Verification of analytic point defect models using SUPREM-IV (dopant diffusion)

Two-dimensional process modeling has been complicated by the increasing complexity of the physical models needed to characterize modern processes. Present physical models for diffusion favor modeling of both the vacancies and interstitials as well as the dopant impurities. these vacancies and interstitials are important in two dimensions since the lateral characteristic lengths are on the order of a few microns, which is comparable to device dimensions. Hence, full two-dimensional solutions of the point defect equations are required to accurately model two-dimensional diffusion. Since the defects diffuse many order of magnitude faster than the impurities, the problem of solving the coupled equations is numerically stiff and the numerical techniques become critical. The numerical methods in SUPREM-IV are described. Since the numerical solutions are time consuming, suitable analytic solutions are considered as an alternative. These analytic solutions are compared to the numerical results to test the limits of the assumptions used in the analytic formulation. The comparison forces the conclusion that analytic solutions are of limited use in prediction of point defect profilesin two dimensions. >