LOCALIZATION AND QUANTUM HALL-EFFECT IN A 2-DIMENSIONAL PERIODIC POTENTIAL

A numerical study has been made of two-dimensional electrons in the lowest Landau level in the presence of a periodic potential and a random potential. The Hall conductance, density of states and density of current-carrying states are calculated when disorder is varied. The results show that the states in a subband carrying quantized Hall conductance t (t > 1 for example, and similarly t < -1) are split into two carrying unit conductance and t - 1, respectively, as disorder is introduced. The states carrying unit Hall current will merge and annihilate with that carrying -1 in the neighbouring subband separated by a small gap when disorder is increased and the gap vanishes. Based on this, a diagram of current-carrying states against disorder has been proposed. There is also a calculation on the localization length, and it is found that extended states appear at singular energies under certain conditions. Futhermore, it has been shown that a plateau in the Hall conductance, rather than a rapid oscillation, can be measured as a function of magnetic field for one-third-filled Landau bands. Similar results have been obtained for tight-binding electrons in a magnetic field with random site energies.