Exploring Heat Transfer at the Atomistic Level for
Thermal Energy Conversion and Management
ARO-M"
by
MASSACHUSETTS INSTITUTE
OF TECI-HNOLOGY
Zhiting Tian
AUG 15 2014
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
LIBRARIES
Doctor of Philosophy in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
Author................................
Department of Mechanical Engineering
Signature redacted
Certified by...........
May 9, 2014
..............
Gang Chen
Carl Richard Soderberg Professor of Power Engineering
Department Head
Thesis Supervisor
Signature redactedr
Accepted by.........
David E. Hardt
Ralph E. and Eloise F. Cross Professor of Mechanical Engineering
Chairman, Department Committee on Graduate Students
Exploring Heat Transfer at the Atomistic Level for
Thermal Energy Conversion and Management
by
Zhiting Tian
Submitted to the Department of Mechanical Engineering
on May 9, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
Abstract
Heat transfer at the scales of atoms plays an important role in many applications such
as thermoelectric energy conversion and thermal management of microelectronic
devices. While nanoengineering offers unique opportunities to manipulate heat to our
advantages, it also imposes challenges on the fundamental understanding of nanoscale
heat transfer. As the characteristic lengths of the system size become comparable to the
mean free paths of heat carriers, macroscopic theories based on heat diffusion are no
longer valid due to size effects. Atomistic level simulation can provide powerful
insights into the microscopic processes governing heat conduction, and is the focus of
this thesis.
In this thesis, we first introduce atomistic techniques to investigate phonon transport
in bulk crystals. We start with normal mode analysis within the classical molecular
dynamics framework to estimate the spectral phonon transport properties. Although it
can provide the detailed phonon properties adequately, classical molecular dynamics
with empirical potentials do not always yield accurate predictions. Then, we move to
first-principles density functional theory (DFT) to compute mode-dependent phonon
properties. Such simulations can well reproduce experimental values of phonon
dispersion and thermal conductivity with no adjustable parameters, establishing
confidence that such an approach can provide reliable information about the
microscopic processes. These detailed calculations not only unveil which phonon
modes are responsible for heat conduction in bulk crystals, but also expand our
fundamental understanding of phonon transport, such as the importance of optical
3
phonons. Next, we study thermal transport across single and multiple interfaces via
the atomistic Green's function method, especially the impact of interface roughness on
phonon transmission across a single interface and coherent phonon transport in
superlattices. Both the DFT and Green's function techniques provide fundamental
parameters that then can be used to understand mesoscale transport. This paves the
way for multiscale modeling from first-principles. Through these multiscale modeling
efforts, we are able to obtain a comprehensive understanding of heat transfer from the
atomistic to the macroscale, with important implications for energy applications.
Complementary
to the theoretical
work,
we measure the
interface thermal
conductance using ultrafast time-domain thermoreflectance experiments, examining
thermal
transport
across
solid-liquid
interfaces
modified
by
self-assembled
monolayers. We find that an extra molecular layer can enhance the thermal transport
across solid-liquid interfaces.
In summary, theoretical, computational and theoretical approaches have been applied
to study heat transfer at the atomistic level. The findings from this thesis have
improved our fundamental understanding of phonon transport properties with
important implications for energy applications and beyond, and build a foundation for
multiscale simulation of phonon heat conduction at the mesoscale.
Thesis Supervisor: Gang Chen
Title: Carl Richard Soderberg Professor of Power Engineering
Department Head
4
Dedication
TO MY PARENTS
For raising me to believe that girls can achieve anything
TO MY HUSBAND
For pushing me beyond my limits
AND TO MY SON
For inspiring me to dream big
5
6
Acknowledgements
This thesis owes its completion to the support of numerous individuals. I would like
to gratefully acknowledge their help here.
First, I would like to thank my advisor, Professor Gang Chen. I am very fortunate to
have Gang as my advisor. He has gone far above and far beyond my expectations of
an advisor. Despite his super busy schedule, he always makes time for his students.
He meets us individually as often as possible, provides comments on manuscripts
within a day, spends huge amount of time teaching us how to deliver a good talk,
and providing invaluable advice on job search. He called me almost every evening
while I was deciding on my faculty job offers. Words cannot express my gratitude to
him. I am particularly thankful to Gang for thinking on my feet while suggesting
research topics to me, which makes my Ph.D. research a coherent story with a
variety of skillsets. Meanwhile, he leaves enough room for me to independently
develop my research. For example, when I wrote a proposal on my own and did
experiments at Argonne National Lab, he was very supportive. Without him, I would
not have been able to secure a faculty position straight from graduate school.
Next, I would like to thank my thesis committee members, Millie Dresselhaus, Bora
Mikic, Jeffery Grossman, and Evelyn Wang. They took significant amounts of time to
give me advice on my research and career paths. They wrote reference letters for my
faculty applications and gave me insightful advice on interview and offer decisions.
I would like to thank people in the heat transfer committee who have greatly
encouraged me. Special thanks go to Alan McGaughey at Carnegie Mellon University
who kept an eye on my progress and offered to write reference letter for my job
search.
I would like to thank several female faculties, in addition to Millie and Evelyn, for
setting up a great role model for me: particularly my mentor Dean Christina Ortiz at
MIT, Jian Cao at Northwestern University, and Pamela Norris at University of
Virginia. The interaction with them in person made me determined to pursue a career
in academia.
I would like to thank my lab mates in the NanoEngineering group. I am very lucky to
work with a great group of people, the NanoEngineering group of MIT. My labmates
were also a great resource, and I greatly benefitted from the communication and
collaboration with them. In particular, I would like to Asegun Henry for his detailed
7
instruction on normal mode analysis, Sheng Shen, Austin Minnich and Amy
Marconnet for their valuable suggestions and help on my job search, Keivan Esfarjani
and Junichiro Shiomi for sharing with me their deep knowledge on phonons, Takuma
Shiga and Jivtesh Garg for DFT calculations, Maria Luckyanova and Kimberlee
Collins for teaching me how to conduct TDTR measurements, Tengfei Luo, Nuo Yang
and Yann Chalopin for discussion on molecular dynamics simulations, Bo Qiu and
Yuan Yang on other projects not covered in this thesis. Moreover, it has been a big
family for me. They witnessed the important moments in my life and shared with me
my happiness: getting married and having a baby. They prepared surprise party for my
birthday which made me burst to tears. George Ni, Wei-Chun Hsu and Bolin Liao not
only helped on my wedding but also help me move my home couple of times.
I am very thankful to DOE S3TEC for supporting my research, Argonne National Lab
for precious beam time and NSF Teragrid for supercomputer time.
My experience at MIT has been made far richer by the terrific friends, especially
through Tsinghua Alumni Association and Chinese Students and Scholars Association.
Among those countless many, I would like to especially thank Jiexi Zhang and
Tengfei Zheng for being my closest friends at MIT and supporting me all the time,
especially throughout my pregnancy. I would like to thank Xian Li and Yu Jiang for
being my very close friends since I came to the United States in 2007.
Finally, above all else, I thank my family whose unwavering love carries me through
all of life's adventures. No matter what happens, they are always there. My
wonderful parents Yuzhong Tian and Tingrong Liao set the bar high for me and
urged me to pursue my dream. My uncle Tingyuan Liao tried his best to support me
choosing my path. My life has been more enjoyable accompanied my lovely cousin
Shiying Liao. Last but not least, my husband Yan Zhou is the most amazing man I
have ever met. His endless love and support makes me happy every day. He pushes
me to reach higher. He sacrifices his job for me to accept my best offer. My son
Austin Zirui Zhou is the most precious gift I ever have. He is my sunshine. I am
extremely lucky to have both Yan and Austin. Having both of them in my life is the
biggest achievement in my Ph.D.
8
Contents
introduction......................................................................................................
15
1.1 H eat Conduction by Phonons........................................................................
15
1.2 Outline of the thesis ......................................................................................
19
2.
Spectral Phonon Properties of Ge Using Normal Mode Analysis ...................
2.1 Introduction....................................................................................................
21
2.2 M ethodology .................................................................................................
22
2.3 Results and D iscussion .................................................................................
24
2.3.1 Phonon Lifetim es...................................................................................
24
2.3.2 Therm al Conductivity Validation...........................................................
25
2.3.3 Contribution from Different Phonon Modes...........................................
27
2.4 Conclusion.....................................................................................................
3.
21
29
Phonon Conduction in PbSe, PbTe and their alloys Using First-Principles
Calculations..................................................................................................................31
3.1 Introduction....................................................................................................
31
3.2 M ethodology .................................................................................................
34
3.2.1 Harm onic Properties ...............................................................................
34
3.2.2 Anharm onic Properties...........................................................................
35
3.2.3 Lattice therm al conductivity ...................................................................
36
3.2.4 Alloy modeling ......................................................................................
37
3.3 Results and D iscussion .................................................................................
38
3.3.1 Com parison w ith experim ental results....................................................
38
3.3.2 Com parison between PbSe and PbTe .....................................................
41
3.3.3 The im portance of optical phonons........................................................
44
3.3.4 The potential im pacts of nanostructuring ...............................................
45
3.3.5 The potential impacts of PbSe-PbTe Alloying ........................................
47
3.4
4.
Conclusion.................................................................................................
48
The Importance of Optical Phonons in Nanostructures ...................................
49
4.1 Introduction....................................................................................................
9
49
5.
4.2 M ethodology .................................................................................................
50
4.3 Results and D iscussion .................................................................................
52
4.4 Conclusion .........................................................................................................
57
Phonon Transmission across a Single Si/Ge Interface using the Green's Function
M ethod ....................................................................................................
... - .......
59
5.1 Introduction........................................................................................................59
5.2 M ethodology .................................................................................................
62
5.3 Results and Discussion .................................................................................
66
5.3.1. Rough interface with random distribution.................................................66
5.3.2. Rough interface with Gaussian distribution...........................................
5.4 Conclusion ......................................................................................................
6.
Phonon Transm ission across Si/Ge Superlattices .............................................
73
77
79
6.1 Introduction....................................................................................................
79
6.2 M ethodology.................................................................................................
81
6.3 Results and D iscussion .................................................................................
83
6.4 Conclusion ......................................................................................................
93
Solid-Liquid Interface Conductance Using Time-Domain Thermoreflectance
95
M easurem ents ..............................................................................................................
7.
7.1 Introduction....................................................................................................
95
7.2 Sam ple Preparation and Experim ental Setup..................................................
96
7.3 Results and D iscussion .................................................................................
98
7.4 Conclusion .......................................................................................................
8.
101
Sum m ary and Future Work................................................................................103
8.1 Sum m ary ..........................................................................................................
103
8.2 Future D irections .............................................................................................
104
10
List of Figures
Figure 2-1 Simulation cell of germanium showing two basis atoms per
primitive cell. 10a x 10a x 10a is used as the simulation domain. 22
Figure 2-2 [100] Phonon dispersion of germanium from lattice dynamics.
Two degenerate transverse acoustic branches, one longitudinal acoustic
branch, one longitudinal optical branch and two degenerate transverse
optical branches. .............................................................................
23
Figure 2-3 From autocorrelation of normal mode energy to phonon lifetimes
........................................................................................................
. . 24
Figure 2-4 Phonon lifetimes of the transverse acoustic (TA), longitudinal
acoustic (LA), transverse optical (TO) and longitudinal optical (LO)
m odes in the [100] direction ..........................................................
25
Figure 2-5 Phonon density of states for each polarization: TAl, TA2, LA,
TO 1, TO2 and LO ..........................................................................
26
Figure 2-6 Ensemble average of EMD simulations using Green-Kubo
form ula............................................................................................
27
Figure 2-7 Contributions from different crystallographic directions and
different polarizations.....................................................................
28
Figure 2-8 Thermal conductivity accumulation with respect to the phonon
m ean free paths..............................................................................
28
Figure 3-1 Phonon dispersion for PbSe and PbTe: red lines: calculated
results; black dots: experimental results ..........................................
39
Figure 3-2 Temperature dependent lattice thermal conductivity of PbSe and
PbTe, red lines: calculated results; black crosses: experimental data..41
Figure 3-3 Frequency dependent phonon lifetimes of PbSe (squares) and
PbTe (crosses) at 300 K: (a) TA, (b) LA, (c) TO, and (d) LO. ......
42
Figure 3-4 Frequency dependent phonon group velocities of PbSe (squares)
and PbTe (crosses) at 300 K: (a) TA, (b) LA, (c) TO and (d) LO........43
Figure 3-5 Thermal conductivity from different polarizations (TAl, TA2,
LA, TOl, T02 and LO) versus temperature for PbSe and PbTe ......... 44
Figure 3-6 Temperature dependence of lattice thermal conductivity without
acoustic-optical scattering: PbSe (black dashed line), PbTe (red dashed
line) and with acoustic-optical scattering: PbSe (black solid line), PbTe
(red solid line) .................................................................................
45
Figure 3-7 Cumulative thermal conductivity with respect to phonon mean
free path at 300 K for PbSe (red dashed line), PbTe (black solid line)
11
and PbTeo.5Seo.s (blue dotted line)....................................................
46
Figure 3-8 Calculated composition dependence of the lattice thermal
conductivity in PbTe..xSex at 300 K (solid line) and 500 K (dashed line)
47
..............................................................................................................
Figure 4-1 Cumulative thermal conductivity with respect to MFPs at 277K
from the 18 x 18 x18 k-mesh data; Inset (a)-Inset (b) Zoomed-in
figures for MFP range of (a) 0-70nm and (b) 0-9nm, respectively......52
Figure 4-2 Thermal conductivity of silicon nanowires for d=37nm, 56nm
and 11 5nm, lines: calculated results; crosses, squares and stars:
experim ental results........................................................................
53
Figure 4-3 Thermal conductivity from different polarizations versus
temperature for d=20nm; (b) normalized optical phonon contributions
to the total thermal conductivity versus temperature for d=1 Onm, 20nm,
55
100nm and lm m ............................................................................
Figure 4-4 (a) The thermal conductivity from different polarizations versus
diameters at 277K, (b) the normalized optical phonon contributions to
the total thermal conductivity vs. diameters at 100K, 177K, 277K, and
56
400K .....................................................................................................
Figure 5-1 The system is divided into three parts: left (L), center (C) and
right (R). The left and right leads are semi-infinite crystal lattices. In
the transverse direction, all the three regions have periodic boundary
conditions imposed to represent the infinitely large lateral dimension.
63
..............................................................................................................
Figure
5-2
Total transmission function,
transmittance
and thermal
conductance as a function of phonon frequency for an ideal Si/Ge
interface (solid black line) and for a random rough Si/Ge interface
(colored dashed or dotted lines): (a) Total transmission based on SW
force constants; (b) Transmittance from Si to Ge based on SW force
constants; (c) Thermal conductance based on SW force constants; (d)
Total transmission based on DFT force constants; (e) Transmittance
from Si to Ge based on DFT force constants; (f) Thermal conductance
based on DFT force constants......................................................... 70
Figure 5-3 Phonon Density of States (DoS) of pure Si (Black solid line),
pure Ge (red dashed line) and Si/Ge 1:1 mixture (green dotted line)
using DFT force constants. ............................................................
72
Figure 5-4 (a)Total transmission function, (b)transmittance, and (c) thermal
conductance as a function of phonon frequency for an ideal Si/Ge
interface (solid black line) and for a rough Si/Ge interface with a
12
Gaussian distribution (dashed blue lines) based on DFT force constants.
Inset of (c): The number of Si atom in each layer for an ideal interface
(solid black) and for a Gaussian rough interface (dashed blue)...........75
Figure 5-5 Thermal conductance ratio of a Gaussian rough interface to an
ideal interface as a function of the mass ratio (lower x-axis) and the
acoustic impedance ratio (upper x-axis) of the two materials using DFT
force constants ................................................................................
76
Figure 6-1 Schematic of the system setup: the left reservoir is pure Si, the
right reservoir is pure Ge, the center region is the Si/Ge superlattice. 82
Figure 6-2 (a) Thermal conductivity of superlattices as a function of number
of periods for smooth and rough superlattices at T=300K. (b)
Transmittance as a function of frequency for superlattices (period =2a)
with smooth interfaces; (c) Transmittance as a function of frequency
for superlattices (period=2a) with rough interfaces........................86
Figure 6-3 Phonon dispersion of Si/Ge superlattice with period length = 2a
in [100] direction.............................................................................
87
Figure 6-4 The total thermal conductivity, contribution from phonons with
frequencies larger than the cutoff frequency, and not larger than the
cutoff frequency at (a) T=20K, (b) 50K and (c) 300K....................89
Figure 6-5 Normalized thermal conductance per interface as a function of
number of periods for rough-interfaced superlattices with period length
1 = 2a. The experimental value is extrapolated from the sample of
period length 1 = 4.4nm and 100 periods.....................................90
Figure 6-6 Thermal conductivity of rough-interfaced superlattices as a
function of superlattice length for period length 1 = a, 2a and 4a at
300K ...............................................................................................
93
Figure 7-1 Schematic of SAMs used in this study. (a) Hexenedithiol
SHCH 2(CH 2) 4CH2SH;
Undecanethiol
(b)
Hexanethiol
CH3(CH 2)9CH2 SH;
and
CH3(CH 2) 4CH2SH;
(d)
(c)
Hexadecanethiol
CH3(CH2)1 4CH2 SH ..........................................................................
96
Figure 7-2 (a) Schematic of sample arrangement; (b) Phase data for samples
without SAM and with hexanedithiol............................................
98
Figure 7-3 Thermal conductance between Au and ethanol with and without
SAMs from TDTR measurements at room temperature..................99
Figure 7-4 Contact angles, 0, of ethanol on Au surface modified by (a)
hexanethiol and (b) hexanedithiol SAMs. .........................................
13
100
14
1.
Introduction
1.1 Heat Conduction by Phonons
Emerging nanotechnological applications have necessitated reliable quantitative
understanding of thermal transport. There are fundamental differences between heat
transfer processes at the nanoscale and the macroscale due to quantum and classical
size effects; for example, both the Fourier law for heat conduction and Planck's law
for blackbody radiation break down in nanostructures. This makes modeling the
nanoscale heat transfer challenging. In semiconductors and insulators, heat is carried
primarily by vibrations in the crystal lattices known as phonons. A comprehensive
understanding of phonon transport will facilitate design of innovative nanostructured
materials and devices for thermal energy applications and beyond.
Understanding heat conduction in crystalline solids started only when the quantum
theory of lattice vibrations was developed and the phonon concept was established.
The basic quantum of crystal lattice vibration is called a phonon. The phonon
scattering originating from the anharmonic interatomic potential can be classified as
either a normal process which conserves crystal momentum or an umklapp process
which does not. It was recognized that the umklapp process creates a resistance to
heat flow while the normal process only redistributes phonons. Peierls I extended the
Boltzmann formulation for phonon transport taking into consideration only the
umklapp process. Under the relaxation time approximation and assuming isotropic
group velocity and phonon lifetime, the Boltzmann equation led to the kinetic theory
expression for thermal conductivity as
Kp=
m. C(o) v(a)A(co)do =
-
C(C)V2(c)(c)do
(1.1)
where CQo) = hOD(o)dfBE / dT is the specific heat per unit frequency interval at
frequency o and temperature T, D(w) is phonon density of states per unit volume
15
and per unit frequency interval,
fBE
is the Bose-Einstein distribution, V is the
phonon group velocity, A = vr is the phonon mean free path (MFP) and r is the
phonon lifetime. Callaway further modified the theory by accounting for both the
normal and umklapp scattering processes, assuming that normal processes lead to a
displaced Bose-Einstein distribution and indirectly affect the umklapp processes. The
derived expression is similar to equation (1.1) except for a modification to phonon
lifetimes. The quantities in equation (1.1) that determine the lattice thermal
conductivity, however, are frequency dependent and difficult to obtain. Different
approximations were made to perform the thermal conductivity integral.
First, the Debye approximation that assumes a linear relation between the phonon
frequency and the propagation wavevector is often used for the phonon group velocity
and density of states, based on the fact that the Debye approximation has been very
successfully used to explain the specific heat of crystalline materials. This left the
phonon lifetime unknown. In the 1950s, Klemens
2
derived expressions using the
quantum perturbation theory, i.e., Fermi's golden rule, for phonon lifetimes due to
different scattering mechanisms. The expressions obtained are again based on the
Debye approximation and contains unknown parameters. The experimental values of
the thermal conductivity, speed of sound, and specific heat are often used to fit the
unknown parameters in the Debye model and in the lifetimes. Such fittings usually
work well at low temperatures. Deviations at high temperatures stimulated more
refined models of dispersion, such as the one developed by Holland
3
that used two
different linear dispersions to better represent the rapid flattening of transverse
acoustic phonons in FCC crystals such as GaAs and Si.
