  We study the quasi-neutral limit in an optimal semiconductor design problem
constrained by a nonlinear, nonlocal Poisson equation modelling the drift
diffusion equations in thermal equilibrium. While a broad knowledge on the
asymptotic links between the different models in the semiconductor model
hierarchy exists, there are so far no results on the corresponding optimization
problems available. Using a variational approach we end up with a bi-level
optimization problem, which is thoroughly analysed. Further, we exploit the
concept of Gamma-convergence to perform the quasi-neutral limit for the minima
and minimizers. This justifies the construction of fast optimization algorithms
based on the zero space charge approximation of the drift-diffusion model. The
analytical results are underlined by numerical experiments confirming the
feasibility of our approach.
