A Hybrid Ballistic-Diffusive Method to Solve the Frequency Dependent Boltzmann
Transport Equation
THESIS
Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in
the Graduate School of The Ohio State University
By
Pareekshith Allu
Graduate Program in Mechanical Engineering
The Ohio State University
2016
Master's Examination Committee:
Prof. Sandip Mazumder, Advisor
Prof. Seung Hyun Kim
Copyright by
Pareekshith Allu
2016
Abstract
Thermal related issues are among the most common causes for semiconductor
device failure. With the current generation microprocessors having an exceedingly high
power density, efficient heat removal from transistor to the chip surface level has been a
major bottleneck in further miniaturization performance improvements. As a result,
understanding the physical mechanisms underlying thermal transport at the device scales,
which usually range from a few tens of nanometers to a few micrometers, is critical to
designing efficient heat removal strategies. At room temperature, the energy carrying wave
packets (or phonons) in silicon have an average mean free path around 300nm, which is
comparable to or larger than the device length scales. Thermal transport in such regimes is
referred to as non-equilibrium transport and the Boltzmann Transport Equation (BTE) is a
powerful tool that can model heat conduction in non-equilibrium conditions. Although the
BTE has witnessed great success, it is an eight-dimensional partial differential equation
which makes it quite challenging and computationally intensive to solve.
In this study, a general-purpose deterministic hybrid ballistic-diffusive method is
developed to enhance the computational efficiency of solving the transient and frequency
dependent (non-gray) BTE. To achieve this, phonon transport is demarcated using a cutoff
Knudsen number into two distinct regimes namely, the ballistic (when phonon mean free
paths are larger than the device length scales) and diffusive (when phonon mean free paths
ii
are smaller than the device length scales) regimes. The original BTE is then discretized in
frequency space, and for all those spectral bands whose Knudsen number is larger than the
cutoff Knudsen number, the CADOM is applied, and in order to increase the efficiency,
the spherical harmonics (P1) approximation is invoked in the remainder of the spectral
bands. The end result is a hybrid methodology that is able to solve the BTE in all transport
regimes more efficiently than previously used methods while compromising little on
accuracy.
The accuracy and efficiency of the hybrid method is tested over a broad temperature
range spanning 195K – 205K, 245K – 255K and 295K – 305K for two- and threedimensional configurations both at steady state and under transient conditions. Overall, it
was found that the hybrid method is more accurate than the diffuse P1 approximation while
being computationally more efficient than the CADOM for transient simulation studies at
all temperature ranges. For steady state simulations, the hybrid method is more accurate
than the P1 method in operating temperature ranges of 195K – 205K and 245K – 255K,
while it performs similar to the P1 approximation in the diffuse regime at 295K – 305K.
Overall, the performance of the hybrid method can be tweaked by adjusting the cutoff
Knudsen number to yield the desired gains in either accuracy or efficiency.
The development of the hybrid method serves as a significant step toward
advancement of methods for solving the phonon BTE. Accurate and efficient simulation
of sub-micron thermal transport in semiconductors ultimately lead to better thermal
management strategies that can aid in the further advancement and miniaturization of
microprocessors in the semiconductor industry.
iii
Dedication
This document is dedicated to mom, dad and brother
iv
Acknowledgments
I would like to express sincere gratitude to my advisor, Prof. Sandip Mazumder,
who is responsible for some of the best education I have ever received throughout my life.
He has taught me the importance of meticulousness in one’s work, and his open door policy
in entertaining questions is truly exceptional. I am indebted to you, Prof. Mazumder, for
taking me on as a student and trusting in me. I would also like to thank Prof. Seung Hyun
Kim, for serving on my Master’s thesis committee.
The financial assistance provided by the Dept. of Mechanical Engineering
throughout my graduate studies has been invaluable and I am deeply grateful to the
Department Chairs for placing their belief in me.
This research would also not have completed in time without the help of my lab
mates, Ashraf and Navni, who were willing to entertain my thoughts and queries. I thank
you. Lastly, as a graduate student, a life outside the lab is often minimal. However, I wish
to express thanks to my dearest friends, Aakanksha, Zubin and Anshuman, for supporting
me and perhaps, more importantly, putting up with me.
v
Vita
May 2013 .......................................................B.Tech in Mechanical Engineering,
Indian Institute of Technology, Madras
June 2014 to present ......................................Graduate Research Associate, Fluid and
Thermal Analysis Lab, The Ohio State
University
Fields of Study
Major Field: Mechanical Engineering
vi
Table of Contents
Abstract ............................................................................................................................... ii
Dedication .......................................................................................................................... iv
Acknowledgments............................................................................................................... v
Vita..................................................................................................................................... vi
List of Tables ...................................................................................................................... x
List of Figures .................................................................................................................... xi
List of Symbols ................................................................................................................. xv
Greek ............................................................................................................................... xvii
Abbreviations ................................................................................................................. xviii
Chapter 1: Introduction ...................................................................................................... 1
1.1 Characterizing Non-Equilibrium Heat transport ....................................................... 4
1.2 Modeling of Non-Equilibrium Heat transport ........................................................... 7
1.3 Molecular Dynamics ................................................................................................. 8
1.4 BTE based methods ................................................................................................... 9
1.5 Numerical methods for solving the BTE................................................................. 11
vii
1.5.1 Monte Carlo Simulations............................................................................... 11
1.5.2 Deterministic Solutions ................................................................................. 12
1.6 Objectives of the thesis ........................................................................................... 20
1.7 Summary ................................................................................................................. 20
Chapter 2: Theory of Phonon Transport and Mathematical Models ............................... 22
2.1 Wave Propagation in Crystals ................................................................................. 22
2.2 Energy Quantization ................................................................................................ 24
2.3 Boltzmann Transport Equation (BTE) for Phonons ................................................ 25
2.4 Phonon Dispersion .................................................................................................. 29
2.5 Phonon Scattering ................................................................................................... 31
2.6 Deterministic Solutions of the BTE ........................................................................ 33
2.6.1 The Control Angle Discrete Ordinates Method (CADOM) .......................... 33
2.6.2 Boundary Conditions for the BTE (CADOM) .............................................. 37
2.6.3 Initial Conditions for the BTE (CADOM) .................................................... 38
2.6.4 Method of Spherical Harmonics.................................................................... 39
2.6.5 Boundary Conditions for the P1 Approximation ........................................... 45
2.6.6 Initial Conditions for the P1 Approximation ................................................. 47
2.6.7 Hybrid Method Formulation ......................................................................... 48
2.6.8 Boundary Conditions for the Hybrid Method Formulation........................... 50
2.6.9 Initial conditions for the Hybrid Method Formulation .................................. 51
2.7 Calculation of Heat Flux and Temperature ............................................................. 51
2.8 Solution Algorithm for hybrid method.................................................................... 55
2.9 Summary ................................................................................................................. 57
Chapter 3: Semi-Analytical Solutions to the One-Dimensional Gray BTE .................... 58
3.1 1D Gray Heat Conduction Problem ........................................................................ 58
3.1.1 Exact Analytical Solution .............................................................................. 59
3.1.2 P1 Approximation ......................................................................................... 69
viii
3.2 Results ..................................................................................................................... 71
3.3 Summary ................................................................................................................. 73
Chapter 4: Multi-dimensional solutions to the BTE ........................................................ 74
4.1 Finite Volume Solution Using the CADOM ........................................................... 74
4.2 Finite Volume Solution Using the P1 approximation ............................................. 77
4.3 Finite Volume Solution Using the Hybrid method ................................................. 81
4.4 Multi-Dimensional Simulations of Non-Gray Phonon BTE ................................... 82
4.4.1 2D Test Case with Structured Mesh .............................................................. 83
4.4.1.1 Steady State Simulations..................................................................... 85
4.4.1.2 Transient Analysis for the Same Test Cases ..................................... 107
4.4.2 2D Test Case with Unstructured Mesh........................................................ 122
4.4.3 3D Test Case with Unstructured Mesh........................................................ 128
4.5 Hybrid Method Applied to Problems with Large Temperature Differences ........ 131
4.6 Summary ............................................................................................................... 137
Chapter 5: Summary and Future Work .......................................................................... 138
5.1 Summary ............................................................................................................... 138
5.2 Future Work .......................................................................................................... 140
Bibiliography .................................................................................................................. 143
ix
List of Tables
Table 2.1 Parameters for curve-fitting the silicon dispersion data ................................... 30
Table 3.1 Non dimensional flux and temperature norm for 1D test case ......................... 72
Table 4.1 Summary of CPU time taken and percentage errors in top and bottom wall
fluxes for different methods ............................................................................................ 106
x
List of Figures
Figure 1.1 Regimes of heat transport .................................................................................. 6
Figure 2.1: Dispersion relations in a three-dimensional crystal ....................................... 23
Figure 2.2: Representation of solid angle with the unit direction vector, ŝ ..................... 28
Figure 2.3: Phonon dispersion data in the (100) direction for silicon. The lines are the
quadratic curve-fits to model the experimental data ..........................................................31
Figure 2.4: Directional nature of intensity shown in a) ballistic transport, and b) diffuse
transport ............................................................................................................................ 36
Figure 2.5: Overall solution algorithm for the hybrid method.......................................... 57
Figure 3.1: A 1-D parallel plate in dimensional coordinates with two isothermal walls.. 59
Figure 3.2: Coordinate system used in the exact analytical solution for the BTE solution
of the 1-D heat conduction problem ................................................................................. 60
Figure 4.1: An unstructured 2D stencil showing relevant variables ................................. 76
Figure 4.2: a) 3D unstructured mesh and b) basic triangular surface element that aid in
cross flux calculation for 3-D geometries ......................................................................... 80
Figure 4.3: Geometry and boundary conditions for Test case 1 ....................................... 83
Figure 4.4: Spectrum of Knudsen numbers at various operating temperatures ................ 86
Figure 4.5: Steady state non-dimensional C.L. temperature for CADOM ....................... 87
Figure 4.6: Error of centerline (C.L.) temperatures in comparison to CADOM enabled
BTE for the P1 and HYBRID methods............................................................................. 88
Figure 4.7: Plot for non-dimensional flux along the top wall ........................................... 90
Figure 4.8: Plot for non-dimensional flux along the bottom wall..................................... 90
xi
Figure 4.9: Centerline temperature errors in comparison to CADOM with nondimensional distance ......................................................................................................... 91
Figure 4.10 Plot for non-dimensional flux distribution along the bottom wall using
different cutoffs for the hybrid method ............................................................................. 92
Figure 4.11: Spectral variation of Knudsen numbers for the 195K- 205K test case. ....... 93
Figure 4.12: Temperature contour plots for all methods used in test case 195K - 205 .... 95
Figure 4.13: Spectral variation of Knudsen numbers for the 245K- 255K test case ........ 96
Figure 4.14: a) Steady state non-dimensional C.L. temperature using CADOM and b)
Error of centerline (C.L.) temperatures in comparison to CADOM for the P1 and hybrid
methods ............................................................................................................................. 97
Figure 4.15: a) Top and b) Bottom flux distributions at steady state using CADOM, P1
and hybrid methods ........................................................................................................... 97
Figure 4.16: (a) Centerline temperature errors in comparison to CADOM, and (b) Plot for
non-dimensional flux distribution along the bottom wall different cutoffs for the hybrid
method............................................................................................................................... 98
Figure 4.17: Temperature contour plots for all methods in test case: 245K - 255K....... 100
Figure 4.18: a) Steady state non-dimensional C.L. temperature using CADOM and b)
Error of centerline (C.L.) temperatures in comparison to CADOM for the P1 and hybrid
methods ........................................................................................................................... 101
Figure 4.19: a) Top and b) Bottom flux distributions at steady state using CADOM, P1
and hybrid methods ......................................................................................................... 102
Figure 4.20: Spectral variation of Knudsen numbers for the 295K- 305K test case ...... 103
Figure 4.21: (a) Centerline temperature errors in comparison to CADOM, and (b) Plot for
non-dimensional flux distribution along the bottom wall different cutoffs for the hybrid
method............................................................................................................................. 104
Figure 4.22: Temperature contour plots for all methods in test case 295K - 305K ........ 105
Figure 4.23: a) Top and b) Bottom flux values at time instant  max 100 using CADOM, P1
and hybrid methods. ........................................................................................................ 108
Figure 4.24: a) Top and b) Bottom flux values at time  max 10 using CADOM, P1 and
hybrid methods................................................................................................................ 109
xii
Figure 4.25: a) Top and b) Bottom flux values at time  max 2 using CADOM, P1 and
hybrid methods................................................................................................................ 109
Figure 4.26: a) Centerline temperatures using the CADOM at all time steps and b) the
center line temperature errors at all time steps for P1 and hybrid methods. ................... 110
Figure 4.27: Contour plots of domain temperature using the hybrid method at various
instances of time for test case 195K – 205K................................................................... 112
Figure 4.28: a) Top and b) Bottom flux values at time  max 100 using CADOM, P1 and
hybrid methods................................................................................................................ 113
Figure 4.29: a) Top and b) Bottom flux values at time  max 10 using CADOM, P1 and
hybrid methods................................................................................................................ 113
Figure 4.30: a) Top and b) Bottom flux values at time  max 2 using CADOM, P1 and
hybrid methods................................................................................................................ 114
Figure 4.31: a) Centerline temperatures using the CADOM at all time steps and b) the
center line temperature errors at all time steps for P1 and hybrid methods. ................... 115
Figure 4.32: Contour plots of domain temperature using the hybrid method at various
instances of time for test case 245K – 255K................................................................... 117
Figure 4.33: a) Top and b) Bottom flux values at time  max 100 using CADOM, P1 and
hybrid methods................................................................................................................ 118
Figure 4.34: a) Top and b) Bottom flux values at time  max 10 using CADOM, P1 and
hybrid methods................................................................................................................ 118
Figure 4.35: a) Top and b) Bottom flux values at time  max 2 using CADOM, P1 and
hybrid methods................................................................................................................ 119
Figure 4.36: a) Centerline temperatures using the CADOM at all time steps and b) the
center line temperature errors at all time steps for P1 and hybrid methods. ................... 119
Figure 4.37: Contour plots of domain temperature using the hybrid method at various
instances of time for test case 295K – 305K................................................................... 121
Figure 4.38: 2D geometry with unstructured mesh ........................................................ 122
Figure 4.39: a) Top and b) Bottom fluxes at steady state for all three methods ............. 123
xiii
Figure 4.40: Contour plots of steady state domain temperature for the hybrid, CADOM
and P1 methods. .............................................................................................................. 124
Figure 4.41: Bottom wall flux evolution with time ........................................................ 125
Figure 4.42: Top wall flux evolution with time .............................................................. 126
Figure 4.43: Contour plots of temperature evolution using the CADOM. ..................... 126
Figure 4.44: Contour plots of temperature evolution using the Hybrid method ............. 127
Figure 4.45: Contour plots of temperature evolution using P1 method .......................... 127
Figure 4.46: Unstructured 3D mesh ................................................................................ 128
Figure 4.47: Two sliced planes depicting mesh details .................................................. 129
Figure 4.48: Contour profiles of temperature with different methods for 3D test case .. 130
Figure 4.49: Transient contour profiles of temperature using hybrid methods for 3D test
case .................................................................................................................................. 130
Figure 4.50: a) Steady state non-dimensional C.L. temperature using CADOM and b)
Error of centerline (C.L.) temperatures in comparison to CADOM for the P1 and hybrid
methods ........................................................................................................................... 132
Figure 4.51: a) Top and b) Bottom wall fluxes at steady state for all three methods ..... 134
Figure 4.52: Spectral variation of Knudsen numbers for the 250K- 350K test case ...... 135
Figure 4.53: Contour plots of domain temperature at steady state for the hybrid, CADOM
and P1 methods. .............................................................................................................. 136
xiv
List of Symbols
a
Lattice constant [m]
A, Af
Area [m2]
b
Reciprocal lattice vector [m-1]
BL
Constant used in relaxation time calculation
BTN
Constant used in relaxation time calculation
BTU
Constant used in relaxation time calculation
C
Heat capacity [J K-1]
D
Density of states [J-1 m3]
f
Distribution function
f0
Equilibrium phonon distribution function
Fij
Interatomic force between atoms i and j [N]
G
Integrated phonon intensity [W m-2]
Dirac constant = 1.0546 x 10-34 [m2 kg.s-1]
I
Phonon intesnsity [W m-2 sr-1]
I lm
Position dependent co-efficients in intensity expansion expression
K
Wave vector [m-1]
J0
Direction independent co-efficient in P1 approximation
J1
Direction dependent co-efficient in P1 approximation
kB
Boltzmann constant = 1.381 x 10-23 [m2 kg s-2 K-1]
Kn
Knudsen number
L
Characteristic length [m]
Le
Length of an edge of a face [m]
lf
Vector joining the centers of two neighboring cells [m]
xv
n̂
Unit surface normal
n
Normal to an edge of a face in 3-D [m]
Nb
Number of spectral (frequency) bins
Ndir
Number of angular directions
N
Angular discretization in the azimuthal direction
N
Angular discretization in the polar direction
p
Phonon polarization
Pl m
Associated Legendre Polynomials
q
Heat flux vector [W m-2]
qgen
Volumetric heat generation rate [W m-3]
r
Position vector [m]
ŝ
Unit direction vector
t
Time [s]
tˆ1 , tˆ2
Unit surface tangents
tf
Tangent to a face in 2-D [m]
T
Absolute temperature [K]
t*
Non-dimensional time
U
Internal energy per unit volume [J m-3]
V
Volume [m3]
vg
Phonon group velocity vector [m s-1]
x, y, z
Global coordinates [m]
x*, y*, z*
Non-dimensional space co-ordinates
xvi
Greek

Degree of specularity
t
Time step [s]
i
Bandwidth of a spectral interval [rad s-1]
k
Thermal conductivity [W m-1 K-1]

Mean free path [m]

Polar angle [rad]

Density [kg m-3]
P
Stefan-Boltzmann constant for phonons [W m-2 K-4]

Overall relaxation time [s]
N
Relaxation time for Normal (N) scattering [s]
 NU
Combined relaxation time for N and U scattering [s]
U
Relaxation time for Umklapp (U) scattering [s]

Angular frequency [rad s-1]

Azimuthal angle [rad]
i, j
Interatomic potential [N m]

Solid angle [sr]

Acoustic thickness

Unit tensor
f
Distance between two-sides straddling face, f along the normal [m]
xvii
Abbreviations
BTE
Boltzmann Transport Equation
BZ
Brillouin Zone
CADOM
Control-Angle Discrete Ordinate Method
CFD
Computational Fluid Dynamics
CPU
Central Processing Unit
DOM
Discrete Ordinates Method
EPRT
Equation of Phonon Radiative Transport
I.T.R.S.
International Technology Roadmap for Semiconductors
LA
Longitudinal Acoustic
LBM
Lattice Boltzmann Method
LO
Longitudinal Optical
MC
Monte-Carlo
MD
Molecular Dynamics
MEMS
Micro-Electro-Mechanical Systems
PN
Nth-order spherical harmonics approximation
SN
Nth-order discrete ordinates method
TA
Transverse Acoustic
TO
Transverse Optical
2-D
Two-dimensional
3-D
Three-dimensional
xviii
Chapter 1: Introduction
The mantra “faster, cheaper and smaller microprocessors every generation,” by
which the semiconductor industry abides, is under severe threat due to device overheating
issues. Semiconductor device engineers have been successful in keeping pace with
Moore’s law, which predicted that every 18 months, chip performance and transistor count
would double. However, the advances made toward cooling techniques for efficient heat
removal strategies have not kept up with the rate of processor advancements and they act
as a severe bottleneck in further miniaturization and performance improvement. Currently,
IBM’s POWER8 processor is believed to generate upwards of 150 W/cm2 making it
approach the heat dissipation rate of a nuclear fission reactor [1]. Thus, inefficient heat
dissipation and thermally induced stresses are critical factors that inhibit scaling-up and
performance of several devices including microprocessors, power semiconductors, light
emitting diodes (LEDs) and microelectromechanical systems (MEMS) [2]. For example,
current investigations in LED research span a range of issues, from understanding reasons
for high operating temperatures to the causes for light degradation with time [3].
Consequently, to design effective cooling techniques it is vital to understand the underlying
physical mechanisms for propagation of heat transfer in semiconductor devices.
With the device length scales ranging from a few tens of nanometers to a few
microns, experimental studies at these length scales to analyze heat conduction are near
1
impossible to perform. Techniques based on Atomic Force Microscopy (AFM) provide
nanometer-scale spatial resolutions which are just beginning to be effective in examining
heat transport in nanoscale structures [4]. Currently, nanosecond thermoreflectance
techniques are gaining popularity, where the thermal properties of metals are characterized
by optically measuring transient responses to nanosecond heating pulses [5]. However,
such experimental techniques are not able to fully characterize sub-micron scale heat
transport and are still in nascent stages.
The semiconductor industry’s attention has dutifully turned to modeling and
simulation to aid the design of thermal management strategies for next generation
micro/nanoscale devices [2, 6-8]. Existing software tools for such analyses use a
combination of electrical, thermal and mechanical modeling. Within the industry, the
packaging engineers utilize continuum-based equations such as the Fourier law of heat
conduction to model heat transport and consequently, design thermal management
strategies at a packaging level. Existing thermal analysis software tailored to the
microelectronics industry, essentially perform these functions and in addition, take into
account heat transfer by conduction, convection and radiation [2]. However, heat
generation occurs at the sub-micron scale within the transistors. As a result, heat
conduction is a multi-level phenomena that spans heat generation at the transistor scale, to
heat removal at the packaging scale in a semiconductor device. As a result, there is a
paramount need to have appropriate modelling and analysis tools at each length scale. In
this study, we focus on developing appropriate modeling techniques to analyze heat
generation in the transistor scale.
2
At the sub-micron scale, simulations to model heat transfer based on the Fourier
law of heat conduction break down and become invalid [9, 10]. To understand the reasons
behind the failure of the Fourier law, it is necessary to examine the underlying mechanisms
of heat propagation at sub-micron length scales.
The primary mode of heat conduction in a semiconductor material is through lattice
vibrations. The lattice vibrations result in travelling waves with variable frequencies and
speeds causing energy to propagate within the solid. These travelling waves, from a
quantum mechanical perspective, are known as phonons. Due to the variable frequencies
and speeds of phonons, the phonons have a spectral mean free path in the range of 10 –
1000nm with an average mean free path of 300nm at room temperature [11]. When the
device length scale is comparable to the phonon mean free path, thermodynamic
equilibrium ceases to exist and continuum-based phenomenological laws break down.
Furthermore, temperature which is only defined at equilibrium, loses its conventional
meaning and may only be interpreted as a measure of the average energy of the system.
Since temperature is not defined at non-equilibrium, bulk thermal conductivity, which is
an input to the Fourier law of heat conduction, is defined only when thermodynamic
equilibrium exists, and so it becomes invalid in characterizing thermal properties of the
material [12]. At these length scales, the boundary scattering of phonons becomes
important forcing the thermal conductivity to depend on the size and shape of the
semiconductor devices, in addition to temperature. Typically, the Fourier law is then
limited to situations where [13]:
a) The continuum-based laws can be used to model heat transfer.
3
b) The energy carrier transport is diffuse. This implies that the interface/boundary
scattering effects for the carriers are insignificant compared to particle scattering
events.
c) Material properties like bulk thermal conductivity,  , are valid.
In summary, when thermodynamic equilibrium cases to exist, heat transfer cannot
be described by continuum-based phenomenological laws, such as the Fourier law, and
thermal transport in such regimes is referred to as non-equilibrium transport.
1.1 Characterizing Non-Equilibrium Heat transport
The parameter used to characterize non-equilibrium heat transport is called the
Knudsen number. The Knudsen number, Kn , is defined as
Kn 
 | vg | 

