An accurate alternative to hybrid functionals for germanium: DFT+$\alpha$

TL;DR

This paper introduces DFT+$\alpha$, a semi-empirical correction method that improves the accuracy of germanium's electronic and structural properties in DFT calculations, outperforming hybrid functionals at lower computational cost.

Contribution

The paper presents DFT+$\alpha$, a novel correction scheme that accurately captures germanium's semiconductor properties by addressing PBE's band gap issues, offering a computationally efficient alternative to hybrid functionals.

Findings

DFT+$\alpha$ restores correct band edge ordering in germanium.

DFT+$\alpha$ achieves accurate lattice and elastic properties.

Hybrid functionals like HSE fail to reproduce germanium's band gaps simultaneously.

Abstract

The accuracy of bulk property predictions in density functional theory (DFT) calculations depends on the choice of exchange-correlation functional. While the Perdew-Burke-Ernzerhof (PBE) functional systematically overestimates lattice parameters and strongly underestimates electronic band gaps, hybrid functionals such as Heyd-Scuseria-Ernzerhof (HSE) offer better overall agreement across a broad range of materials. Using germanium as a critical test case, we challenge the ability of both functionals to capture semiconductor properties. Although HSE improves PBE's gap error, it fails to reproduce germanium's correct $\Gamma$-L indirect and $\Gamma$-$\Gamma$ band gaps simultaneously. Noting that the PBE underestimated energy separation between the 4p valence-band maximum and 4s conduction-band minimum causes unphysical $sp$ mixing, we propose DFT+$\alpha$, a semi-empirical correction scheme applied selectively to 4s-like orbitals. For germanium, DFT+$\alpha$ restores the proper ordering and orbital character of the band edges and yields accurate lattice constant, bulk modulus, elastic constants and phonon frequencies at a fraction of hybrid-functional computational cost.