Vibrational Properties of Disordered Graphene and ...

A Dissertation Submitted to the University of Fukui for the Degree of

Doctor of Engineering

Vibrational Properties of Disordered Graphene and Related 2D Materials (不規則グラフェン及び２次元物質の格子振動に関する研究)

September 2014 Md. Sherajul Islam

Declaration of Authorship I, Md. Sherajul Islam, declare that this thesis titled, ’Vibrational Properties of Disordered Graphene and Related 2D Materials’ and the work presented in it are my own. I confirm that:

This work was done wholly or mainly while in candidature for a research degree at this University.

Where any part of this thesis has previously been submitted for a degree or any other qualification at this university or any other institution, this has been clearly stated.

Where I have consulted the published work of others, this is always clearly attributed.

Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work.

I have acknowledged all main sources of help.

Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself.

Signed:

Date:

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In the name of Allah (GOD), most gracious, most merciful (Bismillah Ar-Rahman Ar-Raheem)

O my lord! Glory be to you, we have no knowledge save(except)what you have taught us. In truth(verily), it is You, the All-knower the All-wise(who are perfect in knowledge and wisdom).

God is Truthful

UNIVERSITY OF FUKUI

Abstract Department of Electrical and Electronic Engineering Doctor of Engineering Vibrational Properties of Disordered Graphene and Related 2D Materials by Md. Sherajul Islam

Graphene, the most promising materials for electronic, mechanical, optical and so other versatile applications due to its novel and unique properties. It has also opened the door for the other two dimensional (2D) materials. Due to the strong couplings between electrons and phonons in graphene related materials, vibrational properties have significant effects on the electronic transport properties. Defects which are inevitably present in graphene and its related materials which modify the vibrational properties greatly. However, in contrast to the intensive research performed on the vibrational properties of pristine graphene, the vibrational properties of disordered graphene has not been well studied. In this thesis, the details of the various types of defects effects on the vibrational properties of graphene and analogous 2D materials such as hexagonal boron nitride (h-BN) has been studied systematically. In the first part of this work, a simple yet reasonably accurate method to calculate the vibrational eigenfrequencies and eigenvectors of 2D disordered systems is developed using the forced vibrational method, which is based on the mechanical resonance to extract the pure vibrational eigenmodes and suitable for very large and complex physical systems. This model is then applied to the study of graphene and h-BN with different types of point defects such as isotopes, substitutional doping, vacancies, edges, combined isotope and vacancies, as well as doping with native vacancies.

This dissertation reports many new findings based on these simulations such as a relatively small concentration of defects may lead to significant and specific changes in the phonon density of states (PDOSs), especially at low frequencies, near the Van Hove points and in the vicinity of the Γ and K points of the Brillouin zone. A linear reduction of the Raman active E2g mode frequencies with an increase of

13 C

concentration has

been observed due to the reduced mass variation of the isotope mixture. A downshift of the E2g mode of 65 cm−1 has been found, which are well agreed with the experimental results and the phonon frequencies described by the simple harmonic oscillator model. A broadening and softening of the Raman active E2g phonon mode with an increase of B and N atoms doped to the graphene has been detected. The armchair-edge and vacancytype defects break down the phonon degeneracy at the Γ point of the longitudinal and transverse optical (LO and TO) mode, distort and shift down the PDOSs significantly. The PDOS peaks for the combined isotope or doping and vacancy-type defects show the remarkable increase in the low-frequency region induced by their defect formations. Moreover, for C doped h-BN network, it is found that the PDOS greatly depends on the C distribution and coverage. A critical value of C concentration for the onset of this C-induced vibrational transition has been determined. The C concentrations of about 10% and higher, the E2g peak of h-BN has been reduced into a shoulder or it has completely disappeared. Due to the scattering by disorder entities, the phonon wave functions become localized in the real space. These numerical experiments reveal that the disordered graphene show the remarkably spatial localization of lattice vibrations. The calculated typical mode patterns for K point in-plane TO mode (iTO) phonons indicate that the features of strongly localized state depend on the defect density, and the phonon is localized strongly within a region of several nanometers in the random percolation network structures. In particular, a typical localization length is on the order of ≈ 7 nm for isotope impurities, ≈ 7.5 nm for B or N doping, ≈ 5 nm for vacancy-type defects, and ≈ 6 nm for combined defects at high defect concentrations of 30%. Phonon localization in the hybrid BCN network is also studied. The typical mode patterns show that LO and TO mode phonons at the Γ point is strongly localized and show a desultory behavior within a region of several nanometers in the BCN structure. In particular, at 1400 cm−1 , a typical localization length is on the order of ≈ 4 nm for randomly distributed C atoms and ≈ 8.5 nm for the regular domains of C of 20% concentration, while at 1590 cm−1 ,

these values are ≈ 2 nm and ≈ 4nm, respectively. Further, a detailed analysis procedure of the edge structures of epitaxial graphene has been developed using the polarized Raman scattering spectra. Based on our present experimental results, a rather precise atomic-scale realistic edge model has been proposed combining the armchair and zigzag edges along with some point defects. These findings demonstrate a route towards further studies aimed at a better understanding of the phenomena related with vibrational properties such as thermal conductivity, specific heat capacity, and electron-phonon interaction as well as for the interpretation of experiments on infrared, Raman, and neutron-diffraction spectra of disordered graphene and related 2D crystals.

Acknowledgements I kneel humbly to ALLAH thanking HIM for showing me the right path. Without HIS help my efforts would have gone astray. May God send His praises upon his Beloved Prophet Muhammad (swt). I would like to express my deepest gratitude to my thesis advisor, Professor Akihiro Hashimoto, for his continuous support, invaluable advice, critical and constructive evaluation towards this research. Without his active involvement this work wouldn’t have been possible. His uncompromising attitude towards quality and professionalism helped me become a professional from a student. His kindness and considerateness to students helped me pass through some of the most difficult times during my study. My group and lab mates add endless joy to my Fukui memory. Brilliant, resourceful and witty, each of them are the very best students of their countries. In particular I would like to thank, Kenji Ushida, Takaki Ishida, Daisuke Tamakawa, Kenji Kodama, Kensuke Kasagi, and Takuya Nambu. My deep sense of gratitude to Professor Dr. Ashraful Ghani Bhuiyan from Khulna University of Engineering and Technology for insightful comments and critiques in the writing of the dissertation. I would also like to express my appreciation to my parents, my beloved wife, my daughter, and my other relatives. My mother in law’s endless inspiration at the early part of my study helped me dream big and eventually obtain the highest degree in academia. This work hadn’t been possible without the sacrifice and tremendous support of my lovely wife. During the course of my PhD, I have been immersed into my studies, while my wife quietly took care of everything and selflessly supported me. No language can express my gratefulness to her. Finally, I would like to thank my little daughter, many days she had to play alone, because her daddy was busy with coding. Finally, my sincere thanks are dedicated to the Japan ministry of Education, Culture, Sports, Science and Technology (MEXT) for funding the PhD scholarship.

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Contents Declaration of Authorship

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Abstract

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Acknowledgements

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Contents

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List of Figures

xvii

List of Tables

xix

Abbreviations

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Physical Constants

xxiii

Symbols

xxv

1 Introduction 1.1 Introduction . . . . . . . . . . . . . . . . . . . 1.2 Overview of Disordered Graphene and h-BN 1.2.1 Types of Defects . . . . . . . . . . . . 1.2.1.1 Isotope Disorder . . . . . . . 1.2.1.2 Substitutional Dopant . . . . 1.2.1.3 Vacancies . . . . . . . . . . . 1.2.1.4 Edges . . . . . . . . . . . . . 1.2.1.5 Hexagonal BCN Network . . 1.3 Motivation . . . . . . . . . . . . . . . . . . . 1.4 Objectives . . . . . . . . . . . . . . . . . . . . 1.5 Synopsis of Dissertation . . . . . . . . . . . .

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1 1 5 5 5 6 6 7 8 8 10 12

2 Fundamentals of Graphene and h-BN 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The sp2 -hybridized Carbon Nanomaterials . . . . . . . . . . . . . . . . . . 2.3 Electronic Properties of Graphene . . . . . . . . . . . . . . . . . . . . . .

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Contents

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2.3.1 Tight Binding Approximation . Vibrational Properties . . . . . . . . . 2.4.1 The Force Constant Model . . Raman Spectroscopy . . . . . . . . . . The sp2 -hybridized BN Nanomaterials

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3 Forced Vibrational Method 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Normal Modes of Vibrations . . . . . . . . . . . . . . . . . 3.3 Forced Vibrational Method . . . . . . . . . . . . . . . . . 3.3.1 Method for the Density of States . . . . . . . . . . 3.3.2 Mode Patterns . . . . . . . . . . . . . . . . . . . . 3.3.3 Numerical Algorithm . . . . . . . . . . . . . . . . . 3.4 Application of FV Method to the Graphene Lattice . . . 3.4.1 Evaluating the Accuracy of Calculated Eigenvalues 3.5 Localized Eigenmodes . . . . . . . . . . . . . . . . . . . . 4 Vibrational Properties of Isotope Disordered Graphene 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Computational Details . . . . . . . . . . . . . . . . . . . . 4.3 Phonon Density of States . . . . . . . . . . . . . . . . . . 4.4 Typical Mode Patterns . . . . . . . . . . . . . . . . . . . . 4.5 Localization Length . . . . . . . . . . . . . . . . . . . . .

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5 Vacancy Effects on Phonon Properties of Graphene Nanoribbons 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Calculation Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Phonon Density of States . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Vibration Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Temperature Dependent Specific Heat Capacity . . . . . . . . . . . . .

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59 59 59 62 64 67

6 Effect of Boron and Nitrogen Doping 6.1 Introduction . . . . . . . . . . . . . . . 6.2 Computational Details . . . . . . . . . 6.3 Results and Discussion . . . . . . . . . 6.3.1 Phonon Density of States . . . 6.3.2 Localized Modes of Phonon . . 6.3.3 Phonon Localization Length .

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7 Polarized Raman Spectroscopy for Edge Structure 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 7.2 Edges of Graphene . . . . . . . . . . . . . . . . . . . 7.3 Raman Spectroscopy on the Edges . . . . . . . . . . 7.4 Experimental Details . . . . . . . . . . . . . . . . . . 7.5 Results and Discussion . . . . . . . . . . . . . . . . . 7.5.1 Generalized Edge Model . . . . . . . . . . . . 7.6 Vibrational Modes on Realistic Edge Structures . . .

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of Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents 8 Vibrational Properties of C-doped Hexagonal BN 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 8.2 Computational Details . . . . . . . . . . . . . . . . . 8.3 Phonon Density of States . . . . . . . . . . . . . . . 8.4 Localization of Eigenmodes . . . . . . . . . . . . . . 8.5 Phonon Localization Length . . . . . . . . . . . . . .

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9 Summary and Outlook 107 9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 9.2 Future Outlooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A Derivation of FV Method 111 A.1 Method for the Density of States . . . . . . . . . . . . . . . . . . . . . . . 112 A.2 Mode Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 B Matlab Code for Phonon Density of States and Mode B.1 Generation of Hexagonal Lattice Spaces for Graphene . B.2 Calculation of Phonon Density of States of Graphene . B.3 Calculation of Mode Patterns . . . . . . . . . . . . . . .

Bibliography

Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures 1.1 1.2 1.3 1.4 1.5

The The The The The

schematic of current and future applications of application chart of graphene companies . . . graphene-based display and electronic devices graphene-based photonics applications . . . . zigzag and armchair edges of graphene . . . .

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The The The The The The The The The The The The The The The

formation of sigma and pi bonds between two carbon atoms . . honey comb lattice of graphene . . . . . . . . . . . . . . . . . . various forms of graphene . . . . . . . . . . . . . . . . . . . . . unit cell and its Brillouin zone of graphene . . . . . . . . . . . . energy dispersion relations for graphene . . . . . . . . . . . . . neighbor atoms of a graphitic plane . . . . . . . . . . . . . . . force constants between the A and B atoms on a graphene sheet phonon dispersion relation and density of states of 2D graphene normal modes of Γ point phonon of 2D graphene . . . . . . . . normal modes of K point phonon of 2D graphene . . . . . . . . typical Raman spectra of disordered graphene . . . . . . . . . . Raman scattering process for the G, D and D’ peaks . . . . . . unit cell and band structure of h-BN . . . . . . . . . . . . . . . allotropes of h-BN . . . . . . . . . . . . . . . . . . . . . . . . . . phonon dispersion relation of 2D h-BN . . . . . . . . . . . . . .

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3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

The The The The The The The The

plot of sync function . . . . . . . . . . . . . . . . . . . . . . . . . direction of force constant tensors up to the third neighbors . . . direction of force constant tensors for the fourth nearest neighbors PDOS of graphene according to the FV and FC models . . . . . PDOS of graphene according to the FV and FC models . . . . . . PDOS of graphene according to the FV and FC models . . . . . . PDOS of graphene according to the FV and FC models . . . . . . system size dependent phonon density of states of graphene . . .

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4.1 4.2 4.3 4.4 4.5 4.6 4.7

The The The The The The The

graphene with isotopes and percolation network structures . . . PDOS of graphene with different concentrations of 13 C isotope. E2g mode frequency as a function of 13 C atoms. . . . . . . . . . PDOS of vacancy-type defective graphene . . . . . . . . . . . . PDOS of combined isotope and vacancy-type defective graphene defect dependent phonon frequencies . . . . . . . . . . . . . . . typical mode pattern of isotope disordered graphene . . . . . .

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graphene . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

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4.8

The localization length as a function of defect density . . . . . . . . . . . 57

5.1 5.2 5.3 5.4 5.5 5.6

The The The The The The

lattice structure of vacancy-type defective AGNR . . . width dependent PDOS of AGNR . . . . . . . . . . . . defects dependent PDOSs . . . . . . . . . . . . . . . . defects dependent LO phonon frequencies . . . . . . . . typical vibration pattern of disordered AGNR . . . . . temperature dependent specific heat capacity of AGNR

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6.1 6.2 6.3 6.4 6.5 6.6 6.7


B- and N-doped graphene . . . . . . . . . . . . . . doping dependent PDOSs . . . . . . . . . . . . . . vacancy induced PDOS of grapene . . . . . . . . . combined doping and vacancy effects on PDOSs . E2g mode frequencies as a function of defect density typical mode patterns of doped graphene . . . . . . doping dependent localization length . . . . . . . .

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7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

The The The The The The The The The The

armchair edge with three orientational segments . . . . . optical micrograph of the measured graphene sample . . . typical Raman spectra from the edges of grpahene . . . . . Raman mapping measurements of the edges . . . . . . . . predicted edge diagram from the polarized Raman results polarization dependent D-band intensity . . . . . . . . . . AFM images of the edges . . . . . . . . . . . . . . . . . . . typical mode pattern for perfect armchair and zigzag edges typical mode pattern for vacancy type disordered graphene typical mode pattern for non-ideal disordered edges . . . .

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examples of several structures of C-doped h-BN . . . . . . . . . . calculated phonon density of states of h-BN sheet . . . . . . . . . PDOSs of BCN network with randomly distributed C atoms . . . carbon concentration dependence of E2g frequencies of BCN sheet PDOSs of regular domains of BN and grphene sheet . . . . . . . . typical mode patterns for BCN network . . . . . . . . . . . . . . . phonon localization length as a function of C content . . . . . . .

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List of Tables 2.1

The allotropes of carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 3.2 3.3 3.4 3.5 3.6

The The The The The The

7.1

The polarization dependent D band intensity ratio of edges . . . . . . . . 86

angle between the A type atom and different neighbors angle between the B type atom and different neighbors force constants tensors fitted to the INS data . . . . . force constants tensors fitted to the RRS data . . . . . force constants tensors fitted to the IXS data . . . . . force constants tensors fitted to the LD data . . . . .

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Abbreviations AFM

Atomic force microscope

AGNR

Armchair graphene nanoribbons

BNNT

Boron nitride nanotube

BZ

Brillouin zone

DFT

Density functional theory

DUV

Deep ultraviolet

EELS

Electron energy loss spectroscopy

FBC

Free boundary condition

FC

Force constant

FV

Forced vibrational

HREELS

High-resolution EELS

h-BN

Hexagonal boron nitride

iLO

In plane longitudinal optical

INS

Inelastic neutron scattering

iTO

In plane transverse optical

IR

Infra red

IXS

Inelastic X ray scattering

PDOS

Phonon density of state

PBC

Periodic boundary condition

SW

Stone Wales

SWCNT

Single walled carbon nanotube

RRS

Resonance Ramnan scattering

2D

Two dimensional

3D

Three dimensional

xxi

Physical Constants Reduced plank constant

~

=

xxiii

1.05457172647 × 10−34 J.s

Symbols a

lattice constant of graphene

nm

ω

wave number

cm−1

xxv

Dedicated to show the Heaven’s Light of all the Atheist on the earth

xxvii

Chapter 1

Introduction 1.1

Introduction

The experimental discovery of graphene [1, 2], the first two dimensional (2D) stable material of sp2 -bonded carbon atoms arranged in a honeycomb network, is a milestone in solid state physics, engineering, and material science. In an era when electronics components are becoming progressively smaller and the limits to Moore’s Law are frequently considered, the isolation of a single graphene layer represents a major technological achievement. This one-atom-thick fabric of carbon uniquely combines many supreme properties that have been exceeded those obtained in any other material, with some reaching theoretically predicted limits: room-temperature electron mobility of 2.5 × 105 cm2 V−1 s−1 (Ref. [3]) (theoretical limit [4] ∼ 2.5 × 105 cm2 V−1 s−1 ); a Young’s modulus of 1 TPa and intrinsic strength of 130 GPa (Ref. [5], very close to that predicted by theory [6]); very high thermal conductivity (above 3,000 WmK−1 ; Ref. [7]); optical absorption of exactly πα = 2.3% (in the infrared limit, where α is the fine structure constant) [8]; complete impermeability to any gases [9], ability to sustain extremely high densities of electric current (a million time than copper) [10]. Another property of graphene, already demonstrated [11–13], is that it can be readily chemically functionalized. These properties suggest that graphene could replace other materials in existing applications [14–24], including high speed and high frequency electronics, optoelectronic devices, touch screens, light emitting diodes, solar cells, sensors, individual gas molecules detectors, the applications where high sensitivity to electric charge, magnetic field and mechanical strength are required, electrode material for capacitors in rechargeable batteries, and reverse osmosis membranes for water purification, for instance. That all the extreme properties are combined in one material means that graphene could also enable several disruptive technologies. The combination of transparency, 1

Chapter 1. Introduction

2

device Advaced Graphene Synthesis

energy

display

bio

flexible

military Large domain / High quality Transfer free / Large scale etc.

communication automobile

Figure 1.1: A schematic showing the high quality large domain graphene by advanced synthesis method along with the current and future applications (Ref. [25]).

conductivity and elasticity will find use in flexible electronics, whereas transparency, impermeability and conductivity will find application in transparent protective coatings and barrier films; and the list of such combinations is continuously growing. A schematic of the high quality large domain graphene by advanced synthesis method along with the current and future applications is shown in Fig. 1.1. An application chart for graphene companies is presented in Fig. 1.2. This information refers to end user markets graphene companies are targeting their products to, by percentage. A road map for potential applications of graphene in electronics and optoelectronics are also shown in Figs. 1.3 and 1.4. Soon after the discovery of graphene it has also opened the gate for the synthesis of a new class of materials with novel physical properties. Mono layer hexagonal boron nitride (h-BN) [28], a structural analog of graphene, has been demonstrated to exhibit the same versatility as its C counterpart. Besides graphene, h-BN has recently attracted increased attention. Even though h-BN has the lattice parameters very close to their C counterparts, different elemental components give it quite different physical properties. It possesses several advantageous properties over graphene. Most notably, exhibit much better chemical and thermal stability compared to graphene, enhanced oxidation resistance [29], good optical properties [30], and at the same time possess a similarly high thermal conductivity and excellent mechanical properties [31, 32]. However, the most significant differences in physical properties of h-BN and graphene structures are


3

Figure 1.2: Applications chart of graphene companies. This information was accrued from a comprehensive survey of graphene companies (Source: Future Markets [26]).

their electrical performances. Hexagonal BN is an electrical insulator with constantly wide band gap (5-6 eV) [30, 33], while graphene is semi-metallic with zero bandgap [34]. These distinctions make h-BN uniquely attractive for applications in electronics, photonics, and nanocomposites [35–38]. For any new materials, the characterization and investigation of their fundamental and application oriented properties are a crucial prerequisite for tomorrow’s application. Vibrational properties are of fundamental importance as they play a key role in so many physical observable, particularly thermal and transport properties. The electron-phonon interaction and the transport properties which have a large impact on the electronic device performances can be understood, in most situations, only with a detailed knowledge of the vibrational properties. Vibrational properties are also of fundamental importance to understand the anharmonicity in the lattice potential energy. The performance of electronic devices (size of a few micro/nanometers) largely depends on the nature of the heat dissipation and hence on the thermal conductivity, and the phonons are the dominant heat carriers in crystalline semiconducting materials. Moreover, Raman spectroscopy is one of the most popular techniques for the characterization


4

Transferred or directly grown large-area graphene (high quality)

Graphene transfer medium quality

High-frequency

transistor

Future devices

Logic transisitor/ thin-film transistor

Touch screen Rollable e-paper Display

Folded OLED

Electronics 2020

2015

2010

2030

2025

2035

Year

Figure 1.3: Graphene-based display and electronic devices. Display applications are shown in green; electronic applications are shown in aqua. Possible application time line, based on projections of products requiring advanced materials such as graphene. The figure gives an indication of when a functional device prototype could be expected based on device roadmaps and the development schedules of industry leaders (Ref. [27]).

Transferred or directly grown graphene

Polarized controller

Isolator

Photodector Modulator

optical optical Interconnect

Mode-locked semiconductor laser

Transferred graphene (high quality)

Solid-state-mode-locked laser THz wave generator

Tunable fiber-mode-locked laser THz wave detector

2010

2015

2020

2025

2030

2035

Year

Figure 1.4: Graphene-based photonics applications. Optical applications are shown in purple; optical interconnect applications are shown in lighter red. Possible application time line, enabled by continued advances in graphene technologies, based on projections of products requiring advanced materials such as graphene. The figure gives an indication of when a functional device prototype could be expected based on device road-maps and the development schedules of industry leaders (Ref. [27]).

of carbon systems [39] and recently it has also been demonstrated for BN materials [40]. To assess the structural properties of BN, infrared (IR) spectroscopy is the most widely used since it is very sensitive to the polar boron-nitride bonds, and the hexagonal forms have distinct infrared features [41–46]. Since the photons of IR or visible light carry a


5

momentum, which is negligible compared to the momentum associated with atomic motion, only phonons with a wave vector (the “point” in reciprocal space) can be excited in first-order Raman and IR absorption processes. For the interpretation of such spectra, an accurate knowledge of the phonon frequencies is also indispensable.

