Theory of Tunneling Effect in 1D AIII-class Topological Insulator (Nanowire) Proximity Coupled with a Superconductor

Abstract

We study the tunneling effect in an AIII-class insulator proximity coupled with a spin-singlet $s$-wave superconductor, in which three phases are characterized by the integer topological invariant $\mathcal{N}$. By solving the Bogoliubov-de Gennes equation explicitly, we analytically obtain a normal reflection coefficient $R_{\sigma\sigma'}$ and an Andreev reflection coefficient $A_{\sigma\sigma'}$, and derive a charge conductance formula,where $\sigma(\sigma')$ is the spin index of a reflected (injected) wave. The resulting conductance indicates a wide variety of line shapes: (i)gap structure without coherence peaks for $\mathcal{N}=0$, (ii)quantized zero-bias conductance peak (ZBCP) with height $2e^{2}/h$ for $\mathcal{N}=1$, and (iii)ZBCP spitting for $\mathcal{N}=2$. At zero bias voltage $eV=0$, $\sum_{\sigma\sigma'} R_{\sigma\sigma'} = \sum_{\sigma\sigma'} A_{\sigma\sigma'}$ is satisfied and the spin direction of an injected electron is rotated at approximately $90^\circ$ for the $\mathcal{N}=1$ state. Meanwhile, $A_{\sigma\sigma'}=0$ is satisfied for the $\mathcal{N}=2$ state, and the spin rotation angle can become $180^\circ$.