INVESTIGATION OF THE THERMAL TRANSPORT IN SUPERCONDUCTING NIOBIUM

AND TANTALUM

By

Peng Xu

A DISSERTATION

Michigan State University

in partial fulﬁllment of the requirements

Submitted to

for the degree of

Mechanical Engineering – Doctor of Philosophy

2019

ABSTRACT

INVESTIGATION OF THE THERMAL TRANSPORT IN SUPERCONDUCTING NIOBIUM

AND TANTALUM

By

Peng Xu

Superconducting radio frequency (SRF) cavities fabricated from bulk high purity niobium (Nb)
are increasingly used for particle accelerators to achieve continuous operation. Even in the su-
perconducting regime, residual resistance and small imperfections on the RF surface can dissipate
energy and cause local heating that leads to cavity quench. Large values of thermal conductivity
can mitigate local temperature excursions and prevent cavity quench, thus improving cavity per-
formance. Understanding thermal transport in bulk and thin ﬁlm superconducting Nb may guide
thermal design of current and next generation SRF cavities.

The thermal conductivity of metals is composed of electronic and lattice (phonon) components.
In normal conductors, the electronic component dominates, and in superconducting metals, as
the temperature drops below the critical temperature, phonons become increasingly important
carriers of thermal energy. A widely used model of thermal conductivity in superconductors
omits explicit accounting of the eﬀect of dislocations, which result from deformation. Here, this
model is extended by accounting for the eﬀects of phonons scattered by dislocations independent
from boundary scattering. This extended model agrees better with measurements of thermal
conductivity in deformed Nb samples, especially at temperatures T less than 3 K. An apparent
threshold of dislocation density Nd is found to be Nd = O(1012) m-2 for Nb and when applied
to tantalum (Ta), it is Nd = O(1011) m-2. There is little contribution to the thermal conductivity
when the dislocation density is less than this threshold. This model can also be used to estimate
the dislocation density by ﬁtting measured values of thermal conductivity.

Examination of thermal conductivity data for superconducting Nb shows that there is often a
local maximum, a so-called phonon peak, kpp. The temperature at which this kpp occurs Tpp is
between 1.72 K and 2.35 K and shifts for samples after deformation. It is well known that the

magnitude of kpp decreases as the material is deformed, and hence with increasing Nd. Less cited is
that Tpp increases with increasing Nd. This may aﬀect the operating temperature of an SRF cavity.
At a certain level deformation (i.e., 4.7% deformation for a residual resistivity ratio RRR=185), the
phonon peak disappears. More deformation is needed for higher RRR, (i.e., greater purity).

The models discussed above require estimating several parameters from thermal conductivity
measurements and may be best suited to explaining the relative importance of the several scattering
mechanisms. For predicting thermal conductivity from basic material variables, the Boltzmann
transport equation (BTE) is solved by two methods to predict the lattice component of thermal
conductivity. One method uses a substitution of variables from frequency to wavevector in the
Callaway model to include the nonlinear phonon dispersion relationship for the longitudinal acoustic
(LA) and transverse acoustic (TA) phonon polarizations. This model incorporates a relaxation time
approximation using Matthiessen’s rule to consider phonon scattering by electrons, boundaries,
and dislocations. Another method to predict the lattice thermal conductivity uses an energy-based,
variance-reduced Monte Carlo (MC) solution to the BTE for phonons. The MC solution allows
more general consideration of the individual scattering mechanisms. It may also be generalized for
more complex geometries. The MC solution technique was ﬁrst veriﬁed by comparing the predicted
thermal conductivity in bulk Si and Si nanowires with experimental results. Both solutions of the
BTE for the lattice thermal conductivity of undeformed and deformed superconducting Nb agreed
well with experimental values. The MC model was also used to demonstrate that interstitial
impurities must be near saturation to change the lattice thermal conductivity of Nb. The MC
solution was also eﬀective in predicting the lattice thermal conductivity of superconducting Ta,
with the appropriate change in dispersion relation and other material parameters.

ACKNOWLEDGEMENTS

This research is supported by the U.S. Department of Energy, Oﬃce of High Energy Physics,
through Grant No. DE-FG02-09ER41638.

I am indebted to my dissertation adviser, Dr. Neil T. Wright, for his outstanding guidance
through my PhD career. He behaves like a father to me, and I highly appreciate his patience and
encouragement when I am struggling with my research. He guided me step by step on technical
writings, oral presentations, and teaching of the heat transfer lab. I am impressed by his broad
knowledge in various ﬁelds, including heat transfer, biological systems, physics, material science,
and many more, and I have beneﬁted a lot by his willing of sharing. I wish to express my deepest
gratitude to him for demonstrating how to think from the audience perspective and how to look at
the larger pictures.

I am grateful to Dr. Thomas R. Bieler for introducing me to the concepts of material science
and metallurgy. His critical thinking and strong knowledge in metals both theoretically and
experimentally has greatly inspired me. I am also thankful for proving me this opportunity to work
on the superconducting materials.

I wish to thank Dr. Andre Bénard and Dr. Rebecca Anthony for their comments and suggestions

on my work, and for being members on my doctoral guidance committee.

I am grateful to my fellow graduate student, Saravan Chandrasekaran, for kindly and patiently

introducing me to the experiments of measuring the thermal conductivity.

The love and encouragement from my parents, Yuegui Xu and Guihua Yao have been the
foundation of my standing today. The love, responsibility, and happy times with my children Jordan
Xu and Ella Xu have supported me through my PhD and will support me during my life. I cannot
express enough my appreciation of my wife, Qing Feng. Her willingness and desire to stand by me
without any doubt, to encourage me during easy and tough times, to take good care of the children
when I am working on my research, were instrumental in me. This dissertation would not have
been possible without her warm love, continued patience, and endless support. Thank you!

iv

TABLE OF CONTENTS

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INTRODUCTION .

LIST OF TABLES .
LIST OF FIGURES .
CHAPTER 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
ix
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CHAPTER 2 REVIEW OF THERMAL CONDUCTIVITY OF NB . . . . . . . . . . . .

4
2.1 Experiments on thermal conductivity of Nb . . . . . . . . . . . . . . . . . . . . . 10
. 23
2.2 Models on thermal conductivity of Nb . . . . . . . . . . . . . . . . . . . . . . .
2.3 Conclusion .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 31

. .

1

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CHAPTER 3 EFFECT OF DEFORMATION ON THE THERMAL CONDUCTIVITY

3.1
3.2 Analysis .

Introduction .
.

3.2.1

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AND PHONON PEAK TEMPERATURE OF NB . . . . . . . . . . . . . . 32
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
. . . . . . . . . . . . . . . . . . . . . . . . . 35
Scaled sensitivity coeﬃcient of boundary scattering . . . . . . . 35
Scaled sensitivity coeﬃcient of normal electron scattering . . . . 35
Scaled sensitivity coeﬃcient of dislocation scattering . . . . . . . 36
. 38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Scaled sensitivity coeﬃcient
3.2.1.1
3.2.1.2
3.2.1.3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

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3.3 Results and Discussion .
3.4 Conclusion .

. .

CHAPTER 4 MONTE CARLO MODEL OF THE LATTICE THERMAL CONDUC-

.

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.

. .

4.3.1

Introduction .

TIVITY OF NB .
.
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Monte Carlo Solution Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Initialization . .
. . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.1.1 Number of particles
Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.1.2
4.3.1.3
Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1.4 Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1.5
Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Phonon-electron scattering . . . . . . . . . . . . . . . . . . . . . 55
Phonon-dislocation scattering . . . . . . . . . . . . . . . . . . . 56
4.3.4 Calculate the thermal conductivity of Nb . . . . . . . . . . . . . . . . . . . 57
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.1 Temperature proﬁle at the diﬀusion limit . . . . . . . . . . . . . . . . . . . 57
4.4.2 Temperature proﬁle at the ballistic limit
. . . . . . . . . . . . . . . . . . . 58
. . . . . . . . . . . . . . . . . . . . . . . 59
4.4.3 Thermal conductivity in bulk Si

4.3.2 Drift process
Scattering .
4.3.3
4.3.3.1
4.3.3.2

4.4 Results and Discussion .

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v

4.4.4
4.4.5

. 60
Size eﬀect on thermal conductivity of Si nanowires . . . . . . . . . . . .
Simulation of the thermal conductivity of Nb . . . . . . . . . . . . . . . . 61
Simulation of the lattice thermal conductivity of undeformed Nb . 62
4.4.5.1
Simulation of the lattice thermal conductivity of deformed Nb . . 64
4.4.5.2
4.5 Conclusion .
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

. .

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CHAPTER 5 EFFECT OF IMPURITIES ON THE LATTICE THERMAL CONDUC-

TIVITY OF NB .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1 Residual resistance ratio (RRR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Eﬀect of oxygen diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Eﬀect of hydrogen diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 72
5.4 Eﬀect of nitrogen diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5 Substitutional atoms of diﬀerent mass
. . . . . . . . . . . . . . . . . . . . . . . . 73
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.6 Point defects in the MC Model
5.7 Monte Carlo simulation considering impurity scattering . . . . . . . . . . . . . .
. 74
5.8 Phonon-impurity scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.9 Eﬀect of Grain Boundary scattering on the lattice thermal conductivity of super-

conducting Nb .
5.9.1 Models
5.10 Conclusion .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
.

CHAPTER 6 CALCULATION OF THE LATTICE THERMAL CONDUCTIVITY OF

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. .

Introduction .

NB CONSIDERING PHONON DISPERSION . . . . . . . . . . . . . . . . 84
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.1
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 Model
Phonon dispersion of Nb . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2.1
Phonon-boundary scattering . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2.2
Phonon-electron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2.3
Phonon-dislocation scattering . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.4
6.2.5 Total relaxation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3 Results and Discussions .
6.4 Conclusion .
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. .

Introduction .

7.1
.
7.2 Monte Carlo simulation .
7.3 Comparison of total k between Nb and Ta . . . . . . . . . . . . . . . . . . . . .

CHAPTER 7 THERMAL CONDUCTIVITY OF SUPERCONDUCTING TANTALUM . 93
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
. 98
7.3.1 Comparison of simulated lattice k between undeformed Nb and Ta . . . . . 99
7.3.2 Comparison of simulated lattice k between deformed Nb and Ta . . . . . . 100
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.4 Conclusion . .

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CHAPTER 8 CONCLUSIONS .
APPENDIX .
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BIBLIOGRAPHY .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

vi

LIST OF TABLES

Table 2.1: Chronology of measurements of the thermal conductivity of superconducting
Nb. RRR values marked with asterisks (*) are estimated using measured
k(4.2 K) × 4 .

. 21
Table 2.2: Numerical integration results for I1(z) and I2(z) [1] . . . . . . . . . . . . . . . . 29
Table 2.3: Research progress on modeling k of superconducting Nb . . . . . . . . . . . . . 30

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

Table 4.1: Parameters for the samples used in the simulation . . . . . . . . . . . . . . . . . 61

Table 5.1: Contributions of diﬀerent impurities to ∂ ρi
∂Ci

[1]

. . . . . . . . . . . . . . . . . . 69

Table 5.2: Typical RRR values of Nb in relation to processings [2], where Cavities and

Samples refer to post puriﬁed Nb origins, Other refers to other preparations. . . . 70

Table 7.1: Parameters for the Ta samples used in the simulation [3], where the size refers
to the diameter of the cylinder. All of the three Ta samples were deformed at
temperature higher or equal to 295 K, therefore they are likely to have edge
dislocations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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vii

LIST OF FIGURES

Figure 1.1: A typical example of the thermal conductivity of single crystal supercon-
ducting Nb sample from 0.3 K to 5 K with diﬀerent levels of uniaxial strain,
replotted from [4]. Undeformed state refers to Cauchy strain of 0%, and 1%,
2.4%, 4.7%, and 8.8% refer to the corresponding amounts of Cauchy strain.
The thermal conductivity at temperature lower than 3 K decreases with the
increase of strain. The phonon peak appears at 0%, 1%, and 2.4% deforma-
tion and disappears after 4.7% deformation. The thermal conductivity at the
phonon peak decreases the most after deformation compared with the thermal
conductivity at other temperatures.

. . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.1: An example of the sketch of a 9-cell TESLA accelerating structure [5]. The
outside surface of the cavity is immersed into liquid He to keep Nb in su-
perconducting temperature. Charged particles move inside the cavity to be
accelerated by the magnetic ﬁled.

. . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.2: Picture of the delamination crack. The crack is located approximately in the

middle of the Nb sheet, spiting the sheet into two parts.

. . . . . . . . . . . . .

Figure 2.3: Microscopic image of the delamination crack. The crack propagates around
25 mm along the circumference, the thickness of the Nb sheet is around 2
mm, the crack splits the Nb sheet by half. . . . . . . . . . . . . . . . . . . . . .

Figure 2.4: The contributions of the normal electron thermal conductivity ke and the
lattice thermal conductivity kg to the overall thermal conductivity k = ke +kg.
At temperature lower than 2 K, kg dominates the total thermal conductivity
and ke dominates the thermal conductivity for temperature higher than 3 K.
For this Nb with relatively few dislocations, there is a signiﬁcant phonon peak. .

2

4

5

5

7

Figure 2.5:

Illustration of the roles of dislocation density and RRR on polycrystalline
[6, 7] and single crystal Nb [4]. Thermal conductivity at temperature T=4.2
K is proportional to the RRR. However, at the temperatures close to 2 K, the
strain has more eﬀect on the thermal conductivity than the RRR. Of particular
note is that a single sample is measured in its as-received state at RRR=390,
after annealing at 1200 ◦C for two hours, but with a Ti getter, the purity is
. . . . . . . . . . . . . . .
reduced to RRR=68, but a phonon peak develops.

.

9

viii

Figure 2.6: A typical example of the thermal conductivity of single crystal supercon-
ducting Nb sample with screw dislocations from 0.3 K to 5 K with diﬀerent
levels of uniaxial strain, replotted from [4]. Undeformed state refers to 0
Cauchy strain, and 3%, 7.3%, and 14.7% deformation correspond to the same
amount of Cauchy strain. The thermal conductivity at temperature lower than
3 K decreases with the increase of strain. The phonon peak appears at 0%
deformation and 3% at about 1.8 K and 2 K separately. There is nearly no
phonon peak after 7.3% deformation and 14.7% deformation. The thermal
conductivity data at temperatures higher than 3.5 K are the same for diﬀerent
amount of deformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 2.7: Comparison between ﬁtting with or without the dislocation term for deformed

sample [4]. The estimated value of dislocation density is Nd = 3.83 × 1014 m−2. 28

Figure 3.1: Measured phonon peak temperatures Tpp for undeformed, deformed, and
annealed specimens as a function of the ratio of the thermal conductivity at
the phonon peak kpp to the local minimum thermal conductivity klm. . . . . . . 33

Figure 3.2: The scaled sensitivity coeﬃcients of phonons scattered by boundary (parame-
ter B) as a function of temperature. The B values vary from 2×103 Wm−2K−4
to 10 × 103 Wm−2K−4.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Figure 3.3: The scaled sensitivity coeﬃcients of phonons scattered by normal electrons
(parameter D) as a function of temperature. The D values vary from 2× 10−3
WK3m−1 to 10 × 10−3 WK3m−1. . . . . . . . . . . . . . . . . . . . . . . . . . 37

Figure 3.4: The scaled sensitivity coeﬃcients of phonons scattered by dislocations (pa-
rameter Nd) as a function of temperature. The Nd values vary from 2 × 1012
m−2 to 10 × 1012 m−2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Figure 3.5: Comparison between ﬁt with or without dislocation term for an undeformed
sample from [4]. The estimated value of dislocation density is Nd = 4.67 ×
1012 m−2. Both the ﬁtted lines for temperatures lower than 1 K and higher
than 2.5 K agree well with the experimental data. At 1.5 K < T < 2 K,
ﬁtting with the dislocation term has k values lower than the experimental
results, while ﬁtting without dislocation term has k values higher than the
experimental results. The ﬁt with or without the dislocation term provides
similar accuracy for undeformed samples.

. . . . . . . . . . . . . . . . . . .

. 39

ix

Figure 3.6: Comparison between ﬁtting with or without dislocation term for deformed
sample [4]. The estimated value of dislocation density is Nd = 3.83 × 1014
m−2. Both the ﬁtted lines for temperatures lower than 1 K and higher than
2.5 K agree well with the experimental data. At 1 K < T < 2.5 K, ﬁtting
with dislocation term matches well with the experimental results, while ﬁtting
without the dislocation term deviates from the experimental results. The ﬁt
with the dislocation term improves the accuracy for samples after deformations.

40

Figure 3.7: Diﬀerences between measurement data and the ﬁt results of thermal conduc-
tivity with or without the dislocation term for an undeformed sample [4].
Here the diﬀerences, which are also called the residuals, are calculated as
km − k f , where km is the measured thermal conductivity and k f is the ﬁtted data. 41

Figure 3.8: Diﬀerences between measurement data and the ﬁt results of thermal con-
ductivity with or without the dislocation term for a sample after 14.7% de-
formation [4]. Here the diﬀerences, which are also called the residuals, are
calculated as km − k f , where km is the measured thermal conductivity and
k f is the ﬁtted data.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

.

.

Figure 3.9: Thermal conductivity of a sample from [4] (an example of predominantly
screw dislocation) using curve ﬁtting including the eﬀect of dislocation. Here
the dislocation density is estimated from Eq. 2.4 for the Cauchy strain  =0%,
3%, 7.3%, 10.3%, and 14.7%, respectively. Wasserbäch [4] provided a mean
value of the dislocation density to be 0.8 − 1 × 1014 m−2, and the dislocation
density measured at 14.7% is on the order of magnitude of 1014 m−2. . . . . . . 43

Figure 3.10: Thermal conductivity of a sample from [4] (an example of predominantly
edge dislocation) using curve ﬁtting including the eﬀect of dislocation. Here
the dislocation density is estimated from Eq.
2.4 for the Cauchy strain
 =0%, 4%, and 8% respectively. Wasserbäch [4] provided a mean value of
the dislocation density to be 1 − 2 × 1014 m−2, and the dislocation density
measured at 8% is on the order of magnitude of 1014 m−2.

. . . . . . . . . . . 44

Figure 3.11: The eﬀect of dislocation density on thermal conductivity of Nb at temperature
between 1.5 K and 4.5 K. Equation 2.4 is used to ﬁt the experimental results
from [8]. The results are obtained by using the ﬁtted parameters (e.g., B, D,
a) and changing Nd. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

.

Figure 3.12: Comparison between phonon boundary thermal resistance and phonon dislo-
cation resistance for diﬀerent dislocation density by using the ﬁtted parameters
(e.g., B, D, a) and changing Nd. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

x

Figure 3.13: The ratio of dislocation density after deformation to that before deformation as
a function of uniaxial straining, estimated by ﬁtting the thermal conductivity
from experimental results [4]. Samples H1, K6, H2, and K10 are deformed
at temperatures of 77 K, 195 K, 295 K, and 1470 K, respectively. H1 and K6
had predominantly screw dislocations, H2 and K10 had predominantly edge
dislocation. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

. .

. .

.

Figure 4.1: Comparison of the temperature proﬁle obtained using the Monte Carlo model
with the usual Fourier conduction results in the diﬀusion limit. Six Monte
Carlo simulations have been carried out to demonstrate the repeatability of
the model, and the scatter in the results are shown. . . . . . . . . . . . . . . . . 58

Figure 4.2: Comparison of the temperature proﬁle by Monte Carlo method with Stefen-
Boltzmann equation in the ballistic limit. Six simulations have been carried
out to demonstrate the repeatability of the Monte Carlo model.

. . . . . . . . . 59

Figure 4.3: Comparison of the thermal conductivity of bulk Si between diﬀerent methods.
The solid line is Eq. 4.22, the circles come from Monte Carlo simulation by
Lacroix et al. [9], and the stars are the current simulation results.

. . . . . . . . 60

Figure 4.4: Comparison of the thermal conductivity of Si nanowire between current
simulations (solid lines) and the data (triangles) from Li et al. [10]. Six
simulations have been carried out and the error bars were plotted for the
Monte Carlo simulation results.

. . . . . . . . . . . . . . . . . . . . . . . . . . 61

Figure 4.5: Phonon dispersion relation of Nb in the [1 0 0] direction, replotted from
reference [11], where a is the lattice parameter, which refers to the physical
dimension of unit cell in a crystal lattice, for Nb, a = 3.3 × 10−10 m.

. . . . . . 62

Figure 4.6: Simulation results of lattice k of undeformed Nb or Nb after heat treatment.
Data used from undeformed samples from reference [4] and the sample after
heat treatment of reference [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 4.7: Simulation results of lattice thermal conductivity of Nb LS 1 from [4], an

example of screw dislocations, for several strains.

. . . . . . . . . . . . . . . . 65

Figure 4.8: Simulation results of lattice thermal conductivity of Nb LS 2, from [4] an

example of edge dislocations, for several strains.

. . . . . . . . . . . . . . . . . 66

Figure 5.1: Typical thermal conductivity data [4, 7, 13] related with RRR and strain,
thermal conductivity increases with RRR at 4.2 K, while at 2 K, there is no
relation between RRR and thermal conductivity. The phonon peak usually
exists for appropriately heat treated sample while strain may destroy the
phonon peak.
(This ﬁgure is a repeat of Fig. 2.5), included here for the
reader’s convenience.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

.

xi

Figure 5.2: Temperature proﬁle of Si for diﬀerent levels of impurities from 1023 m−3 to
1028 m−3. The temperature proﬁle changes gradually from the ballistic limit
for the lowest defect density 1023 m−3 to diﬀusion limit at the highest defect
density 1028 m−3.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

.

.

Figure 5.3: Simulation of the lattice thermal conductivity of Nb with diﬀerent levels of
H impurities compared with ﬁtting results without H impurities. The lattice
thermal conductivity values are similar for low concentrations of H impurities
(i.e. No impurity, 1026 m−3, and 1028 m−3), but at the impurity concentration
of 1029 m−3, the lattice thermal conductivity decreased by 20% at the phonon
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
peak.

.

.

.

.

.

.

.

.

.

.

. 76

Figure 5.4: Simulation of the lattice thermal conductivity of Nb with diﬀerent levels of
N impurities compared with ﬁtting results without N impurities. The lattice
thermal conductivity values are similar for low concentrations of N impurities
(i.e. No impurity, 1026 m−3, and 1027 m−3), but at the impurity concentration
of 1028 m−3, the lattice thermal conductivity decreased by 15% at the phonon
peak.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

.

.

.

.

.

.

.

.

.

.

Figure 5.5: Lattice thermal conductivity of Nb by considering the eﬀect of phonon impu-
rity scattering. This sample was heat treated at 800 ◦C for 2 hours ﬁrst, then
it was infused with hydrogen in a 75% H2, 25% N2 environment of 0.5 atm
pressure at 300 ◦Cfor 1 hour, at last it was heat treated again at 800 ◦C for 2
hours. The last heat treatment to degas the H impurity recovers the thermal
conductivity at phonon peak.

. . . . . . . . . . . . . . . . . . . . . . . . . .

. 78

Figure 5.6: Lattice thermal conductivity of Nb by considering the eﬀect of phonon impu-
rity scattering. This sample was heat treated at 140 ◦C for 48 hours and 1100
◦C for 4 hours ﬁrst, then it was infused with hydrogen in a 75% H2, 25%
N2 environment of 0.5 atm pressure at 300 ◦C for 1 hour, at last it was heat
treated again at 800 ◦C for 2 hours. The last heat treatment has no eﬀect to
degas the H impurity and does not recover the thermal conductivity at phonon
peak.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

.

.

.

.

.

.

.

.

.

Figure 6.1: Phonon dispersion of Nb at [1 0 0] direction, replotted from the experimental

. .

. .

data of [14]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Figure 6.2: Comparision between the calculation results and the data ﬁtting results for an
example sample of screw dislocations [4]for undeformed state and 8% tensile
deformation. The group velocity is taken as vg = dω/dq, the dislocation
density is calculated based on the ﬁtting using Eq. 2.4. . . . . . . . . . . . . . . 89

xii

Figure 6.3: Comparision between the calculation results and the data ﬁtting results for
another example sample of screw dislocations [4]for undeformed state and
several levels of deformation. The group velocity is taken as vg = dω/dq,
the dislocation density is calculated based on the ﬁtting using Eq. 2.4.

. . . . . 90

Figure 6.4: Comparision between the calculation results and the data ﬁtting results for an
example sample of edge dislocations (sample H2) [4]for the undeformed state
and 4% tensile deformation. The group velocity is taken as vg = dω/dq, the
. . . . .
dislocation density was calculated based on the ﬁtting using Eq. 2.4.

. 91

Figure 6.5: Comparision between the calculation results and the data ﬁtting results for
another example sample of screw dislocations (sample K10) [4]for the un-
deformed state and 1% and 2.4% tensile deformation. The group velocity is
taken as vg = dω/dq, the dislocation density was calculated based on the
ﬁtting using Eq. 2.4. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Figure 7.1: The thermal conductivity of Ta is the sum of the contributions of electrons
and phonons [3]. Below 1 K, the electronic contribution is negligible. The
contribution of phonons dominates in this temperature range because of the
condensation of free electrons into Cooper pairs.

. . . . . . . . . . . . . . .

. 94

Figure 7.2: Phonon dispersion relation of Ta in the [1 0 0] direction, replotted from [15],
 = aq/π, where q is the wave vector,  is the reduced wave vector, a is the
lattice parameter, which refers to the physical dimension of a unit cell in a
crystal lattice. For Ta, a ≈ 3.3 × 10−10 m, the same as Nb, LA refers to
longitudinal acoustic phonons, and TA refers to transverse acoustic phonons.

. . 95

Figure 7.3: Comparison between the Monte Carlo simulation results of lattice thermal
conductivity of Ta 1 with the ﬁtting results using Eq. 2.4 from [3], from
undeformed to 2%, 3.1%, and 7.3% deformation. . . . . . . . . . . . . . . . . . 96

Figure 7.4: Comparison between the Monte Carlo simulation results of lattice thermal
conductivity of Ta 2 with the ﬁtting results using Eq. 2.4 from [3], from
undeformed to 0.5%, 1.2%, 2.4%, 3.9%, and 6.2% deformation. . . . . . . . . . 97

Figure 7.5: Comparison between the Monte Carlo simulation results of lattice thermal
conductivity of Ta 3 with the ﬁtting results using Eq. 2.4 from [3], from
undeformed to 1% deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Figure 7.6: Comparison between ﬁtting with the experimental data of the thermal con-
ductivity of undeformed and deformed superconducting Nb [4] and Ta [16].
The dislocation density is calculated during the ﬁtting processes.

. . . . . . . . 99

xiii

Figure 7.7: Comparison between Monte Carlo simulation of the lattice thermal conductiv-
ity with the ﬁtting results using Eq. 2.4 for undeformed Nb [4] and undeformed
Ta [16]. The parameters, including size, RRR, deformation temperature, and
the calculated dislocation density, of the Nb and Ta samples are listed in Table 4.1.100

Figure 7.8: Comparison between Monte Carlo simulation of the lattice thermal conduc-
tivity with the ﬁtting results using Eq. 2.4 of undeformed and deformed Nb [4]
and Ta [16]. The parameters, including size, RRR, deformation temperature,
and the calculated dislocation density, of the Nb and Ta samples are listed in
Table 4.1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

.

.

.

.

.

.

xiv

CHAPTER 1

INTRODUCTION

Particle accelerators are increasingly used in a variety of applications for scientiﬁc research,
medicine, and industrial processing. They accelerate beams of charged particles to high speeds
(e.g., high energy physics applications require particle beams to be accelerated to near the speed of
light.) [17]. Radio frequency (RF) alternating current on the order of GHz provides energy to each
charged particle for acceleration. Copper (Cu) was originally used for normally conducting cavities
because of its easy formability and good electrical conductivity at room temperature. However,
these cavities have high RF losses due to the ﬁnite electrical resistivity of normally conducting Cu,
leading to loss of performance. Even with water cooling, Cu cavities cannot provide continuous
operation because of the temperature excursions due to energy loss.

Alternatively, niobium (Nb) has been widely used to construct superconducting radio frequency
(SRF) cavities in particle accelerators because it has the highest superconducting critical tempera-
ture (Tc = 9.25 K) and the highest thermodynamic critical ﬁeld (Hth = 1.6 × 105 A m−1) of any
element [18]. As a superconductor, it has no electrical DC resistance and little RF resistance [5].
Manufacturing SRF cavities from large-grain Nb may reduce cost and improve the quality factor
as compared with using polycrystalline Nb (i.e., small-grain, typical grain size of about 50 µm)
[13], presumably because fewer grain boundaries exist in large-grain Nb. The resistance caused by
grain boundary scattering may reduce the heat transfer and increase the surface resistance thus to
decrease the performance of cavities. Even in the superconducting regime, residual resistance to
RF energy and small imperfections at the RF surface dissipate energy and cause local heating that
leads to cavity quench [19]. Large values of thermal conductivity can mitigate local temperature
excursions by improving heat transfer to the liquid He bath, thus preventing cavity quench and im-
proving cavity performance [20]. Processing Nb to obtain the largest thermal conductivity possible
is an important strategy for improving performance.

Particle accelerators require the superconducting nature of Nb on the inner surface of a cavity

1

Figure 1.1: A typical example of the thermal conductivity of single crystal superconducting Nb
sample from 0.3 K to 5 K with diﬀerent levels of uniaxial strain, replotted from [4]. Undeformed
state refers to Cauchy strain of 0%, and 1%, 2.4%, 4.7%, and 8.8% refer to the corresponding
amounts of Cauchy strain. The thermal conductivity at temperature lower than 3 K decreases with
the increase of strain. The phonon peak appears at 0%, 1%, and 2.4% deformation and disappears
after 4.7% deformation. The thermal conductivity at the phonon peak decreases the most after
deformation compared with the thermal conductivity at other temperatures.

to a depth of up to only the London penetration depth, which is about 30 nm for Nb [21]. This
suggests that the superconducting properties of the cavity are only needed by a small fraction of
the wall thickness (typically about 3 mm), which makes thin ﬁlm superconducting composites an
opportunity for the next generation of the cavities. The next generation of SRF accelerators may be
developed using Nb clad Cu. This composite structure may take advantage of improved magnetic
performance of thin ﬁlm Nb along with the large values of thermal conductivity at low temperatures
(e.g., thermal conductivity of copper is on the order of magnitude of 103 Wm−1K−1 at 2 K [22]).

2

012345010203040506070Temperature (K)Thermal conductivity (Wm−1K−1)  Undeformed1% deformation2.4% deformation4.7% deformation8.8% deformationThe thermal conductivity of Nb at temperatures lower than Tc is signiﬁcantly inﬂuenced by
the processing history of Nb. A typical ﬁgure of thermal conductivity of Nb as a function of
temperature is replotted from [4], shown in Fig. 1.1, at diﬀerent levels of deformation. Purity and
deformation have been demonstrated to change the thermal conductivity in both polycrystalline
(small grain) and large grain Nb [7, 8]. Deformation is well known to reduce the local maximum
thermal conductivity (also called the phonon peak kpp), but it is less known that it also increases the
temperature at which the phonon peak occurs (Tpp). Several studies have shown that while plastic
deformation may destroy the phonon peak, heat treatment may also restore it [8, 23]. The phonon
peak of a single crystal Nb sample after 2.4% deformation has been measured to decrease by about
69% compared with its undeformed state [4]. The phonon peak temperature of the same specimen
increased from 1.85 K at the undeformed state to 2.18 K after 2.4% deformation. Although the
increase in temperature of the phonon peak is only 0.33 K, energy savings on this temperature
diﬀerence could be signiﬁcant for the working temperature of SRF cavity. This suggests that there
is an optimization problem of balancing better performance of the cavities and more energy savings
in the liquid He bath. Studying the mechanisms of both the thermal conductivity of bulk Nb and
thin ﬁlm Nb will provide guidance for the thermal design of current and next generation of SRF
cavities.

