Probing microscopic structures of quantum Hall incompressible phases: Filling-factor-dependent evolution contributing to quantum Hall effects

The integer quantum Hall (QH) effects characterized by topologically quantized and nondissipative transport are caused by an electrically insulating incompressible phase that prevents backscattering between chiral metallic channels. Using a scanning gate technique incorporating nonequilibrium transport, we mapped the local incompressible region near an integer $\nu$ within a Hall bar. The observed pattern revealed the filling-factor ($\nu$)-dependent evolution of the microscopic incompressible structures located along the edge and in the bulk region, respectively, attributed to incompressible edge strip and bulk localization, which makes a significant contribution to the QH effects observed in the global transport measurements. Further, we found that the $\nu$ dependency of the incompressible patterns is, in turn, destroyed by a large imposed current during the deep QH effect breakdown. These results demonstrate the ability of our method to image the microscopic transport properties of a topological two-dimensional system.