Two Dimensional Concentration Dependent Diffusion

In 1965, Kennedy and O'Brien analytically investigated the impurity atom distribution near the edge of a diffusion mask.1 The geometric effects are significant, and the results of their study are routinely applied in semiconductor device design." The Kennedy and O'Brien model used the simple, linear, constant-coefficient, diffusion equation. In 1968, Hu and Schmidt introduced a concentration dependent diffusion model which included the effects of both the charged vacancy reaction and the impurity-induced electric fields. This model was investigated in one dimension, and it established that, in the regime in which semiconductors are actually fabricated, these nonlinear effects are quite significant for some impurities, such as arsenic. 1 In the decade since these studies, there have been significant improvements in fabrication technologies and lithographic techniques for both bipolar and MOS transistors. Furthermore, the importance of MOS transistors has also greatly increased. In this paper, we investigate the combination of nonlinear and geometric effects on the impurity atom distribution by studying a two-dimensional, concentration dependent diffusion model in a region containing the edge of the diffusion mask. In Section II, the nonlinear diffusion coefficient, including the effects of "autodoping," electric fields, and multiply ionized impurities, is derived in a form suitable for the calculation of the desired impurity profiles. The modeling parameters, the geometries, and the boundary conditions for the partial differential equations are also described.