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A Multi-Scale Electro-Thermo-Mechanical Analysis of
Single Walled Carbon Nanotubes
By
Tarek Ragab
April 2010
A dissertation submitted to the
Faculty of the Graduate School of State
University of New York at Buffalo
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
Department of Civil, Structural, and Environmental Engineering
Copyright by
Tarek Ragab
2010
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ACKNOWLEDGEMENTS
I gratefully acknowledge the assistance of many colleagues and collaborators
during my graduate program. My most important colleague—for he has always treated
me as such—has of course been my advisor, Dr. Cemal Basaran. I thank him for years of
encouragement, advice and support. He has never failed to watch out for me.
Dr. Peihong Zhang generously shared his knowledge on the subject. Meetings
with him were always insightful and pleasant. I sincerely thank him for his support and
for his directions. I am also grateful for Professor David Kofke and Professor Gary
Dargush who served on my committee.
Friends, of course, make it all worthwhile. I am grateful for sharing days and
nights in the lab with Dr. Mohamed Abdel-Hamid, Dr. Shidong Li, Mr. Eray Gunel, Mr.
Bicheng Chen and of course with my colleague Mr. Mike Sellers who made the difficult
time feel much shorter.
Most of all, my heartfelt thanks go out to the most important people in my life
who have never failed to encourage me. I wish my Dad was here to share these moments
with me. I thank my Mother and my siblings. I cannot express enough gratitude to my
beloved wife, Heba and my son Hazim. I love all of you more than I can say.
This dissertation is based on work supported by NSF CMS division grant No.
CMS-0508854 and Office of Naval Research Advanced Electrical Power Systems
Division by program director Terry Ericsen.
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TABLE OF CONTENTS
Acknowledgements ................................................................................................ iii
Table of contents .................................................................................................... iv
List of figures ....................................................................................................... viii
List of tables .......................................................................................................... xii
List of tables .......................................................................................................... xii
Abstract ................................................................................................................ xiii
Chapter 1 ................................................................................................................. 1
Introduction ............................................................................................................. 1
1.1 Motivation ..................................................................................................... 1
1.2 Objectives ..................................................................................................... 3
1.3 Contributions of present research ................................................................. 4
1.4 Outline........................................................................................................... 4
Chapter 2 ................................................................................................................. 6
Literature review ..................................................................................................... 6
2.1 Experimental investigation of CNTs under high current densities ............... 6
2.2 Related analytical studies of electrical and thermal properties ................... 12
2.3 Molecular dynamics simulations for carbon nanotubes.............................. 13
Chapter 3 ............................................................................................................... 16
Molecular dynamics Simulations of CNTs under uniaxial tension ...................... 16
3.1 Introduction ................................................................................................. 16
3.2 Geometry of carbon nanotubes ................................................................... 17
iv
3.3 Molecular Dynamics Simulation details ..................................................... 22
3.3.1 Equations of motion for different ensembles ....................................... 22
3.3.2 Integration algorithm ........................................................................... 24
3.3.3 Boundary and initial conditions ........................................................... 26
3.3.4 Interatomic potentials........................................................................... 27
3.4 Stress calculation ........................................................................................ 34
3.5 Results and discussion ................................................................................ 36
3.5.1 Stress calculations ................................................................................ 37
3.5.2 Influence of displacement increment ................................................... 44
3.5.3 Carbon chain unraveling ...................................................................... 53
3.6 Conclusions ................................................................................................. 54
Chapter 4 ............................................................................................................... 55
The unravelling of open-ended single walled carbon nanotubes using molecular
dynamics simulations ........................................................................................................ 55
4.1 Introduction ................................................................................................. 55
4.2 Molecular dynamics simulation .................................................................. 56
4.3 Behaviour of single atomic chain ............................................................... 57
4.4 Unravelling of nanotubes ............................................................................ 59
4.4.1 Restrained scheme ............................................................................... 59
4.4.2 Restrained scheme ............................................................................... 69
4.5 Conclusions ................................................................................................. 76
Chapter 5 ............................................................................................................... 78
v
Joule heating and electron-induced wind forces using the time relaxation
approximation ................................................................................................................... 78
5.1 Introduction ................................................................................................. 78
5.2 Energy dispersion relation .......................................................................... 80
5.2.1 Electronic structure of carbon nanotubes ............................................. 81
5.2.2 Tight binding method for graphene ..................................................... 84
5.2.3 Band structure of a (10, 10) single walled nanotubes .......................... 91
5.3 Phonon dispersion relation .......................................................................... 96
5.4 Scattering rates .......................................................................................... 101
5.5 Momentum and energy transfer quantum model ...................................... 108
5.6 Results and discussion .............................................................................. 110
Chapter 6 ............................................................................................................. 119
Joule heating and electron-induced wind forces using ensemble Monte Carlo
simulations ...................................................................................................................... 119
6.1 Introduction ............................................................................................... 119
6.2 Monte Carlo Simulation ............................................................................ 120
6.3 Results and Discussion ............................................................................. 125
6.4 Conclusions ............................................................................................... 141
Chapter 7 ............................................................................................................. 143
Conclusions and Recommendations for future research ..................................... 143
7.1 Conclusions ............................................................................................... 143
7.2 Original contributions of this dissertation................................................. 145
7.3 Recommendations for future research ...................................................... 145
vi
Appendices .......................................................................................................... 147
Appendix 1 Matlab code for generating the initial position and velocity of
atoms in perfect CNTs ............................................................................................... 147
Appendix 2 Matlab code for generating the energy dispersion relation and the
energy density of states of CNTs ............................................................................... 151
Appendix 3 Matlab code for generating the phonon dispersion relation of
CNTs .......................................................................................................................... 154
Appendix 4 Matlab code for calculating the scattering rates for CNTs ........ 165
Appendix 5 Matlab code for Ensemble Monte Carlo Simulations ................ 189
References ........................................................................................................... 197
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LIST OF FIGURES
Figure 1 Current saturation in SWCNTs (After Yao et. al. 2000[44]). .................. 7
Figure 2 Sequential failure of individual shells in a MWCNT (After Collins et. al.
2001[13])............................................................................................................................. 8
Figure 3 SEM image of damaged CNTs under high current density (After Collins
et. al. 2001[13]) ................................................................................................................... 8
Figure 4 Resistance stability of CNTs under high current density (After Wei et. al.
2001[14])............................................................................................................................. 9
Figure 5 Failure of CNTs of different lengths (After Javey et. al. 2004[46])....... 11
Figure 6 Graphene as a lattice of unit cells of two atoms ..................................... 19
Figure 7 Classification of Carbon nanotubes according to their chirality ............ 20
Figure 8 Unit cell of (5, 1) carbon nanotube ......................................................... 21
Figure 9 (10, 10) CNT model used in the study ................................................... 37
Figure 10 Virial stress and continuum stresses at the end of convergence period 39
Figure 11 Stress-strain diagram for CNT stretched from both sides. “A” stands for
stresses calculated using virial stress, “B” for stresses calculated using the continuum
mechanics approach .......................................................................................................... 41
Figure 12 Stress-strain diagram for CNT stretched from one side. “A” stands for
stresses calculated using virial stress, “B” for stresses calculated using the continuum
mechanics approach .......................................................................................................... 42
Figure 13 Carbon chain unraveling in CNTs ........................................................ 43
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Figure 14 Stress-strain curves for different simulations with strain rate=1.69E+09
Sec-1 ................................................................................................................................. 47
Figure 15 Stress-strain curves for different simulations with clear length=118.08
Angstrom........................................................................................................................... 48
Figure 16 Stress-strain curves for different simulations with displacement
increment=0.025 Angstrom .............................................................................................. 49
Figure 17 Effect of the displacement increment on the maximum stress in the
simulated CNTs with length equal 118.08 Angstroms ..................................................... 51
Figure 18 Effect of the length on the maximum stress in the simulated CNTs with
displacement increment equal 0.025 Angstroms .............................................................. 52
Figure 19 Effect of the CNT length on the maximum stress during uniaxial
extension with strain rate of 1.69E+09 sec-1 .................................................................... 53
Figure 20 Force-Strain relation for carbon single atomic chain at A. 300 K and B.
at 1200K ............................................................................................................................ 58
Figure 21 Force-Displacement diagram for (10, 10) CNTs at different
temperatures using the restrained scheme......................................................................... 62
Figure 22 The axial stresses at the fixed end of the (10, 10) CNT at 300K ......... 63
Figure 23 The general steps of unraveling in SWCNTs ....................................... 64
Figure 24 Force-Displacement diagram for (18, 0) CNTs at different temperatures
using the restrained scheme. ............................................................................................. 69
Figure 25 Force-Displacement diagram for (10, 10) CNTs at different
temperatures using the unrestrained scheme..................................................................... 73
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Figure 26 Force-Displacement diagram for (18, 0) CNTs at different temperatures
using the unrestrained scheme. ......................................................................................... 75
Figure 27 The formation of sigma and Pi bonds between 2 carbon atoms. .......... 82
Figure 28 Honey comb lattice of graphene and carbon nanotubes ....................... 83
Figure 29 Direct lattice (left), reciprocal lattice and basis vectors in the reciprocal
lattice and the K-points in the 1st Brilluoin zone (right) .................................................. 88
Figure 30 The energy dispersion relation of graphite (after (Minot 2004)). ........ 91
Figure 31 Reciprocal lattice of graphene and CNTs. The dotted hexagons show
the Brillouin zones for graphene, while the solid lines show the different subbands of (10,
10) CNT in the first and the second Brillouin zones. ....................................................... 93
Figure 32 Energy dispersion relation of the valence and conduction bands for (10,
10) CNT in the first and second BZs. ............................................................................... 95
Figure 33 Associate cell for graphene utilizing fourth nearest neighbor
interactions ........................................................................................................................ 98
Figure 34 LA and LO phonon Dispersion relation for (10, 10) CNT in the first
BZ. The lowered labeled subbands are for the LA mode, and the upper unlabeled
subbands are the LO modes. ........................................................................................... 102
Figure 35 Illustration of the scattering mechanisms considered. ........................ 103
Figure 36 Scattering rates for LA and LO modes at different temperatures. A- LA
scattering for subband 10. B- LA scattering for subband 9. C- LO scattering for subband
10. D- LO scattering for subband 9. ............................................................................... 107
Figure 37 Experimental data versus theoretical I-V curves for metallic SWCNTs
at 300K. ........................................................................................................................... 111
x
Figure 38 Theoretical I-V curves at different temperatures................................ 112
Figure 39 The force generated per unit length of a (10, 10) CNT at 300 K. ...... 113
Figure 40 The force generated per unit length of a (10, 10) CNT as a function of
temperatures at different electric field forces. ................................................................ 115
Figure 41 Comparison of theoretical and experimental data of joule heating in
CNT at 300K. .................................................................................................................. 116
Figure 42 Heating power per unit length of CNT at different temperatures ...... 118
Figure 43 Illustration of the algorithm for Ensemble Monte Carlo simulation .. 122
Figure 44 Time evolution of a sample electron location at 300K and 1200K. AWavevector for 300K. B- Wavevector for 1200K. C- BZ index for 300K. D- BZ index
for 1200K. E- Subband index for 300K. F- Subband index for 1200K. ......................... 129
Figure 45 Cumulative momentum transferred from the electron to the lattice
during the simulation time for all the electric fields simulated. ..................................... 132
Figure 46 Electric-induced wind force generated per unit length of (10, 10) CNT
at 300K using different approaches. ............................................................................... 133
Figure 47 Electron-Induced wind forces generated per unit length of (10, 10)
using EMC ...................................................................................................................... 136
Figure 48 Cumulative energy transferred from the electron to the lattice during the
simulation time for all the electric fields simulated. ....................................................... 139
Figure 49 Joule heating power generated in one angstrom length of (10, 10) CNT
using different approaches at different temperatures. The markers are data points
extracted from the EMC simulations. The thin line is for the power generated calculated
using joule’s law based on experimental I-V curve. ....................................................... 140
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LIST OF TABLES
Table 1 Parameters for carbon-carbon pair terms ................................................. 30
Table 2 Values for fitting the parameters for the function “ F ” (after Brenner et.
al. [68]).............................................................................................................................. 33
Table 3 Values for fitting the parameters for the function “ T ”(after Brenner et. al.
[68])................................................................................................................................... 34
Table 4 Description of simulations. "A" stands for stresses calculated using virial
stress, "B" for stresses calculated using continuum mechanics approach ........................ 40
Table 5 Results and simulation details for the displacement increment study ..... 46
Table 6 Angles between the atoms in the core unit cell and different atoms in the
associate cell and indicate there nth nearest to each other. .............................................. 100
Table 7 Values of the parameters used for the force constant tensor in
104dyn/cm[117]............................................................................................................... 100
xii
ABSTRACT
Carbon nanotubes are formed by folding a graphene sheet. They have gained a lot
of attention during the last decade due to their extra ordinary mechanical, thermal and
electrical properties. Molecular dynamics simulations have been used extensively for
studying the mechanical properties of carbon nanotubes. In this thesis, a quantum
mechanics and molecular dynamics level multi-scale modeling and analysis of single
walled carbon nanotubes is presented. This dissertation reports many findings based on
these simulations such as some parameters that affect the correctness of the results
obtained by molecular dynamics simulation like the boundary conditions and the
displacement increment. The effects of the strain rate and the length of the nanotube on
the mechanical properties of carbon nanotubes under uniaxial tension are also reported. A
simplification for calculating the virial stresses with multibody potential is derived to use
for calculating the stresses in carbon nanotubes and compared with the stresses calculated
using continuum mechanics engineering stresses.
Simulation of unraveling of carbon nanotubes during field emission is simulated
using Molecular dynamics simulations. The force required to start the unraveling in
carbon nanotubes with different chiralities is reported as well as the maximum force that
can be sustained by the atomic chain.
Due to the nonlinearity in the current-voltage relation of carbon nanotubes, the
traditional Joule’s law for calculating joule heating in carbon nanotubes can not be used.
In this thesis, the joule heating and the electron-induced wind forces per unit length of
carbon nanotubes are calculated using a quantum mechanical formulation based on the
xiii
energy and momentum exchange between the electrons and the phonons. Two
approaches were used in the calculations; the first one is based on formulating an integral
form that makes use of the relaxation time approximation into the modified Fermi-Dirac
distribution for the electron occupation probability. The other approach uses the
Ensemble Monte Carlo simulations and tracks the energy and the phonon exchange
during the simulation time. The results are used to calculate the effective charge number
in carbon nanotubes at different temperatures.
The methods proposed in this thesis for calculating the joule heating and the
effective charge number can be used for any nanoscale material, and can be extended to
include effects like phonon-phonon interaction and hot phonon effects.
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CHAPTER 1
INTRODUCTION
1.1 Motivation
Carbon Nanotubes (CNTs) are the single atom thick tubes formed by wrapping a
sheet of graphite made out of hexagonally-arranged carbon atoms. In 1952 L.V.
Radushkevich and V. M. Lukyanovich published clear images of 50 nm diameter tubes
made of carbon [1]. It was not untill the experimental reidentification in 1991 [2] that
CNTs have attracted considerable curiosity to investigate their electrical and thermomechanical behaviour. Experiments show that carbon nanotubes have extraordinary
electrical [3-6], thermal [3, 7] and mechanical properties [8-11]. Mechanically, CNTs
have a tensile strength that is twenty times that of high strength steel [8] and a Young’s
modulus in the order of a terapascal [9]. These extraordinary mechanical properties can
easily be explained by the strong hybrid sp 2 carbon-carbon bond which is considered to
be the strongest bond in nature [12]. Electrically, CNTs have shown a high current
density carrying capacity in the order of 109 Amp/Cm2 [13] and very high resistance to
electromigration-induced failure [14]. For these reasons, CNTs have a great potential to
replace traditional metals like aluminum and copper that has a current carrying capacity
on the order of 106 Amp/Cm2 in IC interconnect applications [13, 15, 16]. The attractive
electrical properties of CNTs need a deeper understanding to comprehend. Geometrically,
carbon nanotubes (CNTs) can be classified into single-walled carbon nanotubes
(SWCNTs) formed by folding a single sheet of graphite or multi-walled carbon
1
nanotubes (MWCNTs) that is formed of SWCNTs that are concentrically aligned inside
each other. The direction of folding the graphite sheet is defined by the chiral vector
Ch (n, m) [17], where CNTs can be classified into armchair nanotubes ( n, n) and zigzag
nanotubes (n, 0) or chiral nanotubes (n, m) , where n and m are the chirality vector
indices to be explained in details in chapter 3. Both the mechanical and the electrical
properties of CNTs depend on the chirality of the nanotube.
It is difficult to measure properties of CNTs experimentally due to their nanoscale
dimensions; however, Molecular Dynamics (MD) simulations can serve as a powerful
tool for studying CNTs in different applications [18-21]. However, as also pointed out by
Mylvaganam and Zhang[8] in the literature there is no comparison or clarification on key
parameters like strain rate, displacement increment, length of the nanotube, different kind
of defects and the method for calculating the stresses that influence molecular dynamics
simulations results.
Joule heating, in materials exhibiting non-linear behaviour in its current-voltage
relation at value of high current densities, can not be calculated using Joule’s law due to
the hot electron effect [22], and thus needs to be calculated using a more accurate method
like Monte Carlo simulations. Hot electron effect is defined as the electron distribution in
states far from the thermal equilibrium state.
Until recently, research on the coupling between electrical field and mechanical
forces was only directed toward studying the effect of mechanical forces on the electrical
properties of the CNTs [23-25], however wind forces induced on CNTs due to electron
transport has never been studied with quantum mechanics. This is very important to be
2
able to calculate high current density capacity of CNTs before it fails and the stresses
generated due to the electric field.
1.2 Objectives
In the course of this dissertation, the following objectives were achieved:
1.
Use MD simulations for simulating a (10,10) armchair (SWCNT) under uniaxial
tension until failure, and calculate the stresses using an approach based on virial stress
theorem [26-28] and compare the results with stresses calculated by the widely used [8,
29-34] method based on engineering stresses.
2.
Study the effect of the boundary conditions, displacement increment in MD
simulations on the calculated stress values.
3.
Study the effect of the length and strain rate on the stress strain behaviour of
perfect SWCNTs under uniaxial tension.
4.
Study the mechanism of unravelling in carbon nanotubes during field emission.
5.
Formulate a quantum mechanical model based on the relaxation time
approximation for calculating the joule heating in metallic SWCNTs and use this model
to study the effect of the temperature and the electric field on the joule heating power
generated.
6.
Formulate a similar model to calculate the electron-induced wind forces in
metallic SWCNTs at different temperatures and under different values of electric field.
7.
Develop an Ensemble Monte Carlo (EMC) simulator for calculating the joule
heating and the electron-induced wind forces semi-classically directly without using the
approximation used in 5 and 6 and compare the results to that obtained using the
relaxation time approximation to asses its limitation.
3
8.
Extract the values of the effective charge number in metallic SWCNTs under
different temperatures.
1.3 Contributions of present research
1. The virial stress formula is simplified to ease the calculations of virial stresses in
multibody potentials.
2. A parametric study is performed for molecular dynamics simulations of carbon
nanotubes to quantify the threshold value for the displacement increment used for
carbon nanotubes. This can be used in any other study.
3. The current-voltage relation of carbon nanotubes is calculated based on the relaxation
time approximation and gives satisfactory results in comparison with experimental
data.
4. A semi-classical transport model using Ensemble Monte Carlo simulation model is
developed for calculating the joule heating in carbon nanotubes and can be used to
calculate the joule heating in any other nanoscale material.
5. A new method for calculating the electron-induced wind forces and effective charge
number is formulated and used to calculate the effective charge number in armchair
single-walled carbon nanotubes numerically for the first time. This method is not
limited to carbon nanotubes and can be used for any material.
1.4 Outline
This thesis is organized in seven chapters and six appendices as follows:
Chapter 2 gives a literature review of the experimental investigation of CNTs
under high current densities and electromigration stability and theoretical investigation of
joule heating using EMC simulations in silicon and another quantum mechanical
4
formulation of the joule heating. Also it gives a review of the MD simulations done on
CNTs and the results obtained.
In chapter 3, the Molecular dynamics simulation of (10, 10) SWCNTs failure
under uniaxial tension is presented and a framework for stress calculation using virial
stresses in CNT is derived.
The results obtained in chapter 3 are used in chapter 4 to perform molecular
dynamics simulations of the unravelling of CNT similar to what happens during field
emission.
The relaxation time approximation is used to calculate the joule heating and the
electron-induced wind forces in metallic SWCNTs using a quantum mechanical model in
chapter 5.
Chapter 6 gives the details of the EMC simulations used for calculating the joule
heating and electron-induced wind forces in metallic SWCNTs and compares the results
with that obtained from chapter 5.
Finally, chapter 7 presents the conclusions and proposed future research.
5
CHAPTER 2
LITERATURE REVIEW
2.1 Experimental investigation of CNTs under high current densities
Over the last decade a lot of experimental work has been conducted on CNTs for
the characterization of their electrical properties [35-43]. Only during the last few years
have scientists started to investigate the behavior of carbon nanotubes under high current
densities. An early research that investigated that behavior was that conducted by Yao
and his colleagues [44]. They found that individual SWCNTs can sustain high current
densities of more than 109 A / cm 2 at which the current seems to saturate (Figure 1). They
suggested that the observed current saturation is due to possible scattering mechanisms
and optical or zone-boundary phonon emission by the high-energy electrons and not due
to the depletion of electrons near the Fermi level. The carbon nanotubes that they tested
had a diameter of ~ 1nm and a length in the order of 1 m , and they were tested at roomtemperature. They suggested that for current to saturate at a level of ~ 25 A , phonons of
frequency 1300cm 1 should be emitted which correspond to energy of 0.16eV . They
fitted their experimental results to the Boltzmann Transport Equation (BTE) to calculate
the scattering parameters. From fitting, they found that the mean free path for elastic
scattering is 300 nm, which is equivalent to a scattering rate of 2.66E12 s 1 , and the mean
free path for optical phonon backscattering is 10nm, which is equivalent to a scattering
rate of 8E13 s 1 and no forward scattering.
6
In 2001 a research on the current saturation and electrical breakdown in
(MWCNTs) was conducted at IBM [13]. In that research, it was observed that MWCNTs
do not fail in the continuous accelerating manner typical of electromigration, but instead
they fail in series of sharp current steps assigned to the sequential destruction of
individual nanotube shells (Figure 2), and failure in SWNTs happens in a single sharp
step. They observed that the current is nearly saturated at failure, and thus they suggested
that the saturation and the eventual breakdown process are linked to a common
dissipative process, and they suggested that it most likely involves the excitation of high
energy optical or zone boundary phonons. Figure 3 shows the loss of parts of a CNT
failed under high current density.
Figure 1 Current saturation in SWCNTs (After Yao et. al. 2000[44]).
7
Figure 2 Sequential failure of individual shells in a MWCNT (After Collins et. al.
2001[13])
Figure 3 SEM image of damaged CNTs under high current density (After Collins et. al.
2001[13])
Reliability testing of MWCNTs was carried out in 2001 [14] by passing an
electrical current of ~10mA (corresponding to 1010 A / cm 2 ) in MWCNTs of diameters
8 16nm at 250 C . The tested nanotubes showed stability of resistance for long times
extending to 334 hours without any observable defects (Figure 4), revealing that
nanotubes have very high current density capacity.
8
In 2004 two different researches were conducted to estimate the mean free path
for different electron-phonon scattering mechanisms in carbon nanotubes under low and
high biases. In one of these researches [45] metallic SWCNTs of diameter 1.8nm and
lengths ranging from 50nm to 10 m were tested under ambient conditions and it was
found experimentally that the mean free path for optical and zone-boundary phonons
under high bias is around 180nm and 37nm , respectively and 2.4  m for acoustic
phonons. They also suggested that an electron must first accelerate in a length lT to attain
sufficient excess energy to emit an optical phonon with energy equal to   0.2eV or a
zone-boundary phonon with energy equal   0.16eV , and this length lT is given as
lT 
L
eV
, where L is the length of the tube and V is the potential difference.
Figure 4 Resistance stability of CNTs under high current density (After Wei et. al.
2001[14])
9
In the other research [46], the mean free path for acoustic phonon scattering was
estimated to be lap ~ 300nm , and that for optical phonon scattering is lop ~ 15nm . These
constant values were obtained from fitting the experimental results to a simplified Monte
Carlo simulations, in which it was assumed that the energy dispersion relation is linear
and ignored the details of the phonon dispersion. Also only phonon emission was
included in the simulations. In that research, SWCNTs of diameters ranging from 1.5 to
2.5nm were tested at room temperature and it was found that at high biases, current
saturates at
~ 20  A
for long nanotubes ( L ~ 700,300nm ) before they fail
instantaneously, and reaches 60, 70  A for nanotubes of lengths of 55,10nm respectively
but does not show current saturation (Figure 5). It was also suggested that the channel
conductance under low bias is controlled by lap and that under high bias is controlled by
lop , and the failure in ultra-long nanotubes ( L ~ 3 m ) is due to defect scattering, while
for medium lengths ( L ~ 300nm ) may be due to electron-optical phonon coupling, and
for ultra-short nanotubes failure is suggested to be due to high-field impact ionization
assisted by optical phonon scattering. The difference between the fitted values for the
mean free path in the two studies can be attributed to the difference in the measured
current-voltage relations (i.e. For 50 nm long CNTs at voltage of 0.8 volts the measured
electrical current was ~ 33 A in ref. [45] and ~ 48 A in ref. [46])
The highest current transported through a SWNT was achieved in 2004 [47]and is
equal to 110 A corresponding to  4 109 A / cm2
10
For suspended nanotubes of lengths  400  700nm , it is observed that current
saturates at  8 A [48], which is significantly lower than the saturation currents for
nanotubes with a similar resistance that lie on a substrate as mentioned above. Since
current saturation at high fields in SWNTs is caused by the scattering of optical or zoneboundary phonons; the lower saturation current in a suspended nanotube can be
understood by the lack of a thermally conductive substrate as a heat sink. Electrical
heating is thus rapid in suspended nanotubes and the heat cannot be efficiently conducted
away to the surroundings. It was suggested that this leads to increased acoustic phonon
scattering, which is responsible for the observed negative differential resistance.
Figure 5 Failure of CNTs of different lengths (After Javey et. al. 2004[46])
In 1995, a group of researchers [49] reported on the experimental observation of
carbon chain unraveling from the end of MWCNT that was opened using laser heating by
11
the force of the electric field in an experiment to quantify the field emission in
MWCNTs. They came to the conclusion that the emitting structures were the linear
chains of carbon atoms pulled out from the open edge of the layers of the MWCNT by
the force of the electric field. That was the only reasonable explanation figured out, but
they never quantified the magnitude of those forces due to the electric field as that was
not the scope of the report, and no one to date did so until this dissertation.
2.2 Related analytical studies of electrical and thermal properties
An important research that is related to the Monte Carlo simulations carried out in
this thesis, is that conducted by Pop et. al. in 2005 [50]. In this research the details of the
joule heating in bulk and strained silicon are examined using Monte Carlo simulations.
The detailed phonon generation rates at various electric fields ranging from 5KV/cm to
50 KV/cm were calculated. Also, the integrated net energy generation rates for each
phonon mode were computed. In their simulations, the six conduction X valleys of the
electron energy bands of silicon were modeled with analytical non-parabolic bands.
Interband scattering to higher energy bands were neglected. Longitudinal and transverse
phonon dispersions were modeled with a quadratic analytical approximation.
In 2005, researchers at NASA [51] used the non-equilibrium Green’s function and
Poisson’s equation self-consistently to calculate the current carrying capacity of short
( 100nm ) single-walled metallic zigzag CNTs including the effect of phonon
scatterings. They used the scattering rates calculated in reference [45] for calculating the
deformation potentials for different phonon modes (longitudinal acoustic and longitudinal
optical phonons).
12
It was shown by researchers [52] that calculating joule heating with traditional
methods in nanoscale conductors is inaccurate [53]. Horsfield and co-workers [52, 54-57]
studied the joule heating in nanoscale devices using classical, semi-classical and quantum
mechanical formulations by coupling the electronic and atomic dynamics. This modeling
technique is called the Correlated Electron-Ion Dynamics (CEID). In CEID the power
delivered to the atoms consists of a heating term and a cooling term that come from the
interaction between the electrons and the atoms or the ions of the lattice.
The Monte Carlo simulation method was used extensively by researchers to
identify some electrical properties in single-walled semiconducting CNTs like electro
velocity oscillations [58], mobility and drift velocities [59-63]. In some of these studies
[58, 60, 63], only the lowest energy subbands were included and phonon subbands
essential for intraband scattering and interband scattering between those lowest energy
subbands were only included. In references [60, 63], the energy dispersion relation of the
lowest subbands were analytically fitted to non-parabolic subbands and the acoustic
phonon dispersion relations were analytically linearly-fitted, while the optical phonon
subbands were assumed dispersion-less. In the rest of the studies [59, 61, 62], the full
numerical energy band of the CNTs were included, while the phonon dispersion relations
were approximated to third order and a fifth order polynomial for the longitudinal
acoustic and the longitudinal optical phonons, respectively.
2.3 Molecular dynamics simulations for carbon nanotubes
With the rapidly growing interest in carbon nanotubes and the difficulties in direct
measurements of their properties due to their nano-scale dimensions, molecular dynamics
simulation has been widely used in characterizing the mechanical properties and
13
understanding the mechanisms of deformation [8, 10, 31-33, 64-66]. In these researches,
various time steps, ranging from 0.15 to 15 fs have been used with different
thermostating techniques, and almost all of them use the first generation or the second
generation Reactive Empirical Bond Order (REBO) potentials [67, 68]. Also the TersoffBrenner potential was used [69] to study the brittle and ductile behavior of armchair and
zigzag nanotubes and the nucleation of defects without dealing with the stress-strain
behavior.
Belytschko and his colleagues [29] used a modified Morse potential to study the
fracture of nanotubes. In their work the continuum meachanics engineering stresses
formulation was used to calculate the stresses. It was found that using the Modified
Morse potential in the simulations results in fracture at lower values of strain (around 15
%) compared to the more popular(at that time) Brenner Potential (around 30%). Morse
potential only takes into account the bond stretching and the bond angle bending, but
does not take into account the dihedral angle energetics. Also they assumed a wall
thickness of 0.34 nm and their simulations were carried out for CNTs with length of 4.24
nm.
Also one important finding that was concluded in some studies is that mechanically,
zigzag nanotubes can sustain higher loads than armchair nanotubes [8, 32].
Cornwell and Wille [33] used the whole cross-section of the CNT for calculating the
elastic stress in SWCNT under compression. They used a time step of 15 fs for a total
simulation time of 120 Pico-seconds (Ps) and the first generation REBO potential for
inter-atomic potentials.
14
Yakobson and his colleagues [70] used the first generation REBO potential to
study the effect of the strain rate and temperature on the failure strain of SWCNT and
double-walled CNT with length equal 5nm. The strain rate was varied from 2E8 to 2E9
S 1 and the temperature varied from 75 to 1200 K. They were also the first to report on
the carbon chain unraveling under uniaxial tension. In their study, no results were
presented on the stress-strain behavior of the nanotubes.
One of the important publications addressing details of the molecular dynamics
simulations of CNTs under uniaxial tension is that of Mylvaganam and Zhang [8]. In
their study, they addressed the problem of selecting appropriate interatomic potential,
number and type of thermostat atoms, time and displacement steps and number of
relaxation steps to reach the dynamics equilibrium. They concluded that using the second
generation REBO potential for the interatomic potential, Brendsen thermostat with all
atoms as thermostat atoms and using 50 relaxation steps after each displacement is the
most reasonable and cost effective method. The thickness of the CNT wall was taken
equal to 0.617 Angstrom as recommended by Vodenitcharova and Zhang [71].
15
CHAPTER 3
MOLECULAR DYNAMICS SIMULATIONS OF CNTS
UNDER UNIAXIAL TENSION
3.1 Introduction
Equation Chapter 3 Section 1Historically, the first paper reporting a molecular
dynamics simulation was written by Alder and Wainwright in 1957 [72]. In 1960, the
first example of a molecular dynamics calculation was presented [73], in which a
continuous potential based on a finite difference time integration method is applied. The
method of molecular dynamics (MD) gained popularity in material science and since the
1970s also in biochemistry and biophysics. In physics, MD is used to examine the
dynamics of atomic-level phenomena that cannot be observed directly, such as thin film
growth and ion-sub plantation. It is also used to examine the physical properties of
nanotechnology devices that have not or cannot yet be created.
In this chapter MD is used for simulating a (10,10) armchair SWCNT under
uniaxial tension until failure, and the stresses are calculated using an approach based on
the virial stress theorem [26-28] and compared with stresses calculated by a method
based on continuum mechanics which is commonly used in the literature for CNT stress
calculations [8, 29-34]. Two different boundary conditions are used in the simulation and
the results are compared. The effect of the computational error due to the magnitude of
16
the displacement increment is also studied. Finally the effect of several parameters
(length, strain rate, temperature and defects) on the strength of CNTs is studied.
In the following sections a description of carbon nanotubes geometry as well as
classification of carbon nanotubes according to their geometric properties (section 3.2)
will be presented. In section 3.3, the details of the Molecular Dynamics simulations are
presented. The method of calculating the stresses in the CNT is detailed in section 3.4.
Section 3.5 gives details of the simulations carried out and their results. Conclusions are
presented in section 3.6.
3.2 Geometry of carbon nanotubes
Carbon nanotubes can be simply defined as the tubes that results from folding
around a single layer of graphite sheet (graphene) (Figure 6) to form a single-walled
carbon nanotube (SWCNT). Also several layers of single-walled carbon can be
concentrically nested inside each other forming a Multi-walled carbon nanotube
(MWCNT). Carbon nanotubes can be 0.4 to 100 nanometers in diameter with lengths
ranging from few microns up to 1 millimeter [15].
Graphene is a sheet formed of carbon atoms connected to each other covalently
making the shape of a honeycomb lattice (Figure 6). Each line in Figure 6 represents a
covalent bond of length ao  0.142 ~ 0.144 nanometers between two carbon atoms.
A graphite layer can be viewed as a lattice formed of a unit cells of two adjacent
carbon atoms, A and B, replicated in the direction of the basis vectors a1 , a2 (Figure 6),
where
17
3ao ˆ
3ao ˆ
i
j
2
2
3a
3ao ˆ
a2  o iˆ 
j
2
2
a1 
(3.1)
, and iˆ, ˆj are the unit vectors along the X and Y axes respectively. The position of
any unit cell R on the periodic lattice can be described by the set of integers m, n where
R  ma1  na2
(3.2)
Now that we defined the position vector of any unit cell on the graphite sheet, we
can go further to form a carbon nanotube by rolling the graphite sheet along certain
vector Ch called the chiral vector, where this chiral vector represents the line traveling
around the perimeter of the nanotube, then
Ch  ma1  na2
(3.3)
where m, n represents the number of unit cells along the perimeter of the carbon
nanotube in the direction of the vectors a1 , a2 respectively.
Carbon nanotubes can be mainly classified according to the different values of the
chiral vector Ch (i.e. according to chirality) into achiral (symmorphic) nanotubes or
chiral (non-symmorphic) nanotubes [12]. An achiral nanotube is defined by a carbon
nanotube whose image is identical structure to the original one. The only two cases of
achiral nanotubes are armchair and zigzag nanotubes (Figure 7).
Armchair nanotubes are nanotubes where the chiral vector Ch is parallel to the Xaxis, so that the angle  between the chiral vector and a1 is 30 and m  n .
18
Figure 6 Graphene as a lattice of unit cells of two atoms
Zigzag nanotubes on the other hand are nanotubes where the chiral vector Ch is
parallel to the Y-axis but because of the hexagonal symmetry of the honeycomb lattice
that is equivalent to the case where Ch makes a 30 angle with the X-axis or in another
words the angle  between the chiral vector and a1 is 0 . For this case m is any
positive integer while n must be zero.
For chiral carbon nanotubes the angle  takes any value except 0 ,30 and the
chiral vector Ch can take any positive integer values except for m  n (armchair
nanotubes) and n  0 (Zigzag nanotubes)
19
A. (3, 3) Armchair nanotube
B. (5, 0) Zigzag nanotube
C. (4, 2) Chiral nanotube
Figure 7 Classification of Carbon nanotubes according to their chirality
Another important vector for characterizing CNTs is called the translational
vector T which represents the minimum repetitive length perpendicular to the chiral
vector Ch and can be calculated as [12]
T  t1a1  t2 a2
(3.4)
20
where t1 and t 2 are equal to
2n  m
2m  n
and 
, respectively, with d R is the greatest
dR
dR
common divisor of 2m  n and 2n  m . The translational vector and chiral vector are
shown for a (5, 1) CNT as an example in Figure 8. The rectangle shown in the figure after
folding into a tube along the direction of Ch represents the repetitive unit cell of the
CNT.
Figure 8 Unit cell of (5, 1) carbon nanotube
21
3.3 Molecular Dynamics Simulation details
3.3.1 Equations of motion for different ensembles
Generally in Molecular Dynamics simulation, the evolution of the atomic
trajectories (positions, velocities) is described by Newton’s second law of motion as
ri 
dri pi

