Edge states, band structure, and the Hall effect in two-dimensional lattice structures: quantum dot arrays and the tight-binding model

Using a model that is based on a transfer matrix formalism, we study the electronic structure and transport in two dimensional periodic arrays of quantum dots in magnetic fields. The quantum dots in our model are connected to each other via ballistic constrictions. The spectrum for this system has much in common with that with the tight-binding model. In particular, q bulk bands arise if the normalized magnetic flux per unit cell is p/q, where p and q are coprime integers. Working within an edge-state picture, we investigate if these similarities translate to a correspondence in the transport properties of the two systems. As we shall show, the answer to this question depends very much on the transmission probability of the constrictions. Our analysis also shows that, under certain circumstances, the Hall conductance within the context of the tight-binding model may take on fractional values.