Josephson Effect In Type-? Weyl Semimetals

Weyl semimetal is a new type of topological quantum state,which has been a hot topic for us to explore.Weyl fermions appearing in condensed matter may have a tilted dispersion relationship,breaking the Lorentz invariance.The dispersion relations of tilt and over tilt are called type-I Weyl fermions and type-II Weyl fermions,respectively.For the type-I,the Fermi surface at the Weyl point is point-like,while for the type-II,the Weyl point can also exist on the boundary of the electron and hole pockets.In this paper,we first present a theoretical model of a Weyl semimetals superconductor /type-II Weyl semimetals/Weyl semimetals superconductor(S-N-S)superconducting junction to study the Josephson effect of this superconducting junction.According to the findings,the doped Weyl semimetal promotes two types of superconducting pairing: The inter-node pairing forms a zero-momentum BCS state and the intra-node pairing exhibits a finite-momentum FFLO state.Since the Josephson effect is used as an important tool to test superconducting pairings,theoretical calculations such as Josephson currents were performed for each of these two pairing types.In our calculations,we assume short junction limit and that the temperature is zero temperature.For the BCS state,it is found that the total current of the S-N-S junction depends on the junction length and tilt parameters.Change the tilt parameter and find that when the tilt parameter increases,the current amplitude decreases.The article also studied the dependence of the total current on the junction length,and found that as the junction length increases,the total current decreases,when changing the tilt parameter,it is found that the smaller the tilt parameter,the faster the total current oscillates,and the oscillation amplitude also decreases.Finally,it was found that the Josephson current is dependent on the angle of the Weyl points.In contrast,the FFLO state exhibits a very different Josephson effect.We find that the current is only phase-dependent,independent of both junction length and tilt.When the phase is (?)_t= n?-(?)/2,the current obeys J((?))?sin(?)/2,and when the phase is (?)_t=(n+1)?-(?)/2,the current obeys J((?))?-sin(?)/2,then we can switch the supercurrent direction by changing the phase.In addition,the article also compared the Josephson current performance of the BCS state and the FFLO state,and found that for the FFLO state,the BCS state current amplitude is larger and the phase change period is shorter.It is believed that the research results of this article will help distinguish the different pairing mechanisms in Weyl semimetals in the experiment.