Studies of non-diffusive heat conduction through
spatially periodic and time-harmonic thermal ARCHIVES
excitations
MASSACHUSETTS INSTITUTE
by
APR 152015
Kimberlee Chiyoko Collins
LIBRARIES
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2015
@
Massachusetts Institute of Technology 2015. All rights reserved.
Signature redacted
A uthor ........................
Department of Mechanical Engineering
January 15, 2015
Signature redacted
C ertified by..........................
Gang Chen
Carl Richard Soderberg Professor of Power Engineering
Thesis Supervisor
Accepted by ..............................
Signature redacted
Cadu"Te
ardt
Chairman, Department Committee on Graduate Theses
2
Studies of non-diffusive heat conduction through spatially
periodic and time-harmonic thermal excitations
by
Kimberlee Chiyoko Collins
Submitted to the Department of Mechanical Engineering
on January 15, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
Abstract
Studies of non-diffusive heat conduction provide insight into the fundamentals of
heat transport in condensed matter. The mean free paths (MFPs) of phonons that
are most important for conducting heat are well represented by a material's thermal
conductivity accumulation function. Determining thermal conductivity accumulation
functions experimentally by studying conduction in non-diffusive regimes is a recent
area of study called phonon MFP spectroscopy. In this thesis, we investigate nondiffusive transport both experimentally and theoretically to advance methods for
determining thermal conductivity accumulation functions in materials. We explore
both spatially periodic and time-harmonic thermal excitations as a means for probing
the non-diffusive transport regime, where the Fourier heat diffusion law breaks down.
Boltzmann transport equation calculations of one-dimensional (1D) spatially sinusoidal thermal excitations are performed for gray-medium and fully spectral cases.
We compare our calculations to simplified transport models and demonstrate that a
model based on integrating gray-medium solutions can reasonably model materials
with a narrow range of dominant heat-carrying phonon MFPs. We also consider the
inverse problem of determining thermal conductivity accumulation functions from
experimental measurements of thermal-length-scale-dependent effective thermal conductivity. Based on experimental measurements of Si membranes of varying thickness,
we reproduce the thermal conductivity accumulation function for bulk Si.
To investigate materials with short phonon MFPs, we developed an experimental approach based on microfabricating 1D wire grid polarizers on the surface of
a material under study. This work finds that the dominant thermal length scales
in polycrystalline Bi 2 Te3 are smaller than 100 nm. We also determine that even
small amounts of direct sample optical excitation, which occurs when light transmits
through the grating and directly excites electron-hole pairs in the substrate, can appreciably influence the measured results, suggesting that an alternate approach that
prevents all direct optical excitation is preferable.
To study thermal length scales smaller than 100 nm without the need for microfabrication, we develop a method for extracting high frequency response information
3
from transient optical measurements. For a periodic heat flux input, the thermal
penetration depth in a semi-infinite sample depends on the excitation frequency, with
higher frequencies leading to shallower thermal penetration depths. Prior work using frequencies as high as 200 MHz observed apparent non-diffusive behavior. Our
method allows for frequencies of at least 1 GHz, but we do not observe any deviation from the heat diffusion equation, suggesting that prior observations attributed
to non-diffusive effects were likely the result of transport phenomena in the metal
transducer.
Thesis Supervisor: Gang Chen
Title: Carl Richard Soderberg Professor of Power Engineering
4
Acknowledgments
Many people contributed to the completion of this thesis in both tangible and intangible ways. I am deeply grateful to Prof. Gang Chen, who advised and supported
my graduate studies. I am also grateful to the other members of my Ph.D. thesis
committee, who spent many hours offering constructive feedback on my work: Prof.
Millie Dresselhaus, Prof. Keith Nelson and Prof. Evelyn Wang. I was privileged to
have the opportunity to learn from each of these extraordinary individuals.
I also want to thank my colleagues who collaborated on the work presented in this
thesis. In particular, I am grateful to Dr. Alex Maznev, whose close involvement was
invaluable, and who I view as a scientific mentor. I would also like to acknowledge
the contributions of Dr. John Cuffe, Vazrik Chiloyan, Lingping Zeng, Prof. Austin
Minnich, Sam Huberman, Jeff Eliason, Prof. Keivan Esfarjani, Prof. Jivtesh Garg,
Prof. Zhiting Tian, and Prof. Tony Feng.
Beyond direct collaboration, I benefited from numerous discussions and interactions with my colleagues, especially Daniel Kraemer, Poetro Sambegoro, Dr. Ken
McEnaney, Maria Luckyanova, Dr. Matthew Branham, Edi Hsu, Prof. Selcuk Yerci,
Kara Manke, Sangyeop Lee, Dr. Sveta Boriskina, and Dr. Mayank Bulsara. Every
member of the NanoEngineering Group at MIT impacted me and helped shape the
direction of my graduate studies. I also benefited from those who came before me
in the NanoEngineering Group, especially Prof. Aaron Schmidt, who mentored me
at the start of my graduate career and taught me a great deal. The Spectroscopy
Group within the Solid-State Solar-Thermal Energy Conversion (S 3TEC) Center also
provided a forum for regular and fruitful discussion, and I am grateful to all those
who participated.
The resources offered at MIT also made this work possible. In particular, I would
like to acknowledge the Microsystems Technology Laboratories and the NanoStructures Laboratory at MIT. I was aided especially by Kurt Broderick, Dr. Tim Savas,
Jim Daley, Gary Riggott, Dr. Richard Hobbs, and Mark Mondol. Administrators and
staff in the NanoEngineering Group, in the S 3TEC Center, and within MIT helped in
5
numerous ways both seen and unseen, and for that I am grateful to Ed Jacobson, Mai
Hoang, Mary Ellen Sinkus, Juliette Pickering, Keke Xu, Leslie Regan, Read Schusky,
Pierce Hayward, Mark Ralph, and Dick Fenner.
I would like to thank my network of colleagues, friends and family who supported
me throughout my graduate studies. I was privileged to be a part of MIT's MacGregor House as a Graduate Resident Tutor. I learned a great deal from the MacGregor
community that I will carry with me for a long time. Many individuals and institutions mentored me and gave me opportunities that helped me attend MIT both as an
undergraduate and as a graduate, especially Wes Masuda, Le Jardin Academy, 'lolani
School, Dr. Gerry Luppino, Prof. George Ricker, Dr. Joel Villasenor, MIT Lincoln
Laboratory, John Sultana, Prof. John Heywood, and Prof. Carol Livermore. I am
deeply grateful for the support and friendship of Chris Celio and Dr. Ellen Chen. I
owe thanks to all those I shared good times with, and to all those who helped me
learn and grow.
Finally above all else, I am forever grateful to my family, especially my parents
and my siblings, whose unwavering love carries me through all of life's adventures.
6
Contents
Introduction
. . . . . . . . . . . . . . . . . .
. . .
19
. . . . . . . . .
. . .
21
. . . .
. . .
23
Non-diffusive heat conduction
1.2
Thermal conductivity accumulation functions
1.3
Laser-based thermal property measurement techniques
Frequency-domain thermoreflectance (FDTR)
. . . . .
. . .
24
1.3.2
Time-domain thermoreflectance (TDTR) . . . . . . . .
. . .
25
1.3.3
Heat transfer model for TDTR and FDTR responses
.
. . .
27
1.3.4
Transient thermal grating (TTG) . . . . . . . . . . . .
. . .
29
.
.
.
1.3.1
.
.
.
.
1.1
Organization of the thesis
30
. . . . . . . . . . . . . . . . . . . .
.
1.4
33
Boltzmann transport equation (BTE) overview . . . . . . . . .
. . .
33
2.2
Modeling the TTG experimental geometry . . . . . . . . . . .
. . .
35
2.2.1
Gray-medium BTE . . . . . . . . . . . . . . . . . . . .
. . .
37
2.2.2
Spectrally-dependent BTE . . . . . . . . . . . . . . . .
. . .
42
2.2.3
Frequency-integrated gray-medium model
. . . . . . .
. . .
46
2.2.4
Two-fluid model . . . . . . . . . . . . . . . . . . . . . .
. . .
48
2.2.5
Contribution of long MFP phonons . . . . . . . . . . .
50
Thermal conductivity accumulation function reconstruction . .
51
2.3.1
Theoretical foundation . . . . . . . . . . . . . . . . . .
52
2.3.2
TTG experimental geometry . . . . . . . . . . . . . . .
2.3.3
TTG thin membrane experimental geometry . . . . . .
55
2.3.4
Experimentally deriving heat flux suppression functions
58
.
.
.
.
.
.
.
.
2.3
.
2.1
.
Modeling non-diffusive heat conduction
.
2
19
.
1
7
. . .
53
2.4
. . . . . . . . . . . . . . . . . . . . .
63
3.1
Concept and design criteria for wire grid polarizer . . . . . . . . . .
64
3.2
M icrofabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.2.1
Dry etching Al
71
3.2.2
Polishing polycrystalline Bi 2 Te 3 samples
. . . . . . . . . . .
73
3.2.3
Resulting one-dimensional grating structures . . . . . . . . .
75
.
.
grid polarizers
.
.
.
. . . . . . . . . . . . . . . . . . . . . . . . .
Optical transmission results
. . . . . . . . . . . . . . . . . . . . . .
76
3.4
TDTR results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.4.1
Grating heat transfer model . . . . . . . . . . . . . . . . . .
80
3.4.2
Measurements varying pump laser diameter
81
3.4.3
Measurements varying angle between laser polarization and grat-
.
.
.
3.3
.
. . . . . . . . .
ing transmission axis . . . . . . . . . . . . . . . . . . . . . .
81
Result summary . . . . . . . . . . . . . . . . . . . . . . . . .
84
3.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
.
Future directions
.
3.4.4
4
59
Investigation of non-diffusive conduction with microfabricated wire
.
3
Summary and future directions
Frequency-domain representation of TDTR data, and applications
for studying non-diffusive conduction
91
4.1
. . . . . . . . . . . . . . . . . . . . . . . . . .
94
4.1.1
FD T R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
4.1.2
Single-shot TDTR
. . . . . . . . . . . . . . . . . . . . . . . .
95
4.1.3
TDTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.1.4
Frequency-domain representation of TDTR data . . . . . . . .
98
4.2
Theoretical foundation
Experimental demonstration of fdTDTR
. . . . . . . . . . . . . . . .
99
4.2.1
Data collection
. . . . . . . . . . . . . . . . . . . . . . . . . .
100
4.2.2
Data stitching . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
4.2.3
Fourier series representation . . . . . . . . . . . . . . . . . . .
103
4.2.4
Frequency upper limit
104
4.2.5
Direct comparison of fdTDTR and FDTR data
. . . . . . . . . . . . . . . . . . . . . .
8
. . . . . . . . 107
4.3
5
Thermal model analysis of fdTDTR data . . . . . . . . . . . . . . . .
108
4.3.1
110
Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . .
4.4
Comparison of fdTDTR data for various samples
. . . . . . . . . . .
115
4.5
Summary and future directions
. . . . . . . . . . . . . . . . . . . . .
119
Summary and Outlook
123
9
10
List of Figures
1-1
Conceptual illustration of how non-diffusive heat flux differs from the
predictions of the Fourier heat diffusion equation.
1-2
21
Normalized thermal conductivity accumulation functions for various
materials from first-principles calculations.
1-3
. . . . . . . . . . .
. . . . . . . . . . . . . . .
22
Schematic diagrams of (a) the traditional frequency-domain thermore-
flectance (FDTR) and (b) the broadband FDTR (BB-FDTR) systems
in our lab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-4
Schematic diagram of the time-domain thermoreflectance (TDTR) system in our lab.
1-5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
31
Illustration of TTG crossed laser beam interference, which produces a
spatially-sinusoidal temperature profile that decays in time. . . . . . .
2-2
26
Schematic diagrams of transient thermal grating (TTG) systems for
(a) transmission and (b) reflection measurements.
2-1
25
36
Dimensionless gray-medium Boltzmann transport equation (BTE) thermal decays for a range of dimensionless length scales compared to diffusive limits and Fourier model best fits.
2-3
. . . . . . . . . . . . . . . .
Dimensionless gray-medium BTE thermal decays compared to the ballistic transport lim it. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-4
41
Normalized effective thermal diffusivities found by fitting gray-medium
BTE decay curves with the heat diffusion equation.
2-5
40
. . . . . . . . . .
42
Set of material parameters for Si and PbSe at 300 K from density functional theory (DFT) which were used in our spectral BTE calculations.
11
44
2-6
TTG thermal decays for a range of grating periods calculated from a
numerical solution to the spectral BTE using DFT input parameters
for (a) Si and (b) PbSe at 300 K, compared to the diffusive limit.
2-7
. .
45
Effiective thermal conductivity for various TTG periods calculated
from the spectral BTE for (a) Si and (b) PbSe at 300 K. Comparisons are shown for the gray-medium model using literature values of
gray mean free path (MFP) and best fit gray MFP values.
2-8
. . . . . .
46
Effective thermal conductivity for various TTG periods calculated from
the spectral BTE for (a) Si and (b) PbSe at 300 K. Comparisons are
shown for the best fit gray-medium model and the frequency-integrated
gray medium model.
2-9
. . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Comparison of suppression functions from the two-fluid model and the
frequency-integrated gray-medium model.
. . . . . . . . . . . . . . .
48
2-10 TTG thermal decay calculated from a numerical solution to the spectral BTE using DFT input parameters for Si at 300 K, compared to the
diffusive limit and the approximate solution provided by the two-fluid
m odel.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2-11 Effective thermal conductivity for various TTG periods calculated from
the spectral BTE for (a) Si and (b) PbSe at 300 K. Comparisons are
shown for the best fit gray-medium model as well as the predicted
results using both the gray-medium and the two-fluid heat flux suppression functions.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2-12 Thermal conductivity accumulation functions for different grating periods calculated using DFT parameters for Si at 300 K and the two-fluid
suppression function. . . . . . . . . .. . . . . . . . . . . . . . . . . . .
51
2-13 Percent contribution to effective thermal conductivity from phonons
with mean free paths greater than half the TTG period.
. . . . . . .
52
2-14 Reconstruction of thermal conductivity accumulation functions for (a)
Si and (b) PbSe at 300 K using BTE calculated effective thermal conductivity values as "experimental" inputs.
12
. . . . . . . . . . . . . . .
54
2-15 Calculation of normalized effective thermal conductivity t(L) from reconstructed normalized thermal conductivity accumulation functions
4D(A) to verify agreement with the ,.(L) values used to reconstruct <b(A). 54
55
2-16 Illustration of a TTG measurement on a thin membrane. .......
2-17 TTG measured effective thermal conductivity at 300 K for a range of
Si membrane thicknesses. . . . . . . . . . . . . . . . . . . . . . . . . .
56
2-18 Fuchs-Sondheimer suppression function . . . . . . . . . . . . . . . . .
57
2-19 Reconstructed thermal conductivity accumulation function from the
Fuchs-Sondheimer suppression function and experimentally measured
effective thermal conductivities for thin Si membranes.
. . . . . . . .
57
2-20 (a) Thermal conductivity per MFP and (b) thermal conductivity accumulation function for Si at 300 K from DFT calculations.
. . . . .
59
2-21 Reconstructed heat flux suppression function for thin, diffusely scattering membranes. The exact solution given by the Fuchs-Sondheimer
relationship is shown for comparison.
3-1
. . . . . . . . . . . . . . . . . .
60
Conceptual illustration of a one-dimensional metal grating acting as a
linear polarizer to block pump and probe light from directly exciting
a generic substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3-2
Schematic illustrating the domain used for COMSOL calculations. . .
66
3-3
COMSOL calculations for transmittance and reflectance for 800 nm
and 400 nm light for various grating geometries. . . . . . . . . . . . .
3-4
67
COMSOL calculations for transmittance and reflectance with a constraint on the relationship between L and d such that the gap between
grating lines is held constant at L - d = 100 nm.
3-5
. . . . . . . . . . .
68
COMSOL calculations for transmittance and reflectance with a constraint on the relationship between L and d such that the filling fraction
is kept constant with L = xd, where x is an integer. . . . . . . . . . .
3-6
69
COMSOL transmittance calculations varying the type of metal used
for the one-dimensional grating. . . . . . . . . . . . . . . . . . . . . .
13
70
3-7
COMSOL transmittance and reflectance calculations for a Bi 2 Te 3 substrate patterned with a one-dimensional Al grating. . . . . . . . . . .
70
3-8
One-dimensional grating fabrication process flow.
71
3-9
Scanning electron microscope images of the one-dimensional grating
. . . . . . . . . . .
fabrication process. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3-10 Examples of post-etch corrosion (a) where corrosion products form
clusters and (b) where corrosion results in removed sections of Al.
.
74
3-11 Representative atomic force microscopy measurement of a polished
polycrystalline Bi 2 Te 3 sample. . . . . . . . . . . . . . . . . . . . . . .
76
3-12 Examples of fabricated one-dimensional Al gratings. . . . . . . . . . .
77
3-13 Transmission measurement setup where either the 400 nm pump or the
800 nm probe is used . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3-14 Idealized linear polarizer transmitted intensity as a function of angle.
78
3-15 TDTR grating measurement setup. . . . . . . . . . . . . . . . . . . .
79
3-16 Measured substrate thermal conductivity as a function of pump laser
diameter for (a) fused silica and (b) Si. . . . . . . . . . . . . . . . . .
82
3-17 Reflection results for various angles on (a) Si and (b) fused silica showing both amplitude and phase TDTR data as a function of delay time.
83
3-18 Thermoreflectance signal from bare Si. . . . . . . . . . . . . . . . . .
84
3-19 Reflection results for various angles on Si at (a) high transmission
angles and (b) low transmission angles. . . . . . . . . . . . . . . . . .
85
3-20 Reflection results for angles around the low transmission angle for a
one-dimensional Al grating on Si. . . . . . . . . . . . . . . . . . . . .
86
3-21 Concept for structures that prevent optical transmission into the substrate.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3-22 Using a crude model we extract substrate thermal conductivity values
for substrates of Si and fused silica patterned with grating structures
like that pictured in Fig. 3-21. . . . . . . . . . . . . . . . . . . . . . .
14
88
4-1
Illustration of thermal profiles in a semi-infinite solid that result from
excitation by a sinusoidally periodic heat flux.
4-2
. . . . . . . . . . . . .
Thermal penetration depth for semi-infinite solid slabs of Si, A12 0
92
3
and fused silica heated by a sinusoidal heat flux. . . . . . . . . . . . .
93
4-3
Typical laser heating profiles for (a) FDTR and (b) TDTR measurements. 94
4-4
TDTR experimental diagram.
The additional static delay line com-
bined with the movable delay stage allows for the collection of more
than one full period of delay-time-domain data.
4-5
. . . . . . . . . . . .
101
Example of stitching TDTR data sets collected in two parts by introducing an additional fixed delay for measuring long delay time data
(>6 ns)........
4-6
...................................
103
Room temperature TDTR data for a sample of A1 20 3 with an Al transducer layer, represented in (a) the frequency-domain and (b) the delaytim e-dom ain.
4-7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
A noise estimate is obtained by smoothing the raw delay-time-domain
data, as shown in (a). Subtracting the smoothed curve from the raw
data produces the noise data shown in (b). . . . . . . . . . . . . . . .
4-8
Comparison of the frequency response amplitude obtained from the
raw data in Fig. 4-7(a) and the noise data in Fig. 4-7(b).
4-9
106
. . . . . .
107
Comparison of phase data from a frequency-domain representation of
TDTR data (fdTDTR) and FDTR measurements on the same sample. 109
4-10 Thermal model best fits from simultaneously varying the A1 2 0
3
ther-
mal conductivity, kAl 2 o 3 , and the Al-A1203 thermal interface conductance, G, are shown, assuming either a 10 nm or a 25 nm thick isothermal Al layer, dis.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
4-11 fdTDTR phase data and model curves assuming isothermal Al layer
. . . . . . . . . . . . . . . .
112
4-12 Thermal model sensitivity plots. . . . . . . . . . . . . . . . . . . . . .
113
thicknesses of 25 nm, 10 nm and 40 nm.
4-13 Contours of least squares fitting residuals given a range of kAl 2 o 3 and
G values varied up to 10% about the best fit values. . . . . . . . . . .
15
114
4-14 fdTDTR data from Si, A1 2 0 3 , and fused silica substrates with a 110
nm thick Al transducer.
Lines show thermal model assuming bulk
properties and using di,, = 25 nm.
. . . . . . . . . . . . . . . . . . .
117
4-15 fdTDTR data from Si, A1 2 0 3 , and fused silica substrates with a 60
nm thick Al transducer.
Lines show thermal model assuming bulk
properties and using dis, = 25 nm.
. . . . . . . . . . . . . . . . . . .
118
4-16 fdTDTR data from Si, A1 2 0 3 , and fused silica substrates with a 160
nm Au transducer with a 5 nm Ti stiction layer. Lines show thermal
model assuming bulk properties and using di 5 , = 80 nm.
