Mathematical and Numerical Modelling of Heterostructure Semiconductor Devices: From Theory to Programming

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Mathematical and Numerical Modelling of Heterostructure Semiconductor Devices: From Theory to Programming

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The commercial development of novel semiconductor devices requires that their properties be examined as thoroughly and rapidly as possible. These properties are investigated by obtaining numerical solutions of the highly nonlinear coupled set of equations which govern their behaviour. In particular, the existence of interfaces between different material layers in heterostructures means that quantum solutions must be found in the quantum wells which are formed at these interfaces.This book presents some of the mathematical and numerical techniques associated with the investigation. It begins with introductions to quantum and statistical mechanics. Later chapters then cover finite differences; multigrids; upwinding techniques; simulated annealing; mesh generation; and the reading of computer code in C++; these chapters are self-contained, and do not rely on the reader having met these topics before. The author explains how the methods can be adapted to the specific needs of device modelling, the advantages and disadvantages of each method, the pitfalls to avoid, and practical hints and tips for successful implementation. Sections of computer code are included to illustrate the methods used.Written for anyone who is interested in learning about, or refreshing their knowledge of, some of the basic mathematical and numerical methods involved in device modelling, this book is suitable for advanced undergraduate and graduate students, lecturers and researchers working in the fields of electrical engineering and semiconductor device physics, and for students of other mathematical and physical disciplines starting out in device modelling. <章節目錄>

Part I Overview and physical equations1 Overview of device modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Physical theory and modelling equations . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 Physical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 Modelling equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Mathematical and numerical techniques . . . . . . . . . . . . . . . . . . . . . . . . 141.4 What is in this book, and its limitations . . . . . . . . . . . . . . . . . . . . . . . . 192 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 The physical basis of quantum mechanics . . . . . . . . . . . . . . . . . . . . . . 212.2 The Schrodinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Derivation of the time-dependent Schrodinger equation . . . . 242.2.2 The time-independent Schrodinger equation . . . . . . . . . . . . . . 252.3 Boundary and continuity conditions, and parity. . . . . . . . . . . . . . . . . . 262.3.1 Boundary and continuity conditions . . . . . . . . . . . . . . . . . . . . . 262.3.2 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 The probability current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5 One dimensional motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5.2 Reflection and Transmission coefficients . . . . . . . . . . . . . . . . . 312.5.3 Single finite step forE < V0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5.4 Infinite barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5.5 Infinite square well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5.6 Finite square well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5.7 δ-function potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.5.8 Square potential barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5.9 The sech2 potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.6 Operators and observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.7 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.8 The postulates of quantum mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . 512.9 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.9.1 Solution of the differential equation . . . . . . . . . . . . . . . . . . . . . 542.9.2 The ladder operator method . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.9.3 Oscillations in more than one dimension . . . . . . . . . . . . . . . . . 612.9.4 The displaced harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 632.10 Spherically symmetric potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.10.1 The Schrodinger equation in spherical polar coordinates . . . . 642.10.2 Solution of the angular components . . . . . . . . . . . . . . . . . . . . . 652.10.3 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.10.4 The hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.11 Angular momentum and spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.11.1 The necessity for extra energy levels . . . . . . . . . . . . . . . . . . . . 752.11.2 Generalised angular momentum . . . . . . . . . . . . . . . . . . . . . . . . 762.11.3 Particles with spin 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.11.4 Energy splitting using spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.12 Systems of identical particles: BE and FD statistics . . . . . . . . . . . . . . 822.12.1 Symmetric and antisymmetric wave functions . . . . . . . . . . . . 822.12.2 The Pauli exclusion principle . . . . . . . . . . . . . . . . . . . . . . . . . . 