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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 ii 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. iii 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 vii 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 viii 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 ix 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 xi 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. xiv 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 ma1 na2 (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 bji ] bij (3.14) where the terms bij , bji 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 ( bji 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) n1 96 where [ M ] is the mass matrix, un (t ) , un (t ) are the time-dependent acceleration and displacement of atom n respectively, [ Knn ] 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 ]eiq 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; 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