The shape of the temperature dependence of thermal conductivity is universal among
crystalline solids. At high temperatures, the thermal conductivity usually decreases
with increasing temperature as T-", with theoretically n=I although practically n=1 -1.5.
This is because at high temperatures, phonon specific heat is a constant according to
the Pettit-Delong law, and phonon energy, i.e. the number of phonons, increases
linearly with temperature. Since the scattering rate is proportional to the number of
phonons, the thermal conductivity decreases with increasing temperature. At low
temperatures, the thermal conductivity is usually proportional to T3 . In this regime,
phonon-phonon scattering is weak and phonon mean free paths (MFPs) are longer
than the size of the sample. Phonons scatter more frequently with the boundaries and
the phonon MFP is effectively equal to the sample size, and is independent of
frequency. Thermal conductivity is thus proportional to the specific heat and hence the
T3 behavior. This size effect was first discovered by De Haas and Biermasz 4, and
explained by Casimir
5 (sometimes
called the Casimir regime). Casimir assumed that
16
surfaces of samples are rough, and scatter phonons diffusely, although latter studies
also investigated partially diffuse and partially specular surfaces 6. In between low
and high temperatures, impurity scattering is usually important, and the peak value of
the thermal conductivity depends sensitively on impurity concentrations.
Although the approaches pioneered by Klemens were successful in explaining the
trend of experimental observations, quantitative details of phonon transport may be
far from what the past approaches described above predicted, especially in terms of
the phonon lifetimes and MFPs. The reason that the Debye model together with
Klemens' treatment on scattering can fit experimental data is because the thermal
conductivity integral is quite forgiving. In fact, one can use different sets of
parameters to fit the experimental data based on different approximations and
scattering mechanisms. Extracting the exact phonon transport properties, especially
phonon lifetimes and MFPs, thus remained unsolved.
Since 1980s, efforts started on calculating thermal conductivity of crystals using
molecular dynamics (MD) simulations
711.
In classical MD, the approximate
trajectories of each individual atom within the simulation domain are tracked based on
an empirical interatomic potential and Newton's second law. Two prevailing methods
used to study heat transport are equilibrium molecular dynamics (EMD) and
nonequilibrium molecular dynamics
(NEMD).
EMD
is suitable for transport
properties, whereas NEMD simulates actual transport processes. EMD first obtains
the history of individual particles in an equilibrium system, from which the transport
properties are extracted on the basis of linear response theory. The thermal
conductivity is calculated from the autocorrelation of the instantaneous heat flux
through the Green-Kubo formula
1, 1.
For NEMD, one can either impose a
temperature difference to calculate the heat flux
14,
the resulting temperature distribution
The thermal conductivity is then
15, 16.
or impose a heat flux to calculate
determined by the Fourier law. The NEMD methods are relatively easy to implement
and are usually faster than the EMD methods because the latter requires the
calculation of the autocorrelation function, which can take a long time to decay. In
addition, the EMD method may involve the artificial autocorrelation caused by the
often-used periodic boundary condition. However, NEMD also suffers from several
drawbacks. Firstly, the statistical foundation of NEMD is not as soundly established
as that of EMD
1.
Second, the finite simulation size in NEMD might be shorter than
the MFP of some phonon modes, leading to artificial size effects with boundaries
imposed at the heat reservoirs. Third, a large temperature difference is applied across
a small simulation domain.
17
To extract the phonon lifetimes and MFPs, the phonon spectral energy density
analysis or normal mode analysis 18-21 has been applied in combination with EMD.
The key idea is to project the atomic displacements onto normal mode coordinates
and to determine the phonon lifetimes by tracing the temporal amplitude decay of
each mode or fitting the width of the spectral energy density peaks in the frequency
domain. To apply this approach, one needs to utilize eigenvalues and eigenvectors
from lattice dynamics calculations and to perform separate analysis from the
traditional thermal conductivity calculations using MD.
As a widely adopted simulation tool, MD simulations do not require any a priori
knowledge of the phonon transport properties, are straightforward to implement, and
automatically include the temperature-dependent anharmonicity. Yet MD is only
rigorously applicable to solids above the Debye temperature and being entirely
classical MD assumes each vibrational mode is equally excited. At low temperatures,
quantum corrections for temperature-related effects are needed. The electronic
contribution to thermal transport cannot be assessed in MD. Moreover, the empirical
potentials used in classical MD can cause the thermal conductivity to deviate
significantly from the experimental data 2 2
Conversely, first-principles calculations where the potential is obtained directly from
the electron charge density via density functional theory (DFT) without any adjustable
parameters provide the most reliable simple way of computing the lattice thermal
conductivity. Ab inito MD simulations 23,24 use the forces computed on-the-fly by
DFT and are computationally expensive, though. The alternative way is to extract the
interatomic force constants from DFT calculations for limited atomic configurations
and to then conduct further calculations. After DFT calculations, one can either obtain
the force constants from reciprocal space based on density functional perturbation
theory (DFPT)
25,26
or from real space calculations by fitting the force-displacement
data in a supercell with a polynomial potential
27, 28.
Despite the fact that the real
space approach is simpler yet less precise, both approaches are accurate enough to
reproduce the experimental results for the lattice thermal conductivity. Broido et al. 25
calculated the intrinsic lattice thermal conductivity of Si and Ge using the reciprocal
space approach and obtained excellent agreement with experimental data. The
reciprocal lattice approach was later applied to SixGei.x alloys and superlattices by
Garg et aL.
29,30,
and to PbSe, PbTe and PbTev.xSex by Tian et al.
started the real space approach with Si
half-Heusler
32,
28,
31.
Esfarjani et al.
and then extended the approach to
PbTe 33 and GaAs 34.
Once the harmonic and anharmonic force constants are obtained, one can either
18
perform MD simulations based on the developed Taylor expansion potential
32
or
employ lattice dynamics calculations: first obtain the vibrational eigenmodes based on
then compute the scattering rates of each mode
the harmonic part of the potential,
by treating the anharmonic potential as a perturbation using Fermi's golden rule, and
solve the Boltzmann transport equation (BTE) to find the thermal conductivity 25,28-33
Although MD simulations based on fitted potentials have more flexibility to directly
obtain the lattice thermal conductivity for complicated structures, lattice dynamics
calculations can produce the detailed phonon transport properties without extra efforts
because the thermal conductivities are the integrated quantities over the first Brillion
zone using either the solution of BTE under the single-mode relaxation time
approximation 28,29,
31-33,35
or based on an iterative solution of the integral BTE
25
1.2 Outline of the thesis
This thesis focuses on phonon transport properties using emerging computational
tools at the atomistic level to reveal the microscopic origin of the thermal conductivity
of a solid. Chapters 2 and 3 deal with phonon transport properties in bulk materials,
while
Chapters
5
and
6 focus
on phonon
transmission
across
interfaces.
Complementary to the theoretical work, experimental measurements are covered in
Chapter 7.
In Chapter 2, we introduce normal mode analysis which utilizes a combination of
molecular dynamics and lattice dynamics to capture the spectral phonon transport
properties of germanium. The contribution of different phonon polarizations and
phonons with different mean free paths are estimated.
In Chapter 3, we describe in detail first-principles calculations of phonon conduction
in PbSe, PbTe and their alloys. The calculated phonon dispersion and thermal
conductivity agree very well with experimental data. Mode-dependent phonon
transport properties are extracted. The origin of the low thermal conductivity, the
mean free path distribution and the importance of optical phonons are discussed.
In Chapter 4, we examine the importance of optical phonons to thermal conductivity
in nanostructures. Silicon nanowires are chosen as one example. In nanostructures,
acoustic
phonons
with
long
mean
free
paths
are
strongly
scattered
at
interfaces/boundaries, the optical phonons are much less influenced. This leads to a
rebalance of the relative importance between acoustic and optical phonons.
19
In Chapter 5, we apply the Green's function method to study phonon transmission
across single Si/Ge interfaces. We evaluate the effects of interface roughness and find
that the roughness introduced by atomic mixing can enhance phonon transmission in a
certain range through a smoother transition in the phonon vibrational spectrum.
In Chapter 6, we apply the Green's function method to multiple Si/Ge interfaces,
namely Si/Ge superlattices, and demonstrate that the thermal resistance cannot always
be taken as a sum of individual interfaces. Coherent phonon transport is thus observed
under certain conditions.
In Chapter 7, we discuss about our experimental endeavors on thermal transport
across a solid-liquid interface. The interface thermal conductance at a gold-ethanol
interface modified by self-assembled monolayers is measured using time-domain
thermoreflectance techniques.
Finally, Chapter 8 lists potential future work and concludes the thesis.
20
2. Spectral Phonon Properties of Ge
Using Normal Mode Analysis
2.1 Introduction
Size effects are highly important in the micro/nanoscale regime where the phonon
mean free paths become comparable to the device length and the thermal conductivity
can decrease by several orders of magnitude. 17, 3640 To understand and manipulate
thermal transport at these small length scales, knowledge of phonon mean free paths
is required. However, the direct calculation of phonon mean free paths has been
neglected for many years. The major challenge is to determine phonon lifetimes
because most existing models2 are semi-empirical and a large uncertainty remains in
fitting multiple parameters. To extract phonon lifetimes and MFPs, the phonon
spectral energy density analysis or the normal mode analysis1 8 -21 have been developed.
In the normal mode analysis, atomic vibrations from a MD trajectory are decomposed
into vibrational eigenstates, or into so-called normal mode coordinates via lattice
dynamics (LD) calculation. The lifetimes can then be extracted from the temporal
decay of the normal mode energies. With knowledge of the phonon frequency, group
velocity and lifetime of each mode, we can use the Boltzmann transport equation
(BTE) to calculate the total thermal conductivity. Comparing these results with the
thermal conductivity calculation based on the Green-Kubo formula, the methodology
can be validated. More importantly, based on the mode-dependent lifetimes and mean
free paths, we could fully detail the phonon contributions from different polarizations,
frequencies and wavelengths.
In this chapter, we perform normal mode analysis for germanium, one of the most
important semiconductors, to fully detail the spectral dependence of the phonon
transport properties in bulk germanium. The contributions of different phonon
frequencies and polarizations to the thermal conductivity are discussed.
21
2.2 Methodology
The detailed methodology of normal mode analysis can be found elsewhere.1'19 In
short, the MD trajectory is transformed to normal mode coordinates via LD
calculations. Equation (2.1) is used to calculate the normal mode amplitudes.
A(I, p,t) = (
*(j,kp)-ii(jl,t) -exp(-ik -r(fl))
)11
(2.1)
where e is the polarization vector obtained from LD calculations using GULP, and U'
is the displacement of each atom recorded in EMD using LAMMPS . The normal
mode energy is calculated using equation (2.2)
E(kp,t)
=
-O2A(k, pt)- A*(k, p,t) + -A(k, p,t) - A*(k, p,t)
2
2
(2.2)
The phonon lifetimes are then determined by equation (2.3)7' 18.
f(E(k, p, t)&E(k, p,0))(23
_.(2.3)
r~,p= 0
)
(&E2(k,
p,0))
Detailed simulation procedures are described below. First, we use the StillingerWeber potential 42,
43
and construct a 10a*10a*10a simulation size with periodic
boundary conditions in three dimensions as shown in Fig. 2-1. The system is
maintained at 300K. The two basis atoms,
denoted in different colors, are
distinguished in order to perform the correct projection.
Figure 2-1 Simulation cell of germanium showing two basis atoms per primitive cell.
10a x 10a x 10a is used as the simulation domain.
22
Second, phonon dispersion curves are obtained by LD calculations along three high
symmetry lines [100], [110] and [111] within the First Brillouin Zone, although only
[100] direction is shown in Fig. 2-2. There are six polarizations or phonon branches:
two transverse acoustic (TA) branches, one longitudinal acoustic (LA) branch, two
transverse optical (TO) branches and one longitudinal optical (LO) branch. The
allowed wave vectors are discrete points due to the periodic boundary conditions
applied over a finite number of unit cells.
Third, we trace the evolution of each phonon mode in the EMD simulations. As
shown in Fig. 2-3, from the autocorrelation of the normal mode energy, the frequency
of this mode can be identified
via the Fourier transform.
Because energy
autocorrelation is performed, the frequency thus obtained would be twice the mode
frequency. We could compare this frequency with the eigenvalue we obtained from
LD as a quick check whether this is the desired mode. Then, by fitting the peaks of
the autocorrelation with an exponential function, the denominator of the power gives
phonon lifetime of this mode.
12
..
.-7
10T
8-
LO
S6-
2-
TA
0.5
1
k [2it/a]
Figure 2-2 [100] Phonon dispersion of germanium from lattice dynamics. Two
degenerate transverse acoustic branches, one longitudinal acoustic branch, one
longitudinal optical branch and two degenerate transverse optical branches.
23
Autocorrelation of Normal Mode Energy
0.21
0.15
0)
*--6.69THz
Fourier Transform
E
<0 .5
0.1
0.05
N
0
z
0
0o
S
0
2
50
100
150
-5 Time s
k*1/
S0.8
5
10
15
20
Frequency fTHzI
MD Simulation Data
-Fitted Curve
-
E
0.6
xp(4
N! 0.4
cc
E
05s
00
50
0
05.
1
1.5
2
100
Time rosl
150
200
Figure 2-3 From autocorrelation of normal mode energy to phonon lifetimes
2.3 Results and Discussion
2.3.1 Phonon Lifetimes
We run five independent simulations at T=300K and then take the average. The [100]
relaxation times are shown in Fig. 2-4. Similarly, we obtained the relaxation times for
the [110] and [111] directions. The error bars denote the standard deviation from five
runs. Acoustic relaxation times exhibit strong frequency dependence at the low
frequencies. For low frequency modes, their lifetimes follow the (o-2 , consistent with
Klemens' prediction. 2 For higher frequency acoustic modes and optical phonons,
however, their lifetimes differ significantly from this trend. This reiterates the
importance and necessity of extracting the detailed phonon properties. In addition,
acoustic relaxation times are about one order of magnitude higher than optical
relaxation times.
24
TA
LA
E
2
102
C
0
C
0
101
101
100
Frequency [THz]
10
0A
TO
CL
102
E
L 10
C
0 101
C
0
101
7
10
9
8
11
Frequency [THz]
Figure 2-4 Phonon lifetimes of the transverse acoustic (TA), longitudinal acoustic
(LA), transverse optical (TO) and longitudinal optical (LO) modes in the [100]
direction
2.3.2 Thermal Conductivity Validation
The thermal conductivity is first retrieved using the BTE approach by integrating over
the whole spectrum as expressed in equation (2.4),
k
"d
"CP
P
25
2.--dv
(2.4)
dE
where C(v) =-=
dT
h v,. D(v) -
df0
dT7,
in which
fo
is the Bose-Einstein statistics and
v is the group velocity for each mode which can be calculated from the derivative of
the phonon dispersion. Note that D(v) is the polarization dependent density of
states instead of the overall density of states. We calculated the polarization dependent
density of states, as shown in Fig. 2-5, using lattice dynamics by sampling the whole
first Brillouin zone with 50 x 50 x 50 k-points and we count the number of states in
each frequency interval for each branch. The thermal conductivity obtained from the
BTE approach is 171.5 W/m/K.
0.-02
.i)
-- TA1
- ft-TA2
i
-lLA
0.015
LO
NI
TO1
a)
TO2
0.01
E
0
&
---
E
-a
I
tILI
t!
0.005
0
)
2
8
4
6
Frequency [THz]
1
10
Figure 2-5 Phonon density of states for each polarization: TAl, TA2, LA, TOl, T02
and LO
To validate the results, Green-Kubo simulations12
44
based on linear response theory
are performed. The thermal conductivity is computed via an autocorrelation of the
heat flux using EMD.
For each ensemble, we run 5 ns and calculate the
auto-correlation of heat flux in 200ps. By taking ten ensemble averages, the thermal
conductivity was found to be 175.5 W/m/K as shown in Fig. 2-6.
26
300
E 250200 -
Ensemble Average
150-
.
0
-100-
50-
-
.c
0
0
50
100
150
Time [ps]
200
250
Figure 2-6 Ensemble average of EMD simulations using Green-Kubo formula
The agreement of the two approaches (171.5 W/m/K from BTE and 175.5 W/m/K
from Green-Kubo formula) confirmed the reliability of the calculation procedures.
However, both values are much higher than the experimental result of 60.2 W/m/K
.
due to the inaccurate phonon properties given by the Stillinger-Weber potential22
Although the absolute value is too high, it is still worth looking at the relative values
and defining the contributions from different polarizations.
2.3.3 Contribution from Different Phonon Modes
As shown in Fig. 2-7, the thermal conductivity in the [100] direction is largest while
that in the [111] direction is smallest. Moreover, a major obstacle to the analytical
study of phonon-phonon scattering has been the relative scaling of the contributions
from different polarizations. Fortunately, our BTE approach could provide a clear
picture of the contributions. The LA mode comprises about 40%, while the TAl and
TA2 modes comprise about 50% and the LO comprises about 10%.
With all the relaxation times, we could calculate the phonon mean free paths in the
frequency domain. Thus, the accumulative contribution to the thermal conductivity
from different phonon mean free paths can be determined as shown in Fig. 2-8. At
room temperature, phonons with MFPs between 100 nm and 10 micron, comprise
about 80% of the thermal conductivity.
27
[100]
M[110]
0[111]
100
-
a
50
-
O Average
E
------
0
LA
50.00%
TA1
TA2
TO1
LO
L
40.70%
'
~ ~
T02
Average
22.40% 26.20%
9.45%
0.00%
---J-
LA
TA1
TA2
H-
0.58% 0.25%
LO
TO1 T02
Figure 2-7 Contributions from different crystallographic directions and different
polarizations
1n"
F 10
e
Q
20
300K
k
20
Ns
10
10-2
-02
Mea Fuue PaM (pm)
Figure 2-8 Thermal conductivity accumulation with respect to the phonon mean free
paths
28
2.4 Conclusion
Normal mode analysis is applied to calculate phonon lifetimes while nonlinear
dispersion and temperature dependent anharmonicity are included in the modeling.
The spectral dependence of phonon transport prosperities was fully detailed.
Contributions from different polarizations were provided based on the relaxation
times extracted along different polarizations: LA ~ 40%, TA1+TA2 ~ 50%, LO -10%.
Contributions from phonon mean free paths were exhibited: Phonons with mean free
paths between 1 00nm and 10 micron comprise about 80% of the thermal conductivity.
While these results provide detailed phonon transport properties, it should be
mentioned that the empirical potential limits the accuracy of a quantitative prediction
and thus the results should be taken more qualitatively.
29
30
3. Phonon Conduction in PbSe,
PbTe and their alloys Using
First-Principles Calculations
3.1 Introduction
Although the normal mode analysis is able to provide mode-dependent properties
within the classical molecular dynamics framework,
the empirical potentials
employed in these studies put question marks on the quantitative predictions. Since
empirical potentials are fit to experimental properties of materials i.e., crystal structure,
elastic constants, these potentials do not always yield accurate predictions for the
thermal properties of a specific material. Therefore, there is no assurance that they
would deliver an accurate microscopic picture such as for the mode-dependent
phonon transport properties, as the germanium example shows in the previous
chapter.
The interatomic potential can be obtained directly from the electron charge density via
density functional theory (DFT) without any adjustable parameters, thereby providing
a most reliable way of computing the lattice thermal conductivity. However, ab inito
MD simulations 23, 24 use
the forces computed
on-the-fly by DFT and are
computationally expensive. The alternative way is to extract the interatomic force
constants from DFT calculations for a limited number of atomic configurations and
then to conduct further calculations. After DFT calculations, one can either obtain the
force constants from reciprocal space calculations based on density functional
perturbation theory (DFPT) 25, 26 or from real space calculations by fitting the
force-displacement data in a supercell with a polynomial potential 21,28. Despite that,
the real space approach is simpler yet less precise, and both approaches are accurate
enough to reproduce the experimental results for the lattice thermal conductivity.25,28
31
Thermoelectric materials are of great interest for their potential in converting heat into
electricity 45-49. The efficiency of thermoelectric power generators is determined by
the dimensionless figure-of-merit
zT ( zT = S 2 o.T / k , where
S is the Seebeck
2
coefficient, a is the electrical conductivity, S a is the power factor and k is the
thermal conductivity). First-principles calculations on some thermoelectric materials
show that phonons have a wide mean free path (MFP) distribution, and hence
28
relatively large nanostructures can reduce their lattice thermal conductivity , 32, 49
Semiconducting
lead
chalcogenides,
such as PbSe and PbTe,
are attractive
thermoelectric materials for intermediate temperature (600-800 K) applications47.
