Quantum Charge Fluctuations in a Superconducting Grain

TL;DR

This paper analyzes how quantum charge fluctuations affect the Coulomb blockade staircase in a superconducting grain connected to a normal metal, revealing a transition from discrete steps to a continuous charge with unique capacitance features.

Contribution

It provides an analytical description of the Coulomb blockade staircase shape considering quantum fluctuations at finite conductance and temperature.

Findings

Quantum fluctuations smooth out charge steps at finite conductance.

Differential capacitance exhibits discontinuities at certain gate voltages.

The staircase shape is analytically characterized at non-zero temperatures.

Abstract

We consider charge quantization in a small superconducting grain that is contacted by a normal-metal electrode and is controlled by a capacitively coupled gate. At zero temperature and zero conductance $G$ between the grain and the electrode, the charge $Q$ as a function of the gate voltage $V_g$ changes in steps. The step height is $e$ if $\Delta<E_c$, where $\Delta$ and $E_c$ are, respectively, the superconducting gap and the charging energy of the grain. Quantum charge fluctuations at finite conductance remove the discontinuity in the dependence of $Q$ on $V_g$ and lead to a finite step width $\propto G^2\Delta$. The resulting shape of the Coulomb blockade staircase is of a novel type. The grain charge is a continuous function of $V_g$ while the differential capacitance, $dQ/dV_g$, has discontinuities at certain values of the gate voltage. We determine analytically the shape of the Coulomb blockade staircase also at non-zero temperatures.