L
L
(1.1)
where  is the mean free path of the phonons, which is defined as the average distance
travelled by a phonon prior to interaction with another phonon, L , is the characteristic
length scale of the system (within which heat transport is being considered), vg , is the
phonon group velocity, and  , is the relaxation time constant, which is defined as an
average time taken for the phonon energy distribution to relax to equilibrium. This occurs
via scattering of phonons. The acoustic thickness,  , is another parameter used to
characterize non-equilibrium transport. It is simply the inverse of the Knudsen number and
is a popular form of characterization analogous to the optical thickness used in radiation
literature [12]. It is defined as

1
L
L
 
Kn  | vg | 
4
(1.2)
For this study, the Knudsen number is used as the primary parameter to define
various regimes of heat transport. When Kn  1 , i.e. the mean free path of the phonons,
 , is larger than the characteristic length, L, scattering of the phonons is rare and they
travel between the system’s geometric boundaries largely unhindered. This nature of
phonon transport is termed as “ballistic transport.” The ballistic regime of transport is
typically characterized by high Kn numbers. On the other end of the transport regime is
the so called diffusive regime. This is characterized by small Knudsen numbers, i.e.,
Kn  1. Here, the mean free path of the phonons is small compared to the geometric
dimension of the system and so the transport is characterized by numerous scattering events
thereby taking the system to equilibrium. This regime is termed “diffuse” and the Fourier
law of heat conduction is valid here. These regimes are better represented in Fig. 1.1. In
real materials, the phonon group velocities and relaxation time constants are dependent of
frequency. As a result, the Knudsen numbers are frequency dependent in a real material
and thus show a spectral variation.
Figure 1.1 Regimes of heat transport
5
It is important to capture the detailed physics of phonon transport when equilibrium
ceases to exist. Thus, developing effective models to capture non-equilibrium heat
transport is vital for the instances when we consider [12],

Heat conduction in semiconductor devices, since the geometric length scale, L and
the phonon mean free paths,  are comparable. The subsequent study of test cases
in this thesis mainly deals with this regime.

Processes with low temperatures such as those in the cryogenic range (< 100K),
where the mean free paths of the phonons becomes comparable to length scales of
the physical system.

The relaxation time scale,  , is longer than the time scale for actual heat
propagation, such that, the process is non-local in time. Such processes include
short laser pulse driven manufacturing.
1.2 Modeling of Non-Equilibrium Heat transport
Recall that the Fourier law is written as
q (r )   (T )T (r )
(1.3)
where  is the thermal conductivity, q is the heat flux vector, r is the position vector
and T is the absolute temperature. For no heat generation in a static medium, the First law
of thermodynamics states that
U
  q
t
(1.4)
where U is the internal energy per unit volume of the medium and  q is the divergence
of heat flux. Substituting Eq. (1.3) into Eq. (1.4), we obtain
6
U
  ( T )
t
(1.5)
The transient heat conduction equation in a medium can then be defined as
C
T
  ( T )
t
(1.6)
where C is the heat capacity and the thermodynamic relation, dU  CdT has been utilized
in Eq. (1.6).
Equation (1.6) is a partial differential equation with an infinite wave speed, which
results in an instantaneous transfer of energy. However, since the phonons have a finite
group velocity, vg , with which they propagate, the Fourier law of heat conduction cannot
be used in places where the energy transfer is not instantaneous, since it disregards the
wave nature of phonon propagation. As a result, Eq. (1.6) models heat conduction well in
the diffuse regime, where the scattering time scales are small. However, it falls short in the
ballistic regime.
To overcome the limitations associated with the instantaneous heat transfer as
predicted by the Fourier law of heat conduction, a modification to Eq. (1.6) was proposed
by Catteneo in 1958 [14]. A second order transient term was added to make the equation
hyperbolic and thus impart it a finite wave speed of propagation. The Cattaneo equation is
then given as
 2T
T
C 2  C
  ( T )
t
t
(1.7)
where the value of  imparts the damped-wave equation a finite wave propagation speed,
given by
k
C
. Equation (1.7) assumes that the time scales of interest are of the order of
7
relaxation time, whereas the length scales are much larger than the characteristic size for
local thermodynamic equilibrium. This makes the Cattaneo equation nonlocal in time but
local in space. Therefore, in the high Knudsen number regime, the ballistic behavior of
phonons is still not captured effectively. In addition, the frequency dependent behavior of
phonons also cannot be effectively modelled using Eq. (1.7).
To circumvent the inadequacies of the afore-mentioned models, several approaches
were developed to model phonon transport. The major ones being a) Molecular Dynamics
(MD) simulations, and b) phonon Boltzmann Transport Equation (BTE) based methods. A
brief description of MD simulations and BTE based methods are presented next.
1.3 Molecular Dynamics
Modeling thermal transport in semiconductor devices using classical molecular
dynamics is becoming increasingly popular. We begin by considering a system of N
particles, each with a mass, M , and located at position, ri . Newton’s second law of motion
for each atom yields,
d 2 ri
M 2 
dt
N
F
j , j i
i, j
(1.8)
where Fi , j is the interatomic force between atoms ' i ' and ' j , ' modeled using empirical
potentials. There are two families of MD simulations based on the physical model used,
namely, “classical MD” and “quantum MD.” In the “classical MD” approach, the potentials
are found by treating the particles as classical objects, with each vibrational mode equally
excited and contributing to the energy of the system. The “quantum MD” model on the
other hand is governed by quantum statistical mechanics and the potentials are determined
8
by ab initio or first principal calculations. Though it is computationally more intensive, it
can effectively model physics that is overlooked by the classical model [4, 15-16].
Many thermal transport studies have been published using the Leonard Jones interatomic potentials which yield qualitative insights. For silicon, three-body potentials such
as the Stillinger-Weber potentials have also been used. Once the MD trajectories have been
computed, the thermal conductivity for dielectric materials are derived from two
techniques: (a) Green Kubo (GK) approach which is an equilibrium method and (b) a direct
application of Fourier law which is a steady state non-equilibrium approach [13]. In the
GK method, using classical statistical thermodynamics, a time-correlation expression for
thermal conductivity is developed. Thermal conductivity is related to the time taken for
fluctuations of the heat current vector to dissipate. The “direct method” on the other hand,
mimics an experiment by imposing a temperature gradient on the system and then
determining the thermal conductivity from Fourier’s law.
Although MD simulations are powerful in modeling non-equilibrium transport,
they are computationally intensive and are limited to simulating tens of nanometers in
terms of domain size. For a silicon film of size 50 nm, around a million atoms need to be
tracked simultaneously for accurate modeling and as a result, the inter-atomic force
calculations derived from the multi-body potentials are very expensive [10, 17, 18]. Thus,
MD simulations become impractical for analyses at the device level.
1.4 BTE based methods
The semi-classical Boltzmann Transport Equation has long been used to model
phonon transport. It is popular because of its ability to model heat conduction in nonlocality
9
and non-equilibrium in both space and time [9]. Its application is valid as long as the
phonons are considered to be discrete particles i.e., quantum wave effects can be neglected.
However, when the characteristic length of the system, L , is lower than the de Broglie
wavelength of the atomic nuclei of the material, the particle assumption fails and one would
have to incorporate wave effects, such as interference [10].
As far as analytical solutions to the BTE are concerned, Callaway [19, 20] proposed
a phenomenological model based on a “gray” assumption, where phonon dispersion is not
included and no distinction is made between the relaxation times of the longitudinal
acoustic (LA) and transverse acoustic (TA) phonons. This model was utilized to calculate
the thermal conductivities at low temperatures. Holland [21, 22] worked on it further to
include phonon dispersion effects, as well as distinct frequency and temperature dependent
scattering time scales for the transverse and longitudinal phonons. However, for each
phonon polarization, he assumed a linear dispersion relationship (no dispersion), resulting
in a constant group velocity. On account of the afore-mentioned assumptions, the
formulations gave inaccurate estimates for bulk thermal conductivity, which in turn overpredicted the temperature in comparison to experimental observations. In addition, these
approaches did not analyze the transient behavior of the system.
In order to accurately model the phonon transport mechanisms and to account for
the energy exchange between different phonon polarizations and wave vectors, dispersion
relations for phonon transport were incorporated into transport calculations [16]. For
silicon, Chung et al. [23] described the importance of various curve fits for phonon
dispersion data in predicting the thermal conductivity of materials. He assumed a quadratic
10
curve fit for LA phonons and a cubic curve fit for TA phonons. The frequency dependent
BTE that incorporates phonon dispersion relations is called a non-gray BTE. More on
phonon dispersion shall be described later in Chapter 2. The non-gray phonon BTE is an
eight dimensional partial differential equation, as will be shown later, which renders it
extremely challenging to solve. In the past decade, there have been huge strides in the field
of modern computing which spawned off great interest in solving the non-gray BTE
equations using numerical methods.
1.5 Numerical methods for solving the BTE
Generally, the phonon BTE is numerically solved using two approaches: a)
stochastic methods such as the Monte Carlo method, b) deterministic discretization based
methods.
1.5.1 Monte Carlo Simulations
Prior to solving the phonon BTE, Monte Carlo (MC) simulations have been utilized
in gas dynamics and electron transport simulations. The earliest reported work that
developed MC simulations for phonon BTE solutions was done by Klistner, who worked
on heat transport in the ballistic limit [24]. Peterson [25], then used a 1-D array of 40 cells
to model steady state and transient heat transport incorporating the linear Debye theory.
Subsequently, Mazumder and Majumdar [26] built upon Peterson’s work to include dual
phonon polarizations and non-linear dispersion relations while accounting for transition
between the various phonon polarizations. This study, though pivotal, did fall short in
accounting for energy propagation by optical phonons. Other studies [27] and also Mittal
and Mazumder [28] subsequently included the effects of optical phonon polarization in
11
their MC simulations to better account for energy propagation at higher temperatures. The
study by Mittal and Mazumder [28] had, however, reported that the statistical errors were
largest in the diffuse regime (~300 K for Si) when the number of probability events is
largest, a quantity that is computed in MC simulations. In order to bring down the statistical
errors to 7% from 22%, they reportedly had to increase the number of phonon samples
from 50,000 to 500,000. This increased the computational time required by around a factor
of eight, making it computationally expensive.
Perhaps, the most efficient Monte Carlo simulations for the phonon BTE have been
carried out by Peraud and Hadjiconstnatinou [29]. They reduced statistical uncertainty in
the sampling solution by introducing a variance-reduced formulation. This formulation is
based on the concept of control variates which relies on the fact that signal strength is
linked to deviation from equilibrium. Through this approach, the authors reduced the
computational cost associated with small deviations from equilibrium energy. They were
able to successfully model three-dimensional and transient heat conduction problems.
However, based on computational requirements, the feasibility of MC simulations for
increasingly complex 3-D calculations make them difficult to be implemented. Therefore,
we shall now look at some of the deterministic solutions that were developed to solve the
BTE.
1.5.2 Deterministic Solutions
Before we discuss some of the more widely used deterministic formulations to
solve the BTE, the Lattice Boltzmann Method (LBM) and its application to the BTE is
described first. The LBM was initially used to describe fluid flow by Succi [30]. It was
12
later formulated to solve the phonon BTE by Zhang and Fisher [31]. Their results showed
close agreement with both continuum and sub-continuum solutions of the 1D heat transfer
problem in thin films. Amon and co-workers [32, 33] then utilized several forms of phonon
lattices and made better approximations to simulate phonon transport. However, the
effectiveness of the LBM in modelling phonon transport is still limited to simple 2-D
geometries due to the exponential growth of computational requirements with geometric
sizes.
It can be said that the advent of deterministic solutions to the BTE began in 1993,
when Majumdar proposed the Equation of Phonon Radiative Transport (EPRT) [34] by
drawing an analogy between the BTE and the radiative transport equation [35]. He showed
that by applying the EPRT, in a macroscopic or acoustically thick medium (low Knudsen
number regime), one would recover the Fourier Law. Furthermore, in the acoustically thin
limit, the EPRT would yield the blackbody radiation flux.
Deterministic methods to solve the phonon BTE in multi-dimensional geometry
were first popularized and implemented by Murthy and her group [36, 37]. The Discrete
Ordinates Method (DOM) [38, 39] or the so called SN approximation, which was used to
solve neutron transport and radiation problems, has been adapted by this group to solve the
gray BTE in Field Effect Transistors, so as to predict the thermal field. The phonon BTE
solves for the intensity which is a directional quantity and as a result, angular discretization
is necessary. Over the years, the shift was made from DOM to the Control Angle Discrete
Ordinates Method (CADOM) so as to mitigate so-called “ray effects” and “false scattering”
that were observed while solving the BTE using DOM [40, 41]. Instead of applying a finite
13
differencing of the solid angle as in the DOM, the CADOM uses finite solid angle volumes
for directional discretization [35]. In the high Knudsen number regime, the CADOM
outperforms the DOM, since it is less sensitive to ray effects. There was also further work
done to improve accuracy and convergence through higher order robust spatial
discretization as outlined in the SMART scheme [42].
Though the CADOM is highly accurate in solving the BTE, it is still
computationally expensive. The reasons lie with the BTE being solved by traversing
through each direction band, frequency interval, polarization and spatial coordinate, and
then ultimately being coupled through the lattice temperature, which is obtained by solving
the energy equation. In semiconductor devices, the characteristic lengths typically range
from a few micrometers to a few nanometers. As a result, for most BTE simulations of
semiconductor devices, the spectral Knudsen number varies over a few orders of magnitude
spanning the diffuse, transition and ballistic regimes. In the ballistic or the acoustically thin
regime, there is a need for a high number of angular discretization to resolve the directional
dependence of intensity and obtain sufficient accuracy [16, 36-37]. On the other hand,
within the diffuse or the acoustically thick regimes, poor convergence of the CADOM
results from strong inter-BTE coupling.
With these complications in mind, to effectively solve the BTE in 3D geometries
of practical interest like transistors, Ali et al. [43] have pointed out the necessity for
parallelization. From simulating a full transistor, they estimated that nearly 1600 GB of
runtime memory is required to store all the unknowns in double precision. They also
estimated that for transient simulations of the BTE, around 1017 floating point operations
14
are required, and thus justified the need for parallel computation. Clearly, there is a
paramount need to resort to techniques that either speedup convergence of the BTE
solutions or reduce the total number of equations to be solved without sacrificing accuracy
in the flux and temperature predictions, especially when parallel computing cores are
unavailable.
One such method [44], proposed as an acceleration scheme of the BTE, is based on
a coupled ordinates method that was initially developed by Mathur and Murthy [45] to
solve the radiative transfer equation. In this method, at any location in the physical space,
the intensity equations for all frequency bands and the lattice temperature equation are
solved simultaneously for the BTE. This procedure was stated to improve the coupling
between various frequency bands. To then improve spatial coupling, the authors used the
coupling procedure as a sweep in a coarser grid while employing the geometric multigrid
method. Considerable acceleration in overall convergence was then reported. The method
is apparently best suited to problems with isotropic scattering, however, this is not a
significant disadvantage and is a valid approximation for scattering in pure materials.
The coupled ordinates procedure solves the BTE in its exact form and does provide
considerable acceleration. Simultaneously, there also have been numerous attempts to
solve for an approximate form of the BTE. Such methods attempt to reduce the number of
discretized equations to be solved, so as to improve computational efficiency without
sacrificing accuracy. In the discussion to follow, some of these methods that deal with
approximations to the BTE, are discussed.
15
To circumvent the complications with using high numbers of angular discretization
to resolve the solid angle space in a pure BTE method, the method of spherical harmonics
(or PN approximation) was developed. For example, in a 3D geometry, to effectively
resolve the angular dependence of the phonon intensity, the solid angle space is discretized
into approximately 400 directions. By circumventing this angular discretization, the
number of discretized PDEs to be solved can be drastically reduced especially when the
phonon intensity propagation in diffuse. The method of spherical harmonics uses spherical
basis functions, specifically the Legendre polynomials to analytically model the angular
variation of the intensity. The lowest order of spherical harmonics, namely the P 1
approximation reduces the BTE to a single Helmholtz equation with Robin boundary
conditions. This makes the P1 approximation an attractive choice over the CADOM to
solve the BTE, since it brings down the number of equations to be solved by several
hundreds, through elimination of angular discretization. However, it is only in the low
Knudsen number regime, when the phonon transport is diffuse, that the P1 approximation
is most effective. The method is known to yield unacceptable accuracy for solutions to the
BTE in the intermediate and the high Knudsen number regimes, where the directional
dependence of intensity is strong [31, 46], as will also be demonstrated in this work.
To overcome the shortfalls of the P1 approximation, researchers have developed
various hybrid schemes that tried to combine features of the direct and the modified BTE
methods to reduce the solution time without sacrificing accuracy. One such method is
based on the Modified Differential Approximation (MDA), proposed by Modest for
radiative heat transfer problems [35]. It was first developed by Olfe [47] in 1967 to model
16
radiative transfer of radiation-gas dynamics problems. This hybrid method based on the
MDA aims to remove the flaws of P1 approximation for intermediate and high Knudsen
numbers.
The MDA method splits the intensity of the energy carrier into ballistic and
diffusive components. A surface-to-surface exchange formulation employing geometric
view factors is used to calculate the ballistic component. The diffusive component is
evaluated by invoking the spherical harmonics formulation, generally the P1
approximation. This hybrid formulation which uses separate methodologies for the ballistic
and diffusive components is expected to be accurate at all Knudsen numbers. In the early
2000’s, Chen and co-workers [48-50] utilized the MDA approach to design the BallisticDiffusive Equations (BDE) to describe phonon transport. However, as pointed out by
Mittal and Mazumder [51], their approach has a few shortcomings. For one, they used
material properties like specific heat capacity as an input to the BDE, which makes a
comparison to the BTE difficult since it doesn’t require such inputs. In addition, the
formulation of the boundary conditions is confusing with the introduction of artificial
ballistic and media temperatures, which only serve as mathematical artifacts in the
governing equations. Lastly, the ballistic equation solver which employed the surface to
surface exchange formulation is expensive and cumbersome for complex 3-D geometries,
thus preventing the method from scaling up effectively.
The MDA approach has also inspired the formulation of a hybrid discrete ordinatesspherical harmonics solution by Mittal and Mazumder [51]. They solved the gray phonon
BTE similarly, by splitting the phonon intensities into two components. The ballistic
17
component is then solved using a CADOM based formulation while the diffusive
component is solved with the P1 approximation. Further progress in this work incorporating
non gray effects has not been undertaken because of an apparent violation of energy
conservation, while combining the ballistic and diffusive fluxes, which arises out of
ineffective coupling of the two intensity equations.
Another type of hybrid formulation developed to solve the BTE in a reduced form
was proposed by Loy et al. [52], which is based on a Knudsen number cutoff. The phonon
BTE is discretized in the frequency space and thus has a characteristic Knudsen number
for each spectral band by virtue of a frequency dependent group velocity and a relaxation
time scale. By choosing an appropriate cutoff Knudsen number, the spectral bands of the
BTE are solved either by a direct BTE solver if the spectral band’s Knudsen number is
higher than the imposed cutoff Knudsen number or by a Modified Fourier Equation (MFE)
if the spectral band’s Knudsen number is lower than the cutoff. Though the concept is
novel, there are a few issues that make their formulation confusing. The researchers assign
a phonon frequency band based temperature and also a lattice temperature which are
different from the thermodynamic temperature. Secondly, they also approximate the
energy density functions in the BTE using a linearized temperature difference over which
the specific heat is assumed to be a constant. In addition, while formulating the isothermal
boundary conditions, an effective temperature is introduced at the walls for the MFE to
capture the wall temperature jumps. Lastly, while solving for effective lattice temperature,
the polarization and BTE band/MFE band specific temperature is multiplied with a
dimensionless quantity called the lattice ratio. The lattice temperature is then dependent on
18
the contribution of each band and also on the band specific heat capacity. In spite of these
formulations, the authors still managed to achieve reasonably accurate solutions. In the
solution procedure, the BTEs and the MFEs in the respective bands were solved
sequentially to compute the lattice temperature. To further speed up convergence, the
authors attempted to solve the equations implicitly by coupling the MFE and the lattice
temperature equation using a multigrid approach. This showed better performance than the
sequential method. However, the research was restricted to steady state simulations in
simple 2D geometries.
Despite the use of artificial temperatures and formulations to forcefully include
temperature jumps at the boundaries, the concept behind the Knudsen number cutoff is still
novel and is expected to yield significant acceleration with reasonable accuracy as reported
by the authors. As a result, the philosophy of the Knudsen number cutoff has been utilized
in the present study and further built upon.
Upon discretization of the phonon BTE in frequency space, a hybrid methodology
is developed in this study, which solves for the exact BTE using the CADOM for all those
spectral bands whose Knudsen number is greater than the cutoff Knudsen number, and the
P1 approximation, for the remainder of the spectrally discretized bands. A rigorous analysis
of a gray, 1D BTE helps identify an appropriate cutoff Knudsen number to be used in the
hybrid methodology, as will be explained later. What sets this study further apart is the
development of a hybrid methodology that is capable of performing transient simulations
of sub-micron scale heat transfer in 3D geometries on unstructured meshes, for any
temperature difference imposed on the medium.
19
1.6 Objectives of the thesis
The specific objectives of this thesis are to:

To perform studies aimed at determining an appropriate cutoff Knudsen number to
be utilized in the hybrid method formulation.

