The feasibility of obtaining general impurity distributions by diffusion

Although it is possible to optimize selected characteristics of semiconductor devices by controlling the net impurity distribution, present methods used for fabricating planar devices by diffusion techniques invariably produce profiles which are approximately either Gaussian or complementary error functions. The purpose of this paper is to examine the feasibility of obtaining a general impurity profile within intrinsic material using diffusion techniques. It is shown that by treating the problem as an optimum control problem, one can, in principle, obtain a best-fit approximation to a given profile by controlling the gas stream impurity concentration. A first-order model is developed to characterize the surface interactions between the gas ambient and the solid; the boundary value problem for diffusion is then solved. A computational routine using linear programming techniques is developed which, for a given diffusion time, determines the necessary control function producing a best-fit approximation to the desired profile over the spatial range of interest. The optimum control functions governing the diffusion of boron in silicon for three given impurity profiles are synthesized: a constant profile, an exponential profile, and the minimum transit time profile. The computational results clearly establish the feasibility of obtaining an approximation to a given impurity profile.