Topological phases and surface flat bands in superconductors without inversion symmetry

We examine different topological phases in three-dimensional noncentrosymmetric superconductors with time-reversal symmetry by using three different types of topological invariants. Due to the bulk boundary correspondence, a nonzero value of any of these topological numbers indicates the appearance of zero-energy Andreev surface states. We find that some of these boundary modes in nodal superconducting phases are dispersionless, i.e., they form a topologically protected flat band. The region where the zero-energy flat band appears in the surface Brillouin zone is determined by the projection of the nodal lines in the bulk onto the surface. These dispersionless Andreev surface bound states have many observable consequences, in particular, a zero-bias conductance peak in tunneling measurements. We also find that in the gapless phase there appear Majorana surface modes at time-reversal invariant momenta which are protected by a ${\mathbb{Z}}_2$ topological invariant.

Figure 1

Phase diagram as a function of singlet pairing amplitude ${\Delta }_s$ and SOC $g_2$ [see Eqs. (2) and (3)] for the point group (a) $O$ and (b) $C_{4v}$. The gapped phases are characterized by the winding number $\nu$ with $\nu =0$ (white), $\nu =\pm 1$ (dark and light blue hatched), $\nu =-5$ (light brown hatched), and $\nu =+7$ (dark brown hatched). Gray shaded and dotted regions are nodal superconducting phases with $N_{{\mathcal{C}}_1}=\pm 1$ (red and black dotted), $N_{{\mathcal{C}}_2}=\pm 1$ (light and dark gray), $N_{{\mathcal{C}}_3}=+1$ (white dotted), and $N_{{\mathcal{C}}_4}=+1$ (gray).

Figure 2

Nodal rings (a) and (111) surface states (c),(d) for the point group $O$ with $(g_2,{\Delta }_s)=(0.3,0.5)$. This parameter choice corresponds to the red dotted region in Fig. 1a. (b) Topological invariant $N_{\left(111\right)}$, Eq. (7), as a function of surface momentum ${\mathit{k}}_{\parallel }$. Gray and dark blue indicate $N_{\left(111\right)}=\pm 1$, while light blue is $N_{\left(111\right)}=0$. (c) Band structure for a slab with (111) face as a function of surface momentum $k_2^{\parallel }$ with $k_1^{\parallel }=0.75\pi$. (d) Energy dispersion of the lowest-lying state with positive energy. The color scale is such that black corresponds to zero energy. The states at zero energy in (c) and (d) are localized at the surface. The flat bands in (c) and (d) are singly degenerate (i.e., one branch per surface), whereas the linearly dispersing zero mode at the center of the BZ in (d) is doubly degenerate.

Figure 3

Same as Fig. 2 but for the point group $C_{4v}$, for a slab with (012) face, and with $(g_2,{\Delta }_s)=(0.0,0.5)$. This parameter choice corresponds to the white dotted area in Fig. 1b.

Figure 4

Surface and bulk density of states for the point group (a) $O$ and (b) $C_{4v}$. The surfaces are oriented perpendicular to the (111) and (012) axes, respectively. The employed parameter values are the same as in Figs. 2 and 3.