Visualization of Quantum Hall Bulk and Edge States in Monolayer Graphene by Microwave Impedance Microscopy

The exact quantization of Hall conductance in a two-dimensional (2D) electron gas under high magnetic fields is closely related to the edge structure of the system. Microscopic probing of edge and bulk states in the quantum Hall regime can yield critical information that is not accessible by conventional transport measurements. Using a dilution-fridge-based microwave impedance microscope (MIM), we explored the evolution of local conductivity distribution in a gated monolayer graphene sample around several integer filling factors. Under ultralow temperatures and high magnetic fields, the apparent widths of the conductive edge states exhibit a strong dependence on the size of the energy gap. The transition from bulk-insulating to bulk-conductive states is vividly seen in the MIM data. Moreover, the zero-energy state develops edge-like structures at low magnetic fields and becomes completely insulating at high fields, consistent with the canted antiferromagnetic ground state. Our findings provide a comprehensive microscopic picture of the edge and bulk states in graphene as the Fermi level moves across its quantized energy spectrum.