1.2

Overview of Disordered Graphene and h-BN

Despite being praised for being inert, ultra-strong and impermeable to any gaseous material [47], realistic graphene structures always contain defects [48]. One generally refers to defects in graphene as anything that breaks the symmetry of the infinite carbon honeycomb lattice. Thus, different types of defects can be defined such as isotopes, vacancies, edges, grain boundaries, substitutional doping, implanted atoms and defects associated with a change of carbon-hybridization, for example from sp2 into sp3 .

1.2.1 1.2.1.1

Types of Defects Isotope Disorder

Naturally occurring carbon materials are made up of two stable isotopes, dance 98.9%) and

13 C

(1.1%). Because of the low concentration of

13 C,

12 C

(abun-

we can usually

neglect this isotope effect for discussing the physical properties of graphene-related materials. However, if we consider, for example, the intrinsic spectral line-width in Raman spectroscopy, the isotope effect might be essential as one of the important intrinsic scattering mechanisms. When we intentionally increase the

13 C

isotope concentration in an

sp2 material, the graphene-related material made of varying concentrations of

13 C

has

provided interesting information for better understanding of phonon related properties [49] as well as growth mechanisms [50, 51]. The advantage of the isotope enrichment technique is that only phonon frequencies or thermal properties can be modified without changing the electrical or chemical properties, so that we can distinguish the electronphonon interaction from electronic or electron-electron interactions in making assignments for unassigned optical spectral features [49]. In the case of crystal growth, if we substitute a 12 C atom in a gas source molecule by a 13 C atom, we can get information on how the carbon atoms from the gas molecule are used for the crystal growth of carbon nanotubes [50] and graphene [51].

Chapter 1. Introduction 1.2.1.2

6

Substitutional Dopant

Although graphene has emerged as one of the most promising materials for the next generation electronics by overcoming silicon-based electronics limitations [52], the electronic applications are handicapped by the absence of a bandgap in the pristine graphene [1, 53]. The development of graphene based electronics depends on our ability to open a tunable bandgap. In the quest to opening and tuning an energy gap in graphene, various approaches have been developed to improve the semiconducting properties. The substitutional defects (doping) is one of the most feasible methods which can tailor the electronic band structure of graphene, and open up an energy gap between the valence band and the conduction band. Atoms such as B and N are the ideal candidates as substitutional defects (dopants) in graphene because of their similar atomic size as that of C and of their hole acceptor and electron donor characters for substitutional B- and N-doping, respectively. B- or N-doped graphene-based materials have shown enhanced electrical properties relative to the materials without doping [54–56]. The semiconducting B- and N-doped graphene have also been successfully synthesized by chemical vapor deposition (CVD) and electro-thermal reactions [57–59]. The tunable energy gap opening and doping dependent electronic properties of graphene make the vast application of graphene-based devices even more promising.

1.2.1.3

Vacancies

From a practical point-of-view, when graphene-based materials have been fabricated experimentally, they will have some vacancy and adatom-vacancy defects [60]. Vacancies have a striking effect on the electronic structure. In particular, it is well known that vacancies induce localized states with associated resonant peaks at the Dirac point, and that state in the vicinity of the Dirac point has an enhanced tendency for localization, as revealed by an enhancement of the electronic inverse participation ratio [61]. Recent experimental studies [62–66] imply that some mutual effects between structural defects and dopant atoms are present during the incorporation of heteroatoms into graphene. Hou et al. [67] analyzed the energetic stability of substitutional N dopants in graphene with vacancies and Stone-Wales (SW) defect (e.g., forming non-hexagonal rings by transforming four hexagons into two pentagons and two heptagons [5-7-7-5] defect, or frequently occurring two pentagons and one octagon [5-8-5] defect) [68, 69]. They demonstrated [70] from the total energy calculations based on density functional theory (DFT) that the native point defect and N dopant attract each other, i.e., they display cooperative effect, suggesting that N dopants can prompt the creation of point defects and vice versa. The formation of complexes between native point defect and N dopant strongly modifies the role of N doping regarding the free carrier production in the bulk π bands.


7

Figure 1.5: (a) Zigzag and (b) Armchair edge of graphene.

1.2.1.4

Edges

Since carbon atoms are arranged in a honeycomb lattice, the graphene has two kinds of boundaries, called armchair and zigzag edges (see Fig. 1.5). The borders or the edges of a graphene sheet act like defects in the crystallographic structure and may have great impact in transport properties. For example, the presence of edges in graphene has strong implications for the low-energy spectrum of the π-electrons [71–73]. Recently, graphene nanoribbons (GNRs) have been also receiving remarkable interest [74–76]. The edge geometry is the key parameter which determines the electronic properties of the nanoribbons. Although the two-dimensional graphene is a zero band gap semi-metal, the electronic structure of GNRs depend sensitively on the crystallographic orientation of their edge geometry [73, 77, 78]. It is predicted that GNRs with certain edge chirality would open the band gap [74, 75, 79] and show distinguish optical [80] and superconductive [81] properties. It is also expected that the edge states play an important role for the magnetic properties in nanometer-sized graphene systems, because of their relatively large contribution to the density of states at the Fermi energy [71, 82]. The investigation of the edge structures of graphene is thus of crucial prerequisite for the practical applications and of great general interest for the physical understanding of those structures.

Chapter 1. Introduction 1.2.1.5

8

Hexagonal BCN Network

In the most present years, h-BN is doped with C atoms to explore the properties of this new structure by creating hybrid BCN systems [83]. It is expected that the mixture of B, N, and C atoms form stable structures displaying interesting characteristics [83]. Substitutional C doping of the honeycomb lattices in BN nanosheet was achieved via in situ electron beam irradiation inside the transmission electron microscope (TEM) using a deliberately introduced C feedstock [84]. It has been shown that C doping can transform the h-BN sheets from an electrical insulator to a conductor via recording two-terminal I-V curves of the h-BN before and after doping [84]. Atomic layers with hybridized h-BN and C domains have also been successfully synthesized, and their conductivity has been demonstrated to increase with C content [58]. The preferential doping was found to occur at the sites more vulnerable to electron beam irradiation. This transformed BN structures from electrical insulators to conductors. Moreover, changing the C doping concentration and cluster size, the transparency of h-BN can be modified from the infrared to the ultraviolet, covering the whole visible spectrum [85]. Chen et al. has exposed that the transition-metal Sc dispersed C doped BN nanostructures exhibit good hydrogen adsorption ability, indicating the possibility of fabricating hydrogen storage media with high capacity [86]. The new materials constructed from multiple elements using B, C,and N atoms and their versatile properties could possibly be useful for a wide range of applications, including high-power electronics, nanoelectronic, nanoscale sensing devices, and protective coatings, which are also indispensable materials for nanotechnology and the advancement of science in the twenty-first century.

1.3

Motivation

Graphene has been considered as a prototypical bench-top relativistic quantum system that consists of a one-atom-thick planar sheet of the sp2 -bonded carbon atoms [47]. In sp2 carbons, the phonons, like the electrons, depend on the atomic structure that are the main source of Raman spectra in the literature, and the Raman phonon spectra can be used to study the similarities and differences in the various materials within the sp2 carbon family. The presence of disorder in sp2 hybridized carbon systems leads to rich and intriguing phonon properties. For example, the effect of breaking the translational symmetry of crystal by introducing disorder into the lattice is the break-down of momentum conservation, through the activation of phonons at interior K points of the Brillouin zone. Therefore, strong D band features are observed in the Raman spectra which are generally inactive for perfect graphene [87–89]. Moreover, even small concentrations of


9

defects in graphene-based 1D and 2D nanostructures may alter the vibrational properties significantly and thus change their optical absorption, low temperature specific heat and transport properties. The defect effects in carbon-based materials are significant, because in a real graphene sample, we know that 1.1% of the atoms are

13 C.

The isotopic disorder in a crystal

is a perturbation in which the average atomic mass is changed, and therefore, some physical properties, such as thermal conductivity and phonon scattering are sensitive to isotope mixing [90, 91]. Among the physical properties of crystals, phonons are those which reflect most strongly differences in isotopic mass M , their frequencies being usually proportional to M −1/2 [92]. Hence, the effects of M fluctuations should be best observed by spectroscopic techniques (Raman, infrared (IR) absorption, neutrons, possibly He-atom scattering, electron energy loss spectroscopy (EELS), and so on. Electrons and phonons are strongly coupled in graphene-like sp2 carbon systems [93] ensuing Kohn anomalies [94, 95] and possible soft modes [96, 97]. Because of free carrier-induced many body interactions, one expects the observed phonon energy in a doped system to be different from the bare phonon energy in an undoped system. It has been shown recently that the phonons in graphene are altered interestingly by tuning the applied gate voltage [93, 98] and that the Raman spectroscopy has been used to measure the nature and level of doping-induced carriers. Mechanisms of molecular doping and the related chemical sensor properties of graphene have also been investigated [98, 99]. Electronic band structure and phonon properties of aromatic molecules such as aniline and nitrobenzene using first principles calculations have been reported very recently [100]. It has been found recently that the electron or hole doping shifts the Raman frequency of the G band along with the decrease in its line width in the case of single walled carbon nanotube (SWCNT) [101] and graphene [102]. This shift in frequency is associated with the shifts in Fermi level and charge transfer between graphene and dopant atoms. A detailed understanding of the vibrational properties of doped-graphene with native vacancy defects is thus of fundamental importance as they govern electron transport, and hence the performance of graphene-based electronic devices. Over the past few years, a large number of studies have been devoted to estimate the phonon properties of graphene or h-BN using various theoretical approaches including the force constant fittings [103–105], the density functional theory [94, 96, 106], the tight-binding model [107], and the first-principles calculations [108–112]. The phonon properties of graphite has not been completely resolved by experiment, mostly due to the lack of large enough samples of crystalline quality. It has been partly measured by inelastic neutron scattering(INS), electron-energy loss spectroscopy(EELS), and inelastic x-ray scattering (IXS) [113–118]. Some experimental studies have also been


10

performed on the phonon spectrum of mono-layer film of h-BN and bulk h-BN using high-resolution EELS (HREELS) [119], and IXS measurements [120]. In contrast to the intensive research performed on the phonon properties of pristine graphene or h-BN, too little attention has been paid so far to the description of the direct influence of defects on the phonon spectra. Moreover, there are no studies at all on the vibrational properties, especially considering the combined effects of isotope and vacancies or the doping and vacancies or the h-BCN network, which are the more realistic situation. When impurities present in a system, they may induce the symmetry breakdown of elemental topological arrangements, which generate more complex lattice structures. Therefore, the dynamical matrix technique requires huge computational resources. These long computational times and convergence problems in the dynamical matrix calculations limit the systems of interest to benchmark molecules. However, the finite-size effects influence the result of the vibration modes in the low-frequency regime significantly. A significantly larger scale model is essential for an in-depth understanding of vibrational properties of defective atomic structures. It is thus crucial to develop a reliable model to observe the vibration modes of disordered graphene-related materials directly.

1.4

Objectives

The purpose of this dissertation is to go through systematically the details of the various types of defects effects on the vibrational properties of graphene and h-BCN network. The forced vibrational (FV) method introduced by Williams and Marris [121] has been employed for developing a well defined quantitative model to estimate the vibrational properties of disordered 2D material systems. This method is based on the principle that a linear mechanical system when driven by a periodic external force of frequency Ω will respond with large amplitudes in those eigenmodes close to this frequency. This technique has the following advantages: (i) One can apply this algorithm to a very large, complex, as well as disordered system; (ii) it is possible to calculate quite accurately the DOS in the low-frequency regime; (iii) the magnitude of the DOS at any frequency can be obtained by sweeping the resonant frequency. When some disorders are present in a system, there may appear vibrational modes [122, 123] lying outside of the allowed frequency range of the perfect crystal. These are called localized vibrational modes or local modes because the mode energy is spatially concentrated at the defect site, which is similar to Anderson’s localization for electronic wave functions in the disordered lattice. In general crystal imperfections give rise to localized excitations which can also limit the phonon life time through elastic or inelastic scattering. If the vibration modes or the phonons are localized, the mean free path


11

of the phonon becomes finite and proportional to the square of the localization length. Generally, when the electron–phonon interaction of graphene is discussed [124, 125], the phonon wave function is treated as delocalized in the crystal. However, owing to the presence of defects in the graphene or BN materials, phonons have a finite lifetime and a finite localization length due to the scattering by impurity atoms, which can contribute considerably to the natural line width of the Raman spectra. Although graphene-related materials have excellent intrinsic thermal conductance, disordered regions interrupt the crystallinity and impede the flow of thermal energy. Since localization virtually immobilizes phonons and makes them non-conducting, these phonons lose their nature as heat carriers. To quantify these localization properties is another important objective. In the course of this dissertation, the following objectives were achieved: 1. Develop a reliable model based on the FV technique to determine the vibrational properties of disordered 2D materials, and compare the results with vibrational properties calculated by the widely used dynamically matrix techniques [105, 126]. 2. Report the combined effects of isotope and vacancy-type defects on the phonon properties of graphene for the first time. 3. A more extensive and systematic numerical analyses on the vibrational properties of B- and N-doped graphene with native vacancies has been studied. 4. Study the effect of vacancy-type defects on the vibrational properties of graphene nanoribbons numerically. 5. Develop a detailed analysis procedure for the edge structures of epitaxial graphene at the atomic level using the polarized microscopic laser Raman scattering. 6. Explore the vibrational properties of randomly and regularly distributed C-doped h-BN network for the first time. The above contributions have resulted in the following papers which have already been published. 1. Md.

Sherajul Islam, Disuke Tamakawa, Satoru Tanaka, Takayuki Makino,

Akihiro Hashimoto, Polarized microscopic laser Raman scattering spectroscopy for edge structure of epitaxial graphene and localized vibrational mode, Carbon, 77 (2014) 1073–1081. (Impact Factor: 6.16) 2. Md. Sherajul Islam, Satoru Tanaka, Akihiro Hashimoto, Effect of vacancy defects on phonon properties of hydrogen passivated graphene nanoribbons , Carbon, Accepted (In press) (http://dx.doi.org/10.1016/j.carbon.2014.08.049). (Impact Factor: 6.16)


12

3. Md. Sherajul Islam, Kenji Ushida, Satoru Tanaka, Takayuki Makino, Akihiro Hashimoto, Analysis of vibrational properties of C-doped hexagonal boron nitride (h-BN), Computational Materials Science, Accepted (In press) (http://dx.doi. org/10.1016/j.commatsci.2014.04.047). (Impact Factor: 1.88) 4. Md. Sherajul Islam, Kenji Ushida, Satoru Tanaka, Takayuki Makino, Akihiro Hashimoto, Effect of boron and nitrogen doping with native point defects on the vibrational properties of graphene, Computational Materials Science, Accepted (In press) (http://dx.doi.org/10.1016/j.commatsci.2014.01.040). (Impact Factor: 1.88) 5. Md. Sherajul Islam, Kenji Ushida, Satoru Tanaka, Akihiro Hashimoto,Numerical analysis on vacancy induced vibrational properties of graphene nanoribbons, Computational Materials Science, 79 (2013) 356–361. (Impact Factor: 1.88) 6. Md. Sherajul Islam, Kenji Ushida, Satoru Tanaka, Akihiro Hashimoto, Numerical experiments on phonon properties of isotope and vacancy-type disordered graphene, Diamond and Related Materials 40 (2013) 115–122. (Impact Factor: 1.75) 7. Md. Sherajul Islam, Md. Tawabur Rahman, Ashraful. G. Bhuiyan, Akihiro Hashimoto, Numerical Analysis on Phonon localization of vacancy Type Disordered Graphene, Journal of Circuits, Systems, and Computers, Accepted (In press)

1.5

Synopsis of Dissertation

Chapter 2 gives an overview of the basic concepts for graphene and h-BN, in order to establish a basis for discussing the obtained results in the later chapters. After introducing the different types of carbon nanostructures, emphasis is laid on the electronic and vibrational properties of graphene and their influences on the Raman spectra, followed by a description of the fundamentals of h-BN. Chapter 3 introduces the used computational technique, which allow to calculate the very large and complex disordered systems. After the motivation for the use of the forced vibrational method, its details simulation procedures and validity will be explained. In chapter 4, the vibrational properties of isotope disordered graphene with the aid of the forced vibrational method is presented in detail. This is complemented by the phonon density of states, typical mode patterns and phonon localization length of combined isotope and vacancy-type disordered graphene.


13

Chapter 5 addresses the vibrational properties of B- and N-doped graphene with native vacancies. Subsequently, doping dependent phonon density of states, typical mode patterns, and phonon localization lengths are described, whose aim was to an insightful understanding of how the doping effects combined with vacancies influences the phonon properties of graphene. Chapter 6 focuses on the vacancy-type defects effects on the vibrational properties of GNR, with emphasis of their width and defects effects on the temperature dependent specific heat capacity. Following the structural and the vibrational characterization of graphene or GNRs in the preceding chapters, the topic of Chapter 7 is the developing an atomic scale edge structures of graphene nanostructures. This involves the use of polarized microscopic laser Raman scattering spectroscopy accompanied with simulation models to systematically analyze the details of the edge structures of the transferred epitaxial graphene. Chapter 8 deals with the phonon properties of h-BCN network prepared with random and regular distribution of carbon atoms. Here, the main emphasis is on the determination of the effects of various types of carbon distribution in the h-BN sheet on the vibrational properties. Finally, chapter 9 provides a summary of the most relevant results, combined with an outlook on possible future research directions.

Chapter 2

Fundamentals of Graphene and h-BN 2.1

Introduction

This chapter is intended to provide an introduction and overview of the fundamental concepts of graphene and BN. The basic physical quantities such as the electron energy band diagram, phonon dispersion relations and Raman spectra of graphene and BN have been derived that will be useful to understand the results in the later chapters.

2.2

The sp2 -hybridized Carbon Nanomaterials

Carbon is a group IV element that is very active in producing many molecular compound and crystalline solids. A carbon atom in its ground state has six electrons with the electronic structure 1s2 2s2 2p2 . Generally, four valence electrons in carbon tend to interact with each other to produce the various types of allotrope. When carbon atoms come together to form a crystals, one of the 2s electrons is excited to the 2pz orbital gaining energy from the neighboring nuclei, which has net effect of lowering the overall energy of the system. Interactions or bonding subsequently follow between the 2s and 2p orbitals of neighboring carbon atoms. These interactions or mixing of atomic orbitals is commonly called the hybridization, and the new orbitals that are formed are referred to as hybrid orbitals. Figure 2.1 and 2.2 illustrate the formation of the covalent bond through the hybridization between the carbon atoms in graphene. The sp2 -hybridization of the double occupied 2s orbital and two 2p orbitals (e.g. px and py ) leads to a trigonal planar structure, with an in-plane σ bond between two neighboring carbon atoms. 15

Chapter 2. Fundamentals of Graphene and h-BN

16

Additionally, the singly occupied pz orbital of a carbon atom which is perpendicular to the planar structure binds with the pz orbitals of neighboring carbon atoms leading to the formation of a delocalized π band, which is half filled [127]. This half filled band is the basis for the semi-metallic properties of graphene. The existence of multiple flavors of hybridization in carbon is what leads to the different allotropes shown in Table 2.1. Graphene, a layer of carbon atoms arranged into a two-dimensional hexagonal lattice, is a single layer of graphite (see Fig. 2.3a

A. Electronic structure of a carbon atom in ground state Y

Z

X

1s (2 electrons)

2s 2px (2 electrons) (1 electron)

2py (1 electron)

B. Promotion of one electron from the 2s to 2pz orbital

1s (2 electrons)

2s 2px (2 electrons) (1 electron)

2py (1 electron)

2pz (1 electron)

C. Mixing of the 2s, 2px and 2py orbitals forming three hybridized sp2 orbitals

1s sp2 (2 electrons) (3 electrons)

2pz (1 electron)

D. Sigma and Pi bond between two carbon atoms

Figure 2.1: The formation of sigma and pi bonds thorough the hybridization between two carbon atoms [128].


17

Figure 2.2: Honey comb lattice structure of carbon atoms generating graphene. Table 2.1: Allotropes of sp2 bonded carbon. Dimension

0D

1D

2D

3D

Allotrope Structure Hybridization Electronic properties

C60 bucky ball Spherical sp2 Semiconductor

Carbon nanotube Cylindrical sp2 Metal or Semiconductor

graphene Planar sp2 Semi-metal

graphite Stacked planar sp2 Metal

and b) [1]. Carbon nanotubes (CNTs) represent rolled-up cylinders of graphene (see Fig. 2.3c) [129–131]. Fullerene [132] molecules (e.g. C60 ) are obtained by wrapping graphene, combined with the introduction of pentagons into the hexagonal lattice (see Fig. 2.3d).

2.3 2.3.1

Electronic Properties of Graphene Tight Binding Approximation

The hexagonal graphene lattice can be seen as a triangular lattice with a basis of two carbon atoms per unit cell (highlighted by blue (A) and red (B) color in Fig. 2.4). The carbon atoms in mono-layer graphene are located at the vertices of the hexagons, where a1 and a2 are unit vectors. The real space unit vectors a1 and a2 of the hexagonal lattice are expressed in Cartesian coordinates as:


18

3D Graphite

2D Graphene

(a)

(b) 0D Fullerence

1D Carbon Nanotube

(d)

(c)

Figure 2.3: (a) Mono-layer 2D graphene. (b) Three layer graphene or graphite. (c) Carbon nanotubes are rolled-up cylinders of graphene. (d) Fullerenes (e.g. C60 ) are molecules consisting of wrapped graphene (Ref. [39]).

a1 =

a √ 3, 3 , 2

a2 =

√ a 3, − 3 2

(2.1)

◦

where a = 1.42A is the carbon-carbon distance. Likewise the unit cell in reciprocal space is described by the unit vectors b1 and b2 of the reciprocal lattice given by:

b1 =

√ 2π 2π √ 1, 3 , b2 = 1, − 3 3a 3a

(2.2)

The unit vectors of b1 and b2 of the hexagonal lattice are rotated by 30 ◦ from the unit vectors a1 and a2 in the real space, respectively. The three high symmetry points in the Brillouin zone, Γ, K and M are the center, the corner, and the center of the edge of the hexagon, respectively. The three nearest-neighbor vectors in the real space are given by:

a √ 1, 3 , 2 √ a δ2 = 1, − 3 , 2 δ3 = a (−1, 0) δ1 =

(2.3)


19

Figure 2.4: Unit cell of honeycomb lattice and its Brillouin zone. (left) Lattice structure of graphene is composed of sub-lattices A and B with lattice vectors a1 and a2 . (right) The Brillouin zone of graphene.

