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

TL;DR

This paper studies the limit of an optimal design problem for the quantum drift-diffusion model as quantum effects diminish, establishing convergence of solutions and providing numerical validation for semiconductor devices.

Contribution

It introduces a novel analysis of the semiclassical limit for the QDD model using Gamma-convergence, ensuring the convergence of minimizers and solutions in the absence of uniqueness.

Findings

Convergence of minima and minimizers as Planck constant tends to zero.

Existence of quantum solutions converging to classical solutions.

Numerical optimization results for MESFET devices supporting theoretical findings.

Abstract

We consider an optimal semiconductor design problem for the quantum drift diffusion (QDD) model in the semiclassical limit. The design question is formulated as a PDE constrained optimal control problem, where the doping profile acts as control variable. The existence of minimizers for any scaled Planck constant allows for the investigation of the corresponding sequence. Using the concepts of Gamma-convergence and equi-coercivity we can show the convergence of minima and minimizers. Due to the lack of uniqueness for the state system and optimization problem, it was necessary to establish a new result for the QDD model ensuring the existence of a sequence of quantum solutions converging to an isolated classical solution. As a by-product, we obtain new insights into the regularizing property of the quantum Bohm potential. Finally, we present the numerical optimization of a MESFET device underlining our analytical results.