Josephson $\pi$ state induced by valley polarization

We theoretically explore possible $\pi$-state Josephson junctions made from graphene-like two-dimensional materials (G) with the honeycomb lattice structure. It is shown that the valley polarization in the G sheet could lead to a 0-$\pi$ state transition of the Josephson junction because of the valley-singlet Cooper pairs acquiring a nonzero momentum. When the valley-mixing scattering exists in the interfaces of the junction due to lattice mismatch, an odd-frequency valley-triplet supercurrent flows in the system even though the G sheet is fully valley polarized, and the supercurrent is characterized by a rapid atomic-scale oscillation with a periodicity of three lattice constants.

Figure 1

(a) Schematic of an S/G/S Josephson junction. The Homo-S electrode denotes the superconducting metal deposited on the G sheet and leaking Cooper pairs into the G lattice, while the Bulk-S electrode connects directly with the normal G sheet by a coupling strength $t^{\prime }$. The low-energy dispersion $E$-$q$ relation of the normal G for the $\kappa =+1$ and $\kappa =-1$ valley band with $\lambda =2E_z=1$ in (b) and $\lambda =E_z=1$ in (c), where an arbitrary unit is used and $\delta q$ denotes the Fermi-momentum splitting between the $\kappa =1$ and $\kappa =-1$ valley bands.

Figure 2

Dimensionless Andreev-reflection conductance as a function of bias $eV/\Delta$ in a G/S junction with (a) $\lambda =E_z=0$ and (b) $\lambda =E_z=10\Delta$. Black solid lines are for the Bulk-S case, while red dashed lines are for the Homo-S case. Different lattice site energies $U_i$ taken in the normal G sheet are noted in the figure.

Figure 3

Supercurrent density $I$ in the S/G/S junction within the Homo-S electrodes as functions of (a) the phase difference $\phi$, (b) the G layer length $L$, (c) the staggered potential $E_z$, and (d) the light field strength $\lambda$. Parameters are $\phi =0.5\pi$, $\lambda =E_z=10\Delta$, $U_i=-22\Delta$, and $L/a=1200$, and others are marked in the figure.

Figure 4

Plot of supercurrent density $I$ versus the length $L$ in the S/G/S junction for the Bulk-S case. The site energy of the normal G sheet is taken as (a) $U_i=-22\Delta$ and (b) $U_i=-10\Delta$. A segment in each $I$-$L$ curve is enlarged in the inset. Other parameters are $\lambda =E_z=10\Delta$ and $\phi =0.5\pi$. The red dashed curve in (b) is the supercurrent in the Homo-S junction with the same parameters for comparison.