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Computational Methods for Electromagnetic Phenomena

Book Description

A unique and comprehensive graduate text and reference on numerical methods for electromagnetic phenomena, from atomistic to continuum scales, in biology, optical-to-micro waves, photonics, nanoelectronics and plasmas. The state-of-the-art numerical methods described include: • Statistical fluctuation formulae for the dielectric constant • Particle-Mesh-Ewald, Fast-Multipole-Method and image-based reaction field method for long-range interactions • High-order singular/hypersingular (Nyström collocation/Galerkin) boundary and volume integral methods in layered media for Poisson–Boltzmann electrostatics, electromagnetic wave scattering and electron density waves in quantum dots • Absorbing and UPML boundary conditions • High-order hierarchical Nédélec edge elements • High-order discontinuous Galerkin (DG) and Yee finite difference time-domain methods • Finite element and plane wave frequency-domain methods for periodic structures • Generalized DG beam propagation method for optical waveguides • NEGF(Non-equilibrium Green's function) and Wigner kinetic methods for quantum transport • High-order WENO and Godunov and central schemes for hydrodynamic transport • Vlasov-Fokker-Planck and PIC and constrained MHD transport in plasmas

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright
  4. Dedication
  5. Contents
  6. Foreword
  7. Preface
  8. Part I: Electrostatics in Solvation
    1. 1 - Dielectric Constant and Fluctuation Formulae for Molecular Dynamics
      1. 1.1 Electrostatics of Charges and Dipoles
      2. 1.2 Polarization P and Displacement Flux D
        1. 1.2.1 Bound Charges Induced by Polarization
        2. 1.2.2 Electric Field Epol(r) of a Polarization Density P(r)
        3. 1.2.3 Singular Integral Expressions of Epol(r) inside Dielectrics
      3. 1.3 Clausius–Mossotti and Onsager Formulae for Dielectric Constant
        1. 1.3.1 Clausius–Mossotti Formula for Non-Polar Dielectrics
        2. 1.3.2 Onsager Dielectric Theory for Dipolar Liquids
      4. 1.4 Statistical Molecular Theory and Dielectric Fluctuation Formulae
        1. 1.4.1 Statistical Methods for Polarization Density Change ∆P
        2. 1.4.2 Classical Electrostatics for Polarization Density Change ∆P
        3. 1.4.3 Fluctuation Formulae for Dielectric Constant ∈
      5. 1.5 Appendices
        1. 1.5.1 Appendix A: Average Field of a Charge in a Dielectric Sphere
        2. 1.5.2 Appendix B: Electric Field due to a Uniformly Polarized Sphere
      6. 1.6 Summary
    2. 2 - Poisson–Boltzmann Electrostatics and Analytical Approximations
      1. 2.1 Poisson–Boltzmann (PB) Model for Electrostatic Solvation
        1. 2.1.1 Debye–Hückel Poisson–Boltzmann Theory
        2. 2.1.2 Helmholtz Double Layer and Ion Size Effect
        3. 2.1.3 Electrostatic Solvation Energy
      2. 2.2 Generalized Born (GB) Approximations of Solvation Energy
        1. 2.2.1 Still’s Generalized Born Formulism
        2. 2.2.2 Integral Expression for Born Radii
        3. 2.2.3 FFT-Based Algorithm for the Born Radii
      3. 2.3 Method of Images for Reaction Fields
        1. 2.3.1 Methods of Images for Simple Geometries
        2. 2.3.2 Image Methods for Dielectric Spheres
        3. 2.3.3 Image Methods for Dielectric Spheres in Ionic Solvent
        4. 2.3.4 Image Methods for Multi-Layered Media
      4. 2.4 Summary
    3. 3 - Numerical Methods for Poisson–Boltzmann Equations
      1. 3.1 Boundary Element Methods (BEMs)
        1. 3.1.1 Cauchy Principal Value (CPV) and Hadamard Finite Part (p.f.)