832.12.3 Non-interacting identical particles . . . . . . . . . . . . . . . . . . . . . . 842.13 The Schrodinger equation in device modelling . . . . . . . . . . . . . . . . . . 85Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 Equilibrium thermodynamics and statistical mechanics . . . . . . . . . . . . 893.1 The scope and laws of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 893.1.1 The Zeroth Law of thermodynamics . . . . . . . . . . . . . . . . . . . . 903.1.2 The First Law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . 923.1.3 The Second Law of thermodynamics . . . . . . . . . . . . . . . . . . . . 963.1.4 Properties of the thermodynamic entropy . . . . . . . . . . . . . . . . 993.2 The statistical entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.3 Maximisation of entropy subject to constraints . . . . . . . . . . . . . . . . . . 1023.4 The distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.4.1 The Canonical distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.4.2 The Grand Canonical distribution . . . . . . . . . . . . . . . . . . . . . . . 1063.4.3 The Microcanonical distribution . . . . . . . . . . . . . . . . . . . . . . . . 1073.5 Fermi-Dirac and Bose-Einstein statistics . . . . . . . . . . . . . . . . . . . . . . . 1083.6 The continuous approximation and the Ideal Quantum Gas . . . . . . . . 109Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124 Density of states and applications—1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.1 Electron number and energy densities . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.1.1 Density of states—general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.1.2 Density of states—particles free in three dimensions . . . . . . . 1174.1.3 Density of states—particles free in two dimensions . . . . . . . . 1194.1.4 Density of states—particles free in one dimension . . . . . . . . . 1204.2 Blackbody radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.3 Classical aspects of specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.3.1 The Equipartition of Energy theorem . . . . . . . . . . . . . . . . . . . . 1244.3.2 Examples on the Equipartition of Energy theorem . . . . . . . . . 1254.4 Quantum aspects of specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.4.1 Quantum vibrational aspects: the Einstein solid . . . . . . . . . . . 1274.4.2 Quantum rotational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.4.3 Schottky peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.5 Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.6 Thermionic emission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.7 Semiconductor statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.7.1 Allowed and forbidden bands . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.7.2 The effectivemass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.7.3 Electron and hole densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.7.4 The non-degenerate approximation . . . . . . . . . . . . . . . . . . . . . 140Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415 Density of states and applications—2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.1 Periodic potential: the Bloch theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.2 Heterostructures: position-dependent mass . . . . . . . . . . . . . . . . . . . . . 1475.3 Heterostructures: the effective mass approximation . . . . . . . . . . . . . . 1495.4 Quantum wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.4.1 The general structure of a quantum well . . . . . . . . . . . . . . . . . 1525.4.2 Density of states in quantum wells . . . . . . . . . . . . . . . . . . . . . . 1535.4.3 Quantum wells—particles free in two dimensions . . . . . . . . . 1565.4.4 Quantum wells—particles free in one dimension . . . . . . . . . . 157Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586 The transport equations and the device equations . . . . . . . . . . . . . . . . . . 1616.1 The Boltzmann transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.1.1 Derivation of the BTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.1.2 The relaxation-time approximation . . . . . . . . . . . . . . . . . . . . . 1636.2 Themoments of the BTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1646.2.1 The general moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1656.2.2 First moment: carrier concentration equation . . . . . . . . . . . . . 1666.2.3 Second moment: momentum conservation equation . . . . . . . 1676.2.4 Third moment: energy transport equation . . . . . . . . . . . . . . . . 1686.3 Models based on the BTE moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1696.3.1 The Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.3.2 The simplified energy transport model . . . . . . . . . . . . . . . . . . . 1716.3.3 The drift-diffusion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.4 The Wigner distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.4.1 Definition of the Wigner function . . . . . . . . . . . . . . . . . . . . . . . 1736.4.2 Properties of the Wigner function . . . . . . . . . . . . . . . . . . . . . . . 1736.5 The Wigner transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1746.5.1 Derivation of the Wigner equation . . . . . . . . . . . . . . . . . . . . . . 1756.5.2 Special cases of the Wigner equation . . . . . . . . . . . . . . . . . . . . 