The tight binding Hamiltonian for electrons in graphene can be written as: H=−

~2 2 ∇ + V (r) 2m

(2.4)

In the 2D graphene case, the electron wave function can be written as:

|ψk i = CkA |ψkA i + CkB |ψkB i

(2.5)

where CkA and CkB are complex functions of the quasi-momentum k. Both |ψkA i and |ψkB i are Bloch functions with 1 X ik.ri |ψkα i = √ e φ(r − ri ) N ri ∈α

(α = A, B)

(2.6)

Therefore, the Hamiltonian matrix for electrons can be written as: hψk |H|ψk i = Ek hψk |ψk i

(2.7)


Ek =

=

20

hψk |H|ψk i hψk |ψk i P α β∗ Ck Ck hψk |H|ψk i α,β

P α,β

∗

Ckα Ckβ hψk |ψk i

(2.8)

∗

For a given k value, the coefficient Ckβ is optimized so as to minimize Ek P ∂Ek ∗ ∂Ckβ

=

α,α0 ,β 0

∗

0

0

0

Ckα Ckβ hψkβ |ψkα i[hψkβ |H|ψkα i − EK hψkβ |ψkα i]Ckα [

P

α,β

∗

Ckα Ckβ hψkβ |ψkα i]2

=0

(2.9)

The secular equation of the system can be written as: det[H − ES] = 0

(2.10)

where Hαβ (k) = hψkA |H|ψkB i and Sαβ (k) = hψkA |ψkB i. The Hamiltonian H can be presented as:

H=

HAA HAB

!

HBA HBB

(2.11)

The interaction between A and A atoms become: HAA (k) = hψkA |H|ψkA i 1 X ik.(rA −r0 ) A hφ(r − r 0 )|H|φ(r − r )i = e A A N 0 rA ,rA

= ε2p

(2.12)

where ε2p = hφ(r)|H|φ(r)i, and interaction between A and B atoms become HAB (k) = hψkA |H|ψkB i 1 X ik.(rB −rA ) = e hφ(r − rA )|H|φ(r − rB )i N r ,r A

= tf (k)

B

(2.13)

Chapter 2. Fundamentals of Graphene and h-BN The value of f (k) is

√

f (k) = eikx a/

3

21

√

+ 2e−ikx a/2

3

cos

a 2

(2.14)

HBA and HAB are same and HAA and HBB are same, Sij = hφA |φB i and SAA = SBB = ∗ . Therefore, the value of H can be written as: 1, SAB = sf (k) = SBA

H=

ε2p tf (k)∗

! tf (k)

S=

ε2p

! sf (k)

1 sf (k)∗

1

(2.15)

The total energy of the system can be written as:

ε2p tf (k)∗

! ! tf (k) CkA ε2p

CkB

1

=E

! ! sf (k) CkA

sf (k)∗

1

CkB

(2.16)

The secular equation det[H − ES] = 0, can be written as: ! f (k)(t − sE)

ε2p − E f (k)∗ (t − sE)

Ek =

ε2p − E

=0

ε2p ± tw(k) 1 ± sw(k)

(2.17)

(2.18)

where the value of w(k) is √

s q w(k) = |f (k)2 | =

1 + 4 cos

ky a ky a 3kx a cos + 4 cos2 2 2 2

(2.19)

In Fig. 2.5, the electronic energy dispersion relations for the π-bands of monolayer graphene are shown throughout the two-dimensional first Brillouin zone and the inset shows the energy dispersion relations along the high symmetry axes using the parameters ε2p = 0, t = −3.033 eV, and s = 0.129 [39]. The upper half of the energy dispersion curves describes the π ∗ -energy ”antibonding” band, and the lower half is the π-energy bonding band. The upper π ∗ -band and the lower π-band are degenerate at the K(K 0 ) point through which the Fermi energy passes for an undoped mono-layer graphene sample. Finally, since valence (lower) and conduction (upper) band touch at only six points(K, K 0 and the remaining four equivalent vertex points), that are located precisely at the Fermi level, the intrinsic graphene is an unusual zero gap semiconductor.


22

Figure 2.5: The energy dispersion relations for graphene are shown through the whole region of the Brillouin zone [34]. The lower and the upper surfaces denote the valence π and the conduction π ∗ energy bands, respectively. The √ coordinates of high symmetry points are Γ = (0, 0), K = (0, 2π/3a), and M = (2π/ 3a, 0). The energy values at the K, M, and Γ points are 0, t, and 3t, respectively.

2.4 2.4.1

Vibrational Properties The Force Constant Model

The vibrational properties of graphene can be calculated within a force constant model [34] or by tight binding [133] or ab initio [96] method. In the force constant model, interactions including as many nearest neighbors in the graphene can be considered in order to improve the agreement with experiment. The phonon dispersion relations for monolayer graphene can be measured along high symmetry directions in the Brillouin zone by electron energy loss spectroscopy [114], inelastic neutron scattering [126], and inelastic X-ray scattering [103, 116], The force constants up to the fourth nearest-neighbor were fitted to the phonon frequencies measured by inelastic neutron scattering in graphite [126]. Furthermore, the force constants model using up to 20 nearest neighbor terms were fitted to inelastic X-ray scattering data [134]. In general, the equations of motion for the displacement of the ith atom measured from the equivalent position, ui = (xi , yi , zi ) for N atom in the unit cell is given by:

Mi u¨i =

X j

K (i,j) (uj − ui ), (i = 1, ....., N )

(2.20)


23

where Mi is the mass of the ith atom and K (i,j) represents the 3 × 3 force constant tensor between the ith and the jth atoms. The sum over j in Eq. 2.20 is normally taken over only a few neighbor distances relative to its site, which for a 2D graphene sheet has been carried out up to fourth nearest neighbor interactions [126]. In a periodic system we can perform a Fourier transform of the displacement of the ith atom with the wave (i)

number k 0 to obtain the normal mode displacement uk ui = √

1 X −i(q0 .Ri −ωt) (i) e uq0 NΩ q0

(2.21)

1 X i(q.Ri −ωt) e ui NΩ R

(2.22)

or uq( i) = √

i

in which the sum is taken over all (NΩ ) wave vectors q 0 in the first Brillouin zone and Ri denotes the atomic position of the ith atom in the crystal. When we assume the same eigenfrequencies ω for all ui , that is u¨i = −ω 2 ui , then Eq. 2.20 can formally written by defining a 3N × 3N dynamical matrix D(q) D(q)uq = 0

(2.23)

To obtain the eigenvalues ω 2 (q) for D(q) and nontrivial eigenvectors uq 6= 0, we solve the secular equation detD(q) = 0 for a given q vector. It is convenient to divide the dynamical matrix D(q) into small 3 × 3 matrices Di,j (q), (i, j = 1, ...., N ), where we denote D(q) by {D(i,j) (q)}, and from Eq. 2.23 it follows that D(i,j) (q) is expressed as:   X X 00 0 D(ij) (q) =  K (ij ) − Mi ω 2 (q)I  δij − K (ij ) eiq.∆Rij 0 j 00

(2.24)

j0

in which I is a 3 × 3 unit matrix and ∆Rij = Ri − Rj is the relative coordinates of the ith atom with respect to the jth atom. The vibration of the ith atom is coupled to that of the jth atom through the K ij force constant tensor. In graphene, since there are two distinct carbon atoms A and B in the unit cell, we must consider six coordinates uk (or 6 degrees of freedom) in Eq. 2.24. The secular equation to be solved is thus a 6 × 6 dynamical matrix D. The dynamical matrix D for graphene is written in terms of the 3 × 3 matrices: (1)DAA , (2)DAB , (3)DBA , and (4)DBB for the coupling between (1) A and A, (2) A and B, (3) B and A and several (4) B and B atoms in the various unit cells.

D=

DAA DAB DBA DBB

! (2.25)


24

Figure 2.6: Neighbor atoms of a graphitic plane up to 4th nearest neighbors for an A atom and the B atom at the center denoted by green and red circles. From the 1st to the 4th neighbor atoms, we plot solid and dotted lines for A and B atoms, respectively.

When we consider an A atom, the three nearest neighbor atoms (as shown in Figs. 2.6 and 2.7) are B1 , B2 , and B3 whose contributions to D are contained in DAB , while the six nearest neighbor atoms contributions to D that are contained in DAA and so on. The remaining problem is how to construct the force constant tensor K ij . First the force constant between an A atom and a nearest neighbor B1 atom on the x axis as shown in Fig. 2.7 has been considered. The force constant tensor is given by:

K (A,B1 )

(1)

(1)

  (1) φr 0 0   (1) = 0   0 φti  (1) 0 0 φto

(2.26)

(1)

where φr , φti , and φto represent the force constant parameters in the radial (bondstretching), in plane and out of plane tangential (bond bending direction) of the nearest neighbors, respectively. The force constant matrices for the two other nearest neighbor atoms , B2 and B3 are obtained by rotating the matrix in Eq. 2.26 −1 A,B1 K (A,Bm ) = Um K Um

(m = 2, 3)

(2.27)

where the unitary matrix Um is here defined by a rotating matrix around the z axis in Fig. 2.7, taking the B1 atom into the Bm atom,


25

Figure 2.7: Force constants between the A and B1 atoms on a graphene sheet. Here φr , φti , and φto represent forces for the nearest-neighbor atoms in the radial (bondstretching), in-plane and out-of-plane tangential (bond-bending) directions,respectively. B2 and B3 are nearest neighbors equivalent to B1 , whose force constant tensors are obtained by appropriately rotating the tensor for A and B1 [34].



cos θm

sin θm

 Um =  − sin θm cos θm 0 0

 0  0  1

(2.28)

G D

Figure 2.8: Phonon dispersion relation (left) and density of states (right) of 2D graphene calculated by FC model [126].


26

Figure 2.9: A single graphene sheet. The open and solid dots indicate the A and B sub-lattices, respectively. The arrows show directions of the normal mode displacements for the six phonon modes of the graphene sheet at the Γ point.The dotted and crossed points in (c) and (f) represent the vectors pointing in and out of the image plane (Ref. [39]).

√ To make the method explicit, the force constant matrix for the B2 atom at [−a/(2 3), a/2, 0], and U2 is evaluated assuming θ2 = 2π/3 

K (A,B2 )

(1)

(1)

φr + 3φ √ (1) ti(1) 1 =  3(φti − φr ) 4 0

√

(1)

(1)

3(φti − φr ) (1)

(1)

3φr + φti 0

0



 0   (1) φto

(2.29)

√ and the corresponding phase factor is given by exp[−iqx a/(2 3) + iqy a/2]. Figure 2.8 shows the calculated phonon dispersion curves along with the density of states for a mono-layer graphene sheet using the force constant (FC) model [126]. The modes of the phonons are label as the in-plane and out-of-plane, the longitudinal and transverse, and the acoustic and optical (iLA, iTA, oTA, iLO, iTO, oTO) modes, according to the directions of the atomic displacements within the unit cell. The atomic motion of the normal modes for the Γ and K points are shown in Figs. 2.9 and 2.10a to f.


27

Figure 2.10: A single graphene sheet. The open and solid dots indicate the A and B sub-lattices, respectively. The arrows show directions of the atomic displacements for the six stationary phonon modes of the graphene sheet at the K point.The labels of the phonon modes are explained in the text. The dotted and crossed points in (c) and (f) represent the vectors pointing in and out of the image plane. The large √ and small √ points in (c) and (f) indicate the magnitudes of the vectors equal to 2 and 1/ 2, respectively, of the magnitudes of the vectors in (a), (b), (d), and (e) (Ref. [39]).

2.5

Raman Spectroscopy

Raman spectroscopy has proven to be a powerful technique for probing selected phonons in graphene. It is also useful for identifying the number of layers, detecting the density of defects, determining local doping levels, studying electron-phonon coupling and thus the electronic properties themselves [93, 124, 135–137]. The Raman spectrum of graphene is characterized by four major peaks, namely the D, G, D’, and 2D peak (see the Fig. 2.11). The G peak position (1580 cm−1 ) is independent of the excitation wavelength. It corresponds to the doubly degenerate optical phonon of E2g symmetry (in-plane transverse optical plus longitudinal optical mode) at the Brillouin zone center (see the red circle at the Γ in the Fig. 2.8 [87, 135]). The E2g phonon involves in-plane bond stretching of all pairs of carbon atoms (see the inset of Fig. 2.11). The Raman G peak measurement is a three step process in graphene (Fig. 2.12a): (i) photon absorption leads to the excitation of an electron/hole pair, (ii) relaxation of the electron (or the hole) via emission of a G phonon, and (iii) electron/hole recombination emits a red-shifted photon. As the electron is excited to a real state (the conduction band) rather than a virtual state, one speaks of a resonant Raman process. The D peak (A01 symmetry) is a ring-breathing

Chapter 2. Fundamentals of Graphene and h-BN E2g G Mode

28 A'1 D Mode

Figure 2.11: Typical Raman spectra of disordered graphene. Inset shows the atomic displacements of the G and D mode.

mode (in-plane transverse optic phonon, see the inset of Fig. 2.11), which originates from the Brillouin zone border (K < q < M ). In order to be Raman active, a defect is required to reach a total momentum of approximately zero, as required for Raman scattering [87, 88]. Reference [89] identified double resonance (DR) as the scattering mechanism. DR Raman scattering is a four step process (Fig. 2.12b), comprising (i) photon absorption, (ii) elastic defect scattering, (iii) inelastic electron-phonon scattering, and (iv) electron/hole recombination plus photon emission. The DR condition is reached when the energy is conserved in all these steps. Thus DR links the phonon wave vector to the electronic band structure. Accordingly, the D peak position varies with excitation energy (1320 cm−1 for 633 nm, 1350 cm−1 for 488 nm). For higher excitation energies a larger momentum transfer is needed. Due to the dispersion relation of the iTO phonon, a D phonon with larger momentum has a higher energy (see the red circle in the K point of Fig. 2.8), resulting in a larger redshift for higher excitation energies.

2.6

The sp2 -hybridized BN Nanomaterials

Boron nitride (BN), a III-V compound, is similar to carbon in that it exists in both hexagonal- and cubic-bonded structures. Structural similarities between carbon and BN based materials are well known and supported, for instance, by the theoretical prediction [139] and experimental realization [140] of BN nanotubes. The hexagonal boron nitride (h-BN) phase (sometimes called white graphite) is isostructural to graphite except for the different stacking sequences of the atomic planes: well-crystallized graphite displays


(a) G mode

29

(c) D' mode

(b) D mode

Defect -

e

iTO + LO phonon eK

iTO phonon K

e-

Defect LO phonon

K'

E h+

h+

h

+

K

Figure 2.12: Raman scattering process for (a) the G peak (in-plane transverse optic plus longitudinal optic mode). (b) One-phonon double resonance process for the D peak (intervalley scattering) and (c) the D’ peak (intravalley scattering) [124, 138].

Figure 2.13: (a) The two-atom unit cell of h-BN (in the red box). (b) Corresponding reciprocal lattice and the first Brillouin zone. (c) The band structure of mono-layer h-BN (Ref. [141]).

the Bernal (AB) stacking sequence, while hexagonal boron nitride is stacked with boron on top of nitrogen and vice versa (AAA... stacking). h-BN has recently found to be a promising deep ultraviolet (DUV) light emitter [30] and a new route of synthesis has found to produce high quality crystals [143]. Boron and nitrogen atoms, analogous to carbon atoms in graphite, form in h-BN a 2D honeycomb structure with strong covalent bonds in the plane and weak bonds between different planes, slightly ionic in h-BN. Although the BN and carbon-based materials have very similar crystal structure (aBN =0.250 nm and aC =0.246 nm; cBN =0.666 nm and cC =0.674 nm), their electronic properties are very different, as h-BN is an insulator with a direct energy gap of about 5.9 eV [30, 144], while graphite is a semi-metal. The two-atom unit cell of h-BN with the corresponding reciprocal lattice and the first


30

Figure 2.14: The allotropes of h-BN. 2D Nano sheet of h-BN with armchair and zigzag edges, rolled up cylinders of h-BN sheet forming 1D BNNTs, and 0D bucky ball structure by wrapping h-BN, combined with the introduction of pentagons into the hexagonal lattice (Ref. [142]).

Brillouin zone are shown in Fig. 2.13a and 2.13b, respectively. The band structure of single layer h-BN is displayed in Fig. 2.13c. Analogous to graphene, various types of allotropes of h-BN may also exist [142]. Hexagonal BN nanoshhet, a layer of BN atoms arranged into a two-dimensional hexagonal lattice (Fig. 2.14). Boron Nitride Nanotubes (BNNTs) represent rolled-up cylinders of h-BN (Fig. 2.14 ). Fullerene molecules are obtained by wrapping h-BN, combined with the introduction of pentagons into the hexagonal lattice (Fig. 2.14) [142]. Like graphene, the vibrational properties of 2D h-BN can be obtained by the force constant

Figure 2.15: The phonon dispersion relation of 2D h-BN using the fourth nearest neighbor force constant method (Ref.[105]).


31

method. The calculated phonon dispersion relation using the fourth nearest neighbor force constant method [105] along with experimental results is shown in Fig. 2.15.

Chapter 3

Forced Vibrational Method 3.1

Introduction

The first part of this chapter outlines the numerical algorithm used in the calculation of the later chapters. In the second part, the accuracy of the calculated eigenmodes has been justified by comparing the eigenmodes calculated by the dynamical matrix techniques along with the experiments.

3.2

Normal Modes of Vibrations

Let us consider that a set of N atoms are coupled together by linear springs. The equation of motion of the systems with the scalar displacement of the l th mass, ul (t) is: Ml u¨l +

X

φll0 ul0 (t) = 0

(3.1)

l0

where Ml is the mass of lth atom and φll0 is the strength of the spring between the lth and l0 th atoms. The displacement can be decomposed as: ul (t) =

X λ

el (λ) Qλ (t) √ Ml

(3.2)

where Qλ is the amplitude of the normal mode λ and el (λ) is the displacement pattern or the polarization vector of the mode λ. By putting this value into equaton 3.1

d2 Ml 2 dt

X λ

el (λ) Qλ (t) √ Ml

! +

X l0

33

φll0

X λ

el0 (λ) Qλ (t) √ =0 Ml 0

(3.3)

Chapter 3. Forced Vibrational Method

X

Ml

λ

Dividing this equation by

X

¨ λ (t) e√l (λ) Q Ml

√

34

! +

X

φll0

l0

X λ

el0 (λ) =0 Qλ (t) √ Ml 0

(3.4)

Ml we have:

¨ λ (t)el (λ) + Q

X l0

λ

φll0

X λ

el0 (λ) =0 Qλ (t) √ Ml M l 0

(3.5)

If the amplitude of the normal mode varies as Qλ (t) = Aλ exp(iωλ t) + Bλ exp(−iωλ t), after two times differentiation the equation of motion becomes: −ωλ2

X

(Aλ exp(iωλ t) + Bλ exp(−iωλ t))el (λ) +

X l0

λ

φll0

X

(Aλ exp(iωλ t)

λ

el0 (λ) =0 +Bλ exp(−iωλ t)) √ M l Ml 0 Dividing the Eq. (3.6) by

P λ

(3.6)

(Aλ exp(iωλ t) + Bλ exp(−iωλ t))el (λ), we obtain

X l0

el0 (λ) φll0 √ = ωλ2 el (λ) 0 Ml M l

(3.7)

Thus, to find the frequencies and displacement patterns of the normal modes one has to find the eigenvalues and eigenvectors of an N × N matrix. However, the conventional dynamical matrix techniques require a large amount of computer times and huge memory space. These techniques are also limited for perfect system only. Thus, one has to find out another approach.

3.3

Forced Vibrational Method

The Forced vibrational (FV) method employed in the present work, introduced by Williams and Maris [121], enables us to treat the eigenvalue problem of very large and complex systems. The fundamental idea is based on the physical analogy that a complex mechanical system when driven by a periodic external force of frequency Ω will respond with a large amplitude in those eigenmodes close to this frequency. Because this algorithm requires a memory space of the order of N , one can calculate eigenfrequencies and eigenmodes of very large system.

