$\pi$/2-Josephson junction as a topological superconductor

Superconducting states with broken time-reversal symmetry are rarely found in nature. Here we predict that it is inevitable that the time-reversal symmetry is broken spontaneously in a superconducting Josephson junction formed by two superconductors with different pairing symmetries dubbed as $\pi /2$-Josephson junction. While the leading conventional Josephson coupling vanishes in such an $\pi /2$-Josephson junction, the second-order coupling from tunneling always generates chiral superconductivity orders with broken time-reversal symmetry. Josephson frequency in the $\pi /2$ junction is doubled, namely $\omega =4eV/h$. The result can not only provide a way to engineer topologically trivial or nontrivial time-reversal breaking superconducting states, but also be used to determine the pairing symmetry of unconventional superconductors.

Figure 1

The sketch of a vdW Josephson junction with two SC layers having two different pairing symmetries, e.g., $d_{x^2-y^2}$ and $d_{xy}$.

Figure 2

The sketch of the two-layer model and calculated SC orders from the mean-field calculation: (a) two layer $t\text{−}J$ models with the top layer $J_1$ and bottom layer $J_2$; (b) superconducting orders ${\mathrm{\Delta }}_{xy}$ (red arrow) and ${\mathrm{\Delta }}_{x^2-y^2}$ (blue arrow) as function of exchange strength $J_{1,2}$ with other parameters given in the main text. The magnitudes and phases of pairing orders are represented by the lengths and directions of the arrows; panels (c) and (d) show the real and imaginary parts of $d_{x^2-y^2}+i{d}_{xy}$ order in the first layer with $J_1=3.4$ and $J_2=3$. The Fermi surfaces of the two-layer Hamiltonian are represented by the red dashed lines, which coincide with the arcs in BZ.

Figure 3

The Berry curvature for different TRB orders in the Hamiltonian (3) with parameters $t_1^1=0.88$, $t_1^2=-0.35$, $t_2^1=1.67$, $t_2^2=-0.33$, ${\mu }_1=-0.4$, ${\mu }_2=-1.2$, $\mu =0$, $t=0.4$, ${\mathrm{\Delta }}_{\alpha }=0.8183$, ${\mathrm{\Delta }}_{\beta }=0.5562$, ${\theta }_{\alpha }=0$, and ${\theta }_{\beta }=\pi /2$. Panels (a)–(d) represent $d+i{s}^{\prime }$, $d+i{d}^{\prime }$, $d+is$, and $d^{\prime }+is$ orders, respectively. According to our analysis, the Berry field of $d\pm is\left(s^{\prime }\right)$ belongs to $B_2$ IR, $d^{\prime }\pm is\left(s^{\prime }\right)$ belongs to $B_1$ IR, and only $d+i{d}^{\prime }$ belongs to $A_1$ IR. Only $d\pm i{d}^{\prime }$ in (b) shows a nonzero Chern number.

Figure 4

Edge states of $d+i{d}^{\prime }$ superconductivity in the Hamiltonian (6); the parameter we choose is $J_1=3.4$, $J_2=3$, and the corresponding self-consistent mean-field SC orders are ${\mathrm{\Delta }}_{x^2-y^2}=0.8183$, ${\mathrm{\Delta }}_{xy}=0.5562i$. On the top panel is the band structure with left edge state blue and right red. The bottom panel shows the corresponding distribution of edge states.