where 2𝜃 1 is the angle between two beams, 𝜆 is the optical wavelength, and 𝑑 𝑃𝑀 is
the phase mask period.
The first order diffracted pump beams from the phase mask are focused on the
sample, which can be expressed as
Epu,±1 = 𝜂 𝑃𝑀 𝐸 𝑝𝑢,𝑙𝑎𝑠𝑒𝑟 exp(𝑖k±1 · r − 𝑖𝜔𝑡)

(2.2)

k±1 = ∓𝑘 𝑥 𝑥ˆ + 𝑘 𝑧 𝑧ˆ

(2.3)

|k±1 | 2 = 𝑘 𝑥2 + 𝑘 2𝑧

(2.4)

where Epu±1 is the electric field of the pump beams, 𝜂 𝑃𝑀 is the optical diffraction
efficiency of the phase mask (typically 0.8 for wavelength ∼ 500 nm), k is the wave

10

Figure 2.1: Schematic illustration of the TG. (A) Pump and probe beams diffract
from the phase mask from which first order diffracted beams are created. The
diffracted pump beams pass through telescope lenses and are focused on the sample.
Since two beams are coherent, they interfere on the sample. The interference on the
sample is impulsively absorbed in the sample and create a spatially periodic thermal
gradient. One of the diffracted probe (signal beam) is optically phase adjusted and
diffracts from the grating profile on the sample; the other probe (reference beam)
is attenuated by ND filter and transmitted from the sample. Finally, first order
diffracted signal and the transmitted reference beams are fed into the detector, and
used for detecting pump-induced thermal response. (B) Detailed beam path for the
pump. As an example, measured optical interference of the diffracted pump beams
at phase mask period of ∼ 60 𝜇m is shown. Corresponding profile is a Gaussian
shape with a slow modulation with a period of 15 𝜇m.
vector of the light, and 𝜔 is the optical frequency. The interference of the pump
beams create an optical grating profile with an intensity (𝐼):
𝐼 𝑝𝑢 = |Epu,+1 + Epu,−1 | 2 = 2(𝜂 𝑃𝑀 |𝐸 𝑝𝑢,𝑙𝑎𝑠𝑒𝑟 |) 2 [1 + cos (2𝑘 𝑥 )]

(2.5)

As an example, the actual intensity of the optical interference at a phase mask
period 60 𝜇m is given in the inset of Fig. 2.1. The interference pattern was taken

11
using CCD camera beam profiler (BC106N-VIS, Thorlabs inc.). As in the inset of
Fig. 2.1, the profile is a slowly varying Gaussian profile as well as the modulation
due to |𝐸 𝑝𝑢,𝑙𝑎𝑠𝑒𝑟 | 2 in Eq. 2.5 [40].
The absorption of the optical interference generates the thermal transient grating on
the sample (Δ𝑇)

2𝜋
Δ𝑇 = 𝛼𝐼 𝑝𝑢 ∝ [1 + cos (𝑞𝑥)] = 1 + cos
(2.6)
where 𝛼 is the absorption coefficient, 𝑞(= 2𝑘 𝑥 ) is the grating wave vector of the
sample, and 𝐿 is the grating period on the sample defined by 𝐿 = 2𝜋/𝑞.
We can relate phase mask period (𝑑 𝑃𝑀 ) to 𝐿 using the geometry of the telescope
shown in Fig. 2.1. Consider the telescope lenses with arbitrary focal lengths being
𝑓1 and 𝑓2 ( 𝑓1 > 𝑓2 ). Considering the same numerical aperture of diffracted pump
beams entering the lenses, the distance of the pump beam from the optical axis can
be expressed as:

𝑓1 tan 𝜃 1 = 𝑓2 tan 𝜃 2

(2.7)

where 𝜃 2 is the laser incidence angle to the sample. Using Eq. 2.1, Eq. 2.6, and
Eq. 2.7, we can derive an expression for 𝐿

−1



𝐿 = (𝜆/2) sin tan

 −1
𝑓1
−1
tan sin
𝑓2
𝑑 𝑃𝑀

(2.8)

In the limit of 𝜆
𝑑 𝑃𝑀 , or 𝜃 1 , the expression above is simplified to

𝐿≈

𝑓2
𝑑 𝑃𝑀
2 𝑓1

(2.9)

The grating period on the sample (𝐿) can be confirmed using several methods.
First, the spacing of the burnt grating profile on a metal or dielectric film can be
measured. Second, oscillation frequency ( 𝑓 ) of the acoustic signal from the opaque
sample can be measured. Given the acoustic sound velocity (𝑣), the actual grating
period on the sample can be obtained 𝐿 = 𝑣/ 𝑓 . Third, CCD imaging of the optical
interference can be used; this method is limited by camera pixel resolution, therefore
less accurate than the other two methods especially at lower grating periods.
An example of the grating period confirmation on the de-ionized (DI) water is
illustrated in Fig. 2.2. The optical wavelength of the pump and probe used is 530

12
nm and 514 nm, respectively. To increase the absorption, a commercial food dye
with a red color was dissolved in DI water. The phase mask period was varied from
4 𝜇m to 43 𝜇m. As shown in Fig. 2.2, the frequency of oscillation on the order of
a few hundred MHz was observed, which decreases as the grating period increases.
Figure 2.2B shows measured frequency of the oscillation versus the grating period
on the sample, which agrees with the calculation from Eq. 2.8.

Figure 2.2: Grating period confirmation in TG. The data was taken from the Astrella
laser setup built in the Minnich lab at Caltech. (A) Representative signal traces of the
Acoustic wave obtained from deionized water with food color at several phase mask
periods. Both the sinusoidal curve fitting and the Fourier transform were performed
to obtain the frequency of the oscillation which decreases as the phase mask period
increases. (B) The oscillation frequency versus the grating period on sample. The
frequency was given at (A) whereas the grating period is calculated from Eq. 2.8.
For comparison, the expected acoustic frequency of water using 𝑓 = 𝑣/𝐿 is shown
as a dashed line. Excellent agreement was observed. Inset shows the corresponding
dispersion of the water given by the acoustic wave frequency versus the inverse
grating period.

2.2

Signal monitoring via heterodyne detection

In the preceding discussion, we described the formation of the transient grating from
interference of the pump beams. In TG, corresponding responses can be detected
using a phase sensitive detection [41]. This process is known to be the heterodyne
detection which allows to amplify the signal magnitude as well as to suppress the
unwanted noise components.
More precisely, two phase-controlled probe beams are used for detection, which are
generated from the phase mask. A phase adjustable glass window is placed at one
of the probe path (signal beam), while neutral density (ND) filter placed at the other
probe path (reference beam). Both beams are focused on the sample with Bragg

13
angle; from the sample, the signal beam diffracts and the reference beam transmits
under thin grating condition. The first order diffracted signal beam can be expressed
as

E𝑠𝑖𝑔 = 𝜂 𝑃𝑀 𝐸 𝑝𝑟 exp(𝑖kB · r − 𝑖𝜔𝑡 + 𝜙 𝑠𝑖𝑔 )

(2.10)

kB = ∓𝑘 𝐵 𝑥 𝑥ˆ + 𝑘 𝐵 𝑧 𝑧ˆ

(2.11)

where E𝑠𝑖𝑔 and 𝑛(𝑇, 𝑡) is the complex optical transfer function, 𝜙 𝑠𝑖𝑔 is the optical
phase of the signal, 𝑡 is ratio of the complex optical transfer function, and the kB is
the wave vector at the Bragg angle of incidence. The thermal grating induces the
spatial modulation of the complex refractive index such that [40]

𝑛 = 𝑛 + Δe
𝑛 cos (𝑞𝑠)

(2.12)

Δe
𝑛 = Δ𝑛(𝑇, 𝑡) + 𝑖Δ 𝑘 (𝑇, 𝑡)

(2.13)

where Δ𝑛 and Δ 𝑘 are changes in the real and imaginary parts of the refractive index,
respectively. The spatially modulated refractive index leads to a transfer function
under thin transmission grating condition [40, 42]

𝑡 = 𝑡0 [1 + (Δ 𝑘 (𝑇, 𝑡) + 𝑖Δ𝑛(𝑇, 𝑡))] cos (𝑞𝑥)

(2.14)

where 𝑡0 is the intrinsic sample transmission without the thermal grating. Multiplying Eq. 2.14 with Eq. 2.10, we obtain the expression for the first order diffracted
signal beam from the sample
E𝑠𝑖𝑔 = 𝑡0 (Δ 𝑘 (𝑇, 𝑡) + 𝑖Δ𝑛(𝑇, 𝑡))𝜂 𝑃𝑀 𝐸 𝑝𝑟 exp(𝑖kBs · r − 𝑖𝜔𝑡 + 𝑖𝜙 𝑠 )
kBs = 𝑘 𝐵 𝑥 𝑥ˆ + 𝑘 𝐵 𝑧 𝑧ˆ

(2.15)
(2.16)

where 𝜙 𝑠 accounts for the optical phase introduced both by tilting of the glass
window and the diffraction from the sample. Similarly, the transmitted reference
beam from the sample can be expressed as follows
E𝑟𝑒 𝑓 = 𝑡0 𝑡𝑟 𝜂 𝑃𝑀 𝐸 𝑝𝑟 exp(𝑖kBr · r − 𝑖𝜔𝑡 + 𝑖𝜙𝑟 )

(2.17)

kBr = −𝑘 𝐵 𝑥 𝑥ˆ + 𝑘 𝐵 𝑧 𝑧ˆ

(2.18)

14
where 𝑡𝑟 is the attenuation factor introduced by ND filter, and 𝜙𝑟 is the optical
phase introduced by ND filter and the transmission from the sample. Note that the
attenuation factor is determined by the optical density (OD) of the ND filter. We
examine the intensities of the beams at a fixed temperature. The intensity of the
static reference beam is given by
I𝑟𝑒 𝑓 = |𝑡𝑟 | 2 |𝑡0 | 2 |𝜂 𝑃𝑀 | 2 |𝐸 𝑝𝑟 | 2

(2.19)

Similarly, the intensity of the diffracted signal in the absence of the reference beam
is given by,
I𝑛𝑜𝑛
(2.20)
𝑠𝑖𝑔 = |𝑡 0 | |𝜂 𝑃𝑀 | |𝐸 𝑝𝑟 | Δ 𝑘 (𝑡) + Δ𝑛(𝑡)
The intensity indicated by Eq. 2.20 is called non-heterodyned signal which contains
both real and imaginary parts of the pump induced refractive index changes. Since,
impulsive pump excitation relaxes due to the thermal diffusion, the I𝑛𝑜𝑛
𝑠𝑖𝑔 eventually
approaches zero. As indicated by Eq. 2.17 and Eq. 2.15, the first order diffracted
signal beam and the transmitted reference beam are collinear. Corresponding interference can be expressed as the superposition of the electric field given in Eq. 2.17
and Eq. 2.15.

I𝑠𝑖𝑔 = |𝑡0 | 2 |𝜂 𝑃𝑀 | 2 |𝐸 𝑝𝑟 | 2 Δ𝑛(𝑡) 2 + Δ 𝑘 (𝑡) 2 + 𝑡𝑟2 + 2𝑡𝑟 [Δ 𝑘 cos 𝜙 + Δ𝑛 sin 𝜙]
(2.21)
where 𝜙 = 𝜙 𝑠 − 𝜙𝑟 is the phase difference between signal and the reference.
By removing original static reference intensity (homodyne contribution Eq. 2.19,
see Ref. [43]) as well as the original signal part in Eq. 2.21, one can obtain the
heterodyne part of the superposition
ℎ𝑑
= |𝑡 0 | 2 |𝜂 𝑃𝑀 | 2 |𝐸 𝑝𝑟 | 2 2𝑡𝑟 [Δ 𝑘 cos 𝜙 + Δ𝑛 sin 𝜙]
I𝑠𝑖𝑔

(2.22)

While both the heterodyne signal (Eq. 2.22) and the non-heterodyne signal (Eq. 2.20)
contain time-varying index of refraction changes, heterodyne signal can provide pure
real and imaginary parts of the refractive index by adjusting the optical phase. For
heterodyne signal, if 𝜙 = 𝜙 𝑠 − 𝜙𝑟 becomes the integer multiples of 𝜋 (𝜋/2), one
can measure the response directly proportional to Δ 𝑘 (𝑡) (Δ𝑛(𝑡)); amplitude grating
(phase grating). The phase grating is known to be contributed by temperature grating
and thermally induced stress whereas the amplitude grating contains the response

15
from the temperature grating only [33, 40, 44]. In prior TG experiments, the pure
phase grating was generally observed in dye-incorporated opaque sample including
water, m-xylene, and polyethylene. The pure amplitude grating signal was observed
in crystalline silicon. Another benefit of the heterodyne signal is the presence of
the attenuation factor (𝑡𝑟 ) in heterodyne signal. The attenuation factor substantially
increases the signal-to-noise ratio while not saturating the detector.
2.3

Thermal signal traces and fitting

In the preceding sections, we have discussed formation of the transient grating and its
detection using heterodyne scheme. In this section, we examine decaying dynamics
of the thermal signal. According to the Fourier’s law of heat diffusion, the heat flux
can be expressed as
J = −𝜅∇𝑇
(2.23)
Considering the arbitrary heat source (𝐽𝑔𝑒𝑛 ), the energy transfer over a constant
volume can be related to the internal energy using the first law of the thermodynamics
[22]
𝜕𝑇
(2.24)
[∇ · (𝜅∇𝑇) + 𝐽𝑔𝑒𝑛 ]𝑑𝑉 =
𝜌𝐶 𝑑𝑉
𝜕𝑡
𝜕𝑇
𝜌𝐶
= ∇(𝜅∇𝑇) + 𝐽𝑔𝑒𝑛
(2.25)
𝜕𝑡
where 𝐶 is the heat capacity, 𝐽𝑔𝑒𝑛 is the rate of the heat generation, and 𝜌 is the
density. Since all the experiments presented in this thesis are based on transmission
geometry of the TG, we limit our discussion only in x direction. We consider a
impulsive spatially periodic thermal gradient as a heat source. Corresponding 1-D
heat diffusion can be expressed as
𝜌𝐶

𝜕𝑇
𝑑 2𝑇
= 𝜅 2 + 𝑄 0 cos (𝑞𝑥)ℎ(𝑡)
𝜕𝑡
𝑑𝑥

(2.26)

where 𝑄 0 is the magnitude of the energy given by laser and ℎ(𝑡) is the heat intensity
distribution [45]. We solve the problem using a Fourier transform in space and time
with a solution given by
𝑇 = 𝐼0 exp [𝑖𝜔𝑡 − 𝑖𝑘𝑥]
(2.27)
𝜕𝑇
𝑑 2𝑇
] = 𝜅F [ 2 ] + 𝑄 0 F [cos (𝑞𝑥)ℎ(𝑡)]
(2.28)
𝜕𝑡
𝑑𝑥
Assuming that the temporal width of the laser is sufficiently small compared to
the time scale of the diffusion, the heat source term in Eq. 2.28 can be simplified
to ∼ 𝑄 0 . Considering the space and time dependent term only, Eq. 2.28 can be
simplified to
𝜌𝐶F [

16
F [𝑇 (𝑡, 𝑥)] =

𝑄0
𝑖𝜌𝐶 + 𝜅𝑞 2

(2.29)

Using thermal diffusivity defined by 𝛼 = 𝜅/(𝜌𝐶), the expression for temporally
decaying temperature is given by
2𝜋𝑄 0
𝑇 (𝑡, 𝑥) =
exp [−𝛼𝑞 2 𝑡] ∝ Δ𝑇 exp [−𝑡/𝜏]
(2.30)
𝜌𝐶
where Δ𝑇 is the transient temperature induced by the thermal grating at 𝑡 = 0,
and the 𝜏 = 1/𝛼𝑞 2 is the thermal decay time constant. From Eq. 2.30, we see
features concerning the thermal grating signal. First, the initial signal magnitude is
linearly proportional to the optical energy injected, which is related to the transient
temperature of the sample. Second, the signal decays with a time constant 𝜏 that
depends on both the thermal diffusivity of a material and the grating period.
2.4

TG arrangements

Having demonstrated the fundamental working principles, we now discuss appropriate arrangements and requirements of the TG.
Optical setup
The aforementioned description implies several requirements for TG. First, for probing heat-carrying acoustic vibrations, relevant thermal length scale (∼ 𝐿/2𝜋) should
be offered on the order of hundred nanometer, which will be changed depending on
the frequency of the vibration. Second, typical one-dimensional thermal decaying
dynamics, represented by Eq. 2.30, occurs at a nanosecond to microsecond scale,
meaning that corresponding frequency ranges for detection are on the order of subGHz. Third, the heterodyne detection scheme requires to adopt a phase sensitive
optics. These requirements, albeit stringent, lead to designing the TG as follows.
Figure 2.3 describes the arrangements of the TG setup as a schematic. The optical
power is adjusted using the half-wave plate and the polarizing beam splitter. For
the impulsive heating process, it is desirable having the same optical polarization
for both pump and probe [39], therefore vertical polarized lights for both beam
paths were used in this setup. A great advantage of the TG is the employment of
the phase mask. This makes the first order diffracted beams coherent, resulting in
interference. The resulting period of the interference on the sample position can
be variably tuned by a simple translation of the phase mask. The corresponding

17

Figure 2.3: Schematic of the typical real-time TG setup. The shown is the schematic
of the setup built in the Minnich lab. The setup is identical to what is described in
[37]. The pump pulses pass through multiple optics such as the polarizing optics for
intensity controller, telescopes for the precise beam controller at the sample position.
A CW probe is mechanically chopped to minimize the duty cycle for reducing steady
heating of the sample. After the sample, the heterodyne component of the signal is
delivered to the oscilloscope via the detector. The temporal resolution can be varied
by the choice of the detector depending on the response time. Similarly, spatial
length scale for the thermal gradient can be conveniently varied by translating a
phase mask.

length scale that can be changed is from > 500 nm nanometer to microns, which are
matched with phonon MFPs in many materials.
The second advantage is the employment of the CW laser as a probe and detection via
oscilloscope, providing a variable temporal resolution and time window. Typically in
conventional pump probe spectroscopy, the time window in which one can monitor
the signal is limited by the length of the delay stage. For instance, the maximum
time window is order of ≤ 50 ps for ultrafast pump-probe spectroscopy, or on the
order of ≤ 13 ns for time-domain thermo-reflectance, a time scale not sufficient for

18
decaying dynamics corresponding to Eq. 2.30. To overcome this challenge, in TG,
an oscilloscope rather than the delay stage is typically used to provide a longer time
window than conventional pump probe spectroscopy. In this setting, the temporal
resolution is now limited by the rise time of the detector on the order of 1 ns - 5 ns,
and further can be reduced to ∼ 250 ps by using a fiber-optic detector. Note that the
setup presented in Fig. 2.3 was used to perform the measurements in chapters 4 and
5.
Application to crystalline thin-films
In the preceding section, we discussed the optical configurations and associated
advantages for probing thermal transport using TG. We now present an application
to study the bulk crystalline thin film, a 1 𝜇m-thick crystalline silicon membrane.
The silicon membrane was provided by Prof. Clivia Stotomayor Torres’ group at
ICN2. The membrane was prepared similar to the method described in Ref. [33, 46].
To measure the membrane, we prepared the optical setup as follows. Briefly, a
Ti:sapphire laser (wavelength: 800 nm, repetition rate: 1 kHz, pulse-width: 25 fs,
Astrella, Coherent inc.) was used as a pump seed laser. The pump was then optical
parametric amplified to be converted to 532 nm (power incident on the sample: 5
𝜇J). A CW laser was chosen as probe laser (wavelength: 514 nm, average power:
∼ 5 mW, Genesis MX, Coherent inc.). The CW probe was chopped with ∼ 10%
of duty cycle. We varied the grating period from 1.3 𝜇m to 15 𝜇m. The 1 GHz
free space photodetector (1 GHz bandwidth, Newfocus 1601, Newport inc.) was
used except for 𝐿 = 1.3 𝜇m at which we used 4 GHz fiber-optic detector (4 GHz
bandwidth, Newfocus1591NF, Newport inc.).
Figure 2.4A shows measured signal traces at each heterodyne phase (𝜙). By subtracting the signal at 𝜙 = −𝜋 from 𝜙 = 𝜋, the resulting thermally induced signals
were obtained as shown in Fig. 2.4B. In Fig. 2.4B, we see two different components
of the decay. The signal decays with an initial negative magnitude within 1 ns, followed by slower decay with a time constant around 30 ns. It is established that the
origin of the faster decay is the ambipolar diffusion of the electron-hole pairs, while
the following slower decay is associated with the heat diffusion [33]. Therefore,
following Ref. [33], we fit the signal using bi-exponential model from which we
determined the thermal decay time constant. The corresponding thermal conductivity was calculated using the calculated heat capacity (quasiharmonic approximation
using LDA normconserving pseudopotentials, data provided by Dr. Navaneetha K.