dt mi
(3.5)
pi  mi vi  f i
where, ri , vi , pi are the position, velocity and momentum vectors for atom i respectively,
f i is the force vector exerted on atom i and mi is the mass of the atom i .
The energy, or the Hamiltonian  , of a system composed of N atoms can be
2
N
pi
i 1
2mi
expressed as a sum of the kinetic energy   
, and the potential energy U , where
the kinetic energy is function of the N atomic velocities ( v1 , v2 ,...., vN ) and the potential
energy is a function of the N atomic positions ( r1 , r2 ,...., rN ).
Molecular dynamics simulations are commonly classified into two categories;
equilibrium and non-equilibrium [74]. In equilibrium MD simulations, the system is
completely isolated from its surroundings with a fixed number of atoms, volume and
constant energy. These boundary conditions correspond to the micro-canonical (NVE)
ensemble in statistical mechanics [75]. The equations of motion given by Newton’s
second law satisfy these boundary conditions without any further treatment. In nonequilibrium MD simulations, the system is allowed to interact with the surrounding
environment through either thermal or physical constraints. One way to accomplish this
in molecular dynamics is to introduce the concept of an extended system [76].
22
Essentially, Newton’s equations of motion are augmented and coupled to additional
differential equations that describe the relationship between the system and the
environment. Commonly, molecular dynamics calculations are performed at a constant
temperature (canonical ensemble NVT) or constant pressure (NPT). In our research, we
will use the canonical (NVT) ensemble and non-equilibrium MD Simulations.
To perform calculations in the NVT molecular ensemble, two techniques are
commonly used [77], these are direct velocity rescaling and the extended system
methods. Direct velocity rescaling involves resetting the velocities of the particles at each
time step so that the total kinetic energy of the system remains constant [78, 79]. This
acts like an occasional random coupling with a thermal bath. In the extended system
method, the equations of motion for the system are augmented by a frictional coefficient
 that is allowed to vary with time and couples the system dynamics to an external
temperature reservoir. We use Berendsen method [80] for the extended system, where the
equations of motion are modified as
ri 
pi
mi
pi  fi   pi
 
(3.6)
1 To
(  1)
T T
where  T , is a relaxation time for thermal fluctuations, T is the instantaneous
“mechanical” temperature, To is the temperature of the thermal reservoir towards which
the temperature of the system is adjusted.
23
In equation(3.6), if T  To then the system is hotter than required and  will be
negative forcing the system to cool down, and if the system is cooler than required 
will increase forcing the system to heat up.
It is worth mentioning that for equilibrium MD (micro-canonical NVE systems)
the conserved quantity is the Hamiltonian as implied by equation (3.5), while for nonequilibrium MD (canonical NVT systems) the conserved quantity is the Helmholtz free
energy [81].
3.3.2 Integration algorithm
Molecular dynamics simulation is basically the numerical step-by-step solution of
the equations of motion which are simply a system of coupled ordinary differential
equations. A variety of different numerical methods are available for solving these
equations [82]. Only two classes of method have achieved widespread use; one involving
a predictor-corrector approach (PC), the other uses a low-order time-reversible
integration technique.
Two very simple time-reversible integration schemes that are widely used in MD
are the Leapfrog method [83] and the Verlet method [84-86]. The formulas for these
methods follow immediately from the Taylor expansion of the coordinate as a function of
time.
In the course of this thesis, the third order Nordsieck Predictor-corrector method
is used [87, 88]. Predictor-corrector methods are multiple-value methods, in the sense
that they make use of several items of information computed at earlier time steps [89].
24
The Predictor-corrector method is based on the truncation of Taylor’s expansion
of the position, velocity and acceleration and the derivative of the acceleration of each
atom i . This step is called the Predictor step and is given as
1
1
ri (t  t )  ri (t )  vi (t )t  ai (t ) t 2  dai (t ) t 3
2
6
1 2
vi (t  t )  vi (t )  ai (t )t  dai (t ) t
2
ai (t  t )  ai (t )  dai (t )t
(3.7)
where ai (t ) , dai (t ) are the acceleration and its derivative of atom i at time t and t is
the integration time step. These values of the trajectories are incorrect because they do
not follow the equation of motion (equation (3.5)), and so they are corrected in the
Correction step by the value of the forces generated on the atoms fi (t  t ) resulting
from the interatomic potential and given in section 3.3.4. The correction Factor CF for
each atom is given as
CFi  ai (t  t ) 
fi (t  t )
mi
(3.8)
where mi is the mass of atom i (which is the same for all the carbon atoms in the case of
CNTs). This correction factor is multiplied by certain coefficients that minimize the error
and maximize the stability [88], so the corrected trajectories is given as
1
ri (t  t )  ri (t  t )  CFi
6
5
vi (t  t )  vi (t  t )  CFi
6
ai (t  t )  ai (t  t )  CFi
1
dai (t  t )  dai (t  t )  CFi
3
25
(3.9)
The local error introduced at each time step by the third order Nordsieck
Predictor-corrector algorithm due to the truncation of Taylor series is of order (t 4 ) for
the atomic positions and (t 3 ) for the velocities.
3.3.3 Boundary and initial conditions
In molecular dynamics simulations, periodic boundary conditions are often used
to capture macroscopic properties to eliminate the wall effect, and the system is
equivalent to an infinite system of identical copies of the simulated system. But in nanosystems like carbon nanotubes such boundary conditions are not appropriate since the
actual size of the system is on the order of several hundred nanometers, and thus fixed
boundary conditions should be used. In our research the two ends of modeled nanotubes
are constrained by fixing the positions of several unit cells at each end of the nanotube
throughout the whole simulation time.
The initial conditions in MD simulations are assigned by determining the initial
values for the atomic positions, velocities and accelerations. A simple choice for the
atomic positions is to start with the atoms at the sites of the regular lattice such as the
simple cubic lattice or the face centered cubic lattice which are usually available in most
of the MD software. A carbon nanotube can be considered as one large molecule, and the
initial positions and velocities of the atoms can be generated using the simple Matlab
code given in Appendix 1 and based on what was discussed in section 3.2 depending on
the chirality vector of the nanotube. The initial velocities are assigned in random
directions and a fixed magnitude based on the temperature and they are adjusted to insure
26
that the center of the mass is stationary. Finally the atomic accelerations (which
correspond to the atomic forces) are initialized to be zero for all the atoms.
3.3.4 Interatomic potentials
There are two primary aspects to the practical implementation of molecular
dynamics: (i) the numerical integration of the equations of motion together with the
boundary conditions and any constraints on the system (these have been discussed in the
previous subsections); and (ii) the choice of the interatomic potentials. All of the physics
in the molecular dynamics method is contained in the forces acting on each atom in the
system, which are determined by the interatomic potentials through the following
equation:
fi  

U (r1 , r2 ,...., rN )
ri
(3.10)
and thus the reliability of molecular dynamics simulations depends on the use of
appropriate interatomic potential energies and thus forces. The choice of the appropriate
potential for a molecular dynamics simulation is determined by factors such as the bond
type, the desired accuracy, transferability and the available computational resources.
These interactions are generally described using either analytic potential energy
expressions or semi-empirical electronic structure methods, or obtained from the totalenergy first principle calculations [68].
Generally, four classes of interatomic potentials can be defined. These are pair
potentials, cluster potentials (or many-body interactions), pair functional and cluster
functional [77]. Each class corresponds to an increasing level of complexity in the
potential energy approximation. For pair potentials, such as the Lennard-Jones 12-6
27
potential [90, 91], the force between two atoms is a function of only the distance between
those two atoms, where the position of neighboring atoms does not influence the strength
of the bond. On the other hand the many-body interactions consider both the distance
between atoms and the angles between sets of atoms in the force calculation. Cluster
functional is capable of extending the applicability of pair functional to the angular
dependent space.
In this research, the second-generation reactive empirical bond order (REBO)
potential is used to model the carbon-carbon bonds [68]. This potential was derived
specifically for solid carbon and hydrocarbon molecules. The advantages of this potential
are [68]:
1. It reproduces the bonding characteristics (such as binding energy and bond length) for
solid carbon very well.
2. It allows for covalent bond breaking and forming with appropriate changes in atomic
hybridization.
3. It is not computationally intensive.
4. It models the forces associated with the rotation about dihedral angles for carboncarbon double bonds.
This potential is based on the Abell-Tersoff bond order formalism [92-96]. AbellTersoff formalism is not based on a traditional many-body expansion of potential energy
in bond lengths and angles; instead, a parameterized bond order function is used to
introduce many-body effects and chemical bonding into a pair potential. The total
potential energy U is given as
28
N
U  [U R (rij )  bijU A (rij )]
(3.11)
i 1 j i
where, rij is the distance between atom i and atom j , U R ,U A are the pair-additive
interactions that represent the interatomic repulsions and attractions, respectively, and bij
is the bond order between atom i and atom j .
In equation (3.11) and all the equations that follow, the summation is done only
over the nearest neighbors, where the effect of further atoms is included through the bond
order term, not in this summation. This can be achieved through the switching function
hij ( rij ) which has the form
hij (rij )  1

rij  1.7 Å
1  cos[
(rij  1.7 Å )
0.3 Å
2
]
0
1.7 Å  rij  2 Å
(3.12)
rij  2 Å
where it gives a value of one for the nearest neighbor atoms (at a distance less than 1.7 Å )
and zero for distant atoms (at a distance more than 2Å ).
The pair additive interactions between atom i and atom j , U R ,U A is given as
U R (rij )  hij (rij )[1 
3
Q
 r
] Ae ij
rij
U (rij )  hij (rij ) Bn e
A
(3.13)
  n rij
n 1
with the non-physical parameters Q, A,  , Bn and n given in Table 1.
29
Parameter
Value
Parameter
Value
B1
12388.79197798 eV
1
4.7204523127 Å 1
B2
17.56740646509 eV
2
1.4332132499 Å 1
B3
30.72493208065 eV
3
1.3826912506 Å 1
Q
0.3134602960833 Å
A
10953.544162170 eV

4.7465390606595 Å 1
Table 1 Parameters for carbon-carbon pair terms
The empirical bond order function bij is given as [68]
bij  12 [bij   bji  ]  bij
(3.14)
where the terms bij  , bji  take into account the change in the in-plane angle  between
three carbon atoms, while the term bij takes into account properties that are attributed to
the  bond (will be defined in the next chapter). For carbon nanotubes (only carbon
atoms are simulated, with no hydrogen atoms) bij  ( bji  is the same but switching the
indices i and j can be simplified as
bij  
1
1

k ( i , j )
(3.15)
hik (rik ) g (cos( jik ))
30
The function g (cos( jik )) modulates the contribution that each nearest neighbor
makes to the empirical bond order according to the cosine of the angle between atoms i
and k and atoms i and j . This function is given as
g (cos( jik ))  G (cos( ))  Q( N it )[ (cos( ))  G (cos( ))]
 G (cos( ))
0    109.476
109.476    180
(3.16)
where according to Brenner and his colleagues [68], G (cos( ),  (cos( )) are sixth order
polynomials in cos( ) , but since they only give six fitting parameters for every range of
the angle  , we can conclude that they are fifth order polynomials only and calculate
them as
For 0    109.476 :
G (cos( ))  [0.37545  1.40678cos( )  2.25438cos 2 ( )
 2.03128cos 3 ( )  1.42971cos 4 ( )  0.5024 cos5 ( )]
 (cos( ))  [0.271856  0.488916 cos( )-0.433082 cos 2 ( )
(3.17)
-0.559677 cos 3 ( )  1.272041cos 4 ( )-0.040055cos 5 ( )]
, for 109.476    120 :
G(cos( ))  [0.70728  5.67747 cos( )  24.097212cos 2 ( )
 57.5923095cos3 ( )  71.8834528cos 4 ( )  36.2791415cos 5 ( )]
(3.18)
, and for 120    180
G(cos( ))  [0.0026  1.098cos( )  4.346cos 2 ( )
 6.83cos3 ( )  4.928cos 4 ( ) 1.3424cos5 ( )]
In equation(3.16), the function Q ( N it ) is defined by
31
(3.19)
Q( N it )  1
N it  3.2
1  cos(2 ( N it  3.2))

2
0
3.2  N it  3.7
(3.20)
N it  3.7
where N it is the total number of atoms that are neighbors of atom i and is defined as
N it 
h
k ( i )
ik
(3.21)
(rik )
Brenner and colleagues [68] expanded the function bij as a summation of two
terms  ijRC and bijDH , where the value of the first depends on whether a bond between
atoms i and j has radical character (i.e. whether another atoms are attached to them or
not and thus the  bond would change its position continuously) and is part of a
conjugate system and thus represents the influence of the  bond conjugation on the
bond energies. This term is necessary to account for non-local conjugation effects that
govern the different properties of the carbon-carbon bonds in graphite, while the value of
the second term depends on the dihedral angle for carbon-carbon double bonds.
They proposed a tricubic function to expand  ijRC in the form
 ijRC  Fij ( N it , N tj , N ijconj ) 

lmn
( N it )l ( N tj ) m ( N ijconj ) n
(3.22)
l ,m,n
where
Nijconj  1  [

k ( i , j )
hik (rik )  ( X ik )]2  [

l ( i , j )
h jl (rjl )  ( X jl )]2
 ( X ik )  1
,
, and
(3.23)
X ik  2
1  cos( ( X ik  2))
2
0

