Dual topology and edge-reconstruction in $\alpha$-Sn

TL;DR

This paper models cubic α-Sn using a tight-binding approach based on DFT, revealing dual topological phases, edge states, and the effects of strain and layer thickness on its topological properties.

Contribution

It introduces a variable-bond-angle tight-binding model for α-Sn that captures dual topological phases and edge-reconstruction phenomena under strain and varying layer thickness.

Findings

α-Sn exhibits both $ ext{Z}_2$ topological invariant and mirror Chern number.

A non-trivial quantum spin Hall state appears only under specific compressive strain and layer conditions.

Multiple edge and hinge states are found, originating from a minimal chiral-symmetric model.

Abstract

We formulate the tight-binding model for cubic $\alpha$-Sn based on the DFT calculations. In the model, we incorporate a variable bond angle, which allows us to simulate the effect of the in-plane strain. In the bulk, we demonstrate the presence of the $\mathbb{Z}_2$ topological invariant and a non-zero mirror Chern number, making $\alpha$-Sn one of the rare cases where dual topology can be observed. We calculate the topological phase diagram of multi-layer $\alpha$-Sn as a function of strain and number of layers. We find that a non-trivial quantum spin Hall state appears only for compressive strain above five layers of thickness. Quite surprisingly, both in the trivial and non-trivial phases, we find a plethora of edge-states with energies inside the bulk gap of the system. Some of these states are localized at the side surfaces of the slab, some of them prefer top/bottom surfaces and some are localized in the hinges. We trace the microscopic origin of these states back to a minimal model that supports chiral symmetry and multiple one-dimensional winding numbers that take different values in different directions in the Brillouin zone.