Majorana Fermions and Exotic Surface Andreev Bound States in Topological Superconductors: Application to ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$

The recently discovered superconductor ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ is a candidate for three-dimensional time-reversal-invariant topological superconductors, which are predicted to have robust surface Andreev bound states hosting massless Majorana fermions. In this work, we analytically and numerically find the linearly dispersing Majorana fermions at $k=0$, which smoothly evolve into a new branch of gapless surface Andreev bound states near the Fermi momentum. The latter is a new type of Andreev bound states resulting from both the nontrivial band structure and the odd-parity pairing symmetry. The tunneling spectra of these surface Andreev bound states agree well with a recent point-contact spectroscopy experiment [S. Sasaki et al., Phys. Rev. Lett. 107, 217001 (2011)] and yield additional predictions for low temperature tunneling and photoemission experiments.

Figure 1

(a) Side view of a semi-infinite crystal of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The two relevant $p_z$ orbitals are shown in the zoom-in view of the QL unit cell. (b) Bulk and surface bands of the tight-binding model for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. ${\mu }_1$ and ${\mu }_2$ denote two chemical potentials where the surface states have, respectively, not merged and merged into the bulk bands.

Figure 2

SABS dispersion for the tight-binding model in which (a) $m=-0.3<0$, ${\mu }_1=0.6$ and (b) $m=-0.3<0$, ${\mu }_2=1$. The mirror eigenvalues are displayed near each branch of SABS. The SABS twist with a second crossing near Fermi momentum, as also observed in Ref. 19. The arrow denotes where the dispersion has zero slope, resulting in a van Hove singularity in the density of states.

Figure 3

Tunneling local density of states (arbitrary units) at (a) $T=0$ and (b) finite temperature. In both cases, the chemical potential is ${\mu }_2=1$.