Topological superconductivity in Ni-based transition metal trichalcogenides

Based on a two-orbital honeycomb lattice model and by use of a random phase approximation analysis we investigate the pairing symmetry of Ni-based transition metal trichalcogenides. We find that an $I$-wave $\left(A_{2g}\right)$ state and a chiral $d$-wave $\left(E_g\right)$ state are dominant and nearly degenerate for typical electron and hole dopings. Both pairing states exhibit nontrivial topological properties, which manifest themselves by the chiral edge states for the $d+id$-wave state and dispersionless Andreev bound state at zero energy for the $I$-wave state. We thus show that Ni-based transition metal trichalcogenides provide a promising platform to study exotic topological phenomena emerging from electronic correlation effects.

Figure 1

(a) Crystal structures of the monolayer $\mathrm{NiP}{X}_3(X=$ S, Se) (space group P-31m). (b) Top view of the monolayer $\mathrm{NiP}{X}_3$. (c) Electronic band structure and density of states (DOS) for ${\mathrm{NiPS}}_3$. The orbital characters of bands are represented by different colors. (d) Band dispersion of tight-binding model based on $d_{xz}$ and $d_{yz}$ orbitals of Ni atoms.

Figure 2

Orbital resolved Fermi surface with different electron filling. (a) $\delta =0$, (b) $\delta =-0.2$, (c) $\delta =0.2$, and (d) $\delta =0.4$. The Fermi surfaces around $K$ and $\frac{K}{2}$ are represented by $\alpha$ and $\beta$.

Figure 3

Susceptibilities for different doping. (a) Bare susceptibilities along high-symmetry lines for $\delta =-0.2$ (green line), 0.2 (red line), and 0.4 (blue line). The corresponding RPA spin susceptibilities in the Brillouin zone are shown in (b), (c), and (d) with $U=0.4$ and $J/U=0.2$.

Figure 4

Leading pairing strengths in spin singlet and triplet channels for superconducting states with different doping near the half filling. The pairing strengths as a function of Hund's rule coupling parameter $J/U$ with 0.2 and 0.4 electron doping at $U=0.4$ are plotted in (a) and (c). The pairing strengths as a function of on-site intraorbital Coulomb interaction $U$ with $J/U=0.2$ with 0.2 and 0.4 electron doping are plotted in (b) and (d).

Figure 5

The gap functions of two leading singlet pairings with U=0.4 and J/U=0.2 at 0.2 electron doping. (a) gap function of $A_{2g}(I$-wave) irreducible representation. (b), (c) gap functions of $d_{xy}$ and $d_{x^2-y^2}$ wave in $E_g$ irreducible representation. (d) gap function of $d_{xy}+i{d}_{x^2-y^2}$.

Figure 6

Same as Fig. 5, but at 0.4 electron doping.

Figure 7

The two leading pairing strengths in singlet and triplet channels for superconducting states with 0.2 hole doping. (a) The pairing strengths as a function of Hund's rule coupling parameter $J/U$ with $U=0.4$ at 0.2 hole doping. (b) The pairing strengths as a function of on-site intraorbital coulomb interaction $U$ with $J/U=0.2$ at 0.2 hole doping.

Figure 8

The gap functions of two leading singlet pairings with $U=0.44$ and $J/U=0.2$ at 0.2 hole doping. (a) Gap function of $A_{2g}(I$-wave) irreducible representation. (b), (c) Gap functions of $d_{xy}$ and $d_{x^2-y^2}$ wave in $E_g$ irreducible representation. (d) Gap function of $d_{xy}+i{d}_{x^2-y^2}$.

Figure 9

Zigzag and armchair edge spectra for the chiral $d$-wave (top panel) and $I$-wave pairing states (bottom panel) at 0.2 electron doping. Onsite $d$-wave pairing and NN $I$-wave pairing are adopted in the calculations.

Figure 10

Gap function of $A_{2g}(I$-wave) irreducible representation with $U=0.4$ and $J/U=0.2$ at 0.2 electron doping. The green line is the folded Brillouin zone for the zigzag edge state in momentum space.