Generalized Boltzmann relations in semiconductors including band tails

Arnout Beckers, Dominique Beckers, Farzan Jazaeri, Bertrand Parvais, Christian Enz

Research output: Contribution to journalArticle


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 is timely for the modeling of semiconductor devices at cryogenic temperatures for large-scale quantum computing.

Original languageEnglish
Article number045701
JournalJournal of Applied Physics
Issue number4
Publication statusPublished - 28 Jan 2021

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