Topological superconductor in quasi-one-dimensional ${\mathrm{Tl}}_{2-x}{\mathrm{Mo}}_6{\mathrm{Se}}_6$

We propose that the quasi-one-dimensional molybdenum selenide compound ${\mathrm{Tl}}_{2-x}{\mathrm{Mo}}_6{\mathrm{Se}}_6$ is a time-reversal-invariant topological superconductor induced by intersublattice pairing, even in the absence of spin-orbit coupling (SOC). No noticeable change in superconductivity is observed in Tl-deficient $\left(0\le x\le 0.1\right)$ compounds. At weak SOC, the superconductor prefers the triplet $d$ vector lying perpendicular to the chain direction and two-dimensional $E_{2u}$ symmetry, which is driven to a nematic order by spontaneous rotation symmetry breaking. The locking energy of the $d$ vector is estimated to be weak and hence the proof of its direction would rely on tunneling or phase-sensitive measurements.

Figure 1

Hexagonal lattice and Brillouin zone of ${\mathrm{Tl}}_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$. [(a), (b)] Top and side views of a unit cell. Large pink, median purple, and small green spheres denote Tl, Mo, and Se atoms, respectively. In a unit cell, two Mo triangles, $A$ and $B$, form the (electronic) basis. (c) Wannier functions for $A$ and $B$ sublattices come from $d_{xz}$ orbitals and are $C_3$ invariant. Red and blue colors in the Wannier function stand for opposite signs. (d) The first Brillouin zone and high-symmetry $k$ points. $\mathrm{\Gamma },M,A$, and $L$ are time-reversal invariant momenta.

Figure 2

Band structures of undoped ${\mathrm{Tl}}_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$ with (a) or without (b) SOC. Blue lines represent results from first-principles calculations and red dots are from fitting the tight-binding model. The Fermi energy is set to zero.

Figure 3

(a) Pair susceptibilities of ${\mathrm{Tl}}_{2-x}{\mathrm{Mo}}_6{\mathrm{Se}}_6$ with $x=0$ for relevant channels. Large ${\chi }_0,{\chi }_3$, and ${\chi }_6$ correspond to $A_g,A_u$, and $E_{2u}$ states, respectively. ${\chi }_3$ and ${\chi }_6$ curves almost overlap. (b) Superconducting phase diagram between $A_g$ and $E_{2u}$ states as functions of intra- and intersublattice interaction strengths, $U$ and $V$. Doping has weak effect on the phase boundary. The range of $V$ corresponds to $T_c$ from about 10 to 1000 K (nonlinear relation).

Figure 4

Pair susceptibilities of ${\mathrm{Tl}}_{2-x}{\mathrm{Mo}}_6{\mathrm{Se}}_6$ for all channels for different dopings: (a) $x=0\left(\mu =0\right)$ and (b) $x=0.1(\mu =-0.176$ eV). The most dominant channel ${\chi }_0$ corresponds to the $A_g$ state, followed by ${\chi }_3$ and ${\chi }_6$ for $A_u$ and $E_{2u}$ states, respectively. Differences of ${\chi }_3$ and ${\chi }_6$ are shown in the insets.

Figure 5

Fitting for ${\chi }_6$ and $({\chi }_6-{\chi }_3)$ calculated from the tight-binding model (TB) for $x=0$ [(a), (b)] and $x=0.1$ [(c), (d)]. The fitting formula for ${\chi }_6$ is given by Eq. (B3), but the formula for $({\chi }_6-{\chi }_3)$ is given by $\mathrm{\Delta }{\chi }_0^{\mathrm{fit}}-\mathrm{\Delta }{\mathcal{N}}_{\mathrm{eff}}^{\mathrm{fit}}ln\left(k_{\mathrm{B}}T\right)$. The resulting fitting parameters are given in Table 3. We have chosen to fit the high temperature range above $k_{\mathrm{B}}T={10}^{-3}$ eV due to numerical limit.