Exact description and data fitting of ion-implanted dopant profile evolution during annealing

We solve analytically the problem of dopant redistribution from an arbitrary initial implanted profile. The diffusivity, assumed concentration independent, can be an otherwise arbitrary function of temperature. Next, we derive a closed-form expression for the annealed concentration distribution in the special case of an initial truncated Gaussian. Our solution is valid over a much wider range of experimental conditions than is Seidel and MacRae’s approximation [Trans. AIME 245, 491 (1969)]. We offer a simple, yet precise, criterion for the validity of this approximation, and we guard against its indiscriminate use. Last, we fit short-time annealed P profiles in implanted Si to get an average diffusivity D̄≂3×10−12 cm2/s. We describe simple and accurate one-parameter data-fitting procedures.