19

Figure 2.4: Example of the TG for application to crystalline membrane. A representative of TG signal traces for crystalline silicon measured at 𝐿 = 8.3 𝜇m (A)
The signals at each heterodyne phase. (B) Resulting thermal transient grating signal
obtained from the subtraction of the signal traces in (A). We see a faster decay with
a negative magnitude followed by a slower decay with a positive magnitude. The
faster decay is due to the diffusion of the electron and hole pairs, while the slower
decay is due to the diffusion of the phonons. Corresponding bi-exponential fitting
is indicated as a dashed line.
Ravichandran). The resulting thermal conductivity versus the grating period for the
1 𝜇m crystalline silicon membrane is shown in Fig. 2.5A. The measured thermal
conductivity is ∼ 100 Wm−1 K−1 above 4 𝜇m, below which the value decreases.
The corresponding value of the thermal conductivity for a 1 𝜇m thick membrane
agrees with the literature data given in Fig. 2.5B. The estimated thermal conductivity
versus the grating period is presented as a dashed line in Fig. 2.5A. The estimation
was performed using 𝐹 (Λ) given in Ref. [35] and the thermal conductivity suppression function [34], and the Fuch-Sondheimer MFP reduction factor in the absence
of the specular scattering (𝑝 = 0) [47]. We see a reasonable agreement between the
data and the estimation, but slightly lower than the prediction, which may be due to
the thickness slightly lower than 1 𝜇m.
2.5

Summary

In this chapter, we demonstrated the TG as a tool to study heat conduction in thin
films. We have described the theoretical background along with the principles of
the TG, including the generation of the grating and corresponding detection. We
showed optical setup arrangements that are required for designing.
We have demonstrated an example of its application of TG to measure both bulk
thermal conductivity and grating dependent thermal conductivity. We have shown

20

Figure 2.5: (A) Thermal conductivity versus the grating period for 1 𝜇m thick
silicon membrane (measured: symbol). An estimation using the literature value
of the cumulative thermal conductivity versus MFP [35], and phonon suppression
function [34] is shown as a dashed line. A reasonable agreement is observed. (B)
Literature data of thermal conductivity versus the membrane thickness for crystalline
silicon [48]. Our measured thermal conductivity at 𝐿 = 15 𝜇m is also shown (∼ 100
Wm−1 K−1 ). Note that this value agrees with prior literature experimental data on
the observation of classical size effects and the predictions.

that the bulk value is close to the prior observations.
To conclude this chapter, the TG technique is appropriate for measuring the thermal
transport in many materials due to its adaptiveness. In the next chapter, we will
present an example of tailored TG to study elastic and thermal properties of a
crystalline material simultaneously. We seek to expand its capability of application
to semi-crystalline and amorphous materials to study the thermal transport properties
in them.

21
Chapter 3

IN-PLANE ELASTIC AND THERMAL PROPERTIES OF
MOLYBDENUM DISULFIDE
This chapter has been adapted, in part, from:

Taeyong Kim, Ding Ding, Jong-Hyuck Yim, Young-Dahl Jho, and Austin J. Minnich. Elastic and thermal properties of free-standing molybdenum disulfide membranes measured using ultrafast transient grating spectroscopy. APL Materials, 5
(8):086105, 2017. doi: 10.1063/1.4999225. URL https://doi.org/10.1063/
1.4999225.
T.K. prepared the experimental data, analyzed the results, and participated in the
writing of the manuscript.
In this chapter, we present in-plane elastic and thermal measurements by employing
TG spectroscopy. We begin by introducing and describing the development of the
ultrafast TG experiments. We then demonstrate its capability by applying the TG to
characterize the thermal and elastic properties in a bulk crystalline material, MoS2 .
As mentioned earlier, it is important to prepare an experimental probe that has relevant temporal length scale for the transport of the phonons. Concerning a crystal
that can support higher speed of sound, corresponding oscillation frequency at a
TG wave vector is order of GHz ranges, which is challenging to access using conventional arrangements in TG. Therefore, we develop an ultrafast transient grating
(UTG) technique to examine the in-plane bulk thermal and elastic of crystalline
solids. This technique is advantageous over the conventional TG in that this timeresolved scheme offers temporal resolution ∼ 100 fs, allowing us to simultaneously
probe phonon group velocity as well as fast decaying heat diffusion.
3.1

Ultrafast transient grating

The figure 3.1 shows the schematic of the ultrafast time-resolved transient grating
spectroscopy. The setup was built at Joint Center for Articifical Photosynthesis
(JCAP) building at Caltech. We used a regeneratively amplified Ti:sapphire laser
(Libra, Coherent) that generated 100 fs pulses at 10 kHz with an energy of 0.4
mJ. The 1/𝑒 2 diameter of the laser output is ∼ 8 mm. The pulse train, which was

22
centered at 800 nm, was split into pump and probe beams using a half-wave plate
and polarizing beam splitter.
To spectrally separate pump and probe pulses, a two-tint optical filtering scheme
was used: an optical filter was placed along the pump path (single-band band pass
filter at 786 nm, Semrock Inc.), and along the probe path and before the detector
for suppressing undesirable pump scattering into the detector (RazerEdge long pass
filter at 808 nm, Semrock Inc.) [49].
The pump beam was mechanically chopped at 2.5 kHz with a chopper wheel and
focused onto a translatable optical phase mask to yield two excitation pulses. The
excitation pulses were focused onto a spot with a diameter of 550 𝜇m; their interference generated a periodic heating profile that resulted in thermoelastic standing LA
waves and a spatially periodic thermal gradient. The probe beam was directed to a
mechanical delay stage (ACT115DL, Aerotech) with a silver coated retro-reflector.
The delay stage was arranged to perform a double-pass reflection, providing approximately 4 m of spatial delay corresponding to around 13 ns in time delay. The probe
pulses were then directed to the optical phase mask, yielding signal and reference
beams. The signal beam passed through a glass window to provide optical phase
adjustment and was focused onto the sample; the reference beam was attenuated by
a neutral density filter (ND 3) and focused onto the sample. The diameter of the
focused probe beam was 400 𝜇m. After transmitting through the sample, the overlap of the transmitted reference and first-order diffracted signal beam were directed
to the detector (BPW34, Vishay Inc.). This signal was then balanced by another
photodiode measuring a reference beam and the subtracted signal fed into lock-in
amplifier (SR 830 of Stanford Research Systems). The relative heterodyned phase
difference between the reference beam and the probe beam was controlled by motor
controller (DC Stepper Motorized Actuator, Thorlabs).
Setup modification and improvements
Here, we describe our special efforts that were put into this setup. First, the
improvement in the setup was made in the power adjustment. As mentioned above,
the optical energy from the Libra laser is 0.4 mJ. Such high optical energy can
damage the optical elements. In addition, in TG, the optical energy required for
obtaining adequate signal magnitude (∼ 1 mV) is typically less than less than 20 𝜇J;
we therefore should attenuate the optical energy. One approach is to use the ND filter,
and the other approach is to use polarizing optics. The metallic ND filters are widely

23

Figure 3.1: Schematic of the ultrafast TG in Joint Center for Artificial Photosynthesis
Laboratory at Caltech. The output of the regeneratively amplified Ti:sapphire laser
source with a repetition rate of 10 KHz is split into pump and probe using a halfwave plate and polarizing beam splitter. A two-tint optical coloring using optical
filters was employed to spectrally separate the pump and probe by around 20 nm in
order to suppress pump scattering into the detector. The pump was mechanically
chopped at 2.5 KHz and its first order diffracted beams are focused on the phase
mask (spot diameter 550 𝜇m) and interfere on the sample. The interference creates
a thermoelastic standing LA waves and a spatially periodic thermal gradient. The
probe beam travels an additional ∼ 4 m corresponding to delay time around 13
ns, and are directed to the sample. The reference beam and the signal beams are
separated from phase mask and the relative optical heterodyne phase is controlled
using ND filter and glass window. The glass window is tilted using motor controller
(DC Stepper Motorized Actuator, Thorlabs). The first order diffracted probe beam
and the transmitted reference beams are sent to the fast Si photodiode; the signal is
balanced by another photodiode and fed into the lock-in amplifier.

used in laser experiments because optical power can be attenuated using the light
reflection without changing laser property such as beam collimation. Originally, the
reflective ND filter was used to attenuate the laser beam. Unfortunately, unwanted
non-linear phenomena were observed due to the thermal lensing effect. The thermal
lensing effect has been reported primarily in continuous-wave laser system [50] or

24
high power lasers. Briefly, the laser-heating causes the first-order effect (thermal
gradient) in addition to the second-order effect (thermally induced mechanical stress)
in the medium, eventually changing the refractive index. The first order effect is
known to be the primary source of the focusing, while the second order effect is
attributed to be the source of the aberration. Figure. 3.2 shows several profiles of the
laser beam, propagating after the ND filter. First, We find that the ND filter changes
the undesirable collimation property; the filter focused the beam with a focal length
of around 5 m. We estimated the focal length from thermal lensing effect in our
laser system using
𝑓 −1 =

𝑑𝑛/𝑑𝑇
2𝜅𝑟

(3.1)

where the 𝑑𝑛/𝑑𝑇 is the thermo-optic coefficient, 𝜅 is the thermal conductivity, 𝑟
is the beam radius, 𝑃 is the laser power. Using the property of the N-BK7 glass
(𝑑𝑛/𝑑𝑇 = 7.1 × 10−6 K−1 , 𝜅 = 1.1Wm−1 K−1 ), and the r = 4.4 mm and 3.8 W, we
obtained the focal length of thermal lens 5 m, which agrees with our observation.
Not only the beam focusing, the astigmatism was observed.; initially the beam was
elongated to x-direction, eventually to y-direction due to different focal point as in
Fig. 3.2A-E. The reason for this is that the focal length of the ND filter caused by
the second order effect is different depending on ray bundles in different planes; the
meridional rays (along the y-direction) focuses earlier than that along the sagittal
(along the x-direction) plane as indicated in Fig. 3.2G. We solved this problem
by using half-wave plate and polarizing beam splitter. Furthermore, the pump
and probe power was fine-tuned using slight rotation of the two-tint optical filters
mentioned in previous section.
Second, we use an open loop beam stabilization system to improve the beam stability.
Typically, lasers have spatial fluctuation which are examined as an angular value,
or laser pointing stability. The pointing stability is order of micro to milliradian,
which can influence the pointing accuracy at a longer distance from the laser. As
indicated in Fig. 3.3, is around 3 m; the measured laser position drift near the half
wave plate was around 400 - 800 𝜇m with a frequency of a few Hz. We improved the
beam stability using open-loop piezo controller system (position sensor: PDP90A,
Thorlabs inc.; piezo driver: KPZ 101, Thorlabs inc.; 2 channel piezo adjuster:
Polaris-K1PZ2, Thorlabs inc.).

25

Figure 3.2: Astigmatism caused by the thermal lensing effect. (A - E) The laser
beam profiles at several locations after the ND filters are shown. The images were
taken by CCD camera. Due to the thermal lensing effect laser, the ND filter focuses
the beam with a focal point ∼ 5 m. Also, due to the astigmatism, the beam is initially
elongated to horizontal direction near z= 4995 mm after which is elongated to the
vertical position. Note that the Fig. 3.2A is the profile at 2 inch iris, Fig. 3.2B,C the
profiles at 1 inch iris, and the Fig. 3.2C,D are the profiles at CCD aperture without
lens. (F) A schematic describing the thermal lensing effect. Due to the secondary
effect of the thermal lensing, the different amount of the refractive index changes
are induced depending on the optical axis, therefore making uneven projection of
the light on xy plane.

Lens configuration to control the beam size
Not only adjusting the optical energy as discussed in Sec.3.1, but also controlling
the beam size are significantly important. Unnecessarily high optical energy in unit
area (energy density) would make corresponding thermal response significantly
deviate from the operating temperature, or eventually damage the sample. On the
other hand, a certain amount of the energy density should be provided in order to
obtain adequate signal magnitude without a self-heating of the sample. As shown in
Fig. 3.1, we have used beam compressor to make the beam diameter on the sample
550 𝜇m (400 𝜇m) for pump (probe). Since we use the TG telescope lenses with
𝑓1 = 150 mm, 𝑓2 = 75 mm, the spot diameter at the phase mask position corresponds
to 1100 𝜇m (800 𝜇m) for pump (probe). In our designing of the beam compressor,
we considered the following requirements. First, Galilean telescope rather than the

26

Figure 3.3: (A) Schematic illustration of the open-loop beam stabilization system.
The beam stability was improved using beam position sensor, piezoelectric controller, and the piezoelectric mirror. (B) Actual picture of the beam stabilization
system. The corresponding components are labeled.

Newtonian telescope was considered to avoid non-linear optical effect at the focal
point past the first lens. Second, addition of third lens with a larger depth of focus
was considered. According to Gaussian beam optics, the larger depth of focus
provides longer Rayleigh length, thus corresponding beam sizes are less likely to
diverge at longer distance. Also, the loose-focusing by the third lens significantly
reduces the risk of the non-linear optical effect compared to the tight-focusing.
The 3.4 shows our our calculated Gaussian beam propagation. The calculation
was performed using Matlab, considering lens maker’s formula and the Gaussian
beam optics. Calculated beam diameter at phase mask position is around ∼ 1100
𝜇m, ∼ 800, for pump and probe, respectively. Finally, actual beam size at sample
position was characterized by knife-edge measurement as shown in Fig. 3.5. The
knife-edge measurements confirm 1/𝑒 2 diameter to be ∼ 550 𝜇m, ∼ 400 𝜇m for
pump, and probe, respectively.
Grating period calibration using TG
Despite a special care that may be taken, the actual projected grating on the sample
can be slightly off from the Eq. 2.8. This can be caused by the fact that the lens
have finite thickness, thus the back focal length and the quoted focal length is
different. Another factor is the difference between the lens thickness at the center
and at the edge; the beam overlap position can be different from the nominal focal
point, causing actually projected grating spacing on the sample deviate from Eq. 2.8.

27

Figure 3.4: Calculated Gaussian beam profile of (A) pump (B) probe. The blue
solid lines indicate the location of the optics in which 𝑓 is the focal length of the
lenses, and PM is the phase mask location. For pump, the phase mask was designed
to be at the beam waist position, indicated by red dashed line in (A).

Therefore, the grating period was calibrated by verifying the acoustic velocity of
sound waves in water. Since the water is transparent, we prepare a green dye (weight
≤ 0.4 mg, Epolight 2735, Epolin inc.) The dye was dissolved into ∼ 100 mL of
DI water. To increase the optical absorption,the solution was diluted by 24 times.
Figure 3.6A shows the surface acoustic waves from water at several phase mask
period. The optical power that was used is 24 mW (30 mW) for pump (probe).
The signal was obtained by ∼ 20 averages. The frequency of the oscillation was

28

Figure 3.5: Beam size characterization at sample position for (A) pump (B) probe.
Symbols represent our measured values. The solid line is corresponding erfc fitting,
from which we reconstruct beam profiles shown as dashed lines. Corresponding
1/𝑒 2 diameter is ∼ 550 𝜇m, ∼ 400 𝜇m for pump, and probe, respectively.

obtained by performing the Fourier transform; the frequency was further confirmed
by sinusoidal curve fitting. The resulting acoustic frequency versus grating period
at sample position is shown in Fig. 3.6B. Given the oscillation frequency ( 𝑓 ), the
dashed line indicates the expected grating period using 𝑓 = 𝑣 𝑤 /𝐿 where 𝑣 𝑤 is
sound velocity in water (∼ 1500 𝑚𝑠−1 [51]). Compared to the expected grating
period using 2.8 (calculation in Fig. 3.6A), the actual grating period (calibration in
Fig. 3.6B) was observed to be bigger by factor of 30% at all grating period. This
is mainly caused by combination effect of the non-idealities in the phase mask and
the optics that were used. The quoted phase mask period may be different from
the actual period. Also, the lens parameter of the achromatic doublet lens as a
TG telescope lenses (AC508-150-B and AC508-075-B, Thorlabs Inc.) could be
different from quoted values due to the thickness of the lens (order of 15 mm),
yielding additional uncertainties.
3.2

Material: Molybdenum disulfide

Background
We thus far discussed the designing of the thermal characterization system. In this
section, we discuss the prior knowledges of the thermal transport in a crystalline
material, MoS2 .
Transition-metal dichalcogenides (TMDCs) are of considerable interest due to their
unique layer-thickness-dependent electronic properties and robust thermal stability.
[52–56] In particular, molybdenum disulfide (MoS2 ), a member of the TMDC fam-

29

Figure 3.6: Grating period calibration of the ultrafast TG setup. (A) Normalized
Acoustic signal obtained from deionized water with dye at several phase mask period.
We obtain the frequency of the oscillation by performing both the Fourier transform
and sinusoidal curve fitting. (B) The oscillation frequency versus the grating period.
The frequency was obtained from (A). The calculated grating period using Eq. 2.8 is
represented by blue triangles. The grating period was calibrated (red circles) using
sound velocity of DI water (∼ 1500 m𝑠−1 , dashed line). The difference between the
calculation and the actual grating period is observed to be 30% due to non-ideality
in the optics mentioned in the text. Inset shows the frequency versus the inverse
grating period, indicating the acoustic dispersion of the water.

ily, has been extensively investigated due to its potential applications in electronics,
photovoltaics, and thermoelectrics. [57–60] In addition to its intriguing electronic
properties such as high electron mobility (up to 200 cm2 V−1 s−1 ) and indirect-todirect transition of bandgap with reduced dimensionality [57, 60], MoS2 has further
revealed promising thermoelectric features such as high Seebeck coefficient [61].
Thermal transport in MoS2 is therefore of intense interest. However, theoretical
and experimental studies report a broad range of in-plane thermal conductivity values of MoS2 . For example, calculations of the in-plane thermal conductivity 𝜅 of
monolayer MoS2 vary from ∼ 20 to 155 Wm−1 K−1 . [61–69] Various experimental techniques, including Raman spectroscopy, thermal bridge, and time-resolved
magneto-optical Kerr effect (TR-MOKE), have been applied to measure the 𝜅 of
monolayer, few layer, and bulk MoS2 . The reported values vary widely; from
30.5 - 101 Wm−1 K−1 for monolayers [70–72], 52 - 102 Wm−1 K−1 for few layers
[70, 73, 74], and 85 - 110 Wm−1 K−1 for bulk [75].
At the same time, the elastic and acoustic properties of MoS2 have also been studied,
motivated by applications such as flexible electronics. [76–80] In particular, the
cross-plane longitudinal sound velocity of MoS2 has recently been characterized
by a transient reflection method. [81] However, although ab-initio calculations of

30
in-plane sound velocity have been reported, to the best of our knowledge the value
has not been experimentally confirmed[68, 77]. From the next section, we present
our results on the bulk characterization of the Mo𝑆2 .
Sample preparation and characterization
All the measurements presented in this chapter were performed on free-standing
membrane. The samples were made by our collaborator, Jho research group at
Gwangju Institute of Science and Technology (GIST) in South Korea. We prepared
suspended MoS2 membranes by mechanical exfoliation of bulk crystal MoS2 (2D
Semiconductor Inc.). Silicon-on-insulator (SOI) wafers were wet-etched to create
rectangular apertures of dimension about 1 - 2 mm at the device side of the wafer.
Both SOI wafers and bulk MoS2 were separately cleaned in an ultrasonic bath of
acetone for 5 minutes, then in an ultrasonic bath of methanol for 5 minutes and finally
in an ultrasonic bath of deionized (DI) water for 5 minutes. Then, both wafers and
bulk MoS2 were left in a nitrogen gas flow and annealed at 110 ◦𝐶 during 3 minutes
to dry the samples. Multilayer MoS2 flakes were then cleaved from the bulk MoS2
by repeatedly exfoliating with scotch tape. To fix the MoS2 flake to the SOI wafer,
UV curable epoxy was applied on each corner of the rectangular apertures. The
flakes were then placed over the aperture in the substrate and were left under UV
illumination for 1 minute to cure the epoxy.
The membranes were characterized using optical microscopy, and energy dispersive x-ray spectroscopy (EDXS) and transmission electron microscope (TEM). Figure 3.8A shows a confocal microscope image of a sample prepared with this method.
The image shows that the membrane is intact with some wrinkles over the clear
aperture area. The thickness of the membrane was determined by measuring the
transmitted optical power through the membrane. For an incident power of 3 mW
at 800 nm, 765 𝜇W was transmitted. Using the Beer-Lambert law and the known
dielectric function of MoS2 [82–84], we estimate the thickness as 5 𝜇m. The elemental composition along with their spatial distributions were characterized by the
Energy dispersive x-ray spectroscopy (EDXS) elemental mapping as in Fig. 3.8B.
Spatially uniform distribution of the Molybdenum and the sulfide was observed. We
also used high resolution TEM to verify the single-crystalline nature of the sample
along with selected area electron diffraction of [001] zone axis to verify the lattice
constants, as indicated in Fig. 3.8C. We obtained the magnitude of reciprocal lattice
vector as 3.7 nm−1 , which agrees well with the hexagonal symmetry with lattice
spacing of 0.27 nm along (100) plane as in Fig. 3.8B [85, 86].

31

Figure 3.7: Illustration of the preparation of the thin MoS2 membranes. First,
silicon-on-insulator wafers were wet-etched to create rectangular apertures of dimension 1 - 2 mm. Multi-layer MoS2 flakes were cleaved from the bulk single
crystal by repeated mechanical exfoliation using scotch tape. Then UV curable
epoxy was applied near the rectangular apertures and the MoS2 was fixed on top.
Finally, the device was left under UV illumination to cure the epoxy. Photo credit:
Jho group at GIST

3.3

TG experiments on MoS2

With the Mo𝑆2 prepared, we performed the TG measurement to obtain in-plane
elastic and thermal properties. We conducted measurements with transient grating
periods (𝐿) varying from 1.89 𝜇m to 7.15 𝜇m at the sample surface. Following
the procedures described in preceding sections, experimental data at distinct phase
differences between probe and signal beams 𝜙 𝐻 = 0 and 𝜙 𝐻 = 𝜋 are subtracted
to isolate the thermally induced heterodyned signal. The pump and probe beam
powers were set to 15 mW and 6 mW at the sample, respectively. The corresponding
steady-state and transient temperature rises on the sample are around 6 K and 0.6
K, respectively.
Figure 3.9 shows two representative heterodyned signals at different transient grating
periods. For each measurement, the signals at 𝜙 𝐻 = 0 and 𝜙 𝐻 = 𝜋 are clearly
superimposable by flipping the sign of one measurement, as expected. The signals
at 𝜙 𝐻 = 0 and 𝜙 𝐻 = 𝜋 were then subtracted to yield the final signal. As in Figs. 3.9

32

Figure 3.8: (A) Confocal optical microscope images of a MoS2 membrane suspended
over a rectangular hole in an SOI wafer. The dimensions of the free-standing area
are about 1 × 2 mm2 . The dashed line highlights the area where the membrane is
suspended. (B) Energy dispersive x-ray spectroscopy (EDXS) elemental mapping of
the MoS2 membrane. The purple and the green indicate the Molybdenum (Mo) and
the Sulfide (S), respectively. The EDXS mapping indicates the uniform distribution
of M and S in the sample. (C) High resolution transmission electron microscopy
(HRTEM) image of MoS2 indicating single-crystalline structure and revealing the
hexagonal lattice. The inset of (C) displays the corresponding selected-area electron
diffraction (SAED) pattern along the [001] zone axis. The lattice spacing was
identified to be 0.27 nm which is well-matched with the reciprocal lattice distance
3.7 nm−1 obtained from SAED.

