Using an atomic force microscope as a local voltmeter, we measure the Hall voltage profile in a 2D electron gas in the quantum Hall (QH) regime. We observe a linear profile in the bulk of the sample in the transition regions between QH plateaus and a distinctly nonlinear profile on the plateaus. In addition, localized voltage drops are observed at the sample edges in the transition regions. We interpret these results in terms of theories of edge and bulk currents in the QH regime.PACS numbers: 73.40. Hm, 61.16.Ch, Since the discovery of the quantum Hall (QH) effect [1], the electrical characteristics of quantum Hall conductors (QHCs) have been intensely studied. The universal nature of the quantization in QHCs leads to a resistance that is independent of the microscopic properties of the sample. As a consequence, local properties of a QHC such as the current and voltage distributions within the sample are inaccessible with standard transport measurements. These local properties remain controversial, with some theories suggesting that the current flow and associated voltage drops are concentrated at the edges of the sample [2-5], while others predict a distribution extending throughout the bulk of the sample [6][7][8] The results, while informative, have lacked sufficient spatial and/or energy resolution to determine unambiguously how the current is partitioned between edge and bulk channels and the shape of the associated voltage profile across the QHC.To address these issues, we have used a scanned potential microscope to study the potential distribution in a QHC with submillivolt voltage and submicron spatial resolution. We find that at low magnetic fields the Hall voltage drop is linear, indicating uniform current flow throughout the sample. At high magnetic fields, large voltage drops are seen at the sample edges in the transition regions between QH plateaus, indicating that the current is concentrated near the edges. On the QH plateaus, the current is distributed throughout the bulk in a complex, non-uniform way. These results are in good agreement with existing results for edge and bulk transport in the QH regime.The samples consist of 10-20 µm wide Hall bars defined by wet etching of a GaAs/AlGaAs heterostructure with a 2D electron gas lying 77 nm beneath the surface. Results will be presented on two samples with mobilities of 8 and 70 m 2 /V-s and densities of 2.8 x 10 15 and 2.6 x 10 15 m -2 . All measurements were performed at temperatures between 0.7 and 1.0 K. The samples were also characterized by standard transport measurements.We measure the local voltage with a lowtemperature atomic force microscope (AFM) operating in non-contact mode [18]. As shown schematically in Fig. 1, an AC potential V 0 is applied to the contacts of the sample, producing inside the sample the AC potential V(x,y) whose spatial distribution is to be measured. The local sample potential V(x,y) interacts electrostatically with the sharp, metallized AFM tip, deflecting the AFM cantilever with a force that can be simply modelled...

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