Generalized Boltzmann relations in semiconductors including band tails

TL;DR

This paper generalizes the Boltzmann relations in semiconductors to include exponential band tails, providing simple formulas for carrier densities that are applicable at cryogenic temperatures relevant for quantum computing.

Contribution

It introduces a new analytical approach to incorporate band tail effects into Boltzmann relations using hypergeometric functions and series expansions.

Findings

Derived simple relations involving two exponentials for electron and hole densities.

Relations reduce to classical Boltzmann relations when band-tail parameters are zero.

Applicable to modeling semiconductor devices at cryogenic temperatures.

Abstract

Boltzmann relations are widely used in semiconductor physics to express the charge-carrier densities as a function of the Fermi level and temperature. However, these simple exponential relations only apply to sharp band edges of the conduction and valence bands. In this article, we present a generalization of the Boltzmann relations accounting for exponential band tails. To this end, the required Fermi-Dirac integral is first recast as a Gauss hypergeometric function, followed by a suitable transformation of that special function, and a zeroth-order series expansion using the hypergeometric series. This results in simple relations for the electron and hole densities that each involve two exponentials. One exponential depends on the temperature and the other one on the band-tail parameter. The proposed relations tend to the Boltzmann relations if the band-tail parameters tend to zero. This work comes timely for the modeling of classical semiconductor devices at cryogenic temperatures for large-scale quantum computing.