Evolution of pair correlation symmetries and supercurrent reversal in tilted Weyl semimetals

We study the effective symmetry profiles of superconducting pair correlations and the flow of charge supercurrent in ballistic Weyl semimetal systems with a tilted dispersion relation. Utilizing a microscopic method in the ballistic regime and starting from both opposite-pseudospin (the band degree of freedom) and equal-pseudospin phonon-mediated opposite-spin electron-electron couplings, we calculate the anomalous Green's function to study various superconducting pair correlations that Weyl semimetal systems may develop. The momentum-space profile reveals that by properly manipulating the parameters of Weyl semimetal systems, including the tilting parameter, the effective symmetry class of even-parity $s$-wave (odd-parity $p$-wave) superconducting correlations can be converted into a $d$-wave ($f$-wave) symmetry class that consists of equal-pseudospin and opposite-pseudospin channels. We also find that the supercurrent in a ballistic Weyl Josephson junction can be made to vanish or switch directions, depending on the tilt of the Weyl cones, in addition to the relevant parameters characterizing the Weyl semimetal and junction. We show that inversion-symmetry-breaking terms introduce transitions that result in the appearance of self-biased current at zero difference between the macroscopic phases of the superconducting segments, creating a ${\phi }_0$ Josephson state. Weyl semimetal systems are shown to offer several experimentally tunable parameters to control the induction of higher harmonics into the current phase relations.