Statistics of conductance fluctuations in quantum dots

Abstract

We present a complete analytical theory of conductance fluctuations in quantum dots in the regime of chaotic dynamics. Using the supersymmetry method we calculate for the first time exactly the conductance distribution function for a system of noninteracting electrons. Knowledge of this quantity enables us to obtain information about statistical properties of wave functions of a closed dot. The far tail of the distribution function describes fluctuations of resonance conductance in the Coulomb blockade regime.PACS numbers: 73.40. Gk, 05.45.+b, 72.20.My, 73.20.Dx Recent development of nanometer technology made it possible to study tunneling through very small systems in which quantum effects are extremely important [1]. The tunneling spectroscopy of these novel objects is done by measuring conductance versus gate voltage, magnetic field, or other external parameters. The most striking feature of the experiments on the mesoscopic structures is an irregular dependence of the conductance on the varied parameters [2]. The conductance fluctuations are typically of the same order of magnitude as the average conductance. Therefore, in order to describe an experiment in an adequate way one needs to know not only traditional averages but the whole distribution function. Calculation of such a quantity for mesoscopic objects is a new and not trivial theoretical problem which requires development of new nonperturbative methods.The scheme of analytical description of small systems depends on the relation between intrinsic parameters of the systems and tunneling characteristics of measurement contacts. A connection with external bulk leads results in a level broadening 7. If 7 is large enough, 7 > E c , where E c ^ D/L 2 is the Thouless energy; the conductance fluctuations can be considered within a perturbation theory by summation of a certain class of diagrams (Cooperon and diffusion contributions [3,4]). In modern devices E c can be large due to small sample sizes L and large diffusion coefficient D. At the same time the coupling to the leads can be weak. So, the inequality 7 <C E c can be easily achieved and one encounters completely new interesting phenomena. Below we use the term quantum dot for systems where this inequality is fulfilled.A theoretical description of the quantum dots becomes rather difficult since the conventional diagrammatic technique that was so efficient for large 7 [3,4] no longer works here. In addition, the Coulomb interaction becomes very important and can lead to an appreciable reconstruction of electronic states and to a strong suppression of tunneling. However, for a still large system the Coulomb interaction is reduced primarily to charging effects and, therefore, can be rather simply incorporated in theory [5,6]. In this regime statistical properties of the conductance are mainly determined by chaotic dynamics of noninteracting electrons in a dot.The first work concerning the statistical properties of the conductance of dots has been done by Jalabert et al.[6], who studied the fluctuations ...

Semiclassical theory of Coulomb blockade peak heights in chaotic quantum dots

“…This alternative argument [27] immediately yields the functional form (27). When the closed dot is defined by the Dirichlet boundary conditions, the wave function in the narrow strip S near the "edge" can be represented asψ = zϕ (y) (28) where y is the coordinate along the boundary of the dot and z is in the direction of the normal. In this limit, the correlation function is…”

“…By using the correlation function of a random superposition of plane waves, the probability distributions of level-widths and conductance peaks in the case of multi-mode leads to the quantum dot were found [11,26]. A distribution similar to this ansatz was derived microscopically for disordered systems, a specific kind of chaotic system, using the nonlinear sigma model [28][29][30][31]. The next step was to constrain the correlation function by the short-time classical dynamics.…”

Conductance distribution of a quantum dot with nonideal single-channel leads

“…The theory unifies and extends known results. [2][3][4] The characteristic β-dependence of the distribution function that was found for ideal leads [Eq. (1)] is strongly suppressed for transmission probabilities T larger than Γ 2 . A closely related phenomenon is the nontrivial Γ-dependence of the conductance fluctuations for the three symmetry classes.…”

“…Recently, it was found that a "quantum dot" has a qualitatively different conductance distribution. [2][3][4] A quantum dot is a small confined region, having a large level spacing compared to the thermal energy, which is weakly coupled by point contacts to two electron reservoirs. The classical motion within the dot is assumed to be ballistic and chaotic.…”

Resonance statistics in a microwave cavity with a thin antenna

“…To illustrate how it works, we are going then to apply it to the case of a flat microwave cavity to which a thin antenna is attached. We compute the resonance spacing distribution in this system and show that it reproduces experimental data -within certain range of energies but without introducing any free parameters.Many recent studies involve an analysis, both theoretical and experimental, of spectral and transport properties of systems with a complicated geometry: let us recall various microwave resonators [1,2,3,4,5] or conductance fluctuations in quantum dots -see [6,7,8,9] and references therein. …”