# Computation of the Spatial Distribution of Charge-Carrier Density in Disordered Media

## TL;DR

This paper introduces numerical methods to calculate electron distribution in disordered semiconductors, avoiding complex quantum solutions.

## Contribution

The paper introduces novel numerical techniques for calculating charge-carrier density in disordered media under different statistical conditions.

## Key findings

- Two numerical approaches for degenerate systems with Fermi statistics were developed and validated.
- Two methods for non-degenerate systems with Boltzmann statistics were introduced and tested.
- The approximate calculations showed high accuracy when compared to exact quantum-mechanical solutions.

## Abstract

The space- and temperature-dependent electron distribution n(r,T) determines optoelectronic properties of disordered semiconductors. It is a challenging task to get access to n(r,T) in random potentials, while avoiding the time-consuming numerical solution of the Schrödinger equation. We present several numerical techniques targeted to fulfill this task. For a degenerate system with Fermi statistics, a numerical approach based on a matrix inversion and one based on a system of linear equations are developed. For a non-degenerate system with Boltzmann statistics, a numerical technique based on a universal low-pass filter and one based on random wave functions are introduced. The high accuracy of the approximate calculations are checked by comparison with the exact quantum-mechanical solutions.