MODELLING OF VARIABLE BAND-GAP SUPERLATTICES WITH USING A GENETIC ALGORITHM

Abstract

The numerical calculation of the coordinate distributions of the electrostatic potential in an asymmetric variable-band superlattice with piecewise linear coordinate profiles of the gap width and electron affinity was performed using a genetic algorithm. A mathematical model is formulated, represented by two nonlinear Poisson equations and four boundary conditions, which reflect the continuity of the electrostatic potential and electric field strength at the boundaries of the variable band-gap superlattice, as well as the spatial periodicity of its physical quantities. The solution of the three-point boundary value problem for the electrostatic potential is reduced to the solution in an iterative cycle of two Cauchy problems with initial conditions determined using a genetic algorithm. The search space was chosen in the form of a rectangular region centered at a point that corresponded to the values of the electrostatic potential and electric field strength on one of the boundaries of the superlattice. These values were obtained by analytically solving the corresponding Poisson equations. An initial population of 50-100 trial solutions (chromosomes) was created by selection with equal probability in the search area. The coding of chromosomes and their modification under the action of crossover and mutation operations were used. The genetic algorithm was implemented in the Matlab environment using the Genetic Algorithm package. Numerical experiments showed that the initial population of trial solutions no later than after 20 -30 generations approaches to the desired initial values for the Cauchy problems. The latter were solved by the Runge-Kutta method of the 4th order. The obtained dependences of the electrostatic potential show that its maximum value, which corresponds to the zero intensity of the internal electric field, in contrast to symmetric variable band-gap superlattices, is observed not at the interface of layers, but in a layer of greater thickness. The results of the calculations indicate a substantial quantitative difference in the values of the electrostatic potential for the nonlinear and linearized models. Key words: variable band-gap semiconductors, superlattices, modelling, genetic algorithms.