Topological Phase Diagram in Finite-Thickness Semiconductor Nanowires with Proximity-Induced Chemical Potential, Zeeman Splitting, and Superconducting Gap

Abstract

We study analytically the topological phase diagram in finite-thickness semiconductor nanowire with proximity-induced chemical potential, Zeeman splitting and superconducting gap at its interface. Approximating the electrostatic potential formed inside the semiconductor by the negative gate voltage to a triangular potential, we analytically derive the condition for topological phase transition of the semiconductor nanowire and the gate voltage for which the system can be tuned into the topological nontrivial regime at the minimal Zeeman splitting. We also derive analytically the width of the topological nontrivial region and the interval between them in the topological phase diagram. Our results provide a qualitative and good description of the previous numerical studies and will provide a rationale and general understanding for realizing Majorana fermion in superconducting-ferromagnetic insulator-semiconductor nanowires.