Imaging of Coulomb-Driven Quantum Hall Edge States

The edges of a two-dimensional electron gas (2DEG) in the quantum Hall effect (QHE) regime are divided into alternating metallic and insulating strips, with their widths determined by the energy gaps of the QHE states and the electrostatic Coulomb interaction. Local probing of these submicrometer features, however, is challenging due to the buried 2DEG structures. Using a newly developed microwave impedance microscope, we demonstrate the real-space conductivity mapping of the edge and bulk states. The sizes, positions, and field dependence of the edge strips around the sample perimeter agree quantitatively with the self-consistent electrostatic picture. The evolution of microwave images as a function of magnetic fields provides rich microscopic information around the $\nu =2$ QHE state.

Figure 1

(a) Schematic setup of the microwave microscope and the 3D rendered image of the sample surface. The reflected 1 GHz microwave from the cantilever tip is amplified and demodulated to form imaginary (MIM-Im) and real (MIM-Re) parts of the impedance maps. (b) A line profile of the surface topography through three dots. The 2DEG located 30 nm below the surface is indicated in the plot.

Figure 2

(a) Density profile and (b) energy diagram near the sample edge at the bulk filling factor ${\nu }_b=2.31$. The etched area (I), depletion region (II), metallic (III) and insulating (IV) strips, and the bulk (V) are labeled in the plot. The circles in the energy diagram (filled, half filled, and empty) show the level occupancy. (c) MIM-Im and (d) MIM-Re images at $B=5.4\mathrm{T}$ and $T=2.3\mathrm{K}$. The full color scale corresponds to 0.2 V in MIM-Im and 0.03 V in MIM-Re. The scale bars are $1\mu \mathrm{m}$. (e) Line cuts of the microwave data, labeled in (c). The vertical scales are 40 mV for the MIM-Im (solid line) and 4 mV for the MIM-Re data (dashed line). Rising and falling edges are indicated by arrows. (f) Results of the finite-element modeling, including the 2D axisymmetric analysis (thick solid and dashed lines) for the metallic edge and the bulk and the full 3D simulation (thin solid and dashed lines) for the insulating strip. (g) Idealized conductivity map combining the MIM images and the simulation.

Figure 3

(a) Longitudinal and Hall resistivity as a function of $B$ or $\nu$ at 2 K. The corresponding $B$ fields in (b)–(l) are labeled in the ${\rho }_{xx}$ trace. (b)–(l) Counterclockwise from top left to top right, MIM images at $T=2.3\mathrm{K}$ as the $B$ field increases from 4.8 T (${\nu }_b=2.6$) to 7.3 T (${\nu }_b=1.7$). All scale bars are $1\mu \mathrm{m}$. The full color scales (not shown) are the same as Figs. 2c, 2d. (m) From left to right, schematic density profiles across the center of the dots at ${\nu }_b=2.2$, 2.0, 1.8, and 1.7, respectively. The shaded areas are sketches of the localized band.

Figure 4

(a) Widths of the measured conducting edges (squares) and insulating strips (triangles). The solid and dashed lines are results from the electrostatic calculation [3]. The semiclassical cyclotron diameter with much smaller values and incorrect trend is also plotted for comparison. (b) Comparison between the macroscopic dc transport (solid line) and microscopic microwave (solid circles) conductivity. The ${\sigma }_{xx}(1\mathrm{GHz})$ data between the two dashed lines show large uncertainties due to strong nonuniformity observed in the bulk of the 2DEG.