This thesis examines the model of thermal conductivity of superconducting Nb buy including
the eﬀect of phonon-dislocation scattering. The relationship between kpp and Tpp is found by
examining diﬀerent sets of measured data of the thermal conductivity of Nb. Fitting the data with
the enhanced model leads to more accurate results than neglecting phonon dislocation scattering,
especially at low temperatures(T < 3 K) and after deformation. The energy-based variance reduced
Monte Carlo simulation technique is used to model and predict the lattice thermal conductivity of
Nb before and after deformation using the relaxation time approximation. The simulation results
match well with the ﬁtting results using the enhanced model by considering the phonon-boundary
scattering, phonon-electron scattering and phonon-dislocation scattering for both undeformed and
deformed Nb samples.

3

CHAPTER 2

REVIEW OF THERMAL CONDUCTIVITY OF NB

Nb has great similarity to tantalum (Ta) in its physical and chemical properties, making them diﬃcult
to distinguish. As a refractory metal [24], Nb has a high melting point of 2477 ◦C which makes
it resistant to wear, corrosion, and deformation. Nb has also been widely used as the material for
superconducting cavities in particle accelerators because it has the highest superconducting critical
temperature (Tc = 9.25 K), the greatest thermodynamic critical ﬁeld, and the greatest magnetic
penetration depth [25] of any element. The superconductive properties, and especially the thermal
conductivity are strongly dependent on the purity of Nb metal [26]. Very pure Nb may have
high thermal conductivity at cryogenic temperatures, however, the high purity makes Nb metal
comparatively soft and ductile [27], making it diﬃcult to maintain dimensional stavility of SRF
cavities. An example of a 9-cell SRF cavity developed at TESLA [5] is shown in Fig. 2.1.

The ﬁrst measurements of the thermal conductivity of superconducting Nb focused on un-
derstanding the mechanisms of heat conduction in superconductors [28, 29]1. They compared
the conductivity in the superconducting and normal conducting states at the same temperature by
imposing a magnetic ﬁeld to suppress the superconductivity of Nb below its critical temperature.
Understanding the thermal conductivity of Nb took on a practical purpose as Nb became the mate-
rial of choice for SRF particle accelerators starting in the 1970s [30]. SRF accelerators are typically
comprised of thin-walled channels that are evacuated on the inside and submerged in liquid helium

Figure 2.1: An example of the sketch of a 9-cell TESLA accelerating structure [5]. The outside
surface of the cavity is immersed into liquid He to keep Nb in superconducting temperature.
Charged particles move inside the cavity to be accelerated by the magnetic ﬁled.

1Niobium was originally named columbium (Cb) and some earlier studies used that term.

4

Figure 2.2: Picture of the delamination crack. The crack is located approximately in the middle of
the Nb sheet, spiting the sheet into two parts.

Figure 2.3: Microscopic image of the delamination crack. The crack propagates around 25 mm
along the circumference, the thickness of the Nb sheet is around 2 mm, the crack splits the Nb sheet
by half.

to operate in the temperature range of 1.8 K < T < 4.5 K. Oscillating electric ﬁelds accelerate
charged particles along the axis of the evacuated channel to speeds that are a substantial fraction of
the speed of light.

As a superconductor, Nb has no electrical resistance to direct current ﬁelds and small resistance
to RF ﬁelds [5]. The resistance when subject to RF ﬁelds, as happens in a particle accelerator, leads
to heat generation that must be dissipated to the helium bath surrounding the accelerator. This
heat generation can lead to temperature excursions that may diminished accelerator performance
to breakdown the magnetic ﬁeld [20] and large values of thermal conductivity k at accelerator
operating temperatures are essential for mitigating potential hot spots that would cause quenching
of an accelerator cavity and degraded performance [5, 31, 32]. Thermal breakdown of an SRF cavity
occurs when the RF surface temperature reaches its critical temperature [7]. Recent progresses in
Nb cavities for particle accelerators has resulted in signiﬁcant increase in thermal breakdown ﬁeld
up to 180∼200 mT, which is close to the thermodynamic critical ﬁeld (200 mT) [5, 32]. However,
imperfections such as defects or dislocations can cause to a sharp decrease of thermal breakdown
ﬁeld [33]. Delamination cracks might also be a factor to decrease the ﬁeld because of its resistance

5

to heat conduction between the RF surface and liquid He bath. Recent studies at FRIB of MSU
[34] shows that after EB spot welding, a crack in the 2 mm Nb sheet of the outer conductor was
discovered. The crack is approximately in the middle of the sheet, splitting the sheet in two parts
and propagating around 25 mm along the circumference, the depth of crack cannot be determined.
Fig.2.2 and Fig. 2.3 show macro the picture and microscopic images of the delamination crack,
respectively.

Padamsee correlated the peak magnetic ﬁeld with the purity of Nb as characterized by its

electrical resistivity ratio (RRR) [35].

RRR =

ρ295K
ρ4.2K

(2.1)

where ρ295K and ρ4.2K are the electrical resistivity at 295 K and 4.2 K, respectively.

The thermal conductivity at 4.2 K of superconducting Nb can also be correlated to RRR [36]

as in Eq. 2.2. This illustrates the importance of having a large value of thermal conductivity.

RRR = 4k4.2K

(2.2)

where k4.2K is the thermal conductivity at the temperature of 4.2 K.

Heat conduction in metals occurs by conduction of normal conducting electrons and lattice

vibrations (i.e., phonons) [37]. These two components are additive [38] such that

k = ke + kg

(2.3)

where ke represents heat conduction by electrons and kg represents the heat conduction by phonons.
In normal conducting metals, the electron component dominates and the phonon part is usually
negligible.
In the superconducting regime, however, the condensation of normal conducting
electrons into Cooper pairs leads to a reduction in the electron contribution to energy transport, as
well as an increase in the phonon contribution due to reduced scattering of phonons by electrons [39].
Therefore, the phonon component of heat conduction increases in signiﬁcance in superconducting
materials.

6

Figure 2.4: The contributions of the normal electron thermal conductivity ke and the lattice thermal
conductivity kg to the overall thermal conductivity k = ke + kg. At temperature lower than 2 K, kg
dominates the total thermal conductivity and ke dominates the thermal conductivity for temperature
higher than 3 K. For this Nb with relatively few dislocations, there is a signiﬁcant phonon peak.

Representative contributions of ke and kg for superconducting Nb are shown in Figure 2.4. The
Nb used for illustration has relatively few dislocations, such as a material that has been relatively
undeformed after a proper heat treatment. For T (cid:38) 3 K, ke > kg and the contribution of kg
decreases with increasing T as phonons are increasingly scattered by normal conducting electrons.
For T (cid:46) 2 K, ke is negligible as essentially all of the electrons have formed Cooper pairs [39].
As a result of the reduced scattering of phonons, a phonon peak in the thermal conductivity kpp
is evident at T ≈ 2 K. As the Nb cools below 2 K, kg decreases due to boundary scattering until
vanishing at 0 K.

Experimental illustration of the electrical and lattice contributions to k is given in Fig. 2.5,

7

00.511.522.533.544.5Temperature (K)01020304050Thermal conductivity (Wm-1K-1) k kg kewhich compares six tests on three specimens from three studies [4, 6, 7]. Others have shown
similar responses using other measurements (e.g., [40]). The red circles are for a Nb specimen
with RRR = 1200 [4]. The open circles are for Nb that is undeformed following annealing and
the closed symbols are for the same sample having undergone a uniaxial Cauchy strain of  = 8%.
While the conductivity for T > 3 K is the same before and after deformation of  = 8.5%, the kpp
at T ≈ 2 K disappears after deformation, likely due to the increase in dislocation density. It is worth
noting that k ≈ 175 W m-1 K-1at T = 4.2 K in this specimen, instead of the expected k = 300 W
m-1 K-1, which does not agree with Eq. 2.2. The solid black squares are for Nb with RRR = 390
in its “as-received” condition (i.e., with unknown deformation) [7]. The thermal conductivity for
T > 3 K behaves as expected, and there is no kpp. The open squares are for the same specimen
after it had been annealed, although inadvertently without a titanium getter. Without a getter, the
impurity concentration increased, as exempliﬁed by the RRR decreasing to 68. This suggests that
although the RRR decreased, the dislocation density had also decreased with the result of increasing
k for T ≤ 3 K. The triangles in Fig. 2.5 are for a Nb specimen with RRR = 255 before deformation
and RRR = 237 after deformation, a small change as exempliﬁed by k for T > 3 K [6]. The open
triangles are for the undeformed specimen and ﬁlled symbols are for the Nb after deformation to
 = 8.5%. The undeformed specimen has a phonon peak at about 2 K. After the specimen was
strained to a Cauchy strain of  = 8%, there is no kpp.

Given the practical and fundamental importance of k in superconducting Nb, it has been subject
to numerous studies. Normal conducting electrons largely determine k for Tc/3 (cid:46) T (cid:46) Tc, even
as they condense into Cooper pairs. At these temperatures, the impurity concentration largely
determines k. As the temperature decreases further, the paired electrons no longer conduct thermal
energy and no longer scatter phonons. The lack of electron scattering, can lead to a local maximum
in k, that is kpp. But other mechanisms scatter the phonons, including material boundaries,
dislocations, and grain boundaries. Other eﬀects can greatly alter the thermal response of Nb.
Magnetic ﬁelds, for example, can suppress the superconductivity in type II superconductors, and
leave the normal conducting electrons as the dominant mode of energy transport. The following

8

Figure 2.5: Illustration of the roles of dislocation density and RRR on polycrystalline [6, 7] and
single crystal Nb [4]. Thermal conductivity at temperature T=4.2 K is proportional to the RRR.
However, at the temperatures close to 2 K, the strain has more eﬀect on the thermal conductivity
than the RRR. Of particular note is that a single sample is measured in its as-received state at
RRR=390, after annealing at 1200 ◦C for two hours, but with a Ti getter, the purity is reduced to
RRR=68, but a phonon peak develops.

9

0.511.522.533.544.5Temperature (K)050100150200250Thermal conductivity (Wm-1K-1)RRR=68 Heat TreatedRRR=237 =8.5%RRR=255 =0RRR=390 As Rec'dRRR=1200 =0RRR=1200 =8%sections review the studies that have measured thermal conductivity in superconducting Nb subject
to many external eﬀects.

2.1 Experiments on thermal conductivity of Nb

To put the measurements into context, it is instructive to examine a model for k that is based on
Bardeen-Cooper-Schrieﬀer (BCS) theory [39]. Koechlin and Bonin [41] developed an expression
for k of superconducting Nb by parameterizing the Bardeen-Rickayzen-Tewordt (BRT) [42] model
for k based on BCS theory [39]. Xu et al. [43] added a term to the Koechlin and Bonin model to
account explicitly for phonons scattered by dislocations in the lattice thermal conductivity. This
augmented model can be written as

(cid:21)−1

k = R(y)(cid:104) ρ

+ AT2(cid:105)−1

LT

(cid:20)

+

1

D exp(y)T2 +

1
BΛT3 +

KNd
T2

(2.4)

where the two terms in the ﬁrst brackets account for electrons scattered by impurities and electrons
scattered by phonons, respectively, and the three terms in the second brackets account for phonons
scattered by electrons, phonons scattered by boundaries, and phonons scattered by dislocations,
respectively. The term R(y) quantiﬁes the condensation of normal conducting electrons into Cooper
pairs, ρ is the residual electrical resistivity (ρ = ρ295K/RRR), RRR is the residual resistivity ratio,
ρ295K is the electrical resistivity at 295 K, L ≈ 2.45 × 10−8 W K-2 is the Lorentz number, A is the
coeﬃcient of momentum exchange of electrons with the lattice, D scales phonon-electron scattering,
B scales phonon-boundary scattering, Λ is the phonon mean free path, Nd is the dislocation density
and K is a parameter describing phonon-dislocation scattering. The term R(y) may be calculated
as

(cid:34)

(cid:35)

R(y) =

kes
ken

= f(0)−1

f(−y) + y ln(1 + exp(−y)) +

2

y

(2(1 + exp(y)))

(2.5)

where kes and ken are the electronic thermal conductivity of superconducting state and normal
conducting state, respectively. The ratio of the superconducting energy gap (∆(T)) to thermal
ﬂuctuations of kBT may be written as

y =

∆(T)
kBT

=

∆(T)
kBTc

Tc
T

10

(2.6)

Þ ∞

f(−y) =

zdz

and f(−y) as

(2.7)
For T/Tc < 0.6, y can be approximated as y = αTc/T with α ≈ 1.76 from BCS theory. The
parameters ρ, a, B, D, Nd, and α are typically estimated from measurements of conductivity. The
parameter K is expressed following Klemens [44] for randomly distributed dislocations as

1 + exp(z + y)

0

K =

0.038(¯vh2)b2

3

kB

2

γ

(2.8)

where γ is the Grüneisen constant (γ=1.4 for Nb), b is the Burgers vector, ¯v is the average group
velocity of Nb, and h is the Planck constant. Using values for Nb, an expression for K = 3.05×10−15
m3K3 W-1.

The expression by Koechlin and Bonin [41] has been used successfully to model the thermal
conductivity of superconducting Nb in many circumstances. The extended expression of Eq.
2.4 shows promise for improving modeling of k for superconducting Nb and Ta with substantial
dislocations, especially at T/Tc ≈ 0.22 [43].

To our best understanding, the ﬁrst measurement on the thermal conductivity of Nb is conducted
by Mendelssohn and Olsen at 1950 [28]. They measured the thermal conductivity of Nb rod
from superconducting temperature to normal conducting temperature and found that the thermal
conductivity curve behaved diﬀerently than other superconductors (Ta and Pb-Sn alloys) as k
at superconducting state exceeds the values at normal conducting state for temperatures below
4 K. They explained this feature to be the reason of appearance of circulation heat ﬂow at low
temperatures because of the applied magnetic ﬂux. Two years later, Mendelssohn and Rosenberg
[29] measured the thermal conductivity of 99.99% pure Nb at temperatures from 2 K to 40 K. They
showed similar behavior with reference [28] as the thermal conductivity curve at superconducting
and normal conducting state (because of the applied magnetic ﬂux) crossed over with each other
for the magnetic ﬁeld of 2300 & 3300 Gauss ﬁeld at T < 6 K and T < 3 K, respectively. The reason
is that the decrease of lattice k because of the scattering of phonons by extra electrons produced by
the magnetic ﬁeld was not fully compensated by the increase of electronic k.

11

Mendelssohn [45] investigated the thermal conductivity of several superconductors experimen-
tally below 1 K. He obtained the thermal conductivity of poly-crystal Nb to be proportional to
T3 between 0.4 K and 0.75 K, which agrees with Casimir model [46] for boundary scattering of
dielectric materials. He thus concluded that the thermal conductivity of superconductive metal
near absolute zero behaves like that of a dielectric crystal. The Casimir model [46] for the lattice
thermal conductivity due to crystal boundary scattering was expressed as

(2.9)
where R is the radius for a circular cylindrical specimen, A = Cv/T3 with Cv being the speciﬁc
heat per unit volume, P is expressed as

kb = 2.31 × 103RPA2/3T3

P =

2(vl/vt)2 + 1

[2(vl/vt)3 + 1]2/3

(2.10)

where vl and vt are the longitudinal and transverse velocities of sound in the material, respectively.
In 1958, Mendelssohn [47] was the ﬁrst to quantify the phonon conduction in superconductors
to the number of dislocations by measuring the thermal conductivity of superconducting lead and
niobium. Van Bueren’s formula [48] was used to calculate the dislocation density for face centered
cubic (FCC) lattices, (i.e., lead). For body centered cubic (BCC) materials, (i.e., Nb), Klemens
model [44] was used to estimate the dislocation density. Mendelssohn [47] also mentioned that
the superconductive metal can be rendered normal by applying a low magnetic ﬁeld, and the
strain only aﬀects the thermal conductivity of the superconducting state, not the normal state.
Another observation in this paper is that the zone reﬁned Nb sample showed a maximum in thermal
conductivity in the superconducting state that exceeded that of the normal state at 2 K. After zone
reﬁning process, the thermal conductivity was measured to be 180 Wm−1K−1 at 2 K, whereas before
zone reﬁned process kpp ≈ 75 Wm−1K−1 at 2.4 K, which was less than the thermal conductivity
in the normal state.

Montgomery [49] measured the thermal conductivity of lead between 1 K and 4 K and noted
that below about 1.4 K, the thermal current is carried entirely by lattice waves, but not conduction

12

electrons, which is consistent with [45]. He also found that grain boundaries are responsible only
for a part of the phonon scattering, with sub-grain structure and dislocation networks being of
comparable importance. He suggested that thermal conductivity of superconductors at supercon-
ducting state becomes size dependent at low temperatures. Chaudhuri et al. [50] measured the
thermal conductivity of Nb and V at superconducting temperatures and investigated the eﬀect of
radiation on the thermal conductivity. The eﬀect of neutron irradiation on the thermal conductivity
of superconducting Nb behaves similarly as dislocation scattering to have the thermal resistance W
being proportional to T−2. The explanation is that neutron irradiation introduces interstitials in the
sample, where the condensation of interstitials increases the dislocations. In 1961, Mendelssohn
[51] examined diﬀerent aspects of the thermal conductivity of metals at low temperatures, i.e.,
impurity, strain, and irradiation. Electrons are mainly scattered by impurities [44], therefore they
aﬀect the thermal conductivity in both normal and superconducting state. Phonons are scattered by
dislocations, often formed by plastic deformation, and contribute to the lattice thermal conductiv-
ity. Therefore, strain only aﬀects the thermal conductivity in the superconducting state. The eﬀect
of neutron irradiation on the thermal conductivity of superconducting Nb behaves like phonon-
dislocation scattering. However, in the normal state thermal conductivity can also be reduced by
neutron irradiation, because vacancies were produced by irradiation [50].

Calverley et al. [52] systematically investigated the thermal and magnetic properties of Ta, Nb,
and V at liquid He temperatures. For the eﬀect of strain on thermal conductivity of Nb, they observed
reduced thermal conductivity following deformation. Their ﬁgures of thermal conductivity show
that the phonon peak temperature gradually increased with deformation until the peak disappeared
because of increased dislocation density. Connolly and Mendelssohn [53] measured the thermal
conductivity of Nb between 0.2 K and 4.2 K. They found a local maximum at about 0.2 Tc (T=1.85 K)
and analyzed the electron and lattice components of k. They concluded that contribution of phonon
conduction needs to be considered for T < 0.5Tc. Kuhn [54] measured the thermal conductivity
of Nb and investigated the eﬀect of irradiation, phonons scattered by dislocations, and phonons
scattered by electrons using the BRT [42] model. Thermal conductivity in the superconducting

13

state can exceed that of the normal conducting state for Nb after recrystalization. Wasim and
Zebouni [55] measured the thermal conductivity of Nb of intermediate purity (l/ξ0 ≈ 1, where l is
the electronic mean free path, and ξ0 is the coherence length) in the mixed states. The clean and
dirty limits of superconductors are deﬁned as the comparison between l and ξ0. If l << ξ0, it is
referred as dirty superconductors [56], otherwise, it is referred as clean superconductors. For dirty
superconductors, the impurities are usually high that can scatter with electrons to reduce the thermal
conductivity of superconductors. They found that at the local minimum of thermal conductivity,
both phonon and electron thermal conductivities decrease just above the lower critical ﬁeld (Hc1).
As a type-II superconductor, Nb has two critical magnetic ﬁelds, namely Hc1 and Hc2 [5].
When the applied magnetic DC magnetic ﬁeld exceeds Hc1, the normal conducting magnetic
ﬂux vortices penetrate the Nb surface. However, the Nb material around these vortices is still
in superconducting state. When the applied magnetic DC magnetic ﬁeld is greater than Hc2, the
magnetic ﬂux vortices penetrate the Nb surface completely and transform the superconductor into
normal conductor. When the applied magnetic ﬁeld lies between Hc1 and Hc2, this is the mixed
state. For Nb at 0 K, Hc1 = 1.08 × 104 A/m, Hc2 = 1.91 × 104 A/m [5].

Mittag [57] measured the Kapitza resistance and thermal conductivity of Nb from 1.3 K to 2.1
K, the Kapitza resistance is higher for a sample after annealing and chemical polishing than for a
sample that was only machined. Carlson and Satterthwaite [58] measured the thermal conductivity
of Nb between 0.3 K and 5 K. They found an anomalously high thermal conductivity (a factor
of 5 higher than BCS) between 0.3 K and 0.6 K and proposed it was due to a small second
energy gap. However, Anderson et al., [59] measured the thermal conductivity of single-crystal
Nb from 0.04 K to 4 K and found no evidence of a second energy gap. From another measurement
of four single-crystal Nb samples, Anderson and Smith [23] proposed a resonant like scattering
of thermal phonons lying near 5 × 1010 Hz, which is independent of stain and impurity. They
explained this feature to be a dynamic phonon-dislocation interaction.
In 1974, Anderson and
O’Hara [60] studied the lattice thermal conductivity of Nb as limited by the interaction of phonons
with electrons. They indicated that the relationship between the mean free path of transverse and

14

longitudinal acoustic phonons and temperature are similar, both are about 4 × 10−5T−1. Kes et al.
[61] measured the thermal conductivity of Nb considering crystal defect structures like dislocations,
grain boundaries, and dislocation loops due to neutron irradiation. They studied diﬀerent scattering
mechanisms with relaxation time approximation, including grain boundary scattering, point defect
scattering, dislocation scattering, and phonon electron scattering, based on BRT theory [42] and
Klemens model [44]. They found that dislocation loops introduced by neutron irradiation behave
like clusters of point defects, and irradiation can enhance the original dislocation term. The
relaxation times used in this paper are expressed as the following.

Grain boundary scattering:

−1
GB = 0.03γ
τ

2

2

α

vsNg

(2.11)

where the Grüneisen constant has been deﬁned in Eq.
2.8, α is the angle of tilt of the grain
boundary (α < π/4), vs is the sound velocity, and Ng is the number of grain boundaries crossing a
line of unit length.

Strain ﬁeld scattering:

−1
D = 0.033γ
τ

2b2Ndω

where ω is the phonon frequency.

Point defect scattering:

(2.12)

(2.13)

−1
p =
τ

4

3a3S2
Gπv

ω
3
s

where a3 is the volume per atom, G is the number of items per crystal, and S is the scattering
1 = (∆M/M)2/12, for
amplitude of the point defects. For the diﬀerence in mass of the impurities S2
the lattice distortion by the impurities, S2

2(∆R/R)2, where R is the lattice spacing.

2 = 3γ

Phonons scattered by electrons:

−1
pe = g(x, T)E xT

τ

15

(2.14)

where x = ω
kBT
term g(x, T) is a monotonically decreasing function of x, and E is a ﬁtting parameter.

, with  being the reduced Planck constant, kB being the Boltzmann constant. The

Moore and Satterthwaite [62] measured the thermal conductivity of pure Nb in magnetic ﬁeld
near upper critical ﬁeld Hc2. Measurement results conﬁrmed the theory that the mixed state
thermal conductivity varied with ﬁeld at the magnetic ﬁeld near Hc2, which was also veriﬁed by
Kes et al. [63] by measuring the thermal conductivity of Nb in the mixed state at diﬀerent ﬁeld
orientations. Mamyia et al. [64] measured the thermal conductivity of Nb with diﬀerent RRR of
6300, 2500, and 195. Niobium, as an intermediate coupling superconductor, behaves diﬀerently
to the electronic thermal conductivity due to electrons scattered by phonons from strong coupling
superconductors (e.g. Lead) and weak coupling superconductors (e.g. Tin). They characterized the
slope of the phonon scattering term of reduced thermal conductivity against reduced temperature
for intermediate coupling superconductor Nb to be 2.8, as compared with 1.5 for weak coupling
superconductors. Here the reduced thermal conductivity is deﬁned as kS/kN where kS is the thermal
conductivity at superconducting state and kN is the thermal conductivity at normal conducting
state. And the reduced temperature is deﬁned as T/Tc. Gladun et al. [65] measured the thermal
conductivity of single-crystal Nb of high purity (RRR=33000 2) at the temperature range from 0.05
K to 23 K. They found the thermal conductivity values in the range of six orders of magnitude
(k = 2 × 10−3W m-1 K-1at 0.05 K to k = 2.2 × 103W m-1 K-1at 9 K). The surface roughness
of the sandblasted sample was characterized and they concluded that the diﬀerence in thermal
conductivity of the sandblasted sample below 2 K may be due to the annealing of dislocations in
the surface layer or within the crystal due to sandblasting. Reiche and Pompe [66] investigated the
thermal conductivity and electrical conductivity of bulk niobium and niobium ﬁlms in the normal
state from 9 to 50 K, based on the deviations from the Matthiessen rule. They showed a slight
dependence of the electron-phonon scattering coeﬃcient on the impurity content of the sample.
An additional resistance term proportional to T5.5 was proposed in the temperature range from 20
to 50 K.

2RRR=33000 is the theoretical limit for Nb

16

Figure 2.6: A typical example of the thermal conductivity of single crystal superconducting Nb
sample with screw dislocations from 0.3 K to 5 K with diﬀerent levels of uniaxial strain, replotted
from [4]. Undeformed state refers to 0 Cauchy strain, and 3%, 7.3%, and 14.7% deformation
correspond to the same amount of Cauchy strain. The thermal conductivity at temperature lower
than 3 K decreases with the increase of strain. The phonon peak appears at 0% deformation and
3% at about 1.8 K and 2 K separately. There is nearly no phonon peak after 7.3% deformation and
14.7% deformation. The thermal conductivity data at temperatures higher than 3.5 K are the same
for diﬀerent amount of deformations.

17

00.511.522.533.544.550102030405060708090100Temperature (K)Thermal conductivity (Wm−1K−1)  Undeformed3% deformation7.3% deformation10.3% deformation14.7% deformationWasserbäch systematically measured the thermal conductivity of single-crystal Nb and Ta in
the temperature range 0.3 ≤ T ≤ 20 K . He considered the eﬀects of interstitial and substitutional
impurities [67], plastic deformation [4], phonons scattered by electrons [68], deformation at in-
termediate temperatures [3], and heterogeneity of the dislocation distribution [69] on the lattice
thermal conductivity of Nb. The thermal conductivity of Nb below 1 K was found proportion to T3,
but its magnitude decreased with increasing impurity content [67]. In 1978, Wasserbäch measured
the eﬀect of the temperature at which deformation occurs on the thermal conductivity of Nb spec-
imens with diﬀerent RRR values. He characterized the deformation as weak ( (cid:46) 2%), medium
(2% (cid:46)  (cid:46) 6%) and strong ( (cid:38) 7%), although these varied somewhat. Figure 2.6, similar to Fig.
1.1, shows the typical response of thermal conductivity of Nb to temperature and deformation. For
weak and medium deformation at 77 K and 195 K, the thermal resistance of phonons scattered by
dislocations was proportional to T−3. These dislocations were measured by transmission electron
micrographs (TEM) to be predominantly screw dislocations. Strong deformation at 195 K and
medium to strong deformation at 295 K and 470 K gave rise to thermal resistance proportional
to T−2. Deformation at 295 K and 470 K led to predominantly edge dislocations. Experimen-
tal investigation of phonons scattered by electrons [68] indicated that both the longitudinal and
transverse acoustic phonons were scattered strongly by electrons for Nb, while only longitudinal
acoustic phonons are scattered by electrons in Pippard’s theory [70]. Wasserbäch’s results agree
with the phonon mean free path investigation by Anderson et al. [60]. Single crystal Nb or Ta
plastically deformed at T = 355, 370, or 420 K demonstrated thermal resistance proportional to T−2
[69], which is consistent with [4] and with Klemens’s model [44] of phonon scattering by a static
distortion ﬁeld of dislocations.

Williams et al. [71] measured the thermal conductivity and electrical conductivity of Nb and
Nb-based alloys in the temperature range 80-400 K. They analyzed the data by considering phonon-
phonon and phonon-electron scattering in limiting the phonon thermal conductivity and compared
the phonon-electron thermal resistance with ﬁrst-principles calculation with good agreement. Kraﬀt
[72] measured the surface resistance, thermal conductivity and Kapitza resistance between Nb and

18

liquid He and showed that the above three parameters must be established to make predictions about
the role of thermal transport in determining the performance of SRF cavity. He also mentioned
that high thermal conductivity did stablize the cavity surface against local heating due to defects.
Padamsee [35] studied the inﬂuence of thermal conductivity on the breakdown ﬁeld of Nb cavities
by varying the RRR values in 40 diﬀerent tests. He found that the breakdown ﬁeld of Nb cavity is
proportional to the square root of the RRR. A new puriﬁcation technique [73] was used to improve
the thermal conductivity of Nb by removing the dominant interstitial impurity (oxygen) in Nb.
Hörmann [74] described the manufacturing process for superconducting niobium. He mentioned
that electron beam melting and hot and cold working may increase the commercial Nb RRR from
20-40 to 160-300, and the thermal conductivity from 8-10 W m-1 K-1to around 70 W m-1 K-1.
Kneisel et al. [75, 76] investigated the SRF performance of high ﬁeld cavities by comparing material
properties such as thermal conductivity, residual resistivity, and tensile behavior. Koechlin et al.
[77] measured the thermal conductivity of Nb from 1.5 to 20 K. They concluded that high purity
contributes to good thermal conductivity by electrons when T > 2.5 − 3K, however, low thermal
conductivity in the temperature range of 1.8-2.0 K is due to small grain dimensions. Moderate
temperature annealing (800 ◦C< T < 1200 ◦C) can increase the grain size from 50-70 µm to 400-
500 µm and the thermal conductivity at around 2 K. Boucheﬀa et al. [78] measured the Kapitza
resistance and thermal conductivity of Nb. They showed that thermal conductivity at T > 3 K
increases with RRR, however for T < 3 K, the thermal conductivity behavior becomes dependent
on other factors, particularly the grain size. Also, the Kapitza resistance of RRR≈ 200 is strongly
dependent on its surface treatment. Ikebe [79] measured the thermal conductivity of Nb based
alloys and ﬁtted the data with the Tewordt and Wölkhausen (TW model) [80] with good agreement.
Amrit [81] measured the thermal conductivity of poly-crystal Nb and investigated the eﬀect
of phonon-grain boundary scattering on the thermal conductivity. He showed that the thermal
conductivity decreases strongly as the number of grains increase up to a limit of 10. When the
number of grains exceeds 10, the anisotropy of thermal conductivity disappears. Singer et al. [13]
investigated the SRF performance of large grain cavities using both the basic material properties

19

and material availability, production, and preparation aspects. They obtained a gradient up to 41
MV/m at the quality factor Q0 = 1.4 × 1010.

