Epitaxial binding and strain effects of monolayer stanene on the ${\mathrm{Al}}_2{\mathrm{O}}_3\left(0001\right)$ surface

Stanene, the two-dimensional (2D) monolayer form of tin, has been predicted to be a 2D topological insulator due to its large spin-orbit interaction. However, a clear experimental demonstration of stanene's topologically nontrivial properties has eluded observation, in part because of the difficulty of choosing a substrate on which stanene will remain topologically nontrivial. In this paper, we present first-principles density functional theory calculations of epitaxial monolayer stanene grown on the (0001) surface of alumina, ${\mathrm{Al}}_2{\mathrm{O}}_3$, as well as freestanding decorated stanene under strain. By describing the energetics and nature of how monolayer stanene binds to alumina, we show a strong energetic drive for the monolayer to be coherently strained and epitaxial to the substrate. By analyzing the electronic structure of strained stanene, we find it to be a quantum spin Hall insulator on ${\mathrm{Al}}_2{\mathrm{O}}_3$. We also describe the effect of in situ fluorine decoration on the bound stanene monolayer, including on its potential for mechanical exfoliation.

Figure 1

(a), (d) Top (a) and side (d) views of the bare ${\mathrm{Al}}_2{\mathrm{O}}_3$(0001) slab used as a substrate. The three inequivalent exposed Al atoms are labeled A, B, and C. The blue arrow in (a) indicates the viewing direction of panels (d)–(f). (b), (e) Top (b) and side (e) views of second-most-stable registry choice A/B for stanene on ${\mathrm{Al}}_2{\mathrm{O}}_3$, placing the upper and lower Sn atoms in positions A and B, respectively. Black arrows in (e) label the bond length $d$, the buckling $b$, and the vertical binding distances $h_1$ and $h_2$, whose values are found in Table 2. (c), (f) Top (c) and side (f) views of the most stable registry choice A/C for stanene on ${\mathrm{Al}}_2{\mathrm{O}}_3$, placing the upper and lower Sn atoms in positions A and C, respectively.

Figure 2

(a), (c) Side view of freestanding stanene layer and stanene bound to the ${\mathrm{Al}}_2{\mathrm{O}}_3$(0001) slab, respectively. (b) Side view of bare alumina slab. The yellow features are an isosurface of the local density of states integrated from the Fermi level ($E_F$) to 1.0 eV above $E_F$. (d)–(f) Density of states (DOS) plots for freestanding stanene, bare alumina, and bound stanene on alumina, respectively. In the plot for the full system, the DOS is also projected onto the Löwdin orbitals of the tin atoms (stanene) and the aluminum/oxygen atoms (${\mathrm{Al}}_2{\mathrm{O}}_3)$.

Figure 3

Side view of stanene on alumina showing positive (yellow) and negative (cyan) isosurfaces for electron redistribution. For regions in yellow, the electron density of stanene on alumina is greater than the sum of the electron density of the bare slab and that of the freestanding stanene monolayer, indicating an increase in electron density during binding. For regions in cyan, the reverse is true.

Figure 4

(a) Evolution of the band gap of bare stanene (green), fluorinated stanene (SnF, magenta), and hydrogenated stanene (SnH, black) as a function of the lattice parameter. The band gap is defined as the signed energy difference between the conduction band (CB) minimum and the valence band (VB) maximum, meaning that it is negative for semimetallic materials. The equilibrium lattice parameter of ${\mathrm{Al}}_2{\mathrm{O}}_3$ is indicated with a vertical black line. For each structure, a dashed line indicates a trivial material with a $Z_2$ topological index of 0, while a solid line indicates a nontrivial material with $Z_2$ index of 1. Each material's equilibrium lattice parameter is marked with a large open circle, while the band structures plotted in Fig. 5 correspond to the points marked with small filled circles. (b)–(d) Band structures of bare stanene, fluorinated stanene (SnF), and hydrogenated stanene (SnH) at their equilibrium lattice parameters. Bands are colored by their $s$-orbital (red) and $p$-orbital (blue) characters, and labeled with their parities at time-reveral invariant momenta (TRIMs). On the $x$ axis, each TRIM is labeled with the product of all of its band parities, the quantity called ${\delta }_i$ in the Fu-Kane treatment [40]. The zero of band energy is the Fermi level $E_F$.

Figure 5

Band structure plots showing the change of topological character via band inversions in bare stanene (top row), fluorinated stanene, SnF (middle row), and hydrogenated stanene, SnH (bottom row). The zero of energy in each plot is the Fermi level $E_F$. In each case, a negative-parity $s$ band crosses down from the conduction band into the valence band as the lattice parameter increases. Due to their different equilibrium lattice parameters, freestanding stanene ($a=4.68\text{Å}$) and SnF ($a=5.02\text{Å}$) are topological insulators, while SnH ($a=4.72\text{Å}$) is a trivial insulator, but each material can be tuned to the other regime using strain.

Figure 6

Calculations of epitaxial stanene on alumina. (a) DFT band structure plot: bands are colored by their Sn $s$-orbital (red) and Sn $p$-orbital (blue) characters. The DFT band gap is calculated to be 0.263 eV. (b) Evolution of the Wannier charge centers (WCCs) in the $x\text{−}y$ plane, computed according to the technique in [41]. The winding of the WCCs demonstrates that bound stanene is a topological insulator.

Figure 7

In situ fluorination of the stanene sheet. (a), (f) Top (a) and side (f) views of the half-fluorinated stanene sheet. (b), (g) Top (b) and side (g) views of the higher-energy stanene sheet fluorinated at the bridging site but not at the Al-bound site. (c), (h) Top (c) and side (b) views of the fully fluorinated stanene sheet in the ${\mathrm{Al}}_2{\mathrm{O}}_3$-bound ground state. (d) Illustration of clean SnF exfoliation. (e) Illustration of exfoliation of the half-fluorinated stanene sheet, with the bound F atom left behind. The energy cost for this process is higher than for clean exfoliation.

Figure 8

Orbital-projected hybrid (HSE06) band structures of stanene and SnF; the analogous nonhybrid calculations are shown in Figs. 4 and 4. The zero of energy is the Fermi level $E_F$.

Figure 9

Substrate thickness dependence of the stanene-on-alumina band structure.