Discrete charging of metallic grains: Statistics of addition spectra

We analyse 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) χ_N ≤ E(N + 1) - 2E(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 for single grain quantum dots has been the focus of experimental and theoretical investigations during the last two decades. In the more general case of quantum dots composed of several grains, we provide an algorithm for calculating the distribution function corresponding to χ_N and show that this function is piecewise polynomial.