AMaSiS 2018 Workshop: Abstracts

On the entropy method and exponential convergence to equilibrium for a recombination-drift-diffusion system with self-consistent potential

Klemens Fellner${}^{\mathbf{\text{(1)}}}$, and Michael Kniely${}^{\mathbf{\text{(2)}}}$

(1) University of Graz, Institute for Mathematics and Scientific Computing

(2) Institute of Science and Technology (IST Austria)

We consider a Shockley–Read–Hall recombination-drift-diffusion model coupled to Poisson’s equation and subject to boundary conditions, which imply conservation of the total charge. Our main result is an explicit functional inequality

$$P(n,p,\psi )\ge C_{\mathrm{E}EP}E(n,p,\psi )$$ (1)

between relative entropy $E$ and entropy production rate $P$ (depending on the concentrations of negatively and positively charged species $n$, $p$ and the electrostatic potential $\psi$). From this estimate, we deduce exponential convergence to equilibrium with explicit constant and rate.

In more detail, we prescribe no-flux boundary conditions to the equations for $n$ and $p$ and homogeneous Neumann conditions for $\psi$. The first ones guarantee conservation of charge and, hence, charge neutrality

$${\int }_{\mathrm{\Omega }}(n-p-C)𝑑x=0$$ (2)

(with $C$ being the internal doping concentration) provided the initial states $n_I$, $p_I$ satisfy (2). The charge neutrality constitutes the necessary and sufficient compatibility condition for solving Poisson’s equation

$$-\epsilon \mathrm{\Delta }\psi =n-p-C$$ (3)

on a bounded domain $\mathrm{\Omega }$ with homogeneous Neumann conditions on $\partial \mathrm{\Omega }$.

The employed techniques for proving (1) build on ideas which have been successfully applied to a similar problem in [1], even though the unique solvability of (3) is a consequence of mixed Dirichlet-Neumann boundary conditions in [1]. Nevertheless, we will see that the approach of [1] is applicable as the charge neutrality will not be used explicitly in the proof of (1).

Acknowledgments: The second author has been supported by the International Research Training Group IGDK 1754 “Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures”, funded by the German Research Council (DFG) and the Austrian Science Fund (FWF): [W 1244-N18].