. . . . . . .
119
4-17 fdTDTR data from Si, A12 0 3 , and fused silica substrates with a 55 nm
Au transducer with a 5 nm Ti stiction layer. Lines show thermal model
assuming bulk properties and treating the entire Au layer as isothermal. 120
4-18 Comparison of early delay time TDTR amplitude signal shapes for
different transducer layers on substrates of A1 2 0
3
including (a) 110 nm
Al, (b) 60 nm Al, (c) 160 nm Au with 5 nm Ti, and (d) 55 nm Au with
5 nm T i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4-19 fdTDTR data from a polycrystalline Bi 2 Te 3 sample with a 90 nm Al
transducer layer. Model fits shown for diso = 25 nm and di,,,
16
=
10 nm.
122
List of Tables
. . . . . . . . . . . . .
3.1
Fabricated grating polarizer extinction ratios.
3.2
COMSOL calculated extinction ratios for the fabricated grating polar-
3.3
79
izers in Table 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
TDTR results on grating samples. . . . . . . . . . . . . . . . . . . . .
86
17
i8
Chapter 1
Introduction
Heat conduction at small scales behaves differently from heat conduction at macroscales
due to a lack of a well defined local thermal equilibrium. Phonons are quantized lattice vibrations that transport heat and sound in solids. Diffusive transport theory
assumes a multitude of phonon scattering events which establish local thermal equilibrium, giving rise to a local temperature, and the temperature gradient governs the
heat flux. When conduction length and/or time scales do not allow for many phonon
scattering events, transport is non-diffusive, deviating from Fourier's heat diffusion
law which over-predicts the heat flux in non-diffusive regimes [1, 2]. Studying heat
conduction in non-diffusive regimes provides insight into the underlying nature of the
phonons that carry heat in a material [3, 4], and can inform the design of materials
with tailored thermal properties. The study of non-diffusive heat conduction to extract information about phonons is an emerging area of research called phonon mean
free path spectroscopy [4, 5]. In this thesis, we explore both spatially periodic and
time-harmonic thermal excitations as a means for probing the non-diffusive transport
regime.
1.1
Non-diffusive heat conduction
Phonons travel an average distance A, called the phonon mean free path (MFP),
before being scattered. The MFP relates to the relaxation time
19
T
through the group
velocity v, where A
VT.
For length scales comparable to or smaller than phonon
MFPs, or time scales comparable to or smaller than phonon relaxation times, energy
transport deviates from the predictions of Fourier's law, which assumes local thermal
equilibrium. Establishing local thermal equilibrium requires many local phonon scattering events. Heat transport is non-diffusive in lieu of sufficient scattering events to
produce local thermal equilibrium.
Fourier's law over-predicts the heat flux in non-diffusive transport regimes [1, 2],
as illustrated conceptually in Fig. 1-1. For a material surrounded by parallel plate
heat reservoirs at different temperatures, Fourier's law predicts a steady state heat
flux of q = kAT/L, where L is the distance between the plates, AT is the indicated
temperature difference, and k is the material's thermal conductivity.
As L -+
0,
Fourier's law predicts a divergent heat flux, which is clearly unphysical for a finite
AT. In reality, at some small enough L, the heat flux will reach a maximum value, as
illustrated by the dashed line in Fig. 1-1. In this regime, the phonons travel between
the hot and cold reservoirs totally ballistically without scattering, a phenomenon
also called phonon radiative transport [6, 7]. In the ballistic limit, it is not possible
to define a local temperature anywhere in the material. Between the ballistic limit
and the diffusive limit, there exists an intermediate quasiballistic transport regime
[3, 4], where some phonon scattering occurs, but not enough to define a local thermal
equilibrium.
Heat transport measurements that are performed in non-diffusive regimes are
often interpreted using the Fourier heat diffusion equation, in spite of its lack of
validity [5].
Measurement regimes near the onset of non-diffusive transport retain
diffusive-like behavior [8], and can be modeled by the heat diffusion equation using a
modified effective thermal conductivity, keff. The over-prediction of the heat flux by
Fourier's law results in a lower value of experimentally determined effective thermal
conductivity compared to that of the bulk material. The phonon Boltzmann transport
equation (BTE) [9] is valid in both diffusive and non-diffusive transport regimes
as long as wave effects are unimportant, but because the BTE is difficult to solve
and requires a multitude of input parameters, the BTE is not typically used for
20
ballistic quasiballistic
diffuse
q|
q = kAT/L
L
L
L
Figure 1-1: Conceptual illustration of how non-diffusive heat flux differs from the predictions of the Fourier heat diffusion equation (solid line). For 1D steady conduction
between parallel thermal reservoirs separated by a distance L, the heat diffusion equation predicts a divergent heat flux q as L --+ 0, while in reality, the heat flux will reach
a maximum at some small enough L where the phonons travel totally ballistically, as
illustrated by the dashed line.
directly fitting experimental data. We discuss solutions to the phonon BTE for simple
experimental geometries in Chapter 2, as well as how BTE solutions can help link
experimentally determined keff data to phonon spectral information.
1.2
Thermal conductivity accumulation functions
Crystalline materials have distributions of phonon MFPs. One convenient representation of a phonon MFP distribution in a material is through the thermal conductivity
accumulation function, kaccu, which sums the contributions of the thermal conductiv-
ity per MFP, kaiff [10, 111,
kaccu
=
kdiffdA,
where
21
(1.1)
P
3
L'ifQA(w
dA
(1.2)
Here A is the phonon MFP, w is the radial frequency, p indexes the different phonon
branches, C, is the mode specific heat, and v is phonon group velocity. The integration variable is A, so all properties are treated as functions of A rather than the
more commonly used w dependence. Equations (1.1) and (1.2) assume isotropic dispersion relations and isotropic bulk MFPs [11. Thermal conductivity accumulation
functions for several materials calculated from first-principles simulations are plotted
in Fig. 1-2. From Fig. 1-2, we can see that half of the thermal conductivity of Si
comes from phonons with MFPs > 600 nm, and that other materials like Bi have
much shorter dominant heat carrying MFPs.
1.0
PbT
a)
Bi
~5 .B
E
rCoSb
aA
bS
b
:3 :3 0.6
0 0
S0
4 e
Si
o
W0
0.20.0
T=300 K
1
10
100
1000
10000
phonon mean free path (nm)
Figure 1-2: Normalized thermal conductivity accumulation functions for various materials from first-principles calculations from Ref. [12].
A material's thermal conductivity accumulation function shows which phonon
MFPs dominate heat conduction in that material.
The emerging area of phonon
MFP spectroscopy [4, 5] aims to experimentally determine thermal conductivity accumulation functions of materials indirectly through measurements of the lengthscale-dependent effective thermal conductivity, keff(L), where L is the thermal length
22
scale in the measurement.
Direct measurements of phonon lifetimes and dispersion relations are possible
through inelastic neutron scattering [131, x-ray Raman scattering [14], or high frequency photoacoustics [15, 16]. While powerful as direct methods, neutron scattering
and x-ray Raman methods require expensive equipment, usually in the form of national facilities, and are limited to single crystal samples.
Photoacoustic methods
have been used to measure frequencies up to 2 THz in superlattice structures [15],
but generating high frequency acoustic waves in bulk samples remains a challenging
limitation [17].
The cost and challenges associated with direct methods motivate the use of indirect
methods to measure thermal conductivity accumulation functions.
Measurements
of length-scale-dependent effective thermal conductivity keff(L) data can be related
to thermal conductivity accumulation functions kaccu(A), a process which will be
discussed in Chapter 2.
1.3
Laser-based thermal property measurement techniques
Laser-based, non-contact optical methods provide a convenient tool for studying heat
transport. These techniques use a pulsed and/or modulated laser (pump beam) to
heat the surface of a sample.
The surface temperature variation leads to a change
in optical properties, which is monitored by a probe laser beam. By comparing the
measured response with model calculations, parameters of the sample such as the
thermal conductivity and the thermal interface conductance between sample layers
can be determined.
In this section we introduce three notable measurement tech-
niques that will be relevant for our investigations. Sections 1.3.2 and 1.3.4 discuss
time-domain methods, and Section 1.3.1 discusses a frequency-domain method. Each
of these has been used to study length-scale-dependent thermal conductivity, by varying either beam diameter [4], thermal penetration depth [18], or spatial interference
23
pattern period [19].
1.3.1
Frequency-domain thermoreflectance (FDTR)
The advent of using modulated light to excite a thermal response originates from
the field of photoacoustics [20]. The use of thermoreflectance as a means for detecting a thermal response from a periodic heat input was pioneered by Rosencwaig and
coworkers in 1985 [21], representing the first experimental demonstration of frequencydomain thermoreflectance (FDTR) [21, 22, 23, 18].
When a material undergoes a
change in temperature, its index of refraction, and correspondingly its reflectivity,
changes. Thus, observations of thermoreflectance can be related to a material's thermal response. A schematic of the FDTR system in our lab is shown in Fig. 1-3(a).
A continuous wave (CW) pump laser beam is sinusoidally modulated by an electrooptic modulator (EOM). The pump produces time-harmonic heating on the sample,
resulting in surface temperature oscillations at the modulation frequency, which are
monitored by a CW probe beam. The amplitude and phase of the surface temperature response are measured as functions of the modulation frequency, making this
a frequency-domain measurement. Phase measurements are oftentimes preferred due
to their higher accuracy [18].
The modulation frequency in FDTR typically varies from kHz to -10
MHz [21,
22, 23]. Recently, an extension of the frequency range up to 200 MHz was reported
[18, 24], using a method named broadband FDTR (BB-FDTR) by the developers. A
schematic of the BB-FDTR system in our lab is shown in Fig. 1-3(b). By additionally modulating the reflected probe with a second EOM, BB-FDTR enables lock-in
detection at the combined lower frequency of fo - fi, which allows for modulation
frequencies up to fo = 200 MHz. Since the penetration depth of the temperature
oscillations becomes smaller at higher frequencies, such an extension is beneficial for
studying thermal transport at fine spatial scales.
24
(a)
laser
iCW
488 nm pump
EOM, fo
CW lasrobe
BS
sample
obj.
n
V4
PBS
r_
band pass detector
filter
(b)
CW laser
LJ
488 nm pump
EOM, fo
CW laser
532 nm probe
BS
sample
obj.
X/4
PBS
EOM, f1
band pass detector
filter
Figure 1-3: Schematic diagrams of (a) the traditional frequency-domain thermore-
flectance (FDTR) and (b) the broadband FDTR (BB-FDTR) systems in our lab,
which were based off the design of Ref. [18].
1.3.2
Time-domain thermoreflectance (TDTR)
Time-domain thermoreflectance (TDTR) is a closely related technique to FDTR.
The first TDTR experiment was reported by Paddock and Eesley in 1986 [25], and
notable improvements to the technique were made by Capinski and Maris [26 and
by Cahill and coworkers [271. In TDTR, the pump beam comes from a femtosecond
laser operating at a high repetition rate (typically -80
MHz) and is additionally
modulated by an EOM, as illustrated in Fig. 1-4, which depicts the TDTR setup in
our lab. Unlike FDTR, the heating in TDTR is not time-harmonic but is comprised
of many frequency components. The probe beam is derived from the same laser, and
the probe pulses are delayed with respect to the pump pulses by a mechanical delay
line. Our TDTR setup uses a probe beam that is concentric with the pump beam. To
25
prevent pump light from entering the detector, we frequency-double the pump beam
using a second harmonic generator (SHG) optic. To mitigate overlap alignment errors
as the delay stage is swept, the probe is expanded prior to entering the delay line,
and the focused probe beam diameter on the sample is typically much smaller than
the pump beam diameter.
The thermoreflectance response is measured by a lock-in amplifier with the pump
modulation frequency serving as a reference.
The TDTR signals, i.e. the in-phase
and out-of-phase (quadrature) outputs of the lock-in amplifier as functions of the
delay time, do not, in general, reflect the actual time-domain dynamics of the surface
temperature of the sample. However, just as in FDTR, the response can be compared
to model calculations to determine the properties of the sample such as the thermal
conductivity of the substrate and the thermal boundary resistance between the substrate and the metal film typically used to facilitate both laser-induced heating and
thermoreflectance measurements [28, 29].
fo
red SHG
filter
EOM, fo
4x
compress 44x
pump
400 nm
probe
800 nm
A/2
sample
LLJ U
1Ox dichroic BS
obj.
expnd
exan
f~
111
moable
molasae
delay
stage
aser, fs
blue detector
filter
Figure 1-4: Schematic diagram of the TDTR system in our lab [30, 29].
26
k+
k2
1.3.3
Heat transfer model for TDTR and FDTR responses
Typically TDTR data is interpreted by solving the heat diffusion equation in the
frequency domain to find the temperature response to time-harmonic heating, just as
in FDTR measurements. In order to model a TDTR signal, many frequency responses
are added together [28, 29]. Modeling TDTR and FDTR responses with the thermal
diffusion equation is well documented in the literature [28, 29, 31]. We follow the
methodology of Ref. [29] to model heat transport using a thermal quadrupole [32, 33]
solution to the heat diffusion equation for a multilayer, semi-infinite solid, which
accounts for both radial and anisotropic conduction.
The heat diffusion equation in cylindrical coordinates is
C OT
ot
(1.3)
r Or
r
Or &z2
where T is temperature, k is thermal conductivity, C is volumetric heat capacity,
r is the radial in-plane direction, z is the cross-plane direction, and t is time. The
time-harmonic nature of the heat flux excitation and the radial symmetry of the
problem enable simplifications of the heat diffusion equation with Fourier and Hankel
transforms respectively. Applying Fourier and Hankel transforms to Eq. (1.3) leads
to
iwCT = -krS 2 T + kz
(z2'
(1.4)
where w is the radial frequency Fourier transform variable, s is the spatial Hankel
transform variable, and T is the temperature in Fourier-Hankel space. Rearranging,
2
=-mT,
where m 2 = (kS 2
(1.5)
+ iwC)/kz. The in-plane direction (r-direction) is assumed to be
isotropic, while anisotropy in the cross-plane direction (z-direction) is accounted for.
Equation (1.5) is a second-order ordinary differential equation, and fits nicely into
the thermal quadrupoles framework [32, 33].
27
The thermal quadrupole approach readily adapts to multilayer structures by the
relating heat fluxes and temperatures on the top and bottom surfaces of a multilayer
stack through matrixes of material properties. For a material layer,
cosh(md)
-sinh(md)
M, =,,m
-kn
sinh(md)
(1.6)
cosh(md)
where d is the layer thickness. For an interface layer between materials, the property
matrix becomes
I
G-1
0
1
Ain =
,
(1.7)
where G is the thermal interface conductance, which is the inverse of the interface
resistance. A relationship between the Fourier-Hankel-domain temperatures T and
heat fluxes q on the top and bottom surfaces of a structure with n layers is given by
1A
Tbottom
jI
qo
(1.8)
~qoton
qbottom
=
M7 1M Ai~...Kf 1
[
B
T 0P
C D
qOP
,
where the top layer corresponds to n = 1. For TDTR, the top surface is thermally
excited and probed, and the bottom layer is thick enough to be semi-infinite. For
a semi-infinite substrate, no heat flux exists through the bottom surface, so the top
surface temperature, Ttp relates to the top surface heat flux, qtop, in the FourierHankel-domain by
TOP=
-D ~
C(1.9)
For time-harmonic heating with a Gaussian-shaped pump laser beam, the top
surface heat flux input is
qtOP -
2AO
X
-2r2
28
exp (ioot) ,
(1.10)
where A, is the absorbed pump power, wo is the pump 1/e 2 radiusi, and w, is the
excitation frequency. Taking Fourier and Hankel transforms of Eq. (1.10) leads to
Z Ao
qtop = -- exp
-s,
((8
(1.11)
-w,).
Thus from Eq. (1.9), the surface temperature in Fourier-Hankel space T',,p is
Z
TtOP =
-D Ao
C 27
/s2
xp
8
-
o
).(1.12)
The response is measured by a coaxial probe beam that also has a Gaussian
intensity distribution with a 1/e 2 radius of wi. Taking an inverse Hankel transform
and weighting by the probe intensity distribution leads to a frequency-domain surface
temperature of [28, 29
A
top(w) =
s
6(W - wo)
2.7r
-D)
-s(_82W
C
(1.13)
0exp 9)ds.
8
The frequency response to a time-harmonic heat flux input is given by h(w)
=
D08-s2(
N(W) 0 o
s
+2(W
(_
exp
8
)
T(w)/(w). Thus the measured frequency response h(w) is proportional to [28, 29]
(1.14)
72)ds.
For practical purposes, constants in front of h, such as the absorbed laser power, are
unimportant because they either cancel out, as in the case of analyzing the phase of h,
or they are not experimentally determinable so the data is treated without absolute
units, as in the case of analyzing the amplitude of h.
1.3.4
Transient thermal grating (TTG)
Transient thermal grating (TTG) experiments [34, 35, 36] generate a spatially sinusoidal optical grating by interfering two short pulsed coherent pump beams.
The
angle of interference between the beams 0, and the pump wavelength A, determine
'The 1/e 2 radius defines the radius at which the intensity is reduced to 1/e2
intensity at the center of the beam.
29
=
0.135 of the peak
the spatial grating period L, where L = A/(2 sin(O/2)) in a medium of air. Absorption
of the optical grating induces a periodic temperature profile that causes a spatially
periodic change in the refractive index, which depends on temperature. The resulting
diffraction grating is examined with a probe beam. The temperature profile relaxes
in time as heat flows from the peaks to the nulls of the spatial thermal grating, causing the intensity of the diffracted probe to vary in time. The rate of the transient
grating relaxation
T,
depends on the material thermal diffusivity a, and the grating
9
wavevector q = 27r/L, where T = aq2
Figure 1-5 illustrates TTG experimental setups for both transmission and reflection style measurements [37]. A phase mask is used to split the pump beam into
1
diffraction orders, which are subsequently focused on the sample, producing a spatial
interference pattern. The angle of interference 0 can be adjusted by varying the period of the phase mask. The diffracted probe is mixed with a low intensity reference
beam for phase-controlled heterodyne detection [38], which improves signal-to-noise.
The relative phase of the probe and reference beams are adjusted by rotating a glass
slide in the probe path. The detector signal as a function of time is monitored with
a high resolution oscilloscope.
1.4
Organization of the thesis
In this thesis, we explore various strategies for measuring non-diffusive transport and
how such measurements can be related to thermal conductivity accumulation functions, which provide valuable information about the phonon MFPs which are most
important for transporting heat.
In Chapter 2 we discuss modeling non-diffusive
transport using the Boltzmann transport equation and reconstructing thermal conductivity accumulation functions from experimental measurements in non-diffusive
transport regimes.
Our BTE calculations focus on TTG experimental geometries,
which are especially amenable to theoretical analysis. We explore different simplified
models for approximating the full BTE result, and show that integrating gray-medium
solutions can reasonably reproduce solutions for materials with narrow thermal con-
30
532 nm probe =
515 nm pump
-
(a)
ND filter
sample
detector
phase
phase
mask
probe + reference
adjust
(b) 532 nm probe
detector
-
515 nm pump
ND filter
sample
0.phase
mask
phase
adjust
Figure 1-5: Schematic diagrams of TTG systems for (a) transmission and (b) reflection measurements based off of Ref. [371.
ductivity accumulation functions. We also use measured TTG data on Si membranes
of varying thicknesses to accurately reproduce the thermal conductivity accumulation
function of bulk Si.
Chapter 3 presents our investigation of a method for measuring non-diffusive transport over 100 nm length scales using microfabricated wire grid polarizers. We design
gratings that meet a set of criteria including light blocking, fabrication constraints,
and likely requirements for observing non-diffusive behavior. We fabricate gratings
on substrates of Si, fused silica and polycrystalline Bi2 Te 3. Transmission measurements on the transparent fused silica samples confirm the polarizing characteristics
of our fabricated gratings. TDTR measurements reveal clear non-diffusive behavior
in Si and bulk behavior in fused silica, as expected, although we identify that some
small direct optical excitation of the Si substrate adds to experimental uncertainty.
For polycrystalline Bi 2 Te3 , we do not observe any deviation from bulk thermal con-
31
ductivity, suggesting that the heat carrying phonons in polycrystalline Bi 2 Te 3 have
MFPs of less than 100 nm.
To study even smaller length scales without the need for microfabrication, we develop a method for extracting high excitation frequency information from our TDTR
measurements, which we discuss in Chapter 4. We show that FDTR and TDTR provide identical information, and develop a method for transforming TDTR data into
the frequency response data measured by FDTR. The high time resolution inherent
in TDTR systems enables the extraction of high frequency response information, at
least as high as 1 GHz, which is a regime currently inaccessible to FDTR methods.