Figure 3.9: Signals measured at grating period of 1.89 𝜇m (A, B, C), and 3.95
𝜇m (D, E, F). At each grating period, the signals with optical heterodyne phase of
𝜙 𝐻 = 0 (A, D) and 𝜙 𝐻 = 𝜋 (B, E) are subtracted to isolate thermally induced signal
(C, F).

33
C and F, we observe an initial rise at 0 ns that quickly relaxes away within 2 ns
followed by another slower decay with the opposite amplitude sign. We attribute the
first relaxation to ambipolar diffusion of photo-generated electrons and holes and
the second decay to the thermal and acoustic response of the sample. The thermal
decay becomes slower as the grating period increases, as expected. Considering
the acoustic response, we added additional decay term to fit these experimental data
using the formula:
− 𝑡

− 𝑡

𝑓 (𝑡) = 𝐴𝑒 𝜏𝑇 + 𝐵𝑒 𝜏𝑆 𝑐𝑜𝑠(2𝜋𝜈𝑡 + 𝜙)

(3.2)

where the first (second) term represents the thermal (acoustic) signal with amplitude
𝐴 (𝐵), 𝜏𝑇 and 𝜏𝑆 are the decay time constants for the thermal and acoustic signals,
𝜈 is the frequency of the oscillations, and 𝜙 is the phase of the oscillation [87]. The
frequency of the oscillations 𝜈 was first extracted using a Fourier transform (inset
of Figs. 3.10A and B and substituted into Eq. 3.2. The fitting is performed after
the electronic relaxation ends, corresponding to a thermalization time of around 2
ns but which increases with the grating period [87]. We obtain the other fitting
parameters using least square fitting with Levenberg-Marquardt algorithm.
The results of the fitting for two grating periods are shown in Figs. 3.10A and B. We

Figure 3.10: Signal obtained by the subtraction of heterodyne phase at 𝜙 𝐻 = 0 and
𝜙 𝐻 = 𝜋 versus delay time and the model fit at transient grating period 𝐿 = (A) 1.89
𝜇𝑚 and (B) 3.95 𝜇𝑚. Inset: fast Fourier transform of the data showing acoustic
oscillation at each grating period. (C) Uncertainty in fitting curve with respect to the
variation of thermal decay rate from B. Purple (green) line describes lower (upper)
bound used to determine the uncertainties. The uncertainty is identified to be on the
order of ± 15%.

34
observe good agreement between the model and the data. Figure 3.10C visualizes
the sensitivity of the solutions of fitted curves to the fitted thermal decay rate. We
obtain error bounds on the thermal decay rate by determining by what percentage it
can change while still overlapping the data. We find that the noise in the experimental
measurements yields fitting uncertainties on the order of 15%.
Elastic property of MoS2
The measured frequency of oscillation can be related to the elastic properties of
the MoS2 . We examine the measured acoustic frequency versus grating period,
as shown in Fig.3.11A. The acoustic frequency exhibits a linear trend with inverse
grating period, as expected. A linear fit of these data yields an in-plane sound
velocity of 7000 ± 40 m s−1 and corresponding elastic constant of 248 GPa. The
measured elastic constant agrees well with prior theoretical and experimental reports
which range from 211 - 240 GPa [80]. Since the membrane is suspended, the
corresponding sound velocity is not expected to approach the speed of Rayleigh
mode of the substrate as is the case for a supported membrane. In a free-standing
medium, it has been already established that this Lamb wave mode mainly involves
intrinsic longitudinal displacements [88]. We note that the Lamb mode originates
from thermally induced generation of coherent longitudinal acoustic phonons [33,
44, 88]. Our measurement is reasonably close to prior ab-initio calculations of an
in-plane longitudinal acoustic sound velocity of 6500 - 6700 m s−1 [68, 77, 89].
To illustrate this, the literature data of the ab-initio calculation of the in-plane
phonon dispersion in Ref. [68] is given in Fig. 3.11B. As shown in Fig. 3.11C,
the calculated dispersion along Γ − 𝐾 in the Brillouin zone excellently agrees
with our obtained data. Figure 3.11D shows the calculated dispersion in Ref. [89]
and corresponding INs measurements in which we can see quantitative difference
compared to our measurements. The INS can map out the Brillouin zone over
the higher frequency range. However, since the scattering intensity substantially
decrease at low frequencies [90], it is challenging to obtain the properties of subTHz vibration such as phonon group velocity near Γ Brillouin zone.
Thermal property of MoS2
We now examine the thermal properties. Figure 3.12A shows the thermal decay rate
versus the square of wave vector. As shown in Eq. 2.30, assuming one-dimensional

35

Figure 3.11: (A) Measured acoustic frequency versus inverse grating period. The
sound velocity is calculated to be 7000 ± 40 m s−1 , and the corresponding elastic
constant is 248 GPa. (B) Literature data of ab-initio calculation of the in-plane
phonon dispersion in Ref. [68]. The direction of the wave vector is illustrated as
an inset. (C) Calculated phonon dispersion along Γ − 𝐾 in the Brillouin zone
in Ref. [68] (black dashed line) and our measured acoustic frequency (symbols).
a good agreement is observed. (D) Prior experimental observation of the phonon
dispersion in MoS2 using INS in Ref. [90] (symbols) along with ab-initio calculation
in Ref. [89]. The INS successfully map out the dispersion above 1 THz. However,
scattering intensity decrease at low frequencies limits to study the dispersion at
low frequencies using INS, therefore prevent from deducing the sound velocity as
discussed in Ref. [90].
heat diffusion, this trend should be linear following the relation:
 2
2𝜋
=𝛼
= 𝛼𝑞 2

(3.3)

where 𝜏 is the average thermal decay time, 𝛼 is the in-plane thermal diffusivity, and
𝐿 is the transient grating period. This is a reasonable assumption since the optical
penetration depth of MoS2 at 800 nm (12.5 𝜇m) is longer than the thickness of the
sample (5 𝜇m) [33][82].

36

Figure 3.12: (A) Thermal decay rate, or inverse thermal decay time constant, versus
wave vector squared. The slope of dashed line yields the thermal diffusivity value
from a linear least square fit. Purple (yellow) dashed line denotes the upper (lower)
bound of thermal diffusivity. The value of measured thermal conductivity is 74 ± 21
Wm−1 K−1 . (B) Calculation of ab-initio thermal conductivity accumulation versus
phonon mean free path of MoS2 . The phonon mean free path of MoS2 is spanning
from a few nm to ∼ 2 𝜇m. The black dashed lines indicate thermal length scale
corresponding to the minimum and the maximum grating period, respectively. The
thermal length scale can be approximated as 𝐿/2𝜋. The relevant thermal length
scale is ∼ 0.3 − 1.1 𝜇m for our measurements which corresponds to ∼ 78 − 98% of
the accumulated thermal conductivity.

In Fig. 3.12A, we show a linear least square fit from which we obtain the thermal
diffusivity. Using a heat capacity from Ref. [91], we calculate an in-plane thermal
conductivity value of 74 ± 21 Wm−1 K−1 . Our experimental result of thermal conductivity value is slightly lower than recent experimental measurement in Liu et al.
[75]; which obtained a value of 85 ± 6 Wm−1 K−1 using TR-MOKE. The discrepancy
may be due to the differences in sample quality; TG probes a larger thermal length
scale than does TR-MOKE and thus is sensitive to defects. Additionally, defects may
have been introduced in our sample preparation procedure. Nevertheless, our value
is in reasonable agreement with this prior report [75]. We also tentatively conclude
that phonon mean free paths are smaller than the grating periods considered here as
the linear trend observed in Fig. 3.12A is characteristic of phonon diffusion [66, 75].
Our tentative conclusion can be examined by ab-initio calculation of the bulk thermal
transport in MoS2 . The Figure 3.12B shows ab-initio predicted thermal conductivity
versus the phonon mean free path in MoS2 . Briefly, the calculation was performed
using interatomic force constants from density functional theory as well as solving
the Boltzmann transport equation. The detailed description of the method can be
found in Ref. [92]. As shown in Fig. 3.12B, we see that the the mean free path (MFP)

37
are spanning from a few nm scale to ∼ 2 𝜇m scale. Based on the calculation, we can
quantitatively estimate the expected accumulated thermal conductivity. Roughly,
we estimate the relevant thermal length scale that are comparable to MFP is ∼ 𝐿/2𝜋,
which corresponds to the MFP ∼ 0.3 − 1.1 𝜇m. From the calculation of the thermal
conductivity accumulation, we find that our value of measured thermal conductivity
is 98% at 𝐿 = 7.15 𝜇m, which should decrease to 78 % at 𝐿 = 1.89 𝜇m. However,
as mentioned above, the diffuse boundary scattering of the phonons with MFP can
suppress the their contribution to the thermal conductivity, thus reducing their total
contribution of the thermal conductivity. Our discussion should be re-examined
with measurements of smaller uncertainty than those presented here. Nevertheless,
our experimental observations are well supported by some of previous theoretical
predictions on the thermal conductivity of bulk MoS2 (83 Wm−1 K−1 ) using firstprinciples calculations [68].
Finally, we note that the error bounds on the thermal conductivity obtained here,
on the order of ± 15%, are larger than typically obtained from TG. We attribute
this uncertainty to several factors. First, the laser employed for this work exhibited
substantial intensity noise that affected the signal to noise ratio despite the use of
balanced detection. Second, wrinkles in the sample may have scattered light and
prevented it from reaching the detector. Third, the finite length of delay stage means
that data can only be collected to time delays of 13 ns, leading to reduced sensitivity
to heat diffusion in the observed signal, particularly for longer grating periods. These
effects can be mitigated in the future by improving sample preparation procedures
or using chemical vapor deposition for sample fabrication, using a more stable laser
source, and performing the real-time TG experiment that employs a continuouswave (CW) probe and a fast oscilloscope rather than a lock-in amplifier. These
improvements will be the subject of future work and will lead to substantially
reduced uncertainty.
3.4

Summary

To conclude this chapter, we introduced a tailored TG to study the elastic and thermal
properties of the crystalline TMDC thin films. Before this work, the TG application
was limited mostly to characterize the thermal transport in crystalline silicon, which
we expanded the range of the scope.
Concerning the modification, we have built time-resolved TG to provide a temporal
resolution ∼ 100 fs to simultaneously measure phonon group velocities and the in-

38
plane thermal conductivity. Careful designing of the optics was discussed including
the choice of the polarizing optics as a optical energy controller, piezo-controlled
beam stabilizer, and the design of the telescope for beam size controller.
We have applied the TG to Mo𝑆2 to study the elastic and thermal properties in it.
More precisely, we obtain a in-plane 𝑣 𝐿 𝐴 of 7000 ± 40 m s−1 and a corresponding
c11 elastic constant of 248 GPa, which was not experimentally reported until our
discovery. We have reported measured in-plane thermal conductivity of 74 ± 21
Wm−1 K−1 . Finally, our thermal conductivity results and associated microscopic
heat carrying properties were examined by comparing with more recent ab-initio
calculation and experiments.
We expect that these findings provide insight into in-plane thermal and elastic
transport in MoS2 and set the stage for further studies of other crystalline thin films
such as TMDCs using TG. In the next chapter, we further expand the range of TG
applications to semi-crystalline materials.

39
Chapter 4

HEAT-CARRYING PHONONS WITH MICRON-SCALE MEAN
FREE PATHS IN HIGHLY ORIENTED POLYETHYLENE
This chapter has been adapted, in part, from:

Taeyong Kim, Andrew B. Robbins, Stavros X. Drakopoulos, Ignacio MartinFabiani, Sara Ronca, and Austin J. Minnich. Heat-carrying phonons with micronscale mean free paths in highly oriented polyethylene (in preparation).
T.K. participated in the conception of the project, prepared the experimental data,
analyzed the results, and participated in the writing of the manuscript, 2020.
Stavros X. Drakopoulos, Georgios C. Psarras, Aurora Nogales, Tiberio A. Ezquerra, Taeyong Kim, Andrew B Robbins, Austin J. Minnich, Georgia C. Manika,
Gianfranco Claudio, Ignacio Martin-Fabiani, and Sara Ronca. Gold/ultra high
molecular weight polyethylene nanocomposites for electrical energy storage: Enhanced recovery efficiency and thermal conductivity upon uniaxial deformation
(in review).
T.K. prepared the thermal characterization data, analyzed the thermal characterization results, and participated in the writing of the manuscript, 2020.
In the preceding chapter, we discussed TG’s ability to measure bulk thermal properties of the lattice, beyond conventional applications. As earlier mentioned, there
were a few reports concerning the measurements of the bulk thermal properties
of polymers. In this chapter, we apply the TG spectroscopy to study microscopic
heat-carrying properties in highly oriented semi-crystalline polyethylene (PE) films.
First, we will present our measurements of partially oriented PEs; a qualitative
analysis on the microscopic property based on the combination of the bulk thermal
and structural characterization will follow. Next, we resolve the MFPs of heatcarrying acoustic phonons in highly oriented PE to reveal the origin of the thermally
conductive polymers.
4.1

Background

Thermally conductive polymers are of interest both for fundamental materials science as well as applications such as thermal management [4, 93]. Despite the low
thermal conductivity ∼ 0.1 - 0.6 Wm−1 K−1 of unoriented polymers [4], early works

40
reported orders of magnitude increase in uniaxial thermal conductivity along the
alignment direction of oriented samples, including for polyethylene (PE) [8, 94],
polyacetylene [5], and polypropelene [6]. In particular, the reported thermal conductivity of oriented polyethylene ranged from ∼ 14 Wm−1 K−1 [95] for draw ratio
𝐷 𝑅 = 25 up to ∼ 40 Wm−1 K−1 for solution processed PE with a DR of 350 [96].
The enhancement was attributed to increased chain alignment along the drawing
direction [94], phonon focusing in the elastically anisotropic crystalline phase [97],
and increased crystallinity [98, 99]. Recently, reported thermal conductivities have
increased further, with values of ∼ 50 − 100 Wm−1 K−1 reported in nanocrystalline
fibers. [100, 101] Advances in synthesis have led to the fabrication of disentangled
ultra-high molecular weight PE (UHMWPE) [102] with reported thermal conductivities up to 60 Wm−1 K−1 in bulk films [9, 103]. Recent works have also reported
high thermal conductivity up to 20 - 30 Wm−1 K−1 in a diverse set of polymers
besides PE, including polybenzobisoxazole, [104] polyethylene oxide [105], and
amorphous polythiophene [10].
Diverse methods have been employed to understand the origin of the increased
thermal conductivity in aligned polymers. Numerical studies have provided insights
into the properties of heat-carrying phonons in perfect polymer crystals, including
with empirical [106] and ab-initio [107–109] interatomic potentials, but oriented
polymer samples exhibit a structure far more complex than that of a perfect crystal.
As a result, many studies have focused on understanding the structural changes that
occur during drawing. The accepted picture is that an initial crystalline fraction on
the order of ∼ 60 − 70% exists in nascent polyethylene [96, 110–112]. On initial
drawing, the crystallites begin to align and a periodic lamellar structure, consisting
of alternating crystalline and amorphous domains, develops [113, 114]. Small angle
X-ray Scattering (SAXS) has been used to measure the period of the lamellae as
around ∼ 8 to 40 nm, and that corresponding size of the crystalline domain consisting
of the lamellae is around 10 - 20 nm [115, 116]. For 𝐷 𝑅 < 20, the crystallinity also
increases under drawing as measured using wide angle X-ray scattering (WAXS) up
to ∼ 80 − 90% [111, 117].
Below DR100, although the thermal conductivity increased with drawing, the degree
of the crystallinity and crystallite size did not exhibit as strong a trend with DR.
The crystallinity was reported to increase only by 10 %, despite substantial change
in the DR from 20 to 100. [111, 118] While some prior studies reported increase
in the crystallite size from ∼ 30 nm to ∼ 70 nm, nearly three-fold increase, with

41
drawing from [119], others reported the opposite trend [115, 120]. Furthermore,
the scattered intensity with SAXS was observed to decrease with increasing DR
[121–123], indicating the transformation from the periodically alternating lamellar
structure to a different one such as inter-crystalline bridges [95, 115] or continuously
extended rigid crystal structure [115, 120].
The presence of such an extended crystal was identified by measurements in highly
oriented PE using solid-state NMR [124] along with SAXS [120]. In particular,
Litvinov et al. have measured the dimension of crystalline domain along the chain
direction around 100 - 250 nm, far exceeding the lamella period length (40 nm)
measured from SAXS in the same structure. [120]. Since the upper limit of
the detectable size from NMR is around 200 nm, the actual dimension of such
crystal was thought to be greater than the measured value [120]. The elongated
crystals in highly oriented PE thin films were observed to have length as long as
several microns yet with diameter in the range of 9 - 20 nm (Transmission electron
microscope [125, 126]), 14 - ∼ 20 nm (SAXS and WAXS [119, 120, 127]), and 10
nm (solid-state NMR, [124]).
Various models have been proposed to understand the evolution of the uniaxial thermal conductivity under drawing. Two-phase effective medium models with series
thermal resistances for each phase have been considered, including the Helix-coil
model [98, 128], a modified Maxwell model [129], and the Takayanagi model[130].
Choy et al. have shown that the thermally isotropic Maxwell model best describes
the thermal conductivity in low DR (𝐷 𝑅 ≤ 5) [129]. The Takayanagi model considers an additional thermal conduction channel along the inter-crystalline bridge
[130]; this model better explained the measured thermal conductivity for 𝐷 𝑅 > 5,
possibly due to the presence of such structures in higher DR samples. Xu et al.
used the isotropic helix-coil model and concluded that the thermal conductivity of
the amorphous phase (𝜅 𝑎 ) must be as high as 16 Wm−1 K−1 to explain their thermal
conductivity measurements [9]. Ronca et al. have used the same model, but reported
that the high thermal conductivity originates from the increased thermal conductivity in the crystalline domain (𝜅 𝑐 ) as high as 47 Wm−1 K−1 [103]. The reason for
the discrepancy is that the theory requires the characteristic length of the repeated
periodic structure that may not exist in actual ultra-drawn samples. Also, the bulk
properties that are free parameters in the theory (𝜅 𝑐 , 𝜅 𝑎 ) are challenging to individually measure, hitherto yielding significant differences in measured 𝜅 𝑎 s [9, 103]. In
addition, the theory does not account for possibility of phonons transmitting across

42
domain boundaries, in which case the effective medium treatment is no longer valid.
Evidence of such processes has been reported even in partially oriented PE samples
with low DR ∼ 7.5 using transient grating spectroscopy [131]. As a result, the
microscopic properties of heat-carrying phonons and the origin of the high thermal
conductivity of highly oriented polyethylene remain a topic of debate.
In this chapter, we report that the dominant heat-carrying phonons in highly oriented polyethylene exhibit mean free paths on micron length scales using transient
grating spectroscopy. The mean free paths exceed those in samples of lesser draw
ratio by an order of magnitude, indicating that the high thermal conductivity of the
present samples primarily arises from decreased structural scattering in the crystalline phase. The decreased structural scattering in turn could arise from larger
crystalline domains as well as higher transmissivity across crystalline-amorphous
boundaries in the highly aligned samples. Our work provides insight into the microscopic transport properties of heat-carrying phonons in highly oriented polyethylene
that will support efforts to create PE samples with higher thermal conductivity.
4.2

Bulk thermal property of PE

Sample: polyethylene

Figure 4.1: Samples that were presented in this thesis. The dye was included to
increase the absorption of the visible wavelength. (A) DR4 with AuNP, (B) DR5
with AuNP, (C) DR9 with BPEA, (D) DR196 with AuNP. The concentration of dye
is only 1% to minimize the filler effect on the thermal conductivity, while ensuring
sufficient absorption of the visible light. Note that the diameter of the AuNP is 2-5
nm.
The samples that are presented in this chapter were fabricated by our collaborators at Loughborough University in the UK. Briefly, the samples are disentangled
UHMWPE synthesized using the same procedure described in Ref. [37, 102, 103].
The synthesized samples were drawn by the processes with a particular draw ratio
(DR) given by the length of the film before and after the drawing processes. Typ-

43
ically, partially drawn samples are prepared using compression molding process.
Similarly, our partially oriented samples (DR4, DR5, and DR9) were compression
molded. For the highly oriented sample (DR196), the film was rolled (×7) and
stretched (×28) to achieve the high draw ratio. Since PE is transparent to visible
light, low concentration of the dye was added to minimize the effect of the filler on
the thermal conductivity while enabling the formation of a thermal grating on the
sample. Specifically, 9,10-Bis(phenylethynyl)anthracene (BPEA) 1% was added for
DR9. Similarly, the concentration of added gold nanoparticle (AuNP, diameter of
2-5 nm) for other samples is only 1% [132, 133].
Since the purpose of the drawing is to make the fiber oriented along a certain
direction, we characterized the fiber alignment using scanning electron microscope
(SEM) and helium-ion microscope (HIM). For comparison, an SEM image of
partially oriented sample (DR7.5 with ZnO nanoparticles) is shown in Fig.4.2 in
which we do not see obvious alignment of the fibers. On the other hand, our DR196
samples demonstrated clear aligned fibers that are extended over tens of micron
scales as shown in Fig. 4.2.

