Magnetotransport in a two-dimensional tight-binding model

In a two-dimensional electron system with a lateral superlattice potential in a perpendicular magnetic field, the Landau levels split into a complicated, self-similar, field-dependent spectrum, known as ‘‘Hofstadter’s butterfly.’’ We study transport along a strip of finite width, subject to a magnetic field over a long but finite interval, as a function of field and energy. The fractal structure shows up in the field and energy dependence of the magnetoconductance. The scale of this structure, given by one flux quantum per plaquette, ${\mathit{a}}^2$, is easily accessible with laboratory fields in the case of fabricated superlattices, where a is in the range of a≊1 μm. The three-dimensional butterfly resulting from plotting the conductance as a function of field and energy summarizes a number of well known facts about magnetotransport. Additional features are due to band gaps in the edge state spectrum. The lateral current distribution of the edge states, and downstream from the field region, is calculated. Reflections at the field–no-field boundaries cause Aharonov-Bohm ripples on the conductance plateaus. The amplitudes of these ripples depend in a nontrivial manner on how the magnetic field is switched on and off.