Disorder in Andreev reflection of a quantum Hall edge

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.

Figures (1)

• Figure 1: (a) A chiral edge state with a segment proximitized by a "dirty", grounded superconductor. Electrons are launched towards the segment from an upstream electrode biased by voltage $V$. An electron propagating along the segment converts randomly into a hole over the distance $l_{\rm A}$, which is controlled by disorder in the superconductor, see Eq. \ref{['eq:var']}. (b) Evolution of the electronic wave function, see Eq. \ref{['eq:evol']}, is similar to the motion of a "spin" in a stochastic effective "magnetic field". The conductance $G = I / V$ is determined by the result of a random walk of a point on a Bloch sphere. (c) $G$ is a random quantity that fluctuates upon varying the electron density $n$ in the 2DEG (traces are simulated using Eq. \ref{['eq:evol']}; units of $n$ are the same for the two plots and are otherwise arbitrary). (d) The loss of correlation between the values of $G$ upon a change in $n$ is quantified by function ${\cal C}(\delta n)$, see Eqs. \ref{['eq:corrdef']}--\ref{['eq:ncor']}. The origin of the correlations loss is illustrated by the divergence between two stochastic trajectories on a Bloch sphere. The "spins" corresponding to different values of $n$ experience a different effective "magnetic field", and thus drift apart in the course of evolution. The separation of the "spins" is slower for stronger disorder. As the result, the trace $G(n)$ in panel (c) is smoother for smaller $l_{\rm A}$.