Significant efforts have been made to enhance the zT value of PbTe 46-5. By
introducing resonant states, TI doped p-type PbTe resulted in a high zT value of 1.5
at 773 K48. Non-resonant doping can also lead to zT-1.3 around 700 K in K or Na
doped p-type PbTe 3 . Through band engineering to converge the valence bands, an
extraordinary zT value of 1.8 at about 850 K was reported for doped PbTei.,Sex
alloysso.
Heremans et al.' observed an enhancement of the Seebeck coefficient in
PbTe with nanograins. As the sister material of PbTe, PbSe has received much less
attention although Se is more abundant and PbSe may offer an inexpensive alternative
to PbTe especially for high temperature power generation. A recent calculation by
Parker and Singh 56, predicted that heavily doped PbSe may reach zT ~ 2 at 1000 K
due to the flattening of the valence band. The experiments5 7,5 8 later reported that the
zT values could reach 1.2 and 1.3 at 850 K for heavily doped p-type and Al doped
n-type PbSe, respectively.
Past efforts in increasing the zT of PbTe and PbSe have mostly been based on
improving the power factor S 2 .. Another approach to improve zT -is to reduce the
lattice thermal conductivity without substantially sacrificing the electronic properties.
Previous studies 5 9-6 1 demonstrated the effectiveness of the nanostructuring in
suppressing the lattice thermal conductivity and thus improving zT. Most of the
recent experimental studies on the strong reduction of the lattice thermal conductivity
in nanostructured PbTe 54' 55 emphasized the importance of dislocations, nanoscale
precipitates and strain while pointing out that the mere presence of nanostructuring
cannot sufficiently increase the phonon scattering. He et al.5 2 found that not all
nanostructures
favorably
scatter
phonons.
A
necessary
condition
for
the
nanostructures to be effective in scattering phonons is to have their characteristic
lengths, such as nanoparticle diameter and/or interparticle spacing, to be comparable
or less than the MFP. Recent first principles calculations have shown that the MFP
distribution is much narrower for PbTe3 3 , and thus, further characterizations of the
distributions and the associated detailed heat conduction of lead chalcogenides are
important for better materials' design. For example, the extracted MFPs from our
32
calculation can be combined with the Monte Carlo sampling of phonon free paths 62 to
predict the thermal conductivity of the nanostructures of the lead chalcogenides.
Besides nanostructuring, alloying may be another approach to reduce the lattice
,
thermal conductivity. Previous experimental and theoretical studies on Si-Ge alloys 29
63 have found a dramatic decrease in the lattice thermal conductivity from pure Si
and
Ge. There are still few reports on PbTe.xSe., and they only cover partial composition
ranges (x<0.3). Based on the limited experimental data on the bulk PbSe-PbTe alloy6 4
p-type PbSe-PbTe alloy 0', 6s, and PbSe-PbTe nanodot superlattice66, the reduction in
the lattice thermal conductivity is mild compared to that in Si-Ge alloys. The first
principles calculation of the lattice thermal conductivity for PbSe-PbTe alloys over
the whole composition range would allow us to better estimate the impacts of
alloying.
Despite the highly symmetric rock-salt structure of PbSe and PbTe, the lattice thermal
conductivities reported in experiments were as low as 1.7-2.2 W/mK at 300 K4758
67-69. The first principles calculations are useful to gain insight into the
low heat
conduction, with the capability of accurately capturing the transport properties of each
phonon mode, including the optical modes. In most bulk materials, the optical
phonons are ignored for the lattice thermal conductivity calculation 70 . For instance,
the optical phonons contribute only 5% of the lattice thermal conductivity in bulk
silicon at room temperature
8,25,28,71.
When the system size reaches the nanoscale,
the optical phonons can contribute about 20% as discussed in Chapter 472. Another
perspective to examine the importance of optical phonons is the acoustic-optical
scattering, as described by Ward and Broido 73 . They removed the optical phonons in
their calculations and observed an over three times increase in the lattice thermal
conductivity for Si. The large anharmonicity of optical phonons was emphasized by
the Oak Ridge group in PbTe to address the low thermal conductivity74.
In this chapter, we explore the detailed phonon transport properties in PbSe and PbTe
to gain more guidance for the thermoelectric applications of these materials systems.
We first calculate the harmonic and anharmonic force constants from density
functional perturbation theory (DFPT) calculations75-77. The anharmonic phonon
lifetimes are then obtained based on Fermi's golden rule. The total lattice thermal
conductivity is determined under the relaxation time approximation by summing up
the contribution from each mode. Our results are validated by comparing them with
the reported experimental data. We present a detailed analysis and we quantify
contributions from different phonon modes to the thermal conductivity for both PbSe
and PbTe, and discuss the importance of optical phonons and the potential impacts of
33
nanostructuring and alloying on further lattice thermal conductivity reduction in both
materials systems. The results indicate that: 1) the optical phonons are important not
only because they directly comprise over 20% of the lattice thermal conductivity, but
also because they provide strong scattering channels for acoustic phonons, which is
crucial for the low thermal conductivity; 2) nanostructures of less than -10 nm are
needed to reduce the lattice thermal conductivity for pure PbSe and PbTe; 3) alloying
should be a relatively effective way to reduce the lattice thermal conductivity.
3.2 Methodology
Accurate interatomic force constants (IFCs) are crucial for the lattice thermal
conductivity calculation. We adopt the DFPT approaches for both PbSe and PbTe.
DFPT approaches have demonstrated unparalleled accuracy in reproducing the lattice
thermal conductivity 25 , 29, 30 and are sufficiently computationally affordable for the
simple rock-salt structure with only 2 atoms in the primitive cell. More specifically, in
our work, both the harmonic and anharmonic IFCs are obtained based on DFPT
calculations implemented in the Quantum Espresso package7 1. In the ground-state
calculations, the newly developed norm conserving fully relativistic pseudopotentials
which incorporate the spin-orbit interaction (SOI) effect appropriately are chosen
under the local density approximation (LDA) for the electron exchange-correlation
potential. Through a sensitivity study of the lattice thermal conductivity with SOI and
without SOI, we find that for both PbSe and PbTe, the SOI effect is important and
fully relativistic pseudopotentials are necessary. For example, the phonon lifetimes of
all modes are noticeably larger with SOI, which results in a twice larger thermal
conductivity with SOI than that without SOI at 300K.
3.2.1 Harmonic Properties
The harmonic IFCs are obtained using the primitive cell calculation of 2 atoms. In the
self-consistent calculation of electronic properties, a Monkhorst-Pack 10x 10x 10
mesh79 is used to sample the electronic states in the first Brillouin zone and an energy
cutoff of 60 Ryd (-816 eV) is used for the plane-wave expansion to ensure the force
convergence. In the following DFPT calculation, a Monkhorst-Pack 4 x 4 x 4 q-mesh
is used to calculate the dynamical matrix at each q grid, which, through an inverse
Fourier transform to real space, gives the harmonic IFCs. The harmonic IFCs allow
computation of the dynamical matrix at any q point:
34
(3.1)
,,eiR
D,,",(q)=
VM,,m,,
where
D is the harmonic IFC, m is the atomic mass, R,
of the unit cell
is the translation vector
' , while q specifies the q th atom in the primitive cell, and a, fi
are Cartesian components. The eigenvalues of the dynamical matrix yield the phonon
frequencies and the phonon dispersion, from which the phonon group velocities can
be calculated.
3.2.2 Anharmonic Properties
There are two approaches to calculate the anharmonic IFCs in the reciprocal space.
The results from both approaches are equivalent. One approach is based on 2n+1
theorem 26,8 0 which assumes that the third order IFCs can be obtained from the first
order wave function. It is computationally effective since it does not involve the
supercell calculation, but it is relatively complicated to implement. The other
approach is to calculate the third order IFCs from the second order IFCs using a finite
difference method, which is computationally more expensive
but simpler to
implement. We use the latter approach in this study.
The third order derivatives are determined by taking the derivative of the second order
IFCs through a central difference scheme as below:
a2V
q fya~v
1'17
Oi=~
q~.
a2 V
K
P1 au/Xu7,,'
U
V 17 - au,",,7u ,,.8u ,,.. 2u1",, 2uo"
2uoa720
(3.2)
where
V
is the interatomic potential. We first perform the IF point phonon
calculation in a super cell to generate the harmonic IFCs for two different atomic
configurations namely involving displacement of an atom along positive and negative
Cartesian directions around the equilibrium position. All the required cubic IFCs are
obtained by sequentially changing the displaced atom to be any of the atoms in the
primitive cell. To ensure the accuracy of the cubic IFCs, we test three values for the
displacements. We use a Monkhorst-Pack 4 x 4 x 4 mesh 79 to sample electronic
states with the same energy cutoff of 60 Ryd (-816 eV).
35
The cubic IFCs are needed to compute the three-phonon scattering matrix elements,
which measure the strength of the scattering events and are given by
=/(
V3 (qs,q's',q"s")
i,"I-Rj.
flyi,'-RI.
8NOc9(qs)co(q's')co(qofs
x
,-afl
17 P17 P,,-
1117
e," (qs)e 17 (q's')e,,(q"s"t)
1
?
m,,m,,,m,,,,
(3.3)
where h is the Planck constant divided by 27r, N is the total number of modes in the
first Brillouin zone, and s denotes the different polarizations.
golden rule to the cubic Hamiltonian 81,82 the phonon lifetimes
By applying Fermi's
rqs
due to the normal
and umklapp three-phonon scattering processes can be expressed as
V(qs, q's',q"s")12
= 21, = 7r
x[2(nqs
- nqs,,)S(a)(qs) + w(q's') -(qs"))
qs s"
qS
+(1+nq's +nq',s)8(co(qs) - o)(q's') -(q's"))]
(3.4)
where nqs
is the Bose-Einstein distribution function
nqs =
1/(eh0p IkBT
.1)
The
conservation of momentum requires q+q'+q"=G, where G is a reciprocal lattice
vector, for which G =0 results in the normal processes and G
0 relates to the
umklapp processes. The choices of q" are limited by the choices of q and q', and
thus the summation involves only q'.
3.2.3 Lattice thermal conductivity
We compute
the lattice thermal conductivity
approximation using the well-known formula
36
based on the relaxation time
3QK N=,(3.5)V
3QNo ,,
2Tq 'coqs
aq
aT
where C2 is the volume of the unit cell and vqs is the amplitude of the group
velocity. We use a 30 x 30 x 30 q-mesh within the first Brillouin zone to ensure
convergence. Comparing the total calculated lattice thermal conductivity with the
experimental data serves as a validation of our calculations.
More importantly, decomposition of the total lattice thermal conductivity into each
mode allows us to account for the contributions from phonons with different MFPs
and polarizations, which provides insights into specific thermoelectric applications.
The phonon MFP for each mode is defined as
Aqs = vqs,
(3.6)
One way to quantify the contribution from phonons with various MFPs is to evaluate
the thermal conductivity accumulation with respect to MFPs 18,
83
. By summing the
thermal conductivity contribution coming from modes with MFPs up to A, the
cumulative thermal conductivity can be determined as follows:
1
(A)N= 3
A <A
vqsAqhoqs,
(3.7)
(3.7)
To separate the contribution among the different polarizations, we simply sum the
thermal conductivity of the modes for each polarization s as
1
2nqs
y ,qshoqs
K,= 3QNI
3.2.4 Alloy modeling
37
(3.8)
To take into account alloy effects, we use the virtual crystal approach, first introduced
by Abeles 63, where the disordered crystal is replaced with an ordered one with an
average lattice parameter, atomic mass and a set of force constants which vary
according to the composition. The mass disorder and anharmonicity are both treated
as perturbations. Garg et al.29 has applied this approach to Si-Ge alloys using the force
constants from DFPT and reached excellent with the experimental data by following
this approach.
The effective phonon scattering rate is defined as the sum of the scattering rates due to
mass disorder and anharmonicity:
1
qs
-=
1
Iqs
While the anharmonic phonon lifetimes vr,,-
(9
1
+
(3.9)
qs
are calculated in the same way as the
pure cases except for different input parameters, the harmonic phonon lifetimes due to
mass disorder is given by8:
=
qs
;r
2N
S(
q,
q's9
-
q)sg
where e is the polarization vector, and g 2(-)=
f(a-) and
2
(.)
s(a)eS(a)
f(a)[1
-m
12
(a) / m (-)]2,
(3.10)
in which
m, (a) are the concentration and the atomic mass of ith isotope of the
a atom.
3.3 Results and Discussion
3.3.1 Comparison with experimental results
38
PbSe
r-I
4
0
a)
U)
0
0
0
G
K)
L
G
X(
5
PbTe
0
3
0
00
a)
0r
a)
I-.
LL
1<
.
0
G
K
X
G
L
Figure 3-1 Phonon dispersion for PbSe and PbTe: red lines: calculated results; black
dots: experimental results from neutron scattering
Figure 3-1 shows the phonon dispersion relations of PbSe and PbTe along the high
symmetry directions within the first Brillouin zone of the primitive cell with two
39
atoms. There are six polarizations: two transverse acoustic (TA), one longitudinal
acoustic (LA), two transverse optical (TO) and one longitudinal optical(LO) modes.
The disperion of PbSe agrees reasonably well with the experimental results 8. The
splitting of the LO and TO branches at r
point, which depends on the Born effective
charges and dielectric constants, agrees with that in the experiment. The dispersion of
PbTe matches well with the experiments86 except for the LO branch. The discrepancy
for the TO-LO splitting at the F point comes from the difference in the Born
effective charges. By setting the Born effective charge to the value obtained in the
experiment (6.5 e ), the dispersion meets the experimental data while all other modes
and the total thermal conductivity changes by less than 1%. It has also been claimed
in previous work 32 , 33 that the inclusion of the LO-TO splitting has only negligible
effects. For better comparison in terms of the actual frequency range, we use the tuned
Born effective charge for the latter discussions.
Although the frequency of the TO mode at zone center matches perfectly with the
experimental value measured at room temperature, some uncerntainties exist in the
calculation of this specific mode. As found in previous studies, the TO mode at the
Gamma point is soft and directly relates to the ferroelectric ground state
87, 88
'
. The
ferroelectric mode is difficult to calculate accurately due to its strong temperature and
volume dependences, and different pseudopotentials and lattice constants lead to
different frequencies
87-89
. However, since we focus on the integrated properties of all
the phonon modes, the discrepancy of a single mode or of few modes near the zone
center with very small or even zero group velocity does not make any noticeable
change to our conclusions because they hardly carry any heat.
The dispersion relations of PbSe and PbTe are similar but do not scale with the total
primitive cell mass ratio because they have one element Pb in common. Although the
frequencies of the optical modes of PbTe drop significantly compared to those of
PbSe, the differences between the acoustic modes, especially the TA modes, are much
smaller.
40
PbSe,
2
4-
2
0
-0
-
4-,
PbTe
0 L-
0
200 400 600
Temperature [KI
800
0
200 400 600
Temperature [1
800
Figure 3-2 Temperature dependent lattice thermal conductivity of PbSe and PbTe, red
lines: calculated results; black crosses: experimental data.
We compare the calculated lattice thermal conductivities with experimental results in
figure 3-2. For both PbSe and PbTe, the calculations achieve decent agreement with
experimental values 68, 69 . The small discrepancies of PbSe between 300 K and 400 K
might come from the impurity or defect scattering in the experimental sample, which
becomes inferior to three-phonon scattering at higher temperatures. Above 400 K, the
calculated results lie on top of the experimental data. The agreement for PbTe over the
whole temperature range is excellent. The good agreement bears out the accuracy of
our approach, and the validity of the relaxation time approximation, and supports our
following discussions.
3.3.2 Comparison between PbSe and PbTe
The calculated lattice thermal conductivity of PbSe is 11% higher than that of PbTe at
temperatures of 300 K-700 K. The atomic masses of Pb, Se and Te are 207.2, 78.96
and 127.6, respectively. Se is about 40% lighter than Te, but due to the heavy mass of
Pb, the mass difference for PbTe and PbSe is only 17%. At a first glance, the mass
difference seems to fully explain the thermal conductivity difference. Yet how the
mass difference actually leads to the variance in different quantities is far from a
simple deduction, as we will show below.
41
3
10 3
C)
10
10
TA
-2
T'0) -2
2_
CL)
0
E
:r_
E
JD 10 L
-
2
10
Frequency[THz]
101
10
(A
PbSe
PbTe
10
0 -(b)
10
10
-
(a)
10
10-1
10
LA
-____
10
Frequency[THz]
LO
0~
a-
E 10
-
E 1U
10
S0
-- -
----
(d)
101 0
a
S
01
100
10
Frequency[THz]
Frequency[THz]
Figure 3-3 Frequency dependent phonon lifetimes of PbSe (squares) and PbTe
(crosses) at 300 K: (a) TA, (b) LA, (c) TO, and (d) LO.
We show the phonon lifetimes in figure 3-3. In the low frequency range, the lifetimes
of the acoustic modes exhibit a co- dependence, in agreement with Klemens'
prediction90 . The trends of the lifetimes with respect to frequencies are similar for
PbSe and PbTe. For most of the TA modes, the lifetimes of PbSe are substantially
larger than those of PbTe, while for the LA and optical modes, the lifetimes of PbSe
are not necessarily higher. This is a nontrivial observation since the anharmonicity of
PbSe is normally expected to be larger due to the larger average Griineisen parameter
reported from experiments 67. For the optical modes, the lifetimes of PbTe are
obviously larger.
42
300 0
-- TA- --
40C 0
--
E 200 0
E
0
05
0
30C 0
x
PbSe
PbTe
20C 0
100 0
10C
0
1
Frequency[T Hz]
0
2
-------
40C 0
0
2
Frequency[T Hz]
4
2
4
Frequency[T Hz]
6
300
E 200
E
30OC
0
0@
0
10
(c)
2
3
Frequency[THz]
4
Figure 3-4 Frequency dependent phonon group velocities of PbSe (squares) and PbTe
(crosses) at 300 K: (a) TA, (b) LA, (c) TO and (d) LO.
With heavier mass, PbTe was anticipated to have smaller group velocities in general.
Nevertheless, figure 3-4 shows that for the TA modes, the group velocities of PbSe
and PbTe are almost the same because of the closely matched acoustic dispersions.
Noticeably, these TA modes are fairly soft with maximum value around 2000 m/s. In
terms of the LA modes, the group velocities of PbSe are moderately higher. Between
1 THz and 2 THz, several TO modes of PbTe possess exceptionally high group
velocities (>3500 m/s) and even higher than the TO modes of PbSe. For the LO
modes, the group velocities of PbTe are perceptibly smaller than those of PbSe.
43
%PbSe
TA1
PbTe
---TA2
LA
E 0.6
E 0.6-
0.5
0.5
---- TO2
0.4
LO
0
U
0.3-
: 0.3
0
0.2
-i 0.2 F
-FU
E
-
E 0.1
00
600
500
400
Temperature[K]
.1
700
------
-
00
400
500
600
700
Temperature[K]
Figure 3-5 Thermal conductivity from different polarizations (TAl, TA2, LA, TOl,
T02 and LO) versus temperature for PbSe and PbTe
Integrating the transport properties over the entire first Brillouin zone, we can obtain
the polarization dependent thermal conductivities as shown in figure 3-5. Remarkably,
over a wide temperature range of 300 K to 700 K, the three acoustic branches for
PbTe contribute equally and three optical branches contribute almost evenly to the
thermal conductivity of PbTe. In the case of PbSe, by contrast, the contribution among
acoustic and among optical modes are all distinguishable.
Considering all the differences in phonon frequencies, lifetimes, and group velocities,
it is impossible to identify the decisive one source of the differences between PbTe
and PbSe, despite the simple mass difference argument.
3.3.3 The importance of optical phonons
The normalized optical phonon contributions can be calculated by adding TO and LO
modes together. For the whole temperature range considered (300 K-700 K), the
contributions of optical phonons remain about 25% for PbSe and 22% for PbTe.
These findings are rather surprising especially considering the simple rocksalt crystal
structures of these two materials and the fact that only half of the modes are optical
phonons. Our calculations demonstrate that optical phonons are not always negligible
even in simple crystalline bulk materials.
44
without
acoustic-optical
scattering
E
1
_,010
PbSe
PbTe
with
acoustic-optical
scattering
E
PbSe
10
PbTe
100
200
300
400
500
600
700
800
Temperature [K]
Figure
3-6
Temperature
dependence
of lattice thermal
conductivity
without
acoustic-optical scattering: PbSe (black dashed line), PbTe (red dashed line) and with
acoustic-optical scattering: PbSe (black solid line), PbTe (red solid line).
