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James M. Loy
An Acceleration Technique for the Solution of the Phonon Boltzmann Transport
Equation
Master of Science in Mechanical Engineering
Prof. Jayathi Y. Murthy
Prof. Alina Alexeenko
Prof. Xiulin Ruan
Prof. Jayathi Y. Murthy
Anil K. Bajaj 4/13/2010
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An Acceleration Technique for the Solution of the Boltzmann Transport Equation
Master of Science in Mechanical Engineering
James M. Loy
4/7/2010
3. AN ACCELERATION TECHNIQUE FOR THE SOLUTION OF THE PHONON
BOLTZMANN TRANSPORT EQUATION
A Thesis
Submitted to the Faculty
of
Purdue University
by
James M. Loy
In Partial Fulfillment of the
Requirements for the Degree
of
Master of Science in Mechanical Engineering
May 2010
Purdue University
West Lafayette, Indiana
5. iii
ACKNOWLEDGMENTS
I am fortunate to be able to say that I did not finish this thesis on my own. First, I
would like to thank my colleagues and mentors: Dr. Lin Sun, Chandra Varanasi, Zhen
Huang, Ben Pax, Aarti Chigullapalli, Jose Pascual-Gutierrez, and Shankadeep Das. I
cherish your friendship and support and look forward to working more with you. I
especially would like to thank Dhruv Singh for helping to develop the basis for the hybrid
solver, and Dr. Charles Ni for providing and helping me decipher the heat generation
rates from the electron Monte Carlo. I would also like to thank Drs. Zlatan Aksamija and
Umberto Ravaioli of UIUC for generously providing their e-MC data.
I give my sincere gratitude to my academic advisor, Prof. Jayathi Murthy. You
have been a constant beacon of support ever since I started working with you and I am
deeply appreciative for everything you have done for me. Your laid back approach,
kindness, enthusiasm, determination, and expertise makes you amazing to work with and
gives me great pride as your student. I look forward to working with you in the future.
I would also like to thank Prof Alina Alexeenko and Prof Xiulin Ruan for
agreeing to serve on my master’s committee.
Support Purdue’s PRISM Center under funding from the Department of Energy
(National Nuclear Security Administration) Award Number DE-FC52-08NA28617 is
gratefully acknowledged.
Lastly, I must thank my family, and friends. I especially would like to thank my
parents. I am blessed to have such kind, understanding, and supportive people in my life.
6. iv
TABLE OF CONTENTS
Page
LIST OF TABLES............................................................................................................. vi
LIST OF FIGURES ..........................................................................................................vii
NOMENCLATURE ........................................................................................................... x
ABSTRACT......................................................................................................................xii
CHAPTER 1. INTRODUCTION ....................................................................................... 1
1.1. Motivation ................................................................................................................ 1
1.2. Survey of Literature.................................................................................................. 3
1.2.1. Fourier Models....................................................................................................3
1.2.2. Phonon BTE........................................................................................................4
1.2.3. Acceleration Methods for the BTE .....................................................................7
1.2.4. Monte Carlo Simulations ....................................................................................9
1.2.5. Molecular Dynamics.........................................................................................11
1.2.6. Interface Modeling and Simulation...................................................................13
1.2.7. Thermal Modeling in Microelectronics ............................................................14
1.3. Overview of Proposed Work.................................................................................. 16
CHAPTER 2. A NEW HYBRID FOURIER-BTE MODEL............................................ 19
2.1. Hybrid Fourier-BTE Model.................................................................................... 20
2.2. Non-Gray Phonon Boltzmann Transport Equation ................................................ 21
2.2.1. Boundary Conditions ........................................................................................23
2.3. Modified Fourier Equation..................................................................................... 24
2.3.1. Boundary Conditions ........................................................................................26
2.3.2. Limits of MFE...................................................................................................28
2.4. Lattice Temperature Recovery ............................................................................... 29
2.5. Finite Volume Method for BTE ............................................................................. 29
2.5.1. Boundary Conditions ........................................................................................32
2.6. Finite Volume Method for Modified Fourier Equation.......................................... 33
2.6.1. Boundary Conditions for MFE .........................................................................34
2.7. Dimensionless Parameters...................................................................................... 34
2.8. Solution Algorithm................................................................................................. 34
CHAPTER 3. ALGORITHM PERFORMANCE............................................................. 37
3.1. Verification of Single Band BTE and Hybrid Solvers ........................................... 37
3.2. Performance of Two-Band Model.......................................................................... 41
3.2.1. Results...............................................................................................................43
3.3. Full Band Simulation.............................................................................................. 48
3.3.1. Results...............................................................................................................50
7. v
Page
CHAPTER 4. THERMAL MODELING OF BULK MOSFET....................................... 63
4.1. Domain ................................................................................................................... 63
4.1.1. Simulation of Joule Heating..............................................................................64
4.2. Results .................................................................................................................... 66
4.2.1. Temperature ......................................................................................................66
4.2.2. Heat Flux...........................................................................................................70
4.2.3. Timing...............................................................................................................73
CHAPTER 5. CLOSURE................................................................................................. 74
5.1. Summation.............................................................................................................. 74
5.2. Future Work............................................................................................................ 75
5.3. Closure.................................................................................................................... 77
LIST OF REFERENCES.................................................................................................. 78
APPENDIX. CRYSTAL PROPERTIES OF SILICON AT 300 K.................................. 86
8. vi
LIST OF TABLES
Table Page
3.1 Timing and accuracy results of BTE and hybrid solvers............................................ 51
3.2 Timing results for hybrid simulations......................................................................... 60
3.3 Timing metrics for hybrid solver. ............................................................................... 61
4.1 Percentage of heat generation in each phonon polarization [35]................................ 66
4.2 Branchwise percentage of total heat leaving the boundaries...................................... 73
Appendix Table
A.1 Crystal properties of the Longitudinal Acoustic branch in the [100] direction. ........ 86
A.2 Crystal properties of the Transverse Acoustic branch in the [100] direction. ........... 87
A.3 Crystal properties of the Longitudinal Optical branch in the [100] direction............ 87
A.4 Crystal properties of the Transverse Optical branch in the [100] direction............... 88
9. vii
LIST OF FIGURES
Figure Page
2.1 Dispersion curves for silicon in [100] direction using environment
dependent inter-atomic potentials [28]. ...................................................................... 21
2.2 Unit vector for s. ......................................................................................................... 22
2.3 Thermalizing boundary condition at temperature T1 with an outward
pointing normal n....................................................................................................... 27
2.4 Finite volume discretization in the spatial domain. .................................................... 30
2.5 Finite volume angular discretization........................................................................... 31
2.6 Flow chart of overall solution algorithm. ................................................................... 36
3.1 Domain schematic for phonon transport between parallel plates. .............................. 38
3.2 Dimensionless temperature profile obtained from the BTE solver
compared to the exact solution obtained from [29]. Values in parentheses
are the Knudsen numbers............................................................................................ 39
3.3 Dimensionless temperature profile obtained from the hybrid solver
compared to the exact solution obtained from [29]. ................................................... 40
3.4 Percentage difference in the heat flux predicted by the hybrid solver
compared to [29] and to Fourier’s law........................................................................ 41
3.5 Schematic of the two-dimensional two-band domain................................................. 42
3.6 Hybrid heat flux error plotted against varying BTE band Kn..................................... 44
3.7 Hybrid heat flux error plotted against varying MFE band Kn.................................... 44
3.8 Iteration count for the hybrid and all-BTE solver with varying Knudsen
number and lattice ratios............................................................................................. 45
3.9 Timing results of the hybrid solver compared to the BTE solver............................... 47
3.10 Mean free paths for silicon at 300 K......................................................................... 48
10. viii
Figure Page
3.11 Dispersion curves for silicon in the [100] direction taken from
the environment dependent inter-atomic potential [28].............................................. 50
3.12 Lattice temperature of all-BTE and hybrid solvers along the diagonal from
(x,y)=(0,0) to (x,y)=(L,L). L=100 nm. ...................................................................... 52
3.13 Longitudinal branch temperatures along the diagonal from (x,y)=(0,0)
to (x,y)=(L,L) obtained from BTE and hybrid solver. L=100 nm. .......................... 53
3.14 Branchwise fractional heat flux along the left wall obtained from
the all-BTE and hybrid solvers. L=100 nm. .............................................................. 54
3.15 Total heat flux along the left boundary obtained from the all-BTE and hybrid
solvers. ....................................................................................................................... 55
3.16 Bandwise temperatures (x,y)=(0,0) to (x,y)=(L,L) obtained from
the hybrid solver at L=500 nm................................................................................... 56
3.17 Bandwise temperatures (x,y)=(0,0) to (x,y)=(L,L) obtained from
the hybrid solver at L=1000 nm.................................................................................. 56
3.18 Bandwise temperatures (x,y)=(0,0) to (x,y)=(L,L) obtained from
the hybrid solver at L=3000 nm.................................................................................. 57
3.19 Dimensionless heat flux on the left wall obtained from the hybrid
solver. L=500 nm....................................................................................................... 58
3.20 Dimensionless heat flux on the left wall obtained from the hybrid
solver. L=1000 nm..................................................................................................... 59
3.21 Dimensionless heat flux on the left wall obtained from the hybrid
solver. L=3000 nm..................................................................................................... 59
4.1 Domain used for thermal simulation of MOSFET. .................................................... 64
4.2 Spatial distribution of heat generation due to electron-phonon scattering [35].......... 65
4.3 Contours of lattice temperature (K) calculated by the hybrid solver.......................... 67
4.4 Branchwise temperature plot at position x*=1/6........................................................ 69
4.5 Branchwise temperature plot at position x*=5/6........................................................ 69
4.6 Branchwise temperature plot at position y*=0.85...................................................... 70
4.7 Branchwise dimensionless heat flux along left wall................................................... 71
11. ix
Figure Page
4.8 Branchwise dimensionless heat flux along right wall................................................. 72
4.9 Branchwise dimensionless heat flux along bottom wall............................................. 72
12. x
NOMENCLATURE
C Volumetric specific heat, J/m3
K
D(ω) Density of states, m-3
e” Spectral energy density, J/m3
sr
e0
Equilibrium energy density, J/m3
sr
"e Band average spectral energy density, J/m3
sr
f Particle population, m-3
sr-1
m-1
f0
Equilibrium particle population, m-3
sr-1
m-1
Reduced Plank’s constant, Js
i Unit vector in x direction
j Unit vector in y direction
k Unit vector in z direction
k Thermal conductivity, W/mK
K Wave number, m-1
kb Boltzmann constant, J/K
Kn Knudsen number
Knc Cutoff Knudsen number
L Characteristic length, m
n Outward pointing normal vector
13. xi
q” Heat flux vector, W/m2
r Position vector, m
RL Lattice ratio
s Unit direction vector
t Time, s
T Temperature, K
Greek Symbols
θ Polar angle, rad, or Dimensionless Temperature
τ Relaxation time, s
φ Azimuthal angle, rad
Φ Band spectral perturbation, J/m3
sr
Ψ Dimensionless heat flux
ω Frequency, rad/s
Ω Solid angle, sr
14. xii
ABSTRACT
Loy, James M. M.S.M.E., Purdue University, May, 2010. An Acceleration Technique for
the Solution of the Phonon Boltzmann Transport Equation. Major Professor: Dr. Jayathi
Murthy, School of Mechanical Engineering
Advanced nanofabrication techniques have enabled the creation of novel engineering
devices. As a consequence of shrinking size, new physics must be explored due to sub-
continuum behavior. Specifically, phonons have been identified as a major player in
thermal management in semiconductor devices. Non-gray phonon transport solvers
based on the Boltzmann transport equation (BTE) are frequently employed to simulate
sub-micron thermal transport. Typical solution procedures using sequential solution
schemes encounter numerical difficulties because of the large spread in scattering rates.
