Scaling of sub-gap excitations in a superconductor-semiconductor nanowire quantum dot

TL;DR

This study investigates how sub-gap excitations in a superconductor-semiconductor nanowire quantum dot scale with system parameters, revealing a quantum phase transition and confirming theoretical predictions through experimental and numerical analysis.

Contribution

It provides the first detailed experimental validation of the scaling behavior of Andreev bound states with the Kondo temperature and superconducting gap in such systems.

Findings

The energy of sub-gap states scales with T_K/Δ ratio.

A quantum phase transition occurs at T_K/Δ ≈ 0.6.

Experimental results agree with numerical renormalization group calculations.

Abstract

A quantum dot coupled to a superconducting contact provides a tunable artificial analogue of a magnetic atom in a superconductor, a paradigmatic quantum impurity problem. We realize such a system with an InAs semiconductor nanowire contacted by an Al-based superconducting electrode. We use an additional normal-type contact as weakly coupled tunnel probe to perform tunneling spectroscopy measurements of the elementary sub-gap excitations, known as Andreev bound states or Yu-Shiba-Rusinov states. We demonstrate that the energy of these states, $\zeta$, scales with the ratio between the Kondo temperature, $T_K$, and the superconducting gap, $\Delta$. $\zeta$ vanishes for $T_K/\Delta \approx 0.6$, denoting a quantum phase transition between spin singlet and doublet ground states. By further leveraging the gate control over the quantum dot parameters, we determine the singlet-doublet phase boundary in the stability diagram of the system. Our experimental results show remarkable quantitative agreement with numerical renormalization group calculations.