Moreover,
optical phonons provide
important scattering channels for acoustic
phonons and are essential for the low thermal conductivity of PbSe and PbTe. By
removing the acoustic-optical scattering, the thermal conductivity of PbSe/PbTe
increases dramatically by a factor of six/five over the entire temperature range
investigated here (300 K to 700 K) as shown in figure 3-6. This difference is about
twice larger than that of Si73 . Due to the softening of the optical phonons, the
longitudinal acoustic and transverse optical phonons are strongly coupled, as observed
in PbTe by Delaire et al.74 in the experiment, and by Shiga et al.3 3 in the calculation,
and help lower the lattice thermal conductivity.
3.3.4 The potential impacts of nanostructuring
45
2.
E
2
PbTe
0
1 5
E
SAK-PbSe
Te
1PbSe-V
05
E
0 .57
--,
.-
.
___-
101
100
Phonon Mean Free Path[nm]
10
102
Figure 3-7 Cumulative thermal conductivity with respect to phonon mean free path at
300 K for PbSe (red dashed line), PbTe (black solid line) and PbTeo.5 Seo 5 (blue dotted
line)
The calculated cumulative thermal conductivity with respect to phonon mean free
paths (MFPs) at 500 K is shown in figure 3-7. The total accumulation for PbSe keeps
increasing as the MFPs increase while the accumulation for PbTe gradually
approaches a plateau after the MFPs reach 10 nm. Phonons with MFPs smaller than
10 nm comprise around 80% of the lattice thermal conductivity for PbSe and about
90% for PbTe. In other words, even if the interface backscattered all the ballistic
phonons, the nanostructuring with length scale 10 nm would only potentially reduce
the thermal conductivity by 20% for PbSe and 10% for PbTe at the most. Therefore,
to
significantly
reduce
the
lattice
thermal
conductivity
in
these
materials,
nanostructures with characteristic length smaller than 10 nm are required. Therefore,
smaller scale inhomogeneities and alloying might be more effective in reducing the
lattice thermal conductivity.
46
3.3.5 The potential impacts of PbSe-PbTe Alloying
2.5
L
L-
-E
300K
500K
2
-a 1.5
0
E
a)
1-
0.5'
0
0.2
0.4
0.6
PbTe ixSex
0.8
1
Figure 3-8 Calculated composition dependence of the lattice thermal conductivity in
PbTei.xSex at 300 K (solid line) and 500 K (dashed line)
We plot the lattice thermal conductivity of different composition of PbTei-xSex alloy
in figure 3-8. At x = 0.5, we obtain a maximum decrease in k of 30% (1.46 W/mK)
compared to the average lattice thermal conductivity of PbSe and PbTe (2.1 W/mK) at
300 K. There is no sharp decrease feature in the dilute alloy limit as reported in the
Si-Ge alloy 29 due to the small difference in acoustic impedance between PbSe and
PbTe. As the temperature increases, the phonon-phonon scattering becomes dominant,
and the influence from alloy scattering becomes less important. Therefore, comparing
300 K with 500 K, the reduction of lattice thermal conductivity is greater at 500 K.
The mean free path accumulation of PbTeo.5 Seo.5 is plotted in figure 3-7. The phonons
with high frequencies and short mean free paths are strongly scattered by mass
disorder, while the phonons with small frequencies and long mean free paths are
much less influenced. This leads to a redistribution among different mean free paths
and consequently a shift in the accumulation curve results. Since the accumulation
47
curve of PbTeo.5 Seo.5 is considerably more flat above 10 nm, similar to PbSe and PbTe,
nanostructuring on alloys could not push down lattice thermal conductivity by a
significant amount.
Taking into account the practical difficulty in introducing nanostructures at the scale
of 10 nm and the potential reduction in the lattice thermal conductivity, the simple
alloying approach is more promising in reducing the lattice thermal conductivity
because experimentally grain growth in these materials is a problem with annealing.
3.4
Conclusion
We perform first-principles calculations to detail the spectral phonon transport
properties of PbSe and PbTe. We first extract harmonic and anharmonic force
constants from density functional perturbation theory calculations within a supercell.
We then extract the phonon lifetimes based on Fermi's golden rule and we then
compute the thermal conductivity under the relaxation time approximation. The total
lattice thermal conductivities quantitatively agree with the experimental results.
Comparison of mode-dependence properties between PbSe and PbTe suggests that the
transport properties of these two sister materials are similar in principle but different
in specifics. The optical phonons not only directly contribute a considerable amount
to the total lattice thermal conductivity of bulk PbSe and PbTe but also serve as
important scattering channels for acoustic phonons. Both PbSe and PbTe possess very
low lattice thermal conductivities, which is attractive for thermoelectric applications.
Nanostructuring, however, would be difficult to further reduce the lattice thermal
conductivity unless the characteristic lengths of the nanostructures could be reduced
and maintained to much less than 10 nm.
Alloying, on the other hand, has
advantages over nanostructuring in reducing the lattice thermal conductivity. The
parallel studies of these two materials provide insights into the phonon properties and
may help design better thermoelectric materials.
48
4. The Importance of Optical
Phonons in Nanostructures
4.1 Introduction
It is generally understood that optical phonon contributions to the thermal
conductivity k are small and negligible in bulk materials because of their short
lifetimes and low group velocities. Several recent theoretical efforts 18 ,2 5,71 that fully
detail the spectral phonon transport properties of bulk silicon (Si) all concluded that
the contribution of optical phonons to k is around 5% at room temperature, regardless
of the method used. Chapter 3, however, highlights the importance of optical phonons
for bulk PbSe and PbTe mainly due to the softening of their TO modes and calls for
attention to optical phonons in certain bulk materials. The importance of optical
phonons in nanostructures, on the other hand, is more universal. When the system size
decreases, the contributions
of optical phonons to heat conduction become
increasingly important. While acoustic phonons are strongly scattered at boundaries
and interfaces, optical phonons have short mean free paths (MFPs) and are scattered
much more strongly inside the nanostructures than at the boundaries. Such a
difference in scattering leads to a rebalance of the relative importance of optical
phonons and acoustic phonons to the thermal conductivity of nanostructures. In this
chapter, we examine this shift using Si as a test case, because several puzzling
experimental results for Si nanowires 36 ' 91-93 have yet to be explained satisfactorily,
despite several theoretical and computational studies94-99. Based on first principles
calculations,
we first examine
the cumulative
contributions
to the thermal
conductivity in bulk Si by phonons with different MFPs and polarizations, namely
longitudinal acoustic (LA), transverse acoustic (TA), longitudinal optical (LO) and
transverse optical (TO). We then model the thermal conductivity of Si nanowires
based on the spectral properties in bulk Si and evaluate the contributions of optical
49
phonons as a function of nanowire diameter over a wide temperature range. Our
modeling results show that around room temperature optical phonon contributions can
increase to 18% when the nanowire diameter is reduced to 20 nm.
4.2 Methodology
The detailed methodology and calculation procedures for bulk Si are presented
27,32
. In short, electronic structure calculations based on density functional
elsewhere
theory were applied to extract interatomic force constants via the direct displacement
method.2 7 The cubic anharmonic force constants lead to three-phonon lifetimes. The
phonon lifetimes rk,p due to the normal and umklapp three-phonon scattering
processes have been calculated for each polarization p and each k point sampled in the
first Brillouin zone based on the scattering rate determined from application of
Fermi's golden rule. The thermal conductivity is then computed from the relaxation
time approximation using the well-known formula
(4.1)
1~~k v2_
kP
= ho~T~hA;p
3 QNk t, kT
K~
where 2 is the volume of the unit cell and nk,p is the Bose-Einstein distribution
function. The phonon MFP for each mode is defined as
(4.2)
Ak~p = Vk~prk~p
By sorting the thermal conductivity contribution of each mode according to increasing
MFPs18, 83, the polarization dependent and total cumulative thermal conductivity can
be determined by equation (4.3a) and (4.3b), respectively, as follows:
K,(1A) =
hCo
kp<A
1(A)=
vpAk)wk
k
3QNk
K(A)
=
EZKP(A)
P
50
_
a
(4.3a)
(b
(4.3b)
As the system size decreases, it is often found that a large thermal conductivity
reduction occurs as the nanometer regime is approached. 40' 83, 100-104
Past studies
suggest that this reduction is due mainly to increased boundary scattering. The
effective phonon lifetimes including boundary scattering can be estimated by adding
the scattering rates due to anharmonic and boundary processes:
1
k,p
1I
1
+1I
p-p
B
Tkp
Tk,p
(44)
where for nanowires with diameter d, the Casimir limit gives
1
v
(4.5)
d
kp
which assumes purely diffuse scattering at the boundary. Alternatively, by solving
Boltzmann's equation for an infinite wire, Sondheimer also arrived at a similar result,
and generalized it to the case of boundaries with scattering continuously going from
specular to diffuse. 28
51
4.3 Results and Discussion
160 r-
(a)
E
LOTA1to
140 2 10120
A
5--
-
4-'
100
20
0
0.4
:3
0
-
0
80
TA2
40
60
LA
60
TO1
p.
.0.2-
TA2
T02
40
LO
-
0 I-
-
20 H
0
2
6
4
A [nm]
0
100
.m...
rX
--
8
TA1
ItLA
.
U
Total
I
-A[nm],b
0.4
(b)
-'
E
-c
f
10
102
1
10 3
10 4
Mean Free Path A [nm]
Figure 4-1 Cumulative thermal conductivity with respect to MFPs at 277K from the
18 x 18 x 18 k-mesh data; Inset (a)-Inset (b) Zoomed-in figures for MFP range of (a)
0-70nm and (b) 0-9nm, respectively
The detailed cumulative contributions to the thermal conductivity by phonons of
different MFPs and polarizations at 277K are shown in Fig. 4-1. Note the slope
Inset (a) in
change in the thermal conductivity when the phonon MFPs are -30nm.
Fig. 4-1 shows that this sharp increase is due to the rapid growth in contributions from
the TA phonons. Contributions to the total thermal conductivity for phonons with a
MFP less than -30 nm are mainly due to LA and optical phonons. Although TO
phonons have very short MFPs (less than 9 nm), they are dominant contributors (inset
(b)) to the thermal conductivity in this MFP range due to their large density of states
(DOS).
52
Fig. 4-1 also leads to additional insights on the contributions of acoustic modes to the
thermal conductivity. Below 45 nm, LA modes contribute more to the thermal
conductivity than TA modes. These small MFP modes correspond to phonons near
the first Brillouin zone edge where the group velocities of TA modes are lower than
those of LA modes. The sharp increase of the TA2 accumulation between 40nm and
I00nm is due to larger DOS of TA2 modes.
60
E
7
50-
/~*
~1l5nm
40I
-o
30
a
56nm
0
&
a)
X
/ X*
100
0
37nm
-- - - - - - -- - - -...
.
20 :
50
100
200
250
150
Temperature [K]
300
350
Figure 4-2 Thermal conductivity of silicon nanowires for d=37nm, 56nm and 115nm,
lines: calculated results; crosses, squares and stars: experimental results
Considering the properties of different polarizations in bulk Si presented above, we
investigate how the contributions change in the context of nanowires. To validate our
model, we first compare the thermal conductivities of nanowires with experimental
results. To best represent the experimental sample, we include isotope scattering and
add the scattering rate according to Matthiessen's rule as below:
53
1
1
TkpTP+T
Tk
4
1
B
k,p
+
1
and
1
= Aco4
k,p
+TITi
(4.6)
k~p
Tkp
s' is analytically determined from the isotope concentration
.
where A = 1.32 x 10
k,p
+
Using Matthiessen's rule to include boundary scattering is an approximation and there
are other studies solving BTE directly 0 . Without any fitting parameters, we have
obtained decent agreement with experimental data3 6 for d=l 15nm, 56nm and 37nm as
shown in Fig. 4-2.
The good agreement between experiment and theory supports the
following discussions based on the boundary scattering effect. We do not include
results for the nanowire of 22nm since we could not explain the experimental results
after taking into account the optical phonon contributions.
7-
(a)
67
E
TA2
5
> 4--
E 2-
TO 1&T02
;
0~
0
200
800
600
400
Temperature [K]
54
1000
C
0.25
-b-
o
1 Onm
0.2
0
0.1
00nm
~
|
-
1
~
~
C 0.15 --
1mm
0.0
0
0
200
600
400
Temperature [K]
800
1000
Figure 4-3 Thermal conductivity from different polarizations versus temperature for
d=20nm;
(b)
normalized
optical
phonon
contributions
to
the total
thermal
conductivity versus temperature for d= 1 Onm, 20nm, 1 00nm and 1mm.
The temperature dependence of the relative phonon contributions from the different
polarizations is shown in Fig. 4-3(a) for the d=20nm nanowire. At temperatures below
100K, the three acoustic thermal conductivity contributions exhibit a T 3 dependence,
the same as the temperature dependence of the specific heat, due to the dominance of
boundary scattering. The two TA modes grow more rapidly than the LA modes since
the specific heat of the TA modes rises more rapidly due to their lower frequency and
higher DOS. The LO modes contributions are noteworthy over most of the
temperature
range considered.
The TO modes,
however, make
a negligible
contribution for this temperature range.
Adding LO and TO modes together, the normalized optical phonon contributions are
shown in Fig. 4-3(b) for different nanowire diameters. As the diameter decreases, the
optical phonon contributions relative to the acoustic phonons become larger as
expected.
Between OK and 300K, the optical phonon contributions increase due to
the increase in their specific heat.
55
)
-
3025
20- 20
TA2
-
TA1
15-
LA
10
LO
---
5
TO1 &T02
0
wan
0
50
200
100
150
Diameter d [nm]
aw
250
0.25
M
300
(b)~
0.2-
0.15 \
400K
277K--
0.1
0.05 .
-L
t
--_r , ,I
0L
0
1OOK
-
50
-100
177K
-
150
200
Diameter d [nm]
250
300
Figure 4-4 (a) The thermal conductivity from different polarizations versus diameters
at 277K, (b) the normalized optical phonon contributions to the total thermal
conductivity vs. diameters at 100K, 177K, 277K, and 400K
Fig. 4-4(a) depicts the thermal conductivities from different polarizations for Si
nanowires of different diameters, varying from 5nm to 300nm, at 277K. Acoustic
phonon contributions increase with increasing diameter in the plotted diameter range
(5-300nm), while the optical phonon contributions saturate around 100 nm due to
56
their lower MFP values arising from three phonon scattering processes. Note that the
LO modes contribute much more significantly than the TO modes. The normalized
contributions with the total thermal conductivities at different temperatures are shown
in Fig. 4(b). At 277K, the optical phonon contributions grow from below 10% to 21%
when d decreases from 300nm to 5nm. It suggests that optical phonons can have
significant impact on the thermal conductivity in nanostructures, especially at
temperatures on the order of the Debye temperature or higher.
4.4 Conclusion
In summary, we have used the relaxation times determined from first principles
derived force constants to calculate the thermal conductivity of bulk Si and Si
nanowires. Detailed analysis of the respective contributions shows that optical
phonons comprise up to 20% of the total thermal conductivity in Si nanowires around
room
temperature,
despite
conventional
contributions are usually negligible.
wisdom
which
suggests
that their
This finding brings to light the importance of
optical phonon contributions to heat conduction in nanostructures.
Although Si is
taken as the model material, we expect that similar behavior should exist in many
other materials.
57
58
5. Phonon Transmission across a
Single Si/Ge Interface using the
Green's Function Method
5.1 Introduction
The reduced lattice thermal conductivity observed in many nanostructured materials
has significant implications for applications from thermoelectric energy conversion to
microelectronics thermal management.
The Boltzmann transport equation (BTE)
can be used to accurately model the phonon transport in nanostructures if the input
parameters, such as the phonon mean free paths and interfacial transmission, can be
properly represented.
In recent years, excellent progress has been made in
computing the mode-dependent phonon mean free paths in bulk materials using
first-principles approaches 28, 32,33,106 as covered in Chapter 3.
In contrast, research
on phonon transmission across interfaces is still limited and prior first-principles
studies of phonon interfacial transport are rather scarce.
First-principles based
approaches have been recently applied to nanotubes107108 ; however, their applications
to interfaces between bulk 3D materials are significantly more demanding due to the
large number of transverse wavevectors required.
Interface roughness due to atomic disorder and defects commonly occurs at interfaces
during material synthesis.
A thorough understanding of the influence of interface
roughness on phonon transport is crucial for surface engineering and improved device
design.
It is generally accepted that interface roughness is a very important driving
mechanism for thermal conductivity reduction in different nanostructures such as
nanowires and superlattices.
However, it is not clear how interface roughness affects
interfacial phonon transmission.
et al.1
09
Using a lattice Green's function formalism, Fagas
found that the phonon transmittance is strongly dependent on phonon
59
frequency and the disorder correlation length by varying the atomic masses in a
two-dimensional disordered atomic layer. Following the same approach, Zhao and
Freund" 0 studied the phonon scattering at a rough interface induced by atomic mixing
between two FCC lattices, and found that the transmittance is insensitive to the
roughness parameters.
Using molecular dynamics (MD) simulations, Sun and
Murthy"' focused on the transmittance change as the roughness thickness was
increased.
For long wavelength phonons, they concluded that the transmittance is
For mid-range wavelength phonons, the
independent of roughness thickness.
transmittance is reduced as roughness thickness increases but eventually saturates to
become independent of the roughness. Nevertheless, the above studies have not
drawn a comparison between the ideal and rough interface, and furthermore, the
conclusions were derived from empirical potentials.
112
dynamics model, Kechrakos
Using a simplified lattice
found that the interface conductance can be enhanced
by as much as a factor of three for highly mismatched materials.
The calculation
only included one monolayer roughness and one branch mode.
Stevens et al.113
observed that interface mixing improved thermal transport by nearly a factor of 2
through non-equilibrium molecular dynamics (NEMD) simulations.
using NEMD, English et al.
114
Most recently,
found that by sandwiching an intermediate layer
between two dissimilar materials, the interfacial thermal conductance
enhanced compared to that of the two dissimilar materials.
unable
to unveil
any
information
about
the
can be
NEMD, however, is
mode-dependent
Additionally, an empirical potential was used in their simulations.
transmission.
The behavior of
different phonon modes at a rough interface using reliably accurate force constants
would be preferable, and as we will show in this paper, results can differ by up to 50%
depending on the choice of the force field.
Phonon interface transmittance is critical in determining the interfacial thermal
resistance. Phonon interface transmittance models have yet to reliably predict
experimental observations.
There are two widely used models for the phonon
transmittance at an interface: the acoustic mismatch model (AMM) 115 and the diffuse
mismatch model (DMM)1 16.
As a continuum model, the AMM assumes that
phonons undergo specular reflection or transmission at the interface.
This model
is
valid in the long-wavelength limit, where due to their small details compared to the
incident phonon wavelength, interfaces are seen as sharp.
The DMM, on the other
hand, assumes not only purely diffuse scattering at the interface, but also an
equivalence between phonon reflectance from one side to the transmittance from the
other side.
This model, as opposed to AMM is valid for very rough or dirty
interfaces and short wavelength phonons.
Neither AMM nor DMM consistently
predict interface thermal boundary resistance.
60
Using molecular dynamics (MD)",
17-2,
wave-packets can be created and the phonon transmittance can be obtained by
tracking the energy transmitted and reflected after encountering an interface.
Although easy to implement, it is computationally expensive since one separate MD
simulation is needed for every incoming phonon mode, although using the multiple
phonon wave packets reduces computational intensity' 1.
Additionally, MD
simulations cannot capture wide angles of incidence because it requires a large lateral
size that is difficult to achieve.
Linear lattice dynamics (LD) calculations 122-125 have
been performed to extract the mode-dependent phonon transmittance by solving the
reflected and transmitted wave functions subject to boundary conditions. However,
this method can be difficult to implement for complex atomic structures.
As an
alternative and more straighforward approach, Green's function methods dedicated to
solve for the response from a point source perturbation are employed to compute the
phonon transmission function that can be easily related to transmittance as described
in Sec. II.
The Green's function approach has been described thoroughly for
'
transmission function calculations in electron transport by Datta 126 . Mingo et al.12177
128 applied the approach to deal with phonon transport
within an elastic scattering
domain in nanowires and referred to this method as the atomistic Green's function
(AGF) method.
