Discovering two-dimensional topological insulators from high-throughput computations

We have performed a computational screening of topological two-dimensional (2D) materials from the Computational 2D Materials Database (C2DB) employing density functional theory. A full ab initio scheme for calculating hybrid Wannier functions directly from the Kohn-Sham orbitals has been implemented and the method was used to extract ${\mathbb{Z}}_2$ indices, Chern numbers, and mirror Chern numbers of 3331 2D systems including both experimentally known and hypothetical 2D materials. We have found a total of 48 quantum spin Hall insulators, seven quantum anomalous Hall insulators, and 21 crystalline topological insulators. Roughly 75% are predicted to be dynamically stable and one-third was known prior to the screening. The most interesting of the topological insulators are investigated in more detail. We show that the calculated topological indices of the quantum anomalous Hall insulators are highly sensitive to the approximation used for the exchange-correlation functional and reliable predictions of the topological properties of these materials thus require methods beyond density functional theory. We also performed $GW$ calculations, which yield a gap of 0.65 eV for the quantum spin Hall insulator ${\mathrm{PdSe}}_2$ in the ${\mathrm{MoS}}_2$ crystal structure. This is significantly higher than any known 2D topological insulator and three times larger than the Kohn-Sham gap.

Figure 1

Colored area indicate the unit cell in reciprocal space. For each value of $k_2$, the Berry space is calculated by parallel transporting the Bloch states along $k_1$ (indicated by dashed lines).

Figure 2

Top: structure and Brillouin zone of ${\mathrm{FeBr}}_3$ in the ${\mathrm{BiI}}_3$ crystal structure. Middle: band structure of ${\mathrm{FeBr}}_3$ with colors denoting the expectation value of $S_z$. The band PBE gap is 42 meV. Bottom: Berry phases of the four highest occupied states of ${\mathrm{FeBr}}_3$ calculated as a function of $k$ in the direction of $M_2$.

Figure 3

Top: structure and Brillouin zone of stanene. Middle: band structure of stanene. Bottom: Berry phases of the four highest occupied states of stanene calculated as a function of $k$ in the direction of $M_2$ with colors denoting the expectation value of $S_z$.

Figure 4

Top: structure and Brillouin zone of SnTe in the PbSe crystal structure. Middle: band structure of SnTe. Bottom: Berry phases of the eight highest occupied states of ${\mathrm{FeBr}}_3$ calculated as a function of $k$ in the direction of $Y$ with colors denoting the expectation value of $S_z$.

Figure 5

LDA (dashed) and GW (solid) band structures of ${\mathrm{PdSe}}_2$ in the MoS2 prototype. In both cases the energy of the top of the valence band has been set to zero.

Figure 6

Band gap as a function of magnetization angle with the out-of-plane axis in ${\mathrm{OsO}}_2$. The material undergoes a topological transition for $C=0$ to $C=-2$ when the gap closes in the vicinity of $\pi /4$.