For frequency bands with very low Knudsen numbers, strong coupling between the
directional BTE’s results in slow convergence for sequential solution procedures. In this
M. S. thesis, a hybrid Fourier-BTE model is presented which addresses this issue. By
establishing a phonon group cutoff (say Knc=0.1), phonon bands with low Knudsen
numbers are solved using a modified Fourier equation (MFE) which includes a scattering
term as well as corrections to account for boundary temperature slip. Phonon bands with
high Knudsen numbers are solved using a BTE solver. Once the governing equations are
solved for each phonon group, their energies are then summed to find the total lattice
energy and correspondingly, the lattice temperature. An iterative procedure combining
the lattice temperature determination and the solutions to the modified Fourier and BTE
equations is developed. The procedure is shown to work well across a range of Knudsen
numbers.
15. xiii
To demonstrate the robustness and accuracy, the hybrid solver is used to predict
temperature fields in one- and two-dimensional silicon slabs. Substantial solution
acceleration is shown for a wide range of phonon Knudsen number with very little loss in
accuracy. Joule heating in a metal oxide semiconductor field effect (MOSFET) is also
simulated using phonon emission data taken from an electron Monte Carlo (eMC)
simulation developed by Aksamija. Quantities of interest are compared to an all-BTE
simulation to establish performance and accuracy.
16. 1
CHAPTER 1. INTRODUCTION
1.1. Motivation
Over the last two decades, thermal management has emerged as a critical
bottleneck to the continued scaling of microelectronics [1,2]. The move to finFETs and
thin-body multigate devices has led to improved sub-threshold slopes, but at the cost of
increased thermal resistance [3]. Despite reduction in operating voltage, power density
has been shown to scale inversely with channel length, leading to strongly non-
equilibrium transport in emerging ultrascaled microelectronic devices. Device sizes of 16
nm have already been fabricated [4] in keeping with Moore’s Law. Power dissipation per
unit area continues to increase and self-heating remains a central cause for failure in
microelectronics [2]. Thus, there is interest in understanding thermal transport at the sub-
micron scale. Furthermore, exciting advances in micro- and nanofrabrication have made
possible novel materials and systems. These include new nanocomposites [5],
thermoelectrics [6], photovoltaics [7], as well as engineered microsystems such as micro
electro-mechanical systems (MEMS) and nano-electromechanical systems (NEMS).
These advances have spurred interest in understanding and simulating multiphysics,
multiscale phenomena [8-10] .
Thermal transport in semiconductors and dielectrics is paramount in
characterizing performance and reliability of microsystems. It is known that predicting
thermal behavior on length and time scales of these microsystems using Fourier’s Law
yields erroneous results [11]. Thermal transport in semiconductors and dielectrics is due
quantized lattice vibrations called phonons [12]. Depending on the length scale, phonons
can exhibit particle-like behavior, or wave-like behavior with interference and coherence
effects. Scientists and engineers have employed several strategies to model thermal
behavior with varying success. These include atomistic models of thermal transport
17. 2
using molecular dynamics [13,14] and mesoscale models based on the Boltzmann
transport equation [11,15] . Multiscale models integrating thermal behaviors across
scales have also begun to appear [16]. Furthermore, in most engineered nanosystems,
interfaces play a dominant role because of the increased surface-to-volume ratio at the
nanoscale. A variety of new techniques for determining interface transmissivity at
heterogeneous interfaces have begun to appear in the literature [17-19].
These modeling advances call for a commensurate effort in developing robust,
efficient and accurate simulation methods to support the analysis and design of micro-
and nanosystems and their components. The focus of the present work is to develop
simulation algorithms to efficiently and accurately address the computation of thermal
transport in phononic systems across a range of length scales. The Boltzmann transport
equation is used as the basis. Though the method developed here is applied to phonon
transport, it may be generalized to other types of carrier transport based on the Boltzmann
transport equation, whether it is by phonons, electrons or gas molecules.
The Boltzmann transport equation (BTE) has been used to model phonon
transport in semiconductors and dielectrics and is valid when the characteristic length of
the domain is much larger than the phonon wavelength [11,20,21]. In this regime, wave
effects may be ignored and phonons can be viewed as semi-classical particles where
ballistic transport may be combined with scattering events to model energy transport. A
variety of parameters and models are necessary to provide inputs to the phonon BTE,
including phonon group velocities, scattering rates and interface transmission rates in
heterogeneous systems. Relaxation rates have recently either been computed from
perturbation theory [22] or molecular dynamics [23]. Interface transmissivity has been
computed using molecular dynamics [13] or using the atomistic Green’s function (AGF)
technique [17]. Furthermore, the phonon BTE yields Fourier’s Law at sufficiently low
Knudsen numbers. Thus, the BTE offers an attractive and comprehensive pathway
coupling the atomistic scale to the meso- and macroscales.
Nevertheless, a number of computational challenges must be addressed before
comprehensive simulation methods can be developed which robustly span these scales.
First is the issue of computational cost. Phonon BTE simulations which account for
18. 3
phonon dispersion and polarization are very expensive because of the large phase space
to be resolved. Second, the underlying computational techniques must perform robustly
across a wide range of length scales. It has been known that currently-published solution
techniques for the BTE suffer from slow convergence at low phonon Knudsen numbers
(large domain lengths); numerical solutions are very difficult to obtain at sufficiently
large length scales.
The focus of the present thesis is to develop a computational method for the non-
gray phonon BTE which allows efficient multi-scale simulation. We demonstrate that the
proposed technique significantly reduces computational cost and accelerates solution
convergence across the range of Knudsen numbers, with little or no compromise in
solution accuracy. In the sections below, we first review the major simulation
methodologies in the literature, and their application to microelectronics and sub-micron
thermal transport. An overview of the proposed work is then given.
1.2. Survey of Literature
1.2.1. Fourier Models
In the continuum limit, thermal transport may be modeled using the classical
Fourier conduction equation. A number of models based on this approach have been
used to simulate thermal transport in microelectronics [24-26] . In [25], Majumdar et al.
separated phonons into two groups: a dispersionless longitudinal optical (LO) group, and
a longitudinal acoustic (LA) group. The LO group was responsible for scattering with
electrons and LA phonons, but had no contribution to thermal transport. The LA group
scattered only with LO phonons and was responsible for all thermal conduction. Electron
transport was calculated using the Poisson equation, the drift diffusion equation, and
particle conservation. Electron energy was solved using a hydrodynamic model which
included scattering with LO phonons. The above equations were solved simultaneously
to determine electrical performance and temperature fields in a silicon MOSFET with a
19. 4
source-drain voltage of 3V and zero gate voltage. Results showed extremely high
electron temperatures and modest rises in acoustic and optical phonons. The same
methodology in [25] was applied in [24], but the effect of gate voltage was investigated.
Electrical and temperature fields were examined when a gate voltage of -2V was applied.
It was found that with the applied gate voltage, Joule heating was reduced due to an
effective shortening of the electron channel length, which reduced the current flow
thereby reducing the power dissipation, as well as a decrease of electron drift velocity
due to saturation. However, electron temperatures were higher in certain regions. This
was also due to the shortening of the channel length, which increased the electric field,
thereby energizing electrons much more. Lai [26] used the same model in [24] and [25],
but considered the possibility of low electric fields which caused electrons to scatter with
LA phonons rather than LO phonons. Moreover, Lai made a simplifying assumption in
the electron momentum equation based on the electron Knudsen number. An implicit
relation between the electric field and electron energy is also used in the calculation. The
equation set was solved for a MOSFET with a source-drain voltage of 3.3 V and a gate
bias of 2.5 V.
The main drawback to the three models mentioned above is the characterization
of ballistic effects. Only acoustic phonons are considered to have any effect on thermal
transport, which is not a bad assumption. However, it is assumed that thermal transport is
by diffusion. In most semiconductor materials of interest, acoustic phonons have large
mean free paths which lead to more ballistic transport of energy. Also, boundary and
interface scattering cannot be accounted for directly.
