Existence of solutions and uniform bounds for the stationary semiconductor equations with generation and ionic carriers

TL;DR

This work addresses the stationary drift-diffusion system for semiconductor devices with ionic carriers and external generation, a setting relevant to perovskite solar cells. The authors develop a robust existence theory for weak solutions and derive explicit uniform bounds on carrier densities and potentials by combining truncation, Stampacchia-type energy estimates, and a Leray–Schauder construction. They further establish bounded outward currents at ohmic contacts and illustrate the parameter dependence through numerical simulations of a three-layer PSC and a LBIC setup. The results provide a rigorous foundation for stable simulations of generation-influenced transport with ionic species and offer practical insights into how generation and ionic effects shape device behavior.

Abstract

We consider a stationary drift-diffusion system with ionic charge carriers and external generation of electron and hole charge carriers. This system arises, among other applications, in the context of semiconductor modeling for perovskite solar cells. Thanks to truncation techniques and iterative energy estimates, we show the existence and uniform upper and lower bounds on the solutions. The dependency of the bounds on the various parameters of the model is investigated numerically on physically relevant test cases.

Figures (9)

• Figure 3.1: (a) Schematic of the three-layer solar cell architecture, with the perovskite layer $\Omega_{\text{PVK}}$ sandwiched between the electron transport layer $\Omega_{\text{ETL}}$ and the hole transport layer $\Omega_{\text{HTL}}$. Light enters through the left contact. The boundaries $\Gamma^N$ and $\Gamma^D$ are also indicated. (b) Exponentially decaying photogeneration rate $G$, present only within the absorbing perovskite layer $\Omega_{\text{PVK}}$.
• Figure 3.2: (a) Electric potential $\psi$ (blue), and quasi-Fermi potentials of (b) electrons (green) and (c) holes (red), shown for different values of the photogeneration prefactor $G_0$. Brighter colors correspond to larger values of $G_0$ with arrows indicating the direction of increasing $G_0$. The externally applied voltage, which enters through the Dirichlet boundary conditions, is set to $\overline{V} = 2$.
• Figure 3.3: (a) Electron density (green) and hole density (red) shown for different values of the photogeneration prefactor $G_0$. Brighter colors correspond to larger values of $G_0$ with arrows indicating the direction of increasing $G_0$. (b) $L^{\infty}$ norms of the carrier densities of electrons (green), and holes (red dashed) as well as (c) $L^{\infty}$ norms of the quasi-Fermi potentials of electrons (green) and holes (red dashed), and the electric potential (blue dotted), plotted as a function of the photogeneration prefactor $G_{0}$. The externally applied voltage, which enters through the Dirichlet boundary conditions, is set to $\overline{V} = 2$.
• Figure 3.4: (a) Electric potential $\psi$, and quasi-Fermi potentials of (b) electrons $v_{\text{n}}$ (green), vacancies $v_\text{a}$ (gold) and (c) holes $v_{\text{p}}$, shown for different values of the photogeneration prefactor $G_0$. Brighter colors correspond to larger values of $G_0$ with arrows indicating the direction of increasing $G_0$. The applied voltage is set to $\overline{V} = 1$.
• Figure 3.5: (a) Electron density (green), hole density (red), and vacancy density (gold) shown for different values of the photogeneration prefactor $G_0$. Brighter colors correspond to larger values of $G_0$ with arrows indicating the direction of increasing $G_0$. (b) $L^{\infty}$ norms of the carrier densities of electrons (green), holes (red dashed), and ions (gold) as well as (c) $L^{\infty}$ norms of the quasi-Fermi potentials of electrons (green), holes (red dashed), ions (gold) and the electric potential (blue dotted), plotted as a function of the photogeneration prefactor $G_{0}$. In (b) we see also the scaled total mass $M_\text{a}/| \Omega_{\text{PVK}}|$, which is the same for all $G_0$. The applied voltage is set to $\overline{V} = 1$.

Theorems & Definitions (20)

• Example 1.2: Statistics functions
• Example 1.3: Recombination-generation rates
• Example 1.4: External generation
• Theorem 1.5
• Corollary 1.6
• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof