Ab initio study of the ${\mathrm{PbTaSe}}_2$-related superconducting topological metals

Recent reports have demonstrated the $Z_2$ topological property of the superconductor ${\mathrm{PbTaSe}}_2$ ($T_c\sim 3.8$ K). ${\mathrm{PbTaSe}}_2$ simultaneously exhibits superconductivity and nontrivial $Z_2$ topology without the need for doping or proximity effect. Scanning tunneling microscopy confirms that the topological surface states are gapped by superconductivity, indicating the likelihood of the chiral $p_x+i{p}_y$ pairing mechanism. Motivated by these exciting findings, we predict by means of the first-principles scheme that $AB{\mathrm{Se}}_2$ ($A=\text{Pb}$ or Sn and $B=\text{Ta}$ or Nb) is also a superconductor with nontrivial $Z_2$ topology. Among these, ${\mathrm{SnNbSe}}_2$ shows the highest $T_c\sim 7.0$ K. Due to the fact that the required energy resolution to detect the Majorana bound states is of the order of ${\mathrm{\Delta }}^2/{\varepsilon }_F$, the predicted higher $T_c$ of $AB{\mathrm{Se}}_2$ may help mitigate some experimental challenges and provides a better platform for the exploration of the topological superconductivity.

Figure 1

Top view (left) and side view (right) of $AB{\mathrm{Se}}_2$. The gray, brown, and green spheres represent $A,B$, and Se atoms, respectively.

Figure 2

The bulk band structures of (a) ${\mathrm{PbTaSe}}_2$ (b) ${\mathrm{PbNbSe}}_2$, (c) ${\mathrm{SnTaSe}}_2$, and (d) ${\mathrm{SnNbSe}}_2$. The components of the Pb/Sn-$p$ and Ta/Nb-$d$ orbitals are displayed in red and blue, respectively. The shaded region in each figure indicates the continuous gap below which the corresponding $Z_2$ invariant is computed.

Figure 3

Phonon spectra, Eliashberg spectrum functions ${\alpha }^{2}F\left(\omega \right)$, and the integrated e-ph coupling constants $\lambda \left(\omega \right)$ of (a) ${\mathrm{PbTaSe}}_2$, (b) ${\mathrm{PbNbSe}}_2$, (c) ${\mathrm{SnTaSe}}_2$, and (d) ${\mathrm{SnNbSe}}_2$. The phonon spectra are displayed with the decomposition of the atomic motions and the magnitudes of ${\lambda }_{\mathbf{q}\nu }$ (red circles) in left and middle panels, respectively. ${\alpha }^{2}F\left(\omega \right)$ (blue curve) and $\lambda \left(\omega \right)={\int }_0^{\omega }\lambda \left({\omega }^{\prime }\right)d{\omega }^{\prime }$ (red curve) are shown in the right panel. The magnitude of ${\lambda }_{\mathbf{q}\nu },{\alpha }^{2}F\left(\omega \right)$, and $\lambda \left(\omega \right)$ are displayed with identical scale in all figures for comparison.

Figure 4

The phase evolution of the Wannier centers of $AB{\mathrm{Se}}_2$. The left and right panels represent the phase evolution of the effective 2D system at $k_z=0$ and $\frac{\pi }{2}$, respectively. $Z_2={\delta }_1·{\delta }_2=-1$ for all these materials, indicating the topologically nontrivial state.

Figure 5

The decomposed surface band structures of (a) ${\mathrm{PbTaSe}}_2$, (b) ${\mathrm{PbNbSe}}_2$, (c) ${\mathrm{SnTaSe}}_2$, and (d) ${\mathrm{SnNbSe}}_2$. As denoted on top of each panel, projected $A\text{−}p$ and $B\text{−}d$ orbitals of the surface atoms are shown in the top-left panel in each figure, while the other panels display the projected spin polarizations of bands contributed from the surface atoms. The $\mathrm{\Gamma }\text{−}K$ and $\mathrm{\Gamma }\text{−}M$ directions are chosen to lie along $+x$ and $+y$, respectively, to clearly demonstrate the behavior of spin-momentum lock.