Figure 4.2: High resolution microscopic images of the oriented PE. (A) Literature
data of DR7.5 [134]. The chain alignment is not evidently visible. (B) SEM (C)
HIM image of the highly oriented PE. The evidence of the extended chain over a
micron distance is clearly visible.
As mentioned in the previous chapter, it is important to avoid surface instability
for the application of the TG to the thin films. For PE, experimental challenge
is related to the following factors especially for highly drawn samples. First, the
striations present in the entire sample of DR196, as evidenced in Fig. 4.1D can
inhibit the formations of the thermal grating profile over micron scale distances.
Second, surface inhomogeneity introduced during the drawing process can scatter
the visible light over the length scale comparable to the optical wavelength, which
can substantially decrease the signal-to-noise ratio (SNR). Therefore, we imaged the

44
optical topography as shown in Fig.4.3. The AFM crosscut perpendicular to fiber
alignment direction (dashed line in Fig.4.3. A) is shown in Fig.4.3. B. Calculated
RMS roughness is ∼ 70 nm, and the maximum peak-to-valley difference is ∼ 360
nm. The height difference indicates the surface inhomogeneity over length scales
comparable to the optical wavelength (515 nm) used.

Figure 4.3: AFM characterization of the PE surface. (A) AFM topographic image
of the sample. The x and y direction indicates parallel to, perpendicular to the
fiber alignment, respectively. (B) AFM crosscut along the dashed line in (A). RMS
roughness is calculated to be ∼ 70 nm, and the maximum peak-to-valley difference
is ∼ 360 nm.

4.3

TG application on highly oriented PE

Description of the TG system
We measured the in-plane thermal conductivity of several semi-crystalline polyethylene (PE) films including highly oriented PE using transient grating spectroscopy.
The transient grating setup is identical to that described in Ref. [131]. Briefly, a pair
of pump pulses (wavelength 515 nm, spot diameter 530 𝜇m, pulse duration ∼ 1 ns,
pulse energy 13 𝜇J, repetition rate 200 Hz) is focused onto the sample to impulsively
create a spatially sinusoidal temperature rise of period 𝐿 and wave vector 𝑞 = 2𝜋/𝐿.
The grating relaxes by heat conduction, and its decay is monitored by the diffraction
of a pair of continuous wave laser beams (wavelength 532 nm, spot diameter 470
𝜇m, average power 900 𝜇W, chopped at 3.2 % duty cycle to reduce steady heating
on the sample).

45
Bulk thermal conductivity characterization
First, we examined bulk thermal conductivity. More precisely, uni-directional thermal conductivity and the role of the drawing to the thermal anisotropy were characterized. For this purpose, the TG experiments on several PE films with different
draw ratio were performed. Figure 4.4 shows representative signal traces for samples
with different draw ratios. The sample was placed parallel to the TG axis.

Figure 4.4: Signal magnitude versus the time for several PE films along the transient
grating setup. (A) DR4 with AuNP, (B) DR5 with AuNP, (C) DR9 with BPEA,
(D) DR196 with AuNP. The symbols indicate that the signal is an average of 3×104
repetitions. The dashed line is an exponential fitting with an equation given in
Eq. 4.1.
As discussed in Ref. [134], there are various non-thermal decay components with
different magnitude and time constants in TG measurements on PE. Therefore, we
fit the signal following Ref. [134]
𝑓 (𝑡) = 𝑐 1 + 𝑢(𝑡) 𝑐 2 +
𝐴𝑚 exp[−𝑡/𝜏𝑚 ]
(4.1)
𝑚=1

where 𝑢(𝑡) is Heaviside unit step function, 𝑐 1,2 is the background before and after
𝑡 = 0, A is the initial magnitude of the decay, and 𝜏 is the decay time constant. In

46
Eq. 4.1, the index 𝑚 = 1 indicates the thermal signal component as a dominant decay
component for the signal in Fig. 4.4. The index 𝑚 = 2 (𝑚 = 3) corresponds to the
electronic response (long-term relaxation) with a time-constant on the order of ps
(10 − 30 𝜇s); detailed description on the physical origin of each non-thermal source
can be found in [134]. The dashed lines in Fig. 4.4 show corresponding fits from
Eq. 4.1, which well-describe the measured data. We note that the signal magnitude
as well as the corresponding SNR in Fig. 4.4D is noticeably small compared to
Figs. 4.4A-C due to the surface instability in DR196. As a comparison with prior
TG experiments, the SNRs of the DR196 are generally less than 20, which is about
a third of that from 𝐷 𝑅 = 7.5 reported in Ref. [131].
Next, we measured angle-resolved thermal conductivity for the samples mentioned
above. As mentioned in many prior literatures including Ref. [134], the thermal
conductivity of semi-crystalline PE depends on the angle, with a substantial decrease
at the perpendicular direction due to the thermal anisotropy. We systemically varied
the angle by rotating the sample with respect to the principle TG axis. Examples
of resulting angle-resolved thermal conductivities are shown in Fig. 4.5, in which
0 (90) degree indicates the heat-flow direction along (perpendicular to) the chain
orientation. Following Ref. [134], we fit the thermal conductivity versus the angle
given in Fig. 4.5, using an expression given by
𝜅 = 𝜅 k cos2 𝜃 + 𝜅⊥ sin2 𝜃

(4.2)

where 𝜅 k (𝜅⊥ ) is the thermal conductivity at 0 degree (90 degree). The above
expression is a geometric model considering that the heat flow into each direction
independently contributes to the heat transfer, similar to the assumption in prior
effective medium models used for polymers [98, 128, 130]. The dashed lines in
Fig. 4.5 show that the geometric model can well describe the data.
By comparing the value 𝜅 k , we can estimate the ratio of heat flow anisotropy. The
anisotropy is defined by 𝜅 k /𝜅⊥ . We see that the maximum thermal conductivity
at DR4 is 𝜅 k = 4.2 ± 0.4 Wm−1 K−1 , which decreases to ∼ 1 Wm−1 K−1 with an
anisotropy ∼ 4.2. On the other hand, for DR 196, the 𝜅 k is 42 ± 3 Wm−1 K−1 , and 𝜅 ⊥
is ∼ 0.4 Wm−1 K−1 , resulting in an anisotropy ∼ 105. The draw ratio dependence of
𝜅 k , 𝜅⊥ for DR3 to 196 is shown in Fig.4.6A. The value of 𝜅 k is ∼ 3 Wm−1 K−1 for
DR3, which is around 5 times larger than 𝜅⊥ . The 𝜅 k increases as the DR increases
up to DR66, above which the observed value saturates to ∼ 40 Wm−1 K−1 . The
opposite trend is observed for 𝜅⊥ , which approaches to 0.4 Wm−1 K−1 . Note that

47

Figure 4.5: Thermal conductivity versus angle between draw direction and thermal
gradient defined by a grating period. (A) AuNP4 at 𝐿 = 10.7 𝜇m with 𝜅 k = 4.2 ± 0.4
Wm−1 K−1 ), (B) AuNP5 at 𝐿 = 10.7 𝜇m with 𝜅 k = 4.6 ± 0.3 Wm−1 K−1 ), (C)
BPEA9 at 𝐿 = 6.9 𝜇m with 𝜅 k = 4.4 ± 0.6 Wm−1 K−1 ), (D) AuNP196 𝐿 = 10.7
𝜇m with 𝜅 k = 42 ± 3 Wm−1 K−1 ). The 0 (90) degree indicates heat flow direction
parallel to (perpendicular to) the draw direction. The maximum value of the thermal
conductivity occurs along the chain, while the perpendicular value is comparable to
that in bulk PE. Dashed lines indicate fits using the Eq. 4.2
the value of 𝜅⊥ ∼ 0.4 Wm−1 K−1 is close to the value in unoriented PE, which is
attributed to the heat conduction by interchain van der Waals interactions as in the
bulk PE [129]. Corresponding ratio of the thermal conductivity (𝜅 k /𝜅⊥ ) is shown in
Fig. 4.6B. Above DR66, the measured anisotropy was observed to be ∼ 100 − 150.

48

Figure 4.6: (A) 𝜅 k and 𝜅⊥ versus the draw ratio. The draw ratio is from 3 to 196.
With draw ratio, the 𝜅 k increases from ∼ 3 Wm−1 K−1 to ∼ 40 Wm−1 K−1 up to DR 66,
above which the value saturates. The 𝜅⊥ decreases and approaches to the bulk value
as the draw ratio increases, which is attributed to the heat conduction by interchain
van der Waals interactions as in the bulk [129]. (B) The anisotropy (𝜅 k /𝜅⊥ ) versus
the draw ratio. We see that the anisotropy increases as the draw ratio increases up
to ∼ 100 − 150 above DR66.
Thermal conductivity versus the temperature
Having characterized the effect of the DR on the thermal anisotropy, we now seek for
inferring the microscopic thermal transport properties. In semi-crystalline PE with
low DRs, the structural properties are known to limit the heat-carrying process [37],
we can qualitatively extract the microscopic heat transport by relating temperature
dependent thermal conductivity to the structural property.
First, the thermal conductivity versus the temperature for two samples with similar
DRs is shown in Fig. 4.7AB. Due to the low DRs, thermal conductivity exhibits an
increase with temperature indicating substantial structural scattering, in agreement
with a prior study [37]. The feature ∼ 200 K is attributed to the glass transition [37].
The results from DR196 at 𝐿 = 9.8 𝜇𝑚 is shown in Fig. 4.8. The bulk thermal
diffusivity along the chain direction versus the temperature is plotted in Fig. 4.8A. A
linearly increasing trend with decreasing temperature is observed. The corresponding bulk 𝜅 k versus temperature is shown in Fig. 4.8B. The thermal conductivity
remains constant within the measurement uncertainty from room temperature to
∼ 220 K, below which the thermal conductivity decreases. The trend of the thermal
conductivity below 220 K is characteristic of structural scattering.

49

Figure 4.7: Thermal conductivity versus temperature at 𝐿 = 10.7 𝜇m for (A) DR4
with AuNP, and (B) DR5 with AuNP. With temperature, the thermal conductivity
exhibits a monotonic increase, characteristic of the phonon-domain boundary scattering. A hump near 200 K is associated with the glass transition temperature of the
PE.

Figure 4.8: Temperature dependent bulk thermal properties of DR196 with AuNP.
(A) Thermal diffusivity along the chain axis versus the temperature at a grating
period 𝐿 = 9.8 𝜇𝑚. With decreasing temperature, linear increase in the thermal
diffusivity is clearly observed. (B) Thermal conductivity along the chain versus
the temperature at a grating period 𝐿 = 9.8 𝜇𝑚. The thermal conductivity was
calculated using reported heat capacities for linear PE in Ref. [135] The thermal
conductivity is approximately constant from 300− ∼ 220 K, below which the thermal
conductivity decreases.

The above thermal conductivity versus the temperature can be used to infer the microscopic heat transfer properties. For example, comparing Fig. 4.7AB, while we can
find an evidence of structural scattering, there is not an obvious difference between
them, implying that micro-structures in both samples would be similar. A direct

50
evidence for structural scattering is shown in Fig. 4.7C. Figure 4.9 shows structural
characterization of the PE with AuNP using WAXS and SAXS. The measurements
were performed by our collaborator [133]. The measurements demonstrate that the
crystallinity of ∼ 74% and an average lamella spacing around 8 - 22 nm [132, 133],
and that the micro-structural properties in both DR4 and DR5 are similar. From the
results in Fig. 4.9, we can tentatively conclude that the phonons may propagate a
distance order of 10 nm.

Figure 4.9: SAXS and WAXS characterization in PE with AuNPs. The data were
taken from Ref. [133]. (A) Left axis: Crystallinity of PE with AuNP. For DR4 and 5,
the crystallinity was measured to be 73.5%. Right Axis: The ratio of WAXS intensity
peak over a narrow 𝑞 range near (200) reflection to the (110) refletion from which
the crystallinity was extracted. (B) Thickness-normalized Lorentz corrected SAXS
intensity profiles measured from PE with AuNP. The dashed lines indicate the wave
vector of the intensity bump around 𝑞 = 0.29 nm−1 and 𝑞 = 0.8 nm−1 , respectively.
Corresponding lamella long-period of the crystalline lamella (𝑑 = 2𝜋/𝑞) is around
8 - 22 nm.

However, the above qualitative approach is not applicable for highly oriented samples, since corresponding domain sizes in inaccessible using SAXS and WAXS
[124]. According to the prior reports, the SAXS intensity disappear for DR ≥ 80,
mainly attributed to the marginal fraction of the amorphous fraction or the presence
of the extended crystals [9, 124]. Furthermore, the measurements of temperaturedependent thermal conductivity alone cannot rigorously examine other microstructural properties such as distribution of the phonon MFPs and corresponding transmission and reflection of phonons at the domain boundary.

51
4.4

Grating period dependent thermal conductivity

In the preceding section, we have demonstrated the capability of the TG to measure
angle- and temperature-dependent thermal conductivity and that enables to infer
qualitative information of the microscopic heat-transport properties. In this section,
we systematically vary the relevant thermal length scale comparable to the mean free
path, therefore provide a basis for understanding microscopic transport properties.
We measured the thermal conductivity along the parallel direction versus the grating
period at temperatures of 300 K, 220 K, 100 K, and 30 K. The raw signal traces are
given in Fig. 4.10.

Figure 4.10: Measured TG signal versus time (symbols) at several grating periods
for DR196 at (A) 300 K, (B) 220 K, (C) 100 K, (D) 30 K. The signal shown are
normaliazed by its initial magnitude at t = 0. The dashed lines are fit using Eq. 4.1.
Error bars indicate 95 % confidence intervals obtained using an identical procedure
as that in Ref. [131].
If heat-carrying phonons propagate ballistically over the grating period, the thermal
decay is slower than that predicted from the bulk thermal conductivity [33, 35].
Representative TG signal for a grating period of 𝐿 = 1.5 𝜇𝑚 at 300 K is shown in
Fig. 4.11 A. The decay is clearly slower than expected based on the bulk thermal

52
conductivity value, indicating the presence of ballistic phonons on the length scale
of the grating period. Measurements of the decay rate versus 𝑞 2 for all the grating
periods at 300 K are given in Fig. 4.11. The measured decay rate is close to that
predicted by the bulk thermal conductivity at 𝑞 2 ≤ 0.3𝜇𝑚 −2 , above which the decay
rate is slower by up to a factor of 5 at the smallest grating period.

Figure 4.11: (A) Measured TG signal versus time (symbols) for grating period
𝐿 = 1.5𝜇𝑚, corresponding to 𝑞 2 ∼ 17.5 𝜇𝑚 −2 , along with the best fit (solid
red line) and expected decay using the bulk thermal conductivity (dashed black
line). The signal is an average of 5 × 104 shots measured at a single location.
The signal clearly decays slower than predicted from bulk thermal conductivity,
evidence of the departure from the diffusive thermal transport. (B) Decay rate
versus 𝑞 2 . The measured decay rates for 𝑞 2 & 0.3 𝜇𝑚 −2 deviate from that predicted
from bulk thermal conductivity, indicating that the heat-carrying phonons transport
ballistically.
Corresponding measured thermal conductivity versus grating period at 300 K - 30
K, is shown in Fig. 4.12 A - D. The thermal conductivity is nearly constant at
large grating period (𝐿 ≥ 10 𝜇𝑚), below which a grating period dependence is
observed. Note that the grating period dependence in DR 196 is much stronger
than that reported in partially oriented PE [131], indicating the presence of heatcarrying phonons with longer MFPs in the present highly oriented samples. A
similar grating period dependence was also observed at all the temperatures. Noting
that the apparent thermal conductivity at 𝐿 = 6 𝜇𝑚 is around half of the bulk value,
the measurements imply that heat-carrying phonons with MFPs above ∼ 𝐿/2𝜋 ∼ 1
𝜇𝑚 carry half the heat over 30 - 300 K.

53

Figure 4.12: Thermal conductivity versus grating period at (A) 300 K, (B) 220
K, (C) 100 K, (D) 30 K. The thermal conductivity clearly depends on the grating
period at both temperatures. With decreasing grating period, the measured thermal
conductivity levels off to a constant value at 𝐿 ∼ 10 𝜇m, below which the value
substantially decreases. Particularly at all the temperatures, the value is reduced
nearly 50% from its bulk value at 𝐿 ∼ 5 − 6 𝜇m, which corresponds to thermal
length 𝑑 ∼ 𝐿/2𝜋 ∼ 1 𝜇m. Error bars indicate 95 % confidence intervals obtained
using an identical procedure as that in Ref. [131].

4.5

Ballistic transport of the heat-carrying phonons in DR196

Regarding the grating dependent thermal conductivity, the evidence of the ballistic
transport of heat-carrying phonons can be used to characterize the spectrum of
MFPs of phonons. In this section, we review the approach to reconstruct the MFP
of heat-carrying phonons.
Background
We start from the kinetic theory form of the thermal conductivity. Here, we express
the quantity in the MFP domain. For a bulk substance, the accumulated thermal

54
conductivity at a given MFP (Λ) is defined as
∫ ∞
𝜅 𝑏𝑢𝑙 𝑘 =
𝑓 (Λ)𝑑Λ𝜔

(4.3)

where f(Λ) is a differential form of the thermal conductivity at a given MFP in the unit
of 𝑊𝑚 −2 𝐾 −1 . In TG, as the thermal length scale becomes smaller, the contribution
of the heat-carrying phonons with longer MFP to the thermal conductivity start to
become smaller. Corresponding thermal conductivity is [35]
∫ ∞
𝜅𝑇𝐺 =
𝑓 (Λ)𝑆(𝑥)𝑑Λ𝜔
(4.4)

where 𝑥 = 𝑞Λ = Λ(2𝜋/𝐿) is the ratio of the MFP to the thermal length scale, and
𝑆(𝑥) is the phonon thermal conductivity suppression function. The suppression
function is known to be
tan−1 (𝑥)
𝑆𝑖𝑠𝑜 = 2 1 −
(4.5)
(4.6)
1 + 𝑥2
where 𝑆𝑖𝑠𝑜 is the suppression function for the isotropic material such as silicon [34],
and 𝑆 𝑎𝑟𝑏 is that for the anisotropic material with an arbitrary dispersion [34]. Since
the PE is thermally anisotropic, we use 𝑆 = 𝑆 𝑎𝑟𝑏 in the following analysis.
𝑆 𝑎𝑟𝑏 =

Due to the nature of the cumulative form of the thermal conductivity using an
integration by parts, Eq. 4.4 can be expressed as
∫ ∞
𝜅𝑇𝐺 =
𝑞𝐾 (𝑥)𝐹 (Λ𝜔 )𝑑Λ𝜔
(4.7)

where 𝐾 (𝑥) = −𝑑𝐾/𝑑𝑥 is the Kernel function. The Eq. 4.7 is known to be ill-posed,
therefore cannot be uniquely solved. Nevertheless, Eq. 4.7 is still useful, since the
Eq. 4.7 does not require any prior knowledge concerning the microscopic properties
such as phonon dispersion and frequency dependent scattering mechanism.
A way to solve the problem (Eq. 4.7) is Bayesian inference used in several recent
works [37, 47]. The details are described in [47, 131]. Here, we briefly explain
the Bayesian inference that was used in this thesis. Following the procedures in
Ref [131], we performed iterative 𝑛 steps to search for posterior solution 𝐹𝑝𝑜𝑠𝑡 (Λ)
based on 𝐹𝑝𝑟𝑖𝑜𝑟 (Λ) given the measurements. The following is the inference procedures for sampling 𝐹𝑝𝑜𝑠𝑡 (Λ) in each steps. First, we assumed a profile of cumulative
thermal conductivity versus the mean free path 𝐹 𝑗−1 (Λ) for 𝑗 = 1, 2 · · · 𝑛. Next,

55
we sampled posterior profile 𝐹 𝑗 (Λ) based on the Metropolis-Hasting Markov chain
Monte Carlo algorithm. We then computed the likelihood of the posterior probability (𝑃 𝑝𝑜𝑠𝑡 ) given the experiment
𝑃 𝑝𝑜𝑠𝑡 (𝐹 𝑗 |𝜅 𝑒𝑥 𝑝𝑡 ) ∝ 𝑃(𝜅 𝑒𝑥 𝑝𝑡 |𝐹 𝑗 )𝑃 𝑝𝑟𝑖𝑜𝑟 (𝐹 𝑗 )
∫ 2
𝑑 𝐹𝑗
𝑃 𝑝𝑟𝑖𝑜𝑟 (𝐹 𝑗 ) ∝ exp
𝑑Ω
−2𝛾 2
𝑑Ω2
Õ 𝜅𝑖 − 𝜅 𝑐𝑎𝑙𝑐,𝑖 (𝐹 𝑗 )
𝑃(𝜅|𝐹 𝑗 ) ∝ exp −
2𝜎𝑖2

(4.8)
(4.9)

(4.10)

where 𝛾 is a standard deviation for second drivative, 𝜅𝑖 (𝜎𝑖 ) is the measurements
(experimental error bound), and Ω = log 10(Λ), Finally, we accepted the posterior
profile with an acceptance probability 𝛼 given by
𝑃(𝐹 𝑗 )
(4.11)
𝛼(𝐹 𝑗 , 𝐹 𝑗−1 ) = 𝑚𝑖𝑛 1,
𝑃(𝐹 𝑗−1 )
Otherwise, we update 𝐹 𝑗 = 𝐹 𝑗−1 .
MFP accumulation in DR196 from Bayesian inference
Using the Bayesian inference mentioned in the previous section, the thermal conductivity accumulation (𝐹 (Λ)) versus the phonon mean free path that best explains
the experimental observations in DR196 were extracted.
The reconstructed MFP accumulations are given in Fig. 4.13. In Fig. 4.13, the
intensity of the shaded region corresponds to the density of the posterior distribution,
indicating that the calculated thermal conductivity versus the grating period based
on these MFP profile agrees with the measurement. Solid lines (dashed lines)
indicate means (95 % credible intervals) of the distribution.
In order to confirm that the predicted 𝐹 (Λ) from the Bayesian inference is accurate,
we need to perform the forward calculation using Eq. 4.7 and to compare it with
the experiments. The corresponding thermal conductivity versus the grating period
from the profiles from the Bayesian inference is shown in Fig. 4.14. In Fig. 4.14,
the intensity of the shaded region is the calculation obtained using the MFP profiles,
enclosed by 95 % credible intervals (dashed line). The solid lines correspond to
the calculation using the mean of the MFP profile. At all the temperature ranges
in this work (30 K, 100 K, 220 K, 300 K), we see a good agreement between the
calculation and the experiment.