Gurevich [32] studied the eﬀect of nonlinear surface resistance on the SRF thermal breakdown
and proposed a model of non-uniform thermal breakdown by macroscopic hotspots on the cavity
surface. Aizaz et al. [7] measured the thermal conductivity and Kapitaza conductance of Nb and
studied the eﬀect of plastic deformation on phonon transmission and Kapitza resistance. They
found the absence of phonon peak and the reduced Kapitza conductance are due to the applied
strain. Low temperature annealing (around 750 ◦C) did not restore the phonon peak. Kneisel
et al. [82] tested six single cell Nb cavities at TESLA. After appropriate surface treatments by
buﬀered chemical polishing and eletropolishing, they are able to reach the accelerating gradients
between 38 MV/m and 45 MV/m. Singer et al. [31] measured the RRR of Nb using DC and
AC methods and analyzed the advantages and disadvantages of these two methods. Basically, the
DC method has higher accuracy but is a destructive method only good for quality control of Nb
before start of cavity fabrication. The AC method preformed directly on the cavity surface and
makes it possible to estimate the RRR value, however, the accuracy is less than that of DC method.
Aizaz et al. [83] studied the thermal designs in SRF Nb cavities by considering the phonon peak
and Kapitza conductance. Strain can reduce the thermal conductivity at its phonon transmission
regime by 80%. Although low temperature annealing did not restore the phonon peak [7], moderate
temperature (around 1300 ◦C) annealing during titaniﬁcation process can restore the phonon peak
in the thermal conductivity curve.

Chandrasekaran et al. [8, 84, 85] measured the thermal conductivity of bi-crystal Nb with
diﬀerent RRR values and considered the inﬂuence of grain size and grain orientation, impurity, de-
formation, and heat treatment on the thermal conductivity of Nb. They proposed a novel parameter
estimation method [86] based on temperature and heat ﬂux measurements. The phonon peak was
found at close to 1.8 K [8] for nearly of all the undeformed samples. While plastic deformation
has the ability to destroy the phonon peak [8], heat treatment with appropriate temperature and
duration may restore it. Dhakal et al. [12] measured the thermal conductivity of single crystal

20

Nb and analyzed the parameters by ﬁtting with the model proposed by Koechlin and Bonin [41].
Chandra et al. [87] investigated the magnetic ﬁeld dependence of the thermal conductivity of Nb.
They found that the quasi-particle scattering from vortices dominates in the low ﬁeld, however,
the intervortex quasi-particle tunneling dominated in high ﬁelds. Mondal et al. [88] measured the
thermal conductivity of large grain niobium and investigated its dependence on trapped vortices in
a parallel magnetic ﬁeld. They found that the phonon peak at around 2 K disappeared due to the
presence of trapped vortices because of strong scattering of phonons by vortex cores.

To the best of our knowledge, the details of researchers’ contributions to experiments on thermal

conductivity of Nb are listed in Table 2.1.

Table 2.1: Chronology of measurements of the thermal conductivity of superconducting Nb. RRR
.

values marked with asterisks (*) are estimated using measured k(4.2 K) × 4

Year RRR

99.99%
99.99%
99.99%
36*
48-54*
52*
400*
56-40*
60.5-120

1950
1952
1955
1958
1960
1961
1960
1961
1962
1966 NA
1969
29
196
1970
1971
99.9%

T (K)
2-22
2-22
0.4-0.75
1-4.2
1-4.2
0.4-9
1-4.2
1-4.2
0.2-4.2
0-9.25
1.95-9.0
0.3-6
1.3-2.1

Contributions

w/ & w/o magnetic ﬁeld
w/ & w/o magnetic ﬁeld
k ∝ T3 with magnetic shielding
Dislocation,  = 0, 5%, 13.4%, 19.5%
Fast neutron irradiation
Dislocation, irradiation, RRR, and boundary
Plastic deformation & dislocation density
Measurements in single- and poly-crystal sample
Electronic and lattice contribution
Measurements in normal and superconducting metals
k at mixed state of intermediate purity
Anomalous k for T ≤ 0.6, possible 2nd energy gap
hKapitza, annealed and chemically polished

Ref.

[28]
[29]
[45]
[47]
[50]
[51]
[89]
[52]
[53]
[54]
[55]
[58]
[57]

Table 2.1 (cont’d)

21

Year RRR

26 & 2000

261-961

26-1800
100
15.6-24.3
195
15.6-24.3
8000*
300
185-1200
11.4-4910

1971
1972 NA
1973
1973 NA
1974
1974
1974
1974
1975
1977
1977
1978
1979
1983 NA
1983
500
25-2000
1985
1985
65-300
185-1200
1985
185
1985
1987
185-1200
25-350
1988
1991
40-250
1994
38-270
1996 N.A.

Table 2.1(cont’d)

Contributions
No evidence of 2nd energy gap

Dislocation, resonant phonon scattering
Neutron scattering measurement of phonon line width
kg, longitudinal and transverse phonons
Hydrogen impurity
Dislocation, irradiation, and grain boundary
Phonon scattering of inter. coupling superconductors

T (K)
0.04-4.2
1.79-2.0 Magnetic ﬁelds near HC2, electron mean free path
0.03-4
0-14
0.5-4
0.1-4.2
1-10
1-15
2.13-6.04 Magnetic ﬁeld orientation, mixed state
Surface roughness, phonon scattering
0.05-23
0.3-20
Interstitial and substitutional impurity
Phonon-dislocation scattering
0.3-20
9-50
Bulk Nb and Nb thin ﬁlms
kg at intermediate T
80-400
1.5-4
Thermal transport of metal Kapitza resistance
k on thermal breakdown ﬁeld
1-9
4-9
New puriﬁcation technique to improve k
Electron-phonon scattering
0.3-20
Deformation at intermediate temperatures
0.3-20
0.3-20
Heterogeneity of dislocation distribution
Electron beam melting, hot and cold working
2-10
1.5-20
Anneal, grain size, RRR
hKapitiza and k
1.4-2.15
1.2-4.2
Lattice k of Nb-Ta alloys

Ref.

[59]
[62]
[90]
[91]
[60]
[92]
[61]
[64]
[63]
[65]
[67]
[4]
[66]
[71]
[72]
[35]
[73]
[68]
[3]
[69]
[74]
[77]
[78]
[79]

Table 2.1 (cont’d)

22

Year RRR

2840-10500
673
200-500
230-300
280,470
230-300
104-196
224±8
280
61-158
104-196

2002
2006
2007
2007
2008
2010
2011
2012
2012
2012
2013
2017 N.A.

T (K)
0.03-3
1.6-2.1
0-Tc
1.6-4.2
0-Tc
1.6-4.2
1.5-4.5
1.5-Tc
2-12
1.8-5
1.5-4.5
0.1-0.6

Table 2.1(cont’d)

Contributions

Internal friction
Number of grains, grain boundary hKapitiza
Large grain Nb performance
Plastic deform., low T heat treatment, hKapitiza
Single crystal Nb, SRF performance
Phonon peak, hKapitiza
Heat treatment, phonon scattering
BCP, heat treatment, surface resistance
Magnetic ﬁeld, impurity
Trapped vortice, large grain Nb
Grain size, orientation, impurity, , H.T., kpp
Thin wires, k at low T deviates from BCS

Ref.

[93]
[81]
[6]
[7]
[82]
[83]
[85]
[12]
[87]
[88]
[8]
[94]

2.2 Models on thermal conductivity of Nb

Heat is carried by both the conduction electrons and the lattice vibrations in metals [37]. The
thermal conductivity of Nb can be modeled as consisting of electron transport of energy and
phonon transport of energy. These two components are additive and can be expressed as Eq. 2.3.
In normally conducting metals, the phonon part is usually negligible due to scattering of normal
electrons. However, in the superconducting regime, the formation of free electrons into Cooper
pairs leads to a reduction in the electron contribution to energy transport as well as a reduction in
scattering of phonons by electrons [39]. Therefore, the phonon contribution to thermal conduction
increases in signiﬁcance in superconducting materials.

Callaway [95] developed a phenomenological model for the lattice thermal conductivity at
low temperature by considering the diﬀerent scattering mechanisms of phonons, such as boundary

23

scattering, the normal three phonon process, impurity scattering, and Umklapp process, based on
the Boltzmann transport equation (BTE). At very low temperatures, the boundary scattering is
dominant, the normal process is small and can be neglected. The total lattice thermal conductivity
can be expressed as

T3Þ ΘD/T

0

k =

k3
B
3

kB
2
2π
vg

τ exp(x)x4
(exp(x) − 1)2 dx

(2.15)

where x = ω
is the non-dimensional frequency, vg is the group velocity, here taken as a
kBT
constant with the value of sound velocity (no dispersion considered), τ is the overall relaxation
time of diﬀerent scatterings, including phonon-phonon scattering, phonon-electron scattering,
phonon-impurity scattering, phonon-boundary scattering (phonon-grain boundary scattering), and
phonon-dislocation scattering, and ΘD is the Debye, here taken as ΘD=275 K for Nb.

The Callaway model assumes that all phonon modes are scattered independently of one another
and uses the Debye approximation. The Debye approximation assumes that there is no phonon
dispersion, in other words, the longitudinal and transverse acoustic polarizations behave identically.
Holland [96] extended Callaway’s model by treating transverse and longitudinal acoustic phonons
diﬀerently to estimate the lattice thermal conductivity. This model agrees well with the Callaway
model in the impurity scattering and boundary scattering regions. The Holland model improves
the accuracy of estimating the lattice thermal conductivity by considering the linear dispersion.
However, this model still considers the overall relaxation time. Moreover, as the model is under the
assumption of linear dispersion, it would be diﬃcult to estimate the lattice thermal conductivity of
materials with non-linear dispersion.

Klemens [44] did systematic research on the expressions of scattering relaxation times which
contribute to the lattice thermal conductivity, including phonon-phonon scattering, phonon-impurity
scattering [44], phonon-point defect scattering [97], and phonon-dislocation scattering [44]. Re-
garding to the lattice thermal conductivity of superconductors, Klemens and Tewordt [44] studied
the reduction of lattice thermal conductivity due to point defects.

The relaxation time is related to the phonon mean free path Λp as

24

τ =

Λp
vg

(2.16)

where vg is the phonon group velocity. In the case of phonon-electron scattering, the relaxation
time can be calculated as the phonon mean free path, which is the inverse of attenuation coeﬃcient
β according to Pippard’s theory [70].

τ =

Λp
vg

=

1
vg β

(2.17)

At liquid He temperatures (T < 4 K), the lattice thermal conductivity kg is limited by phonon-
electron scattering and phonon-boundary scattering in defect free single crystals [44, 45]. According
to Klemens [98], the electronic thermal conductivity is limited by electron-phonon scattering and
electron-impurity scattering.

BCS theory [39] is the ﬁrst microscopic theory of superconductivity which describes super-
conductivity as a microscopic eﬀect caused by the condensation of Cooper pairs [99], for which
they won the 1972 Nobel Prize. BRT [42] extended this theory to the thermal conductivity of
superconductors by treating the excited states as quasi-particles to apply to the BTE. Tewordt
[100] investigated the intrinsic electronic thermal conductivity of superconductors by introducing a
nonequilibrium part of the distribution function into the corresponding BTE. He found the ratio of
thermal conductivity in the superconducting state to that in the normal conducting state increases
monotonically with T/Tc with a limiting slope close to 1.62 at Tc. Tewordt and Wölkhausen [80]
established the theory of lattice thermal conductivity of high Tc superconductors by considering
phonon-boundary scattering, phonon-point defect scattering, phonon-strain ﬁeld scattering , and
phonon-electron scattering, based on the BCS theory. The expression is shown in the following.

kg(t) = A1t3Þ ∞

0

dx

x4ex
(ex − 1)2 × [1 + A2t4x4 + A3t4x2 + A4tx + A5txg(x, y)]−1

(2.18)

where t = T/Tc is the reduced temperature, x = ω
is the reduced frequency, A1 accounts for
kBT
the scattering of phonons by boundary, A2 is for the scattering of phonons by point defect, A3

25

represents the scattering of phonons by strain ﬁeld of sheetlike faults, A4 is for the scattering of
phonons by strain ﬁeld of dislocations, and A5 accounts for the scattering of phonons by electrons.
Connolly and Mendelssohn [53] measured the thermal conductivity of Nb and Ta below 1 K and
analyzed their data into its electronic and lattice components. Their analysis agrees well with BCS
theory. Anderson et al. [23] analyzed their thermal conductivity data of Nb by using Kinetic theory
and separating the phonon mean free path into phonon boundary scattering and phonon dislocation
scattering before and after deformation. Anderson and O’Hara [101] investigated the interaction
of phonons by electrons to contribute to the thermal conductivity of Nb, they concluded that the
mean free paths of transverse and longitudinal phonons are similar, which is close to 4 × 10−5T−1
(cm K), in the normal state. Amrit [81] developed a model to investigate the eﬀect of average
grain size, Kapitza resistance at grain boundaries and number of grains on the thermal conductivity
of poly-crystal Nb. The model predicted a strong spatial anisotropy in determining the thermal
conductivity for more than 10 grains.

Koechlin and Bonin [41] modeled the thermal conductivity of superconducting Nb by pa-
rameterizing the BRT [42] expression according to Bardeen-Cooper-Schrieﬀer (BCS) theory [39]
as

k = ke + kg = R(y)(cid:104) ρ

+ AT2(cid:105)−1

LT

(cid:20)

(cid:21)−1

1

1

+

D exp(y)T2 +

BΛT3

(2.19)

where the two terms in ke are due to electron-impurity scattering and electron-phonon scattering,
respectively, and the two terms in kg are due to phonon-electron scattering and phonon-boundary
scattering, respectively. The term R(y) quantiﬁes the condensation of normal conducting electrons
into Cooper pairs [42].

Chandrasekaran et al. [84] improved this model by estimating directly from the temperature and
heat ﬂux measurements instead of estimating from the intermediate thermal conductivity results.
This new estimating method groups coeﬃcient of T n in the model as diﬀerent βs in the following

β1 =

ρ295K
LRRR,

β2 = a,

β3 =

1
D,

β4 =

1
BΛ

,

β5 =

∆(T)
kBTc

(2.20)

Equation 2.19 can be redeﬁned as

26

+ β2T2(cid:21)−1

+

(cid:20)

(cid:20) β1

T

k = R(y)

β3

exp(y)T2 +

β4
T3

(cid:21)−1

with

y = β5

Tc
T

(2.21)

(2.22)

Xu et al. [43] further improved this model by including the phonon dislocation scattering
separately from boundary scattering in examining the lattice thermal conductivity. The model can
be expressed as

k = R(y)(cid:104) ρ

+ AT2(cid:105)−1

LT

(cid:20)

+

1

D exp(y)T2 +

1
BΛT3 + Wd

(2.23)

(cid:21)−1

where Wd = KNd/T2 is the thermal resistance of phonon dislocation scattering expressed in
Eq. 2.4. Based on the properties of Nb, Klemens expressed Wd [44] for randomly distributed
dislocations as

WdT2
Nd

= 3.05 × 10−15 m3K3/W

(2.24)

According to Wasserbäch [4], Bross analyzed Wd in Cu for edge and screw dislocations. Bross
obtained results to be 1.67 and 1.26 greater for edge and screw dislocations, respectively, than
the values evaluated by Klemens of Eq. 2.8 [44]. Then the following equations are assumed for
modeling the eﬀect of dislocations to Nb.

(cid:33)edge
(cid:33)screw

(cid:32)
(cid:32)

WdT2
Nd
WdT2
Nd

= 5.08 × 10−15 m3K3/W

= 3.85 × 10−15 m3K3/W

(2.25)

(2.26)

Comparisons of data from [4] with ﬁtting of the model are shown in Fig. 3.6 for a sample
with RRR=250 after uniaxial straining of 14.7%. Fitting including the dislocation term improves
comparison qualitatively and quantitatively with the experimental results [4], especially for samples
after deformation and at low temperature. This improved agreement suggests that the observation
by Koechlin and Bonin of a deviation in the ﬁtting of their low temperature data using Eq. 2.19

27

Figure 2.7: Comparison between ﬁtting with or without the dislocation term for deformed sample
[4]. The estimated value of dislocation density is Nd = 3.83 × 1014 m−2.

[41] may have been due to the lack of a phonon-dislocation scattering term. It also supports the
idea that phonon-dislocation scattering contributes signiﬁcantly to the thermal conductivity of Nb
at low temperatures.

In Pippard’s theory of free electrons in an isotropic metal, the longitudinal waves interact with
the conduction electrons due to the density variation of the lattice contributing to the electric ﬁeld
of the space charge [102]. The transverse waves cause no density variation, therefore, no space
charge. The attenuation coeﬃcients for longitudinal and transverse waves can be written as

βj =

NmvF

Dmvj ΛeFj(z)

(2.27)

where N is the number of conduction electrons per unit volume, m is the electronic mass, vF
is the Fermi velocity, Dm is the mass density, Λe is the electron mean free path, and Fj(z) is a

28

00.511.522.53Temperature (K)02468101214Thermal conductivity (Wm-1K-1)Fit with dislocation termExperimental data (14.7% deformation)Fit without dislocationTable 2.2: Numerical integration results for I1(z) and I2(z) [1]

TΛe (cm K)

10−8
10−7
7 × 10−7
9 × 10−7
10−6
1.2 × 10−6
1.5 × 10−6
2 × 10−6
3 × 10−6
4.5 × 10−6
6 × 10−6
10−5
10−4

I1(z)
2471
247.3
37.34
29.92
27.36
23.58
19.89
16.30
12.90
10.78
9.786
8.666
7.336

I2(z)
1493
150.9
30.99
28.41
27.77
27.25
27.59
29.57
35.58
46.20
57.44
88.23
79.38

function of phonon wave vector q and Λe (z = qΛe), j denotes the longitudinal (l) or transverse (t)
phonon branch.

For the longitudinal branch

For the transverse branch

Fl(z) =

z2 arctan(z)
3(z − arctan(z))

Ft(z) = 1 − 3

2z2((y

2 + 1)arctan(z)

z

− 1)

(2.28)

(2.29)

With these attenuation coeﬃcients, the lattice thermal conductivity of metals considering

phonon scattering with electrons only is obtained from Eq. 6.1

where

kpe =

Dmk3

BT2

3NmvF2) ∗ Ij(z)

(π

Þ ∞

(2.30)

Ij(z) =

π
6

(2.31)
is the reduced frequency. The integrals Ij(z) have been tabulated by Lindenfeld

0

zFj(z)x3 exp(x)
(exp(x) − 1)2 dx

where x = ω
kBT
and Pennebaker [103] in Table 2.2.

The number of free electrons decrease exponentially with temperature when it comes into
superconducting state, because of the condensation of electrons into Cooper pairs. Assuming the

29

number of free electrons at the normal conducting state is N0, the expression for the number of free
electrons at superconducting state is

Equation 2.30 becomes

N = N0 exp(−y)

Dmk3
B

3N0mvF2) × Ij(z) exp(y)T2

kpe =

(π

(2.32)

(2.33)

Dmk3
B

(π

The front part of the above equation

3N0mvF 2)×Ij(z) corresponds to the parameter D in
Koechlin and Bonin’s model [41] for phonon-electron scattering. The value of this parameter is
calculated to be from 6.85×10−3 to 7.55×10−3 WK3m−1 in the superconducting temperature range.
This value is within 10% of the value D obtained from ﬁtting with experimental data. It should be
mentioned that there is no adjustable parameters in this model.

Table 2.3: Research progress on modeling k of superconducting Nb

.

Year

Principle Investigator

Contributions

Wilson [37]
Klemens [44]

BCS [39]

Klemens [38]
Callaway [95]

1954
1955
1957
1958
1959
1959
1960
1962
1963
1963
1964 Klemens and Tewordt [98] Phonon-defect scattering for superconductors

Theory of thermal conductivity of metals
Phonon-impurity and phonon-dislocation scattering
Superconducting theory
Scattering relaxation time in dielectrics
Model of conduction in dielectrics
Theory of Superconductivity
Phonon-point defect scattering
Electronic and lattice contribution
Consider linear dispersion
Theory of electronic thermal conductivity

Klemens [97]
Connolly [53]
Holland [96]
Tewordt [100]

BRT [42]

Cont’d on next page

30

Table 2.3 – Cont’d from previous page

Year

Principle Investigator

Contributions

1974
1974
1974
1978
1985
1989
1993
1996 Koechlin and Bonin [41]
2006
2010 Chandrasekaran et al.[84] Estimation from heat ﬂux and temperature
2017

Mean free path of scatterings
Longitudinal and Transverse phonons
Details of scattering relaxation time of Nb
Phonon-dislocation scattering
Phonon-electron scattering
Thermal conductivity of high Tc superconductors
Theory of k of High Tc superconductor
Parameter estimation of thermal conductivity of Nb
Grains, Grain boundaries, Kapitza resistance

Phonon-dislocation scattering explicitly

Amrit et al.[81]

Anderson [23]
Anderson [101]

Kes [61]

Wasserbach [4]
Wasserbach [3]
Tewordt [80]
Goodson [104]

Xu et al. [43]

2.3 Conclusion

The thermal conductivities of superconducting Nb have been measured and modeled since
the 1950s. Even so, there are some important omissions in our understanding of the relation-
ships between k and other material properties including the eﬀect of dislocations, impurities,
and boundaries. This thesis is to explicitly examine these eﬀects on the thermal conductivity of
superconducting Nb and Ta.

31

CHAPTER 3

EFFECT OF DEFORMATION ON THE THERMAL CONDUCTIVITY AND PHONON

PEAK TEMPERATURE OF NB

3.1 Introduction

Manufacturing SRF cavities from Nb sheets requires large deformation that increases dislocation
density [105], which has been shown to reduce the thermal conductivity of superconducting large
grain Nb [3, 4, 8, 23, 57, 61]. Mendelssohn et al.[47] were the ﬁrst researchers to attempt to
quantify the phonon conduction in superconductors using the dislocation density Nd by measuring
the thermal conductivity of superconducting Pb and Nb. A model from Van Bueren [48] was used
to calculate the dislocation density for face-centered cubic (FCC) lattices (e.g., Pb), and the model
of Klemens [44] was used to account for the temperature dependent thermal conductivity due to
dislocations for body-centered cubic (BCC) materials (e.g., Nb). He found a local maximum thermal
conductivity at 2.4 K for an unstrained Nb sample and the peak disappeared after 5% deformation.
Wasserbäch measured the thermal conductivity of Nb after uniaxial straining of up to 22.2% [4].
An example of these data for a single specimen for which its conductivity was measured after
each increment of deformation as replotted in Fig. 2.6. The thermal conductivity for temperatures
colder than 3 K decreases with increasing deformation. Of particular note is the phonon peak in
conductivity kpp at approximately 2 K that decreases with increasing deformation. Wasserbäch
examined the eﬀect of deformation on conduction [3] by using a relaxation time approximation
according to the Callaway model [95] with the relaxation time expression from Klemens [44].
Scattering of phonons by electrons, which is a signiﬁcant factor at the working temperature of SRF
cavities, was not included in the analysis. Chandrasekaran [8] measured the eﬀect of deformation
on k and quantiﬁed the role of subsequent heat treatment on the recovery of phonon peak and
the decrease in dislocation density. A phonon peak that has disappeared after deformation can be
partially restored with heat treatment of appropriate temperature and duration [85]. In analyzing

32

Figure 3.1: Measured phonon peak temperatures Tpp for undeformed, deformed, and annealed
specimens as a function of the ratio of the thermal conductivity at the phonon peak kpp to the local
minimum thermal conductivity klm.

the eﬀect of deformation on conductivity, Chandrasekaran used only kpp to estimate the dislocation
density. Koechlin and Bonin [41] used a simpliﬁed equation based on the BRT model [42] (see Eq.
2.19) to ﬁt the experimental results. This equation was reparameterized by Chandrasekaran [84]
for analysis and to reduce uncertainties in estimating the thermal conductivity of Nb. However,
neither of these equations considered the scattering of phonons by dislocations, which might be the
reason that Koechlin and Bonin noted discrepancies in ﬁtting of k at temperatures lower than 2 K
[41].

Figure 3.1 shows reported values of 1.72 ≤ Tpp ≤ 2.35 K for a number of studies of supercon-
ducting Nb [4, 7, 8, 12, 23, 61, 64, 78, 92] as a function of kpp/klm, where klm is deﬁned as the
thermal conductivity at the local minimum, it typically appears at about T=3 K for Nb. Previous
studies have indicated that kpp is a function of dislocation density. This new observation suggests

33

11.522.531.71.81.922.12.22.32.42.5kpp / klmTpp (K)  Wasserbach − UndeformedAizaz − UndeformedChandrasekaran − UndeformedWasserbach − DeformedChandrasekaran − DeformedAnderson − DeformedChandrasekaran − AnnealedAnderson − Annealed Boucheffa − Annealed Mamyia − AnnealedO’Hara − AnnealedDhakal − Annealedthat both kpp and Tpp are functions of the dislocation density, and thus the deformation of Nb. This
further suggests that the scattering of phonons by dislocations should be accounted for explicitly
in the models of thermal conductivity. Figure 3.1 also shows the little discussed observation that
the temperatures at which kpp occurs increases as kpp decreases. Typically, the temperature of the
phonon peak Tpp of superconducting Nb is cited as occurring in the range 1.8 ≤ Tpp ≤ 2 K, but
studies do not note a systematic variation. This addition of the dislocation term, and some of its
implications, is considered here.

3.2 Analysis

The results in Fig. 3.1 represent measurements from several studies for as received, deformed,
and annealed specimens. The temperature Tpp is plotted as a function of the ratio kpp/klm.
Chandrasekaran [8] showed that kpp/k3, where k3 is the thermal conductivity of 3 K, correlates
inversely with dislocation density for a given specimen having undergone deformation or heat
treatment. Although there is scatter in the data in Fig. 3.1, resulting from various sample processing
histories and experimental techniques, there is a clear trend of Tpp decreasing with increasing value
of kpp/klm, and thus with decreasing dislocation density. Samples tested before deformation or
after heat treatment have larger values of kpp/klm. For T > 3 K, the eﬀect of dislocation density
can be neglected because the electron contribution dominates the thermal conductivity at this
temperature range.

Figure 3.1 further shows that undeformed samples typically have colder Tpp, but greater kpp.
For deformation typically less than  =3% strain, kpp decreases and Tpp increases. For uniaxial
straining greater than 3%, the phonon peak disappears in most samples because of the increase
of thermal resistance due to the scattering of phonons by dislocation. High purity samples,
however, maintain a phonon peak after greater deformation (e.g., the sample shown with RRR=1200
maintains a phonon peak after 4% uniaxial straining [4]). Deformation not only aﬀects the value of
thermal conductivity, but also has an apparent eﬀect on Tpp. Heat treatment that causes dislocation
annihilation (recovery) can reduce dislocation content [12] to partially or fully restore the phonon

34

peak, depending on the annealed temperature and duration. Chandrasekaran [8] showed that 1000
◦C for 4 hours is enough to nearly recover the phonon peak to that of the undeformed state.

As mentioned above, the scattering of phonons by dislocations should be included in the model,
especially for Nb following deformation. Inspired by Klemens [44], a dislocation term may be
added to Eq. 2.19 to account for the eﬀect of dislocation as in Eq. 2.4.

3.2.1 Scaled sensitivity coeﬃcient

To examine the eﬀects of diﬀerent scattering mechanisms, the scaled sensitivity coeﬃcients of
phonons scattered by boundaries, normal electrons, and dislocations are plotted with temperature.

3.2.1.1 Scaled sensitivity coeﬃcient of boundary scattering

The scaled sensitivity coeﬃcients are calculated as the product of the scattering parameters with
the partial derivative of thermal conductivity over the parameters. Therefore, the scaled sensitivity
coeﬃcient has the same unit as the the thermal conductivity.

(cid:20)

(cid:21)−2

(cid:21)−2

(3.1)

(3.2)

χB = B ∂k
∂B

=

1

BΛT3

1

D exp(y)T2 +

1
BΛT3 +

KNd
T2

where χB is the scaled sensitivity coeﬃcient due to phonons scattered by boundaries.

3.2.1.2 Scaled sensitivity coeﬃcient of normal electron scattering

Similarly, the scaled sensitivity coeﬃcient of phonons scattered by electrons is shown as the
following

(cid:20)

1

1

χD = D ∂k
∂D

=

D exp(y)T2

D exp(y)T2 +

1
BΛT3 +

KNd
T2

where χD is the scaled sensitivity coeﬃcient due to phonons scattered by electrons.

35

Figure 3.2: The scaled sensitivity coeﬃcients of phonons scattered by boundary (parameter B) as
a function of temperature. The B values vary from 2 × 103 Wm−2K−4 to 10 × 103 Wm−2K−4.

3.2.1.3 Scaled sensitivity coeﬃcient of dislocation scattering

The scaled sensitivity coeﬃcient of phonons scattered by dislocations is shown as

(cid:20)

(cid:21)−2

(3.3)

χN = Nd

∂k
∂Dd

= Wd

1

D exp(y)T2 +

1
BΛT3 +

KNd
T2

where χN is the scaled sensitivity coeﬃcient due to phonons scattered by dislocations.

The scaled sensitivity coeﬃcient of phonons scattered by boundaries is plot in Fig. 3.2. The
values of D and Nd are kept the same and the B values vary from 2 × 103 Wm−2K−4 to 10 × 103
Wm−2K−4 to investigate the eﬀect of phonons scattered by the physical boundaries. Similarly,
for phonons scattered by normal electrons, the values of B and Nd are kept the same and the D
values vary from 2 × 10−3 WK3m−1 to 10 × 10−3 WK3m−1. The scaled sensitivity coeﬃcient
of phonons scattered by boundaries is plot in Fig. 3.3. For phonons scattered by dislocations, the

36

123456789Temperature (K)01020304050Scaled Sensitivity coefficient B (Wm-1K-1)B=2×103 Wm-2K-4B=4×103 Wm-2K-4B=6×103 Wm-2K-4B=8×103 Wm-2K-4B=10×103 Wm-2K-4Figure 3.3: The scaled sensitivity coeﬃcients of phonons scattered by normal electrons (parameter
D) as a function of temperature. The D values vary from 2×10−3 WK3m−1 to 10×10−3 WK3m−1.

values of B and D are kept the same and the Nd values vary from 2 × 1012 m−2 to 10 × 1012 m−2.
Taking the example of screw dislocations, the scaled sensitivity coeﬃcient of phonons scattered by
dislocations is plotted in Fig. 3.4.

Comparing Fig. 3.2 and Fig. 3.3 shows that χB and χD nearly have the same shape and
magnitude, but of the maximums occurs at diﬀerent temperatures. Figure 3.4 shows χN is of
diﬀerent magnitude and sign. The diﬀerent sign is because the lattice k decreases with Nd, while
the lattice k increases with parameter B (i.e., specularity) and D (i.e., free electrons). It can be also
seen from the ﬁgures of scaled sensitivity coeﬃcients that the maximum occurs at about T=1.5 K
for parameter B, higher than 2 K for parameter D, and T ≈ 1.8 K for parameter Nd. This observation
conﬁrms that boundary scattering dominates the lattice k at the lowest temperature, dislocation
scattering aﬀects the lattice k at about the phonon peak temperatures (i.e. 1.6 K < T < 2.2 K), and

37

123456789Temperature (K)05101520253035Scaled Sensitivity coefficient  D (Wm-1K-1)D=2×10-3 WK3m-1D=4×10-3 WK3m-1D=6×10-3 WK3m-1D=8×10-3 WK3m-1D=10×10-3 WK3m-1Figure 3.4: The scaled sensitivity coeﬃcients of phonons scattered by dislocations (parameter Nd)
as a function of temperature. The Nd values vary from 2 × 1012 m−2 to 10 × 1012 m−2.

the phonons scattered by normal electrons start to aﬀect the lattice k at T > 2 K.

