Synthesis of general impurity profiles using a two-step diffusion process

By optimizing selected characteristics, the performance of modern semiconductor devices and integrated circuits could be improved if it were possible to control the impurity profile in given regions of the material. However, 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 demonstrate the feasibility of obtaining a general impurity profile in semiconductor material compatible with the two-step diffusion process. The general boundary value problem for the two-step diffusion process is solved and the impurity concentration in the solid is determined as a function of the impurity concentration of the carrier gas in the diffusion tube. An optimization 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 impurity profile. The optimum control functions governing the diffusion of boron in silicon for two examples are synthesized. The numerical results clearly indicate that a best-fit approximation to a given impurity profile can be realized by controlling the impurity concentration in the carrier gas during the gaseous phase of a two-step diffusion process.