To develop a general purpose hybrid ballistic-diffusive methodology that can
perform full transient simulations of the non-gray phonon BTE in 3D geometries on
unstructured meshes.

To compare the computational efficiency and accuracy of the hybrid method to
the CADOM-based solution of pure BTE as well as the P1 approximation.
1.7 Summary and Organization of Thesis
In this chapter, various methods that have been developed over the years to solve
the phonon BTE are discussed. The rationale behind the development of a hybrid
formulation based on a cutoff Knudsen number is also stated.
In order to achieve the afore-mentioned objectives, the thesis is divided into several
chapters. Chapter 2 deals with phonon transport theory and also the mathematical models
to solve the BTE using CADOM, the P1 approximation and the hybrid methodology are
described in detail. Expressions for phonon dispersion relations and scattering time scales
are also introduced.
In the Chapter 3, a rigorous analysis of the one-dimensional, gray phonon BTE is
performed to identify an appropriate cutoff Knudsen number. Using an analytical solution
and the P1 approximation, the BTE is solved so as to compare the fluxes at several Knudsen
numbers.
20
Chapter 4 introduces the finite volume formulations of the CADOM, P1
approximation and the hybrid methodology in order to solve the BTE. The accuracy and
efficiency of the hybrid methodology is then evaluated with respect to both CADOM and
P1 method for a variety of test cases.
A brief summary of the work is presented in Chapter 6, followed by
recommendations for future work.
21
Chapter 2: Theory of Phonon Transport and Mathematical Models
Phonons are the prevalent carriers of heat in semiconductor materials. Nonequilibrium conduction of heat occurs when the scattering time scales for phonons are
larger than the time required by the phonons to traverse the characteristic length of the
semiconductor device under scrutiny. It is in this regime that the Fourier law of heat
conduction breaks down, and the bulk thermal conductivity expressions used in the Fourier
law of heat conduction become invalid. In such situations, the Boltzmann Transport
Equation (BTE) may be used to model heat conduction effectively. In the subsequent
sections, some of the theory behind phonon transport and the solutions to the BTE are
described.
2.1 Wave Propagation in Crystals
A crystal structure can be modeled as a three-dimensional arrangement of masses
and springs that represent the atoms and the chemical bonds in a lattice respectively.
Energy transport in such crystalline materials takes place by vibrations of atoms in the
lattice. This energy of lattice vibrations is quantized and each quantum of energy is called
a phonon. By assuming a travelling wave equation to describe the vibrations of atoms,
relations between the frequency of vibration,  , and wave vector space, K , can be
obtained. Such a relation is called the dispersion relation. Depending on the degrees of
freedom for each atom in the crystal lattice, phonons can have several polarizations,
22
resulting in as many dispersion relations. In a three dimensional crystal, there are six
phonon polarizations, namely three ‘acoustic’ and three ‘optical,’ as shown in Fig. 2.1.
These relations repeat in the region for | K |   a , where a refers to the physical
dimension of a unit cell, called the lattice constant.
Figure 2.1: Dispersion relations in a three-dimensional crystal [53]
A quantity called the group velocity, vg , which describes the velocity with which
the wave packets (phonons) carry energy within the medium is defined as [54]
vg   K 
(2.1)
If we assume a linear dispersion relation, the group velocity becomes a constant. This is
also referred to as “no dispersion.” This is often referred to as no dispersion. It can also be
seen in Fig. 2.1 that by virtue of Eq. (2.1), the group velocity of the optical branch is nearly
zero or negligible compared to that of the acoustic branch.
23
2.2 Energy quantization
Quantum mechanics restricts the number of energy states that can be occupied
during vibration of atoms in a lattice. As a result, the energy is said to be quantized and
each such quantum (or phonon) carries energy,  . Here,
is the Dirac constant, or the
reduced Planck constant   1.054 1034 J s  . Using Quantum statistics, Bose formulated
the equilibrium distribution function, f 0 , in order to define the equilibrium number of
phonons in a solid at a given temperature, T . The distribution function, f 0 , defines the
number density of phonons in discrete energy states at equilibrium in a crystal lattice for a
particular temperature. f 0 is derived using Bose-Einstein statistics [54], and is given by
f0 
1
 
exp 
 k BT

 1

(2.2)
where k B is the Boltzmann constant ( 1.3806 1023 J s) and  is the angular frequency
in rad s .
The phonon density of states, D  , p  , is defined as the number of vibrational
states per unit frequency per unit volume of the lattice for each polarization. This is an
important quantity used to compute the internal energy of a crystal since it quantifies the
number of states that can be occupied by the phonons in each frequency interval per unit
volume. From the definitions of the equilibrium distribution function and the density of
sates, one can then write the internal energy in a material of volume, V  L3 as
U

1
 f0  1 2
V p 

24
K , p D   , p  d 
(2.3)
where U is the energy per unit volume of the crystal and p refers to the particular branch
of phonon mode or polarization. The quantity, D , p  d , represents the number of
vibrational states between  and   d for any polarization, p . The second term within
the parenthesis on the right side of Eq. (2.3) is the zero-point energy. The wave vector, K ,
takes discrete values in a crystal lattice, but in the limit of a large crystal, the wave-vector
space becomes dense and the discrete summation can then be replaced by an integral. In
deriving the energy equation given by Eq. (2.3), the integration over wave vector space
was replaced by the frequency space by using the dispersion relationship, since it is valid
to assume that the wave vector space is isotropic, i.e., K  K , ,   , where  and  are
the polar and azimuthal angles used to describe a unit direction vector, as will be shown
later.
Though a particle description of phonons with characteristic polarizations and wave
vectors is useful in predicting the crystal energy of the lattice, some aspects of the wave
description such as the dispersion behavior are still required to effectively model phonon
behavior and heat propagation.
2.3 Boltzmann Transport Equation (BTE) for Phonons
Phonons can be modeled using the BTE since the heat carriers interact via short
range forces or scattering processes and also follow the statistical Bose Einstein
distribution. The BTE for phonons is written as
f
 f 
 vg f   
t
 t  scattering
25
(2.4)
where f is the statistical distribution function of an ensemble of phonons, which describes
the number density of phonons at time, t , position vector, r , and wave vector, K , and is
represented as [54]
f (t , r , K )dK  number of phonons in dK
(2.5)
The terms on the left hand side of Eq. (2.4) are called drift terms, whereas the right
 f 
hand side term,  
contributes to restoring the equilibrium Bose-Einstein
 t  scattering
distribution through phonon scattering events. The most widely used approximation to
simplify the scattering term is the single relaxation time approximation and is written as
f f
 f 
 0
 t 

scattering
(2.6)
The BTE under the single relaxation time approximation maybe then be re-written as
f f
f
 vg f  0
t

(2.7)
where  is the overall relaxation time scale of the phonons due to all scattering processes
in combination and f 0 is the aforementioned equilibrium Bose-Einstein distribution.
For an isotropic wave vector space, the distribution function, f , is a function of
eight independent variables, i.e., f  f (t , r , sˆ, , p) , where the wave vector space, K , is
now expressed using the independent variables,  and ŝ for each phonon polarization, p.
Here, the position vector, r has three components in three-dimensional space and the unit
direction vector, ŝ , maybe expressed using two independent variables in spherical
coordinates: the polar angle,  , and the azimuthal angle,  . The polarization, p , is a
26
discrete variable since they represent phonons in various polarized states. The equilibrium
distribution function, f 0 , is independent of direction and polarization and is expressed as
f0  f0 (t , r ,  ). The group velocity, vg is a function of direction, angular frequency and
polarization, i.e., vg  vg (sˆ, , p) , whereas the relaxation time-scale,  is a function of
angular frequency, temperature and polarization, i.e.,    (, T , p) . The single relaxation
time approximation linearizes the BTE and stipulates that a system is restored to
equilibrium following an exponential decay law: f  f 0  exp(t  ) .
Phonons, unlike electrons or molecules, do not follow number conservation.
Instead they follow energy conservation similar to photons. To better represent the energy
carried by phonons, the distribution based phonon BTE is transformed into an equation
containing phonon intensity, I , which is related to the distribution function, f through
[9]
I  I (t , r , sˆ, , p)  | vg |  fD(, p) 4
(2.8)
The spectral equilibrium phonon intensity, I 0 , can also be defined as
I 0  I 0 (t , r , , p)  | vg |  f 0 D(, p) 4
(2.9)
Substituting Equations (2.8) and (2.9) into Eq. (2.7), allows us to express the BTE in terms
of phonon intensity as
I I
I
 vg I  0
t

27
(2.10)
Recognizing that vg is the group velocity vector and maybe written as vg  | vg | sˆ , intensity
is then simply the flux of energy per unit time in the direction of phonon propagation and
per unit frequency interval around  [34]. Equation (2.10) can be re-written as
I I
I
 | vg |   ( Isˆ)  0
t

(2.11)
Since sˆ  sˆ( ,  ) and the variables  ,  , x, y and z are all independent of each other, the
unit direction vector can be taken inside the divergence operator. The physical
interpretation of phonon intensity can be better understood from Fig. 2.2.
Figure 2.2: Representation of solid angle with the unit direction vector, ŝ .
Since, the intensity, I in Eq. (2.8) is directly related to the distribution function,
f , intensity is also a function of eight independent variables namely space, direction,
angular frequency, time and polarization. I 0 , on the other hand is direction independent,
and so, is only a function of space, angular frequency, time and polarization.
28
2.4 Phonon Dispersion
To solve the phonon BTE, the phonon dispersion relationship is required. The
dispersion relationship describes the exact relation between momentum and energy of a
phonon. With the inclusion of dispersion, the phonon mean free paths become dependent
on angular frequency and polarization through non-constant phonon group velocities and
relaxation time scales. Thus, simulating a microscale heat conduction problem would result
in Knudsen numbers becoming spectrally dependent. These spectral Knudsen numbers
vary over a few orders of magnitude, depending on the temperature and geometric
dimensions of the material at hand. This variation of spectral Knudsen numbers indicates
that the heat transport is typically characterized by both diffusive and ballistic transport.
The dispersion curves are obtained prior to solving the BTE, so that the group velocity and
phonon density of states can be extracted and used as inputs to the BTE.
Phonon dispersion curves can either be obtained by neutron scattering experiments
or by ab initio techniques. Experimental dispersion data for bulk silicon has been
documented by Brockhouse [55] and Dolling [56]. Ab initio techniques have been used by
Giannozzi et al. [57] and Wei and Chou [58]. The use of bulk phonon dispersion
relationships becomes invalid in cases where the geometric length scales of the medium
become comparable to the atomic bond length. For silicon, this typically happens when the
device length scales are 10-20 nm thick [59]. However, for the length scales considered in
this study, bulk silicon dispersion relations hold true and are used. Each phonon dispersion
branch is then modeled using a quadratic curve fit expression under the isotropic Brillouin
Zone (BZ) approximation [59]. This quadratic relationship is given by
29
K  0  vs K  c K
2
(2.12)
In Eq. (2.12), for the acoustic modes, the constants 0 , vs and c are chosen to capture the
slope of the dispersion curve near the center of the BZ and also to obtain the maximum
frequency at the edge of the BZ. To evaluate the constants for optical phonons, the
magnitude of the maximum frequency at the BZ boundary and the zero slope at the BZ
boundary have been utilized. The quadratic expressions ensure adequate accuracy in
modeling the dispersion data and are also easily invertible to obtain the phonon wave vector
and group velocity. In Table 2.1, the curve-fit parameters for silicon in the (100) direction
are listed [61]. The dispersion relationships for the various phonon polarizations obtained
using the curve-fits are shown in Fig. 2.3.
Phonon
Polarization
LA
TA
LO
TO
0 (1013 rad / s)
vs (102 m / s)
c(107 m2 s)
0
0
9.88
10.20
9.01
5.23
0.00
-2.57
-2
-2.26
-1.60
1.12
Table 2.1: Parameters for curve-fitting the silicon dispersion data [60].
30
Figure 2.3: Phonon dispersion data in the (100) direction for silicon. The lines are the
quadratic curve-fits for the experimental data [61].
In this study, the role of optical phonons is not considered since they are known to
contribute little to thermal transport by virtue of having a low occupation number and
negligible group velocity, as pointed out by Mazumder and Majumdar [26].
2.5 Phonon Scattering
Phonon scattering maybe of two kinds: (i) elastic scattering where both energy and
momentum are conserved, and (ii) inelastic scattering, where only energy is conserved.
Scattering of phonons can happen due to numerous mechanisms such as lattice
imperfections, electron-phonon interactions, phonon-phonon interactions (or intrinsic
scattering) and interactions with geometries boundaries. Both impurity and boundary
scattering are elastic scattering processes.
The most prevalent scattering process is the intrinsic phonon-phonon scattering
process. Intrinsic scattering involves three or more phonons, which occurs due to the
anharmonicity in the interatomic forces [11]. Three-phonon scattering is broadly
31
categorized into Normal (N) and Umklapp (U) processes. The energy and momentum
conservation rules dictate that [51]
   '   '' ( Normal  Umklapp)
(2.13)
K  K '  K '' ( Normal )
(2.14)
K  K '  K '' b (Umklapp)
(2.15)
where  ,  ' and  '' are the angular frequencies and K , K ' and K '' are the wave vectors
of the interacting phonons. During these processes, two phonons may combine to give a
third or a single phonon can split into two. It is only in the Umklapp process that momentum
is not conserved, and the difference of wave vectors results in the reciprocal lattice vector,
b . On the other hand, momentum is conserved for the Normal process. Energy is conserved
in both processes.
Among the two processes, the momentum conserving Normal processes do not
pose any resistance to the heat flow, and it is only the Umklapp process that is directly
responsible for the thermal resistance. The effect of Normal processes comes into play
indirectly since they may create a sufficient number of high- K phonons to participate in
the U processes [62]. As a result, it is important to consider the contributions of both these
processes in determining the overall scattering time scale. In this study, the scattering time
scales proposed by Holland [21] are used. These expressions, which were developed for
both the acoustic modes, are frequency and temperature dependent.
1
 NU
 BL 2T 3 ( LA, Normal  Umklapp)
(2.16)
 N1  BTN T 4 (TA, Normal )
(2.17)
32
TA,Umklapp for   1/2
0
2
 BTU  / (sinh(  ) / (kBT ))
 U1  
TA,Umklapp for   1/2
(2.18)
where 1/2 is the frequency corresponding to  K Kmax   0.5 and for the material Silicon,
BL  2 1024 K 3rad 2 s,
BTN  9.3 1013 K 4 rad 1
and
BTU  5.5 1018 rad 2 s
are
constants which are obtained through calibration against experimental data [21]. The two
major assumptions made by Holland in formulating these time scale expressions are that
only high frequency TA phonons undergo Umklapp processes and that LA phonons do not
undergo U processes [63]. With these assumptions, the time scales for the N processes are
obtained from Eq. (2.16) and those for the U processes are derived from Eq. (2.17). The
overall relaxation time is then calculated using the Mathiessen’s rule as [9]:
1


1
N

1
U
(2.19)
Now that the two inputs, namely the phonon dispersion relations and the scattering
time scales have been discussed, numerical solution procedures for the BTE are described
next.
2.6 Deterministic Solutions of the BTE
In this section, the CADOM and the P1 approximation for solving the BTE are
discussed. Subsequently, the methodology for the hybrid scheme shall also be introduced.
2.6.1 The Control Angle Discrete Ordinates Method (CADOM)
The control angle discrete ordinates method (CADOM) is an extension of the
discrete ordinates method (DOM), also called the S N method. The discrete ordinates
method was first proposed by the renowned physicist S. Chandrasekhar [64] and further
33
advanced by Fiveland [39, 65-66] and Truelove [67] for radiative transport. The DOM
solves for the directional variation of intensity along a set of discrete directions spanning
the total solid angle, 4 . It is essentially an application of finite differencing in directional
space [12]. However, one of the major drawbacks associated with the DOM is the so called
“ray effect.” Ray effect is characterized by the streaking behavior of directional intensities
in the ballistic regime. Furthermore, the DOM sometimes does not ensure energy
conservation because simple quadrature is used for angular discretization [35]. As a result,
the method was improved by using a finite volume type approach for angular discretization,
which is referred to as the CADOM in this study. The CADOM solves the BTE using exact
integration to evaluate the solid angle integrals and is similar to the finite volume
discretization used in physical space. The CADOM method is described next.
In order to discretize the solid angle, the three dimensional angular space, 4 , is
divided into distinct non overlapping solid angles,  i . The centroids for these discrete
solid angles are denoted by the direction vector, ŝ . The polar angle,  and the azimuthal
angle,  , are then uniformly divided into N and N angles, respectively, in the spherical
coordinate system. This is shown in Fig. 2.2. The direction vector, ŝ is then expressed as
sˆ  sin  sin  iˆ  sin  cos  ˆj  cos  kˆ
(2.20)
and the subtended solid angle,  i is written as
i  
 
2
sin  d d  2sin  sin 
 


i

34



(2.21)
where  and  are the differential extents of the polar and the azimuthal angles
respectively. The discretized form of the BTE using CADOM along with the boundary and
initial conditions are described next. For a detailed explanation, the reader is referred to
Murthy and Mathur [68] or Mittal [12].
Using Eq. (2.11), the BTE at each frequency and phonon polarization can be
expressed as
I , p
t
 | vg | , p   ( I , p sˆ) 
I 0, , p  I , p
, p
(2.22)
where the subscripts,  and p imply that the equation is to be applied at any frequency,
 , and any polarization, p . To discretize the equation in frequency space, each phonon
polarization is discretized into a number of spectral bins. For each such bin, the average
intensity, I pj is computed using Eq. (2.8), which yields
I pj 


| vg | , p  fD( , p)
j
4
d
(2.23)
where the superscript, j is the index of the spectral band under consideration. Equation
(2.22) can then be re-written in terms of the spectral bins for each polarization branch,
regardless of whether they are LA or TA phonons. This implies
I pj
t
 | v |   ( I sˆ) 
j
g p
j
p
I 0,j p  I pj
 pj
j  1, 2,....N P
(2.24)
where N P is the total number of such spectral bins that are considered in each of the
phonon polarizations, and the spectral equilibrium distribution, I 0,j p is defined as
35
I 0.j p 

| vg | , p  f 0 D( , p)
4
 j
d
(2.25)
In the set of Equations (2.23-2.25), each spectral band uses an average value for the
modulus of group velocity, | vg | , p and the phonon relaxation time,   , p depending on the
spectral band,  j .
Discretizing Eq. (2.24) over angular space gives the directionally dependent, and
spectrally discretized intensity equation [12]
I i ,j p
t
 | vg | pj   ( I i ,j p sˆi ) 
I 0,j p  I i ,j p
i  1, 2,....N dir

j  1, 2,....N P
j
p
(2.26)
where the subscript, i , refers to the direction along which the equation is being solved. The
finite volume procedure to spatially discretize and numerically solve Eq. (2.26) for multidimensional problems is later described in Chapter 4.
The spectral integrated intensity field can be computed from the directional
intensity as
N dir
G   I i ,j p
j
p
(2.27)
i 1
Summing up Eq. (2.27) over all spectral bands and polarizations, the total integrated
intensity is defined as
NP
G   G pj
p
j 1
36
(2.28)
The total integrated intensity is required to calculate the divergence of heat flux from which
a pseudo-temperature is estimated, as will be shown later. Next, the boundary and initial
conditions required to solve the phonon BTE are considered.
2.6.2 Boundary Conditions for the BTE (CADOM)
To solve for the intensity in Eq. (2.22), only one boundary condition is required
since it is a first order partial differential equation in space. In the present study, only one
kind of a boundary is simulated, the thermalizing boundary. A thermalizing boundary or
an isothermal wall is analogous to a blackbody in thermal radiation. The thermalizing
boundary absorbs all incoming phonon intensity and emits into the domain an equilibrium
intensity distribution based on the boundary temperature which is diffuse in nature. In
equation form this can be expressed as
I , p (t , rw , sˆ)  I 0, , p (t , rw )
(2.29)
where rw is the position vector of the thermalizing isothermal boundary.
Another kind of boundary frequently encountered is the reflective boundary, or the
adiabatic wall. For such boundaries, the incident phonons can undergo specular, fully
diffuse or even partially diffuse (or partially specular) reflections. For a fully specular
reflection, the reflected intensity is computed from
I , p (t , rw , sˆr )  I , p (t , rw , sˆi )
(2.30)
where the reflected direction vector, sˆr is defined as
sˆr  sˆi  2 | sˆi nˆ | nˆ
37
(2.31)
where n̂ is the unit normal vector pointing out of the surface into the domain. Most often,
real surfaces only exhibit a partially diffuse (or specular) behavior, where only a part of the
incident intensity undergoes a specular reflection. Defining a degree of specularity,  , the
reflective boundary condition becomes
I , p (rw , sˆr )   I , p (rw , sˆi ) 
(1   )