Our measurements to date have not observed behavior that deviates from the heat
diffusion equation, suggesting that earlier reports of non-diffusive transport at high
frequencies [18] may have resulted from transport effects in the metal transducer
layer, which may obscure observations of non-diffusive transport in the substrate
under study.
32
Chapter 2
Modeling non-diffusive heat
conduction
In non-diffusive transport regimes, the assumptions in the Fourier heat diffusion equation break down, as discussed in Section 1.1. A more appropriate model that applies in
both diffusive and non-diffusive regimes is the Boltzmann transport equation (BTE)
[9, 2]. To gain intuition about behavior in non-diffusive transport regimes, we examine BTE solutions for the simple transient thermal grating (TTG) experimental
geometry introduced in Section 1.3.4. We go on to explore methods for connecting
experimentally measured thermal length scale information to thermal conductivity
accumulations functions, which were introduced in Section 1.2. Reliable methods to
measure thermal conductivity accumulation functions are a key challenge for the field
of phonon mean free path spectroscopy.
2.1
Boltzmann transport equation (BTE) overview
A general form of the BTE is given by [9, 2]
Of
Of
Of
=
where
f
scatt
f+v-f+a-
(2.1)
is the distribution function, v is velocity, a is acceleration, and t is time. For
33
phonon heat conduction, the acceleration term drops out because there are no forces
accelerating phonons.
The scattering term can be simplified under the relaxation
time approximation, which is valid for temperatures comparable to or greater than
the Debye temperature [39]. Under the relaxation time approximation, the scattering
term becomes
Of
fo - f
where
T
is the relaxation time.
pation function,
f0 ,
(exp(hw/kBT) -
1
(2.2)
T
Ot)scatt
For bosons (like phonons) the equilibrium occu-
follows the Bose-Einstein distribution function,
f,
=
fBE
=
)-i, where h is the reduced Planck constant, w is angular fre-
quency, kB is Boltzmann's constant, and T is temperature. For heat transport along
one dimension (ID) where boundary scattering is unimportant, the BTE becomes
2o-
Of +
i f =
-fo 0--, f(23
+11
at
OX
T
(2.3)
where x is the spatial dimension and i = cos(6) is the directional cosine.
When using the BTE to describe phonon dynamics, it is convenient to perform a
change of variables, defining the phonon distribution function g as the phonon energy
density per unit frequency interval per unit solid angle, g = hwD(w)f/4Tr, where
D(w) is the phonon density of states. Thus, the phonon BTE (also referred to as the
equation of phonon radiative transport and the Boltzmann-Peierls equation) is given
by [40, 9, 2]
Bg Og
- + pt
at
OX
where v is the phonon group velocity and
g -g
=
T
T
,
is the phonon relaxation time.
(2.4)
The
equilibrium distribution function follows the Bose-Einstein distribution. A first-order
Taylor expansion can be used to further simplify the equilibrium distribution go,
leading to a proportional relationship with temperature [41]
34
1
go(T) = -- hwD(w)fBE eg0 o(T)
47r
1
- 1C
47r
(2-5)
(T - To),
where C, is the differential, frequency-dependent specific heat (also called the mode
specific heat). Such a simplification is reasonable as long as the temperature deviations from the equilibrium temperature, To, are small.
In addition, conservation of energy requires [9, 42, 41
0jwrn
I
f J
(2.6)
didw.
dpdw =
In the gray-medium approximation, where a material contains a single phonon relaxation time and no parameters depend on frequency, the integrals over W cancel,
leading to an even simpler relationship between g and go,
g0 - 1j
2
_
(2.7)
gdp.
Together, Eqs. (2.4) and (2.6) are used to solve for the equilibrium distribution, which
relates to temperature, for all space and time given a set of initial and boundary
conditions.
2.2
Modeling the TTG experimental geometry
One notable experimental method for measuring thermal properties is the transient
thermal grating (TTG) technique [34, 35, 36], which we introduced in Section 1.3.4.
The TTG technique is especially amenable to theoretical analysis because of its inherent spacial symmetry. As shown in Fig. 2-1(a), crossed laser-beam interference
produces a sinusoidal pattern with a period L determined by the laser wavelength A
and the angle of interference 0, where L = A/(2 sin(0/2)) in air.
Absorbed optical energy leads to a sinusoidal temperature excitation that decays
in time following the short pulse laser excitation, as illustrated in Fig. 2-1(b). For
opaque samples, the excitation grating exists near the surface, while for transparent
samples, the grating penetrates throughout the depth of the sample.
35
Our analysis
treats the one-dimensional case of a TTG measurement in a bulk material not influenced by boundary scattering, where the depth of the thermal grating is much greater
than the grating period. Shortly after pulsed laser excitation, the temperature profile
is given by
T(x,t = 0) = Ta.cos(qx),
(2.8)
where T is the temperature deviation from T, the average background temperature,
Tmax is the peak temperature deviation, t is time, x is the spatial variable, and
q = 27r/L is the spatial wave vector. The solution to the heat diffusion equation
yields an exponential decay of the form
T(x, t) = Tma cos(qx)e-a
2
,
(2.9)
where a is the material's thermal diffusivity.
Lt
(b)
(a)
Figure 2-1: (a) TTG crossed laser beam interference produces a sinusoidal interference
pattern that induces (b) a spatially-sinusoidal temperature profile that decays in time.
Non-diffusive phonon transport has been observed when L is within the range
of the mean free paths (MFPs) of the heat carrying phonons in a material [19]. At
small grating periods, Johnson et al. observed deviations from the bulk, diffusive
thermal conductivity of Si membranes. Our theoretical treatment differs from the
experimental geometry of Ref.
[19], where heat transport was influenced by the
boundary scattering in a thin membrane. Rather, our approach is appropriate for
TTG measurements in bulk materials, where the depth of the thermal grating is
36
much greater than the grating period [34]. In order to extend this approach to thin
membranes or strongly absorbing materials in which heat is dissipated into the depth
of a sample, a multidimensional BTE would need to be considered.
We discuss a
boundary scattering problem for thin membranes in Section 2.3.3.
In non-diffusive transport regimes, where the assumptions of the heat diffusion
equation break down, the transport physics are better described by the BTE. We
begin by considering solutions to the gray-medium BTE under the relaxation time
approximation in Section 2.2.1, before moving on to discuss spectral solutions in
Section 2.2.2.
2.2.1
Gray-medium BTE
The simplest approach to solving the BTE is to assume a gray-medium, where only
one phonon MFP exists in a material, and both v and
T
are constants. For the gray-
medium BTE, we derive a dimensionless, semi-analytical solution for the ID TTG
relaxation, and show the limiting behaviors of the decay. As described in Section 2.1,
under the relaxation time approximation, the 1D phonon BTE takes the form of Eq.
(2.4) and obeys Eq. (2.7). By Eq. (2.5), for small temperature excursions relative
to the background temperature T,, the equilibrium distribution g, is proportional to
temperature
g 0 (Xt)
47r
hwD(w)fBE(T)
47r
CwT.
(2.10)
For mathematical convenience, the distribution function g and the temperature T
are defined as deviations from the average background values, which correspond to
thermal equilibrium at the background temperature.
We seek a spatially-periodic solution for g by assuming that g and T are of the
form p(t, p)exp(iqx) and T(t)exp(iqx). Now Eq. (2.4) takes the form of a first-order
ordinary differential equation for g,
OtT
+
go= ,(2.11)
37
with the solution
j(t, pt)
1
-
e-('-')jo(t')dt' + Ae-ft,
(2.12)
(1 + iqprVT)/T and A= j(t = 0,where
[t) = j,(O). Although
I the thermal
grating is one-dimensional, we account for phonons traveling in all directions using
the directional cosine, t = cos(O). Applying Eq. (2.7) leads to
o(t')e(t'-t)/rsinc(qv(t' - t))dt',
jo(t) = jo()sinc(qvt)e-/ + -j
T0
where sinc(x) = sin(x)/x.
We can further generalize Eq.
(2.13)
(2.13) by substituting
nondimensional variables,
t
(=-,
T
27rA )T
y~v
L
L
0 (t)
-
go(0)
_
T(t)
,(.4
(0)(
producing
T(()
sinc(r()eC +
T((')sirnc(j((' - ())e('-C)d(',
(2.15)
which is a nondimensional solution of the gray-medium, one-dimensional phonon BTE
for a sinusoidal temperature profile. Here A is the phonon MFP, which relates to
relaxation time and group velocity through A = vr. 9 is the phonon Knudsen number,
a ratio of MFP to characteristic length. Here the characteristic length is given by the
spatial wave vector of the grating, q = 27/L. Equation (2.15) is a Volterra integral
equation of the second kind, which can be solved using standard numerical techniques
[43].
An alternate approach to deriving a non dimensional, analytical solution to the
ID gray-medium BTE for a TTG temperature profile assumes a spatially-periodic
solution for g and utilizes a Fourier transform in time. Starting with the gray-medium
phonon BTE with a spatially-periodic instantaneous source,
38
=
+p
+
o
(2.16)
A6(t)eiqx,
assuming a periodic spatial solution of the form, exp(iqx), and taking a Fourier
transform in time leads to
where
=
=
z 9
--
(217
( 2.17
A,
T
)
-ivy + iqv p=
f j(t, p)eivtdt. Integrating over p to satisfy energy conservation,
as shown in Eq. (2.7), gives
AT
z
9 qA (tan-' (j-0)
-
(2.18)
1
which is an analytical solution for the equilibrium distribution function in the frequency domain. To obtain a time domain solution, an inverse Fourier transform can
be performed numerically,
- A-T
A
= pdv. (2.19)
27r
exp(-ivt)
-oc
qA (tan-' (
-1
A))
Writing Eq. (2.19) in the nondimensional variables of Eq. (2.14) leads to
T(C)
where
2
=
=
127
d',
_)
7O( tan-
(2.20)
-1
vT.
Equations (2.15) and (2.20) produce identical decay curves, shown by the solid
lines in Fig 2-2.
A range of different 77 values are used to compute T(() decays.
Small rj values show more diffusive behavior, while larger n values produce highly
non-exponential decays.
Decays that oscillate about T = 0 (the n
=
5 curve for
example), indicate that the sinusoidal temperature profile relaxation illustrated in
Fig. 2-1(b) overshoots the T, equilibrium position before settling. The solution to
the heat diffusion equation, given by Eq.
non-dimensional variables as
39
(2.9), can be rewritten in terms of our
T(() = e-O,
where
i
(2.21)
= aq 2T. The diffusion equation predicts an exponential decay described by 0.
In the diffusive limit, abulk = vA/3, and
/
3
ul
=l
rj2 /3. For comparison, the diffusive
limit curves are also plotted in Fig. 2-2. The diffusive limit curves yield faster decays
than the corresponding BTE curves, indicating that the thermal transport at small
length scales slows down compared to Fourier law predictions [9, 1, 2, 8, 411.
-- gray BTE
-- diffusive lim.
0..5
05
=5
0
2
4
6
8
10
Figure 2-2: Dimensionless gray-medium BTE thermal decays for a range of dimensionless length scales (solid lines) compared to diffusive limits (dashed lines) and
Fourier model best fits (dot-dash lines).
At large r7 values, the decay becomes strongly non-exponential and acquires an oscillatory character. The thermal decay for very large r/ values approaches the ballistic
limit, as shown in Fig. 2-3. In the ballistic limit,
T -4
oo, and Eq. (2.15) reduces to
T = sinc(r](). We can understand this oscillatory behavior by considering the case
of purely ballistic transport, which would be equivalent to having non-interacting
particles moving with a constant velocity v, and having an initial density distribution
cos(qx). For a subset of particles whose velocity makes an angle 0 with the x direction, the particle density will oscillate as cos(qx - qvpt). Integrating over all angles
yields a sine function in time, identical to the ballistic limit derived from the BTE.
40
71=
q = 100
10
1
66
-gray
0.5
-
S0
BTE
-ballistic lim.
--
0
2.5
0.5
5 0
- t1T
1
= t1T
Figure 2-3: Dimensionless gray-medium BTE thermal decays compared to the ballistic
transport limit.
We can also gain insight by fitting the gray-medium BTE decays using the heat diffusion solution to find best fit values of /,
or correspondingly, the "effective" thermal
diffusivity. In experiments, analyzing non-diffusive data with the Fourier heat diffusion equation is common practice [37, 19]. Example best fit Fourier curves are shown
in Fig. 2-2. This produces a set of effective values of 0, which are normalized and
plotted in Fig. 2-4. In the limit of small r, which corresponds to large grating periods,
transport is in the diffusive regime, with Oeff /#buk
3 3
/
eff /712 - Ceef/ Ceul = 1, and
Fourier fits are good. At progressively larger values of I, the transport transitions
to the ballistic regime, with BTE curves displaying highly non-exponential behavior
which cannot be captured by the fitted Fourier curves, and the effective diffusivity
approaches zero. This result does not mean that ballistic phonons do not carry heat;
they simply transfer much less heat than diffusion theory predicts.
The curve in Fig. 2-4 is universal due to its dimensionless form, and the graymedium assumption. If a material behaved like a gray-medium, we could predict the
aef values that would be measured in a TTG experiment by scaling the horizontal axis
with the appropriate phonon MFP. In general, however, materials do not behave as
gray-mediums, so it is more accurate to consider a solution to the BTE that accounts
for different phonon modes.
41
-
-
1
0.8-50.6.0
ao0.4
0.20 2
10
-1
10
0
1
10
10
7=27rA/L
2
10
3
10
Figure 2-4: Set of effective thermal diffusivities aeff normalized to the bulk thermal
diffusivity abulk, which were found by fitting the gray-medium BTE decay curves for
different values of q with the exponential solution from the heat diffusion equation.
2.2.2
Spectrally-dependent BTE
Since real materials support a range of phonon MFPs, a solution to the spectrallydependent BTE will provide a more accurate representation of the real thermal relaxation as compared to the simpler gray-medium approximation. Moreover, a solution
that incorporates a realistic phonon density of states and set of phonon relaxation
times would be an improvement over semi-empirical models, like the Callaway [441 or
Holland [45] models. We utilize density of state and relaxation time data for all six
phonon branches calculated from first principles density functional theory (DFT) in
our spectrally-dependent BTE solution [46, 47].
Figure 2-5 shows the DFT parameters utilized in our calculations for Si and PbSb
respectively at 300 K. The range of values plotted arise from different directions in
the Brillouin zone. Our calculation assumes an isotropic material, so we interpolate
the DFT parameters to find the mean value for a given phonon frequency and branch,
and verified that our averaged properties produced literature values for the thermal
conductivities and volumetric specific heats. Si and PbSe are interesting case materials to consider due to their different thermal conductivity accumulation functions. Si
has a thermal conductivity accumulation function spanning a wide range of MFPs,
42
from tens of nanometers to tens of microns [46], while PbSe has a much more narrow
distribution [47].
In the 1D spectrally-dependent treatment, the distribution function depends on
four variables, g
g(x, t, p, w). As in the gray-medium case, we assume a spatial
dependence of the phonon distribution and temperature of the form exp(iqx), producing
0+ qt
ot
-
+ 'tqp/g =
g -gj
(2.22)
2.2
T
T
The equilibrium phonon distribution function is found from Eq. (2.6). To simplify
the analysis of Eq. (2.6), a small temperature rise is assumed, such that
-
.1
o(T) ~~-Cwi,
(2.23)
4fr
leading to a temperature variation of [41]
T
0'
j
-dw
fo
j
f_1 T
(2.24)
dpdw.
The summation over phonon branches is implied in the integrations over W.
Our numerical solution of Eqs.
(2.22)-(2.24) uses an explicit finite difference
scheme. The granularity used in our calculations for the variables in the finite difference solution included at least 32 bins for p and 100 bins for w, with dt < 2 ps.
Convergence was verified by systematically increasing granularity. We further verified
our spectral BTE code by inputting gray-medium parameters and achieving identical
results to the gray-medium model discussed previously. As an additional confirmation, we implemented a solution that did not assume a spatial dependence on g or T,
which produced the same results, but at a much higher computational cost.
Figure 2-6 shows temperature decay curves for Si at 300 K for a range of TTG
periods, and the corresponding diffusive limit decays.
As with the gray-medium
model, the thermal grating decays more slowly than the diffusive model predicts,
even for grating periods as large as L = 20
pm. The decay retains an exponential
behavior even at grating periods as small as L = 1 pm, in contrast to the gray-medium
43
10 8
TA1
TA2
*LA
- LO
10000.
0
4
8000-
00
o
0
8e
00o
a)F
E
- TA2
* LA
- LO
TOl
10
-TO1
***
*
6000-
10
00
1 X10
o TA1
0.
-TO2
0.61
0W
E
en 0 .4
3
-
5
o) (rad/s)
t
10
10-1
0
5
o) (rad/s)
X 1013
TOl
TA1
TA2
LA
9
2NA
-1
2000
^0
04C
C
T021I
-1
4000
10
8-
-
12000
LO
0 .2
0
10
x 1013
5
(0
(rad/s)
10
x 1013
(a)
4500
- TA1
. TA2'
* LA
* TOl
4000
3500
3000
200
%*
10~10
o TA1
TA2
OLA
- T01.
02
N
0*
15T02
1 x 10TA2
10 9
.*0
0.-
0 .8
cc
0
C?
E
Ci)
LA
0 .4
P
101
:!!F
T02
0 .6 TA1
10
>
LO
TOl
2040
0 .2
25-
.
-13LI
(t
0
(rad/s)
X 1013
2
o (rad/s)
4
x 1013
0
0
2
o (rad/s)
4
13
x 10
(b)
Figure 2-5: Set of DFT material parameters for (a) Si and (b) PbSe at 300 K,
including phonon group velocity v, relaxation time T, and mode specific heat C",
for longitudinal acoustic (LA), longitudinal optic (LO), transverse acoustic (TA),
transverse optic (TO) phonon branches as a function of phonon frequency w. Different
data for the same value of w arise from different Brillouin zone directions. DFT data
was provided courtesy of Keivan Esfarjani and Zhiting Tian [46, 47].
44
solution shown in Fig. 2-2, which exhibits non-exponential behavior even for small
values of rq = 27rA/L.
L = 2 rm
L =0.2 Rm
1~
L = 20 sm
Si
I
-spec.
BTE
-- -diffusive
lim.
( 0.5
0
o
0.1
0.2 0
t (ns)
5
t (ns)
10 0
250
t (ns)
500
(a)
L = 100 nm
L=40nm
L=10nm
1
-spec.
BTE
-- -diffusive lim.
PbSe
CD
0.5
0
0.01
t (ns)
0.02 0
0.1
t (ns)
0.2 0
0.5
t (ns)
1
(b)
Figure 2-6: TTG thermal decays for a range of grating periods calculated from a
numerical solution to the spectral BTE using DFT input parameters for (a) Si and
(b) PbSe at 300 K, compared to the diffusive limit.
Again proceeding in accordance with typical experimental methodologies, we fit
the BTE decays using the heat diffusion equation to find effective values of the thermal
diffusivity for each grating period. The resulting effective diffusivities are normalized
and plotted in Fig 2-7(a) for Si and Fig.2-7(b) for PbSe.
Now that we have calculated a spectral BTE solution, we can compare it to the
simpler gray-medium BTE, for which we derived the universal solution plotted in Fig.
2-4. Normally, MFP estimates are obtained from the experimental values of thermal
diffusivity using the expression given by the gray-medium BTE, abuIk = vA/3, and
assuming the Debye model in which v is the branch-average acoustic velocity [2]. This
approach yields MFP values of ~40 nm for Si [2, 48] and -2 nm for PbSe [47, 49], but
these produce effective diffusivity curves shifted towards much lower grating periods
than our spectral BTE calculations, as shown in Fig. 2-7. It has been suggested that
the gray-medium BTE can be made to work better for Si by using a larger MFP value
45
0.8
m
0
0.6
>0.4 -,
>0.4
,|
0.2
0.2
,
010 ----- r
10
10-9 10
<
10~
1
L (m)
Ie
Si, 300 K
S . ,,*. .. *...5.-------3
/
gray, A =1
4mPbSe, 300 K
--
1pc
0.8 .......
+spec, BTE
o.--gray,A=2nm
.. gray, A = 6.5 nm
10 10 110 ~910~ 10~1 10 - 10-5 10~4 10
L(m)
(a)
(b)
Figure 2-7: Effiective thermal conductivity for various TTG periods calculated from
the spectral BTE for (a) Si and (b) PbSe at 300 K (solid markers). Comparisons are
shown for the gray-medium model using literature values of gray MFP (dashed lines)
and best fit gray MFP values (dotted lines).
[50, 48, 18]. We find that the best fit to the spectral BTE results for Si is achieved
with a MFP as large as 1 pm, and even then the fit is quite poor, as can be seen
from the dotted line in Fig. 2-7(a). For PbSe, a MFP of 6.5 nm yields a somewhat
better fit to the spectral BTE results, as shown in Fig. 2-7(b). In the next section,
we discuss approximate models that are able to more accurately match our spectral
BTE results.
