Abstract
We measure the conductance of a quantum point contact (QPC) while the biased tip of a scanning probe microscope induces a depleted region in the electron gas underneath. At finite magnetic field we find plateaus in the real-space maps of the conductance as a function of tip position at integer (\nu=1,2,3,4,6,8) and fractional (\nu=1/3,2/3,5/3,4/5) values of transmission. They resemble theoretically predicted compressible and incompressible stripes of quantum Hall edge states. The scanning tip allows us to shift the constriction limiting the conductance in real space over distances of many microns. The resulting stripes of integer and fractional filling factors are rugged on the micron scale, i.e. on a scale much smaller than the zero-field elastic mean free path of the electrons. Our experiments demonstrate that microscopic inhomogeneities are relevant even in high-quality samples and lead to locally strongly fluctuating widths of incompressible regions even down to their complete suppression for certain tip positions. The macroscopic quantization of the Hall resistance measured experimentally in a non-local contact configuration survives in the presence of these inhomogeneities, and the relevant local energy scale for the \nu=2 state turns out to be independent of tip position.
Highlights
Two-dimensional electron gases at high magnetic fields applied normal to the electron-gas plane exhibit the quantum Hall effect
The integer [1] and fractional [2] quantum Hall effects are two different macroscopic quantum phenomena, which lead to surprisingly similar observations in electron transport experiments
In the integer quantum Hall effect, each occupied bulk Landau level gives rise to a pair of counterpropagating channels at opposite sample edges [3,4]
Summary
Two-dimensional electron gases at high magnetic fields applied normal to the electron-gas plane exhibit the quantum Hall effect. Scanning gate microscopy (SGM) experiments on quantum point contacts in the integer quantum Hall regime have complemented the conventional transport experiments by inducing a local potential perturbation in the electron gas near the quantum point contact with the scanning tip [31,32]. This arrangement leads to tip-controlled spatially resolved selective backscattering of integer edge channels. We report scanning gate experiments at an electron temperature of 170 mK that explore the formation of integer and fractional quantum Hall edge channels in a constriction under the influence of a scanning tip. We find an unprecedented rich structure on the local scale, which we interpret as an interplay between the interaction-driven local formation of correlated states leading to edge reconstruction and small residual potential variations occurring in our high-mobility sample on typical length scales of a few hundred nanometers, far below the zerofield elastic mean free path of electrons
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