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