I , p sˆi nˆ d 
(2.32)
sˆ nˆ  0
By choosing a value of either 0 or 1 for  in Eq. (2.32), the fully diffuse and fully specular
boundary conditions are recovered respectively.
2.6.3 Initial Conditions for the BTE (CADOM)
Since Eq. (2.22) also contains temporal terms, initial conditions are required. The
transient term in Eq. (2.22) is first order. Therefore, only one initial condition for intensity
is required. Generally, for heat conduction problems, the initial temperature distribution is
known. Under the assumption that the physical system is initially at thermodynamic
equilibrium, intensity can be initialized as the equilibrium intensity distribution at each
phonon frequency. This implies
I (t  0, r , sˆ, , p)  I 0 (t  0, r , , p)
(2.33)
Though the CADOM approach has no approximations other than angular
discretization, the computational requirements are often prohibitive, especially while
solving 3D transient problems [43]. Part of the reason lies in the fact that for each spectral
band in Eq. (2.26), there are a total of N dir equations to be solved. For example, in a 3-D
test case, nearly 400 angular (20 azimuthal and 20 polar) discretizations are necessary to
resolve the solid angle space effectively in order to mitigate ray effects. Such large numbers
38
of angular discretization are necessary especially when the intensity propagation is highly
Cold Patch
Cold Patch
directional, as seen in part (a) of Fig. 2.4, where the phonon transport is ballistic.
Hot Patch
Hot Patch
Figure 2.4: Directional nature of intensity shown in a) ballistic transport, and b)
diffuse transport
To circumvent the usage of N dir equations for each spectral band and the associated
computational costs, the method of spherical harmonics was developed as an
approximation to the BTE. This method is especially useful when the propagation of
phonon intensity is diffuse as seen in part (b) of Fig. 2.4.
2.6.4 Method of Spherical Harmonics
Jeans [69] first proposed the method of spherical harmonics to model the equations
of radiative energy transfer in 1917. In the 1950’s, several researchers namely, Kourganoff
[70], Murray [71] and Davidson [72] further developed it and applied the method to
transport problems in neutron diffusion. According to the spherical harmonics theory, the
intensity field, I (r , sˆ, , p) at any point r in the medium, may be visualized as the value
of a scalar function on the surface of a sphere of unit radius, which is centered at the
location r [35]. The intensity field, I is then expressed in terms of a two-dimensional
Fourier series [35] as
39

l
I (r , sˆ, , p)    I l m (r , , p) Yl m ( sˆ, , p)
(2.34)
l 0 m  l
where the intensity field has been split into a) location dependent coefficients, Il m (r , , p),
and b) spherical harmonics namely, Yl m (sˆ, , p) , defined as
1/2
Yl ( sˆ, , p)  (1)
m
( m |m|)/2
 (l  | m |)!  im |m|

 e Pl (cos  , , p)
 (l  | m |)! 
(2.35)
where the unit directional vector, ŝ , has been expressed in terms of the polar and azimuthal
angles,  and  , respectively, and Pl |m| are the associated Legendre polynomials.
The series in Eq. (2.34) is truncated by setting the coefficients, I l m to zero for all
l  1 to derive the lowest order of spherical harmonics, namely the P1 (P1) approximation.
With this approximation, the lowest order spherical harmonics function corresponding to
N  1 results in a single partial differential equation as described in detail by Modest [35]
and Mittal [12], which the reader is referred to. The P1 approximation is given as
I , p (t , r , sˆ)  J 0, , p (t , r )  J1, , p (t , r )  sˆ
(2.36)
Prior to application of the P1 approximation to the phonon BTE, Eq. (2.22) is nondimensionalized to aid in the derivation of the integrated intensity equation using the P1
approximation. Multiplying Eq. (2.22) throughout by   , p we obtain
, p
I , p
t
   , p vg
, p
   I , p sˆ   I 0, , p  I , p
Next, the non-dimensional variables are defined as,
40
(2.37)
t* 
t

, p





 , p 
t   , p t *
t t *
(2.38)
dx


and
   , p | vg | , p
 | vg | , p
x *
x *
0 , p
(2.39)
and
dx
dx* 
  , p | vg | , p
x
 x*  
since the spatial and temporal variation of   , p has to be accounted for. This transforms
Eq. (2.37) into a non-dimensional form written as
I , p
t *
  *   I , p sˆ   I 0, , p  I , p
(2.40)
Equation (2.40) is the modified Boltzmann Equation in the spatial coordinates, x*,
y*, z* and time, t*. Therefore, the intensity, I , p is written as, I , p  I , p (t*, r *, sˆ ) where,
t * is the non-dimensional time and r * is the non-dimensional position vector. Hereon,
the ' ' sign is dropped to simplify the notations. The equations in the remainder of the subsection (2.6.4) without the starred variables shall therefore be considered in nondimensional form, until the equations later revert to the original dimensional form. Next,
the P1 approximation is applied to simplify the Boltzmann transport equation at each
frequency and polarization. Substituting Eq. (2.36) in Eq. (2.40) yields

( J 0, , p  J1, , p  sˆ)    ( J 0, , p  J1, , p  sˆ) sˆ   I 0, , p  ( J 0, , p  J1, , p  sˆ)
t
(2.41)
At this point, for simplicity of notation, the '  ' and the ' p ' subscripts shall also be dropped
from Eq. (2.41) and therefore, J 0 and J1 are used instead of J 0, , p and J1, , p . Next, by
integrating Eq. (2.41) over the solid angle, 4 , we obtain
41
4

0
4

( J 0  J1  sˆ) d      ( J 0  J1  sˆ) sˆ  d  
t
0
4
  I
0
 ( J 0  J1  sˆ)  d 
(2.42)
0
To simplify Eq. (2.41), the following solid angle integration formulae for the direction
vector are used [35]
ˆ 0
  sd
4

4
ˆˆ  
ssd
4

3
(2.43)
where  is a third-order unit tensor, and J1   J1 . This reduces Eq. (2.42) to

1
 J 0      J1   I 0  J 0
t
3 
(2.44)
To obtain another equation in terms of J o and J1 , Eq. (2.41) is multiplied with ŝ to yield

ˆˆ)    ( J 0  J1  sˆ) sˆsˆ   I 0 sˆ  ( J 0 sˆ  J1  sˆˆs)
( J 0 sˆ  J1  ss
t
(2.45)
Integrating Eq. (2.45) over the solid angle, 4 , yields
  4 
4
 4

J1     
J 0   
J1

t  3 
3
 3

(2.46)
Equation (2.46) is further simplified to yield
J1
 J 0   J1
t
(2.47)
 J 
   1     J 0      J1 
 t 
(2.48)
Divergence of Eq. (2.47) then yields
which is further simplified to obtain
42
  J1
  2 J 0    J1
t
(2.49)
Next, Eq. (2.44) is differentiated with respect to non-dimensional time to yield
2
1

J 
  J1   I 0  J 0 
2  0
t
3 t
t
 
(2.50)
where I 0 depends on temperature, T , which is a function of time and space, i.e.,
T  T (t , r ).
From Eq. (2.49), an expression for
  J1
is obtained, which is then substituted into Eq.
t
(2.50) to yield
I J
2
1
J     J1  2 J 0   0  0
2  0
t
3
t
t
(2.51)
Moreover, Eq. (2.44) can be rearranged to write

1 
   J1   I 0  J 0   J 0 
t
3 
(2.52)
Substituting Eq. (2.52) into Eq. (2.51) yields
2
 

 I J
 1
J    I 0  J 0   J 0    2 J 0   0  0
2  0
t
t
t
 3
 
 t
(2.53)
which is simplified as
2 J0
J 1
I
 2 0  2 J 0  J 0  I 0  0
2
t
t 3
t
(2.54)
where, in Eq. (2.54), the vector, J1 , has been eliminated. Lastly, the integrated intensity
field at any frequency can then be found using the expression
43
 Id     J
G
4
 J1  sˆ  d   4 J 0
0
(2.55)
4
Substituting Eq. (2.55) into Eq. (2.54), followed by reversion back into the dimensional
form yields,
 2G , p
t 2

2 G , p | vg | , p

  (  , p | vg | , p G , p )
  , p t
3  , p

1
, p2
G , p 
4
, p2
4 I 0, , p
I 0, , p 
  , p t
(2.56)
wherein the subscripts, '  ' and ' p ' are brought back to indicate that the governing
equation, Eq. (2.56) is satisfied at every frequency and polarization. Multiplying Eq. (2.56)
throughout by   , p 2 , we obtain
, p
2
 2G , p
t 2
 2  , p
G , p
t

  , p | vg | , p
3
  (  , p | vg | , p G , p )
G , p  4 I 0, , p  4  , p
I 0, , p
(2.57)
t
The spectral integrated intensity is then obtained from Eq. (2.57), by spectrally discretizing
Eq. (2.57) as
G pj 
 G
,p
d .
(2.58)
 j
The total integrated intensity is obtained by summing over all spectral bins and
NP
polarizations Eq. (2.58), to yield G   G pj . Next, the boundary and initial conditions
p
j 1
for the governing equation, Eq. (2.56) are discussed.
44
2.6.5 Boundary Conditions for the P1 Approximation
The formulation of a thermalizing isothermal boundary condition is not
straightforward in the P1 approximation. The governing equation for the P1 approximation
is in terms of the integrated intensity, G , p , and so, the boundary condition needs to be
formulated accordingly. A popular method called Marshak’s procedure is adopted, which
essentially satisfies flux conservation at the boundaries [35]. We begin by expanding the
formulation for spectral heat flux into total heat flux at a boundary,
q  nˆ     q , p  nˆ  d
(2.59)
p 
where the definition for the spectral heat flux, q , p in terms of phonon intensity, I , p
comes from Eq. (2.80), as will be shown later in sub-section (2.7). The spectral heat flux
normal to a boundary is then determined by taking into account both the equilibrium
distribution of phonons emitted from the boundary and the incoming phonons absorbed at
the boundary as
q , p  nˆ 

I 0, , p sˆ  nˆ d  
sˆn 0

I , p sˆ  nˆ d 
(2.60)
sˆn 0
which is further simplified as
q , p  nˆ   I 0, , p 

I , p sˆ  nˆ d 
(2.61)
sˆn  0
Next, the term

I , p sˆ  n d  is evaluated using the P1 approximation, which is followed
sˆn  0
by substitution of Eq. (2.85) to yield,
45

I , p sˆ  n d  
sˆn  0
 J
0, , p

 J1, , p sˆ  nˆ d 
sˆn  0
2


   J 0, , p 
J1, , p  nˆ 
3


1


   J 0, , p  q , p  nˆ 
2


(2.62)
Substituting Equations (2.55) and (2.62) into Eq. (2.61), yields an expression for heat flux
at every frequency and polarization as
q , p  nˆ  2 I 0, , p  2 J 0, , p  2 I 0, , p 
G , p
2
(2.63)
Next, substituting Equations (2.56) and (2.85) into the non-dimensional Eq. (2.47) yields
q , p
1
 G , p  q , p
t
3
(2.64)
which on performing a dot product with the unit surface normal vector, n̂ yields
  q , p  nˆ  1
 G , p  nˆ  q , p  nˆ
t
3
(2.65)
Substituting Eq. (2.63) into Eq. (2.65), we obtain
G , p  1
G , p 


 2 I 0, , p 
  nˆ  G , p    2 I 0, , p 

t 
2  3
2 

(2.66)
which is simplified further to yield
4
I 0, , p
t

G , p
t
2
 nˆ  G , p  4 I 0, , p  G , p
3
For walls that are at a fixed temperature,
G , p
t
I 0,
t
 0 . Therefore, Eq. (2.67) reduces to
2
 nˆ  G , p  4 I 0, , p  G , p
3
46
(2.67)
(2.68)
Reversion of the non-dimensional Eq. (2.68) into the dimensional form yields
, p
G , p
t
2
   , p | vg | , p nˆ  G , p  4 I 0, , p  G , p
3
(2.69)
Thus, a boundary condition for the thermalizing isothermal wall has been derived
in terms of the integrated intensity, G , p . The boundary condition for an adiabatic wall is
discussed next.
The net normal heat flux in an adiabatic wall is zero since all the phonons striking
the boundary are absorbed and then immediately re-emitted. This implies
q  nˆ  0
(2.70)
Substituting Eq. (2.59) into Eq. (2.70), we obtain,
   q
,p
 nˆ  d  0
(2.71)
p 
For an adiabatic wall, substituting Eq. (2.63) into Eq. (2.71), then yields
  G
,p
(t , rw )  4 I 0, , p (t , rw )  d  0
(2.72)
p 
Equation (2.72) is used to compute the temperature of the boundary at an adiabatic
wall, where the net heat flux is zero. This sub-section concludes the evaluation of boundary
conditions for the P1 approximation.
2.6.6 Initial Conditions for the P1 Approximation
Since Eq. (2.58) is second order in time, two initial conditions are required. As
described for the CADOM, the initial temperature distribution is generally known for heat
conduction problems. Under the assumption that, initially, the entire system is in
47
thermodynamic equilibrium, the spectral integrated intensity, G , p has the following
relations valid at time t  0
G , p (t  0, r )  4 I 0, , p (t  0, r )
G , p
t
(t  0, r )  0
(2.73)
(2.74)
Since the isothermal wall boundary condition in Eq. (2.69) is also first order in
time, an initial value for the spectral integrated intensity at the boundary is required
G , p (t  0, rw )  4 I 0, , p (t  0, rw )
(2.75)
With that the entire P1 formulation to approximate the BTE has been discussed. Next, the
hybrid method formulation is introduced.
2.6.7 Hybrid Method Formulation
As previously stated, for intermediate and high Knudsen numbers, the P1 method
results in inaccurate flux and temperature analyses. At these Knudsen numbers, the P1
approximation is unable to effectively capture the directional variation of intensity and thus
leads to erroneous results. However, the P1 approximation is expected to yield an accurate
description of phonon transport at low Knudsen numbers for microscale heat conduction
problems (as will be investigated in Chapter 3). On the other hand, the CADOM gives
accurate solutions at all Knudsen numbers, but suffers from being computationally
expensive because of high numbers of discretization required in the angular space.
In Section 2.4, the spectral variation of Knudsen numbers, Kn , p , which arises out
of dependence of the phonon mean free path on spectral group velocities and relaxation
time scales, was discussed. In a microscale heat conduction problem, when the heat
48
transport is characterized by both ballistic and diffusive regimes, it is justifiable to establish
a demarcation (or a cutoff Knudsen number), that separates the heat transport into the two
distinct regimes. Consequently, we can solve the original BTE with full angular
discretization using the CADOM in the ballistic regime while invoking the P1
approximation in the diffuse regime. Since the P1 approximation eliminates the need to
solve the directional BTE, it is expected to significantly improve computational efficiency
without sacrificing the accuracy as long as it is used only for cases where the spectral
Knudsen number is small. The cutoff Knudsen number, Knc , is chosen by rigorously
analyzing the effectiveness of the P1 approximation in solving a 1D gray BTE at several
Knudsen numbers, which is subsequently discussed in chapter 3.
The choice of a cutoff Knudsen number then aids in formulating a hybrid method
which then utilizes the CADOM for solving the phonon BTE at those Knudsen numbers
higher than the cutoff, and the P1 approximation to solve the BTE for the remainder of the
spectral bands. Using Equations (2.22) and (2.57), the governing equations for the hybrid
method can then be formulated as
I
I
 I , p
 | vg | , p   ( I , p sˆ)  0, , p  , p

, p
 t


2
G , p
 2  G , p
  , p t 2  2  , p t

  , p | vg | , p


  (  , p | vg | , p G , p )

3

I

 G , p  4 I 0, , p  4  , p 0, , p
t

49
if
Kn , p  Knc
if
Kn , p  Knc
(2.76)
Once the spectral integrated intensities are computed using either the CADOM or
the P1 method, they are summed up over all phonon frequencies and polarizations to find
the total integrated intensity. The boundary and initial conditions for the hybrid method are
setup based on the cutoff Knudsen number, by using the CADOM or the P 1 method
formulations accordingly. Once the integrated intensities have been evaluated, the energy
equation is utilized to compute the temperature of the domain.
The concept of a cutoff Knudsen number is not new and, as previously stated, was
first used by Loy et. al. [52]. However, what sets this study apart is i) An appropriate cutoff
Knudsen number is chosen by rigorously analyzing the solutions to a 1D gray phonon BTE,
and ii) the hybrid methodology developed here is capable of simulating transient heat
conduction problems in both 2D and 3D geometries using unstructured meshes for large
temperature differences.
2.6.8 Boundary Conditions for the Hybrid Method Formulation
The boundary conditions for a thermalizing isothermal wall can be formulated from
the CADOM and P1 approximations, using Equations (2.29) and (2.69), to yield
if
 I , p (t , rw )  I 0, , p (t , rw )

G , p 2

   , p | vg | , p nˆ  G , p  4 I 0, , p  G , p if
  , p
t
3

Kn , p  Knc
Kn , p  Knc
(2.77)
For an adiabatic wall, the boundary conditions are formulated using Equations (2.32) and
(2.72), to yield
(1   )

 I , p (rw , sˆi )   I , p (rw , sˆr )  
 I , p sˆ nˆ d 
sˆ nˆ  0

G (t , r )  4 I
0, , p (t , rw )  0
 , p w
50
if
if
Kn , p  Knc
Kn , p  Knc
(2.78)
where in Eq. (2.78), the degree of specularity,  varies as 0    1 . These boundary
conditions are formulated in terms of the spectral intensities and integrated intensities
based on the governing equation formulation in Eq. (2.76).
2.6.9 Initial Conditions for the Hybrid Method Formulation
The initial conditions for the hybrid method are obtained from Equations (2.33) and
(2.73-2.75), to yield









I , p (t  0, rw , sˆ)  I 0, , p (t  0, rw )
if
G , p (t  0, r )  4 I 0, , p (t  0, r ) 

G , p

(t  0, r )  0

t

G , p (t  0, rw )  4 I 0, , p (t  0, rw ) 
Kn , p  Knc
(2.79)
if
Kn , p  Knc
This section concludes the formulation of the boundary and initial conditions for
the hybrid method. The energy equation formulations and the calculation of temperature
from the integrated intensities are listed next.
2.7 Calculation of Heat Flux and Temperature
Up until this point in the chapter, only the calculations for integrated intensities
using the CADOM, P1 approximation and the hybrid methods have been discussed.
However, the primary interest is to evaluate the normal heat flux at the boundaries and the
spatial temperature distribution of the medium under consideration. This section presents
the procedure to calculate these quantities from the integrated intensities of the phonons
that were calculated in each of the methods described earlier.
The spectral heat flux is related to the spectral intensity for any phonon polarization
through
51
q , p 
 I
,p
ˆ 
sd
(2.80)
4
To obtain the temperature distribution, one has to implement the first law of
thermodynamics, previously stated in Eq. (1.4). This equation is modified to account for
the spectral variation of heat flux, as
U
     q , p d
t
p 
(2.81)
To evaluate the divergence of spectral heat flux,   q , p for the CADOM, Eq. (2.22) is
integrated over the solid angle, 4 , to yield
I
I

