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

TL;DR

The paper questions the reliability of standard semilocal PBE and hybrid HSE functionals for germanium, showing PBE severely underestimates band gaps and HSE cannot reproduce both $Γ$-$Γ$ and $Γ$-$L$ gaps consistently. It introduces DFT+$α$, a semi-empirical scheme that selectively shifts $4s$-like states to fix the problematic $4s$-$4p$ energy alignment and remove unphysical $sp$ mixing. With $α ≈ 1.4$, DFT+$α$ reproduces the experimental lattice constant, bulk modulus, elastic constants, phonon frequencies, and both gaps within a few percent, at a fraction of the cost of hybrid functionals. This approach offers a practical alternative to hybrids for germanium and potentially other semiconductors, particularly when accurate bulk properties are required at lower computational expense, and the authors provide data and code access.

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 $Γ$-L indirect and $Γ$-$Γ$ 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+$α$, a semi-empirical correction scheme applied selectively to 4s-like orbitals. For germanium, DFT+$α$ 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.

Figures (3)

• Figure 1: DFT calculations of germanium's properties using the HSE functional, displayed as a function of the exchange mixing parameter. (a) The $\Gamma$-$\Gamma$ and $\Gamma$-$\text{L}$ band gaps calculated at the optimized lattice parameter. Experimental values are taken from ref Madelung:2004 and previous results from refs. Martin:2006Hummer:2009Deak:2010Weber:2013Pasquarello:2016. (b) The optimized lattice parameter. Exp. from ref. Hu:2003 and previous results from refs. Martin:2006Hummer:2009Weber:2013Rasander:2015Pasquarello:2016. (c) The elastic constants $B_0$, $C_{11}$, $C_{12}$ and $C_{44}$. Experimental constants are taken from ref. Madelung:2004 and previous results from refs. Hummer:2009Weber:2013Rasander:2015Pasquarello:2016.
• Figure 2: DFT+$\alpha$ calculation of germanium's properties using the PBE functional displayed as a function of the $\alpha$ parameter. (a) The $\Gamma$-$\Gamma$ and $\Gamma$-$\text{L}$ band gaps at the optimized lattice parameter for each $\alpha$ value. Yellow arrows mark the transition from a closed band gap to an open one, as well as the inversion of the $\Gamma$-$\Gamma$ and $\Gamma$-L band gaps. Experimental values are taken from ref. Madelung:2004. (b) The optimized lattice parameter. The experimental lattice parameter is reported in ref. Hu:2003. (c) The elastic constants, $B_0$, $C_{11}$, $C_{12}$ and $C_{44}$ at the optimized lattice parameter for each $\alpha$ value. Experimental constants are collected from ref. Madelung:2004.
• Figure 3: DFT+$\alpha$ calculation of germanium's properties for the chosen value $\alpha = 1.4$ at its optimized lattice constant, $a = 5.676$ Å. (a) Electronic band structure within the DFT+$\alpha$ approach is shown in black. For comparison, the standard DFT calculation using the PBE functional has been performed at the same lattice parameter and is shown in red. Gray-filled areas correspond to occupied states or bands 1-4. Experimental values for the $\Gamma$-$\Gamma$ and $\Gamma$-L band gaps at cryogenic temperatures are taken from ref. Madelung:2004. The projected electronic density of states (DOS) onto the 4s and 4p atomic orbitals for the DFT+$\alpha$ calculation is also included. (b) Phonon dispersion curves of Ge, computed using standard DFT (blue), DFT+$\alpha$ (red), and HSE (black). For each calculation, the corresponding optimized lattice constant is used. Experimental values are taken from ref. Nilson:1971.