Chapter 3. Forced Vibrational Method

3.3.1

35

Method for the Density of States

The density of states can be found by the following procedure. The initial set at t = 0 is prepared in which all atoms are at rest and have zero displacements. A random force is applied to each atom at rest, which is given by:

Fl cos(Ωt) = F0

p Ml cos(φl )cos(Ωt)

(3.8)

Where F0 is a constant and φl varies from 0 to 2π. The corresponding equation of motion of the system becomes: Ml u¨l +

X

φll0 ul0 (t) = F0

p

Ml cos(φl )cos(Ωt)

(3.9)

l0

The total energy of the system can be written as the summation of kinetic and potential energy as follows: E(t) = K + U 1 XX 1X = Ml u˙ 2l (t) + ul (t)φll0 u0l (t) 2 2 l l l0 X X X 1 2 0 0 ul (t)φll ul (t) = Ml u˙ l (t) + 2 0 l

l

(3.10)

l

Thus, for large times the periodic external force excites only those modes whose frequency is close to Ω. When we average all possible values of φl and use the orthonormality of the eigenvectors el (λ), the average value of energy hEi becomes (the detailed derivation can be seen in appendix A)

hEi =

F0 2 X sin2 {(Ω − ωλ )/2 t} 4 (Ω − ωλ )2

(3.11)

λ

The modes which contribute to the sum in Eq. 3.11 are those whose frequencies lie within about ±2π/t of Ω as shown in Fig. 3.1. If we choose t such that

1 4πN 0 & M(i,j-1)>0; if M(i-1,j) > M(i,j-1); M(i,j)=M(i,j-1); elseif M(i-1,j) 0; M(i,j)=M(i,j-1); end end if mod(j,2)==1 & mod(i,3)==1; if M(i-1,j-1)==0; M(i,j)=k; k=k+1; elseif M(i-1,j-1)>0; M(i,j)=M(i-1,j-1); end end if mod(j,2)==0 & mod(i,3)==0; if M(i-1,j)==0;

120

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns M(i,j)=k; k=k+1; elseif M(i-1,j)>0; M(i,j)=M(i-1,j); end end end end end RM=zeros(n,n); RRM=zeros(n,n); Mmin=zeros(n,n); L=true; while L RM=M; for i=2:n-1; for j=2:n-1; if M(i,j)>0; if mod(j,2)==1 & mod(i,3)==2; Mmin(i,j)=RM(i,j); if (M(i-1,j)>0) & (RM(i-1,j)0) & (RM(i,j-1)0) & (RM(i,j+1)0; RRM(i-1,j)=Mmin(i,j); end if M(i,j-1)>0; RRM(i,j-1)=Mmin(i,j); end

121

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns if M(i,j+1)>0; RRM(i,j+1)=Mmin(i,j); end end if mod(j,2)==0 & mod(i,3)==2; Mmin(i,j)=RM(i,j); if (M(i,j-1)>0) & (RM(i,j-1)0) & (RM(i+1,j)0) & (RM(i,j+1)0); RRM(i,j-1)=Mmin(i,j); end if (M(i+1,j)>0); RRM(i+1,j)=Mmin(i,j); end if (M(i,j+1)>0); RRM(i,j+1)=Mmin(i,j); end end if mod(j,2)==1 & mod(i,3)==1; Mmin(i,j)=RM(i,j); if (M(i-1,j-1)>0) & (RM(i-1,j-1)0) & (RM(i+1,j)0) & (RM(i-1,j+1)0);

122

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns RRM(i-1,j-1)=Mmin(i,j); end if (M(i+1,j)>0); RRM(i+1,j)=Mmin(i,j); end if (M(i-1,j+1)>0); RRM(i-1,j+1)=Mmin(i,j); end end if mod(j,2)==0 & mod(i,3)==0; Mmin(i,j)=RM(i,j); if (M(i+1,j-1)>0) & (RM(i+1,j-1)0) & (RM(i-1,j)0) & (RM(i+1,j+1)0); RRM(i+1,j-1)=Mmin(i,j); end if (M(i-1,j)>0); RRM(i-1,j)=Mmin(i,j); end if (M(i+1,j+1)>0); RRM(i+1,j+1)=Mmin(i,j); end end end end end L=0; for i=1:n; for j=1:n;

123

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns while M(i,j)>RRM(i,j); M(i,j)=RRM(i,j); L=L+1; end end end L>0; end sizemax=0; totalmax=0; for k=1:n*n; total=0; for i=1:n; for j=1:n; if M(i,j)==k; total=total+1; end end end if total>totalmax; totalmax=total; sizemax=k; end end for i=1:n; for j=1:n; if M(i,j)~=sizemax; M(i,j)=0; end end end for i=1:n; for j=1:n;

124

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns if M(i,j)~=0; M(i,j)=1; end end end A = sqrt(3) / 2; [X Y] = meshgrid(0:1:n); N = size(X,1); X = A * X; Y = Y + repmat([0 0.5],[N,N/2]); for i=1:n+1; for j=1:n+1; if M(i,j)==0; X(i,j) =

0;

Y(i,j) =

0;

end end end for i=2:n-1; for j=2:n-1; if mod(j,2)==1 & mod(i,3)==2; if M(i,j)==1 & M(i,j-1)==0; M(i,j-1)=2; end if M(i,j)==1 & M(i,j+1)==0; M(i,j+1)=2; end if M(i,j)==1 & M(i-1,j)==0; M(i-1,j)=2; end end end end

125

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns for i=2:n-1; for j=2:n-1; if mod(j,2)==0 & mod(i,3)==2; if M(i,j)==1 & M(i,j-1)==0; M(i,j-1)=2; end if M(i,j)==1 & M(i,j+1)==0; M(i,j+1)=2; end if M(i,j)==1 & M(i+1,j)==0; M(i+1,j)=2; end end end end for i=2:n-1; for j=2:n-1; if mod(j,2)==1 & mod(i,3)==1; if M(i,j)==1 & M(i-1,j-1)==0; M(i-1,j-1)=2; end if M(i,j)==1 & M(i-1,j+1)==0; M(i-1,j+1)=2; end if M(i,j)==1 & M(i+1,j)==0; M(i+1,j)=2; end end end end for i=2:n-1; for j=2:n-1; if mod(j,2)==0 & mod(i,3)==0; if M(i,j)==1 & M(i+1,j-1)==0; M(i+1,j-1)=2; end if M(i,j)==1 & M(i+1,j+1)==0;

126

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns

127

M(i+1,j+1)=2; end if M(i,j)==1 & M(i-1,j)==0; M(i-1,j)=2; end end end end LN=0; for i=1:n; for j=1:n; if M(i,j)==1; LN=LN+1; end end end save(’gra_per_10500_9_exact’, ’M’, ’LN’, ’X’, ’Y’) plot(X,Y,’k.’,’MarkerSize’,5) axis equal, axis([-1 150 -1 150]) toc;

B.2

Calculation of Phonon Density of States of Graphene

clc; clear; tic; name1=’gra_DOS_10000_WR_forth_9_exact.txt’; name2=’gra_DOS_10000_DOS_forth_9_exact.txt’; name3=’gra_DOS_10000_k_forth_9_exact.txt’; fid1 = fopen(name1,’a’); fid2 = fopen(name2,’a’); fid3 = fopen(name3,’a’); FORMAT = ’%g ’;

load(’gra_per_10500_9_exact’, ’M’, ’LN’, ’X’, ’Y’);

% loading lattice states


128

LX=126;

% X direction lattice point number

LY=126;

% Y direction lattice point number

%Force constants until forth nearest neighbor atoms taken from Jishi et al.[124] Kin=245; Kra=365; Kout=98.2; K2in=-32.3; K2ra=88; K2out=-4; K3in=-52.5; K3ra=30; K3out=1.5; K4in=22.9; K4ra=-19.2; K4out=-5.8; M1=1.993e-26;

% mass of carbon

F0=1; TOTAL_ATOM=LN Kin1=zeros(LY+8,LX+8); Kin2=zeros(LY+8,LX+8); Kin21=zeros(LY+8,LX+8); Kin22=zeros(LY+8,LX+8); Kin23=zeros(LY+8,LX+8); Kin31=zeros(LY+8,LX+8); Kin32=zeros(LY+8,LX+8); Kin411=zeros(LY+8,LX+8); Kin412=zeros(LY+8,LX+8); Kin413=zeros(LY+8,LX+8); Kin421=zeros(LY+8,LX+8); Kin422=zeros(LY+8,LX+8); Kin423=zeros(LY+8,LX+8); Kin431=zeros(LY+8,LX+8); Kin432=zeros(LY+8,LX+8); Kin433=zeros(LY+8,LX+8); Kin441=zeros(LY+8,LX+8); Kin442=zeros(LY+8,LX+8); Kin443=zeros(LY+8,LX+8);

% TOTAL ATOM

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns Kra1=zeros(LY+8,LX+8); Kra2=zeros(LY+8,LX+8); Kra21=zeros(LY+8,LX+8); Kra22=zeros(LY+8,LX+8); Kra23=zeros(LY+8,LX+8); Kra31=zeros(LY+8,LX+8); Kra32=zeros(LY+8,LX+8); Kra411=zeros(LY+8,LX+8); Kra412=zeros(LY+8,LX+8); Kra413=zeros(LY+8,LX+8); Kra421=zeros(LY+8,LX+8); Kra422=zeros(LY+8,LX+8); Kra423=zeros(LY+8,LX+8); Kra431=zeros(LY+8,LX+8); Kra432=zeros(LY+8,LX+8); Kra433=zeros(LY+8,LX+8); Kra441=zeros(LY+8,LX+8); Kra442=zeros(LY+8,LX+8); Kra443=zeros(LY+8,LX+8); Kout1=zeros(LY+8,LX+8); Kout2=zeros(LY+8,LX+8); Kout21=zeros(LY+8,LX+8); Kout22=zeros(LY+8,LX+8); Kout23=zeros(LY+8,LX+8); Kout31=zeros(LY+8,LX+8); Kout32=zeros(LY+8,LX+8); Kout411=zeros(LY+8,LX+8); Kout412=zeros(LY+8,LX+8); Kout413=zeros(LY+8,LX+8); Kout421=zeros(LY+8,LX+8); Kout422=zeros(LY+8,LX+8); Kout423=zeros(LY+8,LX+8); Kout431=zeros(LY+8,LX+8); Kout432=zeros(LY+8,LX+8); Kout433=zeros(LY+8,LX+8); Kout441=zeros(LY+8,LX+8); Kout442=zeros(LY+8,LX+8); Kout443=zeros(LY+8,LX+8);

129


130

for X=2:LX+8; for Y=2:LY+7; if M(Y,X)0 | M(Y-1,X-1)>0 | M(Y+1,X-1)>0 ; Kin2(Y,X)=Kin; Kra2(Y,X)=Kra; Kout2(Y,X)=Kout; else Kin2(Y,X)=0; Kra2(Y,X)=0; Kout2(Y,X)=0; end end end end

for X=1:LX+8; for Y=2:LY+8; if M(Y,X)0; Kin1(Y,X)=Kin; Kra1(Y,X)=Kra; Kout1(Y,X)=Kout; else Kin1(Y,X)=0; Kra1(Y,X)=0; Kout1(Y,X)=0; end

% spring constant % atom is not lattice point


131

end end end for X=3:LX+8; for Y=1:LY+8; if M(Y,X)0; Kin22(Y,X)=K2in; Kra22(Y,X)=K2ra; Kout22(Y,X)=K2out; else Kin22(Y,X)=0; Kra22(Y,X)=0; Kout22(Y,X)=0; end end end end for X=2:LX+8; for Y=1:LY+6; if M(Y,X)0 | M(Y+2,X-1)>0 ; Kin21(Y,X)=K2in; Kra21(Y,X)=K2ra; Kout21(Y,X)=K2out;

else Kin21(Y,X)=0; Kra21(Y,X)=0; Kout21(Y,X)=0;


132

end end end end for X=2:LX+8; for Y=3:LY+8; if M(Y,X)0 | M(Y-1,X-1)>0; Kin23(Y,X)=K2in; Kra23(Y,X)=K2ra; Kout23(Y,X)=K2out;

else Kin23(Y,X)=0; Kra23(Y,X)=0; Kout23(Y,X)=0; end end end end

for X=1:LX+8; for Y=3:LY+8; if M(Y,X)0; Kin31(Y,X)=K3in; Kra31(Y,X)=K3ra; Kout31(Y,X)=K3out; else Kin31(Y,X)=0;

% spring constant % atom does not exist lattice point


133

Kra31(Y,X)=0; Kout31(Y,X)=0; end end end end for X=3:LX+8; for Y=2:LY+7; if M(Y,X)0 | M(Y+1,X-2)>0; Kin32(Y,X)=K3in; Kra32(Y,X)=K3ra; Kout32(Y,X)=K3out;

else Kin32(Y,X)=0; Kra32(Y,X)=0; Kout32(Y,X)=0; end end end end for X=3:LX+8; for Y=1:LY+6;

% spring constant

if mod(X,2)==1 & mod(Y,3)==2; if M(Y,X)0 ; Kin411(Y,X)=K4in; Kra411(Y,X)=K4ra; Kout411(Y,X)=K4out;

% spring constant k=0


134

else Kin411(Y,X)=0; Kra411(Y,X)=0; Kout411(Y,X)=0; end end end end end for X=4:LX+8; for Y=1:LY+8;

% spring constant



% spring constant



% spring constant



% spring constant


136



% spring constant


else Kin423(Y,X)=0; Kra423(Y,X)=0; Kout423(Y,X)=0; end end end



137

end end for X=2:LX+8; for Y=1:LY+6;

% spring constant



% spring constant


else Kin432(Y,X)=0;



138

Kra432(Y,X)=0; Kout432(Y,X)=0; end end end end end for X=3:LX+8; for Y=3:LY+8;

% spring constant



% spring constant

if mod(X,2)==0 & mod(Y,3)==0; if M(Y,X)0 ;



139

Kin441(Y,X)=K4in; Kra441(Y,X)=K4ra; Kout441(Y,X)=K4out; else Kin441(Y,X)=0; Kra441(Y,X)=0; Kout441(Y,X)=0; end end end end end for X=4:LX+8; for Y=1:LY+7;

% spring constant



% spring constant


else Kin443(Y,X)=0; Kra443(Y,X)=0; Kout443(Y,X)=0; end end end end end TAU=0.01e-13; SQM=sqrt(M1); TM=TAU/M1; WS=3.51e14; P1=8; WE=5.7e14; WSTEP=5e12; DIV=5e12; %MAIN ROUTINE for W=WS:WSTEP:WE; %W=1e12; DN0=W/DIV; DNA=round(DN0); DN=DNA+1; ECST=2.0*pi*DN/(W*TAU); U011=zeros(LY+8,LX+8); U111=zeros(LY+8,LX+8);


140

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns VD11=zeros(LY+8,LX+8); V11=zeros(LY+8,LX+8); U012=zeros(LY+8,LX+8); U112=zeros(LY+8,LX+8); VD12=zeros(LY+8,LX+8); V12=zeros(LY+8,LX+8); U021=zeros(LY+8,LX+8); U121=zeros(LY+8,LX+8); VD21=zeros(LY+8,LX+8); V21=zeros(LY+8,LX+8); U022=zeros(LY+8,LX+8); U122=zeros(LY+8,LX+8); VD22=zeros(LY+8,LX+8); V22=zeros(LY+8,LX+8); U041=zeros(LY+8,LX+8); U141=zeros(LY+8,LX+8); VD41=zeros(LY+8,LX+8); V41=zeros(LY+8,LX+8); U042=zeros(LY+8,LX+8); U142=zeros(LY+8,LX+8); VD42=zeros(LY+8,LX+8); V42=zeros(LY+8,LX+8); U043=zeros(LY+8,LX+8); U143=zeros(LY+8,LX+8); VD43=zeros(LY+8,LX+8); V43=zeros(LY+8,LX+8); U044=zeros(LY+8,LX+8); U144=zeros(LY+8,LX+8); VD44=zeros(LY+8,LX+8); V44=zeros(LY+8,LX+8); U045=zeros(LY+8,LX+8); U145=zeros(LY+8,LX+8); VD45=zeros(LY+8,LX+8); V45=zeros(LY+8,LX+8); U046=zeros(LY+8,LX+8); U146=zeros(LY+8,LX+8); VD46=zeros(LY+8,LX+8); V46=zeros(LY+8,LX+8); U0411=zeros(LY+8,LX+8);

141

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns U1411=zeros(LY+8,LX+8); VD411=zeros(LY+8,LX+8); V411=zeros(LY+8,LX+8); U0421=zeros(LY+8,LX+8); U1421=zeros(LY+8,LX+8); VD421=zeros(LY+8,LX+8); V421=zeros(LY+8,LX+8); U0431=zeros(LY+8,LX+8); U1431=zeros(LY+8,LX+8); VD431=zeros(LY+8,LX+8); V431=zeros(LY+8,LX+8); U0441=zeros(LY+8,LX+8); U1441=zeros(LY+8,LX+8); VD441=zeros(LY+8,LX+8); V441=zeros(LY+8,LX+8); U0451=zeros(LY+8,LX+8); U1451=zeros(LY+8,LX+8); VD451=zeros(LY+8,LX+8); V451=zeros(LY+8,LX+8); U0461=zeros(LY+8,LX+8); U1461=zeros(LY+8,LX+8); VD461=zeros(LY+8,LX+8); V461=zeros(LY+8,LX+8); U0X=zeros(LY+8,LX+8); U1X=zeros(LY+8,LX+8); VDX=zeros(LY+8,LX+8); VX=zeros(LY+8,LX+8); U0Y=zeros(LY+8,LX+8); U1Y=zeros(LY+6,LX+8); VDY=zeros(LY+8,LX+8); VY=zeros(LY+8,LX+8); U0Z=zeros(LY+8,LX+8); U1Z=zeros(LY+8,LX+8); VDZ=zeros(LY+8,LX+8); VZ=zeros(LY+8,LX+8); FLX=zeros(LY+8,LX+8); FLY=zeros(LY+8,LX+8); FLZ=zeros(LY+8,LX+8);

142


143

for X=1:LX+8; for Y=1:LY+8; FLX(Y,X)=F0*SQM*cos(2.0*pi*rand(1)); FLY(Y,X)=F0*SQM*cos(2.0*pi*rand(1)); FLZ(Y,X)=F0*SQM*cos(2.0*pi*rand(1)); end end %caluculation of the displacement and velocity of each atom of the system %

N=10;

for N=0:ECST; for X=4:LX+5; for Y=4:LY+5; if M(Y,X)>0; VDX(Y,X)=FLX(Y,X)*cos(W*N*TAU); VDY(Y,X)=FLY(Y,X)*cos(W*N*TAU); VDZ(Y,X)=FLZ(Y,X)*cos(W*N*TAU); if mod(X,2)==1 & mod(Y,3)==2; VDX(Y,X)=VDX(Y,X)+Kra2(Y,X)*(U011(Y,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra2(Y,X+1)*(U021(Y,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kin2(Y,X)*(U012(Y,X-1)-U012(Y,X))*(-0.5)... +Kin2(Y,X+1)*(U022(Y,X+1)-U022(Y,X))*(0.5)... +Kin1(Y,X)*(U0X(Y-1,X)-U0X(Y,X))... +Kra21(Y,X)*(U022(Y+1,X-1)-U022(Y,X))*(0.5)... +Kin21(Y,X)*(U021(Y+1,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))... +Kra23(Y,X)*(U012(Y-2,X-1)-U012(Y,X))*(-0.5)... +Kin23(Y,X)*(U011(Y-2,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra21(Y-2,X+1)*(U022(Y-2,X+1)-U022(Y,X))*(0.5)... +Kin21(Y-2,X+1)*(U021(Y-2,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X+2)*(U0X(Y,X+2)-U0X(Y,X))... +Kra23(Y+1,X+1)*(U012(Y+1,X+1)-U012(Y,X))*(-0.5)... +Kin23(Y+1,X+1)*(U011(Y+1,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kra32(Y,X)*(U021(Y-1,X-2)-U021(Y,X))*(sqrt(3)/2)... +Kin32(Y,X)*(U022(Y-1,X-2)-U022(Y,X))*(0.5)... +Kra32(Y-1,X+2)*(U011(Y-1,X+2)-U011(Y,X))*(-sqrt(3)/2)... +Kin32(Y-1,X+2)*(U012(Y-1,X+2)-U012(Y,X))*(-0.5)... +Kin31(Y+2,X)*(U0X(Y+2,X)-U0X(Y,X))... +Kra411(Y,X)*(U041(Y+2,X-2)-U041(Y,X))*(-sqrt(3/7))...


144

+Kin411(Y,X)*(U0411(Y+2,X-2)-U0411(Y,X))*(sqrt(4/7))... +Kra412(Y,X)*(U042(Y,X-3)-U042(Y,X))*(-sqrt(27/28))... +Kin412(Y,X)*(U0421(Y,X-3)-U0421(Y,X))*(sqrt(1/28))... +Kra413(Y,X)*(U043(Y-3,X-1)-U043(Y,X))*(-sqrt(3/28))... +Kin413(Y,X)*(U0431(Y-3,X-1)-U0431(Y,X))*(sqrt(25/28))... +Kra433(Y+2,X+2)*(U044(Y+2,X+2)-U044(Y,X))*(sqrt(3/7))... +Kin433(Y+2,X+2)*(U0441(Y+2,X+2)-U0441(Y,X))*(sqrt(4/7))... +Kra422(Y,X+3)*(U045(Y,X+3)-U045(Y,X))*(sqrt(27/28))... +Kin422(Y,X+3)*(U0451(Y,X+3)-U0451(Y,X))*(sqrt(1/28))... +Kra421(Y-3,X+1)*(U046(Y-3,X+1)-U046(Y,X))*(sqrt(3/28))... +Kin421(Y-3,X+1)*(U0461(Y-3,X+1)-U0461(Y,X))*(sqrt(25/28)); VDY(Y,X)=VDY(Y,X)+Kra2(Y,X)*(U011(Y,X-1)-U011(Y,X))*(0.5)... +Kra2(Y,X+1)*(U021(Y,X+1)-U021(Y,X))*(0.5)... +Kin2(Y,X)*(U012(Y,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin2(Y,X+1)*(U022(Y,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kra1(Y,X)*(U0Y(Y-1,X)-U0Y(Y,X))... +Kra21(Y,X)*(U022(Y+1,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y,X)*(U021(Y+1,X-1)-U021(Y,X))*(0.5)... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))... +Kra23(Y,X)*(U012(Y-2,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y,X)*(U011(Y-2,X-1)-U011(Y,X))*(0.5)... +Kra21(Y-2,X+1)*(U022(Y-2,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y-2,X+1)*(U021(Y-2,X+1)-U021(Y,X))*(0.5)... +Kin22(Y,X+2)*(U0Y(Y,X+2)-U0Y(Y,X))... +Kra23(Y+1,X+1)*(U012(Y+1,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y+1,X+1)*(U011(Y+1,X+1)-U011(Y,X))*(0.5)... +Kra32(Y,X)*(U021(Y-1,X-2)-U021(Y,X))*(0.5)... +Kin32(Y,X)*(U022(Y-1,X-2)-U022(Y,X))*(-sqrt(3)/2)... +Kra32(Y-1,X+2)*(U011(Y-1,X+2)-U011(Y,X))*(0.5)... +Kin32(Y-1,X+2)*(U012(Y-1,X+2)-U012(Y,X))*(-sqrt(3)/2)... +Kra31(Y+2,X)*(U0Y(Y+2,X)-U0Y(Y,X))... +Kra411(Y,X)*(U041(Y+2,X-2)-U041(Y,X))*(sqrt(4/7))... +Kin411(Y,X)*(U0411(Y+2,X-2)-U0411(Y,X))*(sqrt(3/7))... +Kra412(Y,X)*(U042(Y,X-3)-U042(Y,X))*(sqrt(1/28))... +Kin412(Y,X)*(U0421(Y,X-3)-U0421(Y,X))*(sqrt(27/28))... +Kra413(Y,X)*(U043(Y-3,X-1)-U043(Y,X))*(-sqrt(25/28))... +Kin413(Y,X)*(U0431(Y-3,X-1)-U0431(Y,X))*(-sqrt(3/28))... +Kra433(Y+2,X+2)*(U044(Y+2,X+2)-U044(Y,X))*(sqrt(4/7))...


