The semiclassical tool in mesoscopic physics

Semiclassical methods are extremely valuable in the study of transport and thermodynamical properties of ballistic microstructures. By expressing the conductance in terms of classical trajectories, we demonstrate that quantum interference phenomena depend on the underlying classical dynamics of non-interacting electrons. In particular, we are able to calculate the characteristic length of the ballistic conductance fluctuations and the weak localization peak in the case of chaotic dynamics. Integrable cavities are not governed by single scales, but their non-generic behavior can also be obtained from semiclassical expansions (over isolated trajectories or families of trajectories, depending on the system). The magnetic response of a microstructure is enhanced with respect to the bulk (Landau) susceptibility, and the semiclassical approach shows that this enhancement is the largest for integrable geometries, due to the existence of families of periodic orbits. We show how the semiclassical tool can be adapted to describe weak residual disorder, as well as the effects of electron-electron interactions. The interaction contribution to the magnetic susceptibility also depends on the nature of the classical dynamics of non-interacting electrons, and is parametrically larger in the case of integrable systems.Comment: Latex, Cimento-varenna style, 82 pages, 21 postscript figures; lectures given in the CXLIII Course "New Directions in Quantum Chaos" on the International School of Physics "Enrico Fermi"; Varenna, Italy, July 1999; to be published in Proceeding

Unbounded fluctuations in transport through an integrable cavity

We derive a semiclassical scheme for the conductance through a rectangular cavity. The transmission amplitudes are expressed as a sum over families of trajectories rather than a sum over isolated trajectories. The contributing families are obtained from the evaluation of a finite number of continued fractions. We find that, contrary to the chaotic case, the conductance fluctuations increase with the incoming energy and the correlation function exhibits a singularity at the origin.Comment: 9 pages + 3 figures, accepted for Eur. Phys. J.

Quantum Mesoscopic Scattering: Disordered Systems and Dyson Circular Ensembles

We consider elastic reflection and transmission of electrons by a disordered system characterized by a $2N\!\times\!2N$ scattering matrix $S$. Expressing $S$ in terms of the $N$ radial parameters and of the four $N\!\times\!N$ unitary matrices used for the standard transfer matrix parametrization, we calculate their probability distributions for the circular orthogonal (COE) and unitary (CUE) Dyson ensembles. In this parametrization, we explicitely compare the COE--CUE distributions with those suitable for quasi--$1d$ conductors and insulators. Then, returning to the usual eigenvalue--eigenvector parametrization of $S$, we study the distributions of the scattering phase shifts. For a quasi--$1d$ metallic system, microscopic simulations show that the phase sift density and correlation functions are close to those of the circular ensembles. When quasi--$1d$ longitudinal localization breaks $S$ into two uncorrelated reflection matrices, the phase shift form factor $b(k)$ exhibits a crossover from a behavior characteristic of two uncoupled COE--CUE (small $k$) to a single COE--CUE behavior (large $k$). Outside quasi--one dimension, we find that the phase shift density is no longer uniform and $S$ remains nonzero after disorder averaging. We use perturbation theory to calculate the deviations to the isotropic Dyson distributions. When the electron dynamics is noComment: 39 pages, 14 figures available under request, RevTex, IPNO/TH 94-6

Semiclassical analysis of level widths for one-dimensional potentials

We present a semiclassical study of level widths for a class of one-dimensional potentials in the presence of an ohmic environment. Employing an expression for the dipole matrix element in terms of the Fourier transform of the classical path we obtain the level widths within the Golden rule approximation. It is found that for potentials with an asymptotic power-law behavior, which may in addition be limited by an infinite wall, the width that an eigenstate of the isolated system acquires due to the coupling to the environment is proportional to its quantum number.Comment: 8 pages, 2 figures, RevTe

Transmission phase of a quantum dot and statistical fluctuations of partial-width amplitudes

Experimentally, the phase of the amplitude for electron transmission through a quantum dot (transmission phase) shows the same pattern between consecutive resonances. Such universal behavior, found for long sequences of resonances, is caused by correlations of the signs of the partial-width amplitudes of the resonances. We investigate the stability of these correlations in terms of a statistical model. For a classically chaotic dot, the resonance eigenfunctions are assumed to be Gaussian distributed. Under this hypothesis, statistical fluctuations are found to reduce the tendency towards universal phase evolution. Long sequences of resonances with universal behavior only persist in the semiclassical limit of very large electron numbers in the dot and for specific energy intervals. Numerical calculations qualitatively agree with the statistical model but quantitatively are closer to universality.Comment: 8 pages, 4 figure

Partial local density of states from scanning gate microscopy

Scanning gate microscopy images from measurements made in the vicinity of quantum point contacts were originally interpreted in terms of current flow. Some recent work has analytically connected the local density of states to conductance changes in cases of perfect transmission, and at least qualitatively for a broader range of circumstances. In the present paper, we show analytically that in any time-reversal invariant system there are important deviations that are highly sensitive to imperfect transmission. Nevertheless, the unperturbed partial local density of states can be extracted from a weakly invasive scanning gate microscopy experiment, provided the quantum point contact is tuned anywhere on a conductance plateau. A perturbative treatment in the reflection coefficient shows just how sensitive this correspondence is to the departure from the quantized conductance value and reveals the necessity of local averaging over the tip position. It is also shown that the quality of the extracted partial local density of states decreases with increasing tip radius.Comment: 16 pages, 9 figure

From the Fermi glass towards the Mott insulator in one dimension: Delocalization and strongly enhanced persistent currents

When a system of spinless fermions in a disordered mesoscopic ring becomes instable between the inhomogeneous configuration driven by the random potential (Anderson insulator) and the homogeneous one driven by repulsive interactions (Mott insulator), the persistent current can be enhanced by orders of magnitude. This is illustrated by a study of the change of the ground state energy under twisted boundary conditions using the density matrix renormalization group algorithm.Comment: 4 pages, 5 figures; RevTe

Spin-orbit effects in nanowire-based wurtzite semiconductor quantum dots

We study the effect of the Dresselhaus spin-orbit interaction on the electronic states and spin relaxation rates of cylindrical quantum dots defined on quantum wires having wurtzite lattice structure. The linear and cubic contributions of the bulk Dresselhaus spin-orbit coupling are taken into account, along with the influence of a weak external magnetic field. The previously found analytic solution for the electronic states of cylindrical quantum dots with zincblende lattice structures with Rashba interaction is extended to the case of quantum dots with wurtzite lattices. For the electronic states in InAs dots, we determine the spin texture and the effective g-factor, which shows a scaling collapse when plotted as a function of an effective renormalized dot-size dependent spin-orbit coupling strength. The acoustic-phonon-induced spin relaxation rate is calculated and the transverse piezoelectric potential is shown to be the dominant one.Comment: 12 pages, 5 figure