Compressibility images of the quantum Hall state

We have imaged the electronic compressibility of a two-dimensional electron gas in the integer quantum Hall regime. The compressibility images show quasi-insulating `incompressible' strips that separate region of electron liquid of near integer filling. The strips follow the contours of constant electron density that match exact occupancy of the Landau levels, and shift accordingly with magnetic field and electron density changes. The surface electrostatic potential distribution has an overall correlation with the compressibility patterns and the density contours marked by them and an abrupt step, at the boundaries, that provides a direct measure of the energy gap between the Landau levels.