3.3 Results and Discussion

Comparisons of data from [4] with ﬁtting of Eq. 2.4 are shown in Fig. 3.5 for an undeformed
sample with RRR=250 and Fig. 3.6 for the same sample after uniaxial straining of 14.7%. The
diﬀerences in thermal conductivity between measurement data and the ﬁtting results are also plotted
for samples before and after deformation, to examine the accuracy of the model, as shown in Fig.
3.7 and Fig. 3.8. The residuals are similar in magnitude for the undeformed specimen in Fig. 3.7,
because the thermal resistance due to phonons scattered by dislocations is negligible compared to
that for phonons scattered by boundaries for samples before deformation. However, the residuals are
signiﬁcant in Fig. 3.8, but the ﬁt that includes the dislocation term has residuals close to 0. The ﬁt

38

123456789Temperature (K)-20-15-10-50Scaled Sensitivity coefficient N (Wm-1K-1)Nd=2×1012 /m2Nd=4×1012 /m2Nd=6×1012 /m2Nd=8×1012 /m2Nd=10×1012 /m2Figure 3.5: Comparison between ﬁt with or without dislocation term for an undeformed sample
from [4]. The estimated value of dislocation density is Nd = 4.67× 1012 m−2. Both the ﬁtted lines
for temperatures lower than 1 K and higher than 2.5 K agree well with the experimental data. At
1.5 K < T < 2 K, ﬁtting with the dislocation term has k values lower than the experimental results,
while ﬁtting without dislocation term has k values higher than the experimental results. The ﬁt
with or without the dislocation term provides similar accuracy for undeformed samples.

with the dislocation term included improves qualitatively and quantitatively with the experimental
results [4], especially for samples after deformation and at low temperature. This improved
agreement suggests that the observation by Koechlin and Bonin of a deviation in the ﬁtting of their
low temperature data [41] may have been due to the lack of a phonon-dislocation scattering term in
their model. It also supports the idea that phonon-dislocation scattering contributes signiﬁcantly
to the thermal conductivity of Nb at low temperatures.

Several other sets of data from Wasserbäch [4] were ﬁtted using the above method to examine
the role of dislocation density and its eﬀect on the thermal conductivity of Nb. These ﬁts are shown

39

00.511.522.53Temperature (K)0102030405060Thermal conductivity (Wm-1K-1)Fit with dislocation termUndeformed experimental dataFit without dislocation termFigure 3.6: Comparison between ﬁtting with or without dislocation term for deformed sample
[4]. The estimated value of dislocation density is Nd = 3.83 × 1014 m−2. Both the ﬁtted lines
for temperatures lower than 1 K and higher than 2.5 K agree well with the experimental data.
At 1 K < T < 2.5 K, ﬁtting with dislocation term matches well with the experimental results,
while ﬁtting without the dislocation term deviates from the experimental results. The ﬁt with the
dislocation term improves the accuracy for samples after deformations.

in Fig. 3.9 for screw dislocations and Fig. 4.8 for edge dislocations. Wasserbäch [4] calculated
the dislocation density for sample K6 to be on the order of magnitude of 1014 m−2 after the
last deformation (14.7%), without including phonon-electron scattering. The dislocation density
extracted from the ﬁt for the 14.7% deformation case is 3.83 × 1014 m−2. The dislocation density
reaches 1014 m−2 after 10% deformation 1.57×1014 m−2. These are in reasonable agreement with
values given by Wasserbäch [4], where he mentioned that the mean value of the dislocation density
after transmission electron microscopy (TEM) measurement is in the range of 0.8 to 1 ×1014 m−2.
Using the enhanced model of Eq. 2.4, the eﬀect of the magnitude of dislocation density on the

40

00.511.522.53Temperature (K)02468101214Thermal conductivity (Wm-1K-1)Fit with dislocation termExperimental data (14.7% deformation)Fit without dislocationFigure 3.7: Diﬀerences between measurement data and the ﬁt results of thermal conductivity with
or without the dislocation term for an undeformed sample [4]. Here the diﬀerences, which are also
called the residuals, are calculated as km − k f , where km is the measured thermal conductivity and
k f is the ﬁtted data.

thermal conductivity is studied. It is found that the phonons scattered by dislocations only aﬀects
the thermal conductivity when the dislocation density reaches a certain magnitude, e.g., 1012 m−2,
shown in Fig. 3.11 by ﬁtting the experimental results from [8] and varing the dislocation density in
the model. The best ﬁt is at Nd < 1012 m−2. However, the plots in Fig. 3.11 almost overlap with
each other for Nd < 1012 m−2, indicating the scattering of phonons by dislocations is not important
at these low values of dislocation density. The reason appears to be that phonon-boundary scattering
dominates the heat conduction at low temperatures [46], and that the thermal resistance of phonons
scattered by dislocations for Nd < 1012 m−2 is much smaller than that of boundary scattering, as
shown in Fig. 3.12. A threshold of dislocation density for contributing the thermal conductivity of

41

00.511.522.53Temperature (K)-8-6-4-202468Residuals of k  (Wm-1K-1)Fit w/ Nd termFit w/o Nd termFigure 3.8: Diﬀerences between measurement data and the ﬁt results of thermal conductivity with
or without the dislocation term for a sample after 14.7% deformation [4]. Here the diﬀerences,
which are also called the residuals, are calculated as km − k f , where km is the measured thermal
conductivity and k f is the ﬁtted data.
Nb is established here to be Nd = 1012 m−2 using the proposed model.

The ratio of dislocation density after deformation Nd to that before deformation Nd0 is plotted as
a function of deformation (i.e., uniaxial straining), shown in Fig. 3.13 for samples of varying RRR
from [4]. This ratio increases with increasing deformation, the slope for samples with predominantly
edge dislocations is larger than that for those with predominantly screw dislocations, implying
that edge dislocations have greater inﬂuence on thermal conductivity. Note that the undeformed
dislocation density is diﬀerent for each sample, perhaps due to diﬀerent RRR. For example, the best
ﬁt of the undeformed dislocation density for sample H2 (RRR=1200) is 6.11× 1010 m−2, however,
the values for the other three samples are approximately 1012 m−2 with similar RRR values for

42

00.511.522.53Temperature (K)-1-0.500.511.52Residuals of k (Wm-1K-1)Fit w/ Nd termFit w/o Nd termFigure 3.9: Thermal conductivity of a sample from [4] (an example of predominantly screw
dislocation) using curve ﬁtting including the eﬀect of dislocation. Here the dislocation density is
estimated from Eq. 2.4 for the Cauchy strain  =0%, 3%, 7.3%, 10.3%, and 14.7%, respectively.
Wasserbäch [4] provided a mean value of the dislocation density to be 0.8 − 1 × 1014 m−2, and the
dislocation density measured at 14.7% is on the order of magnitude of 1014 m−2.

185, 250, and 350.

3.4 Conclusion

For high performance of SRF applications, large values of thermal conductivity are desired for
improved thermal stability. However, higher working temperature might yield signiﬁcant savings
in the total energy required for SRF cavity operations. Therefore, optimization of the phonon peak
temperature and the value of thermal conductivity at that temperature becomes necessary for the
thermal design of SRF cavities.

43

012345020406080100Temperature (K)Thermal conductivity (Wm−1K−1)  Undeformed Nd=0.467×1013 /m23% deformation Nd=1.47×1013 /m27.3% deformation Nd=6.46×1013 /m210.3% deformation Nd=15.7×1013 /m214.7% deformation Nd=38.3×1013 /m2Figure 3.10: Thermal conductivity of a sample from [4] (an example of predominantly edge
dislocation) using curve ﬁtting including the eﬀect of dislocation. Here the dislocation density is
estimated from Eq. 2.4 for the Cauchy strain  =0%, 4%, and 8% respectively. Wasserbäch [4]
provided a mean value of the dislocation density to be 1− 2× 1014 m−2, and the dislocation density
measured at 8% is on the order of magnitude of 1014 m−2.

Analysis of the thermal conductivity of superconducting Nb shows that in addition to kpp
decreasing after deformation, there is an increase in Tpp. The proposed model adds a phonon–
dislocation scattering term that improves the accuracy of ﬁts to experimental results, especially for
samples after deformation. The proposed model can also be used to infer the dislocation density
from measurements of k, as shown in Fig. 3.13. Results show that a threshold of dislocation density
exists below which there is no signiﬁcant eﬀect on the thermal conductivity. A dislocation density
smaller than 1012 m−2 has little contribution to the thermal conductivity of superconducting Nb,
buy boundary scattering dominates at low temperatures, as shown in Fig. 3.11.

44

01234550100150200250300350Temperature (K)Thermal conductivity (Wm−1K−1)  Undeformed Nd=0.0455×1013 /m2 4% deformation Nd=1.93×1013 /m2 8% deformation Nd=19.3×1013 /m2Figure 3.11: The eﬀect of dislocation density on thermal conductivity of Nb at temperature between
1.5 K and 4.5 K. Equation 2.4 is used to ﬁt the experimental results from [8]. The results are obtained
by using the ﬁtted parameters (e.g., B, D, a) and changing Nd.

45

1.522.533.544.50510152025303540Temperature (K)Thermal conductivity (Wm−1K−1)  Experimental dataNd = 1×1010 /m2Nd = 1×1011 /m2Nd = 1×1012 /m2Nd = 1×1013 /m2Nd = 1×1014 /m2Figure 3.12: Comparison between phonon boundary thermal resistance and phonon dislocation
resistance for diﬀerent dislocation density by using the ﬁtted parameters (e.g., B, D, a) and changing
Nd.

46

01234510−610−410−2100102104Temperature (K)Thermal resistance (mKW −1)  Boundary resistanceDislocation resistance Nd=1×1010 /m2Dislocation resistance Nd=1×1011 /m2Dislocation resistance Nd=1×1012 /m2Dislocation resistance Nd=1×1013 /m2Dislocation resistance Nd=1×1014 /m2Figure 3.13: The ratio of dislocation density after deformation to that before deformation as a
function of uniaxial straining, estimated by ﬁtting the thermal conductivity from experimental
results [4]. Samples H1, K6, H2, and K10 are deformed at temperatures of 77 K, 195 K, 295
K, and 1470 K, respectively. H1 and K6 had predominantly screw dislocations, H2 and K10 had
predominantly edge dislocation.

47

0510152025100101102103104Deformation (%)Nd / Nd0  Sample H1   RRR=350Sample K6   RRR=250Sample H2   RRR=1200Sample K10 RRR=185MONTE CARLO MODEL OF THE LATTICE THERMAL CONDUCTIVITY OF NB

CHAPTER 4

4.1 Introduction

The BTE has been successfully used to model phonon transport in both macro- and micro-
scale heat conduction [95, 96, 106–108]. Solutions of the BTE are either through deterministic
approaches [109–112] or stochastic methods (i.e., Monte Carlo method) [9, 113, 114]. However,
in the deterministic methods, the dispersion relationship has been omitted and the transverse and
longitudinal polarization of the acoustic phonons has been replaced by a single eﬀective polarization
branch. The dispersion relation is important for modeling thermal transport because it provides
details of the distribution of the properties of each phonon (i.e., frequency, wave vector, and
group velocity).
In addition, in the deterministic methods, complex geometries result in high
computational costs and the computational time increases exponentially with increasing dimension
(i.e., from 2D to 3D). Moreover, nonlinear scattering events are also diﬃcult to incorporate without a
relaxation time approximation because the individual scattering events cannot be treated in isolation,
where the relaxation time is deﬁned as a time constant during which the distribution function relaxes
to its equilibrium state. The stochastic Monte Carlo method is robust for modeling the dispersion
of phonon transport from micro-scale to macro-scale. It is well suited for complex structures and
multi-dimensions, and compared with molecular dynamics, it can reduce computational cost with
large model size [115].

The Monte Carlo method has been widely used for phonon and electron transport [9, 113, 114,
116–128]. This method was ﬁrst used to solve the BTE in modeling hot electrons in metal ﬁlms
[129]. Since then, it has been widely used in studying drift velocity and diﬀusion coeﬃcients
of charge transport in semiconductors [130], electron-electron interactions in GaAs [131], and
thermal conductivity of thin ﬁlms [114, 124], nanowires [118, 122], and nanotubes [116]. The
Monte Carlo method has been successfully used to study ballistic phonon transport based on the

48

Debye model by assuming the same propagating speed and average lifetime interactions among
phonons [113]. The temperature proﬁle as a function of time evolution was predicted using these
assumptions. Mazumder and Majumdar [114] presented a comprehensive solution of the BTE
by considering phonon dispersion and polarization. Their simulation results agree well with the
experimental results for Si at temperatures between 20 K and 300 K. Mittal and Mazumder [124]
extended this model by including optical phonons, resulting in better matching with experimental
observations for the thermal conductivity of Si thin-ﬁlms. McGaughey and Jain [123] predicted
the thermal conductivity of nanoscale Si thin ﬁlms with arbitrary geometries. Lacroix et al. [9]
developed a distribution function by considering the collision processes with respect to energy
conservation during phonon scattering events in thin ﬁlms. Their results provided good agreement
with experimental results for phonon transport in both the ballistic and diﬀusion regimes. Chen
et al. [118] modeled the phonon transport in single crystal Si nanowires by using a genetic algo-
rithm including energy and momentum conservation in modeling the phonon-phonon normal and
Umklapp scattering processes. Jeng et al. [120] modeled the thermal conductivity in nanoparticle
composites of Si-Ge. They reproduced the earlier reported BTE results to validate their Monte
Carlo model and then used the validated simulation method to study nanoparticle composites and
showed that the thermal conductivity of nanoparticles can be smaller than that of the bulk value.
Péraud et al. [126] proposed an energy-based variance-reduced Monte Carlo method. This method
only needs to model the deviation in the energy of a particle population from its nearby equilibrium,
with the equilibrium described analytically. This technique saves signiﬁcant computational cost as
compared with standard Monte Carlo methods.

4.2 Boltzmann Transport Equation

The general form of BTE for phonon can be written as [114]

(cid:20) ∂ f

∂t

+ vg · ∆ f =

∂ f
∂t

(cid:21)

scat

49

where f(r, q, t) is the distribution function of an ensemble of phonons, which is related with
position r, time t, and wave vector q, and vg is the phonon group velocity. The left hand side of

(4.1)

this equation represents the change of the distribution function due to motion or drift, and the right
hand side represents the change due to collision or scattering. However, due to the nonlinear nature
of scattering events, the relaxation time approximation is often used to simplify the scattering term
by writing

(cid:20) ∂ f

(cid:21)

∂t

scat

f − f0
τ

=

(4.2)

where τ is the overall relaxation time due to various scattering mechanisms, such as phonon-
phonon scattering, phonon-impurity scattering, phonon-boundary scattering, and phonon-electron
scattering, and f0 is the phonon distribution function at equilibrium. Here phonon-impurity
scattering refers to phonons scattered by impurities, phonon-boundary scattering refers to phonons
scattered by physical boundaries, and phonon-electron scattering refers to phonons scattered by
normal electrons.

4.3 Monte Carlo Solution Technique

The Monte Carlo method is used to solve the BTE under stochastic simulation and the relaxation
time approximation. The energy-based variance-reduced technique [126] is used to model the lattice
thermal conductivity of superconducting Nb by considering the phonon dispersion relation and the
scattering of phonons by boundaries, normal electrons, and dislocations. The major steps for Monte
Carlo simulation are outlined below.

4.3.1

Initialization

As the geometry of the studied material does not matter in the Monte Carlo simulation [9], a simple
cubic cell stacks is chosen. The ﬁrst step in the simulation, once medium, geometry, and mesh
haven been chosen, is to initialize the state of an ensemble of phonons within each cell describing
the material.

50

4.3.1.1 Number of particles

E = 

(cid:18)


p=L A,T A

q

(cid:19)

1
2

Using an integrated distribution function, the total vibrational energy of the crystal may be obtained
as [9]

(cid:104)nq,p(cid:105) +

ω

(4.3)

where  is the reduced Planck constant, ω is the phonon frequency, (cid:104)nq,p(cid:105) is the local thermodynamic
phonon population with polarization p and wave vector q. The longitudinal and transverse acoustic
branches are represented by LA and TA, respectively. The phonon population is assumed to follow
the Bose-Einstein distribution function, that is

1
exp( ω
kBT ) − 1
Introducing a density of state function of phonons, Dp,

(cid:104)nq,p(cid:105) =

Dp(ω)dω =

Vq2dq
2
2π

(4.4)

(4.5)

where V is the volume of the three-dimensional crystal. The group velocity of the phonons is
deﬁned based on the dispersion relation as

vg =

dω
dq

Then the total number of phonons in a given volume may be expressed as


Nbin

p=L A,T A

b=1

N = V

[

1
exp( ωb,p
kBT ) − 1

]

q2
b,p
2
vgb,p

2π

gp∆ω

(4.6)

(4.7)

where gp is the degeneracy of either the LA (gp = 1) or TA (gp = 2), Nbin is the number of
spectral bins (chosen here to be 1000) in the range from 0 to ωmax
, as in reference [114], and
L A
∆ω = ωmax

L A /1000.

Even for a small sample of material (e.g., 10 nm silicon cube) at 300 K, there are approximately
5 × 105 phonons [114]. Simulation can be reasonably accomplished only when the temperature
is cold and the dimension size is small. Therefore, a weighting factor W is often used to save

51


Nbin

i=1

N = V

p=L A,T A





computational cost by considering a bundle of phonons having the same properties. Then, the
number of particles in the statistical computation N∗ can be expressed as the actual number N
divided by this weighting factor N∗ = N/W. Péraud and Hadjiconstantinou [126] developed
an eﬃcient simulation method to model the deviation energy from the equilibrium by setting a
hypothetical equilibrium temperature as the reference temperature and calculating the diﬀerences
between the particle energy and corresponding equilibrium energy, making the computation time
about 15 times faster to reach steady state. They also considered the energy-based method to
provide a convenient and rigorous way to conserve energy automatically.
In this method, each
computational particle has the same amount of energy. The number of particles is deﬁned as

ω

1
exp( ω
kBT ) − 1

−

1
exp( ω
kBTeq

) − 1

q2
2
2π
vg

gps(i)∆ω

(4.8)

where Teq is the chosen reference temperature, s(i) is the sign of the particle, which is positive if
T > Teq and negative if T < Teq to ensure that N is positive. Then, the energy is conserved by
conserving the number of simulated particles.

During the initialization process, the ﬁrst cell is set to the hot temperature Th and the last cell
to the cold temperature Tc, while all the other cells are also initially set to the cold temperature.
The energy associated with each cell equals the number of particles in each cell as calculated by
Eq. 4.8 and should match the calculated energy such that

Ncell

N∗

c=1

i=1

N =

W ∗ s(i)

(4.9)

4.3.1.2 Frequency

The number of particles in the ith spectral bin per unit volume is deﬁned as

Ni(T) = (cid:104)n(ω0,i, L A)(cid:104)D(ω0,i, L A)∆ω + 2(cid:104)n(ω0,i, T A)(cid:104)D(ω0,i, T A)∆ω

(4.10)

where the equilibrium distribution is evaluated at the central frequency of the spectral bin.

Once the number of particles is initialized, a frequency is assigned to each particle by a
normalized number density function FD. According to Mazumder and Majumdar [114], this

52

function is written using the cumulative summation of the number of particles Ni(T) from Eq. 4.10,
that is

(4.11)

i
Nbin
j=1 Nj(T)
j=1 Nj(T)

FDi(T) =

Then, a random number R1 is selected from 0 to 1 and compared with the value of FDi(T) in each
bin. A bisection algorithm [114] is used to choose the ith bin to satisfy FDi−1 (cid:54) R1 (cid:54) FDi. The
actual frequency of the phonon is randomly chosen from another random number R2 (0 < R2 < 1)
so that the frequency is linearly distributed between ω0 − ∆ω/2 and ω0 + ∆ω/2, that is

ωi = ω0i + (2R2 − 1) ∆ω
2

(4.12)

where ω0i is the central frequency of the ith spectral bin.

4.3.1.3 Polarization

The polarization (either LA or TA) is found from the ratio of the number of LA particles to the
number of TA particles using Eq. 4.8 and the dispersion relation. The associated probability is
expressed as

PL A(ωi) =

NL A(ωi)

(4.13)
where N(ωi) is the number of particles in the branch of LA or TA with respect to the frequency.
Then, a third random number R3 is selected from 0 to 1. If R3 < PL A(ωi), the phonon is on the

NL A(ωi) + NT A(ωi)

LA branch, otherwise, it belongs to the TA branch.

4.3.1.4 Direction

Again, using the dispersion relation, the phonon group velocity can be determined. Assuming
directional isotropy, the spherical coordinates, including the two angles θ and φ, can be chosen so
that the corresponding directions are uniformly distributed in the 4π full-space solid angle. The

53

direction Ω can be determined using two random numbers R4 and R5 from 0 to 1, such that



sin θ cos φ

sin θ sin φ

cos θ

(4.14)

Ω =

where cos θ = 2R4 − 1, and φ = 2πR5

4.3.1.5 Location

The last part of the initialization is to set a random location of each particle in each cell. Using the
same method as before, the location of the nth particle in cell u is expressed as

rn,u = ru + Ldx R6i + LdyR7 j + LdzR8k

(4.15)

where ru are the coordinates of the cell u, i, j, k are coordinate directions, R6, R7, R8 are random
numbers, and Ldi is the size of the cube at each coordinate.

4.3.2 Drift process

g

g

to consider all scattering events, and to avoid ballistic jumps over the cells, vmax

After initialization comes the drift process, during which each particle will move according to its
group velocity vg and timestep ∆t, such that rnew = rold + vg∆t. The timestep here is chosen so that
∆t < Lcx/vmax
is
the maximum group velocity. The ﬁrst and last cells are kept at constant hot and cold temperatures,
and each particle coming into these two cells will be absorbed and re-sampled in order to keep the
boundary cell acting as black-bodies. The rest of the medium is divided into several cells in the
x-direction with the boundaries in the y- and z-directions being assumed to be adiabatic. When a
particle reaches one of these adiabatic boundaries, it will be reﬂected either specularly (i.e., angle
of incidence equal to angle of reﬂection) or diﬀusively (i.e., a random angle of reﬂection) according
to the degree of surface specularities p (0 (cid:54) p (cid:54) 1, p = 0 purely specular, p = 1 purely diﬀuse).
Thus, a random number R9 is drawn, when R9 < p, a new particle drift direction is assigned using

54

Eq. 4.14, otherwise, the particle reﬂects specularly. After the drift process, the energy at each
cell is calculated by the summation of the number of particles, and the temperature of each cell is
computed by numerical inversion of Eq. 4.8.

4.3.3 Scattering

Phonons move inside a crystal structure with group velocity vg and engage with scattering due
to boundary, lattice structure, impurities, electrons, and defects. In the Monte Carlo simulation,
the phonon boundary scattering process is carried out during the drift process, and the other
scattering processes are treated independently from the drift. As phonons are mainly scattered by
boundaries, electrons, and dislocations in Nb at superconducting temperatures [43], the phonon-
phonon scattering and impurity scattering are neglected in this paper [95].

4.3.3.1 Phonon-electron scattering

Phonon-electron scattering is modeled according to Kes [61] by considering the condensation
of electrons into Cooper pairs. Thus, the relaxation time for the phonon-electron scattering is
computed as

−1
pe = g(x, T)E xT

τ

(4.16)
where g(x, t) quantiﬁes the ratio of the normal-conducting relaxation time to superconducting
relaxation time and is a monotonically decreasing function of the non-dimensional frequency x
(x = w/(kBT)), T is the temperature and E is a ﬁtting parameter found to be in the range of
1.4 − 3 × 109 K−1s−1.

Assuming g(x, T) = g(0, T)(1 − x/(8xm)), where g(0, T) = 2(exp(y) − 1)−1 [61], xm is the
value of x when x4 exp(x)(exp(x) − 1)2 has a maximum. Here xm is calculated as 3.83. Then, the
relaxation rate of phonon-electron scattering is expressed as

pe = 2(exp(y) − 1)−1(cid:16)1 −

−1

τ

55

(cid:17)

x

8 × 3.83

E xT

(4.17)

The probability of phonon scattered by electrons is

Ppe = 1 − exp

(cid:19)

(cid:18)

− ∆t
τpe

(4.18)

where ∆t is the time step of Monte Carlo simulation.

A random number R10 is sampled to compare with the scattering probability Ppe. If R10 < Ppe,
the phonon will be scattered by electrons. As the phonon-electron scattering is considered to be
inelastic scattering, the phonon will be re-sampled after scattering, thus the frequency, wave vector,
and velocity will be reassigned based on the dispersion relation and the number density function.

4.3.3.2 Phonon-dislocation scattering

Phonon-dislocation scattering plays an important role in contributing to the thermal conductivity of
superconducting Nb, especially after deformation [43]. The expression of the phonon-dislocation
scattering relaxation rate is

−1
τ
pd

= A0γ

2b2Nd w

(4.19)

where γ is the Grüneison constant (γ = 1.4 for Nb [61]), b is the Burgers vector, and A0 is a constant.
By applying the relaxation time of phonon-dislocation scattering into the Callaway model [95] and
comparing the expression of the resistance in the Klemens model [44], A0 is evaluated to be
1.1398 for edge dislocations and 0.8639 for screw dislocations. Screw dislocations usually appear
after deformation at temperatures lower than room temparure, whereas edge dislocations appear
following high temperature deformation [4].

The probability of phonons scattered by dislocations is

Ppd = 1 − exp(− dt
τpd

)

(4.20)

Similar to phonon-electron scattering, a random number R11 is assigned to compare with the
probability Ppe. If R11 < Ppd, the phonon will be scattered by dislocations. Here, the phonon-
dislocation scattering is considered to be elastic scattering, so that energy will be conserved after

56

scattering. Therefore, only the direction of velocity will be randomly assigned, with the frequency,
wave vector, and magnitude of velocity remaining unchanged.

4.3.4 Calculate the thermal conductivity of Nb

In the present study, the thermal conductivity is calculated knowing the heat ﬂux of phonons through
the medium for a given temperature gradient by directly applying Fourier’s law. The temperature
diﬀerence is preset so as to determine average thermal conductivity. The phonon heat ﬂux is
calculated along the temperature gradient as

N

(cid:48)(cid:48)

=

q

i=1
where vgx is the group velocity in x coordinate.

Ws(i)vgx

(4.21)

4.4 Results and Discussion

4.4.1 Temperature proﬁle at the diﬀusion limit

Comparison of the temperature proﬁle at the diﬀusion limit was performed in a one-dimensional
Si thin ﬁlm with the boundary temperatures set as Th = 310 K and Tl = 290 K. The equilibrium
temperature was chosen to be the mean of Th and Tl as Teq = 300 K to reduce the computational
cost. The medium was divided into 20 cells, with each cell being 5 × 10−7 m by 5 × 10−7 m by
5× 10−8 m, so the total length in the heat ﬂux direction is 20× 5× 10−8 = 1× 10−6 m = 1 µm. The
time step was chosen to be ∆t = 1 ps and there were 1000 spectral bins for sampling the frequency.
The weighting factor was taken as W = 0.5 × 10−16. First, the diﬀusion limit (Λ << Lc) was
calculated at steady state. At the diﬀusion limit, phonon transport follows Fourier’s law and the
temperature proﬁle is linear at steady state, as shown in Fig. 4.1.
It shows that the simulation
method successfully models the diﬀusion limit. One may notice that the simulated temperature is
slightly lower than the diﬀusion temperature at T < 300 K and slightly greater than the diﬀusion
temperature at T > 300 K. This is the result of the thermal conductivity of Si being temperature

57

Figure 4.1: Comparison of the temperature proﬁle obtained using the Monte Carlo model with
the usual Fourier conduction results in the diﬀusion limit. Six Monte Carlo simulations have been
carried out to demonstrate the repeatability of the model, and the scatter in the results are shown.

dependent, it changes slightly from Tl = 290 K to Th = 310 K. This is unaccounted for in the simple
diﬀusion model.

4.4.2 Temperature proﬁle at the ballistic limit

Comparison at the ballistic limit was also performed for the one-dimensional Si thin ﬁlm. To
enable a simulation using the same geometry of the material, the boundary cell temperatures
were set to be Th = 20 K and Tl = 10 K, because at lower temperatures, the phonon mean free
path of Si increases and becomes too long for the physical geometry of the model. Keeping
the size of the ﬁlm unchanged, at low enough temperatures, the phonon mean free path will be
comparable to the thickness of the ﬁlm, causing the temperature distribution to reach the ballistic

58

00.20.40.60.81Dimensionless Size (x/L)290295300305310Temperature (K)Monte Carlo methodTheoretical Diffusion LimitFigure 4.2: Comparison of the temperature proﬁle by Monte Carlo method with Stefen-Boltzmann
equation in the ballistic limit. Six simulations have been carried out to demonstrate the repeatability
of the Monte Carlo model.

+ T4

limit. The temperature at the ballistic limit was calculated using Stephen-Boltzmann equation as
T = ((T4
l )/2)0.25 = ((204 + 104)/2)0.25 = 17.1 K. Figure 4.2 shows good agreement between
h
the Stephen-Boltzmann equation and the Monte Carlo method. The maximum error between the
two solutions is approximately 2 percent. At low temperatures, the equilibrium distribution of
phonons is nonlinear [132]. The small error in the Monte Carlo solution is caused by the non-linear
distribution being approximated by a linear function of Nbin = 1000 segments.

4.4.3 Thermal conductivity in bulk Si

Comparison of the thermal conductivity in bulk Si was conducted using a 2 µm length Si rectangle
with the other two sides chosen as 500 nm × 500 nm. Simulation results of the thermal conductivity

59

00.20.40.60.81Dimensionless size (x/L)101214161820Temperature (K)Monte Carlo methodTheoretical Ballistic LimitFigure 4.3: Comparison of the thermal conductivity of bulk Si between diﬀerent methods. The
solid line is Eq. 4.22, the circles come from Monte Carlo simulation by Lacroix et al. [9], and the
stars are the current simulation results.

of Si, as shown in Fig. 4.3, are compared with the Monte Carlo simulation results of Lacroix et al.
[9] and the analytic equation for bulk Si [9] as
exp(12.570)

(200K (cid:54) T (cid:54) 600K)

kSi =

T1.326

(4.22)

4.4.4 Size eﬀect on thermal conductivity of Si nanowires

Comparison of the thermal conductivity of Si nanowires is carried out for diﬀerent diameters of
the nanowires (37 nm, 56 nm, and 115 nm). Monte Carlo simulation results are compared with the
experimental data from Li et al. [10], as shown in Fig. 4.4. There is good agreement at the size
of 37 nm and 56 nm. However, there are some discrepancies at d=115 nm for temperatures higher
than 100 K, due to the fact that the same weighting factor was used for diﬀerent sizes.

60

100150200250300350400450500Temperature (K)0100200300400500600700800900Thermal conductivity (Wm-1K-1)Data from Lacroix. et al.Curent Monte Carlo resultsAnalytical resultFigure 4.4: Comparison of the thermal conductivity of Si nanowire between current simulations
(solid lines) and the data (triangles) from Li et al. [10]. Six simulations have been carried out and
the error bars were plotted for the Monte Carlo simulation results.

