Quasi-classical theory of tunneling spectroscopy in superconducting topological insulator

We develop a theory of tunneling spectroscopy in superconducting topological insulator (STI), i.e. superconducting state of a carrier-doped topological insulator. Based on the quasi-classical approximation, we obtain an analytical expression of the energy dispersion of surface Andreev bound states (ABSs) and a compact formula for tunneling conductance of normal metal/STI junctions. The obtained compact formula of tunneling conductance makes the analysis of experimental data easy. As an application of our theory, we study tunneling conductance in Cu$_x$Bi$_2$Se$_3$. We reveal that our theory reproduces the previous results by Yamakage {\it et al} [Phys. Rev. B 85, 180509(R) (2012)]. We further study magneto-tunneling spectroscopy in the presence of external magnetic fields, where the energy of quasiparticles is shifted by the Doppler effect. By rotating magnetic fields in the basal plane, we can distinguish between two different topological superconducting states with and without point nodes, in their magneto-tunneling spectra.