Coulomb blockade threshold in inhomogeneous one-dimensional arrays of tunnel junctions

TL;DR

This paper derives a general expression for the Coulomb blockade threshold in inhomogeneous 1D tunnel junction arrays, showing how background charges and array size influence the threshold voltage.

Contribution

It provides a new general formula for the free energy change during tunneling in arrays with arbitrary capacitances and background charges, and analyzes the N-dependence of the threshold voltage.

Findings

Background charges significantly affect threshold voltage scaling.

Threshold voltage scales as N^a, with a crossing from 1/2 to 1 at a critical size.

Inhomogeneity in capacitances influences Coulomb blockade behavior.

Abstract

A general expression is given for the change in free energy when a charge tunnels through a junction in a one-dimensional array of N metallic islands with arbitrary capacitances and arbitrary background charges. This is used to obtain expressions for the (average) threshold voltage of the Coulomb blockade for a few characteristic geometries. We find that including random background charges has a large effect on the N-dependence of the threshold voltage: In an array with identical junction capacitances C and gate capacitances C_g, the threshold voltage, averaged over the background charge, is proportional to N^a, where {a} crosses over from 1/2 to 1 when N becomes larger than 2.5\sqrt{C/C_g}.