Numerical analysis of a finite volume scheme for charge transport in perovskite solar cells

TL;DR

This paper develops and analyzes a finite volume numerical scheme for a drift-diffusion model of charge transport in perovskite solar cells, incorporating nonlinear effects and volume exclusion, with proven stability and verified simulations.

Contribution

It introduces a finite volume scheme with entropy-dissipation properties for a complex charge transport model in perovskite solar cells, including nonlinear and volume exclusion effects.

Findings

Proven discrete entropy-dissipation inequality.

Existence of discrete solutions via fixed point theorem.

Numerical simulations show exponential decay and realistic device behavior.

Abstract

In this paper, we consider a drift-diffusion charge transport model for perovskite solar cells, where electrons and holes may diffuse linearly (Boltzmann approximation) or nonlinearly (e.g. due to Fermi-Dirac statistics). To incorporate volume exclusion effects, we rely on the Fermi-Dirac integral of order -1 when modeling moving anionic vacancies within the perovskite layer which is sandwiched between electron and hole transport layers. After non-dimensionalization, we first prove a continuous entropy-dissipation inequality for the model. Then, we formulate a corresponding two-point flux finite volume scheme on Voronoi meshes and show an analogous discrete entropy-dissipation inequality. This inequality helps us to show the existence of a discrete solution of the nonlinear discrete system with the help of a corollary of Brouwer's fixed point theorem and the minimization of a convex functional. Finally, we verify our theoretically proven properties numerically, simulate a realistic device setup and show exponential decay in time with respect to the L^2 error as well as a physically and analytically meaningful relative entropy.