Weyl superconductors

We study the physics of the superconducting variant of Weyl semimetals, which may be realized in multilayer structures comprising topological insulators and superconductors. We show how superconductivity splits each Weyl node into two. The resulting Bogoliubov-Weyl nodes can be pairwise independently controlled, allowing to access a set of phases characterized by different numbers of bulk Bogoliubov-Weyl nodes and chiral Majorana surface modes. We analyze the physics of vortices in such systems, which trap zero-energy Majorana modes only under certain conditions. We finally comment on possible experimental probes, thereby also exploiting the similarities between Weyl superconductors and two-dimensional $p+ip$ superconductors.

Figure 1

The proposed heterostructure for an experimental realization of Weyl superconductors. Magnetically doped layers of a topological insulator (TI) are alternated with layers of an $s$-wave superconductor (SC). The period of the superlattice is $d$. The arrows in the TI layers depict their magnetization, which is along the superlattice axis.

Figure 2

Evolution of the masses $M_{-}^{+\Delta }$ (upper curve) and $M_{-}^{-\Delta }$ (lower curve) defined in Eq. (14) upon increasing $\left|\Delta \right|$. For $\left|\Delta \right|=0$ and $m_{c1}<m<m_{c2}$, the system has two Weyl nodes of chiral electrons, located at the sign changes of $M_{-}^{\pm \Delta }$. With superconductivity, each Weyl nodes splits into two Bogoliubov-Weyl nodes of equal chirality and opposite particle-hole symmetry. Their separation grows with increasing $\left|\Delta \right|$ from (a) to (d).

Figure 3

Phase diagram of a Weyl superconductor in a TI/SC heterostructure as a function of Zeeman gap $m$ and proximity induced superconducting order parameter amplitude $\left|\Delta \right|$. The values of $m_{c1}$ and $m_{c2}$ are set by the tunneling amplitudes between the surface Dirac layers of the heterostructure depicted in Fig. 1, see Eq. (10). Each phase is characterized by $n_b$, the number of pairs of bulk Bogoliubov Weyl nodes, and $n_s$, the number of two-dimensional Majorana surface modes. The phases are labeled as $n_b,n_s$. The black dots locate the different subfigures of Fig. 2.

Figure 4

The two classes of vortices in Weyl superconductors. (a) sketches a vortex along the superlattice axis $\widehat{z}$, with bound states along a tube through the whole sample. (b) depicts a vortex perpendicular to $\widehat{z}$. Whereas there are no states bound to the vortex, the surface states can be used for Majorana interferometry (thick line).

Figure 5

Sketch of the anomalous thermal Hall effect in Weyl superconductors. The sample is shown from above. A thermal gradient $\nabla T$ is applied across the sample. The upper surface is at a temperature $T_{>}$ larger than the temperature $T_{<}$ of the lower surface. This leads to a net heat current from side $A$ to side $B$ perpendicular to the temperature gradient transported by the surface modes running around the sample.