56

Figure 4.13: Thermal conductivity accumulation versus the mean free path of
the thermal phonons in highly oriented PE at (A) 300 K, (B) 220 K, (C) 100 K,
and (D) 30 K. The intensity of the shaded region indicates the likelihood of the
reconstructed profile. The solid lines (dashed lines) correspond to the mean (95%
credible intervals). We observe that the mean free path in this sample is on the order
of micron and that the fraction of the heat carried by the mean free path above 𝜇𝑚
is close to 50% at these temperatures.

We examine the properties of the heat carrying phonon using the obtained MFP
distribution in Fig.4.13. At 300 K, nearly 50% of the heat is carried by phonon
with MFP above 1 𝜇m, as expected from the rough estimation described above. The
contribution of these phonon to the thermal conductivity does not exhibit strong
temperature dependence as in Figs. 4.13B - D. Moreover, the contribution to the
thermal conductivity for phonons with MFP below 500 nm is less than 0 - 20%,
considering the uncertainties determined from the 95% credible interval. On the
other hand, the percentage of the heat carried by MFP below 100 nm is less than
1%. Prior characterization on the MFP has shown that the range of the MFP in
partially oriented PE (DR = 7.5%) is in hundreds nanometer range, nearly 200 nm
[131]. Our results suggest that the highly oriented PE can support the MFP of the
thermal phonon in the range of hundred nanometer to a few micrometer scale as in

57

Figure 4.14: The calculation of the apparent thermal conductivity versus the grating
period at (A) 300 K, (B) 220 K, (C) 100 K, and (D) 30 K. The intensity of the shaded
region is the calculation using the MFP profiles obtained from the Bayesian inference. The dashed (solid) lines corresponds to the calculation using 95 % credible
interval (mean) of the MFP distribution described in the main text. The calculation
excellently matches with the experiment, indicating that the reconstructed MFP is
the most probable solution.

many other covalent crystals that possess high thermal conductivity such as silicon
[35]. Furthermore, the fraction of the heat carried by phonons with MFPs < 100 nm
is less than 5% at all temperatures, confirming that the phonons are not scattering at
the nano crystalline domain size as high as 100 nm. If the phonon-phonon scattering
mechanism would dominate, one would expect that corresponding domain of the
MFP should significantly increased, which is not the cases here. In other words,
for instance, the reduction of thermal conductivity at low temperatures shown in
Fig. 4.12 would have occured at grating periods that are longer than the values
presented in Fig. 4.12. Therefore, we tentatively conclude that the phonons are still
limited by the domain boundary order of the micron length scale, a scale close to
the reported extended crystalline domain size [115, 120].

58
4.6

Low energy Debye model

As mentioned, the analysis in the MFP domain is beneficial in that the reconstruction
does not require prior knowledge of the phonon properties. However, use of an approximate model describing the low energy excitations of PE can help to understand
the damping mechanisms. As discussed in prior literature [33, 47], this procedure
requires the knowledge of the acoustic dispersion of PE.
4.7

Modeling

For an anisotropic solid, the grating dependent thermal conductivity in Eq.4.7 can
be re-written as
Õ ∫ 𝜔𝑚
𝜅𝑖 =
𝑆𝑛 (𝑥𝑖 ) [𝐶𝑛 (𝜔)𝑣 𝑛 Λ𝑛 (𝜔)] 𝑑𝜔
(4.12)

where n is the index of the polarization, and 𝜔𝑚,𝑠 is the cutoff frequency for phonon
that transports heat. The mode-specific heat capacity is expressed as 𝐶𝑠 (𝜔) =
ℏ𝜔𝐷 𝑠 (𝜔)𝑑𝑓 𝐵𝐸 /𝑑𝑇 where the 𝐷 𝑠 (𝜔) is the density of states (DOS), and 𝑑𝑓 𝐵𝐸 /𝑑𝑇 is
the temperature derivative of the Bose-Einstein distribution function.
We note from Fig 4.13 that phonons with MFPs on the order of hundreds of nanometers carry the majority of the heat. From the calculations of Ref. [107], the only
phonons with such long MFPs are from the LA branch. Therefore, the weakly
damped heat-carrying vibrations are likely from this branch. The marked elastic
anisotropy of PE can be accounted for to good approximation by assuming that
the group velocities all point along the chain axis [108, 136–138]. Therefore, the
measured thermal conductivity versus the grating period (𝑘 𝑖 ) should be simplified
as
∫ 𝜔𝑚
𝑆(𝑥𝑖 ) [𝐶 (𝜔)𝑣Λ(𝜔)] 𝑑𝜔

𝜅𝑖,𝐿 𝐴 =

(4.13)

For our calculation of Eq. 4.13, the ab-initio values of the mode-dependent density of
states and dispersive group velocity for LA branch was used. Following Ref. [107],
the ab-initio properties were computed using temperature-dependent effective potential (TDEP) with the force constants generated at 300 K [108]. Corresponding
DOS and dispersion relation for LA branch is shown in Fig. 4.15. From the dispersion relation, we calculated the dispersive phonon group velocity, which is 17000
𝑚𝑠−1 near the Brillouin zone center. Due to finite grid size, extrapolated values of
DOS and the group velocity were used in our actual calculation for Eq. 4.13.

59

Figure 4.15: Calculated LA phonon properties along the chain obtained using TDEP
with force constants generated at 300 K. (A) Density of states. (B) Dispersion
relation (solid line). The sound velocity near the Brillouin zone center is around
17000 𝑚𝑠−1 (dhhsed line). The TDEP results agree with prior IXS measurements
on PE with DR5.5 (symbols).

We now seek to identify the function Λ(𝜔) that best explains the temperature dependence and grating dependence of the thermal conductivity. Prior works provide
constraints on the values of the MFPs at different frequencies. Inelastic x-ray scattering (IXS) experiments on DR 5.5 have shown that the MFP is around 10 nm at
at ∼ 6 THz. At ∼ 1 THz, the dominant LA phonon approximation on prior thermal
measurements indicates that the MFPs are ranging from 10 - 50 nm [26, 139–
141] as shown in Fig. 4.16. The estimated MFP of LA phonons was obtained
using 𝑙 𝐿 𝐴 = 𝜅/𝐶𝑣,𝐿 𝐴 /𝑣 𝐿 𝐴 where 𝐶𝑣,𝐿 𝐴 (𝑣 𝐿 𝐴 ) was calculated (extrapolated) from
the properties shown in Fig. 4.15. First, since the temperature dependent thermal
conductivities were measured at relatively low draw ratio compared to our DR196,
the corresponding MFP below 1 THz for DR196 should be bigger than order of 10
nm. Second, the cumulative thermal conductivity versus the mean free path lacks
of marked temperature dependence as shown in Fig. 4.13, corresponding frequency
resolved mean free path should be independent to the temperature. Third, low frequency phonons will experience geometric scattering or boundary scattering due
to long-wavelength. Corresponding scattering cross-section forces the lifetime to
be constant. Given these constraints, we assumed constant lifetime which yields
Λ(𝜔) = 𝑣 𝐿 𝐴 𝜏.
Based on the constraints, two profiles of Λ(𝜔) versus the frequency that is compatible
with the data (profile 2) was considered as shown in Fig. 4.16. We found that the
frequency dependent mean free path, Λ(𝜔) = 𝑣 𝐿 𝐴 𝜏 ∼ 500 − 600 nm best explains

60

Figure 4.16: Mean free path profile of LA phonons that are compatible with our
thermal data along with the literature data. Solid line is mean free path profile 2
discussed in the main text. Profile 2 is with 1D DOS and 3D DOS. For the calculation of low energy Debye model using profile 2, 3D DOS was used above 3
THz, below which 1D DOS was used. Symbols are the literature data from thermal
characterizations using dominant LA phonon approximation (upward-pointing triangles, crystallinity 83%, Ref. [141]; circles, crystallinity 73%, Ref. [139]; squares,
DR150, Ref. [142]; dashed line, crystallinity 77%, Ref. [140]) and IXS, DR5.5,
Ref. [26]. The prior data provided constraints for our modeling.

the thermal conductivity versus the grating period. We first consider the calculation
based on Λ(𝜔) = 𝑣 𝐿 𝐴 𝜏 ∼ 500 − 600 nm using ab-initio predicted 3D DOS for
frequency 0 - 1.5 THz. As shown in Fig. 4.17A, the calculated thermal conductivity
versus the temperature at 𝐿 = 9.8 𝜇m agrees with our measurement up to around
100 K, below which a significant deviation was observed. Corresponding calculated
thermal conductivity versus the grating period is shown in Fig. 4.17B-E, with which
we found a disagreement at 30 K.
Therefore, we used profile 2 in Fig. 4.16 with 1D DOS (= 1/𝜋𝑣 𝐿 𝐴 𝜋𝑟 2 ) below ∼ 2.9
THz, above which we used ab-initio predicted 3D DOS. The 𝑟 is the radius of
the extended PE crystal, which we assume 10 nm, following Ref. [119, 119, 120,
120, 124–127, 127]. For this calculation, 3D DOS was used for frequency above
3 THz. The calculated thermal conductivity at 9.8 𝜇𝑚 using profile 2 excellently
agrees with our measurements as shown in Fig. 4.17A. Furthermore, corresponding
thermal conductivity versus the grating period is compatible with the data as shown
in Figs. 4.17B-E.

61

Figure 4.17: Calculation of the thermal conductivity based on profiles 1 and 2. (A)
Thermal conductivity versus the temperature 𝐿 = 9.8 𝜇𝑚. While the calculation
from profile 2 excellently agrees with the data, that from profile 1 deviates from the
data below 100 K. Calculated thermal conductivity versus the grating period at (B)
300 K, (C) 220 K, (D) 100 K, and (E) 30 K along with the measurements. The
calculated thermal conductivity from profile 1 at 30 K is overestimated at all grating
periods.

The above results imply the presence of the 1D transport of the phonons in highly

62
oriented PE. Based on our observation, we applied similar analysis to prior measurements on partially oriented PE (DR7.5) in Ref. [37]. Under the same constraints,
we found that the frequency dependent mean free path, Λ(𝜔) = 𝑣 𝐿 𝐴 𝜏 ∼ 80 nm best
explains the thermal conductivity versus the grating for DR8. Based on these, we
considered profile 1 (3D DOS for 0 - 2 THz), profile 2 (1D DOS for 0 - 3.9 THz, 3D
DOS for frequency above 3.9 THz). Figure 4.18B-E shows our calculation. We find
that only profile 2 provides a calculation that excellently agrees with the data. By
comparing crossover frequency, we find ∼ 1 THz difference compared to the results
from DR196. In fact, the results in Ref. [131] were obtained considering heat capacity of crystalline phases only. In semi-crystalline polymers, it was reported that
the heat capacity of the amorphous fraction for temperatures 100 - 300 K is higher
than that of crystalline fraction [143], with a maximum difference by a factor of 1.5
- 2 near 250 K. If the heat capacity was considered for measurements in Ref. [131],
corresponding low temperature thermal conductivity would increase, indicating that
the transition frequency for 1D to 3D would decrease. Therefore, we believe that
the 1D to 3D transition frequency exists near 2-3 THz. Further measurements are
underway to provide evidence for this claim.
4.8

Discussion

Concerning the polymers, it is of significant importance to understand the microstructuretransport relationship as highlighted in prior studies. Therefore, the results presented
in this chapter can be interpreted as follows.
From the results of the grating dependence and reconstructed cumulative thermal
conductivity, the corresponding domain size of the extended crystal is comparable to
1 𝜇m. Since the thermal conductivity lacks of an obvious temperature dependence,
it is obvious that the heat-carrying phonons are scattered at the boundary of the
extended crystal. If the phonons would be fully diffused scattered at the boundary,
corresponding MFPs would be limited by size of the diameter of on order of 10 nm.
Considering the frequency resolved MFPs, however, we observe that the MFPs of the
phonons are an order of a few hundred nanometer up to around 3 THz, above which
the value of MFP is negligibly small, indicating that the low frequency phonons are
scattered specularly rather than diffusively.
The specular scattering of low frequency acoustic phonons as a dominant mechanism
can be evidenced by comparing the wavelength of the LA phonons to the fiber
diameter dimension. At around 2 THz, the wavelength of the LA phonons is an

63

Figure 4.18: Calculation of the phonon transport properties based on literature
data of DR7.5 in Ref [37]. (A) Mean free path profile of LA phonons that are
compatible with literature thermal data of DR7.5 along with the other literature data
of MFPs. Solid line is the profile 2 accounting for 3D and 1D DOS. Symbols are
from the literature data mentioned in Fig. 4.16, which provided the constraints. (B)
Calculated thermal conductivity at 𝐿 = 9.8 𝜇m. While the calculation from profile
2 excellently agrees with the data, that from profile 1 deviates from the data below
100 K. Calculated thermal conductivity versus the grating period at (C) 300 K, and
(D) 100 K along with the experiments is shown. In DR7.5, the calculated thermal
conductivity from profile 1 at 100 K is overestimated at all grating periods.

order of 8 nm, which is comparable or greater than the reported radius of the
extended fiber (diameter: ∼ 10 nm) from prior studies. According to the theory of
the phonon-boundary scattering, it is well-known that the long wavelength phonons
are more likely to scatter specularly, resulting in a coherent thermal conduction
in PE. More precisely, since the group velocity of the acoustic phonons in PE are
as high as 17000 𝑚𝑠−1 , the phonons propagate coherently over the micron length
scale with its reduced density. Therefore, we attribute our observation to onedimensional coherent thermal conduction within the extended fibers, as an origin of
actual thermally conductive polymers.
However, the presented results here are different from prior predictions on the
thermal transport in thermally conductive polymers. First, limited computational

64
domain size in ab-initio calculation hinders the accurate prediction of the transport of
the low frequency phonons that are known to provide a non-negligible contribution
to the heat. Second, as mentioned earlier, the understanding of the thermal transport
properties in PE has been limited only in a perfect micro-structures (a repeated
coil structure), which is far from actual substances. The discrepancy between these
predictions and experiments highlights that the understanding correlation between
the local domain arrangements and heat-carrying acoustic phonon scattering is
important, and that this should be addressed under the consideration of the phonon
transport that is subject to the realistic structure.
4.9

Summary

In conclusion, we have applied TG to study the origin of the actual thermally conductive polymers. The mean free paths exceed those in samples of lesser draw
ratio by an order of magnitude, indicating that the high thermal conductivity of the
present samples primarily arises from decreased structural scattering in the crystalline phase. The decreased structural scattering in turn could arise from larger
crystalline domains as well as higher transmissivity across crystalline-amorphous
boundaries in the highly aligned samples. We find evidence of one-dimensional density states of phonons with a frequency less than 2THz. The transition frequency is
consistent with prior observations in the highly oriented PE; stiff longitudinal branch
with LA group velocity ∼ 17000 m𝑠−1 makes LA phonons to have a wavelength of
8 nm, which is comparable to a fiber diameter of 10nm. Therefore, the wavelength
of low-frequency heat-carrying acoustic phonons exceed fiber diameter dimensions.
These observations improve the understanding of microscopic transport properties
of heat-carrying acoustic phonons in highly oriented polyethylenes. Our results, in
turn, will improve the understanding of the actual upper-limit of thermally conductive polymers, which will be useful for next-generation low-weight and cost-effective
thermal management materials.

65
Chapter 5

ORIGIN OF ACOUSTIC EXCITATIONS IN AMORPHOUS
SILICON
This chapter has been adapted, in part, from:

Taeyong Kim, Jaeyun Moon, and Austin J. Minnich. Origin of micron-scale
propagation lengths of heat-carrying acoustic excitations in amorphous silicon.
July 2020. URL https://arxiv.org/abs/2007.15777v1.
T.K. participated in the conception of the project, prepared the experimental data,
analyzed the results, and participated in the writing of the manuscript.
So far, we have applied TG to study microscopic thermal transport properties in
solids containing crystalline phases. In this chapter, we further expand the application scope of the TG by applying the technique to amorphous silicon.
5.1

Motivation

The collective acoustic excitations of amorphous solids are of fundamental interest
due to their anomalous properties compared to those of crystalline solids, including an excess heat capacity at cryogenic temperatures [144, 145] and damping
by two-level systems [146–150]. The dispersion and damping of acoustic excitations responsible for heat transport have been extensively explored in many glasses
using experimental methods such as inelastic scattering [28, 151–155], tunnel junction spectroscopy [156], Brillouin scattering [157–159], and picosecond acoustics
[160, 161], among others. These studies have generally found that excitations with
well-defined frequency and wave vectors are supported up to ∼ 1 THz. In vitreous
silica, a relative of amorphous silicon (aSi), the attenuation exhibits several different
regimes, yielding different power-law frequency dependencies. For frequencies below ∼ 600 GHz, the damping scales as 𝜔−2 corresponding to anharmonic damping
and thermally activated two-level system relaxation. Between 600 GHz and 1 THz,
a Rayleigh scattering trend of 𝜔−4 is observed, followed by a return to 𝜔−2 scaling
[28]. At still higher frequencies, Kittel proposed that attenuation is independent
of frequency if the wavelength is comparable to the interatomic length scale [17].
Considering these different regimes, the general trend of MFP versus frequency of

66
acoustic excitations in glasses has been presented in Fig. 7 of Ref. [162] and Fig. 3
of Ref. [163], among others [18, 164].
Amorphous silicon is an anomalous glass for several reasons. First, at ultrasonic
frequencies, attenuation by two-level systems is observed in vitreous silica but not
in aSi, suggesting a low density of these systems in aSi [12, 165]. Second, thermal
transport measurements indicate that the thermal conductivity of aSi can be higher
than those of most glasses [166–169] and that heat-carrying acoustic excitations
travel distances on the order of one micron at room temperature despite the atomic
disorder [11, 169, 170]. This value is far larger than a few nanometer value inferred
for vitreous silica at room temperature [18, 163].
Experimentally resolving the attenuation coefficients by frequency in the sub-THz
frequency band would help to understand the origin of these properties, but probing acoustic excitations in this regime is a long-standing experimental challenge.
Amorphous silicon is synthesized in small volumes as a thin film of at most a few
microns, precluding the use of high-resolution inelastic neutron scattering. Even
if sufficient volumes were available, kinematic constraints complicate the measurement of the dispersion and broadening of low energy excitations. While inelastic
x-ray scattering has been successfully applied to study THz excitations in aSi [151],
the energy resolution is not sufficient to resolve sub-THz excitations. Picosecond
acoustics and Brillouin scattering are generally unable to access frequencies above
∼ 100 GHz, while tunnel junction spectroscopy and cryogenic thermal conductivity
measurements require films with a thickness of several millimeters.
As a result, studies of the acoustic excitations in aSi have relied on numerical
simulations based on normal mode analysis. Feldman and Allen classified the
excitations in aSi as propagons, diffusons, and locons according to the qualities
of the normal mode eigenvectors [14]. Fabian and Allen computed anharmonic
decay rates of the normal modes of aSi, predicting that they should exhibit a clear
temperature dependence [16]. Other molecular dynamics simulations based on
normal mode analysis have predicted that the MFPs decrease as 𝜔−2 with increasing
frequency for excitations of a few THz frequency, leading to the conclusion that they
are damped by anharmonicity [171, 172]. However, some of these predictions are
not supported by experiment. For instance, the predicted temperature dependence
of THz excitations in Ref. [16] is not observed experimentally using inelastic x-ray
scattering [151]. In the hypersonic frequency band ∼ 100 GHz, the measured values
of attenuation are lower than those predicted by anharmonic damping [15].