        2. 3.1.2 Surface Integral Equations for the PB Equations
        3. 3.1.3 Computations of CPV and Hadamard p.f. and Collocation BEMs
      2. 3.2 Finite Element Methods (FEMs)
      3. 3.3 Immersed Interface Methods (IIMs)
      4. 3.4 Summary
    4. 4 - Fast Algorithms for Long-Range Interactions
      1. 4.1 Ewald Sums for Charges and Dipoles
      2. 4.2 Particle-Mesh Ewald (PME) Methods
      3. 4.3 Fast Multipole Methods for N-Particle Electrostatic Interactions
        1. 4.3.1 Multipole Expansions
        2. 4.3.2 A Recursion for the Local Expansions (0 → L-level)
        3. 4.3.3 A Recursion for the Multipole Expansions (L → 0-level)
        4. 4.3.4 A Pseudo-Code for FMM
        5. 4.3.5 Conversion Operators for Electrostatic FMM in ℝ3
      4. 4.4 Helmholtz FMM of Wideband of Frequencies for N-Current Source Interactions
      5. 4.5 Reaction Field Hybrid Model for Electrostatics
      6. 4.6 Summary
  9. Part II: Electromagnetic Scattering
    1. 5 - Maxwell Equations, Potentials, and Physical/Artificial Boundary Conditions
      1. 5.1 Time-Dependent Maxwell Equations
        1. 5.1.1 Magnetization M and Magnetic Field H
      2. 5.2 Vector and Scalar Potentials
        1. 5.2.1 Electric and Magnetic Potentials for Time-Harmonic Fields
      3. 5.3 Physical Boundary Conditions for E and H
        1. 5.3.1 Interface Conditions between Dielectric Media
        2. 5.3.2 Leontovich Impedance Boundary Conditions for Conductors
        3. 5.3.3 Sommerfeld and Silver–Müller Radiation Conditions
      4. 5.4 Absorbing Boundary Conditions for E and H
        1. 5.4.1 One-Way Wave Engquist–Majda Boundary Conditions
        2. 5.4.2 High-order Local Non-Reflecting Bayliss–Turkel Conditions
        3. 5.4.3 Uniaxial Perfectly Matched Layer (UPML)
      5. 5.5 Summary
    2. 6 - Dyadic Green’s Functions in Layered Media
      1. 6.1 Singular Charge and Current Sources
        1. 6.1.1 Singular Charge Sources
        2. 6.1.2 Singular Hertz Dipole Current Sources
      2. 6.2 Dyadic Green’s Functions ḠE (r|r′) and ḠH (r|r′)
        1. 6.2.1 Dyadic Green’s Functions for Homogeneous Media
        2. 6.2.2 Dyadic Green’s Functions for Layered Media
        3. 6.2.3 Hankel Transform for Radially Symmetric Functions
        4. 6.2.4 Transverse Versus Longitudinal Field Components
        5. 6.2.5 Longitudinal Components of Green’s Functions
      3. 6.3 Dyadic Green’s Functions for Vector Potentials ḠA (r|r′)
        1. 6.3.1 Sommerfeld Potentials
        2. 6.3.2 Transverse Potentials
      4. 6.4 Fast Computation of Dyadic Green’s Functions
      5. 6.5 Appendix: Explicit Formulae
        1. 6.5.1 Formulae for G̃1, G̃2, and G̃3, etc.
