Majorana bound states in semiconducting nanostructures

TL;DR

This tutorial introduces Majorana bound states in semiconducting nanostructures, covering theoretical models, topological phase transitions, and experimental platforms, aimed at graduate students and researchers new to the field.

Contribution

It provides a pedagogical overview of minimal models for Majorana bound states, including explicit wave functions and mechanisms in various nanostructures, bridging theory and experiment.

Findings

Explicit Majorana wave functions in strong SOI limit

Identification of topological phase transition in Rashba nanowires

Discussion of Majorana corner states in 2D topological superconductors

Abstract

In this Tutorial, we give a pedagogical introduction to Majorana bound states (MBSs) arising in semiconducting nanostructures. We start by briefly reviewing the well-known Kitaev chain toy model in order to introduce some of the basic properties of MBSs before proceeding to describe more experimentally relevant platforms. Here, our focus lies on simple `minimal' models where the Majorana wave functions can be obtained explicitly by standard methods. In a first part, we review the paradigmatic model of a Rashba nanowire with strong spin-orbit interaction (SOI) placed in a magnetic field and proximitized by a conventional $s$-wave superconductor. We identify the topological phase transition separating the trivial phase from the topological phase and demonstrate how the explicit Majorana wave functions can be obtained in the limit of strong SOI. In a second part, we discuss MBSs engineered from proximitized edge states of two-dimensional (2D) topological insulators. We introduce the Jackiw-Rebbi mechanism leading to the emergence of bound states at mass domain walls and show how this mechanism can be exploited to construct MBSs. Due to their recent interest, we also include a discussion of Majorana corner states in 2D second-order topological superconductors. This Tutorial is mainly aimed at graduate students -- both theorists and experimentalists -- seeking to familiarize themselves with some of the basic concepts in the field.