Table 4.1: Parameters for the samples used in the simulation

RRR Dislocation Deformation T

Heat Treatment

Sample
LS 1
LS 2
LS 3

Size
3.9mm 350
3.2mm 1200
2mm
224

Screw
Edge
NA

195 K
295 K
NA

NA
NA

3hr@800◦C & @140◦C

4.4.5 Simulation of the thermal conductivity of Nb

As the ability to characterize the thermal conductivity has been successfully demonstrated, it is
next applied to model the lattice thermal conductivity of superconducting Nb. Samples used in the
model are listed in Table. 4.1

To model the thermal transport of Nb at low temperatures, where phonons dominate the thermal
conductivity, the phonon dispersion relation of Nb is replotted from Jani et al. [11] in Fig. 4.5 for

61

050100150200250300350Temperature (K)0102030405060Thermal conductivity (Wm-1K-1)d=56 nmd=115 nmd=37 nmFigure 4.5: Phonon dispersion relation of Nb in the [1 0 0] direction, replotted from reference [11],
where a is the lattice parameter, which refers to the physical dimension of unit cell in a crystal
lattice, for Nb, a = 3.3 × 10−10 m.

their experimental work. The phonon dispersion characterizes the relationship between frequency
ω, wavevector q, and group velocity vg. It is the key to accurately model phonon transport and
needs to be considered especially for this non-linear phonon dispersion.

4.4.5.1 Simulation of the lattice thermal conductivity of undeformed Nb

According to BCS theory [39], lattice thermal conductivity of superconductors consists of phonon-
boundary scattering and phonon-electron scattering, where the phonon-defect scattering (i.e.,
phonon-dislocation scattering) is included in the phonon-boundary scattering as the form of
Wd ≈ T−3 [41]. Considering phonon-dislocation scattering as the form of Wd ≈ T−2 [98] in
an extra term [43] demonstrated better ﬁtting with the experimental results. Thus the phonon-

62

00.20.40.60.8101234567ε = aq/πFrequency (1012 /s)  LATAFigure 4.6: Simulation results of lattice k of undeformed Nb or Nb after heat treatment. Data used
from undeformed samples from reference [4] and the sample after heat treatment of reference [12].

dislocation scattering is included in the simulation to model the lattice thermal conductivity of Nb.
Dislocation density is introduced by deformation and it increases with the amount of deformation
[51]. For undeformed Nb, phonon-dislocation scattering can be neglected because of the low dis-
location density. The overall relaxation time in undeformed Nb τscat consists of phonon-boundary
scattering and phonon-electron scattering.

−1
scat = τ
τ

−1
pb

−1
pe

+ τ

(4.23)

Simulation results of the lattice thermal conductivity of undeformed Nb sample and heat-treated
sample are shown in Fig. 4.6 and compared with the ﬁtting results using Eq. 2.4 of the phonon
contributions. Heat treatment helps restore the phonon peak and increase the thermal conductivity
because of the recovery (annihilation) of dislocations during heat treatment [8], where the phonon
peak is the point at the local maximum thermal conductivity, which usually appears at around

63

11.522.533.5Temperature (K)010203040506070Lattice k (Wm-1K-1)Simulation LS 1Data Fitting LS 1Simulation LS 2Data Fitting LS 2Simulation LS 3Data Fitting LS 32 K for Nb [85]. Therefore, the phonon-dislocation scattering can be neglected for the samples
after appropriate heat treatment [12]. Simulation results agree well with the ﬁtting results over the
temperature range of SRF cavities when including phonon-boundary scattering, phonon-electron
scattering, but neglecting phonon-dislocation scattering.

4.4.5.2 Simulation of the lattice thermal conductivity of deformed Nb

For as-received Nb samples or deformed samples, the dislocation density is typically large thus the
phonon-dislocation scattering cannot be neglected because it contributes signiﬁcantly to phonon
scattering in superconducting Nb [43]. In general, there are two basic types of dislocations, the
edge dislocations and the screw dislocations. The diﬀerence between edge and screw dislocation
is that the edge dislocation occurs when an extra half-plane of atoms exists in the middle of the
crystal lattice whereas the screw dislocation occurs when the planes of atoms in the crystal lattice
trace a helical path around the dislocation line [132]. According to reference [4], edge dislocations
aﬀect the thermal conductivity approximately 1.5 times more than screw dislocations because of
their scattering mechanisms with phonons (there is a well-deﬁned geometrical energy minimum
due to the disturbed lattice positions, and the dislocation can easily wiggle along the line length;
screw dislocation distort the lattice to a smaller extent). The overall relaxation time is expressed
as the sum of phonon-boundary scattering, phonon-electron scattering, and phonon-dislocation
scattering.

−1
scat = τ
τ

−1
pb

−1
pe + τ

−1
pd

+ τ

(4.24)

where τpd is the phonon-dislocation scattering relaxation time either predominantly by edge dislo-
cations or by screw dislocations.

Simulation results of the lattice thermal conductivity of Nb LS 1, an example of screw dis-
locations, have been compared with the ﬁtting results for a sample before deformation, after 3%
deformation, 7.3% deformation, and 10.3% deformation. The parameters are extracted from ﬁtting
and are included in the relaxation times of the simulation to consider diﬀerent scattering mecha-

64

Figure 4.7: Simulation results of lattice thermal conductivity of Nb LS 1 from [4], an example of
screw dislocations, for several strains.

nisms separately. The simulation results match well with the ﬁtting results, as shown in Fig. 4.7.
The phonon peak occurs at around 1.8 K in the simulation for sample LS 1 before deformation
and it shifts to about 1.9 K after 3% deformation. After 7.3% deformation, the peak is diﬀuse and
moves to 2.2 K. There is nearly no phonon peak after 10.3% deformation for this sample. Lattice
thermal conductivity at temperatures lower than the phonon peak temperature is dominated by
phonon-boundary scattering and follows the T3 relation. At temperature higher than the phonon
peak, phonon-dislocation scattering is starting taking over for sample after large amount of defor-
mation. Phonon-electron scattering contributes to the lattice thermal conductivity above 2 K and
is dominant above 3 K.

Simulation results of the lattice thermal conductivity of Nb LS 2, an example of edge dislo-
cations, have been compared with the ﬁtting results for a sample before deformation, after 4%

65

11.522.533.5Temperature (K)01020304050Lattice k  (Wm-1K-1)Fitting =0MC =0Fitting =3%MC =3%Fitting =7.3%MC =7.3%Fitting =10.3%MC =10.3%NbFigure 4.8: Simulation results of lattice thermal conductivity of Nb LS 2, from [4] an example of
edge dislocations, for several strains.

deformation, and 8% deformation, as shown in Fig. 4.8 The simulation results match well with the
ﬁtting results of both undeformed and deformed Nb samples by including the eﬀects of phonon-
dislocation scattering. Similar to sample LS 1, the steep phonon peak of the simulation results of
sample LS 2 occurs at about 1.7 K. It moves to be around 2.2 K after 4% deformation with smooth
peak. The phonon peak is much more diﬀuse after 8% deformation.

4.5 Conclusion

Monte Carlo solutions of the BTE have been successfully performed to model the lattice thermal
conductivity of superconducting Nb given the phonon dispersion relationship of Nb. The validity
of the simulation code was ﬁrst demonstrated at the diﬀusion and ballistic limits. It was further
demonstrated by comparing the simulation results of the thermal conductivity of bulk and nanowire

66

11.522.533.5Temperature (K)010203040506070Lattice k (Wm-1K-1)Fitting =0MC =0Fitting =4%MC =4%Fitting =8%MC =8%NbSi with analytical and experimental results [9, 10]. Then, the simulation method, incorporating the
Nb dispersion relation and phonon-boundary, phonon-electron, and phonon-dislocation scattering,
was used to calculate the lattice thermal conductivity of Nb for samples before deformation, after
deformation to several strains, and after appropriate heat treatment. The simulation results agree
well with the results extracted from experiments [4, 12]. The results show that boundary scattering
dominates for temperatures below 2 K, where the phonon mean free path is comparable to the size of
the sample, and phonon-electron scattering is important above 2 K and dominates the conductivity
above 3 K. The phonon peak appears at temperatures of approximately 1.8 K in undeformed
samples and shifts to a higher temperature after a small amount of deformation. Phonon-dislocation
scattering decreases the lattice thermal conductivity and a large amount of deformation destroys the
phonon peak. Appropriate heat treatment may restore the phonon peak by reducing the dislocation
density. This simulation method may be extended to predict the thermal conductivity of Nb thin
ﬁlms in the application of Nb cladding on Cu to provide guidance for designing future generations
of SRF cavities. In addition, with appropriate dispersion relations, this modeling technique may
be used to model the lattice thermal conductivity of other superconductors, such as Ta and Nb3Sn,
where conductivity is determined by considering the phonon scattering mechanisms.

67

EFFECT OF IMPURITIES ON THE LATTICE THERMAL CONDUCTIVITY OF NB

CHAPTER 5

Thermal conductivity due to the contribution of electrons in metals consists of two forms of
resistances to its ﬂow. The ﬁrst is from the interaction of electrons with interstitial and substitutional
impurities in the metal (Wei), and the second is from the momentum exchange of the electrons with
the lattice (Wep). These resistances act in series and the total resistance is shown as

We = Wei + Wep

(5.1)

The Wiedeman-Franz law relates the impurity limited electronic thermal conductivity kei and

the electrical conductivity σ of the metal to the temperature as

kei
σ

= LT

(5.2)

where L is the Lorenz constant and the electric resistivity ρ = σ−1

5.1 Residual resistance ratio (RRR)

Large values of thermal conductivity are required for good performance of SRF cavities to
transfer the dissipated RF power to the liquid helium bath and prevent hot spots [31]. For bulk
Nb cavities, this demands high purity. Because the thermal conductivity is usually related to the
residual resistance ratio, which is an empirical indicator of the impurity concentration [36], at 4.2
K as

where the RRR is deﬁned as

k4.2K =

RRR

4

RRR =

ρ295K
ρ4.2K

(5.3)

(5.4)

where ρ295K and ρ4.2K are the resistivity of Nb at room and liquid helium saturation temperature
at standard atmospheric pressure, respectively [31].

68

Table 5.1: Contributions of diﬀerent impurities to ∂ ρi
∂Ci

[1]

Impurity
× 1010
∂ ρi
∂Ci

N
5.2

O
4.5

C
4.3

H Hf
0.8
1.4

Ti
1.4

Zr

W

0.6-1.4

0.4-1.1

Ta Mo
0.25
0.21

It is generally recognized that the resistivity of metals at low temperature is described as

ρ(T) = ρres + ρph(T) + ρm

(5.5)

is the residual resistivity at T≈ 0 K, which consists of electron-
where ρres = ρimp + ρde f
impurity scattering and the scattering by lattice defects; ρph(T) represents the temperature depen-
dent electron-phonon scattering, and ρm is the resistivity term in a magnetic ﬁeld. For recrystallized
niobium, the ρde f contribution is small. In the absence of a magnetic ﬁeld, the resistivity at low
temperature is simpliﬁed as:

(5.6)
where ρph(T) is also usually referred as the temperature dependent ideal resistivity. The resistivity
due to the impurity contribution is expressed as

ρ(T) = ρimp + ρph(T)

ρ(imp) = ∂ ρi

∂Ci

Ci

(5.7)

where Ci is the impurity concentration in atomic ppm and ∂ ρi
∂Ci
Therefore, Eq. 5.4 is updated to the following equation as

has units of ohm-cm/at.-ppm.

RRR =

ρ295K

ρph(4.2K) + ∂ ρi
∂Ci

Ci

(5.8)

The impurity content is typically consists of interstitial impurity atoms (e.g. H, N, O, and C)
and substitutional impurity atoms (also called metallic impurities in Nb), including Ta, Mo, Zr, W,
Ti, and Hf. The contributions of diﬀerent impurities to ∂ ρi
∂Ci

are listed in Table 5.1.

The interstitial impurities aﬀect the RRR more signiﬁcantly than the substitutional impurities.
Therefore, attention should be paid to the chemical processing of the cavities. For example, the
etching process will bring H impurities to the surface and the N baking process will add N impurities
to the cavity. Typical RRR values for various types of Nb are listed in Table 5.2.

69

Table 5.2: Typical RRR values of Nb in relation to processings [2], where Cavities and Samples

refer to post puriﬁed Nb origins, Other refers to other preparations.

Origin Commercial RF Application Cavities Samples Other
RRR
5-6000

600-800 ≤ 1800

30-50

200-300

Theoretical
≈ 33000

Figure 5.1: Typical thermal conductivity data [4, 7, 13] related with RRR and strain, thermal
conductivity increases with RRR at 4.2 K, while at 2 K, there is no relation between RRR and
thermal conductivity. The phonon peak usually exists for appropriately heat treated sample while
strain may destroy the phonon peak. (This ﬁgure is a repeat of Fig. 2.5), included here for the
reader’s convenience.

The predominant impurities in Nb from commercial cavities and post-puriﬁed cavities are
interstitial impurities (including H, N, O, and C) [2]. The substitutional impurities contribute to
higher RRR cavities from post-puriﬁed cavities below the theoretical limit, when there is 100%
pure Nb.

Below the superconducting transition temperatures, the number of free electrons decreases

70

0.511.522.533.544.5Temperature (K)050100150200250Thermal conductivity (Wm-1K-1)RRR=68 Heat TreatedRRR=237 =8.5%RRR=255 =0RRR=390 As Rec'dRRR=1200 =0RRR=1200 =8%exponentially because of the formation of Cooper pairs [39]. Therefore, the phonon contribu-
tion will dominate the thermal conductivity at low temperatures (T < 0.2Tc) and the electronic
contribution is not important so that the eﬀect of impurities can be neglected. Typical values of
thermal conductivity with diﬀerent RRR, strain, and heat treatment are plotted in Fig. 5.1. Thermal
conductivity at 4.2 K increases with the increase of RRR values because the impurity atoms able
to scatter with the free electrons decreases with the increase of RRR and the relation RRR = 4k4.2
is descriptive. However, at temperatures lower than 3 K, especially at 2 K where the phonon peak
usually appears, there is no relation between the thermal conductivity and RRR. The reason is that
at this temperature, the phonon dominated thermal conductivity consists of the resistances due to
phonon-boundary scattering, phonon-electron scattering, and phonon-dislocation scattering [43].
Phonons usually do not scatter with impurities at the lowest temperature because the sizes of the
impurities are much smaller than the phonon mean free path of Nb at superconducting temperatures.
Fabrication of SRF cavities require surface processing, which includes buﬀered chemical pol-
ishing (BCP) and electropolishing (EP), and plastic deformation. These treatments increase the
interstitial impurity content including O, H, and N and also form oxides, hydrides, and nitrides.
These impurities aﬀect the surface resistance of the SRF cavity to decrease the SRF performance
[5]. Appropriate heat treatment might help decrease the near-surface concentration of impurities
by dilution and also decrease the dislocation density. The eﬀect of diﬀerent types of impurities is
discussed in the following.

5.2 Eﬀect of oxygen diﬀusion

During low temperature baking (100 ◦C to 200 ◦C) of Nb SRF cavities, the oxygen diﬀuses
from the surface oxide or an oxygen-rich interface layer just below the oxide into Nb, causing a
decrease in the BCS surface resistance [5]. The average diﬀusion length of oxygen in Nb (xd) is
expressed as

xd =(cid:112)2DO exp(−E/RT)t

(5.9)

71

where DO = 0.015 cm2s−1 is the diﬀusion coeﬃcient for oxygen in Nb [133], E=112890 Jmol−1 is
the activation energy, R=8.31 Jmol−1K−1 is the universal gas constant, and t is the diﬀusion time.
After baking at 145 ◦C for 45 hours with a surface atomic oxygen concentration as 0.33%,
which is close to the solubility of oxygen in Nb at 145 ◦C [5], the diﬀusion depth of oxygen
spreads over 150 nm and drops by a factor of two within 50 nm. This thickness is comparable to
the magnetic penetration depth of Nb, which is around 30 nm.

5.3 Eﬀect of hydrogen diﬀusion

During fabrication process of SRF cavities, the BCP or EP processes are regularly used to etch
the inner surface of the cavity to make the surface smooth and remove surface irregularities [5].
Niobium-hydride precipitates typically form at 100-150 K when the amount of hydrogen dissolved
in niobium is signiﬁcant, e.g., the acid gets hot (T > 20 ◦C) during etching or too much H arrives
at the cavity surface during EP process. This Nb-H eﬀect is often called as the H-related Q-disease
[134].

Chandrasekaran measured the thermal conductivity of superconducting Nb infused with satu-
rated hydrogen (41.2% atomic concentration) [8]. He found a 25% reduction in thermal conductivity
at the phonon peak for the a Nb bi-crystal exposed to 300 ◦C for 1 h in a 75% hydrogen, 25%
nitrogen atmosphere at 0.5 atm., and a 15% reduction at the phonon peak for the bi-crystal specimen
under uncooled BCP etch condition [8]. Several niobium-hydrogen phases have been identiﬁed
depending on the concentration of hydrogen in Nb and the temperature of the specimen based on
the Nb-H phase diagram [135]. The the Nb-H system is in the α(cid:48) phase after the hydrogen infusion
at 300 ◦C. It transformed from α(cid:48) to α(cid:48) + β(cid:48) during cooling at about 115 ◦C, and to α + β at
around 85 ◦C. In the α and α(cid:48) phases, the hydrogen atoms are disordered and behaves like a gas
[136]. However, in the β phase, hydrogen is ordered with well deﬁned spacing [136] and behaves
like a solid. Therefore, the mobility of the hydrogen atoms in the β phase is about two orders
of magnitude lower than that in the α(cid:48) phase [137, 138]. After an 800 ◦C heat treatment for 2h
the phonon peak was restored for one of the samples (heat treated at 800 ◦C for 2 hours before

72

hydrogen infusion) because of degassing the hydrogen [139].

5.4 Eﬀect of nitrogen diﬀusion

To reduce the cryogenic operating cost accelerators while providing reliable operation, it is
critical to improve the SRF performance of cavities, and thus improve the quality factor [5]. Nitrogen
doping is an eﬃcient way to improve this quality factor. Grassellino et al. [140] investigated the
eﬀect of nitrogen and argon doping of niobium on SRF performance. They found that 800 ◦C
and 1000 ◦C nitrogen doping leads to low values of the microwave surface resistance (about
1.5 nΩ), and an improvement in the eﬃciency of the accelerating structures up to a factor of 3.
The reason for this enhancement of SRF performance is the formation of niobium nitride phases
leading to a higher critical temperature than Nb. Dhakal et al. [141] also investigated the eﬀect
of low temperature (120-140 ◦C) nitrogen baking on the SRF performance of cavities and found
an improvement in the quality factor after annealing at 800 ◦C for 3 hours followed by baking at
120–140 ◦C in a low partial pressure of nitrogen. However, the temperature has to be controlled
carefully because a larger reduction (about 35%) of the quench ﬁeld was observed when the baking
temperature was 160 ◦C.

The interaction of impurities with the lattice typically appears in semiconductors or dielectric
materials. Klemens [44] did a systematic investigation of impurity scattering by the lattice and it is
discussed brieﬂy in this chapter.

5.5 Substitutional atoms of diﬀerent mass

In the case of an isolated impurity, the relaxation time of phonons scattered by an impurity is

expressed as

−1 =

τ

a3
G

( ∆M
M

4
)2 ω
4πv

3

(5.10)

where a is the lattice parameter, G is the number of atoms in the crystal, ∆M is the diﬀerences of
mass between impurity and the lattice, M is the mass of lattice, and v is the group velocity based
on the phonon dispersion.

73

5.6 Point defects in the MC Model

In the case of point imperfection, the relaxation time is expressed as

−1
pi =
τ

3a3
G

4
3

S2 ω
πv

where S2 = S2

1 + S2

2 + S2
3.

S2
1 =

∆M
M

1
√
3

2

S2
2 =

2
δv
2

v

1√
6

3 = −Qγ
S2

∆R
R

(cid:114)2

3

where Q is a constant, R is the atomic radius, and γ is the Grüneisen constant.

Mazumder and Majumdar [114] simpliﬁed this relaxation time as the following:

−1
pi = ασ ρ|Vg|

τ

(5.11)

(5.12)

(5.13)

where α is constant with the order of unity, ρ is the impurity concentration of point defect, and σ
is the scattering cross-section expressed as

4

σ = πR2( χ
χ

4 + 1)

(5.14)

where χ = R|q|.

The eﬀect of impurity concentration on the temperature proﬁle of Si thin ﬁlm is shown in Fig.
5.2 by the Monte Carlo simulation [114]. The specularity is set to be unity so that the boundary
scattering is neglected in the simulation. It is shown from the ﬁgure that with increasing defect
density, the temperature drop across the ﬁlm increases.

5.7 Monte Carlo simulation considering impurity scattering

The eﬀect of interstitial impurities (H and N) on the lattice thermal conductivity of supercon-

ducting Nb is investigated using the Monte Carlo simulation.

Simulation of the lattice thermal conductivity of Nb is shown in Fig. 5.3 for diﬀerent levels
of H concentration and in Fig. 5.4 for diﬀerent levels of N concentration. The ﬁtting results
were conducted using data from reference [4] using the enhanced model [43] without considering

74

Figure 5.2: Temperature proﬁle of Si for diﬀerent levels of impurities from 1023 m−3 to 1028 m−3.
The temperature proﬁle changes gradually from the ballistic limit for the lowest defect density 1023
m−3 to diﬀusion limit at the highest defect density 1028 m−3.

impurity scattering. The simulation results with no impurity atoms are close to the ﬁtting results.
There are no signiﬁcant diﬀerences between the simulated lattice thermal conductivity from phonon-
impurity scattering until the concentration reaches a certain amount. To signiﬁcantly aﬀect the
lattice thermal conductivity of Nb, the concentration has to reach 1028 m−3 and 1027 m−3 for H
and N impurity atoms, respectively. In these cases, Nb is saturated with H or N, and the atomic
ratio of the impurity to Nb lattice is close to 1.

Although k is correlated with RRR for T (cid:38) 0.3Tc (speciﬁcally k(4.2 K) = RRR/4), a few
measurements [4, 8] suggest that kpp may be correlated with RRR. See, for example, Fig. 2.5. This
result is unexpected based on the BRT model of k of superconductors [42] and the theory of metals
[37]. The extended model (Eq. 2.4) oﬀers an explanation based on the results of Wasserbäch [4]

75

00.20.40.60.81Dimensionless Size(x/L)1011121314151617181920Temperature(K)1023 m-31024 m-31025 m-31026 m-31027 m-31028 m-3Defect DensityFigure 5.3: Simulation of the lattice thermal conductivity of Nb with diﬀerent levels of H impurities
compared with ﬁtting results without H impurities. The lattice thermal conductivity values are
similar for low concentrations of H impurities (i.e. No impurity, 1026 m−3, and 1028 m−3), but at
the impurity concentration of 1029 m−3, the lattice thermal conductivity decreased by 20% at the
phonon peak.

and Chandrasekaran [8]. The coeﬃcient of the phonon-boundary scattering term varies with RRR.
In samples with large kpp, the dislocations have little eﬀect and the value of the kpp is a result of
the balance of the phonon-boundary and phonon-electron scattering.

Simulations of the lattice thermal conductivity of Nb were also conducted for two diﬀerent
samples from reference [8]. The ﬁrst sample was heat treated at 800 ◦C for 2 hours ﬁrst, then it
was infused with hydrogen in a 75% H2, 25% N2 environment of 0.5 atm pressure at 300 ◦C for
1 hour, and then heated again at 800 ◦C for 2 hours. The simulated lattice thermal conductivity
without considering impurity atoms matches with the ﬁtting results with no impurities, as shown

76

1.51.61.71.81.922.1Temperature (K)2224262830323436Lattice  k (Wm-1K-1)Fitting no impurityMC no impurityImpurity 1026 m-3Impurity 1028 m-3Impurity 1029 m-3Figure 5.4: Simulation of the lattice thermal conductivity of Nb with diﬀerent levels of N impurities
compared with ﬁtting results without N impurities. The lattice thermal conductivity values are
similar for low concentrations of N impurities (i.e. No impurity, 1026 m−3, and 1027 m−3), but at
the impurity concentration of 1028 m−3, the lattice thermal conductivity decreased by 15% at the
phonon peak.

in Fig. 5.5. After H infusion, the concentration of H is estimated to be around 3.854 × 1028m−3.
The simulated lattice thermal conductivity agrees well with the data extracted from the experiment
results, however, the H impurity concentration has to reach 2.2 × 1029m−3 in the simulation, which
is 6 times greater than the estimated concentration, to match with the data, shown in Fig. 5.5. The
data extraction process is to obtain the lattice thermal conductivity of Nb considering impurity
scattering from the diﬀerence between the total measured thermal conductivity and the electronic
thermal conductivity, as shown in Eq. 5.15. Since the electronic thermal conductivity, which
dominates at T > 3K, does not change with impurity content for the ﬁrst sample above 3 K, the
main inﬂuence of H impurity will be on the lattice thermal conductivity.

77

1.51.61.71.81.922.1Temperature (K)2224262830323436Lattice  k (Wm-1K-1)Fitting no impurityMC no impurityImpurity 1026 m-3Impurity 1027 m-3Impurity 1028 m-3Figure 5.5: Lattice thermal conductivity of Nb by considering the eﬀect of phonon impurity
scattering. This sample was heat treated at 800 ◦C for 2 hours ﬁrst, then it was infused with
hydrogen in a 75% H2, 25% N2 environment of 0.5 atm pressure at 300 ◦Cfor 1 hour, at last it was
heat treated again at 800 ◦C for 2 hours. The last heat treatment to degas the H impurity recovers
the thermal conductivity at phonon peak.

kp = k − ke

(5.15)

5.8 Phonon-impurity scattering

Imperfections typically aﬀect the lattice thermal conductivity, however this phenomenon changes
at the lowest temperatures [22]. The reason can be explained as follows. When the temperature
decreases, the wavelength of the lattice increases to be much greater than the atomic spacing, e.g.,
the dominant wavelengths reach around 100 times the atomic radius at 1 K. These long wavelengths

78

11.522.533.5Temperature (K)0510152025303540Lattice  k ( Wm-1K-1)Simualtion No impurityFitting No impuritySimualtion i=2.2×1029 m-3Data extraction with impurityFigure 5.6: Lattice thermal conductivity of Nb by considering the eﬀect of phonon impurity
scattering. This sample was heat treated at 140 ◦C for 48 hours and 1100 ◦C for 4 hours ﬁrst, then
it was infused with hydrogen in a 75% H2, 25% N2 environment of 0.5 atm pressure at 300 ◦C for
1 hour, at last it was heat treated again at 800 ◦C for 2 hours. The last heat treatment has no eﬀect
to degas the H impurity and does not recover the thermal conductivity at phonon peak.

are nearly unaﬀected by the imperfections on the order of the atomic scale, but are aﬀected by the
physical boundary of a crystal. However, the wavelengths of electrons depend little on the tem-
perature and are on the same order as the atomic scale. Therefore, the imperfections will be the
dominant scatter for electrons at low temperatures.

Moreover, at unrealistic high concentration of impurities, the phonons will be scattered by the
imperfections and aﬀect the lattice thermal conductivity. The expression of the relaxation time of
phonon-impurity scattering τpi is deﬁned as

79

11.522.533.5Temperature (K)0102030405060Lattice  k ( Wm-1K-1)Simulation No impurityFitting No impurityData extraction with impuritySimulation i=8×1028 m-3where A1 is constant expressed as

−1 = A1ω

4

τpi

(5.16)

(5.17)

where Γ =i fi( ∆M

A1 =

V Γ
3
4πv
s

M )2, fi is the atomic fraction of the impurity, M is the mass of the host atom

and ∆M is the mass diﬀerences between host atom and impurity, and V is the volume.

The second sample was heat treated at 1100 ◦C for 4 hours ﬁrst, then it was infused with
hydrogen in a 75% H2, 25% N2 environment of 0.5 atm pressure at 300 ◦C for 1 hour, at last it
was heat treated again at 800 ◦C for 2 hours. The simulated lattice thermal conductivity without
considering impurities matches well with the ﬁtting results of no impurities, as shown in Fig. 5.6.
After H infusion, the concentration of H impurity is estimated to be around 3.9 × 1028m−3. The
simulated lattice thermal conductivity agrees well with the data extracted from the experiment
results, and also with the ﬁtting results of considering phonon-impurity scattering. In this sample,
the H impurity concentration needs to reach 8 × 1028 m−3 in the simulation, which is about two
times greater than the estimated concentration, to match with the data extraction and ﬁtting results,
as shown in Fig. 5.6.

5.9 Eﬀect of Grain Boundary scattering on the lattice thermal conductivity

of superconducting Nb

A grain boundary is deﬁned as the interface between two grains, or crystallites, in a polycrys-
talline material. Grain boundaries are considered as 2D defects in the crystal structure, and tend to
decrease the electrical and thermal conductivity of the material [98].

5.9.1 Models

Nan and Birringer investigated the inﬂuence of grain size and grain–boundary thermal resistance
(Kapitza resistance RK) on the eﬀective thermal conductivity kE of poly-crystals. They obtained

80

the normalized thermal conductivity (kE/k0) in terms of the Kapitza length LK and the grain size
d. For an isotropic poly-crystalline solid composed of spherical crystals of equal size, they showed

(cid:18)

(cid:19)−1

kE
k0

=

1 +

φLK
d

(5.18)

where k0 is the bulk thermal conductivity of a single crystal, φ is a constant depending on
the aspect ratio of the crystallite (φ ≈ 2). This equation is powerful in that it states that the
normalized thermal conductivity (kE/k0) must scale as a function of (LK/d).
It also implies
that the diﬀerent microscopic scattering mechanisms take place in poly-crystals, that is, phonon
scattering at grain–grain interfaces (via RK) and the phonon–phonon, phonon–boundary, and
phonon–defect scattering.

Yang et al.[5] developed a simple heat transfer model to analyze their measurements of the
thermal conductivity of nano-crystalline yttria-stabilized zirconia (YSZ). They investigated exper-
imentally grain sizes ranging from 10 to 100 nm using the 3ω method at 300 K. In their model,
they ﬁrst considered the case when λ < Λm f p < d where λ and Λm f p are the phonon wavelength
and mean free path, respectively. If ∆T0 is the temperature jump over the bulk single crystal, then
= k0T0/d. In reality, the single crystal has an eﬀective thermal
the heat ﬂux is simply given by q
conductivity k due to grain boundary scattering, which gives rise to a grain– grain boundary thermal
= ∆Tgb/RK, where
(Kapitza) resistance. This latter is related to the heat ﬂux by the equation q
∆Tgb is the temperature jump at a grain–grain boundary. The following expression is obtained

(cid:48)(cid:48)

(cid:48)(cid:48)

= (1 +

LK
d

+ µ)−1

kE
k0

(5.19)
where µ = ∆B/∆T0 is an artiﬁcially introduced parameter which accounts for the situation where
λ > d, ∆T0 is the temperature jump across a single crystal and ∆B is a temperature correction to
∆T0.

This expression is similar to Eq. 5.18 except for the aspect ratio γ and the artiﬁcially introduced
parameter µ. The latter accounts for the situation when λ ≥ d. Yang et al.[5] ﬁtted their data with

81

the above expression, using a unique value of RK in the entire d range. The excellent agreement
testiﬁes that the reduction in the conductivity is due to decreasing grain sizes.