2  X ik  3
(3.24)
X ik  3
X ik  N kt  hik (rik )
(3.25)
32
Values in Table 2 are used to fit the parameters lmn in equation (3.22)
Table 2 Values for fitting the parameters for the function “ F ” (after Brenner et. al. [68])
In equation (3.23) Nijconj takes on a value of 1 if all the carbon atoms that are
bonded to a pair of carbon atoms i and j have four or more neighbors (i.e. the bond
between these atoms is not considered to be a part of a conjugated system).
Finally, the term bijDH is given as
bijDH  Tij ( Nit , N tj , Nijconj )[
  (1  cos (
2
k ( i , j ) l ( i , j )
ijkl
))hik (rik )h jl (rjl )]
(3.26)
where Tij ( Nit , N tj , Nijconj ) is also a tricubic function and the values in Table 3 are used to fit
its coefficients, and the  is the dihedral angle and can be calculated easily as
33
ijkl  [(ri  rj )  (rk  ri )]  [(rj  ri )  (rl  rj )]
(3.27)
Table 3 Values for fitting the parameters for the function “ T ”(after Brenner et. al. [68])
3.4 Stress calculation
In the literature [8, 29, 31-34], for calculating the stresses in CNTs under uniaxial
loading, an engineering stress concept is commonly used, where the forces over the fixed
atoms at the end of the nanotube are added and divided by the cross-sectional area.
Engineering stress disregards molecular degree of freedom of the matter. Therefore, we
believe this engineering stress computation is incorrect, simply because a CNT does not
satisfy the basic continuum mechanics requirements, therefore it can not be treated as
such. Although engineering stresses in CNT is not perfect, however it is used due to
difficulty in calculating the virial stresses in the case of many-body potentials like the
second generation REBO potential used for the carbon atoms. In this section, in order to
be able to use the virial stress theorem, a direct simplification of virial stresses is
proposed. Starting with the familiar virial stress formula given by [26-28],
34
     kin    int
  int 
1
( fij rij  )
2V i V j
  kin 
1
( mi vi vi )
V i V
(3.28)
where   is the total virial stress,   kin is the kinetic part of the virial stresses,  int is
the internal part of the virial stresses, V is the volume used to calculate the stresses, fij
is the force acting on atom i due to its interaction with atom j acting in the direction of
 , rij  is the component of the distance between atom i and j in the  direction, mi is
the mass of atom i and vi is the velocity of atom i in the  direction. rij  can be
expanded as (rj  ri ) , where ri  and rj are the position of atoms i and j on  axis,
respectively, with respect to the same coordinate system giving
  int 
1
1
( fij (rj  ri )  ) 
( ( f ij rj   f ij ri  )
2V i V j
2V i V j
(3.29)
Since the force acting on atom j due to its interaction with atom i ( f ji ) is equal
in magnitude and opposite in direction to f ij , equation (3.29) can be rewritten a
  int 
1
1
( ( f ji rj   fij ri  ) 
( ( fij ri   f ji rj  )
2V i j
2V i j
(3.30)
Writing this as two separate summations, we have
  int 
1
( fij ri    f ji rj  )
2V i j
j
i
(3.31)
Since the total force acting on atom i along direction  can be calculated as the
summation of forces due to its interaction with all the atoms along that direction, we can
write
35
 f   f
ij
j
i
and  f ji  f j
(3.32)
i
Thus using equation(3.32), equation (3.31) can be written as
  int 
1
( fi ri    f j rj  )
2V i
j
(3.33)
Changing the indices for the second term, thus  int can be given by
  int 
1
 ( fi ri  )
V i
(3.34)
Because of this simplification, the virial stresses are much easier to implement in
the molecular dynamics simulations.
A wide range of nanotube wall thickness values are used in the literature for
calculating the stresses in a CNT ranging from 0.617 angstrom to 3.4 angstrom ([8, 29,
31, 64, 97]. In this thesis, the thickness of the nanotube was taken as 0.617 angstrom as
this was the only value that its calculation was referenced [71].
3.5 Results and discussion
In this chapter, simulations are performed only for (10, 10) armchair SWCNTs of
diameter 13.6 angstrom and 40 atoms per unit cell and different lengths (Figure 9). The
focus on the armchair SWCNTs is due to their suitability to be used as interconnect
material in nanoelectronics as they are always metallic[12]. As a boundary condition, all
the atoms in the two unit cells on either side of the CNT are not allowed to move freely
and are used for applying prescribed displacements in order to simulate the stretching of
the carbon nanotube, while the rest of the atoms are allowed to move freely through
forces that result from the interatomic potential.
36
Figure 9 (10, 10) CNT model used in the study
The integration time step in the simulations is 0.5 femto-second (Fs) which is less
than 10% of the vibration period of a carbon atom [8].
Two different prescribed displacements were applied on the nanotube. In the first
one, two unit cells at the left end of the CNT were fixed, and the prescribed displacement
was applied by moving the atoms in the two unit cells at the other (right) end of the
nanotube along the axis of the nanotube. In the second one, the CNT was stretched
axially from both ends in the opposite directions simultaneously. The strain rate can be
the same for both loadings. In both procedures, the nanotube was left to stabilize for a
certain number of time steps after applying each displacement increment until
convergence is achieved.
3.5.1 Stress calculations
In this section, the axial stress is calculated using the formulation given above
(equations (3.28)-(3.34)) and the engineering stress method used in the literature. In order
to compare the convergence of the two methods, a simulation is carried out for a 246
angstrom long nanotube (4000 atoms) being stretched from both sides at a displacement
37
rate of 200 m/s by applying a displacement increment of 0.025 angstroms every 50 time
steps at each side up to a strain of 20%. Then, the axial stress is calculated at every time
step using both approaches for 800 time steps, which are shown in Figure 10. Results
show that convergence of the stresses calculated by the virial theorem from the beginning
of the measurement of the stress, on the other hand the engineering stresses shows
fluctuations of about 20% during the 800 time steps monitored. Therefore, it is fair to
state that the virial stresses converges much faster than the engineering stresses approach.
It is not clear if engineering stresses in CNTs ever approach a stable value, just like virial
stresses.
38
Figure 10 Virial stress and continuum stresses at the end of convergence period
In order to quantify the difference between the stresses calculated based on both
approaches a group of different simulations were performed. These simulations were
performed on a nanotube 246 angstrom long with 100 repetitive unit cells along its
length. Loading is done at two different displacement rates of 200 m/s and 400 m/s which
are equivalent to displacement increments of 0.05 angstrom and 0.025 angstrom,
respectively, applied every 50 time steps if the CNT is stretched from both sides. For
calculating the virial stresses, all the atoms were included in the calculation, while for the
39
engineering stresses only the boundary atoms were included. Table 4 gives a description
of the different simulations performed. The axial stress is calculated at every time step
and then averaged over each discrete displacement increment and plotted against the
engineering strain. The simulations were run for 150,000 time steps.
Displacement
Simulation
rate m/s
A-1/B-1
Ultimate
strain at
Number of Displacement Ultimate
Boundary
continuum Failure the start
stabilization
increment
virial stress
condition
stress
strain of bond
steps
(angstrom)
(pascal)
(pascal)
breaking
200
A-2/B-2
400
A-3/B-3
400
A-4/B-4
200
A-5/B-5
200
streching
from both
sides
streching
from one
side
50
0.025
1.68E+12
1.36E+12
0.459
0.39
50
0.050
1.16E+12
9.17E+11
0.483
0.335
25
0.025
1.77E+12
1.37E+12
0.431
0.395
50
0.050
1.09E+12
9.17E+11
0.385
0.33
25
0.025
1.74E+12
1.33E+11
0.415
0.39
Table 4 Description of simulations. "A" stands for stresses calculated using virial stress,
"B" for stresses calculated using continuum mechanics approach
The results in Figure 11 and Figure 12, show that the ultimate stress (at strain of
33~40%) using the virial stress theorem is about 1670 GPa (A-3) which is 35% higher
than the one estimated by continuum mechanics approach (B-3). It is worth mentioning
that the effect of the boundary atoms could lead, in some cases where the momentum part
is large compared to the interatomic part, to an increased virial stresses[98, 99].
Molecular dynamics simulations in this study show that carbon chain unravelling initiates
at a strain of 0.395 (Figure 13). The same unravelling behaviour was also reported in the
literature [70]. The difference between the strain at the ultimate stress and the failure
40
strain is due to the molecular chain unravelling, which the engineering stress approach
does not (and can not) take into account.
Figure 11 Stress-strain diagram for CNT stretched from both sides. “A” stands for
stresses calculated using virial stress, “B” for stresses calculated using the continuum
mechanics approach
41
Figure 12 Stress-strain diagram for CNT stretched from one side. “A” stands for stresses
calculated using virial stress, “B” for stresses calculated using the continuum mechanics
approach
42
Figure 13 Carbon chain unraveling in CNTs
43
Also from the results it can be observed that when the same strain rate is used
with smaller displacement increments (0.025 Angstrom in simulation A-1/B-1 rather than
0.050Angstrom in simulation A-4/B-4), the ultimate stress is dramatically increased.
Results also indicate that for the same strain rate and for the same displacement
increment and changing the number of time steps does not change the results much
(comparing simulationsA-1/B-1 with A-5/B-5). Also by comparing simulations with
different strain rates and for the same displacement increments (A-1/B-1 with A-3/B-3
and A-2/B-2 with A-4/B-4), it can be observed that the engineering stresses is insensitive
to strain rate. However, displacement increment has more influence on the ultimate stress
than strain rate. Influence of displacement increment on MD simulations of CNTs is also
reported by Mylvaganam and Zhang [8]. This is probably due to influence of
displacement increment on the kinematic behaviour of the atoms, and thus we dedicate
the following section to perform convergence study to better understand the influence of
the magnitude of the displacement increment on the maximum stress and maximum strain
that can be reached in SWCNTs
3.5.2 Influence of displacement increment
In order to apply a mechanical strain on a nanotube in MD simulations,
displacement increments are applied to the ends of the nanotube at discrete time steps,
which is not the case in real life where the displacement is applied in a continuous
smooth manner. Thus one can expect that any dependence of the mechanical behaviour of
CNTs on the magnitude of the displacement increment is completely computational and
should not be taken seriously. Also one would expect that as the displacement increment
decreases, the simulation becomes closer to reality, and the results would be more
44
accurate. However, decreasing the displacement increment demands more computational
resources, and thus it is important to figure out an optimum magnitude for the
displacement increment beyond which the change in the mechanical behaviour can be
neglected.
In this section in order to study the effect of the magnitude of the displacement
increment, a convergence study is carried out by changing three parameters in the
simulations; the length of the nanotube, the strain rate and the magnitude of the
displacement increment. This parametric study is essential because this information is not
available in the literature. In order to achieve this objective, 14 different MD simulations
where carried out with CNT lengths ranging from 12.3 angstrom to 1180.8 angstrom, and
strain rates ranging three orders of magnitude and displacement increments ranging from
0.00025 angstrom to 0.25 angstrom. In all these simulations, the nanotubes were
stretched from both sides and were left to stabilize for 50 time steps between every two
consecutive load increments. The details of these simulations and their results are shown
in Table 5. In these simulations, the stress is calculated using the virial stress calculations
presented in the previous section. The stress-strain curves for all the simulations are
shown in Figure 14 through Figure 16. In Figure 17, the maximum stress for CNTs with
length of 118.08 Angstroms is plotted against the displacement increment (on logarithmic
scale). Keeping the length of the nanotube and the time step constant in the different
simulations and varying the displacement increment results in a change in the strain rate.
Therefore in Figure 17, the X-axis is for both the displacement increment and the strain
rate. Studying Figure 15, Figure 18 and Figure 19, one can infer that the strain rate has
the least influence on the maximum stress value among the three parameters.
45
Strain rate
unrestrained Displacement
Total length
(per
Simulation
length
increment
(angstrom)
second)
(angstrom)
(angstrom)
Ultimate
stress
(pascal)
Failure
strain
C-chain unravelling strain
start of bond
breaking
chain
formation
C-1
22.151
12.3
0.025
1.47E+12
7.546
0.443
1.504
C-2
127.92
118.08
0.250
6.33E+11
0.2752
0.165
0.2625
C-3
127.92
118.08
0.100
1.17E+12
0.7622
0.3353
0.45054
C-4
68.89
59.04
0.025
1.66E+12
NA
0.402
0.469
C-5
127.92
118.08
0.050
1.77E+12
0.4971
0.4014
0.4463
C-6
127.92
118.08
0.025
1.72E+12
0.5407
0.3974
0.4347
C-7
305.05
295.2
0.025
1.75E+12
0.40176
0.3924
0.4
C-8
127.92
118.08
0.010
1.69E+12
0.422
0.3936
0.42
C-9
1191.242
1180.8
0.025
1.57E+12
0.375
0.3726
0.374
C-10
305.05
295.2
0.00625
1.65E+12
0.403
0.3879
0.395
C-11
127.92
118.08
0.0025
1.60E+12
0.386
0.3836
0.3857
C-12
68.89
59.04
0.00125
1.56E+12
0.8677
0.413
0.415
C-13
22.151
12.3
0.00025
1.50E+12
NA
0.439
NA
C-14
127.92
118.08
0.00025
1.55E+12
0.3808
0.38
0.3803
1.69E+11
6.76E+10
3.38E+10
1.69E+10
6.67E+09
1.69E+09
1.69E+08
Table 5 Results and simulation details for the displacement increment study
From Figure 17 it can be concluded that when the displacement increment is
0.025 Angstrom or less (which is 1.76% of the equilibrium unstrained bond length
between two sp 2 -bonded carbon atoms) maximum stress value observed during uniaxial
tension test does not change. When the displacement increment is larger results do not
converge to a asymptotic value. For values of displacement increment less than 0.025
Angstrom, it is fairly to state that the maximum stress level obtained from the simulations
is linearly proportional to the logarithm of the displacement increment.
46
Figure 14 Stress-strain curves for different simulations with strain rate=1.69E+09 Sec-1
Comparing the stress- strain curves for the different simulations shown in Figure
15, it is clear that the initial elastic modulus is the same for all and the only difference is
the level of stress at which the nanotube breaks. In Figure 14 and Figure 16, increasing
the length of the CNT from 12.3 Angstrom to 1180.08 Angstrom was accompanied with
increasing the initial modulus of elasticity from 2.6 GPa to 4.26 GPa. This observation
can be used as an evidence that although both the magnitude of the displacement
47
increment and the strain rate are changing, only the displacement increment has the most
influence on the magnitude of the maximum stress in the stress-strain behaviour.
Figure 15 Stress-strain curves for different simulations with clear length=118.08
Angstrom
Simulations were carried out to study the effect of the length of the nanotube and
the strain rate on the mechanical behaviour of the tubes are done with displacement
increment less than or equal 0.025 Angstrom (Figure 14, Figure 16, Figure 18 and Figure
19). In Figure 18, the maximum stress is plotted against the length of the nanotube and
48
the strain rate (in logarithmic scale) for simulations with displacement increment equal
0.025 Angstrom. Figure 19 gives the relation between maximum stress and the length of
the nanotube along with the displacement increment (in logarithmic scale) for simulations
with strain rate equal 1.69E+09 sec-1.
Figure 16 Stress-strain curves for different simulations with displacement
increment=0.025 Angstrom
Comparing the curves in Figure 18 and Figure 19, it is obvious that they almost
have the same behavior, and since the length of the nanotube is the only common
49
parameter in these two figures, then this behavior can be attributed only to the change in
the length of the nanotube, while the change in the strain rate has smaller effect on the
maximum stress in the nanotubes. The lower stress values in Figure 19 are due to the
effect of displacement increment. These findings are not in agreement with observations
reported by Mylvaganam and Zhang [8], who reported that for the armchair tube
variation in the displacement step did not show any significant difference in the stressstrain relationship. However authors also state that to get the same results they varied the
time step according to the displacement step used. However, for the zigzag tube they
report that when the displacement increment was varied and time step was kept constant,
stress-strain response varied significantly.
50
Figure 17 Effect of the displacement increment on the maximum stress in the simulated
CNTs with length equal 118.08 Angstroms
51
Figure 18 Effect of the length on the maximum stress in the simulated CNTs with
displacement increment equal 0.025 Angstroms
52
Figure 19 Effect of the CNT length on the maximum stress during uniaxial extension
with strain rate of 1.69E+09 sec-1
3.5.3 Carbon chain unraveling
Carbon chain unraveling has been observed in simulations C-1, 3, 4, 5, 6, 12 and
13 (Figure 13). It is obvious that as the nanotube gets shorter and the strain rate is higher
and the displacement increment is smaller, chain unraveling behaviour is observed in
simulations. We believe the computational error due to large displacement increments
prevents capturing the chain unraveling behaviour. Carbon chain unraveling is
53
responsible for delaying the complete failure of the CNT which is shown in Figure 14
through Figure 16. For short CNTs like that in C-1 and C-13, the tube structure after the
ultimate stress changes to a group of acetylene-like bonds, which is responsible for the
residual stress carrying capability until the failure of these structures. Acetylene-like
bonds are capable of supporting a stress level of 0.4GPa with very large strains. Complete
failure of the CNT in simulations C-4 and C-13 was not reached during the simulation up
to strain of 82%. Failure in CNTs simulated in C-2 and C-5 occurred almost at the bonds
connecting the boundary atoms with the free moving ones, which agrees with what we
mentioned before that the failure at these larger magnitudes of displacement increment is
computational and not real.
3.6 Conclusions
A molecular dynamics simulation procedure has been proposed to calculate
stresses in carbon nanotubes up to failure point, taking into account chain unravelling
behaviour. It is shown that using engineering stress formulation significantly
underestimates ultimate stresses and completely ignores chain unravelling behaviour.
Applied displacement increments can affect the results dramatically. A displacement
increment less than 1.76% of the unstrained equilibrium sp 2 bond length is
recommended. The strain rate has a weak effect on the mechanical behaviour of armchair
single-walled carbon nanotubes, while the length of the nanotube can affect the value of
the maximum stress by around 10%. Unravelling of molecular chains is especially
important for short CNTs.
54
CHAPTER 4
THE UNRAVELLING OF OPEN-ENDED SINGLE WALLED
CARBON NANOTUBES USING MOLECULAR DYNAMICS
SIMULATIONS
4.1 Introduction
Equation Chapter (Next) Section 1After the first proposals of the usage of
MWCNT as field emitters in 1995 [49, 100] a lot of research was directed to study their
applicability [6, 101, 102], and showed a lot of success. In ref. [49], the enhanced field
emission of a single MWCNT was attributed to the unravelled atomic chain from the
open-ended nanotube. It was proposed that the electric field generated the forces that
caused the unravelling process. In 1997 [103], ab initio density functional formalism was
used to simulate the unravelling process in double-walled CNTs as proposed in ref. [49]
In this chapter Molecular Dynamics (MD) Simulation is used to investigate the
mechanical unravelling of (10,10) armchair and (18, 0) zigzag SWCNT till failure, using
different mechanical schemes at different temperatures. MD simulations can serve as a
powerful tool for studying CNTs that allows for the investigation of the applied atomic
forces and stresses as well as the atomic trajectories during the course of the simulation.
55
4.2 Molecular dynamics simulation
The second generation Reactive Empirical Bond Order (REBO) potential [67, 68]
based on the Abell-Tersoff potential [92, 96] is used to represent the covalent bonding
between the carbon atoms (taking into account different possible hybridizations).
The simulations were performed in the canonical (NVT) ensemble where the
temperature was kept constant at 300, 600, 900 and 1200۫۫ Kelvin using Brendsen
thermostat technique [80] for all moving atoms. The integration time step in the
simulations is 0.5 femtosecond(FS) which is less than 10% of the vibration period of a
carbon atom [8]. The third order predictor-corrector Nordsieck algorithm is used for
integrating the equations of motion.
In this chapter, MD simulations were performed on a single carbon atomic
structure and CNTs. Displacement was applied on one side of the simulated structure and
the stresses and the forces were calculated due to the inscribed displacement. The
displacement is applied in increments every 50 simulation time steps and the forces and
the stresses were calculated as the average value over these 50 time steps. The value of
the displacement increment is proven to be crucial and has a major effect on the
kinematic behaviour of the atoms at failure [104]. In the previous chapter, we have shown
that a value of 0.025 angstroms or less is required for the displacement increment to
avoid any error in the simulated behaviour of a carbon nanostructure, thus through out
this paper we use a displacement increment of 0.0125 angstrom.
56
4.3 Behaviour of single atomic chain
The behaviour of a single atomic chain and its force-displacement relation is
required to fully understand and explain the behaviour of the unravelling of CNTs. In this
section we calculate the force-strain relation of a single atomic chain of carbon using MD
simulations till the failure of the chain at different temperatures. Due to the difference in
the hybridization between a single atomic chain structure and CNTs ( sp in a single
atomic chain with 2  bonds per atom versus sp 2 in CNT with 1  bond per atom), the
equilibrium bond length used in CNTs (1.42 angstroms) can not be used. For calculating
the equilibrium bond length, a 0۫۫ Kelvin molecular mechanics simulation for a 105
atoms long chain is run for 100000 time steps. The first and the last atoms in the chain
were restrained in the 2 directions perpendicular to the axial direction of the chain to
insure the straightness of the chain and free to move in the axial direction. The rest of the
atoms were free to move in all 3 directions. For the first half of the simulation time,
atoms were left to relax to their equilibrium position without any trajectory tracked. In
the second half of the simulation, the distance between the atoms were averaged over all
the time steps used for the calculation and yielded an equilibrium interatomic distance of
1.292 Angstroms which is expected compared to that of CNTs due to the stronger bond.
For calculating the force-strain relation for the atomic chain, the same structure
used for the energy minimization was used but with fully restraining 4 atoms at one end
of the chain and using the last atom on the other end of the chain for the application of
the displacement increments as described in the previous section. The force-strain
relation is plotted in Figure 20.
57
Figure 20 Force-Strain relation for carbon single atomic chain at A. 300 K and B. at
1200K
58
It is clear from the figure that the thermal fluctuation in the calculated axial force
at 1200K increases significantly compared to that calculated at 300K and is the reason for
the change of the maximum average force that can be sustained in the chain from
16.7eV/angstrom at 300K to 14eV/angstrom at 1200K. But it is clear from the figure that
the absolute maximum force is the same at both temperatures with a value of
18.6eV/angstroms.The maximum strain in the chain is 32% which is equivalent to 1.7
angstrom bond length at failure.
4.4 Unravelling of nanotubes
4.4.1 Restrained scheme
In this section, simulations are performed for two types of CNTs; (10, 10) armchair
SWCNTs of diameter 13.6 angstrom and 40 atoms per unit cell with 52 unit cells
resulting in a total length of 128 angstroms and (18, 0) zigzag SWCNTs of diameter 14.1
angstroms and 72 atoms per unit cell with 32 unit cells resulting in a total length of 136.4
angstroms. These two selected CNTs are always metallic [12]. As a boundary condition,
all the atoms in the two unit cells on one side of the CNT are completely fixed in all three
directions. In both (10, 10) and (18, 0) CNTs the geometry of the CNT was designed to
have one dangling atom on the free end of the nanotube. The coordinates of this atom is
kept fixed through out the simulation time in the directions perpendicular to axial
direction of the nanotube and used for applying the prescribed displacements in the axial
direction by changing its axial coordinate every 50 time steps as described earlier. The
rest of the atoms are allowed to move freely (but with keeping the temperature constant)
under the effect of the forces that result from the interatomic potential.
59
The force acting on the terminal atom of the chain is calculated and is averaged every
50 time steps. The absolute resultant of the two forces acting in the plan perpendicular to
the axial direction is also calculated. The axial force and the absolute resultant force for
(10, 10) CNT are plotted against the displacement at the end of the chain in Figure 21.
The axial stress at the fixed end of the nanotube is plotted against the displacement in
Figure 22. Figure 23 shows a schematic of the steps of the unravelling in SWCNTs.
60
61
Figure 21 Force-Displacement diagram for (10, 10) CNTs at different temperatures using
the restrained scheme.
In Figure 21-a, for the (10, 10) CNT at 300K, the axial force builds up till
reaching a value of 16eV/angstroms with a displacement of 5.2 angstroms then an atom
unravels from the tube to the chain causing part of the force to relax immediately as
shown in the figure. This is also accompanied with a peak in the in-plan force, which
raises the question of whether restraining the terminal atom in the in-plan direction has
any effect on the mechanism of the unravelling, thus, leading us to use another scheme to
study this effect in the next section.
62
Figure 22 The axial stresses at the fixed end of the (10, 10) CNT at 300K
After the unravelling of the first atom, stresses build up again with increasing the
displacement and relax several times, but not all of these relaxations are due to the
addition of new atom to the chain; some of them are only due to an internal relaxation at
the end of the nanotube itself by the formation and breakage of several bonds. These is in
contrast to the simple continuous radial unravelling of the end atoms without any change
in the structure of the nanotube itself suggested earlier [49, 103]. It is also important to
note that the force required to unravel more atoms or cause the internal relaxation is
independent of the force required to start the unravelling and can sometimes be larger or
smaller than the initial unravelling force.
63
Figure 23 The general steps of unraveling in SWCNTs
64
At a displacement of 12.7 angstroms two hexagons from the body of the nanotube
unravel together and the end of the nanotube at the position of its connection to the
atomic chain starts to have some curvature similar to that in closed ended nanotubes. The
force required to start the unravelling process decreases slightly with the increase of the
simulation temperature and it is clear from Figure 21 that at temperatures of 300K and
600K the failure of unravelled chain only occurs when the direct force at the end of the
chain exceeds the maximum force allowed in an atomic chain calculated in the previous
section, but at higher temperatures the plotted value of the force at failure is less than
these value. This can be attributed to a local instantaneous indirect increase in the stress
in the atomic chain near the nanotube during the addition of a new atom to the chain
assisted by the thermal fluctuations. It is important to note that in all the simulations, the
direction of the unravelling from the end of the nanotube is not radial as suggested earlier
[49, 103] and that the source of the atoms feeding the chain does not rotate around the
circumference of the nanotube. At 900K and 1200K, initially, the unravelling starts in the
radial direction but soon tends to stop. This is the reason for the delay of the failure at
those temperatures to displacements of 26~33 angstroms as we observed that the failure
happens only after the formation of a partial cap at the end of the nanotube and the
formation of this cap is due to the depletion of the atoms at a certain position at the end of
the nanotube which will not happen if the unravelling continues radially. The formation
of the partial cap is also a result of the pulling force causing the walls of the nanotube to
fold onto itself. In Figure 22, the maximum axial stresses built in the body of the (10, 10)
nanotube at 300K due to the unravelling is 150 GPa which is about 10% of the capacity
of the perfect nanotubes [104], and thus failure in the body of the nanotube can never
65
happen under these loading conditions. In the figure, it is clear that there is a time lag
between the actions taking place in the force-displacement diagram and the stressdisplacement diagram as it take a few time steps for the effect to reach the fixed end of
the nanotube. The same level of stresses is also generated at the other temperatures
simulated.
Figure 24 shows the force-displacement relation for the simulated (18, 0) zigzag
nanotubes at different temperatures. Unlike the (10, 10) CNTs, the force required to start
the unravelling in (18, 0) CNT is in the order of 10 eV/angstrom with a slight decrease
with the increase in the simulation temperature but after the unravelling of the first atom,
the force require to unravel more atoms is in the same range as the (10, 10) CNTs. This is
in agreement with the density functional calculations carried out by Lee and his
colleagues [103], where they found that unravelling happens in zigzag nanotubes at an
electric field of 2eV/angstrom while armchair nanotubes unravel at an electric field of
3eV/angstrom which is the same ratio in this work. The same ratio also holds for the
maximum stress in perfect CNT and thus the lower unravelling force in the zigzag
nanotubes is due to the easier to break zigzag oriented bonds. For 600K, it appears in the
figure that the force required to start the unravelling is larger than that at 300K. Actually,
the force causing the unravelling is not larger as the unravelling in both simulations
happens at the same displacement level and the extra force is a reflex force after the
unravelling already started. Regarding the failure of the atomic chain, at 300K, although
the start of the curvature of the end of the nanotube and the formation of the partial cap
happens as early as a displacement of 12.8 angstroms, but the unravelling continues till a
displacement value of around 35 angstroms. This is due to that the location of the
66
connection between the atomic chain and the end of the nanotube is not at the center of
the cap as in the previous simulations but moves the edge of the cap. The same behaviour
is observed at 600K delaying the failure to a displacement value beyond the maximum
displacement value simulated of 50 angstroms. This also happens at 1200K, and the delay
is increased by the delay of the formation of the cap itself due to a few radial unravelling
at the beginning of the simulation.
67
68
Figure 24 Force-Displacement diagram for (18, 0) CNTs at different temperatures using
the restrained scheme.
4.4.2 Restrained scheme
In this section, the same CNTs simulated in the previous section are simulated with
exactly the same details except for the boundary conditions of the terminal atom of the
atomic chain used to apply the displacement. The terminal atom is allowed to move
freely in the two directions perpendicular to the axial direction of the tube. This can be
more close to the case of unravelling during field emission where there are no restrains
on the end of the tube.
69
For (10, 10) CNT, the displacement to failure generally increased compared to the
restrained scheme except for the 600K simulation. In the 600K simulation, a special
failure occurred, where the failure is not due to the curvature of the end of the nanotube,
as no curvature was observed till failure in this simulation due to the unravelling being in
the radial direction for the first few atoms. In this simulation, failure happened due to the
separation of the whole atomic chain from the tube end assisted by an increased indirect
reflex force, and not due to the breakage in the atomic chain itself. At 1200K, the
duration of the unravelling is even extended more by the change of the position of the
connection between the end of the tube and the atomic chain to the end of the cap instead
of its center.
For the (18, 0) CNT, at 300K, increases a little bit to 12eV/angstrom due to the change
of the restraining condition. At 300K, 600K and 1200K the failure happens due to the
increase of the direct force in the chain above its maximum capacity without the
availability of a relaxation mechanism. At 900K, two different random phenomena
occurred. First, unravelling started at a smaller displacement value corresponding to a
force of 5.6 eV/angstroms. This is due to an internal change in the structure near the
unravelled atom assisted by the thermal fluctuations, before reaching the force required
for unravelling. Second, failure was delayed beyond the simulation time by a repetitive
movement of the connection of the chain to the tube to the side of the partial cap at
displacements of 35 and 42 angstroms.
Generally, comparing the kinematics of the atomic systems in the restrained scheme
with the unrestrained scheme, it is observed that the radial force is not strong enough to
force the unravelling to the radial direction. It can be concluded that the jumps in the
70
radial force is not due to the restraining of the unravelling atom in the radial direction; as
it is present in both schemes, but rather can be attributed to the stress trying to force the
unravelling toward the radial direction, and since it is not strong enough to cause that
change in the unravelling direction, it relax through the atomic movements immediately.
71
72
Figure 25 Force-Displacement diagram for (10, 10) CNTs at different temperatures using
the unrestrained scheme.
73
74
Figure 26 Force-Displacement diagram for (18, 0) CNTs at different temperatures using
the unrestrained scheme.
75
4.5 Conclusions
Two molecular dynamics simulation procedures have been proposed to simulate
the unravelling of single walled (10, 10) and (18, 0) carbon nanotubes (SWCNTs) till
failure of the atomic chain and compared to the capacity of an atomic chain structure.
From the simulations the following can be concluded:
-
The maximum force that can be supported in a single atomic chain structure is
18.6eV/angstroms.
-
Generally, using the unrestrained scheme delays the failure of the atomic chain
specially in the armchair carbon nanotubes, but has no significant effect on the
magnitude of the unravelling forces
-
The unravelling force is in the order of 15 eV/angstroms for armchair nanotubes
and 10eV/angstroms for zigzag nanotubes. The start of the unravelling can rarely
start at a lower force if an internal mutation forms near the unravelling atom.
-
The force required to continue the unravelling changes according to the evolution
of the atomic coordinates after the unravelling of the first atom and can be higher
or lower than the force required to unravel the first atom, thus it is recommended
to apply a force level only 1eV/angstrom less than the failure force to insure a
maximum level of unravelling for the best field emission behaviour.
-
Failure of the unravelled structure is redundant and can be due to the (listed
according to the probability of occurrence) indirect failure of the atomic chain due
76
to a reflex force generated during the addition of an additional atom to the chain,
direct failure of the atomic chain due to the exceedance of the force in the chain.
-
Failure usually happens after the formation of a partial cap at the end of the
nanotube which would not happen in a multi-walled carbon nanotube (MWCNT)
due to the presence of internal walls, and thus failure in MWCNTs is expected to
be delayed compared to SWCNTs.
77
CHAPTER 5
JOULE HEATING AND ELECTRON-INDUCED WIND
FORCES USING THE TIME RELAXATION
APPROXIMATION
5.1 Introduction
Equation Chapter (Next) Section 1 Insatiable demand for miniaturization in the
electronics industry requires decreasing the size of the interconnects which leads to
increased current densities in these components that are far beyond the current carrying
capacity of traditional metals and semiconductors currently used [105, 106]. Research has
shown that carbon nanotubes (CNTs) are a strong candidate for replacing traditional
metals and semiconductors [13] due to their high current carrying capacity and
insensitivity to failure mechanisms like electromigration and thermomigration [14].
However, due the novelty of the material and the nanoscale dimensions of CNTs,
important properties affecting the electromigration and thermomigration reliability of
CNTs like the effective charge number Z * (which gives the force induced on the atomic
lattice due to a unit electric field force) is still not yet established. Until recently research
on the coupling between electrical field and mechanical forces was only directed toward
studying the effect of mechanical forces on the electrical properties of the CNTs [23-25],
however wind forces induced on CNTs due to electron transport has never been studied
with quantum mechanics. This is very important to be able to calculate high current
78
density capacity of CNTs before it fails. The effective charge number Z * is a constant
value that is field-independent. To be able to calculate the electron wind forces in CNTs
Z * is essential. For interconnect metals, this has been predicted either based on
experimental observations [107-109] or based on an analytical model [110].
In this chapter, the forces acting on the atoms due to the momentum transferred to
the lattice during electron-phonon scattering for metallic single-walled CNTs is
calculated. This is done through a quantum mechanical formulation incorporating the
energy and the phonon dispersion relations of the CNT under consideration as well as the
scattering probabilities calculated using Fermi’s golden rule. The energy dispersion
relation is calculated based on a tight-binding model. Scattering with longitudinal
acoustic and optical phonons are considered. The calculations are done for (10, 10)
armchair CNT at various electric fields and at temperatures ranging from 300K to 1800K.
The current-voltage characteristics are also calculated using the same model and
compared with experimental results to validate the proposed quantum mechanics
formulation. The occupation probability of the electron states in the presence of an
electric field is approximated using a modified Fermi-Dirac distribution [111].
The effect of the temperature generated due to joule heating on the electrical
transport properties of metallic Single-Walled Carbon Nanotubes (SWCNTs) has been
investigated in some recent studies [3-7]. In these studies, the joule heating is identified
to be dependent on the macroscopic “bulk” resistance of the CNT under investigation
without any consideration for the underlying quantum mechanics of the joule heating.
Horsfield and co-workers [52, 54-57] studied the joule heating in nanoscale
devices using classical, semi-classical and quantum mechanical formulations by coupling
79
the electronic and atomic dynamics using the CEID model. In this chapter, similar to the
CEID model, the electrons are excited by the electrical current, and then the joule heating
is identified from the response of the ionic motion to this excitation of the electrons. A
similar quantum mechanical formulation used for predicting the electron-induced wind
forces is used to calculate the joule heating in SWCNTs based on the energy transfer to
the ionic motion in response to the excitation of the electrons under electrical current.
The only connection between the formulation used in this paper and the CEID model is
the concept that the joule heating can be calculated from the ionic response to the
electrical current without any further relations. Our formulation starts from the basic
concepts of quantum mechanics to write a formula for the joule heating. As a case study,
the effect of the temperature and the electric field on the power generated in a (10, 10)
single-walled armchair CNT is studied.
The energy and phonon dispersion relations taken into account are presented in
sections 5.2 and 5.3 respectively. The calculations of the scattering rates are presented in
section 5.4. The Models used in the study for calculating the electron-induced wind
forces and the joule heating are given in section 5.5. In section 5.6, the electron-induced
wind forces and joule heating power are calculated for an armchair metallic (10, 10)
single-walled CNT at different electric field forces and temperatures. Discussion of the
results is also presented.
5.2 Energy dispersion relation
In this section, we calculate the energy dispersion relation starting by a
description of the electronic structure of graphene and CNTs.
80
5.2.1 Electronic structure of carbon nanotubes
As mentioned in section 3.2, carbon nanotubes are formed from the rolling of the
graphene sheet formed from carbon atoms that are bonded covalently. A carbon atom in
its ground state has six electrons with the electronic structure 1s 2 2s 2 2 p 2 . Two of these six
electrons are in the first energy level (the 1s orbital) and these electrons are strongly
bound to the nucleus and thus called the core electrons. The other four electrons occupy
the second energy level; two occupy the 2s orbital while the other two occupy the 2 p
orbital with each electron in a different sub-orbital 2 px , 2 p y (Figure 27-a). So every
carbon atom has four valence electrons in the outermost (second) energy level, which
means that each carbon atom has to form four covalent bonds with the neighboring atoms
to achieve energetic stability.
The formation of the covalent bond between the carbon atoms in graphene can be
illustrated through the following steps (Figure 27):
1. First, one of the two 2s electrons gets promoted to the empty 2 pz orbital. That is
only possible due to the slight difference in energy between the 2s and the 2 p
orbitals.
2. Since the energy difference between the 2s and the 2 p orbitals in carbon is small
compared with the binding energies between carbon atoms, the electronic wave
functions for the 2s , 2 px and 2 p y electrons mathematically mixes together to form
three sp 2 with a new probability wave function of finding the electrons. This process
of orbital mixing is called hybridization. The probability wave function for these sp 2
orbitals can be represented by three coplanar lobes (all lie in X-Y plane) at 120 to
each other. The fourth valence electron is still in the 2 pz orbital with its probability
81
wave function can be presented by a lobe perpendicular to the plane of the other three
electrons.
Figure 27 The formation of sigma and Pi bonds between 2 carbon atoms.
82
Figure 28 Honey comb lattice of graphene and carbon nanotubes
3. The three sp 2 electrons are used to form strong covalent bonds with three different
neighboring carbon atoms with angles of 120 , thus resulting in the familiar honeycomb shape of the graphene (Figure 28). These strong covalent bonds are called the
“sigma (  ) bond”.
4. The 2 pz electron (perpendicular to the plane of the graphite sheet) is shared with one
of the three carbon atoms bonded to its carbon atom, forming what is known as “Pi
(  ) bond”. Contrarily to the sigma bond, the Pi bond is a weak delocalized covalent
bond that easily breaks and changes position from one carbon atom to another.
Based on what is mentioned above two important points can be deduced:
83
1. The electrical conductance of graphite or carbon nanotubes is attributed to the 2 pz 
electron, while the other three valence electrons play no role in the electrical transport
process. Thus, in the following section tight binding calculations are used to solve the
Schrödinger equation for the  electron in periodic lattice to find the band structure
for the  electron, where the variance of the electrical properties of carbon nanotubes
can be explained based on their band structure.
2. As shown in Figure 28, the probability wave function for the 2 pz electron is oriented
outside the plane of the graphite sheet itself (X-Y plane), with one half above the
graphite sheet and the other half beneath it. Thus we can conclude (without any
mathematical basis) that during the motion of the  electron during electric current
flow, the probability of scattering with the carbon atoms or even with the other
electrons that do not contribute to the electrical current flow (the 1s and sp 2
electrons) is too low, thus long electron mean free paths would be expected which
agrees with the experimental results reported in the literature [45, 46] and ballistic
and quasi-ballistic transport in short carbon nanotubes is reported [38, 39, 42, 44,
112]. In other words, because of long mean free path, conduction in CNTs is usually
classified as ballistic or quasi-ballistic.
5.2.2 Tight binding method for graphene
According to basic quantum mechanics, an electron moving in a periodic potential
V ( r ) is governed by Schrödinger’s equation
[
where,
2
2m
 2  V (r )] i (r )  Ei i (r )
is the modified Planck’s constant,  i (r ) are
the
(5.1)
wavefunctions
or
eigenfunctions for the electron, Ei are the energy eigenvalues and m is the mass of the
electron.
84
A common method for solving the former Schrödinger’s equation for a crystalline
solid is the tight binding method [113]. In the tight binding method, the electronic
wavefunction is approximated as a linear combination of Bloch functions as follows [12]:
n
 i (k , r )   Cij (k ) j (k , r )
(5.2)
j 1
where, k is the wavevector,  j (k , r ) is the j th Bloch function and Cij (k ) are the
weighting coefficients.
A Bloch function  j (k , r ) is given as the weighted summation of the j th atomic
wavefunction  j (r ) in the unit cell at the different atoms in the lattice. Assuming that
there are n atomic wave functions per unit cell and N unit cells in the solid lattice, then
there are n Bloch functions with the j th Bloch function given by
1
N
 j (k , r ) 
N
e
l 1
ik  Rl
 j (r  Rl )
(5.3)
where Rl is the position of the l th atom.
To find the weighting coefficients Cij (k ) , the values of the allowed energy Ei (k )
are minimized. According to quantum mechanical principles, Ei (k ) is given by the
following equation:
Ei (k ) 
i  i
i i
   dr

