${\mathrm{BaSn}}_2$: A wide-gap strong topological insulator

${\mathrm{BaSn}}_2$ has been shown to form as layers of buckled stanene intercalated by barium ions. However, despite an apparently straightforward synthesis and significant interest in stanene as a topological material, ${\mathrm{BaSn}}_2$ has been left largely unexplored, and has only recently been recognized as a potential topological insulator. Belonging to neither the lead nor bismuth chalcogenide families, it would represent a unique manifestation of the topological insulating phase. Here we present a detailed investigation of ${\mathrm{BaSn}}_2$, using both ab initio and experimental methods. First-principles calculations demonstrate that this overlooked material is indeed a strong, wide-gap topological insulator with a bulk band gap of 200 meV. We characterize the surface state dependence on termination chemistry, providing guidance for experimental efforts to measure and manipulate its topological properties. Additionally, through ab initio modeling and synthesis experiments, we explore the stability and accessibility of this phase, revealing a complicated phase diagram that indicates a challenging path to obtaining single crystals.

Figure 1

${\mathrm{BaSn}}_2$ is composed of alternating layers of tin and barium. The tin layer forms a buckled honeycomb lattice, essentially forming stanene. Barium forms a flat triangular network, with barium atoms vertically aligned with the hollows of the stanene plaquettes.

Figure 2

(a) The bulk band structure of ${\mathrm{BaSn}}_2$ computed using the PBE functional. The topological band inversion is seen to occur at A=(0,0,$\pi$), with spin-orbit splitting opening a gap of $\sim 200\mathrm{meV}$ along A-H. However, the conduction band at M sits at the same energy as the HOMO maximum near A. Furthermore, the direct gap near A actually closes away from of the high-symmetry line at $\mathbf{k}=\mathbf{2}\pi (\mathbf{0}.\mathbf{12},\mathbf{0}.\mathbf{00},\mathbf{0}.\mathbf{69})$ (b) The bulk bandstructure computed with the HSE hybrid functional. The major effect of the inclusion of exact exchange is to widen the gap at M moving the conduction band clear of the Fermi energy. Additionally, the band gap is increased to $\sim 200\mathrm{meV}$ and occurs as a direct gap at $\mathbf{k}=\mathbf{2}\pi (\mathbf{0}.\mathbf{10},-\mathbf{0}.\mathbf{02},\mathbf{0}.\mathbf{59})$.

Figure 3

In the center, the band structure along the line from $\mathrm{\Gamma }$ to A is shown, with parity eigenvalues at the end points marked in blue ($+1$) and green ($-1$). On either side, the wave functions are shown for the bands that invert, and the bands are colored according to their projection (in the Wannier basis) onto the conduction and valence bands at $\mathrm{\Gamma }$. This makes it clear that these are the bands that have inverted; the orbital character of the marked valence (conduction) band at A is qualitatively the same as that of the conduction (valence) band at $\mathrm{\Gamma }$.

Figure 4

Comparison of band structures for 50 layer slabs computed using the Wannier tight-binding model generated from (a)PBE and (b) HSE functionals. The effect of HSE on the surface states is to increase the splitting between the conduction and valence bands, and lift the degeneracy at $\mathrm{\Gamma }$ in energy, nearly into the bulk gap. Purple intensity denotes contribution of the Wannier functions localized to the top- and bottom-most layers.

Figure 5

PBE band structures for three different terminating compositions. (a) Barium layer termination, but with potassium substituted for barium. The resulting surface states lie deep within the valence manifold, and are nearly degenerate. (b) Inclusion of a monolayer of chlorine above a full layer of barium raises and splits the surface bands. (c) Substituting thalium for bismuth raises the surface bands even further, into the conduction manifold, while substantially increasing the surface band splitting.

Figure 6

(a) Calculated formation enthalpies for relevant Ba-Sn phases at $T=0$ K. Circles and diamonds show the results at 0 and 6 GPa, respectively. As described in the text, the shown hP12-${\mathrm{BaSn}}_2$ phase could be a preferred starting material for obtaining crystalline hP3-${\mathrm{BaSn}}_2$. (b,c) Calculated Gibbs energies of hP3-${\mathrm{BaSn}}_2$ and hP12-${\mathrm{BaSn}}_2$ as a function of $T$ or $P$ relative to the combination of hP8-${\mathrm{BaSn}}_3$ and oS32-${\mathrm{Ba}}_3{\mathrm{Sn}}_5$.

Figure 7

Powder XRD of ${\mathrm{BaSn}}_3$ crystals and Sn flux obtained from ${\mathrm{Ba}}_{30}{\mathrm{Sn}}_{70}$ growth (black line) and powder XRD after low-temperature annealing showing appearance of ${\mathrm{BaSn}}_2$ phase (red line).

Figure 8

Distributions of enthalpies during evolutionary structure optimization runs at $P=10$ GPa for unit cells with 2:6, 4:8, 6:10, and 4:6 Ba and Sn atoms. The best member in each generation is shown in red. The structures shown above each panel were found to have the lowest enthalpy at the corresponding composition. In the hP12-${\mathrm{BaSn}}_2$ phase, expected to be stable above $\sim 4$ GPa, the Sn framework intercalated by Ba atoms comprises the kagome lattice bridged by Sn2 atoms.