[Paper Review] Disorder in Andreev reflection of a quantum Hall edge

Vladislav D. Kurilovich, Zachary M. Raines+1 more|arXiv (Cornell University)|Jan 2, 2022

Quantum and electron transport phenomena4 citations

TL;DR

This paper develops a theory of conductance fluctuations in a quantum Hall edge proximitized by a disordered superconductor, showing that disorder enables efficient Andreev reflection without magnetic field fine-tuning. The conductance becomes a stochastic quantity with zero average for long segments, and its statistical distribution depends on electron density, magnetic field, and temperature, with fluctuations suppressed more weakly than in conventional conductors due to chiral edge states.

ABSTRACT

We develop a theory of charge transport along the quantum Hall edge proximitized by a "dirty" superconductor. Disorder randomizes the Andreev reflection rendering the conductance of a proximitized segment a stochastic quantity with zero average for a sufficiently long segment. We find the statistical distribution of the conductance and its dependence on electron density, magnetic field, and temperature.

Motivation & Objective

• To understand how disorder in a superconductor enables robust Andreev reflection in quantum Hall edges without magnetic field fine-tuning.
• To explain the experimentally observed stochastic conductance with zero average in long proximitized segments.
• To develop a quantitative theory of conductance fluctuations in chiral, one-dimensional edge states, accounting for electron density, magnetic field, and temperature dependence.
• To address the breakdown of standard mesoscopic transport theory due to the chiral nature of edge states.
• To predict the statistical distribution of conductance and its correlation functions under varying control parameters.

Proposed method

• Formulate an effective Hamiltonian for chiral edge states in a quantum Hall system coupled to a disordered superconductor via tunneling at the interface.
• Model the stochastic conversion of electrons into holes over a characteristic length scale $ l_{ m A} $, determined by disorder in the superconductor.
• Map the evolution of the Andreev reflection amplitude onto a random walk on a Bloch sphere, representing the stochastic effective magnetic field acting on a pseudospin.
• Derive the conductance $ G $ as a function of electron density $ n $, using the stochastic pseudospin evolution and calculating the correlation function $ ilde{ ho}( ilde{n}) $.
• Compute the correlation length $ n_{ m cor} $ for conductance fluctuations, showing that correlations are lost when $ ilde{n} \gtrsim n_{ m cor} $, with $ n_{ m cor} \propto l_{ m A}/L $.
• Analyze finite-temperature effects by considering both thermal smearing $ T_{ m sm} \sim \Delta (l_{\rm A}/L)^{1/4} $ and inelastic scattering $ T_{ m in} \propto L^{-1/2} $, with $ T_{ m in} $ depending on disorder strength and material parameters.

Experimental results

Research questions

• RQ1 How does disorder in a superconductor enable efficient Andreev reflection in a quantum Hall edge without requiring fine-tuned magnetic fields?
• RQ2 What is the statistical distribution of conductance in a long proximitized quantum Hall edge segment, and how does it depend on electron density and magnetic field?
• RQ3 Why does the average conductance vanish for sufficiently long segments, and what determines the fluctuation amplitude?
• RQ4 How do conductance fluctuations correlate with changes in electron density, and what is the characteristic correlation length $ n_{\rm cor} $?
• RQ5 How do finite temperature effects suppress conductance fluctuations, and how does the chiral nature of edge states alter the suppression compared to conventional conductors?

Key findings

• For long segments ($ L \gg l_{\rm A} $), the average conductance $ \langle G \rangle $ vanishes due to stochastic Andreev reflection, while individual realizations fluctuate within $ \pm 2e^2/h $.
• The conductance correlation length $ n_{\rm cor} \propto l_{\rm A}/L $, meaning fluctuations are correlated only over small electron density changes; larger changes lead to uncorrelated conductance values.
• Abrupt conductance jumps occur when a vortex enters the superconductor, with a jump amplitude $ \mathcal{C}_{\rm jump}(d) \sim \langle\langle G^2 \rangle\rangle $ at $ d \sim l_{\rm A} $, matching experimental observations.
• Thermal smearing suppresses fluctuations at $ T_{\rm sm} \sim \Delta (l_{\rm A}/L)^{1/4} $, a much weaker dependence than in conventional conductors due to chiral edge transport.
• Inelastic scattering provides an additional suppression mechanism at $ T_{\rm in} \propto L^{-1/2} $, with $ T_{\rm in} \sim \hbar\omega_{\rm c} [\kappa\hbar v/e^2] (l_{\rm A}/L)^{1/2} $, depending on disorder and material parameters.
• The theory explains the zero-average, stochastic conductance observed in experiments [Zhao et al., 2020], including the absence of fine-tuning and the dependence on control parameters.