Majorana and non-Majorana modes in a nanowire in partially proximity to a superconductor

We investigate the Majorana and non-Majorana modes in a nanowire in partial proximity to a superconductor, in which the gapped superconductor will play different roles in different topological regimes. In the gapped topological superconducting phase, it plays the role of a topological barrier, which confines some localized edge modes in the quantum dot (QD) region. The probability for the wave function in this region can approach unity by tuning the system parameters. These low-lying localized modes exhibit linear spectra with equal energy level spacing, with eigenvalues ε n = v F n π / ( 2 L ), where v F is the Fermi velocity, L is the size of the QD region, and n ∈ Z. We demonstrate these features using a spinless nanowire in proximity to a p-wave superconductor and a spin–orbit coupled semiconductor nanowire in proximity to a s-wave superconductor. A simple picture is proposed to understand the behavior of these results. However, in the trivial superconducting phase when both bands are occupied in the spin–orbit coupled mode, we observe some non-Majorana modes, with complicated low-lying excited spectra, which resembles that reported in experiments. These differences are rooted deeply in the bulk-edge correspondence. These observations may be able to facilitate the identification of Majorana zero modes in experiments.