I , p d  | vg | , p   (  I , p sˆ d )   0, , p  , p d 

t 4
, p
4
4
(2.82)
Using Equations (2.27) and (2.80), Eq. (2.82) can then be re-formulated as
G , p
t
 | vg | , p   q , p 
4 I 0, , p  G , p
, p
(2.83)
or
  q , p 
1 G , p G , p  4 I 0, , p

| vg | , p t
| vg | , p   , p
(2.84)
Equation (2.84) yields an expression for the divergence of heat flux using the CADOM.
Next, using the P1 approximation, Eq. (2.80) is written as
q , p 
 I
,p
ˆ 
sd
4
4
J1, , p
3
(2.85)
This implies the divergence of spectral heat flux can be written as
  q , p 
4
  J1, , p
3
52
(2.86)
Substituting Equations (2.56) and (2.86) into Eq. (2.44), we once again end up with the
same expression for the divergence of spectral heat flux given by Eq. (2.84). This implies
that Eq. (2.84) is valid for both the CADOM and the P1 approximation, and consequently,
the hybrid method. Substituting Eq. (2.84) into Eq. (2.80) yields
 1 G , p G , p  4 I 0, , p
U
 


t
t
| vg | , p   , p
p   | vg | , p

 d

(2.87)
Under the assumption that a local thermodynamic equilibrium exists, the internal
energy of the medium is expressed in terms of the medium temperature using the BoseEinstein distribution. By inverting this relation, the medium temperature is calculated. The
internal energy of the crystalline material, U without considering the zero-point energy is
re-written using Eq. (2.3) as


1
U  
  D( , p) d



p   exp 
k BT   1 
(2.88)
Equation (2.87) is then inverted to compute a so called pseudo-temperature by substituting
Eq. (2.88) in Eq. (2.87). After evaluating the integrated intensity, G , p using any of the
CADOM, P1 or hybrid methods, the corresponding domain temperatures can then be
calculated and compared.
The numerical procedure to evaluate the temperature after substituting the
expression for U from Eq. (2.88) into Eq. (2.87) is described next. An implicit
methodology is applied to evaluate the term,
This yields
53
U
at any time step, ' n  1' in Eq. (2.87).
t
U
t
n 1

U n1  U n
t
(2.89)
The term, U n , in Eq. (2.89) is evaluated at a known temperature, T n from the
previous time step. To obtain the temperature, T n1 , the quantity, U n 1 is linearized using
a Taylor Series approximation. Using an implicit backward Euler formulation again, this
time in temperature, we obtain
U
n 1
(Tnew )  U
n 1
U
(Told )  (T )
T
n 1
 2U
 (T )
T 2
2
n 1
 
(2.90)
where, U n1 (Told ) is evaluated using the temperature at the previous iteration within the
n 1
n 1
 Told
same time step as will be shown later and T  Tnew
.
Since the internal energy, U and the equilibrium intensity distribution, I 0 are
directly related through Eq. (2.3) and Eq. (2.9), upon substituting Eq. (2.9) into Eq. (2.88),
and discretizing the angular frequency space into spectral bands, similar to the procedure
followed in Equations (2.23) and (2.25), we obtain,
NP
4 I 0,j p
j 1
| vg | pj
U  
p
(2.91)
Equation (2.91) states that the total internal energy, U can be expressed as a
summation of discretized spectral equilibrium intensities, I 0,j p . In addition, the quantity,
U
T
n 1
I
in Eq. (2.90), can be expressed in terms of 0
T
n 1
. Differentiating Eq. (2.9) with
respect to temperature and spectrally discretizing it, we then obtain for every time step,
54
I 0,j p
T
n 1

8 3 k BT 2
 | K |

 exp   1
exp
2
1


2
k BT n

 j
2
d
(2.92)
k BT n
where the temperature in Eq. (2.92) is approximated as T n at time step, n , and | K | is the
wave vector magnitude which is obtained by inverting Eq. (2.12). The integration in Eq.
(2.92) is carried out using a 20-point Gaussian quadrature rule. Equation (2.91) is also used
to evaluate U n (T n ) and U n1 (Told ) as
NP

p
4 I 0,j ,pn (T n )
| vg | pj
j 1
and
NP
4 I 0,j ,pn 1 (Told )
j 1
| vg | pj

p
respectively. Substituting these expressions and Equations (2.91) and (2.92) into Eq.
(2.92), we obtain the change in temperature, T at time step, n  1 as
 4 I 0,j ,pn
 1 1  G pj ,n
 1 1 
j , n 1

G
 4 I 0,n 1  j   
 j   

p
j 
j


p j 1 | vg | p 
 p
  p t  t
 p t  
NP
T
n 1

1
j
4 I 0, p

j
p j 1 | vg | p T
NP
n 1
 1 1 
  j 
 t  p 
(2.93)
Once T for the time step, n  1 has been evaluated, the time step can be then be updated
to evaluate the new temperature.
Next, the general solution algorithm and the steps for the hybrid methodology are
presented.
2.8 Solution Algorithm for the Hybrid Method
This section outlines the general strategy implemented to solve the BTE using a
hybrid approach as depicted in Fig. 2.5.
The various steps involved are:
55
1. Set the initial temperature (internal energy) and also the integrated intensity, G for
the P1 method and intensity, I for the CADOM appropriately based on the cutoff
Knudsen number as given by Eq. (2.79). The energy Eq. (2.87) needs to be satisfied
at t = 0.
2. Guess the temperature distribution within the whole computational domain so that
the equilibrium intensity distribution, I 0 can be estimated through Eq. (2.25).
3. Calculate the spectral Knudsen numbers using Eq. (1.1) and compare with the
cutoff Knudsen number to invoke either the P1 or the CADOM within the hybrid
method as needed.
4. Solve for the intensity and ultimately the spectral integrated intensity using the
CADOM if the spectral Knudsen number is greater than the cutoff. If the spectral
number Knudsen number is lesser than the cutoff, then solve for the spectral
integrated intensity using the P1 approximation. The governing equations are listed
in Eq. (2.76) with boundary conditions provided by Eq. (2.77).
5. Solve the overall energy balance equation (Eq. 2.87) by summing the spectral
integrated intensities calculated from the previous step, so as to estimate the new
temperature distribution from Eq. (2.93).
6. Repeat steps 3-5 until convergence so that the temperature of the medium doesn’t
change within the particular time step, this constitutes as an outer iteration.
7. Compute heat fluxes at the boundaries using Eq. (2.80).
8. Proceed to the next time step and repeat steps 2-8.
56
Initialize T and initialize G , p
& I , p based on Kn c
Start of time marching
Guess T of whole computational domain
If Kn , p  Kn c
If Kn , p  Kn c
Inner
Iterations
Evaluate I sˆ , , p
using CADOM
Evaluate G , p using P1
approximation
Evaluate G , p
Inner
Iterations
Outer iterations
New time step
Evaluate Kn  , p for every spectral band
Combine to apply First law and update
T of computational domain
YES
Converged?
NO
Figure 2.5: Overall solution algorithm for the hybrid method
2.9 Summary
This chapter introduced the theory of phonon transport and also discussed the
formulation of the CADOM and the P1 approximation to solve the phonon BTE
numerically. Using the concept of a cutoff Knudsen number, the governing equations for a
hybrid method are then derived along with the corresponding boundary and initial
conditions. In Chapter 3, the procedure behind choosing an appropriate cutoff Knudsen
number is described.
57
Chapter 3: Semi-Analytical Solutions to the One-Dimensional Gray BTE
The hybrid methodology proposed in Section 2.6.7 of Chapter 2 to solve the BTE
is based on the determination of an appropriate cutoff Knudsen number. To determine such
a cutoff Knudsen number, the frequency independent or gray 1D BTE is solved with and
without invoking the P1 approximation for various Knudsen numbers. While using the P1
approximation, an error analysis of the boundary fluxes in comparison with the exact
solution is then conducted at several Knudsen numbers. Tabulating the fluxes with respect
to the Knudsen numbers, we determine an appropriate cutoff Knudsen number based on
the percentage error in the boundary fluxes. In this chapter, an analytical solution to solve
the gray 1-D BTE is developed and also the application of the P1 approximation to the 1D BTE is shown. The theory and results obtained from the application of these methods
are described below.
3.1 1D Sub-Micron Heat Conduction Problem
The problem considered here is pictorially shown in Fig. 3.1. The steady state
temperature distribution in the domain is sought, when the two walls are isothermally
maintained at two different temperatures. The Knudsen number is varied in steps from 0.1
to 10 in the study. Prior to discussing the solution strategy and implementation, the theory
behind the numerical procedures to solve the gray BTE using the exact analytical method
and the P1 approximation are discussed.
58
T2
X= L
Isothermal
walls
X
X= 0
T1
Figure 3.1: A 1-D parallel plate in dimensional coordinates with two isothermal walls
3.1.1 Exact Analytical Solution
Under the gray (frequency independent) assumption, the BTE at steady state [Eq.
(2.22)] reduces to
vg    Isˆ  
I0  I

(3.1)
The quantities,  and vg , which denote the relaxation time and the phonon group velocity,
respectively, are assumed to be frequency independent in a gray BTE. For a 1D problem,
Eq. (3.1) can be re-written as
 vg
The term,
dI
 I0  I
ds
(3.2)
dI
refers to the directional derivative of I along ŝ as shown in Fig. 3.2.
ds
Since the problem is one-dimensional, the direction vector, ŝ can be expressed in terms of
 , which is the cosine of the angle between the phonon propagation direction and the Xaxis, the coordinate direction perpendicular to the walls.
59


x
s
x 1
x'
I 2 ( )
s'
s'
x
s

s'
`
I 1 ( )
A2
`
I  ( x , )
`
A2
s'
x'
A1
A1
I  ( x , )
`
`

Figure 3.2: Coordinate system used in the exact analytical solution for the BTE solution
of the 1-D heat conduction problem [35]
Equation (3.2) is then expressed in terms of the x-coordinate to yield
 vg
dI
 I0  I
dx
(3.3)
where   cos  , as shown in Fig. 3.2.
Non-dimensionalizing Eq. (3.3) using the length scale, L , i.e., x*  x / L we obtain
 Kn
dI
 I0  I
dx*
(3.4)
where the non-dimensional parameter, Kn is given by Eq. (1.1). Hereon, the x* notation
will be replaced by x, to denote non-dimensional length. Equation (3.4) then reduces to
dI I 0  I

dx  Kn
(3.5)
Integrating Eq. (3.5) from x '  0 to any point in the medium, x '  x, we obtain
x  x

I 0  x 
 x' 
 x' 
d
I
exp

exp 




 dx '
x '0 

  Kn   x0  Kn
  Kn 
x ' x
which is further simplified as
60
(3.6)
x  x
I 0  x 
 x 
 x' 
I  x,   exp 

I
x

0

exp 



 dx '

 Kn
  Kn 
  Kn 
x  0
(3.7)
At the bottom wall, x  0 in Eq. (3.7), we can write I  x  0   I 0  x  0  since it is a
thermalizing isothermal boundary at constant temperature. Equation (3.7) then reduces to
x  x
I 0  x 
   x  x  
 x 
I  x,    I 0  x  0  exp 
exp 
 dx '
 
  Kn  x0  Kn
  Kn 
(3.8)
Equation (3.8) is the forward propagating component of the intensity at any nondimensional location, x, and is therefore denoted by I  x,    I   x,   , as shown in Fig.
3.2. The intensity, I   x,   is a result of the intensity emitted from the point shown on the
bottom wall in the ŝ direction, and augmented by scattering of the phonons within the
medium in the direction shown. Re-writing Eq. (3.8), we obtain
x  x
I 0  x 
   x  x  
 x 
I   x,    I 0  x  0  exp 

exp 
 dx '
 
  Kn  x0  Kn
  Kn 
0   1
(3.9)
The backward propagating intensity, I  ( x) is then the intensity emanating from a point on
the top wall, and is similarly augmented by scattering within the medium in the ŝ direction.
Thus, we obtain,
x '1
 1 x 
 x ' x   x 
I  x,    I 0  x  1 exp 

   I 0  x  exp 
d 
  Kn  x ' x
  Kn   Kn 

1    0
(3.10)
Next, the total integrated intensity, G( x) , which was calculated previously using
Eq. (2.28), in the one-dimensional, frequency independent form reduces to
1
0
1
1
G  x   2  I  x,   d   2  I

61
1
 x,   d   2 I   x,   d 
0
(3.11)
As previously stated in Section 2.6.1, the integrated intensity is necessary to evaluate the
divergence of heat flux and obtain the domain temperature using the energy equation. The
procedure to evaluate the domain temperature is described next.
To further simplify Eq. (3.11), the integral containing the forward propagation
intensity on the right hand side is evaluated as
 x  x 
x
1
1
1 x  x


I  x   Kn


2 I   x,   d   2  I 0  x  0  e  Kn d     0
e
dx ' d  
 Kn
0
0 x  0



0
(3.12)
which is simplified as
x
 x 
 x  x   x  
2 I  x,   d   2 I 0  x  0  E2 
  2 I 0  x  E1 
d 

 Kn 
 Kn   Kn 
0
0
1

(3.13)
where the En functions are the exponential integral functions of order n , defined as

En ( x)   exp   xt 
1
 x 
dt
   n 2 exp   d 
n
t
  
0
1
(3.14)
A detailed explanation of exponential integral functions and their relationships for various
orders can be found in Appendix E of [35]. Similarly, the other integral on the right hand
side of Eq. (3.11) can be simplified to yield
1
 1 x 
 x ' x   x 
2  I  x,   d   2 I 0  x  1 E2 
  2 I 0  x  E1 
d 

 Kn 
 Kn   Kn 
1
x
0

(3.15)
Substituting Equations (3.13) and (3.15) into Eq. (3.11), the total integrated intensity is
then evaluated as
62

 x 
 1 x 
G  x   2  I 0  x  0  E2 
  I 0  x  1 E2 

 Kn 
 Kn 


1
 x  x   x 
 x ' x   x  

I
x
E
d

0 0   1  Kn   Kn  x I 0  x E1  Kn  d  Kn  
x
(3.16)
At steady state, with no heat generation, the energy equation given by Eq. (1.4) reduces to
q  0
(3.17)
The divergence of heat flux at steady state is expressed in terms of the integrated intensity,
G and equilibrium intensity distribution, I 0 , using Eq. (2.84) as
q 
4 I 0  G
 | vg |
(3.18)
Under the assumption of a linear dispersion relationship, the internal energy of a crystalline
material using Eq. (2.88) reduces to [8]
4 PT 4
U
| vg |
(3.19)
where under the Debye assumption, the expression for the phonon density of states,
D( ) 
2
2 2 | vg |3
has been used in the derivation and  P is the Stefan-Boltzmann constant
for phonons given by [54]
P 
 2 kB 4
40
3
| vg |2
(3.20)
It is noteworthy that the Stefan-Boltzmann constant is only applicable when there is a linear
dispersion which results in constant phonon group velocity. It is thus not a true constant.
63
Next, the internal energy in Eq. (3.19) is related to the equilibrium intensity distribution,
I 0 through Eq. (2.91), which, under the gray assumption yields
 PT 4
I0 

(3.21)
Substituting Eq. (3.18) and Eq. (3.21) into Eq. (3.17), the integrated intensity, G , can be
obtained in terms of the domain temperature as
G( x)  4 I 0 ( x)  4 PT 4 ( x)
(3.22)
Substituting Eq. (3.22) into Eq. (3.16), the governing equation can then be formulated in
terms of the domain temperature as
T 4  x 
1
 x  x
1 4  x 
 1 x 
4
4
T
E

T
E

1
2
 2 2
 T  x  E1 

2
 Kn 
 Kn  0
 Kn
  x  
d 

  Kn  
(3.23)
Equation (3.23) is the Fredholm Integral Equation [35]. The integral on the right hand side
is difficult to integrate directly because of a singularity at x  x , which results in
E1  0    . In order to remove the singularity, Eq. (3.23) is modified. First, we break up
the integration interval at x  x and revert back to the original formulation listed in Eq.
(3.16). The integral in Eq. (3.23) can then be expressed as
 x  x   x  x 4
 x  x   x 
0T  x E1  Kn  d  Kn   0T  x E1  Kn  d  Kn  
1
 x ' x   x 
4

T
x
E


1

d 

x
 Kn   Kn 
1
4
64
(3.24)
With this formulation, the singularity moves from somewhere in the middle of the domain,
 x  x '  0 , to the upper limit of the first integral and the lower limit of the second integral.
We now discretize the domain [0, 1 ] into N-1 intervals, such that Eq. (3.24) becomes
 x  x   x  N 1 u , j 4
 x  x   x 


T
x
E
d

T
x
E







1
1



d 

0

 Kn   Kn 
 Kn   Kn  xj 1x ' xl , j
x
1
4
xu , j
 x ' x   x 
T 4  x  E1 

d 


 Kn   Kn 
j 1 xl , j
N 1
(3.25)
x x '
where xl , j and xu , j are the lower and upper limits of the j-th interval, respectively. Here,
j varies from 1 to N-1. The number of intervals in Eq. (3.25) is N-1, which implies that
we have a total of N nodes with two boundary nodes (i = 1 and i = N), and N-2 interior
nodes.
Next, by assuming that T 4 varies weakly compared to E1 within each interval, T 4
may then be assumed to be a constant, resulting in
 x  x
0T  x E1  Kn
1
4
  x  N 1 4 xl , j  xu , j u , j  x  x   x 
d 
   T ( 2 )  E1 
d 

Kn
Kn
Kn






j

1

xl , j
x x '
x
N 1
T
4
(
xl , j  xu , j
j 1
x x '
2
xu , j
)
xl , j
 x ' x   x 
E1 
d 

 Kn   Kn 
(3.26)
The remaining integrals can now be performed analytically, resulting in
 x  x
4
0T  x E1  Kn
1
  x  N 1 4 xl , j  xu , j
d 
   T ( 2 )[ E2 ( x  x j 1 )  E2 ( x  x j )] 
  Kn  j 1
x x '
(3.27)
N 1
T
4
(
xl , j  xu , j
j 1
x x '
65
2
)[ E2 ( x j  x)  E2 ( x j 1  x)]
Next, the term T 4 (
xl , j  xu , j
2
) is approximated using [T 4 ( xl , j )  T 4 ( xu , j )] / 2 , which is similar
to using the trapezoidal rule that introduces an error of the order, x 2 . Thus, the solution
accuracy will depend on the number of nodes, N, being used. Substituting Eq. (3.27) into
Eq. (3.23), we obtain

 x 
 1  x 
4
2T 4  x   T14 E2 
  T2 E2 

 Kn 
 Kn  

1 N 1
  [T 4 ( xl , j )  T 4 ( xu , j )][ E2 ( x  x j 1 )  E2 ( x  x j )]
2 j 1
(3.28)
x x '

1 N 1 4
[T ( xl , j )  T 4 ( xu , j )][ E2 ( x j  x)  E2 ( x j 1  x)]

2 j 1
x x '
Equation (3.28) is then satisfied at as many collocation points as used on the right
hand side integral. The expression then reduces to algebraic equations of the form
1 i 1 4
2Ti  E2 ( xi )   [T j  T 4 j 1 ][ E2 ( xi  x j 1 )  E2 ( xi  x j )]
2 j 1
4
1 N 1
  [T j4  T j41 ][ E2 ( x j  xi )  E2 ( x j 1  xi )]
2 j i
(3.29)
i  2,3,..., N  1
Equation (3.29) can then be rearranged in a matrix form as, [ A][T 4 ]  [ R], such that for all
the interior nodes, i, the source matrix, Ri , and the coefficient matrix, Ai , become
 x 
 1  xi 
Ri  T14 E2  i   T24 E2 

 Kn 
 Kn 
66
(3.30)
1

Wi , j
j  1

2

1
Ai , j   (Wi , j 1  Wi , j ) 2 i , j j  2,3,..., N  1
2
1

Wi , j 1 j  N

2

i  2,3,..., N  1
(3.31)
where  i , j is the Kronecker delta, and
[ E2 ( xi  x j 1 )  E2 ( xi  x j )]
Wi , j  
[ E2 ( x j  xi )  E2 ( x j 1  xi )]
j  i  1
j  i
(3.32)
The equations corresponding to i  1 and i  N are set up such that the following boundary
conditions are recovered.
T 4 ( x  0)  T14
T 4 ( x  1)  T2 4
(3.33)
Solving this set of N simultaneous equations gives the solution to T 4 at all the interior
nodes.
Another goal of the 1D heat conduction analysis is to estimate the heat flux. Using
Eq. (2.80) in Chapter 2, the one-dimensional heat flux at any location, x in terms of the
direction cosine can be written as
1
q( x)  2  I ( x,  )  d 
(3.34)
1
Substituting the expressions derived for I from Eq. (3.9-3.10) into Eq. (3.34) and upon
further simplification, we obtain
67

 x 
 1 x 
q  x   2  I 0  x  0  E3 
  I 0  x  1 E3 

 Kn 
 Kn 

1
 x  x   x 
 x ' x   x  

I
x
E
d

0 0   2  Kn   Kn  x I 0  x E2  Kn  d  Kn 
x
(3.35)
Substituting Eq. (3.22) into Eq. (3.35), we obtain the heat flux in terms of temperature as
1
 x  x

 x 
 1 x 
4
4

q  x    2 PT14 E3 

2

T
E

2

T
x
E



P 2
3
P
2


0
 Kn 
 Kn 

 Kn
  x  
d 

  Kn  
(3.36)
The heat flux at the boundary wall, x  0 , can then be formulated using Eq. (3.36) as
1
 x   x  
4
4

q   2 PT1  2 PT2 E3 (1)  2  PT 4  x  E2 
d 

 Kn   Kn  
0
(3.37)
For ease of comparison between the various methods considered in this study, we define
a quantity called the non-dimensional emissive power as
( x) 
T 4 ( x)  T2 4
T14  T2 4
(3.38)
The non-dimensional heat flux,  at the x  0 boundary wall, is then defined as
 x 0 
1
q
 x   x 

