Discovering topological phases in gray-Tin

TL;DR

The paper addresses predicting topological phases in the narrow-gap diamond-lattice material α-Sn, where small lattice distortions and spin–orbit coupling drive transitions among semimetal, Dirac semimetal, TI, and trivial insulator phases. It applies fully self-consistent relativistic GW (scGW) with exact two-component (X2C) SOC to map the phase diagram under uniform and anisotropic strain, removing dependence on the initial DFT functional. A robust diagnostic based on band- and orbital-occupation dispersions in the symmetrized atomic orbital (SAO) basis is proposed for identifying topology in correlated calculations. Key findings include a Dirac semimetal at the phase boundary and TI and Dirac-like TI phases realized by strain, as well as a trivial insulating phase under other lattice parameters, illustrating strain-engineered topological phase transitions. The work demonstrates predictive first-principles band engineering in α-Sn and offers a general diagnostic framework for correlated topological materials.

Abstract

Non-trivial topological phases often emerge in narrow-gap semiconductors with a delicate blend of spin-orbit coupling and electron correlation. The diamond-lattice allotrope of Sn ($α$-Sn) exemplifies this behavior, hosting multiple topological phases that can be tuned by small distortions in the lattice. Despite rapid experimental progress, theoretical descriptions of $α$-Sn lack predictive power and rely mainly on tight-binding models and density functional theory with uncontrolled approximations. We employ first-principles fully self-consistent, relativistic GW (scGW) to overcome these limitations. The scGW recovers the experimentally observed zero-gap semiconductor and the strain-induced topological insulator and Dirac semimetal phases, while also predicting new trivial and topological insulators and a Dirac semimetal phase, further demonstrating the versatility of $α$-Sn for band engineering. Additionally, we propose a robust diagnostic of topological behavior based on a combined analysis of band and orbital-occupation dispersions, tailored for correlated methods where standard mean-field-based topological invariants fall short. Our findings pave the way for studying a broad class of topological materials using accurate first-principles methods beyond density functional theory.

Figures (3)

• Figure 1: (a) PBE and (b) sc$GW$ Band structures for $\alpha$-Sn for lattice constants ranging $a=6.4$ and $6.7~\mathrm{\AA}$ along the $L$-$\Gamma$-$X$-$K$-$\Gamma$-$K$ high-symmetry path.
• Figure 2: Influence of lattice parameter on the electronic structure of $\alpha$-Sn. (a) PBE and (b) sc$GW$ band structures along the $K$–$\Gamma$–$X$ high-symmetry path for lattice constants ranging from $6.4$ to $6.7~\mathrm{\AA}$. (c) sc$GW$ orbital occupation trends for the $5s$ and $5p$ symmetrized atomic orbitals (SAOs). (d) Schematic illustration of the band evolution at the $\Gamma$ point, highlighting the distinct electronic phases that emerge as the lattice parameter varies. In the native semimetallic phase of $\alpha$-Sn, the double inversion between the $5s$ and $5p$ orbitals gives rise to two topological surface states rogalev_double_2017.
• Figure 3: Effect of uniaxial strain along the [001] axis in bulk $\alpha$-Sn: (a) sc$GW$ band structure along the $Z$-$\Gamma$-$X$ path; (b)-(c) spectral weight contribution from the $5s$ and $5p$ SAOs; and (d) corresponding orbital occupation trends. While the stretched lattice exhibits a Dirac semimetal, exemplified by a Dirac point along the $Z$-$\Gamma$ path, the squeezed lattice develops into a topological insulator, visibly clearly under 5% strain. All results are shown near the $\Gamma$ point, with $a = 6.7,\mathrm{\AA}$ as the reference lattice parameter.