General Theory of Impurity Diffusion in Semiconductors via the Vacancy Mechanism

A general theory is developed for impurity diffusion in semiconductors via the vacancy mechanism, which introduces and unifies a number of new and existing concepts into a self-consistent phenomenological formalism. The thermodynamics of the vacancy-impurity-semiconductor system is first analyzed based on an energy-band model, in which the activity coefficients of the vacancy and of the impurity and the concentration of the vacancy-impurity pairs are obtained. The relationship between the diagonal and the off-diagonal phenomenological coefficients is discussed. The diagonal elements are then determined from random-walk theory, using an appropriate atomistic model. A special treatment is given to a tight-binding approximation. It is shown that the validity of Seitz's analysis of the "chemical pump effect" depends primarily on the correlation effect which he neglected. A vacancy flux from the interior, however, will be induced in the initial period as a consequence of the lowering of the vacancy activity coefficient in the donor-doped region. On the assumption of a quasiequilibrium vacancy concentration and a moderately low impurity concentration, the general theory reduces to a particularly simple form $J_A=-{D_A}^{*}{{\gamma }_v}^{-1\mathrm{}}(1+\frac{\partial \ln {\gamma }_A}{\partial \ln N_A})\frac{\partial N_A}{\partial x},$ where $J_A$ and $N_A$ are, respectively, the flux and the concentration of the impurity; ${D_A}^{*}$ is its diffusivity under intrinsic condition; and ${\gamma }_v$ and ${\gamma }_A$ are the activity coefficients of the vacancy and the impurity, respectively.