Random matrix model for quantum dots with interactions and the conductance peak spacing distribution

Abstract

We introduce a random interaction matrix model (RIMM) for finite-size strongly interacting fermionic systems whose single-particle dynamics is chaotic. The model is applied to Coulomb blockade quantum dots with irregular shape to describe the crossover of the peak spacing distribution from a Wigner-Dyson to a Gaussian-like distribution. The crossover is universal within the random matrix model and is shown to depend on a single parameter: a scaled fluctuation width of the interaction matrix elements. The crossover observed in the RIMM is compared with the results of an Anderson model with Coulomb interactions.PACS numbers: 73.23. Hk, 05.45+b, 73.20.Dx, The transport properties of semiconductor quantum dots can be measured by connecting the dots to leads via point contacts [1]. When these point contacts are pinched off, effective barriers are formed between the dot and the leads, and the charge on the dot is quantized. Adding an electron to the dot requires a charging energy E C to overcome the Coulomb repulsion with electrons already in the dot. This repulsion can be compensated by modifying the gate voltage V g on the dot. For temperatures below E C , a series of Coulomb blockade oscillations is observed in the linear conductance as a function of V g . For temperatures much smaller than the mean level spacing ∆, the conductance is dominated by resonant tunneling and the Coulomb blockade oscillations become a series of sharp peaks.In dots with irregular shapes, the classical singleelectron dynamics is mostly chaotic. Quantum mechanically, chaotic systems are expected to exhibit universal fluctuations that are described by random matrix theory (RMT). The distributions [2] and parametric correlations [3] of the Coulomb blockade peak heights in quantum dots have been derived using RMT, and these predictions have been confirmed experimentally [4,5].Another quantity of recent experimental and theoretical interest is the peak spacing statistics. The peak spacing ∆ 2 can be expressed as a second order difference of the ground state energy E (n) g.s. of the n-electron dot as a function of the number of electrons:Using the constant interaction model (which ignores interactions except for a classical Coulomb energy of n 2 E C /2), and assuming a single-particle spectrum that is independent of n, ∆ 2 = E n+1 − E n + E C , where E n is the n-th single-particle energy. Within this model, RMT suggests a Wigner-Dyson distribution of the peak spacings with a width of ∼ ∆/2. However recent experiments find a distribution that is Gaussian-like and has a larger width [6][7][8][9]. This observation underlines the limitations of the constant interaction model and the importance of electron-electron interactions beyond an average Coulomb energy. Some observed features of the peak spacing distribution have been reproduced using exact numerical diagonalization of small disordered dots (n < ∼ 10) with Coulomb interactions [6,10]. The width of the distribution is found to increase monotonically with the gas parameter r s . Analytic RPA e...

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