1

2
( x ') E2 
d 

4
4

 P (T1  T2 )
Kn

  Kn 
0
(3.39)
This concludes the discussion of the analytical solution to the steady state 1D gray
BTE, where the non-dimensional heat flux at the x  0 boundary wall has been estimated.
Next, the P1 approximation has been utilized to estimate the non-dimensional heat flux at
the boundary walls and subsequently, compare the results obtained using the two methods
at different Knudsen numbers.
68
3.1.2 P1 Approximation
In Section 2.6.4, the lowest order spherical harmonics, namely the P1
approximation, was applied to derive the frequency-dependent form of the transient BTE
in multidimensional geometries. To reduce the governing equation, namely Eq. (2.56), into
a frequency independent form, we utilize a linear dispersion relation that results in a
constant Knudsen number, as obtained from Eq. (1.1). By neglecting the transient terms in
Eq. (2.57), we obtain the final gray form of the steady state, P1 approximated BTE for a 1D
geometry as,
G
Kn2 d 2G
 4 I o
3 dx 2
(3.40)
Utilizing Eq. (3.21), we can once again express the equilibrium intensity, I 0 in terms of
the domain temperature, as
G ( x) 
Kn2 d 2G
 4 P T ( x)4
2
3 dx
(3.41)
For this particular problem, the thermalizing isothermal boundary conditions can be
implemented by reducing Eq. (2.68) into a gray, steady state 1-D formulation, given by
2
dG
Kn
3
dx
2
dG
Kn
3
dx
 Gx 0  4 P T14
x 0
(3.42)
 Gx 1  4 P T2 4
x 1
where Gx 0 and Gx 1 represent the integrated intensities at the boundary walls with T1 and
T2 being their isothermal wall temperatures respectively. From Eq. (3.22), it was seen that
the integrated intensity, G , can be expressed in terms of the domain temperature, T, as a
69
direct consequence of the energy equation, Eq. (2.84). Thus, by utilizing the fact that,
G  4 PT 4 , Eq. (3.41) can be further simplified to yield
d 2G
0
dx 2
(3.43)
which can then be analytically solved to obtain,
G  C1 x  C2
(3.44)
where C1 and C2 are constants of integration. Simultaneously satisfying the governing
equation and the boundary conditions yield expressions for C1 and C2 , as
4 P (T2 4  T14 )
4
Kn  1
3
2
C2  4 P T14  KnC1
3
C1 
(3.45)
To find the magnitude of the non-dimensional flux at either x  0 or x  1 , the
expression, q nˆ  
Kn  G
is utilized, which is obtained by dimensionalizing Eq. (2.64).
3
Thus, Eq. (3.42) can be re-written as
2qx 0  Gx 0  4 P T14
2qx 1  Gx 1  4 P T2 4
(3.46)
Substituting the governing equation for G from Eq. (3.44) into the boundary condition
given by, Eq. (3.46), we obtain
2qx 0  4 P T14  C2
2qx 1  4 P T2 4  C1  C2
70
(3.47)
At steady state, for the 1D heat conduction problem, qx 0 and qx 1 would be equal in
magnitude but opposite in sign. Upon further simplification using Eq. (3.45), an expression
for the magnitude of q is derived as,
qx  0
4
Kn P (T2 4  T14 )
 3
4
Kn  1
3
(3.48)
The non-dimensional flux is then obtained as
 x 0 
1
3
1
4 Kn
(3.49)
Now that expressions for the non-dimensional fluxes have been derived in Eq.
(3.39) and Eq. (3.49), a parametric study at several Knudsen numbers is performed for the
1D heat conduction problem, using both the exact BTE and the P1 approximation methods.
By comparing the magnitude of non-dimensional fluxes obtained at the boundaries, a
suitable Knudsen number is then chosen based on a prescribed percentage error for the P 1
method.
3.2 Results
The 1D gray BTE is solved exactly as well as by using the P1 approximation in
conjunction with the 1D energy equation. Comparisons in the non-dimensional fluxes at
the wall boundary, x  0 and the L2 norm of the error in domain temperatures using the
two methods are listed in Table 3.1.
71
Table 3.1 Non dimensional flux and temperature norm for 1D test case
As the Knudsen number is decreased from 1.5 to 0.1, the percentage error in the P1
flux consistently decreases to less than 1%. At low Knudsen numbers, it is well known that
the P1 method accurately solves the BTE which results in accurate wall fluxes and domain
temperature predictions. A similar trend can also be observed in the L2 norm of the domain
temperature, indicating that the temperature predictions for the P1 and the analytical
method get closer at low Knudsen numbers.
However, increasing the Knudsen number beyond 1.5 results in the percentage error
of the P1 non-dimensional fluxes to decrease. This is because as the Knudsen number
reaches  , the wall non-dimensional flux ultimately has to reach the physical limit of 1.
Thus, both the analytical and P1 fluxes would then converge to 1 and the percentage error
72
would actually decrease for further increasing Knudsen numbers (beyond 10) as seen in
the ballistic regime. In the ballistic limit, the phonons travel unhindered from the hot wall
to the cold wall and as a result, the flux magnitude would default to that value prescribed
by T4 laws.
This analysis provides a strong basis to develop a hybrid BTE method for multidimensional geometries, which utilizes the accurate CADOM to solve the BTE in the
ballistic regimes and the P1 method to solve the BTE in the diffuse regimes of phonon
transport, based on a cutoff Knudsen number equal to 0.1.
3.3 Summary
Solutions for the 1D gray BTE have been developed using an analytical method
and the P1 approximation. The results for the non-dimensional fluxes at the boundary walls
and L2 norms of the domains temperature have been listed and compared. The basis for
choosing a Knudsen number cutoff of 0.1 has been justified and thus the hybrid solution to
the multidimensional heat conduction problem using the frequency dependent BTE is
developed. In a non-gray phonon BTE simulation, the spectral Knudsen numbers computed
indicate that phonon transport ranges from the diffusive to ballistic regimes. Thus, for all
those spectral bands with a Knudsen number above 0.1, the BTE is solved directly using
the CADOM and the remaining bands are then solved using the P1 approximation, as will
be shown in the subsequent chapter.
73
Chapter 4: Multi-Dimensional Solutions to the BTE
In Chapter 3, solutions to the gray, one-dimensional, steady state BTE using the
exact solution and P1 approximation were introduced. In Chapter 4, we present the finite
volume procedure adopted to discretize and numerically solve the multi-dimensional nongray BTE using the CADOM, P1 approximation and hybrid methods. Subsequently, the
accuracy and efficiency of the hybrid method is tested by varying the cutoff Knudsen
number for simple 2D test cases spanning a range of operating temperatures. The
performance of the hybrid method is then compared to the CADOM and P1 approximations
for both steady state and transient simulations. Finally, the hybrid method is extended to
solving the BTE in 3D geometries with unstructured meshes.
4.1 Finite Volume Solution Using the CADOM
The BTE, after angular and spectral discretization, was derived previously in Eq.
(2.26), and is re-stated here
I i ,j p
t
 | vg | pj   ( I i ,j p sˆi ) 
I 0,j p  I i ,j p
i  1, 2,....N dir

j  1, 2,....N P
j
p
where the subscript, i , refers to the directional dependence, the superscript, j , refers to
the spectral band of the corresponding phonon polarization, p, in the BTE. Equation (2.26)
is a set of N dir  N P , 4-dimensional (time and spatial coordinates) PDEs for each phonon
polarization. However, the angular discretization utilized in Eq. (2.26) is of the finite
74
differencing form. Subsequently, we shall introduce the CADOM version of the BTE and
thus, for now, we shall neglect the angular discretization. Applying the finite volume
procedure in space and integrating Eq. (2.26) over an arbitrary control volume of volume,
VO , yields
I pj ,O
t
   (I
VO  | v |
j
g p
j
p
sˆ)dV 
I 0,j p ,O  I pj ,O
 pj ,O
VO
VO
(4.1)
where an average cell center value for the relaxation time-scale is used over the control
volume for each spectral band. Applying the Gauss divergence theorem to Eq. (4.1), yields
I pj ,O
t
VO  | v |  I ( sˆ nˆ )dA 
j
g p
j
p
I 0,j p ,O  I pj ,O
S
 pj ,O
VO
(4.2)
where the volume integral in Eq. (4.1) has been converted to a surface integral. Here, S is
the surface area of the control volume and n̂ is the local outward pointing surface normal
of the control surface. By assuming that the control volume under consideration is a convex
polyhedron bounded by discrete faces, the surface integral in Eq. (4.2) can be reduced to a
discrete summation, to yield
I pj ,O
t
VO  | v |
j
g p
I
j
p, f
( sˆ nˆ f ) Af 
I 0,j p ,O  I pj ,O
f
 pj ,O
VO
(4.3)
where the subscript, f refers to the control face being considered, A f is the surface area
and nˆ f is the surface normal of the control face. The unknown face intensity, I pj , f is
approximated using the upwind cell’s intensity as
I
j
p, f
j

 I p ,O
 j

 I p,N
if sˆ nˆ f  0
if sˆ nˆ f  0
(4.4)
where the cell corresponding to ' N ' is a neighbor to the cell ' O ' as shown in Fig. 4.1.
75
Figure 4.1: An unstructured 2D stencil showing relevant variables [11]
By integrating Eq. (4.3) over a solid angle, i , and dividing throughout by VO i , one
obtains the complete discretized version of the BTE, written as
I i ,j p ,O
t

| vg | pj
iVO
 I
i
j
p, f
( sˆ nˆ f ) Af di 
f
I 0,j p ,O  I i ,j p ,O
 pj ,O
(4.5)
Equation (4.5) can be re-written by interchanging the integral and the discrete summation,
so that we obtain
I i ,j p ,O
t

| vg | pj
iVO
j
j
j
ˆ  nˆ A  I 0, p ,O  I i , p ,O
I
S
 i, p, f i f f
j
f


 p ,O
i  1, 2,....N dir
j  1, 2,....N P
(4.6)
where
Sˆi  
i
ˆ i
sd
(4.7)
has been used. Substituting the formulations for ŝ and i from Eq. (2.20) and Eq. (2.21)
respectively into Eq. (4.7), we obtain,
76
Si 
  
sin i sin 
    cos 2i sin   iˆ
 2 
  
 cos i sin 
    cos 2i sin   ˆj
 2 
  
ˆ
 
 sin 2i sin  k
 2 
(4.8)
The integrated intensity field is then calculated as
N P N dir
G   I i ,j p
p
(4.9)
j 1 i 1
which is simply a summation of the phonon intensity, I i ,j p over all directions, spectral
bands and phonon polarizations. For a more detailed approach, the reader is referred to
Mittal [12]. Next, the finite volume formulation for the P1 approximation is described.
4.2 Finite Volume Solution Using the P1 approximation
Spectral discretization of the governing equation in Eq. (2.57), with the subsequent
application of finite volume integration and Gauss divergence theorem yields,

j 2
p ,O O
V
 2G pj ,O
t
2
 2
j
p ,O O
V
G pj ,O
t

( pj ,O | vg | pj )
3

j
p, f
| vg | pj (nˆ f  G pj , f )
f
G pj ,OVO  4 I 0,j p ,OVO  4 pj ,OVO
I 0,j p ,O
(4.10)
t
where the subscript, O , indicates the cell center value, the subscript, f , indicates that the
face value for the quantity under consideration, as previously shown in Fig. 4.1. The
superscript, j , refers to the spectral band of the corresponding phonon polarization.
Equation (4.10) can be further simplified by expressing, G pj , f in terms of quantities
defined at cell centers [11]. We start with the vector identity for G pj , which can be written
as
77
Gpj  (nˆ f  Gpj )nˆ f  (nˆ f  Gpj )  nˆ f
(4.11)
For a 2D geometry, such as the one shown in Fig. (4.1), Eq. (4.11) can be further simplified
to yield,
Gpj  (nˆ f  Gpj )nˆ f  (Gpj  tˆf )tˆf
(4.12)
Taking the dot product of Eq. (4.12) with l f , where l f is a vector connecting the cell
centers of the cells, N and O, as seen in Fig. (4.1), we obtain
Gpj  l f  (nˆ f  Gpj )nˆ f  l f  (Gpj  tˆf )tˆf  l f
(4.13)
where tˆf is the unit vector along the face of the cell. A quantity,  f is defined as the
distance along the normal between the cell centers, N and O , i.e.,  f  nˆ f  l f and also
using the fact that Gpj  l f  Gpj , N  Gpj ,O , we can simplify Eq. (4.13) to obtain
Gpj , N  Gpj ,O  (nˆ f  Gpj ) f  (Gpj  tˆf )tˆf  l f
(4.14)
Equation (4.14) can now be re-written as
nˆ f  G 
j
p
G pj , N  G pj ,O
f

(G pj  tˆf )tˆf  l f
f
(4.15)
Now, the expression (G pj  tˆf ) can be expressed in terms of G pj at the vertices a and b ,
by using distance weighted interpolation of values at cell centers and then obtaining
expressions for G pj ,a and G pj ,b . Equation (4.15) can be written as
G  tˆf 
j
p
(G pj ,a  G pj ,b )
| ab |
(4.16)
where | ab | is the straight line distance between the nodes, a and b as depicted in the Fig.
4.1. This is sometimes expressed as | ab | Af , where A f is the area of the face, f.
78
Essentially, from Equations (4.15) and (4.16), the flux normal to a cell has been separated
into a flux along the direction joining the cell centers that bound the face, and a tangential
flux along the face. Substituting Eq. (4.15) and Eq. (4.16) into Eq. (4.11) and multiplying
with the appropriate coefficients on both sides, yields
( pj ,O | vg | pj )
3VO

j
p, f
| vg | pj (nˆ f  G pj , f ) 
f
( pj ,O | vg | pj )
3VO

f
j
p, f
 G pj , N  G pj ,O (G pj ,a  G pj ,b )tˆf  l f
|v | 


f
Af

j
g p

Af


(4.17)
where  pj , f at the face is evaluated by using distance weighted interpolation of values at
cell centers.
For a 3-D geometry, such as the one shown in Fig. 4.2, the basic surface element is
a triangle. The application of the finite volume formulation remains the same with the only
change arising in the calculation of the cross flux term (nˆ f  Gpj )  nˆ f in Eq. (4.11). A
procedure similar to the one listed in the derivation of the 2D tangential flux is followed
and so Eq. (4.11) is dotted with l f , to yield
Gpj , N  Gpj ,O  (nˆ f  Gpj ) f  (nˆ f  Gpj )  nˆ f   l f
79
(4.18)
a)
b)
Figure 4.2: a) 3D unstructured mesh and b) basic triangular surface element that
aid in cross flux calculation for 3-D geometries
The last term or the cross flux term in Eq. (4.18) is then computed as [73]
1
(nˆ f  G pj )  nˆ f   l f 
Af
G
j
p ,e
Le nˆe  l f
(4.19)
e
where the summation is calculated over all the edges of the ABC triangle as depicted in
Fig. 4.2 and Af , refers to the area of the face ABC. The unit vector, nˆe is located in the
triangular plane and is perpendicular to each side of the triangle. It is the edge surface
normal and is calculated using the edge tangent as
nˆe  tˆe  nˆ f
(4.20)
Substituting Eq. (4.20) into Eq. (4.19), we obtain
1
(nˆ f  G pj )  nˆ f   l f 
Af
 G   L tˆ  nˆ
j
p ,e
e e
e
f
  l f

(4.21)
Upon further simplification and utilizing the fact that te  Letˆe , we obtain
1
(nˆ f  G pj )  nˆ f   l f 
Af
80
G
e
j
p ,e
t  nˆ
e
f
 l f 

(4.22)
In Eq. (4.22), the quantity nˆ f  l f is independent of summation over the edges and thus,
needs to be computed only once per face instead of computing it at every edge of the face.
For the interior nodes, the edge value of G pj ,e is the average of the values of G pj at the
vertices. For nodes which are on the boundary face, the values of G pj at the centers of the
surrounding boundary faces are used.
4.3 Finite Volume Solution Using the Hybrid Method
The hybrid solution methodology is a combination of the finite volume procedures
applied to the CADOM and P1 approximation. Based on the spectral Knudsen number,
either one of the finite volume procedures are adopted as previously stated in the Chapter
3. The final discretized forms of the governing equations after the application of the finite
volume procedure are listed here
 I i ,j p ,O | vg | pj ,O
I 0,j p ,O  I i ,j p ,O
j
ˆ
ˆ

I
S

n
A


 i, p, f i f f
iVO f
 pj ,O
 t


 j 2  2G pj ,O
G pj ,O
j
 p ,O
 2 p ,O

t 2
t

j
j
  ( p | vg | p )O  j | v | j (nˆ  G j )
f p, f g p f p, f

3VO

I 0,j p ,O

j
j
j
  G p ,O  4 I 0, p ,O  4 p ,O
t



if
Kn pj  Knc
if
Kn pj  Knc
(4.23)
Now that the finite volume procedures for all the methods have been described, the
performance of the hybrid method in comparison to CADOM and the P1 approximation is
evaluated for simulations of the non-gray BTE in multi-dimensional geometries.
81
4.4 Multi-Dimensional Simulations of Non-Gray Phonon BTE
In a silicon block, the mean free paths of the phonons generally range from a few
tens of nanometers to a few tens of microns. Therefore, for a sub-micron scale geometry,
the spectral Knudsen numbers vary over a few orders of magnitude. We now consider
simple 2-D and 3-D test cases geometries for transient and steady state simulations of the
non-gray phonon BTE using the hybrid, CADOM, and P1 approximation methods for both
structured and unstructured meshes. The simulations performed in all the subsequent
sections are for pure silicon and so, the appropriate phonon dispersion relations developed
in section 2.4 are used as inputs to the BTE. As previously stated, the performance and
accuracy of the hybrid method at various transport regimes is evaluated by changing the
average operating ranges of temperature from 200K to 300K in the 2-D test cases. In
addition, we also perform parametric studies of the cutoff Knudsen number and its effect
on the accuracy and efficiency of the hybrid methodology at varying operating
temperatures.
Extensive studies are initially performed to evaluate the hybrid methodology for a
difference in operating temperature of only 10K. As a result, the relaxation time scales can
be safely assumed to be spatially independent and in fact, an average value of the domain
temperature is used for calculation of the relaxation time scale. By doing so, we establish
a global splitting of spectral bands for the entire domain. This approximation is made so as
to study the effectiveness of the hybrid method by neglecting the error introduced by
ignoring the spatial variation of the relaxation time scale. This error stems from the fact
that the spectral Knudsen numbers would vary spatially and as a result, the number of
82
spectral bands solved by CADOM or the P1 approximation should ideally be varying
spatially for a fixed cutoff Knudsen number. However, in such a scenario the hybrid
method would be rendered incapable of performing simulations. We can then ignore the
errors associated with neglecting the spatial variation of the relaxation time as long as the
temperature differences imposed are small. Later, we also simulate a 2D test case with a
100K temperature difference imposed on the medium to isolate the additional error
associated with spatially varying relaxation time scales and study its effect on the hybrid
method.
4.4.1 2D Test Case with Structured Mesh
The first test case chosen to demonstrate the effective working of the hybrid method
is a simple 2D square plate, with a hot patch centered in the bottom wall as shown in Fig.
4.3. Its primary purpose is to test the accuracy of the hybrid method, which later can be
extended to complicated geometries.
Figure 4.3: Geometry and boundary conditions for Test case 1
The lengths of the sides are chosen to be 1 micron each. Yang et al. [50] and Mittal
[12] used a similar setup in their respective BTE simulation studies. All the walls shown
83
are isothermal with the heater depicted as a bold patch, centered on the bottom wall. The
patch is 0.1 micron in length. Non-gray phonon transport is considered by simulating the
2-D test case at three different temperature ranges. The phonon relaxation times and group
velocities, which were described earlier in Chapter 2 serve as inputs to the BTE. These
calculations are performed on a 100 X 100 structured mesh. The frequency domain is
discretized into 40 bands (20 LA + 20 TA phonons). In addition to this, 100 solid angles
were used to discretize the directional dependence for the CADOM. The solution is
converged when the temperature norm of the domain decreased by 5 orders of magnitude.
All simulations are performed on a 2.33 GHz Intel Core 2 Duo processor. The list of studies
performed for this particular test case are detailed below:
1) Three different temperature ranges are chosen to study the effectiveness of the
hybrid method with a cutoff of 0.1 in comparison to the CADOM and the P1
approximation for steady state simulations.
a) Case 1: Hot patch at 205 K and rest of domain at 195 K.
b) Case 2: Hot patch at 255 K and rest of domain at 245 K.
c) Case 3: Hot patch at 305 K and rest of domain at 295 K.
2) A parametric study of the cutoff Knudsen number is performed to evaluate its
effects on the hybrid method performance and accuracy at all three temperature
ranges.
3) Transient simulations for a structured mesh are performed with a fixed cutoff
Knudsen number chosen from the parametric study listed in bullet 2. This is
84
repeated at all three temperature ranges and once again comparisons with the
CADOM and the P1 approximation are made.
The three different temperature ranges are chosen so as to increase the nature of transport
from predominantly ballistic at 195K – 205K to predominantly diffuse at 295K – 305K.
Increasing the operating temperature shortens the mean free path, thus decreasing the
Knudsen number spectrum. In Cases 1-3, the temperature differences imposed on the
domain is only 10K.
Hereon, the three methods used to solve the BTE are referred to as CADOM,
Hybrid and P1. They imply the following:

CADOM – The BTE in its original form is solved for all spectral bands using the
CADOM.

