Mesoscopic Fluctuations in the Ground State Energy of Disordered Quantum Dots

Universal Voltage Fluctuations in Disordered Superconductors

The Aharonov-Casher effect is the analogue of the Aharonov-Bohm effect that applies to neutral particles carrying a magnetic moment. This effect can be manifested by vortices or fluxons flowing in trajectories that encompass an electric charge. These vortices have been predicted to result in a persistent voltage that fluctuates for different sample realizations. Here, we show that disordered superconductors exhibit reproducible voltage fluctuation, which is antisymmetrical with respect to the magnetic field, as a function of various parameters such as the magnetic field amplitude, field orientations, and gate voltage. These results are interpreted as the vortex equivalent of the universal conductance fluctuations typical of mesoscopic disordered metallic systems. We analyze the data in the framework of random matrix theory and show that the fluctuation correlation functions and curvature distributions exhibit behavior that is consistent with Aharonov-Casher physics. The results demonstrate the quantum nature of the vortices in highly disordered superconductors, both above and below T_{c}.

Development of effective stochastic potential method using random matrix theory for efficient conformational sampling of semiconductor nanoparticles at non-zero temperatures

The relationship between structure and property is central to chemistry and enables the understanding of chemical phenomena and processes. Need for an efficient conformational sampling of chemical systems arises from the presence of solvents and the existence of non-zero temperatures. However, conformational sampling of structures to compute molecular quantum mechanical properties is computationally expensive because a large number of electronic structure calculations are required. In this work, the development and implementation of the effective stochastic potential (ESP) method is presented to perform efficient conformational sampling of molecules. The overarching goal of this work is to alleviate the computational bottleneck associated with performing a large number of electronic structure calculations required for conformational sampling. We introduce the concept of a deformation potential and demonstrate its existence by the proof-by-construction approach. A statistical description of the fluctuations in the deformation potential due to non-zero temperature was obtained using infinite-order moment expansion of the distribution. The formal mathematical definition of the ESP was derived using the functional minimization approach to match the infinite-order moment expansion for the deformation potential. Practical implementation of the ESP was obtained using the random-matrix theory method. The developed method was applied to two proof-of-concept calculations of the distribution of HOMO-LUMO gaps in water molecules and solvated CdSe clusters at 300 K. The need for large sample size to obtain statistically meaningful results was demonstrated by performing 10⁵ ESP calculations. The results from these prototype calculations demonstrated the efficacy of the ESP method for performing efficient conformational sampling. We envision that the fundamental nature of this work will not only extend our knowledge of chemical systems at non-zero temperatures but also generate new insights for innovative technological applications.

Universality classes in Coulomb blockade conductance peak-height statistics

We investigate, using exact diagonalization techniques, the distribution of conductance peak heights in Coulomb blockade regime of a quantum dot connected to two leads under generic dot conditions. The study reveals a three-parametric dependence of the distribution: (i) two dot-lead contact characteristics and (ii) the complexity parameter, mimicking the combined effect of all dot conditions. This also indicates the presence of an infinite range of universality classes of conductance statistics, dominantly characterized just by the complexity parameter and global symmetry constraints.

Electron counting spectroscopy of CdSe quantum dots

We report on the electronic properties of semiconducting CdSe quantum dots that can be filled or emptied with many electrons. To accomplish that, we employ a device layout where the investigated quantum dot is attached to only one electrode, a carbon nanotube. Measurements consist of detecting individual electrons transferred onto the quantum dot (by monitoring the nanotube resistance) while sweeping the electrochemical potential of the dot with a gate. This technique allows us to detect the energy gap of the semiconducting quantum dot and to access many electronic levels. We exploit the latter finding to study the statistical aspects of the spectrum of the quantum dot. The measured spectrum distribution approaches the bimodal Wigner distribution, which is the most basic prediction of the random matrix theory applied to quantum dots.

Semiclassical theory of nonlocal statistical measures: residual Coulomb interactions

In a recent paper [Phys. Rev. Lett. 100, 164101 (2008)] and within the context of quantized chaotic billiards, random plane-wave and semiclassical theoretical approaches were applied to an example of a relatively new class of statistical measures, i.e., measures involving both complete spatial integration and energy summation as essential ingredients. A quintessential example comes from the desire to understand the short-range approximation to the first-order ground-state contribution of the residual Coulomb interaction. Billiards, fully chaotic or otherwise, provide an ideal class of systems on which to focus as they have proven to be successful in modeling the single-particle properties of a Landau-Fermi liquid in typical mesoscopic systems, i.e., closed or nearly closed quantum dots. It happens that both theoretical approaches give fully consistent results for measure averages, but that somewhat surprisingly for fully chaotic systems the semiclassical theory gives a much improved approximation for the fluctuations. Comparison of the theories highlights a couple of key shortcomings inherent in the random plane-wave approach. This paper contains a complete account of the theoretical approaches, elucidates the two shortcomings of the oft-relied-upon random plane-wave approach, and treats non-fully-chaotic systems as well.

Study of single silicon quantum dots' band gap and single-electron charging energies by room temperature scanning tunneling microscopy

Scanning tunneling spectroscopy in the shell-filling regime was carried out at room temperature to investigate the size dependence of the band gap and single-electron charging energy of single Si quantum dots (QDs). The results are compared with model calculation. A 12-fold multiple staircase structure was observed for a QD of about 4.3 nm diameter, reflecting the degeneracy of the first energy level, as expected from theoretical calculations. The systematic broadening of the tunneling spectroscopy peaks with decreasing dot diameter is attributed to the reduced barrier height for smaller dot sizes and to the splitting of the first energy level.