1.2.2. Phonon BTE
When phase coherence effects are not important, a particle approach to transport
is viable, and has been successfully applied to electrons and phonons as well as gas
molecules. The Boltzmann transport equation is shown below: [12]
coll
f f
f f
t t
r vv (1.1)
20. 5
Here f is a distribution of particles with a given wave vector, position, and time, v is the
velocity vector associated with the particles, and α is an acceleration vector applied to the
particles. The source term on the right hand side is the rate at which the distribution
function is changed due to scattering events, due to particle-particle, particle-boundary or
other interactions. For phonon transport, the acceleration term in Eq. (1.1) is zero, and
for steady transport, the time derivative term is zero as well. Noting that the energy
associated with a phonon is ħω, we may rewrite Eq. (1.1) in energy form as follows:
,
, ,
, ( )
p
p p
coll
p
e
e
t
e f D d
v
(1.2)
Here, ,pe is the average energy associated with a frequency band , D(ω) is the
density of states of phonons of frequency and vω,p is the group velocity computed from
the dispersion relation. It may be computed as the gradient of the dispersion relations,
which are computed using lattice dynamics for a given material [12,27,28] .
Eq. (1.2) is difficult to solve analytically, even for the simplest cases [29].
Numerical methods have been able to obtain solutions to Eq. (1.2) with varying levels of
complexity. The simplest of these approximations is the gray phonon model which
assumes no dispersion or polarization dependence, isotropic crystal properties, and
represents the right hand side through a single-mode relaxation time approximation. The
single-mode relaxation time approximation within the framework of the gray BTE
computes the scattering term by assume that phonons relax to an equilibrium bath over a
relaxation time τ; the equilibrium energy density is given by the Bose-Einstein
distribution. The gray relaxation time approximation form of the BTE is shown below:
0
0
"
"
1
"
4
L ref
e e
v e
e e d C T T
(1.3)
Here, e0
is the equilibrium energy density given by the Bose-Einstein distribution. v is
the phonon group velocity, and is typically chosen to represent the most dominant
phonons in the problem under consideration. The scattering rate, τ, is typically found by
21. 6
requiring that the gray BTE recover the measured bulk thermal conductivity n the thick
limit:
21
3
k Cv (1.4)
Here C is the specific heat and k is the bulk thermal conductivity. Using this relatively
inexpensive version of the BTE, Fourier conduction is obtained in the limit of low
Knudsen number (Kn=vτ/L) with t>>τ; this model also recovers the bulk thermal
conductivity. However, the gray BTE fails to account for any granularity in phonon
properties, and therefore cannot resolve highly non-equilibrium situations where hot
phonon effects may be important. Such a model is unsuitable for modeling self-heating
in microelectronics; here electrons generally scatter preferentially to high frequency
phonons with low group velocities, rather than to all phonons equally. It is also incapable
of resolving frequency and mode-dependent filtering that occurs at heterogeneous
interfaces [30].
To combat the lack of granularity found in the gray model, the so-called “two-
fluid” model was developed [31]. Here, all phonons are split into two groups: a
propagation group and a reservoir group. The propagation group is solely responsible for
energy transport, while the reservoir group is responsible for accounting for the
capacitive nature of the lattice and is given a zero group velocity. Some granularity is
captured; however the use of a single group velocity and a single relaxation time may still
lead to erroneous results. If the relaxation time is found by matching bulk k, as with the
gray model, the time scale for optical-to-acoustic mode scattering is found to be far
longer than generally estimated. Such long times between scattering events leads to an
unphysical rise in temperature when predicting Joule heating [20].
To better capture the differences between different phonon branches, a so-called
“full dispersion” model was developed [15,32]. Here multiple phonon bands are defined
which resolve the dispersion curves of silicon, albeit under an isotropic assumption.
Each branch is discretized into phonon bands. The optical modes are assumed to have
negligible group velocities so the ballistic term is ignored. All interactions between
22. 7
different phonon bands are accounted for through an interband scattering rate based on
the work of Klemens [33] and modeled on the relaxation time approximation.
The “full-scattering” BTE model is the most rigorous of this class of BTE models,
and accounts for dispersion, polarization and crystal anisotropy [34-36]. Furthermore,
the scattering rate is computed by careful consideration of all three-phonon interactions,
based on the perturbation theory of Klemens [33]. Wang [34] developed a search scheme
which found all three phonon interactions between all phonon groups. From there,
perturbation theory was used to calculate frequency and direction dependent scattering
rates. Dispersion in all directions is also calculated. The resulting model, though
rigorous, is computationally expensive because the scattering rate uses the non-
equilibrium distributions of the interacting phonons. Thus millions of three-phonon
interactions must be computed each iteration of the numerical solution. In [35], effective
single-mode relaxation times were computed based on the three-phonon scattering rates
in [34]. These relaxation times could be pre-computed and stored, greatly reducing the
time-to-solution while still maintaining direction and frequency dependence. Using this
scheme, coupled electro-thermal modeling in a bulk field effect transistor (FET) was
conducted. The heat source due to self-heating was obtained from an electron-Monte
Carlo simulator, and coupled one-way to the phonon transport simulation. The result
showed that optical phonons are crucial in determining hotspot temperature rise due to
the small group velocity.
1.2.3. Acceleration Methods for the BTE
Though detailed non-gray phonon transport models can capture the details of
thermal transport well, they are difficult to solve because of the large spread in relaxation
times. In a typical transport problem in room-temperature silicon, single-mode relaxation
times may range over several orders of magnitude, rendering some phonon groups nearly
ballistic, while others may be nearly diffuse. Typical discrete ordinates or finite volume
solution methods, borrowed from the radiation literature [37], employ sequential solution
techniques whereby the BTEs in different directions, frequency bands and polarizations
are solved in turn. The BTEs are coupled by the lattice temperature, which is determined
23. 8
from the total energy of all phonon groups. If the Knudsen number is low, inter-BTE
coupling becomes too strong for such a sequential procedure to be tenable, and slow
convergence rates result. Furthermore, careful resolution of exponentially-thin energy
density profiles is necessary at given-temperature boundaries to capture heat fluxes
accurately.
A number of publications in the thermal radiation literature have sought to
address the convergence issue through the development of solution acceleration schemes
for the radiative transfer equation (RTE). The RTE is similar in structure to the BTE for
isotropic scattering. One popular strategy is to advance the angular-average of the
radiative intensity as a way to improve inter-directional coupling. Chui and Raithby [38]
proposed a multiplicative correction of the average intensity in the context of the finite
volume scheme. However, the scheme was not uniformly convergent. Fiveland and
Jessee [39] proposed and evaluated a number of acceleration strategies for the discrete
ordinates method. These included the successive over-relaxation method, the mesh
rebalance method, and the synthetic acceleration method. The mesh rebalance method,
which is similar to [38], was found to perform the best, but its performance deteriorated
as the mesh-based optical thickness decreased. The method had to be modified to
perform rebalance on a coarser mesh than that for the actual solution, so as to keep the
rebalance mesh optical thickness (inverse of Knudsen number) greater than unity.
Mathur and Murthy [40] proposed a point-coupled multigrid technique to significantly
accelerate solution convergence; however, the method is best suited to isotropic
scattering problems, and may become too complex for arbitrary scattering kernels. More
recently, Hassanzadeh [41] developed the QL algorithm, in which an equation for the
average radiative intensity was used to better couple directional intensities. Significant
solution acceleration was reported for radiative equilibrium problems. Mathur and
Murthy [42] proposed modifications to the scheme based on a two-level angular
multigrid idea which alleviated the loss of overall energy balance in the QL algorithm.
Though these new methods hold much promise for phonon transport as well, we are not
aware of any use of these acceleration schemes in the BTE literature.
24. 9
1.2.4. Monte Carlo Simulations
Monte Carlo (MC) simulations have been very popular in recent years to simulate
charge transport [43-46] in semiconductor devices. Because of its relatively long history,
several variations of electron Monte Carlo (eMC) exist. Mainly, their differences lie in
the representation of the electron dispersion and level of inclusion of electron-phonon
scattering, which will be discussed in Section 1.2.7. In general, an eMC simulation starts
with the initialization of N number of electrons. Quantities of interest including position
in real space, position in K-space, etc. are chosen randomly according to the appropriate
distribution. The electrons are then allowed to travel undisturbed for a time step Δt. The
state of the electron is determined and a scattering probability is found based on this
energy. The state of the electron is then determined for the beginning of the next time
step based on whether scattering occurred. After a long enough simulation time, overall
quantities of interest can be extracted from the simulation. MC methods have also been
used to simulate radiative heat transport in nonparticipating media [47], as well as
participating media [48].
Only in the past decade has MC been applied to simulate thermal transport
through tracking phonon behavior. Mazumdar and Majumdar [49] outline the technique
for phonon MC considering only acoustic phonons. Mittal and Mazumdar [50] modified
the technique found in [49] to include optical phonons. A brief explanation of the
solution procedure is given now.
The physical space is discretized in to arbitrary polygons (2-D) or polyhedral (3-
D). The frequency spectrum is discretized in to several frequency bins, Nb. The initial
number of phonons at a given polarization and frequency band is determined using the
relation below [12,49]:
, ,
1
i
Nb
p p i i
i
N n D
(1.5)
Here, Nω,p is the phonon number density for a given polarization, p, and spectral width,
Δωi. D(ω) is the density of states which is multiplied by the equilibrium phonon
distribution nω,p. Generally, Nω,p is too large for practical computation, so a scaling factor
is introduced, whereby one particle in the simulation is used to represent multiple
25. 10
particles in reality. Once the number density is determined, the particle position in real
space is randomly calculated within each control volume. The frequency, polarization,
and wave vector of each particle is then determined with the use of random number
generation and appropriate statistical distributions. After all initialization is complete, the
simulation starts with allowing all phonons to engage in free flight for a time step Δt, the
velocity of which is determined from the dispersion relation. At the end of the time step,
the energy of each phonon is calculated. With the current energy, perturbation theory
[33,51,52] is used to calculate the probability of scattering. If the phonon is scattered, the
frequency is found from the equilibrium distribution at the current temperature. From
this, the wave vector can be found. The polarization is determined from a random
number and a statistical distribution. An important point, compared with eMC, is that
phonons are created and destroyed during the scattering process. Therefore, the number
of phonons in a domain is never conserved, rather, the summed energy is conserved by
adding or subtracting phonons to guarantee that energy is not lost or created through
scattering.
