Quantum states in disordered media. II. Spatial charge carrier distribution

TL;DR

This paper introduces two innovative methods to determine the spatial and temperature-dependent electron distribution in disordered semiconductors, simplifying quantum calculations and outperforming existing theories at higher temperatures.

Contribution

The paper presents two novel techniques, RWF and ULF, that approximate quantum charge distributions without solving the Schrödinger equation, improving accuracy over localization landscape theory.

Findings

Both methods accurately reproduce quantum solutions.

Methods outperform localization landscape theory at elevated temperatures.

Numerical verification confirms the effectiveness of the approaches.

Abstract

The space- and temperature-dependent electron distribution $n(\mathbf r,T)$ is essential for the theoretical description of the opto-electronic properties of disordered semiconductors. We present two powerful techniques to access $n(\mathbf r,T)$ without solving the Schr\"odinger equation. First, we derive the density for non-degenerate electrons by applying the Hamiltonian recursively to random wave functions (RWF). Second, we obtain a temperature-dependent effective potential from the application of a universal low-pass filter (ULF) to the random potential acting on the charge carriers in disordered media. Thereby, the full quantum-mechanical problem is reduced to the quasi-classical description of $n(\mathbf r,T)$ in an effective potential. We numerically verify both approaches by comparison with the exact quantum-mechanical solution. Both approaches prove superior to the widely used localization landscape theory (LLT) when we compare our approximate results for the charge carrier density and mobility at elevated temperatures obtained by RWF, ULF, and LLT with those from the exact solution of the Schr\"odinger equation.