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
The time evolution of a wave packet within a disordered quantum dot is investigated. It is shown that although the disorder-averaged electron density is nearly homogeneous at times of the order of the difFusive time across the dot, at larger times there is a remarkable evolution towards a state with increasing correlation with the original state (i.e., an echo appears). At long times compared to the inverse mean level spacing the density distribution becomes time independent and preserves a memory of the original state. These efFects are shown to reveal themselves in the relaxation of current through a quantum dot weakly coupled to reservoirs.PACS numbers: 73.40.Gk, 05.45.+b, 72.20. My, 73.20.Dx In this Letter we discuss the dynamics of a quantum particle (electron) inside a quantum dot. Namely, we consider the time evolution of the probability to find the particle at a point rz if it was located at a point ri at time t = 0. We assume that the host potential of the dot contains a random component large enough for the mean free path of the particle within the dot, l, to be much smaller than the size of the dot, L. On the other hand, let l be much larger than the wavelength of the particle A so that the dot is "metallic"; i.e. , the relevant quantum states of the particles are extended within the dot.In this case A « l « L we can distinguish between four difFerent time regimes for the evolution of the wave packet.(i) At the shortest times t & r = l/v (v is the velocity of the particle) the evolution is ballistic and the size of the wave packet R increases linearly with time.(ii) When time t & r (or R & l) the particle dynamics becomes difFusive. The crossover from the ballistic to the difFusive regimes is well known in the theory of disordered metals. It was discussed, probably for the first time, in connection with the kinetic theory of gazes by Boltzmann who invented the "stosszahl ansatz" [1]. In the difFusive regime the size of the packet increases as Qt and the probability to find the particle at the original point is proportional to t dlz where d is the dot dimensionality. Since the size of the quantum dot is finite this behavior lasts only until the size of the wave packet becomes comparable with L. This happens when t L /D = tl. , where D = vlld is the difFusion constant.(iii) Classical dynamics appears at t & t1, and is not very interesting: The particle has already spread over all the system and the difFerence in the probabilities to find it at the original point and at any other point decreases exponentially with increasing t. In the quantum case the dynamics at the large times (t & tL, ) is much less trivial and has not been previously discussed. This is the subject of our paper. We will see that the quanturn interference and the discreteness of the exact energy levels of the particle in the dot lead to a nontrivial time dependence for t » t1, . The inhomogeneous part of the electron density distribution will increase linearly with time for t & tH = h/6 where b is the mean spacing between the energy ...
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