2.2.3
Frequency-integrated gray-medium model
A single-MFP gray-medium model is not able to accurately describe the
(L) that
we calculate from the full spectral BTE. We can improve on the single-MFP model
by assuming that phonons of frequency w, for a given phonon branch, contribute
to the thermal conductivity according to the gray-medium model with MFP A(w).
The effective thermal conductivity is found by summing over the phonon spectrum
as follows:
keff
K
where the function
Sgray
kbf1
kwuhr
1
1
f"
3kbutko
Jo
Sgray
ovAdw,
(2.25)
describes how the contributions of phonons are reduced
46
compared to the predictions of the diffusive heat diffusion equation. Sgray = aeff/abuk
is shown in Fig. 2-4.
This approach might be reasonable if phonons of different frequencies did not
interact such that phonons at each frequency, for a given phonon branch, obeyed the
gray-medium BTE. At room temperature, phonon scattering is dominated by phononphonon interactions, in which case Eq. (2.25) lacks a solid foundation. Nevertheless,
one can hope that it will yield an improvement over the single-MFP gray-medium
model, and indeed we observe that this "frequency-integrated gray-medium" approach
does yield better results, as shown by the dash-dot lines in Fig. 2-8. In fact, for PbSe
the dependence of the effective diffusivity on grating period is reasonably reproduced
over a wide range of grating periods.
The thermal conductivity accumulation function for PbSe is more akin to a graymedium than the accumulation function for Si, which spans a much broader range of
phonon MFPs. To describe the behavior in Si, we require a model that accounts for
the interactions of different phonon modes.
1
0.8
-
-
spec. BTE
--- gray, A= 1
1
-'
..-
0.8
_+0.6
10.6
-9
-5
0.4
I
0.4
-
.-
0.2----10
1
10
1
L (m)
1 Si, 300 K
10
i
PbSe, 300 K
*0m
-- freq-int gray-med.
+0
.
spec. BTE
gray, A = 6.5 nm
-freq.-nt. gray-med.
.
5
10
10
1
1
1104
L (m)
10
(b)
(a)
Figure 2-8: Effective thermal conductivity for various TTG periods calculated from
the spectral BTE for (a) Si and (b) PbSe at 300 K (solid markers). Comparisons are
shown for the best fit gray-medium model (dotted lines) and the frequency-integrated
gray medium model (dash-dot lines).
47
2.2.4
Two-fluid model
Maznev et al. derived an approximate solution to the spectral BTE in the ID TTG
geometry by focusing on the onset of non-diffusive transport, where the TTG period is
much larger than the MFPs of high-frequency phonons that are responsible for most of
the specific heat [8]. Accordingly, those high-frequency phonons are assumed to obey
the diffusion model, whereas the low-frequency phonons are analyzed with the BTE.
Within this two-fluid approach, it was found that the TTG decay remains exponential,
as in 2.9, with the thermal conductivity modified by a suppression function as follows:
keff
kbulk
1
_
3
kbu1k
J
fWmax
(2.26)
with
Stwo-fluid
where q = 27rA/L.
4
(
Figure 2-9 compares
--
,
tan'
Sto fluid
to
(2.27)
Spay
from our frequency-
integrated gray-medium model.
1
-sgray
C/)
c 0.8
two-fluid
0.6
0
0.6
U)
1
-1
0
Uc
-2
10
0
100
10
102
103
fl = 27cA/L
Figure 2-9: Comparison of suppression functions from the two-fluid model and the
frequency-integrated gray-medium model.
A comparison of our calculated spectral BTE decays for Si at 300 K and the
48
exponential TTG decays predicted by Ref. [8] shows good agreement down to L =1
pm, where the spectral BTE yields a nearly exponential decay, and remains reasonable
even at L = 0.2 pm, as shown by the dotted lines in Fig. 2-10. In Ref. [8] it was
suggested that for Si at 300 K, the approximate solution would be expected to work
for L > 1 pm. We see that in fact it works quite well for a much wider range of TTG
periods.
L
0.2 [m
BTE
-spec.
-- diffusive lim.
- two-fluid
L=1 m
.
0
1
1L
0.1
t [ns]
0.20
=2im
1
t [ns]
L =20
2
m
0.5
0
5
t [ns]
100
250
t [ns]
500
Figure 2-10: TTG thermal decay calculated from a numerical solution to the spectral
BTE using DFT input parameters for Si at 300 K (solid lines), compared to the
diffusive limit (dashed lines) and the approximate solution provided by the two-fluid
model (dotted lines).
The two-fluid model proposed by Ref.
[8] closely predicts the dependence of
effective diffusivity on TTG period for Si at 300 K, as shown by the dashed line in
Fig. 2-11(a). The agreement is quite good in Si for L > 1 pm, and is reasonable in
PbSe for L > 200 nm. The assumptions in the derivation of the two-fluid model are
only valid during the onset of non-diffusive transport, where the grating period is large
compared to the MFPs of high frequency phonons that are primarily responsible for
specific heat. These high frequency phonons are modeled as a thermal reservoir that
obeys the heat diffusion equation, while low frequency phonons, which are mainly
responsible for thermal conductivity, are modeled with the BTE. Thus, the two-fluid
model is only expected to be valid for large grating periods, which agrees with the
49
findings in Fig. 2-11.
spec. BTE
0.8
e
PbSe, 300 K
------ gray, A= 1 stm
---- freq.-int. gray-med.
--- two-fluid
0.8,4/
0.6
/
_0.6
0.4
0.4
0.2
0.2
i
+ spec. BTE
10~
//- freq.-int. gray-med.
10
10 810
10
L (m)
gray, A = 6.5 nm
/...
Si3 K
two-fluid
'--
Si, 300 K
10~5 10 -4
10
10-9 10
10~ 10L (m)
10-5 104 10-
(b)
(a)
Figure 2-11: Effective thermal conductivity for various TTG periods calculated from
the spectral BTE for (a) Si and (b) PbSe at 300 K (solid markers). Comparisons
are shown for the best fit gray-medium model (dotted lines) as well as the predicted
results using both the gray-medium (dot-dash lines) and the two-fluid (dashed lines)
heat flux suppression functions.
2.2.5
Contribution of long MFP phonons
Experimental observations of large deviations from the heat diffusion equation do
not necessarily indicate that long MFP phonons are contributing significantly to the
observed heat flux reduction.
In fact, for typical TTG periods (L > 1 pim), most
of the reduced heat flux can be attributed to phonons with A < L/2. As a first
approximation, we define long MFP phonons as those with A > L/2 and consider the
contributions of these phonons to reductions in thermal conductivity.
The thermal conductivity accumulation function, kaccu is defined as
kaccu(A) =
SovA
3
dA
dA,
(2.28)
where S is a function describing how the phonon modes are reduced relative to the
predictions of the diffusive heat diffusion equation as a function of experimental length
scale. In Section 2.2.4, we demonstrated that the two-fluid model is in good agreement
with the full spectral BTE calculations for Si at 300 K, so for simplicity we use
50
S =
Stwo-fluid,
given by Eq. (2.27). Figure 2-12 shows kaccu(A) for several different
TTG periods using DFT input parameters for Si at 300 K. Each curve asymptotes at
the effective thermal conductivity plotted in Fig. 2-11(a) for a given grating period.
150
.6
bulk
L= 100 pm
100-
L= 10pm
E
a 5Q
.L=1
pm
CO)
L=0.1 pm
10
10
10
10
10
MFP,A ([im)
10
10
Figure 2-12: Thermal conductivity accumulation functions for different grating periods calculated using DFT parameters for Si at 300 K and the two-fluid suppression
function (see Eq. (2.28)). Dotted lines indicate A = L/2.
We examine the contribution to kaccu provided by phonons with A > L/2. Figure
2-13 shows the percentage contribution of phonons with A > L/2 to kaccu as a function
of grating period. The contribution of these long MFP phonons is small until the TTG
period is smaller than 100 nm, even though significant reductions in keff compared to
kbulk
2.3
are observed at grating periods as large as 50 pfm, as shown in Fig. 2-11(a).
Thermal conductivity accumulation function
reconstruction
In Sections 2.2.4 and 2.2.3 we discussed the forward problem of predicting effective
thermal conductivity values for different TTG periods from known material parameters and a model of heat flux suppression. More useful is the inverse problem of
predicting thermal conductivity accumulation functions from measurements of lengthscale dependent
effective thermal conductivity and a model of heat flux suppression.
Solving the inverse problem allows experimental measurements to be predictive of
51
; 0.8
0 0
~0.6-
o
o
Z"0.4A
0.20
LI-
*-" 2--3o
10-3
2 --
10-2
1
0
-
102
10-1
100
101
TTG period, L ([m)
103
Figure 2-13: Percent contribution to keff from phonons with A > L/2.
material properties, namely thermal conductivity accumulation functions, which have
been shown to be important for analyzing thermal transport in bulk materials and
nanostructures [10, 11].
2.3.1
Theoretical foundation
Minnich [51] suggested a way to reconstruct the thermal conductivity accumulation
function, 4D(A) = kaccu(A)/kbulk, using non-diffusive measurements of normalized effective thermal conductivity, i(L) = keff(L)/kbulk, and an appropriate suppression
function, S, as follows:
(L) =
OS( )(A)dA=
JOJ
0
D(A)dA.
(2.29)
#(A)
through <D(A) =
d77 dA
Here FD(A) is related to the thermal conductivity per MFP
f #(A')dA'.
S is a function of a dimensionless length scale (oftentimes the phonon
Knudsen number) q = A/L, where L is the thermal length scale in the experiment
and A is the phonon MFP. A similar equation appears in the analysis of thermal
conductivity size effects in nanostructures, with the nanostructure dimension defining
L [11].
52
Even though such a reconstruction is an ill-posed problem, given the limited number of
4(A).
K
measurements, progress can be made if certain constraints are imposed on
Minnich [51] showed that if o(A) is a smooth function that monotonically
increases from 0 to 1, convex optimization [52, 53] can be used to reasonably estimate
D(A).
2.3.2
TTG experimental geometry
Following Minnich's approach [51], we reconstruct the thermal conductivity accumulation functions for Si and PbSe at 300 K. For K(L), we use our full spectral BTE
solutions plotted in Fig. 2-11. For S(r), we test both the two-fluid model,
Stw,_flu2d
(given by Eq. (2.27)) and the frequency-integrated gray-medium model, Sray (shown
in Fig. 2-4). For both suppression function models, the dimensionless length scale is
given by y = 27rA/L, where L is the TTG period.
The resulting reconstructions are shown in Figs. 2-14(a) and 2-14(b). For the
reconstruction, we used a smoothing factor of 1, and set the length of P to 200
elements.
We found that modifying the smoothing factor or the length of (D by
a factor of 2 had little effect on the resulting reconstruction.
The reference D(A)
distributions are calculated from the same DFT dispersion and relaxation time data
[46, 47] that we used for our spectral BTE calculations.
We can verify the reconstruction by using the resulting 4(A) to calculate K(L)
from Eq. (2.29). The resulting calculations, compared to spectral BTE solutions for
K(L), are shown in Fig. 2-15. The reasonable agreement provides confidence in the
convergence of the convex optimization algorithm.
As expected from the results plotted in Fig. 2-11, using Stofluiid produces a more
accurate MFP distribution reconstruction for Si and using Sgray produces a more accurate reconstruction for PbSe. The approximations in the two-fluid approach of Ref.
[8] lead to a more accurate result for low-frequency, long MFP phonons, and indeed,
we observe that the two-fluid model well reproduces the long MFP thermal conductivity accumulation function for both Si and PbSe. Our frequency-integrated graymedium approach is more appropriate for materials that approximate gray-mediums
53
.-------
-
1
Si, 300 K
PbSe, 300 K
1a
0.8
0.8
0
0
0.6
.9
0.4
00
0.6
0
0
0.4
-
0.2
)-reference
refe rence a
o 4)fr oom S,,
o ( fr oom Stwo-fluid
0.2
-
n
1-9
1-
1
0 -6
-7
Do 0
0-4
0-5
o
rom S gray
a 4 from
10 -3
A (m)
10__11
10 -10
10-9
10
A [m]
(a)
10
10
1-7
(b)
Figure 2-14: Reconstruction of thermal conductivity accumulation ftnctions for (a) Si
and (b) PbSe at 300 K using BTE calculated effective thermal conductivity values as
"experimental" inputs. Reconstructions using a gray-medium (open circles) or a twofluid (open squares) heat flux suppression function are shown. Thermal conductivity
accumulation functions calculated from DFT are also shown for reference (solid lines).
1
0.
.
1'
*
spec. BTE
o from S
o3 from S
0. 8
S
0
0. 6-
0. 6
0. 4
-
0. 4-
0
0. 2
10
10-
10^
10-5
10-4
+ spec. BTE
fromSgray 4) reconst.
O
0. 2
S
Si, 300 K
0
10~10 10
PbSe, 300 K
D reconst.
40 reconst.
3 from Stflud 4P reconst.
*
10 -3
L (m)
'-
10
10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
L (m)
(a)
(b)
Figure 2-15: Calculation of normalized effective thermal conductivity i(L) from reconstructed normalized thermal conductivity accumulation functions <D(A) to verify
agreement with the ri(L) values used to reconstruct <D(A) (closed diamonds). Verification is shown for <b(A) determined from Sgray (open circles) and from StOaflicd
(open squares) for both (a) Si and (b) PbSe.
54
with step-like thermal conductivity accumulation functions, and hence works better
for PbSe than for Si.
2.3.3
TTG thin membrane experimental geometry
A convenient geometry for TTG measurements is that of a thin membrane, where the
optical penetration depth of the pump and probe lasers is longer than the membrane
thickness, allowing for a transmission-style measurement. Such a scheme is illustrated
in Fig. 2-16.
pump beams
Figure 2-16: Illustration of a TTG measurement on a thin membrane [54].
Measurements on membranes have the advantage of a well defined thermal length
scale, determined by the membrane thickness, that can be varied over a wide range.
Recent TTG measurements on Si membranes ranging in thickness from 15 nm to 1.5
pm were reported [54]. The data spans a wide range of effective thermal conductivity,
making it especially amenable to reconstructing the thermal conductivity accumulation function. The experimentally measured normalized effective thermal conductivity values as a function of membrane thickness are plotted in Fig. 2-17. This data
was collected at a large TTG period, such that the dominant thermal length scale
was the membrane thickness, not the grating period. Measurements with different
grating periods produced the same results and are included in the plotted error bars.
55
1
0.8
=3
*0. 6
0.4-
0.2
00-9
10
10
-8
-7
10
d(m)
10
-6
10
-5
Figure 2-17: TTG measured effective thermal conductivity at 300 K for a range of Si
membrane thicknesses [54].
A membrane geometry is also amenable for theoretical analysis. The analytical
Fuchs-Sondheimer suppression function was derived from the BTE for electron transport in thin films [55, 56], and has been adapted for phonon transport [19, 54]. The
Fuchs-Sondheimer suppression function is given by
I
-I)e-x'7dx,
S(-)
(2.30)
1-
q+
j
where r = A/d and d is the membrane thickness. The suppression function is shown
in Fig. 2-18.
Using SFS in Eq. (2.29), we reconstruct the bulk thermal conductivity accumulation function in Si, and achieve good agreement with the results from DFT, as shown
in Fig. 2-19.
Since there is some variation in the measured data, as shown by the
error bars in Fig. 2-17, we perform a series of reconstructions using all the possible
combinations of maximum and minimum K data error bars to determine the resulting
restriction variation, which is plotted as error bars in Fig. 2-19.
56
1
0.8
Cl)
UCD)
0.6
-
-3
-
0
0.4
0.2
0
1ic -4
10
10
10
10
10
10
10
10
1 = Aid
Figure 2-18: Fuchs-Sondheimer suppression function, SFS, where d is the membrane
thickness and A is phonon MFP.
1
0.8-
--I-- & 12
-refere nce D
o D frorr SFS
0.60.4-
0.2[
10
If .90.- -8 -7 -6 -5 -4 -3 -2
10
10
10
10 10
A(m)
10
10
10
Figure 2-19: Reconstructed thermal conductivity accumulation function <((A) (open
circles) from the Fuchs-Sondheimer suppression function SFS and experimentally measured K for thin Si membranes (see Fig. 2-17). The thermal conductivity accumulation
function from DFT calculations [46] is shown for comparison (solid line). Error bars
in the reconstructed 4D come from reconstructions based on error bars in the measured
K data.
57
2.3.4
Experimentally deriving heat flux suppression functions
If a heat flux suppression function exists, mathematically it should be possible to
extract it based on the first half of Eq. 2.29, given K and
#.
DFT calculations on a
material such as Si can produce 0, and /- can be obtained through experimental measurements over a range of thermal length scales. The prospect of extracting S from
experimental measurements is inviting, due to the inherent difficulty in calculating S
by solving the BTE.
For this experimental extraction scheme to work, S must be reformulated as a
monotonically increasing function, and as an array rather than a matrix. A change
of integration variable from A to 77, where now q is defined as
=
j(d)
S(n)#(A) (-)
=
d/A leads to
dq.
(2.31)
Here d is the thermal length scale in the measurement, A is the phonon mean free
path, /-, is the normalized effective thermal conductivity, and
ductivity per mean free path. Defining
7 as
#
is the thermal con-
d/A makes S an increasing function of n.
This reformulation makes it possible to treat S as the unknown array in the convex
optimization, which uses the constraints that S increases smoothly and monotonically
from 0 to 1.
The reconstruction of S is complicated by the sharp features in O(A), which require
a fine spacing in A for proper integration. O(A) for Si at 300 K is plotted in Fig.
2-20(a) [46]. The thermal conductivity accumulation function <b(A), shown in Fig.
2-20(b), is related to O(A) through <b(A) = f
#(A')dA'.
To test the viability of reconstructing S based on experimental / and known
#,
we
consider the measurement on Si membranes discussed in Section 2.3.3. Si membranes
form a nice test case for several reasons: DFT calculations for
#
Fig. 2-20(a)); we have a wide length scale range of measured
for Si are available (see
K
data (see Fig. 2-17);
and the correct suppression function is known to be given by the Fuchs-Sondheimer
relationship (see Eq. (2.30)), so the quality of our S reconstruction can be evaluated.
58
x
7
-6-
10,
3
20.8
6-
U
S0.6
4-6 0.40
0
0
2-
. 0.2
E
L_(D
C
10~9
10
10
10
A [M]
10 5
10~-
10
10
10
10~-
10-5
104
A [M]
(b)
(a)
Figure 2-20: (a) Thermal conductivity per MFP and (b) thermal conductivity accumulation function for Si at 300 K from DFT calculations [46].
We proceed to reconstruct S for thin membranes with diffusely scattering boundaries, as shown in Fig. 2-21. To properly integrate 0, we used 1600 elements in the
array of S, and we verified that this discretization could reproduce <b. The large
number of elements in S required a higher smoothing factor to produce a smooth
curve. We used a smoothing factor of 100.
The resulting reconstructed suppression function shown in Fig. 2-21 differs somewhat from the exact Fuchs-Sondheimer suppression function, but can still effectively
reconstruct the Si thermal conductivity accumulation function from measured Si
membrane data. Determining suppression functions in this manner, while attractive
due to the difficultly of solving the BTE, requires further study to verify viability. For
example, the choice of smoothing factor influences the result due to the large length
of the S array required to integrate
2.4
#
reliably.
Summary and future directions
We have presented gray-medium and spectral solutions to the one-dimensional phonon
BTE corresponding to the spatially-sinusoidal temperature profile in a TTG experiment.
Our gray-medium analysis yielded an analytical solution that approached
59
1
1
[***~.~**
CD)
-
0.8
-
S0.6
C
- 0.4
~0.
0.2
C/)
--- reconst.
1
~.-SFS
C-4
-3
-2
-1
0
1
2
.3
10 10 10 10 10 10 10 10
,q = d/A
4
10
Figure 2-21: Reconstructed heat flux suppression function for thin, diffusely scattering membranes (dashed line). The exact solution given by the Fuchs-Sondheimer
relationship (see Eq. (2.30)) is shown for comparison (solid line).
the diffusive limit for grating periods that were large compared to the gray-medium
phonon MFP, and approached the ballistic limit for small grating periods.
Spec-
tral BTE solutions were found for Si and PbSe at 300 K using phonon dispersions
and lifetimes for all six phonon branches from DFT calculations. We compared the
spectral BTE decays to several approximate models: a single-MFP BTE solution, a
frequency-integrated gray-medium BTE model, and a two-fluid model from Ref. [8]
that combines the BTE with the diffusion equation. We found that the spectral BTE
results for Si were well reproduced by the two-fluid model from Ref. [8], and that
PbSe was reasonably modeled using our proposed frequency-integrated gray-medium
BTE approach. We also showed that the contribution of ballistic phonons is small
even for large reductions in keff compared to
kbu1k.
