ar X iv : c on d - m at / 9 60 71 21 v 1 1 7 Ju l 1 99 6 Mesoscopic conductance fluctuations in dirty quantum dots with single channel leads

We consider a distribution of conductance fluctuations in quantum dots with single channel leads and continuous level spectra and we demonstrate that it has a distinctly non-Gaussian shape and strong dependence on time-reversal symmetry, in contrast to an almost Gaussian distribution of conductances in a disordered metallic sample connected to a reservoir by broad multichannel leads. In the absence of time-reversal symmetry, our results obtained within the diagrammatic approach coincide with those derived within nonperturbative techniques. In addition, we show that the distribution has lognormal tails for weak disorder, similar to the case of broad leads, and that it becomes almost lognormal as the amount of disorder is increased towards the Anderson transition. PACS numbers: 72.15.-v, 73.20.Dx, 73.20.Fz Typeset using REVTEX 1 Recently it has been shown that the conductance of clean quantum dots with point-like external contacts (lead width w ≈ h̄k F , where h̄k −1 F is the Fermi wavelength) have nonGaussian distribution functions [1,2]. Weak transmission through the contacts means that the electrons typically spend more time in the system than that required to cross it so that the energy level broadening due to inelastic processes in the dot γ ≪ Ec where Ec ≈ h̄D/L 2 is the Thouless energy. This inequality corresponds to the zero mode regime which allows the use of non-perturbative techniques including random matrix theory [3] and the zero dimensional supersymmetric σ model [4]. In contrast, it is known that the distribution function of the conductance is mainly Gaussian [5] in a weakly disordered open sample connected to a reservoir by broad external contacts of width w > l, where l is the elastic mean free path. Inelastic scattering processes in the reservoir result in a level broadening γ ∼ Ec. In the present paper the aim is to determine the distribution function of a dirty quantum dot with two point contacts (which allow a single transport channel). We will describe the regime of continuous energy levels, γ > ∼ ∆, where γ is the broadening due to inelastic scattering in the dot and ∆ is the mean level spacing. This overlaps with the supersymmetric (SUSY) calculations [1] in the ergodic regime, ∆ < ∼ γ < Ec. While the SUSY approach is valid also in the quantum regime, γ < ∆, a perturbative diagrammatic approach applied below can be used also for γ > ∼ Ec. In this case all diffusion modes contribute to the conductance rather than a single homogeneous “zero” mode which is the only mode taken into account within the nonperturbative SUSY calculations. The conductance distribution function has a non-Gaussian shape also for such a strong level broadening so that the nonGaussian shape is due to geometric factors – namely, the point-like structure of contacts, rather than due to the dominance of the zero mode. In addition, we will use a standard renormalisation group technique to consider the rôle of increasing disorder in the dot. Traditionally the conductance of a system with broad, spatially homogeneous contacts is considered by means of the Kubo formula [6]. For a lead geometry which involves spatially inhomogeneous currents, however, the conductance is often more conveniently expressed via scattering probabilities using the Landauer-Büttiker formula [7]. In this paper, we start by writing the conductance in terms of Green’s functions with the help of the Landauer-Büttiker formula. Then we will determine the conductance distribution in the case of a continuous energy levels spectrum, γ > ∼ ∆, finding the moments of conductance by diagrammatic perturbation expansion in the parameter (γ/∆). Finally we will use an effective functional of the non-linear σ model as a framework for the renormalisation group analysis necessary to describe dependence of the moments of conductance (and thus of the distribution) on