Dependence of the Diffusion Coefficient on the Fermi Level: Zinc in Gallium Arsenide

Abstract

The experimental variation of the diffusion coefficient $D$ with Zn concentration $C_s$ has been determined at 1000, 900, 800, and 700°C from radioactive ${}_{}{}^{65}{}_{}{}^{}\mathrm{Zn}$ diffusion profiles by a Boltzmann-Matano analysis. With interstitial Zn as the dominant diffusing species and its concentration controlled by the interstitial-substitutional equilibrium in which the singly ionized interstitial donor reacts with a neutral Ga vacancy to form a singly ionized substitutional acceptor and two holes, the effective diffusion coefficient is described by $D=D^{*}{C_s}^2{{\gamma }_{\mathrm{p}}}^2[1+(\frac{C_s}{2{\gamma }_{\mathrm{p}}}\left)\right(\frac{d{\gamma }_{\mathrm{p}}}{d{C}_s}\left)\right]$, where ${\gamma }_{\mathrm{p}}$ is the hole activity coefficient. The term $D^{*}$ equals $\frac{2D_i}{K_1{p_{\mathrm{As}4}}^{\frac{1}{4}}}$, where $D_i$ is the interstitial diffusion coefficient, $K_1$ the reaction equilibrium constant, and $p_{\mathrm{As}4}$ the ${\mathrm{As}}_4$ pressure. The relationship between ${\gamma }_p$ and the Fermi level $E_f$ is given by ${\gamma }_p=\left(\frac{A}{p}\right)\exp \left(\frac{E_f}{\mathrm{kT}}\right)$, where $A$ is a constant dependent only on temperature and $p$ is the hole concentration. This derivation for $D$ has extended previous analyses to include both the built-in field and the nonideal behavior of holes which occurs when the impurity level broadens into an impurity band and merges with the valence band to form impurity-band tails at high Zn concentrations. The observed nonmonotonic dependence of the Zn diffusion coefficient on its concentration is a consequence of the nonideal behavior of holes at high concentrations. Quantitative comparison of $D$ with the experimental concentration dependence has permitted the determination of ${\gamma }_p$ and $E_f$ as functions of the hole concentration.