Later, Zhang et al.1 29 extended the method to phonon transport in
3D structures. They calculated the phonon transmission across the Si-Ge interface
using an empirical interatomic potential and investigated the strain effect on
interfacial transport.
A general formulation and full derivation have been detailed by
'
Zhang et al. 26 and Mingo' 30 31 1 . Several other studies utilize the same framework 0 7
108, 132-134 including the only first-principles based calculations with the AGF
method
in 1D structures 10'108.
Here we incorporate the first-principles force constants into
AGF and demonstrate the importance of using accurate force constants.
Without any
fitting to experimental data, the force constants from first-principles calculations
demonstrated the ability to accurately reproduce the lattice thermal conductivity of
bulk materials
25 28 32 33
, , , ,106.
These force constants can also improve the quantitative
prediction for interfacial phonon transport.
In this Chapter, we employ the AGF method to study the interface roughness
stemming from atomic mixing between Si and Ge interfaces. Although thermal
conductivity reduction in nanostructured materials can usually be described by
phonon scattering due to interface roughness, we show how a Green's function
method in conjunction with the Laudauer formalism suggests that interface roughness
induced by atomic mixing can increase phonon transmission and interfacial thermal
conductance.
This is the first attempt to incorporate first-principles force constants
derived from ab initio density functional theory (DFT) into Green's function
calculation for infinitely large 3D crystal structure.
61
We also demonstrate the
importance of accurate force constants by comparing the phonon transmission and
thermal conductance using force constants obtained from the semi-empirical Stillinger
and Weber (SW) potential, and first-principles DFT calculations.
5.2 Methodology
2
The detailed methodology of AGF has been presented elsewhere '
2
.
In short, the
system is partitioned into three regions: the left lead, the central region (also known as
the scattering region) and the right lead, as shown in Fig. 5-1.
The advantage of the
Green's function approach lies in its ability to replace the infinite leads by finite leads
with self-energies 126.
The self-energy 1a describes the effect of the lead a on the
central block and is defined as
(5.1)
Ea =#CagaOCa
where a stands for left (L) or right (R), C stands for center; the
#
's are the harmonic
force constant matrices divided by their corresponding atomic masses:
#aa means
onsite force constants of a block in lead a,
#' is the
two neighboring blocks within lead a and
#a
means the
the hopping matrices between
complex conjugate of
#
; g is
the surface Green's function defined by:
=
2 _
a
+ ]'
(5.2)
The surface Green's function corresponds to the uncoupled semi-infinite system and
135
The coupled Green's function for the
-
L -
is solved iteratively using a fast algorithm3.
central region is expressed as:
GR
= [C2_
62
R
(5.3)
where the superscript R stands for retarded, o
is phonon frequency, and $c
represents the onsite force constants of the central region.
*
** 000
**
0
*
LO
*******
00
0
0
0
OCO
e
0
00000
0
0
oo
ORO
0000000 e
0
0
0
0
Figure 5-1 The system is divided into three parts: left (L), center (C) and right (R).
The left and right leads are semi-infinite crystal lattices. In the transverse direction, all
the three regions have periodic boundary conditions imposed to represent the
infinitely large lateral dimension.
To tackle the infinitely large size in the transverse direction, a Fourier transform is
performed parallel to the interface to decouple the infinite degrees of freedom into
independent transverse wavevectors, k, assuming ideal translational invariance. We
can then treat them as independent one dimensional chains with different transverse
As the phonon frequency and transverse momentum are conserved
wavectors.
across the interface, the transmission function, E(c,k), as a function of these
parameters is given as a trace over the Green's function of the center and coupling
terms between the leads and the center:
E(o, k) =Tr[FL (
where Ia
=i[ER
_
EA]
,)GR
k)R(co, k )G^(
)]
describes the rate at which phonons enter and exit the leads.
The retarded Green's function, GR, and retarded self-energy,
ER
, are the Hermitian
conjugate of the advanced Green's function, GA , and advanced self-energy,
respectively.
(5.4)
IA,
The total transmission at a given frequency is simply the sum of the
transmission function of different transverse wavevectors normalized by the total
number of transverse k points:
E(I/, k )
E(c)=1/N
k,
63
.
While the phonon
frequency and transverse wavevector are conserved, mode conversion is allowed and
the longitudinal wavevector can change. In other words, the phonons can elastically
scatter into different directions at rough interfaces.
The thermal conductance per unit area, -, based on the total transmission function,
E(co), is calculated using Landauer's formula136
I 1 "0 af(oj, T)
TE(co)dco
c-(T)=- x - f h C
s 21 0
aT
(5.5a)
f
is the Bose-Einstein distribution and s is the cross-sectional area of the
simulation cell perpendicular to the direction of the heat flow direction. Note that
this definition yields a finite thermal conductance in the limit of an identical material
because the temperature drop, AT, used to derive equation (5.5a) is between the
reservoir temperatures, T, and T2, instead of the temperature drop across the
interface. In other words, equation (5.5a) is the formula corresponding to a
where
two-probe setup where the thermometer probes the bulk phonons incident on the
interface'1.
If a thermometer probes the temperature drop right across the interface (this
17
corresponds to a four-probe setup), equation (5.5a) needs to be modified ' 137
Despite the highly nonequilibrium distribution near the interface, we can define two
equivalent equilibrium temperatures, Tei and Te2, as proposed by Chen . The
equivalent equilibrium temperature corresponds to the final equilibrium temperature
of these phonons if we assume they adiabatically approach equilibrium. Then we
could use the Bose-Einstein distribution as a function of the equivalent equilibrium
temperature to represent the local energy density. On the other hand, we can express
the local energy density as a summation of the phonons emitted from both ends with
the reservoir temperatures. By equating the two approaches, we obtain the relation
between the equivalent equilibrium temperature and the heat reservoir temperature as
Te = T, + (T 2 - T 1)a/(2a1 ) and Te 2 = T2 - (T 2 - T1)a/(2a 2). Finally, we reach
a modified expression for the thermal conductance as
-I(T) = a(T) x
1-
where o, and
1
( a(T)+ (T))
-2 (T)
2 a-1 (T)
(5.5b)
-2 are the "thermal conductance" of pure material 1 and pure
64
material 2 using equation (5.5a), respectively, with E(w) equaling the number of
phonon bands at the frequency co. For a pure material, equation (5.5b) gives infinite
thermal conductance as there is no temperature drop across the virtual interface. In the
limit of low conductance (a- <<-i, a-<<a 2 ), equation 5.5a and 5.5b reach the same
value as the denominator approaches 1. In the following discussion (Sec. III),
equation 5.5b is applied.
The transmittance can be related to the transmission function as
'T 2 (0)=
(5.6)
T21()=
(CO)
2(CO)
where r,(c2) is the transmittance from material 1 to material 2, while r2 ()
transmittance from material 2 to material 1.
Transmittance describes the fraction of
the incident phonons of frequency o that is transmitted.
between zero and unity.
is the
Consequently, its value lies
The transmission function, on the other hand, can exceed
unity because it describes the number of modes transmitted at a specific frequency.
The maximum value of the transmission function at a certain frequency would be the
total number of phonon modes available at that frequency.
Although the
transmission function from either side is identical, the requirement of detailed balance
requires the transmittance to have a directional dependence.
In this study, we first construct an ideal Si/Ge interface as shown in Fig. 5-1 with Si
on the left of the interface and Ge on the right of the interface, using the lattice
constant of Si.
a=5.43
A
Lattice constants for the SW potential and DFT potential for Si are
and a=5.3976 A, respectively.
The transverse direction of all the
three regions is set to be 3a x 3a, which has converged by comparing to the results of
the 6a x 6a simulation size.
transverse directions.
Periodic boundary conditions are imposed in the
The longitudinal length of the central region is 2a, which
equals the largest thickness of the rough region investigated in this study.
For
simplicity, we use the force constants obtained from Si throughout the system since
those of Ge are very similar in magnitude. The major factor affecting the phonons of
Si and Ge are their very different masses.
28.0855 and 72.63 respectively.
41
and DFT, LAMMPS
The atomic masses for Si and Ge are
To obtain the force constants from the SW potential
and Quantum Espresso 7 8 are used to record the force and
displacement data, respectively.
For our DFT calculation, we use the local density
approximation of Perdew and Zunger 138 with a cutoff energy of 40 Ryd and
4 x 4 x 4 k-points for a 2 x 2 x 2 supercell of 64 atoms.
65
By fitting the general
expression of the Taylor expansion of the interatomic potential to the set of
force-displacements obtained from different atomic configurations , we extract the
harmonic force constants that are input into our transmission calculation. We take
exactly the same parameters as Esfarjani et al.28 used where they obtained excellent
agreement with experimental data for the phonon dispersion and thermal conductivity
of Si. This gives us confidence in the DFT force constants and corresponding
phonon properties. The harmonic
force constants that determine the phonon
are essential for the transmission and thermal
conductance. To calculate the total transmission, the number of transverse k points
within the Brillouin zone is chosen to be 10 x 10 to ensure convergence. A similar
frequencies and eigenvectors
procedure has been followed for rough interfaces except for the system setup that
obtains the force constants. For rough interfaces, the atoms in the interface region are
assigned to one of the two atomic masses according to some probability (uniform or
Gaussian), constrained by the thickness of the rough region, and then the effective
4
were obtained by dividing the Si force constants by the newly
assigned masses. Lattice mismatch between Si and Ge, i.e. strain effects, and
As observed by the NEMD
anharmonicity are not included in this study.
force constants
simulations 139, anharmonic effects were not important for temperatures lower than
500 K.
To first validate our methodology, we compare our calculated thermal conductance of
an ideal Si/Ge interface using the SW potential and equation (5.5a) with available data
in the literature.
2
Our result yields 2.8 x 108 W/m K at 300 K, which is close to
124
3.1 x 108 W/m 2 K from the lattice dynamics calculation by Zhao and Freund , and
(3.2
0.2) x 108 W/m 2 K
McGaughey13 9 .
from
the
NEMD
calculation
by
Landry
and
We can then focus on the discussion on rough interfaces using
equation (5.5b).
5.3 Results and Discussion
5.3.1. Rough interface with random distribution
To create random atomic mixing, we select a certain number of layers (2, 4, 6, and 8)
in the central region and randomly shuffle the atoms within these layers. Three
independent configurations are constructed for each roughness thickness and
calculations are conducted for each configuration. The average value is plotted for
each thickness of the rough region. The total transmission function, transmittance and
66
thermal conductance are plotted in Fig. 5-2. The total transmission function,
transmittance and thermal conductance of the ideal interface are plotted in Fig. 5-2 as
a reference.
14 rS
1 2C
0
C.,
C
...
\
1 0--
Ideal
--- 2-layer Ran dom
4-layer Ran dom
--- 6-layer Ran dom
-8-layer Ran dom
(a)
Rough
Rough
Rough
Rough
8
0
cc
4.
p
2
0
0
100
300
200
Frequency[cm-1]
67
400
500
&
09
0017
[>iI9jne-iedwe
00C
ON~
001
ij6no~j wopue~j Jee-q
q6no~j wopue~j JeAel-9
0
0
-..-
q~no>.j wopue~j jael-t...
C)
Lfbfo~j wopue>j JeAel-Z..
0
CL
---
-
--
p
CD
-
- am
ow -
-
-
0
W" -
son-
~
--
-I
on ago
ts10 . X
0 '09
EL-wo]Aouenbij
ooC
o0z
00t,
00M
MAS ?o0
-~
4
q1 o~ wpe~ j~eopu~j e~e-gI
q~n~j
j~elt,,
q~noj wpuej
wou
qKo~
CD)
Iaeleei
(q)
G)
L0Oc
i
--
~1
'p
6'0
**:~
~1
-
12
DFT
-
10
0
C)
8-
0
6-
C:
4
(d)
Ideal
--- 2-layer Random
----- 4-layer Random
6-layer Random
--- 8-layer Random
Rough
Rough
Rough
Rough
20
0
100
300
200
Frequency[cm-1]
400
500
0.9
(e)
Ideal
Rough
--- 2-layer Random
..... 4-layer Random Rough
0.8
---
6-layer Random Rough
-8-layer
Random Rough
0.7a)
C
E
CO
0.6
0.50.4
A
0.3
0.2
0.1
DFT
0
100
300
200
Frequency[cm-1]
69
400
500
x
108
L
f
3 DFT
2 .5-
00
c\E
1 2
0
-Ideal
Random Rough
4-layer Random Rough
-- 6-layer Random Rough
- -8-layer
Random Rough
0-'(.)
f---2-layer
0
0 .5
0
100
200
300
Temperature[K]
400
5 )0
Figure 5-2 Total transmission function, transmittance and thermal conductance as a
function of phonon frequency for an ideal Si/Ge interface (solid black line) and for a
random rough Si/Ge interface (colored dashed or dotted lines): (a) Total transmission
based on SW force constants; (b) Transmittance from Si to Ge based on SW force
constants;
(c) Thermal conductance
based on SW force constants; (d) Total
transmission based on DFT force constants; (e) Transmittance from Si to Ge based on
DFT force constants; (f) Thermal conductance based on DFT force constants.
One counter-intuitive finding, arguably the most important highlight, from Fig. 5-2 is
that the phonon transmission across a rough Si/Ge interface can be higher than the
ideal Si/Ge interface for certain frequencies, contributing to a larger thermal
conductance at certain roughness thicknesses.
In the low frequency limit, the long
wave-length phonons do not sense the interface roughness and propagate through the
interface as if they are traveling across an ideal sharp interface. Due to its short
length scale, atomic roughness has negligible influence on the long-wavelength
phonons.
In the high frequency limit, the transmission is zero because there are no
70
available states on the Ge side.
The most interesting phenomena are observed for
the phonons with mid-range frequencies, where the atomic roughness could play a
role in enhancing the transmission.
The roughness softens the abrupt change of the
acoustic impedance at the interface and facilitates phonon propagation. Surface
roughness can also allow phonons with large incidence angles, which would
otherwise be internally reflected at the interface, to be transmitted. More specifically,
this can be understood by investigating the phonon density of states (DoS) of the two
materials where incident and outgoing phonons are contained, and the interfacial
region where reflection and transmission happens.
As shown in Fig. 5-3, the phonon
DoS of pure Si and Ge are quite different, while the Si/Ge mixture has an intermediate
DoS which serves to bridge the gap between Si and Ge. Therefore, phonons that
originally cannot propagate across the Si/Ge interface can now be transmitted via new
elastic scattering channels created in the Si/Ge mixture.
Accordingly, the phonon
transmission and transmittance are boosted in the 200 to 300 /cm frequency range
where the overlap of the two DoS is enhanced.
This frequency range corresponds to
the top of the TA branches close to the zone boundary, where the typical phonon
wavelength is a few lattice constants at the most.
Although one configuration of a
Si/Ge mixture is used in Fig. 5-3, it can represent the trend for general Si/Ge mixtures
at the interface since the atomic ratio of all the configurations involved in our
calculation is 1:1 with the only difference being the atomic positions.
been well-known that interface roughness
photons 140-143 and electrons 144-147.
In fact, it has
can increase the transmittance
of
For phonons, interface roughness leads to a
reduction in thermal conductivity in nanowires 3,91,12 because of back scattering and
in superlattices'48-150 due to the loss of coherence.
But for an individual interface,
interface roughness is able to increase transmittance. This has not received much
attention before.
71
0.09
-Si-
0.08 -
--- Ge
Si/Ge mixture
I"
0.07
0.06S0.05-
-
0.04
0.03-
0.020.01-
0
100
400
300
200
500
600
Frequency[cm-1]
Figure 5-3 Phonon Density of States (DoS) of pure Si (Black solid line), pure Ge (red
dashed line) and Si/Ge 1:1 mixture (green dotted line) using DFT force constants.
For the 2-layer rough configuration, SW predicts a -20% increase in the thermal
conductance at 300K (Fig. 5-2(c)), while DFT predicts a -30% increase, compared to
Empirical potentials can qualitatively capture the
trend, but are unable to quantitatively predict the difference. As the thickness of the
rough region increases, the transmission does not keep increasing, which is consistent
with earlier observations"' 14 There are two competing factors: 1) overlapping
perfect interfaces (Fig. 5-2(f)).
DoS which increases transmission; 2) diffuse scattering at the rough interface which
reduces transmission. As observed in the SW case (Fig. 5-2(a)), the 2-layer rough
Above a thickness of two layers,
diffuse scattering becomes the more significant mechanism that affects thermal
conductance. In the DFT case (Fig. 5-2(d)), however, the 4-layer rough configuration
configuration gives the highest transmission.
gives the highest transmission at a frequency of around 120 cm~ and 2-layer
roughness gives highest transmission between 230 cm~1 and 300 cm~I, which leads to
fairly close thermal conductance between the 2-layer rough configuration and the
This finding cannot be
4-layer rough configuration as shown in Fig. 5-2(f).
represented by the calculation using SW prediction partly because their phonon
bandwidths are different from DFT. Compared to the ideal interface, the thermal
conductance is larger when the rough region is thinner than 6 layers using the SW
force constants and up to 8 layers using the DFT force constants. This discrepancy
reiterates the necessity of adopting the DFT force constants to provide precise
72
guidance in practical applications.
constants results are presented.
In the following discussion, only the DFT force
As thickness increases even further, the thermal
conductance decreases below that of the ideal interface.
understood by considering the limiting case.
increases
to infinity,
diffuse
This can be easily
As the thickness of rough region
scattering becomes
dominant
and the thermal
conductance should approach the alloy limit.
5.3.2. Rough interface with Gaussian distribution
To mimic atomic diffusion at an interface, we also create an atomic profile for one
type to obey a half Gaussian distribution as shown in the Fig. 5-4(c) inset.
The
phonon transmission, phonon transmittance and thermal conductance are plotted in
Fig. 5-4.
constants.
A significant increase in phonon transmission is observed using DFT force
At 300K, there is 32.6% increase for a 6 layer roughness thickness.
For
the same roughness thickness, the Gaussian distribution shows more enhanced
transmission compared to the random roughness distribution.
Comparison with experimental data is difficult since there is no experimental data on
a single Si/Ge interface.
On the other hand, several experiments had reported
reduced thermal conductivity on Si/Ge superlattices 148 149.
If we assume that the
measured thermal conductivity is due to interfacial resistances only, as one would
expect in the very thin limit when phonon transport is completely incoherent15 1 and
yet ballistic through individual layers of the superlattice, the extrapolated thermal
conductance is 2 x 101 W/m 2 K 148 (period = 3 nm) and 1.8 x 10 W/m 2 K 149
(period = 4.4 nm) at 300 K. Both the extrapolated values are close but about one order
of magnitude larger than our calculated value of 2 x 108 W/m 2 K for an ideal
interface and 2.8 x 108 W/m 2 K for a Gaussian rough interface based on DFT force
constants. The higher than predicted value is actually consistent with recent
experimental observation152 that long wavelength phonons maintain their coherence in
thermal transport in superlattices, and hence lead to a higher conductance value than
that of a single interface as we calculated.
73
(a)
Ideal
--- 6-layer Gaussia i Rough
1
0
r
0
C,
0
C,)
6-i
it
E
2-
0
0
100
200
300
Frequency[cm-1]
400
1
A
0.8
500
(b)
-Ideal
-- 6-Iayi er Gaussian Rough
I
I
0.6
CD
A
1
0.4
10
00
FrqecIm1
30
0.2
0O
0
400
74
500
x10
-Ideal
- -
-layer
(c)
Gaussian Rough
-
---
-
3
2.51
2
20
1.5
0
0$
~0
0
-nterface
;;Z 15.
40'- 10
1
.E 5.
z0
0.5
-4-3-2-1 1 2 34
0
Figure
Layer Number
C
5-4 (a)Total
100
200
300
Terrperature[K]
transmission function,
400
(b)transmittance,
and
500
(c) thermal
conductance as a function of phonon frequency for an ideal Si/Ge interface (solid
black line) and for a rough Si/Ge interface with a Gaussian distribution (dashed blue
lines) based on DFT force constants. Inset of (c): The number of Si atom in each layer
for an ideal interface (solid black) and for a Gaussian rough interface (dashed blue).
75
Z221
1.41
.41
2
2.45
2.83
3.16
8
10
1.35
1.3-
S1.2
$
1.25\
b 1.15
1.1
Emmmmmm0.
1.05
2
4
6
m 2/m
1
Figure 5-5 Thermal conductance ratio of a Gaussian rough interface to an ideal
interface as a function of the mass ratio (lower x-axis) and the acoustic impedance
ratio (upper x-axis) of the two materials using DFT force constants
To explore the generality of the transmission enhancement between different materials,
we keep the Gaussian rough configuration and vary the mass of the atoms on the Ge
sites from 1.25 times that of Si to 10 times that of Si, corresponding to acoustic
mismatch values from 1 to 3.16. The thermal conductance ratio of a Gaussian
interface over an ideal interface is plotted in Fig. 5-5 as a function of the
mass/acoustic impedance ratio of the two materials on both sides of the interface.