67
An experimental approach to measure the damping of sub-THz phonons versus
frequency and thereby resolve this discrepancy is needed. For solids like aSi with
MFPs in the micron range, transient grating (TG) is a tabletop experimental method
that is capable of measuring the MFP accumulation function, or the cumulative
thermal conductivity distribution versus MFP [36, 37, 173]. The technique relies on
observations of non-diffusive thermal transport to constrain this function; knowledge
of the acoustic dispersion of the solid then provides the MFP versus frequency. In
the case of amorphous silicon, the dispersion is known and isotropic [151, 174],
simplifying the analysis further. Combined with picosecond acoustics and inelastic
x-ray scattering for the low and high frequency limits of the attenuation coefficient,
respectively, TG measurements could constrain the frequency dependence of the
damping in the sub-THz frequency band and thus resolve the discrepancy.
5.2

Anomalous thermal properties observed in amorphous materials

As briefly described in 5.1, many amorphous materials possess universal anomalies
that are absent in crystals: thermal conductivity, heat capacity, and the acoustic
attenuation. In this section, we present experimental observations that were made
for such anomalies.
Thermal conductivity of amorphous materials
Figure 5.1A shows the thermal conductivity of SiO2 both in crystalline and amorphous forms. The thermal conductivity in 𝛼-Quartz increases as the temperature
decreases above 10 K, below which the value decreases; the temperature dependence
is attributed to phonon-phonon scattering followed by characteristic boundary scattering. On the other hand, the thermal conductivity of vitreous silica monotonically
decreases with a decreasing temperature with smaller values compared to that in
𝛼-Quartz. For instance, the thermal conductivity of vitreous silica at in the room
temperature is ∼ 1.3 Wm−1 K−1 , a value lower by factor of ∼ 6 compared to that of
𝛼-Quartz. The reported thermal conductivities versus the temperature from glassy
substances are shown in Fig. 5.1B. Both the thermal conductivity and its trend of the
temperature dependencies are similar both qualitatively and quantitatively, which
was supported by other earlier works on many glasses [175, 176].
Excess heat capacity
Another unique feature is excess heat capacities at low temperatures in amorphous
materials [18, 148, 179, 180]. At low temperatures, according to the Debye theory,

68

Figure 5.1: (A) The thermal conductivity versus the temperature for 𝛼-Quartz
(crystalline SiO2 ) and for vitreous silica (amorphous SiO2 ) in Ref. [18]. The thermal
conductivity in 𝛼-Quartz increases as the temperature decreases above 10 K, below
which the value decreases. With decreasing temperature, the increasing trend is due
to reduced phonon-phonon scattering whereas the decreasing trend is due to phonon
boundary scattering. In vitreous silica, the thermal conductivity monotonically
decreases with decreasing temperature with values smaller typically more than one
order of magnitude compared to that in 𝛼-Quartz. (B) The thermal conductivity
versus the temperature for several amorphous materials in Ref. [177]. All the glassy
materials show that the trend of the thermal conductivity monotonically decreases
with decreasing temperature. The black dashed line indicates glassy-range below 1
K in Ref. [178], indicating that the reported values below 1K are within this range.
the specific heat would be proportional to 𝑇 3 . The theory had been considered to be
applicable on any solid, since the wavelength of the dominant heat-carrying vibrations would far exceed the characteristic length of disorder. However, experimental
evidences indicate that the heat capacities in amorphous solids at low temperatures
deviate from the Debye prediction. Figure 5.2 shows literature data of low temperature heat capacities of some glassy substances. In silica, measured heat capacity far
exceeds the Debye calculation of the heat capacity with its linear dependence of the
temperature as shown in Fig. 5.2A. The presence of the excess heat capacities with
its linear temperature dependence is evidenced in other glassy materials such as Se
and Cer-Vit as shown in Fig. 5.2. To explain the linear temperature dependence of
the heat capacity, earlier works have proposed a model assuming the existence of
two level states with a constant density of states [149].
Damping of acoustic excitations in amorphous materials
Figure 5.3 shows the literature data of MFP versus the frequency for heat-carrying
vibrations in vitreous silica in Ref. [163]. As suggested by other earlier works on

69

Figure 5.2: Literature data of low temperature heat capacities of several amorphous
materials. (A) Heat capacity of SiO2 versus the temperature below 1 K [148]. Below
1 K, the heat capacity is approximately linearly proportional to the temperature.
Observed heat capacity exceeds the Debye specific heat of vitreous silica (dashed
line). (B) Scaled heat capacity versus the temperature for Se [179] and Cer-vit [180].
Shown is the heat capacity divided by T3 . Peak of the heat capacity is observed at
∼ 5 K (∼ 15 K) for Se (Cer-Vit). The heat capacity of Se exceeds the calculation
using the Debye prediction (dotted line). For Cer-Vit, the heat capacity demonstrates
a temperature dependence below 4 K with a best fit of 5𝑇 + 0.8𝑇 3 (dashed line),
indicating a deviation from the Debye prediction.

amorphous materials [149, 181, 182], it is considered that there are three different
frequency dependent regimes in amorphous materials. At the high frequency regime,
the MFP is nearly independent of the temperature with a value less than 1 nm,
due to the strong scattering by atomic disorder. As the frequency decreases, the
MFP exhibits a rapid increase with a frequency dependence Λ ∝ 𝜔−4 , analogous
to Rayleigh scattering of the phonons and photons. In this frequency range, prior
literature concluded that the wavelength is sufficiently longer than the disorder length
scale, yielding a Rayleigh-type trend of the scattering. At very low frequencies, for
example 𝜔 < 0.01𝜔 𝐷 in Fig. 5.3, the frequency dependence becomes weaker with
Λ ∝ 𝜔−2 . For these frequencies, it is discussed that the excitations are damped
by structural relaxation of two level systems, an asymmetric double-well potential
induced by the impurities [146, 183].
Figure 5.4 shows the internal friction of several amorphous materials versus the
temperature measured using torsional oscillator [184]. Here the internal friction
is proportional to the ratio of the attenuation coefficient to the frequency of kHz
acoustic excitations. With a decreasing temperature, the magnitude of the acoustic
damping in glassy substances decreases above 100 - 300 K, below which that

70

Figure 5.3: The mean free path of vibration (Λ) versus the frequency normalized by
𝜔 𝐷 in vitreous silica in Ref. [163]. From high frequency above 0.1𝜔 𝐷 , the MFP is
order of sub-nm with independence of the frequency due to atomic disorder. Below
0.1𝜔 𝐷 , a Rayleigh-type trend of the MFP (Λ ∝ 𝜔−4 ) is observed up to 𝜔 = 0.04𝜔 𝐷 ,
below which Λ is proportional to 𝜔−2 , indicating different scattering mechanisms
of the acoustic excitation. Λ ∝ 𝜔−2 is due to the damping via structural relaxation
by two-level system.

increases. In this temperature range, the damping of the vibration is highly dependent
on the frequency [148] and the chemical composition [184]. As in Fig. 5.4, for
instance, the attenuation of ∼ 100 kHz vibrations reaches its minimum at ∼ 300
K, which increases up to ∼ 100 K for sub-kHz vibrations. Below 100 - 300 K,
as the temperature decreases, the magnitude of the damping again decreases with
lesser apparent temperature dependence compared to the higher temperature range
as shown in 5.4. At these temperatures, the damping is known to be independent
of the chemical composition; it was reported that the magnitude of the damping
falls into similar values, is separated by less than a factor of 20 as indicated by
the glassy range as shown in Fig.5.4. The trend of the constant magnitude of the
acoustic damping continues up to ∼ 1 K, below which the damping again possesses
temperature and frequency dependence.

71

Figure 5.4: The internal friction versus the temperature for several glassy materials
in Ref. [184]. The internal friction (𝑄 −1 ) is defined by 𝑣𝛼/𝜔, where 𝑣 is the sound
velocity, 𝜔 is the acoustic frequency, and the 𝛼 is the attenuation coefficient [178]. At
a high temperature up to 100 - 300 K, the internal friction exhibits a clear temperature
dependence, which varies depending on the chemical composition. Below 100 - 300
K, as the temperature decreases, hence the internal friction decreases with weaker
temperature dependence. The values of the internal friction in this temperature
range for amorphous materials are reported to fall into similar values, separated
within a factor of 20 as indicated by the glassy range (dashed line) [178] . For
temperatures less than 1 K, the temperature dependence of the internal friction
again shows marked temperature dependence.

Several empirical models have been proposed to explain the anomalies mentioned
above. The minimum thermal conductivity model was employed to explain the
thermal transport amorphous materials [185, 186]. The two level system (TLS) was
developed to explain both the anomalies of the excess heat capacity and the damping
mechanism in amorphous materials [146, 149, 183].
5.3

Prior empirical models

As discussed in 5.1, several empirical models have been proposed to explain the
anomalies present in amorphous materials. In this section, we briefly discuss

72
previous descriptions.
Minimum thermal conductivity
The minimum thermal conductivity is one of the empirical approaches to explain
the thermal transport in glassy substances. The proof of concept was proposed by
Einstein [187], and the theory was applied to glasses by Cahill [185, 186]. The
minimum thermal conductivity assumes the heat transfer process as a random walk
of the heat energy. The minimum thermal conductivity assumes that the atomic
vibrations oscillate near the equilibrium point and that they have an incoherent
phase. Corresponding distance resulting from the random walk is Λ = 𝜋/𝜔 which
gives the expression for the lower limit to the thermal conductivity
 𝜋  1/3
Õ  𝑇  2 ∫ 𝜃 𝑖 /𝑇 𝑥 3 𝑒 𝑥
2/3
𝑘 𝐵𝑛
𝑣𝑖
(5.1)
𝜅 𝑚𝑖𝑛 =
𝑥 − 1) 2
(𝑒
where 𝜃 𝑖 is the Debye temperature, and 𝑛 is the number density. It was discussed
that this theory is not applicable to the crystals, since periodic arrangements of the
atoms are known to produce a coherence between the vibrations [186]. On the
other hand, Ref. [186] demonstrated that the prediction from the minimum thermal
conductivity agrees with experimental observations in amorphous materials.
However, the applicability of this theory should be re-examined. In fact, the minimum thermal conductivity is not favorable for explaining the transport of the low
energy excitations. First, analogous to acoustic phonons in crystals, the wavelength of the low energy excitations is relatively long, indicating that the excitations
propagate collectively rather than randomly diffuse within disordered length scales.
Second, the frequency dependent transport properties are missing in the minimum
thermal conductivity theory. Vibrations in solids are dispersive and corresponding transport properties are different depending on the frequency. Therefore, if
the mechanism of the damping of the excitations would be different at higher frequencies, the corresponding frequency dependence of the MFP should be different,
which the minimum thermal conductivity model cannot precisely capture.
Two-level system (TLS)
Earlier works have described the linear temperature dependence of the heat capacity by the two-level system (TLS) damping of the low energy excitations, initially
proposed by Anderson, Halperin, and Varma [149]. According to this model, the
damping of the acoustic excitation is due to resonant absorption at very low temperatures (typically less than 5 K), above which is due to thermally activated relaxation

73
[146, 183]. In the following, we briefly describe the frequency dependence of the
MFP induced by resonant absorption.

Figure 5.5: Assymetric potential well versus the position proposed by Anderson,
Halperin, and Varma [149]. The figure was taken from Ref. [149], and modified.
Briefly, this theory assumes that the impurities induce an asymmetric double-well
potential system with a constant density as follows
𝑘 𝐵𝑇
(5.2)
𝐶𝑇 𝐿𝑆 ∝
(ℏ𝜔)𝑛(𝜔)𝑑𝜔 ∝
The theory of the TLS assumes that the elastic waves near TLS deform the doublewell potential, which causes the tunneling of acoustic excitation through the potentials. Here, we describe an earlier mathematical framework proposed by Anderson,
Halperin, and Varma [149]. The interaction between local TLS and the phonon can
be described by the first-order perturbation theory where the Hamiltonian can be
described by
𝛿𝜖 0
1 𝜖1 Δ
𝐻 = 𝐻0 + 𝐻1 =
(5.3)
2 Δ 𝜖2
0 𝛿𝜖
where the 𝜖1 (𝜖2 ) is the local minima of the asymmetric potential, 𝛿𝜖 is the deformation potential, and Δ is the zero-point energy due to the overlap of the wavefunction
at the local minima. The mean free path of the acoustic excitation for ℏ𝜔 > 𝑘 𝐵𝑇
Λ(𝜔) = (𝜎h𝜏 −1 i) −1

(5.4)

74
where 𝜎 = 4𝜋𝑣 2 𝜔2 is the scattering cross section, and h𝜏 −1 i is the ensemble averaged
transition rate between the eigenstates of the matrix above. Applying a Fermi-golden
rule under an approximation of ℏ𝜔 = 𝜖 12 + 𝜖22 (resonant absorption), the transition
rate is given by
Δ2
3𝜋𝐵
𝜏 −1 =
(5.5)
𝑚𝑣 2 𝜔3𝐷 ℏ
−1
h𝜏 i =
𝑝(𝜖 1 − 𝜖2 ; Δ; 𝐵)𝜏 −1 𝑑𝐵𝑑Δ 𝑑 (𝜖1 − 𝜖2 )
(5.6)
where 𝑝 is the probability distribution over ensemble of levels, and 𝐵 is the differences in the constants for the deformation potential. Corresponding MFP can be
simplified to
Λ ∝ 2 𝜔−1
(5.7)
Finally, considering the temperature dependence, an empirical factor was added
such that
ℏ𝜔
1 −1
(5.8)
Λ𝑟𝑒𝑠 ∝ 2 𝜔 coth
2𝑘 𝐵𝑇
where Λ𝑟𝑒𝑠 is the MFP due to resonant absorption. For 𝜔
𝑘 𝐵𝑇/ℏ, Eq. 5.8 can be
simplified to 𝑙 ≈ 𝜔2 /𝑇 for 𝜔
𝑘 𝐵𝑇/ℏ. The theory of the TLS has shown excellent
agreement with the low-temperature thermal conductivity as well as the attenuation
of the low frequency excitations below a few K [148].
Density fluctuation
The density fluctuation is based on the structural inhomogeniety as a primary mechanism for the scattering of the acoustic excitations [182]. The model considers free
excess sub-volumes in an amorphous structure [188]. The re-distribution of the
sub-volumes would occur, resulting in the local density deviating from the average
density. The local density deviation becomes a scattering cross-section for the excitations, similar to point defect phonon scattering. Here, we briefly describe the
frequency dependence shown in Ref. [182].
We consider a sub-volume 𝑣 with a different density (𝜌) in an amorphous structure.
The corresponding scattering cross-section for an acoustic excitation is given by
𝜎1 ≈

1 4 2 Δ𝜌 2
𝑞 𝑣 (
12𝜋

𝜎2 ≈ 𝑣 −1/3 (

Δ𝜌 2

(𝑞𝑣 1/3 < 1)
(𝑞𝑣 1/3 > 1)

(5.9)
(5.10)

75
where 𝑞 is the wave vector of the acoustic excitation, and 𝜌 is the density deviation.
The above expression is analogous to the scattering cross-section for Rayleigh
scattering and geometric scattering, respectively.
First, considering a short wave vector excitations, the corresponding mean free path
of the excitation from Eq. 5.9 is
Λ−1
1 =

𝑃(Δ 𝜌)

Δ𝜌

1 4 2 Δ 𝜌2
1 4 2 Δ𝜌 2
𝑞 𝑣 (
) =
𝑞 𝑣
12𝜋
12𝜋
𝜌2

Δ 𝜌2

𝜌2

𝑉𝑝
𝑣𝑁 (1 − 𝑝)

(5.11)

(5.12)

where 𝑃 is the probability of the given density fluctuation, Δ 𝜌 2 is the mean square
deviation of the density, 𝜌 2 is the average density, N is the number of the subvolumes, and (𝑝) is the probability for density deviation. For the volumes from 𝑣
to 𝑣 + 𝑑𝑣, using Eq. 5.12 and Eq. 5.11, the corresponding MFP of the excitation is
given by

Λ−1
1 =

𝑞4 𝑝
𝑑𝑣
12𝜋 1 − 𝑝

(5.13)

which yields Λ−1 ∝ 𝑞 4 for short wave vector limit.
Similarly, for 𝑞𝑎 > 1,

Λ−1
2 =

𝑃(Δ 𝜌)𝜎2 ≈ 𝑣 −4/3

Δ𝜌

𝑑𝑣
1− 𝑝

(5.14)

The integration over 𝑑𝑣 gives an expression for the MFP

−1

−1
Λ−1
(𝑣,
𝑞)
(𝑣,
𝑞)
𝑑𝑣 ∝ 𝑞

(5.15)

The above expression for the MFP can be re-written as the MFP versus the frequency
at different frequency range
Λ ∝ 𝜔−4

(5.16)

for high frequency, whereas for low frequency,
Λ ∝ 𝜔−1

(5.17)

76
The density fluctuation theory has shown a good agreement with the thermal conductivity data [18, 182], provided that the information of the density deviation is
well-defined. However, the theory may not be applicable to the damping of the
excitation below 1 K, since the extremely large length scale of the fluctuation is
required for explaining the damping at very low temperatures.
5.4

Picosecond sound acoustics (PSA) on aSi

Having discussed the transport of heat-carrying acoustic excitations in amorphous
materials, we now present our novel approach for measuring frequency resolved
MFPs of acoustic excitations in amorphous silicon. We first describe the measurements of 100 GHz excitation using the picosecond acoustics. Then we describe a
table-top approach to measure the MFP over the frequency range from ∼ 0.1 − 3
THz.
Background
Here, we present a pump-probe scheme used for measuring the damping of the
∼ 100 GHz excitations in aSi. The attenuation coefficient of hypersonic waves in
aSi films on sapphire substrates was measured using picosecond acoustics (PSA).
The PSA was developed to generate and detect wave packets of the vibration over a
wide range of the frequencies and the temperatures [189]. Such an objective led to
a development of an experimental setup with sub-ps temporal resolution. Since the
details were discussed in many prior works, here we only briefly discuss the PSA
[189]. The principle of the PSA is similar to that of time domain thermoreflectance
(TDTR), therefore we prepared a PSA optical system that is identical to TDTR.
Briefly, a train of the pulses (repetition rate: 76 MHz, wavelength 785 nm) is split
into pump (1/e2 diameter: 13.7 𝜇m, power: ∼ 10 mW) and probe (1/e2 diameter:
12.4 𝜇m, power: ∼ 3 mW) using a two-tint color method [49]. Sharp-edge filter
(short pass filter, Semrock inc.) was used to spectrally separate the pump and probe
at the wavelength near 785 nm. The pump is amplitude-modulated at 9.2 MHz and
focused onto the sample. The pump absorption induces a thermal expansion of a
transducer, thermoelastically launching a longitudinal strain pulse.
Considerations in the setup
As in TDTR, it is important to control the beam size in PSA. In PSA, the actual
signal is the mixture of the thermal background signal and the acoustic echo signal.
Since the magnitude of the acoustic echo signal to that of the thermal background

77

Figure 5.6: Schematic of PSA measurement system in the Minnich lab at Caltech.
Pump and probe pulses are spectrally separated at the wavelength near 785 nm,
using a sharp-edge short pass filter. Pump pulse train is amplitude-modulated
using an electro-optic modulator, and impulsively induces a thermal expansion of
a transducer, thermoelastically launching a longitudinal strain wave packet. The
probe travels additional 4 m to provide a maximum temporal delay ∼ 13 ns with
sub-ps temporal resolution. The acoustic strain wave packet is monitored by probe
as a transient change of the reflectivity.

is typically less than a few percent, therefore we put our effort to reduce laser
spot diameter to be comparable to 10 𝜇m. The small spot diameter enables to
minimize the uncertainties of measurements that may be arisen from the variations
in the sample thickness over the laser spot area. For this purposes, we used a 10X
objective lens with a working distance of 33.5 mm (10X EO M plan apo, Edmund
optics inc.). Next, we discuss the choice of the optimal optical energy and the
pump modulation frequency. Considering the measurement system as a linear-time
invariant (LTI) system [190], the corresponding thermal transfer function is
𝑧(𝑡) = 𝑍 (𝜔0 ) exp[𝑖𝜔0 𝑡]
𝑍 (𝜔0 ) ∝ 𝑄 𝑝𝑢𝑚 𝑝 𝑃 𝑝𝑟𝑜𝑏𝑒
𝐻 (𝜔0 𝑛)

(5.18)
(5.19)

78

Figure 5.7: Spot radius of (A) pump and (B) probe in PSA setup. The symbols
represent the knife-edge measurement and the dashed lines fit. Corresponding spot
diameter at the sample position is 13.7 𝜇m and 12.4 𝜇m for pump and probe,
respectively.

where 𝑧(𝑡) (𝑍 (𝜔)) is the thermal response function in the time domain (frequency
domain), 𝜔0 = 2𝜋 𝑓0 is the angular modulation frequency, and 𝐻 is the heat impulse
response. From Eq. 5.19, we see important factors which determine the magnitude of the signal. First, the signal is proportional to pump energy and the probe
power. Second, since the magnitude of the impulse response becomes smaller as
the modulation frequency increases, the corresponding signal magnitude at higher
modulation frequency decreases. Therefore, it is necessary to decrease the modulation frequency to increase the signal magnitude. However, it is known that the
electronic noise (1/ 𝑓 noise) simultaneously increases as the modulation frequency
decreases [190]. Therefore, to maximize both the signal magnitude with ample
signal-to-noise ratio (SNR), we systematically change the pump power and the
pump modulation frequency. Figure 5.8A and B shows the measured average signal
magnitude and corresponding SNR versus pump power from aSi on different substrates. As expected, the signal magnitude increases with increasing pump power,
and the corresponding SNR increases up to around 10 mW, above which the value
saturates. Figure 5.8C and D shows the signal magnitude and the corresponding
SNR versus the pump modulation frequency. We find a substantial increase in SNR
at modulation frequency of around 9.2 MHz, while its signal magnitude is relatively
smaller than that lower modulation frequency. Considering these observations, a
pump power of 10 mW and the modulation frequency of 9.2 MHz were used in our
actual experiments.
Lastly, we employed a fast-scanning system with a modification in Lab-view programming. Concerning typical PSA signal traces in aSi, the acoustic echoes that

79

Figure 5.8: Average signal magnitude and corresponding SNR versus the pump
power in aSi on (A) Tungsten substrate and (B) Sapphire substrate. While the
average signal magnitude increases as the pump power increases, the corresponding
value of the SNR saturates at 10 mW. Average signal magnitude and corresponding
SNR versus the modulation frequency in aSi on (C) Tungsten substrate and (D)
Sapphire substrate. The average signal magnitude decreases at higher modulation
frequencies with substantial increase in SNR. The average signal magnitude was
obtained from 10 averages. Then the average signal was divided by the subtraction
between each signal and the average from which we determined the SNR.

are generated by the strain pulses have a temporal duration less than ∼ 40 ps. For a
500 nm thick aSi sample, the temporal separation between each echo is > 100 ps,
which increases to ∼ 1 ns for a 4 𝜇m thick sample. This means that a huge amount
of the data would be unnecessarily acquired for a long time without adjusting the
delay step [191]. Therefore, we employed adjustable temporal delay stepping using Labview programming. The adjustable delay stepping is advantageous in that
acoustic echoes can be measured with a fine linear time step, while unwanted signal
components can be captured using a logarithmic time step. Figure 5.9A shows our
Labview designing which enables a sequential linear and logarithmic time stepping.
The signal measured using this program is shown in Fig. 5.9B. To describe the signal
acquisition shown in Fig. 5.9B, initially, the data were collected with a linear step for
time delay ±20 ps for correcting instrumental phase offset. Acoustic echoes were
measured using subseqauent linear scans, while other unwanted signal components

80

Figure 5.9: (A) Image of PSA data acquisition software (a panel of Labview). The
software was programmed to perform linear to logarithmic sequential scanning,
therefore enabled to selectively scan acoustic echo signals (highlighted as a red
rectangle) (B) Representative signal traces measured in a 500 nm thick aSi sample
using the program shown in (A). The data was collected with 1 ps time step for -20
- 20 ps (phase offset part), 125 - 160 ps (1st echo component), 250 - 300 ps (2nd
echo component). Unwanted signal components were collected using logarithmic
time step with ∼ 20 ps spacing.

were collected using logarithmic scans.
PSA sample preparation
To measure the damping of 100 GHz acoustic excitation, we prepared the PSA
sample using plasma-enhanced chemical vapor deposition (PECVD) described in
Ref. [151]. The aSi film was deposited on a sapphire substrate (University Wafer inc.)
and a variable deposition time to prepare three samples with different thicknesses: ∼
500 nm, ∼ 1 𝜇m, and ∼ 4 𝜇m. Al with 15 nm was deposited afterwards by electron
beam evaporation to serve as a transducer.
PSA measurements and data analysis
The damping of 100 GHz excitation was characterized with the sample prepared with
a method in Sec. . A 15 nm Al film was chosen as a transducer to generate an acoustic
of pulse around ∼ 100 ± 20 GHz that propagates through the aSi as in Ref. [15].
The strain pulse experiences multiple reflections between the Al transducer and the
sapphire substrate, producing changes in reflectance of the transducer film. Sapphire
was chosen as a substrate to enhance the magnitude of the echo signals based on the
acoustic mismatch model [15] (for sapphire, 𝜌 = 3.98 g𝑐𝑚 −3 , v 𝐿 𝐴 ∼ 11000 m𝑠−1 )
[192]. The resulting signal is detected using a lock-in amplifier.

