Anomalous Josephson Hall effect in doped topological insulators with nematic superconductivity

We study the physics of the Josephson effect in the odd-parity nematic superconductors within the Ginzburg-Landau approach. The order parameter is a two-component vector that transforms as a coordinate vector in the $(x,y)$ plane under rotation and its direction is usually refer to as the nematicity direction. A nontrivial interplay between the nematicity and crystallographic axes of the superconductors that form the junction makes the Josephson effect quite unusual. We derive current-phase relations for different configurations of the junction, crystallographic axes of the sample, and the nematicity direction in the superconductors. We obtain that the Meissner kernel in the considered samples has off-diagonal components and the transverse phase difference across the junction can induce a Josephson current that flows along the contact. We show that such an anomalous Josephson Hall effect can be observed without any magnetization. We calculate the magnetic field dependence of the maximum current through the junction. We find that the period of the Fraunhofer oscillations of the maximum Josephson current depends on the geometry of the junction, the direction of the magnetic field, and the nematicity directions. The obtained results can be generalized to other superconductors with nondiagonal Meissner kernels.