1776.5.3 Moments of the Wigner equation . . . . . . . . . . . . . . . . . . . . . . . 1786.6 Description of a typical device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.7 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.7.1 GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.7.2 AlGaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.7.3 InGaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.7.4 The electron mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.8 The Schrodinger equation applied to the HEMT . . . . . . . . . . . . . . . . . 1836.9 The overall nature of the modelling equations . . . . . . . . . . . . . . . . . . . 184Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185Part II Mathematical and numerical methods7 Basic approximation and numerical methods . . . . . . . . . . . . . . . . . . . . . . 1897.1 Reading the C programmes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897.2 Finite differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1937.2.1 Description of themesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1937.2.2 Numerical differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1937.2.3 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1967.2.4 Discretisation of the Poisson and Schrodinger equations . . . . 1987.3 Solution of simultaneous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1997.3.1 Linear equations: direct method . . . . . . . . . . . . . . . . . . . . . . . . 2007.3.2 Linear equations: relaxation method . . . . . . . . . . . . . . . . . . . . 2027.3.3 The Newton method: a brief introduction . . . . . . . . . . . . . . . . 2067.4 Time discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2077.4.1 Explicit and implicit schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 2087.4.2 The ADImethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.5 Function updating and fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2137.5.1 Updating due to altered boundary conditions . . . . . . . . . . . . . 2137.5.2 Discretising mixed boundary conditions . . . . . . . . . . . . . . . . . 2167.5.3 Modelling abrupt junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2188 Fermi and associated integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218.1 Definition of the Fermi integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218.1.1 The standard Fermi integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 2228.1.2 The associated Fermi integrals . . . . . . . . . . . . . . . . . . . . . . . . . 2228.2 Approximation of the associated integrals . . . . . . . . . . . . . . . . . . . . . . 2238.3 Implementation of the approximation scheme . . . . . . . . . . . . . . . . . . . 2258.3.1 Method of implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2258.3.2 Results of the implementation. . . . . . . . . . . . . . . . . . . . . . . . . . 2268.3.3 Improvements to the scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 2278.4 Calculation of the standard Fermi integrals . . . . . . . . . . . . . . . . . . . . . 227Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2289 The upwinding method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2299.1 Description of the upwinding approach . . . . . . . . . . . . . . . . . . . . . . . . 2299.2 Upwinding applied to device equations . . . . . . . . . . . . . . . . . . . . . . . . 2309.3 Upwinding in terms of the C-function . . . . . . . . . . . . . . . . . . . . . . . . . 2339.3.1 Definition of the C-function . . . . . . . . . . . . . . . . . . . . . . . . . . . 2339.3.2 Properties of the C-function and related functions . . . . . . . . . 2349.4 Upwinding using the C-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2379.5 Numerical diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2399.6 The limit of uniformtemperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24410 Solution of equations: the Newton and reduced method . . . . . . . . . . . . . 24710.1 The Newtonmethod for one variable . . . . . . . . . . . . . . . . . . . . . . . . . . 24710.2 Error analysis of the Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . 24910.3 The multi-variable Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25110.4 The reduced Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25310.5 The Newton method applied to device modelling . . . . . . . . . . . . . . . . 25410.5.1 The reduced method applied to device modelling . . . . . . . . . . 25410.5.2 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25711 Solution of equations: the phaseplane method . . . . . . . . . . . . . . . . . . . . . 25911.1 The basis of the phaseplane method . . . . . . . . . . . . . . . . . . . . . . . . . . . 25911.2 The phaseplane method for one variable . . . . . . . . . . . . . . . . . . . . . . . . 26011.3 Discretisation of the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26111.4 The condition for a stable solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26211.5 Exact correspondence between the differential and discretisedequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26511.6 A one-variable example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26611.7 The phaseplane equations for several variables . . . . . . . . . . . . . . . . . . 26711.8 Connection with the Newton and SOR/SUR schemes . . . . . . . . . . . . 27011.9 Error analysis of the phase plane method . . . . . . . . . . . . . . . . . . . . . . . 27111.9.1 One-variable case using exact derivatives . . . . . . . . . . . . . . . . 