Since the roughness is caused by the mass difference, when the mass ratio is 1, there
is no atomic mixing and no roughness at the interface. As the mass ratio increases,
the phonon dispersions of the two materials begin to differ from each other and the
roughness favors phonon propagation via graded acoustic impedances at the interface.
The thermal conductance ratio reaches its maximum at 2.586, which happens to be the
mass ratio of Si to Ge. As the mass ratio increases even further, the phonon
dispersions of two materials fall further apart from each other and it becomes less
effective to bridge the large gap through the effects of roughness. Therefore, the
thermal conductance ratio drops and flattens out with increasing mass ratio.
Nevertheless, the thermal conductance ratio is kept over unity up to a mass ratio of 10
76
and will stay above unity in the infinite mass mismatch limit since such an interface
provides a smooth transition for intermediate frequency phonons to transmit thermal
energy across the interface.
Although there are variations in the extent to which
surface roughness increases thermal conductance, the enhancement generally holds.
5.4 Conclusion
In summary, we apply the atomistic Green's function method to calculate the phonon
transmission across an ideal and rough Si/Ge interface.
The atomistic roughness can
increase the phonon transmission across two dissimilar materials if the roughness
thickness and profile are properly controlled, contrary to the commonly held notion
that rougheness reduces transmission.
This effect is more pronounced if the acoustic
mismatch between the two materials is moderately large.
new design considerations for interface engineering.
This finding elucidates
As our contribution to the AGF
framework, we incorporate the first-principles force constants determined from DFT
into the AGF method for phonon transport in an infinitely large 3D structure.
The
comparison between the results from SW force constants and those from DFT force
constants demonstrates that DFT force constants are necessary in reliable predictions.
Since interface transmission is crucial for bridging the calculation of pure materials to
nanocomposites, we can now integrate the interfacial transmission and the bulk mean
free paths, both calculated from first-principles DFT, to accurately model heat
transport in complex nanostructured materials.
77
78
6. Phonon Transmission across
Si/Ge Superlattices
6.1 Introduction
As mentioned in Chapter 5, the calculated thermal conductance across a single Si/Ge
interface is one order of magnitude lower than the extrapolated value from
experimental data on Si/Ge superlattices. This suggests that phonon transport across
multiple interfaces is not a simple summation of individual interfaces and coherent
phonon transport is expected to exist in Si/Ge superlattices. In this chapter, we
investigate phonon transmission across Si/Ge superlattices using the Green's function
method with first-principles force constants derived from ab initio density functional
theory.
The thermal properties of semiconductor superlattices have been under intense
investigation due to their potential uses in thermoelectric energy conversion37,39,153
and optoelectronic devices. 154
The thermal conductivity of superlattices can be even
lower than their alloy counterparts.148,
15-157 Although diffuse scattering at interfaces
is responsible for the remarkable thermal conductivity reduction,51, 158 coherent
phonon transport has been experimentally observed in GaAs/AlAs superlattices1 59 and
for perovskite oxides 60. To further reduce the thermal conductivity for thermoelectric
applications, it is crucial to understand and control the different phonon transport
modes in superlattices. Phonon heat conduction in superlattices can be attributed to
incoherent and coherent phonon modes. Coherent modes preserve their phase as they
propagate through multiple interfaces. For these phonons, Bloch mode extends
through the whole structure and the superlattices can be treated as a homogeneous
material with its own unit cell. If interfaces destroy the superposition of waves, due to
roughness or other structures, phonon modes lose their phase information and their
transport is incoherent. For these modes, superlattices act as a composite made of a
79
stack of two alternating materials.
Previous theoretical studies on superlattices have focused on changing the periodicity.
Most common theories developed to understand phonon transport in superlattices fall
into one of two pictures: the incoherent particle picture which is rooted in solving the
Boltzmann transport equation,151 ,
161
and the coherent wave picture where lattice
dynamics calculations were employed.
162-164
Either picture could fully explain the
experimentally observed thermal conductivity trend as a function of period length in
both in-plane and cross-plane directions, though. 39
A combination of both pictures is
desired. Lattice dynamics based on damped wavefunctions was used to predict a
165 66
minimal in the thermal conductivity of superlattices in the cross-plane direction. ,1
More recently, a perturbation method based on the Fermi golden rule 30, 159,
167
was
developed but the method may have limitations on treating interface scattering, as
strong scattering may not be captured by perturbation. One alternative approach is to
use molecular dynamics simulations,1 50"6 11 which do not assume the nature of phonon
transport but on the other hand are classical in nature. Yet the empirical potentials
involved in molecular dynamics limit the accuracy and it is difficult to explore the
detailed phonon mode behavior.
The Green's function method has been applied to study phonon transport across single
and multiple Si/Ge interfaces. For single Si/Ge interfaces, effects of strain 129, lattice
mismatch' 6 9 and interface roughness"o,
170
on phonon transmission have been
investigated. Green's function study on Si/Ge superlattices, however, is scarce. Zhang
et. aL.129 briefly discussed the impact of the number of interfaces on the overall
thermal resistance across multiple Si/Ge interfaces while the transmission function
was not detailed. In this study, we use the Green's function method to investigate the
coherent phonon transport across Si/Ge superlattices. First-principles force constants
have been incorporated as our previous work on single Si/Ge interfaces
170
In this chapter, we calculate the phonon transmission and corresponding thermal
conductivity of Si/Ge superlattices with varying interface roughnesses. By keeping
the period thickness fixed while changing the number of periods, we show that
interface roughness partially destroys coherent phonon transport, especially at high
temperatures. The competition between the low-frequency coherent modes and
high-frequency incoherent modes leads to an optimum period length for the minimum
thermal conductivity. To destroy the coherence of the low frequency modes, a
scattering length scale on the order of period length is required. This finding is useful
to guide the design of superlattices to reach even lower value of the thermal
conductivity.
80
6.2 Methodology
We follow the same atomistic Green's function method1 27,
170
single Si/Ge interface.
129
as we applied for a
The only difference is that, in this study, we use the Si/Ge
superlattices as the center region as shown in Fig. 6-1. As a brief overview, we
employ the force constant,
#c,
from ab initio density functional theory into the
Green's function to determine the transmission function. The retarded Green's
function is given by
GR
where
GR
=[)2
_CO
1
C
-YL (C)-YR (OT
is the retarded Green's function,
0 is the phonon frequency,
represents the onsite force constants of the center region and the self-energy
6.1)
#c
E,
describes the effect of the lead a on the center block. The transmission function,
E(o), is given as a trace over the Green's function of the center region and the
coupling terms between the leads and the center:
E(c) = Tr[FL (w)GR ()R (o)G A (c)]
where
=
-a] .describes
(6.2)
the rate at which phonons enter and exit the leads.
In these calculations transverse periodic boundary conditions are assumed and the
above formulas hold for every single transverse momentum, over which a final
summation needs to be performed in order to obtain the total transmission.
The interface transmittance is then defined as
r(W) = -
()
(6.3)
)
.pure((
In our system setup, we use Em
pure (to) = ESi (to)
The 2-probe thermal conductance per unit area, a, based on the total transmission
function, E(eo), is calculated using Landauer's formula 136
81
a(T)
=
s
hc af(co, T) E()dco
I 1
2;r 0
T
(6.4a)
where f is the Bose-Einstein distribution function and s is the cross-sectional area
of the simulation cell perpendicular to the direction of heat flow. The 4-probe
conductance can then be written as1 70
-
1
( (T) +
2 o-,(T)
(6.4b)
)
-'(T) = o-(T) x
c 2 (T)
Although the difference of thermal conductance between 2-probe formula and 4-probe
formula becomes small as the number of interfaces increases, we use 4-probe formula
throughout this study to be consistent with our previous calculation of single interface
and experiments.
The thermal conductivity of a sample length L is defined to be L times the 4-probe
conductance:
(6.5)
k ='L
s
k
sts4
"4
G
Figure 6-1 Schematic of the system setup: the left reservoir is pure Si, the right
reservoir is pure Ge, the center region is the Si/Ge superlattice.
The calculations in this paper do not include phonon-phonon scattering. According to
experimental' 4 8'
155
and modeling 30 SO,
151
results on Si/Ge superlattices, anharmonic
effects are not important for temperatures below 500K. The anharmonicity would
become important when the phonon mean free path due to anharmonicity becomes
smaller than the superlattices length L. In the harmonic regime, specular scattering
leads to coherent wave effects171-1
73
while diffuse scattering could destroy coherence.
82
6.3 Results and Discussion
For incoherent transport, the interfaces behave like a series of thermal resistors and
the effective thermal conductivity becomes independent of the number of periods. For
coherent transport, the thermal resistance keeps constant with respect to the number of
periods and the thermal conductivity increases linearly with an increasing number of
periods. The thermal conductivities for smooth- and rough-interfaced superlattices are
shown in Fig. 6-2a. The thermal conductivity of smooth-interfaced superlattices
demonstrates a linear increase with respect to the number of periods, indicating
coherent transport at 300K although ultimately anharmonicity limits the number of
periods
over
which
transport
is
coherent.
The
thermal
conductivity
of
rough-interfaced superlattices increases more slowly than linear, indicating partially
coherent and partially incoherent transport. Another noticeable point is that roughness
increases the thermal conductivity of small-period superlattices, in contrast to
conventional wisdom. This is because atomic roughness generates a smoother change
of density of states between layers
170
We then compare the transmittance across smooth-interfaced superlattices (Fig. 6-2b)
with a rough-interfaced one (Fig. 6-2c). When the number of periods equals 1, it
reverts to the single Si/Ge interface as we investigated before 17 0 , which we include as
a reference. What we are mainly interested in here are multiple interfaces. For a
number of periods > 1, there are clearly two frequency regimes: the low frequency
regime and the higher frequency regime separated by the vertical lines at 55.6 cm-.
The low frequency regime is defined as the region where the transmittance does not
change as the number of periods is increased.
This indicates that low frequency,
long wavelength phonons pass through the entirety of the superlattices as if it is a
homogeneous medium. They form passing bands and transport phonon waves
coherently. The low frequency regime is the same for both smooth and rough
superlattices. In the rough case, the constancy of the transmittance versus the number
of periods for low-frequency phonons is due to the fact that such phonons have
wavelengths larger than the roughness scale and thus see an effectively homogeneous
interface of atoms with an effective mass intermediate between Si and Ge. As such,
they do not get scattered by the roughness at the interface and thus, similar to ideal
interfaces, their transmittance does not change with the number of periods. In the
higher frequency regime, the transmittance for smooth-interfaced superlattices no
longer changes as the number of periods becomes larger than 5, suggesting the
formation of minibands. Because the superlattice eigenstates are formed from the
constructive interference between all the multiple reflected waves, the wave needs to
83
go a few periods away and be reflected back for a few times in order to get a coherent
eigenstate of the superlattice. In contrast, the transmittance for rough-interfaced
superlattices keeps dropping due to more diffuse scattering at the interfaces. In other
words, roughness destroys coherence especially for higher frequency modes.
2
(a)
E 1.5
J
3
-o
1
E 0.5-
-u-*
Smooth Interfaces
-- +- Rough Interfaces
r-
0
0
5
10
15
Number of Periods
84
20
1
I'
--
---- 3 pd
....... 5 pd
..-- 10 pd
20 pd
0.8r
a)
C
1 pd
0.6
E
0.4
'
C
0.20
(b)
0
100
200
Frequency [cm-1]
300
400
1
-- 1
d
---- 3 pd
..-. 5 pd
----- 10 pd
20 pd
0.8 L
a)
CU)
0.6
\
EW,
0.4-
I'I....'
-
0.2
0
100
200
Frequency [cm~ 1]
85
300
400
Figure 6-2 (a) Thermal conductivity of superlattices as a function of the number of
periods for smooth and rough superlattices at T=300K. (b) Transmittance as a
function of frequency for superlattices (period =2a) with smooth interfaces; (c)
Transmittance as a function of frequency for superlattices (period=2a) with rough
interfaces.
To unveil the cutoff frequency, wcutoff, of the low frequency regime, we plot the
phonon dispersion of the SiGe superlattice with period length 1 = 2a in [100]
direction (Fig. 6-3). The zone boundary frequency of the lowest acoustic branch is
55.6 cm 1 . It is intriguing that the cutoff frequency is the lowest acoustic phonon
branch at the folded Brillouin zone edge.
Although some of the higher frequency
phonons have a long wavelength in the folded zone representation, they are unable to
maintain their coherence. Therefore, phonon wavelengths of higher frequency modes
in the folded zone do not matter. We expect that this argument generally holds for
different materials. As the period length increases, the first Brillouin zone becomes
shorter since the edge of the folded zone is proportional to the inverse of the period
length. Thus, the cutoff frequency of the totally coherent regime is determined by the
reduced first Brillouin zone, or the period length. It is, therefore, difficult to destroy
the coherence of the low frequency modes unless scattering length scale comparable
to the period length can be introduced.
86
500
400
E200
100
0
X
Figure 6-3 Phonon dispersion of Si/Ge superlattice with period length
2a in [100]
direction
We then explore the temperature dependence of coherent and incoherent phonon
transport. This temperature dependence comes only from phonon occupation or heat
capacity. At all temperatures, phonons with frequencies smaller than the cutoff
frequency yield a linearly increasing thermal conductivity as a function of number of
periods as shown in Fig. 6-4. To illustrate this effect, we choose temperatures of 20,
50 and 300K which correspond to frequencies of 13.9, 34.7 and 208.1 cm~respectively. These are to be compared with the cutoff frequency of 55.6 cm- . At low
temperatures, only low frequency modes are excited, thus the phonon transport is
mostly coherent. As the temperature increases, more and more high frequency modes
are excited and incoherent phonon transport plays a more and more important role.
Correspondingly, we observe that the thermal conductivity increases more slowly than
linear and becomes flatter as the number of periods increases.
87
0.1
E
-
(a)
0.08
L__
_
T=20K
-+-Total
--&-) > OCUtoff
0.06-
.-
..
4-a
0
0..<= o cutoff
0.04-
E
g......-
. 0.02
0-
--------------
5
0
0.4
20
15
10
Number of Periods
(b)
T=50K
,.0.35
0.3L
0.25.
-
0.2'
0
o
-
Total
--e-- (0 > (Oof
,'ctf
0.15
-U
0
cutoff
..---
0.05
0
5
15
10
Number of Periods
88
20
(C)
T=300K
0 .80.6--0 0.4-
-E
Total
---
O > 0) cutoff
E0
cutoff
p0.2
0
5
15
10
Number of Periods
20
Figure 6-4 The total thermal conductivity, contribution from phonons with frequencies
larger than the cutoff frequency, and not larger than the cutoff frequency at (a) T=20K,
(b) 50K and (c) 300K.
In our previous paper on single Si/Ge interfaces 170,
a
we found that the conductance at
single Si/Ge interface is an order of magnitude lower than the extracted
experimental thermal conductance from Si/Ge superlattices assuming the thermal
resistance only happens at the interfaces. We predicted that the discrepancy comes
from the long-wavelength phonons which maintain their coherence. Now with
coherent transport, the calculated thermal conductance per interface increases with the
number of periods and matches well with experiments (Fig. 6-5). Although the size of
the superlattices is much smaller than that of the experimental sample due to
computational limitations, we can at least see that the trend is consistent. It states that
the thermal conductance is not intrinsic to the interface, but depends on what exists on
both sides of the interface.
89
Extrapolated Experimental Value
(9
_1.5~
-E
-U
(0U
C
0
E
I-
0
10
20
30
Number of Periods
40
Figure 6-5 Normalized thermal conductance per interface as a function of number of
periods
for
rough-interfaced
superlattices
with
period
length
1 = 2a . The
experimental value is extrapolated from the sample of period length 1 = 4.4nm and
100 periods.
To destroy coherence in rough superlattices for the purpose of reducing the thermal
conductivity, there are two competing effects as the period length increases: 1) the
low frequency regime with totally coherent transport shrinks, which is beneficial; and
2) the interface density decreases and the importance of interface roughness decreases,
which is detrimental. We plot the thermal conductivity as a function of superlattice
length for period length I = a, 2a and 4a, respectively in Fig. 6-6. At the same
superlattice
length,
superlattices
with
period
length
1 = 2a = 1nm
possess
minimum thermal conductivity. The crossover from the coherent to incoherent regime
is naturally included in our formulation, and we thus observe the minimum thermal
conductivity
of superlattices 60' 166' 167 under the
atomistic
Green's function
framework.
We then introduce a simple model below to identify the dependence of the thermal
90
conductivity on period length. We write the thermal conductivity as a sum over the
contribution of acoustic (the three low-lying folded acoustic phonons) and optical
(rest of the bands) phonons:
k = kac + kop
=
cutoff
f(
C (&v)D(c)v(&v)A(co)d(A + fwmax Cj,(w)D(cv)v(co)A(w )dw
cutoff
(6.6)
where &max represents the highest phonon frequency. For acoustic phonons, we
2
assume that the specific heat C,(o) = kB, that the density of states D (CO) = A&)
(A is a coefficient which can be determined from the Debye approximation or lattice
dynamics calculations), and v(cv) = c (average speed of sound of acoustic modes),
and A(w) = L (in the absence of anharmonicity, scattering occurs at the sample
boundaries). Then,
Acv
since f''toff
0
2
AW 2 kB cLdcd =
fOC'f
dc =
3
-cell
and
Qceul
k CL
(6.7)
= 1 Atrans, with Atrans being the area of
the unit cell in the transverse direction. For optical phonons, we use C,(c) = hcv
,
kac =
D(c) = D (average density of states in a given volume per unit frequency interval),
v(cv) = c0 p (average speed of sound for optical modes, which is much smaller than
the speed of sound) and A(co) = 12 1-p 1 where p accounts for the probability of pure
specular scattering at each interface, and we have assumed that the thickness of each
medium is equal to half the superlattice period.
kop
hcvL)Dcop
'=f""
8T
cutoff
'a'
If the further assumption of
-I-
kBT
i
2 1-p
d
~:-
-Ocell
kL cop niL
2 1-p
(6.8)
«1 is made. Here N is the total number of atoms in
a period. Therefore, the overall conductivity (neglecting anharmonicity) would be a
sum of
91
k =
3
1Atrans
kc L +
'
1cell
kEcop 1+p
2 1-p
(6.9)
Equation (6.9) has a minimum at an optimum period length of
l
'opt
where
t2
M3cellcL(1-p)
(3N-3)Atranscop(1+p)
fDcelloc~L-p)
NOAtranscop(1p)
(6.10)
cello and No are the volume and number of atoms in a unit cell, respectively.
It is noteworthy that the optimum period length depends on the superlattice length
because the relative contribution from coherent and incoherent phonons would vary as
the superlattices length changes. When designing superlattices to reduce the thermal
conductivity, the optimum period length would be desirable. For SiGe superlattices
considered in this work, No = 8, -celo = a3 , Atrans = 3a x 3a and a = 0.54nm.
We assume c = 5400m/s (speed of sound for germanium), cop = 100m/s, and
p = 0.5. This leads to lopt = 0.82nm at L = Snm, lopt = 1.16nm at L = 10nm,
and lopt = 1.64nm at L = 20nm. All the optimum period lengths are close to
2a = 1.08nm, which is consistent with our calculations using the atomistic Green's
function method. In strongly anharmonic materials, however, kop would be
independent of 1 and there is no minimum thermal conductivity.
92
.
1
0.9
00 0
0.8 -
0
0
0 00 00 0t 0t 00
0.7
-.- Period: =a
a)
=2a
Period: =4a
-
25
20
15
10
5
Length [nm]
Figure 6-6 Thermal conductivity of rough-interfaced superlattices as a function of
superlattice length for a period length 1 = a, 2a and 4a at 300K.
6.4 Conclusion
In summary, we apply the atomistic Green's function method to calculate the phonon
transmission across Si/Ge superlattices. We focus our discussion on coherent vs.
incoherent phonon transport in superlattices. We show examples of totally coherent
phonon transport in smooth-interfaced superlattices, and partially coherent and
partially
incoherent
demonstrate
that
phonon
the
in
transport
contribution
from
rough-interfaced
coherent
superlattices.
phonons
We
decreases
in
rough-interfaced superlattices as the temperature increases. To obtain lowest thermal
conductivity, there is an optimum length resulting from the competition between
coherence of low-frequency phonons and incoherence of high frequency phonons
caused by interface scattering when anharmonicity is negligible. Our theoretical study
complements earlier experiments, providing guidance for the design of superlattices.
