Vacancy Transients During Impurity Diffusion in Semiconductors

Abstract

Vacancy transients associated with the diffusion of donor atoms in semiconductors are examined for a model of strong vacancy-impurity association. The causes for the vacancy transients are the drift of charged vacancies in the internal field and the shifting of the local thermal-equilibrium condition of vacancy concentration. The impurity is considered to diffuse into a semi-infinite solid with a constant surface concentration. Simultaneous equations of continuity for the impurity and the vacancy are obtained from phenomenological flux expressions, and their Boltzmann transforms are solved numerically. For the example studied, which is typical of donor diffusions in silicon, the maximum vacancy undersaturation was found to be 4.17\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}7}$. An inequality estimation gives $\ensuremath{\sigma}\ensuremath{\lesssim}\frac{8{D}_{A}(0)}{{D}_{\ensuremath{\nu}}}$, where $\ensuremath{\sigma}$ is the vacancy supersaturation (negative in the present case), ${D}_{A}(0)$ is the impurity diffusivity at the surface, and ${D}_{v}$ is the vacancy diffusivity. It is concluded that, during impurity diffusion, the departure of the vacancy concentration from its equilibrium value is entirely negligible for the model concerned. It is also pointed out that the cross coefficients of the phenomenological flux expressions are dependent on an atomistic model. For example, it is shown that if Seitz's chemical-pump model is assumed, then there will be significant vacancy nonequilibrium.