Hybrid – The Hybrid method is employed to solve the BTE using the CADOM for
all bands with spectral Knudsen numbers above the cutoff Knudsen number and the
rest of the spectral bands solved by the P1 method.

P1 – The BTE is solved in all bands using the P1 approximation.
4.4.1.1 Steady State Simulations
Instead of time-marching a transient simulation to steady state, we perform a
steady-state analysis by removing the transient terms in the appropriate governing
equations. To compare the effectiveness of the various methods, the flux distributions at
the top and bottom walls and the centerline temperature profiles are computed and shown.
Before, we take a look at the results for the simulations, the spectrum of Knudsen numbers
at various temperatures is plotted in Fig. 4.4.
85
Figure 4.4: Spectrum of Knudsen numbers at various operating temperatures.
As the operating temperatures increase from Case 1 to Case 3, the Knudsen number plots
for both LA and TA phonon polarizations shift down. As a result, the number of spectral
bands below the Knudsen number cutoff of 0.1, are increasingly being solved by the P1
approximation with respect to the CADOM, for increasing temperatures.
The temperature range of 195K – 205K is considered first. Solutions using the
CADOM, P1 and Hybrid methods are listed below. The cutoff Knudsen number is set to
0.1 for the Hybrid method. This results in a split of 30 spectral bands solved using the
CADOM and only 10 spectral bands solved with the P1 approximation. To compare the
effectiveness of the three methods, the bottom wall and top wall flux distributions and the
centerline temperature distributions are studied both qualitatively and quantitatively using
the three methods.
86
The non-dimensional centerline temperature profile along x* = 0.495 obtained by
employing a CADOM based solution is shown in Fig. 4.5. The non-dimensional
temperature is computed as
T* 
T  Tcold
Thot  Tcold
(4.24)
A large temperature slip is evident near the wall because of the ballistic nature of
phonon transport at 200 K. To analyze the error in the centerline (C.L.) temperatures using
the P1 and the Hybrid methods, only the absolute error profiles are computed and displayed
subsequently, since the actual contour profiles nearly overlap and not much information
can be discerned.
Figure 4.5: Steady state non-dimensional C.L. temperature for CADOM
Figure 4.6 shows the deviation of the centerline temperature for the P1
approximation and the hybrid methods from the CADOM solution. This error is calculated
87
by the subtracting the centerline temperatures for the hybrid and the P1 approximation from
the CADOM solution and is expressed as
 THybrid / P   TCADOM  THybrid / P
1
1
(4.25)
Clearly, at steady state, the P1 approximation under predicts the cell temperatures
closest to the hot patch. This demonstrates the inability of the P1 approximation in
estimating accurate temperature contours, when the phonon transport is dominated by
ballistic behavior. The hybrid error on the other hand is much smaller than the P 1 error
since it retains the CADOM enabled solution for all those spectral bands with a Knudsen
number above the cutoff of 0.1, which are significant in number at this temperature range.
Figure 4.6: Error of centerline (C.L.) temperatures in comparison to CADOM for the P1
and HYBRID methods
88
Next, the non-dimensional fluxes for the top and the bottom walls are also evaluated
and compared using all the three methods as shown in Figures 4.7 and 4.8. The nondimensional flux,  is computed as

q nˆ
  I 0,hot  I 0,cold 
(4.26)
where the heat flux, q is computed using Eq. (2.80) and has particular expressions
depending on the solution procedure. The denominator in Eq. (4.26) refers to the maximum
possible heat flux between the hot and cold walls in the domain. For a chosen cutoff
Knudsen number of 0.1, it can be seen in Fig. 4.8 that the non-dimensional flux for the
bottom wall using the hybrid method is closer in magnitude to the CADOM solution and
is bounded by CADOM and the P1 solutions. Furthermore, at this temperature range, the
P1 approximation actually over-predicts the non-dimensional fluxes and overshoots the
upper physical bound of 1. The top wall non-dimensional flux distributions are shown in
Fig. 4.7. The top wall flux distribution for the hybrid method closely agrees with the
CADOM flux distribution, though the P1 approximation also seems accurate.
To quantify the error in the bottom patch flux for the P1 and the hybrid methods,
the percentage error of only the hot patch flux with respect to the CADOM approach is
computed and averaged. This is expressed as,
 Hybrid / P 
1
top / bot
 Hybrid / P  CADOM 

