Fluctuations in quantum dot charging energy

TL;DR

This paper explains the quasi-periodic fluctuations in the charging energy of small chaotic quantum dots as arising from strongly scarred states linked to classical periodic orbits, supported by self-consistent calculations.

Contribution

It demonstrates that scarred states cause fluctuations in charging energy and Coulomb oscillation spacings, aligning theoretical results with experimental observations.

Findings

Scarred states lead to large fluctuations in E_C.

Gate voltage dependence enhances Coulomb oscillation fluctuations.

Temperature scaling of fluctuations matches Delta/T, not E_C/T.

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.