We went on to consider the inverse problem of reconstructing thermal conductivity
accumulation functions from measured effective thermal conductivities and modeled
suppression functions. While the suppression function from Ref. [8] produced better
results for Si, the suppression function from our frequency-integrated gray-medium
BTE approach produced reasonable results, and, in fact, worked better for PbSe.
We anticipate that the latter approach, applied to different experimental geometries,
60
may offer reasonable estimations for modeling non-diffusive thermal transport, and
extracting phonon spectral information from experimental measurements. Additionally, we demonstrated that the Fuchs-Sondheimer relationship could be used as a
suppression function for reconstructing the thermal conductivity accumulation function for bulk Si from TTG measurements on Si membranes.
Finally we explored
the viability of deriving heat flux suppression functions experimentally, using the Si
membrane data as an ideal test case, and found that such an approach is complicated
by the sharp features in the thermal conductivity per MFP function.
Our theoretical work demonstrated that TTG can be a useful tool for studying
non-diffusive thermal transport. However, to study low thermal conductivity materials with short phonon MFPs, it is necessary to generate thermal length scales smaller
than the diffraction limit of optical light. Membranes are one approach for generating small thermal length scales, but while Si membranes are commonly produced,
thin membranes of other more exotic materials are challenging to fabricate. These
factors motivated us to consider alternate methods for experimentally generating and
measuring heat transport over small thermal length scales.
61
62
Chapter 3
Investigation of non-diffusive
conduction with microfabricated
wire grid polarizers
Prior works have made progress in developing methods to observe non-diffusive heat
conduction, but have been restricted in thermal length scale by the optical diffraction
limit [4, 19], or have been limited to transparent substrates [3, 57, 58]. Diffraction
limited methods have used laser diameter [4] or transient thermal grating period
[19] as variable heating scales. Microfabticated metal heaters patterned on optically
transparent substrates have been used to go beyond the diffraction limit [3, 57, 58].
In addition, thin Si membranes have been used [59, 54] as discussed in Section 2.3.3,
but fabricating thin membranes of other materials is challenging.
To study non-diffusive transport in generic opaque materials with short phonon
MFPs, we require a method that can achieve small thermal length scales and detect
the resulting heat transport. In this chapter, we explore one such method that uses a
microfabricated wire grid linear polarizer on the surface of a generic sample of interest
in conjunction with TDTR measurements.
optical excitation of the underlying sample.
63
The polarizer is designed to minimize
3.1
Concept and design criteria for wire grid polarizer
To achieve length scales below the diffraction limit of visible light and to minimize
direct optical excitation of the sample under study, we explored using a wire grid linear
polarizer (LP) fabricated on the surface of the sample, as illustrated in Fig. 3-1. Timedomain thermoreflectance, which was described in Section 1.3.2, is then performed
with linearly polarized pump and probe beams. When the electric field directions of
the laser beams are perpendicular to the transmission axis of the linear polarizer, a
minimal amount of optical intensity will be transmitted into the underlying sample.
The LP transmission axis is perpendicular to the metal wires, as indicated in Fig. 3-1.
When the electric field direction is aligned parallel to the metal wires, free electrons
in the metal act to absorb the light, but when the electric field is perpendicular to
the lines where electrons are not as free to move, a maximum amount of light is
transmitted through the grating. The metal wires act both as localized heaters and
as transducers for detecting the change in thermoreflectance resulting from transient
temperature relaxation as the delay time between the pump and probe beams is
varied. The thermal length scale in the measurement will depend on the grating line
width and period.
We use COMSOL electromagnetic wave simulation software to design the geometry of the ID metal grating that acts to minimize light transmission into the
underlying substrate. We have several design criteria. The grating must minimize
transmission of the 800 nm probe and 400 nm pump beams into the underlying substrate. We also desire a structure that has a high likelihood of observing non-diffusive
behavior, so we prefer small thermal length scales. Since the thermal length depends
both on the grating line width and period, we desire small line widths with large periods to minimize the spreading of heat between neighboring lines. The grating should
also be reasonable to fabricate. A schematic of the two-dimensional (2D) finite element simulation space is shown in Fig. 3-2, which consists of a single grating period
L. Periodic boundary conditions are used to simulate an infinite 1D grating.
64
The
top view:
generic sample
Al
side view:
transmission
axis of LP
E, 400 nm pump
E, 800 nm probe
Figure 3-1: Conceptual illustration of a ID metal grating acting as a linear polarizer
(LP) to block pump and probe light from directly exciting a generic substrate. For
minimum transmission through the grating polarizer, the pump and probe are linearly
polarized with their electric fields aligned perpendicular to the polarizer's transmission
axis.
electric field of the light is aligned along the metal wires, perpendicular to the grating
transmission axis, and the light propagates from the air towards the substrate.
Figure 3-3 shows the transmittance and reflectance results for wavelengths A of
400 nm and 800 nm for various grating geometries, and provides some intuition about
the geometries that are more effective at light blocking. For these calculations, the
substrate material is A12 0 3 and the metal is Al. Shorter period (smaller L), higher
aspect ratio (larger h) gratings have lower transmittance, and longer wavelengths are
more easily blocked than shorter wavelengths.
The range of grating line widths d
with low transmittance is the largest for Fig. 3-3d, which combines small periods
with high aspect ratios.
A large range of d with low transmittance can be achieved by keeping the gap
between the metal lines, L - d, constant and smaller than the wavelength, as shown
in Fig.
3-4, which uses a 100 nm gap width.
65
Even though this constant small
L
air
3L
Figure 3-2: Schematic illustrating the domain used for COMSOL calculations.
gap approach achieves small optical transmission, it may be difficult to observe nondiffusive transport at larger values of d. The thermal length scale depends both on
the period and the line width. For larger filling fractions, which correspond to larger
ratios of d/L, non-diffusive effects are less apparent [60].
For interpreting measured non-diffusive effects, it is intuitive to keep the filling
fraction constant [60]. Figure 3-5 shows transmittance and reflectance results where
the filling fraction is held constant at L = xd, where x is an integer. Only calculations
for A
=
400 nm, which will have higher transmission than A = 800 nm, are shown.
Larger values of x will result in less thermal information sharing between neighboring
metal wires making the observation of non-diffusive transport more apparent, but
smaller values of x result in improved light blocking.
Figure 3-5(a) shows that a
transmittance of less than 10% is possible for d as large as 100 nm for x = 2, and Fig.
3-5(b) shows that higher aspect ratio lines (larger h) further reduce the transmittance.
Thus far, we have determined that L = 2d with h
=
100 nm is a feasible design for
minimizing transmission into the sample, and that h = 200 nm would be preferable
if fabrication constrains allow. For h > 200 nm, in addition to fabrication challenges,
there could be issues with TDTR signal-to-noise, whereby the measurement is not
66
L = 200 nm, h =100 nm
(a) CD
8
(b)
1-
U)
CO
C:
E
0.6
W
0.4
0
0.6
-
+
C)
L = 100 nm, h= 100 nm
(D
orA
I
W 0.4
C 04
0.2
as0. 2
4-1
0
-0
10
-9
-8
10
10
d (m)
-7
10
-10
10
-9
0-8
d (m)
U)
10
""
1-7
1)
(c)
L = 200 nm, h = 200 nm
(D
(d)
1
1
C
L = 100 nm, h = 200 nm
E 0.8
-
E 0.8.0.6
0
CU 0.4-
M
(D
-
0.4
I-,Y
0
-
c 0.2-
10
-
0
10
9OeE
0.6
10
10
d (m)
"
~"
S
10-7
0
-10
10
10
10
d (m)
107
Figure 3-3: COMSOL calculations for transmittance (solid lines) and reflectance
(dashed lines) for 800 nm (square markers) and 400 nm (circle markers) light for
various grating geometries. For these calculations, the electric fields are aligned in
the direction of the metal wires, the substrate material is A12 0 3 and the metal is Al.
67
L-d= 100 nm, h =100 nm
8 100
CO
E 10
C
O 10
Ca
() -2
0 10
Cz
U
10-
-0
10
d (m)
Figure 3-4: COMSOL calculations for transmittance (solid lines) and reflectance
(dashed lines) for 800 urn (square markers) and 400 nm (circle markers) light with
a constraint on the relationship between L and d such that the gap between grating
lines is held constant at L - d = 100 nm. For these calculations, the electric fields
are aligned in the direction of the metal wires, the substrate material is A12 0 3 and
the metal is Al.
very sensitive to the thermal conductivity of the underlying substrate. The preceding
calculations used Al as the metal, which is preferred due to its high thermorefeletance
response at our 800 nm probe wavelength [61]. Figure 3-6 shows some transmittance
results for different metals. Since Al has a lower or comparably low transmittance to
other metals, in conjunction with its favorable thermoreflectance response, we choose
Al as the metal for our grating design. We also verify our transmittance calculations
using Bi 2 Te3 as the substrate material, as shown in Fig. 3-7.
3.2
Microfabrication
In Section 3.1 we determined that a ID Al grating with a thickness of 100-200 nm, a
filling fraction of 50%, and a line width of 100 nm or less, would have lower than 10%
transmittance.
To fabricate these structures, we use a combination of interference
lithography and reactive ion etching. Interference lithography is a convenient method
for rapidly patterning large areas, and reactive ion etching can produce high aspect
ratio structures with vertical sidewalls.
68
L = xd, h = 100 nm
..
w.
10
10
x =4
10
x=3
10x =2
10
--8
10
d (m)
(a)
10
L = xd, h = 200 nm
-- ------
10
10
--
--
--
+-
10
10-1
10
10
10
x=2
1 ) --8
d (i)
10
(b)
Figure 3-5: COMSOL calculations for transmittance (solid lines) and reflectance
(dashed lines) for 400 nm light with a constraint on the relationship between L and
d such that the filling fraction is kept constant with L = xd, where x is an integer.
Grating line heights of (a) 100 nm and (b) 200 nm are considered. For these calculations, the electric fields are aligned in the direction of the metal wires, the substrate
material is A1 2 0 3 and the metal is Al.
Figure 3-8 illustrates the basic process flow, and Fig. 3-9 shows some scanning
electron microscope (SEM) images of various steps along the process flow. Fabrication
was carried out in the NanoStructures Laboratory (NSL) at MIT, with extensive
process development consultation from NSL personnel. The starting wafer substrate
is coated in Al followed by SiO 2 using electron beam evaporation. The SiO 2 will serve
69
L = 2d, h = 100 nm
Ag
. 10C -3
10
E
Cr
10
10
Al
108
d (m)
10-
.
Figure 3-6: COMSOL transmittance calculations for 400 nm light, varying the type of
metal used for the one-dimensional grating. For these calculations, the electric fields
are aligned in the direction of the metal wires and the substrate material is A12 03
= 2d, h =100 nm
)L
C 100...................
C',
-
E 10
C
(',
0
c
CD
10
-1
-2
10
-4
10 10
d (m)
Figure 3-7: COMSOL transmittance (solid line) and reflectance (dashed line) calculations for a Bi 2 Te 3 substrate patterned with a one-dimensional Al grating, using 400
nm light with an electric field in the direction of the Al wires.
as a mask for etching the Al. An antireflection coating (ARC), which prevents back
reflections during the lithography process, is spun on. A thin SiO 2 layer is evaporated
on the ARC to serve as an etch mask. Since SiO 2 is a hard mask, a thicker layer of
ARC may be etched than if photoresist alone served as the mask. Photoresist (PR)
is spun on, and the wafer is baked prior to exposure.
70
The PR is exposed with a Lloyd's mirror interference lithography (IL) system [62]
using a 325 nm laser source. The patterned period is set by the angle of the mirror,
which divides the source into two overlapping coherent beams.
The source is also
expanded and made to have a uniform intensity over the exposure area, which can be
as large as a 3 in wafer, using a lens and pinhole assembly. After exposure, the wafer
is baked to harden the resist before development. The resulting developed resist is
shown in Fig. 3-9(a).
The developed resist is used as the first mask in a multi-step reactive ion etching
(RIE) process.
RIE is a dry etch process that directs ions in an energetic plasma
towards the wafer plate using a large voltage difference.
The etch process occurs
through a combination of ion bombardment, which mechanically removes material,
and chemical reactions, which speed material removal. The SiO 2 layers are etched
with a CF 4 plasma, and the ARC is etched with an 02 plasma. The resulting SiO 2
to be used for Al RIE is shown in Fig.3-9(b). RIE of Al is a challenging process, and
is discussed further in Section 3.2.1. After Al RIE, the remaining SiO 2 mask shown
in Fig. 3-9(c) needs to be removed. This is done by spinning a thick layer of ARC
that covers the grating structures, as shown in Fig.3-9(d).
Another RIE process is
used, etching the ARC, SiO 2 and ARC, to produce the finished Al grating shown in
Fig.3-9(e).
ARC
IL
RIE
spin
RIE
Figure 3-8: One-dimensional grating fabrication process flow.
3.2.1
Dry etching Al
RIE of Al is a challenging process that had not previously been developed at MIT's
NSL. Through systematic iteration, we developed a reliable process for etching Al in
71
(a)
PR
Si02
ARC
Si02
Al
Si
(b)
~I
(c)
(d)
(e)
Figure 3-9: SEM images of the 1D grating fabrication process: (a) post IL exposure
and PR development, (b) the SiO 2 mask before Al RIE and (c) post Al RIE, (d) spun
ARC for SiO 2 mask removal, and (e) the completed grating structures.
72
the NSL's inductively coupled plasma (ICP) RIE. Our recipe uses equal amounts of
C1 2 and N 2 gases at a low pressure (1.6 mT) with a high RF bias (300 W) and little or
no ICP coil power. C12 reacts chemically with Al to form AlCl 3 , even in the absence
of a plasma. Low pressure helps promote the removal of AlCl 3 products and improves
the anisotropy of the etch. The C12/N
2
plasma is sparked at a high pressure (15 mT)
for 5 seconds, and subsequently lowered to 1.6 mT, where the remaining etch time
for a 100 nm thick Al layer is -2 min. The electrostatic chuck holding the carrier
wafer during the etch is maintained at 35 C.
After removing etched Al from the vacuum chamber, the Al undergoes a post-etch
corrosion process upon exposure to the atmosphere since adhered molecules of AC1 3
undergo hydrolysis, producing HCl. Some examples of post-etch corrosion are shown
in Fig. 3-10. To prevent this corrosion process, we developed a two step post-etch
treatment. After Al etching, but before venting the vacuum chamber, a high density,
low RF bias N 2 plasma is sparked and allowed to bombard the sample for 5 mins. This
helps to remove adhered molecules of Al etch products. Immediately after venting the
vacuum chamber, the sample is immersed in deionized (DI) water, and then rinsed in
DI water for ~2 mins. This post-etch treatment prevented any observable corrosion
over the course of weeks following Al etching.
3.2.2
Polishing polycrystalline Bi 2 Te 3 samples
Bi 2Te 3 is a promising thermoelectric material due to favorable electrical properties
and low thermal conductivity. The thermal conductivity can be reduced by nanostructuring [63]. We wanted to investigate whether we could observe any size-dependent
thermal conductivity using our wire grid polarizer approach. To do so, we had to
fabricate ID Al grating structures on the surface of polycrystalline Bi 2Te 3 samples.
Polycrystalline Bi 2 Te 3 samples were provided to us by Professor Zhifeng Ren from
the University of Houston. These were fabricated from powdered forms of Bi 2Te 3 that
were hot pressed into 2 cm diameter cylinders with grain sizes of tens of nanometers.
The cylinders were sliced into 1-2 mm thick pieces with a diamond saw. These small
wafers had to be polished before we could microfabricate our grating structures.
73
(a)
(b)
Figure 3-10: Examples of post-etch corrosion (a) where corrosion products form clusters and (b) where corrosion results in removed sections of Al.
A polishing process for polycrystalline Bi 2Te 3 had been developed previously in
our group [64], but required multiple complex steps and proprietary chemicals. We
developed a simpler process that reliably achieves 2 nm RMS surface roughnesses
with commercially available polishing products. We use a commercial benchtop motorized polishing machine (South Bay Technology model 920 lapping and polishing
machine) fitted with a central rotary plate ("wheel") and a yoke ("arm") for additionally rotating the sample holder. Starting with a rough, as-diced polycrystalline
Bi 2Te 3 wafer, we use a 6 Mim diamond or A1 2 0
3
suspension on a hard polishing cloth
(Buehler Trident or TexMet C) to planarize the surface, followed by a 1 pim A12 0
74
3
suspension to reduce the surface roughness. Polishing suspensions are introduced at
a rate of -1-2 drops/min, and DI water at -1-2 drops/sec is used for lubrication. For
this initial planarizing and smoothing, the polishing wheel and arm speeds are fast
and contra rotating. The planarizing step takes -2-5
mins and the rough smoothing
step takes -5-10 mins. Between each polishing step, the sample is throughly rinsed
with DI water.
Fine polishing is achieved in two subsequent steps.
diluting commercially available Buehler 50 nm A1 2 0
the solution has a pH of -7.5.
3
A suspension is made by
solution with DI water until
The diluted suspension is introduced at a rate of -1-2
drops/sec. A hard polishing cloth (Buehler Trident) is used, with fast wheel and arm
speeds that are counter rotating. After -10-15
mins, the sample has a mirror finish
with some visible uniform shallow scratches. The final polishing step uses the diluted
50 nm A12 0
3
suspension introduced at a rate of -2
drops/s, and a soft polishing
cloth (Buehler Microcloth) with fast wheel and arm speeds that are counter rotating.
A 250 gram weight is added to provide more downward force on the sample.
The
total time for this step is critical, and -30 s. Polishing too long will result in pits
from corrosion or an orange-peel texture on the surface, and polishing for too short a
time will not remove all the scratches. After -30 s, a copious amount of DI water is
introduced for -15
s while spinning polishing wheel and sample are still in contact,
which helps to remove polishing slurry from the sample surface. After this step, the
sample is cleaned in warm 45'C DI water in an ultrasonic cleaner to remove any
remaining polishing debris. The finished sample has a mirror finish, and atomic force
microscope (AFM) measurements over multiple 100 pm 2 measurement areas indicate
an RMS surface roughness of -2 nm, as shown in Fig. 3-11.
3.2.3
Resulting one-dimensional grating structures
Examples of final fabricated gratings are shown in Fig. 3-12. Figure 3-12(a) shows a
thinner line width grating with a higher aspect ratio than Fig. 3-12(b). These grating
-
structures were fabricated on wafers of Si, fused silica, and polycrystalline Bi2 Te3
Subsequent sections describe our TDTR measurement results on these structures.
75
10 nm
5 nm
__
0 nm
10 pm
Figure 3-11: Representative AFM measurement of a polished polycrystalline Bi 2 Te 3
sample. The RMS surface roughness over the imaged 100 pm 2 area is 1.6 nm. Measurements over several locations on the sample showed comparable roughness. AFM
scan courtesy of Lingping Zeng.
3.3
Optical transmission results
To evaluate the polarizing efficiency of our fabricated gratings, we measure the extinction ratios of the gratings on fused silica substrates, which are transparent to the
wavelengths in our TDTR system. A schematic for the transmission measurement
setup is shown in Fig. 3-13. The linearly polarized pump and probe beams are circularly polarized by a quarter-wave plate (A/4) at 450 , and then linearly polarized
by a linear polarizer (LP) in a rotational stage.
The pump and probe beams are
linearly polarized in the same direction, and focused onto the fused silica sample with
the patterned Al grating structure. The focusing angle for the beams is < 4'. The
transmitted light intensity is collected by a detector.
By rotating the LP, the angle between the electric fields of the pump and probe
beams and the grating transmission axis, 6, can be varied. 6
transmission Tmin, and 0
=
=
900 gives the minimum
00 gives the maximum transmission Tmax. Figure 3-14
76
(a)
(b)
Figure 3-12: Examples of fabricated 1D Al gratings. (a) Grating with a 200 nm pitch
and an 85 nm line width, that is 160 nm thick. (b) Grating with a 200 nm pitch and
a 105 nm line width, that is 85 nm thick.
plots the ratio of the transmitted intensity I to the incident intensity 1, as a function
of 0, assuming an ideal linearly polarizing grating.
Measured extinction ratios, Tmin/Tma, for the 400 nm pump beam and 800 nm
probe beam for various grating geometries are shown in Table 3.1. The extinction
ratio for the 85 nm line width, 85 nm thick grating was > 10% for 400 nm light,
and increasing the aspect ratio by increasing the thickness to 160 nm improved the
extinction ratio substantially. Low extinction ratios indicate that our fabricated gratings are indeed acting as reasonable polarizers. For comparison, COMSOL calculated
77
pump
probe
1
.j
PIN
detector
lens sample lens
LP. in M/4 cold
rot. stage
mirror
B.S
Figure 3-13: Transmission measurement setup where either the 400 nm pump or the
800 nm probe is used.
