Maximally localized Wannier functions for describing a topological phase transition in stanene

Starting from first-principles calculations on pristine stanene and using maximally localized Wannier functions (MLWFs), we analyze the different phases of the system when it is driven to a topological phase transition. The transition is achieved by a continuous parameter represented by an external electric field ${\mathcal{E}}_z$ as a generic inversion-symmetry-breaking term. We also compare the results with those of a multiorbital tight-binding (TB) model for stanene, whose hopping integrals are determined by Slater-Koster parameters. The system is in a topological (trivial) phase for ${\mathcal{E}}_z<{\mathcal{E}}_c$ (${\mathcal{E}}_z>{\mathcal{E}}_c$). We obtain that the critical field is ${\mathcal{E}}_c\simeq 0.69\mathrm{eV}/\AA$ for the more realistic MLWF compared to ${\mathcal{E}}_c\simeq 0.15\mathrm{eV}/\AA$ in the TB model. This suggests a larger stability of the topological phase of stanene when deposited on inert substrates.

Figure 1

Top: top and side view of the atomic structure of stanene obtained from DFT calculations including relaxation. Bottom: Brillouin zone, special points, and basis vectors mentioned in the text.

Figure 2

(a) Maximally localized Wannier functions (MLWF) centered in one of the Sn atoms. (b) Band structure of stanene obtained with Wannier interpolation (dashed red) compared with first-principles DFT results (black).

Figure 3

Band structure of stanene with the multiorbital tight binding of Hattori et al. compared with ab initio DFT results. Black: DFT results. Red: Multiorbital TB.

Figure 4

Band gap as a function of applied electric field ${\mathcal{E}}_z$, normalized by the critical value ${\mathcal{E}}_c$. Black circles: DFT calculations (${\mathcal{E}}_c=0.690\mathrm{eV}/\AA$). Red down triangles: multiorbital TB model with ${\xi }_0=0.8$ eV (${\mathcal{E}}_c=0.149\mathrm{eV}/\AA$). Blue up triangles: multiorbital TB model with ${\xi }_0=0.644$ eV (${\mathcal{E}}_c=0.102\mathrm{eV}/\AA$).

Figure 5

Calculated values of the absolute value of the Pfaffian, $\left|P\right(K\left)\right|$, for the multiorbital model given in Eq. (2) at the $K$ point as a function of ${\mathcal{E}}_z/{\mathcal{E}}_c$, being ${\mathcal{E}}_c$ the critical value of ${\mathcal{E}}_z$ that closes the gap. Black line: using all eight occupied orbitals. Red dashed line: using the last two occupied orbitals.

Figure 6

Orbital weights for the whole set of eigenstates at the $K$ point in the multiorbital TB model, $|u_n\left(K\right)\rangle$, with $n=1,\cdots ,16$. (a) ${\mathcal{E}}_z<{\mathcal{E}}_c$, within the topological phase. (b) ${\mathcal{E}}_z>{\mathcal{E}}_c$, within the trivial phase.

Figure 7

Different phase factors ${\theta }_m\left(k_2\right)$ (see text) as a function of $k_2$ for (a) the multiorbital TB model; (b) DFT with Wannier interpolation. Top panels: topological nontrivial phase with $\mathcal{E}=0.023\mathrm{eV}/\AA$ ($\mathcal{E}=0.05\mathrm{eV}/\AA$) for the TB model (DFT). ${\theta }_m\left(k_2\right)$ crosses the reference ${\theta }_0=\pi /2$ value (red dashed line) once, an odd number of times, and therefore $Z_2=1$. Lower panels: trivial state with $\mathcal{E}=0.23\mathrm{eV}/\AA$ ($\mathcal{E}=1.00\mathrm{eV}/\AA$) for the TB model (DFT). ${\theta }_m\left(k_2\right)$ crosses the reference line twice (none) in the TB model (DFT), an even number, and therefore $Z_2=0$.

Figure 8

$Z_2$ topological invariant using Wannier interpolation (black circles) and the multiorbital TB model (red dashed line). In each case, the critical value ${\mathcal{E}}_c$ corresponds to the electric field where the gap closes.