Transport properties of the two-dimensional electron gas (2DEG) are considered in the presence of a perpendicular magnetic field B and of a weak two-dimensional (2D) periodic potential modulation in the 2DEG plane. The symmetry of the latter is rectangular or hexagonal. The well-known solution of the corresponding tight-binding equation shows that each Landau level splits into several subbands when a rational number of flux quanta $h/e$ pierces the unit cell and that the corresponding gaps are exponentially small. Assuming the latter are closed due to disorder gives analytical wave functions and simplifies considerably the evaluation of the magnetoresistivity tensor ${\rho }_{\mu \nu }.$ The relative phase of the oscillations in ${\rho }_{\mathrm{xx}}$ and ${\rho }_{\mathrm{yy}}$ depends on the modulation periods involved. For a 2D modulation with a short period $<~100\mathrm{nm},$ in addition to the Weiss oscillations the collisional contribution to the conductivity and consequently the tensor ${\rho }_{\mu \nu }$ show prominent peaks when one flux quantum$h/e$ passes through an integral number of unit cells in good agreement with recent experiments. For periods 300–400 nm long used in early experiments, these peaks occur at fields 10 to 25 times smaller than those of the Weiss oscillations and are not resolved.

• Received 30 July 2003

DOI:https://doi.org/10.1103/PhysRevB.69.035331