COMPUTATIONAL METHODS FOR ELECTROMAGNETIC PHENOMENA ELECTROSTATICS IN SOLVATIONPDF电子书下载

Part Ⅰ Electrostatics in solvation 1

1 Dielectric constant and fluctuation formulae for molecular dynamics 3

1.1 Electrostatics of charges and dipoles 3

1.2 Polarization P and displacement flux D 5

1.2.1 Bound charges induced by polarization 6

1.2.2 Electric field Epol（r） of a polarization density P（r） 7

1.2.3 Singular integral expressions of Epol（r） inside dielectrics 9

1.3 Clausius-Mossotti and Onsager formulae for dielectric constant 9

1.3.1 Clausius-Mossotti formula for non-polar dielectrics 9

1.3.2 Onsager dielectric theory for dipolar liquids 11

1.4 Statistical molecular theory and dielectric fluctuation formulae 16

1.4.1 Statistical methods for polarization density change ΔP 18

1.4.2 Classical electrostatics for polarization density change ΔP 20

1.4.3 Fluctuation formulae for dielectric constant e 21

1.5 Appendices 23

1.5.1 Appendix A： Average field of a charge in a dielectric sphere 23

1.5.2 Appendix B： Electric field due to a uniformly polarized sphere 24

1.6 Summary 25

2 Poisson-Boltzmann electrostatics and analytical approximations 26

2.1 Poisson-Boltzmann （PB） model for electrostatic solvation 26

2.1.1 Debye-Hiickel Poisson-Boltzmann theory 27

2.1.2 Helmholtz double layer and ion size effect 30

2.1.3 Electrostatic solvation energy 34

2.2 Generalized Born （GB） approximations of solvation energy 36

2.2.1 Still’s generalized Born formulism 37

2.2.2 Integral expression for Born radii 37

2.2.3 FFT-based algorithm for the Born radii 39

2.3 Method of images for reaction fields 44

2.3.1 Methods of images for simple geometries 45

2.3.2 Image methods for dielectric spheres 47

2.3.3 Image methods for dielectric spheres in ionic solvent 53

2.3.4 Image methods for multi-layered media 55

2.4 Summary 59

3 Numerical methods for Poisson-Boltzmann equations 60

3.1 Boundary element methods （BEMs） 60

3.1.1 Cauchy principal value （CPV） and Hadamard finite part（p.f.） 61

3.1.2 Surface integral equations for the PB equations 65

3.1.3 Computations of CPV and Hadamard p.f. and collocation BEMs 71

3.2 Finite element methods （FEMs） 82

3.3 Immersed interface methods （IIMs） 85

3.4 Summary 88

4 Fast algorithms for long-range interactions 89

4.1 Ewald sums for charges and dipoles 89

4.2 Particle-mesh Ewald （PME） methods 96

4.3 Fast multipole methods for N-particle electrostatic interactions 98

4.3.1 Multipole expansions 98

4.3.2 A recursion for the local expansions （0 -→ L-level） 102

4.3.3 A recursion for the multipole expansions （L→0-level） 104

4.3.4 A pseudo-code for FMM 104

4.3.5 Conversion operators for electrostatic FMM in R3 105

4.4 Helmholtz FMM of wideband of frequencies for N-current source interactions 107

4.5 Reaction field hybrid model for electrostatics 110

4.6 Summary 116

Part Ⅱ Electromagnetic scattering 117

5 Maxwell equations， potentials， and physical/artificial boundary conditions 119

5.1 Time-dependent Maxwell equations 119

5.1.1 Magnetization M and magnetic field H 120

5.2 Vector and scalar potentials 122

5.2.1 Electric and magnetic potentials for time-harmonic fields 123

5.3 Physical boundary conditions for E and H 125

5.3.1 Interface conditions between dielectric media 125

5.3.2 Leontovich impedance boundary conditions for conductors 127

5.3.3 Sommerfeld and Silver-Müller radiation conditions 129

5.4 Absorbing boundary conditions for E and H 132

5.4.1 One-way wave Engquist-Majda boundary conditions 132

5.4.2 High-order local non-reflecting Bayliss-Turkel conditions 134

5.4.3 Uniaxial perfectly matched layer （UPML） 138

5.5 Summary 144

6 Dyadic Green’s functions in layered media 145

6.1 Singular charge and current sources 145

6.1.1 Singular charge sources 145

6.1.2 Singular Hertz dipole current sources 147