   dr
*
i
i
*
i
(5.4)
i
where  is the Hamiltonian. Thus substituting equation (5.2) into equation (5.4) and
differentiating with respect to Cij (k ) and equating by zero, we get that:
[ H ]Ci  Ei (k )[S ]Ci
(5.5)
85
where, Ci is a vector of the n coefficients, [ H ] and [ S ] are called the integral and
overlap matrices, respectively and given as:
H ij   i   j
(5.6)
Sij   i  j
By solving the eigenvalue problem given by equation (5.5), we get the required
energy dispersion relation. Noting that the eigenvalues Ei (k ) are periodic function in the
reciprocal lattice ( k x , k y , k z ), then it can be fully described just within the first Brillouin
zone.
Thus the calculations of the tight binding method can be summarized in the
following steps [12]:
1. Specify the unit cell, the basis vectors ( a1 , a2 , a3 ) and the coordinates of the atoms in
the unit cell.
2. Select n atomic orbitals to be considered for the calculations.
3. Specify the Brillouin zone and the reciprocal lattice vectors ( A1 , A2 , A3 ).
4. Calculate the transfer and overlap matrix elements.
5. Solve the eigenvalue problem and obtain the energy dispersion relation for the chosen
atomic orbitals.
Now we can use these five steps to calculate the energy dispersion relation of a
graphite layer as follows:
1. As shown in section 3.2, a graphite layer can be viewed as a lattice formed of a unit
cells of two adjacent carbon atoms, A and B, replicated in the direction of the basis
vectors a1 , a2 (Figure 6), where a1 , a2 are given by equation (3.1)
86
2. As discussed in section 5.2.1, the electric properties of graphene depend only on the
2 pz electron, thus only two  electrons in a unit cell (one electron for each atom) are
considered for the calculations and thus two atomic orbitals are taken as basis for
Bloch’s functions.
3. To find the reciprocal lattice and the first Brillouin zone for graphene, the procedure
given by Datta [114] is used where the points on the reciprocal lattice in the k x  k y
plane are given by
K  M A1  NA2
(5.7)
where M , N  are integers and A1 , A2 are the basis vectors in the reciprocal lattice
which are determined as
A1 
2 ˆ 2 ˆ
i
j
3ao
3ao
(5.8)
2 ˆ 2 ˆ
A2 
i
j
3ao
3ao
where, iˆ, ˆj are the unit vectors along the k x , k y axis of the reciprocal space. Using
these basis vectors the reciprocal lattice can be constructed as shown in Figure 29.
The first Brillouin zone for graphene is then obtained by drawing the perpendicular
bisectors of the lines joining the origin to the neighboring points on the reciprocal
lattice.
87
Figure 29 Direct lattice (left), reciprocal lattice and basis vectors in the reciprocal lattice
and the K-points in the 1st Brilluoin zone (right)
4. For calculating the elements of the transfer matrix [ H ] and overlap matrix [ S ] , we
have two Bloch functions, and thus these matrices are two by two matrices. The two
Bloch functions are given as
 j (k , r ) 
1
N
N
e
l 1
ik  Rl
 j (r  Rl )
(5.9)
where  j (r ) is the 2 pz electronic wave function for the electrons in the A and B
atoms when j  1, 2 respectively.
The H11 element of the transfer matrix is calculated by substituting equation
(5.9) (with j  1 ) into equation(5.6), thus we have:
88
H11 
1
N

1
N
N
N
 e
ik .( Rl  Ro )
L 1 o 1
N

L  o 1
2 p 
z
1
N
1 (r  Ro )  1 (r  Rl )
N
 e
ik .( Rl  Ro )
L 1 o  L
1 (r  Ro )  1 (r  Rl )
(5.10)
  2 pz  terms with Rl  Ro  3ao
  2 pz
In equation (5.10) the maximum contribution to the matrix element H11
comes from terms with L  o , which, neglecting the crystal potential in the
Hamiltonian, gives the 2 pz orbital energy (  2 pz ). The other terms in equation (5.10)
(with the distance between two different “A” atoms equal to or larger than 3ao ) can
be neglected [12]. By the same way H 22 gives the same value  2 pz .
For the off-diagonal matrix element H12 (and similarly for H 21 ) only the three
nearest neighbor “B” atoms relative to an “A” atom are considered. More distant
atoms will have much less contribution and can be neglected. These atoms are
denoted by the vectors R1 , R2 , R3 as shown in Figure 29. Using the same procedure
used for calculating H11 we have
1
H12 
N
N
N
 e
l 1 o 1
ik .( Rl  Ro )
1 (r  Ro )  2 (r  Rl )
3
  1 (r  R)  2 (r  ( R  Ri )) eik .Ri  terms with Rl  Ro  ao
i 1
(5.11)
3
 t e
ik . Ri
i 1
 terms with Rl  Ro  ao
3
 t  eik .Ri
i 1
where the transfer integral t is given by:
t  1 (r  R)  2 (r  R)
(5.12)
89
For calculating the elements of the overlap matrix [ S ] , the same procedure is
used
S11  S 22  1
(5.13)
3
S12  S 21  s  eik Ri
i 1
where the overlap integral s is given by:
s  1 (r  R) 2 (r  R)
(5.14)
5. Solving the eigenvalue problem (equation (5.5)) for calculated values of [ H ],[ S ]
yields
E (k ) 
 2 p  tw(k )
(5.15)
z
1  sw(k )
Where w(k )  1  4cos(
3k y ao
3k y ao
3k x ao
) cos(
)  4cos 2 (
)
2
2
2
For further approximation s can be taken equal to zero [12] (this
approximation will have a great impact on the energy dispersion of the valence band
but will only have a minor effect on the conduction band [12] which is the only band
included in this thesis and thus makes it acceptable to be used) and thus the energy
dispersion relation for graphite is given as
E (k )   2 pz  tw(k )
(5.16)
Equation (5.16) is plotted in Figure 30. Figure 30-a. shows the high energy
E   2 pz and low energy E   2 pz states that make up the conduction and the valence
bands of graphene [115]. In this plot, the conduction and valence bands meet at
certain points in the k-space. These special points, where the conduction and the
90
valence bands are degenerate, are called the “K-points” with E   2 pz . These points
are shown in Figure 29.
In the following sub-section the energy dispersion relation for the graphite layer is
used to find the electrical band structure of carbon nanotubes and classify them into
metallic and semi-conducting nanotubes according to their chiral vector.
Figure 30 The energy dispersion relation of graphite (after (Minot 2004)).
5.2.3 Band structure of a (10, 10) single walled nanotubes
In the previous sub-section the allowed energy levels of graphene were expressed
in terms of the wave-vector k through the energy dispersion relation. The graphite sheet
91
was assumed to be extending infinitely in both directions, and that leads to a continuous
band structure. But once a sheet of graphite is rolled up into a nanotube, the allowed
values of E (k ) are constrained by the imposition of periodic boundary conditions along
the circumferential direction of the nanotube which is given by the chiral vector Ch , and
thus the wave-vector k becomes quantized. The periodic boundary condition is imposed
by [114]:
k  Ch  (k xiˆ  k y ˆj )  (ma1  na2 )
3a
3ao
 k x (m  n) o  k y (m  n)
 2
2
2
(5.17)
where  is an integer.
Equation (5.17) defines a series of parallel lines, each corresponding to a different
integer value of  . Thus the energy dispersion relation for carbon nanotubes is defined
by the set of the one-dimensional dispersion relations ( E (k ) , one for each sub-band  )
along these lines. These one-dimensional sub-bands are just cross-sections of the
dispersion relation of graphene. This method is called the zone folding technique [12] and
is used to obtain the energy and the phonon dispersion relations for the CNT from that of
graphene.
For convenience, in the case of CNTs, the wavevector k (k x , k y ) is represented as
k (k , k )  k ( . k , k )
(5.18)
where the value of  k depends on the chirality of the CNT and k  , k are the reciprocal
lattice vectors along the CNT circumference and axis respectively. We show the first and
92
the second Brillouin zones (BZ) of a (10, 10) armchair CNT, which is the case modeled
in this study, along with the BZ of graphene in Figure 31.
Figure 31 Reciprocal lattice of graphene and CNTs. The dotted hexagons show
the Brillouin zones for graphene, while the solid lines show the different subbands of (10,
10) CNT in the first and the second Brillouin zones.
93
The BZ of the (10, 10) CNT consists of 20 different subbands (  9  10 ). It is
clear from the figure that subbands n and –n are degenerate due to the hexagonal
symmetry of the BZ of graphene, and thus only 11 of the twenty subbands are distinct
and all of them are symmetric about k  0 . Also, from the figure, it is clear that subband n
in the first BZ is the mirror copy of subband 10  n in the second BZ. These above
mentioned properties are of great interest as memory requirements and computational
processing is reduced by 7/8 compared to computation of the full band in the first and
second BZs.
Using equation (5.18) along with equation (5.16), the energy dispersion relation
for the different subbands for (10, 10) CNT can be written as

E (k , )  t 1  4cos(
10
) cos(
3ka0
3ka0
)  4cos 2 (
)
2
2
(5.19)
These are plotted in Figure 32, where the positive energies give the conduction
bands and the negative energies give the valence bands with no energy gap for subband
  10 thus giving the (10, 10) CNT its metallic characteristics. The Matlab code used
for generating the energy band structure for CNTs is presented in Appendix 2
94
Figure 32 Energy dispersion relation of the valence and conduction bands for (10,
10) CNT in the first and second BZs.
The classification of carbon nanotubes into metallic and semi-conducting
nanotubes depends on whether the resulting sub-band dispersion relations will show an
energy gap or not. It is clear that the resulting energy dispersion relations will not show
an energy gap (and thus the attributed carbon nanotube is metallic) if one of the lines
defined by equation (5.17) passes through one of the six K-points (Figure 30). But since
these K-points differ only by a reciprocal lattice vector, we can focus only on one pair of
95
them k1 , k2 , with coordinates (0, 
4
) and thus for a given sub-band to pass through
3 3ao
point k1 , equation (5.17) becomes:
4
3 3ao
3ao
(m  n)  2
2
(5.20)
( m  n)


3
and thus for equation (5.20) to be satisfied (i.e. the CNT is metallic), m  n must be a
multiple of three, otherwise the carbon nanotube will be semi-conducting. Thus it is
concluded that armchair carbon nanotubes are always metallic regardless of the value of
their chiral vector ( m  n is always equal to zero), while for zigzag and other chiral
nanotubes their electrical conductance and classification into metallic or semi-conducting
nanotubes depends on their chiral vector.
5.3 Phonon dispersion relation
For the phonon dispersion relation, the parameters  , q are used instead of  , k
that were used for the energy dispersion relation to differentiate them from each other. To
find the phonon dispersion relation of graphene, a Laplace transform of time and a
discrete Fourier transform of the lattice equation of motion in the real space given by
equation (5.21), yields equation (5.22) in the frequency and wavevector domain
N
[ M ]{un (t )}   [ K n n ]{un (t )  un (t )}
(5.21)
( 2 (q )[ M ]  [ Kˆ (q )])uˆ (q ,  )  0
(5.22)
n1
96
where [ M ] is the mass matrix, un (t ) , un (t ) are the time-dependent acceleration and
displacement of atom n respectively, [ Knn ] is the stiffness matrix (which is diagonal in
the local coordinates of the bond between atom n and n ),  ( q ) is the frequency of the
phonon mode and [ Kˆ ( q )] is the discrete Fourier transform of the stiffness matrix given as
N
[ Kˆ (q )]   [ K n 0 ]eiq
rn
(5.23)
n 0
, where rn is the position vector of atom n .
Now, the associate cell should be determined. The associate cell in lattice
mechanics is the smallest part of the lattice that fully determines its mechanical
properties. According to Aizawa et. al. [116], interactions of a carbon atom with its fourth
nearest neighbor atoms are necessary for accurately predicting the properties of graphite
[117]. This level of interaction was sufficient to replicate the phonon dispersion measured
experimentally. Figure 33 shows the associate cell that takes into account the fourth
nearest neighbor interaction into account for both atoms A and B. In the figure, the doted
circles show the first, second, third and fourth nearest atoms for atom A0, while the solid
circles show them for atom B0. From the figure it is clear that the total number of unit
cells (N) that should be considered to represent the associate cell is seventeen. This unit
cells are numbered in the figure and the indices m, n are indicate for each of them.
97
Figure 33 Associate cell for graphene utilizing fourth nearest neighbor interactions
According to Jishi et. al. [117] the interaction between two carbon atoms is given
by the force constant tensor K given as
  nr

K  0
 0

0
 tin
0
0 

0 
 ton 
(5.24)
where the two atoms are the nth nearest atoms to each other and  nr ,  tin and  ton are the
radial, tangential in-plane, and tangential out-of-plane interactions between the two
atoms.
98
This K matrix gives the interaction locally between the atoms and thus should be
transformed to a global coordinate system which is given in Figure 6. Due to the
periodicity of the lattice, only interactions between the core unit cell 0 and all other unit
cells in the associate cell is of interest.
Table 6 gives the values of the angles (in degrees) between atoms A0 and B0 in
the core unit cell and the different atoms in the associate cells; also it gives how close
these atoms are to each other. The values of the parameters  nr ,  tin and  ton are listed in
Table 7 according to reference [117].
The interaction between the core unit cell 0 and the Nth unit cell is given by the
following 6x6 matrix
K
K N (m, n)   A0 AN
 K B 0 AN
K AoBN 

K B 0 BN 
where K gives the global force constant tensor between atom  and atom  .
99
(5.25)
A0
unit cell
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B0
A
angle
0
-150
150
90
30
-30
-90
B
n-nearest
0
2
2
2
2
2
2
>4
>4
>4
>4
>4
>4
>4
>4
>4
>4
angle
0
-120
120
60
19
-19
-60
180
139
101
-101
-139
A
n-nearest
1
1
1
3
4
4
3
3
4
4
4
4
>4
>4
>4
>4
>4
angle
180
-161
161
120
60
-60
-120
0
-41
-79
79
41
B
n-nearest
1
4
4
3
1
1
3
>4
>4
>4
>4
>4
3
4
4
4
4
angle
0
-150
150
90
30
-30
-90
n-nearest
0
2
2
2
2
2
2
>4
>4
>4
>4
>4
>4
>4
>4
>4
>4
Table 6 Angles between the atoms in the core unit cell and different atoms in the
associate cell and indicate there nth nearest to each other.
n=1
n=2
n=3
n=4
 nr
36.5
8.8
3
-1.92
 tin
24.5
-3.23
-5.25
2.29
 ton
9.82
-0.4
0.15
-0.58
Table 7 Values of the parameters used for the force constant tensor in 104dyn/cm[117]
Solving the eigenvalue problem of equation(5.22) using equations (5.23)-(5.25),
gives the phonon dispersion relation of graphene. Since there are two atoms per unit cell
of graphene with three degrees of freedom at each, there are six distinct phonon branches.
Out of these six branches only the longitudinal (Acoustic LA and Optical LO) modes are
of interest because they have larger effect on the energy dispersion relation than the effect
100
caused by the transverse modes [118, 119]. The polarization of each mode is used to
distinguish between the different modes. The zone folding technique is then used to find
the phonon dispersion relation of the LA and LO phonons for the (10, 10) CNT (Figure
34). The code used for generating the phonon dispersion relation of CNTs is shown in
Appendix 3. Using the zone folding technique has proven to give accurate results for the
longitudinal modes of the CNT but have some deficiencies in predicting the transverse
modes [12] which are not used in this study. Similar to the energy dispersion relation,
only 11 of the 20 phonon subbands are distinct.
5.4 Scattering rates
For calculating the scattering rates of an electron in a certain state, both LA and
LO phonon absorption and emission are allowed as well as forward scattering and
backward scatterings, thus giving 8 different scattering mechanisms for interaction with a
specific phonon branch  . Moreover, an electron in a specific subband is allowed to
scatter to a state in any of the other subbands (interband scattering), thus for the case of
the (10, 10) CNT under consideration in this study, there are a total of 160 scattering
events to be taken into account. These are illustrated in Figure 35. Also electrons in the
first BZ are allowed to scatter to both states in the first BZ (normal scattering) as well as
states in the second BZ (Umklapp scattering).
In this study, the CNT is assumed to be infinitely long, perfect and un-doped (i.e.
No additions are added that would change the charge carrier concentration), and thus
scattering off potential barriers and scattering with defects or impurities are not included.
Moreover, electron-electron scattering as well as scattering with transverse phonons are
ignored because electron-electron scattering has no effect on the momentum or the
101
energy transferred to the lattice and scattering with transverse phonons is less likely to
happen in CNT as well as it has trivial effect on the electronic structure of the CNT[118120].
Figure 34 LA and LO phonon Dispersion relation for (10, 10) CNT in the first
BZ. The lowered labeled subbands are for the LA mode, and the upper unlabeled
subbands are the LO modes.
The selection rules are that, for an electron in state ( k , ) with energy
E ( k , ) scattering to another state ( k ,  ), it should emit or absorb a phonon with energy
E p (q,  ) that satisfies energy conservation as well as momentum conservation in both
102
the axial and circumferential directions. These are imposed through the following
equations , respectively
E (k , )  E (k , )   E p (q,  )
(5.26)
k  k   q
(5.27)
     