Phonon MC has many advantages. The most obvious is the simplicity of the
implementation. The solution procedure is simplified greatly when considering gray
phonon transport. Parallel implementation of phonon MC is also very easily admitted.
Because of the calculation of scattering solution procedure, phonon MC is also easily
coupled with electron transport. This makes it the optimal choice for simulating coupled
electro-thermal transport.
Just as with a deterministic solution of the BTE, a MC technique experiences
computation difficulty when considering low Knudsen number phonons. This is due to
the large amount of scattering events that takes place during each time step. Also, the
number of phonons in the simulation increases as the domain length increases in order to
obtain statistically significant results. Both of these issues make phonon MC very
difficult to implement for multiscale simulations, such as very large scale integrated
(VLSI) circuits. An inherent noisiness is also plagues phonon MC. This is because of
the stochastic nature of the solution procedure. This noise makes it difficult to couple
phonon MC to finite volume solvers. A more subtle disadvantage of phonon MC is the
26. 11
conservation of phonon momentum. As mentioned, care is taken to conserve energy by
adding and subtracting phonons. However, no such care is taken to conserve phonon
momentum. Thus momentum conservation rules are not guaranteed to be satisfied for
either N or U processes in current implementations.
1.2.5. Molecular Dynamics
Classical molecular dynamics (MD) simulations treat each molecule as a solid
particle which is coupled to its neighbors through forces determined from an inter-atomic
potential. For solid state MD, the motion of each molecule is limited to small
perturbations about a given lattice site. Many inter-atomic potentials in use today are
determined using empirical curve fits to experimental data. One of the first potentials
used is the Lennard-Jones (LJ) potential [53]. Other potentials include the Stillinger-
Weber potential [54], the Tersoff potential [55], and the environment dependent inter-
atomic potential (EDIP) [28]; these have been widely used for silicon and carbon. An
MD simulation begins with the initialization of each molecule at a time t0. A velocity is
chosen based on the initial temperature and a Maxwellian distribution. With the
potentials, Newton’s second law is used to calculate the acceleration of the particle.
After all the accelerations for each molecule are found, the position of each molecule is
found at a new time t0+Δt. This procedure is repeated until a prescribed time is reached.
The thermal conductivity of a material is commonly predicted using one of two
methods: the Green-Kubo method [56], or from non-equilibrium molecular dynamics
(NEMD) [57-59]. The Green-Kubo method for determining thermal conductivity, also
called equilibrium molecular dynamics (EMD), is based on the fact that a system in
equilibrium has zero net heat flux at any given location. After the system is initialized,
the domain is allowed to reach equilibrium. The thermal conductivity is related to time
needed for the system to reach zero heat flux at all locations. Using NEMD to determine
thermal conductivity is based on imposing a thermal gradient in the domain and using
Fourier’s law to calculate the thermal conductivity. Studies have compared EMD and
NEMD [60] and have found consistent results between the two.
27. 12
MD simulations have also been used to calculate quantities related to phonon
transport. Because MD simulations perform calculations in real space, post processing
must be used to find momentum space information such as phonon density of states,
dispersion, and mode-wise relaxation times. The phonon density of states can be found
from finding the Fourier transform of the velocity-velocity autocorrelation [61].
McGaughey and Kaviany [62] found the relaxation times and phonon dispersion relations
using MD simulations. This was accomplished by using the normal mode to find the
temporal decay of the autocorrelation of the total energy. In [62] they made a single
mode relaxation time approximation, so the decay of the autocorrelation of the total
energy is related to the relaxation time. A non-localized wave perturbation method can
also be used to characterize multi-phonon interactions [63]. This is accomplished by
perturbing a system and analyzing how the perturbation decays into other modes.
Transmissivities can also be found using a localized wave packet method [64]. A wave
packet is created in a material and allowed to propagate until it encounters an interface.
The energy on either side of the interface is found after the encounter. From there,
transmissivities can be found.
MD simulations offer several advantages. Because of the direct description, there
are no fitting parameters needed besides the inter-atomic potentials. Also a result of the
direct description, anharmonicities are automatically accounted for. MD simulations are
inherently parallelizable, making them attractive for high performance parallel computing
(HPPC). Disadvantages in MD simulations are either computational or as a result of the
classical description. The computational disadvantage results from the large number of
atoms needed to simulate domains of practical interest. Simulation results rely heavily on
the interatomic potential assumed, which is generally empirical, and not always available
for heterogeneous systems. Furthermore, because Newton’s second law is the only
means by which atoms interact with each other, quantum effects are completely ignored.
The features limit MD simulations to small scales and domains where quantum
effects can be ignored. However, MD calculations are still useful for determining
properties which can be used in meso-scale simulations. Relaxation times, dispersion
28. 13
relations, transmission coefficients, etc. which are difficult to obtain from experiments
can be found using MD.
1.2.6. Interface Modeling and Simulation
Interface-related phenomena become important in microelectronics when dealing
with devices which contain heterogeneous junctions. As the characteristic length
decreases, the surface to volume ratio increases, which causes surface physics to play a
more dominant role in carrier transport. One of the earliest models for interface transport
is the acoustic mismatch model (AMM) [65,66]. In the AMM, continuum phonons are
assumed to be plane waves in a perfect lattice. With this, reflection and transmission
coefficients can be determined, being only a function only of direction and carrier
velocity. Scattering at the interface is not taken into account. The AMM produces
results which are in agreement with experiment at low temperatures when interface
scattering is minimal and the perfect lattice assumption is valid. However, at higher
temperatures, scattering begins to play a more dominant role and the AMM produces
thermal resistances which are several orders of magnitude lower than experiments. To
remedy these shortcomings, the diffuse mismatch model (DMM) was suggested by
Swartz [67]. In the DMM, reflection and transmission coefficients are determined by
considering only the carrier density of states and carrier velocity on either side of the
interface. At low temperatures, the DMM yields results very close to the AMM. When
scattering becomes more relevant, the DMM predicts values for thermal boundary
resistance which can be an order of magnitude higher than the AMM. MD simulations
have been used to detail interactions between two materials with success [18]. The
atomistic Green’s function (AGF) method has also been used to predict thermal transport
between interfaces [17,68]. AGF also suffers a domain size limitation, but moreover,
AGF can only be used at interfaces and is not applicable for bulk thermal transport
because many current implementations ignore anharmonicity. However, AGF captures
wave and coherence effects which make it attractive in situations where the length scale
is competitive with the wave length and bulk scattering can be ignored.
29. 14
1.2.7. Thermal Modeling in Microelectronics
A central aspect of thermal modeling in microelectronics is electron-phonon
coupling. A common method is the use of electron Monte Carlo simulation (eMC). In
general, electron Monte Carlo methods either use an analytic parabolic or non-parabolic
band shape or a semi-empirical pseudopotential calculation to model electron transport.
The main variation is usually associated with the representation of the phonon dispersion.
The simplest methods [69] include analytic parabolic band structures for electron
transport and fixed intervalley acoustic scattering with a dispersionless longitudinal
branch. Similar representations for the phonon scattering combined with more complex
representations for the electron transport may be found in [43,70]. In [43] a non-
parabolic analytic band model is assumed while in [70] an empirical pseudo-potential
band structure is used. [70] also includes possibility of impact ionization, which is
necessary when the device potential difference is larger than the material band gap.
However, this is not important for modern devices which are small and have a device
potential which is smaller than the material band gap. More complex representations of
electron-phonon scattering may be found in [45,71]. Both [45,71] use a full anisotropic
description of phonon dispersion using an adiabatic bond charge model. Electron-phonon
scattering rates are also calculated as a function of energy and wave vector.
As electron and phonon representations become more sophisticated,
computational costs become prohibitive. Consequently, an effort was made to create a
more efficient Monte Carlo scheme which accounted for only the necessary
characteristics [72]. In [72] an analytic non-parabolic energy band profile was assumed
as well as an isotropic analytic phonon dispersion model which differentiates between
optical and acoustic, as well as longitudinal and transverse phonons. The method
developed in [72] was used to examine the mobility in strained and unstrained silicon and
does not give information regarding the phonon generation due to electron-phonon
collision. Aksamija [73] used the method developed in but tracked the phonon
generation due to electron-phonon scattering.
The energy scattered to the phonon field must be transported by phonons.
Electrons that scatter energy to the lattice typically scatter with high frequency optical
30. 15
phonons. The final device lattice temperature depends on the rate of energy scattered to
the various phonon groups, and the rate at which they transport energy to the boundaries.
Sinha et al. [74] used MD to examine the decay of a certain type of phonon which
scatters with electrons. The g-process phonon causes an electron to scatter from one
conduction band valley, to the corresponding conduction band valley on the same axis. It
was seen that the g process LO phonon decayed to a zone edge TA phonon and an LA
phonon. The resulting phonon distribution, which may depart significantly from
equilibrium, must alter the electron-phonon scattering rates. Thus, the electron and
phonon problems are coupled. Recently, both one-way and two-way coupled models
have been developed [3,35] .
In one-way coupling, information regarding the Joule heating is computed before
hand and included in the phonon transport model as a source term. Such a technique has
been employed in [35] by Ni. The simplest approach entails assuming a hotspot size and
volumetric heat generation within the hotspot. The heat generation is assumed to be
evenly distributed among all bands, frequencies, and directions, and the lattice
temperature predicted. Since a uniform distribution of heat generation among all phonon
modes is not what physically occurs, the predicted temperature rise is expected to be
lower than that which would actually occur. The next level of sophistication entails
performing an electron Monte Carlo (eMC) which produces phonon generation data
dependent on wave vector and energy. Then a one-way coupling is assumed and the
results of the eMC are used to create the heat generation term in the BTE. This approach
has been shown in [35] to produce hot spot temperatures which are higher than those
obtained using a Fourier model, but far lower than those predicted using two-temperature
models.
Rowlette and Goodson [3] developed a two way coupled method for predicting
electron and phonon transport in a 320 nm n+/n/n+ FET. A six valley, analytic, non-
parabolic, single conduction band was used to model electron dispersion. Fitted
quadratic equations in the [100] direction were used to model phonon dispersion. An
eMC procedure developed in [72] was used to solve electron transport. A split flux
phonon BTE (p-SFBTE) developed by Sinha et al. [75] is used to solve phonon transport.
