The energy spectra of spin-$\frac{1}{2}$ electrons under two-dimensional magnetic field modulations are calculated beyond the one-band approximation. Our formulation is generally applicable to a modulation field with a rectangular lattice symmetry. The field distribution within a plaquette is otherwise arbitrary. The spectra being obtained are qualitatively different from their electric-modulated counterparts. Peculiar features of the spectra are that, for an electron with a $g$ factor precisely equal to 2, no matter how strong the modulation is, the zero-energy level seems to be unaffected by the modulation and is separated from higher energy levels with a nonzero energy gap. Moreover, there is a twofold degenerancy for all states with positive energies with respect to spin flip. These features agree with earlier analytical studies of the periodically magnetic-modulated systems.

• Received 17 November 1997

DOI:https://doi.org/10.1103/PhysRevB.57.13002