A procedure is described for estimating various parameters governing the diffusion of impurities in semiconductors; these parameters are required for a number of explicit numerical models of non-linear diffusion in III-V crystals. The method is based on an analytical solution of the continuum equivalent of a discrete numerical model due to Zahari and Tuck and provides a systematic procedure for analysing experimental data to yield predictions for the coefficient of diffusion of the impurity, the coefficient of self-diffusion of the host material, the bulk equilibrium vacancy concentration and, under conditions of 'dissociation' pressure, the surface vacancy concentration. Application of the procedure to two sets of independent experimental data provided reasonably consistent values of the parameters.
1. Introduction
A useful and effective numerical model for non-linear diffusion in III-V semiconductors is that first proposed by Zahari and Tuck' in order to describe certain diffusion systems in which the concentration of the diffusing species is large compared with the vacancy concentration, and the self-diffusion coefficient of the host atoms is small compared with the diffusion coefficient of the diffusant. The model is based on a simple substitutional diffusion mechanism in which the number of (impurity or host) atoms moving onto neighbouring lattice planes depends not only on the number available to jump but also on the number of vacancies available to accept them. The resulting discrete model yields an explicit numerical algorithm which is simple to use and is obtained without recourse to differential equations. Indeed it is little more complicated than the classical finite difference solution2 of Fick's laws. Furthermore Zahari and Tuck' showed that this model can predict the anomalous 'double' profiles6 which is often observed in semiconductor systems, but which cannot be predicted by linear Fickian diffusion. Subsequently Tuck and his co-workers have extended the model to include an interstitial-substitutional me~hanism,~.' and to predict diffusion profiles in the more difficult context of Czochralski growth of doped semiconductor crystals.8.9 In all cases the results obtained are consistent with published experimental diffusion work for relevant semiconductor systems.
As for most numerical models, one of the principal difficulties in the use of this method is in estimating the values of the parameters describing the system. Whilst in theory it is possible to systematically vary the various parameters so as to obtain 'best-fit' agreement with experimental results, in practice it is usually either impracticable or, at best, inefficient to do so.
How to overcome this difficulty is the subject of the present paper, in which we demonstrate how an analytical solution of the governing differential equations can be used to provide a straightforward method of accurately determining the relevant parameters from experimental observations. The analysis treats only the first Zahari-Tuck model, but similar methods can be applied to the other, interstitial-substitutional, models. The parameters so found are shown to give excellent agreement with experimental data.
In the next section we briefly review the discrete model proposed by Zahari and Tuck, together with the apposite numerical algorithm. This is equivalent to a pair of simultaneous difference equations, which can be interpreted as an appropriate generalization of the explicit finite difference form of the partial differential equation form of Fick's laws. The continuum form of the discrete