A Variational Formulation for Modeling a Phosphorus Doped Silicon Semiconductor Through a Multi‐Well Approach

Abstract This article develops a variational formulation for modeling a silicon semiconductor through a multi‐well approach utilizing phosphorus atoms as a dopant substance. The results are based on standard tools of calculus of variations and optimization theory. It is worth highlighting, the variational formulation here developed may be used to find an optimal phosphorus density distribution concerning an originally silicon density, in order to maximize the electrical conductivity of such a sample. Finally, in the last section, a numerical example is presented to illustrate the applicability of such results developed in the previous sections.