Amrit [81] modiﬁed Eq. 5.18 and Eq. 5.19 by considering an arrangement of single crystals to

account for the spatial anisotropy of poly-crystals:

ni − 1
ni

)−1

LK
d

= (1 +

kE
k0

(5.20)
where ni is the number of grains in the direction i, and ni adjacent grains have ni − 1 interfaces.
In this equation, the thermal conductivity not only depends on the Kapitza length (LK) and grain
size, but also is a function of the number of grains. He concluded that the thermal conductivity
decreased strongly with the increase of the number of grains up to 10. He referred to this change
of thermal conductivity as anisotropy [81]. When the number of grains exceed 10, the eﬀect of the
anisotropy of thermal conductivity disappeared.

5.10 Conclusion

Fabrication of SRF cavities requires surface processing (i.e., BCP, EP, and plastic deformation)
which increases the interstitial impurity content. These impurities aﬀect the surface resistance of the
SRF cavity to decrease the SRF performance [5]. RRR values are a function of diﬀerent interstitial
impurities and substantial impurities, and RRR is an indicator of the thermal conductivity at 4.2 K
based on RRR = 4× k4.2K. Because the impurity atoms typically scatter electrons for metals rather
than phonons, it aﬀects the electronic thermal conductivity, but not the lattice thermal conductivity,
except for special conditions (i.e. the impurities are saturated in the Nb sample). Monte Carlo
simulations of the eﬀect of H impurity and N impurity atoms on the lattice thermal conductivity
show that only at the highest concentrations of impurity (i.e., 1028m−3 for H and 1027m−3 for N), the
lattice thermal conductivity will decrease with increasing impurity concentration. This is similar to
the deﬁnition of the threshold of dislocation density. Here the threshold of impurity concentration
is calculated as 1028m−3 and 1027m−3 for H and N impurities, separately. Simulations of the
lattice thermal conductivity of H atomic concentration at 41.2% match the the ﬁtting results

82

with experimental data considering the eﬀect of phonon-impurity scattering. However, there are
still some discrepancy on the impurity concentration. The reason might be that the measured
thermal conductivity above 3 K remained the same after saturation of H atoms, which does not
agree with the BRT theory in the electronic dominated temperature range, where electrons are
mainly scattered with impurities and the thermal conductivity will decrease with the increase of
impurity concentration. Therefore the data extraction overestimate the eﬀect of impurities on the
lattice thermal conductivity. Various scattering relaxation time expressions for phonon-impurity
scattering have been listed for point defect, substantial impurity at diﬀerent mass, separately. The
eﬀect of grain boundary scattering is also studied and it will aﬀect the lattice thermal conductivity
for poly-crystal Nb samples because of the increasing number of grains than single crystal samples.

83

CHAPTER 6

CALCULATION OF THE LATTICE THERMAL CONDUCTIVITY OF NB

CONSIDERING PHONON DISPERSION

6.1 Introduction

Lattice thermal conductivity of superconducting Nb consists of phonon-boundary scattering
and phonon-electron scattering for samples before deformation, without the eﬀect of dislocations.
A term considering phonon-dislocation scattering has been added in the model from Koechlin and
Bonin [41] to account for the eﬀect of deformation [43] on the lattice thermal conductivity of
superconductors. This eﬀect was studied in Chapter 3. Numerical simulation using Monte Carlo
simulation of the lattice thermal conductivity of superconducting Nb is robust and takes account of
individual scattering mechanisms as well as including the phonon dispersion. It is also convenient
for complex structures. This model was studied in Chapter 4. However, this method is not a
closed form solution to the BTE so will have some statistic errors. Investigation of the closed form
solution of the BTE is critical to understand the mechanism in contributing to the lattice thermal
conductivity of superconductors.

6.2 Model

The Callaway model [95] has been widely used to model the thermal conductivity of dielectric
materials and semiconductors with the relaxation time approximation. The TW model [80] which
considers the condensation of electrons into Cooper pairs is often used for high Tc superconductors.
These two models are similar, and the general form is expressed as

T3Þ ΘD/T

0

kg =

k3
B
3

kB
2
2π
vg

τ exp(x)x4
(exp(x) − 1)2 dx

(6.1)

where x = ω
is the non-dimensional frequency, vg is the group velocity, here taken as a constant
kBT
with the value of sound velocity (no dispersion considered), τ is the overall relaxation time of

84

Figure 6.1: Phonon dispersion of Nb at [1 0 0] direction, replotted from the experimental data of
[14]

diﬀerent scattering mechanisms, including phonon-phonon scattering, phonon-electron scattering,
phonon-impurity scattering, phonon-boundary scattering (phonon-grain boundary scattering), and
phonon-dislocation scattering, and ΘD is the Debye temperature, here taken as ΘD=275 K for Nb.
The model in Eq. 6.1 does not include phonon dispersion and assumes isotropy in the longitudi-
nal and transverse acoustic directions. The following model includes phonon dispersion and treats
the phonon ensemble separately to calculate the lattice thermal conductivity of superconducting
Nb and is expressed as the following

kg =

1
2kBT2

2π

0

where ω is the phonon frequency, q is the wave vector.

85

Þ qmax

(ω)2

ω

2

(cid:17)
τ exp(cid:16) ω
kBT ) − 1(cid:105)2 dq
(cid:104)exp( ω

kBT

(6.2)

00.511.52Wavevector (m-1)101001234567Frequency (Hz)1012LA BranchTA Branch6.2.1 Phonon dispersion of Nb

To account for the eﬀect of phonon dispersion, two polarization branches have been included in
the integration, namely longitudinal acoustic (LA) branch and transverse acoustic (TA) branch,
where LA branch has one direction and TA branch has two directions. The experimental phonon
dispersion of Nb is replotted from reference [14], as shown in Fig. 6.1. The optical direction is
neglected in the expression because of its low group velocity to contribute to the lattice thermal
conductivity [114]. For each wave vector q, the group velocity vg is related to ω following the
relation of dω/dq = vg. This allows consideration of variable phonon group velocities.

The relaxation times of diﬀerent scattering mechanisms are expressed as the following to be

inserted into the extended Callaway model of Eq. 6.2.

6.2.2 Phonon-boundary scattering

The phonon-boundary scattering accounts for the traveling of phonon particles when they hit the
physical boundary of the specimen. This scattering is considered as elastic scattering, therefore
the energy is conserved by the scattering event. The properties (frequency, wave vector, and
the magnitude of group velocity) of the scattered phonons will remain the same. The phonons
will reﬂect back either specularly or diﬀusively depending on the specularity of the surface. The
relaxation time τpb is expressed as

(6.3)

−1
τ
pb

=

v(1 + p)
l(1 − p)

where p is the specularity of the surface which relates the scattering directionality to the surface
roughness. If p = 0, it means purely diﬀusive reﬂection, p = 1 refers to purely specular reﬂection.

6.2.3 Phonon-electron scattering

Phonon-electron scattering occurs when the traveling phonons collide with the normal electrons
(note that electrons which form in Cooper pairs do not scatter phonons). This scattering is
considered as unharmonic collision process, which is also called the inelastic scattering. Therefore,

86

the energy is not conserved after the scattering, so the properties (frequency, wave vector, and group
velocity) of the scattered phonons will be changed. In the Monte Carlo simulation, this process is
carried out during the re-sampling process of the phonons using random numbers. A new phonon
will be drawn in the corresponding temperature range to replace the old phonon with new properties.
The expression of the relaxation time of phonon-electron scattering (τpe) is
8 × 3.83)E xT

−1
pe = g(x, T)E xT = 2(exp(y) − 1)−1(1 −

τ

x

(6.4)

where g(x, t) quantiﬁes the ratio of the normal-conducting relaxation time to superconducting
relaxation time and is a monotonically decreasing function of the non-dimensional frequency x
(x = w/(kBT)), E is a ﬁtting parameter that is found to be in the range of 1.4 − 3 × 109 K−1s−1
[61]. The constant 8 in the expression is obtained by comparing the integrated thermal conductivity
due to phonon-electron scattering only with the thermal conductivity from the model of reference
[41].

6.2.4 Phonon-dislocation scattering

The phonon-dislocation scattering happens when the traveling phonons collide with the dislocations,
either screw dislocations or edge dislocations. This scattering process is also considered as elastic
scattering, therefore the energy is conserved by the scattering event. The properties (frequency,
wave vector, and the magnitude of group velocity) of the scattered phonons will remain the same,
while the direction of the group velocity is resampled using random numbers in the Monte Carlo
simulation. The expression for the phonon-dislocation scattering relaxation time τpd is

−1
τ
pd

= A0γ

2b2Nd w

(6.5)

where γ is the Grüneison constant (γ = 1.4 for Nb [61]), b is the Burgers vector, and A0 is
a constant related to the dislocation mechanisms (i.e., edge dislocations or screw dislocations).
By applying the relaxation time of phonon-dislocation scattering to the Callaway model [95] and
comparing the expression of the resistance in the Klemens model [44], A0 is calculated to be

87

1.1398 for edge dislocations and 0.8639 for screw dislocations. Screw dislocations usually appear
after deformation at temperatures lower than room temperature, whereas edge dislocations appear
following high temperature deformation [4].

6.2.5 Total relaxation time

The total relaxation time of diﬀerent scattering mechanisms studied above is obtained by using the
Matthiessen’s rule as

−1
−1 = τ
pb

τ

−1
pe + τ

−1
pd

+ τ

(6.6)

This total relaxation time in Eq. 6.6 is inserted in the extended Callaway of Eq. 6.2 by considering
the phonon dispersion. The lattice thermal conductivity kg is calculated by integrating over the
wave vector from 0 to the maximum values. However, this model is limited by the requirement that
all of the relaxation times being considered to be of the same form as shown in Eq. 6.6.

6.3 Results and Discussions

Calculation of lattice thermal conductivity of superconducting Nb was performed for both

undeformed and deformed samples to account for the diﬀerent scattering mechanisms.

Calculation results agreed well with the data ﬁtting results of measurement for an example
sample of screw dislocations [4] before deformation and after 8% deformation, as shown in Fig. 6.2.
Here the parameters in the relaxation times, including specularity p in phonon-boundary scattering,
E in phonon-electron scattering and dislocation density Nd in phonon-dislocation scattering, are
extracted from ﬁtting the experimental data using Eq. 2.4. This data extraction process in Eq. 2.4
is to obtain the lattice thermal conductivity from the total thermal conductivity to compare with
the calculation results, and it is shown in Chapter 3 of the extended model by explicitly including
the phonon-dislocation scattering. Phonon peak thermal conductivity kpp decreases by about 50%
and the phonon peak temperature Tpp moves from 1.8 K before deformation to about 2 K after 8%
deformation.

88

Figure 6.2: Comparision between the calculation results and the data ﬁtting results for an example
sample of screw dislocations [4]for undeformed state and 8% tensile deformation. The group
velocity is taken as vg = dω/dq, the dislocation density is calculated based on the ﬁtting using Eq.
2.4.

This calculation was performed for another example of predominantly screw dislocations [4].
The calculation results agreed well with the data ﬁtting results of measurement for diﬀerent levels of
deformation, including undeformed, 3% deformation, 7.3% deformation, and 10.3% deformation,
as shown in Fig. 6.3. Here, the parameters in the relaxation times, including specularity p in
phonon-boundary scattering, E in phonon-electron scattering and dislocation density Nd in phonon-
dislocation scattering, are extracted from ﬁtting the experimental data using Eq. 2.4. A phonon
peak appears at about 1.8 K for the undeformed state, and it shifts to 1.9 K after 3% deformation,
2 K after 7.3% deformation, and 2.2 K after 10.3% deformation. In addition, kpp is about 60 W
m-1 K-1before deformation, it decreases to about 40 W m-1 K-1after 3% deformation, it further
decreases to about 17 W m-1 K-1after 7.3% deformation and about 8 W m-1 K-1after 10.3%

89

012345Temperature (K)010203040Lattic  k  (Wm-1K-1)Undeformed CalculationUndeformed Fitting8% Deformation Calculation8% Deformation FittingFigure 6.3: Comparision between the calculation results and the data ﬁtting results for another
example sample of screw dislocations [4]for undeformed state and several levels of deformation.
The group velocity is taken as vg = dω/dq, the dislocation density is calculated based on the ﬁtting
using Eq. 2.4.

deformation.

Calculations of the lattice thermal conductivity of materials with of predominantly edge dislo-
cations were performed for the two samples from reference [4], separately. The calculation results
agreed well with the data ﬁtting results of measurement at diﬀerent levels of deformation, including
undeformed and 4% deformation for sample H2 and undeformed, 1% deformation, 2.4% deforma-
tion for sample K10, as shown in Fig. 6.4 and Fig. 6.5. Here, the parameters in the relaxation times
are also extracted from ﬁtting the experimental data using Eq. 2.4 to obtain the lattice thermal
conductivity. The phonon peak is at about 1.8 K for undeformed state, and it shifts to 1.9 K after
4% deformation for sample H2 or after 1% deformation for sample K10. The Tpp moves to about

90

012345Temperature (K)0102030405060Lattice  k  (Wm-1K-1)Undeformed CalculationUndeformed Fitting3% Deformation Calculation3% Deformation Fitting7.3% Deformation Calculation7.3% Deformation Fitting10.3% Deformation Calculation10.3% Deformation FittingFigure 6.4: Comparision between the calculation results and the data ﬁtting results for an example
sample of edge dislocations (sample H2) [4]for the undeformed state and 4% tensile deformation.
The group velocity is taken as vg = dω/dq, the dislocation density was calculated based on the
ﬁtting using Eq. 2.4.

2.4 K after 2.4% deformation for sample K10. In addition, kpp is about 85 W m-1 K-1for sample
H2 before deformation, and it decreases to about 22 W m-1 K-1after 4% deformation. For sample
K10, the lattice thermal conductivity for sample K10 before deformation is around 57 W m-1 K-1,
it decreases to be about 33 W m-1 K-1after 1% deformation, and it further decreases to about 10 W
m-1 K-1after 2.4% deformation.

6.4 Conclusion

Lattice thermal conductivity dominates the total thermal conductivity at low temperatures
(T < 2 K) for superconducting Nb. An enhanced closed form model based on Callaway is proposed
to include the phonon dispersion relation of Nb and also consider diﬀerent scattering mechanisms,

91

012345Temperature (K)020406080100Lattice  k (Wm-1K-1)Undeformed CalculationUndeformed Fitting4% Deformation Calculation4% Deformation FittingFigure 6.5: Comparision between the calculation results and the data ﬁtting results for another
example sample of screw dislocations (sample K10) [4]for the undeformed state and 1% and 2.4%
tensile deformation. The group velocity is taken as vg = dω/dq, the dislocation density was
calculated based on the ﬁtting using Eq. 2.4.

including phonon-boundary scattering, phonon-electron scattering, and phonon-dislocation scat-
tering. Calculation of the lattice thermal conductivity of superconducting Nb match well with the
ﬁtting results using Eq. 2.4 for undeformed samples and deformed samples, separately. For the
comparison of deformed samples, both edge dislocations and screw dislocations are considered. A
phonon peak appears at around 1.8 K for undeformed samples and shifts to warmer temperature
with the increase of deformation, while the lattice thermal conductivity kpp decreases with the
increase of deformation. Edge dislocations seem to decrease the lattice thermal conductivity more
than the screw dislocations.

92

012345Temperature (K)0102030405060Lattice  k  (Wm-1K-1)Undeformed CalculationUndeformed Fitting1% Deformation Calculation1% Deformation Fitting2.4% Deofrmation Calculation2.4% Deformation FittingCHAPTER 7

THERMAL CONDUCTIVITY OF SUPERCONDUCTING TANTALUM

7.1 Introduction

Tantalum (Ta) is a part of the refractory metals group and also a superconducting metal, similar
to Nb. Because Nb and Ta are found together, there is always Ta present in SRF cavities, so
this provides some intrinsic interest and motivation for understanding the thermal conductivity
of Ta. Ta and Nb behave like twins in nature and they have very similar physical and chemical
properties. For example, both of them are body-center cubic materials and have similar thermal
conductivity at room temperature. However, the transition temperature (Tc) from normal conductor
to superconductor of Ta is 4.48 K, which is about one half of that of Nb (Tc = 9.25 K) [67].

Thermal conductivity of Ta also consists of the phonon and electronic contributions. A typical
ﬁgure of these contributions to the total thermal conductivity of superconducting Ta is shown in Fig.
7.1. It shows that the electron contribution to the thermal conductivity of Ta is negligible below 1
K. Therefore, phonon dominated thermal conductivity (lattice thermal conductivity) can represent
the total thermal conductivity in this temperature range. A number of factors, including sample
size, specularity, free electrons, impurity concentration, and dislocation density Nd, determine
kg. Plastic deformation, which occurs in many manufacturing processes, increases Nd, while heat
treatments can reduce Nd [8].
7.2 Monte Carlo simulation

Lattice thermal conductivity of superconducting Nb before and after deformation have been
simulated in the previous chapter using the Monte Carlo method.
It is used here model the
lattice thermal conductivity of Ta. Similarly, the phonon dispersion relationship is needed for the
simulation.

To model the thermal transport of Ta at low temperatures, where phonons dominate the thermal

93

Figure 7.1: The thermal conductivity of Ta is the sum of the contributions of electrons and phonons
[3]. Below 1 K, the electronic contribution is negligible. The contribution of phonons dominates
in this temperature range because of the condensation of free electrons into Cooper pairs.

Table 7.1: Parameters for the Ta samples used in the simulation [3], where the size refers to the
diameter of the cylinder. All of the three Ta samples were deformed at temperature higher or

equal to 295 K, therefore they are likely to have edge dislocations.

RRR Deformation T

Sample
Ta 1
Ta 2
Ta 3
Nb 1
Nb 2
Nb 3

Size
3.8 mm 185
3.3 mm 111
3.9 mm 60
3.2 mm 350
3.2 mm 1200
3.9 mm 250

420 K
355 K
295 K
77 K
295 K
195 K

Dislocation type
Edge dislocations
Edge dislocations
Edge dislocations
Screw dislocations
Edge dislocations
Screw dislocations

conductivity, the phonon dispersion relation of Ta is plotted in Fig. 7.2 [15]. The samples used
for comparing with the simulation results are listed in Table 7.1, where the conditions before
deformation and after several levels of deformation are described for samples Ta 1, Ta 2, and Ta 3.
The relaxation times used in the simulation are similar to those of Nb. That is, for undeformed

94

00.511.52Temperature (K)0510152025Thermal conductivity (Wm-1K-1)Phonon contribution to kElectron contribution to kTotal thermal conductivityFigure 7.2: Phonon dispersion relation of Ta in the [1 0 0] direction, replotted from [15],  = aq/π,
where q is the wave vector,  is the reduced wave vector, a is the lattice parameter, which refers to
the physical dimension of a unit cell in a crystal lattice. For Ta, a ≈ 3.3 × 10−10 m, the same as
Nb, LA refers to longitudinal acoustic phonons, and TA refers to transverse acoustic phonons.

sample, the phonon-boundary scattering and phonon-electron scattering are considered, but the
phonon-dislocation scattering is neglected. For samples after deformation, all of the three scattering
mechanisms, including phonon-boundary, phonon-electron, and phonon-dislocation scattering, are
considered in the simulation.

Simulation results of the lattice thermal conductivity of undeformed Ta and samples after several
levels of deformation are compared using measurements in Figs. 7.3-7.5, separately. Measurements
were ﬁtted using Eq. 2.19 for undeformed samples and Eq. 2.4 for samples after deformation, and
the phonon contributions are shown for the lattice thermal conductivity of Ta. The phonon peak
thermal conductivity for Ta typically appears at around 1 K [4], which is approximately half that of

95

0246810Wave vector (m-1)1090123456Frequency (Hz)1012LATAFigure 7.3: Comparison between the Monte Carlo simulation results of lattice thermal conductivity
of Ta 1 with the ﬁtting results using Eq. 2.4 from [3], from undeformed to 2%, 3.1%, and 7.3%
deformation.

Nb. The simulation results agree well with the ﬁtted data for undeformed samples Ta 1, Ta 2, and Ta
3 when including phonon-boundary scattering, phonon-electron scattering, but neglecting phonon-
dislocation scattering. For samples after deformation, i.e., 2% deformation and 3.1% deformation
for sample Ta 1, 0.5%, 1.2%, 2.4%, 3.9%, and 6.2% deformation for Ta 2, and 1% deformation for
sample Ta 3, the simulation results also match well with the ﬁtted data. The phonon peak occurs at
around 0.9 K in the simulation for all of the three samples Ta 1, Ta 2, and Ta 3 before deformation and
it shifts to about 1 K after 2% deformation for Ta 1, 0.5% deformation for Ta 2, and 1% deformation
for Ta 3. After 3.1% deformation for sample Ta 1 and 1.2% deformation for Ta 2, the peaks move
to around 1.1 K. The phonon peak is broad and low after 7.3% deformation for Ta 1 and after
2.4% deformation for Ta 2. The reason for the existence of a phonon peak after large deformation

96

00.511.522.5Temperature (K)024681012Lattice k (Wm-1K-1)MC =0Fitting =0MC =2%Fittng =2%MC =3.1%Fitting =3.1%MC =7.3%Fitting =7.3%TaFigure 7.4: Comparison between the Monte Carlo simulation results of lattice thermal conductivity
of Ta 2 with the ﬁtting results using Eq. 2.4 from [3], from undeformed to 0.5%, 1.2%, 2.4%, 3.9%,
and 6.2% deformation.

is because of the phonon contributions only, the total thermal conductivity will lose the phonon
peak after large deformation (e.g., 7.3% deformation for Ta 1). Lattice thermal conductivity at
temperatures lower than the phonon peak temperature is dominated by phonon-boundary scattering
and follows the T3 relation. At temperatures above the phonon peak, phonon-dislocation scattering
is taking over with larger amounts of deformation. Phonon-electron scattering is contributing to
the lattice thermal conductivity after 1 K and will dominate the conductivity after 1.5 K. These
results are similar to that of Nb for the contributions from diﬀerent scattering mechanisms.

97

00.511.522.5Temperature (K)02468101214Lattice k (Wm-1K-1)Fitting =0MC =0Fitting =0.5%MC =0.5%Fitting =1.2%MC =1.2%Fitting =2.4%MC =2.4%Fitting =3.9%MC =3.9%Fitting =6.2%MC =6.2%TaFigure 7.5: Comparison between the Monte Carlo simulation results of lattice thermal conductivity
of Ta 3 with the ﬁtting results using Eq. 2.4 from [3], from undeformed to 1% deformation.

7.3 Comparison of total k between Nb and Ta

It has been demonstrated that the model of Eq. 2.4 matches better with the experimental data
than Eq. 2.19 of K&B model [41], it is used to model the thermal conductivity of both Nb and Ta
in the following.

Comparison of data from [4, 16] with ﬁtting of the model (Eq. 2.4) are shown in Fig. 7.6 for
superconducting Nb and superconducting Ta, both before and after deformation. The ﬁgures show
that after deformation, thermal conductivity of both Nb and Ta decrease because of the increase of
dislocation density, where the temperatures at the phonon peak (Tpp) increase with deformation for
both Nb and Ta. The phonon peak disappears after a certain amount of deformation. For example,
the phonon peak disappears after 7.3% deformation for these Nb and Ta samples. After 10.3%

98

00.511.522.5Temperature (K)0246810Lattice k (Wm-1K-1)Fitting =0MC =0Fitting =1%MC =1%Figure 7.6: Comparison between ﬁtting with the experimental data of the thermal conductivity
of undeformed and deformed superconducting Nb [4] and Ta [16]. The dislocation density is
calculated during the ﬁtting processes.

deformation for this Nb sample, the thermal conductivity increases proportionally with temperature.
This temperature relation is not with T3 because both boundary scattering and dislocation scattering
are important to contribute to the lattice thermal conductivity, and boundary scattering contributes
to kg as T3 while dislocation scattering contributes to kg as T2. Another interesting similarity
between Nb and Ta is that kpp occurs at T/Tc ≈0.2 and klm (local minimum) occurs at T/Tc ≈0.3
for both Nb and Ta. However, at T/Tc ≈0.4, the thermal conductivity of Ta is greater than the value
at the phonon peak, while the thermal conductivity of Nb is smaller than the value at its phonon
peak.

7.3.1 Comparison of simulated lattice k between undeformed Nb and Ta

SRF cavities can be formed using large grain Nb sheet. These Nb sheets are deformed and then
welded to make the cavities and then heat treated to get rid of the impurities and also the dislocations.
Before deformation, the dislocation density is usually low and the deformation will increase the
dislocation density.

Monte Carlo simulation results of the lattice thermal conductivity of superconducting Nb
and Ta samples before deformation are shown in Fig. 7.7 by comparing with the model of Eq.
2.4. The simulation results match well with the ﬁtted model by including the eﬀect of phonon-

99

00.10.20.30.4T/Tc01020304050Thermal conductivity (Wm-1K-1)=0 Data=0 Nd=5.1×1012=3% Data=3% Nd=8.6×1012=7.3% Data=7.3% Nd=4.8×1013=10.3% Data=10.3% Nd=1.6×1014NbT=3.70 K00.10.20.30.4T/Tc051015Thermal conductivity (Wm-1K-1)=0 Data=0 Nd=2.2×1010=2% Data=2% Nd=1.2×1013=3.1% Data=3.1% Nd=2.4×1013=7.3% Data=7.3% Nd=9.4×1013TaT=1.79 KFigure 7.7: Comparison between Monte Carlo simulation of the lattice thermal conductivity with
the ﬁtting results using Eq. 2.4 for undeformed Nb [4] and undeformed Ta [16]. The parameters,
including size, RRR, deformation temperature, and the calculated dislocation density, of the Nb
and Ta samples are listed in Table 4.1.

boundary scattering, phonon-electron scattering, and phonon-dislocation scattering. Here the
phonon-dislocation scattering is less important for undeformed samples with the other two scattering
mechanisms because of the low dislocation density before deformation. The phonon peak appears
at about 1.8 K for Nb and 0.9 K for Ta for all of the three samples. The temperature at the phonon
peak is close to 0.2Tc. Therefore the phonon peak at the undeformed Nb and Ta has small variations.

7.3.2 Comparison of simulated lattice k between deformed Nb and Ta

At undeformed states, the dislocation density is usually low so that it will not aﬀect the lattice thermal
conductivity eﬀectively compared with the other scattering mechanisms, i.e.phonon-boundary
scattering and phonon-dislocation scattering. However, the dislocation density will increase with
deformation and becomes more and more important in its eﬀect on the lattice thermal conductivity.
Simulation results of the lattice thermal conductivity of superconducting Nb and Ta samples
after deformation are shown in Fig. 7.8 by comparing with the model of Eq. 2.4. The simulation
results also match well with the ﬁtting model by including the eﬀect of phonon-boundary scatter-
ing, phonon-electron scattering and phonon-dislocation scattering. Here the phonon-dislocation
scattering is more important for deformed samples than the other two scattering mechanisms, due

100

11.522.533.5Temperature (K)010203040506070Lattice k (Wm-1K-1)H1 Fitting =0H1 MC =0H2 Fitting =0H2 MC =0K6 Fitting =0K6 MC =0Nb00.511.522.5Temperature (K)02468101214Lattice k (Wm-1K-1)Ta1 Fitting =0Ta1 MC =0Ta2 Fitting =0Ta2 MC =0Ta3 Fitting =0Ta3 MC =0TaFigure 7.8: Comparison between Monte Carlo simulation of the lattice thermal conductivity with
the ﬁtting results using Eq. 2.4 of undeformed and deformed Nb [4] and Ta [16]. The parameters,
including size, RRR, deformation temperature, and the calculated dislocation density, of the Nb
and Ta samples are listed in Table 4.1.

to the increase of dislocation density after deformation. Wasserbäch observed that sample Nb1
has predominantly screw dislocations while Nb2 has predominantly edge dislocations. He also
found low T (T<295 K) deformation leads to predominant screw dislocations and intermediate
temperature (T>295 K) results in predominant edge dislocations [4]. As the Ta samples were
deformed at T>295 K, which falls into the intermediate temperature range, they are likely to have
edge dislocations [16]. It is shown from the ﬁgure that kpp decreases with deformation and will
eventually disappear after large deformation, while Tpp increases with deformation for both Nb
and Ta. These results can account for the eﬀect of dislocations. It can be also seen that the lattice
thermal conductivity of Nb has a greater value than Ta in similar conditions. Also, for a given
deformation, the ratio of the thermal conductivity to the undeformed state (k/k=0) for Ta is smaller
than that in Nb.

7.4 Conclusion

Monte Carlo solutions of the BTE have been successfully performed to model the lattice thermal
conductivity of superconducting Ta before and after deformation given the phonon dispersion
relationship of Ta. Diﬀerent scattering mechanisms, including phonon-boundary, phonon-electron,
and phonon-dislocation scattering, which contribute to the lattice thermal conductivity of Ta,

101

11.522.533.5Temperature (K)01020304050Lattice k  (Wm-1K-1)Fitting =0MC =0Fitting =3%MC =3%Fitting =7.3%MC =7.3%Fitting =10.3%MC =10.3%Nb00.511.522.5Temperature (K)024681012Lattice k (Wm-1K-1)MC =0Fitting =0MC =2%Fittng =2%MC =3.1%Fitting =3.1%MC =7.3%Fitting =7.3%Tawere used in the simulation to calculate the lattice thermal conductivity of Ta for samples before
and after several deformations. The simulation results agree well with the results extracted from
experiments [3] for diﬀerent samples both before and after deformation. The results show that
boundary scattering dominates for temperatures smaller than 0.5 K, where the phonon mean free
path is comparable to the size of the sample, and that phonon-electron scattering is important when
the temperature is greater than 1 K and dominates the conductivity for temperatures greater than
1.3 K. Phonon-dislocation scattering aﬀects the lattice thermal conductivity around the phonon
peak temperatures (between 0.7 K and 1.1 K) and large amount of deformation leads to a lower and
broader peak in this temperature range. The phonon peak appears at temperatures of approximately
0.9 K in undeformed samples and shifts to higher temperature after small amount of deformation.
Phonon-dislocation scattering decreases the lattice thermal conductivity and a large amount of
deformation destroys the phonon peak.

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CHAPTER 8

CONCLUSIONS

An extended model based on that of the Koechlin and Bonin model [41] has demonstrated that
dislocation density Nd needs to be considered explicitly, especially in deformed Nb (a type II
superconductor) and Ta (a type I superconductor). This model is based on BCS [39] (and BRT
[42]) theory, and the dislocation term is based on the work of Klemens [44]. Examining the thermal
conductivity of Nb and Ta using this extended model suggests that normal electrons determine k
for T (cid:38) 0.3Tc, while from 0.1Tc (cid:46) T (cid:46) 0.25Tc, phonons scattered by dislocation dominate k, and
phonons scattered by boundaries dominate k for T (cid:46) 0.1Tc. The extended model also shows that
there are threshold dislocation densities for Nb and Ta. Below these thresholds, the contributions
from dislocation scattering of phonons can be neglected in the calculation of lattice k even in the
range of 0.1Tc (cid:46) T (cid:46) 0.25Tc. Furthermore, k demonstrates a local maximum, often called the
phonon peak kpp at a temperature Tpp. The magnitude of kpp decreases with increasing Nd while
Tpp increases with increasing Nd, until kpp disappears.

