Josephson currents on the surface of line Weyl semimetals

In line Dirac semimetals (SMs) the bulk conduction and valence bands touch along a line, typically a ring. One of the key properties of line Dirac SMs is to have "drumhead'' surface states, i.e. surface states with a very large effective mass. In the presence of certain types of spin-rotation symmetry breaking perturbations the Dirac rings split into Weyl rings. In this work, we study the properties of Josephson junctions formed by two superconductors separated by a line Weyl SM in the limit in which the effect of the bulk states of the Weyl SM can be neglected. In this limit the drumhead states mediate the Josephson current between the two superconductors. We first investigate the Andreev reflection between drumhead surface states and a superconductor and show the relation between the strength of the Andreev reflection and the effective mass of the drumhead states. We then obtain the Josephson current for both short and long junctions for the
different types of spin-rotation symmetry breaking perturbation that lead to the splitting of the Dirac rings into Weyl rings.