E. A. B. Cole

Genetic algorithms and simulated annealing

Many applications require either the maximisation or minimisation of a function. For example, in many fields of theoretical physics, a sum of least squares must be minimised, or the energy of a system must be minimised. Many of the standard numerical techniques which exist for the optimisation of functions apply to functions which can be specified in a relatively straightforward manner. However, many of the functions which arise in the modelling of semiconductor devices cannot be specified in this manner, and some of these functions can only be evaluated using many subsidiary functions and integrals. Genetic algorithms allow such sets of functions to be optimised using random processes to “breed” generations of solutions which should converge to the optimal solution. This chapter presents an introduction to genetic algorithms and simulated annealing. Sections of code written in C++ are used to illustrate the implementation of aspects of the subjects. The coefficients involved in the numerical approximation of the associated Fermi integrals are evaluated using these methods.

Basic approximation and numerical methods

There exists such a vast literature on the subjects of approximation and numerical methods that it is only possible to include a very small portion in this book. The contents of this chapter will be concerned with only some of the techniques which will be found most useful in applications to device modelling. Later, some sections of programme code written in the C programming language (with a small subset of C++) are given to illustrate the theory, and Sect. 7.1 contains an explanation of some aspects of this language for readers whose first programming language is not C. Later sections cover finite differences, solution of simultaneous equations, time discretisation, phase plane methods, and other techniques which will be useful specifically for device modelling.

Equilibrium thermodynamics and statistical mechanics

The subject of thermodynamics deals with general systems and the relationships between the different macroscopic variables which describe each system. The first three Laws of thermodynamics will be introduced. In particular, the Second Law of thermodynamics leads to the related ideas of absolute temperature and entropy. Using the fact that the entropy of a system cannot decrease as it approaches equilibrium, the statistical entropy is introduced with properties which are related to those of the thermodynamic entropy. This statistical entropy is based on the probabilities of finding the system in its various microscopic states, and is shown to have the same form as Shannon’s information content. The maximisation of this entropy, subject to certain constraints, leads to Bose-Einstein and Fermi-Dirac statistics. The applications of thermodynamics and statistical mechanics are rich and varied, but in Chapters 4 and 5 we will concentrate only on those aspects which are directly relevant to device modelling.

Solution of equations: the phaseplane method

The phaseplane method is an iterative method which allows the defining equations to be easily modified without having to do much preparatory work, in the form of re-writing of code, for the resulting numerical solutions. Extra variables are defined to enable the system of second order differential equations to be written as a system of first order equations, and the enlarged system is iterated in the phase plane which is spanned by this enlarged set of variables. It is shown how this method can be tailored to the specific needs of device modelling.

Solution of equations: the multigrid method

The equations governing the modelling of many semiconductor devices, including the HEMT, are differential equations with associated boundary conditions. These equations are discretised on a grid which is fine enough to preserve all of the necessary physical detail. In an iterative solution of the equations on this fine grid, the high frequency components of the errors are rapidly eliminated, but the convergence then becomes slow. In the multigrid method, the equations and their partial solutions are moved up and down through a succession of grids, from the initial fine grid to a very coarse grid, and different error frequencies are eliminated on different grids. This method allows the equations to be solved on a fine grid with a significant decrease in computing time. An introduction to the multigrid method is given, and some of the specific problems which arise in the application of this method to device modelling are discussed.

Solution of equations: the Newton and reduced method

The Newton method is an iterative method for the solution of a set of nonlinear equations. In the case of a single variable, the method involves division by a derivative, but in the case of more than one variable the method requires the inversion of the Jacobian matrix, and this can be very time consuming. An introduction will be given to the Newton method. It will be shown that the direct Newton method is equivalent to iterating a set of equations to a time steady state. This correspondence will be used to suggest a reduced Newton method in which matrix inversion is kept to a minimum. It is shown how the variables which arise in device modelling can be grouped conveniently when solving the associated equations using the Newton method.

Density of states and applications—1

The Fermi-Dirac and Bose-Einstein distributions which were derived in Chapter 3 will now be used to describe a number of physical situations, including blackbody radiation, classical and quantum aspect of specific heat, Bose-Einstein condensation, thermionic emission, and a brief introduction to semiconductor statistics.

Fermi and associated integrals

It has been shown in Chapters 4 and 5 how the Fermi integrals play an important role in the evaluation of electron number and energy densities. In particular, the associated Fermi integrals which were introduced in those chapters are used in the evaluation of these quantities in quantum wells. These integrals must be evaluated as rapidly and accurately as possible in any device simulation. It is shown how numerical approximations can be made to these integrals in order to achieve this goal.

The upwinding method

It has already been shown in Chapter 7 that, when using the von Neumann stability analysis on the time-discretisation scheme, the magnitude and direction of the electric field has an important bearing on the stability of the scheme. The upwinding method is a method of discretisation which utilises the “flow” of an influence from neighbouring spatial points. This method has previously been applied to the electron continuity equation by making use of Bernoulli functions, but it will be seen how the method can be generalised to use with the energy transport equation using the C-functions which will be introduced in this chapter.

Approximate and numerical solutions of the Schrödinger equation

In practically all realistic physical situations, the hamiltonian operator is such that it is not possible to specify the potential as a single function. In this case, it is not possible to calculate the exact eigensolutions of the Schrodinger equation. Even when the potential is specified as a functional form, it is possible to solve the Schrodinger equation only for a very limited number of artificial potentials. Many analytical and numerical approximation methods are available for the solution of the equation, and in this chapter some of the main techniques will be described which will be of relevance to device modelling. These will include the WKB approximation, both time independent and time dependent perturbation theory, the variational method, and numerical methods.