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

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, 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 that 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.

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

“…So far, we considered here degenerate systems with Fermi statistics. In Section 3, we address a complementary case of a non-degenerate system with Boltzmann statistics, following two recent studies [13,14]. While in those studies, hundreds of equations were used to treat the non-degenerate case, we demonstrate in this paper that the essence of the theory for Boltzmann statistics can be introduced using only several equations.…”

“…In the LLT, the random potential is converted into some effective potential, drastically simplifying the calculations. However, recent studies [13,14] revealed substantial problems of the LLT. For instance, the effective potential in the LLT lacks the temperature dependence, which is necessary to describe n(r, T) appropriately.…”

“…We also describe two recently developed computational techniques for calculations of n(r, T) in non-degenerate systems controlled by Boltzmann statistics [13,14]. One algorithm is based on applying a temperature-dependent universal low-pass filter (ULF) to the random potential V(r).…”

“…This procedure offers several advantages over the widely used LLT. Unlike the LLT, which relies on an adjustable parameter that can only be determined through comparison with the exact solution [13,14], the RWF scheme does not involve adjustable parameters, and it works across all temperatures. Additionally, the accuracy of the RWF approach in computing n(r, T) can systematically be improved, whereas the accuracy of the LLT is inherently limited.…”

Empirical tight-binding method for large-supercell simulations of disordered semiconductor alloys

“…It should be noted that there are indirect methods investigating the essential aspects of electron states in disordered structures without solving explicitly the Schrödinger equation, such as the localization landscape theory [25][26][27] or the low-pass filter approach [28,29]. However, since the energy band structure is one of our interests here, we focus on direct band-structure calculation methods.…”