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 ${J}_{A}=\ensuremath{-}{{D}_{A}}^{*}{{\ensuremath{\gamma}}_{v}}^{\ensuremath{-}1}(1+\frac{\ensuremath{\partial}\mathrm{ln}{\ensuremath{\gamma}}_{A}}{\ensuremath{\partial}\mathrm{ln}{N}_{A}})\frac{\ensuremath{\partial}{N}_{A}}{\ensuremath{\partial}x},$ 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 ${\ensuremath{\gamma}}_{v}$ and ${\ensuremath{\gamma}}_{A}$ are the activity coefficients of the vacancy and the impurity, respectively.

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