145

+Kin433(Y+2,X+2)*(U0441(Y+2,X+2)-U0441(Y,X))*(-sqrt(3/7))... +Kra422(Y,X+3)*(U045(Y,X+3)-U045(Y,X))*(sqrt(1/28))... +Kin422(Y,X+3)*(U0451(Y,X+3)-U0451(Y,X))*(-sqrt(27/28))... +Kra421(Y-3,X+1)*(U046(Y-3,X+1)-U046(Y,X))*(-sqrt(25/28))... +Kin421(Y-3,X+1)*(U0461(Y-3,X+1)-U0461(Y,X))*(sqrt(3/28));

VDZ(Y,X)=VDZ(Y,X)+Kout2(Y,X)*(U0Z(Y,X-1)-U0Z(Y,X))... +Kout2(Y,X+1)*(U0Z(Y,X+1)-U0Z(Y,X))... +Kout1(Y,X)*(U0Z(Y-1,X)-U0Z(Y,X))... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))... +Kout22(Y,X)*(U0Z(Y+1,X-1)-U0Z(Y,X))... +Kout23(Y,X)*(U0Z(Y-2,X-1)-U0Z(Y,X))... +Kout21(Y,X+2)*(U0Z(Y,X+2)-U0Z(Y,X))... +Kout22(Y+1,X+1)*(U0Z(Y+1,X+1)-U0Z(Y,X))... +Kout23(Y-2,X+1)*(U0Z(Y-2,X+1)-U0Z(Y,X))... +Kout32(Y,X)*(U0Z(Y-1,X-2)-U0Z(Y,X))... +Kout32(Y-1,X+2)*(U0Z(Y-1,X+2)-U0Z(Y,X))... +Kout31(Y+2,X)*(U0Z(Y+2,X)-U0Z(Y,X))... +Kout411(Y,X)*(U0Z(Y+2,X-2)-U0Z(Y,X))... +Kout412(Y,X)*(U0Z(Y,X-3)-U0Z(Y,X))... +Kout413(Y,X)*(U0Z(Y-3,X-1)-U0Z(Y,X))... +Kout433(Y+2,X+2)*(U0Z(Y+2,X+2)-U0Z(Y,X))... +Kout422(Y,X+3)*(U0Z(Y,X+3)-U0Z(Y,X))... +Kout421(Y-3,X+1)*(U0Z(Y-3,X+1)-U0Z(Y,X)); end if mod(X,2)==0 & mod(Y,3)==2; VDX(Y,X)=VDX(Y,X)+Kra2(Y,X)*(U021(Y,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra2(Y,X+1)*(U011(Y,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kin2(Y,X)*(U022(Y,X-1)-U022(Y,X))*(0.5)... +Kin2(Y,X+1)*(U012(Y,X+1)-U012(Y,X))*(-0.5)... +Kin1(Y+1,X)*(U0X(Y+1,X)-U0X(Y,X))... +Kra21(Y,X)*(U022(Y+2,X-1)-U022(Y,X))*(0.5)... +Kin21(Y,X)*(U021(Y+2,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))... +Kra23(Y,X)*(U012(Y-1,X-1)-U012(Y,X))*(-0.5)... +Kin23(Y,X)*(U011(Y-1,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra21(Y-1,X+1)*(U022(Y-1,X+1)-U022(Y,X))*(0.5)...


146

+Kin21(Y-1,X+1)*(U021(Y-1,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X+2)*(U0X(Y,X+2)-U0X(Y,X))... +Kra23(Y+2,X+1)*(U012(Y+2,X+1)-U012(Y,X))*(-0.5)... +Kin23(Y+2,X+1)*(U011(Y+2,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kra32(Y,X)*(U011(Y+1,X-2)-U011(Y,X))*(-sqrt(3)/2)... +Kin32(Y,X)*(U012(Y+1,X-2)-U012(Y,X))*(-0.5)... +Kra32(Y+1,X+2)*(U021(Y+1,X+2)-U021(Y,X))*(sqrt(3)/2)... +Kin32(Y+1,X+2)*(U022(Y+1,X+2)-U022(Y,X))*(0.5)... +Kin31(Y,X)*(U0X(Y-2,X)-U0X(Y,X))... +Kra421(Y,X)*(U046(Y+3,X-1)-U046(Y,X))*(sqrt(3/28))... +Kin421(Y,X)*(U0461(Y+3,X-1)-U0461(Y,X))*(sqrt(25/28))... +Kra422(Y,X)*(U045(Y,X-3)-U045(Y,X))*(sqrt(27/28))... +Kin422(Y,X)*(U0451(Y,X-3)-U0451(Y,X))*(sqrt(1/28))... +Kra423(Y,X)*(U044(Y-2,X-2)-U044(Y,X))*(sqrt(3/7))... +Kin423(Y,X)*(U0441(Y-2,X-2)-U0441(Y,X))*(sqrt(4/7))... +Kra413(Y+3,X+1)*(U043(Y+3,X+1)-U043(Y,X))*(-sqrt(3/28))... +Kin413(Y+3,X+1)*(U0431(Y+3,X+1)-U0431(Y,X))*(sqrt(25/28))... +Kra412(Y,X+3)*(U042(Y,X+3)-U042(Y,X))*(-sqrt(27/28))... +Kin412(Y,X+3)*(U0421(Y,X+3)-U0421(Y,X))*(sqrt(1/28))... +Kra441(Y-2,X+2)*(U041(Y-2,X+2)-U041(Y,X))*(-sqrt(3/7))... +Kin441(Y-2,X+2)*(U0411(Y-2,X+2)-U0411(Y,X))*(sqrt(4/7)); VDY(Y,X)=VDY(Y,X)+Kra2(Y,X)*(U021(Y,X-1)-U021(Y,X))*(0.5)... +Kra2(Y,X+1)*(U011(Y,X+1)-U011(Y,X))*(0.5)... +Kin2(Y,X)*(U022(Y,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin2(Y,X+1)*(U012(Y,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kra1(Y+1,X)*(U0Y(Y+1,X)-U0Y(Y,X))... +Kra21(Y,X)*(U022(Y+2,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y,X)*(U021(Y+2,X-1)-U021(Y,X))*(0.5)... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))... +Kra23(Y,X)*(U012(Y-1,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y,X)*(U011(Y-1,X-1)-U011(Y,X))*(0.5)... +Kra21(Y-1,X+1)*(U022(Y-1,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y-1,X+1)*(U021(Y-1,X+1)-U021(Y,X))*(0.5)... +Kin22(Y,X+2)*(U0Y(Y,X+2)-U0Y(Y,X))... +Kra23(Y+2,X+1)*(U012(Y+2,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y+2,X+1)*(U011(Y+2,X+1)-U011(Y,X))*(0.5)... +Kra32(Y,X)*(U011(Y+1,X-2)-U011(Y,X))*(0.5)... +Kin32(Y,X)*(U012(Y+1,X-2)-U012(Y,X))*(-sqrt(3)/2)...


147

+Kra32(Y+1,X+2)*(U021(Y+1,X+2)-U021(Y,X))*(0.5)... +Kin32(Y+1,X+2)*(U022(Y+1,X+2)-U022(Y,X))*(-sqrt(3)/2)... +Kra31(Y,X)*(U0Y(Y-2,X)-U0Y(Y,X))... +Kra421(Y,X)*(U046(Y+3,X-1)-U046(Y,X))*(-sqrt(25/28))... +Kin421(Y,X)*(U0461(Y+3,X-1)-U0461(Y,X))*(sqrt(3/28))... +Kra422(Y,X)*(U045(Y,X-3)-U045(Y,X))*(sqrt(1/28))... +Kin422(Y,X)*(U0451(Y,X-3)-U0451(Y,X))*(-sqrt(27/28))... +Kra423(Y,X)*(U044(Y-2,X-2)-U044(Y,X))*(sqrt(4/7))... +Kin423(Y,X)*(U0441(Y-2,X-2)-U0441(Y,X))*(-sqrt(3/7))... +Kra413(Y+3,X+1)*(U043(Y+3,X+1)-U043(Y,X))*(-sqrt(25/28))... +Kin413(Y+3,X+1)*(U0431(Y+3,X+1)-U0431(Y,X))*(-sqrt(3/28))... +Kra412(Y,X+3)*(U042(Y,X+3)-U042(Y,X))*(sqrt(1/28))... +Kin412(Y,X+3)*(U0421(Y,X+3)-U0421(Y,X))*(sqrt(27/28))... +Kra441(Y-2,X+2)*(U041(Y-2,X+2)-U041(Y,X))*(sqrt(4/7))... +Kin441(Y-2,X+2)*(U0411(Y-2,X+2)-U0411(Y,X))*(sqrt(3/7)); VDZ(Y,X)=VDZ(Y,X)+Kout2(Y,X)*(U0Z(Y,X-1)-U0Z(Y,X))... +Kout2(Y,X+1)*(U0Z(Y,X+1)-U0Z(Y,X))... +Kout1(Y+1,X)*(U0Z(Y+1,X)-U0Z(Y,X))... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))... +Kout22(Y,X)*(U0Z(Y+2,X-1)-U0Z(Y,X))... +Kout23(Y,X)*(U0Z(Y-1,X-1)-U0Z(Y,X))... +Kout21(Y,X+2)*(U0Z(Y,X+2)-U0Z(Y,X))... +Kout22(Y+2,X+1)*(U0Z(Y+2,X+1)-U0Z(Y,X))... +Kout23(Y-1,X+1)*(U0Z(Y-1,X+1)-U0Z(Y,X))... +Kout32(Y,X)*(U0Z(Y+1,X-2)-U0Z(Y,X))... +Kout32(Y+1,X+2)*(U0Z(Y+1,X+2)-U0Z(Y,X))... +Kout31(Y,X)*(U0Z(Y-2,X)-U0Z(Y,X))... +Kout421(Y,X)*(U0Z(Y+3,X-1)-U0Z(Y,X))... +Kout422(Y,X)*(U0Z(Y,X-3)-U0Z(Y,X))... +Kout423(Y,X)*(U0Z(Y-2,X-2)-U0Z(Y,X))... +Kout413(Y+3,X+1)*(U0Z(Y+3,X+1)-U0Z(Y,X))... +Kout412(Y,X+3)*(U0Z(Y,X+3)-U0Z(Y,X))... +Kout441(Y-2,X+2)*(U0Z(Y-2,X+2)-U0Z(Y,X)); end if mod(X,2)==1 & mod(Y,3)==1; VDX(Y,X)=VDX(Y,X)+Kra2(Y,X)*(U021(Y-1,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra2(Y-1,X+1)*(U011(Y-1,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kin2(Y,X)*(U022(Y-1,X-1)-U022(Y,X))*(0.5)...


148

+Kin2(Y-1,X+1)*(U012(Y-1,X+1)-U012(Y,X))*(-0.5)... +Kin1(Y+1,X)*(U0X(Y+1,X)-U0X(Y,X))... +Kra21(Y,X)*(U022(Y+1,X-1)-U022(Y,X))*(0.5)... +Kin21(Y,X)*(U021(Y+1,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))... +Kra23(Y,X)*(U012(Y-2,X-1)-U012(Y,X))*(-0.5)... +Kin23(Y,X)*(U011(Y-2,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra21(Y-2,X+1)*(U022(Y-2,X+1)-U022(Y,X))*(0.5)... +Kin21(Y-2,X+1)*(U021(Y-2,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X+2)*(U0X(Y,X+2)-U0X(Y,X))... +Kra23(Y+1,X+1)*(U012(Y+1,X+1)-U012(Y,X))*(-0.5)... +Kin23(Y+1,X+1)*(U011(Y+1,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kra32(Y,X)*(U011(Y+1,X-2)-U011(Y,X))*(-sqrt(3)/2)... +Kin32(Y,X)*(U012(Y+1,X-2)-U012(Y,X))*(-0.5)... +Kra32(Y+1,X+2)*(U021(Y+1,X+2)-U021(Y,X))*(sqrt(3)/2)... +Kin32(Y+1,X+2)*(U022(Y+1,X+2)-U022(Y,X))*(0.5)... +Kin31(Y,X)*(U0X(Y-2,X)-U0X(Y,X))... +Kra431(Y,X)*(U046(Y+2,X-1)-U046(Y,X))*(sqrt(3/28))... +Kin431(Y,X)*(U0461(Y+2,X-1)-U0461(Y,X))*(sqrt(25/28))... +Kra432(Y,X)*(U045(Y-1,X-3)-U045(Y,X))*(sqrt(27/28))... +Kin432(Y,X)*(U0451(Y-1,X-3)-U0451(Y,X))*(sqrt(1/28))... +Kra433(Y,X)*(U044(Y-2,X-2)-U044(Y,X))*(sqrt(3/7))... +Kin433(Y,X)*(U0441(Y-2,X-2)-U0441(Y,X))*(sqrt(4/7))... +Kra443(Y+2,X+1)*(U043(Y+2,X+1)-U043(Y,X))*(-sqrt(3/28))... +Kin443(Y+2,X+1)*(U0431(Y+2,X+1)-U0431(Y,X))*(sqrt(25/28))... +Kra442(Y-1,X+3)*(U042(Y-1,X+3)-U042(Y,X))*(-sqrt(27/28))... +Kin442(Y-1,X+3)*(U0421(Y-1,X+3)-U0421(Y,X))*(sqrt(1/28))... +Kra411(Y-2,X+2)*(U041(Y-2,X+2)-U041(Y,X))*(-sqrt(3/7))... +Kin411(Y-2,X+2)*(U0411(Y-2,X+2)-U0411(Y,X))*(sqrt(4/7)); VDY(Y,X)=VDY(Y,X)+Kra2(Y,X)*(U021(Y-1,X-1)-U021(Y,X))*(0.5)... +Kra2(Y-1,X+1)*(U011(Y-1,X+1)-U011(Y,X))*(0.5)... +Kin2(Y,X)*(U022(Y-1,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin2(Y-1,X+1)*(U012(Y-1,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kra1(Y+1,X)*(U0Y(Y+1,X)-U0Y(Y,X))... +Kra21(Y,X)*(U022(Y+1,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y,X)*(U021(Y+1,X-1)-U021(Y,X))*(0.5)... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))... +Kra23(Y,X)*(U012(Y-2,X-1)-U012(Y,X))*(-sqrt(3)/2)...


149

+Kin23(Y,X)*(U011(Y-2,X-1)-U011(Y,X))*(0.5)... +Kra21(Y-2,X+1)*(U022(Y-2,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y-2,X+1)*(U021(Y-2,X+1)-U021(Y,X))*(0.5)... +Kin22(Y,X+2)*(U0Y(Y,X+2)-U0Y(Y,X))... +Kra23(Y+1,X+1)*(U012(Y+1,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y+1,X+1)*(U011(Y+1,X+1)-U011(Y,X))*(0.5)... +Kra32(Y,X)*(U011(Y+1,X-2)-U011(Y,X))*(0.5)... +Kin32(Y,X)*(U012(Y+1,X-2)-U012(Y,X))*(-sqrt(3)/2)... +Kra32(Y+1,X+2)*(U021(Y+1,X+2)-U021(Y,X))*(0.5)... +Kin32(Y+1,X+2)*(U022(Y+1,X+2)-U022(Y,X))*(-sqrt(3)/2)... +Kra31(Y,X)*(U0Y(Y-2,X)-U0Y(Y,X))... +Kra431(Y,X)*(U046(Y+2,X-1)-U046(Y,X))*(-sqrt(25/28))... +Kin431(Y,X)*(U0461(Y+2,X-1)-U0461(Y,X))*(sqrt(3/28))... +Kra432(Y,X)*(U045(Y-1,X-3)-U045(Y,X))*(sqrt(1/28))... +Kin432(Y,X)*(U0451(Y-1,X-3)-U0451(Y,X))*(-sqrt(27/28))... +Kra433(Y,X)*(U044(Y-2,X-2)-U044(Y,X))*(sqrt(4/7))... +Kin433(Y,X)*(U0441(Y-2,X-2)-U0441(Y,X))*(-sqrt(3/7))... +Kra443(Y+2,X+1)*(U043(Y+2,X+1)-U043(Y,X))*(-sqrt(25/28))... +Kin443(Y+2,X+1)*(U0431(Y+2,X+1)-U0431(Y,X))*(-sqrt(3/28))... +Kra442(Y-1,X+3)*(U042(Y-1,X+3)-U042(Y,X))*(sqrt(1/28))... +Kin442(Y-1,X+3)*(U0421(Y-1,X+3)-U0421(Y,X))*(sqrt(27/28))... +Kra411(Y-2,X+2)*(U041(Y-2,X+2)-U041(Y,X))*(sqrt(4/7))... +Kin411(Y-2,X+2)*(U0411(Y-2,X+2)-U0411(Y,X))*(sqrt(3/7)); VDZ(Y,X)=VDZ(Y,X)+Kout2(Y,X)*(U0Z(Y-1,X-1)-U0Z(Y,X))... +Kout2(Y-1,X+1)*(U0Z(Y-1,X+1)-U0Z(Y,X))... +Kout1(Y+1,X)*(U0Z(Y+1,X)-U0Z(Y,X))... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))... +Kout22(Y,X)*(U0Z(Y+1,X-1)-U0Z(Y,X))... +Kout23(Y,X)*(U0Z(Y-2,X-1)-U0Z(Y,X))... +Kout21(Y,X+2)*(U0Z(Y,X+2)-U0Z(Y,X))... +Kout22(Y+1,X+1)*(U0Z(Y+1,X+1)-U0Z(Y,X))... +Kout23(Y-2,X+1)*(U0Z(Y-2,X+1)-U0Z(Y,X))... +Kout32(Y,X)*(U0Z(Y+1,X-2)-U0Z(Y,X))... +Kout32(Y+1,X+2)*(U0Z(Y+1,X+2)-U0Z(Y,X))... +Kout31(Y,X)*(U0Z(Y-2,X)-U0Z(Y,X))... +Kout431(Y,X)*(U0Z(Y+2,X-1)-U0Z(Y,X))... +Kout432(Y,X)*(U0Z(Y-1,X-3)-U0Z(Y,X))... +Kout433(Y,X)*(U0Z(Y-2,X-2)-U0Z(Y,X))...


150

+Kout443(Y+2,X+1)*(U0Z(Y+2,X+1)-U0Z(Y,X))... +Kout442(Y-1,X+3)*(U0Z(Y-1,X+3)-U0Z(Y,X))... +Kout411(Y-2,X+2)*(U0Z(Y-2,X+2)-U0Z(Y,X)); end if mod(X,2)==0 & mod(Y,3)==0; VDX(Y,X)=VDX(Y,X)+Kra2(Y,X)*(U011(Y+1,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra2(Y+1,X+1)*(U021(Y+1,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kin2(Y,X)*(U012(Y+1,X-1)-U012(Y,X))*(-0.5)... +Kin2(Y+1,X+1)*(U022(Y+1,X+1)-U022(Y,X))*(0.5)... +Kin1(Y,X)*(U0X(Y-1,X)-U0X(Y,X))... +Kra21(Y,X)*(U022(Y+2,X-1)-U022(Y,X))*(0.5)... +Kin21(Y,X)*(U021(Y+2,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))... +Kra23(Y,X)*(U012(Y-1,X-1)-U012(Y,X))*(-0.5)... +Kin23(Y,X)*(U011(Y-1,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra21(Y-1,X+1)*(U022(Y-1,X+1)-U022(Y,X))*(0.5)... +Kin21(Y-1,X+1)*(U021(Y-1,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X+2)*(U0X(Y,X+2)-U0X(Y,X))... +Kra23(Y+2,X+1)*(U012(Y+2,X+1)-U012(Y,X))*(-0.5)... +Kin23(Y+2,X+1)*(U011(Y+2,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kra32(Y,X)*(U021(Y-1,X-2)-U021(Y,X))*(sqrt(3)/2)... +Kin32(Y,X)*(U022(Y-1,X-2)-U022(Y,X))*(0.5)... +Kra32(Y-1,X+2)*(U011(Y-1,X+2)-U011(Y,X))*(-sqrt(3)/2)... +Kin32(Y-1,X+2)*(U012(Y-1,X+2)-U012(Y,X))*(-0.5)... +Kin31(Y+2,X)*(U0X(Y+2,X)-U0X(Y,X))... +Kra441(Y,X)*(U041(Y+2,X-2)-U041(Y,X))*(-sqrt(3/7))... +Kin441(Y,X)*(U0411(Y+2,X-2)-U0411(Y,X))*(sqrt(4/7))... +Kra442(Y,X)*(U042(Y+1,X-3)-U042(Y,X))*(-sqrt(27/28))... +Kin442(Y,X)*(U0421(Y+1,X-3)-U0421(Y,X))*(sqrt(1/28))... +Kra443(Y,X)*(U043(Y-2,X-1)-U043(Y,X))*(-sqrt(3/28))... +Kin443(Y,X)*(U0431(Y-2,X-1)-U0431(Y,X))*(sqrt(25/28))... +Kra423(Y+2,X+2)*(U044(Y+2,X+2)-U044(Y,X))*(sqrt(3/7))... +Kin423(Y+2,X+2)*(U0441(Y+2,X+2)-U0441(Y,X))*(sqrt(4/7))... +Kra432(Y+1,X+3)*(U045(Y+1,X+3)-U045(Y,X))*(sqrt(27/28))... +Kin432(Y+1,X+3)*(U0451(Y+1,X+3)-U0451(Y,X))*(sqrt(1/28))... +Kra431(Y-2,X+1)*(U046(Y-2,X+1)-U046(Y,X))*(sqrt(3/28))... +Kin431(Y-2,X+1)*(U0461(Y-2,X+1)-U0461(Y,X))*(sqrt(25/28)); VDY(Y,X)=VDY(Y,X)+Kra2(Y,X)*(U011(Y+1,X-1)-U011(Y,X))*(0.5)...


