Superconductivity in type-II Weyl semimetals

We study superconductivity in a Weyl semimetal with a tilted dispersion around two Weyl points of opposite chirality. In the absence of any tilt, the state with zero momentum pairing between two Fermi sheets enclosing each Weyl point has four point nodes in the superconducting gap function. Moreover, the surface of the superconductor hosts Fermi arc states and Majorana flat bands. We show that a quantum phase transition occurs at a critical value of the tilt, at which two gap nodes disappear by merging at the center of the first Brillouin zone, or by escaping at its edges, depending on the direction of the tilt. The region in the momentum space that the Majorana flat band occupies is found to increase as the tilt parameter is made larger. Additionally, the superconducting critical temperature and electronic specific heat can be enhanced in the vicinity of the quantum phase transition due to the singularity in the electronic density of states.

Figure 1

Tilt parameter $\beta$ dependence of (a) order parameter $\left|\mathrm{\Delta }\right|/|{\mathrm{\Delta }}_0|$ at zero temperature and (b) critical temperature $T_c/T_{c0}$, which are normalized by their values at $\beta =0$.

Figure 2

Band structure of the bulk Weyl superconductor as a function of momentum $k_z$ for different values of the tilt parameter $\gamma$ [see Eq. (12)]. The energy is normalized by the hopping energy integral $t$. The $x$ axis is directed normal to the two parallel plane boundaries of the superconductor, so that the longitudinal momenta $k_y$ (set to zero here), and $k_z$ are good quantum numbers. The left panel shows the band structure of a Weyl superconductor without tilting, where $\gamma =0$. The $\gamma$ parameter is increased from left to right as follows: $\gamma =0,1.5t,2.5t$, and $3t$. The chemical potential is fixed at $\mu =0.5t$ and the superconducting gap is set at $\mathrm{\Delta }=0.2t$. As seen, the region of the Majorana zero energy flat band, bounded to the nodes in the superconducting gap, increases as $\gamma$ increases. When $\gamma =2.0t$ there is a quantum phase transition, whereby two out of the four gap nodes migrate to the edges of the first Brillouin zone and then reappear at the center when $\gamma \approx 2.7t$. With further increase of $\gamma$, the Majorana flat band region increases.

Figure 3

Band structure of the Weyl superconductor as a function of $k_y$ for $\gamma =3t$ [corresponding to Fig. 2] and for different values of $k_{z}d/\pi \in [0.0,0.25,0.37,0.5]$. Other parameters are set equal to those of Fig. 2.

Figure 4

Two-dimensional momentum space map of the band structure at low energies close to zero $E(k_y,k_z)\ll 1$ as a function of $k_y$ and $k_z$. The node titling parameter $\gamma$ evolves from 0 to $3.0t$ corresponding to Figs. 2 and 3. The chemical potential and superconducting gap are fixed at $\mu =0.5t$ and $\mathrm{\Delta }=0.2t$, respectively.