31. 16
To incorporate interparticle coupling, each iteration consisted of a solution of the eMC,
sending relevant scattering information to the p-SFBTE, a solution of the p-SFBTE, and
then sending phonon populations to the eMC. They found that this iterative procedure
usually required around five iterations to obtain convergence. From this simulation, it
was found that thermal conductance in increasingly small devices will continue to be
reduced due to size confinement and low thermally conducting oxides. They, however,
did not foresee electrical conductance decreasing appreciably.
The above examples of electron-phonon simulation have brought great insight
which has helped our understanding of non-equilibrium processes in micro- and
nanoelectronics. However, fully-coupled three dimensional simulations with complete
resolution of the phonon spectrum are still very expensive. MC and MD codes are best
for acting as a parameter filter which scientists and engineers can use to create more
efficient simulation methods.
1.3. Overview of Proposed Work
The focus of this thesis is the development of a computational technique to
significantly reduce the cost of non-gray phonon transport simulations and to
substantially accelerate their convergence in multiscale transport problems involving a
large range of Knudsen numbers. We wish to accomplish while retaining the accuracy of
existing non-gray phonon BTE solution techniques.
Most semiconductor and dielectric materials of industrial interest (Si, Ge, GaAs)
have mean free paths which span several orders of magnitude. The average mean free
path for room-temperature silicon is about 300 nm. Furthermore, since the BTE is valid
for domain length scales larger than the phonon wavelength (1-2 nm in Si at 300 K),
typical mean free paths may range from tens of nanometers to hundreds of microns in
typical non-gray simulations. A single simulation may thus involve phonon bands with
Knudsen numbers that are 4-5 orders of magnitude apart. When phonon Knudsen
numbers are low, the scattering term dominates. In this limit, sequential solution
procedures such as those in [19,20,76], which loop over each directional BTE in turn,
become extremely slow to converge because of the predominance of inter-equation
32. 17
coupling [40]. Furthermore, to accurately capture scattering and transport behavior, it is
necessary to employ tens of BTE bands with tens of angular directions. Thus, any
technique that can reduce the number of BTE solutions would significantly reduce
computational cost.
The contributions of the thesis are summarized below.
1. A hybrid Fourier-BTE model is developed to reduce the number of BTEs that are
solved. The phonon spectrum is divided into frequency bands. Phonon groups with
low-enough Knudsen numbers are approximated by a modified Fourier equation,
while the other phonon groups are solved using a BTE. Since the main solution cost is
due to the BTE, this is shown to significantly reduce the overall computational cost.
2. A modified Fourier equation is developed for the low-Knudsen number bands which
accounts for phonon-phonon scattering as well as near-wall slip. These
approximations are shown not to compromise the accuracy of the overall model in
comparison to an all-BTE model.
3. A semi-implicit solution procedure is developed which drastically reduces iteration
count compared to standard BTE solution methods. Here, the tight coupling between
the Fourier bands is accounted for by a block-coupled solution procedure while the
BTE bands are solved sequentially as before. Since the BTE bands are not tightly
coupled because of their high Kn values, a low-memory sequential procedure is ideal.
The resulting algorithm is shown to reduce computational cost over and above that
obtained by reducing the number of BTE bands.
4. The proposed hybrid model is benchmarked for accuracy and speed and detailed
metrics of performance are provided. The model is then applied to the problem of
self-heating in a bulk metal-oxide-semiconductor field effect transistor (MOSFET).
Phonon generation rates resolved with respect to the phonon wave vector are obtained
from [35] and used to predict hot-spot temperature and heat fluxes. Comparisons are
made with an all-BTE solution wherever possible and the advantages of the hybrid
model established.
This paper contains five chapters. The first chapter is an overview of the state of
the art. The second chapter describes the proposed acceleration scheme. Within chapter
33. 18
2, the reader is acquainted with the BTE which will be used in this paper, and shown the
derivation of the diffusion equation which will approximate phonon transport. Chapter 3
presents accuracy and solution acceleration results. Chapter 4 applies the proposed
acceleration method to model thermal behavior in a bulk Si MOSFET. Results are
compared to standard BTE solutions as well as results found in literature [35]. Chapter 5
summarizes the proposed work and suggests directions for future research.
34. 19
CHAPTER 2. A NEW HYBRID FOURIER-BTE MODEL
In this chapter, we present a new hybrid model for phonon transport which
significantly reduces the computational cost of non-gray phonon BTE computations, with
little or no cost in accuracy. As noted earlier, the phonon Boltzmann transport equation
encounters computational difficulty when the characteristic length is much larger than the
phonon mean free path, i.e., at low Knudsen numbers Kn. The difficulty is specific to
sequential solution procedures which are the norm in this area. In the relaxation time
approximation, all phonon groups scatter to a common reservoir, the lattice, whose
energy determines the lattice temperature. In typical sequential solution procedures, the
lattice temperature (and the corresponding lattice energy) is assumed temporarily known,
and each phonon group is solved in turn. The lattice temperature is then updated, and the
procedure continues until convergence. Phonon groups with low Kn are tightly coupled
to the lattice by scattering, and this explicit update significantly slows convergence.
Since nearly all non-gray simulations span 4-5 orders of magnitude in relaxation time
[77], thick phonon bands are frequently encountered, even in domains a small as 100 nm.
The strategy to combat this computational obstacle is to divide phonon groups or
bands into two categories, those with Kn below a certain cutoff, Knc, and those with Kn
above the cutoff. Solution acceleration and computational savings are achieved by (i) the
development of a diffusion equation valid for low Knudsen number phonon bands which
removes angular dependence, thereby reducing computational effort, and (ii) the creation
of a semi-implicit solution scheme for these “thick” bands which simultaneously solves
multiple phonon bands, thereby improving inter-band coupling and reducing iteration
count.
In the sections that follow, we first describe the phonon Boltzmann transport
equation, which will be used to solve high Knudsen number phonon bands. We then
35. 20
develop a modified Fourier equation, which will be used to solve low Knudsen number
phonon bands. An explanation of lattice temperature recovery will be given as well. A
finite volume discretization scheme is then developed for both the Fourier equation and
the BTE. A semi-implicit solution procedure is then developed for solving the coupled
Fourier and BTE bands.
2.1. Hybrid Fourier-BTE Model
In this section we describe the new hybrid Fourier-BTE model described above.
We assume an isotropic Brilloiun zone, though this is not integral to the formulation.
Typically, dispersion curves in a high symmetry direction, say [100], are chosen as
shown in Figure 2.1, and assumed the same for all directions. The frequency spectrum is
discretized into bands Δω, and each band is identified by its frequency and polarization
p. The corresponding wave vector band is ΔK. Associated with each phonon band is a
band Knudsen number Knω,p given by:
, ,
,
p p
p
v
Kn
L
(2.1)
where tau is the effective scattering rate (i.e., 1 1 1 1
...eff U N imp
),vω,p is the phonon
group velocity taken from the dispersion relation, and L is the characteristic length of the
domain. If Knω,p >Knc, the cutoff Knudsen number, a non-gray phonon BTE, described
in Section 2.2 is used. If Knω,p < Knc, a modified Fourier equation, described in Section
2.3 is used. The coupling between these two descriptions is through the lattice
temperature, which is defined in Section 2.4.
36. 21
2.2. Non-Gray Phonon Boltzmann Transport Equation
Figure 2.1 Dispersion curves for silicon in [100] direction using environment dependent
inter-atomic potentials [28].
The steady-state, non-gray BTE for a phonon band of width Δω centered about
frequency and with polarization p is given by [76]:
0
, ,
, ,
,
"
"
p p
p p
p
e e
v e
s (2.2)
The relaxation time approximation is assumed. Here vω,p is the phonon group velocity
and τω,p is the corresponding relaxation time. 𝑒" 𝜔,𝑝 is the volumetric energy density per
unit solid angle at a given frequency and polarization, and 𝑒 𝜔,𝑝
0
is the corresponding
equilibrium energy density given by a Bose-Einstein distribution [12]. The vector s is the
unit vector associated with a given direction and is shown in Figure 2.2.
37. 22
Figure 2.2 Unit vector for s.
In Cartesian unit vectors, s is defined as
sin sin sin cos cos s i j k (2.3)
For problems involving relatively small lattice temperature differences or if the
temperature is above the Debye temperature, the phonon specific heat may be assumed
temperature-independent 𝑒 𝜔,𝑝
0
is given by
,0
,
4
p
p L ref
C
e T T
(2.4)
Here, TL is the lattice temperature. C, p is the specific heat associated with the band, and
is given by:
0
2
, 4p
K
f
C K dK
T
(2.5)
where ΔK is the wave vectore band associated with frequency ω and polarization p.
Furthermore, the total energy associated with the band is
, , , ,
4
" 4 "p p p ref pe d C T T e
(2.6)
38. 23
where T,p is the equilibrium “temperature” associated with the average energy density,
, pe
of the band. It should be noted that Tω,p has no thermodynamic meaning and is
merely a measure of the average spectral energy density of the band.
2.2.1. Boundary Conditions
In this thesis, three boundary conditions used in the simulation: diffusely-
reflecting, specularly-reflecting and thermalizing. They are explained in the following
sections.
2.2.1.1. Diffusely-Reflecting Boundary Condition
Assuming elastic boundary scattering, the energy density of phonons leaving the
boundary from the domain is given by [35]:
, ,0
0
1
, "r p pe e e d
s n
s n
s r s n (2.7)
Here, n is the outward pointing normal to the boundary. For directions outgoing to the
domain, the energy density is assumed to remain invariant along the ray direction, and
Eq. 2.8 applies.