151

+Kra2(Y+1,X+1)*(U021(Y+1,X+1)-U021(Y,X))*(0.5)... +Kin2(Y,X)*(U012(Y+1,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin2(Y+1,X+1)*(U022(Y+1,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kra1(Y,X)*(U0Y(Y-1,X)-U0Y(Y,X))... +Kra21(Y,X)*(U022(Y+2,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y,X)*(U021(Y+2,X-1)-U021(Y,X))*(0.5)... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))... +Kra23(Y,X)*(U012(Y-1,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y,X)*(U011(Y-1,X-1)-U011(Y,X))*(0.5)... +Kra21(Y-1,X+1)*(U022(Y-1,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y-1,X+1)*(U021(Y-1,X+1)-U021(Y,X))*(0.5)... +Kin22(Y,X+2)*(U0Y(Y,X+2)-U0Y(Y,X))... +Kra23(Y+2,X+1)*(U012(Y+2,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y+2,X+1)*(U011(Y+2,X+1)-U011(Y,X))*(0.5)... +Kra32(Y,X)*(U021(Y-1,X-2)-U021(Y,X))*(0.5)... +Kin32(Y,X)*(U022(Y-1,X-2)-U022(Y,X))*(-sqrt(3)/2)... +Kra32(Y-1,X+2)*(U011(Y-1,X+2)-U011(Y,X))*(0.5)... +Kin32(Y-1,X+2)*(U012(Y-1,X+2)-U012(Y,X))*(-sqrt(3)/2)... +Kra31(Y+2,X)*(U0Y(Y+2,X)-U0Y(Y,X))... +Kra441(Y,X)*(U041(Y+2,X-2)-U041(Y,X))*(sqrt(4/7))... +Kin441(Y,X)*(U0411(Y+2,X-2)-U0411(Y,X))*(sqrt(3/7))... +Kra442(Y,X)*(U042(Y+1,X-3)-U042(Y,X))*(sqrt(1/28))... +Kin442(Y,X)*(U0421(Y+1,X-3)-U0421(Y,X))*(sqrt(27/28))... +Kra443(Y,X)*(U043(Y-2,X-1)-U043(Y,X))*(-sqrt(25/28))... +Kin443(Y,X)*(U0431(Y-2,X-1)-U0431(Y,X))*(-sqrt(3/28))... +Kra423(Y+2,X+2)*(U044(Y+2,X+2)-U044(Y,X))*(sqrt(4/7))... +Kin423(Y+2,X+2)*(U0441(Y+2,X+2)-U0441(Y,X))*(-sqrt(3/7))... +Kra432(Y+1,X+3)*(U045(Y+1,X+3)-U045(Y,X))*(sqrt(1/28))... +Kin432(Y+1,X+3)*(U0451(Y+1,X+3)-U0451(Y,X))*(-sqrt(27/28))... +Kra431(Y-2,X+1)*(U046(Y-2,X+1)-U046(Y,X))*(-sqrt(25/28))... +Kin431(Y-2,X+1)*(U0461(Y-2,X+1)-U0461(Y,X))*(sqrt(3/28)); VDZ(Y,X)=VDZ(Y,X)+Kout2(Y,X)*(U0Z(Y+1,X-1)-U0Z(Y,X))... +Kout2(Y+1,X+1)*(U0Z(Y+1,X+1)-U0Z(Y,X))... +Kout1(Y,X)*(U0Z(Y-1,X)-U0Z(Y,X))... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))... +Kout22(Y,X)*(U0Z(Y+2,X-1)-U0Z(Y,X))... +Kout23(Y,X)*(U0Z(Y-1,X-1)-U0Z(Y,X))... +Kout21(Y,X+2)*(U0Z(Y,X+2)-U0Z(Y,X))...

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns +Kout22(Y+2,X+1)*(U0Z(Y+2,X+1)-U0Z(Y,X))... +Kout23(Y-1,X+1)*(U0Z(Y-1,X+1)-U0Z(Y,X))... +Kout32(Y,X)*(U0Z(Y-1,X-2)-U0Z(Y,X))... +Kout32(Y-1,X+2)*(U0Z(Y-1,X+2)-U0Z(Y,X))... +Kout31(Y+2,X)*(U0Z(Y+2,X)-U0Z(Y,X))... +Kout441(Y,X)*(U0Z(Y+2,X-2)-U0Z(Y,X))... +Kout442(Y,X)*(U0Z(Y+1,X-3)-U0Z(Y,X))... +Kout443(Y,X)*(U0Z(Y-2,X-1)-U0Z(Y,X))... +Kout423(Y+2,X+2)*(U0Z(Y+2,X+2)-U0Z(Y,X))... +Kout432(Y+1,X+3)*(U0Z(Y+1,X+3)-U0Z(Y,X))... +Kout431(Y-2,X+1)*(U0Z(Y-2,X+1)-U0Z(Y,X)); end VX(Y,X)=VX(Y,X)+VDX(Y,X)*TM; U1X(Y,X)=U0X(Y,X)+VX(Y,X)*TAU; VY(Y,X)=VY(Y,X)+VDY(Y,X)*TM; U1Y(Y,X)=U0Y(Y,X)+VY(Y,X)*TAU; V11(Y,X)=VX(Y,X)*(-sqrt(3)/2)+VY(Y,X)*(0.5); U111(Y,X)=U1X(Y,X)*(-sqrt(3)/2)+U1Y(Y,X)*(0.5); V12(Y,X)=VX(Y,X)*(-0.5)+VY(Y,X)*(-sqrt(3)/2); U112(Y,X)=U1X(Y,X)*(-0.5)+U1Y(Y,X)*(-sqrt(3)/2); V21(Y,X)=VX(Y,X)*(sqrt(3)/2)+VY(Y,X)*(0.5); U121(Y,X)=U1X(Y,X)*(sqrt(3)/2)+U1Y(Y,X)*(0.5); V22(Y,X)=VX(Y,X)*(0.5)+VY(Y,X)*(-sqrt(3)/2); U122(Y,X)=U1X(Y,X)*(0.5)+U1Y(Y,X)*(-sqrt(3)/2); V41(Y,X)=VX(Y,X)*(-sqrt(3/7))+VY(Y,X)*(sqrt(4/7)); U141(Y,X)=U1X(Y,X)*(-sqrt(3/7))+U1Y(Y,X)*(sqrt(4/7)); V411(Y,X)=VX(Y,X)*(sqrt(4/7))+VY(Y,X)*(sqrt(3/7)); U1411(Y,X)=U1X(Y,X)*(sqrt(4/7))+U1Y(Y,X)*(sqrt(3/7)); V42(Y,X)=VX(Y,X)*(-sqrt(27/28))+VY(Y,X)*(sqrt(1/28)); U142(Y,X)=U1X(Y,X)*(-sqrt(27/28))+U1Y(Y,X)*(sqrt(1/28)); V421(Y,X)=VX(Y,X)*(sqrt(1/28))+VY(Y,X)*(sqrt(27/28)); U1421(Y,X)=U1X(Y,X)*(sqrt(1/28))+U1Y(Y,X)*(sqrt(27/28)); V43(Y,X)=VX(Y,X)*(-sqrt(3/28))+VY(Y,X)*(-sqrt(25/28)); U143(Y,X)=U1X(Y,X)*(-sqrt(3/28))+U1Y(Y,X)*(-sqrt(25/28)); V431(Y,X)=VX(Y,X)*(sqrt(25/28))+VY(Y,X)*(-sqrt(3/28)); U1431(Y,X)=U1X(Y,X)*(sqrt(25/28))+U1Y(Y,X)*(-sqrt(3/28)); V44(Y,X)=VX(Y,X)*(sqrt(3/7))+VY(Y,X)*(sqrt(4/7)); U144(Y,X)=U1X(Y,X)*(sqrt(3/7))+U1Y(Y,X)*(sqrt(4/7));

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Appendix B. Matlab Code for Phonon Density of States and Mode Patterns V441(Y,X)=VX(Y,X)*(sqrt(4/7))+VY(Y,X)*(-sqrt(3/7)); U1441(Y,X)=U1X(Y,X)*(sqrt(4/7))+U1Y(Y,X)*(-sqrt(3/7)); V45(Y,X)=VX(Y,X)*(sqrt(27/28))+VY(Y,X)*(sqrt(1/28)); U145(Y,X)=U1X(Y,X)*(sqrt(27/28))+U1Y(Y,X)*(sqrt(1/28)); V451(Y,X)=VX(Y,X)*(sqrt(1/28))+VY(Y,X)*(-sqrt(27/28)); U1451(Y,X)=U1X(Y,X)*(sqrt(1/28))+U1Y(Y,X)*(-sqrt(27/28)); V46(Y,X)=VX(Y,X)*(sqrt(3/28))+VY(Y,X)*(-sqrt(25/28)); U146(Y,X)=U1X(Y,X)*(sqrt(3/28))+U1Y(Y,X)*(-sqrt(25/28)); V461(Y,X)=VX(Y,X)*(sqrt(25/28))+VY(Y,X)*(sqrt(3/28)); U1461(Y,X)=U1X(Y,X)*(sqrt(25/28))+U1Y(Y,X)*(sqrt(3/28)); VZ(Y,X)=VZ(Y,X)+VDZ(Y,X)*TM; U1Z(Y,X)=U0Z(Y,X)+VZ(Y,X)*TAU; end end end

for X=5:LX+4; for Y=5:LY+4; if M(Y,X)==1; U0X(Y,X)=U1X(Y,X); U0Y(Y,X)=U1Y(Y,X); U011(Y,X)=U111(Y,X); U012(Y,X)=U112(Y,X); U021(Y,X)=U121(Y,X); U022(Y,X)=U122(Y,X); U041(Y,X)=U141(Y,X); U042(Y,X)=U142(Y,X); U043(Y,X)=U143(Y,X); U044(Y,X)=U144(Y,X); U045(Y,X)=U145(Y,X); U046(Y,X)=U146(Y,X); U0411(Y,X)=U1411(Y,X); U0421(Y,X)=U1421(Y,X); U0431(Y,X)=U1431(Y,X); U0441(Y,X)=U1441(Y,X); U0451(Y,X)=U1451(Y,X); U0461(Y,X)=U1461(Y,X);

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Appendix B. Matlab Code for Phonon Density of States and Mode Patterns U0Z(Y,X)=U1Z(Y,X); end end end end %caluculation of the total energy of the system ENERGY=0;

%Total energy of the syste /* MAKE THIS RED */

ENERGY1=0; %Kinectic energy of the system ENERGY2=0; %Interaction energy of the system for X=4:LX+5; for Y=4:LY+5; if M(Y,X)>0; ENERGY1=ENERGY1+M1*VX(Y,X)*VX(Y,X); ENERGY1=ENERGY1+M1*VY(Y,X)*VY(Y,X); ENERGY1=ENERGY1+M1*VZ(Y,X)*VZ(Y,X); if mod(X,2)==1 && mod(Y,3)==2; ENERGY2=ENERGY2+Kra2(Y,X)*(U011(Y,X-1)-U011(Y,X))^2... +Kin2(Y,X)*(U012(Y,X-1)-U012(Y,X))^2... +Kin1(Y,X)*(U0X(Y-1,X)-U0X(Y,X))^2... +Kra1(Y,X)*(U0Y(Y-1,X)-U0Y(Y,X))^2... +Kout2(Y,X)*(U0Z(Y,X-1)-U0Z(Y,X))^2... +Kout1(Y,X)*(U0Z(Y-1,X)-U0Z(Y,X))^2... +Kra21(Y,X)*(U022(Y+1,X-1)-U022(Y,X))^2... +Kin21(Y,X)*(U021(Y+1,X-1)-U021(Y,X))^2... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))^2... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))^2... +Kra23(Y,X)*(U012(Y-2,X-1)-U012(Y,X))^2... +Kin23(Y,X)*(U011(Y-2,X-1)-U011(Y,X))^2... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))^2... +Kout22(Y,X)*(U0Z(Y+1,X-1)-U0Z(Y,X))^2... +Kout23(Y,X)*(U0Z(Y-2,X-1)-U0Z(Y,X))^2... +Kra32(Y,X)*(U021(Y-1,X-2)-U021(Y,X))^2... +Kin32(Y,X)*(U022(Y-1,X-2)-U022(Y,X))^2... +Kout32(Y,X)*(U0Z(Y-1,X-2)-U0Z(Y,X))^2... +Kra411(Y,X)*(U041(Y+2,X-2)-U041(Y,X))^2... +Kin411(Y,X)*(U0411(Y+2,X-2)-U0411(Y,X))^2...

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Appendix B. Matlab Code for Phonon Density of States and Mode Patterns +Kra412(Y,X)*(U042(Y,X-3)-U042(Y,X))^2... +Kin412(Y,X)*(U0421(Y,X-3)-U0421(Y,X))^2... +Kra413(Y,X)*(U043(Y-3,X-1)-U043(Y,X))^2... +Kin413(Y,X)*(U0431(Y-3,X-1)-U0431(Y,X))^2... +Kout411(Y,X)*(U0Z(Y+2,X-2)-U0Z(Y,X))^2... +Kout412(Y,X)*(U0Z(Y,X-3)-U0Z(Y,X))^2... +Kout413(Y,X)*(U0Z(Y-3,X-1)-U0Z(Y,X))^2; end if mod(X,2)==0 & mod(Y,3)==2; ENERGY2=ENERGY2+Kra2(Y,X)*(U021(Y,X-1)-U021(Y,X))^2... +Kin2(Y,X)*(U022(Y,X-1)-U022(Y,X))^2... +Kout2(Y,X)*(U0Z(Y,X-1)-U0Z(Y,X))^2... +Kra21(Y,X)*(U022(Y+2,X-1)-U022(Y,X))^2... +Kin21(Y,X)*(U021(Y+2,X-1)-U021(Y,X))^2... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))^2... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))^2... +Kra23(Y,X)*(U012(Y-1,X-1)-U012(Y,X))^2... +Kin23(Y,X)*(U011(Y-1,X-1)-U011(Y,X))^2... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))^2... +Kout22(Y,X)*(U0Z(Y+2,X-1)-U0Z(Y,X))^2... +Kout23(Y,X)*(U0Z(Y-1,X-1)-U0Z(Y,X))^2... +Kra32(Y,X)*(U011(Y+1,X-2)-U011(Y,X))^2... +Kin32(Y,X)*(U012(Y+1,X-2)-U012(Y,X))^2... +Kin31(Y,X)*(U0X(Y-2,X)-U0X(Y,X))^2... +Kra31(Y,X)*(U0Y(Y-2,X)-U0Y(Y,X))^2... +Kout32(Y,X)*(U0Z(Y+1,X-2)-U0Z(Y,X))^2... +Kout31(Y,X)*(U0Z(Y-2,X)-U0Z(Y,X))^2... +Kra421(Y,X)*(U046(Y+3,X-1)-U046(Y,X))^2... +Kin421(Y,X)*(U0461(Y+3,X-1)-U0461(Y,X))^2... +Kra422(Y,X)*(U045(Y,X-3)-U045(Y,X))^2... +Kin422(Y,X)*(U0451(Y,X-3)-U0451(Y,X))^2... +Kra423(Y,X)*(U044(Y-2,X-2)-U044(Y,X))^2... +Kin423(Y,X)*(U0441(Y-2,X-2)-U0441(Y,X))^2... +Kout421(Y,X)*(U0Z(Y+3,X-1)-U0Z(Y,X))^2... +Kout422(Y,X)*(U0Z(Y,X-3)-U0Z(Y,X))^2... +Kout423(Y,X)*(U0Z(Y-2,X-2)-U0Z(Y,X))^2; end if mod(X,2)==1 & mod(Y,3)==1; ENERGY2=ENERGY2+Kra2(Y,X)*(U021(Y-1,X-1)-U021(Y,X))^2...

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Appendix B. Matlab Code for Phonon Density of States and Mode Patterns +Kin2(Y,X)*(U022(Y-1,X-1)-U022(Y,X))^2... +Kout2(Y,X)*(U0Z(Y-1,X-1)-U0Z(Y,X))^2... +Kra21(Y,X)*(U022(Y+1,X-1)-U022(Y,X))^2... +Kin21(Y,X)*(U021(Y+1,X-1)-U021(Y,X))^2... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))^2... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))^2... +Kra23(Y,X)*(U012(Y-2,X-1)-U012(Y,X))^2... +Kin23(Y,X)*(U011(Y-2,X-1)-U011(Y,X))^2... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))^2... +Kout22(Y,X)*(U0Z(Y+1,X-1)-U0Z(Y,X))^2... +Kout23(Y,X)*(U0Z(Y-2,X-1)-U0Z(Y,X))^2... +Kra32(Y,X)*(U011(Y+1,X-2)-U011(Y,X))^2... +Kin32(Y,X)*(U012(Y+1,X-2)-U012(Y,X))^2... +Kin31(Y,X)*(U0X(Y-2,X)-U0X(Y,X))^2... +Kra31(Y,X)*(U0Y(Y-2,X)-U0Y(Y,X))^2... +Kout32(Y,X)*(U0Z(Y+1,X-2)-U0Z(Y,X))^2... +Kout31(Y,X)*(U0Z(Y-2,X)-U0Z(Y,X))^2... +Kra431(Y,X)*(U046(Y+2,X-1)-U046(Y,X))^2... +Kin431(Y,X)*(U0461(Y+2,X-1)-U0461(Y,X))^2... +Kra432(Y,X)*(U045(Y-1,X-3)-U045(Y,X))^2... +Kin432(Y,X)*(U0451(Y-1,X-3)-U0451(Y,X))^2... +Kra433(Y,X)*(U044(Y-2,X-2)-U044(Y,X))^2... +Kin433(Y,X)*(U0441(Y-2,X-2)-U0441(Y,X))^2... +Kout431(Y,X)*(U0Z(Y+2,X-1)-U0Z(Y,X))^2... +Kout432(Y,X)*(U0Z(Y-1,X-3)-U0Z(Y,X))^2... +Kout433(Y,X)*(U0Z(Y-2,X-2)-U0Z(Y,X))^2; end if mod(X,2)==0 & mod(Y,3)==0; ENERGY2=ENERGY2+Kra2(Y,X)*(U011(Y+1,X-1)-U011(Y,X))^2... +Kin2(Y,X)*(U012(Y+1,X-1)-U012(Y,X))^2... +Kin1(Y,X)*(U0X(Y-1,X)-U0X(Y,X))^2... +Kra1(Y,X)*(U0Y(Y-1,X)-U0Y(Y,X))^2... +Kout2(Y,X)*(U0Z(Y+1,X-1)-U0Z(Y,X))^2... +Kout1(Y,X)*(U0Z(Y-1,X)-U0Z(Y,X))^2... +Kra21(Y,X)*(U022(Y+2,X-1)-U022(Y,X))^2... +Kin21(Y,X)*(U021(Y+2,X-1)-U021(Y,X))^2... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))^2... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))^2... +Kra23(Y,X)*(U012(Y-1,X-1)-U012(Y,X))^2...

