ENERGY-SPECTRUM OF ELECTRONS ON 3-DIMENSIONAL LATTICE IN EXTERNAL MAGNETIC-FIELD

Energy spectrum of electrons on three-dimensional lattice (two-dimensional lattices coupled with interlayer hoppings) in the magnetic field along the (1, 1, 1) direction is investigated. The two-dimensional Hofstadter's gaps become small as the electron hoppings between layers are increased. It is found that the critical value of the interlayer hoppings at which the largest gap is zero is close to the value of hoppings in the layer except for the small q, where q is the denominator of the flux through each unit square, phi = p/q. This means that a finite gap is opened by a small anisotropy in the simple cubic lattice. The effects of the next-nearest-neighbor and third-nearest-neighbor (diagonal) hoppings are also investigated.