        2. 6.5.2 Closed-Form Formulae for ψ̃(kρ)
      6. 6.6 Summary
    3. 7 - High-Order Methods for Surface Electromagnetic Integral Equations
      1. 7.1 Electric and Magnetic Field Surface Integral Equations in Layered Media
        1. 7.1.1 Integral Representations
        2. 7.1.2 Singular and Hyper-Singular Surface Integral Equations
      2. 7.2 Resonance and Combined Integral Equations
      3. 7.3 Nyström Collocation Methods for Maxwell Equations
        1. 7.3.1 Surface Differential Operators
        2. 7.3.2 Locally Corrected Nyström Method for Hyper-Singular EFIE
        3. 7.3.3 Nyström Method for Mixed Potential EFIE
      4. 7.4 Galerkin Methods and High-Order RWG Current Basis
        1. 7.4.1 Galerkin Method Using Vector–Scalar Potentials
        2. 7.4.2 Functional Space for Surface Current J(r)
        3. 7.4.3 Basis Functions over Triangular–Triangular Patches
        4. 7.4.4 Basis Functions Over Triangular–Quadrilateral Patches
      5. 7.5 Summary
    4. 8 - High-Order Hierarchical Nédélec Edge Elements
      1. 8.1 Nédélec Edge Elements in H(curl)
        1. 8.1.1 Finite Element Method for E or H Wave Equations
        2. 8.1.2 Reference Elements and Piola Transformations
        3. 8.1.3 Nédélec Finite Element Basis in H(curl)
      2. 8.2 Hierarchical Nédélec Basis Functions
        1. 8.2.1 Construction on 2-D Quadrilaterals
        2. 8.2.2 Construction on 2-D Triangles
        3. 8.2.3 Construction on 3-D Cubes
        4. 8.2.4 Construction on 3-D Tetrahedra
      3. 8.3 Summary
    5. 9 - Time-Domain Methods – Discontinuous Galerkin Method and Yee Scheme
      1. 9.1 Weak Formulation of Maxwell Equations
      2. 9.2 Discontinuous Galerkin (DG) Discretization
      3. 9.3 Numerical Flux h(u–, u+)
      4. 9.4 Orthonormal Hierarchical Basis for DG Methods
        1. 9.4.1 Orthonormal Hierarchical Basis on Quadrilaterals or Hexahedra
        2. 9.4.2 Orthonormal Hierarchical Basis on Triangles or Tetrahedra
      5. 9.5 Explicit Formulae of Basis Functions
      6. 9.6 Computation of Whispering Gallery Modes (WGMs) with DG Methods
        1. 9.6.1 WGMs in Dielectric Cylinders
        2. 9.6.2 Optical Energy Transfer in Coupled Micro-Cylinders
      7. 9.7 Finite Difference Yee Scheme
      8. 9.8 Summary
    6. 10 - Scattering in Periodic Structures and Surface Plasmons
      1. 10.1 Bloch Theory and Band Gap for Periodic Structures
        1. 10.1.1 Bloch Theory for 1-D Periodic Helmholtz Equations
        2. 10.1.2 Bloch Wave Expansions
        3. 10.1.3 Band Gaps of Photonic Structures
        4. 10.1.4 Plane Wave Method for Band Gap Calculations
        5. 10.1.5 Rayleigh–Bloch Waves and Band Gaps by Transmission Spectra
      2. 10.2 Finite Element Methods for Periodic Structures
        1. 10.2.1 Nédélec Edge Element for Eigen-Mode Problems
        2. 10.2.2 Time-domain Finite Element Methods for Periodic Array Antennas
      3. 10.3 Physics of Surface Plasmon Waves
        1. 10.3.1 Propagating Plasmons on Planar Surfaces
        2. 10.3.2 Localized Surface Plasmons
      4. 10.4 Volume Integral Equation (VIE) for Maxwell Equations
      5. 10.5 Extraordinary Optical Transmission (EOT) in thin Metallic Films
      6. 10.6 Discontinuous Galerkin Method for Resonant Plasmon Couplings
      7. 10.7 Appendix: Auxiliary Differential Equation (ADE) DG Methods for Dispersive Maxwell Equations
        1. 10.7.1 Debye Material
        2. 10.7.2 Drude Material
      8. 10.8 Summary
    7. 11 - Schrödinger Equations for Waveguides and Quantum Dots
      1. 11.1 Generalized DG (GDG) Methods for Schrödinger Equations
        1. 11.1.1 One-Dimensional Schrödinger Equations
        2. 11.1.2 Two-Dimensional Schrödinger Equations
      2. 11.2 GDG Beam Propagation Methods (BPMs) for Optical Waveguides
        1. 11.2.1 Guided Modes in Optical Waveguides
        2. 11.2.2 Discontinuities in Envelopes of Guided Modes
