Optimal Control of the Stationary Quantum Drift-Diffusion Model

A. Unterreiter, S. Volkwein
2007 Communications in Mathematical Sciences  
In this work an optimal control problem for a stationary quantum drift diffusion (QDD) model is analyzed. This QDD model contains four space-dependent observables: The nonnegative particle density of electrons, the electrostatic potential, the quantum quasi-Fermi potential and the current density. The goal is to optimize the shape of quantum barriers in a quantum diode. Existence of optimal solutions is proved. Moreover, first-order necessary optimality conditions are derived. For the notion of
more » ... . For the notion of Lebesgue and Sobolev spaces we refer the reader to [1], for instance. The quantum quasi-Fermi potential F acts as the electron's velocity potential, Corresponding to the conservation of mass the current density is divergence free: The electrostatic potential V is self-consistent, i.e., where ≈ 11.6 · 10 −12 As Vm is the -assumingly: constant -dielectricity constant of the underlying crystal, and the quantum quasi-Fermi potential F is where ≈ 6.626 · 10 −34 Js is Planck's constant, k B ≈ 1.38 · 10 −23 J K is Boltzmann's constant and q ≈ 1.6 · 10 −19 As is the elementary charge. The term − 2 6m ∆ √ n √ n is Bohm's
doi:10.4310/cms.2007.v5.n1.a4 fatcat:s77ikbuse5fa5pjh76xyjvkvlm