Although k is correlated with RRR for T (cid:38) 0.3Tc (speciﬁcally k(4.2 K) = RRR/4), a few
measurements [4, 8] have suggested that kpp may be correlated with RRR. See, for example, Fig.
2.5. This result is unexpected based on the BRT model of k of superconductors [42] and the theory
of metals [37]. The extended model oﬀers an explanation based on the results of Wasserbäch [4]
and Chandrasekaran [8]. The coeﬃcient of the phonon-boundary scattering term varies with RRR.
In samples with large kpp, the dislocations have little eﬀect and the value of the kpp is a result of
the balance between phonon-boundary and phonon-electron scattering.

The Callaway model [95] for the lattice thermal conductivity has been extended to include
a phonon dispersion relation. This allows consideration of variable phonon group velocities.
However, this model is limited by the requirement that all of the relaxation times being considered
to be of the same form. The Callaway model is also most suited to simple regular geometries, e.g.,
rectangular, spherical, or cylindrical geometries. Hence, an energy-based, variance-reduced Monte

103

Carlo solution [126] of the BTE for phonons has been developed, incorporating a relaxation time
approximation and the phonon dispersion relation. The Monte Carlo simulation has the advantage
of being able to model the scattering mechanisms individually, as well as the possibility to be applied
to more complex geometries. Simulations of lattice k of undeformed and deformed Nb and Ta match
well with the results of the extended Koechlin and Bonin model [41] (Eq. 2.4) when including the
relaxation times for phonon-boundary, phonon-electron, and phonon-dislocation scattering. The
Monte Carlo simulation has also be used to investigate the eﬀect on the lattice k of saturating Nb
with hydrogen. The simulation results match well with measured results for Nb saturated with H,
bolstering the argument that H impurity aﬀect the lattice k at near saturation.

Future work includes measuring the dislocation density of predominant screw and edge dis-
locations to verify using the extended K& B model (Eq. 2.4) to estimate the dislocation density
for various levels of deformation. It will be also important to apple the developed Monte Carlo
simulation method to superconducting thin ﬁlm Nb and other superconductors with higher critical
temperature, e.g.. Nb3Sn and NbN. After this, measuring the thermal conductivity of Nb with
diﬀerent levels of deformation and also Nb thin ﬁlms will greatly help verify the extended model
and the Monte Carlo simulation.

104

APPENDIX

105

A.1 Main program

1 clc; %% Clear

2 tic; %% Calculate computing time

3 %%%%%%%%%%%% Parameters

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq Et;

%% Global values

5 l=9.75*10^(-5); a=5*10^(-6); b=5*10^(-6);

% dimension of the cell ...

unit: m

% Planck constant

% Boltzmann constant

% Volume, consider a rectangle

6 V=l*a*b;
7 A=a*b; % Surface area of the first and last wall (consider isothermal)
8 h=6.63*10^(-34);
9 hbar=h/(2*pi); % Reduced Planck constant
10 kb=1.38*10^(-23);
11 vs=[909 384.6]*2*pi;
12 c=[-3.04*10^(-8) -5.52*10^(-10)]*2*pi;
13 W=0.3*10^(-17); % scaling factor
14 qmax=1.9*10^10; % Maximum Wavenumber
15 wmax=[6.795*10^12; 7.1075*10^12]*2*pi;
16 qh=0.5*qmax;
17 wth=vs(2)*qh+c(2)*qh^2; % w1/2=w(1/2*qmax) for TA phonons
18 dw=wmax(2)/1000;

% sound velocity in LA and TA phonons

% constant to calculate velocity

% Maximum Frequency

% Half maximum Frequency

% Frequency intervel

19 g=[1 2];

20 Th=2.05;

21 Tc=1.95;

22 Teq=2.0;

23 for i=1:956

% Degenarations, LA or TA

% Hot side Temperature

% Cold side Temperature

% Reference Temperature

w1(i)=i*dw; % Discreate frequency LA 1000 intervels (1-956)

24

25 end

26 for i=957:1:1000

w1(i)=wmax(1)-(i-956)*dw;

% intervels (957-1000)

27

28 end

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29 for i=1:1000

w2(i)=i*dw; % Discreate frequency TA 1000 intervels

30

31 end

32 dt=1*10^(-8);
33 for i=1:40

% time step

Td(i)=-(Th-Tc)/39*i+Th+(Th-Tc)/39;

% Diffusion limit of T profile

Tb(i)=((Th^4+Tc^4)/2)^0.25; % Ballistic limit of T profile

34

35

36 end

37 %%%%%%%%%% end of parameters

38 %%%%%%%%%%%%% Initialization

39 for i=1:1:956

q1(i)=(-vs(1)+sqrt(vs(1)^2+4*c(1)*w1(i)))/(2*c(1)); % Wavenumber ...

distribution 1-956

v1(i)=vs(1)+2*c(1)*q1(i);

% Group Velocity distribution 1-956

40

41

42 end

43 for i=957:1:1000

44

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q1(i)=(-vs(1)-sqrt(vs(1)^2+4*c(1)*w1(i)))/(2*c(1)); % Wavenumber ...

distribution 957-1000

v1(i)=abs(vs(1)+2*c(1)*q1(i));

% Wavenumber distribution 957-1000

46 end

47 for i=1:1000

q2(i)=(-vs(2)+sqrt(vs(2)^2+4*c(2)*w2(i)))/(2*c(2)); % Wavenumber ...

distribution 1-1000

v2(i)=vs(2)+2*c(2)*q2(i);

% Velocity distribution

1-1000

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50 end

51 %%%%%%%%%%%%% hot side initialization

52 for i=1:1000

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nlh(i)=hbar*w1(i)*(1/W)*V*(1/(exp((hbar*w1(i)/(kb*Th)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i)^2)/(2*pi^2*v1(i)))*dw;

...

% LA number for hot side, positive energy

nth(i)=hbar*w2(i)*(1/W)*V*(1/(exp((hbar*w1(i)/(kb*Th)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q2(i)^2)/(2*pi^2*v2(i)))*dw*2; ...

% TA number for hot side, positive energy

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nh(i)=nlh(i)+nth(i);

% Total number for hot side

Ph(i)=nlh(i)/nh(i); % probability of LA for hot side

57

58

59 end

60 Nhh=sum(nh);

61 Nh=floor(sum(nh)); % number of phonons for hot side

62 Mh(1)=nh(1);

63 Fh(1)=0;

64 for i=2:1:1000

Mh(i)=Mh(i-1)+nh(i);

% summation of ith phonon numbers

Fh(i)=Mh(i)/Nhh;

% determine which intervel

65

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67 end

68 %%%%%%%%%%% Hot side only has one cell

69 for j=1:Nh

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R=rand; % random number

a1=1;

% frequeny invervel No. 1

b1=1000;

% frequeny invervel No. 1000

while((R-Fh(a1))*(Fh(a1+1)-R)<0)

% bisection method to ...

determine which interval

mm=round((a1+b1)/2);

% get integer

if (abs(R-Fh(a1))<abs(R-Fh(b1)))

if (R>Fh(mm))

a1=mm;

b1=mm;

else

end

else

if (R<Fh(mm))

b1=mm;

a1=mm;

else

end

end

end

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R1=rand;

%random number

R2=rand;

%random number

f(j)=w1(a1)+(2*R2-1)*dw/2;

% bisection method to determine ...

which interval

if (R1<Ph(a1)) % Polarization (LA)

ml(j)=(-vs(1)+sqrt(vs(1)^2+4*c(1)*f(j)))/(2*c(1));

% LA ...

wavevector (assume isotropy)

mt(j)=0;

% TA wavevector=0

vl(j)=vs(1)+2*c(1)*ml(j);

% LA magnitude group velocity

vt(j)=0;

% TA velocity=0

else

% Polarization (TA)

mt(j)=(-vs(2)+sqrt(vs(2)^2+4*c(2)*f(j)))/(2*c(2));

% TA ...

wavevector (assume isotropy)

ml(j)=0;

% LA wavevector=0

vt(j)=vs(2)+2*c(2)*mt(j);

% TA magnitude group velocity

vl(j)=0;

% LA velocity=0

end

v(j)=(vt(j)+vl(j)); % total velocity

m(j)=mt(j)+ml(j);

% total wave number

R1=rand;

% random number

R2=rand;

% random number

R3=rand;

% random number

R4=rand;

% random number

R5=rand;

% random number

K2=2*pi*R4; % angle 1

psi

K1=2*R5-1; % angle 2

cos(theta)

vg(j,2)=v(j)*sqrt(1-K1^2)*cos(K2);

% direction of velocity ...

sin(theta)*cos(psi)

vg(j,3)=v(j)*sqrt(1-K1^2)*sin(K2);

% direction of velocity ...

sin(theta)*sin(psi)

vg(j,1)=v(j)*K1;

% direction of velocity cos(theta)

x1(j)=(R1)*l;

% location of particles in cell 1

x2(j)=(R2)*a;

% location of particles in cell 1

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x3(j)=(R3)*b;

% location of particles in cell 1

s(j)=1; % Sign of particles 1(T>Teq), -1(T<Teq)

120 %%%%%%%%%%%%% end of hot side initialization

121 %%%%%%%%%%%% cold side initialization, cold side has 19 cells

122 for i=1:1000

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nlc(i)=-hbar*w1(i)*1/W*V*(1/(exp((hbar*w1(i)/(kb*Tc)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i)^2)/(2*pi^2*v1(i)))*dw;

...

% LA number for hot side

ntc(i)=-hbar*w2(i)*1/W*V*(1/(exp((hbar*w1(i)/(kb*Tc)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q2(i)^2)/(2*pi^2*v2(i)))*dw*2; ...

% TA number for hot side

nc(i)=nlc(i)+ntc(i); % Total number for hot side

Pc(i)=nlc(i)/nc(i); % probability of LA for hot side

129 end

130 Ncc=sum(nc);

131 Nc=floor(sum(nc)); % number of phonons for hot side

132 Mc(1)=nc(1);

133 Fc(1)=0;

134 for i=2:1:1000

Mc(i)=Mc(i-1)+nc(i);

% summation of ith phonon numbers

Fc(i)=Mc(i)/Ncc;

% to determine phonon belong to which intervel

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137 end

138 N=Nh+Nc*39; % Total number of phonons
139 %%%%%%%%%%% cold side has 39 cells

140 for i=2:1:40

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for j=1:1:Nc

R=rand; %random number

a1=1;

% frequeny invervel No. 1

b1=1000; % frequeny invervel No. 1000

while((R-Fc(a1))*(Fc(a1+1)-R)<0)

mm=round((a1+b1)/2);

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if (abs(R-Fc(a1))<abs(R-Fc(b1))) % bisection method to ...

determine which interval

if (R>Fc(mm))

a1=mm;

b1=mm;

else

end

else

if (R<Fc(mm))

b1=mm;

a1=mm;

else

end

end

end

R1=rand;

R2=rand;

f(Nh+(i-2)*Nc+j)=w1(a1)+(2*R2-1)*dw/2;

if (R1<Pc(a1)) % Polarization (LA)

ml(Nh+(i-2)*Nc+j)=(-vs(1)+sqrt(vs(1)^2+4*c(1)*f(Nh+(i-2)*Nc+j)))/(2*c(1)); ...

% LA wavevector (assume isotropy)

mt(Nh+(i-2)*Nc+j)=0;% TA wavevector=0

vl(Nh+(i-2)*Nc+j)=vs(1)+2*c(1)*ml(Nh+(i-2)*Nc+j);

% LA ...

magnitude group velocity

vt(Nh+(i-2)*Nc+j)=0;

% TA velocity=0

else

end

mt(Nh+(i-2)*Nc+j)=(-vs(2)+sqrt(vs(2)^2+4*c(2)*f(Nh+(i-2)*Nc+j)))/(2*c(2)); ...

% TA wavevector (assume isotropy)

ml(Nh+(i-2)*Nc+j)=0; % LA wavevector=0

vt(Nh+(i-2)*Nc+j)=vs(2)+2*c(2)*mt(Nh+(i-2)*Nc+j); % TA ...

magnitude group velocity

vl(Nh+(i-2)*Nc+j)=0; % LA velocity=0

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v(Nh+(i-2)*Nc+j)=vt(Nh+(i-2)*Nc+j)+vl(Nh+(i-2)*Nc+j); % total ...

velocity

m(Nh+(i-2)*Nc+j)=mt(Nh+(i-2)*Nc+j)+ml(Nh+(i-2)*Nc+j); % total ...

wave number

R1=rand; % random number

R2=rand; % random number

R3=rand; % random number

R4=rand; % random number

R5=rand; % random number

K2=2*pi*R4; % angle 1

psi

K1=2*R5-1; % angle 2

cos(theta)

vg(Nh+(i-2)*Nc+j,2)=v(Nh+(i-2)*Nc+j)*sqrt(1-K1^2)*cos(K2); % ...

direction of velocity sin(theta)*cos(psi)

vg(Nh+(i-2)*Nc+j,3)=v(Nh+(i-2)*Nc+j)*sqrt(1-K1^2)*sin(K2); % ...

direction of velocity sin(theta)*sin(psi)

vg(Nh+(i-2)*Nc+j,1)=v(Nh+(i-2)*Nc+j)*K1; % direction of ...

velocity cos(theta)

x1(Nh+(i-2)*Nc+j)=(i-1+R1)*l; % location of particles in cell 2-40

x2(Nh+(i-2)*Nc+j)=(R2)*a; % location of particles in cell 2-40

x3(Nh+(i-2)*Nc+j)=(R3)*b; % location of particles in cell 2-40

s(Nh+(i-2)*Nc+j)=-1; % Sign of particles 1(T>Teq), -1(T<Teq)

end

192 end

193 %%%%%%%%%%% end of cold side initialization

194 %%%%%%% Drift process

195 u1=0; % iteration times

196 d=0.996; % specularity

197 Ttol=1500*dt; % Total time
198 for t=dt:dt:Ttol

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for i=1:40

Et(i)=0; % Total energy=0

NNN(i)=0; % Number of particles in each cell before drifting

end

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Nd1=0; % number of particles to be deleted in cell 1

Nd40=0; % number of particles to be deleted in cell 40

for j=1:1:N % drift process

x1(j)=x1(j)+vg(j,1)*dt; % update coordinate 1

x2(j)=x2(j)+vg(j,2)*dt; % update coordinate 2

x3(j)=x3(j)+vg(j,3)*dt; % update coordinate 3
if (x2(j)≤0) % reach bottom boundary

R=rand; % random number

if (R<d) % specular reflection

x2(j)=-x2(j); % update coordinate 2

vg(j,2)=-vg(j,2); % particles bounce back

else

% diffusive reflection

R1=rand; % random number

R2=rand; % random number

K1=2*R1-1; % angle 1

K2=2*pi*R2; % angle 2

ddt=abs((x2(j))/(vg(j,2))); % time for reflection

x1(j)=x1(j)-ddt*vg(j,1); % update coordinate 1

x3(j)=x3(j)-ddt*vg(j,3); % update coordinate 2

x2(j)=x2(j)-ddt*vg(j,2); % update coordinate 3

vg(j,1)=v(j)*K1;% velocity random direction

vg(j,2)=abs(v(j)*sqrt(1-K1^2)*sin(K2)); % velocity ...

random direction

vg(j,3)=v(j)*sqrt(1-K1^2)*cos(K2); % velocity random ...

direction

x1(j)=x1(j) + ddt*vg(j,1); % update coordinate 1

x3(j)=x3(j) + ddt*vg(j,3); % update coordinate 1

x2(j)=x2(j) + ddt*vg(j,2); % update coordinate 1

end

elseif (x2(j)≥a) % reach top boundary

R=rand; % random number

if (R<d) % specular reflection

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x2(j)=2*a-x2(j); % update coordinate 2

vg(j,2)=-vg(j,2); % particles bounce back

else

% diffusive reflection

R1=rand; % random number

R2=rand; % random number

K1=2*R1-1; % angle 1

K2=2*pi*R2; % angle 2

ddt=abs((a-x2(j))/(vg(j,2))); % time for reflection

x1(j)=x1(j)-ddt*vg(j,1); % update coordinate 1

x3(j)=x3(j)-ddt*vg(j,3); % update coordinate 2

x2(j)=x2(j)-ddt*vg(j,2); % update coordinate 3

vg(j,1)=v(j)*K1; % velocity random direction

vg(j,2)=-abs(v(j)*sqrt(1-K1^2)*sin(K2)); % velocity ...

random direction

vg(j,3)=v(j)*sqrt(1-K1^2)*cos(K2); % velocity random ...

direction

x1(j)=x1(j) + ddt*vg(j,1); % update coordinate 1

x3(j)=x3(j) + ddt*vg(j,3); % update coordinate 1

x2(j)=x2(j) + ddt*vg(j,2); % update coordinate 1

end

end

if (x3(j)≤0) % reach left boundary

R=rand; % random number

if (R<d) % specular reflection

x3(j)=-x3(j); % update coordinate 3

vg(j,3)=-vg(j,3); % particles bounce back

else

% diffusive reflection

R1=rand; % random number

R2=rand; % random number

K1=2*R1-1; % angle 1

K2=2*pi*R2; % angle 2

ddt=abs(x3(j)/(vg(j,3))); % time for reflection

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x1(j)=x1(j)-ddt*vg(j,1); % update coordinate 1

x3(j)=x3(j)-ddt*vg(j,3); % update coordinate 2

x2(j)=x2(j)-ddt*vg(j,2); % update coordinate 3

vg(j,1)=v(j)*K1; % velocity random direction

vg(j,2)=v(j)*sqrt(1-K1^2)*sin(K2); % velocity random ...

direction

vg(j,3)=abs(v(j)*sqrt(1-K1^2)*cos(K2)); % velocity ...

random direction

x1(j)=x1(j) + ddt*vg(j,1); % update coordinate 1

x3(j)=x3(j) + ddt*vg(j,3); % update coordinate 1

x2(j)=x2(j) + ddt*vg(j,2); % update coordinate 1

end

elseif (x3(j)≥b) % reach right boundary

R=rand; % random number

if (R<d) % specular reflection

x3(j)=2*b-x3(j); % update coordinate 3

vg(j,3)=-vg(j,3); % particles bounce back

else

% diffusive reflection

R1=rand; % random number

R2=rand; % random number

K1=2*R1-1; % angle 1

K2=2*pi*R2; % angle 2

ddt=abs((b-x3(j))/(vg(j,3))); % time for reflection

x1(j)=x1(j)-ddt*vg(j,1); % update coordinate 1

x3(j)=x3(j)-ddt*vg(j,3); % update coordinate 2

x2(j)=x2(j)-ddt*vg(j,2); % update coordinate 3

vg(j,1)=v(j)*K1; % velocity random direction

vg(j,2)=v(j)*sqrt(1-K1^2)*sin(K2); % velocity random ...

direction

vg(j,3)=-abs(v(j)*sqrt(1-K1^2)*cos(K2)); % velocity ...

random direction

x1(j)=x1(j) + ddt*vg(j,1); % update coordinate 1

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x3(j)=x3(j) + ddt*vg(j,3); % update coordinate 1

x2(j)=x2(j) + ddt*vg(j,2); % update coordinate 1

end

end

if (x1(j)<l) % reach left (delete this phonon)

Nd1=Nd1+1; % put the deleted particle in cell 1

elseif (x1(j)>39*l) % reach right (delete this phonon)

Nd40=Nd40+1; % put the deleted particle in cell 40

end

Ndd=Nd1+Nd40; % total particle to be deleted

end

306 %%%%%%%%%%%%%%%%%%% end of drift process

307 %%%%% reinitialization

308 %%%%%%%%%%%%%%%%%%% Determine the properties of hot cell

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for j=1:Nh

R=rand; % random number

a1=1;

% frequeny invervel No. 1

b1=1000; % frequeny invervel No. 1000

while((R-Fh(a1))*(Fh(a1+1)-R)<0)

mm=round((a1+b1)/2); % get integer

if (abs(R-Fh(a1))<abs(R-Fh(b1))) % bisection method to ...

determine which interval

if (R>Fh(mm))

a1=mm;

b1=mm;

else

end

else

if (R<Fh(mm))

else

b1=mm;

a1=mm;

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end

end

end

R1=rand; %random number

R2=rand; %random number

f(N+j)=w1(a1)+(2*R2-1)*dw/2; % assign the frequency

if (R1<Ph(a1)) % Polarization (LA)

ml(N+j)=(-vs(1)+sqrt(vs(1)^2+4*c(1)*f(N+j)))/(2*c(1)); % LA ...

wavevector (assume isotropy)

mt(N+j)=0; % TA wavevector=0

vl(N+j)=vs(1)+2*c(1)*ml(N+j); % LA magnitude group velocity

vt(N+j)=0; % TA velocity=0

else % Polarization (TA)

mt(N+j)=(-vs(2)+sqrt(vs(2)^2+4*c(2)*f(N+j)))/(2*c(2)); % TA ...

wavevector (assume isotropy)

ml(N+j)=0; % LA wavevector=0

vt(N+j)=vs(2)+2*c(2)*mt(N+j); % TA magnitude group velocity

vl(N+j)=0; % LA velocity=0

end

v(N+j)=vt(N+j)+vl(N+j); % total velocity

m(N+j)=mt(N+j)+ml(N+j); % total wave number

R1=rand; % random number

R2=rand; % random number

R3=rand; % random number

R4=rand; % random number

R5=rand; % random number

K2=2*pi*R4; % angle 1

psi

K1=2*R5-1; % angle 2

cos(theta)

vg(N+j,2)=v(N+j)*sqrt(1-K1^2)*cos(K2); % direction of velocity ...

sin(theta)*cos(psi)

vg(N+j,3)=v(N+j)*sqrt(1-K1^2)*sin(K2); % direction of velocity ...

sin(theta)*sin(psi)

vg(N+j,1)=v(N+j)*K1; % direction of velocity cos(theta)

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end

x1(N+j)=R1*l; % location of particles in cell 1

x2(N+j)=R2*a; % location of particles in cell 1

x3(N+j)=R3*b; % location of particles in cell 1

s(N+j)=1; % Sign of particles 1(T>Teq), -1(T<Teq)

360 %%%%%%%%%%

Determine the properties of cold wall

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for j=1:Nc

R=rand; % random number

a1=1; % frequeny invervel No. 1

b1=1000; % frequeny invervel No. 1000

while((R-Fc(a1))*(Fc(a1+1)-R)<0) % bisection method to ...

determine which interval

mm=round((a1+b1)/2); % get integer

if (abs(R-Fc(a1))<abs(R-Fc(b1))) % bisection method

if (R>Fc(mm))

a1=mm;

b1=mm;

else

end

else

if (R<Fc(mm))

b1=mm;

a1=mm;

else

end

end

end

R1=rand; %random number

R2=rand; %random number

f(N+Nh+j)=w1(a1)+(2*R2-1)*dw/2; % assign the frequency

if (R1<Pc(a1)) % Polarization (LA)

ml(N+Nh+j)=(-vs(1)+sqrt(vs(1)^2+4*c(1)*f(N+Nh+j)))/(2*c(1)); ...

% LA wavevector (assume isotropy)

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mt(N+Nh+j)=0; % TA wavevector=0

vl(N+Nh+j)=vs(1)+2*c(1)*ml(N+Nh+j); % LA magnitude group ...

velocity

vt(N+Nh+j)=0; % TA velocity=0

else % Polarization (TA)

mt(N+Nh+j)=(-vs(2)+sqrt(vs(2)^2+4*c(2)*f(N+Nh+j)))/(2*c(2)); ...

% TA wavevector (assume isotropy)

ml(N+Nh+j)=0; % LA wavevector=0

vt(N+Nh+j)=vs(2)+2*c(2)*mt(N+Nh+j); % TA magnitude group ...

velocity

vl(N+Nh+j)=0; % LA velocity=0

end

v(N+Nh+j)=(vt(N+Nh+j)+vl(N+Nh+j)); % total velocity

m(N+Nh+j)=mt(N+Nh+j)+ml(N+Nh+j); % total wave number

R1=rand; % random number

R2=rand; % random number

R3=rand; % random number

R4=rand; % random number

R5=rand; % random number

K2=2*pi*R4; % angle 1

psi

K1=2*R5-1; % angle 2

cos(theta)

vg(N+Nh+j,2)=v(N+Nh+j)*sqrt(1-K1^2)*cos(K2); % direction of ...

velocity sin(theta)*cos(psi)

vg(N+Nh+j,3)=v(N+Nh+j)*sqrt(1-K1^2)*sin(K2); % direction of ...

velocity sin(theta)*sin(psi)

vg(N+Nh+j,1)=v(N+Nh+j)*K1; % direction of velocity cos(theta)

x1(N+Nh+j)=(39+R1)*l; % location of particles in cell 40

x2(N+Nh+j)=(R2)*a; % location of particles in cell 40

x3(N+Nh+j)=(R3)*b; % location of particles in cell 40

s(N+Nh+j)=-1; % Sign of particles 1(T>Teq), -1(T<Teq)

end

412 %%%%%%%%%%%%%% begin reassignment of the phonons

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Nmax=N+Nh+Nc; % maximum number of particles

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N=Nmax-Ndd; % real number of particles

i=0; % indication of which particle to be reassigned
if (Ndd≤(Nh+Nc)) % If deleted particles ≤ reassigned particles

for j=1:N

if (x1(j)<l || x1(j)>39*l) % if phonons are out of boundary ...

cell, reassign the reinitialized phonons to them

x1(j)=x1(Nmax-i); % update location

x2(j)=x2(Nmax-i); % update location

x3(j)=x3(Nmax-i); % update location

vg(j,1)=vg(Nmax-i,1); % update velocity 1

vg(j,2)=vg(Nmax-i,2); % update velocity 2

vg(j,3)=vg(Nmax-i,3); % update velocity 3

f(j)=f(Nmax-i); % update frequency

ml(j)=ml(Nmax-i); % update wave numeber LA

mt(j)=mt(Nmax-i); % update wave numeber TA

vl(j)=vl(Nmax-i); % update velocity LA

vt(j)=vt(Nmax-i); % update velocity TA

s(j)=s(Nmax-i); % update sign of particles

m(j)=ml(j)+mt(j); % update total wave number (either LA ...

or TA)

v(j)=vl(j)+vt(j); % update total velocity (either LA or TA)

i=i+1;

end

end

else % If deleted particles > reassigned particles

k=1;

for j=1:N

if (x1(j)<l || x1(j)>39*l) % if phonons are out of boundary ...

cell, reassign the reinitialized phonons to them
if (i≤(Nh+Nc))

% If the reassigned particles ≤ Nh+Nc

x1(j)=x1(Nmax-i); % update location

x2(j)=x2(Nmax-i); % update location

x3(j)=x3(Nmax-i); % update location

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vg(j,1)=vg(Nmax-i,1); % update velocity 1

vg(j,2)=vg(Nmax-i,2); % update velocity 2

vg(j,3)=vg(Nmax-i,3); % update velocity 3

f(j)=f(Nmax-i); % update frequency

ml(j)=ml(Nmax-i); % update wave numeber LA

mt(j)=mt(Nmax-i); % update wave numeber TA

vl(j)=vl(Nmax-i); % update velocity LA

vt(j)=vt(Nmax-i); % update velocity TA

s(j)=s(Nmax-i); % update sign of particles

m(j)=ml(j)+mt(j); % update total wave number ...

(either LA or TA)

v(j)=vl(j)+vt(j); % update total velocity (either ...

LA or TA)

i=i+1;

else % If the reassigned particles > Nh+Nc

while(x1(N+k)<l || x1(N+k)>39*l)

% if the rest ...

phonons are out of boundary cell, reassign the ...

reinitialized phonons to them

k=k+1;

end

x1(j)=x1(N+k); % update location

x2(j)=x2(N+k); % update location

x3(j)=x3(N+k); % update location

vg(j,1)=vg(N+k,1); % update velocity 1

vg(j,2)=vg(N+k,2); % update velocity 2

vg(j,3)=vg(N+k,3); % update velocity 3

f(j)=f(N+k); % update frequency

ml(j)=ml(N+k); % update wave numeber LA

mt(j)=mt(N+k); % update wave numeber TA

vl(j)=vl(N+k); % update velocity LA

vt(j)=vt(N+k); % update velocity TA

s(j)=s(N+k); % update sign of particles

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m(j)=ml(j)+mt(j); % update total wave number ...

(either LA or TA)

v(j)=vl(j)+vt(j); % update total velocity (either ...

