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 Schr\"odinger 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.