(5.28)
Figure 35 Illustration of the scattering mechanisms considered.
In equation(5.26), the positive and negative signs stand for phonon absorption and
emission, respectively, while they stand for forward and backward scattering,
103
respectively, in equation(5.27). The rate of an electron in state ( k , ) to scatter to another
state ( k ,  ) by virtue of the scattering mechanism m ( Sm ((k , ),(k , )) ) is calculated
using the first order perturbation theory from Fermi’s Golden Rule as [121]
Sm ((k , ), (k , )) 
2
 k ,  Hˆ ep  k ,
2
 ( E (k , )  E (k , )   )
(5.29)
where, Hˆ ep is the electron-phonon interaction operator,  k and  k  are the initial and
final states respectively and   is the phonon energy emitted or absorbed during the
scattering event, respectively. Fermi’s Golden Rule comes from the solution of the timedependent Schrodinger equation where the perturbed wavefunctions are expanded as a
linear combination of the unperturbed wavefunctions, and gives the scattering rate from
an initial state to a final state as a function of the scattering potential itself. Fermi’s
Golden rule conserves the energy during the scattering event. Using the deformationpotential approximation, the scattering rates due to LA and LO phonons can be simplified
to [58-60, 122]
2
DLA
(q 2  (2 / d )2 )
1 1 dE
S ((k , ), (k , )) 
( N ( E pLA (q,  ))   )
LA
2  E p ( q,  )
2 2 dk
2
DLO
1 1 dE
S ((k , ), (k , )) 
( N ( E pLO (q,  ))   )
LO
2  E p ( q,  )
2 2 dk
1
(5.30)
( k ,  )
1
(5.31)
( k ,  )
where DLA , DLO are the deformation potential constants for the LA and LO phonons,
respectively and are assumed equal to 14 eV for LA phonons [122] and 25.6
dE
eV/Angstrom for LO phonons [45],  is the linear mass density of the CNT,
dk
104
1
is
( k ,  )
the density of the final state (its calculation is shown in Appendix 2), N ( E p (q,  )) is the
Bose-Einstein occupation number for a phonon in the state (q,  ) , and the positive and
negative signs indicates emitting or absorbing a phonon, respectively. The total scattering
rates due to LA, and LO phonons calculated using equations (5.30) and (5.31) are plotted
in Figure 36 for subbands  10, 9 , where most of the electrons are expected to be
populated. The Matlab code used for calculating the scattering rates is presented in
Appendix 4. These calculations are in agreement with scattering rates obtained by fitting
experimental data [46], which have an average values in the order of 1013 and 1014
second-1 for LA and LO phonons backscattering, respectively. The peaks in the scattering
rates shown in Figure 36 are due to electrons scattering near the bottom of a subband
whether this scattering is inter-subband or intra-subband scattering. The density of the
final states was calculated numerically using the central difference method at all the
points and using the forward difference method at the points where the slope is zero at the
final states to avoid the singularities corresponding to scattering to states at the bottom of
the subbands.
Also from Figure 36, it is clear that increasing the temperature results in an
increase in the scattering rate due to the increase in the Bose-Einstein occupation number,
but the percentage of that increase is not equal for all the states. For 300 K, for subband
10, the scattering rate at the bottom of the scattering-well centered about the zero energy
point at k  0.8515 Å is about 17 and 666 times less than the value just outside the well
for LA and LO phonons, respectively. These factors are only 2.4 and 6 for 1200 K, thus
the preference for electrons to stay in the states in the scattering-well is less for higher
105
temperatures than for lower temperatures. As a result, at high temperatures, the electrons
are spread over more final states that are on subbands, other than subband 10.
106
Figure 36 Scattering rates for LA and LO modes at different temperatures. A- LA
scattering for subband 10. B- LA scattering for subband 9. C- LO scattering for subband
10. D- LO scattering for subband 9.
107
5.5 Momentum and energy transfer quantum model
Only electrons scattering with LA or LO phonons (which represent the vibration of the
lattice atoms) gives rise to the electron wind force exerted on the atoms. For an electron
initially in state ( k , ) and scattering by LA or LO phonons to a final state ( k ,  ), the
momentum transferred to or from the lattice in the circumferential direction in this event
will be
(  ) k , while the momentum transferred in the direction of the tube axis
k  k  . Thus from Newton’s second law, the average force, referred as
would be
“electron wind force”, acting on the lattice in the longitudinal direction (momentum
transferred
per
unit
time)
due
to
this
scattering
event
can
be
written
as ( k  k )  S ((k , ), ( k , )) . Taking into account the probability that state ( k , ) is
occupied and the probability that state ( k ,  ) is empty and integrating over all the states
in the first BZ, the total force per unit length ( F ) of CNT can be expressed as
F
1


(

k  k )  Sm ((k , ), (k , ))  f (k , )  (1  f (k , ))dk
(5.32)
m
, where f ( k , ) is the electron occupation probability calculated according to a
modified Fermi-Dirac distribution function in the presence of an electric field E using
the relaxation time approximation (
f (k )
f (k )  f 0 (k )

) and Taylor’s expansion and
t

is given as [111]
f (k , )  f 0 ( E(k , )  e (k , ) vk ,  E)
108
(5.33)
, where f 0 ( E (k , )) is Fermi-Dirac distribution function, v k , is the group velocity vector
of wave packet centered about the state ( k , ) and  ( k , ) is the relaxation time for an
electron in that state which is given as [121]
 (k , ) 
1
 Sm ((k , ), (k , ))
(5.34)
k , 
However, it is important to note that the distribution function given by equation
(5.33) is an approximation of the exact occupation probabilities that can not be calculated
in a direct manner. As the temperature and the applied electric field increase, the
occupation probabilities from equation (5.33) diverge from the exact value [111].
In this study, only electric fields applied along the direction of the tube axis is
considered. In equation (5.32), only the integration over the initial states in the axial
direction of the CNT is performed, while integration over the final states dk  is replaced
by summation over all the scattering mechanisms m . This is because for every initial
electron state ( k , ) interacting with a phonon subband  , there can be only a maximum
of one final state, for each of the 8 scattering mechanisms mentioned above.
Joule heating is the heat generated in a conductor due to the flow of electricity
within it, and losing part of its energy to the vibration of the lattice (phonons). Similar to
the electron-induced wind forces, we calculate the joule heating power as the energy
transferred to the lattice from electron scattering between different states multiplied by
the rate by which these scattering can occur. This should be integrated over all the
electron states to yield the total joule heating power. Hence, we can rewrite equation
(5.32) for the specific (10, 10) CNT to get the joule heating power per unit length as
109
w
1

160
10
  ( E (k , )
m 1 9
m
 E (k , ))  S m ((k , ), (k , ) m )  f (k , )  (1  f ((k , ) m )) dk (5.35)
5.6 Results and discussion
Before presenting the results calculated from equation (5.32) and equation(5.35),
the current-voltage relationship for the (10, 10) CNT is calculated using the same
principles of our proposed model for calculating the induced electron wind forces and
joule heating. Here, the electric current passing through CNTs can be computed quantum
mechanically by integrating the state’s group velocity multiplied by the probability that
the state is occupied over all the states, and can be written as
I
e
 f (k , ) 
  