LA or TA)

k=k+1;

i=i+1;

end

end

end

end

480 %%%%%%%%% Calculate number of energy in each cell

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for j=1:N

for i=1:40

if (x1(j)>(i-1)*l && x1(j)<(i)*l)

NNN(i)=NNN(i)+s(j); % number of particles (energy) in ...

each cell

end

end

end

488 %%%%%%%%% Calculate target energy for each cell

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for i=1:40

Et(i)=NNN(i)*W; % target energy of each cell

end

492 %%%%%%%% end of reassignment of the phonons

493 %%%%%%%%%%%%%%%%%%%%%% Calculating the temperature (Newton-Raphson method)

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T1(1)=100; % initial temperature (T0) of cell 1 iteration 1

T1(2)=T1(1)-z1(T1(1))/(z1dot(T1(1))); % T0 of cell 1 iteration 2

j=1;

while (abs(T1(j+1)-T1(j))>0.01) % Iteration process

j=j+1;

T1(j+1)=T1(j)-z1(T1(j))/(z1dot(T1(j)));

end

T(1)=T1(j+1); % converge, solve T

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503 %%%%%%%% Process is similar to get the following temperatures

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T2(1)=100;

T2(2)=T2(1)-z2(T2(1))/(z2dot(T2(1)));

j=1;

while (abs(T2(j+1)-T2(j))>0.01)

j=j+1;

T2(j+1)=T2(j)-z2(T2(j))/(z2dot(T2(j)));

end

T(2)=T2(j+1);

T3(1)=100;

T3(2)=T3(1)-z3(T3(1))/(z3dot(T3(1)));

j=1;

while (abs(T3(j+1)-T3(j))>0.01)

j=j+1;

T3(j+1)=T3(j)-z3(T3(j))/(z3dot(T3(j)));

end

T(3)=T3(j+1);

T4(1)=100;

T4(2)=T4(1)-z4(T4(1))/(z4dot(T4(1)));

j=1;

while (abs(T4(j+1)-T4(j))>0.01)

j=j+1;

T4(j+1)=T4(j)-z4(T4(j))/(z4dot(T4(j)));

end

T(4)=T4(j+1);

T5(1)=100;

T5(2)=T5(1)-z5(T5(1))/(z5dot(T5(1)));

j=1;

while (abs(T5(j+1)-T5(j))>0.01)

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j=j+1;

T5(j+1)=T5(j)-z5(T5(j))/(z5dot(T5(j)));

end

T(5)=T5(j+1);

T6(1)=100;

T6(2)=T6(1)-z6(T6(1))/(z6dot(T6(1)));

j=1;

while (abs(T6(j+1)-T6(j))>0.01)

j=j+1;

T6(j+1)=T6(j)-z6(T6(j))/(z6dot(T6(j)));

end

T(6)=T6(j+1);

T7(1)=100;

T7(2)=T7(1)-z7(T7(1))/(z7dot(T7(1)));

j=1;

while (abs(T7(j+1)-T7(j))>0.01)

j=j+1;

T7(j+1)=T7(j)-z7(T7(j))/(z7dot(T7(j)));

end

T(7)=T7(j+1);

T8(1)=100;

T8(2)=T8(1)-z8(T8(1))/(z8dot(T8(1)));

j=1;

while (abs(T8(j+1)-T8(j))>0.01)

j=j+1;

T8(j+1)=T8(j)-z8(T8(j))/(z8dot(T8(j)));

end

T(8)=T8(j+1);

T9(1)=100;

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T9(2)=T9(1)-z9(T9(1))/(z9dot(T9(1)));

j=1;

while (abs(T9(j+1)-T9(j))>0.01)

j=j+1;

T9(j+1)=T9(j)-z9(T9(j))/(z9dot(T9(j)));

end

T(9)=T9(j+1);

T10(1)=100;

T10(2)=T10(1)-z10(T10(1))/(z10dot(T10(1)));

j=1;

while (abs(T10(j+1)-T10(j))>0.01)

j=j+1;

T10(j+1)=T10(j)-z10(T10(j))/(z10dot(T10(j)));

end

T(10)=T10(j+1);

T11(1)=100;

T11(2)=T11(1)-z11(T11(1))/(z11dot(T11(1)));

j=1;

while (abs(T11(j+1)-T11(j))>0.01)

j=j+1;

T11(j+1)=T11(j)-z11(T11(j))/(z11dot(T11(j)));

end

T(11)=T11(j+1);

T12(1)=100;

T12(2)=T12(1)-z12(T12(1))/(z12dot(T12(1)));

j=1;

while (abs(T12(j+1)-T12(j))>0.01)

j=j+1;

T12(j+1)=T12(j)-z12(T12(j))/(z12dot(T12(j)));

end

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T(12)=T12(j+1);

T13(1)=100;

T13(2)=T13(1)-z13(T13(1))/(z13dot(T13(1)));

j=1;

while (abs(T13(j+1)-T13(j))>0.01)

j=j+1;

T13(j+1)=T13(j)-z13(T13(j))/(z13dot(T13(j)));

end

T(13)=T13(j+1);

T14(1)=100;

T14(2)=T14(1)-z14(T14(1))/(z14dot(T14(1)));

j=1;

while (abs(T14(j+1)-T14(j))>0.01)

j=j+1;

T14(j+1)=T14(j)-z14(T14(j))/(z14dot(T14(j)));

end

T(14)=T14(j+1);

T15(1)=100;

T15(2)=T15(1)-z15(T15(1))/(z15dot(T15(1)));

j=1;

while (abs(T15(j+1)-T15(j))>0.01)

j=j+1;

T15(j+1)=T15(j)-z15(T15(j))/(z15dot(T15(j)));

end

T(15)=T15(j+1);

T16(1)=100;

T16(2)=T16(1)-z16(T16(1))/(z16dot(T16(1)));

j=1;

while (abs(T16(j+1)-T16(j))>0.01)

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j=j+1;

T16(j+1)=T16(j)-z16(T16(j))/(z16dot(T16(j)));

end

T(16)=T16(j+1);

T17(1)=100;

T17(2)=T17(1)-z17(T17(1))/(z17dot(T17(1)));

j=1;

while (abs(T17(j+1)-T17(j))>0.01)

j=j+1;

T17(j+1)=T17(j)-z17(T17(j))/(z17dot(T17(j)));

end

T(17)=T17(j+1);

T18(1)=100;

T18(2)=T18(1)-z18(T18(1))/(z18dot(T18(1)));

j=1;

while (abs(T18(j+1)-T18(j))>0.01)

j=j+1;

T18(j+1)=T18(j)-z18(T18(j))/(z18dot(T18(j)));

end

T(18)=T18(j+1);

T19(1)=100;

T19(2)=T19(1)-z19(T19(1))/(z19dot(T19(1)));

j=1;

while (abs(T19(j+1)-T19(j))>0.01)

j=j+1;

T19(j+1)=T19(j)-z19(T19(j))/(z19dot(T19(j)));

end

T(19)=T19(j+1);

T20(1)=100;

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T20(2)=T20(1)-z20(T20(1))/(z20dot(T20(1)));

j=1;

while (abs(T20(j+1)-T20(j))>0.01)

j=j+1;

T20(j+1)=T20(j)-z20(T20(j))/(z20dot(T20(j)));

end

T(20)=T20(j+1);

T21(1)=100;

T21(2)=T21(1)-z21(T21(1))/(z21dot(T21(1)));

j=1;

while (abs(T21(j+1)-T21(j))>0.01)

j=j+1;

T21(j+1)=T21(j)-z21(T21(j))/(z21dot(T21(j)));

end

T(21)=T21(j+1);

T22(1)=100;

T22(2)=T22(1)-z22(T22(1))/(z22dot(T22(1)));

j=1;

while (abs(T22(j+1)-T22(j))>0.01)

j=j+1;

T22(j+1)=T22(j)-z22(T22(j))/(z22dot(T22(j)));

end

T(22)=T22(j+1);

T23(1)=100;

T23(2)=T23(1)-z23(T23(1))/(z23dot(T23(1)));

j=1;

while (abs(T23(j+1)-T23(j))>0.01)

j=j+1;

T23(j+1)=T23(j)-z23(T23(j))/(z23dot(T23(j)));

end

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T(23)=T23(j+1);

T24(1)=100;

T24(2)=T24(1)-z24(T24(1))/(z24dot(T24(1)));

j=1;

while (abs(T24(j+1)-T24(j))>0.01)

j=j+1;

T24(j+1)=T24(j)-z24(T24(j))/(z24dot(T24(j)));

end

T(24)=T24(j+1);

T25(1)=100;

T25(2)=T25(1)-z25(T25(1))/(z25dot(T25(1)));

j=1;

while (abs(T25(j+1)-T25(j))>0.01)

j=j+1;

T25(j+1)=T25(j)-z25(T25(j))/(z25dot(T25(j)));

end

T(25)=T25(j+1);

T26(1)=100;

T26(2)=T26(1)-z26(T26(1))/(z26dot(T26(1)));

j=1;

while (abs(T26(j+1)-T26(j))>0.01)

j=j+1;

T26(j+1)=T26(j)-z26(T26(j))/(z26dot(T26(j)));

end

T(26)=T26(j+1);

T27(1)=100;

T27(2)=T27(1)-z27(T27(1))/(z27dot(T27(1)));

j=1;

while (abs(T27(j+1)-T27(j))>0.01)

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j=j+1;

T27(j+1)=T27(j)-z27(T27(j))/(z27dot(T27(j)));

end

T(27)=T27(j+1);

T28(1)=100;

T28(2)=T28(1)-z28(T28(1))/(z28dot(T28(1)));

j=1;

while (abs(T28(j+1)-T28(j))>0.01)

j=j+1;

T28(j+1)=T28(j)-z28(T28(j))/(z28dot(T28(j)));

end

T(28)=T28(j+1);

T29(1)=100;

T29(2)=T29(1)-z29(T29(1))/(z29dot(T29(1)));

j=1;

while (abs(T29(j+1)-T29(j))>0.01)

j=j+1;

T29(j+1)=T29(j)-z29(T29(j))/(z29dot(T29(j)));

end

T(29)=T29(j+1);

T30(1)=100;

T30(2)=T30(1)-z30(T30(1))/(z30dot(T30(1)));

j=1;

while (abs(T30(j+1)-T30(j))>0.01)

j=j+1;

T30(j+1)=T30(j)-z30(T30(j))/(z30dot(T30(j)));

end

T(30)=T30(j+1);

T31(1)=100;

130

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T31(2)=T31(1)-z31(T31(1))/(z31dot(T31(1)));

j=1;

while (abs(T31(j+1)-T31(j))>0.01)

j=j+1;

T31(j+1)=T31(j)-z31(T31(j))/(z31dot(T31(j)));

end

T(31)=T31(j+1);

T32(1)=100;

T32(2)=T32(1)-z32(T32(1))/(z32dot(T32(1)));

j=1;

while (abs(T32(j+1)-T32(j))>0.01)

j=j+1;

T32(j+1)=T32(j)-z32(T32(j))/(z32dot(T32(j)));

end

T(32)=T32(j+1);

T33(1)=100;

T33(2)=T33(1)-z33(T33(1))/(z33dot(T33(1)));

j=1;

while (abs(T33(j+1)-T33(j))>0.01)

j=j+1;

T33(j+1)=T33(j)-z33(T33(j))/(z33dot(T33(j)));

end

T(33)=T33(j+1);

T34(1)=100;

T34(2)=T34(1)-z34(T34(1))/(z34dot(T34(1)));

j=1;

while (abs(T34(j+1)-T34(j))>0.01)

j=j+1;

T34(j+1)=T34(j)-z34(T34(j))/(z34dot(T34(j)));

end

131

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T(34)=T34(j+1);

T35(1)=100;

T35(2)=T35(1)-z35(T35(1))/(z35dot(T35(1)));

j=1;

while (abs(T35(j+1)-T35(j))>0.01)

j=j+1;

T35(j+1)=T35(j)-z35(T35(j))/(z35dot(T35(j)));

end

T(35)=T35(j+1);

T36(1)=100;

T36(2)=T36(1)-z36(T36(1))/(z36dot(T36(1)));

j=1;

while (abs(T36(j+1)-T36(j))>0.01)

j=j+1;

T36(j+1)=T36(j)-z36(T36(j))/(z36dot(T36(j)));

end

T(36)=T36(j+1);

T37(1)=100;

T37(2)=T37(1)-z37(T37(1))/(z37dot(T37(1)));

j=1;

while (abs(T37(j+1)-T37(j))>0.01)

j=j+1;

T37(j+1)=T37(j)-z37(T37(j))/(z37dot(T37(j)));

end

T(37)=T37(j+1);

T38(1)=100;

T38(2)=T38(1)-z38(T38(1))/(z38dot(T38(1)));

j=1;

while (abs(T38(j+1)-T38(j))>0.01)

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j=j+1;

T38(j+1)=T38(j)-z38(T38(j))/(z38dot(T38(j)));

end

T(38)=T38(j+1);

T39(1)=100;

T39(2)=T39(1)-z39(T39(1))/(z39dot(T39(1)));

j=1;

while (abs(T39(j+1)-T39(j))>0.01)

j=j+1;

T39(j+1)=T39(j)-z39(T39(j))/(z39dot(T39(j)));

end

T(39)=T39(j+1);

T40(1)=100;

T40(2)=T40(1)-z40(T40(1))/(z40dot(T40(1)));

j=1;

while (abs(T40(j+1)-T40(j))>0.01)

j=j+1;

T40(j+1)=T40(j)-z40(T40(j))/(z40dot(T40(j)));

end

T(40)=T40(j+1);

854 %%%%%%%%%%%%%%%%%% End of calculating the temperature

855 %%%%%%% Number density function for updated T

856

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862

for i=1:1:40

for j=1:1:1000

if (T(i)≥Teq)

nl(i,j)=hbar*w1(j)*1/W*V*(1/(exp((hbar*w1(j)/(kb*T(i))))-1)-

1/(exp((hbar*w1(j)/(kb*Teq)))-1))*((q1(j)^2)/(2*pi^2*v1(j)))*dw; ...

% LA number for hot side

nt(i,j)=hbar*w2(j)*1/W*V*(1/(exp((hbar*w2(j)/(kb*T(i))))-1)-

1/(exp((hbar*w2(j)/(kb*Teq)))-1))*((q2(j)^2)/(2*pi^2*v2(j)))*dw*2; ...

% TA number for hot side

133

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else

nl(i,j)=-hbar*w1(j)*1/W*V*(1/(exp((hbar*w1(j)/(kb*T(i))))-1)-

1/(exp((hbar*w1(j)/(kb*Teq)))-1))*((q1(j)^2)/(2*pi^2*v1(j)))*dw; ...

% LA number for hot side

nt(i,j)=-hbar*w2(j)*1/W*V*(1/(exp((hbar*w2(j)/(kb*T(i))))-1)-

1/(exp((hbar*w2(j)/(kb*Teq)))-1))*((q2(j)^2)/(2*pi^2*v2(j)))*dw*2; ...

% TA number for hot side

end

n(i,j)=nl(i,j)+nt(i,j); % LA+TA

P(i,j)=nl(i,j)/n(i,j); % Polarization

end

NN(i)=floor(sum(n(i,:)));

M(i,1)=n(i,1);

F(i,1)=0;

for j=2:1:1000

M(i,j)=M(i,(j-1))+n(i,j); % summation of ith phonon numbers

F(i,j)=M(i,j)/NN(i);% determine which frequency intervel

end

end

880 %%%%%%%%%%%% Phonon-electron scattering process

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893

u1=u1+1 % iteration

dd(u1)=0; % number of particles under phonon-electron scattering

for j=1:N

for i=1:40

if (x1(j)>(i-1)*l && x1(j)<(i)*l)

yy(j)=1.76*9.25/T(i);

E0=3.71e8; % constant

xm(j)=3.83; % coefficient for reduced frequency

if (mt(j)==0)

% LA phonon

xf(j)=hbar*f(j)/(kb*T(i));

taol(j)=(2*E0*xf(j)*T(i)*exp(-yy(j))*(1-xf(j)/(8*xm(j)))).^(-1);

taot(j)=0;

else % TA phonon

134

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xf(j)=hbar*f(j)/(kb*T(i));

taot(j)=(2*E0*xf(j)*T(i)*exp(-yy(j))*(1-xf(j)/(8*xm(j)))).^(-1);

taol(j)=0;

end

tao(j)=taol(j)+taot(j); % Relaxation time

Ppe(j)=1-exp(-dt/tao(j)); % scattering probability

R=rand; % Random number

if (R<Ppe(j))

dd(u1)=dd(u1)+1; % particle is scattered, reassignment

R1=rand; % random number

a1=1; % frequeny invervel No. 1

b1=1000; % frequeny invervel No. 1000

while((R1-F(i,a1))*(F(i,a1+1)-R1)<0)

mm=round((a1+b1)/2); % get integer

if (abs(R1-F(i,a1))<abs(R1-F(i,b1))) % ...

bisection method to determine which interval

if (R1>F(i,mm))

a1=mm;

b1=mm;

else

end

else

if (R1<F(i,mm))

b1=mm;

a1=mm;

else

end

end

end

R2=rand; %random number

R3=rand; %random number

f(j)=w1(a1)+(2*R2-1)*dw/2; % reset the frequency

if (R3<P(i,a1)) % Polarization (LA)

135

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ml(j)=(-vs(1)+sqrt(vs(1)^2+4*c(1)*f(j)))/(2*c(1)); ...

% LA wavevector

mt(j)=0;

% TA wavevector=0

vl(j)=vs(1)+2*c(1)*ml(j); % LA magnitude group ...

velocity

vt(j)=0; % TA velocity=0

else % Polarization (TA)

mt(j)=(-vs(2)+sqrt(vs(2)^2+4*c(2)*f(j)))/(2*c(2)); ...

% TA wavevector (assume isotropy)

ml(j)=0; % LA wavevector=0

vt(j)=vs(2)+2*c(2)*mt(j); % TA magnitude group ...

velocity

vl(j)=0; % LA velocity=0

end

v(j)=(vt(j)+vl(j)); % total velocity

m(j)=mt(j)+ml(j); % total wave number

R1=rand; % random number

R2=rand; % random number

K1=2*R1-1; % angle 1

psi

K2=2*pi*R2; % angle 2

cos(theta)

vg(j,2)=v(j)*sqrt(1-K1^2)*cos(K2); % direction of ...

velocity sin(theta)*cos(psi)

vg(j,3)=v(j)*sqrt(1-K1^2)*sin(K2); % direction of ...

velocity sin(theta)*sin(psi)

vg(j,1)=v(j)*K1; % direction of velocity cos(theta)

end

end

end

end

pe(u1)=dd(u1)/N; % ratio of scattered particles

950 %%%%% phonon dislocation scattering

951

952

A0=0.8639; % constant for screw dislocations

Nd=4.76e13; % dislocation density

136

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gama=1.4; % Gruneissen constant

b0=2.85e-10; % Burger's vector

md(u1)=0; % number of particles under phonon-dislocation scattering

for j=1:N

for i=1:40

if (x1(j)>(i-1)*l && x1(j)<(i)*l)

tau_d(j)=(A0*gama^2*b0^2*Nd*f(j))^(-1); % relacation time

Pd(j)=1-exp(-dt/tau_d(j)); % scattering probability

R=rand; % random number

if (R<Pd(j))

md(u1)=md(u1)+1; % particle is scattered, only ...

change velocity direction

R1=rand; % random number

R2=rand; % random number

K1=2*R1-1; % angle 2

K2=2*pi*R2; % angle 1

vg(j,2)=v(j)*sqrt(1-K1^2)*cos(K2); % direction of ...

velocity sin?*cos?

vg(j,3)=v(j)*sqrt(1-K1^2)*sin(K2); % direction of ...

velocity sin?*sin?

vg(j,1)=v(j)*K1;

% direction of velocity sin?

end

end

end

end

975 %%%%%% initialized sign of particles

976

977

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979

980

981

982

for i=1:40

Np(i,u1)=0;

Nn(i,u1)=0;

end

for j=1:N

for i=1:40

if (x1(j)>(i-1)*l && x1(j)<i*l)

137

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if s(j)==1

Np(i,u1)=Np(i,u1)+1; % number of positive particles

Nn(i,u1)=Nn(i,u1)+1; % number of positive particles

else

end

end

end

end

991 %%%%%%%%%%% calculate the heat flux in each cell at each time step

992

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1000

1001

1002

1003

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1009

for i=1:40

qqq(i,u1)=0;

end

for j=1:N

for i=2:39

if (x1(j)>(i-1)*l && x1(j)<i*l)

qqq(i,u1)=qqq(i,u1)+vg(j,1)*(s(j)*W)/V; % total heat flux

end

end

end

qq(u1)=sum(qqq(:,u1))/38; % average heat flux

kk(u1)=qq(u1)*37*l/(T(2)-T(39)); % k at each timestep

for i=1:40

Tm1(u1,i)=T(i); % temperature of each cell at each timestep

Em1(u1,i)=Et(i); % energy of each cell at each timestep

Nm(u1,i)=NNN(i); % number of particles (energy) in each cell

end

Ett(u1)=sum(Et); % total energy at each timestep

1010 end

1011

1012 %%% Calculate Standard Deviation

1013 STD=0;

1014 for i=1:40

1015

STD=STD+(T(i)-yy(i))^2;

138

1016 end

1017 STD=sqrt(STD/40);

1018 kkk=0;

1019 for j=(u1-100):1:u1

kkk=kkk+kk(j); % summation of k for the last 100 timesteps

1020

1021 end

1022 kkk=kkk/101 % display average k

1023 Teq % display temperature

1024 plot(T) % plot temperature profle at the last step

1025 tim = toc;

1026 fprintf('operation time %15.5f.\n',tim); % calculate the operation time

1027 %%%%% Plat transient temperature profile

1028 i=1:40;

1029 plot(i,Tm1(1,i),'r',i,Tm1(100,i),'g',i,Tm1(500,i),'b',i,Tm1(1000,i),'m',i,Tm1(2000,i),'k')

1030 xlabel('Cell Number','FontSize',12,'FontName','Times New Roman');

1031 ylabel('Temperature/K','FontSize',12,'FontName','Times New Roman');

1032 legend('1*dt','100*dt','1000*dt','2000*dt','3000*dt')
1033 title('Energy-Based Variance-Reduced Method

50 nm/cell with Hao ...

scattering')

A.1.1 Child functions

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z1=ff(T)

%%%Function of z1

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

8

9

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

139

%%%% Energy of LA polarization

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z1=ztol-Et(1); %%%Function of z1 is related to Et.1

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z2=ff(T)

%%%Function of z2

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z2=ztol-Et(2); %%%Function of z2 is related to Et.2

140

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z3=ff(T)

%%%Function of z3

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z3=ztol-Et(3); %%%Function of z3 is related to Et.3

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z4=ff(T)

%%%Function of z4

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

141

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z4=ztol-Et(4); %%%Function of z4 is related to Et.4

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z5=ff(T)

%%%Function of z5

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

142

18

19 z5=ztol-Et(5); %%%Function of z5 is related to Et.5

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z6=ff(T)

%%%Function of z6

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z6=ztol-Et(6); %%%Function of z6 is related to Et.6

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z7=ff(T)

%%%Function of z7

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

143

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z7=ztol-Et(7); %%%Function of z7 is related to Et.7

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z8=ff(T)

%%%Function of z8

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

14

15

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

144

%%%% Energy of TA polarization

16

17 end

18

19 z8=ztol-Et(8); %%%Function of z8 is related to Et.8

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z9=ff(T)

%%%Function of z9

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z9=ztol-Et(9); %%%Function of z9 is related to Et.9

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z10=ff(T)

%%%Function of z10

145

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z10=ztol-Et(10); %%%Function of z1 is related to Et.10

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z11=ff(T)

%%%Function of z11

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

146

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z11=ztol-Et(11); %%%Function of z11 is related to Et.11

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z12=ff(T)

%%%Function of z12

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z12=ztol-Et(12); %%%Function of z12 is related to Et.12

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

147

2

3 function z13=ff(T)

%%%Function of z13

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z13=ztol-Et(13); %%%Function of z13 is related to Et.13

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z14=ff(T)

%%%Function of z14

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

148

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z14=ztol-Et(14); %%%Function of z14 is related to Et.14

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z15=ff(T)

%%%Function of z15

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z15=ztol-Et(15); %%%Function of z15 is related to Et.15

20 end

149

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z16=ff(T)

%%%Function of z16

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z16=ztol-Et(16); %%%Function of z16 is related to Et.16

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z17=ff(T)

%%%Function of z17

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

8

9

10

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

150

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z17=ztol-Et(17); %%%Function of z17 is related to Et.17

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z18=ff(T)

%%%Function of z18

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z18=ztol-Et(18); %%%Function of z18 is related to Et.18

20 end

151

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z19=ff(T)

%%%Function of z19

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z19=ztol-Et(19); %%%Function of z19 is related to Et.19

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z20=ff(T)

%%%Function of z20

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

8

9

10

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

152

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z20=ztol-Et(20); %%%Function of z20 is related to Et.20

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z21=ff(T)

%%%Function of z21

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z21=ztol-Et(21); %%%Function of z1 is related to Et.21

20 end

153

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z22=ff(T)

%%%Function of z22

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z22=ztol-Et(22); %%%Function of z22 is related to Et.22

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z3=ff(T)

%%%Function of z23

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

8

9

10

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

154

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z23=ztol-Et(23); %%%Function of z23 is related to Eq.23

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z24=ff(T)

%%%Function of z24

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z24=ztol-Et(24); %%%Function of z24 is related to Eq.24

20 end

155

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z25=ff(T)

%%%Function of z25

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z25=ztol-Et(25); %%%Function of z25 is related to Eq.25

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z26=ff(T)

%%%Function of z26

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

8

9

10

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

156

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z26=ztol-Et(26); %%%Function of z26 is related to Eq.26

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z27=ff(T)

%%%Function of z27

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z27=ztol-Et(27); %%%Function of z1 is related to Eq.27

20 end

157

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z28=ff(T)

%%%Function of z28

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z28=ztol-Et(28); %%%Function of z28 is related to Eq.28

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z29=ff(T)

%%%Function of z29

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

8

9

10

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

158

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z29=ztol-Et(29); %%%Function of z29 is related to Eq.29

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z30=ff(T)

%%%Function of z1

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z30=ztol-Et(30); %%%Function of z30 is related to Eq.30

20 end

159

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z31=ff(T)

%%%Function of z31

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z31=ztol-Et(31); %%%Function of z31 is related to Eq.31

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z32=ff(T)

%%%Function of z32

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

8

9

10

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

160

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z32=ztol-Et(32); %%%Function of z32 is related to Eq.32

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z33=ff(T)

%%%Function of z33

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z33=ztol-Et(33); %%%Function of z33 is related to Eq.33

20 end

161

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z34=ff(T)

%%%Function of z34

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z34=ztol-Et(34); %%%Function of z34 is related to Eq.34

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z35=ff(T)

%%%Function of z35

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

8

9

10

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

162

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z35=ztol-Et(35); %%%Function of z35 is related to Eq.35

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z36=ff(T)

%%%Function of z36

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z36=ztol-Et(36); %%%Function of z1 is related to Eq.36

20 end

163

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z37=ff(T)

%%%Function of z37

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z37=ztol-Et(37); %%%Function of z1 is related to Eq.37

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z38=ff(T)

%%%Function of z38

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

8

9

10

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

164

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z38=ztol-Et(38); %%%Function of z1 is related to Eq.38

20 end

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z39=ff(T)

%%%Function of z39

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z39=ztol-Et(39); %%%Function of z1 is related to Eq.39

20 end

165

1 %%%%%% Child functions to calculate T using Newton-Raphson methos

2

3 function z40=ff(T)

%%%Function of z40

4 global V hbar kb dw w1 w2 q1 q2 v1 v2 g Teq; %%%%% Global variables

5 global Et; %%%% Global variable of total energy

6 ztol=0; %%%% Initial enengy is set to 0

7 for i=1:1000

ztol=ztol+g(1)*hbar*w1(i)*V*(1/(exp((hbar*w1(i)/(kb*T)))-1)-

1/(exp((hbar*w1(i)/(kb*Teq)))-1))*((q1(i).^2)/(2*pi^2*v1(i)))*dw;

%%%% Energy of LA polarization

8

9

10

11 end

12

13 for i=1:1000

ztol=ztol+g(2)*hbar*w2(i)*V*(1/(exp((hbar*w2(i)/(kb*T)))-1)-

1/(exp((hbar*w2(i)/(kb*Teq)))-1))*((q2(i).^2)/(2*pi^2*v2(i)))*dw;

%%%% Energy of TA polarization

14

15

16

17 end

18

19 z40=ztol-Et(40); %%%Function of z1 is related to Eq.40

20 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z1dot=ff(T) %%%Function of z1dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

166

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z1dot=ztoldot; %%%Function of z1dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z2dot=ff(T) %%%Function of z2dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z2dot=ztoldot; %%%Function of z2dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z3dot=ff(T) %%%Function of z3dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

167

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z3dot=ztoldot; %%%Function of z3dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z4dot=ff(T) %%%Function of z4dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z4dot=ztoldot; %%%Function of z4dot is related to the derivative

17 end

168

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z5dot=ff(T) %%%Function of z5dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z5dot=ztoldot; %%%Function of z5dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z6dot=ff(T) %%%Function of z6dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

12

13

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

169

%%%% TA polarization

14

15 end

16 z6dot=ztoldot; %%%Function of z6dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z7dot=ff(T) %%%Function of z7dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z7dot=ztoldot; %%%Function of z7dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z8dot=ff(T) %%%Function of z8dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

6

7

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

170

%%%% LA polarization

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z8dot=ztoldot; %%%Function of z8dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z9dot=ff(T) %%%Function of z9dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z9dot=ztoldot; %%%Function of z9dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

171

2 function z10dot=ff(T) %%%Function of z10dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z10dot=ztoldot; %%%Function of z10dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z11dot=ff(T) %%%Function of z1dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

12

13

14

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

172

15 end

16 z11dot=ztoldot; %%%Function of z11dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z12dot=ff(T) %%%Function of z12dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z12dot=ztoldot; %%%Function of z12dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z13dot=ff(T) %%%Function of z13dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

6

7

8

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

173

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z13dot=ztoldot; %%%Function of z13dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z14dot=ff(T) %%%Function of z14dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z14dot=ztoldot; %%%Function of z14dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z15dot=ff(T) %%%Function of z15dot

174

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z15dot=ztoldot; %%%Function of z15dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z16dot=ff(T) %%%Function of z16dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

175

16 z16dot=ztoldot; %%%Function of z16dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z17dot=ff(T) %%%Function of z17dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z17dot=ztoldot; %%%Function of z17dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z18dot=ff(T) %%%Function of z18dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

176

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z18dot=ztoldot; %%%Function of z18dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z19dot=ff(T) %%%Function of z19dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z19dot=ztoldot; %%%Function of z19dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z20dot=ff(T) %%%Function of z20dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

177

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z20dot=ztoldot; %%%Function of z20dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z21dot=ff(T) %%%Function of z21dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z21dot=ztoldot; %%%Function of z21dot is related to the derivative

178

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z22dot=ff(T) %%%Function of z22dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z22dot=ztoldot; %%%Function of z22dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z23dot=ff(T) %%%Function of z23dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

179

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z23dot=ztoldot; %%%Function of z23dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z24dot=ff(T) %%%Function of z24dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z24dot=ztoldot; %%%Function of z24dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z25dot=ff(T) %%%Function of z25dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

180

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z25dot=ztoldot; %%%Function of z25dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z26dot=ff(T) %%%Function of z26dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z26dot=ztoldot; %%%Function of z26dot is related to the derivative

17 end

181

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z27dot=ff(T) %%%Function of z27dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z27dot=ztoldot; %%%Function of z27dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z28dot=ff(T) %%%Function of z28dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

12

13

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

182

%%%% TA polarization

14

15 end

16 z28dot=ztoldot; %%%Function of z28dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z29dot=ff(T) %%%Function of z29dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z29dot=ztoldot; %%%Function of z29dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z30dot=ff(T) %%%Function of z30dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

6

7

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

183

%%%% LA polarization

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z30dot=ztoldot; %%%Function of z30dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z31dot=ff(T) %%%Function of z31dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z31dot=ztoldot; %%%Function of z31dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

184

2 function z32dot=ff(T) %%%Function of z32dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z32dot=ztoldot; %%%Function of z32dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z33dot=ff(T) %%%Function of z33dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

12

13

14

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

185

15 end

16 z33dot=ztoldot; %%%Function of z33dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z34dot=ff(T) %%%Function of z34dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z34dot=ztoldot; %%%Function of z34dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z35dot=ff(T) %%%Function of z35dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

6

7

8

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

186

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z35dot=ztoldot; %%%Function of z35dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z36dot=ff(T) %%%Function of z36dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z36dot=ztoldot; %%%Function of z36dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z37dot=ff(T) %%%Function of z37dot

187

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z37dot=ztoldot; %%%Function of z37dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z38dot=ff(T) %%%Function of z38dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

188

16 z38dot=ztoldot; %%%Function of z38dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z39dot=ff(T) %%%Function of z39dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z39dot=ztoldot; %%%Function of z39dot is related to the derivative

17 end

1 %%%%%% Child dot functions to calculate T using Newton-Raphson methos

2 function z40dot=ff(T) %%%Function of z40dot

3 global V hbar kb dw w1 w2 q1 q2 v1 v2 g;

%%%%% Global variables

4 ztoldot=0; %%%% Initial zdot is set to 0

5 for i=1:1000

ztoldot=ztoldot+g(1)*V*hbar/(2*pi^2)*dw*(w1(i)*q1(i)^2/v1(i))*(1/(exp

(hbar*w1(i)/(kb*T))-1))^2*exp(hbar*w1(i)/(kb*T))*hbar*w1(i)/(kb*T.^2);

%%%% LA polarization

6

7

8

9 end

189

10

11 for i=1:1000

ztoldot=ztoldot+g(2)*V*hbar/(2*pi^2)*dw*(w2(i)*q2(i)^2/v2(i))*(1/(exp

(hbar*w2(i)/(kb*T))-1))^2*exp(hbar*w2(i)/(kb*T))*hbar*w2(i)/(kb*T.^2);

%%%% TA polarization

12

13

14

15 end

16 z40dot=ztoldot; %%%Function of z40dot is related to the derivative

17 end

190

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