Topologically protected flat zero-energy surface bands in noncentrosymmetric superconductors

Nodal noncentrosymmetric superconductors (NCS) have recently been shown to be topologically nontrivial. An important consequence is the existence of topologically protected flat zero-energy surface bands, which are related to the topological characteristics of the line nodes of the bulk gap via a bulk-boundary correspondence [A. P. Schnyder and S. Ryu, arXiv:1011.1438]. In this Rapid Communication, we examine these zero-energy surface bands using a quasiclassical theory. We determine their spectrum and derive a general condition for their existence in terms of the sign change of the gap functions. A key experimental signature of the zero-energy surface bands is a zero-bias peak in the tunneling conductance, which depends strongly on the surface orientation. This can be used as a fingerprint of a topologically nontrivial NCS.

Figure 1

Panels (a), (c) and (e): Variation of the sign of $\Delta {}_{\mathbf{k}}^{-}$ over the Fermi surface as seen along the $(111)$ direction for (a) the $C_{4v}$ point group with $q=0.5$, (c) the $O$ point group with $q=0.35$, and (e) the $T_d$ point group with $q=0.4$. Dark red indicates $\text{sgn}(\Delta {}_{\mathbf{k}}^{-})=1$, whereas light blue is $\text{sgn}(\Delta {}_{\mathbf{k}}^{-})=-1$. Panels (b), (d) and (f): Surface bound states at the $(111)$ face corresponding to the same parameters as in (a), (c) and (e), respectively. The color scale of each plot is normalized by the maximum bound state energy, (b) $E_{\max }=0.333\hspace{0.5em}{\Delta }_0$, (d) $E_{\max }=0.211\hspace{0.5em}{\Delta }_0$, and (f) $E_{\max }=0.421\hspace{0.5em}{\Delta }_0$. White space indicates the absence of a bound state, and the circle is the projection of the Fermi surface, i.e., $\left|{\mathbf{k}}_{\parallel }\right|=k_F$.

Figure 2

Panels (a), (c) and (e): Normalized conductance spectra at the $(111)$ face for an NCS with point group (a) $C_{4v}$, (c) $O$, and (r) $T_d$ and various values of $0⩽q={\Delta }_s/{\Delta }_t⩽1$. In all panels we take $Z=3$ and $T=0$ K. Curves are vertically shifted by multiples of $0.2$. Panels (b), (d) and (f): ZBCP as a function of surface orientation for an NCS with point group (b) $C_{4v}$ with $q=0.5$, (d) $O$ with $q=0.35$, and (f) $T_d$ with $q=0.4$. The color at each point on the sphere indicates the height of the ZBCP for the corresponding normal vector. Panel (g): Variation of ZBCP height at the $(111)$ face as a function of $q$.