93
94
7. Solid-Liquid Interface
Conductance Using Time-Domain
Thermoreflectance
Measurements
7.1 Introduction
Interfacial thermal conductance has been a subject of fundamental and practical
interest for many years. Usually, extra molecular layers at an interface add to the total
thermal resistance network and reduce the thermal conductance, especially for
solid-solid
interfaces17 4 . Chemical
functionalization,
however,
has
significant
influence on the interface thermal conductance between solid-solid interfaces via the
interfacial bonding mechanism
175, 176
Recent studies have shown that covalent
.
chemical bonding at solid-solid interfaces using self-assembled monolayers (SAMs)
can improve the interfacial thermal conductance
77
. Compared to solid-solid interfaces,
the thermal conductance across solid-liquid interfaces has received limited attention.
Better understanding of solid-liquid interfacial transport is important for different
applications
such
cancer
as
treatment
nanoparticles1 78, solar thermal heating
17 9
based
on
thermal
therapeutics
and
80 8 2
, colloids and nanofluids1 -1 . Experiments
on thermal conductance of solid-liquid interfaces typically employ suspensions of
metal nanorods in water or organic solvents 183-187. Although planar solid-liquid
interfaces
modified
with
experimentally studied'
88
and
hydrophilic
hydrophobic
SAMs
have
been
, there have been no controlled studies of solid-liquid
interfaces with SAMs and without SAMs.
In this chapter, we systematically study the thermal conductance dependence on the
functional end groups using the same class of SAM, as well as the dependence on
95
We show that the addition of the extra SAM layer between the planar
chain length.
Au and ethanol enhances the thermal transport as SAM serves as a transitional layer.
This shares the same idea as discussed in Chapter 5 that a smoother transition favors
phonon transport across an interface. Specifically, we find that increasing the chain
length does not adversely impact the interfacial thermal conductance, while different
functional end groups enhance the thermal conductance to different extents.
7.2 Sample Preparation and Experimental Setup
Alkanethiol and alkanedithiol SAMs are formed on a gold surface using the standard
wet chemical preparation method8 9 . Specifically, Au coated glass slides (purchased
from Phasis Sirl) are immersed in a dilute (-2 mM) ethanolic solution of the thiols
(purchased from Sigma-Aldrich) for 18-24 hours at room temperature. Excess thiol
molecules not bonded to the surfaces are removed by cleaning with ethanol and a
nitrogen gun. Molecular schematics of the four different SAMs grown for this study
are shown in Fig. 7-1. All the SAMs have an oil-like chain (-(CH 2),-) as a molecular
backbone, a head group containing a sulfur atom which can strongly bond to the Au
surface, and a terminal end group. Hexanedithiol and hexanethiol have the same
alkane chain length but a different functional group (thiol group -SH vs methyl group
-CH 3). In contrast, hexanethiol, undecanethiol and hexadecanethiol have the same end
group (-CH 3) but different alkane chain lengths.
-(CH2) n-
(a)
Figure
7-1
(b)
Schematic
SHCH 2(CH 2)4 CH2SH;
of
(b)
(d)
(c)
SAMs
study.
(a)
Hexenedithiol
CH3(CH 2) 4CH2SH;
(c)
Undecanethiol
used
Hexanethiol
in
this
CH 3(CH 2)9CH 2SH; and (d) Hexadecanethiol CH3(CH 2)1 4CH2SH
For characterization of the thermal interface conductance between the Au and ethanol,
the Au slides (with and without SAMs) are placed in contact with half of a
demountable cuvette with a 1 mm thick channel, which is then filled with ethanol, as
96
shown in Fig. 7-2(a). The Au film, on which the SAMs are grown, serves as the
transducer layer for time-domain thermoreflectance (TDTR) measurements of the
interfacial thermal conductance. The detailed TDTR methodology can be found
elsewhere190 ' 191. In brief, pump laser pulses (~150 fs pulse-width, 80.7 MHz
repetition rate) with a spot diameter of 60 im pass through the glass slide and are
absorbed by the 100 nm Au film, heating the sample.
Optically time-delayed probe
pulses coaxial with the pump beam with a diameter of 12 gm measure the temperature
decay after the pump pulses through the change in reflectivity. The amplitude of the
pump pulse train is modulated at 9 MHz to allow for lock-in detection of the
thermoreflectance response. The sample properties, including the interface thermal
conductance, impact the cooling curve and are extracted by fitting the data with a
diffusive heat transfer model 190. Typical phase data are shown in Fig. 7-2(b).
(a)
Glass
A
-SAM
Ethanol
97
(b)
-40k
-
with Hexanedithiol
5
-45
Cu
-50
without SAM
-55
500
1000
2000
1500
2500
3000
3500
Delay (ps)
Figure 7-2 (a) Schematic of sample arrangement; (b) Phase data for samples without
SAM and with hexanedithiol.
7.3 Results and Discussion
To measure the Au-ethanol interface thermal conductance, we first calibrate the
thermal properties of the glass slides and ethanol using a clean gold-coated glass slide.
The thermal conductivities of glass and ethanol are found to be 1.25 W/m/K and 0.17
W/m/K, respectively, and the measured thermal conductance between glass and Au is
51 MW/m 2/K, consistent with textbook values. These values are kept constant and in
all the subsequent fittings we only fit the thermal conductance between Au and
ethanol for the samples with SAMs.
The thermal conductance between Au and
ethanol at room temperature are shown in Fig. 7-3. The error bars represent the
standard deviation between 4-5 different locations on the same sample. The error bars
are smaller for the sample without SAMs, due to the greater uniformity of the bare Au
surface. The spot-to-spot variation may be indicative of non-uniform SAM coverage
due to the polycrystalline nature of the Au surface.
98
120
100I
801
E
60T
40-
gL
20-
Without SAMs
Hexanethiol
Hexanedithiol
Undecanethiol Hexadecanethiol
Figure 7-3 Thermal conductance between Au and ethanol with and without SAMs
from TDTR measurements at room temperature.
Counterintuitively, the interface conductance between Au and ethanol is improved by
the existence of an extra molecular layer in all cases, as shown in Fig. 7-3. While
hexanedithiol improves the thermal conductance by a factor of -5,
hexanthiol,
undecanethiol and hexadecanethiol all improve it by only a factor of-2. For all SAMs
studied, this improvement is likely due to the strong chemical bond formed between
Au and SAMs and between SAMs and ethanol. At one end of the oily chain, all the
SAMs have one thiol group covalently bonded to the gold surface. At the other end,
hexanedithiol
has a thiol group exposed to the ethanol, while hexanethiol,
undecanethiol and hexadecanethiol have a methyl functional group exposed to the
ethanol. Since ethanol itself contains both a polar group (-OH) and a nonpolar methyl
group, both the thiol end group and the methyl end group could form stronger
chemical bonds with ethanol than the bare gold surface in contact with ethanol.
In
other words, the SAMs serve as a bridge between Au and ethanol and facilitate
thermal transport.
While the thermal conductance of the hexanedithiol SAM significantly outperforms
the hexanethiol SAM, they differ only by the functional end group. We measure the
advancing contact angle at three different locations on each sample by using a contact
angle
goniometer.
The measured contact angles
of ethanol on hexanethiol,
undecanethiol and hexadecanethiol are 0 = 340 4', 0 = 350 30 and 0 = 410 2*
respectively, while the contact angle on hexanedithiol is too small to be measured, as
shown in Fig. 7-4. This implies the stronger bonding between hexanedithiol and
99
ethanol, which likely leads to the higher thermal conductance than the sample with
hexanethiol1 9 2 . These results agree with earlier experiments, which showed that
hydrophilic SAMs produced higher interface conductances than hydrophobic SAMs
for interfaces between water and metal 8 8 . In addition to changing the bonding
strength, the SAM layers may impact the acoustic mismatch between solid and liquid.
Specifically, SAM surfaces which demonstrate smaller contact angles may reduce the
acoustic mismatch between the gold and ethanol, by making the interface more
solid-like '93
Figure 7-4 Contact angles, 0, of ethanol on Au surface modified by (a) hexanethiol
and (b) hexanedithiol SAMs.
We use optical spectroscopic ellipsometry to estimate the SAM thickness with a
Cauchy model. The measured thicknesses for hexanethiol, undecanethiol and
hexadecanethiol
are 0.943 0.057
nm,
1.389 0.054 nm and 2.284+0.061
100
nm
.
19 4
respectively, in good agreement with previous measurements on hexadecanethiol
Moreover, the measured thickness increases almost linearly as the number of carbon
atoms increases from 6, to 11, to 16. The tilt angles are 0*-35'. Despite the difference
in chain length and film thickness, there is no observable difference in the thermal
conductance for hexanethiol, undecanethiol and hexadecanethiol. This indicates that
phonon transport is ballistic along the alkane chain, as well as that the phonon
vibrational spectra match regardless of the chain length investigated in this work. This
is consistent with earlier experiments 174, 195 and simulations 196, 197 for
solid-SAM-solid interfaces.
7.4 Conclusion
In summary, we use TDTR measurements to study the thermal conductance between
Au and ethanol with various interfacial SAMs. We show that the SAMs enhance the
thermal transport from Au to ethanol. The interfacial thermal conductance is
insensitive to the length of the alkane chain length, but strongly dependent on the
functional group. Our results shed lights on strategies to further tune the interfacial
conductance for practical applications.
101
102
8. Summary and Future Work
8.1 Summary
This thesis explored nanoscale heat transfer using both atomistic simulations and
ultrafast laser measurements. The detailed calculations expand our fundamental
understanding of the transport processes and provide guidance for practical materials
design to meet different needs. The first portion of the thesis focuses on phonon
transport properties in bulk materials while the second portion focuses on phonon
transport across interfaces.
Chapter 2 describes how to obtain spectral phonon transport properties in germanium.
Using a combination of molecular dynamics and lattice dynamics, we can extract the
information of phonon lifetimes by tracing the temporal amplitude decay of each
mode. With the knowledge of group velocity from phonon dispersion, we can obtain
phonon mean free paths and which phonons carry most heat. The empirical potential
employed, however, leads to a large discrepancy on the total thermal conductivity
between calculation and experiment. A better potential is thus needed.
Chapter 3 introduces in detail how to obtain phonon properties in PbSe, PbTe and
their alloys using first-principles calculations. Density functional theory is used to
compute the electronic band structure and derive more accurate force constants. The
excellent agreements with experimental values for the phonon dispersion and thermal
conductivity demonstrate the predictive power of this approach.
The phonon
properties including lifetimes, group velocities and mean free paths are presented. In
addition, the importance of optical phonons is emphasized, which is crucial for the
low thermal conductivity of these two materials and their alloys.
Chapter 4 discusses the importance of optical phonons in silicon nanowires. Since the
long mean free path acoustic phonons get strongly scattered at interfaces/boundaries,
optical phonons are much less influenced. This leads to a redistribution of the relative
importance between acoustic and optical phonons.
Chapter 5 investigates phonon transmission across a single Si/Ge interface using the
Green's function method. Phonon transmission can be enhanced by interface
103
roughness introduced by atomic mixing. This can attribute to a smoother transition in
the phonon vibrational spectrum.
Chapter 6 looks into coherent phonon transport in Si/Ge superlattices using the
Green's function method. Interface roughness can destroy coherence of high
frequency phonon modes. To destroy coherence of low frequency modes, a scattering
length scale on the order of a period length is needed.
Chapter 7 examines the influence of chemical bonding on the solid-liquid interface
conductance using time-domain thermoreflectance measurements. Self-assembled
monolayers are used to modify the gold-ethanol interface. Enhanced thermal
conductance
is
observed
compared
to
the
gold-ethanol
interface
without
self-assembled monolayers. Certain self-assembled monolayers are more effective
than others.
These atomistic level studies not only improve our fundamental understanding of
nanoscale thermal transport but also pave the way for multiscale simulation from
first-principles.
8.2 Future Directions
One natural extension of this thesis work is to pursue multiscale modeling of thermal
transport. Understanding of heat conduction in nano to meso-scale structures from
multiscale
thermal
modeling could guide the practical materials
design for
applications from thermoelectrics to the thermal management of electronic systems.
Using the extracted mode-dependent phonon bulk mean free paths and interfacial
transmission, both from first-principles approaches, and by integrating them into the
Boltzmann transport equation or Monte Carlo simulation, we can predict the thermal
conductivity of various nanocomposites.
Furthermore, developing an improved
effective medium model based on the knowledge of multiscale simulation would be
useful for practical simulations.
The other direction would be look further into the thermal transport of soft materials.
On the fundamental side, mechanisms of thermal transport in soft materials are of
great interest. Significant progress has been made in understanding thermal transport
properties in crystalline solids. The knowledge of thermal transport in soft materials,
however, is presently falling far behind. While the periodicity in crystals with
well-defined vibrational or phonon modes has been actively studied, the lack of
104
periodicity in soft materials imposes a big challenge onto the fundamental description
of the vibration modes, let alone their properties. The vibrational modes are no longer
pure plane-waves, except in the low-frequency limit. The transport mechanisms are
still under debate. For amorphous materials, fracton transport was proposed and this
approach attributes the increase in thermal conductivity above the plateau, in part, to
anharmonic fracton hopping. 198 Yet Freeman and Anderson discussed that it was not
clear which type of excitation transport heat above the plateau.1 99 Cahill and Pohl
argued that their experimental data did not support the fracton theory. 200 Using normal
mode analysis, Larkin and McGaughey 201 showed that the lifetimes of amorphous
materials show little frequency dependence and a significant number of modes fall
below the loffe-Regel limit. More studies along this line would be useful to gain a
comprehensive
understanding.
In addition, thermal
transport across interfaces
between hard and soft materials needs more investigation. It would be interesting to
study how the vibrational modes couple to each other at the interface and how much
one can tune the coupling to engineer the interface conductance. On the applications
side, a deep understanding of thermal transport in soft materials and soft/hard
interfaces promises applications ranging from photothermal nanotheuraputics to
battery thermal management.
105
106
Biography
1
R. Peierls, Annalen Der Physik 3, 1055 (1929).
2
P. G. Klemens, Solid State Physics-Advances in Research and
Applications 7,
5
1(1958).
M. G. Holland, Physical Review 132, 2461 (1963).
W. J. de Haas and T. Biermasz, Physica 5, 619 (1938).
H. B. G. Casimir, Physica 5, 495 (1938).
6
J. M. Ziman, Electrons and phonons; the theory of transportphenomena in
3
4
solids (Clarendon Press, Oxford, 1960).
7
8
9
10
11
12
13
A. J. C. Ladd, B. Moran, and W. G. Hoover, Physical Review B 34, 5058
(1986).
S. G. Volz and G. Chen, Physical Review B 61, 2651 (2000).
J. Li, L. Porter, and S. Yip, Journal of Nuclear Materials 255, 139 (1998).
J. W. Che, T. Cagin, W. Q. Deng, and W. A. Goddard, Journal of Chemical
Physics 113, 6888 (2000).
C. Oligschleger and J. C. Schon, Physical Review B 59, 4125 (1999).
M. S. Green, Journal of Chemical Physics 22, 398 (1954).
R. Kubo, M. Yokota, and S. Nakajima, Journal of the Physical Society
of
Japan 12, 1203 (1957).
14
W. G. A. Hoover, W.T., in Theoretical Chemistry: Advances and Perspectives,
edited by H. Eyring and D. Henderson (Academic Press, New York, 1975),
Vol. 1, p. 1.
15
S. Kotake and S. Wakuri, JSME International Journal Series B-Fluids and
16
Thermal Engineering 37, 103 (1994).
T. Ikeshoji and B. Hafskjold, Molecular Physics 81, 251 (1994).
17
G. Chen, Nanoscale energy transportand conversion
: a paralleltreatment of
electrons, molecules, phonons, and photons (Oxford University Press, New
York, 2005).
18
19
20
21
A. S. Henry and G. Chen, Journal of Computational and Theoretical
Nanoscience 5, 141 (2008).
A. J. H. McGaughey and M. Kaviany, Physical Review B 69, 094303 (2004).
J. E. Turney, E. S. Landry, A. J. H. McGaughey, and C. H. Amon, Physical
Review B 79, 064301 (2009).
B. Qiu, H. Bao, G.
Q. Zhang, Y
Wu, and X. L. Ruan, Computational Materials
107
Science 53, 278 (2012).
22
D. A. Broido, A. Ward, and N. Mingo, Physical Review B 72, 014308 (2005).
23
M. E. Tuckerman, Journal of Physics-Condensed Matter 14, R1297 (2002).
H. Kim and M. Kaviany, Physical Review B 86, 045213 (2012).
D. Broido, M. Malorny, G. Birner, N. Mingo, and D. Stewart, Applied Physics
24
25
28
Letters 91, 231922 (2007).
G. Deinzer, G. Birner, and D. Strauch, Physical Review B 67, 144304 (2003).
K. Esfarjani and H. Stokes, Physical Review B 77, 144112 (2008).
K. Esfarjani, G. Chen, and H. Stokes, Physical Review B 84, 085204 (2011).
29
J. Garg, N. Bonini, B. Kozinsky, and N. Marzari, Physical Review Letters 106,
30
045901 (2011).
J. Garg, N. Bonini, and N. Marzari, Nano Letters 11, 5135 (2011).
Z. T. Tian, J. Garg, K. Esfarjani, T. Shiga, J. Shiomi, and G. Chen, Physical
26
27
31
32
Review B 85, 184303 (2012).
J. Shiomi, K. Esfarjani, and G. Chen, Physical Review B 84, 104302 (2011).
33
T. Shiga, J. Shiomi, J. Ma, 0. Delaire, T. Radzynski, A. Lusakowski, K.
Esfarjani, and G. Chen, Physical Review B 85, 155203 (2012).
34
T. Luo, J. Garg, J. Shiomi, K. Esfarjani, and G. Chen, Europhysics Letters 101,
35
16001 (2013).
A. J. Minnich, J. A. Johnson, A. J. Schmidt, K. Esfarjani, M. S. Dresselhaus, K.
A. Nelson, and G. Chen, Physical Review Letters 107, 095901 (2011).
36
37
D. Y. Li, Y. Y. Wu, P. Kim, L. Shi, P. D. Yang, and A. Majumdar, Applied
Physics Letters 83, 2934 (2003).
M. S. Dresselhaus, G. Chen, M. Y Tang, R. G. Yang, H. Lee, D. Z. Wang, Z. F.
38
A. J. Minnich, M. S. Dresselhaus, Z. F. Ren, and G. Chen, Energy
&
Ren, J. P. Fleurial, and P. Gogna, Advanced Materials 19, 1043 (2007).
Environmental Science 2, 466 (2009).
39
40
Z. T. Tian, S. Lee, and G. Chen, Journal of Heat Transfer 135, 061605 (2013).
J. Chen, G. Zhang, and B. W. Li, Nano Letters 10, 3978 (2010).
41
S. Plimpton, Journal of Computational Physics 117, 1 (1995).
42
J. Zi, K. M. Zhang, and X. D. Xie, Physical Review B 41, 12915 (1990).
43
44
F. H. Stillinger and T. A. Weber, Physical Review B 31, 5262 (1985).
R. Kubo, Journal of the Physical Society of Japan 12(1957).
45
H. J. Goldsmid, Introduction to Thermoelectricity (Springer, 2010).
46
G. J. Snyder and E. S. Toberer, Nature Materials 7, 105 (2008).
47
J. R.
Sootsman, D.
Y
Chung,
and M. G. Kanatzidis,
Angewandte
Chemie-International Edition 48, 8616 (2009).
48
J. P. Heremans, V. Jovovic, E. S. Toberer, A. Saramat, K. Kurosaki, A.
108
Charoenphakdee, S. Yamanaka, and G. J. Snyder, Science 321, 554 (2008).
49
M. Zebarjadi, K. Esfarjani, M. S. Dresselhaus, Z. F. Ren, and G. Chen, Energy
& Environmental Science 5, 5147 (2012).
50
51
Y. Z. Pei, X. Y. Shi, A. LaLonde, H. Wang, L. D. Chen, and G. J. Snyder,
Nature 473, 66 (2011).
J. P. Heremans, C. M. Thrush, and D. T. Morelli, Physical Review B 70,
115334 (2004).
Q.
He, J. Androulakis, M. G. Kanatzidis, and V. P. Dravid, Nano Letters 12,
52
J.
53
343 (2012).