1
1 D grating
-
1Dg raing
linear polanizer
0.8
0.6
0
0
o 0.4
Qo. aarSM
xIs-
1/0
0.2
0
10
20
30
40
50
0 (deg.)
60
70
80
90
Figure 3-14: Idealized linear polarizer transmitted intensity as a function of angle.
extinction ratios are shown in Table 3.2 for the same grating geometries as those in
Table 3.1.
The fabricated gratings (see Table 3.1) have extinction ratios 1.3 to 4
times greater than the idealized geometries assumed in our COMSOL calculations
(see Table 3.2).
3.4
TDTR results
For reflection measurements, we use the setup illustrated in Fig. 3-15, which is similar
to our transmission setup illustrated in Fig. 3-13, with the exception of the detector
78
Table 3.1: Measured extinction ratio, Tmin/Tma, for various fabricated gratings for
the 400 nm pump and 800 nm probe beams. All gratings have a period of 200 nm
and were fabricated on substrates of fused silica.
grating geometry
800 nm 400 nm
line width 105 nm, thickness 85 nm
line width 85 nm, thickness 85 nm
line width 85 nm, thickness 160 nm
1%
1.7%
0.2%
4.8%
16.6%
3.6%
Table 3.2: Extinction ratios, Tmin/Tmax, from COMSOL calculations corresponding
to the fabricated grating geometries in Table 3.1.
grating geometry
800 nm 400 nm
line width 105 nm, thickness 85 nm
line width 85 nm, thickness 85 urn
line width 85 mu, thickness 160 nm
0.5%
1.3%
0.06%
3.6%
9.1%
0.9%
position and an added color filter for preventing pump light from entering the detector.
This is the same as a standard TDTR setup, like that discussed in Section 1.3.2, with
the addition of a linear polarizer in a rotation stage. The pump beam is modulated,
and the reflected probe beam is collected. The signal from the detector is mixed with
the reference modulation profile in a lock-in amplifier, and the resulting in-phase and
out-of-phase amplitudes for a given delay time are evaluated with a thermal model
to extract model parameters of interest. In our experiment, the unknown parameter
of interest is the substrate thermal conductivity.
pump
sample lens
pro be
L.P. in k/4 cold
rot. stage
mirror
B .S.
lens blue PIN
filter detector
Figure 3-15: TDTR grating measurement setup.
79
3.4.1
Grating heat transfer model
We introduced the heat transfer model for TDTR in Section 1.3.3.
Modeling the
case where the transducer is a 1D grating requires a slight modification. We follow
the approach of Minnich [57], who developed a model for a transducer formed from
a 2D grating. We assume that the metal layer cannot conduct heat in the in-plane
xy-direction, and can only conduct in the cross-plane z-direction.
This approach
also assumes infinite pump and probe beam diameters, and neglects any Gaussian
intensity variation.
The heat diffusion equation solution in Section 1.3.3 assumes cylindrical symmetry,
but for the ID grating geometry, it is more convenient to solve the heat diffusion
equation in Cartesian coordinates and to use a spatial Fourier transform instead of a
Hankel transform. The solution takes the form of Eq. (1.5) with
IM
2
=
k.,Y2
+ iWC
.
(3.1)
Here kxy is the in-plane thermal conductivity which is considered isotropic, k, is
the cross-plane thermal conductivity, sx is the x-direction spatial Fourier transform
variable, and w is the radial frequency temporal Fourier transform variable.
The time harmonic heating at the top surface qtop has a spatial profile of a ID
square wave, assuming that only the grating wires are heated. The spatial square wave
is represented by a Fourier series. In the spatial and temporal transform domains the
top surface heat flux qtop becomes
qtOP(Q) = q,6(w - w,) E
an6(Q
-
n
),
(3.2)
n=-oo
with Fourier series coefficients an of
an
if n 0,
sin(nrQd/2)/(n7r) if n 7 0,
d/L
(3.3)
where the spatial frequency is Q, = 27r/L. Here L is the 1D grating period, d is the
80
grating line width, w, is the time-harmonic excitation frequency, and q is the pump
intensity. The top surface temperature (see Eq. (1.9)) is weighted by the same spatial
square wave probe profile, and the resulting frequency response h(w) is given by [57]
h(W) cX E
|X|
3.4.2
.
C-D
(3.4)
n
n
Measurements varying pump laser diameter
Our thermal model assumes an infinite laser heating size, and as a result, a large
enough pump size is necessary to achieve reasonably accurate results. Figure 3-16
shows fitted measurement results for a 105 nm line width, 200 nm period gratings
on fused silica and Si substrates.
For the fused silica substrate, which has a low
thermal conductivity, we observe a dependence on the pump laser spot size, which we
attribute to the large heating diameter assumption inherent in our thermal model.
At small pump sizes, heat spreading along the metal wires, which is not captured in
our thermal model, becomes appreciable. At sufficiently large pump diameters, the
measured substrate thermal conductivity asymptotes to the bulk value of the thermal
conductivity of fused silica.
Measurements on a Si substrate are not sensitive to pump spot size, because Si
has a higher substrate thermal conductivity. The measured Si thermal conductivity,
however, is much lower than the bulk value, indicating non-diffusive transport. At
a thermal length scale of 105 nm, we expect to see size effects in Si based on the
distribution of heat-carrying phonon MFPs [48, 46].
3.4.3
Measurements varying angle between laser polarization
and grating transmission axis
Varying the angle between the laser polarization and the grating transmission axis influences the transmitted intensity, as shown in Fig. 3-14. The minimum transmission
occurs at 0 = 900 and the maximum transmission occurs at 0 = 00. Our reflection
measurement setup (see Fig. 3-15) allows us to easily vary 0 by rotating the linear
81
50
,40I
-
1.5
-
30
1-3
_020-
_0
C
C
-Fz0.5 -C
E
E 10-
a)
a)
80
100
120
140
pump diameter ([tm)
90
160
(a)
90
100
110
120
pump diameter ([rm)
130
(b)
Figure 3-16: Measured substrate thermal conductivity as a function of pump laser
diameter for (a) fused silica and (b) Si. ID Al gratings with 200 nm periods, 105
nm line widths, and 85 nm thicknesses were fabricated on each substrate, and TDTR
measurements were performed with linearly polarized pump and probe beams aligned
perpendicular to the grating transmission axes (see Fig. 3-15).
polarizer.
The pump and probe beams are polarized in the same direction.
The
measured reflectance signal from the lock-in amplifier is shown in Fig. 3-17 for 0 =
00, 450 and 90'. The in-phase x, and out-of-phase y, lock-in outputs are combined
to form amplitude R, and phase
#,
data as a function of delay time T between the
#
pump and probe pulses. R has relative units, and can be scaled arbitrarily, while
has absolute units of angle.
For the fused silica substrate, the phase signal is constant regardless of 0, and
the amplitude signal is constant except for an offset, as shown in Fig. 3-17(b). The
oscillations in the 0 = 0' and 45' curves result from surface acoustic waves. Some
probe light is reflected off the surface of the fused silica substrate and interferes with
probe probe light reflected off the surface of the metal grating.
As the substrate
and grating thermally expand, the interference can be constructive or destructive,
resulting in oscillations in the measured signal on top of the thermal decay profile.
Since fused silica is transparent to the pump and probe wavelengths, the measured
thermal response should be the same regardless of 0.
For Si, pump absorption in the substrate will influence the heating profile, and the
82
-00
--
..... 90*
-0
.0
-- 45*
510
-9Q*
--
-
100
450
-
r --------- ----- -
-1.. . . . .
10
-20
100
50
-40
--- 60
-
-50
-100
0
1
2
3
-(ns)
4
5
2
1
0
6
3
t (ns)
4
5
6
(b)
(a)
Figure 3-17: Reflection results for various angles 0 on (a) Si and (b) fused silica
showing both amplitude R and phase q TDTR data as a function of delay time T.
probe reflection from the substrate will influence the reflectance signal. As 0 is varied,
we observe dramatically different R and
#
signals from the Si substrate sample, as
shown in Fig. 3-17(a). The minimum transmission angle (0 = 90') produces a familiar
thermal decay profile, while the maximum transmission angle (9 = 0) produces a
much faster R decay, more indicative of bare Si. For reference, the TDTR response
from bare Si with no transducer layer is shown in Fig. 3-18.
In Fig. 3-19, we examine the TDTR signal from a range of angles on a Si substrate
sample. Figure 3-19(a) shows angles near the maximum transmission angle, and Fig.
3-19(b) shows angles near the minimum transmission angle. Figure 3-20 more closely
examines angles near 9 = 90', looking at variations of 5' and 15' on either side. We
observe good agreement between 0 = 850 and 95' and between 9 = 750 and 105',
providing confidence in our determination of 9 = 900.
clear difference that 50 makes in the shape of R and
83
#.
Furthermore, we note the
Figure 3-14 shows that going
100
E-
10
0
-10
-* -15.-20-25
0
1
2
3
x (ns)
4
5
6
Figure 3-18: Thermoreflectance signal from bare Si.
from 0
90' to 6 = 85' corresponds to less than 1% in transmitted intensity, which
suggests that even a small amount of transmitted light influences the TDTR signal
in an appreciable way.
3.4.4
Result summary
The substrate thermal conductivity results obtained from fitting the thermoreflectance
response measured at 0
=
900 for various samples are shown in Table 3.3. For fused
silica and polycrystalline Bi 2 Te 3 substrates, no measurable deviation from bulk values
84
100
10
100
0
100.
-20
50
50
a0
0
0
-50-
-40-
*15*
-25 .
-60-
-- 35*
-~-5*
r
--5
-80
.. 75*
-Mo
201
-1001
0
-
-100
1
2
3
-r (ns)
4
5
--
0
6
1
2
3
-r (ns)
4
5
6
(b)
(a)
Figure 3-19: Reflection results for various angles on Si at (a) high transmission angles
and (b) low transmission angles.
is observed, suggesting diffusive transport behavior. From the Si substrate, we observe a clear deviation from the bulk thermal conductivity, but note that our results
are likely influenced by some signal contribution from the Si substrate, so while a nondiffusive effect is likely, the precise fitted thermal conductivity values are unreliable.
A lack of observed size effect in polycrystalline Bi 2 Te 3 suggests that the dominant
phonon mean free paths (MFPs) are smaller than 100 nm in that material, which has
been confirmed by recent density functional theory calculations that suggest that the
heat-carrying phonon MFPs in crystalline Bi 2 Te 3 are < 10 nm [65].
3.5
Future directions
Since even a small amount of optical transmission into the underlying substrate con-
tributes to experimental uncertainty, it may be advantageous to pursue an approach
85
-750
-- 85*
-- 90"
- -950
100
-
-105*
10
-
-
-
0
-10-20C)
0,
V
-30-40-50
I
-60
0
2
1
3
4
5
6
-c (ns)
Figure 3-20: Reflection results for angles around the low transmission angle for a
one-dimensional Al grating on Si.
Table 3.3: Measured effective thermal conductivity in units of W/mK for substrates
of fused silica, Si and polycrystalline Bi 2 Te 3 with two different Al grating geometries
compared to the bulk substrate thermal conductivity. The 105 nm grating line width
sample had a thickness of 85 nm, and the 85 nm line width sample had a thickness
of 160 nm.
line width 105 nm line width 85 nm
kbu1k
substrate
fused silica
Si
1.4
142
1.36
45
polycrystalline Bi 2 Te3
1.3
1.4
1.4
33
that guarantees no direct optical excitation of the substrate. One approach could be
to fabricate metal gratings with a capping metal layer, as shown in Fig. 3-21. The
86
gratings are fabricated as before, and the gaps are filled with a low thermal conductivity polymer, like the antireflection coating material used for mask removal in Section
3.2. The structure is then coated in an optically thick layer of Al, which acts as a
transducer for the TDTR measurement.
Most of the heat should conduct through
the Al capping layer to the Al wires, circumventing the low conductivity polymer
areas. Properly interpreting the TDTR data would require a refined thermal model,
but preliminary testing suggests that non-diffusive transport can be observed for Si
substrates, and that diffusive transport is observed in fused silica substrates, as expected.
Figure 3-22(a) shows fitted substrate thermal conductivities for structures
like that shown in Fig. 3-21, where the pitch of the grating structures was 180 nm
and the line widths were smaller than 80 nm. The model used for fitting was rather
crude, treating the structure with three layers: an Al top layer, an interface layer
consisting of the grating, polymer and bounding interfaces, and a substrate layer.
The unknown fitting parameters were the interface layer resistance and the substrate
thermal conductivity. In spite of crude modeling, we observe apparent non-diffusive
transport in Si and diffusive transport in fused silica substrates.
Figure 3-21: Concept for structures that prevent optical transmission into the substrate. The gaps between the Al grating lines are filled with polymer (ARC) as shown
on the left, and then an Al capping layer is added as shown on the right.
,
The primary findings of this work are twofold. First, for polycrystalline Bi 2Te 3
length scales smaller than 100 nm are required to possibly observe non-diffusive ef-
87
102
E
-------------------------10
.
.......... .........74.....
72.....
74
72
76
78
.
0
.1007
0
linewidth (um)
(a)
-20
-40
(D
-
-
-60
-Si
-80[
0
1
2
3
-r (ns)
4
5
'
6
-100
S
0
71 nm'
Si, 76. nm
- fsed s76.5 a 71n
-... fused silica, 71 nm
-ue
1
-iia
-6
2
4
3
-c (ns)
-
.0
10
5
6
(b)
Figure 3-22: (a) Using a crude model, where the grating-polymer layer and bounding
interfaces are treated as having some unknown lumped resistance value, we extract
substrate thermal conductivity values for substrates of Si (open circles) and fused
silica (open squares) patterned with grating structures like that pictured in Fig. 321. The dashed line shows the bulk thermal conductivity of Si and the dotted line
shows the bulk thermal conductivity of fused silica. (b) For reference, raw TDTR
phase data traces, 0(r), collected with a pump modulation frequency of 9 MHz, are
shown for samples corresponding to the fits in (a).
fects. Length scales as small as ~20 nm could be achieved with ebeam lithography
methods. Another approach might be fabricating thin membranes of Bi 2Te 3 , where
the membrane thickness would serve as the thermal length scale. The second finding of this study was that even small amounts of optical transmission (increases of
1%) into the substrate influence the TDTR signal.
Optical transmission could be
further reduced by increasing the wavelengths used in the measurement, for example,
by implementing a two-tint TDTR approach that uses near infrared light for both
88
the pump and the probe [66]. To fully eliminate direct excitation uncertainty, the
substrate would either have to be transparent to the wavelengths used, or some light
blocking structure like that of the cap structure described above would be needed.
89
90
Chapter 4
Frequency-domain representation
of TDTR data, and applications for
studying non-diffusive conduction
Since the study of phonons in low thermal conductivity materials such as thermoelectrics likely requires thermal length scales on the order of 10 nm or smaller, we are
motivated to develop an alternate experimental approach. The challenges and cost
associated with microfabrication at 10 nm length scales, which is the current state of
the art, are prohibitive.
Recent work [18] suggested that thermal penetration depth could serve as an effective means of probing small thermal length scales without the need for microfabrication. Periodic heating on a semi-infinite substrate results in a periodic temperature
response that decays with depth. Solving the heat diffusion equation in one dimension, X, for a semi-infinite material subject to a heat flux input of q = q0 cxp(iwt)
results in a transient temperature profile of
T(x, t) =
O
k V/iw/a
C"e--"
eiw,
which has a characteristic thermal penetration depth of
91
(4.1)
k
rCf
dTPD -
(4.2)
Here, k is the substrate thermal conductivity, C is the volumetric heat capacity,
a = k/C is the thermal diffusivity, q, is the magnitude and
f
is the frequency of
the sinusoidal heat flux input, and w = 27rf. Figure 4-1 illustrates the transient
temperature profile, along with the thermal penetration depth. At dTPD, the temperature envelope has decayed to ~37% of the surface value. At 2 dTPD, the temperature
envelope will have decayed to -13.5% of the surface value.
T(x,t)1/f
qo
To
ILL
t
-AdD
X
Figure 4-1: Illustration of thermal profiles, T(x, t), in a semi-infinite solid that result
from excitation by a sinusoidally periodic heat flux q(t). The dashed line shows the
definition for the thermal penetration depth dTPD given in Eq. (4.2).
The thermal penetration depth decreases with an increasing frequency of the periodic heat flux. As a first order estimate for the frequencies required to achieve 10 nm
length scales in various materials, Fig. 4-2 shows dTPD(f) from Eq. (4.2) for three
substrate materials which span a range of thermal diffusivities: Si, A12 0 3 , and fused
silica. Figure 4-2 suggests that frequencies as high as 3 GHz are needed to achieve
10 nm thermal penetration depths in low thermal conductivity materials like fused
silica. Experimentally generating and measuring such high frequency responses is
challenging, but recent advances in optical-thermal measurement systems have made
progress [18, 24].
Frequency-domain thermoreflectance (FDTR), which we introduced in Section
92
104
-Si
- -A 2 O3
--. glass
103
E
10
2
10
100
102
101
3
10 4
f (MHz)
Figure 4-2: Thermal penetration depth dTPD for semi-infinite solid slabs of Si, A12 0 3
and fused silica heated by a sinusoidal heat flux of frequency f (see Eq. (4.2)).
1.3.1, uses a sinusoidally modulated continuous wave (CW) pump beam to produce
time-harmonic heating resulting in surface temperature oscillations at the modulation
frequency, which are monitored by a CW probe beam. The modulation frequency in
FDTR typically varies from kHz to -10
MHz [21, 22, 23]. Recently, an extension
of the frequency range up to 200 MHz was reported, along with a reduction in the
measured effective thermal conductivity of Si at high frequencies [18, 24].
Unlike FDTR, time-domain thermoreflectance (TDTR), which we introduced in
Section 1.3.2, exhibits heating that is not time-harmonic but is comprised of many
frequency components. The frequency content of TDTR measurements includes high
frequency information, principally limited only by the laser pulse duration (typically
~200 ps). It would be advantageous if frequency components of the TDTR response
could be separated and represented in a form similar to FDTR, i.e., in terms of the
amplitude and phase of the surface temperature response to time-harmonic heating.
We demonstrate that such a representation is indeed possible, and we extract frequency data of up to 1 THz. Our method not only allows a direct comparison of
TDTR and FDTR data, but also enables measurements at high frequencies currently
93
not accessible to FDTR.
4.1
Theoretical foundation
We briefly review the signal formation in FDTR and TDTR and we demonstrate
that the frequency components in a TDTR signal can be separated and represented
in a form identical to FDTR, i.e., in terms of the amplitude and phase of the surface
temperature response to time-harmonic heating.
1/f0
1/10
1/f
(a)
-+
)+-
(b)
Figure 4-3: Typical laser heating profiles for (a) FDTR and (b) TDTR measurements.
4.1.1
FDTR
In FDTR, the heat input supplied by the pump laser is a simple sinusoid with an
angular frequency w, = 27rf0 , as illustrated in Fig. 4-3(a), which may be expressed
as
q(t) = qe"Aot,
(4.3)
where q, is the pump power modulation amplitude. This periodic heat input results
in surface temperature oscillations with the same periodicity,
0(t) = qoe"wth(wo).
(4.4)
While the frequency is set by the pump, the magnitude and phase of the surface tem94
perature with respect to the pump are determined by sample properties. The complex
function h(w) describes the frequency-domain response of the sample, and depends
on parameters such as the substrate thermal conductivity and interface conductance.
In practice, often only the phase of h is measured as w, is varied, because accurate
measurements of the magnitude are more difficult [18].
h is typically modeled by
solving the heat diffusion equation to find the sample's transient surface temperature
response to a sinusoidal heat input [28, 29, 31], which is described Section 1.3.3. In
FDTR, sweeping through all values of w, is necessary to fully determine h(w).
4.1.2
Single-shot TDTR
If instead of time-harmonic excitation, the sample were heated by a short laser pulse
approximating a delta function, the surface temperature would be described by the
time-domain response, h(t).
In the linear regime,' h is related to the frequency-
domain response h by a Fourier transform, h(w)
=
f h(t) exp(-iwt)dt. By measuring
h(t), one would fully determine h(w).
To implement the single pulse excitation in an experiment, one would need to
use a low laser pulse repetition rate to make sure that h decays to zero before the
next laser pulse strikes the sample.
In such an experiment, which could be called
"single-shot TDTR," h, and consequently h, could be determined in a single measurement, provided that the signal-to-noise ratio was high enough. TDTR is typically
performed at a high repetition rate (-80 MHz), which improves signal-to-noise, but
results in accumulative effects of multiple pulses that complicate the data analysis.