, pe = 0
s (2.8)
2.2.1.2. Specularly-Reflecting Boundary Condition
At specularly-reflecting boundaries, phonons are assumed to undergo mirror
reflection. Thus phonons leaving the boundary and entering the domain in a direction s
have the same energy as those arriving at the boundary from the domain interior in the
corresponding specular direction sr. The frequency and polarization are assumed
unchanged in the process of reflection [76]. Thus
, ," , " ,p p re e s r s r (2.9)
39. 24
Here sr is the direction which corresponds to a reflection from the direction s and is given
by
For directions outgoing from the domain, the upwind condition (Eq. (2.8)) is applied.
2.2.1.3. Thermalizing Boundaries
A typical thermalizing boundary at temperature T1 is assumed to be diffusely
emitting. The energy emitted from the wall is the energy corresponding to the
equilibrium distribution of phonons at the given wall temperature. For phonons going
into the domain from the boundary, the energy density for a band of frequency and
polarization p is given by:
,
1 1"
4
p
ref
C
e T T
(2.11)
For directions outgoing from the domain, the upwind condition (Eq. (2.8)) is applied.
2.3. Modified Fourier Equation
For the phonon bands with Kn<Knc, a diffuse approximation may be made. The
BTE is integrated over the sphere of 4π under an isotropic assumption to yield
,
, ,
,
p
p L p
p
C
T T
q (2.12)
Eq. (2.12) essentially states that the efflux of phonon energy from the phonon band must
be accounted for by scattering to the lattice. However, we must find an expression for the
heat flux vector in order to make Eq. (2.12) useful. To do this, we will make use of a
first- order perturbation. Let the spectral energy density for a phonon band be equivalent
to the band average energy density plus a small perturbation, Φ:
, , , ,p p pe e
s r (2.13)
If we insert Eq. (2.13) into Eq. (2.2), we can solve for Φ by assuming the divergence of Φ
to be much smaller than the divergence of the spectral energy density. Thus we obtain
2r s s s n n (2.10)
40. 25
,
, , , , ,,
4
p
p L p p p p
C
T T v T
s r s (2.14)
Using Eq. (2.14) and Eq. (2.13), we can write a new definition of the spectral energy
density as follows:
, ,
,, , , , , ,
4 4
p p
pp p ref L p p p p
C C
e T T T T v T
s (2.15)
Combining terms in the above allows us to write our spectral energy density in a more
compact form, as shown below:
, , ,0
, , ,
4
p p p
p p p
C v
e e T
s (2.16)
Thus, the spectral energy density is a small perturbation over the equilibrium energy
density; the amount of perturbation is proportional to the local “temperature” gradient
associated with the band “temperature” Tω,p. With this we can now find a formulation for
the heat flux vector at any given point in terms of the lattice temperature and the phonon
band temperature gradient. In the phonon BTE, the heat flux vector at any given location
is given by
, , ,
4
p p pv e d
q s (2.17)
Now we substitute Eq. (2.16) in Eq. (2.17). Since the integral of
0
, pe
over the sphere is
zero, we see that the heat flux is caused solely by second term in Eq. (2.16). Therefore
2
, , ,
, ,
4
4
p p p
p p
C v
T d
q s s (2.18)
Under an isotropic assumption, the integration yields
, , ,
2
, , ,
,
3
p p p
p p p
p
k T
C v
k
q
(2.19)
41. 26
Upon inserting Eq. (2.19) into Eq. (2.12), we obtain the modified Fourier equation
(MFE).
,
, , ,
,
p
p p L p
p
C
k T T T
(2.20)
2.3.1. Boundary Conditions
We now derive boundary conditions for the MFE. As before, we consider
diffusely and specularly-reflecting boundaries, and thermalizing boundaries.
2.3.1.1. Diffusely and Specularly-Reflecting Boundaries
We first start with a specularly reflecting boundary. At such a boundary, we are
given the value of the spectral energy density entering the domain from the boundary in
terms of the energy falling on the boundary from the domain interior, Eq. (2.9) for
specularly reflecting. If we take the dot product of our heat flux vector (Eq. (2.17)) with
the outward pointing normal of the wall and substitute our specularly reflecting boundary
condition, we arrive at the relation
, , , , ,
0 0
p p p p rv e d e d
s n s n<
q n s n s n (2.21)
The integrals in the bracket are equal and opposite in sign, leading to the adiabatic
boundary condition , p 0
q n = .
For a diffusely reflecting boundary condition, we again substitute our boundary
condition Eq. (2.7) into Eq. (2.17) dotted with the outward-pointing normal vector to
obtain
, , , ,
0 0 0
1
"p p p pv e d e d d
s n s n< s n
q n s n s n s n (2.22)
Again, since the net incoming and outgoing energies at boundary must be equal, the
right-hand side of the above equation is zero, leading to an adiabatic boundary condition
, p 0
q n = .
42. 27
2.3.1.2. Thermalizing Boundary
We now consider a thermalizing boundary with given temperature T1. The
boundary has an outward-pointing normal n. Figure 2.3 shows the energies incident on
the boundary (left half) and emitted by the boundary (right half).
Figure 2.3 Thermalizing boundary condition at temperature T1 with an outward pointing
normal n.
Starting with Eq. (2.6) and Eq. (2.11), we may write the band-averaged energy at the wall
as
,
, , , , ,
4
, , , ,0
, , , 1
0 0
1
4 4
1
4 4 4
p
w p w p ref p
p p p p
p w p ref
C
e T T e d
C v C
e T d T T d
s n s n<
s
(2.23)
43. 28
Here, Tw,ω,p is the effective wall temperature associated with the Fourier band. This is
distinct from the temperature imposed on the boundary, T1. Substituting Eq. (2.4) for
0
, pe we obtain
, ,1
, , , ,
0
2 4
p pL
w p w p
vT T
T T d
s n
s (2.24)
If we multiply both sides by , ,
4
3
p pC v , evaluate the integral on the RHS and group terms
together, we may cast Eq. (2.24) into a typical Robbins boundary condition. Thus
, , , , inf
, ,
1
inf
4
3
2
w p w p
p p
L
h T T
h C v
T T
T
q n
(2.25)
2.3.2. Limits of MFE
It is important to understand the limit approached by the MFE as 0Kn . To
establish this limit, we non-dimensionalize the result obtained in Eq. (2.13). Using θ=
(T1-T)/(T1-T2), and Kn=vω,pτω,p/L it is possible to show that the dimensionless
perturbation is as follows:
,* *
,
, 1 2
4 p
p L
p
Kn
C T T
s (2.26)
As 0Kn , we may cancel the gradient term in Eq. (2.26). Furthermore, as 0Kn a
local equilibrium exists between diffuse bands and the lattice temperature. We can
therefore assume that θ≈ θL. Therefore, *
, 0p as 0Kn and Eq. (2.20) establishes
that each phonon band would satisfy the Fourier conduction equation.
When we non-dimensionalize the thermalizing boundary condition, we also have
a dependence on Kn. It can be shown that Eq. (2.25) is written in dimensionless form as
follows.
44. 29
*
2
w
Kn
n+ = (2.27)
As Kn becomes very small, our boundary condition simply becomes a Dirichlet boundary
condition, as physically expected.
2.4. Lattice Temperature Recovery
The scattering terms in the BTE and the modified Fourier equations must be
purely redistributive in the aggregate, and no energy may be created or lost due to
scattering processes. Thus, the net scattering term, summed over all bands and
polarizations, must be zero:
0
, ,
, ,
0p p
p p
e e
(2.28)
Using Eqs. (2.4), and (2.6), we can rearrange to find our lattice temperature definition.
1
, , ,
, ,
, , ,, , ,
BTE Fourier
p p p
L p p
p p pp p p
C C C
T T T
(2.29)
The first summation in the bracket is over all phonon groups which are solved by the
BTE and the second summation is over all phonon groups which are solved by the MFE.
2.5. Finite Volume Method for BTE
The finite volume method [76] has been widely used in fluid flow and heat
transfer. The ability to employ unstructured discretization has given this class of methods
a distinct advantage over finite difference schemes, though work is underway on
unstructured finite difference [79]. The finite volume method has the advantage of
guaranteeing energy conservation, regardless of mesh size. Numerical solutions for
radiative heat transfer have employed the finite volume method [80] with success. Since
the BTE and the radiative transfer equation (RTE) are similar for isotropic scattering, a
number of published BTE solution techniques have also utilized the finite volume method
[20,21,76].
45. 30
The spatial domain is discretized into Nx x Ny control volumes as shown in Figure
2.4. The discretization need not be uniform or structured, though these assumptions are
made in the present thesis for clarity. An overview of the solution method using an
unstructured mesh is outlined in [21]. In the frequency domain, we discretize each
polarization into separate frequency bands, each with the specified scattering rate,
specific heat, and group velocity. The angular domain is divided into Nθ x Nφ control
volumes as shown in Figure 2.5. The unknown values of , pe
are stored at the cell
centroids and the boundary face centroids.
Figure 2.4 Finite volume discretization in the spatial domain.
46. 31
Figure 2.5 Finite volume angular discretization.
The BTE in any given direction s, frequency band ω and polarization p is integrated over
the control angle ΔΩ and the spatial control volume ΔV to give
0
, ,
, ,
,, ,
p p
p p
pV V
e e
v e dVd dVd
s (2.30)
We apply the divergence theorem to the integral on the LHS and rearrange to yield
, , , ,p p p p
A A
v e dA d v e dA d
s n n s (2.31)
We may evaluate the second argument in the dot product on the right hand side of Eq.
(2.31) analytically. Using Eq. (2.3) and integrating over the extent of the control volume
ΔΩ, we may define the vector S as
47. 32
sin sin 0.5 cos 2 sin
cos sin 0.5 cos 2 sin
0.5 sin 2 sin
d W
i
s + j
+ k
S = (2.32)
where W is a weight factor. In a two-dimensional domain W=2 is used, and only one-half
of the sphere is discretized; in 3D, W=1 is used, and the full sphere is discretized.
Substituting Eq. (2.32) into Eq. (2.31) and evaluating the integrals at the faces e,w,n,s in
Figure 2.4, we arrive at the final form of the finite volume equations.
0
, ,
, , ,
,
p p
p p f f
f p P
e e
v e A V
fn S (2.33)
Here Af is the area associated with the given face, nf is the outward pointing normal
associated with the given face and , ,p fe
is the face value of the spectral energy density.
