Edge channels of broken-symmetry quantum Hall states in graphene visualized by atomic force microscopy

The quantum Hall (QH) effect, a topologically non-trivial quantum phase, expanded the concept of topological order in physics bringing into focus the intimate relation between the “bulk” topology and the edge states. The QH effect in graphene is distinguished by its four-fold degenerate zero energy Landau level (zLL), where the symmetry is broken by electron interactions on top of lattice-scale potentials. However, the broken-symmetry edge states have eluded spatial measurements. In this article, we spatially map the quantum Hall broken-symmetry edge states comprising the graphene zLL at integer filling factors of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\nu }}={{0}},\pm {{1}}$$\end{document}ν=0,±1 across the quantum Hall edge boundary using high-resolution atomic force microscopy (AFM) and show a gapped ground state proceeding from the bulk through to the QH edge boundary. Measurements of the chemical potential resolve the energies of the four-fold degenerate zLL as a function of magnetic field and show the interplay of the moiré superlattice potential of the graphene/boron nitride system and spin/valley symmetry-breaking effects in large magnetic fields.


Supplementary fig. 1 | SPM navigation to central device area. a,
Optical image of the probe tip and its reflection as it is aligned onto the graphene device. The fan-shaped runway is artificially highlighted in yellow. b, Optical image of the fan-shaped runway and device area (on the left). The red arrow indicates the navigation direction and pink dashed line shows the navigating pathway. c, Zoomed-in optical image of a part of the navigation path. An AFM image and AFM "W"-shaped scans along the navigation path are superimposed. As shown, the automated algorithm repeats "W"-shaped AFM linescans to sense the ridge edge and re-calculate the next X and Y piezo-motor steps to adjust the direction. Along the navigation path, several full-frame scans are taken as shown to reveal markers to ensure accurate alignment of the navigation system to the optical image. d, Optical image of the contact edge of the runway displaying the final navigation to the graphene device area. The AFM image showing the contact-graphene area is superimposed on the optical image. AFM topographs in (c) and (d) are measured at 2 nm oscillation amplitude and set frequency shift of −30 mHz.

Supplementary fig. 2 | AFM frequency shift measurement of broken-symmetry states. a,
Raw frequency shift data as a function of sample bias and local back gate G1; the global back gate G2 is set to zero density. The frequency shift changes due to broken-symmetry states, seen as sharp spikes in the middle of the image, are superimposed on a pronounced background. The background is dominated by large steps corresponding to chemical potential changes at filling factors of = ±2 along the gate axis, and electrostatic forces along the sample bias axis. AFM data is measured at = 15 T, with a constant excitation of 520 mV. The blue and red dashed lines indicate the positions for the linescans in (b). b, Linescans from (a) along the gate axis at aa' and b-b', corresponding to sample biases of B = −1.0 V, and − 1.3 V, respectively. The solid back line is a smooth background calculated for each linescan using a Gaussian filter with = 0.2 V. c, Residuals obtained from subtracting the background curves shown in (b) from the raw data. d, Frequency shift map obtained from the background-corrected linescans as in (c) for each sample bias, to highlight the changes due to the broken-symmetry states. The white line reflects the gate-dependent zero-contact potential, as obtained following the procedure in ( Supplementary Fig. 3). The white numerals indicate the filling factor. The large steps are due to Landau levels at filling factors of = ±2, while smaller steps are seen inside the = 0 Landau level due to broken-symmetry states. Note that the chemical potential has a sign change along the sample bias axis as shown in Fig. 3 of the main text, where the KPFM data is obtained with lock-in measurements for improved signal to noise.

Measurements
The dispersion of the edge channels can also be mapped using the dissipation signal in the AFM measurement, which gives more contrast to the edge channel features. The dissipation signal is the excitation voltage needed to keep a constant oscillation amplitude of 2 nm of the AFM qPlus sensor. (Supplementary Fig. 4) (Supplementary Fig. 4) would correspond to the data along the horizontal line in Fig. 5b, but now scanned along the X-axis in the dissipation channel.
The dispersion of broken-symmetry state = −1 dispersing to below = 0 in Fig. 5a-b is seen as the dissipative channel shifting down with decreasing gate voltage in the first row of images (note the Y-axis is horizontal in Fig. 5). Similarly, the dispersion of the = +1 to above = 0 in Fig. 5 is seen as the dissipative channel shifting up with increasing gate voltage in the third row of images in (Supplementary Fig. 4). Overall, the XY-maps in (Supplementary Fig. 4) show the dispersion identical to Fig. 5a-b, progressing uniformly along the QH boundary edge. Since the dissipation measurements are operated off balance, a finite tip gating occurs and filamentary arcs are observed similar to reported in scanning capacitance measurements of AlGaAs/GaAs 2DEG 1 .
Supplementary fig. S4 | Spatial mapping of graphene broken-symmetry quantum Hall edge states using AFM dissipation measurements. Series of spatial AFM excitation channel maps (for constant sensor oscillation amplitude of 2 nm) across the QH edge boundary (indicated in the black circle in Fig. 2a) as a function of the local back gate potential G1. The gate potential outside the boundary was set to = 0, an electronic insulator. (first row) The incompressible = −1 edge state appears out of the boundary and drifts up with increasing gate voltage. (second row) The = 0 appears at the local gate voltage of G1 = 110 mV. (third row) The incompressible = +1 edge state appears from the top of the frame and drifts down with increasing gate voltage. AFM settings: = −600 mV, ∆ = −2 Hz.