J. Androulakis, I. Todorov, D. Y. Chung, S. Ballikaya, G. Y. Wang, C. Uher,
54
and M. Kanatzidis, Physical Review B 82, 115209 (2010).
J. Q. He, J. R. Sootsman, S. N. Girard, J. C. Zheng, J. G. Wen, Y. M. Zhu, M.
G. Kanatzidis, and V. P. Dravid, Journal of the American Chemical Society
132, 8669 (2010).
55
J. Q. He, S. N. Girard, M. G. Kanatzidis, and V. P. Dravid, Advanced
Functional Materials 20, 764 (2010).
56
57
D. Parker and D. J. Singh, Physical Review B 82, 035204 (2010).
Q. Y. Zhang, H. Wang, W. S. Liu, H. Z. Wang, B. Yu, Q. Zhang, Z. T. Tian, G.
Ni, S. Lee, K. Esfarjani, G. Chen, and Z. F. Ren, Energy & Environmental
58
59
Science 5, 5246 (2012).
H. Wang, Y. Z. Pei, A. D. LaLonde, and G. J. Snyder, Advanced Materials 23,
1366 (2011).
B. Poudel, Q. Hao, Y. Ma, Y. C. Lan, A. Minnich, B. Yu, X. A. Yan, D. Z.
Wang, A. Muto, D. Vashaee, X. Y Chen, J. M. Liu, M. S. Dresselhaus, G.
60
61
62
63
64
Chen, and Z. F. Ren, Science 320, 634 (2008).
G. Joshi, X. Yan, H. Z. Wang, W. S. Liu, G. Chen, and Z. F. Ren, Advanced
Energy Materials 1, 643 (2011).
X. A. Yan, G. Joshi, W. S. Liu, Y C. Lan, H. Wang, S. Lee, J. W. Simonson, S.
J. Poon, T. M. Tritt, G. Chen, and Z. F. Ren, Nano Letters 11, 556 (2011).
A. J. H. McGaughey and A. Jain, Applied Physics Letters 100, 061911 (2012).
B. Abeles, Physical Review 131, 1906 (1963).
A. F. Ioffe, S. V. Airapetiants, A. V. Ioffe, N. V. Kolomoetz, and L. S. Stilbans,
Doklady Akademii Nauk Sssr 106, 981 (1956).
65
I. Kudman, Journal of Materials Science 7, 1027 (1972).
66
Y. K. Koh, C. J. Vineis, S. D. Calawa, M. P. Walsh, and D. G. Cahill, Applied
Physics Letters 94, 153101 (2009).
67
Y. I. Ravich, B. A. Efimova, and I. A. Smirnov, Semiconducting Lead
Chalcogenides (Plenum, New York, 1970).
109
68
A. A. El-Sharkawy, A. M. Abou El-Azm, M. I. Kenawy, A. S. Hillal, and H. M.
Abu-Basha, International Journal of Thermophysics 4, 9 (1983).
69
70
E. D. Devyatkova and I. A. Smirnov, Sov. Phys. Solid State 3(1962).
T. Beechem, J. C. Duda, P. E. Hopkins, and P. M. Norris, Applied Physics
Letters 97, 061907 (2010).
71
D. P. Sellan, J. E. Turney, A. J. H. McGaughey, and C. H. Amon, Journal of
72
Applied Physics 108, 113524 (2010).
Z. T. Tian, K. Esfarjani, J. Shiomi, A. S. Henry, and G. Chen, Applied Physics
7
Letters 99, 053122 (2011).
A. Ward and D. A. Broido, Physical Review B 81, 085205 (2010).
0. Delaire, J. Ma, K. Marty, A. F. May, M. A. McGuire, M. H. Du, D. J. Singh,
74
A. Podlesnyak, G. Ehlers, M. D. Lumsden, and B. C. Sales, Nature Materials
10, 614 (2011).
75
S. Baroni, P. Giannozzi, and A. Testa, Physical Review Letters 58, 1861
(1987).
76
A. Debernardi, S. Baroni, and E. Molinari, Physical Review Letters 75, 1819
(1995).
X. Gonze, Physical Review A 52, 1086 (1995).
78
P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D.
Ceresoli, G. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli,
S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. KokaUj, M.
Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A.
Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. Seitsonen,
A. Smogunov, P. Umari, and R. Wentzcovitch, Journal of Physics-Condensed
Matter 21, ARTN 395502 (2009).
79
80
81
H. J. Monkhorst and J. D. Pack, Physical Review B 13, 5188 (1976).
X. Gonze and J. P. Vigneron, Physical Review B 39, 13120 (1989).
A. A. Maradudin and A. E. Fein, Physical Review 128, 2589 (1962).
82
R. A. Cowley, Reports on Progress in Physics 31, 123 (1968).
83
C. Dames and G. Chen, in Thermoelectrics handbook: macro to nano edited
by Rowe, David Michael (CRC/Taylor & Francis, Boca Raton, 2006).
85
S. Tamura, Physical Review B 27, 858 (1983).
P. R. Vijaraghavan, S. K. Sinha, and P. K. Iyengar, Proc.Nuc.Phys.and Solid
State Symp. 16(1973).
86
W. Cochran, R. A. Cowley, G. Dolling, and M. M. Elcombe, Proc. R. Soc.
84
87
Lond. A 293, 8 (1966).
J. M. An, A. Subedi, and D. J. Singh, Solid State Communications 148, 417
(2008).
110
88
0. Kilian, G. Allan, and L. Wirtz, Physical Review B 80, 245208 (2009).
89
Y. Zhang, X. Z. Ke, C. F. Chen, J. Yang, and P. R. C. Kent, Physical Review
B
80, 024304 (2009).
90
P. G. Klemens, Proceedings of the Royal Society of London Series
a-Mathematical and Physical Sciences 208, 108 (1951).
91
92
93
A. I. Boukai, Y Bunimovich, J. Tahir-Kheli, J. K. Yu, W. A. Goddard, and J. R.
Heath, Nature 451, 168 (2008).
A. I. Hochbaum, R. K. Chen, R. D. Delgado, W. J. Liang, E. C. Garnett, M.
Najarian, A. Majumdar, and P. D. Yang, Nature 451, 163 (2008).
K. Hippalgaonkar, B. Huang, R. Chen, K. Sawyer, P. Ercius, and A. Majumdar,
Nano Letters 10, 4341 (2010).
94
95
96
97
S. G. Volz and G. Chen, Applied Physics Letters 75, 2056 (1999).
N. Mingo, L. Yang, D. Li, and A. Majumdar, Nano Letters 3, 1713 (2003).
P. G. Murphy and J. E. Moore, Physical Review B 76, 155313 (2007).
I. Ponomareva, D. Srivastava, and M. Menon, Nano Letters 7, 1155 (2007).
99
R. Chen, A. I. Hochbaum, P. Murphy, J. Moore, P. D. Yang, and A. Majumdar,
Physical Review Letters 101, 105501 (2008).
D. Donadio and G. Galli, Physical Review Letters 102, 195901 (2009).
100
G. Chen, T. Zeng, T. Borca-Tasciuc, and D. Song, Materials Science and
98
Engineering a-Structural Materials Properties Microstructure and Processing
101
102
103
104
105
292, 155 (2000).
D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J.
Maris, R. Merlin, and Phillpot, Sr., Journal of Applied Physics 93, 793 (2003).
C. Dames and G. Chen, Journal of Applied Physics 95, 682 (2004).
L. H. Liang and B. W. Li, Physical Review B 73, 153303 (2006).
S. K. Bux, R. G. Blair, P. K. Gogna, H. Lee, G. Chen, M. S. Dresselhaus, R. B.
Kaner, and J. P. Fleurial, Advanced Functional Materials 19, 2445 (2009).
Y. F. Chen, D. Y Li, J. R. Lukes, and A. Majumdar, Journal of
Heat
Transfer-Transactions of the Asme 127, 1129 (2005).
106
Z. Tian, J. Garg, K. Esfarjani, T. Shiga, J. Shiomi, and G. Chen, Phys. Rev. B
85, ARTN 184303 (2012).
107
N. Mingo, D. Stewart, D. Broido, and D. Srivastava, Physical Review
B 77,
033418 (2008).
108
109
D. Stewart, I. Savic, and N. Mingo, Nano Letters 9, 81 (2009).
G. Fagas, A. Kozorezov, C. Lambert, J. Wigmore, A. Peacock, A. Poelaert,
and
110
R. den Hartog, Physical Review B 60, 6459 (1999).
H. Zhao and J. Freund, Journal of Applied Physics 105, 013515 (2009).
II
L. Sun and J. Murthy, Journal of Heat Transfer-Transactions of the ASME 132,
111
102403 (2010).
113
D. Kechrakos, Journal of Physics-Condensed Matter 3, 1443 (1991).
R. J. Stevens, L. V. Zhigilei, and P. M. Norris, International Journal of Heat
114
and Mass Transfer 50, 3977 (2007).
T. English, J. Duda, J. Smoyer, D. Jordan, P. Norris, and L. Zhigilei, Physical
112
117
Review B 85, ARTN 035438 (2012).
W. Little, Canadian Journal of Physics 37, 334 (1959).
E. Swartz and R. Pohl, Reviews of Modern Physics 61, 605 (1989).
P. Schelling, S. Phillpot, and P. Keblinski, Applied Physics Letters 80, 2484
118
(2002).
C. Kimmer, S. Aubry, A. Skye, and P. Schelling, Physical Review B 75, ARTN
115
116
119
120
144105 (2007).
P. Schelling and S. Phillpot, Journal of Applied Physics 93, 5377 (2003).
Z. Tian, S. Kim, Y. Sun, and B. White, Proceedings of the ASME InterPACK
125
Conference 2009, Vol 1, 607 (2009).
N. Zuckerman and J. Lukes, Physical Review B 77, 094302 (2008).
D. Young and H. Maris, Physical Review B 40, 3685 (1989).
S. Pettersson and G. Mahan, Physical Review B 42, 7386 (1990).
H. Zhao and J. Freund, Journal of Applied Physics 97, 024903 (2005).
J. Wang and J. Wang, Physical Review B 74, 054303 (2006).
126
S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University
127
Press, Cambridge, UK, 1997).
N. Mingo and L. Yang, Physical Review B 68, ARTN 245406 (2003).
N. Mingo and L. Yang, Physical Review B 70, ARTN 249901 (2004).
121
122
123
124
128
129
W. Zhang, T. Fisher, and N. Mingo, Journal of Heat Transfer-Transactions of
the ASME 129, 483 (2007).
130
W. Zhang, T. Fisher,
and N. Mingo, Numerical
Heat Transfer
Part
B-Fundamentals 51, 333 (2007).
131
S. Volz, Thermal Nanosystems and Nanomaterials 118, 1 (2009).
132
A. Dhar and D. Roy, Journal of Statistical Physics 125, 805 (2006).
133
P. Hopkins, P. Norris, M. Tsegaye, and A. Ghosh, Journal of Applied Physics
134
106, 063503 (2009).
X. Li and R. Yang, Journal of Physics-Condensed Matter 24, 155302 (2012).
135
M. Sancho, J. Sancho, and J. Rubio, Journal of Physics F-Metal Physics 15,
136
137
851 (1985).
R. Landauer, Philosophical Magazine 21, 863 (1970).
J. Katerberg, C. Reynolds, and A. Anderson, Physical Review B 16, 673
(1977).
112
138
139
140
141
J. P. Perdew and A. Zunger, Physical Review B 23, 5048 (1981).
E. S. Landry and A. J. H. McGaughey, Physical Review B 80, 165304 (2009).
Y. Kanamori, M. Sasaki, and K. Hane, Optics Letters 24, 1422 (1999).
Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y Peng, T. A. Liu, Y. K. Hsu, C. L.
142
Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen,
Nature Nanotechnology 2, 770 (2007).
K. Choi, S. H. Park, Y M. Song, Y T. Lee, C. K. Hwangbo, H. Yang, and H. S.
143
Lee, Advanced Materials 22, 3713 (2010).
K. C. Park, H. J. Choi, C. H. Chang, R. E. Cohen, G. H. McKinley, and G.
144
Barbastathis, Acs Nano 6, 3789 (2012).
K. Tai, L. Yang, Y Wang, J. Wynn, and A. Cho, Applied Physics Letters 56,
2496 (1990).
145
M. Peters, B. Thibeault, D. Young, J. Scott, F. Peters, A. Gossard, and L.
Coldren, Applied Physics Letters 63, 3411 (1993).
146
M. Hong, J. Mannaerts, J. Hong, R. Fischer, K. Tai, J. Kwo, J. Vandenberg, Y.
147
Wang, and J. Gamelin, Journal of Crystal Growth 111, 1071 (1991).
S. Chalmers, K. Lear, and K. Killeen, Applied Physics Letters 62, 1585
(1993).
148
S. M. Lee, D. G. Cahill, and R. Venkatasubramanian, Applied Physics Letters
149
70, 2957 (1997).
T. Borca-Tasciuc, W. L. Liu, J. L. Liu, T. F. Zeng, D. W. Song, C. D. Moore, G.
Chen, K. L. Wang, M. S. Goorsky, T. Radetic, R. Gronsky, T. Koga, and M. S.
Dresselhaus, Superlattices Microstruct. 28(2000).
150
151
152
E. S. Landry and A. J. H. McGaughey, Physical Review B 79, 075316 (2009).
G. Chen, Physical Review B 57, 14958 (1998).
M. N. Luckyanova, Garg, Jivtesh, Esfarjani, Keivan, Jandl, Adam, Bulsara,
Mayank T., Schmidt, Aaron J., Minnich, Austin J., Chen, Shuo, Dresselhaus,
Mildred S., Ren, Zhifeng, Fitzgerald, Eugene A., and Chen, Gang, 338, 936
(2012).
153
G. Chen, M. S. Dresselhaus, G. Dresselhaus, J. P. Fleurial, and T. Caillat,
International Materials Reviews 48, 45 (2003).
154
C. L. Tien and G. Chen, Journal of Heat Transfer 116, 799 (1994).
T. Borca-Tasciuc, W. L. Liu, J. L. Liu, T. F. Zeng, D. W. Song, C. D. Moore, G.
Chen, K. L. Wang, M. S. Goorsky, T. Radetic, R. Gronsky, T. Koga, and M. S.
155
Dresselhaus, Superlattices and Microstructures 28, 199 (2000).
156
W. S. Capinski, H. J. Maris, T. Ruf, M. Cardona, K. Ploog, and D. S. Katzer,
Physical Review B 59, 8105 (1999).
157
J. C. Caylor, K. Coonley, J. Stuart, T. Colpitts, and R. Venkatasubramanian,
113
Applied Physics Letters 87, 023105 (2005).
158
159
G. Chen, Journal of Heat Transfer-Transactions of the Asme 124, 320 (2002).
M. N. Luckyanova, J. Garg, K. Esfarjani, A. Jandl, M. T. Bulsara, A. J.
Schmidt, A. J. Minnich, S. Chen, M. S. Dresselhaus, Z. F. Ren, E. A.
Fitzgerald, and G. Chen, Science 338, 936 (2012).
160
J. Ravichandran, A. K. Yadav, R. Cheaito, P. B. Rossen, A. Soukiassian, S. J.
Suresha, J. C. Duda, B. M. Foley, C.-H. L. Ye Zhu, A. W. Lichtenberger, J. E.
Moore, D. A. Muller, D. G. Schlom, P. E. Hopkins, A. Majumdar, R. Ramesh,
and M. A. Zurbuchen, Nature Materials, 168 (2013).
161
162
163
164
165
166
167
168
169
170
171
172
173
174
G. Chen and M. Neagu, Applied Physics Letters 71, 2761 (1997).
S. Y Ren and J. D. Dow, Physical Review B 25, 3750 (1982).
P. Hyldgaard and G. D. Mahan, Physical Review B 56, 10754 (1997).
D. A. Broido and T. L. Reinecke, Physical Review B 70, 081310 (2004).
B. Yang and G. Chen, Physical Review B 67, 195311 (2003).
M. V. Simkin and G. D. Mahan, Physical Review Letters 84, 927 (2000).
J. Garg and G. Chen, Physical Review B 87, 140302 (2013).
S. Volz, J. B. Saulnier, G. Chen, and P. Beauchamp, High Temperatures-High
Pressures 32, 709 (2000).
X. B. Li and R. G. Yang, Physical Review B 86, 054305 (2012).
Z. T. Tian, K. Esfarjani, and G. Chen, Physical Review B 86, 235304 (2012).
P. E. Hopkins and J. R. Serrano, Physical Review B 80, 201408 (2009).
L. Hu, L. F. Zhang, M. Hu, J. S. Wang, B. W. Li, and P. Keblinski, Physical
Review B 81, 235427 (2010).
Z. T. Tian, B. E. White, and Y. Sun, Applied Physics Letters 96, 263113
(2010).
R. Y Wang, R. A. Segalman, and A. Majumdar, Applied Physics Letters 89,
173113 (2006).
175
176
177
P. E. Hopkins, M. Baraket, E. V. Barnat, T. E. Beechem, S. P. Kearney, J. C.
Duda, J. T. Robinson, and S. G. Walton, Nano Letters 12, 590 (2012).
K. C. Collins, S. Chen, and G. Chen, Applied Physics Letters 97, 083102
(2010).
M. D. Losego, M. E. Grady, N. R. Sottos, D. G. Cahill, and P. V. Braun,
Nature Materials 11, 502 (2012).
178
S. Lal, S. E. Clare, and N. J. Halas, Accounts of Chemical Research 41, 1842
(2008).
179
180
o. Neumann,
A. S. Urban, J. Day, S. Lal, P. Nordlander, and N. J. Halas, Acs
Nano 7, 42 (2013).
J. W. Gao, R. T. Zheng, H. Ohtani, D. S. Zhu, and G. Chen, Nano Letters 9,
114
181
4128 (2009).
R. T. Zheng, J. W. Gao, J. J. Wang, S. P. Feng, H. Ohtani, J. B. Wang, and G.
Chen, Nano Letters 12, 188 (2012).
182
R. Prasher, P. E. Phelan, and P. Bhattacharya, Nano Letters 6, 1529 (2006).
183
J. Y. Huang, J. Park, W. Wang, C. J. Murphy, and D. G. Cahill, Acs Nano 7,
184
185
186
589 (2013).
J. Park, J. Y Huang, W. Wang, C. J. Murphy, and D. G. Cahill, Journal of
Physical Chemistry C 116, 26335 (2012).
A. J. Schmidt, J. D. Alper, M. Chiesa, G. Chen, S. K. Das, and K.
Hamad-Schifferli, Journal of Physical Chemistry C 112, 13320 (2008).
0. M. Wilson, X. Y Hu, D. G. Cahill, and P. V. Braun, Physical Review B 66,
224301 (2002).
187
Z. B. Ge, D. G. Cahill, and P. V. Braun, Journal of Physical Chemistry B 108,
188
Z. B. Ge, D. G. Cahill, and P. V. Braun, Physical Review Letters 96, 186101
18870 (2004).
189
190
191
192
(2006).
J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides,
Chemical Reviews 105, 1103 (2005).
A. Schmidt, M. Chiesa, X. Y. Chen, and G. Chen, Review of Scientific
Instruments 79, 064902 (2008).
H. K. Lyeo and D. G. Cahill, Physical Review B 73, 144301 (2006).
D. D. Gandhi, M. Lane, Y Zhou, A. P. Singh, S. Nayak, U. Tisch, M.
Eizenberg, and G. Ramanath, Nature 447, 299 (2007).
193
S. Murad and I. K. Puri, Applied Physics Letters 92, 133105 (2008).
194
F. Bordi, M. Prato, 0. Cavalleri, C. Cametti, M. Canepa, and A. Gliozzi,
Journal of Physical Chemistry B 108, 20263 (2004).
195
Z. H. Wang, J. A. Carter, A. Lagutchev, Y. K. Koh, N. H. Seong, D. G. Cahill,
and D. D. Dlott, Science 317, 787 (2007).
196
T. F. Luo and J. R. Lloyd, Journal of Heat Transfer-Transactions of the Asme
132, 032401 (2010).
197
J. C. Duda, C. B. Saltonstall, P. M. Norris, and P. E. Hopkins, Journal of
Chemical Physics 134, 094704 (2011).
198
S. Alexander, 0. Entinwohlman, and R. Orbach, Physical Review B 34, 2726
(1986).
199
J. J. Freeman and A. C. Anderson, Physical Review B 34, 5684 (1986).
200
D. G. Cahill and R. 0. Pohl, Physical Review B 35, 4067 (1987).
J. M. Larkin and A. J. H. McGaughey, Journal of Applied Physics 114, 023507
201
(2013).
115