Landau quantization of nearly degenerate bands

Chong Wang, Wenhui Duan, Leonid Glazman, A. Alexandradinata

Published 2018-10-03Version 1

In the semiclassical theory of Landau levels, a Bloch electron is driven by a magnetic field along constant-energy band contours known as cyclotron orbits. We introduce a quantization rule for Landau levels of nearly-degenerate orbits whose differential area ($\delta S$) in $\boldsymbol{k}$-space is much smaller than their average area $S$. The splitting of two nearly-degenerate orbits commonly originates from spin-orbit coupling, or from the breaking of a crystalline point-group symmetry. By treating both $\delta S/S$ and $1/l^2S$ ($l$ being the magnetic length) as small parameters with finite ratio $l^2\delta S$, our perspective diverges from and generalizes previous semiclassical theories. $1/l^2\delta S$ measures the hybridization of split orbits due to the Zeeman interaction; in the opposing regimes $l^2\delta S {{\ll}} 1$ and $l^2\delta S{\gg}1$, we recover the known Onsager-Lifshitz-Roth rules for nondegenerate and exactly-degenerate orbits, respectively. In the absence of crystalline point-group symmetries, three tunable real parameters are needed to attain a spin-degeneracy between two Landau levels; we have exhaustively identified all symmetry classes of orbits for which this number is reduced from three. In particular, only one parameter is needed for rotational-symmetric orbits; this parameter may be the magnitude or orientation of the field, or the bias voltage in tunneling spectroscopy. A signature of single-parameter tunability is a smooth crossover between period-doubled and -undoubled quantum oscillations in the low-temperature, Schubnikov-de Haas effect. We demonstrate the utility of our quantization rule, as well as the tunability of Landau-level degeneracies, for the Rashba two-dimensional electron gas -- subject to an arbitrarily oriented magnetic field, and possibly augmented by the Dresselhaus spin-orbit interaction.