The 3D transient semiconductor equations with gradient-dependent and interfacial recombination

Abstract

We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.

Interpolation Theory for Sobolev Functions with Partially Vanishing Trace on Irregular Open Sets

“…There, the linear second order operator usually admits a proper theory of weak solutions on Sobolev spaces of type W 1,p carrying the boundary conditions, but in controlling non-linear terms of high order and reaction processes on lower-dimensional substructures of the boundary simultaneously, interpolation spaces of order s ∈ (0, 1) are most appropriate. We refer to [13,Sec. 4.1] for a detailed account on this paradigm in the context of dynamics in a semiconductor device, see also [25,Sec.…”

“…In particular, we confirm the formula for the complex interpolation spaces that was conjectured in connection with fractional powers of divergence form operators in [3,Rem. 10.5] and listed as an open problem in [13,Sec. 5.3].…”

Extrapolated Elliptic Regularity and Application to the van Roosbroeck system of Semiconductors

“…Let us also note that already (1.1) for q > d itself has turned out to be an extremely valuable and well suitedone might even say, indispensable-property in the treatment of nonlinear and/or coupled systems of evolution equations with highly nonsmooth data arising in real-life problems, see e.g. [13,29,39,40]. We next motivate why we need also the optimal regularity result (1.2) for q > d in the fractional Sobolev scales.…”

“…6]), X should be chosen an as interpolation space of the form [L q (Ω), W −1,q D (Ω)] 1−s with parameters q > d and s ∈ (0, 1 − d q ); this was observed in [25,Sect. 6], see also [13,Sect. 4.1].…”

“…This is made more precise in Section 4 below, where the model and the involved quantities are also explained in detail. We refer to the introduction of [13] for a comprehensive collection of related literature. In fact, the van Roosbroeck system (1.4) was treated under similar assumptions recently in [13]; however, the analysis there is quite involved since the system need be reformulated "globally" in the quasi Fermi levels.…”