Ising superconductivity and Majorana fermions in transition-metal dichalcogenides

In monolayer transition-metal dichalcogenides (TMDs), electrons in opposite $K$ valleys are subject to opposite effective Zeeman fields, which are referred to as Ising spin-orbit coupling (SOC) fields. The Ising SOC, originating from in-plane mirror symmetry breaking, pins the electron spins to the out-of-plane directions, and results in Ising superconducting states with strongly enhanced upper critical fields. Here, we show that the Ising SOC generates equal-spin-triplet Cooper pairs with spin polarized in the in-plane directions. Importantly, the spin-triplet Cooper pairs can induce superconducting pairings in a half-metal wire placed on top of the TMD and result in a topological superconductor with Majorana end states. Direct ways to detect equal-spin triplet Cooper pairs and the differences between Ising superconductors and Rashba superconductors are discussed.

Figure 1

(a) The electrons near the $K$ and $-K$ valleys subject to the Ising SOC fields in opposite directions. The effective Zeeman gap is $2{\beta }_{\text{so}}$, as defined in Eq. (1). (b) A half-metal wire is placed on top of a superconducting monolayer TMD. Majorana fermions (red dots) appear at the ends of wire when the spin polarization $\mathit{h}$ of the half metal has in-plane components. (c) The energy spectrum of the setup in (b) as a function of the chemical potential of the wire ${\mu }_w$, using the tight-binding model in Eq. (8). The red line highlights the topological regime with Majorana modes. (d) The zero energy ground state wave function of the system in the topological regime in (c). Evidently, two Majorana fermions reside at the ends of the wire. The parameters of (c) and (d) are given in the Supplemental Material [29].

Figure 2

(a) A half-metal lead is attached to an Ising superconductor. The spin polarization direction of the half metal is $\mathit{h}$. Both spin-singlet and spin-triplet Cooper pairs can exist in Ising superconductors due to Ising SOC. Equal-spin-triplet Cooper pairs have electron spins pointing to the in-plane directions. Schematic band structures are shown with the horizontal dashed line representing the chemical potential. (b) and (c) The tunneling conductance at a half-metal/Ising superconductor interface. ${\beta }_{\text{so}}=0$ in (b) and ${\beta }_{\text{so}}\ne 0$ in (c). In (c), the tunneling conductance decreases when $\mathit{h}$ deviates from the in-plane directions. The parameters of (b) and (c) are given in the Supplemental Material [29].

Figure 3

The spectral function of a half-metal wire in proximity to a monolayer superconducting TMD. (a) The spin polarization $\mathit{h}$ of the wire is in plane. An energy gap is induced on the wire. (b) $\mathit{h}$ is perpendicular to the plane. The superconducting TMD cannot induce equal-spin pairing on the wire and the wire is gapless. The parameters of the model are given in the Supplemental Material [29].