Effect of magnetic modulation on Bloch electrons on a two-dimensional square lattice

The energy spectrum of Bloch electrons confined to a thin layer was calculated by Hofstadter in a uniform perpendicular magnetic field for which the flux ratio through a unit cell was a rational fraction p/q. The well-known Harper equation in the tight-binding, single-band approximation was obtained by using the Peierls substitution. Recently, studies on the effects of magnetic modulation on free electrons have attracted much attention. Both transport and optical properties under magnetic modulation have been studied experimentally and theoretically. In this paper, a formula for calculating the effect of magnetic modulation on Bloch electrons is derived. The modified Harper equation is obtained for arbitrary wave numbers by using a q×q matrix approach. The symmetry breakings in the flux ratio and between the positive and negative energy, as well as the change in the energy dispersion, are found as the magnetic modulation is introduced. Also, the symmetry between the positive and negative wave number in the y direction is broken by the one-dimensional magnetic modulation.