81
The amplitude signal versus the delay time for ∼ 500 nm and ∼ 4 𝜇m samples
are plotted in Figs. 5.10 and 5.11. The resultant magnitude of the signal after one
(two) round trip(s) are shown as Δ 𝑅1 (𝑡) (Δ 𝑅2 (𝑡)). Following Ref. [15], we analyze
the data using Fourier analysis. First, we remove the background signal in Δ 𝑅1 (𝑡)
and Δ 𝑅2 (𝑡). Then, Δ 𝑅2 (𝑡) is normalized by the maximum magnitude of Δ 𝑅1 (𝑡).
Next, the signals are zero-padded to improve the frequency resolution of the fast
Fourier transform (FFT). Finally, the time domain signals are windowed using a
Hann window and the FFT is performed.

Figure 5.10: (A) (Left) Measured amplitude signal from ∼ 500 nm aSi versus
delay time using picosecond acoustics at 300 K. The thermal background signals
have been removed. Echo 1 occurs at the Al transducer after a round trip between
the transducer and the substrate. The echo 2 is the attenuated signal from echo
1 after one more round trip. The thickness of the aSi film was determined from
the time difference between the echoes and the known LA sound velocity. (Right)
Fast Fourier transform (FFT) magnitude of echo 1 and echo 2 at 300 K. The peak
of the FFT for the first echo occurs at ∼ 100 GHz, indicating that a 15 nm Al
transducer can generate acoustic waves with frequencies ∼ 100 GHz as expected.
Note that the attenuation for the 500 nm film is due to the reflection or interface loss
at the boundary between the aSi and the sapphire. Amplitude signal along with the
corresponding magnitude of the Fourier transform measured at (B) 220 K, (C) 150
K, (D) 80 K, (E) 50 K, and (F) 30K.

82

Figure 5.11: (Left) Measured thermoreflectance signal from a 4 𝜇m sample versus
delay time using picosecond acoustics at (A) 300 K, (B) 220 K, (C) 150 K, (D) 80
K, (E) 50 K, and (F) 30 K. The thermal background has been removed. (Right)
Magnitude of Fourier transform of echo 1 and echo 2 at the same temperatures.
The resulting Fourier power spectra of each echo is plotted in Figs. 5.10 and 5.11.
The peak of the FFT occurs at ∼ 100 GHz, indicating that a 15 nm Al transducer can
generate acoustic wave with a frequency ∼ 100 GHz as expected. As discussed in
Ref. [193], the PSA signals consist of wave packets, meaning that the corresponding
response is frequency dependent. Therefore, following Ref. [15, 193, 193, 194],
rather than treating each Fourier amplitude separately, we compare Fourier amplitude at 100 GHz in Fig. 5.12A. As in Ref. [15], we find that the ratio of the
Fourier magnitude at 100 GHz for 500 nm thickness is close to what is predicted by the acoustic mismatch model (AMM) between aSi and the sapphire,
𝑟 = (𝜌 𝑠𝑎 𝑝 𝑝ℎ𝑖𝑟𝑒 𝑣 𝑠𝑎 𝑝 𝑝ℎ𝑖𝑟𝑒 − 𝜌𝑎𝑆𝑖 𝑣 𝑎𝑆𝑖 )/(𝜌 𝑠𝑎 𝑝 𝑝ℎ𝑖𝑟𝑒 𝑣 𝑠𝑎 𝑝 𝑝ℎ𝑖𝑟𝑒 + 𝜌𝑎𝑆𝑖 𝑣 𝑎𝑆𝑖 ) = 0.42 = 2.4−1 ,
indicating that the loss for 500 nm thickness is dominated by transmission into the
sapphire substrate. The attenuation coefficient for 100 GHz vibrations is obtained
from
𝑟Δ 𝑅1 (𝜔)
(5.20)
𝛼 = ln
Δ 𝑅2 (𝜔)

83

Figure 5.12: (A) Ratio of the Fourier magnitude at 100 GHz measured at temperatures from 30 K to 300 K. The black dashed line indicates the predicted loss at
the boundary between aSi and the sapphire from the AMM. Following Ref. [15],
external losses were experimentally deduced from the ratio at ∼ 500 nm. Internal
damping was obtained from the values at ∼ 4 𝜇m using Eq. 5.20. The film thicknesses were obtained from the time difference between the echoes. The error bar
indicates standard deviation determined from multiple measurements. (B) Attenuation coefficients versus temperature using the 500 nm data for reflection loss and
∼ 4 𝜇m data for attenuation. (C) Mean free path versus temperature for the 100
GHz vibrations.
where 𝑑 is the round trip distance, 𝑟 is the inverse of the value from 500 nm aSi in
Fig. 5.12A, and Δ 𝑅1 (𝜔)/Δ 𝑅2 (𝜔) is the value from the 4 𝜇m thickness sample in
Fig. 5.12A.
The calculated attenuation coefficient is shown in Fig. 5.12B. The measured coefficient is ∼ 102 - 103 𝑐𝑚 −1 with its maximum value at 300 K (987 ± 197 𝑐𝑚 −1 ). The
measured value at room temperature is close to that reported from PSA in aSi (780
± 160 𝑐𝑚 −1 )[15], but substantially lower than that of vitreous silica (∼ 10000 cm−1
at ∼ 100 GHz) [195]. Prior studies have reported that fourfold-coordinated covalent
bonded material tends to exhibit a weaker damping near 5 kHz which may be the
reason for our results. [15, 178]. The mean free path of 100 GHz vibrations versus
temperature is shown in Fig. 5.12C. Our measured value is ∼ 10 − 20 𝜇m from 30 K
(∼ 17.5 ± 4 𝜇m) to 300 K (∼ 10.5 ± 2.2 𝜇m) with a weak temperature dependence.
5.5

Thermal characterization of aSi

In the preceding section, we characterized the damping of the 100 GHz acoustic
excitation using PSA. In the following section, we utilize our PSA results to relate
it to the thermal transport properties.

84
Preparation of 500 nm amorphous silicon membrane
We prepared free-standing membrane of aSi using the following method. First, the
amorphous silicon was deposited on silicon nitride handle wafer using PECVD with
silane gas dilute (5%) in argon gas at a deposition table temperature ∼ 473 K [151].
The thickness of the aSi layer was 500 nm. Finally, the handle wafer was dry-etched
to create a rectangular aperture with a 1 mm2 cross-sectional area. A scanning
electron microscope image of the sample indicates clean and smooth surface over
the membrane area, which is given in Fig. 5.13A.
Self-heating and optical powers
As mentioned in the previous section, care was taken to minimize the steady state
and transient temperature rise of the sample while maintaining a signal magnitude
of at least 300 - 400 𝜇V. Here, we present an estimation of the steady and transient
temperature rises that occurred during the experiment and justify the chosen optical
powers.
Estimating the self-heating requires knowledge of the optical absorptance of the
sample. We measured temperature-dependent reflectance and transmittance from
which we estimated the absorptance of the amorphous silicon film. Table 5.1 shows
the measured values at the cryostat temperture (T𝑐 ).
T𝑐 (𝐾)
300
210
135
60
40

Reflectance (%)
39
35
34
32
36

Transmittance (%)

Absorptance (%)
58
62
61
63
59

Table 5.1: Optical properties of aSi measured at various temperatures.
Accounting for optical reflections at various optical elements, the absorptance is
around 50 - 55% at the temperatures in this study. For simplicity, we use 60% for
the calculations below.
Next, estimate the steady heating due to the pump and probe pulses using a thermal
resistor model in cylindrical coordinates. Consider the pump and probe beams
as heat sources of radius 𝑟 𝑝𝑢𝑚 𝑝 ≡ 𝑟 1 . The outer radius 𝑟 2 ≈ 500 𝜇m of the
membrane is fixed at the cryostat temperature. The cylindrical conduction resistance
is 𝑅 = ln(𝑟 2 /𝑟 1 )/(2𝜋𝜅𝑑) where 𝜅 is the bulk thermal conductivity of the membrane.
The temperature rise is then

85

Δ𝑇 = 𝑃𝑎𝑏𝑠,𝑎𝑣𝑔

ln(𝑟 2 /𝑟 1 )
2𝜋𝜅𝑑

(5.21)

where 𝑃𝑎𝑏𝑠,𝑎𝑣𝑔 is the absorbed average power of the laser beam. The absorbed
average power was calculated using 𝑃𝑎𝑏𝑠,𝑎𝑣𝑔 = 𝛼𝑃𝑖𝑛𝑐,𝑎𝑣𝑔 = 𝛼𝑃 𝑝𝑒𝑎𝑘 𝐷 where P𝑖𝑛𝑐,𝑎𝑣𝑔
is the incident average power on the sample, P 𝑝𝑒𝑎𝑘 is the peak power incident on the
sample, and 𝐷 is the duty cycle of the laser beam (1 for pump and 0.032 for probe
from a chopper wheel).
Table 5.2 shows the quantities used for steady temperature rise estimation. In Tab. 5.3
𝑇ℎ is the cryostat temperature, 𝜅 is the thermal conductivity used for calculation,
and 𝑃 𝑝𝑢,𝑖𝑛𝑐,𝑎𝑣𝑔 (𝐸 𝑝𝑟,𝑖𝑛𝑐,𝑎𝑣𝑔 ) is the incident average power of the pump (probe) on
the sample. The corresponding estimated steady temperature is shown in Tab. 5.3
where Δ𝑇𝑝𝑢 (Δ𝑇𝑝𝑟 ) is the steady temperature rise due to pump (probe), Δ𝑇𝑡𝑜𝑡𝑎𝑙 is
the estimated total steady temperature rise of the sample, and 𝑇𝑠𝑡 is the estimated
temperature of the sample.
𝑇ℎ (𝐾)
300
210
135
60
40

𝜅 (Wm−1 𝐾 −1 )
1.5
0.85
0.7
0.55

𝑃 𝑝𝑢,𝑖𝑛𝑐,𝑎𝑣𝑔 (𝜇W)
41
27
48
48
44

𝑃 𝑝𝑟,𝑖𝑛𝑐,𝑎𝑣𝑔 (𝜇W)
134
88
71
61
46

Table 5.2: Quantities used for the estimation of the steady heating. The optical
powers are actually used values for the measurements.
𝑇ℎ (𝐾)
300
210
135
60
40

Δ𝑇𝑝𝑢 (𝐾)
3.4
3.4
7.0
8.5
10.1

Δ𝑇𝑝𝑟 (𝐾) Δ𝑇𝑡𝑜𝑡𝑎𝑙 (𝐾)
11.2
14.6
11.0
14.4
10.5
17.5
10.9
19.4
10.5
20.6

𝑇𝑠𝑡 (𝐾)
∼ 315
∼ 225
∼ 150
∼ 80
∼ 60

Table 5.3: Estimation of the steady heating of the sample based on the quantities
given in Tab.5.2.
Finally, we estimate the impulsive temperature rise induced by the absorbed optical
energy. The temperature rise Δ𝑇 can be estimated from
Δ𝑇 =

𝐸 𝑎𝑏𝑠
𝐶𝑣 𝑉

(5.22)

86
where 𝐸 𝑎𝑏𝑠 is absorbed laser energy, 𝑉 is the volume of the sample illuminated by
the beam, and 𝐶𝑣 is the volumetric heat capacity [196]). The volume 𝑉 = 𝜋𝑟 2 𝑑
where 𝑑 ≈ 500 nm is the estimated thickness of the film, and 𝑟 is the 1/𝑒 2 beam
radius (pump: 260 𝜇𝑚, and probe: 235 𝜇𝑚).
The absorbed laser energy was calculated using 𝐸 𝑎𝑏𝑠 = 𝛼𝐸𝑖𝑛𝑐 where 𝛼 is the
measured absorptivity and 𝐸𝑖𝑛𝑐 is the laser energy. The incident laser energy was
𝑖𝑛𝑐
𝑖𝑛𝑐
𝑖𝑛𝑐
𝑖𝑛𝑐
obtained using 𝐸 𝑖𝑛𝑐
𝑝𝑢 = 𝑃 𝑝𝑢 / 𝑓 for pump and 𝐸 𝑝𝑟 = 𝑃 𝑝𝑟 𝑡 𝑐ℎ𝑜 𝑝 for probe where 𝐸 𝑝𝑢
𝑖𝑛𝑐
𝑖𝑛𝑐
(𝐸 𝑖𝑛𝑐
𝑝𝑟 ) is the pump (probe) energy incident on the sample, 𝑃 𝑝𝑢 (𝑃 𝑝𝑟 ) is the incident
pump (peak probe) power, 𝑓 is the repetition rate of the pump (200 Hz), and the
𝑡 𝑐ℎ𝑜 𝑝 is the time duration of the probe (160 𝜇s).
𝑇𝑠𝑡 (K) 𝐶𝑣 (J𝑐𝑚 −3 )
315
2.27
225
150
1.5
80
0.73
60
0.52

𝐸 𝑖𝑛𝑐
𝑝𝑢 (𝜇J)
0.2
0.1
0.2
0.2
0.2

𝐸 𝑖𝑛𝑐
𝑝𝑟 (𝜇J)
0.7
0.4
0.4
0.3
0.2

Δ𝑇𝑝𝑢 (K)
0.5
0.4
0.9
1.9
2.4

Δ𝑇𝑝𝑟 (K)
2.0
1.5
1.6
2.9
3.1

Δ𝑇𝑡𝑜𝑡𝑎𝑙 (K)
∼3
∼2
∼3
∼5
∼6

Table 5.4: Estimation of the transient heating of the sample at several temperatures.

Table 5.4 shows the estimated transient temperature rise where 𝑇𝑠𝑡 is the estimated
temperature accounting for the steady temperature rise, 𝐶𝑣 is the heat capacity at
𝑇 = 𝑇𝑠𝑡 in Ref. [196], Δ𝑇𝑝𝑢 (Δ𝑇𝑝𝑟 ) is the transient temperature rise due to pump
(probe), and Δ𝑇𝑡𝑜𝑡𝑎𝑙 is the overall transient temperature rise of the sample.
Bulk thermal conductivity of aSi
We conducted TG measurements versus grating period at several temperatures and
versus temperature at a fixed grating period of 𝐿 = 10.7 𝜇m. In our experiment,
the maximum grating period used is 15.7 𝜇𝑚. Figure 5.13B shows representative
normalized signal traces at ∼ 315 K. Shown are average heterodyne signals (104
averages) taken at a single location. Note that measured diffusivities in the main text
were obtained by measuring multiple locations from which we determined the mean
value and error bars. Additional data along with fits are shown in Fig. 5.14. The
decay exhibits a single exponential profile with a time constant in the range of tens
of nanoseconds to microseconds due to thermal transport. The thermal diffusivity is
obtained by extracting the time constant of the single exponential decay; the thermal
conductivity is then computed using the measured heat capacity from Ref. [196].

87

Figure 5.13: (A) Schematic of the TG experiment. Two pump laser pulses are
interfered on the sample, impulsively creating a spatially periodic temperature rise.
The probe beams diffract from the transient grating, monitoring the thermal relaxation. A scanning electron microscope image of the free-standing amorphous
silicon membrane is shown. (B) Representative measured signals versus time and
single exponential fits for aSi at 315 K for various grating periods.

Figure 5.14: Measured TG signal versus time from 60 K to 315 K at several grating
periods.

88

Figure 5.15: Measured thermal conductivity versus temperature for 𝐿 = 10.7
𝜇m. Literature data are 3𝜔 measurements on a 520 nm thick film (open squares,
Ref. [197]), and thin film microcalorimetry measurements on a 278 nm thick film
(open stars, Ref. [198]).

Within the uncertainty of the measurement, the thermal diffusivity does not vary
with a grating period above 10.7 𝜇𝑚, so we take the value at 10.7 𝜇𝑚 to be the
bulk value. The thermal conductivity versus temperature for 10.7 𝜇𝑚 is given in
Fig. 5.15 along with literature measurements; each point represents the average of
∼ 104 measurements taken at multiple locations on the membrane. Our measured
values are generally in good agreement with prior reports.
Ballistic transport of acoustic excitations in aSi
The measured TG signal for a grating period of 𝐿 = 754 nm is shown in Fig. 5.16A.
The actual thermal decay is clearly slower than that expected based on the bulk
thermal conductivity value, indicating the presence of acoustic excitations with
MFPs comparable to the grating period. Measurements of the decay rate versus 𝑞 2
for all the grating periods at 80 K are given in Fig. 5.16B. The measured decay rate is
close to that predicted by the bulk thermal conductivity up to around 𝑞 2 ∼ 4.6 𝜇𝑚 −2
(𝐿 ∼ 3 𝜇m), above which the decay rate becomes smaller. The maximum relative
deviation in time constant is on the order of 50% as shown in Fig. 5.16C. Figure
5.16D shows the measured thermal conductivity versus grating period obtained
from these time constants at two of the five temperatures considered in this study.
As the grating period becomes comparable to some MFPs, the effective thermal

89
conductivity varies with the grating period.

Figure 5.16: (A) Measured TG signal versus time (symbols) for grating period
𝐿 = 754 nm, corresponding to 𝑞 2 ∼ 70 𝜇𝑚 −2 , along with the best fit (solid red
line) and expected decay from the diffusion theory (dashed black line). The actual
signal decays slower than predicted, indicating a departure from diffusive thermal
transport. (B) Inverse time constant versus 𝑞 2 . The measured decay rates follow a
linear trend of diffusive thermal transport for 𝑞 2 . 4.6 𝜇𝑚 −2 (𝐿 & 3 𝜇m), above
which a clear deviation is observed. (C) Relative difference (𝑡 𝑎𝑐𝑡𝑢𝑎𝑙 − 𝑡 𝑏𝑢𝑙 𝑘 )/𝑡 𝑏𝑢𝑙 𝑘 in
the thermal decay time constant versus the grating period. (D) Measured thermal
conductivity versus grating period at 315 K and 60 K. The thermal conductivity
exhibits a dependence on the grating period, indicating the presence of heat-carrying
acoustic excitations with MFPs comparable to the grating period.

