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\ifmmode^\circ\else\textdegree\fi{}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 $D={D}^{*}{{C}_{s}}^{2}{{\ensuremath{\gamma}}_{\mathrm{p}}}^{2}[1+(\frac{{C}_{s}}{2{\ensuremath{\gamma}}_{\mathrm{p}}})(\frac{d{\ensuremath{\gamma}}_{\mathrm{p}}}{d{C}_{s}})]$, where ${\ensuremath{\gamma}}_{\mathrm{p}}$ is the hole activity coefficient. The term ${D}^{*}$ equals $\frac{2{D}_{i}}{{K}_{1}{{p}_{\mathrm{As}4}}^{\frac{1}{4}}}$, 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 ${\ensuremath{\gamma}}_{p}$ and the Fermi level ${E}_{f}$ is given by ${\ensuremath{\gamma}}_{p}=(\frac{A}{p})\mathrm{exp}(\frac{{E}_{f}}{\mathrm{kT}})$, 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 ${\ensuremath{\gamma}}_{p}$ and ${E}_{f}$ as functions of the hole concentration.