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Appendix B. Matlab Code for Phonon Density of States and Mode Patterns +Kin23(Y,X)*(U011(Y-1,X-1)-U011(Y,X))^2... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))^2... +Kout22(Y,X)*(U0Z(Y+2,X-1)-U0Z(Y,X))^2... +Kout23(Y,X)*(U0Z(Y-1,X-1)-U0Z(Y,X))^2... +Kra32(Y,X)*(U021(Y-1,X-2)-U021(Y,X))^2... +Kin32(Y,X)*(U022(Y-1,X-2)-U022(Y,X))^2... +Kout32(Y,X)*(U0Z(Y-1,X-2)-U0Z(Y,X))^2... +Kra441(Y,X)*(U041(Y+2,X-2)-U041(Y,X))^2... +Kin441(Y,X)*(U0411(Y+2,X-2)-U0411(Y,X))^2... +Kra442(Y,X)*(U042(Y+1,X-3)-U042(Y,X))^2... +Kin442(Y,X)*(U0421(Y+1,X-3)-U0421(Y,X))^2... +Kra443(Y,X)*(U043(Y-2,X-1)-U043(Y,X))^2... +Kin443(Y,X)*(U0431(Y-2,X-1)-U0431(Y,X))^2... +Kout441(Y,X)*(U0Z(Y+2,X-2)-U0Z(Y,X))^2... +Kout442(Y,X)*(U0Z(Y+1,X-3)-U0Z(Y,X))^2... +Kout443(Y,X)*(U0Z(Y-2,X-1)-U0Z(Y,X))^2; end ENERGY=ENERGY1+ENERGY2; end end end %caluculate DOS DOS=0; ENERGY; WR=2.0*sin(W*TAU/2.0)/TAU; k=WR*3.335640952e-11/(2.0*pi) DOS=0.5*8.0*ENERGY*cos(W*TAU/2.0)/(pi*N*TAU*F0*F0*LN) fprintf(fid1,FORMAT,WR); fprintf(fid2,FORMAT,DOS); fprintf(fid3,FORMAT,k); fprintf(fid1,’\n’); fprintf(fid2,’\n’); fprintf(fid3,’\n’); end fclose(fid1); fclose(fid2); fclose(fid3);

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toc;

B.3

Calculation of Mode Patterns

%For Mode pattern calculation, until the main routine of phonon density% % of states calculation is same. The cahning parts start from the main% %routine are given below% %MAIN ROUTINE W=2.54414; DN0=W/DIV; DNA=round(DN0); DN=DNA+1; ECST=2.0*pi*DN/(W*TAU); for P=0:P1; U011=zeros(LY+8,LX+8); U111=zeros(LY+8,LX+8); VD11=zeros(LY+8,LX+8); V11=zeros(LY+8,LX+8); U012=zeros(LY+8,LX+8); U112=zeros(LY+8,LX+8); VD12=zeros(LY+8,LX+8); V12=zeros(LY+8,LX+8); U021=zeros(LY+8,LX+8); U121=zeros(LY+8,LX+8); VD21=zeros(LY+8,LX+8); V21=zeros(LY+8,LX+8); U022=zeros(LY+8,LX+8); U122=zeros(LY+8,LX+8); VD22=zeros(LY+8,LX+8); V22=zeros(LY+8,LX+8); U041=zeros(LY+8,LX+8); U141=zeros(LY+8,LX+8); VD41=zeros(LY+8,LX+8); V41=zeros(LY+8,LX+8);

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns U042=zeros(LY+8,LX+8); U142=zeros(LY+8,LX+8); VD42=zeros(LY+8,LX+8); V42=zeros(LY+8,LX+8); U043=zeros(LY+8,LX+8); U143=zeros(LY+8,LX+8); VD43=zeros(LY+8,LX+8); V43=zeros(LY+8,LX+8); U044=zeros(LY+8,LX+8); U144=zeros(LY+8,LX+8); VD44=zeros(LY+8,LX+8); V44=zeros(LY+8,LX+8); U045=zeros(LY+8,LX+8); U145=zeros(LY+8,LX+8); VD45=zeros(LY+8,LX+8); V45=zeros(LY+8,LX+8); U046=zeros(LY+8,LX+8); U146=zeros(LY+8,LX+8); VD46=zeros(LY+8,LX+8); V46=zeros(LY+8,LX+8); U0411=zeros(LY+8,LX+8); U1411=zeros(LY+8,LX+8); VD411=zeros(LY+8,LX+8); V411=zeros(LY+8,LX+8); U0421=zeros(LY+8,LX+8); U1421=zeros(LY+8,LX+8); VD421=zeros(LY+8,LX+8); V421=zeros(LY+8,LX+8); U0431=zeros(LY+8,LX+8); U1431=zeros(LY+8,LX+8); VD431=zeros(LY+8,LX+8); V431=zeros(LY+8,LX+8); U0441=zeros(LY+8,LX+8); U1441=zeros(LY+8,LX+8); VD441=zeros(LY+8,LX+8); V441=zeros(LY+8,LX+8); U0451=zeros(LY+8,LX+8); U1451=zeros(LY+8,LX+8); VD451=zeros(LY+8,LX+8);

159


160

V451=zeros(LY+8,LX+8); U0461=zeros(LY+8,LX+8); U1461=zeros(LY+8,LX+8); VD461=zeros(LY+8,LX+8); V461=zeros(LY+8,LX+8); U0X=zeros(LY+8,LX+8); U1X=zeros(LY+8,LX+8); VDX=zeros(LY+8,LX+8); VX=zeros(LY+8,LX+8); U0Y=zeros(LY+8,LX+8); U1Y=zeros(LY+6,LX+8); VDY=zeros(LY+8,LX+8); VY=zeros(LY+8,LX+8); U0Z=zeros(LY+8,LX+8); U1Z=zeros(LY+8,LX+8); VDZ=zeros(LY+8,LX+8); VZ=zeros(LY+8,LX+8); FLX=zeros(LY+8,LX+8); FLY=zeros(LY+8,LX+8); FLZ=zeros(LY+8,LX+8); for X=1:LX+8; for Y=1:LY+8; FLX(Y,X)=F0*SQM*cos(2.0*pi*rand(1)); FLY(Y,X)=F0*SQM*cos(2.0*pi*rand(1)); FLZ(Y,X)=F0*SQM*cos(2.0*pi*rand(1)); end end %caluculation of the displacement and velocity of each atom of the system %

N=10;

for N=0:ECST; for X=4:LX+5; for Y=4:LY+5; if M(Y,X)>0; VDX(Y,X)=FLX(Y,X)*cos(W*N*TAU); VDY(Y,X)=FLY(Y,X)*cos(W*N*TAU); VDZ(Y,X)=FLZ(Y,X)*cos(W*N*TAU); if mod(X,2)==1 & mod(Y,3)==2;


161

VDX(Y,X)=VDX(Y,X)+Kra2(Y,X)*(U011(Y,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra2(Y,X+1)*(U021(Y,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kin2(Y,X)*(U012(Y,X-1)-U012(Y,X))*(-0.5)... +Kin2(Y,X+1)*(U022(Y,X+1)-U022(Y,X))*(0.5)... +Kin1(Y,X)*(U0X(Y-1,X)-U0X(Y,X))... +Kra21(Y,X)*(U022(Y+1,X-1)-U022(Y,X))*(0.5)... +Kin21(Y,X)*(U021(Y+1,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))... +Kra23(Y,X)*(U012(Y-2,X-1)-U012(Y,X))*(-0.5)... +Kin23(Y,X)*(U011(Y-2,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra21(Y-2,X+1)*(U022(Y-2,X+1)-U022(Y,X))*(0.5)... +Kin21(Y-2,X+1)*(U021(Y-2,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X+2)*(U0X(Y,X+2)-U0X(Y,X))... +Kra23(Y+1,X+1)*(U012(Y+1,X+1)-U012(Y,X))*(-0.5)... +Kin23(Y+1,X+1)*(U011(Y+1,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kra32(Y,X)*(U021(Y-1,X-2)-U021(Y,X))*(sqrt(3)/2)... +Kin32(Y,X)*(U022(Y-1,X-2)-U022(Y,X))*(0.5)... +Kra32(Y-1,X+2)*(U011(Y-1,X+2)-U011(Y,X))*(-sqrt(3)/2)... +Kin32(Y-1,X+2)*(U012(Y-1,X+2)-U012(Y,X))*(-0.5)... +Kin31(Y+2,X)*(U0X(Y+2,X)-U0X(Y,X))... +Kra411(Y,X)*(U041(Y+2,X-2)-U041(Y,X))*(-sqrt(3/7))... +Kin411(Y,X)*(U0411(Y+2,X-2)-U0411(Y,X))*(sqrt(4/7))... +Kra412(Y,X)*(U042(Y,X-3)-U042(Y,X))*(-sqrt(27/28))... +Kin412(Y,X)*(U0421(Y,X-3)-U0421(Y,X))*(sqrt(1/28))... +Kra413(Y,X)*(U043(Y-3,X-1)-U043(Y,X))*(-sqrt(3/28))... +Kin413(Y,X)*(U0431(Y-3,X-1)-U0431(Y,X))*(sqrt(25/28))... +Kra433(Y+2,X+2)*(U044(Y+2,X+2)-U044(Y,X))*(sqrt(3/7))... +Kin433(Y+2,X+2)*(U0441(Y+2,X+2)-U0441(Y,X))*(sqrt(4/7))... +Kra422(Y,X+3)*(U045(Y,X+3)-U045(Y,X))*(sqrt(27/28))... +Kin422(Y,X+3)*(U0451(Y,X+3)-U0451(Y,X))*(sqrt(1/28))... +Kra421(Y-3,X+1)*(U046(Y-3,X+1)-U046(Y,X))*(sqrt(3/28))... +Kin421(Y-3,X+1)*(U0461(Y-3,X+1)-U0461(Y,X))*(sqrt(25/28)); VDY(Y,X)=VDY(Y,X)+Kra2(Y,X)*(U011(Y,X-1)-U011(Y,X))*(0.5)... +Kra2(Y,X+1)*(U021(Y,X+1)-U021(Y,X))*(0.5)... +Kin2(Y,X)*(U012(Y,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin2(Y,X+1)*(U022(Y,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kra1(Y,X)*(U0Y(Y-1,X)-U0Y(Y,X))... +Kra21(Y,X)*(U022(Y+1,X-1)-U022(Y,X))*(-sqrt(3)/2)...


162

+Kin21(Y,X)*(U021(Y+1,X-1)-U021(Y,X))*(0.5)... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))... +Kra23(Y,X)*(U012(Y-2,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y,X)*(U011(Y-2,X-1)-U011(Y,X))*(0.5)... +Kra21(Y-2,X+1)*(U022(Y-2,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y-2,X+1)*(U021(Y-2,X+1)-U021(Y,X))*(0.5)... +Kin22(Y,X+2)*(U0Y(Y,X+2)-U0Y(Y,X))... +Kra23(Y+1,X+1)*(U012(Y+1,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y+1,X+1)*(U011(Y+1,X+1)-U011(Y,X))*(0.5)... +Kra32(Y,X)*(U021(Y-1,X-2)-U021(Y,X))*(0.5)... +Kin32(Y,X)*(U022(Y-1,X-2)-U022(Y,X))*(-sqrt(3)/2)... +Kra32(Y-1,X+2)*(U011(Y-1,X+2)-U011(Y,X))*(0.5)... +Kin32(Y-1,X+2)*(U012(Y-1,X+2)-U012(Y,X))*(-sqrt(3)/2)... +Kra31(Y+2,X)*(U0Y(Y+2,X)-U0Y(Y,X))... +Kra411(Y,X)*(U041(Y+2,X-2)-U041(Y,X))*(sqrt(4/7))... +Kin411(Y,X)*(U0411(Y+2,X-2)-U0411(Y,X))*(sqrt(3/7))... +Kra412(Y,X)*(U042(Y,X-3)-U042(Y,X))*(sqrt(1/28))... +Kin412(Y,X)*(U0421(Y,X-3)-U0421(Y,X))*(sqrt(27/28))... +Kra413(Y,X)*(U043(Y-3,X-1)-U043(Y,X))*(-sqrt(25/28))... +Kin413(Y,X)*(U0431(Y-3,X-1)-U0431(Y,X))*(-sqrt(3/28))... +Kra433(Y+2,X+2)*(U044(Y+2,X+2)-U044(Y,X))*(sqrt(4/7))... +Kin433(Y+2,X+2)*(U0441(Y+2,X+2)-U0441(Y,X))*(-sqrt(3/7))... +Kra422(Y,X+3)*(U045(Y,X+3)-U045(Y,X))*(sqrt(1/28))... +Kin422(Y,X+3)*(U0451(Y,X+3)-U0451(Y,X))*(-sqrt(27/28))... +Kra421(Y-3,X+1)*(U046(Y-3,X+1)-U046(Y,X))*(-sqrt(25/28))... +Kin421(Y-3,X+1)*(U0461(Y-3,X+1)-U0461(Y,X))*(sqrt(3/28));

VDZ(Y,X)=VDZ(Y,X)+Kout2(Y,X)*(U0Z(Y,X-1)-U0Z(Y,X))... +Kout2(Y,X+1)*(U0Z(Y,X+1)-U0Z(Y,X))... +Kout1(Y,X)*(U0Z(Y-1,X)-U0Z(Y,X))... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))... +Kout22(Y,X)*(U0Z(Y+1,X-1)-U0Z(Y,X))... +Kout23(Y,X)*(U0Z(Y-2,X-1)-U0Z(Y,X))... +Kout21(Y,X+2)*(U0Z(Y,X+2)-U0Z(Y,X))... +Kout22(Y+1,X+1)*(U0Z(Y+1,X+1)-U0Z(Y,X))... +Kout23(Y-2,X+1)*(U0Z(Y-2,X+1)-U0Z(Y,X))... +Kout32(Y,X)*(U0Z(Y-1,X-2)-U0Z(Y,X))... +Kout32(Y-1,X+2)*(U0Z(Y-1,X+2)-U0Z(Y,X))...


163

+Kout31(Y+2,X)*(U0Z(Y+2,X)-U0Z(Y,X))... +Kout411(Y,X)*(U0Z(Y+2,X-2)-U0Z(Y,X))... +Kout412(Y,X)*(U0Z(Y,X-3)-U0Z(Y,X))... +Kout413(Y,X)*(U0Z(Y-3,X-1)-U0Z(Y,X))... +Kout433(Y+2,X+2)*(U0Z(Y+2,X+2)-U0Z(Y,X))... +Kout422(Y,X+3)*(U0Z(Y,X+3)-U0Z(Y,X))... +Kout421(Y-3,X+1)*(U0Z(Y-3,X+1)-U0Z(Y,X)); end if mod(X,2)==0 & mod(Y,3)==2; VDX(Y,X)=VDX(Y,X)+Kra2(Y,X)*(U021(Y,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra2(Y,X+1)*(U011(Y,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kin2(Y,X)*(U022(Y,X-1)-U022(Y,X))*(0.5)... +Kin2(Y,X+1)*(U012(Y,X+1)-U012(Y,X))*(-0.5)... +Kin1(Y+1,X)*(U0X(Y+1,X)-U0X(Y,X))... +Kra21(Y,X)*(U022(Y+2,X-1)-U022(Y,X))*(0.5)... +Kin21(Y,X)*(U021(Y+2,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))... +Kra23(Y,X)*(U012(Y-1,X-1)-U012(Y,X))*(-0.5)... +Kin23(Y,X)*(U011(Y-1,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra21(Y-1,X+1)*(U022(Y-1,X+1)-U022(Y,X))*(0.5)... +Kin21(Y-1,X+1)*(U021(Y-1,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X+2)*(U0X(Y,X+2)-U0X(Y,X))... +Kra23(Y+2,X+1)*(U012(Y+2,X+1)-U012(Y,X))*(-0.5)... +Kin23(Y+2,X+1)*(U011(Y+2,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kra32(Y,X)*(U011(Y+1,X-2)-U011(Y,X))*(-sqrt(3)/2)... +Kin32(Y,X)*(U012(Y+1,X-2)-U012(Y,X))*(-0.5)... +Kra32(Y+1,X+2)*(U021(Y+1,X+2)-U021(Y,X))*(sqrt(3)/2)... +Kin32(Y+1,X+2)*(U022(Y+1,X+2)-U022(Y,X))*(0.5)... +Kin31(Y,X)*(U0X(Y-2,X)-U0X(Y,X))... +Kra421(Y,X)*(U046(Y+3,X-1)-U046(Y,X))*(sqrt(3/28))... +Kin421(Y,X)*(U0461(Y+3,X-1)-U0461(Y,X))*(sqrt(25/28))... +Kra422(Y,X)*(U045(Y,X-3)-U045(Y,X))*(sqrt(27/28))... +Kin422(Y,X)*(U0451(Y,X-3)-U0451(Y,X))*(sqrt(1/28))... +Kra423(Y,X)*(U044(Y-2,X-2)-U044(Y,X))*(sqrt(3/7))... +Kin423(Y,X)*(U0441(Y-2,X-2)-U0441(Y,X))*(sqrt(4/7))... +Kra413(Y+3,X+1)*(U043(Y+3,X+1)-U043(Y,X))*(-sqrt(3/28))... +Kin413(Y+3,X+1)*(U0431(Y+3,X+1)-U0431(Y,X))*(sqrt(25/28))... +Kra412(Y,X+3)*(U042(Y,X+3)-U042(Y,X))*(-sqrt(27/28))...


164

+Kin412(Y,X+3)*(U0421(Y,X+3)-U0421(Y,X))*(sqrt(1/28))... +Kra441(Y-2,X+2)*(U041(Y-2,X+2)-U041(Y,X))*(-sqrt(3/7))... +Kin441(Y-2,X+2)*(U0411(Y-2,X+2)-U0411(Y,X))*(sqrt(4/7)); VDY(Y,X)=VDY(Y,X)+Kra2(Y,X)*(U021(Y,X-1)-U021(Y,X))*(0.5)... +Kra2(Y,X+1)*(U011(Y,X+1)-U011(Y,X))*(0.5)... +Kin2(Y,X)*(U022(Y,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin2(Y,X+1)*(U012(Y,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kra1(Y+1,X)*(U0Y(Y+1,X)-U0Y(Y,X))... +Kra21(Y,X)*(U022(Y+2,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y,X)*(U021(Y+2,X-1)-U021(Y,X))*(0.5)... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))... +Kra23(Y,X)*(U012(Y-1,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y,X)*(U011(Y-1,X-1)-U011(Y,X))*(0.5)... +Kra21(Y-1,X+1)*(U022(Y-1,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y-1,X+1)*(U021(Y-1,X+1)-U021(Y,X))*(0.5)... +Kin22(Y,X+2)*(U0Y(Y,X+2)-U0Y(Y,X))... +Kra23(Y+2,X+1)*(U012(Y+2,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y+2,X+1)*(U011(Y+2,X+1)-U011(Y,X))*(0.5)... +Kra32(Y,X)*(U011(Y+1,X-2)-U011(Y,X))*(0.5)... +Kin32(Y,X)*(U012(Y+1,X-2)-U012(Y,X))*(-sqrt(3)/2)... +Kra32(Y+1,X+2)*(U021(Y+1,X+2)-U021(Y,X))*(0.5)... +Kin32(Y+1,X+2)*(U022(Y+1,X+2)-U022(Y,X))*(-sqrt(3)/2)... +Kra31(Y,X)*(U0Y(Y-2,X)-U0Y(Y,X))... +Kra421(Y,X)*(U046(Y+3,X-1)-U046(Y,X))*(-sqrt(25/28))... +Kin421(Y,X)*(U0461(Y+3,X-1)-U0461(Y,X))*(sqrt(3/28))... +Kra422(Y,X)*(U045(Y,X-3)-U045(Y,X))*(sqrt(1/28))... +Kin422(Y,X)*(U0451(Y,X-3)-U0451(Y,X))*(-sqrt(27/28))... +Kra423(Y,X)*(U044(Y-2,X-2)-U044(Y,X))*(sqrt(4/7))... +Kin423(Y,X)*(U0441(Y-2,X-2)-U0441(Y,X))*(-sqrt(3/7))... +Kra413(Y+3,X+1)*(U043(Y+3,X+1)-U043(Y,X))*(-sqrt(25/28))... +Kin413(Y+3,X+1)*(U0431(Y+3,X+1)-U0431(Y,X))*(-sqrt(3/28))... +Kra412(Y,X+3)*(U042(Y,X+3)-U042(Y,X))*(sqrt(1/28))... +Kin412(Y,X+3)*(U0421(Y,X+3)-U0421(Y,X))*(sqrt(27/28))... +Kra441(Y-2,X+2)*(U041(Y-2,X+2)-U041(Y,X))*(sqrt(4/7))... +Kin441(Y-2,X+2)*(U0411(Y-2,X+2)-U0411(Y,X))*(sqrt(3/7)); VDZ(Y,X)=VDZ(Y,X)+Kout2(Y,X)*(U0Z(Y,X-1)-U0Z(Y,X))... +Kout2(Y,X+1)*(U0Z(Y,X+1)-U0Z(Y,X))...


165

+Kout1(Y+1,X)*(U0Z(Y+1,X)-U0Z(Y,X))... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))... +Kout22(Y,X)*(U0Z(Y+2,X-1)-U0Z(Y,X))... +Kout23(Y,X)*(U0Z(Y-1,X-1)-U0Z(Y,X))... +Kout21(Y,X+2)*(U0Z(Y,X+2)-U0Z(Y,X))... +Kout22(Y+2,X+1)*(U0Z(Y+2,X+1)-U0Z(Y,X))... +Kout23(Y-1,X+1)*(U0Z(Y-1,X+1)-U0Z(Y,X))... +Kout32(Y,X)*(U0Z(Y+1,X-2)-U0Z(Y,X))... +Kout32(Y+1,X+2)*(U0Z(Y+1,X+2)-U0Z(Y,X))... +Kout31(Y,X)*(U0Z(Y-2,X)-U0Z(Y,X))... +Kout421(Y,X)*(U0Z(Y+3,X-1)-U0Z(Y,X))... +Kout422(Y,X)*(U0Z(Y,X-3)-U0Z(Y,X))... +Kout423(Y,X)*(U0Z(Y-2,X-2)-U0Z(Y,X))... +Kout413(Y+3,X+1)*(U0Z(Y+3,X+1)-U0Z(Y,X))... +Kout412(Y,X+3)*(U0Z(Y,X+3)-U0Z(Y,X))... +Kout441(Y-2,X+2)*(U0Z(Y-2,X+2)-U0Z(Y,X)); end if mod(X,2)==1 & mod(Y,3)==1; VDX(Y,X)=VDX(Y,X)+Kra2(Y,X)*(U021(Y-1,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra2(Y-1,X+1)*(U011(Y-1,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kin2(Y,X)*(U022(Y-1,X-1)-U022(Y,X))*(0.5)... +Kin2(Y-1,X+1)*(U012(Y-1,X+1)-U012(Y,X))*(-0.5)... +Kin1(Y+1,X)*(U0X(Y+1,X)-U0X(Y,X))... +Kra21(Y,X)*(U022(Y+1,X-1)-U022(Y,X))*(0.5)... +Kin21(Y,X)*(U021(Y+1,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))... +Kra23(Y,X)*(U012(Y-2,X-1)-U012(Y,X))*(-0.5)... +Kin23(Y,X)*(U011(Y-2,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra21(Y-2,X+1)*(U022(Y-2,X+1)-U022(Y,X))*(0.5)... +Kin21(Y-2,X+1)*(U021(Y-2,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X+2)*(U0X(Y,X+2)-U0X(Y,X))... +Kra23(Y+1,X+1)*(U012(Y+1,X+1)-U012(Y,X))*(-0.5)... +Kin23(Y+1,X+1)*(U011(Y+1,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kra32(Y,X)*(U011(Y+1,X-2)-U011(Y,X))*(-sqrt(3)/2)... +Kin32(Y,X)*(U012(Y+1,X-2)-U012(Y,X))*(-0.5)... +Kra32(Y+1,X+2)*(U021(Y+1,X+2)-U021(Y,X))*(sqrt(3)/2)... +Kin32(Y+1,X+2)*(U022(Y+1,X+2)-U022(Y,X))*(0.5)... +Kin31(Y,X)*(U0X(Y-2,X)-U0X(Y,X))... +Kra431(Y,X)*(U046(Y+2,X-1)-U046(Y,X))*(sqrt(3/28))...


