Effective theory approach to the Schrodinger-Poisson problem in semiconductor Majorana devices

TL;DR

This paper introduces an efficient method for solving the Schrodinger-Poisson problem in 3D semiconductor Majorana devices, enabling detailed electrostatic and quantum analysis crucial for device design and understanding Majorana physics.

Contribution

The authors develop a novel approach that reduces a complex 3D problem to an effective 1D model using a Green's function scheme and self-consistent molecular orbitals, improving computational efficiency.

Findings

Electrostatic effects partially suppress Majorana energy splitting oscillations.

Different low-energy bands exhibit similar effective couplings, leading to a single energy scale for induced gaps.

Position-dependent work function differences can create trivial low-energy states mimicking Majorana modes.

Abstract

We propose a method for solving the Schrodinger-Poisson problem that can be efficiently implemented in realistic 3D tight-binding models of semiconductor-based Majorana devices. The method is based on two key ideas: i) For a given geometry, the Poisson problem is only solved once (for each local orbital) and the results are stored as an interaction tensor; using this Green's function scheme, the Poisson component of the iteration procedure is reduced to a few simple summations. ii) The 3D problem is mapped into an effective multi-orbital 1D problem with molecular orbitals calculated self-consistently as the transverse modes of an infinite wire with the same electrostatic potential as the local electrostatic potential of the finite 3D device. To demonstrate the capabilities of our approach, we calculate the response of the system to an external magnetic field, the dependence of the effective chemical potential on the work function difference, and the dependence of the effective semiconductor-superconductor coupling on the applied gate potential. We find that different low-energy bands are characterized by similar effective couplings, which results in induced gap features characterized by a single energy scale. We also find that electrostatic effects are responsible for a partial suppression of the Majorana energy splitting oscillations. Finally, we show that a position-dependent work function difference can produce a non-homogeneous effective potential that is not affected by the screening due to the superconductor and is only partially suppressed by the charge inside the wire. In turn, this potential can induce trivial low-energy states that mimic the phenomenology of Majorana zero modes. Thus, any position-dependent work function difference (even at the 1% level) along the nanowire must be avoided through carefully engineered semiconductor-superconductor interfaces.