6.2 Dyadic Green’s functions GE（r│r’） and GH（r│r’） 148

6.2.1 Dyadic Green’s functions for homogeneous media 149

6.2.2 Dyadic Green’s functions for layered media 150

6.2.3 Hankel transform for radially symmetric functions 150

6.2.4 Transverse versus longitudinal field components 152

6.2.5 Longitudinal components of Green’s functions 153

6.3 Dyadic Green’s functions for vector potentials GA（r│r’） 157

6.3.1 Sommerfeld potentials 158

6.3.2 Transverse potentials 160

6.4 Fast computation of dyadic Green’s functions 160

6.5 Appendix： Explicit formulae 165

6.5.1 Formulae for G1， G2， and G3， etc. 165

6.5.2 Closed-form formulae for ψ（kρ） 167

6.6 Summary 169

7 High-order methods for surface electromagnetic integral equations 170

7.1 Electric and magnetic field surface integral equations in layered media 170

7.1.1 Integral representations 170

7.1.2 Singular and hyper-singular surface integral equations 175

7.2 Resonance and combined integral equations 182

7.3 Nystr?m collocation methods for Maxwell equations 185

7.3.1 Surface differential operators 185

7.3.2 Locally corrected Nystr?m method for hyper-singular EFIE 186

7.3.3 Nystr?m method for mixed potential EFIE 190

7.4 Galerkin methods and high-order RWG current basis 191

7.4.1 Galerkin method using vector-scalar potentials 191

7.4.2 Functional space for surface current J（r） 192

7.4.3 Basis functions over triangular-triangular patches 194

7.4.4 Basis functions over triangular-quadrilateral patches 198

7.5 Summary 203

8 High-order hierarchical Nédélec edge elements 205

8.1 Nédélec edge elements in H（curl） 205

8.1.1 Finite element method for E or H wave equations 206

8.1.2 Reference elements and Piola transformations 208

8.1.3 Nédélec finite element basis in H（curl） 209

8.2 Hierarchical Nédélec basis functions 217

8.2.1 Construction on 2-D quadrilaterals 218

8.2.2 Construction on 2-D triangles 219

8.2.3 Construction on 3-D cubes 222

8.2.4 Construction on 3-D tetrahedra 223

8.3 Summary 227

9 Time-domain methods - discontinuous Galerkin method and Yee scheme 228

9.1 Weak formulation of Maxwell equations 228

9.2 Discontinuous Galerkin （DG） discretization 229

9.3 Numerical flux h（u-， u＋） 230

9.4 Orthonormal hierarchical basis for DG methods 234

9.4.1 Orthonormal hierarchical basis on quadrilaterals or hexahedra 234

9.4.2 Orthonormal hierarchical basis on triangles or tetrahedra 234

9.5 Explicit formulae of basis functions 236

9.6 Computation of whispering gallery modes （WGMs） with DG methods 238

9.6.1 WGMs in dielectric cylinders 238

9.6.2 Optical energy transfer in coupled micro-cylinders 239

9.7 Finite difference Yee scheme 242

9.8 Summary 245

10 Scattering in periodic structures and surface plasmons 247

10.1 Bloch theory and band gap for periodic structures 247

10.1.1 Bloch theory for 1-D periodic Helmholtz equations 248

10.1.2 Bloch wave expansions 250

10.1.3 Band gaps of photonic structures 250

10.1.4 Plane wave method for band gap calculations 252

10.1.5 Rayleigh-Bloch waves and band gaps by transmission spectra 253

10.2 Finite element methods for periodic structures 257

10.2.1 Nédélec edge element for eigen-mode problems 257

10.2.2 Time-domain finite element methods for periodic array antennas 261

10.3 Physics of surface plasmon waves 265

10.3.1 Propagating plasmons on planar surfaces 265

10.3.2 Localized surface plasmons 268

10.4 Volume integral equation （VIE） for Maxwell equations 270

10.5 Extraordinary optical transmission （EOT） in thin metallic films 273

10.6 Discontinuous Galerkin method for resonant plasmon couplings 274

10.7 Appendix： Auxiliary differential equation （ADE） DG methods for dispersive Maxwell equations 276

10.7.1 Debye material 277

10.7.2 Drude material 282

10.8 Summary 283

11 Schr?dinger equations for waveguides and quantum dots 284

