Fluctuation of inverse compressibility for electronicsystems with random capacitive matrices

Abstract

This article is concerned with the statistics of the addition spectra of certain many-body systems of identical particles. In the first part, the pertinent system consists of N identical particles distributed among K<N independent subsystems, such that the energy of each subsystem is a quadratic function of the number of particles residing on it with random coefficients. On a large scale, the ground-state energy E(N) of the whole system grows quadratically with N, but in general there is no simple relation such as EN = aN+bN 2. The deviation of E(N) from exact quadratic behaviour implies that its second difference (the inverse compressibility) XN ≡E(N+1)−2E(N)+E(N−1) is a fluctuating quantity. Regarding the numbers XN as values assumed by a certain random variable X, we obtain a closed-form expression for its distribution F (X). Its main feature is that the corresponding density P (X)=dF (X)/d X has a maximum at the point X=0. As K→∞ the density is Poissonian, namely, P(X)→e−X This result serves as a starting point for the second part, in which coupling between subsystems is included. More generally, a classical model is suggested in order to study fluctuations of Coulomb blockade peak spacings in large two-dimensional semiconductor quantum dots. It is based on the electrostatics of several electron islands among which there are random inductive and capacitive couplings. Each island can accommodate electrons on quantum orbitals whose energy depends also on an external magnetic field. In contrast to a single-island quantum dot, where the spacing distribution between conductance peaks is close to Gaussian, here the distribution has a peak at small spacing value. The fluctuations are mainly due to charging effects. The model can explain the occasional occurrence of couples or even triples of closely spaced Coulomb blockade peaks, as well as the qualitative behaviour of peak positions with the applied magnetic field.