Magnetic properties of type-I and type-II Weyl semimetals in the superconducting state

Superconductivity was observed in certain range of pressure and chemical composition in Weyl semimetals of both type I and type II (when the Dirac cone tilt parameter $\ensuremath{\kappa}>1$). Magnetic properties of these superconductors are studied on the basis of microscopic phonon-mediated pairing model. The Ginzburg-Landau effective theory for the order parameter is derived using the Gorkov approach and used to determine anisotropic coherence length, the penetration depth determining the Abrikosov parameter for a layered material and applied to recent extensive experiments on ${\text{MoTe}}₂$. It is found that superconductivity is of second kind near the topological transition at $\ensuremath{\kappa}=1$. For a larger tilt parameter, superconductivity becomes first kind. For $\ensuremath{\kappa}<1$, the Abrikosov parameter also tends to be reduced, often crossing over to the first kind. For the superconductors of the second kind, the dependence of critical fields ${H}_c2$ and ${H}_c1$ on the tilt parameter $\ensuremath{\kappa}$ (governed by pressure) is compared with the experiments. Strength of thermal fluctuations is estimated and it is found that they are strong enough to cause Abrikosov vortex lattice melting near ${H}_c2$. The melting line is calculated and is consistent with experiments provided the fluctuations are three dimensional in the type-I phase (large pressure) and two dimensional in the type-II phase (small pressure).