Electrons in 2D periodic magnetic and electric fields: Avoided band crossings, metal-insulator transitions, and changes in the spreading direction

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We study the energy spectrum and wave packet dynamics of electrons in a two-dimensional periodic potential subject to a perpendicular spatially modulated magnetic field. We observe avoided band crossings, metal-insulator transitions, and changes in the spreading direction of wave packets in two new situations: (i) for a purely magnetic modulation and (ii) in the limit of weak electric and magnetic modulations, where the Landau band coupling can be neglected. In the second case the corresponding classical system is no longer chaotic but integrable, whereby these phenomena are not expected. We argue that they are due to tunneling.