1 E
dk
k
(5.36)
Equation (5.36) is used to calculate the intensity of the current at different electric
field forces and results are plotted against the experimental measurements of Park et al
[45] in Figure 37. The lower saturation current value measured from the experiments can
be attributed to the scattering with impurities or defects along the length of the CNT,
which is the reason for higher saturation current in shorter CNT [45, 46]. We should also
point out that the chirality, defect and impurity state of the tested CNT is unknown [45].
Moreover the experimental data represent resistance of the composite system of CNT and
gold pads, not the CNT alone. It is also clear from the experimental results [45] that as
the length of the nanotubes increases the slope of the curve becomes steeper, and thus we
would expect that for the limit of an infinite CNT the initial slope would tend to that
calculated from equation(5.36). The slope of the curve at higher electric fields can be
realized from the fact that curves calculated from equation (5.36) were calculated at
110
constant temperatures, but as time evolves the temperature will change due to joule
heating and thus the slope will decrease (Figure 38). This was also proposed and proven
theoretically by Kuroda and Leburton [4]. Thus, one can state that, if there is no contact
resistance, the CNT has no defects and is infinitely long, the theoretical calculations
results shown in Figure 37 would match well with the measured experimental data.
Figure 37 Experimental data versus theoretical I-V curves for metallic SWCNTs
at 300K.
111
Figure 38 Theoretical I-V curves at different temperatures.
Also comparing our calculated I-V curve with that calculated theoretically using
the non-equilibrium Green’s function, we find the results for long CNTs are almost the
same [51]. In reference [51], the non-equilibrium Green’s function was solved selfconsistently with Poisson equations and the effect of electron-phonon scattering was
included. In that reference, a current saturation of 25 micro-Amperes level was predicted
for long CNTs, which is exactly the same value calculated in this dissertation. Based on
the latter premise we can state that calculations of the scattering rates, occupation
probabilities in the presence of an electric field and the relaxation times we utilize for the
calculation of the electron-induced wind forces and the joule heating are correct.
112
Figure 39 presents the electron wind force induced per unit length of the (10, 10)
CNT along the direction of the nanotube axis, calculated from the model derived in
equation (5.32).
Figure 39 The force generated per unit length of a (10, 10) CNT at 300 K.
Results shows that Z * (which is represented by the slope of the curve) can be
assumed constant up to an electric field force of 2KV/cm. After reaching that critical
value as the electric field increases the force induced remains almost constant until a
value of 6KV/cm. For an explanation for this unusual behavior, a deep insight into
equation (5.32) is required. First before applying an electric field, the electrons will be
mostly populated in the vicinity of the bottom of subband 10, as the electric field
increases, the probability of the occupation of the right moving electron states
113
( f (k  0.855,10) ) starts to increase, while the probability of the occupation of the left
moving electron states ( f (k  0.855,10) ) decreases, and thus the integral in equation
(5.32) will increase rapidly, giving the initial ramp in Figure 39. By increasing the
electric field force, f (k  0.855,10) continue decreasing and f (k  0.855,10) continue
increasing until they saturate at an electric field force around 2KV/cm. Saturation for
f (k  0.855,10) happens when it reaches a value of zero and thus can not decrease
anymore, while saturation for f (k  0.855,10) is due to the higher scattering rates at
values of wave vector higher than 0.875 /Angstrom (Figure 36). These higher scattering
rates are due to scattering to the bottom of subband 10. Increasing the electric field force
more than 2KV/cm, has a little impact on f ( k ) and thus on the integral in equation(5.32)
, resulting in a plateau. Increasing the electric field more than 6KV/cm starts to increase
the probability f (k ,10) more over the range of the next peaks in the scattering rates
around the wave vector values of 0.875 and 1 Angstrom-1. These jumps are due to
scattering to states near the bottom of subbands -9, +9 that have energy values that are
close to that of the initial states, and thus the increase in the probability f (k ,10) at these
states is compensated by the nearly equal decrease in the probability (1  f (k , 9)) , thus
preventing the force from growing exponentially to extremely high value. This behavior
should be captured for any armchair CNT regardless of the diameter due to exhibiting the
same energy and phonon dispersions but with different number of subbands.
Finally in Figure 40, the values of the force per unit length that would be
generated in a (10, 10) CNT is plotted as a function of the temperature at all the simulated
values of the electric field forces. From the figure, it is fair to state that the effect of the
electric field force using the integral form of equation (5.32) is almost negligible in
114
comparison with the effect of the temperature and as the temperature increases the
scattering rates would increase and thus increasing the force that would be generated in
the CNT.
It is clear from this discussion, that it is important to operate the CNTs at
temperatures as low as possible to control the current induced forces. Also it is not
recommended to apply an electric field force higher than 6KV/cm.
Figure 40 The force generated per unit length of a (10, 10) CNT as a function of
temperatures at different electric field forces.
Regarding the joule heating, Figure 41 presents the joule heating power per unit
length of the (10, 10) CNT calculated from the model derived in equation (5.35) and it is
plotted against classical power law ( P  I .V ) (equation (5.36) ) and also against values
115
computed from I-V experimental data [45]. It should be pointed out that
P  I .V calculated from experimental data is not measured joule heating data. It is clear
from Figure 41 that the quantum mechanical model gives joule heating power that is two
orders of magnitude less than what is predicted by P  I .V .
Figure 41 Comparison of theoretical and experimental data of joule heating in
CNT at 300K.
This finding is also supported by measurements reported by Deshpande et. al.
[123]; authors explicitly state that “the lattice cannot dissipate 6 W of power without
heating to extremely high temperatures. However, such lattice heating is not supported by
116
our observation”. This is because in the quantum mechanical model, only scattering
events which involve energy transfer are considered to contribute to the joule heating
while the rest of the scattering events that would contribute to the resistance but would
not contribute to the joule heating, are not considered. These scattering events include
electron-phonon scattering that only involves momentum transfer which is considered in
this study, as well as scatterings with impurities and defects that are not included in this
study. Thus omitting the electron scattering with impurities and defects would only
effects the I-V characteristics of the CNT, but should have no effect on the joule heating
power generated (equation (5.35)). Also in Figure 41, the amount of joule heating power
generated seems to come to a constant value at an electric field of around 2KV/cm after
which increasing the electric field has no significant effect on the amount of power
generated. This can be explained by the fact that at after that upper limit electric field
force, the lowest energy subband becomes saturated. Finally in Figure 42, the simulation
for the joule heating power that would be generated in a (10, 10) CNT is plotted as a
function of the temperature. From the figure it is fair to state that the temperature has
more significant effect on the heat generated than the electric field force. As expected, as
the temperature increases the scattering rates would increase and thus increasing the
amount of heat that would be generated in the CNT. This is more significant at lower
temperature as a small increase in that range would have a significant effect on the BoseEinstein distribution.
117
Figure 42 Heating power per unit length of CNT at different temperatures
118
CHAPTER 6
JOULE HEATING AND ELECTRON-INDUCED WIND
FORCES USING ENSEMBLE MONTE CARLO
SIMULATIONS
6.1 Introduction
Equation Chapter (Next) Section 1 In the previous chapter, the joule heating and
the electron-induced wind forces were calculated using an integral quantum mechanical
form. However, it is important to note that the occupation distribution function given by
equation (5.33) is an approximation of the exact occupation probabilities that can not be
calculated in a direct manner. As the temperature and the applied electric field increase,
the occupation probabilities given by equation (5.33) diverges from the exact value [111].
In order to calculate the occupation probability deterministically, two methods in the
literature are always used, namely the Boltzmann Transport Equation (BTE) and the
Ensemble Monte Carlo (EMC) simulation. Using these two methods for calculating the
occupation probability is well established and there is no need to argue about their
validity. Thus, in order to quantify the integrals given by equations (5.32) and (5.35), we
use an EMC simulator (section 6.2) to integrate the equations stochastically to eliminate
the errors from roughly approximating the electron occupation probability without the
need to calculate the occupation probability initially in a separate step. Using the EMC
simulation has two advantages over using the BTE, these are, eliminating the need to
119
calculate the occupation probability initially and then feed it into the integral, and
secondly, using the EMC give a better description of the electron dynamics, and thus help
us to have a better understanding of the physical phenomenon occurring.
Monte Carlo simulations have known to be a reliable method for accurately
solving the Boltzmann Transport Equation (BTE) for calculating the electron state
occupation probability in semiconductor CNTs [58-60, 62, 63, 122, 124]. Javey et al [46]
used EMC simulations to calculate the mean free path of electrons in metallic CNTs,
where only the lowest energy subband was included with an assumed linear relation and
scatterings to higher subbands were neglected. Also the full details of the phonon
dispersion relation were ignored.
6.2 Monte Carlo Simulation
The MC simulation method is a semi-classical transport method for simulating the
electron dynamics. In EMC simulations, the trajectories (position, momentum ( k )) of a
large number of charge carriers are tracked over a long period of time that can reach
nanoseconds. During the EMC simulations, the charge carriers drift under the effect of
the electric field classically according to Newton’s second law of motion for a selected
short time step in the order of FS as
d
k  Fdrift  eE
dt
(6.1)
At the end of these time steps, the charge carriers can scatter off phonons,
impurities or other charge carriers or start a new classical free drift in the electrical field
without scattering. This is decided randomly through the generation of a random number
and comparing it to the “total” scattering rate of the electron state at the end of the free
120
drift (calculated according to quantum mechanics in section 5.4) multiplied by the
duration of the free drift ( td  S ((k , ), (k , )) ), where td is the total duration of the
m
free drift of the charge carrier under consideration. The random number is uniformly
distributed on the range from zero to one. If the random number is smaller than this
factor, then the charge carrier will scatter through one of the scattering mechanisms and
thus change its state immediately. On the other hand, if the random number is larger than
that factor, the charge carrier does not scatter and starts its new free drift classically
without changing its state. If it is decided that the charge carrier will scatter, the
scattering mechanism is selected also randomly from all the scattering mechanisms
allowed at this state. This is done through assigning a slot of the “total” scattering rate at
each state for each allowed scattering mechanism that is proportional to its individual
scattering rate. Then, a random number from zero to one is uniformly generated and
multiplied by the total scattering rate at the state under consideration. The slot that this
factor lies in, its scattering mechanism is considered. Knowing the specific scattering
mechanism at the end of the free drift, the new state of the charge carrier is determined
and then the charge carrier starts its new free drift. It is important to mention, that all the
scattering mechanisms considered conserve energy and momentum. This process is
repeated until reaching the desired simulation time. During the duration of the simulation,
various quantities can be collected to study different material properties like drift velocity
and occupation probabilities.
For the simulations carried out in this thesis, the electric field is applied only
along the direction of the nanotube axes, and the full numerical energy dispersion and
phonon dispersion calculated in sections 5.2 and 5.3 are considered.
121
Initiate the states of the
charge carriers according to
Fermi-Dirac Distribution
(k , ), E ((k , ))
Drift each electron classically in the electric
field for time step td and update its
wavevector at the end of the step as
eE
k k
td
For each electron, generate a random
number “rand” from zero to one
No
Rand<total scattering rate X free drift time
Yes: electron will scatter
Generate a new random number “ran”
SP=ran X Total Scattering rate
For each scattering mechanism “i”
If
i 1
i
m 1
m 1
No: try next “i”
 scattering rate  SP   scattering rate
Yes: Choose this mechanism
Update the state of the electron
No
Simulation time ended
End Simulation
Figure 43 Illustration of the algorithm for Ensemble Monte Carlo simulation
122
For each of the EMC simulations carried out, 100 charge carriers were simulated
for 100,000 time steps. Time steps were 0.1 Femto Second (FS) which is one tenth of the
shortest scattering time calculated in section 5.4. The electric charge of these charge
carriers were chosen to represent only the number of electrons in the conduction band of
1 Å long (10, 10) CNT. The initial distribution of the electrons (or the charge carriers)
among the different states in the EMC simulations is calculated according to Fermi-Dirac
distribution function. Therefore the initial probability for the electrons to be located in the
scattering-well (Figure 36) is decreased. Therefore, including all the subbands in the
EMC simulations is critical for obtaining accurate results at high temperatures.
Electrons were only simulated in the conduction band, and no scattering or
drifting to or from the valence band was taken into account. From Figure 32, it is clear
that only electrons at the top of subband 10 of the valence band can scatter or drift to
states in the conduction band or the opposite (electrons at the bottom of subband 10 of the
conduction band can scatter or drift to states in the valence band). The justification for
not including these mechanisms is that, for drifting, it is clear that electrons can move
from the valence to the conduction band and vice versa, due to the degeneracy at
wavevector value k  0.8515 Å . In CNTs, this degeneracy is lifted, thus prohibiting
electrons from drifting from the valence to the conduction band and vice versa. The
lifting of the degeneracy in CNTs is because the transverse acoustic phonon mode in
CNTs, unlike graphene, does not preserve symmetry, and thus zone folding gives rise to
a small energy gap at these k points in the order of 1meV at zero temperature
accompanied by a parabolic curving of the subbands [125, 126]. On the other hand, for
123
scattering between the valence and the conduction bands, as mentioned before, only
scattering with LA and LO phonons are considered which have a maximum energy of
0.196 eV (Figure 34). For approximation, in that energy range, we can assume that
scatterings of electrons from the valence band to the conduction band is nearly
compensated by scatterings in the opposite direction. This is because the probability of a
state in the conduction band being full fC ( E  0  0.196eV ) is nearly one order of
magnitude less than the probability of a state in the valence band being full
fV ( E  0  0.196eV ) , also the probability of a state in the valence band being empty
(1  fV ) is one order of magnitude less than the probability of a state in the conduction
band being empty (1  fC ) , but on the other hand, the probability of an electron in the
valence band scattering to a state in the conduction band S (V  C ) is two orders of
magnitude less than the opposite way scattering S (C  V ) because the scattering rate
when absorbing a phonon is proportional to N ( E p (q,  )) the Bose-Einstein occupation
number for that phonon while it is proportional to N ( E p (q,  ))  1 for the case of emitting
a phonon. Thus due to these three factors, it is fair to state that both way scattering
compensate for each other.
For calculating the electron-induced wind forces, the momentum difference
before and after any scattering events at each time step is added up for all the charge
carriers and its cumulative value is plotted against the simulation time for the different
temperatures. By definition, the slope of these lines for each temperature and electric
filed yield the electron-induced wind forces per unit length of CNT.
124
For calculating the joule heating generated, the energy difference before and after
any scattering events at each time step is added up for all the charge carriers and its
cumulative value is plotted against the simulation time for the different temperatures. By
definition, the slope of these lines for each temperature and electric filed yield the joule
heating power per unit length of CNT.
The EMC Matlab code generated for the simulations is attached in Appendix 5.
The values of the electron-induced wind forces and the joule heating for (10, 10)
armchair CNT are presented and discussed in the following section.
6.3 Results and Discussion
EMC simulations were carried out at temperatures of 300oK, 600oK, 900oK and
1200oK and for electric fields of 0.25, 0.5, 0.75, 1, 1.25, 1.5, 2, 4, 6, 8, 10, 15 and 20
KV/cm for each temperature. Electrons were allowed to scatter to states in the second
BZ. Figure 44 shows the time history of the location of a charge carrier in the wavevector
space at the two extreme temperatures simulated. From parts A and B of the figure,
which shows the evolution of the wavevector of the sample charge carrier condensed to
the equivalent state in the first BZ, it is clear that electrons at 300K is limited to states in
the scattering-well of Figure 36 (a and c) and that electrons at 1200K are distributed over
the wider scattering-well from 0.75 to 0.95 Å 1 . This later finding supports the argument
presented earlier in section 5.4. Parts C and D show the location of the sample charge
carrier among the first and second BZs as a function of time. This shows that importance
of accounting for scatterings to the second BZ that can not be ignored. As the temperature
increase to 1200K the frequency of scattering across the two BZs increases dramatically
compared to that at 300K. Parts E and F show the location of the sample electron among
125
the different subbands (reduced to the first BZ). At 300K all the electrons are strictly
distributed among the lowest subband (  10 ) but as the temperature increases electrons
scatters to higher subbands and stay there for short periods of time. Thus, it is important
to include the higher energy subbands for high temperatures, while for 300K including all
the energy subbands is just a computational burden without any increase in the accuracy
of the results, but since the electrons scatter to the states in subband 0 in the second BZ
(Equivalent to subband 10 in the first BZ) both subbands 0 and 10 of the phonon
dispersion relation must be taken into account to allow for this type of scattering, even at
300K.
A
126
B
C
127
D
E
128
F
Figure 44 Time evolution of a sample electron location at 300K and 1200K. AWavevector for 300K. B- Wavevector for 1200K. C- BZ index for 300K. D- BZ index
for 1200K. E- Subband index for 300K. F- Subband index for 1200K.
The cumulative momentum transferred to a unit length of the lattice k  k  for
all the simulated pseudo-electrons is plotted as a function of the simulation time and
presented in Figure 45 for different temperatures. In this figure, it is clear that as the
temperature increases, the amplitude of the fluctuations in the momentum transferred per
unit length is increased. This is due to the increase in the scattering rates for both forward
and backward scatterings. At higher temperatures, the momentum transferred to the
lattice increases. This is due to the fact that at higher temperatures the difference between
forward and backward scattering increase. Also from the figure, it can be observed that
129
the relationship between the cumulative momentum transferred per unit length and
simulation time is linear (excluding the consistent fluctuations) for 300K, 600K and
900K. For 1200K, this linearity is interrupted by some random bumps due to the stepwise transfer of an electron that reached subband zero of the second BZ to higher
subbands by emitting optical phonons till reaching subband 10 or vice versa. This
mechanism is illustrated in Figure 44-F at around 80% of the simulation time.
A
130
B
C
131
D
Figure 45 Cumulative momentum transferred from the electron to the lattice
during the simulation time for all the electric fields simulated.
For calculating the force generated per unit length of the CNT, the curves plotted
in Figure 45 should be differentiated with respect to time. In order to eliminate the
fluctuations in the curves, they were linearly fitted and the slopes of the curves were used
to determine the electron wind forces. The calculated force per unit length of the CNT is
plotted for 300K and compared with that calculated using equation (5.32) at different
electric fields in Figure 46.
132
Figure 46 Electric-induced wind force generated per unit length of (10, 10) CNT
at 300K using different approaches.
The non-linear behavior of the results obtained by the integral form can be
explained that, initially, as the electric field increases, according to the approximation
given by equation (5.33), the probability of the occupation of the right moving electron
states ( f (k  0.855,  10) ) starts to increase, while the probability of the occupation of
the left moving electron states ( f (k  0.855,10) ) decreases, and thus force will increase
rapidly, explaining the behavior until an electric field force of around 1.5 KV/cm. By
increasing the electric field force more, f (k  0.855,10) continue decreasing and
f (k  0.855,10) continue increasing till they saturate at an electric field force around
133
2KV/cm. Saturation for f (k  0.855,10) happens when it reaches a value of zero and
thus can not decrease anymore, while saturation for f (k  0.855,10) is due to the step in
the scattering rates at 0.875 Å 1 shown in Figure 36, thus the increase of the electric field
force at that point has a little impact on f ( k ) and thus on the calculated value of the
force, resulting in the observed saturation between 2 and 6KV/cm. Increasing the electric
field more than 6KV/cm starts to increase the probability f (k ,10) more over the range of
the next peaks in the scattering rates around the wave vector values of 0.875 and 1 Å 1 .
These jumps are due scattering to states near the bottom of subbands -9, +9 that have
energy values that are close to that of the initial states, and thus the increase in the
probability f (k ,10) at these states is compensated by the nearly equal decrease in the
probability (1  f (k , 9)) , thus preventing the force from growing exponentially to
extremely high value. From this explanation it is clear that the approximation of the
electron occupation probability has a major effect on the calculation of the electroninduced wind forces.
On the other hand, the linear behavior exhibited by the EMC simulations can be
explained through equation (6.1) where as an electric field is applied, the electrons at the
bottom of subband 10 start to drift toward the states of higher wavevector until they reach
0.875 Å 1 . After that value, the scattering rate increases by orders of magnitude thus
increasing the probability of the electrons scattering by phonons (mostly backwards), thus
gives rise to the induced forces transferred to the lattice. Increasing the electric field will
only make the electrons reach that scattering step faster and thus backward scatter more
frequently and thus increasing the induced forces and since the rate of change in the
wavevector of the electrons is linearly proportional to the electric field force; the change
134
in the induced forces will also be linearly proportional to it. The linear behavior observed
in Figure 46 and Figure 47 is the same as electron wind force formulation suggested by
Ficks in 1959 [127], and widely used in electromigration literature since, where the
electron wind force in metals is given by Z *eE , where Z * is the effective charge number.
The EMC simulations give a more correct view of the electron dynamics than the
approximation used in the integral form, and it is clear that using the modified FermiDirac distribution approximation is not capable of completely capturing the correct
behavior. This behavior calculated here for the (10, 10) CNT should be the same for any
armchair CNT regardless of the diameter due to exhibiting the same energy and phonon
dispersions but with different number of subbands.
For studying the effect of the temperature, the electron-induced forces are plotted
at different electric fields for the temperatures simulated in Figure 47. All of the curves
show the same linear behavior, but the noise in the data increases as the temperature
increase. This noise can be eliminated by extending the simulation time. Also, as the
temperature increases, the forces increase due to the increase in the difference between
the backward and the forward scattering rates.
Finally, the value of the effective charge number at different temperatures ( Z * )
can be calculated from the slopes of the curves in Figure 47 giving values of 3.465E-3,
9.186E-3, 0.0127 and 0.015 Å 1 for 300, 600, 900, 1200K, respectively. This is the first
time that Z * is calculated experimentally or analytically for CNTs.
135
Figure 47 Electron-Induced wind forces generated per unit length of (10, 10)
using EMC
For calculating the joule heating, the energy difference before and after any
scattering events at each time step is added up for all the charge carriers and its
cumulative value is plotted against the simulation time (Figure 48) for the different
temperatures. Similar to the momentum transferred, the consistently linear relation
observed for the temperatures 300K, 600K and 900 K is interrupted by some random
humps for the simulations at 1200K. Comparing Figure 45 and Figure 48, it is clear that
the temperature has less effect on the fluctuations in the energy transferred than its effect
on the fluctuations in the momentum transferred. This can be explained that as the
temperature increase, the scattering rates for mechanisms involving phonon emission
increase dramatically compared to those mechanisms involving phonon absorption as
136
they are proportional to N ( E p (q,  ))  1 and N ( E p (q,  )) , repespectively, while for the
case of momentum transfer the temperature has the same effect on mechanisms involving
forward scattering and backward scattering. By definition, the slope of the lines in Figure
48 for each temperature and electric field is extracted to yield the joule heating power per
unit length of CNT and are plotted in Figure 49. It is clear from the figure that at low
temperature and low electric field force the noise in the extracted results are high due to
the randomness and uncertainty in the EMC simulations which is a drawback that can not
be overcome in the simulation itself.
137
138
Figure 48 Cumulative energy transferred from the electron to the lattice during the
simulation time for all the electric fields simulated.
For comparison, the EMC results are compared with the quantum mechanical
integral form presented in Chapter 5. Results using this integral form as well as the joule
heating power calculated using joule’s law with current and voltage values measured
experimentally [45] are also plotted in Figure 49. Joule heating plotted from experimental
data is based on measured current and voltage data not actual heat dissipation
measurements. Temperature of the sample during the experiment is also not reported. As
the temperature increases, the integral form tends to be closer to the EMC simulation due
to the increased effect of the scattering rates ( S ) on the generated power which is the
same in both approaches. For lower temperatures (300K), the occupation probability has
139
more significant effect, and its calculation is completely different in the two approaches.
Generally, MC simulations is the most accurate way for predicting the occupation
probability and have gained a wide acceptance [121] and thus the results presented
obtained with this method are more accurate than the results based on the integral form,
the only disadvantage for the EMC simulations is the noise discussed earlier at low
temperatures and electric fields.
Figure 49 Joule heating power generated in one angstrom length of (10, 10) CNT
using different approaches at different temperatures. The markers are data points
extracted from the EMC simulations. The thin line is for the power generated calculated
using joule’s law based on experimental I-V curve.
140
It is important to note that the joule heating calculation are carried out at constant
temperature with respect to time, but an important advantage of the model proposed in
here is that it can be extended with an extensive calculation of the scattering rates at a
fine mesh of temperature to follow the evolution of temperature with respect to time and
its effect on the overall joule heating.
6.4 Conclusions
In this chapter, an analytical method using the ensemble Monte Carlo simulations
is presented for calculating the effective charge number and the electron-induced wind
forces in armchair (10, 10) carbon nanotubes. It was found that for 300oK including the
lowest energy subband along with the lowest and the highest phonon subbands is enough
for the simulations. For 1200oK, it is important to include all the energy subbands. The
electron-induced wind force is found to be linearly dependent on the electric field. The
effect of the temperature on the effective charge number Z * was studied. The unit-less
effective charge number value for CNT was found to vary between 4.65E-3 and 15E-3.
Results were compared with the integral solution based on the approximation of the
