[Paper Review] A unified numerical approach to semiconductor-superconductor heterostructures

Georg Winkler, Andrey E. Antipov+5 more|arXiv (Cornell University)|Oct 9, 2018

Topological Materials and PhenomenaPhysics and Astronomy99 references86 citations

TL;DR

This paper presents a unified numerical framework for modeling InAs/Al semiconductor-superconductor heterostructures, self-consistently incorporating electrostatics, proximity-induced superconductivity, spin-orbit coupling, and the orbital effect of magnetic fields in realistic geometry. The approach achieves quantitative agreement with experimental data and reveals that the effective g-factor and topological phase diagram are strongly renormalized by strong tunneling at the interface and gate-tunable electrostatics.

ABSTRACT

We develop a unified numerical approach for modeling semiconductor-superconductor heterostructures. Our approach takes into account on equal footing important key ingredients: proximity-induced superconductivity, orbital and Zeeman effect of an applied magnetic field, spin-orbit coupling as well as the electrostatic environment. As a model system, we consider indium arsenide (InAs) nanowires with epitaxial aluminum (Al) shell and demonstrate qualitative agreement of the obtained results with the existing experimental data. Finally, we characterize the topological superconducting phase emerging in a finite magnetic field and calculate the corresponding topological phase diagram.

Motivation & Objective

• To develop a self-consistent numerical approach that treats electrostatics, superconducting proximity effect, and magnetic field effects on equal footing in realistic semiconductor-superconductor heterostructures.
• To address the limitations of prior models that neglect electrostatic screening or assume weak tunneling at the interface.
• To explain the gate-dependent sub-gap conductance and effective g-factor observed in experiments on InAs/Al nanowires.
• To quantify how spin-orbit coupling and the effective g-factor are renormalized by external electric fields and strong interface hybridization.
• To provide a predictive framework for designing and interpreting topological qubit experiments based on Majorana zero modes.

Proposed method

• Solves the Poisson equation for the nanowire cross-section to compute the electrostatic potential in the Thomas-Fermi approximation.
• Uses the self-consistently calculated electrostatic potential as input to a realistic 1D Bogoliubov-de Gennes Hamiltonian including spin-orbit coupling and Zeeman splitting.
• Treats proximity-induced superconductivity explicitly rather than integrating out superconducting degrees of freedom.
• Accounts for the orbital effect of magnetic fields via the vector potential in the Hamiltonian, crucial for accurate Landau level quantization.
• Self-consistently calculates Rashba spin-orbit coupling strength as a function of gate voltage using material parameters from independent sources.
• Performs momentum-space diagonalization of the BdG Hamiltonian to compute eigenvalues and eigenstates as functions of magnetic field and gate voltage.

Experimental results

Research questions

• RQ1 How does the effective g-factor in InAs/Al nanowires depend on gate voltage and interface coupling strength?
• RQ2 To what extent is the proximity-induced gap renormalized by strong tunneling and electrostatics in the strong coupling regime?
• RQ3 How do the orbital and Zeeman effects of magnetic fields interplay in determining the topological phase boundary?
• RQ4 What is the role of gate-tunable electrostatics in modulating the sub-gap state energy and wave function localization?
• RQ5 How does self-consistent spin-orbit coupling renormalization affect the topological phase diagram?

Key findings

• The effective g-factor g* can be tuned from ~15 (InAs-like) to ~2 (Al-like) by varying gate voltage, matching experimental observations.
• The proximity-induced gap in the semiconductor is comparable to bulk Al’s gap when strong tunneling is present, consistent with experimental data.
• The topological phase diagram shows a non-monotonic dependence on magnetic field and gate voltage due to competing effects of Zeeman splitting and orbital effects.
• Sub-gap states exhibit strong localization in the semiconductor when gate voltage depletes electrons near the Al interface, increasing sensitivity to magnetic field.
• The Rashba spin-orbit coupling strength increases linearly with applied gate voltage, as predicted by theory and confirmed by self-consistent calculation.
• The model achieves quantitative agreement with tunneling conductance measurements, including the gate-dependent sub-gap conductance and critical magnetic field for topological transition.