Fluctuations in quantum dot charging energy

Abstract

We show that quasi-periodic fluctuations in the charging energy E_C of small, chaotic quantum dots result from strongly scarred states which are the remnant of periodic orbits in the classical confining potential. We perform self-consistent, density functional and spin density functional calculations for the dots used in the recent experimental studies of Sivan et al. [PRL 77, 1123 (1996)]. We directly compute the direct Coulomb matrix elements W_{pq}, screened by the gates, between self-consistent states. We show that diagonal elements (denoted U_p) are uniformly larger than off diagonal elements, resulting in spin polarization of the dot. We show that strongly scarred states, which are quasi-one dimensional, have particularly large values of U_p which causes them to remain, partially occupied, at the Fermi surface as gate voltage V_g is swept and more homogeneous, ``chaotic'' states pass through. We show that ultimate double-filling of these scars results in large upward fluctuations of E_C. In addition, we show that V_g dependence of capacitances and level energies substantially enhances fluctuations in the gate voltage spacing dV_g of Coulomb oscillations, as compared to and distinct from fluctuations in the ``inverse compressibility'' (i.e. E_C). Moreover, these dependences modify the distribution of dV_g fluctuations so that purely upward fluctuations of E_C produce symmetric fluctuations in dV_g. Consequently, gate voltage fluctuations (normalized by the appropriate capacitance ratio) are substantially greater than the single particle level spacing, Delta, but, as we show by direct calculation, their temperature T scaling is consistent with Delta/T and not E_C/T, in complete agreement with experiment.