Effect of the type-I to type-II Weyl semimetal topological transition on superconductivity

The influence of recently discovered topological transition between type-I and type-II Weyl semimetals on superconductivity is considered. A set of Gorkov equations for weak superconductivity in Weyl semimetal under topological phase transition is derived and solved. The critical temperature and superconducting gap both have spikes in the transition point as functions of the tilt parameter of the Dirac cone determined, in turn, by the material parameters like pressure. The spectrum of superconducting excitations is different in two phases: The sharp cone pinnacle is characteristic for type I, while two parallel almost flat bands, are formed in type II. Spectral density is calculated on both sides of transition to demonstrate the different weights of the bands. The superconductivity thus can be used as a clear indicator for the topological transformation. Results are discussed in the light of recent experiments.

Figure 1

Spectrum of normal Weyl semimetal. (a) Type I $\left(w/v=0.5\right)$. (b) Strongly tilted Dirac cone for the type-II semimetals $\left(w/v=1.2\right)$.

Figure 2

Spectrum of superconducting quasiparticles as function of momentum (in units of $\hslash \mathrm{\Omega }/v$) for (a) type-I ($w/v=0.5$) and (b) type-II ($w/v=1.2$) Weyl semimetal for chemical potential $\mu =3$ and energy gap $\mathrm{\Delta }=0.25$ (in units of phonon energy $\hslash \mathrm{\Omega }$).

Figure 3

Spectral density $A\left(\mathbf{p},E\right)$ for type-I (a) $\left(w/v=0.5\right)$ and type-II (b) $\left(w/v=1.2\right)$ superconducting semimetals. Here $p_x=1,p_y=0.1$ (green), 0.4 (red), $0.8$ (blue) in units of $\hslash \mathrm{\Omega }/v$. Here $\mu =3, \mathrm{\Delta }=0.25$ (in the units of cut off phonon energy $\hslash \mathrm{\Omega }).$

Figure 4

Equipotential surfaces of the electron energy for different ratio $w/v$ (pairs of the figures from top to bottom): top row, $w/v=0.01$ (left), $w/v=0.4$; middle row, $w/v=0.91$ (left), $w/v=1.004$; and bottom row, $w/v=1.1$ (left), $w/v=1.4.$

Figure 5

The superconducting gap at zero temperature versus cone tilted parameter $\kappa$. (The dashed line marks the region beyond assumption $\mu \gg \mathrm{\Omega }\gg T_c.)$ Cutoff parameter: $\delta =0.01,\mu =3,\lambda =0.15.$