Energy level statistics of quantum dots

We investigate the charging energy level statistics of disordered interacting electrons in quantum dots by numerical calculations using the Hartree approximation. The aim is to obtain a global picture of the statistics as a function of disorder and interaction strengths. We find Poisson statistics at very strong disorder, Wigner-Dyson statistics for weak disorder and interactions, and a Gaussian intermediate regime. These regimes are as expected from previous studies and fundamental considerations, but we also find interesting and rather broad crossover regimes. In particular, intermediate between the Gaussian and Poisson regimes we find a two-sided exponential distribution for the energy level spacings. In comparing with experiment, we find that this distribution may be realized in some quantum dots.

Scar intensity statistics in the position representation

We obtain general predictions for the distribution of wave function intensities in position space on the periodic orbits of chaotic ballistic systems. The expressions depend on effective system size , instability exponent of the periodic orbit, and proximity to a focal point of the orbit. Limiting expressions are obtained that include the asymptotic probability distribution of rare high-intensity events and a perturbative formula valid in the limit of weak scarring. For finite system sizes, a single scaling lambdaN variable describes deviations from the semiclassical N--> infinity limit.

Random matrices with correlated elements: a model for disorder with interactions

The complicated interactions in the presence of disorder lead to a correlated randomization of states. The Hamiltonian as a result behaves like a multiparametric random matrix with correlated elements. We show that the eigenvalue correlations of these matrices can be described by the single parametric Brownian ensembles. The analogy helps us to reveal many important features of the level statistics in interacting systems, e.g., a critical point behavior different from that of noninteracting systems, the possibility of extended states even in one dimension, and a universal formulation of level correlations.

Quantum dots with disorder and interactions: a solvable large-g limit

We analyze the problem of interacting electrons on a ballistic quantum dot with chaotic boundary conditions, where the effective interactions at low energies are characterized by Landau parameters. When the dimensionless conductance g of the dot is large, the disordered interacting problem can be solved in a saddle-point approximation which becomes exact as g --> infinity (as in a large-N theory), leading to a phase transition in each Landau interaction channel. In the weak-coupling phase constant charging and exchange interactions dominate the low-energy physics, while the strong-coupling phase displays a spontaneous distortion of the Fermi surface, smeared out by disorder.

Spin and conductance-peak-spacing distributions in large quantum dots: a density-functional theory study

We use spin-density-functional theory to study the spacing between conductance peaks and the ground-state spin of 2D model quantum dots with up to 200 electrons. Distributions for different ranges of electron number are obtained in both symmetric and asymmetric potentials. The even/odd effect is pronounced for small symmetric dots but vanishes for large asymmetric ones, suggesting substantially stronger interaction effects than expected. The fraction of high-spin ground states is remarkably large.

Interactions and disorder in quantum dots: instabilities and phase transitions

Using a fermionic renormalization group approach, we analyze a model where the electrons diffusing on a quantum dot interact via Fermi-liquid interactions. Describing the single-particle states by random matrix theory, we find that interactions can induce phase transitions (or crossovers for finite systems) to regimes where fluctuations and collective effects dominate at low energies. Implications for experiments and numerical work on quantum dots are discussed.

Conductance peak motion due to a magnetic field in weakly coupled chaotic quantum dots

We study the influence of moderate exchange interactions on the behavior of the peaks in the conductance of single electron transistors. We numerically reproduce recently observed features of the magnetic field dependence of both the peak positions and heights. These features unambiguously identify the total spin S of each ground state. We evaluate the probability of each S as a function of the exchange strength J, and external magnetic field B. The expressions involve only J and the g factor as adjustable parameters. For a broad parameter range these probabilities are determined by a linear combination of J and gB.

Signatures of spin pairing in chaotic quantum dots

Coulomb blockade resonances are measured in a GaAs quantum dot in which both shape deformations and interactions are small. The parametric evolution of the Coulomb blockade peaks shows a pronounced pair correlation in both position and amplitude, which is interpreted as spin pairing. As a consequence, the nearest-neighbor distribution of peak spacings can be well approximated by a modified bimodal Wigner surmise, in which interactions are taken into account beyond the constant interaction model.

Spin degeneracy and conductance fluctuations in open quantum dots

The dependence of conductance fluctuations on parallel magnetic field is used as a probe of spin degeneracy in open GaAs quantum dots. The variance of fluctuations at high parallel field is reduced from the low-field variance (with broken time-reversal symmetry) by factors ranging from roughly 2 in a 1 microm (2) dot to greater than 4 in 8 microm (2) dots. The factor of 2 is expected for Zeeman splitting of spin-degenerate channels. A possible explanation for the larger suppression based on field-dependent spin-orbit scattering is proposed.

Periodic orbit effects on conductance peak heights in a chaotic quantum dot

We study the effects of short-time classical dynamics on the distribution of Coulomb blockade peak heights in a chaotic quantum dot. The location of one or both leads relative to the short unstable orbits, as well as relative to the symmetry lines, can have large effects on the moments and on the head and tail of the conductance distribution. We study these effects analytically as a function of the stability exponent of the orbits involved, and also numerically using the stadium billiard as a model. The predicted behavior is robust, depending only on the short-time behavior of the many-body quantum system, and consequently insensitive to moderate-sized perturbations and interactions.