11.1 Generalized DG （GDG） methods for Schr?dinger equations 284

11.1.1 One-dimensional Schr?dinger equations 284

11.1.2 Two-dimensional Schr?dinger equations 287

11.2 GDG beam propagation methods （BPMs） for optical waveguides 289

11.2.1 Guided modes in optical waveguides 289

11.2.2 Discontinuities in envelopes of guided modes 294

11.2.3 GDG-BPM for electric fields 296

11.2.4 GDG-BPM for magnetic fields 299

11.2.5 Propagation of HE11 modes 301

11.3 Volume integral equations for quantum dots 302

11.3.1 One-particle Schr?dinger equation for electrons 302

11.3.2 VIE for electrons in quantum dots 304

11.3.3 Derivation of the VIE for quantum dots embedded in layered media 306

11.4 Summary 309

Part Ⅲ Electron transport 311

12 Quantum electron transport in semiconductors 313

12.1 Ensemble theory for quantum systems 313

12.1.1 Thermal equilibrium of a quantum system 313

12.1.2 Microcanonical ensembles 315

12.1.3 Canonical ensembles 316

12.1.4 Grand canonical ensembles 319

12.1.5 Bose-Einstein and Fermi-Dirac distributions 320

12.2 Density operator ρ for quantum systems 324

12.2.1 One-particle density matrix ρ（x，x’） 328

12.3 Wigner transport equations and Wigner-Moyal expansions 329

12.4 Quantum wave transmission and Landauer current formula 335

12.4.1 Transmission coefficient T（E） 335

12.4.2 Current formula through barriers via T（E） 337

12.5 Non-equilibrium Green’s function （NEGF） and transport current 341

12.5.1 Quantum devices with one contact 342

12.5.2 Quantum devices with two contacts 346

12.5.3 Green’s function and transport current formula 348

12.6 Summary 348

13 Non-equilibrium Green’s function （NEGF） methods for transport 349

13.1 NEGFs for 1-D devices 349

13.1.1 1-D device boundary conditions for Green’s functions 349

13.1.2 Finite difference methods for 1-D device NEGFs 351

13.1.3 Finite element methods for 1-D device NEGFs 353

13.2 NEGFs for 2-D devices 354

13.2.1 2-D device boundary conditions for Green’s functions 354

13.2.2 Finite difference methods for 2-D device NEGFs 357

13.2.3 Finite element methods for 2-D device NEGFs 359

13.3 NEGF simulation of a 29 nm double gate MOSFET 361

13.4 Derivation of Green’s function in 2-D strip-shaped contacts 363

13.5 Summary 364

14 Numerical methods for Wigner quantum transport 365

14.1 Wigner equations for quantum transport 365

14.1.1 Truncation of phase spaces and charge conservation 365

14.1.2 Frensley inflow boundary conditions 367

14.2 Adaptive spectral element method （SEM） 367

14.2.1 Cell averages in k-space 368

14.2.2 Chebyshev collocation methods in x-space 372

14.2.3 Time discretization 372

14.2.4 Adaptive meshes for Wigner distributions 374

14.3 Upwinding finite difference scheme 375

14.3.1 Selections of Lcoh， Ncoh，Lk，and Nk 375

14.3.2 Self-consistent algorithm through the Poisson equation 376

14.3.3 Currents in RTD by NEGF and Wigner equations 377

14.4 Calculation of oscillatory integrals On（z） 378

14.5 Summary 379

15 Hydrodynamic electron transport and finite difference methods 380

15.1 Semi-classical and hydrodynamic models 380

15.1.1 Semi-classical Boltzmann equations 380

15.1.2 Hydrodynamic equations 381

15.2 High-resolution finite difference methods of Godunov type 388

15.3 Weighted essentially non-oscillatory （WENO） finite difference methods 392

15.4 Central differencing schemes with staggered grids 396

15.5 Summary 400

16 Transport models in plasma media and numerical methods 402

16.1 Kinetic and macroscopic magneto-hydrodynamic （MHD） theories 402

16.1.1 Vlasov-Fokker-Planck equations 402

16.1.2 MHD equations for plasma as a conducting fluid 404

16.2 Vlasov-Fokker-Planck （VFP） schemes 410

16.3 Particle-in-cell （PIC） schemes 413

16.4 ？·B＝0 constrained transport methods for MHD equations 414

16.5 Summary 418

References 419

Index 441