Impact of stoichiometry and strain on Ge$_{1-x}$Sn$_{x}$ alloys from first principles calculations

TL;DR

This study uses first-principles calculations to explore how stoichiometry and strain influence the electronic properties of GeSn alloys, revealing ways to engineer their transition from semimetallic to semiconducting states.

Contribution

The paper provides a detailed first-principles analysis of GeSn alloys, including strain effects, critical thickness estimates, and strategies for electronic structure engineering.

Findings

Tensile strain reduces Sn composition needed for zero direct band gap.

Compressive strain has less impact on the band gap at b1.

Predicted critical thickness and amorphous transition range.

Abstract

We calculate the electronic structure of germanium-tin (Ge$_{1-x}$Sn$_{x}$) binary alloys for $0 \leq x \leq 1$ using density functional theory (DFT). Relaxed alloys with semiconducting or semimetallic behaviour as a function of Sn composition $x$ are identified, and the impact of epitaxial strain is included by constraining supercell lattice constants perpendicular to the [001] growth direction to the lattice constants of Ge, zinc telluride (ZnTe), or cadmium telluride (CdTe) substrates. It is found that application of 1% tensile strain reduces the Sn composition required to bring the (positive) direct band gap to zero by approximately 5% compared to a relaxed Ge$_{1-x}$Sn$_{x}$ alloy having the same gap at $\Gamma$. On the other hand, compressive strain has comparatively less impact on the alloy band gap at $\Gamma$. Using DFT calculated alloy lattice and elastic constants, the critical thickness for Ge$_{1-x}$Sn$_{x}$ thin films as a function of $x$ and substrate lattice constant is estimated, and validated against supercell DFT calculations. The analysis correctly predicts the Sn composition range at which it becomes energetically favourable for Ge$_{1-x}$Sn$_{x}$/Ge to become amorphous. The influence of stoichiometry and strain is examined in relation to reducing the magnitude of the inverted (``negative'') $\Gamma_{7}^{-}$-$\Gamma_{8}^{+}$ band gap, which is characteristic of semimetallic alloy electronic structure. Based on our findings, strategies for engineering the semimetal-to-semiconductor transition via strain and quantum confinement in Ge$_{1-x}$Sn$_{x}$ nanostructures are proposed.