Simulations for Solid State Physics

An interactive resource for students and teachers

Robert H. Silsbee and Jörg Dräger Cornea University

CAMBRIDGE UNIVERSITY PRESS

Contents

Preface xiii

S y s t e m requirements xvi i

1 H o w t o use th is book 1

1.1 First look at the simulations 2 1.2 Using this guidebook 4 1.3 Who the package is designed for 7

2 "bravais" - Crysta l s tructure and x-ray diffraction 9 2.1 Introduction 10 2.2 Crystal lattice versus crystal structure 10 2.3 Lattice planes 12 2.4 Reciprocal lattice 14 2.5 Diffraction conditions 16 2.6 Atomic form factor 19 2.7 Structure factor : 21 2.8 Summary 26 2.A Deeper exploration 27 2.B "bravais" - the program 27

3 "laue" - Diffraction in perfect arid imperfect crysta ls 30 3.1 Introduction 31 3.2 Bragg condition 31 3.3 Atomic form factor . . . . ; 34 3.4 Structure factor 38 3.5 Order-disorder* 43 3.6 Atomic displacements and thermal disorder 45 3.7 Quasicrystals* 51 3.8 Summary 52 3.A Deeper exploration 53 3.B Periodically modulated crystal 53

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viii Contents

3.C "laue" - the program 54

4 "born" — Lat t i ce d y n a m i c s in one d i m e n s i o n 61 4.1 Introduction 62 4.2 Plane waves - infinite medium 63 4.3 Normal modes of finite chain 68 4.4 Wave packets 70 4.5 Pulse propagation* 75 4.6 Impurities 75 4.7 Anharmonicity** 78 4.8 Summary 80 4.A Deeper exploration 80 4.B "born" - the program 81

5 "debye" - Lat t ice d y n a m i c s and heat capaci ty 87 5.1 Introduction 88 5.2 Brillouin zone 88 5.3 Dispersion relations 90 5.4 Density of states 95 5.5 Thermal energy 102 5.6 Heat capacity 103 5.7 Summary 106 5.A Deeper exploration 106 5.B "debye" - the program 107

6 "drude" - D y n a m i c s of t h e classical free e lec tron gas 113 6.1 Introduction 114 6.2 No applied fields 114 6.3 Electric field (DC) 116 6.4 Noise* 119 6.5 Magnetic field 121 6.6 Crossed fields 123 6.7 AC electric field 129 6.8 Summary 131 6.A Deeper exploration 132 6.B "drude" - the program 132

7 "Sommerfeld" - D y n a m i c s of q u a n t u m free e lec trons 137 7.1 Introduction 138 7.2 Ground State 138 7.3 Electric field 141 7.4 Crossed E- and B-fields 148 7.5 Heating of the electron gas** 149

Contents ix

7.6 Summary 152 7.A Deeper exploration 152 7.B "sommerfeld" - the program 152

8 "bloch" - E lec tron energy bands and s ta t e s 158 8.1 Introduction 159 8.2 Solutions in one dimension 159 8.3 In-band states 162 8.4 Gap states* 166 8.5 Nearly free electrons 167 8.6 Tight binding (TB) 169 8.7 'Atomic' and 'molecular' potentials 172 8.8 Summary 173 8.A Deeper exploration 174 8.B "bloch" - the program 175

9 "ziman" — D y n a m i c s of a s ingle band-e l ec tron 182 9.1 Introduction 183 9.2 £(k) and the contour plot 183 9.3 Group velocities 184 9.4 Acceleration in electric fields 185 9.5 Magnetic field 189 9.6 Crossed fields** 193 9.7 Summary 193 9.A Deeper exploration 194 9.B "ziman" - the program 194

10 "peierls" - Transport in a part ial ly füll e l ec tron band 197 10.1 Introduction 198 10.2 Nearly empty band 198 10.3 Half-full band 199 10.4 Nearly füll band: holes 202 10.5 Anisotropy* 207 10.6 Summary - 208 10.A Deeper exploration 208 10.B Preferred Fermi energies 208 10.C "peierls" - the program 209

11 "fermi" — Carrier dens i t ies in s emiconductors 213 11.1 Introduction 214 11.2 Pure (intrinsic) material 214 11.3 Doped (extrinsic) material 218 11.4 Summary 223

X Contents

I L A Deeper exploration 223 11.B "fermi" - the program 224

12 "poisson" - B a n d b e n d i n g in s emiconductors 229 12.1 Introduction 230 12.2 pn-junction 231 12.3 Metal-insulator-semiconductor (MIS) diode 239 12.4 Contact potential and work functions** 245 12.5 Summary 248 12.A Deeper exploration 248 12.B "poisson" - the program 249

13 "ising" - Is ing m o d e l and ferromagnet i sm 255 13.1 Introduction 256 13.2 Ising Simulation 256 13.3 High temperature susceptibility 258 13.4 Ferromagnetic state 261 13.5 Coarsening and nucleation 263 13.6 Critical fmctuations* 265 13.7 Correlation lengths* 266 13.8 Thermodynamic limit* 266 13.9 Summary 267 13.A Deeper exploration 268 13.B "ising" - the program 269

14 "neel" — Ant i f erromagne t i sm and m a g n e t i c d o m a i n s 274 14.1 Introduction 275 14.2 Antiferromagnetism 276 14.3 Ferromagnetic domains 283 14.4 Summary 288 14.A Deeper exploration 288 14.B Dipolar and field energies 289 14.C "neel" - the program 291

15 "burgers" — D i s locat ions and plast ic flow 295 15.1 Introduction 296 15.2 About the "burgers" model 296 15.3 Perfect crystal 298 15.4 Single dislocation 299 15.5 Critical shear stress with dislocations 303 15.6 Impurity pinning 306 15.7 Dislocation interactions 308 15.8 Frank-Read source 308

Contents xi

15.9 Summary 310 15.A Deeper exploration 310 15.B "burgers" - the program 311

Epilogue 314

Appendices 316

A How to run and install the SSS programs 317 A.l Information for all platforms 318 A.2 Microsoft Windows 318 A.3 Apple Power Macintosh 321 A.4 Versions of Unix 322 A.5 Compiling the simulations for yourself 323

B Detailed System requirements 324 B.l Introduction 325 B.2 For all platforms 325 B.3 Microsoft Windows 326 B.4 Apple MacOS 327 B.5 Versions of Unix 327

C Presets and file editing 329 C.l Presets provided with SSS package 330 C.2 Customized presets 331 C.3 Exercises and other editing 334

Bibliography 336

Index 344