Second-order topological insulators in two-dimensional monolayers of the 1T-phase ${\mathrm{PtSe}}_2$ material class

Two-dimensional (2D) second-order topological insulators (SOTIs) have been extensively studied due to their unique feature of fractional charge at corners. In order to realize such kind of SOTI in natural materials, we reveal a class of experimentally synthesized 1T-phase transition metal dichalcogenides (TMDs) monolayers as candidates of SOTI. Taking the monolayer of 1T-${\mathrm{PtSe}}_2$ as an example, we identify its second-order topology by determining the nonzero fractional corner charge using first-principles calculations and symmetry analysis. Furthermore, we emphasize the role of crystalline symmetry in the emergence of corner states based on an effective edge theory. Due to the same symmetry and similar band structure, our analysis can be directly applied to other 1T-TMD monolayers. Our findings uncover the previously overlooked higher-order topology in 2D 1T-TMD materials, which may draw immediate experimental attention.

Figure 1

Atomic and band structure of monolayer ${\mathrm{PtSe}}_2$. (a) Crystal structure of monolayer ${\mathrm{PtSe}}_2$ from top and side view. The symbols Se and ${\mathrm{Se}}^{\prime }$ denote the selenium atoms situated in the upper and lower layers, respectively. The $x$ direction is along the twofold rotation axis of the system. The $z$ direction is perpendicular to the atomic plane. (b) Orbital-resolved band structure of monolayer ${\mathrm{PtSe}}_2$ without SOC. The size of blue and red dots represents the contribution from $p_{x,y}$, and $p_z$ orbitals of Se atoms. $\pm$ marks Bloch states having opposite parities with respect to inversion operations at $\mathrm{\Gamma }$ and $M$. The size of green, violet, and orange dots represents the contribution from $d_{xy,x^2-y^2}, d_{xz,yz}$, and $d_{z^2}$ orbitals of Pt atoms, respectively.

Figure 2

Corner charges of monolayer ${\mathrm{PtSe}}_2$. (a), (b), (c) Top view of the real-space charge distributions of corner states around the Fermi level for the hexagonal (553 atoms, isosurface level = 0.0006 electron ${\mathrm{bohr}}^{-3}$), rhombus (407 atoms, isosurface level = 0.001 electron ${\mathrm{bohr}}^{-3}$), and triangle (358 atoms, isosurface level = 0.0003 electron ${\mathrm{bohr}}^{-3}$) nanodisks. The gray (green) balls represent Pt (Se) atoms. (d), (e) Energy spectrum of hexagonal and rhombus nanodisks. The red circles represent corner states. (f) Schematic illustration of the corner charge at the intersection of two edges. The new coordinate system $x_1\text{−}{x}_2$ is rotated counterclockwise by $\theta =\frac{\pi }{3}$ relative to the original $xy$ coordinates. The two edges are cut along the $y$ and $x_2$ axes, respectively.

Figure 3

Energy spectrum of ${\mathrm{PtSe}}_2$ nanoribbons with (a) zigzag and (b) armchair edges. The red color represents the edge spectrum. The insets in each panel show the structure of nanoribbons from the top view.