The grating period dependence of the thermal conductivity provides constraints on
the frequency-dependence of the MFPs of sub-THz excitations. Following Ref. [35],
the measured thermal conductivity 𝜅𝑖 can be expressed as
Õ ∫ 𝜔𝑚,𝑠
(5.23)
𝜅𝑖 =
𝑆(𝑥𝑖,𝑠 ) 𝐶𝑠 (𝜔)𝑣 𝑠 Λ𝑠 (𝜔) 𝑑𝜔 + 𝜅 𝐼 𝑅
where 𝑠 indexes the polarization, 𝑞𝑖 = 2𝜋𝐿 𝑖−1 , 𝑥𝑖,𝑠 = 𝑞𝑖 Λ𝑠 (𝜔), 𝑆(𝑥𝑖,𝑠 ) is the isotropic
suppression function in Refs. [34, 35], 𝜔𝑚,𝑠 is the cutoff frequency for collective

90
acoustic excitations, and 𝜅 𝐼 𝑅 is the contribution from excitations above the IoffeRegel (IR) cutoff frequency. The Debye heat capacity 𝐶𝑠 is calculated from the
group velocities 𝑣 𝑠 which are known, isotropic, and independent of temperature
[151, 174]. The first term of Eq. 5.23 is a Debye model for the thermal conductivity
of an isotropic solid that includes the effect of non-diffusive thermal transport over
a grating period.
The desired quantity is Λ𝑠 (𝜔), or the MFP versus frequency for LA and TA polarizations. Additional measurements can be used to further constrain Λ𝑠 (𝜔) before the
TG measurements are used. First, the linewidths of LA excitations at frequencies
above 3.7 THz are known from IXS measurements and are independent of temperature [151]. As the TA linewidths are not accessible with IXS, we use the values
from MD simulations as they quantitatively agreed with the measurements for the
LA branch. These values allow the thermal conductivity of collective excitations
above ∼ 3.7 THz to be obtained from Eq. 5.23; at room temperature this contribution
is ∼ 0.5 𝑊𝑚 −1 𝐾 −1 .
Second, the attenuation lengths at hypersonic frequencies can be obtained from
picosecond acoustics. The room temperature value is available from Ref. [15].
As presented in Sec. 5.4, we performed additional measurements of the acoustic
attenuation at temperatures from 30 - 300 K. The values are on the order of 10 - 20
𝜇m in this temperature range.
The lack of temperature dependence of the damping at both 100 GHz and ∼ 3 THz
suggests that the MFPs at intermediate frequencies should also be independent of
temperature. This requirement, the PSA and IXS measurements, and the measured
thermal conductivity versus grating period at the 5 temperatures in this study impose tight constraints on the frequency-dependence of the damping in the sub-THz
frequencies. Figure 5.17A shows two candidate MFP profiles that satisfy these
constraints for the LA and TA branches. Power law dependencies Λ ∼ 𝜔−𝑛 are assumed and combined using Matthiessen’s rule. Profile 1 transitions from a constant
value to 𝑛 = 4 corresponding to Rayleigh scattering, while profile 2 transitions from
constant to 𝑛 = 4 at ∼ 2 − 3 THz and then to 𝑛 = 1 at ∼ 200 − 300 GHz.
A representative plot of thermal conductivity versus grating period using Eq. 5.23
at 60 K is given in Fig. 5.17B. In this figure, 𝜅 𝐼 𝑅 has been taken as a parameter to
give the best fit for each profile. Profile 1 fails to reproduce the trend with grating
period. An alternate profile that increases as 𝑛 = 4 immediately at 3.7 THz yields a
thermal conductivity that exceeds the experimental values at all temperatures (not

91
shown). We find that only profiles consistent with profile 2 are able to explain the
magnitude and grating period dependence of the thermal conductivity. Specifically,
the MFPs must remain constant as frequency decreases and then increase rapidly as
𝑛 = 4. To agree with the PSA data, the trend must then switch to 𝑛 = 1 or 𝑛 = 2.
From the TG data, we are unable to determine this latter trend owing to the influence
of boundary scattering in the 500 nm thick membrane. The specularity parameter
required to produce MFPs on the order of microns with this membrane thickness
is ∼ 95%, a value that is consistent with prior studies of phonons in crystalline Si
in this frequency band [19]. For concreteness, we choose 𝑛 = 1 in the following
analysis and neglect the influence of boundary scattering.
Given these constraints, the MFPs are characterized by two parameters: the transition frequencies from 𝑛 = 4 to 𝑛 = 1 for both acoustic polarizations, 𝜔𝑚,𝐿 and
𝜔𝑚,𝑇 . The remaining unknown parameter is 𝜅 𝐼 𝑅 , which may depend on temperature. We identify the parameters that best explain the TG data by numerically
optimizing 𝜔𝑚,𝐿 , 𝜔𝑚,𝑇 , and 𝜅 𝐼 𝑅 (𝑇) to best fit the experimental data. The resulting
thermal conductivity versus grating period predicted by Eq. 5.23 with these parameters at each temperature is shown in Fig. 5.17, demonstrating good agreement at
all temperatures.
The choices of 𝜔𝑚,𝐿 , 𝜔𝑚,𝑇 , and 𝜅 𝐼 𝑅 that are compatible with the data were determined
as follows. The calculated relative differences between experiment 𝜅 𝑒𝑥 𝑝𝑡 and the
calculation 𝜅 𝑐𝑎𝑙𝑐 at 60 K using different 𝜅 𝐼 𝑅 values 𝜅 𝐼 𝑅 = 0 − 0.2 𝑊𝑚 −1 𝐾 −1 are
shown in the left of Fig. 5.18. Blue regions in the left of Fig. 5.18 A - E indicate the
cutoff frequencies that minimize the relative difference. We find that the optimized
frequencies are in the range of 2-3 THz for LA and 1-2 THz for TA for all cases.
As shown in the center column of Fig. 5.18 A - E, 𝜅 𝑐𝑎𝑙𝑐 at 60 K agrees with 𝜅 𝑒𝑥 𝑝𝑡
for 𝜅 𝐼 𝑅 ≤ 0.2 𝑊𝑚 −1 𝐾 −1 above which the trend of 𝜅 𝑐𝑎𝑙𝑐 starts to deviate from that
of 𝜅 𝑒𝑥 𝑝𝑡 . The right column of Fig. 5.18 shows optimized 𝜅 𝐼 𝑅 for 80 - 315 K using
the optimized frequencies. The cutoff frequencies above were selected to yield the
overall best fit of thermal conductivity versus the grating period at all temperatures.
Therefore, although different choices for cutoff frequency and 𝜅 𝐼 𝑅 are possible, the
frequency-dependence of the MFPs remains unchanged. A steep slope (𝑛 = 4) in
the sub-THz frequencies is necessary to explain the observed grating dependence
of thermal conductivity. We find that the trend of MFP with frequency remains the
same for all of these parameter sets.
With optimal 𝜔𝑚,𝐿 , 𝜔𝑚,𝑇 , the optimal value of 𝜅 𝐼 𝑅 , the thermal conductivity above

92

Figure 5.17: (A) Candidate MFP profiles versus frequency: 𝜔−4 , constant (profile
1: green dotted lines); 𝜔−1 , 𝜔−4 , constant (profile 2: green solid lines). Upper
and lower curves indicate the LA and TA MFPs, respectively. Literature data are
IXS (diamonds, Ref. [151]), PSA at 300 K (asterisks, Ref. [15]), and molecular
dynamics (circles, Ref. [174]). (B) Predicted thermal conductivity versus grating
period from each candidate profile along with the measured data at 60 K. Only profile
2 reproduces the trend of the measured thermal conductivity with grating period.
Measured thermal conductivity versus grating period (symbols) and predicted value
from optimized profile 2 (total, dashed yellow line; LA, dashed blue line; TA, dashed
red line) for (C) 315 K, (D) 225 K, (E) 150 K, and (F) 80 K. Good agreement is
observed at all temperatures with MFPs that are independent of temperature.

the Ioffe-Regel crossover frequency ∼ 10 THz for LA and ∼ 5 THz for TA [174])
was deduced. Figure 5.19A shows 𝜅 𝐼 𝑅 versus temperature for each profile along
with the prediction from Allen-Feldman theory. The theoretical prediction was
computed using the expression for mode diffusivity in Ref. [199] and the quantum-

93

Figure 5.18: (Left) Calculated relative differences at 60 K for (A) 𝜅 𝐼 𝑅 = 0 𝑊𝑚 −1 𝐾 −1 ,
(B) 𝜅 𝐼 𝑅 = 0.03 𝑊𝑚 −1 𝐾 −1 , (C) 𝜅 𝐼 𝑅 = 0.06 𝑊𝑚 −1 𝐾 −1 , (D) 𝜅 𝐼 𝑅 = 0.1 𝑊𝑚 −1 𝐾 −1 , and
(E) 𝜅 𝐼 𝑅 = 0.2 𝑊𝑚 −1 𝐾 −1 . The blue area indicates the range of optimized LA /
TA crossover frequency which minimize the relative difference. (Center) Calculated thermal conductivity versus grating period (𝜅 𝑐𝑎𝑙𝑐 ) using optimized crossover
frequencies and experiments at 60 K. As the 𝜅 𝐼 𝑅 increases, the slope of the 𝜅 𝑐𝑎𝑙𝑐
becomes less steep. (Right) Optimized 𝜅 𝐼 𝑅 at higher temperatures using optimized
crossover frequency determined from minimum relative difference at 60 K.

corrected specific heat. The temperature dependence of the measurements generally
agrees with the theory although the theory slightly underestimates the actual values.

94
Figure 5.19B shows 𝜅 𝐼 𝑅 normalized to its room temperature value along with the
normalized heat capacity. The 𝜅 𝐼 𝑅 trend with temperature matches that of the heat
capacity.

Figure 5.19: (A) Thermal conductivity (𝜅 𝐼 𝑅 ) contributed by excitations above the IR
frequency versus temperature for different MFP profiles discussed in the main text.
The red dashed line is a prediction using harmonic approximation in disordered solid
shown in Ref. [199, 200] for frequencies ≥ 10 THz for LA and ≥ 5 THz for TA. (B)
The values in A normalized by their 315 K values along with the normalized heat
capacity versus temperature (dashed blue line, Ref. [196]). The thermal conductivity
(𝜅 𝐼 𝑅 ) follows the temperature dependence of the heat capacity.

5.6

Unusually weak damping of acoustic excitation in aSi

The extracted MFPs versus frequency for LA and TA branches are shown in Fig. 5.20
along with prior measurements on vitreous silica. The trend for aSi agrees well with
these and other measurements in that similar power-law dependencies are observed
[28, 156, 158–161, 164, 201]. The 𝑛 = 4 Rayleigh scattering trend for both glasses
occurs in the 1 − 3 THz range with a transition to a weaker power law in the sub-THz
frequencies. Thus qualitatively, acoustic damping in aSi is not so different from
those of other glasses.
However, comparing the attenuation between vitreous silica and aSi, differences
also emerge. Comparing the LA MFPs at ∼ 1 − 2 THz, the attenuation due to
Rayleigh scattering is weaker in aSi by around a factor of 5, expected as aSi is a
monatomic glass with less atomic disorder. Further, at room temperature the 𝑛 = 4
trend yields to a 𝑛 ≈ 1 − 2 trend at ∼ 700 GHz in vitreous silica while the same
transition occurs at ∼ 300 − 400 GHz in aSi. At cryogenic temperatures ∼ 1 K for
vitreous silica, the transition frequencies in both materials are comparable. This
difference indicates weaker damping by mechanisms such as two-level systems or

95
Akhiezer damping in amorphous Si and has an important consequence: excitations
with MFPs in the micron range occur at frequencies of ∼ 200 − 1000 GHz in
aSi versus . 100 GHz in vitreous silica at room temperature owing to the steep
𝑛 = 4 slope of Rayleigh scattering. The heat capacity of excitations in the former
frequency range is larger by a factor of (𝜔𝑎𝑆𝑖 /𝜔 𝑆𝑖𝑂 2 ) 2 ∼ 100. The result is that in
aSi, heat-carrying excitations at room temperature have micron-scale MFPs, while
the MFPs of excitations in the same frequency band for vitreous silica are smaller by
an order of magnitude. Excitations in vitreous silica with micron-scale MFPs have
too low frequency to transport substantial heat at room temperature. Remarkably,
the attenuation observed in aSi up to room temperature is of the same order as that
measured in vitreous silica at 1 K, highlighting the unusually weak acoustic damping
in aSi.

Figure 5.20: Mean free path versus frequency for thermal acoustic excitations in aSi
at 300 K along with literature data for vitreous silica from inelastic x-ray scattering
(diamonds, Ref. [28, 154]), picosecond acoustics (downward pointing triangles,
Ref. [161]), Brillouin light scattering (left-pointing triangles, Ref. [158] and 6pointed stars, Ref. [159]), tunnel junction spectroscopy (5-pointed stars, Ref. [156]),
a multi-pulse optical technique (upward pointing triangles, Ref. [195]), and from
transport measurements (dashed line, Ref. [18]).

Our conclusions on the origin of damping of sub-THz excitations in aSi are consistent
with these prior studies of other glasses, and the extracted 𝜅 𝐼 𝑅 (𝑇) for excitations
above the IR frequency is compatible with Allen-Feldman theory. However, our

96
results are not consistent with the conclusions of numerical studies of excitations
below the IR frequency [13, 16, 171, 172, 202]. In these studies, the Hamiltonian
for atoms in a supercell is diagonalized in the harmonic approximation to yield
normal modes. The original studies of Allen and Feldman used these normal modes
to classify excitations in glasses as propagons, diffusons, and locons depending on
the properties of the eigenvectors [200]. The lifetimes of these modes are obtained
using normal mode decomposition and molecular dynamics [203, 204]. With these
approaches, these studies have generally concluded that damping in aSi varies as
𝜔−2 for frequencies around ∼ 1 − 2 THz and below. From this trend, the damping
mechanism has been postulated to involve anharmonicity [171, 172] and to exhibit
a temperature dependence [16].
Discussion
We first address the classification of acoustic excitations. Various numerical [14,
16, 171] and experimental works [170] have noted a transition in the character of
vibrations in aSi around ∼ 1 − 2 THz, leading to the introduction of “diffusons”
as non-propagating yet delocalized vibrations in Refs. [14, 16]. In contrast, our
work attributes this change to a frequency-dependent damping of collective acoustic
excitations. The crossover from propagons and diffusons at ∼ 1 − 2 THz coincides
with the transition from Rayleigh scattering to the Kittel regime in the present work
and thus can be explained without the definition of a new type of vibration. The
IR crossover for the transition from collective excitations to incoherent excitations,
which occurs well above 1-2 THz in amorphous Si, is sufficient to describe the
different characters of excitations in glasses.
The second inconsistency is the prediction by normal-mode analysis of the frequencydependence (𝑛 = 2) and anharmonic origin of damping in the few THz frequency
range. Specifically, the MFPs predicted from normal mode analysis are on the
order of 20 nm at ∼ 1 THz (smaller by factor of 5 compared to present value of
∼ 100 nm) and vary as 𝜔−2 (see Fig. 4B of Ref. [171]), which cannot explain the
measurements of the present work. Here, the inconsistency appears to arise from
the implicit assumption of the normal mode decomposition that the heat-carrying
excitations in glasses are the normal modes of the supercell. This assumption is not
compatible with basic many-body physics and scattering theory, which instead gives
the proper definition and lifetime of a collective excitation of a many-body system
using the self-energy and the single-particle Green’s function [205]. Rather than
normal modes, a physical picture of acoustic excitations of a glass that is compatible

97
with this framework is that originally postulated by Kittel [17] in the continuum
limit and later by Zeller and Pohl [18], in which a glass is imagined to consist of
a fictitious atomic lattice along with perturbations representing the mass and force
constant disorder in the actual glass. The undamped excitations of the fictitious
atomic lattice acquire a lifetime owing to the disorder of the actual glass. The
dispersion and lifetimes of these excitations can be measured experimentally using
inelastic scattering, as has been performed for many glasses in the past decades
[28, 151–155]. In contrast, the lifetimes of normal modes do not appear to be
experimentally accessible or physically meaningful as they are unable to explain the
thermal conductivity measurements presented here.
5.7

Summary

In summary, we have obtained measurements of the frequency-dependent MFPs of
sub-THz acoustic excitations in amorphous silicon using picosecond acoustics and
transient grating spectroscopy. The MFPs lack a temperature dependence and exhibit
a trend characteristic of structural scattering by point defects and density fluctuations.
This result is at variance with numerical studies based on normal mode analysis but
is broadly consistent with prior studies of vitreous silica and other glasses. The
micron-scale MFPs of heat-carrying excitations at room temperature are found to
arise primarily from the weak anharmonic and two-level system damping of subTHz excitations, leading to room temperature attenuation coefficients comparable
to those of other glasses at cryogenic temperatures.

98
Chapter 6

CONCLUSION AND FUTURE WORKS
6.1

Summary

The rapid experimental advancement in recent years has promised better understanding of the microscopic heat transport properties in materials. In this context,
the focus of this thesis is on addressing a critical problem in heat conduction: understanding and characterizing microscopic transport that drives the bulk thermal
properties in thermally conductive polymers and amorphous materials. The elucidation of the mechanism can resolve long-standing questions over decades, and
provide useful insights for their applications. Using thermal transient grating (TG)
as a tool for MFP spectroscopy, we introduced a new picture and route for better
understanding of the transport of the heat-carrying vibrations in these substances.
In chapter 2, we introduced the TG and described its leverage for measuring the
thermal transport in crystalline thin films. We have motivated by introducing the
TG and explaining the key requirements for the TG, which can improve its ability
of adaptation for experimental probe with relevant length and time scales. Finally,
we validated TG’s capability by demonstrating measurements of the bulk thermal
properties of crystalline silicon thin-film.
In chapter 3, we presented in-plane vibrational properties of free-standing MoS2
membranes. We have described the adaptiveness of the TG by introducing a timeresolved transient grating with a temporal resolution less than 100 fs for measuring
thermal and elastic properties, concurrently. We first experimentally reported inplane sound velocity of 7000 ± 40 m s−1 , which demonstrated an excellent agreement
with ab-initio prediction of the phonon group velocity along the Γ−𝐾 Brillouin zone.
We also reported a value of in-plane thermal conductivity of 74 ± 21 Wm−1 K−1 ,
which we do not observe a grating period dependent thermal conductivity. We
explained our observation based on the ab-initio calculation of the accumulated
thermal conductivity versus the phonon mean free path; mean free paths of the
thermal acoustic phonons are smaller than the thermal length scale considered in
this experiment.
In chapter 4, we discussed our measurements of TG to resolve the microscopic
transport of heat carrying acoustic phonons in highly oriented semi-crystalline

99
polyethylene (PE). For this purpose, we began with bulk measurements concerning
the in-plane thermal anisotropy along with temperature dependent thermal conductivity; qualitative description of the mesoscopic transport phenomena in partially
oriented PE was provided. Next, we presented our measurements of the mean free
path of heat-carrying acoustic phonons in actual samples of highly oriented polyethylene. For extracting accumulated thermal conductivity, a Bayesian inference was
applied; we showed that phonons with mean free paths (MFP) on micron length
scales are the dominant heat carriers in highly oriented polyethylene. We further
showed a frequency resolved MFP using a low-energy anisotropic Debye model;
we found evidence of one-dimensional density states of phonons with a frequency
less than ∼ 2 THz. The transition frequency is consistent with prior observations in
highly oriented PE;the stiff longitudinal branch makes the acoustic phonons to have
a wavelength of 8 nm which is comparable to prior reports of fiber diameter ∼ 10
nm. Therefore, the wavelength of low-frequency heat-carrying acoustic phonons
exceeds the fiber diameter dimension. Our results provide insights into the actual
upper-limit of the thermal conductivity in highly oriented polymers that will prove
useful for developing thermally conductive polymers.
In chapter 5, we described the microscopic origin of the heat-carrying acoustic excitation in amorphous materials. Over the decade, the transport of the excitations in
amorphous silicon has been contentious; numerous predictions showed contradictions with experimental results. To address this, we presented a novel approach in
which we probed thermal conduction at sub-micron length scales in thin amorphous
Si films using transient grating spectroscopy and picosecond acoustics. We provided
an evidence of ballistic excitations over the grating period that allows us to discover
the primary damping mechanisms versus frequency from 0.1 - 3 THz. We found that
the primary damping mechanism in amorphous Si is Rayleigh scattering, contrary
to the predictions of normal mode analysis; the absence of anharmonic or two-level
system damping enables micron-scale mean free paths at room temperature. We
provided a clear picture of acoustic damping in amorphous Si and our approach thus
allows us to confirm or refute this widely-used theory by understanding vibrations
as acoustic excitation rather than propagons, diffusons, and locons.
6.2

Outlook and future insight

Concerning the heat transport in semicrystalline polymers and amorphous materials,
there exist numerous questions that remain unresolved.

100
In polymers, one future work is decreasing a considerable knowledge gap between
computational predictions and experiments. As earlier mentioned, the computations cannot accurately capture the role of the microstructures on the transport of
heat-carrying acoustic phonons. In fact, our understandings from the computations
are limited to smaller computational domains (single chain or perfect molecular
crystal), therefore explicit structure affected phonon properties was less described.
For instance, despite experimental evidences regarding phonon-interchain interactions, there is a lack of quantitative description on them. In collaboration with
improved computations, our experimental findings in chapter 4 will aid to establish a comprehensive picture for microscopic heat transport that are subject to the
microstructure.
In this regard, considering the importance of the impact of the structure to the
transport of the heat-carrying acoustic phonons, a systematic experimental study at
different draw ratio (DR) needs to be done. The draw ratios of the polymers that
were analyzed in this thesis are limited: DR4, 5, 7.5, and 200. A systematic study for
semi-crystalline polymers with various DR will further improve our understanding
of the heat transport in polymers.
Regarding amorphous silicon, the understanding of the transport of the excitations has relied on a classification introduced by Allen and Feldman; a concept of
propagons, diffusons, locons was proposed based on normal mode analysis. As
earlier mentioned, according to this theory, propagons are phonon-like, delocalized,
propagating vibrations; locons as localized modes; and diffusons as delocalized
yet non-propagating vibrations. Dozens of papers have been written on the atomic
vibrations of glasses with this framework as a starting point, but have faced discrepancies with experiments. Therefore, the criteria of the classification of the normal
mode analysis should be examined again. We believe that our experimental results
in chapter 5 will set the stage for establishing a clear picture in a basis of collective
acoustic excitations rather than propagon, diffuson, and locons.
On the experimental side, the transient grating as a tool for MFP spectroscopy can
be further improved, we highlight that the results of TG can be better explained with
a combination of other technique, which can potentially provide tight constraints
for the frequency-resolved MFPs. For example, in this thesis, we presented a
table-top approach using the results from a dominant phonon approximation on
prior results or picosecond acoustics measurements. To move forward in aSi,
results of higher frequency excitations from in-elastic x-ray scattering and in-elastic

101
neutron scattering can further provide the constraints, therefore enabling to probe
the phonon transport in a wide range of frequencies. Second, to study heat transport
of disordered materials, the relevant length and time scales should be provided. In
particular, the measurement of transport in amorphous regions is difficult without
further enhancing the spatial resolution. The minimum thermal length scale shown
in this thesis is an order of 100 nm, which may be still far longer than the MFP of
heat-carrying acoustic excitations in amorphous materials with a strong damping.
This can be addressed by employing extreme UV as a source, which is just being
developed.
Another future work is experimentally exploring a new transport regime using TG.
The transport regimes that are presented in this thesis are weak quasi-ballistic and
diffusive regimes. On the other hand, numerical predictions have suggested that
there are other transport regimes including collective-diffusive, strongly ballistic,
and hydrodynamic transport regimes. Further improvements in TG combined with
advances in the sample preparation can confirm the numerical suggestions.
To conclude, we have utilized the capabilities of TG to provide useful insights for the
damping of the heat-carrying acoustic excitations in semi-crystalline polymers and
amorphous materials. We believe that our approaches and results here will provide
fundamental insights into the transport of the heat-carrying acoustic excitations in
these substances, which were challenging with various conventional methods. A
better understanding of heat transport in these substances will eventually provide an
opportunity for promising applications beyond conventional use.

102

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