Discrete charging of metallic grains: Statistics of addition spectra

TL;DR

This paper investigates the statistical properties of addition spectra in metallic grain quantum dots with random capacitance matrices, providing an algorithm to compute the distribution of inverse compressibility, which is key to understanding Coulomb blockade phenomena.

Contribution

It introduces a novel algorithm for calculating the distribution of inverse compressibility in a system of metallic grains with random capacitance matrices, advancing the theoretical understanding of addition spectra.

Findings

Distribution function is piecewise polynomial.

Algorithm efficiently computes addition spectra statistics.

Provides insights into Coulomb blockade peak spacings.

Abstract

We analyze the statistics of electrostatic energies (and their differences) for a quantum dot system composed of a finite number $K$ of electron islands (metallic grains) with random capacitance-inductance matrix $C$, for which the total charge is discrete, $Q=Ne$ (where $e$ is the charge of an electron and $N$ is an integer). The analysis is based on a generalized charging model, where the electrons are distributed among the grains such that the electrostatic energy E(N) is minimal. Its second difference (inverse compressibility) $\chi_{N}=E(N+1)-2 E(N)+E(N-1)$ represents the spacing between adjacent Coulomb blockade peaks appearing when the conductance of the quantum dot is plotted against gate voltage. The statistics of this quantity has been the focus of experimental and theoretical investigations during the last two decades. We provide an algorithm for calculating the distribution function corresponding to $\chi_{N}$ and show that this function is piecewise polynomial.