arXiv subjects
Publications and source records attributed to A. Douglas Stone.
We develop a quantitative semiclassical formula for the resonant tunneling current through a quantum well in a tilted magnetic field. It is shown that the current depends only on periodic orbits within the quantum well. The theory explains the puzzling evolution of the tunneling spectra near a tilt angle of $30^{\circ}$ as arising from an exchange bifurcation of the relevant periodic orbits.
The anomalously strong scarring of wavefunctions found in numerical studies of quantum wells in a tilted magnetic field is shown to be due to special properties of the classical dynamics of this system. A certain subset of periodic orbits are identified which are nearly stable over a very large interval of variation of the classical dynamics; only this subset are found to exhibit strong scarring. Semiclassical arguments shed further light on why these orbits dominate the experimentally observed tunneling spectra.
A semiclassical theory is developed and compared to experiments on the tunneling resonance spectrum for a quantum well in magnetic field tilted with respect to the tunneling direction. As the tilt angle is increased from zero the classical mechanics of an electron trapped within the well undergoes a smooth transition from integrable to chaotic dynamics. Perturbation theory is invalid for most of the regime of experimental interest, motivating a semiclassical treatment based on short periodic orbits within the well. In this paper we present a unified theory of all the periodic orbits within the well which are of relevance to experiments and show that they are all related to bifurcations of the period-one traversing orbits. An analytic theory is derived for the period and stability of these traversing orbits. An unusual feature of the classical mechanics of this system is the existence of certain important periodic orbits only in finite energy bands. We calculate the widths of these bands and relate them to experimental data. In the forthcoming paper the results for these short periodic orbits are used in conjunction with a novel semiclassical tunneling formula to calculate the magnetotunneling current, which is then compared with experiments.
We consider the bilocal conductivity tensor, the two-probe conductance and its fluctuations for a disordered phase-coherent two-dimensional system of non-interacting electrons in the presence of a magnetic field, including correctly the edge effects. Analytical results are obtained by perturbation theory in the limit $σ_{xx} \gg 1$. For mesoscopic systems the conduction process is dominated by diffusion but we show that, due to the lack of time-reversal symmetry, the boundary condition for diffusion is altered at the reflecting edges. Instead of the usual condition, that the derivative along the direction normal to the wall of the diffusing variable vanishes, the derivative at the Hall angle to the normal vanishes. We demonstrate the origin of this boundary condition from different starting points, using (i) a simplified Chalker-Coddington network model, (ii) the standard diagrammatic perturbation expansion, and (iii) the nonlinear sigma-model with the topological term, thus establishing connections between the different approaches. Further boundary effects are found in quantum interference phenomena. We evaluate the mean bilocal conductivity tensor $σ_{μν}(r,r')$, and the mean and variance of the conductance, to leading order in $1/σ_{xx}$ and to order $(σ_{xy}/σ_{xx})^2$, and find that the variance of the conductance increases with the Hall ratio. Thus the conductance fluctuations are no longer simply described by the unitary universality class of the $σ_{xy}=0$ case, but instead there is a one-parameter family of probability distributions. In the quasi-one-dimensional limit, the usual universal result for the conductance fluctuations of the unitary ensemble is recovered, in contrast to results of previous authors. Also, a long discussion of current conservation.
We investigate the spin-magnetism of mesoscopic metallic grains. In the average response of an ensemble of grains there are corrections to macroscopic behaviour due to both spectral fluctuations and electron-electron interactions. These corrections are a non-linear function of the magnetic field. Their temperature dependence is calculated numerically and analytically. An experiment is proposed to measure the unknown interaction coupling constant in the cooper channel. For a single sample the magnetization is found to fluctuate reproducibly about the mean. These fluctuations directly probe the energy level statistics.
We define a current-conserving approximation for the local conductivity tensor of a disordered system which includes the effects of weak localization. Using this approximation we show that the weak localization effect in conductance is not obtained simply from the diagram corresponding to the coherent back-scattering peak observed in optical experiments. Other diagrams contribute to the effect at the same order and decrease its value. These diagrams appear to have no semiclassical analogues, a fact which may have implications for the semiclassical theory of chaotic systems. The effects of discrete symmetries on weak localization in disordered conductors is evaluated and and compared to results from chaotic scatterers.
Linear response theory for open (infinite) systems leads to an expression for the current response which contains surface terms in addition to the usual bulk Kubo term. We show that this surface term vanishes identically if the correct order of limits is maintained in the derivation: the system size, L, must be taken to infinity before the adiabatic turn-on rate $δ$ of the perturbation is taken to zero. In contrast to recent claims this shows that linear response theory for open systems is gauge-invariant without modification of the continuity equation. We show that a simpler derivation of the Landauer-Büttiker equations may be obtained consistently from the bulk Kubo term, noting that surface terms arising here are non-vanishing because they involve the opposite order of limits, $δ\to 0$, then $L \to \infty$.