New degeneracies and modification of Landau levels in the presence of a parallel linear electric field

TL;DR

This paper studies how a linear electric field parallel to a magnetic field modifies Landau levels in a 3D electron system, revealing degeneracy jumps at rational frequency ratios and deriving analytical formulas for energy levels and degeneracies.

Contribution

It introduces a novel analysis of degeneracy modifications in Landau levels under combined magnetic and linear electric fields, including analytical formulas and perturbation effects.

Findings

Degeneracy varies with energy when frequency ratio is rational.

Degeneracy increases when the two frequencies are equal.

Energy levels form bands with intersections indicating residual degeneracy.

Abstract

We consider a three-dimensional system where an electron moves under a constant magnetic field (in the z-direction) and a \textit{linear} electric field parallel to the magnetic field above the z=0 plane and anti-parallel below the plane. The linear electric field leads to harmonic oscillations along the z-direction. There are therefore two frequencies characterizing the system: the usual cyclotron frequency $\omega_c$ corresponding to motion along the x-y plane and associated with Landau levels and a second frequency $\omega_z$ for motion along the z-direction. Most importantly, when the ratio $W=\omega_c/\omega_z$ is a rational number, the degeneracy of the energy levels does not remain always constant as the energy increases. At some energies, the degeneracy jumps i.e. it increases. In particular, when the two frequencies are equal, the degeneracy increases with each energy level. This is in stark contrast to the usual Landau levels where the degeneracy is independent of the energy. We derive compact analytical formulas for the degeneracy. We also obtain an analytical formula for the energy levels and plot them as a function of $W$. The increase in degeneracy can readily be seen in the plot at points where lines intersect. For concreteness, we consider the electric field produced by a uniformly charged ring. Besides a linear electric field in the z direction the ring produces an extra electric field in the xy plane which we treat via perturbation theory. The Landau degeneracy is now lifted and replaced by tightly spaced levels that come in "bands". The plot of the energy levels shows that there is still a degeneracy where the bands intersect.