Electrical imaging of the quantum Hall state

Microscopic images of the local electron compressibility, electrostatic potential, and current-induced Hall voltage of a two-dimensional sheet of electrons in the quantum Hall regime are acquired using a single-electron transistor as a scanned probe. Regions identified as differing in quantum state (Landau level) occupancy of the electrons appear in interrelated ways in all of these properties. The compressibility images show quasi-insulating `incompressible' strips, associated with edge states that bound these regions. These strips follow the contours of a constant electron density that match exact occupancy of Landau levels, and shift accordingly with magnetic field and electron density changes. The potential distribution has an overall correlation with the density contours marked by the incompressible strips and a step across the strips that provides a direct measure of the energy gap between Landau levels. The Hall voltage and the inferred current flow patterns are also guided by these strips.