  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.
