The Semi-Classical Limit of an Optimal Design Problem for the Stationary Quantum Drift-Diffusion Model

摘要

We consider an optimal semiconductor design problem for the quantum driftdiffusion (QDD) model in the semiclassical limit. The design question isformulated as a PDE constrained optimal control problem, where the dopingprofile acts as control variable. The existence of minimizers for any scaledPlanck constant allows for the investigation of the corresponding sequence.Using the concepts of Gamma-convergence and equi-coercivity we can show theconvergence of minima and minimizers. Due to the lack of uniqueness for thestate system and optimization problem, it was necessary to establish a newresult for the QDD model ensuring the existence of a sequence of quantumsolutions converging to an isolated classical solution. As a by-product, weobtain new insights into the regularizing property of the quantum Bohmpotential. Finally, we present the numerical optimization of a MESFET deviceunderlining our analytical results.