        3. 11.2.3 GDG-BPM for Electric Fields
        4. 11.2.4 GDG-BPM for Magnetic Fields
        5. 11.2.5 Propagation of HE11 Modes
      3. 11.3 Volume Integral Equations for Quantum Dots
        1. 11.3.1 One-Particle Schrödinger Equation for Electrons
        2. 11.3.2 VIE for Electrons in Quantum Dots
        3. 11.3.3 Derivation of the VIE for Quantum Dots Embedded in Layered Media
      4. 11.4 Summary
  10. Part III: Electron Transport
    1. 12 - Quantum Electron Transport in Semiconductors
      1. 12.1 Ensemble Theory for Quantum Systems
        1. 12.1.1 Thermal Equilibrium of a Quantum System
        2. 12.1.2 Microcanonical Ensembles
        3. 12.1.3 Canonical Ensembles
        4. 12.1.4 Grand Canonical Ensembles
        5. 12.1.5 Bose–Einstein and Fermi–Dirac Distributions
      2. 12.2 Density Operator ρˆ for Quantum Systems
        1. 12.2.1 One-Particle Density Matrix ρ(x, x′)
      3. 12.3 Wigner Transport Equations and Wigner–Moyal Expansions
      4. 12.4 Quantum Wave Transmission and Landauer Current Formula
        1. 12.4.1 Transmission Coefficient T(E)
        2. 12.4.2 Current Formula Through Barriers Via T(E)
      5. 12.5 Non-Equilibrium Green’s Function (NEGF) and Transport Current
        1. 12.5.1 Quantum Devices with one Contact
        2. 12.5.2 Quantum Devices with Two Contacts
        3. 12.5.3 Green’s Function and Transport Current Formula
      6. 12.6 Summary
    2. 13 - Non-Equilibrium Green’s Function (NEGF) Methods for Transport
      1. 13.1 NEGFs for 1-D Devices
        1. 13.1.1 1-D Device Boundary Conditions for Green’s Functions
        2. 13.1.2 Finite Difference Methods for 1-D Device NEGFs
        3. 13.1.3 Finite Element Methods for 1-D Device NEGFs
      2. 13.2 NEGFs for 2-D Devices
        1. 13.2.1 2-D Device Boundary Conditions for Green’s Functions
        2. 13.2.2 Finite Difference Methods for 2-D Device NEGFs
        3. 13.2.3 Finite Element Methods for 2-D Device NEGFs
      3. 13.3 NEGF Simulation of a 29 nm Double Gate MOSFET
      4. 13.4 Derivation of Green’s Function in 2-D Strip-Shaped Contacts
      5. 13.5 Summary
    3. 14 - Numerical Methods for Wigner Quantum Transport
      1. 14.1 Wigner Equations for Quantum Transport
        1. 14.1.1 Truncation of Phase Spaces and Charge Conservation
        2. 14.1.2 Frensley Inflow Boundary Conditions
      2. 14.2 Adaptive Spectral Element Method (SEM)
        1. 14.2.1 Cell Averages in k-space
        2. 14.2.2 Chebyshev Collocation Methods in x-space
        3. 14.2.3 Time Discretization
        4. 14.2.4 Adaptive Meshes for Wigner Distributions
      3. 14.3 Upwinding Finite Difference Scheme
        1. 14.3.1 Selections of Lcoh, Ncoh, Lk, and Nk
        2. 14.3.2 Self-Consistent Algorithm Through the Poisson Equation
        3. 14.3.3 Currents in RTD by NEGF and Wigner Equations
      4. 14.4 Calculation of Oscillatory Integrals On(z)
      5. 14.5 Summary
    4. 15 - Hydrodynamic Electron Transport and Finite Difference Methods
      1. 15.1 Semi-Classical and Hydrodynamic Models
        1. 15.1.1 Semi-Classical Boltzmann Equations
        2. 15.1.2 Hydrodynamic Equations
      2. 15.2 High-Resolution Finite Difference Methods of Godunov Type
      3. 15.3 Weighted Essentially Non-Oscillatory (WENO) Finite Difference Methods
      4. 15.4 Central Differencing Schemes with Staggered Grids
      5. 15.5 Summary
    5. 16 - Transport Models in Plasma Media and Numerical Methods
      1. 16.1 Kinetic and Macroscopic Magneto-Hydrodynamic (MHD) Theories
        1. 16.1.1 Vlasov–Fokker–Planck Equations
        2. 16.1.2 MHD Equations for Plasma as a Conducting Fluid
      2. 16.2 Vlasov–Fokker–Planck (VFP) Schemes
      3. 16.3 Particle-In-Cell (PIC) Schemes
      4. 16.4 ∇ · B = 0 Constrained Transport Methods for MHD Equations
      5. 16.5 Summary
  11. References
  12. Index