Magnetotunneling through a quantum well in a tilted field I: Periodic orbit theory

TL;DR

This paper develops a semiclassical theory based on periodic orbits to explain magnetotunneling spectra in a quantum well under a tilted magnetic field, revealing a transition from integrable to chaotic dynamics.

Contribution

It introduces a unified semiclassical approach using short periodic orbits related to bifurcations, applicable to experimental tunneling spectra in tilted magnetic fields.

Findings

Classical electron dynamics transition from integrable to chaotic with tilt angle.

Periodic orbits are linked to bifurcations of traversing orbits.

Finite energy bands host important periodic orbits, matching experimental data.

Abstract

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.