Numerical solution to the general one‐dimensional diffusion equation in semiconductor heterostructures

Abstract

A method of solving numerically the one-dimensional diffusion equation for arbitrary profiles and arbitrary functional dependencies of the diffusion coefficient on the position, diffusant concentration and the time, is described. The technique is applied to a variety of diffusion problems in semiconductor quantum wells to illustrate its power and versatility. In particular solutions are shown for diffusion of graded interfaces, a concentration dependent diffusion coefficient, and the effect of a depth and time dependent diffusion coefficient on a superlattice, as occurs in ion implantation and the subsequent annealing out of the resulting radiation damage. The depth dependence of the intermixing due to ion implantation and subsequent rapid thermal anneal, of a GaAs/Gal-,Al,As multiple-quantum-well structure, is deduced from the broadening of the low temperature photoluminescence emission.

Numerical simulation of inhomogeneous and nonlinear diffusion

“…Figure 2 shows the behavior of error variation for the present method (SCM) as a function of N for different ∆ t used in the Runge-Kutta procedure. We have also included here the finite-difference based method [14] similar to the one used by Harrison [11]. In the case of the SCM, for all ∆t values error decreases sharply up to the point where either it approaches machine precision (~1 6 …”

“…For these computations we have used 2L = 80 nm and N = 400. In the second case [11], we consider diffusion of a graded gap QW, consisting of a 30 nm layer of A 0.9 B 0.1 C followed by a 10 nm layer of A 1-y B y C with graded profile and finally 30 nm of A 0.9 B 0.1 C. The initial condition for this case is: [11]. It can also be seen from Figs.…”

“…In such cases, the diffusion coefficient would be a function of position x. We consider here the example discussed in [11]. In this example, diffusion of B is considered from the step alloy heterojunction.…”

“…Next we consider two more cases with constant diffusion coefficient but with graded initial profile [11]. Here we consider a heterostructure made of a ternary alloy say A 1-y B y C. We are interested in studying the evolution of the profile of specie B as diffusion proceeds under various conditions.…”

“…These are the parameters used by Harrison [11] in his computations for the "satisfactory" results; therefore, we assume that an accuracy of 8 …”