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 $R_{\sigma {\sigma }^{\prime }}$ and an Andreev reflection coefficient $A_{\sigma {\sigma }^{\prime }}$, and derive a charge conductance formula,where $\sigma ({\sigma }^{\prime })$ 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 $eV=0$, ${\sum }_{\sigma {\sigma }^{\prime }}{R}_{\sigma {\sigma }^{\prime }}={\sum }_{\sigma {\sigma }^{\prime }}{A}_{\sigma {\sigma }^{\prime }}$ is satisfied and the spin direction of an injected electron is rotated at approximately $90^{\circ }$ for the $\mathcal{N}=1$ state. Meanwhile, $A_{\sigma {\sigma }^{\prime }}=0$ is satisfied for the $\mathcal{N}=2$ state, and the spin rotation angle can become $180^{\circ }$.