Model Calculations of Magnetic Band Structure

Simple model effective Hamiltonians are used to examine the structure of magnetic sub-bands. The positions of the discrete Landau levels are shown to be accurately predicted by the first-order correction to the Onsager quantization rule given by Roth, even for strong magnetic fields. The influence of the rational field condition, occuring when $\frac{l}{N}$ flux quanta pass through a unit cell, is investigated. The effect is pronounced when intraband or interband tunneling occurs. In this case the broadened sub-bands split into a cluster of $l$ smaller sub-bands. This fine structure may be roughly accounted for by use of an effective Hamiltonian based on the sub-band energy function $E\left(\mathbf{q}\right)$.