Computer simulations and edge-state analysis of the Hall effect in two-dimensional quantum-dot arrays connected to phase-randomizing reservoirs

Using numerical simulations based on a scattering matrix approach, we investigate the lateral quantum transport properties of a finite ballistic array of two-dimensional quantum dots. The internal state structure of such an array includes edge states. These edge states are shown to obey current amplitude orthogonality relations. If connected to ideal reservoirs, the edge states of the array are fully populated and extracted, yielding plateaus in the Hall resistance. In order to significantly populate more than just one of these edge states in an array, several of the dots must be open to one or more of the leads and the leads must be phase randomizing. Under appropriate conditions, the ideal case can be realized. The occupation of the edge states in the array can be determined with a decomposition technique that utilizes the orthogonality relations. In some cases, Hall plateaus can still be present even when edge states are only partially occupied and absorbed by the leads. In other cases, the partial population and extraction of edge states leads to unusual behavior in the Hall resistance. There can also be contributions to the Hall current from evanescent modes, as well as scattering between edge states.