166

+Kin431(Y,X)*(U0461(Y+2,X-1)-U0461(Y,X))*(sqrt(25/28))... +Kra432(Y,X)*(U045(Y-1,X-3)-U045(Y,X))*(sqrt(27/28))... +Kin432(Y,X)*(U0451(Y-1,X-3)-U0451(Y,X))*(sqrt(1/28))... +Kra433(Y,X)*(U044(Y-2,X-2)-U044(Y,X))*(sqrt(3/7))... +Kin433(Y,X)*(U0441(Y-2,X-2)-U0441(Y,X))*(sqrt(4/7))... +Kra443(Y+2,X+1)*(U043(Y+2,X+1)-U043(Y,X))*(-sqrt(3/28))... +Kin443(Y+2,X+1)*(U0431(Y+2,X+1)-U0431(Y,X))*(sqrt(25/28))... +Kra442(Y-1,X+3)*(U042(Y-1,X+3)-U042(Y,X))*(-sqrt(27/28))... +Kin442(Y-1,X+3)*(U0421(Y-1,X+3)-U0421(Y,X))*(sqrt(1/28))... +Kra411(Y-2,X+2)*(U041(Y-2,X+2)-U041(Y,X))*(-sqrt(3/7))... +Kin411(Y-2,X+2)*(U0411(Y-2,X+2)-U0411(Y,X))*(sqrt(4/7)); VDY(Y,X)=VDY(Y,X)+Kra2(Y,X)*(U021(Y-1,X-1)-U021(Y,X))*(0.5)... +Kra2(Y-1,X+1)*(U011(Y-1,X+1)-U011(Y,X))*(0.5)... +Kin2(Y,X)*(U022(Y-1,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin2(Y-1,X+1)*(U012(Y-1,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kra1(Y+1,X)*(U0Y(Y+1,X)-U0Y(Y,X))... +Kra21(Y,X)*(U022(Y+1,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y,X)*(U021(Y+1,X-1)-U021(Y,X))*(0.5)... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))... +Kra23(Y,X)*(U012(Y-2,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y,X)*(U011(Y-2,X-1)-U011(Y,X))*(0.5)... +Kra21(Y-2,X+1)*(U022(Y-2,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y-2,X+1)*(U021(Y-2,X+1)-U021(Y,X))*(0.5)... +Kin22(Y,X+2)*(U0Y(Y,X+2)-U0Y(Y,X))... +Kra23(Y+1,X+1)*(U012(Y+1,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y+1,X+1)*(U011(Y+1,X+1)-U011(Y,X))*(0.5)... +Kra32(Y,X)*(U011(Y+1,X-2)-U011(Y,X))*(0.5)... +Kin32(Y,X)*(U012(Y+1,X-2)-U012(Y,X))*(-sqrt(3)/2)... +Kra32(Y+1,X+2)*(U021(Y+1,X+2)-U021(Y,X))*(0.5)... +Kin32(Y+1,X+2)*(U022(Y+1,X+2)-U022(Y,X))*(-sqrt(3)/2)... +Kra31(Y,X)*(U0Y(Y-2,X)-U0Y(Y,X))... +Kra431(Y,X)*(U046(Y+2,X-1)-U046(Y,X))*(-sqrt(25/28))... +Kin431(Y,X)*(U0461(Y+2,X-1)-U0461(Y,X))*(sqrt(3/28))... +Kra432(Y,X)*(U045(Y-1,X-3)-U045(Y,X))*(sqrt(1/28))... +Kin432(Y,X)*(U0451(Y-1,X-3)-U0451(Y,X))*(-sqrt(27/28))... +Kra433(Y,X)*(U044(Y-2,X-2)-U044(Y,X))*(sqrt(4/7))... +Kin433(Y,X)*(U0441(Y-2,X-2)-U0441(Y,X))*(-sqrt(3/7))... +Kra443(Y+2,X+1)*(U043(Y+2,X+1)-U043(Y,X))*(-sqrt(25/28))...


167

+Kin443(Y+2,X+1)*(U0431(Y+2,X+1)-U0431(Y,X))*(-sqrt(3/28))... +Kra442(Y-1,X+3)*(U042(Y-1,X+3)-U042(Y,X))*(sqrt(1/28))... +Kin442(Y-1,X+3)*(U0421(Y-1,X+3)-U0421(Y,X))*(sqrt(27/28))... +Kra411(Y-2,X+2)*(U041(Y-2,X+2)-U041(Y,X))*(sqrt(4/7))... +Kin411(Y-2,X+2)*(U0411(Y-2,X+2)-U0411(Y,X))*(sqrt(3/7)); VDZ(Y,X)=VDZ(Y,X)+Kout2(Y,X)*(U0Z(Y-1,X-1)-U0Z(Y,X))... +Kout2(Y-1,X+1)*(U0Z(Y-1,X+1)-U0Z(Y,X))... +Kout1(Y+1,X)*(U0Z(Y+1,X)-U0Z(Y,X))... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))... +Kout22(Y,X)*(U0Z(Y+1,X-1)-U0Z(Y,X))... +Kout23(Y,X)*(U0Z(Y-2,X-1)-U0Z(Y,X))... +Kout21(Y,X+2)*(U0Z(Y,X+2)-U0Z(Y,X))... +Kout22(Y+1,X+1)*(U0Z(Y+1,X+1)-U0Z(Y,X))... +Kout23(Y-2,X+1)*(U0Z(Y-2,X+1)-U0Z(Y,X))... +Kout32(Y,X)*(U0Z(Y+1,X-2)-U0Z(Y,X))... +Kout32(Y+1,X+2)*(U0Z(Y+1,X+2)-U0Z(Y,X))... +Kout31(Y,X)*(U0Z(Y-2,X)-U0Z(Y,X))... +Kout431(Y,X)*(U0Z(Y+2,X-1)-U0Z(Y,X))... +Kout432(Y,X)*(U0Z(Y-1,X-3)-U0Z(Y,X))... +Kout433(Y,X)*(U0Z(Y-2,X-2)-U0Z(Y,X))... +Kout443(Y+2,X+1)*(U0Z(Y+2,X+1)-U0Z(Y,X))... +Kout442(Y-1,X+3)*(U0Z(Y-1,X+3)-U0Z(Y,X))... +Kout411(Y-2,X+2)*(U0Z(Y-2,X+2)-U0Z(Y,X)); end if mod(X,2)==0 & mod(Y,3)==0; VDX(Y,X)=VDX(Y,X)+Kra2(Y,X)*(U011(Y+1,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra2(Y+1,X+1)*(U021(Y+1,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kin2(Y,X)*(U012(Y+1,X-1)-U012(Y,X))*(-0.5)... +Kin2(Y+1,X+1)*(U022(Y+1,X+1)-U022(Y,X))*(0.5)... +Kin1(Y,X)*(U0X(Y-1,X)-U0X(Y,X))... +Kra21(Y,X)*(U022(Y+2,X-1)-U022(Y,X))*(0.5)... +Kin21(Y,X)*(U021(Y+2,X-1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X)*(U0X(Y,X-2)-U0X(Y,X))... +Kra23(Y,X)*(U012(Y-1,X-1)-U012(Y,X))*(-0.5)... +Kin23(Y,X)*(U011(Y-1,X-1)-U011(Y,X))*(-sqrt(3)/2)... +Kra21(Y-1,X+1)*(U022(Y-1,X+1)-U022(Y,X))*(0.5)... +Kin21(Y-1,X+1)*(U021(Y-1,X+1)-U021(Y,X))*(sqrt(3)/2)... +Kra22(Y,X+2)*(U0X(Y,X+2)-U0X(Y,X))...


168

+Kra23(Y+2,X+1)*(U012(Y+2,X+1)-U012(Y,X))*(-0.5)... +Kin23(Y+2,X+1)*(U011(Y+2,X+1)-U011(Y,X))*(-sqrt(3)/2)... +Kra32(Y,X)*(U021(Y-1,X-2)-U021(Y,X))*(sqrt(3)/2)... +Kin32(Y,X)*(U022(Y-1,X-2)-U022(Y,X))*(0.5)... +Kra32(Y-1,X+2)*(U011(Y-1,X+2)-U011(Y,X))*(-sqrt(3)/2)... +Kin32(Y-1,X+2)*(U012(Y-1,X+2)-U012(Y,X))*(-0.5)... +Kin31(Y+2,X)*(U0X(Y+2,X)-U0X(Y,X))... +Kra441(Y,X)*(U041(Y+2,X-2)-U041(Y,X))*(-sqrt(3/7))... +Kin441(Y,X)*(U0411(Y+2,X-2)-U0411(Y,X))*(sqrt(4/7))... +Kra442(Y,X)*(U042(Y+1,X-3)-U042(Y,X))*(-sqrt(27/28))... +Kin442(Y,X)*(U0421(Y+1,X-3)-U0421(Y,X))*(sqrt(1/28))... +Kra443(Y,X)*(U043(Y-2,X-1)-U043(Y,X))*(-sqrt(3/28))... +Kin443(Y,X)*(U0431(Y-2,X-1)-U0431(Y,X))*(sqrt(25/28))... +Kra423(Y+2,X+2)*(U044(Y+2,X+2)-U044(Y,X))*(sqrt(3/7))... +Kin423(Y+2,X+2)*(U0441(Y+2,X+2)-U0441(Y,X))*(sqrt(4/7))... +Kra432(Y+1,X+3)*(U045(Y+1,X+3)-U045(Y,X))*(sqrt(27/28))... +Kin432(Y+1,X+3)*(U0451(Y+1,X+3)-U0451(Y,X))*(sqrt(1/28))... +Kra431(Y-2,X+1)*(U046(Y-2,X+1)-U046(Y,X))*(sqrt(3/28))... +Kin431(Y-2,X+1)*(U0461(Y-2,X+1)-U0461(Y,X))*(sqrt(25/28)); VDY(Y,X)=VDY(Y,X)+Kra2(Y,X)*(U011(Y+1,X-1)-U011(Y,X))*(0.5)... +Kra2(Y+1,X+1)*(U021(Y+1,X+1)-U021(Y,X))*(0.5)... +Kin2(Y,X)*(U012(Y+1,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin2(Y+1,X+1)*(U022(Y+1,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kra1(Y,X)*(U0Y(Y-1,X)-U0Y(Y,X))... +Kra21(Y,X)*(U022(Y+2,X-1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y,X)*(U021(Y+2,X-1)-U021(Y,X))*(0.5)... +Kin22(Y,X)*(U0Y(Y,X-2)-U0Y(Y,X))... +Kra23(Y,X)*(U012(Y-1,X-1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y,X)*(U011(Y-1,X-1)-U011(Y,X))*(0.5)... +Kra21(Y-1,X+1)*(U022(Y-1,X+1)-U022(Y,X))*(-sqrt(3)/2)... +Kin21(Y-1,X+1)*(U021(Y-1,X+1)-U021(Y,X))*(0.5)... +Kin22(Y,X+2)*(U0Y(Y,X+2)-U0Y(Y,X))... +Kra23(Y+2,X+1)*(U012(Y+2,X+1)-U012(Y,X))*(-sqrt(3)/2)... +Kin23(Y+2,X+1)*(U011(Y+2,X+1)-U011(Y,X))*(0.5)... +Kra32(Y,X)*(U021(Y-1,X-2)-U021(Y,X))*(0.5)... +Kin32(Y,X)*(U022(Y-1,X-2)-U022(Y,X))*(-sqrt(3)/2)... +Kra32(Y-1,X+2)*(U011(Y-1,X+2)-U011(Y,X))*(0.5)... +Kin32(Y-1,X+2)*(U012(Y-1,X+2)-U012(Y,X))*(-sqrt(3)/2)...


169

+Kra31(Y+2,X)*(U0Y(Y+2,X)-U0Y(Y,X))... +Kra441(Y,X)*(U041(Y+2,X-2)-U041(Y,X))*(sqrt(4/7))... +Kin441(Y,X)*(U0411(Y+2,X-2)-U0411(Y,X))*(sqrt(3/7))... +Kra442(Y,X)*(U042(Y+1,X-3)-U042(Y,X))*(sqrt(1/28))... +Kin442(Y,X)*(U0421(Y+1,X-3)-U0421(Y,X))*(sqrt(27/28))... +Kra443(Y,X)*(U043(Y-2,X-1)-U043(Y,X))*(-sqrt(25/28))... +Kin443(Y,X)*(U0431(Y-2,X-1)-U0431(Y,X))*(-sqrt(3/28))... +Kra423(Y+2,X+2)*(U044(Y+2,X+2)-U044(Y,X))*(sqrt(4/7))... +Kin423(Y+2,X+2)*(U0441(Y+2,X+2)-U0441(Y,X))*(-sqrt(3/7))... +Kra432(Y+1,X+3)*(U045(Y+1,X+3)-U045(Y,X))*(sqrt(1/28))... +Kin432(Y+1,X+3)*(U0451(Y+1,X+3)-U0451(Y,X))*(-sqrt(27/28))... +Kra431(Y-2,X+1)*(U046(Y-2,X+1)-U046(Y,X))*(-sqrt(25/28))... +Kin431(Y-2,X+1)*(U0461(Y-2,X+1)-U0461(Y,X))*(sqrt(3/28)); VDZ(Y,X)=VDZ(Y,X)+Kout2(Y,X)*(U0Z(Y+1,X-1)-U0Z(Y,X))... +Kout2(Y+1,X+1)*(U0Z(Y+1,X+1)-U0Z(Y,X))... +Kout1(Y,X)*(U0Z(Y-1,X)-U0Z(Y,X))... +Kout21(Y,X)*(U0Z(Y,X-2)-U0Z(Y,X))... +Kout22(Y,X)*(U0Z(Y+2,X-1)-U0Z(Y,X))... +Kout23(Y,X)*(U0Z(Y-1,X-1)-U0Z(Y,X))... +Kout21(Y,X+2)*(U0Z(Y,X+2)-U0Z(Y,X))... +Kout22(Y+2,X+1)*(U0Z(Y+2,X+1)-U0Z(Y,X))... +Kout23(Y-1,X+1)*(U0Z(Y-1,X+1)-U0Z(Y,X))... +Kout32(Y,X)*(U0Z(Y-1,X-2)-U0Z(Y,X))... +Kout32(Y-1,X+2)*(U0Z(Y-1,X+2)-U0Z(Y,X))... +Kout31(Y+2,X)*(U0Z(Y+2,X)-U0Z(Y,X))... +Kout441(Y,X)*(U0Z(Y+2,X-2)-U0Z(Y,X))... +Kout442(Y,X)*(U0Z(Y+1,X-3)-U0Z(Y,X))... +Kout443(Y,X)*(U0Z(Y-2,X-1)-U0Z(Y,X))... +Kout423(Y+2,X+2)*(U0Z(Y+2,X+2)-U0Z(Y,X))... +Kout432(Y+1,X+3)*(U0Z(Y+1,X+3)-U0Z(Y,X))... +Kout431(Y-2,X+1)*(U0Z(Y-2,X+1)-U0Z(Y,X)); end VX(Y,X)=VX(Y,X)+VDX(Y,X)*TM; U1X(Y,X)=U0X(Y,X)+VX(Y,X)*TAU; VY(Y,X)=VY(Y,X)+VDY(Y,X)*TM; U1Y(Y,X)=U0Y(Y,X)+VY(Y,X)*TAU; V11(Y,X)=VX(Y,X)*(-sqrt(3)/2)+VY(Y,X)*(0.5);

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns U111(Y,X)=U1X(Y,X)*(-sqrt(3)/2)+U1Y(Y,X)*(0.5); V12(Y,X)=VX(Y,X)*(-0.5)+VY(Y,X)*(-sqrt(3)/2); U112(Y,X)=U1X(Y,X)*(-0.5)+U1Y(Y,X)*(-sqrt(3)/2); V21(Y,X)=VX(Y,X)*(sqrt(3)/2)+VY(Y,X)*(0.5); U121(Y,X)=U1X(Y,X)*(sqrt(3)/2)+U1Y(Y,X)*(0.5); V22(Y,X)=VX(Y,X)*(0.5)+VY(Y,X)*(-sqrt(3)/2); U122(Y,X)=U1X(Y,X)*(0.5)+U1Y(Y,X)*(-sqrt(3)/2); V41(Y,X)=VX(Y,X)*(-sqrt(3/7))+VY(Y,X)*(sqrt(4/7)); U141(Y,X)=U1X(Y,X)*(-sqrt(3/7))+U1Y(Y,X)*(sqrt(4/7)); V411(Y,X)=VX(Y,X)*(sqrt(4/7))+VY(Y,X)*(sqrt(3/7)); U1411(Y,X)=U1X(Y,X)*(sqrt(4/7))+U1Y(Y,X)*(sqrt(3/7)); V42(Y,X)=VX(Y,X)*(-sqrt(27/28))+VY(Y,X)*(sqrt(1/28)); U142(Y,X)=U1X(Y,X)*(-sqrt(27/28))+U1Y(Y,X)*(sqrt(1/28)); V421(Y,X)=VX(Y,X)*(sqrt(1/28))+VY(Y,X)*(sqrt(27/28)); U1421(Y,X)=U1X(Y,X)*(sqrt(1/28))+U1Y(Y,X)*(sqrt(27/28)); V43(Y,X)=VX(Y,X)*(-sqrt(3/28))+VY(Y,X)*(-sqrt(25/28)); U143(Y,X)=U1X(Y,X)*(-sqrt(3/28))+U1Y(Y,X)*(-sqrt(25/28)); V431(Y,X)=VX(Y,X)*(sqrt(25/28))+VY(Y,X)*(-sqrt(3/28)); U1431(Y,X)=U1X(Y,X)*(sqrt(25/28))+U1Y(Y,X)*(-sqrt(3/28)); V44(Y,X)=VX(Y,X)*(sqrt(3/7))+VY(Y,X)*(sqrt(4/7)); U144(Y,X)=U1X(Y,X)*(sqrt(3/7))+U1Y(Y,X)*(sqrt(4/7)); V441(Y,X)=VX(Y,X)*(sqrt(4/7))+VY(Y,X)*(-sqrt(3/7)); U1441(Y,X)=U1X(Y,X)*(sqrt(4/7))+U1Y(Y,X)*(-sqrt(3/7)); V45(Y,X)=VX(Y,X)*(sqrt(27/28))+VY(Y,X)*(sqrt(1/28)); U145(Y,X)=U1X(Y,X)*(sqrt(27/28))+U1Y(Y,X)*(sqrt(1/28)); V451(Y,X)=VX(Y,X)*(sqrt(1/28))+VY(Y,X)*(-sqrt(27/28)); U1451(Y,X)=U1X(Y,X)*(sqrt(1/28))+U1Y(Y,X)*(-sqrt(27/28)); V46(Y,X)=VX(Y,X)*(sqrt(3/28))+VY(Y,X)*(-sqrt(25/28)); U146(Y,X)=U1X(Y,X)*(sqrt(3/28))+U1Y(Y,X)*(-sqrt(25/28)); V461(Y,X)=VX(Y,X)*(sqrt(25/28))+VY(Y,X)*(sqrt(3/28)); U1461(Y,X)=U1X(Y,X)*(sqrt(25/28))+U1Y(Y,X)*(sqrt(3/28)); VZ(Y,X)=VZ(Y,X)+VDZ(Y,X)*TM; U1Z(Y,X)=U0Z(Y,X)+VZ(Y,X)*TAU; end end end

170

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns for X=5:LX+4; for Y=5:LY+4; if M(Y,X)==1; U0X(Y,X)=U1X(Y,X); U0Y(Y,X)=U1Y(Y,X); U011(Y,X)=U111(Y,X); U012(Y,X)=U112(Y,X); U021(Y,X)=U121(Y,X); U022(Y,X)=U122(Y,X); U041(Y,X)=U141(Y,X); U042(Y,X)=U142(Y,X); U043(Y,X)=U143(Y,X); U044(Y,X)=U144(Y,X); U045(Y,X)=U145(Y,X); U046(Y,X)=U146(Y,X); U0411(Y,X)=U1411(Y,X); U0421(Y,X)=U1421(Y,X); U0431(Y,X)=U1431(Y,X); U0441(Y,X)=U1441(Y,X); U0451(Y,X)=U1451(Y,X); U0461(Y,X)=U1461(Y,X); U0Z(Y,X)=U1Z(Y,X); end end end end

Fxmax=0.0; Fymax=0.0; Fzmax=0.0;

for X=1:LX+8; for Y=1:LY+8; if M(Y,X)==1; FLX(Y,X)=U0X(Y,X)*M(Y,X)*M1; if Fxmax < abs(FLX(Y,X)); Fxmax = abs(FLX(Y,X));

171

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns end end end end for X=1:LX+8; for Y=1:LY+8; if M(Y,X)==1; FLY(Y,X)=U0Y(Y,X)*M(Y,X)*M1; if Fymax < abs(FLY(Y,X)); Fymax = abs(FLY(Y,X)); end end end end for X=1:LX+8; for Y=1:LY+8; if M(Y,X)==1; FLZ(Y,X)=U0Z(Y,X)*M(Y,X)*M1; if Fzmax < abs(FLZ(Y,X)); Fzmax = abs(FLZ(Y,X)); end end end end

for X=1:LX+8; for Y=1:LY+8; FLX(Y,X) = FLX(Y,X)/Fxmax; FLY(Y,X) = FLY(Y,X)/Fymax; FLZ(Y,X) = FLZ(Y,X)/Fzmax; end end end Fmax=0.0; FXmax=0.0;

172

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns FYmax=0.0; FZmax=0.0; U0=zeros(LY+8,LX+8); for X=1:LX+8; for Y=1:LY+8; % U0(Y,X)=sqrt((U0X(Y,X))^2+(U0Y(Y,X))^2+(U0Z(Y,X))^2); % Gamma-point iLO and TO Mode % U0(Y,X)=sqrt((U0X(Y,X))^2+(U0Y(Y,X))^2); end end

for X=1:LX+8; for Y=1:LY+8; if M(Y,X)==1; if Fmax < abs(U0(Y,X)); Fmax = abs(U0(Y,X)); end end end end for X=1:LX+8; for Y=1:LY+8; if M(Y,X)==1; if FXmax < abs(U0X(Y,X)); FXmax = abs(U0X(Y,X)); end end end end for X=1:LX+8; for Y=1:LY+8; if M(Y,X)==1; if FYmax < abs(U0Y(Y,X));

173

Appendix B. Matlab Code for Phonon Density of States and Mode Patterns FYmax = abs(U0Y(Y,X)); end end end end for X=1:LX+8; for Y=1:LY+8; if M(Y,X)==1; if FZmax < abs(U0Z(Y,X)); FZmax = abs(U0Z(Y,X)); end end end end U=zeros(LY+8,LX+8); UX=zeros(LY+8,LX+8); UY=zeros(LY+8,LX+8); UZ=zeros(LY+8,LX+8); for X=1:LX+8; for Y=1:LY+8; U(Y,X) = U0(Y,X)/Fmax; end end for X=1:LX+8; for Y=1:LY+8; UX(Y,X) = U0X(Y,X)/FXmax; end end for X=1:LX+8; for Y=1:LY+8; UY(Y,X) = U0Y(Y,X)/FYmax; end end

174


175

for X=1:LX+8; for Y=1:LY+8; UZ(Y,X) = U0Z(Y,X)/FZmax; end end save(’graphene_modepattern_9_1350’,’U’,’UX’,’UY’,’UZ’,’U0’,’U0X’,’U0Y’,’U0Z’) toc;

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