1
 Avg 
100

CADOM


top / bot
(4.27)
The average error for the P1 method is then obtained as 48.6% whereas for the Hybrid
method with a cutoff of 0.1, it is 13.50%. A similar error analysis is done for the top wall
89
flux as well, and this yields an average error of 7.89% for the P1 analysis and an error of
1.53% for the Hybrid method.
Figure 4.7: Plot for non-dimensional flux along the top wall
Figure 4.8: Plot for non-dimensional flux along the bottom wall
The hybrid method is thus more accurate at this temperature range in comparison with a P1
method.
90
Next, to better understand how the choice of the cutoff Knudsen number affects the
hybrid method, simulations were also performed by changing the cutoff to 0.05 and 0.2 in
the hybrid method. For a cutoff of 0.05, CADOM is used in 37 spectral bands and P1
approximation is used in the remaining 3 bands, while for a cutoff of 0.2, the split results
in CADOM being used for 21 bands, with the P1 method being used in 19 bands. These
results are shown in Figures 4.9 and 4.10. The error plots for steady state centerline
temperatures demonstrate that decreasing the Knudsen number cutoff improves the
accuracy of the solution at this temperature range. The solution with a cutoff of 0.05 is
generally more accurate among the three solutions obtained, except for the slip at the wall,
which is over-estimated.
Figure 4.9: Centerline temperature errors in comparison to CADOM with nondimensional distance
91
Figure 4.10: Plot for non-dimensional flux distribution along the bottom wall using
different cutoffs for the hybrid method
With regards to the non-dimensional flux distribution in the bottom wall, the hybrid
method with a cutoff of 0.05 performs best and it is closest to the CADOM solution.
Maximum deviation from the CADOM solution is observed in the hybrid method with a
0.2 cutoff. The average error of the hot patch flux computed using Eq. (4.27) for the hybrid
method with a cutoff of 0.05 is obtained as 1.78% and for a cutoff of 0.2, it is 27.92%. The
percentage errors for the top wall flux distributions are also computed. For the hybrid
method with a cutoff of 0.05, it stands at 0.72% and for the 0.2 cutoff case it is 1.39%.
Since the magnitude of the top wall fluxes are themselves small in magnitude, significant
conclusions can’t be made by analyzing the top wall flux errors alone. However, the
general observation is that by decreasing the cutoff Knudsen number, a hybrid method that
is more accurate may be obtained at this temperature range. The parametric study
92
elucidates that as the number of spectral bands, which are being solved by the CADOM as
compared to the P1 approximation are increased, the hybrid method tends to give more
accurate results. To achieve an increased CADOM enabled solution within the hybrid
method, the cutoff Knudsen number can be decreased. To better understand the reasons
behind this, the spectral variation of Knudsen number is once again considered and plotted
in Fig. 4.11. The Knudsen numbers vary from 0.022 to 149.410 at this temperature range.
Figure 4.11: Spectral variation of Knudsen numbers for the 195K- 205K test case
Though it might be tempting to decrease the cutoff Knudsen number and make the
hybrid method more accurate, it is important to realize that the additional accuracy comes
at the expense of increased computational costs. The CADOM which is the most accurate
solution that is being considered in this study takes a total CPU time of 9 hours and 42
minutes to attain convergence. The P1 method on the other hand is the fastest, and computes
in just 2 hours and 27 minutes. The hybrid solution takes an intermediate amount of time.
93
For a cutoff of 0.2, the CPU time taken is 5 hours and 42 minutes, whereas for a cutoff of
0.1, a total of 8 hours and 36 minutes is taken and for the 0.05 cutoff, a CPU time of 9
hours and 12 minutes is clocked. Thus, if the cutoff Knudsen number is decreased, larger
computational times are required. In short, based on the computational time and solution
accuracy requirements, a decision is made to determine the appropriate cutoff. The domain
temperature contour profiles for the various methods are displayed in Figure 4.12 below.
The next section deals with comparison studies for the two test cases of 245K –
255K and 295K – 305K.
94
(a) CADOM
(b) P1
(d) hybrid Knc = 0.05
(c) hybrid Knc = 0.1
(e) hybrid Knc = 0.2
Figure 4.12: Temperature contour plots for all methods used in test case at 195K - 205K
95
For a temperature range of 245K – 255K, the phonon transport becomes more
diffuse than the 195K – 205K temperature range and the spectral Knudsen numbers now
range between 0.009 and 76.497 as shown in Fig. 4.13. Once again, similar studies are
performed by first setting a Knudsen number cutoff of 0.1 and subsequently varying the
Knudsen number cutoff parametrically.
Figure 4.13: Spectral variation of Knudsen numbers for the 245K- 255K test case
The increased diffusiveness in phonon transport causes the boundary slip to be
lower as seen in the C.L. temperature in part (a) of Fig. 4.14 as compared to the 195K –
205K case. In part (b) of Fig. (4.14), it can be seen that the cell temperature closest to the
hot patch is over predicted by the hybrid method. The P1 method on the other hand
erroneously deviates in the remainder of the domain. However, other than the boundary
slip, the centerline temperature predicted by the hybrid method is closer to the CADOM
solution than the P1 method.
96
a)
b)
Figure 4.14: a) Steady state non-dimensional C.L. temperature using CADOM and b)
Error of centerline (C.L.) temperatures in comparison to CADOM for the P1 and hybrid
methods
a)
b)
Figure 4.15: (a) Top and (b) bottom wall flux distributions at steady state using CADOM,
P1 and hybrid methods.
In Fig 4.15, the top wall fluxes are nearly identical for the P1 and hybrid methods
and they differ the most from CADOM at the center of the top wall, whereas the bottom
flux values are over-predicted by the P1 method in comparison to the CADOM. The hybrid
solution is still bounded by the P1 and the CADOM solutions. It is seen that the hybrid
97
method with a cutoff of 0.1 deviates from CADOM by 5.65% for the top wall flux error
and 12.32% for the hot patch flux error, calculated using Eq. (4.24). The errors for the P1
method on the other hand, stand at 4.52% and 25.07% for the top wall and the hot patch
flux respectively. While the P1 method shows an improved accuracy over the previous
temperature range, the superiority of the hybrid method is apparent especially in accurately
estimating the bottom patch flux and the centerline temperature distributions.
a)
b)
Figure 4.16: a) Centerline temperature errors in comparison to CADOM, and
b) Plot for non-dimensional flux distribution along the bottom wall different cutoffs for
the hybrid method.
The comparison studies in Fig. 4.16 indicate that decreasing the Knudsen number
to include more CADOM enabled spectra bands in the hybrid method has a marginal
increase in terms of improving the accuracy of solution. The average error in top wall flux
distribution using the hybrid method with a cutoff of 0.05 is 4.78% and for a cutoff of 0.2,
it is 5.53%. Though not displayed, the average error for the hot patch flux stands at 4.84%
for a 0.05 cutoff and 18.83% for a 0.2 cutoff. Once again, decreasing the cutoff Knudsen
98
number would increase the accuracy of the hot patch flux distribution, however, the
accuracy for the top wall flux distribution is largely unaffected. The slip in the boundary
temperature is also over predicted by decreasing the cutoff Knudsen number. The decision
to use an appropriate cutoff then comes down to the level of accuracy desired in the
solutions. A Knudsen number cutoff of 0.1 gives adequately accurate solutions within a
certain tolerance especially if the top wall fluxes and the centerline temperature
distributions are also considered to be important. Thus, its choice over the P1 method can
also be justified at this temperature range.
The computational time for the CADOM solution at this temperature range stands
at 26 hours and 58 minutes. The P1 method only takes 3 hours and 38 minutes. The hybrid
method with a cutoff of 0.05 takes 20 hours and 51 minutes, with a cutoff of 0.1 takes 16
hours and 42 minutes and for a cutoff of 0.2 takes 11 hours and 35 minutes. This is a direct
testament to the decreasing computational time taken for an increasing Knudsen number
cutoff. The major reason for this drastic increase in computational time for the CADOM
enabled solution over the previous temperature range is attributed to strong inter-coupling
between low Knudsen number bands. For a solution to converge, by sequentially solving
each spectral band, the method is rendered slow, especially at the low Knudsen numbers
when the coupling is strongest. The steady state temperature contour profiles are also
displayed in Fig. 4.17.
99
(a) CADOM
(b) P1
(d) hybrid Knc = 0.05
(c) hybrid Knc = 0.1
(e) hybrid Knc = 0.2
Figure 4.17: Temperature contour plots for all methods in test case: 245K - 255K.
100
At the temperature range of 295K – 305K, the phonon transport becomes largely
diffuse because of the lower relaxation time scales. Once again similar studies are
performed by first setting a Knudsen number cutoff of 0.1 in the hybrid method and then
varying the cutoff Knudsen number parametrically.
a)
b)
Figure 4.18: a) Steady state non-dimensional C.L. temperature using CADOM and b)
Error of centerline (C.L.) temperatures in comparison to CADOM enabled BTE for the P1
and hybrid methods
The increased diffusiveness in phonon transport causes the boundary slip to be
lowest at this temperature range as shown in part (a) of Fig. 4.18. In part (b) of Fig. 4.18,
it can be seen that the slip in the boundary temperature is over predicted by both hybrid
and P1 methods. Beyond the cell temperatures closest to the patch, the hybrid centerline
temperatures are closer to the CADOM predictions in comparison to the P1 method.
101
a)
b)
Figure 4.19: a) Top and b) Bottom wall flux distributions at steady state using CADOM,
P1 and hybrid methods
In Fig. 4.19, the top flux distributions are once again nearly identical for the P1 and
hybrid methods, and they differ the most from the CADOM at the center of the top wall.
At this temperature range, the hybrid method borrows most of its characteristics from the
P1 approximation, as will be seen from the spectral Knudsen number plot. Furthermore,
with a highly diffuse transport nature, most of the phonons don’t reach the opposite wall
from the bottom patch and are scattered within the medium. This is the reason that both the
hybrid and P1 methods under predict the top wall flux distribution. The bottom wall fluxes
are accurately predicted by the P1 and hybrid methods in comparison to CADOM. It is seen
that the hybrid method with a cutoff of 0.1 deviates from CADOM predicted values by
11.91% for the top wall flux and 6.63% for the hot patch flux. As for the P1 method, these
errors stand at 10.84% and 9.14% for the top wall flux and the hot patch flux respectively.
In this temperature range, the choice of a hybrid method for BTE analysis cannot be
102
justified, since the P1 method itself performs an accurate job of predicting the fluxes and
temperature distributions.
The spectral Knudsen numbers range from 0.005 to 44.269 at this temperature
range. The contribution of the P1 approximation to the hybrid method is highest in this case
with a large number of spectral bands having a Knudsen number less than 0.1, as seen in
Fig. 4.20.
Figure 4.20: Spectral variation of Knudsen numbers for the 295K- 305K test case
Performing the parametric study, it is seen from Fig. 4.21 that accuracy is not
improved for the hybrid method, especially while predicting the bottom flux distribution.
Calculating the average bottom flux patch error, it is seen that for a cutoff of 0.05, the error
in fact increases to 7.28% and for a cutoff of 0.2, the error is at 6.93%. The top wall flux
errors now stand at 11.21% for a cutoff of 0.05 and at 11.80% for a cutoff of 0.2. Thus,
decreasing the Knudsen number cutoff in the hybrid method actually deteriorated the
solution and in fact, registered a larger error in predicting the boundary temperature slip,
103
as seen in part (a) of Fig. 4.21. Due to the high errors obtained, especially for the hybrid
method with a cutoff Knudsen number of 0.05, the validity of a Hybrid method at this range
is questionable and the P1 approximation itself might be more accurate in analysis.
a)
b)
Figure 4.21: a) Centerline temperature errors in comparison to CADOM, and
b) Plot for non-dimensional flux distribution along the bottom wall different cutoffs for
the hybrid method.
This is the temperature range with the strongest inter coupling of the spectral bands
with low Knudsen numbers. This directly translates into large computational time
requirements for the CADOM approach which takes 60 hours and 18 minutes to attain
convergence. The P1 approximation on the other hand takes a modest 6 hours and 38
minutes. The Hybrid method with cutoffs of 0.05, 0.1, and 0.2 take 38 hours and 55
minutes, 27 hours and 55 minutes, and 20 hours and 50 minutes respectively. The steady
state temperature contour profiles are also displayed in Fig. 4.22 below.
104
(a) CADOM
(b) P1
(c) hybrid Knc = 0.1
(d) hybrid Knc = 0.05
(e) hybrid Knc = 0.2
Figure 4.22: Temperature contour plots for all methods used in test case at 295K - 305K.
105
A brief summary of the findings described above is tabulated and displayed in Table 4.1.
Table 4.1: Summary of CPU time taken and percentage errors in top and bottom wall
fluxes for different methods
106
4.4.1.2 Transient Analysis for the same Test Cases
With the notion that a cutoff Knudsen number of 0.1 is conservative in estimating
wall fluxes and the temperature distributions accurately, a transient analysis of the same
geometric test case is now performed at the three different temperature ranges to further
test the effectiveness of the hybrid method. The procedure followed here is similar to the
previous analysis performed in Section 4.4.1.1. In all the transient simulations, three
different instances of time are chosen at which the accuracy of both the P 1 and hybrid
methods are compared with CADOM solution. This analysis is detailed below.
For a temperature range of 195K – 205K, the time step chosen in the transient
simulation is dt   max 100 , where  max corresponds to the time scale for that spectral band
with the highest Knudsen number, 149.41. This results in a t = 0.16 ns. The top and
bottom wall flux comparisons are shown at three particular instances, namely
and
 max
2
 max
100
,
 max
10
, using all the three methods, in order to draw conclusions. The C.L. temperature
using the CADOM approach and the error in C.L. temperature profiles using the hybrid
and the P1 methods are later shown in a single graph at the end of the sub-section as part
of the analyses.
From Fig. 4.23, it can be seen that the P1 method largely under predicts the top wall
flux distribution and overstates the hot patch flux distribution at the first time instance. In
fact, it yields an unphysical estimate of the bottom patch flux distribution. The hybrid
method on the other hand, yields a more accurate estimate of the top wall flux distribution
and also the bottom patch flux distribution. It is important to realize that at
107
 max
100
, not
enough time has elapsed for all the phonons from the bottom patch to scatter and propagate
into the medium. At the top wall, the hybrid method’s predicted flux is accurate because it
accounts for the contributing phonons through the directionally dependent CADOM
enabled solution in some of the spectral bands. The P1 method on the other hand, assumes
that the intensity propagation is diffuse, which leads to inaccurate estimations of the top
wall flux, since the phonons emitted from the bottom patch do not reach the top wall for
small amounts of elapsed time.
a)
Figure 4.23: a) Top and b) Bottom wall fluxes at time
hybrid methods.
At time,
 max
10
 max
b)
100 using CADOM, P1 and
, in Fig. 4.24 the system is actually close to steady state as evident
from the top wall and bottom wall flux distributions. The CADOM, P1 and the hybrid
methods yield nearly coincident top wall flux distributions. The hybrid method predicts a
flux distribution that is bounded by the CADOM and P1 solutions.
108
a)
Figure 4.24: a) Top and b) Bottom wall fluxes at time
hybrid methods.
 max
a)
10
b)
using CADOM, P1 and
b)
Figure 4.25: a) Top and b) Bottom wall fluxes at time
methods.
109
 max
2
using CADOM, P1 and hybrid
In Fig. 4.25, at
 max
2
, the system has effectively time marched to steady state. At
this instant of time, the flux distributions coincide with the ones observed at steady state
from sub-section 4.4.1.1. The time marched steady state simulation and the actual steady
state simulation for the 195 K – 205 K temperature range give the same results for flux and
temperature distributions. Thus, the time marched solution is also accurate in predicting
the flux distributions at steady state.
a)
b)
Figure 4.26: a) Centerline temperatures using the CADOM at all time steps and b) the
center line temperature errors at all time steps for P1 and hybrid methods.
In part (a) of Fig. 4.26, the evolution of the C.L. temperature using the CADOM
solution is shown. The temperature profiles at
 max
10
and
 max
2
nearly coincide with the
steady state distribution. In part (b), the errors in boundary temperature slip for the P 1 and
the hybrid methods at different time instances are shown. At time
 max
100
, the error in
temperature slips for the P1 and the hybrid methods are at their respective maximum,
though the hybrid method is more accurate. Most of the shortcomings of the P1 method can
110
be attributed to the fact that enough time has not elapsed for the phonon intensity from the
bottom patch to propagate into the medium. Thus, the P1 method is not accurate for a
transient analysis at the 195K – 205K range and in fact, the hybrid method performs
superiorly. As time progresses, the transient error reduces and the temperature profiles
revert to the steady state distributions.
The time-dependent contour profiles of domain temperature using the hybrid
method are shown next in Fig. 4.27. As time elapses, the phonon intensity propagation
causes the temperature to increase in the medium.
111
(a)
(c)
 max
 max
(b)
100
 max
10
(d) steady state
2
Figure 4.27: Contour plots of domain temperature using the hybrid method at various
instances of time for test case 195K – 205K.
112
 max at a temperature range of 245K – 255K is 8.57ns. This implies that a time
step of ∆t = 0.085ns is chosen for time marching and results are analyzed at
and
 max
2
 max
100
,
 max
10
.
a)
Figure 4.28: a) Top and b) Bottom wall fluxes at time
hybrid methods
 max
100
a)
b)
using CADOM, P1 and
b)
Figure 4.29: a) Top and b) Bottom wall fluxes at time
hybrid methods
113
 max
10
using CADOM, P1 and
,
In Fig. 4.28, it is evident that the P1 method is inaccurate in predicting the top and
bottom wall flux distributions at time,
 max
100
. The hybrid method on the other hand,
performs better and yields physically meaningful results. At time  max 10 , as seen in Fig. 4.29,
the P1 flux decreases from an unphysical limit previously predicted at
approaches the fluxes predicted by the CADOM and hybrid solutions. At
 max
 max
10
100
and
, the hybrid
method starts deviating from CADOM in predicting the top wall flux distribution and is in
fact sandwiched between the P1 and CADOM estimated fluxes, indicating a greater
contribution of the P1 enabled solution. For both the top and bottom wall flux distributions,
the hybrid method is enveloped by the CADOM and P1 method. Thus, the hybrid method
performs better than the P1 approximation at
 max
10
as well.
a)
b)
Figure 4.30: a) Top and b) Bottom wall fluxes at time
methods
114
 max
2
using CADOM, P1 and hybrid
From Fig. 4.30, it is evident that as the system reaches steady state, the P1 and the
hybrid methods limit to their respective steady state magnitudes as previously shown in
section 4.4.1.1. At
 max
2
, the P1 and the hybrid methods almost give the same top wall flux
distributions indicating a large contribution of diffuse behavior in phonon transport.
However, the hybrid method is still more accurate in predicting the bottom patch flux
distribution.
At higher temperatures, more time is often required for the medium to reach steady
state. This is because comparatively more energy is now carried by the intermediate-high
Knudsen number bands as well and sufficient time is required for these phonons to scatter
and transport the heat across the medium. This however is compensated by the fact that the
overall relaxation time scales have decreased due to an increase in temperature. Overall, a
combination of these effects comes into play.
a)
b)
Figure 4.31: a) Centerline temperatures using the CADOM at all time steps and b) the
center line temperature errors at all time steps for P1 and hybrid methods.
115
In Fig. 4.31, the centerline temperature plots demonstrate the propagation of the
system to steady state. The errors are once again highest at the first time instance when the
P1 method hasn’t had sufficient time to account for phonon intensity propagation into the
medium. This reflects in the hybrid solution as well, though it is comparatively more
accurate than the P1 method at every time instance. There is also no temporal accumulation
of errors as the system approaches steady state and in fact by
 max
2
, the system has almost
reached steady state.
The time dependent contour profiles of the domain temperature using the
hybrid method are shown next in Fig. 4.32.
116
(a)
(c)
 max
 max
(b)
100
 max
10
(d) steady state
2
Figure 4.32: Contour plots of domain temperature using the hybrid method at various
instances of time for test case 245K – 255K.
117
 max at a temperature range of 295K – 305K is 4.96ns. This implies that a time step
of ∆t = 0.049ns is chosen for time marching and results are analyzed at
 max
2
 max
100
,
 max
10
, and
.
a)
Figure 4.33: a) Top and b) Bottom wall fluxes at time
hybrid methods
 max
100
a)
b)
using CADOM, P1 and
b)
Figure 4.34: a) Top and b) Bottom wall fluxes at time
hybrid methods
118
 max
10
using CADOM, P1 and
a)
b)
Figure 4.35: a) Top and b) Bottom wall fluxes at time
methods.
a)
 max
2
using CADOM, P1 and hybrid
b)
Figure 4.36: a) Centerline temperatures using the CADOM at all time instances and b)
the center line temperature errors at all time instances for P1 and hybrid methods.
The trends for the 295K – 305K case are shown in Figures 4.33 to 4.36. Here, the
top wall flux estimated by the hybrid method closely agrees with the flux predicted by
119
CADOM at
 max
100
, and it is also enveloped by the CADOM and P1 fluxes at
 max
10
. The P1
method once again grossly under predicts the top flux distribution for small elapsed
intervals of time, since it doesn’t account for directional propagation of intensity in this
largely diffuse transport regime. However, the flux profiles for the hybrid and P1 methods
overlap as the system approaches steady state. The bottom flux prediction by the Hybrid
method is mostly sandwiched between flux predicted by the CADOM and the P1 methods,
although at
 max
100
accurate. Beyond
the fluxes estimated by the P1 and the hybrid methods are not quite
 max
10
, the bottom patch flux distributions for the three methods closely
agree with each other.
In the preceding sub-section, we saw that the average percentage error in the flux
distribution for steady state analysis of the 295K – 305 K temperature case was least for
the P1 method. However in a transient simulation, the hybrid method actually performs
better, especially to analyze the evolution of flux distributions with time. Furthermore, in
this temperature range, the system has not yet reached steady state at
 max
2
. The
intermediate-high Knudsen numbers bands now carry more energy, which in turn require
more time to equilibrate and reach steady state.
Temperature contour profiles for the transient simulation using a hybrid approach
are displayed in Fig. 4.37.
120
(a)
(c)
 max
 max
(b)
100
 max
10
(d) steady state
2
Figure 4.37: Contour plots of domain temperature using the hybrid method at various
instances of time for test case 295K – 305K.
121
4.4.2 2D Test Case with Unstructured Mesh
To demonstrate the working of the hybrid method for an unstructured mesh,
transient and steady state simulations of the same 2D test case are performed to evaluate
the flux distributions and the temperature profiles. The unstructured mesh generated
contains 22,896 triangular cells such that there are exactly 100 nodes equidistantly placed
on each side of the test case, as seen in Fig. 4.38. The spatial and angular discretizations
are similar to those in structured mesh simulations.
Figure 4.38: 2D geometry with unstructured mesh
122
Figure 4.39: a) Top and b) Bottom wall fluxes at steady state for all three methods
Figure 4.39 displays the top and bottom flux distributions using the three methods,
and once again at this temperature range, both the fluxes predicted by the hybrid method
are enveloped by the CADOM and P1 predicted fluxes. These results are similar
quantitatively and qualitatively to the results for the structured mesh simulation in Section
4.4.1.1.
The steady state contour plots for the domain temperatures using all three methods
are shown in Fig. 4.40. Though not much information can be discerned by analyzing the
contour plots using the three methods, the test case was chosen to demonstrate the efficient
functioning of the hybrid method for unstructured meshes.
With regards to computational time, the CADOM took 133 hours and 1 minute of
CPU time, whereas the Hybrid method took 99 hours and 10 minutes and the P 1 method
took a modest 14 hours and 11 minutes. This significant increase in computational time
was observed due to the high number of cell count in the mesh formulation.
123
a) hybrid
a) CADOM
b) P1
Figure 4.40: Contour plots of steady state domain temperature for the hybrid, CADOM
and P1 methods.
124
For transient simulations at the temperature range, 245K - 255K, the evolution of
the contour plots and the top and bottom flux profiles are displayed at
 max
where  max equals 8.57ns.
a)
 max
b)
100
 max
10
Figure 4.41: Bottom wall flux evolution with time
a)
 max
b)
100
 max
Figure 4.42: Top wall flux evolution with time
125
10
100
and
 max
10
,
The plots in Figures 4.41 and 4.42 are similar to the flux distributions obtained in
the transient simulations for a structured mesh as shown previously in sub-section 4.4.1.2.
However, the smoothness of the top wall flux distribution is deteriorated since the fluxes
are small in magnitude and the mesh would require further refinement to introduce
smoother profiles. On the other hand, the bottom wall flux distributions match the
structured mesh simulations exactly.
a) CADOM
 max
b) CADOM
100
 max
10
Figure 4.43: Contour plots of temperature evolution using the CADOM.
126
a) hybrid
 max
b) hybrid
100
 max
10
Figure 4.44: Contour plots of temperature evolution using the Hybrid method.
a) P1
 max
b) P1
100
 max
10
Figure 4.45: Contour plots of temperature evolution using P1 method.
The contour plots for the three methods listed in Figures 4.43 to 4.45 demonstrate
the time evolution of domain temperature. These figures illustrate the efficient functioning
127
of the hybrid method in performing transient simulations in an unstructured mesh. Similar
studies are now performed for the 3D unstructured meshes as well.
4.4.3 3D Test Case with Unstructured Mesh
The 3D geometry considered here is a rectangular block with a circular hot patch
on the top wall. An unstructured mesh is generated using 21,978 tetrahedron cells as shown
in Figures 4.46 and 4.47. 40 spectral bands are used to discretize the frequency domain and
10x10 (polar and azimuthal angles) are used to discretize the solid angle. The temperature
range considered is 195K – 205K. To improve the mesh resolution, the length of the block
in the z-axis is shortened to 0.3 microns, while the lengths along X and Y axes are fixed at
1 micron each. A circular hot patch of radius 0.05 microns shown in red is centered on the
Z = 0.3 micron plane.
Figure 4.46: Unstructured 3D mesh
128
Figure 4.47: Two sliced planes depicting mesh details
The contour plots of domain temperature at steady state using the three methods
are shown next.
b) CADOM – Sliced planes contours
a) CADOM full contours
Figure 4.48: Contour profiles of temperature using different methods for 3D test
case (CONTINUED)
129
Figure 4.48: (CONTINUED)
c) Hybrid – Sliced plane contours
d) P1 – Sliced plane contours
In Fig. 4.48, the steady state temperature contours using the three methods are
displayed. The contour profile for the 3D test case in part (a) indicates that the mesh
resolution is insufficient to generate smoother profiles. However with the current memory
limits, no further mesh refinement is possible. To overcome this, the code needs to be
parallelized and solved with HPC which is beyond the scope of this work. Upon closer
examination of the sliced plane contour plots, it is seen that the contour plots for the Hybrid
method are sandwiched by the P1 and the CADOM contour plots as expected. Finally, the
transient simulations using the hybrid method for the same 3D test case are analyzed and
displayed in Fig. 4.49. Once again, ∆t = 16 ns is considered for transient simulations.
130
a) Sliced plane contours at
 max
b) Sliced plane contours at
100
 max
10
Figure 4.49: Transient contour profiles of temperature using hybrid methods for
3D test case
4.5 Hybrid Method Applied to Problems with Large Temperature Difference
Thus far, the solution of the non-gray BTE for small temperature differences (~10
K) imposed on the medium using the hybrid methodology has been described. Since the
imposed temperature differences were small, the relaxation time was assumed to
independent of temperature and was evaluated at an average temperature, namely
Tavg 
Thot  Tcold
, for the entire medium. With this approximation, the spectral Knudsen
2
numbers need to be calculated just once for the entire solution procedure, and the
discretized spectral bands would then either follow the CADOM or the P1 methodology
until convergence.
However, if a large temperature difference (~100 K) is imposed on the medium, we
cannot guess an average temperature of the medium a priori. In that case we re-evaluate an
131
average temperature at every iteration to establish new global spectral Knudsen numbers
as the solution progresses. By doing so, we hypothesize that the distribution of spectrally
discretized bands into CADOM and P1 procedures, would then be a more accurate
representation for the entire domain. However, this method is still not completely errorfree because of the fact that the spectral Knudsen numbers would still be spatially varying
(since the relaxation time scale now cannot be treated as spatially independent), and we
only take an average variation into account. To better visualize this, it is helpful to
remember that the spectral bands split into either CADOM or P1 approximation methods
based on an average temperature for the entire domain. However, for a fixed cutoff
Knudsen number of say, 0.1, the actual split should be dependent on the local temperature,
since the relaxation time is spatially dependent. It is this local splitting that cannot be taken
into account and thus, results in additional errors.
In this study, the same 2-D test case with a 250K – 350K temperature difference
imposed on the medium is considered. The large temperature range necessitates that the
hybrid method’s global spectral Knudsen number distribution be re-evaluated using an
average domain temperature at every iteration. The test case is solved using the CADOM
and P1 approximation methods as well. Finally, a comparison is made between the three
methods.
132
a)
b)
Figure 4.50: a) Steady state non-dimensional C.L. temperature using CADOM and b)
Error of centerline (C.L.) temperatures in comparison to CADOM for the P1 and hybrid
methods
Figure 4.51: a) Top and b) Bottom wall fluxes at steady state for all three methods
The non-dimensional centerline temperature for the CADOM approach is shown in
part (a) of Fig. 4.50. The transport in this regime is generally diffuse and so a great deal of
boundary slip is not expected. However, from part (b) of Fig. 4.50, it is seen that there are
133
considerable errors in the centerline temperatures predicted by the P1 and hybrid methods.
Relatively speaking though, for a 100K difference in temperature, these errors are of the
same order of what was seen previously in the ~10K range. The hybrid method still
performs better in predicting the domain temperature and the boundary slip at the walls.
Since the phonon transport is predominantly diffuse, the P1 method is itself quite
accurate in predicting the bottom patch flux as seen in part (b) of Fig. 4.51. The hybrid flux
is not completely bounded by the P1 and CADOM fluxes and actually under predicts the
flux at the central region of the bottom patch. Overall though, the average error associated
with the bottom patch flux distribution for the hybrid method is 5.68% in comparison to
10.04% for the P1 method. With regards to the top patch flux distribution, the P1 and hybrid
methods perform similarly and are not very accurate as seen in part (a) of Fig. 4.51. The
average errors are 10.84% and 11.91% for the P1 and hybrid methods respectively.
The global spectral Knudsen number variation for the hybrid method calculated at
an average temperature of 252.6384 K is shown in Fig. 4.52. This is also the average
temperature of the domain at steady state. At this temperature range there are a total of 19
bands solved by CADOM and 21 spectral bands solved by the P1 approach within the
hybrid method. Thus, a strong P1 character is seen imparted to the hybrid solution.
134
Figure 4.52: Spectral variation of Knudsen numbers for the 250K- 350K test case
Next, the contour profiles for the steady state domain temperature using the three
methods are shown in Fig. 4.53. The hybrid contours are enveloped by the contours for the
CADOM and P1 approximation.
At this temperature range, the additional errors that are introduced due to the spatial
variation of the Knudsen number do not significantly affect the hybrid solution. As a result,
the hybrid method can be used to model microscale heat transfer more accurately than a
pure P1 approximation and more efficiently than a pure CADOM solution. However, it
would ultimately be up to the user to tweak the cutoff Knudsen number to ensure a tradeoff
between the accuracy and efficiency of the solution desired.
135
a) hybrid
a) CADOM
b) P1
Figure 4.53: Contour plots of domain temperature at steady state for the hybrid, CADOM
and P1 methods.
136
4.6 Summary
This chapter presented the discussion and analysis of the results obtained by solving
the non-gray BTE using the CADOM, hybrid and P1 approximation methods. The decision
to apply the hybrid method largely depends on the temperature range being considered. It
was seen that for the ballistic regime (~ 200K–250K), the hybrid method does provide a
more accurate description of heat transfer than the P1 analysis. However, in the diffuse
regime (~ 300K), the performances of both the hybrid and P1 methods are nearly identical
and there is really no justification in choosing the hybrid method over the P1 method. As
far as transient analysis is concerned, nearly all of the simulations performed by the hybrid
method is more accurate than the P1 method. With regards to computational time, the CPU
times clocked by the hybrid method are significantly lower than the CADOM approach.
The user is then left with a choice to implement an appropriate cutoff Knudsen number for
the hybrid method so as to balance the outcomes based on accuracy and efficiency of the
simulation. This is achieved by choosing an appropriate cutoff Knudsen number that varies
the percentage of CADOM and P1 enables solutions within the hybrid method.
In the subsequent chapter, a brief summary of the entire thesis is presented and
also the future work that can be undertaken is presented.
137
Chapter 5: Summary and Future Work
5.1 Summary
Chapter 1 introduced the various numerical approaches that have been developed
over the years to effectively solve the phonon BTE. The specific objectives of this thesis
were also stated in Chapter 1 as: (a) to perform studies in order to determine an appropriate
cut-off Knudsen number ( Knc ) by solving a 1-D, frequency independent BTE, (b) to
develop a general purpose deterministic hybrid method based on the cutoff Knudsen
number to enable transient and frequency dependent (non-gray) simulations of the BTE in
complex 3D geometries with unstructured meshes, and (c) to compare and analyze the
results of the hybrid method to the well-known CADOM and P1 approximation for a variety
of test cases.
In order to realize these objectives, an overview of phonon transport theory is
discussed in Chapter 2. Curve-fits for phonon dispersion relations and expressions for
relaxation time approximations were also described, which serve as inputs to the phonon
BTE. Furthermore, the CADOM and P1 approximation techniques to solve the non-gray
phonon BTE were discussed. The reasoning behind the development of a hybrid method
to solve the BTE was described based on the limitations and shortcomings of the CADOM
and P1 approximation techniques. The hybrid method then solves for the BTE using the
CADOM, for those spectral bands with Kn > Knc (or the ballistic regime), and the P1
approximation, for those spectral bands with Kn < Knc (or the diffuse regime).
138
Chapter 3 was dedicated to solving a 1-D, frequency independent or gray BTE so
as to obtain an appropriate cutoff Knudsen number. The non-dimensional fluxes in a 1-D,
plane parallel medium with two isothermal walls, were evaluated using an analytical
approach and the P1 approximation, at several Knudsen numbers. An appropriate cutoff
Knudsen number is then chosen based on the accuracy of the P1 solution in comparison to
the analytical solution.
In Chapter 4, finite volume formulations for the multidimensional CADOM, P1
approximation and hybrid techniques were discussed. The methodologies are then applied
to a variety of 2D and 3D test cases that include steady state and transient simulations for
both structured and unstructured meshes. Initially, the simulations were performed
assuming that the relaxation time scale was spatially independent, in order to avoid those
errors that originate from neglecting the spatial variation of spectral Knudsen numbers. The
results indicate that for steady state simulations, the hybrid method is much more accurate
than the P1 approximation for the low-intermediate temperature ranges of 200K–250K and
also requires significantly lesser computational time than a pure CADOM-enabled
solution. At high temperature ranges on the other hand, the P1 approximation performs
similar to the hybrid approach and there is really no strong justification that warrants the
choice of a hybrid method. As far as transient simulations are concerned, the hybrid method
is more accurate than the P1 method at all intermediate time steps until steady state is
achieved. This is seen across all the temperature ranges that were discussed and the
accuracy is evident from the centerline temperature plots and non-dimensional flux
distributions at the boundaries. Furthermore, the hybrid method was extended to a transient
139
analysis of a simple 3D geometry with an unstructured mesh as well, where the results
show great promise for further analysis.
Lastly, the hybrid method with a spatially dependent relaxation time scale was
implemented. In this case, the global spectral Knudsen number distribution was recalculated based on an evolving average temperature of the domain until convergence.
However, errors due to the spatial variation of relaxation time still come into play since the
local spectral Knudsen numbers do not follow the global distribution of the spectral
Knudsen numbers imposed on the entire medium. Overall, the errors due to spatial
variation of the relaxation time are insignificant and the hybrid method still performs better
in comparison to a pure P1 formulation especially in predicting the domain temperature and
average boundary heat fluxes.
In summary, this work justified and presented a hybrid method based on a cutoff
Knudsen number to simulate the full transient and non-gray phonon BTE in 3D geometries
with unstructured meshes. The present work will allow further investigation of the physical
phenomena underlying heat conduction in sub-micron scale devices. This work might just
be a nascent but an important step towards devising full-fledged thermal management
strategies to mitigate common artefacts like temperature hotspots, high heat fluxes, etc.,
which would ultimately aid in the further advancement and miniaturization of
microprocessors in the semiconductor industry.
5.2 Future work
To simulate complex 3D geometries for real semiconductor devices, parallel
implementation of the hybrid code is necessary. Parallelization has already been achieved
140
for the CADOM solution [43] and it would be a simple matter of extending it to the hybrid
method, so as to enable more accurate and real time simulations of semiconductor devices
of practical interest.
Furthermore, the errors introduced as a result of the spatial variation of the
relaxation time scale are not completely negligible for the hybrid method. For a fixed cutoff
Knudsen number, the actual split of the spectral bands into CADOM and P1 approximation
methods would be spatially varying. Since this variation cannot be taken into account, it
might be beneficial to break the entire domain into smaller sub-regions so that a local
average temperatures can be computed and used to implement the hybrid method in each
of these regions. Such a local average temperature might be a better representation of the
spectral band split that can better capture the local physics of phonon transport.
It might also be advantageous to break up the domain into several regions, each
with its own distinct cutoff Knudsen number. For regions prevalent with high temperatures,
since the phonon transport is diffuse, increasing the cutoff Knudsen number so as to
introduce a more P1 enabled solution within the hybrid method would be more beneficial.
On the other hand, introducing a lower Knudsen number cutoff for low temperature regions
so as to predominantly retain the CADOM character of the solution, would be more
accurate. Furthermore, for those regions with smaller temperature gradients, the hybrid
methodology can also be solved by assuming a spatially independent relaxation time scale.
With these incorporations, the overall temperature distribution can then be predicted.
141
Overall, optimizing the accuracy of the solution with respect to computational
requirements is left to discretion of the user. There doesn’t exist a most appropriate cutoff
Knudsen number that would could be guaranteed as the most accurate.
142
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