electron occupation probability given in the previous chapter, and showed that the
modified Fermi-Dirac approximation in not reliable in calculating effective charge
number and the electron-induced wind forces.
The same approach is used for calculating the joule heating using the Ensemble
Monte Carlo simulations for the case study of (10,10) SWCNTs. Using joule’s law
neglects the effect of hot electrons for the cases of high nonlinearity in the currentvoltage curve of CNTs and overestimates the correct values by two orders of magnitude
141
at low temperatures. Also, it is concluded that in CNTs as the temperature increase the
joule heating increase exponentially, and thus it is important to control the temperature to
stop further heat generation.
The model presented in this chapter can be extended to take into account the hot
phonon effect (the hot phonon effect is the thermally out-of-equilibrium occupation
distribution for specific phonon modes) through continuous update of the phonon
occupation number and the scattering rates in every simulation step based on the previous
scattering events.
142
CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS FOR
FUTURE RESEARCH
7.1 Conclusions
1. Using the engineering stresses in atomistic systems like carbon nanotubes can lead to
an error in the values of the stresses that can reach values up to 35% of that calculated
using the virial stress theorem.
2. A value of 1.76% or less of the unrestrained bond length (equivalent to 0.025
angstroms) is recommended for the displacement increment applied on the end of the
carbon nanotubes every step used to apply uniaxial tension on carbon nanotubes.
Using a displacement increment value larger than the recommended one will lead to
an enormous computational error, while using values smaller than the recommended
value will lead to unnecessary computational burden with no significant impact of the
computed stresses or the mechanism of failure.
3. The strain rate has a minor effect on the mechanical properties of carbon nanotubes
compared to the effect of displacement increment.
4. Applying the displacement on one side of the carbon nanotube or both sides of the
nanotube does not make any difference in the deformation mechanism of the
nanotube or the strength of the nanotube, and thus stretching the nanotube from both
sides is recommended to save the computational effort.
5. The quantum mechanical integral form implementing the time relaxation
approximation for calculating the electron occupation probability using a modified
Fermi-Dirac distribution for calculating the current-voltage characteristics of metallic
143
single-walled armchair carbon nanotubes is fair enough to obtain results that are in
good agreement with the experimental results.
6. When calculating the joule heating generated in carbon nanotubes using the
relaxation time approximation results are in the same order of magnitude as those
calculated using Ensemble Monte Carlo simulations (which is more accurate because
is exact only initially when no electrical field is applied), but the error is significant
that can not be ignored, so it can be concluded that using the quantum mechanical
integral form is only recommended if it is required to find a rough estimate of the
value of the generated power, but if accurate value is required, one should use
Ensemble Monte Carlo simulations.
7. Ensemble Monte Carlo simulations for calculating joule heating in carbon nanotubes,
shows that as the temperature increases the power generated in the nanotubes
increases exponentially and thus it is not recommended to operate carbon nanotubes
under high temperatures.
8. Using Joule’s law for calculating the joule heating in carbon nanotubes is not correct
due to the nonlinearity in the current-voltage relation of the nanotubes.
9. For Monte Carlo simulations of carbon nanotubes, it is important to include all the
energy subbands when simulating carbon nanotubes at high temperatures, while for
room temperature, including the lowest energy subband along with the lowest and
highest phonon subband is sufficient.
10. Calculating the electron wind forces in carbon nanotubes using the relaxation time
approximation gives misleading results and is not a reliable tool for calculating the
effective charge number in metallic carbon nanotubes.
11. Ensemble Monte Carlo simulation for calculating the electron induced wind forces
linearly changes with the electric field and this is in agreement with Fick’s equation
( F  Z *eE ).
144
12. Values of the effective charge number ( Z * ) in single-walled armchair carbon
nanotubes are calculated and have values of 4.65E-3 to 15E-3 for temperatures
ranging from 300K to 1200K.
7.2 Original contributions of this dissertation
6. A Simplification for the virial stress formula is derived to ease the calculations of
virial stresses in multibody potentials.
7. A parametric study was performed for molecular dynamics simulations of carbon
nanotubes to quantify the threshold value for the displacement increment used for
carbon nanotubes. This can be used in any other study.
8. A method is proposed to compute the current-voltage relation of carbon nanotubes
based on the relaxation time approximation and gives satisfactory results in
comparison with experimental data.
9. A semi-classical transport model using Ensemble Monte Carlo simulation model is
developed for calculating the joule heating in carbon nanotubes and can be used to
calculate the joule heating in any other nanoscale material.
10. A new method for calculating the electron-induced wind forces and effective charge
number is formulated and used to calculate the effective charge number in armchair
single-walled carbon nanotubes numerically for the first time. This method is not
limited to carbon nanotubes and can be used for any material.
7.3 Recommendations for future research
1. The methods developed in this thesis for calculating the joule heating and the
electron-induced wind forces can be extended to take into account the hot phonon
effect through the continuous update of the scattering rate through using the updated
phonon occupation number at each step or group of steps not the Bose-Einstein
distribution.
145
2. Also, the model can be extended to take into account the phonon-phonon scattering
through coupling the Ensemble Monte Carlo simulation to wavelength-domain
molecular dynamics simulations.
3. The methods developed in this thesis can be used to calculate the joule heating and
the effective charge number in semiconducting carbon nanotubes and nanotubes with
different chiralities.
4. Results obtained from this thesis can be integrated with other material properties for
carbon nanotubes to formulate a complete constitutive model for simulating carbon
nanotubes at a larger macroscopic scale in composites using finite element method.
5. The ensemble Monte Carlo simulation developed in this thesis uses a phonon
dispersion relation calculated using a force constant tensor. A new phonon dispersion
relation based on a reliable interatomic potential like the second generation reactive
empirical bond order that is discussed in this thesis can be derived and used instead.
6. Include scattering with defects and impurities in the calculation of the scattering rates.
146
APPENDICES
Appendix 1 Matlab code for generating the initial position and velocity
of atoms in perfect CNTs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This code generates the coordinates file for REBO-MD code
% This code generates the coordinates of atoms on a carbon nanotube
% based on the chiral vector Ch(n,m).
% Written by Tarek Ragab, 7/16/2007
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
clc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Part one
% input n, m and output the basic parameters of the carbon nanotube
%
%%%%%%%%%%%%%%
% Input Data %
%%%%%%%%%%%%%%
n=10;
%input n
m=10;
%input m
a0=1.421;
%distance between two carbon atoms
nucells=52;
%Number of unit cells in the Z direction
nfixed=2;
%Number of unit cells to fix at each end
type=1;
%Type of atoms..0->moving atoms, 1-thermostat
temp=1200;
%Temperature(for calculating the velocities)
boltz=8.314277134E-7;
%Boltzman constant in AMU.(Ang)2/((Fs)2.K)
ttime=50000;
%total analysis time in fs
delta=0.5;
%time increment for MD analysis
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
d=gcd(n,m);
%d= the greatest common divisor of n , m
if mod((n-m),3*d)==0
dr=3*d;
else
dr=d;
end
a=sqrt(3)*a0;
%lattice constant
l=a*sqrt(n^2+m^2+n*m);
%perimeter of the nanotube
dt=l/pi;
%diameter of the nanotube
rt=dt/2;
%radius of the nanotube
t1=(2*m+n)/dr;
%t1 for the translational vector
t2=-(m+2*n)/dr;
%t2 for the translational vector
t=sqrt(3)*l/dr;
%length of t (translational vector)
nn=2*(n^2+m^2+n*m)/dr;
%total no. of hexagons in a unit of a CNT
ntotal=2*nn*nucells;
%total number of atoms in the CNT
147
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% find p,q for the rotational vector R %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ichk=0;
if t1==0
n60=1;
else
n60=t1;
end
for p=-abs(n60):abs(n60)
for q=-abs(t2):abs(t2)
j2=t1*q-t2*p;
if j2==1
j1=m*p-n*q;
% M
if (j1>0) & (j1<nn)
ichk=ichk+1;
np(ichk)=p;
nq(ichk)=q;
end
end
end
end
if ichk==0
'no p,q generated'
stop
else
if ichk>=2
'more than one p,q generated'
stop
end
end
mmm=j1;
p=np(1);
q=nq(1);
r=a*sqrt(p^2+q^2+p*q);
% length of vector R
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Part two
% Generates the coordinates of atoms in the unit cell for SWCNT
%
theta1=atan(sqrt(3)*m/(2*n+m));
%angle theta between Ch and a1
theta2=atan(sqrt(3)*q/(2*p+q));
% angle between R and a1
theta3=theta1-theta2;
% angle between Ch & R
theta4=2*pi/nn;
%a period of an angle for atom A
theta5=a0*cos((pi/6)-theta1)/l*2*pi; % difference between atom A & B
h1=abs(t)/abs(sin(theta3));
h2=a0*sin((pi/6)-theta1);
% projection of T on R
% Delta Z between atom A & B
for i=1:nn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% generate the coordinates for atom A %
148
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
k=fix((i-1)*abs(r)/h1);
x(2*i-1)=rt*cos((i-1)*theta4);
y(2*i-1)=rt*sin((i-1)*theta4);
z(2*i-1)=((i-1)*abs(r)-k*h1)*sin(theta3);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% generate the coordinates for atom B %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
z3=((i-1)*abs(r)-k*h1)*sin(theta3)-h2;
x(2*i)=rt*cos((i-1)*theta4+theta5);
y(2*i)=rt*sin((i-1)*theta4+theta5);
if (z3 > t)
z(2*i)=z3-t;
elseif (z3 < 0)
z(2*i)=z3+t;
else
z(2*i)=z3;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Part three
% Replicates the coordinates to multiple unit cells in the Z-direction
%
for i=1:2*nn
for j=0:(nucells-1)
iii=i+j*2*nn;
x(iii)=x(i);
y(iii)=y(i);
z(iii)=z(i)+j*t;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Writting the coordinates to a .d file
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fid=fopen('coord.d', 'w'); % results will be stored in 'coord.d'
fprintf(fid, '%2i x%2i CNT with length%7.4f nanometer and
diameter%4.2f nanometer\n',m,n,t*nucells/10,dt/10);
fprintf(fid, '%7i\n',ntotal);
fprintf(fid, '%6.1f
%4.2f\n',ttime,delta);
fprintf(fid, '10E25
10E25
10E25\n'); %periodic cube
for i=1:nfixed*2*nn
fprintf(fid, '%7i 6 %10.6E %10.6E %10.6E 2\n',i,x(i),y(i),z(i));
end
for i=nfixed*2*nn+1:(2*nn*nucells-nfixed*2*nn)
fprintf(fid,'%7i 6 %10.6E %10.6E %10.6E
%1i\n',i,x(i),y(i),z(i),type);
end
for i=(2*nn*nucells-nfixed*2*nn+1):2*nn*nucells
fprintf(fid, '%7i 6 %10.6E %10.6E %10.6E 2\n',i,x(i),y(i),z(i));
149
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Part four
% Generates the velocity for each atom as function of the temperature
% and set the 2nd and 3rd and 4th derivatives of the locations to zero
%
mass=12;
%mass of carbon atom in AMU
vel=sqrt(3*boltz/mass*temp*(2*nn*nucells-1)/(2*nn*nucells));
%velocity in angstrom/Fs
for i=1:2*nn*nucells
s=2;
while s>1
x1=2*rand(1)-1;
y1=2*rand(1)-1;
s=x1^2+y1^2;
end
Z1(i)=(1-2*s)*vel;
s=2*sqrt(1-s);
X1(i)=s*x1*vel;
Y1(i)=s*y1*vel;
fprintf(fid, '%7i %10.6E %10.6E %10.6E\n',i,X1(i),Y1(i),Z1(i));
end
for j=1:3
for i=1:2*nn*nucells
fprintf(fid, '%7i
0.00 0.00 0.00\n',i);
end
end
status = fclose(fid);
150
Appendix 2 Matlab code for generating the energy dispersion relation
and the energy density of states of CNTs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%This file generates the electronic sub-bands for a given CNT index m,n
% Created by Tarek Ragab on 6-19-2008
% Electronic packaging lab-UB
% Last edited on October-07-2008
% Update from last version: DOS is NOT the absolute value of de/dk
% Units used: energy:ev Length:angstrom
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clc;
clear all;
% List of parameters to be used in the calculations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
t=3.033;
%transfer integral or hopping integral for the Pi-bond
s=0.129;
%overlap integral
a0=1.42;
%distance between two carbon atoms
% Input Data for carbon nanotube:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%
m=10;
%chiral vector index m
n=10;
%chiral vector index n
ques=0;
%if you want to calculate the energy dispersion relation
%for graphite too set to one. Set to zero to skip
% Calculating and plotting the energy dispersion relation of graphene
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% creating the mesh for the plot
if (ques==1)
refine=1000;
%number of mesh points in both directions
counter=0;
kymin=-8*pi/(3*sqrt(3)*a0);
%ky negative range
kymax=8*pi/(3*sqrt(3)*a0);
%ky positive range
kxmin=-4*pi/(3*a0);
%kx negative range
kxmax=4*pi/(3*a0);
%ky positive range
ygrid=(kymax-kymin)/(refine-1);
xgrid=(kxmax-kxmin)/(refine-1);
for i=1:refine
kx=kxmin+xgrid*(i-1);
for j=1:refine
ky=kymin+ygrid*(j-1);
w=sqrt(1+4*cos(3*kx*a0/2)*cos(ky*a0/2)+4*(cos(ky*a0/2))^2);
%egcon(i,j)=t*w/(1+s*w);
%using the exact formula
%egval(i,j)=-t*w/(1-s*w);
%using the exact formula
egcon(i,j)=t*w;
%using the approximate formula
counter=counter+1;
percent=counter/refine^2
end
end
else
151
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculating the translational vector parameters T
d=gcd(n,m);
%d= the greatest common divisor of n , m
if mod((n-m),3*d)==0
dr=3*d;
else
dr=d;
end
a=sqrt(3)*a0;
%lattice constant
l=a*sqrt(n^2+m^2+n*m);
%perimeter of the nanotube
dt=l/pi;
%diameter of the nanotube
rt=dt/2;
%radius of the nanotube
t1=(2*m+n)/dr;
%t1 for the translational vector
t2=-(m+2*n)/dr;
%t2 for the translational vector
T=sqrt(3)*l/dr;
%length of T (translational vector)
nn=2*(n^2+m^2+n*m)/dr;
%total no. of hexagons in a unit of a CNT
% Calculating the reciprocal lattice vectors b1,b2
b1x=2*pi/(sqrt(3)*a);
b1y=2*pi/a;
b2x=2*pi/(sqrt(3)*a);
b2y=-2*pi/a;
% generating the reciprocal lattice vectors k1,k2 for the specified NT
k1x=1/nn*(-t2*b1x+t1*b2x);
k1y=1/nn*(-t2*b1y+t1*b2y);
k2x=1/nn*(m*b1x-n*b2x);
k2y=1/nn*(m*b1y-n*b2y);
K2=sqrt(k2x^2+k2y^2);
%Magnitude(or length) of k2 vector
% Generating the sub-bands
refine=1000;
%number of points in every sub-band
kmin=0;
kmax=pi/T;
kgrid=(kmax-kmin)/(refine-1);
for i=0:(nn/2)
% i is the sub-band index
for j=1:refine
k=kmin+kgrid*(j-1);
e(j,1)=k;
kx=k/K2*k2x+i*k1x;
ky=k/K2*k2y+i*k1y;
w=sqrt(1+4*cos(sqrt(3)*kx*a/2)*cos(ky*a/2)+4*(cos(ky*a/2))^2);
e(j,(i+2))=t*w;
end
end
% Calculating dE/dK for every point
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i=1:(nn/2+1)
for j=1:refine
dedk(j,1)=e(j,1);
if j==1
%use forward difference method
dedk(j,i+1)=(e(2,i+1)-e(1,i+1))/kgrid;
elseif j==refine
%use backward difference method
dedk(j,i+1)=(e(j,i+1)-e(j-1,i+1))/kgrid;
152
else
%use centeral difference method
dedk(j,i+1)=(e(j+1,i+1)-e(j-1,i+1))/(2*kgrid);
end
end
end
% Calculating the total density of state
(DOS)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%for i=1:nn
%
for j=1:refine
%
if j==1
%use forward difference method
%
dos(j,2*i)=1/abs((e(2,i+1)-e(1,i+1))/kgrid);
%
dos(j,2*i-1)=e(j,i+1);
%
elseif j==refine
%use backward difference method
%
dos(j,2*i)=1/abs((e(j,i+1)-e(j-1,i+1))/kgrid);
%
dos(j,2*i-1)=e(j,i+1);
%
else
%use centeral difference method
%
dos(j,2*i)=1/abs((e(j+1,i+1)-e(j-1,i+1))/(2*kgrid));
%
dos(j,2*i-1)=e(j,i+1);
%
end
%
end
% end
end
153
Appendix 3 Matlab code for generating the phonon dispersion relation
of CNTs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This file is used to calculate the phonon dispersion of an(n,m)
% CNT using the zone folding technique
% written by Tarek Ragab in 6/20/2008
% units are cm (some times angstrom when idicated),second,gram, dyne
% Units of angstroms are used for the reciprocal lattice
% The ANGULAR frequencies are normalized by the velocity of sound
% units of normalized angular frequenies are /cm
% or can be given in ev
% Last edited on August 7th 2008
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clc
clear all
syms theta px py;
%theta is the angle between the atoms in degrees
% Input Data for carbon nanotube:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
m=10;
%chiral vector index m
n=10;
%chiral vector index n
%Kn is the interaction of unit cell 0 with unit cell n
%t is the rotation matrix
%KKn is the local n force constant tensor
t=[cos(theta*pi/180)
sin(theta*pi/180)
0;
-sin(theta*pi/180)
cos(theta*pi/180)
0;
0
0
1];
KK1=[36.5e4
0
0
0
24.5e4
0
0;
0;
9.82e4];
KK2=[8.8e4
0
0
0
-3.23e4
0
0;
0;
-0.4e4];
KK3=[3e4
0
0
0
-5.25e4
0
0;
0;
0.15e4];
KK4=[-1.92e4
0
0
0
2.29e4
0
0;
0;
-.58e4];
%%%%%%%%%%%%%%%%% K1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Kaa=t.'*KK2*t;
154
aa=subs(Kaa,theta,-150);
aa1=-1*aa;
Kab=t.'*KK1*t;
ab=subs(Kab,theta,-120);
ab1=-1*ab;
Kba=t.'*KK4*t;
ba=subs(Kba,theta,-161);
ba1=-1*ba;
Kbb=t.'*KK2*t;
bb=subs(Kbb,theta,-150);
bb1=-1*bb;
K1=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Kaa=t.'*KK2*t;
aa=subs(Kaa,theta,150);
aa2=-1*aa;
Kab=t.'*KK1*t;
ab=subs(Kab,theta,120);
ab2=-1*ab;
Kba=t.'*KK4*t;
ba=subs(Kba,theta,161);
ba2=-1*ba;
Kbb=t.'*KK2*t;
bb=subs(Kbb,theta,150);
bb2=-1*bb;
K2=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Kaa=t.'*KK2*t;
aa=subs(Kaa,theta,90);
aa3=-1*aa;
Kab=t.'*KK3*t;
ab=subs(Kab,theta,60);
ab3=-1*ab;
Kba=t.'*KK3*t;
ba=subs(Kba,theta,120);
155
ba3=-1*ba;
Kbb=t.'*KK2*t;
bb=subs(Kbb,theta,90);
bb3=-1*bb;
K3=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Kaa=t.'*KK2*t;
aa=subs(Kaa,theta,30);
aa4=-1*aa;
Kab=t.'*KK4*t;
ab=subs(Kab,theta,19);
ab4=-1*ab;
Kba=t.'*KK1*t;
ba=subs(Kba,theta,60);
ba4=-1*ba;
Kbb=t.'*KK2*t;
bb=subs(Kbb,theta,30);
bb4=-1*bb;
K4=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Kaa=t.'*KK2*t;
aa=subs(Kaa,theta,-30);
aa5=-1*aa;
Kab=t.'*KK4*t;
ab=subs(Kab,theta,-19);
ab5=-1*ab;
Kba=t.'*KK1*t;
ba=subs(Kba,theta,-60);
ba5=-1*ba;
Kbb=t.'*KK2*t;
bb=subs(Kbb,theta,-30);
bb5=-1*bb;
K5=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
156
%%%%%%%%%%%%%%%%% K6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Kaa=t.'*KK2*t;
aa=subs(Kaa,theta,-90);
aa6=-1*aa;
Kab=t.'*KK3*t;
ab=subs(Kab,theta,-60);
ab6=-1*ab;
Kba=t.'*KK3*t;
ba=subs(Kba,theta,-120);
ba6=-1*ba;
Kbb=t.'*KK2*t;
bb=subs(Kbb,theta,-90);
bb6=-1*bb;
K6=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
aa=[0
0
0;
0
0
0;
0
0
0];
aa7=-1*aa;
Kab=t.'*KK3*t;
ab=subs(Kab,theta,180);
ab7=-1*ab;
ba=[0
0
0
0
0
0
ba7=-1*ba;
0;
0;
0];
bb=[0
0
0
0
0
0
bb7=-1*bb;
0;
0;
0];
K7=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
aa=[0
0
0;
0
0
0;
0
0
0];
aa8=-1*aa;
Kab=t.'*KK4*t;
ab=subs(Kab,theta,139);
157
ab8=-1*ab;
ba=[0
0
0
0
0
0
ba8=-1*ba;
0;
0;
0];
bb=[0
0
0
0
0
0
bb8=-1*bb;
0;
0;
0];
K8=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
aa=[0
0
0;
0
0
0;
0
0
0];
aa9=-1*aa;
Kab=t.'*KK4*t;
ab=subs(Kab,theta,101);
ab9=-1*ab;
ba=[0
0
0
0
0
0
ba9=-1*ba;
0;
0;
0];
bb=[0
0
0
0
0
0
bb9=-1*bb;
0;
0;
0];
K9=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% K10 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
aa=[0
0
0;
0
0
0;
0
0
0];
aa10=-1*aa;
Kab=t.'*KK4*t;
ab=subs(Kab,theta,-101);
ab10=-1*ab;
ba=[0
0
0;
0
0
0;
0
0
0];
ba10=-1*ba;
158
bb=[0
0
0;
0
0
0;
0
0
0];
bb10=-1*bb;
K10=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K11 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
aa=[0
0
0;
0
0
0;
0
0
0];
aa11=-1*aa;
Kab=t.'*KK4*t;
ab=subs(Kab,theta,-139);
ab11=-1*ab;
ba=[0
0
0;
0
0
0;
0
0
0];
ba11=-1*ba;
bb=[0
0
0;
0
0
0;
0
0
0];
bb11=-1*bb;
K11=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
aa=[0
0
0;
0
0
0;
0
0
0];
aa12=-1*aa;
ab=[0
0
0;
0
0
0;
0
0
0];
ab12=-1*ab;
Kba=t.'*KK3*t;
ba=subs(Kba,theta,0);
ba12=-1*ba;
bb=[0
0
0;
0
0
0;
0
0
0];
bb12=-1*bb;
159
K12=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
aa=[0
0
0;
0
0
0;
0
0
0];
aa13=-1*aa;
ab=[0
0
0;
0
0
0;
0
0
0];
ab13=-1*ab;
Kba=t.'*KK4*t;
ba=subs(Kba,theta,-41);
ba13=-1*ba;
bb=[0
0
0;
0
0
0;
0
0
0];
bb13=-1*bb;
K13=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K14 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
aa=[0
0
0;
0
0
0;
0
0
0];
aa14=-1*aa;
ab=[0
0
0;
0
0
0;
0
0
0];
ab14=-1*ab;
Kba=t.'*KK4*t;
ba=subs(Kba,theta,-79);
ba14=-1*ba;
bb=[0
0
0;
0
0
0;
0
0
0];
bb14=-1*bb;
K14=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K15 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
aa=[0
0
0;
0
0
0;
160
0
0
0];
aa15=-1*aa;
ab=[0
0
0;
0
0
0;
0
0
0];
ab15=-1*ab;
Kba=t.'*KK4*t;
ba=subs(Kba,theta,79);
ba15=-1*ba;
bb=[0
0
0;
0
0
0;
0
0
0];
bb15=-1*bb;
K15=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K16 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
aa=[0
0
0;
0
0
0;
0
0
0];
aa16=-1*aa;
ab=[0
0
0;
0
0
0;
0
0
0];
ab16=-1*ab;
Kba=t.'*KK4*t;
ba=subs(Kba,theta,41);
ba16=-1*ba;
bb=[0
0
0;
0
0
0;
0
0
0];
bb16=-1*bb;
K16=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% K0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Kab=t.'*KK1*t;
ab=subs(Kab,theta,0);
ab0=-1*ab;
Kba=t.'*KK1*t;
ba=subs(Kba,theta,180);
ba0=-1*ba;
161
aa=aa1+aa2+aa3+aa4+aa5+aa6+aa7+aa8+aa9+aa10+aa11+aa12+aa13+aa14+aa15+aa
16+ab0+ab1+ab2+ab3+ab4+ab5+ab6+ab7+ab8+ab9+ab10+ab11+ab12+ab13+ab14+ab1
5+ab16;
bb=ba0+ba1+ba2+ba3+ba4+ba5+ba6+ba7+ba8+ba9+ba10+ba11+ba12+ba13+ba14+ba1
5+ba16+bb1+bb2+bb3+bb4+bb5+bb6+bb7+bb8+bb9+bb10+bb11+bb12+bb13+bb14+bb1
5+bb16;
K0=[aa ab;
ba bb];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% calculating K(p1,p2) (i.e. fourier transform) %%%%%
a0=1.42
%distance between two carbon atoms in angstrom
a1=3/2*a0*px+sqrt(3)/2*a0*py;
%px,py are the reciprocal axis
a2=3/2*a0*px-sqrt(3)/2*a0*py;
h=6.58479278e-16
%modified plank's constant in ev.second
kp=K0*exp(-1*(a1*0+a2*0)*i)+K1*exp(-1*(a1*-1+a2*0)*i)+K2*exp(1*(a1*0+a2*-1)*i)+K3*exp(-1*(a1*1+a2*-1)*i)+K4*exp(1*(a1*1+a2*0)*i)+K5*exp(-1*(a1*0+a2*1)*i)+K6*exp(-1*(a1*1+a2*1)*i)+K7*exp(-1*(a1*-1+a2*-1)*i)+K8*exp(-1*(a1*0+a2*2)*i)+K9*exp(-1*(a1*1+a2*-2)*i)+K10*exp(-1*(a1*-2+a2*1)*i)+K11*exp(1*(a1*-2+a2*0)*i)+K12*exp(-1*(a1*1+a2*1)*i)+K13*exp(1*(a1*0+a2*2)*i)+K14*exp(-1*(a1*-1+a2*2)*i)+K15*exp(-1*(a1*2+a2*1)*i)+K16*exp(-1*(a1*2+a2*0)*i);
%%%%%%%%%%%%%% mass matrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
avo=6.0221415e23;
%Avogadro's number
mass=-12/avo;
%mass of one carbon atom
V=3.00e10;
%Speed of light(cm/sec)
M=[mass
0
0
0
0
0;
0
mass
0
0
0
0;
0
0
mass
0
0
0;
0
0
0
mass
0
0;
0
0
0
0
mass
0;
0
0
0
0
0
mass];
% Calculating the translational vector parameters T distances in A %%%%
d=gcd(n,m);
%d= the greatest common divisor of n , m
if mod((n-m),3*d)==0
dr=3*d;
else
dr=d;
end
a=sqrt(3)*a0;
%lattice constant
l=a*sqrt(n^2+m^2+n*m);
%perimeter of the nanotube
dt=l/pi;
%diameter of the nanotube
rt=dt/2;
%radius of the nanotube
t1=(2*m+n)/dr;
%t1 for the translational vector
t2=-(m+2*n)/dr;
%t2 for the translational vector
T=sqrt(3)*l/dr;
%length of T (translational vector)
nn=2*(n^2+m^2+n*m)/dr;
%total no. of hexagons in a unit of a CNT
162
% Calculating the reciprocal lattice vectors b1,b2
b1x=2*pi/(sqrt(3)*a);
b1y=2*pi/a;
b2x=2*pi/(sqrt(3)*a);
b2y=-2*pi/a;
% generating the reciprocal lattice vectors q1,q2 for the specified NT
q1x=1/nn*(-t2*b1x+t1*b2x);
q1y=1/nn*(-t2*b1y+t1*b2y);
q2x=1/nn*(m*b1x-n*b2x);
q2y=1/nn*(m*b1y-n*b2y);
Q2=sqrt(q2x^2+q2y^2);
%Magnitude(or length) of q2 vector
% Generating the sub-bands (i.e. solving the eigen value problem %%%%%%
refine=500;
%number of points in every sub-band
counter=0;
qmin=0;
qmax=pi/T;
qgrid=(qmax-qmin)/(refine-1);
for jj=1:refine
q=qmin+qgrid*(jj-1);
wavevec(jj,1)=q;
w1(jj,1)=q;
eigv1(jj,1)=q;
w2(jj,1)=q;
eigv2(jj,1)=q;
w3(jj,1)=q;
eigv3(jj,1)=q;
w4(jj,1)=q;
eigv4(jj,1)=q;
w5(jj,1)=q;
eigv5(jj,1)=q;
w6(jj,1)=q;
eigv6(jj,1)=q;
end
for ii=0:(nn/2)
% ii is the sub-band index
for jj=1:refine
q=qmin+qgrid*(jj-1);
qx=q/Q2*q2x+ii*q1x;
qy=q/Q2*q2y+ii*q1y;
Kmmm=subs(kp,px,qx);
K=subs(Kmmm,py,qy);
[eigv,w]=eig(K,M);
% Sorting the eigenvalues and the eigenvectors
for mm=1:5
for mmm=(mm+1):6
if abs(w(mm,mm)) > abs(w(mmm,mmm))
swap=w(mm,mm);
w(mm,mm)=w(mmm,mmm);
w(mmm,mmm)=swap;
for xxx=1:6
swapv(xxx)=eigv(xxx,mm);
eigv(xxx,mm)=eigv(xxx,mmm);
eigv(xxx,mmm)=swapv(xxx);
end
end
163
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Normalizing the eigenvectors
for mm=1:6
sum=0;
for mmm=1:6
sum=sum+(abs(eigv(mmm,mm)))^2;
end
sum=sqrt(sum);
for mmm=1:6
eigv(mmm,mm)=eigv(mmm,mm)/sum;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%w1(jj,(ii+nn/2+1))=real(sqrt(w(1,1)))/(V*2*pi);
%units(/cm)
w1(jj,(ii+2))=real(sqrt(w(1,1)))*h;
%units(ev)
for mm=1:6
eigv1(jj,ii*6+mm+1)=abs(eigv(mm,1));
end
%w2(jj,(ii+nn/2+1))=real(sqrt(w(2,1)))/(V*2*pi);
%units(/cm)
w2(jj,(ii+2))=real(sqrt(w(2,2)))*h;
%units(ev)
for mm=1:6
eigv2(jj,ii*6+mm+1)=abs(eigv(mm,2));
end
%w3(jj,(ii+nn/2+1))=real(sqrt(w(3,1)))/(V*2*pi);
%units(/cm)
w3(jj,(ii+2))=real(sqrt(w(3,3)))*h;
%units(ev)
for mm=1:6
eigv3(jj,ii*6+mm+1)=abs(eigv(mm,3));
end
%w4(jj,(ii+nn/2+1))=real(sqrt(w(4,1)))/(V*2*pi);
%units(/cm)
w4(jj,(ii+2))=real(sqrt(w(4,4)))*h;
%units(ev)
for mm=1:6
eigv4(jj,ii*6+mm+1)=abs(eigv(mm,4));
end
%w5(jj,(ii+nn/2+1))=real(sqrt(w(5,1)))/(V*2*pi);
%units(/cm)
w5(jj,(ii+2))=real(sqrt(w(5,5)))*h;
%units(ev)
for mm=1:6
eigv5(jj,ii*6+mm+1)=abs(eigv(mm,5));
end
%w6(jj,(ii+nn/2+1))=real(sqrt(w(6,1)))/(V*2*pi);
%units(/cm)
w6(jj,(ii+2))=real(sqrt(w(6,6)))*h;
%units(ev)
for mm=1:6
eigv6(jj,ii*6+mm+1)=abs(eigv(mm,6));
end
counter=counter+1;
percent=counter/(refine*(nn/2+1))
end
end
164
Appendix 4 Matlab code for calculating the scattering rates for CNTs
% This program calculates the joule heating for an (n,n)CNT
% Written by Tarek Ragab on July 28th 2008
% Last edited on December 30th 2008
% Remarks:1-DOS of the electron in the final state is only considered
%
2-ALL the nn phonon subbands are devided in 2 separate loops;
%
one from 0 to nn/2, the other from -nn/2+1 to -1 as backward
%
scattering in the subband. Both of them can not be in the
%
same loop due to unkown numerical error !!
%
3- The scattering rates and the final states after scattering
%
are given for each event separetley (explained below)
%
% Units:
%
Energy: ev
%
Wavevector: Angstrom-1
%
Time: Seconds
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clc;
clear all;
% Input data
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
nn=20;
%Total number of subbands including degenerates(Have to be
%consistent to the energy and phonon dispersion relation
uploaded)
temp=[100 300 600 900 1200 1500 ]; %Temperature in Kelvin
eE=[1e-5 1e-4 1e-3 ]; %Electric field force(ev/Angstrom)
% Loading the mesh for the energy dispersion relation and the LA,LO
% phonon
% dispersion relation
% Loading the derivatives of the energy dispersion and the LA and LO
% branches
% The location of the files containing these meshes should be specified
open E:\(10,10)\LA.mat;
%Mesh for LA branches
LA=ans;
LA=LA.LA;
open E:\(10,10)\LO.mat;
%Mesh for LO branches
LO=ans;
LO=LO.LO;
open E:\(10,10)\e.mat; %Mesh for electron-Energy branches
E=ans;
E=E.e;
open E:\(10,10)\LAdedq.mat;
%Mesh for LA branches
LAdedq=ans;
LAdedq=LAdedq.LAdedq;
open E:\(10,10)\LOdedq.mat;
%Mesh for LO branches
LOdedq=ans;
LOdedq=LOdedq.LOdedq;
open E:\(10,10)\dedk.mat; %Mesh for electron-Energy branches
dedk=ans;
dedk=dedk.dedk;
165
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Constants
h=6.58479278e-16; %modified plank's constant in ev.second
boltz=8.61734315e-5; %Boltzmann constant in ev/K
DLA=14;
%deformation potential for LA phonon(ev)(Pennington-2007)
DLO=25.6;
%Deformation potential for LO (ev/Angstrom) (Park-2004)
a=sqrt(3)*1.42; %Distance between two A-atoms in adjecent cells(ang)
avo=6.0221415e23; %Avogadro's number
me=0.510999e6/(3e18)^2;
%Electron mass in ev.sec2/Angstrom2
mass=12/avo/1000; %mass of one carbon atom in Kilograms
ro=mass*2*nn/a*6.24150965e-2;
%ONLY VALID FOR ARMCHAIR NANOTUBES
%Linear mass density of the nanotube in ev.sec2/angstrom3
d=sqrt(3)*nn*a/2/pi; %Diameter of the armchair CNT
hDLAro=h*DLA^2/2/ro;
hDLOro=h*DLO^2/2/ro;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Changing the 1st value of the energy at q=0 for sub-band 0 in the LA
% branch to eliminate the singularity at this point due to self
% scattering
LA(1,2)=LA(2,2);
%choose any value you like :)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
W=zeros(length(E),nn/2+2,8*nn,length(temp));
%The probablilty of scattering for each of
%the 8*nn scattering events given below
% for i=1:8*nn
%
for j=1:length(temp)
%
W(:,1,i,j)=E(:,1,j);
%
end
% end
kfinal=zeros(length(E),nn/2+2,8*nn);
% The final state after scattering for each of the scattering events
% projected to the equivalent position in the postive part of the first
% Brillouin zone
for i=1:8*nn
kfinal(:,1,i)=E(:,1);
end
mfinal=zeros(length(E),nn/2+2,8*nn);
%The final subband after scattering for
%each of the scattering events(0 to nn/2
for i=1:8*nn
mfinal(:,1,i)=E(:,1);
end
mom=zeros(length(E),nn/2+2,8*nn);
%The momentum lost or gained for each
%of the scattering events devided by h
for i=1:8*nn
166
mom(:,1,i)=E(:,1);
end
energy=zeros(length(E),nn/2+2,8*nn);
%The energy lost or gained for each of the scattering events
for i=1:8*nn
energy(:,1,i)=E(:,1);
end
WtotLA=zeros(length(E),nn/2+2,length(temp));
%The total scattering probapbility with LA
for i=1:length(temp)
WtotLA(:,1,i)=E(:,1);
end
WtotLO=zeros(length(E),nn/2+2,length(temp));
%The total scattering probapbility with LO
for i=1:length(temp)
WtotLO(:,1,i)=E(:,1);
end
Wtot=zeros(length(E),nn/2+2,length(temp));
%The total scattering probapbility with LO&LA
for i=1:length(temp)
Wtot(:,1,i)=E(:,1);
end
j1=zeros(length(temp),length(eE));
%Current density for different temp, electric field
j2=zeros(length(temp),length(eE));
%Current density for different temp, electric field
j3=zeros(length(temp),length(eE));
%Current density for different temp, electric field
j4=zeros(length(temp),length(eE));
%Current density for different temp, electric field
mLA=zeros(4*nn,1);
%The indices m associated with LA phonon scattering
mLO=zeros(4*nn,1);
%The indices m associated with LA phonon scattering
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%
kmax=E(length(E),1);
%Limit of the wavevector in the 1st BZ
kgrid=kmax/(length(E)-1); %kgrid value for electrons energy dispersion
% i=11;
for i=1:(nn/2+1)
%Loop the energy dispersion subabands
i
%
ii=1;
for ii=1:length(E)
%number of points in the energy subband
%
ii
%
j=11;
m=0;
%An index given for the scattering mechanism
mLAindex=0;
mLOindex=0;
for j=1:(nn/2+1)
%Loop the phonon dispersion subbands
%1-Absorbtion of LA phonon &Forward scattering&subband forward
167
m=m+1;
mLAindex=mLAindex+1;
mLA(mLAindex,1)=m;
iiindex=0;
for jj=1:length(LA) %Number of points in the phonon subband
kf=E(ii,1)+LA(jj,1); %final longitudinal wavevector
mf=(i-1)+(j-1);