The right hand side of Eq. (2.33) is found by assuming no spatial and angular variation of
the source term in the control volumes, and is evaluated at the cell centroid P. In the
present work , ,p fe
is evaluated using an upwind difference scheme [78] for simplicity,
though higher-order discretization is easily applied. The resulting discrete equation set is
solved by traversing the structured mesh in a “streamwise” fashion, dictated by the group
velocity vector, shown in Figure 2.4.
2.5.1. Boundary Conditions
The following is an overview of the finite volume discretization of the boundary
conditions presented in section 2.2.1.
2.5.1.1. Diffuse Boundaries
We can easily recast the diffuse boundary condition given in section 2.2.1 using
Eq. (2.32) and our finite volume discretization.
, ,0
0
1
,r p pe e e
s n
s n
s r S n (2.34)
48. 33
Here the summation is over all the discrete solid angles for rays which are leaving the
domain.
2.5.1.2. Specular Boundaries
The specular boundary condition is implemented by finding the incoming
specular direction (Eq. (2.10)) corresponding to the direction vector s. Once the specular
angle is found, the energy associated with that angle is assigned to the energy in the
present direction s.
2.5.1.3. Thermalizing Boundaries
Thermalizing boundary conditions are implemented as Dirichlet conditions in the
manner described in [78,80].
2.6. Finite Volume Method for Modified Fourier Equation
The reader is referred to [78] for further explanation of finite volume methods for
solutions to diffusive scalar transport. A brief explanation is given here for the sake of
completeness.
For a given polarization and frequency band, the domain is divided into spatial
control volumes as shown in Figure 2.4. An energy balance is performed over the control
volume and balances the band-wise energy transfer rate leaving the control volume faces
and the net generation of band energy in the control volume by scattering. Source
linearization procedures described in [78] are used to address the scattering terms in the
MFE. With this, the linear system is easily solvable using prevailing values of TL.
However, we wish to take advantage of the block structure of the equation set describing
the MFE and lattice temperature equation. Therefore, a block structured line by line
tridiagonal matrix algorithm (LBL-TDMA) is implemented to simultaneously solve the
MFE and lattice temperature equations, while holding the energy density of the BTE
bands at prevailing values. Details are given in Section 2.8.
49. 34
2.6.1. Boundary Conditions for MFE
As stated earlier, diffuse and specular boundary conditions both yield adiabatic
conditions while the thermalizing boundary conditions yield a Robbins boundary
condition. Both of these are implemented using standard finite volume practice [78].
2.7. Dimensionless Parameters
The governing dimensionless parameters are:
, , ,
, , ,
,
,
p p ,p p
p L p
,p p
p
v C
Kn , R
L C
(2.35)
for each band. The first dimensionless group is the Knudsen number of the band. The
Knudsen number may be obtained by non-dimensionalizing the BTE. The band Knudsen
number is a measure of the strength of the scattering term in the BTE. Bands with low
Knω,p are dominated by scattering and exhibit diffuse-like behavior, while those with high
Knω,p exhibit ballistic behavior.
The second dimensionless group, which we call the lattice ratio, results from the
lattice temperature equation, Eq. (2.29). To understand the lattice ratio, it is best to non-
dimensionalize the lattice temperature equation as shown in Eq. (2.36) below:
, ,
BTE Fourier
L L i i L i i
i i
R R (2.36)
The dimensionless lattice temperature, θL on the left hand side is a function of the
dimensionless temperature of each band, θi , weighted and summed on the right side.
The lattice ratio, RL,i, is the weighting factor for a given band i and determines the extent
to which the band contributes to the determination of the lattice temperature.
2.8. Solution Algorithm
Since the Fourier and BTE equations are linear, a direct solution of Eqs. (2.2),
(2.20) and (2.29) would produce the final solution in one iteration. However, the
50. 35
memory required for the large number of BTE equations is too large to permit direct
solutions. Typical solution procedures published for the BTE employ a sequential
procedure whereby the BTE in each frequency band and direction is solved sequentially,
assuming prevailing values for TL in evaluating the energy density 0
, pe . Such a procedure
has low memory requirements, and a similar one could be employed here. We would
start with a guess of the lattice temperature TL and solve the BTE equations sequentially
over the spatial domain, keeping the lattice temperature fixed at its prevailing value.
Then the modified Fourier equations would be solved sequentially, again keeping TL at
prevailing values. Eq. (2.27) would then be used to update TL. The procedure would be
repeated until convergence.
This type of sequential procedure was found to be extremely slow, and was
impeded primarily by the explicit update of TL. To circumvent this problem, we have
developed a partially-implicit procedure. This procedure employs a simultaneous
solution of the modified Fourier and lattice temperature equations first, followed by the
sequential solution of the BTE equations. The computation is initiated with a guess of
the lattice temperature which is used only to initialize the BTE bands. Once this is
complete, the modified Fourier and lattice equations are solved using a block tri-diagonal
solver, with the BTE band energies being determined from prevailing values. The
simultaneous solution of the MFE temperatures and the lattice temperature yields a new
lattice temperature field, which is used to update the values of 0
, pe . The BTE bands are
then solved sequentially, visiting each band in turn, and solving over all directions in the
band. Finally, the residual of the lattice temperature equation, Eq. (2.29), is calculated.
The procedure is repeated until a prescribed convergence criterion is met. Figure 2.6
shows a flow chart of the solution procedure.
This type of partially-implicit solution is feasible because the number of modified
Fourier equations is relatively small. As will be seen in the next section, the implicit
nature of the MFE-lattice temperature coupling produces significant solution acceleration
by decreasing the iteration count. This is because TL is determined in large part by the
low Knω,p bands which are solved using the MFE. Therefore, the simultaneous solution
of the modified Fourier equations and TL yields a lattice temperature very close to the
51. 36
correct value. Furthermore, the BTE bands are necessarily acoustically thin, and are only
loosely coupled to each other. Thus a sequential solution procedure is sufficient to solve
them. Additional computational savings also result because the MFE is much less costly
to solve than the BTE.
Figure 2.6 Flow chart of overall solution algorithm.
52. 37
CHAPTER 3. ALGORITHM PERFORMANCE
For a given iteration of a BTE solver, the number of equations to be solved is
NxNyNθNφNT, where NT is the total number of phonon groups. The number of equations
per iteration of a hybrid solver is NxNy(NθNφNb+Nf), where Nf is the total number of
modified Fourier bands and Nb is the total number of BTE bands. If we use the same
spatial discretization for a hybrid solver as we do for a BTE solver, the ratio of hybrid
equations to BTE equations, which is inversely proportional to the speedup1
expected per
iteration is given by
1 1
1 1 1f f
T T
N N
Speedup
N N N N
(3.1)
If the angular discretization is fine enough, the second term in parentheses may be
ignored and the final approximation is valid. Eq. (3.1) is the minimum expected speedup
per iteration.
3.1. Verification of Single Band BTE and Hybrid Solvers
Our first task is to demonstrate the accuracy of the BTE solver and the new hybrid
solver. The first test case is shown in Figure 3.1. Here a square domain is bounded by
two walls, held at T1 and T2 as shown; the top and bottom boundaries are specularly
reflecting boundaries. A solution to the problem has been given by Heaslet and Warming
[29]. We compare the dimensionless temperature profiles θ=(T1-T)/(T1-T2), as well as the
1
When we say speedup, we mean how much faster the hybrid solver is than the BTE.
That is, if the speedup is a number x, and the time required to run the hybrid solver is t,
then the time to run the BTE solver is xt.
53. 38
dimensionless heat flux Ψ computed by an all-BTE solver and the new hybrid solver with
the published solution. The dimensionless heat flux is given by
1 2
4
ballistic
ballistic
q
q
Cv
q T T
(3.2)
Figure 3.1 Domain schematic for phonon transport between parallel plates.
The spatial discretization for both the hybrid and BTE solvers is 100x5 cells in the
x- and y-directions respectively. The angular discretization is 4x4 in each octant. In
keeping with the exact solution, we also assume gray phonon transport, so that only one
phonon band is simulated, described by a single Knudsen number, Kn.
Figure 3.2 shows the dimensionless temperature profile obtained using an all-BTE
solver for varying Knudsen numbers plotted versus dimensionless position. Also on
Figure 3.2 are the exact profiles taken from [29]. The temperature profiles are seen to
match the exact solution well. The error in the heat flux for a Knudsen number of 10 is
0.23%, that for a Knudsen number of 1.0 is 0.20%, while that for a Knudsen number of
0.1 is 5.02%.
54. 39
Figure 3.2 Dimensionless temperature profile obtained from the BTE solver compared to
the exact solution obtained from [29]. Values in parentheses are the Knudsen numbers.
Figure 3.3 shows the dimensionless temperature profiles for varying Knudsen
numbers obtained using the hybrid solver and compared to the exact solution from [29].
In this example, the hybrid solver always employs the MFE for the single band under
consideration. We use the same spatial discretization as that for the all-BTE solution; the
MFE does not require any angular discretization. Excellent agreement is seen even at
Knudsen numbers as high as 1. However, the most telling comparison is shown in Figure
3.4. The lower line in Figure 3.4 indicates the percentage difference in heat flux with
respect to the exact solution in [29]. The heat flux difference is found to be under 10%
for Kn=1/3, and under 5% is observed for Kn=0.1. This gives us an idea of where our
cutoff Knudsen number should be in order to obtain accurate results in a multiband
simulation. The upper line in Figure 3.4 shows the percentage difference between heat
flux predicted by the hybrid solver and that predicted by Fourier’s law. The MFE is seen
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00 0.20 0.40 0.60 0.80 1.00
DimensionlessTemperature
x/L
BTE (10)
Heaslet, et al (10)
BTE (1)
Heaslet, et al (1)
BTE (0.1)
Heaslet, et al (0.1)
55. 40
to approach Fourier’s law for low Knudsen numbers, as expected, but departs
significantly from the Fourier law solution for higher Kn; the Fourier solution is of course
erroneous in the high Kn limit.
