Semiconductor dielectric function, excitons and the Penn model

TL;DR

This paper improves the calculation of the dielectric function by including excitonic effects, providing new expressions that satisfy fundamental relations and closely match numerical results, with applications to metal-insulator transitions.

Contribution

It introduces an improved dielectric function model considering excitonic effects, with new analytical expressions and validation against existing models and numerical data.

Findings

New expressions for dielectric function components are consistent with Kramers-Kronig relations.

The model's coefficient for small gap approximation is 2/3, close to numerical results.

Application to hydrogen suggests accurate critical densities for metal-insulator transition.

Abstract

Improved computation of the dielectric function considering excitonic effects and long wavelength is performed and compared with the nearly free electron band approximation, similarly with the Penn's model case. New expressions for the real and imaginary part of the dielectric function are presented and the real part compared with the Penn's result. The obtained functions satisfy the Kramers-Kr\"onig relations, in contrast with earlier results in the literature. In addition, our improved dielectric function presents a coeficient of 2/3 for small gap approximation (different from the value of 1 in the original Penn model) is very close to the value 0.62 obtained in [Can. J. Phys.53,(1975) p.2549] from pure numerical procedures. The obtained dielectric function also is used in a rough and stimative analysis of the metal-insulator transition in molecular hydrogen being the critical densities determined near the experimental values for the hydrogen coming from other approach. The approximated expressions and critical values are given and the usefulness of the rough methods involved in the determination of the critical points briefly discussed.