Electronic states of wires and slabs of topological insulators : Quantum Hall effects and edge transport

We develop a simple model of surface states for topological insulators, developing matching relations for states on surfaces of different orientations. The model allows one to write simple Dirac Hamiltonians for each surface, and to determine how perturbations that couple to electron spin impact them. We then study two specific realizations of such systems: quantum wires of rectangular cross-section and a rectangular slab in a magnetic field. In the former case, we find a gap at zero energy due to the finite size of the system. This can be removed by application of exchange fields on the top and bottom surfaces, which lead to gapless chiral states appearing on the lateral surfaces. In the presence of a magnetic field, we examine how Landau level states on surfaces perpendicular to the field join onto confined states of the lateral surfaces. We show that an imbalance in the number of states propagating in each direction on the lateral surface is sufficient to stabilize a quantized Hall effect if there are processes that equilibrate the distribution of current among these channels.