Figure 3.3 Dimensionless temperature profile obtained from the hybrid solver compared
to the exact solution obtained from [29].
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00 0.20 0.40 0.60 0.80 1.00
DimensionlessTemperature
x/L
Hybrid (1)
Heaslet, et al. (1)
Hybrid (0.2)
Heaslet, et al. (0.2)
Hybrid (0.1)
Heaslet, et al. (0.1)
56. 41
Figure 3.4 Percentage difference in the heat flux predicted by the hybrid solver compared
to [29] and to Fourier’s law.
3.2. Performance of Two-Band Model
The results from the one-dimensional, one-band case show that by itself the MFE
does an excellent job re-creating the solution to the BTE. Now we wish to characterize
not only the accuracy, but also the performance of the hybrid solver with the inclusion of
inter-band behavior. To do this we use two different domains: (i) the same domain used
in the single band case (Figure 3.1) and (ii) a two dimensional domain with Dirichlet
boundaries (Figure 3.5). With this we can easily examine the accuracy and performance
with respect to the two governing parameters: the Knudsen number, and the lattice ratio.
0
10
20
30
40
50
60
0.01 0.1 1
PercentDifference
Kn
Error with respect to Heaslet, et al.
Error with respect to Fourier's Law
57. 42
Figure 3.5 Schematic of the two-dimensional two-band domain.
First we address the issue of accuracy in the hybrid solver in the domain given in
Figure 3.1. When using our hybrid solver with two bands, one of the bands (the band
with a higher Knudsen number) is solved using a BTE and the remaining band is solved
using the MFE. We denote the band being solved with the BTE as the BTE band and the
band being solved by the MFE as the MFE band. By changing the Knudsen number of
both these bands we may gain insight into the performance and accuracy of the hybrid
solver as we change the cutoff Knudsen number. We examine two different trends in the
heat flux error: (i) we hold the MFE band at Kn=0.1 while varying the BTE Knudsen
number, and (ii) we hold the BTE band at Kn=2 while varying the MFE Knudsen
number. We do not, however, vary the lattice ratio. The lattice ratio for the MFE and
BTE bands are held at 0.90 and 0.10, respectively.
To characterize the performance of the hybrid solver, nine scenarios are used. We
use the domain pictured in Figure 3.5, and only two bands, one MFE and one BTE, as
58. 43
before. To capture the change in performance with respect to the change in lattice ratio,
three lattice ratios are chosen for the MFE band: 0.900, 0.990, and 0.999. Within each
lattice ratio, several variations with the Knudsen number are examined. The same
strategy is employed, i.e, we hold fixed the MFE band and vary the BTE band Knudsen
number, and vice versa. In total, 18 scenarios for each solver were examined: 3 different
lattice ratios, and 6 different Knudsen number arrangements. The results are presented in
the next subsection.
3.2.1. Results
The results obtained using the domain pictured in Figure 3.1 are summarized in
Figure 3.6 and Figure 3.7. Figure 3.6 shows the heat flux error with respect to the all-
BTE solution for the case when the Knudsen number of the MFE band is held fixed at
0.1, while that of the BTE band is varied from 10 to 0.2. The percentage error is found to
be less than 2% for the range acoustic thicknesses considered. However, we note that
there is an increase in error as the BTE band Knudsen number is decreased. This may be
explained in the following way. When we have a high Knudsen number in the BTE
band, the overall heat flux is almost entirely dominated by the BTE band, which is the
more accurate computation. The errors in the MFE band computation do not affect the
overall outcome to a significant degree. However, as the BTE band Knudsen number
decreases, the overall heat flux falls, and the MFE band contributes proportionally more
to the total heat flux. The heat flux error of the MFE band thus become more apparent
and the overall heat flux error increases.
59. 44
Figure 3.6 Hybrid heat flux error plotted against varying BTE band Kn.
Figure 3.7 Hybrid heat flux error plotted against varying MFE band Kn.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 2 4 6 8 10 12
PercentErrorFromAll-BTE(heatflux)
BTE Band Knudsen Number
0
2
4
6
8
10
12
14
16
0 0.2 0.4 0.6 0.8 1 1.2
PecentErrorFromAll-BTE(heatflux)
MFE Band Knudsen Number
60. 45
In Figure 3.7 the BTE band is held fixed at a Knudsen number of 2 while the
MFE band Knudsen number is varied from 0.1 to 1. Again, the heat flux error with
respect to the all-BTE solution is plotted. For a high Knudsen number in the MFE band
(Kn=1), the overall error is entirely dominated by the error in the MFE band solution,
which is relatively high. However, less than a Knudsen number of 0.2 or so, the heat flux
error is seen to fall to values well below 2%, and decreases as the MFE band Knudsen
number decreases. This is because the assumptions made in deriving the MFE become
increasingly true as its Knudsen number decreases. These results indicate that as long as
the cutoff Knudsen number is chosen judiciously, solutions with low error may be
obtained with the hybrid model.
Figure 3.8 Iteration count for the hybrid and all-BTE solver with varying Knudsen
number and lattice ratios.
Figure 3.8 and Figure 3.9 summarize the performance results obtained by using
the domain pictured in Figure 3.5. The ranges of lattice ratio and Kn are chosen to
61. 46
correspond to those typical for silicon. In general, we see that the hybrid solver performs
better than the all-BTE solver, with acceleration factors ranging from 2-200.
The overall behavior of the hybrid scheme is best explained by first considering
the case of a high lattice ratio in the MFE band (RL=0.999, Sections E & F in Figure 3.8).
In this limit, the lattice temperature (Eq. (2.29)) is determined almost entirely by the MFE
band. Thus, a semi-implicit solution of the modified Fourier and lattice temperature
equations is expected to confer a significant advantage and significant solution
acceleration over the all-BTE solution results. Furthermore, because the lattice
temperature is determined almost wholly by the MFE band, changes in the Knudsen
number of the BTE band do not influence the iteration count of both the hybrid solver
and the all-BTE solver. This is demonstrated by Section F in Figure 3.8. Conversely,
when the Knudsen number of the MFE band (RL=0.999) is changed, iteration count is
affected only in the all-BTE solver. The hybrid solver is totally unaffected when the
MFE band is changed because of the implicitness which is inherent in its solution
process. From this result, we can gather that as long as the lattice ratio is high in the
MFE band, the Knudsen number of the BTE band is inconsequential for both the hybrid
solver and the all-BTE solver. However, the all-BTE solver suffers greatly when the
MFE band Knudsen number becomes small because of the explicit dependence on the
lattice temperature in the all-BTE solver. This behavior is demonstrated in Section E in
Figure 3.8. It will be shown, however, that with many phonon groups the hybrid solver
does suffer when the Knudsen number of the MFE band is decreased, but that this is only
an artifact of the linear solver.
We now consider the case of a lower lattice ratio in the MFE band (RL=0.900,
Sections A and B in Figure 3.8). For these cases, the BTE band has a larger influence on
the lattice temperature because of its larger lattice ratio. As shown, a high Kn in the BTE
band is seen to increases convergence rates. Furthermore, this is true for both the hybrid
solver and the all-BTE solver. This is because of two factors. First, the Knudsen number
determines how much the band is influenced by the lattice temperature, while the lattice
ratio determines how much the band influences the lattice temperature. A band with a
high Knudsen number and a high lattice ratio is relatively uninfluenced by the lattice
62. 47
temperature but has a relatively high influence on the lattice temperature. Second, the
BTE band solution, whether in the hybrid solver or the all-BTE solver, is very efficient at
high Kn because it is essentially decoupled from the other band. Therefore, increasing
the Knudsen number of a band which has a significant lattice ratio will decrease iteration
count.
We now turn our attention to Section A of Figure 3.8. We see the hybrid solver is
relatively unaffected by Kn changes in the MFE band. The same is true for Sections A, C
and E of Figure 3.8. This result is expected because the MFE is solved simultaneously
with the lattice temperature equation (Eq. (2.29)) and the performance of this coupled
solver is insensitive to the MFE band Kn. The same explanations used for Sections A
and B, and Sections E and F may be applied to Sections C and D in Figure 3.8.
Figure 3.9 shows CPU time acceleration obtained using the hybrid solver as
compared to the all-BTE solver time-to-solution. In general, the hybrid solver confers a
significant advantage, with acceleration factors ranging from 2-200. The overall trends
are explained by the factors discussed above.
Figure 3.9 Timing results of the hybrid solver compared to the BTE solver.
1
10
100
1000
1-10
0.1-10
0.01-10
0.02-10
0.02-1
0.02-0.1
1-10
0.1-10
0.01-10
0.02-10
0.02-1
0.02-0.1
1-10
0.1-10
0.01-10
0.02-10
0.02-1
0.02-0.1
0.90-0.10 0.99-0.01 0.999-0.001
Kn
(MFE-BTE)
Lattice Ratio
63. 48
3.3. Full Band Simulation
We now turn to the simulation of thermal transport in silicon. The simulation
now involves many frequency bands and polarizations and large spread in phonon mean
free path. Figure 3.10 shows the variation of phonon mean free path (= , ,p pv ) for
silicon for the longitudinal acoustic (LA), transverse acoustic (TA), longitudinal optical
(LO) and transverse optical (TO) modes, obtained from [77] and [28]. A variation of
nearly about 4 orders of magnitude is seen. Thus, even for domains in the 100 nanometer
range, a range of high and low Kn bands would exist.
Figure 3.10 Mean free paths for silicon at 300 K.
The all-BTE solver experiences great difficulties with convergence with such a
wide range of Knudsen numbers. The hybrid solver, however, is extremely robust and
can address this range easily. Therefore, we simulate a domain length L of 100 nm (see
Figure 3.5) for comparison of the hybrid solver with the all-BTE solver. Additional
domain lengths of 500 nm, 1000 nm, and 3000 nm are used to examine the accuracy and
performance of the hybrid solver. For these cases, the all-BTE solver is unable to
produce results, and thus none are presented.
0.1
1
10
100
1000
10000
100000
0 20 40 60 80 100
MeanFreePath(nm)
Frequency (1012 rad/s)
LA
TA
LO
TO