Practical difficulties would need to be overcome to implement a single-shot TDTR
measurement. The laser repetition rate would need to be < 80 MHz (40 MHz would
likely suffice), and the delay length required to capture the full transient temperature
IThe linear response model normally used in the analysis of TDTR and FDTR measurements
[28, 29, 31] is valid as long as temperature variations are small compared to the background temperature. In TDTR, this assumption may be inaccurate at early times (typically < 1 ps) when the
non-equilibrium electron temperature rise in the metal film may be significant even for a moderate
excitation fluence [67, 68]. We assume that the linear response model holds for slower dynamics
determining frequency components of the thermoreflectance response below 1 GHz. However, establishing the domain of validity of the linear response model in TDTR requires further investigation.
95
response would be long (7.5 m in the case of a 40 MHz repetition rate). Aligning long
movable delay lines is challenging, as it is necessary to prevent the beam from walking
or changing diameter as the delay line is swept. Signal-to-noise would also be a challenge, requiring averaging at each delay position. TDTR is more easily implemented
than single-shot TDTR, but the data analysis is more involved.
4.1.3
TDTR
The analysis of TDTR data requires treating the accumulative effects of multiple pulse
excitations. The heat input supplied by a TDTR pump beam can be approximated
by a train of delta pulses modulated by a sinusoid as illustrated in Fig. 4-3(b), which
can be expressed as
q(t) =
Qoe
i~j~t
0 t -27k(45
---
oo
,
(4.5)
k=-co
where
Q,
is the absorbed pump pulse energy and the period between pulses is 2wr/ws.
To convert this transient heat flux into a frequency-domain expression, we utilize
the fact that q(t) is a product between an impulse train and a sinusoidal modulation function.
6( )
E 0
First, we consider the Fourier transform of the impulse train: 2
correin the 6time-domain
S (w - kwo). A product 0_0
sponds to a convolution in the frequency-domain, and the special case of a product
with a sinusoidal function in the time-domain leads to a shift in the frequency-domain:
exp(iwet)x(t) -*(w
- w,). Thus, in the frequency-domain, Eq. (4.5) becomes
00
4(w Q~~ E
6(w
-
kw,
-
w,,).
(4.6)
The surface temperature 6(w) relates to the heat input by 0(w) = h(w)d(w), hence
O(w) = h(w)Qow,
6(w - kw - w,).
(4.7)
k=-oo
2
Note that the summation from k = -oo to oo is important, because the Fourier transform of an
impulse train is derived using a Fourier series representation, which depends on having a periodic
function.
96
In TDTR measurements, the probe is derived from the same laser as the pump,
and can be treated as a train of delta pulses at the laser repetition frequency, delayed
relative to the pump pulses by a delay time r. The incident probe power is given by
qi t) =Q,
>]
M=00
DO t -27rm-T(48
6 (t -2-rr
,
(4.8)
which in the frequency-domain becomes
Vj(w) = Qiw
(4.9)
6(w - mws)e-i""r,
E
m=-OC
where
Qj is
the probe pulse energy. Equation (4.9) is derived using the theorem that
a shift in the time-domain leads to a modulation in the frequency-domain: x(t - t,) <exp(-iwt0 )z(w).
The reflected probe power, which will be collected by the detector,
is given by the product of the sample's thermoreflectance response and the incident
probe power,
q,(t) = Cth(t)qi(t),
where Cet is the thermoreflectance coefficient which
relates the change in reflectance to the change in temperature.
The power of the
reflected probe beam is assumed to vary proportionally to the change in surface
temperature, which is a valid assumption as long as the change in surface temperature
is small [281. In the frequency-domain, the product becomes a convolution, hence
Ct
27r
Y
ChQiw )
27r
w
CthQiQo=2
-
Ct
ei""f
i(w - mWS)
3 3ei"wrh(w
-
mw)6(w - (k + m)w, - w,).
(4.10)
m=-oo k=-oo
Typically, TDTR employs lock-in detection in order to improve signal-to-noise.
The lock-in mixes the signal from the detector with a reference signal at w, and
with a reference signal with a 900 phase offset, to find in-phase and out-of-phase
97
(quadrature) responses, which filters out all frequencies except for a narrow band'
around w 0, which leads to k = -m for all non-zero parts. The complex amplitude of
the lock-in response z(T) for a given delay time T is given by [28, 29]
z(T) =
4CtQ2O
eCikwsh(kws
2
+ w,),
(4.11)
k=-oo
where g represents gain in the detection electronics. The measured data in TDTR
are the in-phase, x, and quadrature, y, components of the lock-in amplitude as the
delay time is varied,
z(r) = x(T) + iy(r).
4.1.4
(4.12)
Frequency-domain representation of TDTR data
As evident in Eq. (4.11), the measured signal in a TDTR experiment is a periodic
function of the delay time T, which is represented in the form of a Fourier series. The
Fourier series coefficients
ak
are given by
ak
=
/S7r/w
2
.,
[27r /oWZ(T)eiksT
-
gChQoQiw2
Sh(kw,
47 2
=
From Eq.
+ w0 ).
(4.13)
(4.13), we can see that the magnitudes of the Fourier coefficients are
proportional to h(w) for a discrete set of w values: w = kws + w0 , where k is an
integer from -oc to oc. Since the impulse response h(t) is a real function, we can
invoke the complex conjugate relation h(w) =*(-w), to find responses h(nws - wO).
Thus the frequency responses can be obtained from the Fourier series coefficients as
follows,
3
The DC portion of the surface temperature response is rejected by the lock-in, which only detects
signal frequency components at the reference frequency.
98
h(nw, + w,) =
h(nw. - w,) =
47r 2
gChQ0
Qiw,2
47r 2
4
2
gCthQQw
a.,
4.-.
(4.14)
(414
where n is a positive integer. By determining the Fourier coefficients from a measured
TDTR response represented as a complex function of the delay time, one is able to
determine the frequency response h(w) for w = nw
w0 , which is equivalent to
determining h(w) from individual FDTR measurements at these frequencies. By Eq.
(4.13), determining the Fourier coefficients is possible given a full period of delaytime-domain TDTR data. We refer to this method of extracting frequency response
information from TDTR data as "frequency-domain TDTR (fdTDTR)."
In some implementations of TDTR, the pump pulses are delayed with respect
to the probe [27]. If the delay line is placed after the modulator used to modulate
the pump beam at w 0 , an additional phase lag [28] given by exp(iwT) is introduced.
In this case, the lock-in response is no longer a periodic function of
T.
Multiplying
the response by exp(-iw0 T) removes the additional phase lag and yields a periodic
function described by Eq. (4.11), after which Eq. (4.14) can be used to find frequencydomain responses.
A similar problem of finding frequency responses from a lock-in output arises in
analyzing acoustic waves measured in a femtosecond pump-probe experiment with a
high repetition rate, and an analogous methodology for extracting acoustic frequency
responses from the lock-in output has been developed [69, 70].
4.2
Experimental demonstration of fdTDTR
We modify our TDTR setup to enable the collection of a full period of delay time
data so that the frequency response analysis described in Eq. (4.14) is possible. We
demonstrate the data collection and processing of fdTDTR data using a sample of
A1 2 0 3 coated with a thin Al transducer layer (typically -100
99
nm). We go on to show
that fdTDTR and FDTR indeed produce the same frequency response information
by measuring a sample of A1 2 0 3 coated with a thin Au-Ti transducer layer using the
recently completed broadband FDTR system in our laboratory.
4.2.1
Data collection
The typical experimental arrangement used in TDTR has been described in numerous
works [25, 26, 27, 29].
We introduced our setup [29], which uses a pulsed laser
oscillator operating at a center wavelength of 800 nm, with a pulse width of ~200 fs,
and a repetition rate of
f,
= 81 MHz, in Section 1.3.2. An electro-optic modulator
(EOM) sinusoidally modulates the pump beam at a frequency
f', which we vary
from
2 to 12 MHz.
To determine the frequency responses of the surface temperature given by Eq.
(4.14), we need a full period of delay-time-domain data from
T =
0 ns to
T
1/fs
12.3 ns. Our existing TDTR setup uses a motorized mechanical delay stage with
a 0.5 m travel distance, and passes the probe beam through this delay four times,
resulting in a maximum probe delay time of around 7 ns. To obtain an additional
6 ns of delay time, we introduce an additional fixed delay, as illustrated in Fig. 4-4.
Thus, the full period of delay time data is collected in two sets. The necessary delay
length in the experiment is some amount longer than 1/fe, because measuring the
reflectance signal peak at zero delay for both data sets is essential for "stitching"
the data sets, as described in Section 4.2.2. To mitigate optical alignment errors and
minimize divergence issues, we expand the beam diameter by 4x before the optical
delay line, to -8 mm.
Accurately extracting high frequency data requires high time resolution at early
delay times, where the peak in thermoreflectance occurs. To achieve high time resolution near the thermoreflectance peak, while minimizing data collection time, we vary
the speed of our movable delay stage such that it moves slowly for delay times near
the peak and more quickly for delay times far away from the peak. Sharp features
in the thermoreflectance response only occur near the peak in thermoreflectance, so
sub-picosecond resolution is only necessary at these short times.
100
fo
additional fixed delay
.
*
red SHG
filter
^4
EOM,f.
4x
compress
pump
400 nm
probe
800 nm
PBS
A/2
expand
%
filter
eND
40
movable
delay stage
laser, fs
1ox dichroic
obj.
BS
blue detector
filter
Figure 4-4: TDTR experimental diagram. The additional static delay line combined
with the movable delay stage allows for the collection of more than one full period of
delay-time-domain data.
To further aid the stitching process, the signal-to-noise levels in the two traces are
roughly equalized by introducing a neutral density (ND) filter into the path of the
probe for the short delay time data set. Due to reflections off of more mirrors, the
longer delay time data set has a lower probe power, and hence a lower signal level.
The neutral density filter reduces the probe power in the early time trace such that
the signal levels in both traces are more matched. Furthermore, the lock-in settings
are kept the same for both traces.
To demonstrate the fdTDTR data collection and processing, we use a sample that
consists of a crystalline substrate of (0001) A1 2 0 3 that was deposited with a 110 nm
thick layer of Al using electron beam evaporation. The Al transducer layer thickness
was verified with atomic force microscopy by scratching away a small area of Al from
2
the substrate and measuring the step height. The Gaussian pump and probe 1/e
radii used were 28 pm and 5 pm respectively. Measurements with larger pump radii,
up to 55 pm, produced the same results, but with lower signal-to-noise. The pump
radius is chosen to be much larger than the probe radius to mitigate overlap alignment
errors.
101
4.2.2
Data stitching
The short and long delay data sets are stitched to form the full period of delay time
data in Fig. 4-5, which shows both the in-phase, x, and quadrature, y, parts of the
complex lock-in amplitude signal as a function of delay time, (see Eq. (4.12)). To
combine the delay time data sets, several post-processing steps are used. A peak in
the thermoreflectance signal occurs when the pump and probe beams arrive at the
sample simultaneously, because at that delay time the probe measures the maximum
temperature rise. The long delay time data set is shifted so that the peak in the signal
magnitude occurs at
T
= 1/hf. The short delay time data set is similarly shifted so
that the peak occurs atr
=
0.
Each data set is independently phase-corrected by requiring that the quadrature
lock-in signal component, y, does not experience a jump [28 at
T =
0 or r
This procedure removes any phase that may be added by the electronics.
=
1/f,.
We find
that the necessary phase correction is roughly the same in both data sets to within 1
degree.
The data sets are scaled so that the magnitudes at
T =
0 and
T
= 1/f, match.
The same scaling factor is used on both x and y to preserve the phase information.
This scaling is performed to compensate for the reduction in signal magnitude caused
by the added optics in the long delay data set.
Even though the signal levels in
both data sets are roughly equalized by introducing a neutral density filter into the
probe path of the short delay time data set, small differences in the signal magnitudes
persist and need to be corrected. After scaling, the data sets are shifted so that the
values of x and y just before
T =
0 and T = 1/f match.
The quality of the data set stitching can be evaluated by observing the delay
time region around 6 ns, where the data sets overlap, as shown in Fig. 4-5. Good
overlap provides confidence in the data stitching procedure, and in the alignment of
the optics during the experiment, demonstrating that the probe does not walk or
diverge significantly as the mechanical delay stage is swept.
102
1
0.8
-a, 0.6
X 0.4
0.2
-
-
--
0
Im
-0.1
CO
-0.12
-0.14
-1ek
0
2
4
6
8
10
12
14
- (ns)
Figure 4-5: Example of stitching TDTR data sets collected in two parts by introducing
an additional fixed delay for measuring long delay time data (> 6 ns). The data sets
have been independently phase corrected, shifted and scaled. x and y are the in-phase
and quadrature output of the lock-in amplifier respectively, and r is the delay time
of the probe with respect to the pump.
4.2.3
Fourier series representation
A numerical fast Fourier transform operation on z(T), from T
duces Fourier series coefficients
values, f
nf,
f0,
ak,
=
0 to
T =
1/f, pro-
which are proportional to h at discrete frequency
according to Eq. (4.14). We present our frequency-domain data
in terms of the amplitude and phase of the surface temperature frequency response,
Re(h) 2 + Im(h) 2 and
#
= tan- 1 (Im(A)/Re(h)). R has a relative mag-
nitude with arbitrary units, while
#
is the absolute phase with units of angle. 0(f)
where R =
and R(f) can be directly compared to an equivalent FDTR measurement.
Figure 4-6(a) presents R and
4 obtained from the lock-in output data shown in
Fig. 4-6(b), which were collected at three modulation frequencies: 4, 8 and 12 MHz.
A continuous frequency dependence could be obtained [69 if
103
f,
could be varied up
to f,/2.
Signal-to-noise issues with our existing TDTR system limit our maximum
pump modulation frequency to -12
Fig. 4-6(a) have frequency gaps.
MHz, so the frequency-domain data curves in
However, since thermal responses typically lack
sharp resonant features, filling in the gaps is not crucial for practical purposes.
Representing TDTR data in the frequency-domain leads more readily to a physical
interpretation than a conventional delay-time-domain representation such as that of
Fig. 4-6(b). In the delay-time-domain, each different pump modulation frequency
results in a different curve which must be evaluated separately or using global fitting
strategies, while in the frequency-domain, all data from different
collapse into a single curve.
f,
measurements
The amplitude data depend on the amplitude factor
that generally varies as we change
f, because of the variations in the modulation
efficiency of the EOM and the sensitivity of the detection electronics, as well as drift
of the laser energy. The delay-time-domain signal magnitude jump at
T
=0
should
be independent of the pump modulation frequency. We normalize each delay-timedomain curve to the magnitude jump at
T
= 0, using the same normalization factor
on both x and y to preserve absolute phase information. Thus, amplitude data from
multiple
f, measurements form a single curve seen in the top panel of Fig. 4-6(a). The
phase data shown in the bottom panel form a single curve without any calibration
effort.
4.2.4
Frequency upper limit
The high time resolution inherent in TDTR (typically limited by the pulse width)
enables the extraction of very high frequency data components. However, at high
frequencies, the Fourier coefficients are reduced, resulting in a poorer signal-to-noise
ratio. Analyzing the signal-to-noise ratio in our frequency-domain data is challenging.
As a first approximation, we estimate the noise in our delay-time-domain data by
smoothing the measured data using a moving average filter as shown in Fig. 4-7(a).
The smoothed data curve is subtracted from the raw data to obtain the noise data
shown in Fig. 4-7(b).
The frequency spectrum of the noise data in Fig. 4-7(b) is found in the same way
104
(a)
100
o
4MHz
S8 MHz
o 12 MHz
Ca
CU
-1
-30
-50
17
-70
-9 ('~1.
-
0
200
400
600
f (MHz)
800
1000
(b)
------ 4 MHz
-- 8 MHz
12 MHz
1
x
0.5
0
U
.................
12 MHz
-0.121-O
-
-0.
0
2
6
-r (ns)
4
8
10
12
Figure 4-6: Room temperature TDTR data for a sample of A12 0 3 with an Al transducer layer, represented in (a) the frequency-domain and (b) the delay-time-domain.
Data are shown for pump modulation frequencies of 4, 8 and 12 MHz.
105
(a)
-raw
data
---smoothed data.
1
x 0.51
0
-0.1
-0.12
cd
-0.14
110
10-2
10 -1
- (ns)
100
101
101 -1
100
101
0.021
(b)
0.01
x
0
-0.01'
0.0
0.00 2-
0
-0.00 2-
-00.nn
10
-
-
10
-r (ns)
Figure 4-7: A noise estimate is obtained by smoothing the raw delay-time-domain
data, as shown in (a). Subtracting the smoothed curve from the raw data produces
the noise data shown in (b). The delay-time-domain data shown here was collected
with f0 = 8 MHz on the Al-coated A1 2 0 3 sample discussed above.
106
as the frequency spectrum of the raw delay-time-domain data in Fig. 4-7(a), and
the two spectra are compared in Fig. 4-8. For frequencies below 1 GHz, the data
amplitude is >2 orders of magnitude larger than the noise amplitude. At 10 GHz,
the data amplitude is ~1 order of magnitude larger than the noise amplitude.
100
10-1 2
- ---
c
-E
a
-2b
10
10 -3
10
noise spectrum
10-7 0
_0- data spectrum
10
10
10
10
10
10
101
f (Hz)
Figure 4-8: Comparison of the frequency response amplitude obtained from the raw
data in Fig. 4-7(a) and the noise data in Fig. 4-7(b).
Determining the true upper limit of our frequency-domain data will require further
modeling work.
response regime.
One of the key assumptions of our analysis was that of a linear
At early times (<5 ps) following short pulse heating in metals,
electrons are excited to high energy levels and have temperatures of several thousand
degrees, resulting a highly non-linear regime. Early time data has the most influence
on high frequency data, but will have some small influence on lower frequency data
as well. As a conservative upper limit estimate, we report our frequency-domain data
out to 1 GHz.
4.2.5
Direct comparison of fdTDTR and FDTR data
Using a recently completed broadband FDTR system in our lab, based on the design
of Ref. [18], we make a direct comparison of FDTR and fdTDTR data. The sample
107
studied was a substrate of crystalline (0001) A1 2 0 3 , coated by electron beam evaporation with 5 nm of Ti followed by 160 nm of Au. Ti was used as a stiction layer to
improve the adhesion and uniformity of the Au layer. Au was selected as a transducer
material due to signal-to-noise issues with our FDTR system. -The high thermoreflectance response of Au at our FDTR probe laser wavelength of 532 nm results in
a good signal-to-noise ratio. Testing on samples with Al transducer layers with our
FDTR system resulted in a low thermoreflectance response, and correspondingly a
low signal-to-noise ratio.
Another signal-to-noise constraint with our FDTR system is the need for small
pump and probe diameters, typically <3 pm, to achieve a high enough pump fluence.
The peak power output of our FDTR pump laser is ~150 mW. Given the spot size
constraints on our FDTR system, to directly compare data sets we needed to adapt
our TDTR system to be able to collect data at comparably small pump and probe
diameters.
Replacing our 10x microscope objective shown in Fig.
4-4 with a 50x
microscope objective, along with careful alignment of all the optics, was sufficient
to achieve Gaussian 1/e2 TDTR pump and probe diameters of 3 pm and 2 pm
respectively. The FDTR data was collected with equal pump and probe diameters of
2 pm.
Figure 4-9 shows a comparison of fdTDTR and FDTR data collected on the
same sample of A1 2 0 3 with a Ti-Au transducer layer. We find reasonable agreement
between the <(f) data sets in the overlapping frequency range.
4.3
Thermal model analysis of fdTDTR data
In Section 1.3.3, we outlined a solution to the thermal diffusion equation that is widely
used for analyzing TDTR and FDTR responses [28, 29, 31].
Our model outputs
the frequency-domain response of the surface temperature, h, which we compare
to our measured frequency-domain data.
We fit our data simultaneously for the
substrate thermal conductivity, kaub, and the thermal interface conductance between
the substrate and the transducer film, G, using a least squares fitting routine. All
108
0
c
-20-
FDTR
o fdTDTR
0
0 0
-40
-60
-80
0
-100 0
2
10
10
10
3
10
f (MHz)
Figure 4-9: Comparison of phase data from fdTDTR and FDTR measurements on
the same sample as described in the text. FDTR data courtesy of Samuel Huberman.
other material parameters are set to literature values. Either the relative amplitude,
R(f), or absolute phase,
#(f),
data sets may be used for fitting.
We begin by examining the fdTDTR data we obtained for a crystalline substrate
of (0001) A12
03
that was deposited with a 110 nm layer of Al. The top part of the Al
film is modeled as an isothermal layer to mimic energy deposition into a finite depth.
The isothermal layer is modeled as having no radial thermal conductivity and a high
cross-plane thermal conductivity. We begin by choosing an isothermal layer thickness
of 10 nm, as was done in Ref. [28], which is comparable to the optical skin depth in
Al. By fitting the data up to 200 MHz, as shown by the dotted lines in Fig. 4-10, we
find best fit values of kAIos3
33.7 W/mK and G = 100 MW/m 2 K from R(f), and
kA 2 o 3 = 38.6 W/mK and G
105 MW/m2 K from 4(f). The literature value [71]
for kAl 2o3 in the (0001) direction is 41.7 W/mK.
