Dual topology and edge-reconstruction in $α$-Sn

TL;DR

This work develops a DFT-informed tight-binding framework for cubic $α$-Sn, introducing a bond-angle variable to model in-plane strain and revealing a dual topological character: a $\mathbb{Z}_2$ topological invariant in the bulk and a nonzero mirror Chern number under certain strains. In multilayer configurations, a quantum spin Hall phase emerges for compressive strain when the number of layers exceeds a threshold, while a profusion of in-gap edge states appears across both trivial and nontrivial regimes. A chiral-symmetric limit exposes winding-number topologies that account for these edge modes and reveal hinge-like states tied to mirror topology. The results highlight a rich, strain- and thickness-dependent topological landscape in $α$-Sn and point to candidate van der Waals systems where similar dual-topology physics could be realized and studied experimentally.

Abstract

We formulate the tight-binding model for cubic $α$-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 $α$-Sn one of the rare cases where dual topology can be observed. We calculate the topological phase diagram of multi-layer $α$-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.

Figures (9)

• Figure 1: Relativistic band structure of $\alpha$-Sn using the tight-binding model reported in Table \ref{['table:1']}. (a) Band structure in the energy range from -3.5 eV to +4 eV. (b) Band structure in an extended energy range. The X-point is at the coordinates ($\frac{1}{2}$,0,0), while the L-point is at the coordinates ($\frac{1}{2}$,$\frac{1}{2}$,$\frac{1}{2}$). $\Gamma_8$ and $\Gamma_7$ represent the p-states at the $\Gamma$ point while $\Gamma_6$ represents the s-state.
• Figure 2: Schematic illustration of the crystal lattice of $\alpha$-Sn consisting of two layers of biatomic unit cells stacked along the $z$-direction. The Sn atoms are represented as balls and nearest-neighbor bonds $h_\alpha$ ($\alpha=A, B, C, D$) as tubes connecting them. Each site supports one $s$-orbital and three $p$-orbitals.
• Figure 3: Schematic illustration of bonds of $\alpha$-Sn in case of compressive and tensile strain along $x$ and $y$ axes and in case without a strain. In the unstrained case, all bonds are equal, while in the strained case, there are two different sets of bonds.
• Figure 4: The response of the bulk band structure around the $\Gamma$-point to the tensile (left) and compressive (right) in-plane strain. This strain is modeled as a small change in the polar angle $\theta$ of the bond between the sublattices. The unstrained $\alpha$-Sn bond angle $\theta=\theta_0=0.955$ (black dashed line in both panels) serves as a reference. The increase/reduction of $\theta$ indicates in-plane stretching/compression of the sample. The comparison indicates that only a compressive strain can induce band inversion at the $\Gamma$-point.
• Figure 5: Phase diagram of a layered $\alpha$-Sn system. On the horizontal axis is $N$, the number of layers along the $\vec{n}_3$ direction. On the vertical axis is the bond polar angle between the sublattices divided by the unstrained angle $\theta_0$. Values larger (smaller) than one indicate tensile (compressive) in-plane strain. The color scheme denotes the value of the direct gap between the conduction and valence bands at the high-symmetry points. The blue and white regions are regions without a direct gap. The additional dark shading represents the regions where the system is topologically trivial.