%final circumferncial wavevector Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)+LA(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);%Locatn of kf at this stp
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));%sign of the error at this q
signj=sign(error(jj-1));
%sign of the error at previousq
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj,j+1)/(boltz*temp(mm)))1)+0)/LA(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LA(jj,1);
energy(ii,i+1,m)=LA(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj1,1)^2+(2*(j-1)/d)^2)*(1/(exp(LA(jj-1,j+1)/(boltz*temp(mm)))1)+0)/LA(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LA(jj-1,1);
energy(ii,i+1,m)=LA(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
168
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%2-Absorbtion of LA phonon &forward scattering&subband backward
if j~=(nn/2+1) & j~=1
m=m+1;
mLAindex=mLAindex+1;
mLA(mLAindex,1)=m;
iiindex=0;
for jj=1:length(LA) %# of points in the phonon subband
kf=E(ii,1)+LA(jj,1); %final longitudinal wavevector
mf=(i-1)-(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)+LA(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);%kf Location at this
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));
%sign of the error at this q
signj=sign(error(jj-1)); %sign at previousq
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj,j+1)/(boltz*temp(mm)))1)+0)/LA(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LA(jj,1);
energy(ii,i+1,m)=LA(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj1,1)^2+(2*(j-1)/d)^2)*(1/(exp(LA(jj-1,j+1)/(boltz*temp(mm)))1)+0)/LA(jj-1,j+1)/abs(dedk(iipre,mf+2));
169
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LA(jj-1,1);
energy(ii,i+1,m)=LA(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%3-Absorbtion of LA phonon&Backward scattering& subband forward
m=m+1;
mLAindex=mLAindex+1;
mLA(mLAindex,1)=m;
iiindex=0;
for jj=1:length(LA) %Number of points in the phonon subband
kf=E(ii,1)-LA(jj,1); %final longitudinal wavevector
mf=(i-1)+(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)+LA(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2); %Location of kf at this
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));%sign of the error at this q
signj=sign(error(jj-1));
%sign of the error at previousq
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj,j+1)/(boltz*temp(mm)))1)+0)/LA(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LA(jj,1);
energy(ii,i+1,m)=LA(jj,j+1);
170
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj-1,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj-1,j+1)/(boltz*temp(mm)))-1)+0)/LA(jj1,j+1)/abs(dedk(iipre,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LA(jj-1,1);
energy(ii,i+1,m)=LA(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%4-Absorbtion of LA phonon&Backward scattering&subband backward
if j~=(nn/2+1) & j~=1
m=m+1;
mLAindex=mLAindex+1;
mLA(mLAindex,1)=m;
iiindex=0;
for jj=1:length(LA) %# of points in the phonon subband
kf=E(ii,1)-LA(jj,1); %final longitudinal wavevector
mf=(i-1)-(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)+LA(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%kf Location at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));
%sign of the error at this q
signj=sign(error(jj-1)); %sign at previousq
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
171
W(ii,i+1,m,mm)=hDLAro*(LA(jj,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj,j+1)/(boltz*temp(mm)))1)+0)/LA(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LA(jj,1);
energy(ii,i+1,m)=LA(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj1,1)^2+(2*(j-1)/d)^2)*(1/(exp(LA(jj-1,j+1)/(boltz*temp(mm)))1)+0)/LA(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LA(jj-1,1);
energy(ii,i+1,m)=LA(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%5-Emission of LA phonon &Forward scattering & subband forward
m=m+1;
mLAindex=mLAindex+1;
mLA(mLAindex,1)=m;
iiindex=0;
for jj=1:length(LA) %Number of points in the phonon subband
kf=E(ii,1)+LA(jj,1); %final longitudinal wavevector
mf=(i-1)+(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)-LA(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%Location of kf at this step
172
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));%sign of the error at this q
signj=sign(error(jj-1));
%sign of the error at previousq
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj,j+1)/(boltz*temp(mm)))1)+1)/LA(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LA(jj,1);
energy(ii,i+1,m)=-LA(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj-1,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj-1,j+1)/(boltz*temp(mm)))-1)+1)/LA(jj1,j+1)/abs(dedk(iipre,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LA(jj-1,1);
energy(ii,i+1,m)=-LA(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%6-Emission of LA phonon &forward scattering&subband backward
if j~=(nn/2+1) & j~=1
m=m+1;
mLAindex=mLAindex+1;
mLA(mLAindex,1)=m;
iiindex=0;
for jj=1:length(LA) %# of points in the phonon subband
kf=E(ii,1)+LA(jj,1); %final longitudinal wavevector
mf=(i-1)-(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
173
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)-LA(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%kf Location at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));
%sign of the error at this q
signj=sign(error(jj-1)); %sign at previousq
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj,j+1)/(boltz*temp(mm)))1)+1)/LA(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LA(jj,1);
energy(ii,i+1,m)=-LA(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj1,1)^2+(2*(j-1)/d)^2)*(1/(exp(LA(jj-1,j+1)/(boltz*temp(mm)))1)+1)/LA(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LA(jj-1,1);
energy(ii,i+1,m)=-LA(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%7-Emission of LA phonon & Backward scattering& subband forward
m=m+1;
mLAindex=mLAindex+1;
mLA(mLAindex,1)=m;
iiindex=0;
for jj=1:length(LA) %Number of points in the phonon subband
kf=E(ii,1)-LA(jj,1); %final longitudinal wavevector
mf=(i-1)+(j-1);
%final circumferncial wavevector
174
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)-LA(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%Location of kf at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));%sign of the error at this q
signj=sign(error(jj-1));
%sign of the error at previousq
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj,j+1)/(boltz*temp(mm)))1)+1)/LA(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LA(jj,1);
energy(ii,i+1,m)=-LA(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj-1,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj-1,j+1)/(boltz*temp(mm)))-1)+1)/LA(jj1,j+1)/abs(dedk(iipre,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LA(jj-1,1);
energy(ii,i+1,m)=-LA(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%8-Emission of LA phonon & Backward scattering&subband backward
175
if j~=(nn/2+1) & j~=1
m=m+1;
mLAindex=mLAindex+1;
mLA(mLAindex,1)=m;
iiindex=0;
for jj=1:length(LA) %# of points in the phonon subband
kf=E(ii,1)-LA(jj,1); %final longitudinal wavevector
mf=(i-1)-(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)-LA(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%kf Location at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));
%sign of the error at this q
signj=sign(error(jj-1)); %sign at previous q
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj,1)^2+(2*(j1)/d)^2)*(1/(exp(LA(jj,j+1)/(boltz*temp(mm)))1)+1)/LA(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LA(jj,1);
energy(ii,i+1,m)=-LA(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLAro*(LA(jj1,1)^2+(2*(j-1)/d)^2)*(1/(exp(LA(jj-1,j+1)/(boltz*temp(mm)))1)+1)/LA(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLA(ii,i+1,mm)=WtotLA(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LA(jj-1,1);
energy(ii,i+1,m)=-LA(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
176
end
break
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%9-Absorbtion of LO phonon &Forward scattering&subband forward
m=m+1;
mLOindex=mLOindex+1;
mLO(mLOindex,1)=m;
iiindex=0;
for jj=1:length(LO) %Number of points in the phonon subband
kf=E(ii,1)+LO(jj,1); %final longitudinal wavevector
mf=(i-1)+(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)+LO(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%Location of kf at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));%sign of the error at this q
signj=sign(error(jj-1));
%sign of the error at previous q
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj,j+1)/(boltz*temp(mm)))1)+0)/LO(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LO(jj,1);
energy(ii,i+1,m)=LO(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj1,j+1)/(boltz*temp(mm)))-1)+0)/LO(jj-1,j+1)/abs(dedk(iipre,mf+2));
177
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LO(jj-1,1);
energy(ii,i+1,m)=LO(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%10-Absorbtion of LO phonon &forward scattering&subband backward
if j~=(nn/2+1) & j~=1
m=m+1;
mLOindex=mLOindex+1;
mLO(mLOindex,1)=m;
iiindex=0;
for jj=1:length(LO) %# of points in the phonon subband
kf=E(ii,1)+LO(jj,1); %final longitudinal wavevector
mf=(i-1)-(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)+LO(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%kf Location at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));
%sign of the error at this q
signj=sign(error(jj-1)); %sign at previous q
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj,j+1)/(boltz*temp(mm)))1)+0)/LO(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LO(jj,1);
178
energy(ii,i+1,m)=LO(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj1,j+1)/(boltz*temp(mm)))-1)+0)/LO(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LO(jj-1,1);
energy(ii,i+1,m)=LO(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%11-Absorbtion of LO phonon&Backward scattering& subband forward
m=m+1;
mLOindex=mLOindex+1;
mLO(mLOindex,1)=m;
iiindex=0;
for jj=1:length(LO) %Number of points in the phonon subband
kf=E(ii,1)-LO(jj,1); %final longitudinal wavevector
mf=(i-1)+(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)+LO(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%Location of kf at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));%sign of the error at this q
signj=sign(error(jj-1));
%sign of the error at previous q
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
179
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj,j+1)/(boltz*temp(mm)))1)+0)/LO(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LO(jj,1);
energy(ii,i+1,m)=LO(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj1,j+1)/(boltz*temp(mm)))-1)+0)/LO(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LO(jj-1,1);
energy(ii,i+1,m)=LO(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%12-Absorbtion of LO phonon&Backward scattering&subband backward
if j~=(nn/2+1) & j~=1
m=m+1;
mLOindex=mLOindex+1;
mLO(mLOindex,1)=m;
iiindex=0;
for jj=1:length(LA) %# of points in the phonon subband
kf=E(ii,1)-LO(jj,1); %final longitudinal wavevector
mf=(i-1)-(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)+LO(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%kf Location at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
180
if jj > 1
signjj=sign(error(jj));
%sign of the error at this q
signj=sign(error(jj-1)); %sign at previousq
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj,j+1)/(boltz*temp(mm)))1)+0)/LO(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LO(jj,1);
energy(ii,i+1,m)=LO(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj1,j+1)/(boltz*temp(mm)))-1)+0)/LO(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LO(jj-1,1);
energy(ii,i+1,m)=LO(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%13-Emission of LO phonon &Forward scattering & subband forward
m=m+1;
mLOindex=mLOindex+1;
mLO(mLOindex,1)=m;
iiindex=0;
for jj=1:length(LO) %Number of points in the phonon subband
kf=E(ii,1)+LO(jj,1); %final longitudinal wavevector
mf=(i-1)+(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
181
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)-LO(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%Location of kf at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));%sign of the error at this q
signj=sign(error(jj-1));
%sign of the error at previous q
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj,j+1)/(boltz*temp(mm)))1)+1)/LO(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LO(jj,1);
energy(ii,i+1,m)=-LO(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj1,j+1)/(boltz*temp(mm)))-1)+1)/LO(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LO(jj-1,1);
energy(ii,i+1,m)=-LO(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%14-Emission of LO phonon &forward scattering&subband backward
if j~=(nn/2+1) & j~=1
m=m+1;
mLOindex=mLOindex+1;
mLO(mLOindex,1)=m;
iiindex=0;
for jj=1:length(LO) %# of points in the phonon subband
kf=E(ii,1)+LO(jj,1); %final longitudinal wavevector
mf=(i-1)-(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
182
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)-LO(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%kf Location at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));
%sign of the error at this q
signj=sign(error(jj-1)); %sign at previousq
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj,j+1)/(boltz*temp(mm)))1)+1)/LO(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LO(jj,1);
energy(ii,i+1,m)=-LO(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj1,j+1)/(boltz*temp(mm)))-1)+1)/LO(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=LO(jj-1,1);
energy(ii,i+1,m)=-LO(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%15-Emission of LO phonon & Backward scattering& subband forward
m=m+1;
mLOindex=mLOindex+1;
mLO(mLOindex,1)=m;
iiindex=0;
for jj=1:length(LO) %Number of points in the phonon subband
183
kf=E(ii,1)-LO(jj,1); %final longitudinal wavevector
mf=(i-1)+(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)-LO(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%Location of kf at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));%sign of the error at this q
signj=sign(error(jj-1));
%sign of the error at previous q
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj,j+1)/(boltz*temp(mm)))1)+1)/LO(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LO(jj,1);
energy(ii,i+1,m)=-LO(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj1,j+1)/(boltz*temp(mm)))-1)+1)/LO(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LO(jj-1,1);
energy(ii,i+1,m)=-LO(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
break
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
184
%16-Emission of LO phonon & Backward scattering&subband backward
if j~=(nn/2+1) & j~=1
m=m+1;
mLOindex=mLOindex+1;
mLO(mLOindex,1)=m;
iiindex=0;
for jj=1:length(LO) %# of points in the phonon subband
kf=E(ii,1)-LO(jj,1); %final longitudinal wavevector
mf=(i-1)-(j-1);
%final circumferncial wavevector
%Adjusting for Umklapp process
if abs(mf) > nn/2
mf=nn-abs(mf);
elseif mf < 0
mf=abs(mf);
end
if abs(kf) > kmax
kf=2*kmax-abs(kf);
mf=nn/2-mf;
elseif kf < 0
kf=abs(kf);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ef1=E(ii,i+1)-LO(jj,j+1);
iipre=iiindex;
%Location of kf at previous step
iiindex=floor(kf/kgrid+1+0.2);
%kf Location at this step
Ef=E(iiindex,mf+2);
error(jj)=Ef1-Ef;
if jj > 1
signjj=sign(error(jj));
%sign of the error at this q
signj=sign(error(jj-1)); %sign at previous q
if signjj~=signj
if abs(error(jj)) < abs(error(jj-1))
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj,j+1)/(boltz*temp(mm)))1)+1)/LO(jj,j+1)/abs(dedk(iiindex,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LO(jj,1);
energy(ii,i+1,m)=-LO(jj,j+1);
kfinal(ii,i+1,m)=iiindex;
mfinal(ii,i+1,m)=mf+2;
else
for mm=1:length(temp)
W(ii,i+1,m,mm)=hDLOro*(1/(exp(LO(jj1,j+1)/(boltz*temp(mm)))-1)+1)/LO(jj-1,j+1)/abs(dedk(iipre,mf+2));
WtotLO(ii,i+1,mm)=WtotLO(ii,i+1,mm)+W(ii,i+1,m,mm);
end
mom(ii,i+1,m)=-LO(jj-1,1);
energy(ii,i+1,m)=-LO(jj-1,j+1);
kfinal(ii,i+1,m)=iipre;
mfinal(ii,i+1,m)=mf+2;
end
185
break
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
end
end
end
m
% Calculating the total scattering rate for each k-state
for i=1:(nn/2+1)
for ii=1:length(E)
for mm=1:length(temp)
Wtot(ii,i+1,mm)=WtotLA(ii,i+1,mm)+WtotLO(ii,i+1,mm);
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Calculating the rate of change of energy due to LA&LO phononscattering
% for different temperatures and different electric fields.
% In the following 1 stands for group velocity=h*K/me
%
2 stands for group velocity=1/h*dedk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dedt1=zeros(length(temp),length(eE));
dedt2=zeros(length(temp),length(eE));
for i=1:(nn/2+1)
%Loop the energy subabands
for ii=1:length(E)
%points in the energy subband
for mm=1:length(temp)
for mmm=1:length(eE)
%Fermi occupation probability for electron when this state is +ve
fkp1=1/(exp((E(ii,i+1)eE(mmm)*(h*E(ii,1)/me)/Wtot(ii,i+1,mm))/(boltz*temp(mm)))+1);
fermip1(ii,i)=fkp1;
fkp2=1/(exp((E(ii,i+1)eE(mmm)*(abs(dedk(ii,i+1))/h)/Wtot(ii,i+1,mm))/(boltz*temp(mm)))+1);
fermip2(ii,i)=fkp2;
%Fermi occupation probability for electron when this state is -ve
fkn1=1/(exp((E(ii,i+1)+eE(mmm)*(h*E(ii,1)/me)/Wtot(ii,i+1,mm))/(boltz*t
emp(mm)))+1);
fermin1(ii,i)=fkn1;
fkn2=1/(exp((E(ii,i+1)+eE(mmm)*(abs(dedk(ii,i+1))/h)/Wtot(ii,i+1,mm))/(
boltz*temp(mm)))+1);
fermin2(ii,i)=fkn2;
% Calculating the current density using 4 different
% approaches. This have to be multiplied by e/pi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
186
if i==1 | i==nn/2+1
j1(mm,mmm)=j1(mm,mmm)+fkp1*(h*E(ii,1)/me)fkn1*(h*E(ii,1)/me);
j2(mm,mmm)=j2(mm,mmm)+fkp2*abs(dedk(ii,i+1))/hfkn2*abs(dedk(ii,i+1))/h;
j3(mm,mmm)=j3(mm,mmm)+fkp1*abs(dedk(ii,i+1))/hfkn1*abs(dedk(ii,i+1))/h;
j4(mm,mmm)=j4(mm,mmm)+fkp2*(h*E(ii,1)/me)fkn2*(h*E(ii,1)/me);
else
j1(mm,mmm)=j1(mm,mmm)+2*fkp1*(h*E(ii,1)/me)fkn1*(h*E(ii,1)/me);
j2(mm,mmm)=j2(mm,mmm)+2*fkp2*abs(dedk(ii,i+1))/hfkn2*abs(dedk(ii,i+1))/h;
j3(mm,mmm)=j3(mm,mmm)+2*fkp1*abs(dedk(ii,i+1))/hfkn1*abs(dedk(ii,i+1))/h;
j4(mm,mmm)=j4(mm,mmm)+2*fkp2*(h*E(ii,1)/me)fkn2*(h*E(ii,1)/me);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for m=1:8*nn
if kfinal(ii,i+1,m)~=0 %i.e. there is a final state
%Probability of finding the electron in this final
% state coming from the +ve state
fkfinp1=1/(exp((E(kfinal(ii,i+1,m),mfinal(ii,i+1,m))eE(mmm)*(h*(E(ii,1)+mom(ii,i+1,m))/me)/Wtot(kfinal(ii,i+1,m),mfinal(ii,
i+1,m),mm))/(boltz*temp(mm)))+1);
fkfinp2=1/(exp((E(kfinal(ii,i+1,m),mfinal(ii,i+1,m))eE(mmm)*(abs(dedk(kfinal(ii,i+1,m),mfinal(ii,i+1,m)))/h)/Wtot(kfinal(ii
,i+1,m),mfinal(ii,i+1,m),mm))/(boltz*temp(mm)))+1);
if i==1 | i==nn/2+1
dedt1(mm,mmm)=dedt1(mm,mmm)+energy(ii,i+1,m)*fkp1*(1fkfinp1)*W(ii,i+1,m,mm);
dedt2(mm,mmm)=dedt2(mm,mmm)+energy(ii,i+1,m)*fkp2*(1fkfinp2)*W(ii,i+1,m,mm);
else
dedt1(mm,mmm)=dedt1(mm,mmm)+2*energy(ii,i+1,m)*fkp1*(1fkfinp1)*W(ii,i+1,m,mm);
dedt2(mm,mmm)=dedt2(mm,mmm)+2*energy(ii,i+1,m)*fkp2*(1fkfinp2)*W(ii,i+1,m,mm);
end
%Probability of finding the electron in this final state coming
%from the -ve state
fkfinn1=1/(exp((E(kfinal(ii,i+1,m),mfinal(ii,i+1,m))+eE(mmm)*(h*(E(ii,1
)+mom(ii,i+1,m))/me)/Wtot(kfinal(ii,i+1,m),mfinal(ii,i+1,m),mm))/(boltz
*temp(mm)))+1);
fkfinn2=1/(exp((E(kfinal(ii,i+1,m),mfinal(ii,i+1,m))+eE(mmm)*(abs(dedk(
187
kfinal(ii,i+1,m),mfinal(ii,i+1,m)))/h)/Wtot(kfinal(ii,i+1,m),mfinal(ii,
i+1,m),mm))/(boltz*temp(mm)))+1);
if i==1 | i==nn/2+1
dedt1(mm,mmm)=dedt1(mm,mmm)+energy(ii,i+1,m)*fkn1*(1fkfinn1)*W(ii,i+1,m,mm);
dedt2(mm,mmm)=dedt2(mm,mmm)+energy(ii,i+1,m)*fkn2*(1fkfinn2)*W(ii,i+1,m,mm);
else
dedt1(mm,mmm)=dedt1(mm,mmm)+2*energy(ii,i+1,m)*fkn1*(1fkfinn1)*W(ii,i+1,m,mm);
dedt2(mm,mmm)=dedt2(mm,mmm)+2*energy(ii,i+1,m)*fkn2*(1fkfinn2)*W(ii,i+1,m,mm);
end
end
end
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
188
Appendix 5 Matlab code for Ensemble Monte Carlo Simulations
% This program runs a Monte Carlo device simulation for an (n,n)CNT
% Written by Tarek Ragab on February 26th 2009
% Last edited on August 11th 2009
%
% Units:
%
Energy: ev
%
Wavevector: Angstrom-1
%
Time: Seconds
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tic
clc;
clear all;
% loading Input data
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The workspace resulting from running the file scattering8.m should be
% saved in the file input.mat to be loaded here
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
display('reading input data')
load ('input.mat')
% Input data
%%%%%%%%%%%%%
n=100;
%number of electrons to be simulated
L=1;
%simulated length of the nanotube in angstrom
eE=0.25e-5;
%Electric field force in ev/Angstrom
dt=0.1e-15; %timestep in sec (has to be less than the scattering time)
t=10e-12;
%total simulation time
nstep=t/dt; %total number of simulation steps
sample=1000; %time used to average the occupation probability
temperature=300; %choose only a value given in the array temp(in kel)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Constants
h=6.58479278e-16; %modified plank's constant in ev.second
boltz=8.61734315e-5; %Boltzmann constant in ev/K
T=2.4595;
%Length of translational vector of (n,n)CNT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculating the location of the temperature required in the temp
array
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tsize=1:length(temp);
tindex = interp1(temp,tsize,temperature);
% changing the structure of W to be compatible with the MC simulation
% Gives the commulative rate Instead of the individual scattering rate
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
display('processing input data for simulation')
sizeW=size(W);
189
% W=zeros(sizeW(1),sizeW(2),sizeW(3),sizeW(4));
for i=1:sizeW(1)
for j=2:sizeW(2)
for ii=1:sizeW(4)
sum=0;
for jj=1:sizeW(3)
W(i,j,jj,ii)=W(i,j,jj,ii)+sum;
sum=W(i,j,jj,ii);
end
end
end
end
% Initializing variables
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dti=zeros(n,1);
%the FREE drift time for each individual electron
flag=0;
check2=0; %total no. of electrons in +ve & -ve energy subbands per A
ntot=0;
%total actual no. of electron in the +ve energy subband
nactp=zeros(nn/2+1,1); %actual no. of electrons in the +ve energy
subbands
nactn=zeros(nn/2+1,1); %actual no. of electrons in the -ve energy
subbands
counter=1; %counter for the electrons distributed in the initial
states
error=0;
%comenergy=zeros(nstep,n); %energy lost at different times
%commom=zeros(nstep,n); %momentum lost at different times
totenergy=zeros(nstep,1); %Commulative energy lost
totmom=zeros(nstep,1);
%Commulative momentum lost
fp=zeros(78,nstep/sample+2); %Occupation probability for +ve states
fn=zeros(78,nstep/sample+2); %Occupation probability for -ve states
for i=1:78
fp(i,1)=E(626+i,1);
fn(i,1)=-1*E(626+i,1);
fp(i,2)=12/(exp(E(626+i,12)/boltz/temperature)+1);
% fermi-dirac distribution is multiplied by a factor of 12 for
% normalization with the rest of the results
end
k=zeros(n,1);
m=zeros(n,1);
BZindex=zeros(n,1);
momentum=zeros(nstep,n); %the wavevector history for all electrons
BZ=zeros(nstep,n);
% the Brillioun zone history for all electrons
meo=zeros(nstep,n); % the subband index history for all electrons
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% initializing k,m (meo) for every simulated electron based on fermi% dirac distribution
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
display('Initializing the states of the electrons in the wavevector
space')
% Calculating the actual number of electrons in the simulated length
check1=2*nn/T %total no. of electrons in +ve & -ve energy subbands/A
for i=1:(nn/2+1)
%Loop the energy subabands
for ii=1:length(E)
%points in the energy subband
190
if i==1||i==nn/2+1
nactp(i)=nactp(i)+(1/pi*kgrid)*2/(exp(E(ii,i+1)/(boltz*temperature))+1)
;
nactn(i)=nactn(i)+(1/pi*kgrid)*2/(exp(1*E(ii,i+1)/(boltz*temperature))+1);
else
nactp(i)=nactp(i)+(1/pi*kgrid)*4/(exp(E(ii,i+1)/(boltz*temperature))+1)
;
nactn(i)=nactn(i)+(1/pi*kgrid)*4/(exp(1*E(ii,i+1)/(boltz*temperature))+1);
end
end
end
for i=1:(nn/2+1)
check2=check2+nactp(i)+nactn(i);
ntot=ntot+nactp(i);
end
check2
ntot=ntot*L;
%Distributing the electrons simulated over the lowest subband only
effch=ntot/n; %The effective weight(charge) of one simulated electron
for ii=1:length(E)-1
nelec=(L/effch)*(1/pi*kgrid)*((1/(exp(E(ii,nn/2+2)/(boltz*temperature))
+1))+(1/(exp(E(ii,nn/2+2)/(boltz*temperature))+1)))/2;
nelec=nelec+error; %Adding the residual from last increment
nele=round(nelec);
error=nelec-nele;
if nele ~=0
for j=1:nele
k(counter)=E(ii,1)+rand*kgrid;
m(counter)=nn/2;
BZindex(counter)=1; % 1..electron in 1st BZ, 2.. in 2nd BZ
counter=counter+1;
k(counter)=-1*(E(ii,1)+rand*kgrid);
m(counter)=nn/2;
BZindex(counter)=1;
counter=counter+1;
end
end
end
% k(1)=0.81;
% m(1)=10;
% BZindex(1)=1;
% %k(2)=-0.81;
% %m(2)=10;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Run MC Simulation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
display('Start MC Simulation')
for i=1:nstep
display(i)
191
if i~=1
totenergy(i)=totenergy(i-1);
totmom(i)=totmom(i-1);
end
for j=1:n
momentum(i,j)=k(j);
BZ(i,j)=BZindex(j);
meo(i,j)=m(j);
%drift the electron in the electric field eE
if BZindex(j)==1 %The electron is in the first BZ
k(j)=k(j)+dt*eE/h;
dti(j)=dti(j)+dt;
%Adjusting for k in the second Brillouin zone
% i.e. it is moved to the complementary k in the 1st BZ
if k(j) > kmax
k(j)=2*kmax-k(j);
m(j)=nn/2-m(j);
BZindex(j)=2;
elseif k(j) < -1*kmax
k(j)=-2*kmax-k(j);
m(j)=nn/2-m(j);
BZindex(j)=2;
end
else
%The Electron is in the second Brillioun zone
k(j)=k(j)-dt*eE/h;
dti(j)=dti(j)+dt;
%Adjusting for k in the second Brillouin zone
% i.e. it is moved to the complementary k in the 1st BZ
if k(j) > kmax
k(j)=2*kmax-k(j);
m(j)=nn/2-m(j);
BZindex(j)=1;
elseif k(j) < -1*kmax
k(j)=-2*kmax-k(j);
m(j)=nn/2-m(j);
BZindex(j)=1;
end
end
%interpolating the scattering rate for the k state after draft
scatt = interp1(Wtot(:,1,1),Wtot(:,m(j)+2,tindex),abs(k(j)));
%checking if the electron will scatter or not
if dti(j)*scatt > rand
%the electron will scatter
%Select the scattering mechanism involved
scattprob=scatt*rand;
for ii=1:sizeW(3) %Loop all the scattering mechanisms
if
(interp1(Wtot(:,1,1),W(:,m(j)+2,ii,tindex),abs(k(j))) >= scattprob)
%if (interp1(Wtot(:,1,1),W(:,m(j)+2,ii,tindex),abs(k(j))) <= scattprob)
& (interp1(Wtot(:,1,1),W(:,m(j)+2,ii,tindex),abs(k(j))) ~= 0)
%Choose this event................Update the wavevector
if
mfinal(floor(abs(k(j))/kgrid+1),m(j)+2,ii)~=mfinal(ceil(abs(k(j))/kgrid
+1),m(j)+2,ii)
if
(mfinal(floor(abs(k(j))/kgrid+1),m(j)+2,ii)==0)
m(j)=mfinal(ceil(abs(k(j))/kgrid+1),m(j)+2,ii)-2;
192
if sign(k(j))~=0
%commom(i,j)=commom(i,j)+(2*BZindex(j)+3)*sign(k(j))*mom(ceil(abs(k(j))/kgrid+1),m(j)+2,ii);
%comenergy(i,j)=comenergy(i,j)+energy(ceil(abs(k(j))/kgrid+1),m(j)+2,ii
);
totenergy(i)=totenergy(i)+effch*energy(ceil(abs(k(j))/kgrid+1),m(j)+2,i
i);
totmom(i)=totmom(i)+effch*h*(2*BZindex(j)+3)*sign(k(j))*mom(ceil(abs(k(j))/kgrid+1),m(j)+2,ii);
k(j)=sign(k(j))*(abs(k(j))+mom(ceil(abs(k(j))/kgrid+1),m(j)+2,ii));
else
%commom(i,j)=commom(i,j)+(2*BZindex(j)+3)*mom(ceil(abs(k(j))/kgrid+1),m(j)+2,ii);
%comenergy(i,j)=comenergy(i,j)+energy(ceil(abs(k(j))/kgrid+1),m(j)+2,ii
);
totenergy(i)=totenergy(i)+effch*energy(ceil(abs(k(j))/kgrid+1),m(j)+2,i
i);
totmom(i)=totmom(i)+effch*h*(2*BZindex(j)+3)*mom(ceil(abs(k(j))/kgrid+1),m(j)+2,ii);
k(j)=k(j)+mom(ceil(abs(k(j))/kgrid+1),m(j)+2,ii);
end
elseif
(mfinal(ceil(abs(k(j))/kgrid+1),m(j)+2,ii)==0)
m(j)=mfinal(floor(abs(k(j))/kgrid+1),m(j)+2,ii)-2;
if sign(k(j))~=0
%commom(i,j)=commom(i,j)+(2*BZindex(j)+3)*sign(k(j))*mom(floor(abs(k(j))/kgrid+1),m(j)+2,ii);
%comenergy(i,j)=comenergy(i,j)+energy(floor(abs(k(j))/kgrid+1),m(j)+2,i
i);
totenergy(i)=totenergy(i)+effch*energy(floor(abs(k(j))/kgrid+1),m(j)+2,
ii);
totmom(i)=totmom(i)+effch*h*(2*BZindex(j)+3)*sign(k(j))*mom(floor(abs(k(j))/kgrid+1),m(j)+2,ii);
k(j)=sign(k(j))*(abs(k(j))+mom(floor(abs(k(j))/kgrid+1),m(j)+2,ii));
else
%commom(i,j)=commom(i,j)+(2*BZindex(j)+3)*mom(floor(abs(k(j))/kgrid+1),m(j)+2,ii);
%comenergy(i,j)=comenergy(i,j)+energy(floor(abs(k(j))/kgrid+1),m(j)+2,i
i);
totenergy(i)=totenergy(i)+effch*energy(floor(abs(k(j))/kgrid+1),m(j)+2,
ii);
totmom(i)=totmom(i)+effch*h*(2*BZindex(j)+3)*mom(floor(abs(k(j))/kgrid+1),m(j)+2,ii);
k(j)=k(j)+mom(floor(abs(k(j))/kgrid+1),m(j)+2,ii);
193
end
else
m(j)=mfinal(round(abs(k(j))/kgrid+1),m(j)+2,ii)-2;
if sign(k(j))~=0
%commom(i,j)=commom(i,j)+(2*BZindex(j)+3)*sign(k(j))*mom(round(abs(k(j))/kgrid+1),m(j)+2,ii);
%comenergy(i,j)=comenergy(i,j)+energy(round(abs(k(j))/kgrid+1),m(j)+2,i
i);
totenergy(i)=totenergy(i)+effch*energy(round(abs(k(j))/kgrid+1),m(j)+2,
ii);
totmom(i)=totmom(i)+effch*h*(2*BZindex(j)+3)*sign(k(j))*mom(round(abs(k(j))/kgrid+1),m(j)+2,ii);
k(j)=sign(k(j))*(abs(k(j))+mom(round(abs(k(j))/kgrid+1),m(j)+2,ii));
else
%commom(i,j)=commom(i,j)+(2*BZindex(j)+3)*mom(round(abs(k(j))/kgrid+1),m(j)+2,ii);
%comenergy(i,j)=comenergy(i,j)+energy(round(abs(k(j))/kgrid+1),m(j)+2,i
i);
totenergy(i)=totenergy(i)+effch*energy(round(abs(k(j))/kgrid+1),m(j)+2,
ii);
totmom(i)=totmom(i)+effch*h*(2*BZindex(j)+3)*mom(round(abs(k(j))/kgrid+1),m(j)+2,ii);
k(j)=k(j)+mom(round(abs(k(j))/kgrid+1),m(j)+2,ii);
end
end
else
m(j)=interp1(mfinal(:,1,1),mfinal(:,m(j)+2,ii),abs(k(j)))-2;
if sign(k(j))~=0
%commom(i,j)=commom(i,j)+(2*BZindex(j)+3)*sign(k(j))*interp1(mom(:,1,1),mom(:,m(j)+2,ii),abs(k(j)
));
%comenergy(i,j)=comenergy(i,j)+interp1(energy(:,1,1),energy(:,m(j)+2,ii
),abs(k(j)));
totenergy(i)=totenergy(i)+effch*interp1(energy(:,1,1),energy(:,m(j)+2,i
i),abs(k(j)));
totmom(i)=totmom(i)+effch*h*(2*BZindex(j)+3)*sign(k(j))*interp1(mom(:,1,1),mom(:,m(j)+2,ii),abs(k(j)
));
k(j)=sign(k(j))*(abs(k(j))+interp1(mom(:,1,1),mom(:,m(j)+2,ii),abs(k(j)
)));
else
%commom(i,j)=commom(i,j)+(2*BZindex(j)+3)*interp1(mom(:,1,1),mom(:,m(j)+2,ii),abs(k(j)));
%comenergy(i,j)=comenergy(i,j)+interp1(energy(:,1,1),energy(:,m(j)+2,ii
),abs(k(j)));
194
totenergy(i)=totenergy(i)+effch*interp1(energy(:,1,1),energy(:,m(j)+2,i
i),abs(k(j)));
totmom(i)=totmom(i)+effch*h*(2*BZindex(j)+3)*interp1(mom(:,1,1),mom(:,m(j)+2,ii),abs(k(j)));
k(j)=k(j)+interp1(mom(:,1,1),mom(:,m(j)+2,ii),abs(k(j)));
end
end
%Adjusting for k in the second Brillouin zone
% i.e. it is moved to the complementary k in the 1st BZ
% No adjustment for the subband index m is required because
% this is already done in scattering8.m
if k(j) > kmax
if BZindex(j)==1
BZindex(j)=2;
else
BZindex(j)=1;
end
k(j)=2*kmax-k(j);
elseif k(j) < -1*kmax
if BZindex(j)==1
BZindex(j)=2;
else
BZindex(j)=1;
end
k(j)=-2*kmax-k(j);
end
dti(j)=0; %reset the free drift time
break
end
if ii==sizeW(3)
flag=flag+1;
end
end
end
end
end
% Calculate the occupation distribution function
display('Calculating the occupation distribution function')
for ii=1:nstep
sumBZ=0;
for i=1:n
if BZ(ii,i)==1
sumBZ=sumBZ+1;
end
end
if sumBZ==0
display('error..all electrons are in the 2nd BZ')
end
factor=n/sumBZ;
for i=1:n
if BZ(ii,i)==1
if sign(momentum(ii,i))==1
for j=1:78
if momentum(ii,i)<fp(j,1)
195
fp(j,ceil(ii/sample)+2)=fp(j,ceil(ii/sample)+2)+1/sample*factor;
break
end
if j==78
fp(j,ceil(ii/sample)+2)=fp(j,ceil(ii/sample)+2)+1/sample*factor;
end
end
else
for j=1:78
if momentum(ii,i)>fn(j,1)
fn(j,ceil(ii/sample)+2)=fn(j,ceil(ii/sample)+2)+1/sample*factor;
break
end
if j==78
fn(j,ceil(ii/sample)+2)=fn(j,ceil(ii/sample)+2)+1/sample*factor;
end
end
end
end
end
end
toc
display('run1')
196
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