Josephson junction of nodal superconductors with a Rashba and Ising spin-orbit coupling

We study the effect of a Rashba spin-orbit coupling on the nodal superconducting phase of an Ising superconductor. Such nodal phase was predicted to occur when applying an in-plane field beyond the Pauli limit to a superconducting monolayer transition metal dichalcogenides (TMD). Generically, Rashba spin orbit is known to lift the chiral symmetry that protects the nodal points, resulting in a fully gapped phase. However, when the magnetic field is applied along the $x$ direction, a residual vertical mirror symmetry protects a nodal crystalline phase. We study a single-band tight-binding model that captures the low-energy physics around the $\mathrm{\Gamma }$ pocket of monolayer TMD. We calculate the topological properties, the edge state structure, and the current phase relation in a Josephson junction geometry of the nodal crystalline phase. We show that while the nodal crystalline phase is characterized by localized edge modes on non-self-reflecting boundaries, the current phase relation exhibits a trivial $2\pi$ periodicity in the presence of Rashba spin-orbit coupling.

Figure 1

Schematic diagram of the triangular lattice used for the tight-binding model showing the lattice vectors ${\mathbf{a}}_{\mathbf{1},\mathbf{2}}$ and the hopping amplitudes corresponding to the Ising and Rashba SOC. This hopping profile results in the Hamiltonian $H_I$ and $H_R$ in Eqs. (6) and (9) respectively.

Figure 2

Energy dispersion of the two low-energy bands of Eq. (10) in the vicinity of the $\mathrm{\Gamma }$ point. The parameters used are $m=1, \tilde{\mu }=-0.3, {\lambda }_{SO}=0.15, h=0.1, \mathrm{\Delta }=0.06$, and ${\alpha }_R=0.02$. The white line marks the zero-energy contour. The four gap-closing points are shifted away from the $E=0$ plane and lie on the $k_x=0$ line.

Figure 3

The evolution of the mirror eigenvalues at the reflection symmetric momenta $k_x=0,\pi$ of the two lowest energy levels for the lattice toy model. (The lines are shifted vertically for clarity.) The dispersion of the two low-energy bands at $k_x=0$ ($k_x=\pi$) is shown in solid (dashed) gray lines, indicating the location of the nodal points. The parameters used are $m=1, \tilde{\mu }=-0.3, {\lambda }_{SO}=0.15, h=0.1, \mathrm{\Delta }=0.06$, and ${\alpha }_R=0.02$.

Figure 4

Energy spectrum for a nanoribbon with open boundary conditions for (a) ${\alpha }_R=0$ and (b) ${\alpha }_R=0.01$ obtained by diagonalizing Eq. (15). The Rashba SOC breaks chiral symmetry and moves the nodal points away from zero energy. Midgap states localized at the open boundaries of the ribbon are marked in red. These states decay exponentially into the bulk for momenta $k_y$ between pairs of nodal points.

Figure 5

Semi-log plot of the decay length of the “particle up” component $|u_{\uparrow }{|}^2$ of the edge-state spinor as a function of position at momentum $k_y\approx 0.75$ for different values of Rashba SOC strength ${\alpha }_R$, while keeping all the other parameters unchanged. Here the parameters are $m=1, \tilde{\mu }=-0.3, {\lambda }_I=0.15, h=0.1$, and $\mathrm{\Delta }=0.06$.

Figure 6

Schematic showing the torus geometry used to study the Josephson junction. The $y$ direction has periodic boundary conditions, making $k_y$ a good quantum number. The change in the phase of the superconducting pairing corresponds to a flux $\mathrm{\Phi }$ through this torus. The $1\text{D}$ chain along the $x$ direction has twisted boundary conditions, i.e., all hopping parameters across the insulating weak link shown in white, acquire a phase $\varphi =\mathrm{\Phi }/{\mathrm{\Phi }}_0, {\mathrm{\Phi }}_0$ being the flux quantum, and are attenuated by a factor proportional to the strength of the insulating barrier.

Figure 7

Energy-phase relation of a Josephson junction obtained by solving the lattice model on a torus geometry of Fig. 6. The Rashba SOC is (a) ${\alpha }_R=0$ and (b) ${\alpha }_R=0.01$. We have chosen the momentum $k_y=0.75$, which lies between two nodal points. In (a) we find that there are zero crossings at $\varphi =\pi ,3\pi$ as is clear from the inset. This means $E_J\left(\varphi \right)$ has a $4\pi$ periodicity. However in (b) the Josephson energy-phase relation is no longer symmetric about $E=0$. Additionally, there is no crossing at $\varphi =\pi ,3\pi$ and therefore only a $2\pi$ periodicity in $E\left(\varphi \right)$, the parameters are $m=1, \tilde{\mu }=-0.3, {\lambda }_I=0.15, h=0.1$, and $\mathrm{\Delta }=0.06$

Figure 8

Spectrum of the bare 1D effective Hamiltonian Eq. (B1) consists of two spin-orbit bands (marked by $\pm$) that cross the Fermi level ${\mu }_{k_y}=0$ at the Fermi momenta $k_F=0,\pm k_{so}$.

Figure 9

Bulk and edge state spectrum for $\alpha =-0.01, \tilde{\mu }=-0.25 {\mu }_l=-0.175, m=1, {\lambda }_{SO}=0.15, h=0.1$, and $\mathrm{\Delta }=0.06$. ${\mu }_l\ne 0$ on either of the two edges, breaks the reflection symmetry and lifts the degeneracy of the edge states (shown as solid red lines).