In spite of recovering close to the literature value of A1 2 0 3 thermal conductivity,
the model fails to capture the phase behavior at high frequencies, suggesting the need
to properly model the transport processes in the Al. Fast non-equilibrium electronic
diffusion during -1
ps following short-pulse excitation deposits the pump energy over
109
a much larger depth than the optical skin depth of ~7 nm [72]. For the purposes
of this work, with the main focus on the experimental methodology rather than on
non-equilibrium dynamics in a metal following a femtosecond excitation [67, 68], we
proceed by finding the isothermal layer thickness that most closely matches our phase
data at high frequencies. We find that an isothermal layer thickness of 25 nm produces
a good fit to our high frequency data, as shown by the solid lines in Fig. 4-10. A
fit with a 25 nm isothermal layer yields best fit values of kAl 2 o 3 = 34.6 W/mK and
G = 103 MW/m2 K from R(f), and kAl 2 o 3 = 40 W/mK and G = 110 MW/m 2 K from
0(f).
The choice of isothermal layer thickness is important for modeling the high
frequency responses, but yields nearly the same thermal conductivity values, because
low frequency data are more sensitive to the substrate thermal conductivity, as will
be shown in Section 4.3.1. The values of kAl 2 o 3 and G obtained by fitting data in the
frequency-domain representation also provide good fits to delay-time-domain data.
The influence of how the metal layer is modeled is apparent in the high frequency
phase data. As shown in Fig. 4-11(a), the effect of reducing the isothermal layer
thickness to 10 nm or increasing it to 40 nm is negligible below 100 MHz, but becomes
increasingly important at high frequencies.
In fact, isothermal layers of 10 and 40
nm thicknesses fail to describe the high frequency data even if we allow both kA1 2 o 3
and G to vary, as shown in Fig. 4-11(b). Admittedly, accounting for non-equilibrium
electronic diffusion in Al with an isothermal layer is a crude approximation.
More
accurate modeling of the heat transport in the metal transducer layer, for example
with a two-temperature model [73], would be the next logical step for improving the
accuracy of modeling high frequency responses. The importance of such analysis in
interpreting high-frequency FDTR responses in terms of possible non-diffusive effects
has been pointed out in a concurrent study [74].
4.3.1
Sensitivity analysis
We can quantify the thermal model sensitivity to a particular parameter, 13, such as
the substrate thermal conductivity or the interface conductance, by considering the
logarithmic derivative of the model's response with respect to that parameter [75],
110
10
c 10
10-2
0
-6-70-
-900
10
10
10
2
10
3
f (MHz)
Figure 4-10: Thermal model best fits from simultaneously varying the A12 0 3 thermal
conductivity and the Al-A1 2 0 3 thermal interface conductance are shown, assuming
either a 10 nm (dotted lines) or a 25 nm (solid lines) isothermal Al layer. Frequencydomain surface temperature amplitude, R, and phase, <5, response derived from room
temperature TDTR measurements (open symbols).
111
-30
(a)
-50-
-70
-90
--30
(b)
-50
CD
-70
..... ........
.............
-90
0
200
600
400
f (MHz)
800
1000
Figure 4-11: <(f) data and model curves assuming isothermal Al layer thicknesses of
25 nm (solid lines), 10 nm (dotted lines) and 40 nm (dashed lines). (a) Curves derived
from only varying the isothermal layer thickness, holding all other model parameters
constant, and (b) best fit model curves allowing both kA 2 o 3 and G to vary.
112
S-dln R
=
d ln #
S
d#-ch
=
'
.n
d In 3
(4.15)
Sensitivity curves for our Al coated A12 0 3 sample are plotted in Fig. 4-12, showing
the thermal model sensitivity to kAl 2 0 3 , G, and the isothermal Al layer thickness,
di,,. Figure 4-12 shows that the thermal model is reasonably sensitive to kAI20 3 out
to -20 MHz in R and -200 MHz in
#,
and at high frequencies, the thermal interface
conductance has a more dominant contribution to the signal than the A12 0 3 thermal
conductivity.
0.8
0.6
kAl o
0.4
ISO
d
2
3
/
CU
-G
C
0.2
C )I
201
,-.
~%..
/
15F
101
-6-
NN
-
5
N
------------ -
100
-
-
-
.
N. N
.
101
*~~--~
-
-
U)
102
103
104
f (MHz)
Figure 4-12: Thermal model sensitivity plots as per Eq. (4.15), using G = 110
MW/m 2 K, kAl 2 o 3 = 41.7 W/mK, an isothermal Al layer thickness of di , = 25 nm,
a non-isothermal Al thickness of 85 nm with kAl = 237 W/mK, and literature values
of volumetric specific heats.
In addition to model sensitivity, we can evaluate the quality of the model fit by
considering the summed squares of the residuals between the model and the data,
x 2 , for
a range of kAl 2 0 3 and G values. Figure 4-13 plots
variations in the best fit values of kA1203 and G, where
113
(X 2
_
/
xmin is the best fit
for 10%
2
value.
mu~J1
The x 2 contours indicate that R(f) data is more sensitive to G, while
#(f)
data is
more sensitive to kAl 2 3- This explains why the phase data yield a significantly better
-
accuracy in measuring kAl 2 o 3
(a)
110
cm
E 105
100
0.5
95
kA1 2 o3
38
36
34
32
(b)
(W/mK)
120
2
115
0.5
0.1
E
110
0.2
0.01
105
100
36
38
kA1203
40
(W/mK)
42
44
i given a range of kAl 2 o3 and G values varied
Figure 4-13: Contours of (x2 _
up to 10% about the best fit values. Our model uses an isothermal Al layer thickness
of 25 nm and includes data up to 1 GHz for (a) the frequency response amplitude
R(f) and (b) the phase of the frequency response 0(f).
114
4.4
Comparison of fdTDTR data for various samples
To further test our fdTDTR method, and to evaluate the utility of using this technique
for phonon mean free path spectroscopy, we studied three substrate materials (fused
silica, (0001) A1 2 0 3 , and (100) Si) spanning a range of thermal diffusivities, and two
different transducer materials (Al and Ti-Au). There have been conflicting reports
in the literature regarding the observation of frequency-dependent substrate thermal
conductivity based on the transducer layer material used. A BB-FDTR study found
a frequency dependent behavior for a Si substrate with a 50 nm thick Au transducer
with a 5 nm Cr stiction layer [18], while another FDTR study found no frequency
dependence using an 80 nm thick Al transducer [74].
Our measurements with Al transducers do not indicate non-diffusive behavior, in
agreement with the observations of Ref. [741. We tested both 110 nm thick (see Fig.
4-14) and 60 nm thick (see Fig. 4-15) Al layers. The thermal model curves shown in
Figs. 4-14 to 4-17 assume bulk values for the substrate thermal conductivities, where
the only fitting parameter was the thermal interface conductance between the metal
transducer and the substrate.
In addition to Al transducers, we also investigated Au-Ti transducers.
Ti was
used as a stiction layer to promote adhesion of the Au. Measuring fdTDTR data on
Au transducers is particularly challenging given our pump and probe wavelengths. A
sharp hot electron peak is apparent in the first 2 ps of TDTR data, as shown in Figs.
4-18(c) and 4-18(d). This sharp peak complicates the phase correction and stitching
procedures that we discussed in Section 4.2.2, necessitating extra care during data
collection and processing. It is important to scan the delay stage slowly enough to
fully resolve the electronic peak, and to keep the signal-to-noise levels in both the
long and shot delay time data traces nearly equivalent.
For a thick Au transducer that consisted of 160 nm of Au with a 5 nm Ti stiction
layer, we observe comparable non-diffusive behavior to that observed for Al transducers (see Fig. 4-16). We find that using a 80 nm thick isothermal surface layer in
115
the Au produces reasonable fits to our high frequency data if we assume bulk values
of substrate thermal conductivity. Measurements on a thiner Au transducer (55 nm
of Al with 5 nm of Ti) similar to that of Ref. [18], however, produced poor fits, even
assuming a 55 nm thick isothermal surface layer (see Fig. 4-17). The thickness of
the Au in this case was thinner than the hot electron diffusion length in Au, which
has been reported to be 100 nm [76]. The Ti stiction layer was likely heated by these
non-equilibrium hot electrons before they deposited their energy to the Au lattice.
This supposition is supported by the shape of the early delay time TDTR data shown
in Fig. 4-18. We observe a dip followed by a rise in the in the TDTR amplitude signal
at early delay times for the 55 nm Au with a 5 nm Ti transducer layer (see the inset
of Fig. 4-18(d)), but not for the 160 nm Au with a 5 nm Ti transducer layer (see the
inset of Fig. 4-18(c)). Such a signal (see inset of Fig. 4-18(d)) could arise from the
excitation of phonons in the Ti layer by hot electrons in the Au [68]. Au has weak
electron-phonon coupling, whereby it takes a longer time for hot electrons to relax to
phonons compared to other metals like Ti which have stronger electron-phonon coupling. Thus, the hot electrons in the Au have time to excite electrons and phonons
in the Ti layer before relaxing to phonons in the Au. The hot Ti phonons will also
conduct heat to the Au layer. Thus, the temperature of the Au phonons increases
before decreasing again as heat conducts into the underlying substrate. This behavior
needs to be accounted for in thermal modeling of TDTR and FDTR data that uses
multilayer metallic transducers where one of the metals has weak electron-phonon
coupling [68].
By not including this effect in our thermal model, we observe poor
agreement with our fdTDTR data (see Fig. 4-17). A thicker Au layer like that in
Fig.
4-16 seems to mitigate this effect, allowing for hot Au electrons to relax by
interacting with Au phonons before heat conducts to the Ti and into the substrate.
Figure 4-19 shows fdTDTR data collected for a polycrystalline Bi 2Te 3 sample
coated in a 90 nm Al transducer layer using a pump beam 1/e 2 diameter of 55 nm
and a probe diameter of 10 pm. Thermal model fits using isothermal layer thicknesses
of 25 nm and 10 nm are also shown. With di, = 10 nm, the model poorly fits the
phase data, but with di,
= 25 nm the model fits well and produces a reasonable
116
100
.
e
-1
C
110 nm Al transducer
ofused silica
1-2
SAl O
2 3
o Si
10-3
-20
-40
U>
-V
-60
~..
~
-801
-100' 0
1i
101
10
10
f (MHz)
Figure 4-14: fdTDTR data from Si, A1 2 0 3 , and fused silica substrates with a 110 nm
thick Al transducer. Lines show thermal model assuming bulk properties and using
dj = 25 nm.
bulk thermal conductivity of kBi2 Te3 = 1.32 W/mK. Previous TDTR measurements
on semiconductor alloys have reported a thermal conductivity dependence on pump
modulation frequency, which was attributed to non-diffusive transport in the alloy
[77].
Recent theoretical and experimental work has suggested that observations of an
apparent frequency dependence could result from an anisotropic failure of the Fourier
law, improper modeling of electron-phonon coupling in the metal transducer, or from
117
100
101
0~
It
0-2
60 nm Al transducer
03
0
fused silica
Al 2 0
3
Si
10-3
-20
-40
0D
-e-
-60
-80
0-
-100'100
101
102
f (MHz)
103
Figure 4-15: fdTDTR data from Si, A1 2 0 3 , and fused silica substrates with a 60 nm
thick Al transducer. Lines show thermal model assuming bulk properties and using
dis, = 25 nm.
nonequilibrium transport near the interface that renders the usual radiative boundary
condition inadequate [73, 74]. We find that accounting for a finite skin depth of optical
penetration as well as some length of electron superdiffusion with the crude model of
an isothermal layer at the top surface of the metal transducer produces a reasonable
fit to our fdTDTR data with bulk material properties for all the substrates we have
examined. Our findings support recent works that advocate for improved modeling of
nonequilibrium electron and phonon transport in the analysis of FDTR and TDTR
118
100
o. a.
~-1
10
10-
-
160 nm Au, 5 nm Ti transducer
fused silica
SAl O
2 3
10
310Si
-40 - ... ......
o
)
-20
-o -60 .,
-1000
10
-
-80.
10
10
2
10
3
f (MHz)
Figure 4-16: fdTDTR data from Si, A1 2 0 3 , and fused silica substrates with a 160 nm
Au transducer with a 5 nm Ti stiction layer. Lines show thermal model assuming
bulk properties and using diso = 80 nm.
data [73, 74].
4.5
Summary and future directions
The frequency-domain representation helps uncover aspects of the measurement physics
which remain obscured in a traditional TDTR measurement, such as the importance
of modeling the details of the heat transport in the metal transducer film for analyzing
119
100
-1
-Z
10-2 55 nm Au, 5 nm Ti transducer
13 fused silica
o A 2O 3
10-
-20
Al
-40-
0-
0
1000
10
10
10
2
10
3
f (MHz)
Figure 4-17: fdTDTR data from Si, A12 0 3 , and fused silica substrates with a 55 nm
Au transducer with a 5 nm Ti stiction layer. Lines show thermal model assuming
bulk properties and treating the entire Au layer as isothermal.
high frequency responses.
We have detailed a modified TDTR technique that allows for transforming TDTR
data collected in the delay-time-domain into the frequency-domain, a representation equivalent to that of FDTR techniques. A single TDTR measurement provides
the same information as sweeping through many different modulation frequencies in
FDTR. The high time resolution inherent in TDTR measurements enables the extraction of very high frequency content, up to 1 GHz or more, which goes well beyond
120
(b)
(a)
100
100
10,
-
0
0
1
0
Ce
Ce
2
3
x (ns)
4
0
100
50
-r (ps)
5
0
6
2
3
t(ns)
100
50
(ps)
C4-C
5
4
6
(d)
(c)
100
100
100
100
0
100
5
x
0
1
2
3
-r (ns)
4
5
50
(ps)
(ps)
6
100
0
1
2
3
-r(ns)
4
5
6
Figure 4-18: Comparison of early delay time TDTR amplitude signal shapes for
different transducer layers on substrates of A12 0 3 including (a) 110 nm Al, (b) 60 nm
Al, (c) 160 nm Au with 5 nm Ti, and (d) 55 nm Au with 5 nm Ti. Insets zoom in on
the peak near T = 0 where the pump and probe pulses arrive at the sample surface
simultaneously.
the current capabilities of FDTR techniques [18, 24]. The method only requires a
small modification of a conventional TDTR experiment, i.e., the extension of the optical delay range up to a full repetition rate period, and can be easily implemented
in any laboratory possessing a standard femtosecond pump-probe apparatus with a
high repetition rate. The frequency-domain representation has revealed that while
the standard heat diffusion equation model works well at frequencies below ~200
MHz, higher frequency responses are affected by electron superdiffusion in the metal
transducer film. This effect will be even more pronounced for metals with weaker
electron-phonon coupling such as gold
[741.
The described methodology not only al-
lows a direct comparison of TDTR and FDTR data and yields frequency responses
121
10 U
10
10
10
-40
-60-80
-100
100
10
10
72
103
f (MHz)
Figure 4-19: fdTDTR data (open symbols) from a polycrystalline Bi 2Te 3 sample
with a 90 nmr Al transducer layer. Model fits shown for di,; = 25 nm (solid lines) and
di,,.= 10 nm (dashed lines). #(f) fit with dir8 = 25 nm gives kBi 2Te 3 = 1.32 W/mK
and G = 17 MW/m 2 K.
at hitherto unattainable high frequencies, but also provides a physically intuitive way
of analyzing TDTR measurements.
122
Chapter 5
Summary and Outlook
In this thesis we explore both spatially periodic and time-harmonic excitations for
measuring non-diffusive conduction heat transport.
We utilize an indirect method
for determining which phonons are important for transporting heat by measuring
thermal conductivity as a function of thermal length scale keff(L) in quasiballistic
transport regimes.
Measurements of keff(L) can be linked to thermal conductivity
accumulation functions through modeling with heat flux suppression functions.
We numerically solved the full spectral Boltzmann transport equation (BTE) for
transient thermal grating experiments on substrates of Si and PbSe at 300 K using
phonon dispersion relations and lifetimes for all six phonon branches from density
functional theory calculations. We also solved the gray-medium problem analytically.
Our simulations reveled that an approximate model based on summing gray-medium
solutions could reasonably model the behavior in PbSe, and that the two-fluid model
from Ref. [8] effectively describes Si [78]. We went on to solve the inverse problem of
reconstructing thermal conductivity accumulation functions from measured effective
thermal conductivities and modeled heat flux suppression functions. We found that a
suppression function derived from the gray-medium BTE could reasonably reconstruct
the thermal conductivity accumulation function for PbSe, and even that of Si to some
extent [78].
We also demonstrated that a Fuchs-Sondheimer suppression function
could be used to reconstruct the thermal conductivity accumulation function of bulk
Si from TTG measurements on Si membranes of various thicknesses [54].
123
Finally
we explored the viability of deriving heat flux suppression functions experimentally,
using the Si membrane data as an ideal test case and found that such an approach is
complicated by the sharp features in the differential thermal conductivity function.
To measure conduction over length scales of 100 nm, we explored a technique
that entails fabricating a ID wire grid polarizer on the surface of a sample of interest. The 1D metal grating acts to minimize light transmission into the underlying
substrate, so that primarily only the metal wires are heated during a time-domain
thermoreflectance (TDTR) measurement with linearly polarized pump and probe
beams aligned perpendicular to the grating transmission axis. We developed a set
of design criteria and performed electromagnetic wave finite element simulations to
design a suitable grating. We then fabricated gratings on substrates of Si, fused silica
and polycrystalline Bi 2Te 3 . Transmission measurements on the transparent fused silica substrate showed that the gratings performed well as linear polarizers for our 800
nm probe and 400 nm pump beams. TDTR measurements indicated non-diffusive
transport in Si, and diffusive transport in fused silica and polycrystalline Bi 2Te 3 , indicating that the heat carrying phonons in polycrystalline Bi2 Te 3 have mean free paths
(MFPs) of less than 100 nm. We also identified that even small amounts of transmitted pump and probe light that directly excites electron-hole pairs in the substrate
and probes the substrate's reflectance response can have an appreciable effect on the
data, adding to experimental uncertainty. An alternate approach that eliminates any
direct optical excitation of the substrate would be preferable.
To study length scales smaller than 100 nm without the need for microfabrication, we developed a method for extracting high frequency response information from
TDTR data. Our approach allows TDTR data to be represented in a form equivalent
to frequency-domain thermoreflectance (FDTR) data [791. At high excitation frequencies, which correspond to shallow thermal penetration depths, FDTR results including
frequencies up to 200 MHz have been reported to exhibit non-diffusive behavior that
enables the reconstruction of thermal conductivity accumulation functions from experimental measurements [18]. To date, our TDTR measurements, which allow for
frequencies of 1 GHz, have not exhibited deviation from the Fourier heat equation.
124
We suspect that prior observations did not account for electron superdiffusion in the
metal transducer, which could explain the observations of apparent non-diffusive behavior in the substrate. Nevertheless, our method of analyzing TDTR data allows
for a direct comparison of FDTR and TDTR data, and provides a more physically
intuitive representation of TDTR data in the form of the frequency response. Future modeling work may help reveal regimes where information about non-diffusive
transport in the substrate can be extracted from high frequency response data. The
high frequency data could also be useful for studying electron transport in the metal
transducer and transport across the transducer-substrate interface.
Future work to advance phonon mean free path spectroscopy should focus on a
few key areas.
Methods for interpreting experimental data should be reevaluated
with regards to the applicability utilizing the heat equation when non-equilibrium
effects like hot electron superdiffusion and non-diffusive phonon transport are present.
The indirect approach of determining thermal conductivity accumulation functions
from measurements of length scale dependent thermal conductivity using heat flux
suppression functions [51] has not been rigorously proven, although applications of
this method have produced good agreement with first principles calculations [51, 78,
80, 81]. Defining heat flux suppression functions can prove difficult, particularly for
experimental length scales that depend on thermal conductivity, such as the thermal
penetration depth in FDTR measurements. Future experimental efforts should not
only focus on generating and measuring small thermal length scales, but should also
emphasize how amenable the measurement geometry is to extracting phonon spectral
information as unambiguously as possible on the basis of present knowledge.
For
example, theoretical analysis of the thin membrane geometry with diffusely scattering
boundaries is rigorous [54], and the thermal length scale (the membrane thickness)
is well defined. In contrast, experiments with multiple thermal length scales [58] or
length scales that depend on thermal diffusivity [18] generate more uncertainty in
interpretation.
Additionally, improper modeling of interface transport can lead to
observations of apparent, but not actual, non-diffusive effects [74], motivating the
need for further studies of transport near interfaces and physical boundaries.
125
126
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