Diffusion coefficients varying with a power of the concentration: Convenient solutions and a reexamination of Zn in GaAs

Abstract

Diffusion, at constant surface concentration, with coefficient varying to the positive n^th power is considered. It is shown that the diffusion equation for these cases has a unique solution whose leading term is concentration ∝Z^1/n, where z = 1−(x/x₊), x being the distance from the surface and x₊ the distance at which the concentration just reaches zero. Numerical solutions can be obtained by standard methods of solution; explicit expressions are given for n = 1, 2, and 3. It is also shown that plots of log(concentration) versus logz give rapid reliable estimates of n from usual diffusion profiles. These methods are applied to previously analyzed data for diffusion of Zn in GaAs at 1000 °C. This reanalysis suggests that (i) the diffusion coefficient for Zn in GaAs is accurately given by the ``interstitial‐substitional'' mechanism at the surface of GaAs, (ii) for surface concentrations of $\mathrm{Zn}{\text{⪞5×10}}^{19}\hspace{0.17em}{\mathrm{cm}}^{\text{−3}}$ , the diffusion is influenced by a limited flux of gallium vacancies from the surface, and (iii) the diffusion coefficient for gallium vacancies is roughly 5×10<sup>−9</sup> cm<sup>2</sup>/sec at 1000°C. It is suggested that ``double'' profiles, often found for surface Zn concentration ${\text{⪞10}}^{20}\hspace{0.17em}{\mathrm{cm}}^{\text{−3}}$ , are due to dual vacancy sources: the free surface and a dislocation net occurring at positions of large Zn gradient.