Linear-Network Model for Magnetic Breakdown in Two Dimensions

A heuristic model is set up for magnetic breakdown in a two-dimensional rectangular lattice at right angles to an applied magnetic field. The model is an adaptation of a method due to Pippard to a set of coupled ordinary differential equations derived from the Schrödinger equation. A linear-chain network is set up. This network can also be derived from Pippard's two-dimensional network and it suggests a simple way to compute the energy bands not only when the number $F$ of flux quanta through a unit cell is the reciprocal of an even integer (the case treated by Pippard) but also when $F$ is any rational fraction. Numerical computations of the energy bands suggest that in the latter case the electron wave moves on large orbits which might be called "hyperorbits." These hyperorbits may be open in a rectangular lattice and may give a resonant open-orbit ultrasonic attenuation. It is also found that when a free-electron Landau level is broadened by the lattice, it splits into two bands separated by a gap. This gap moves through the states from the Landau level as $F$ is changed and may give rise to new de Haas-van Alphen periods. The physical cause of this gap is discussed.