Imaging Quantum Hall Edge States by Microwave Impedance Microscopy

Microwave impedance microscopy (MIM) has been widely used for the imaging of topological edge states. To date, however, the apparent widths of quantum Hall edge channels measured by MIM are typically on the order of 1 μm, which was attributed to the excitation of edge magnetoplasma (EMP) modes in a recent study. In this work, we quantitatively evaluate the effect of EMP on MIM imaging by finite-element simulations that incorporate the conductivity tensor and realistic sample-tip electrostatics. The results indicate that the micron-scale edge widths in the published MIM data are mostly due to the lack of a nearby grounding plane or the presence of finite bulk conductivity. For back-gated 2D electron gas with a thin gate dielectric, the edge width is essentially determined by the tip diameter rather than the EMP resonance for quantum Hall states with vanishing bulk conductivity. The simulation is further corroborated by MIM measurements on a gated graphene device at filling factor ν = 2, where the measured edge width is ~ 100 nm and independent of the microwave frequency.