Quantum transport of a two-dimensional electron gas in a spatially modulated magnetic field

Electrical transport properties of a two-dimensional electron gas are studied in the presence of a perpendicular magnetic field B modulated weakly and periodically along one direction, B=(B+${\mathit{B}}_0$cosKx)z^, with ${\mathit{B}}_0$≪B, K=2π/a, and a being the period of the modulation. ${\mathit{B}}_0$ is taken constant or proportional to B. The Landau levels broaden into bands and their width, proportional to the modulation strength ${\mathit{B}}_0$ oscillates with B and gives rise to oscillations in the magnetoresistance at low B. These oscillations reflect the commensurability between the cyclotron diameter at the Fermi level and the period a and consequently they are distinctly different from the Shubnikov–de Haas ones, at higher B, in period and temperature dependence. The bandwidth at the Fermi energy can be one order of magnitude larger, at low B, than that of the electric case for equal modulation strengths. The resulting magnetoresistance oscillations have a much higher amplitude than those of the electric case with which they are out of phase. Explicit asymptotic expressions are derived for the temperature dependence of the transport coefficients. The case when both electric and magnetic modulations are present is also considered. The position of the resulting oscillations depends on the ratio δ between the two modulation strengths. When the modulations are out of phase there is no shift in the position of the oscillations when δ varies and for a particular value of δ the oscillations are suppressed.