27111.9.2 One-variable case using central difference derivatives . . . . . . 27211.9.3 Multi-variable case using exact derivatives . . . . . . . . . . . . . . . 27211.9.4 Multi-variable case using central difference derivatives . . . . . 27511.9.5 Multi-variable case using forward difference derivatives . . . . 27611.10 The phaseplane method applied to device modelling . . . . . . . . . . . . . 27711.11 Case study: a four-layer four-contact HEMT . . . . . . . . . . . . . . . . . . . 28112 Solution of equations: the multigrid method . . . . . . . . . . . . . . . . . . . . . . . 28312.1 Description of the multigrid method . . . . . . . . . . . . . . . . . . . . . . . . . . 28312.2 Moving between grids: restriction and prolongation . . . . . . . . . . . . . 28612.3 Implementation of the multigrid method . . . . . . . . . . . . . . . . . . . . . . . 28712.4 Efficiency of the multigrid scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29012.5 Multigrids applied to device modelling . . . . . . . . . . . . . . . . . . . . . . . . 29212.5.1 Case study 1: application to a one-dimensional device . . . . . 29312.5.2 Case study 2: application to a two-dimensional device . . . . . 29913 Approximate and numerical solutions of the Schrodinger equation . . 30313.1 The WKB approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30313.1.1 The basis of theWKB method . . . . . . . . . . . . . . . . . . . . . . . . . 30313.1.2 The limit of the approximation . . . . . . . . . . . . . . . . . . . . . . . . . 30513.1.3 The connection formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30613.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30913.2 Time independent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . 31313.2.1 The first order non-degenerate case . . . . . . . . . . . . . . . . . . . . . 31413.2.2 The second order non-degenerate case . . . . . . . . . . . . . . . . . . . 31513.2.3 The degenerate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31613.2.4 Example: linear perturbation to the harmonic oscillator . . . . 31613.3 Time dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 31813.4 The Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32013.5 Discretisation of the Schrodinger equation . . . . . . . . . . . . . . . . . . . . . 32213.5.1 Discretisation in two dimensions . . . . . . . . . . . . . . . . . . . . . . . 32313.5.2 Discretisation in one dimension . . . . . . . . . . . . . . . . . . . . . . . . 32413.6 Numerical solution: the iteration method. . . . . . . . . . . . . . . . . . . . . . . 32513.6.1 The basis of the iterationmethod . . . . . . . . . . . . . . . . . . . . . . . 32513.6.2 Numerical implementation of the iteration method . . . . . . . . 32613.7 Numerical solution: the trial function method . . . . . . . . . . . . . . . . . . . 32813.7.1 The basis of the trial function method . . . . . . . . . . . . . . . . . . . 32913.7.2 Choice of trial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33313.7.3 Numerical implementation of the trial function method . . . . 33313.8 Numerical solution: the matrix method . . . . . . . . . . . . . . . . . . . . . . . . 33614 Genetic algorithms and simulated annealing . . . . . . . . . . . . . . . . . . . . . . 33914.1 How genetic algorithms work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34014.2 Chromosome representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34314.3 The genetic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34514.3.1 Stage 1: Roulette wheel selection . . . . . . . . . . . . . . . . . . . . . . . 34614.3.2 Stage 2: Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34814.3.3 Stage 3:Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35014.4 The multivariable and multifunction cases . . . . . . . . . . . . . . . . . . . . . 35114.4.1 Themultivariable case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35214.4.2 The multifunction case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35314.4.3 Example: maximising a function of two variables . . . . . . . . . 35414.5 Refinements to the GA approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35514.5.1 Differentialmutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35514.5.2 Contractive mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35614.5.3 Range refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35814.6 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35814.6.1 How simulated annealing works . . . . . . . . . . . . . . . . . . . . . . . . 35914.6.2 Acceptance function based on MB statistics . . . . . . . . . . . . . . 36214.6.3 Acceptance function based on Tsallis statistics . . . . . . . . . . . . 36314.6.4 Acceptance function based on BE statistics . . . . . . . . . . . . . . . 36514.6.5 The cooling schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36814.7 Application: approximation to the Associated Fermi integrals . . . . . 37214.7.1 Approximationmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37214.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37415 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37715.1 Overview of grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37815.2 Functions of a grid generation programme . . . . . . . . . . . . . . . . . . . . . 37915.3 Results from the programme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385A The theory of contractive mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401