Types of topological surface states in nodal noncentrosymmetric superconductors

Nodal noncentrosymmetric superconductors have topologically nontrivial properties manifested by protected zero-energy surface states. Specifically, it was recently found that zero-energy surface flat bands of topological origin appear at their surface. We show that the presence of certain inversion-type lattice symmetries can give rise to additional topological features of the gap nodes, resulting in surface states forming one-dimensional arcs connecting the projections of two nodal rings. In addition, we demonstrate that Majorana surface states can appear at time-reversal-invariant momenta of the surface Brillouin zone, even when the system is not fully gapped in the bulk. Within a continuum theory we derive the topological invariants that protect these different types of zero-energy surface states. We independently derive general conditions for the existence of zero-energy surface bound states using the complementary quasiclassical scattering theory, explicitly taking into account the effects of spin-orbit splitting of the bands. We compute surface bound-state spectra for various crystal point-group symmetries and orbital-angular-momentum pairing states. Finally, we examine the signatures of the arc surface states and of the zero-energy surface flat bands in tunneling-conductance spectra and dicuss how topological phase transitions in noncentrosymmetric superconductors could be observed in experiments.

Figure 1

Comparison of the Fermi surfaces in the $x$-$z$ plane of the approximate dispersion Eq. (32) (solid lines) and the exact dispersion Eq. (31) (dashed lines) for $\lambda k_F/2\mu =0.1$.

Figure 2

Surface bound-state spectra at the (100) face of a $C_{4v}$ point-group NCS as a function of surface momentum ${\mathbf{k}}_{\parallel }=(k_y,k_z)$ with (a) $(s+p)$-wave, (b) $(d_{x^2-y^2}+f)$-wave, and (c) $(d_{xy}+p)$-wave pairing symmetry. Here we set $\lambda k_F/\left(2\mu \right)=0.1$ and $r=0.5$. The color scale indicates the energy: black represents zero energy while yellow represents the maximum energy $E_{\max }$. The black (gray) line shows the extent of the projected negative-helicity (positive-helicity) Fermi surface. (d) Winding number $W_{\left(100\right)}$, Eq. (24), at the (100) face corresponding to the same parameters as in panel (c). Black (white) indicates $W_{\left(100\right)}=+2$ ($-2$), dark blue (gray) corresponds to $W_{\left(100\right)}=+1$ ($-1)$, while light blue is $W_{\left(100\right)}=0$. The red dashed (green solid) lines represent the nodal lines on the negative-helicity (positive-helicity) Fermi surface.

Figure 3

Surface bound-state spectra at the (101) face of a $C_{4v}$ point-group NCS as a function of surface momentum ${\mathbf{k}}_{\parallel }$ with (a) $(s+p)$-wave, (c) $(d_{x^2-y^2}+f)$-wave, and (e) $(d_{xy}+p)$-wave pairing symmetry. Here we set $\lambda k_F/\left(2\mu \right)=0.1$ and $r=0.5$. The color scale is the same as in Figs. 2a, 2b, 2c. The black (gray) line shows the extent of the projected negative-helicity (positive-helicity) Fermi surface. Panels (b), (d), and (f) show the winding number $W_{\left(101\right)}$ at the (101) face corresponding to the same parameters as in panels (a), (c), and (e), respectively. Dark blue (gray) indicates $W_{\left(101\right)}=+1$ ($-1)$, while light blue is $W_{\left(101\right)}=0$. The red dashed (green solid) lines represent the nodal lines on the negative-helicity (positive-helicity) Fermi surface.

Figure 4

Schematic phase diagram for an NCS with cubic point group $O$ as a function of the ratio $r={\Delta }_s/{\Delta }_t$ of the singlet and triplet gaps. Here, the second-order SOC $g_2$ [see Eq. (6b)] is assumed to be negative.

Figure 5

(a)–(e) Surface bound-state spectra for the (100) face of an NCS with point group $O$, $g_2=-1.5$, and $\lambda =0$, as a function of surface momentum ${\mathbf{k}}_{\parallel }$ for (a) $r=0$, (b) $r=0.15$, (c) $r=r_c=0.25$, (d) $r=0.6$, and (e) $r=0.9$. The color scale is the same as in Figs. 2a, 2b, 2c. [(f)–(j)] Same as panels (a)–(e) but for the (110) face. [(k)–(o)] Same as panels (a)–(e) but for the (111) face. [(p)–(t)] Winding number $W_{\left(111\right)}$ for the (111) face corresponding to the same parameters as in panels (k)–(o). Dark blue (gray) indicates $W_{\left(111\right)}=+1$ ($-1$), while light blue is $W_{\left(111\right)}=0$. The red dashed lines represent the projections of nodal lines.

Figure 6

(a)–(e) Surface bound-state spectra for the (100) face of an NCS with point group $T_d$ and $\lambda =0$ as a function of the surface momentum ${\mathbf{k}}_{\parallel }$ for (a) $r=0$, (b) $r=0.2$, (c) $r=r_c=4\sqrt{2}/9$, (d) $r=0.75$, and (e) $r=0.9$. The color scale is the same as in Figs. 2a, 2b, 2c. [(f)–(j)] Same as panels (a)–(e) but for the (110) face. [(k)–(o)] Same as panels (a)–(e) but for the (111) face. [(p)–(t)] Winding number $W_{\left(111\right)}$ for the (111) face corresponding to the same parameters as in panels (k)–(o). Dark blue (gray) indicates $W_{\left(111\right)}=+1$ ($-1$), while light blue is $W_{\left(111\right)}=0$. The red dashed lines represent the projections of nodal lines.

Figure 7

Tunneling-conductance spectra for the (100) and (101) interfaces of a $C_{4v}$ NCS with $r=0.5$, $Z=3$, and (a) $(s+p)$-wave gap symmetry [$f\left(\mathbf{k}\right)=1$], (b) $(d_{x^2-y^2}+f)$-wave gap symmetry [$f\left(\mathbf{k}\right)=(k_x^2-k_y^2)/k_F^2$], and (c) $(d_{xy}+p)$-wave gap symmetry [$f\left(\mathbf{k}\right)=2k_x{k}_y/k_F^2$]. In all panels the thick lines show the results for finite spin-orbit splitting $\lambda k_F/2\mu =0.1$, while the thin lines show the results for degenerate helical Fermi surfaces.

Figure 8

Variation of the zero-bias conductance-peak height at the (111) face as a function of singlet-to-triplet ratio $r$ for the point group $O$. The black, downward-pointing arrows indicate topological phase transitions between fully gapped and nodal phases. The red, upward-pointing arrow marks an inflection point in ${\sigma }_S\left(\mathrm{eV}\right)$, where the character of the nodes on ${\Delta }_{\mathbf{k}}^{-}$ changes qualitatively. In this figure we set $g_2=-1$, $Z=3$, and $T=0\mathrm{K}$.