Abstract
Moiré systems made from stacked two-dimensional materials host correlated and topological states that can be electrically controlled with applied gate voltages. One prevalent form of topological state that can occur are Chern insulators that display a quantum anomalous Hall effect. Here we manipulate Chern domains in an interaction-driven quantum anomalous Hall insulator made from twisted monolayer–bilayer graphene and observe chiral interface states at the boundary between different domains. By tuning the carrier concentration, we stabilize neighbouring domains of opposite Chern number that then provide topological interfaces devoid of any structural boundaries. This allows the wavefunction of chiral interface states to be directly imaged using a scanning tunnelling microscope. Our theoretical analysis confirms the chiral nature of observed interface states and allows us to determine the characteristic length scale of valley polarization reversal across neighbouring Chern domains.
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Data availability
Additional data are available from the corresponding authors upon request. Source data are provided with this paper.
Code availability
The computer codes that support the plots within this paper and the findings of this study are available from the corresponding authors upon request.
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Acknowledgements
The authors thank Y.-M. Lu for helpful discussion and S. Stolz for technical support. This research was primarily supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under contract no. DE-AC02-05CH11231 within the van der Waals Heterostructures programme KCWF16 (STM spectroscopy). Work at the Molecular Foundry (graphene characterization) was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under contract no. DE-AC02-05CH11231. Support was also provided by the National Science Foundation Award DMR-2221750 (device fabrication and testing). K.W. and T.T. acknowledge support from JSPS KAKENHI (grant nos. 20H00354, 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan (hBN crystal synthesis and characterization). C.Z. acknowledges support from a Kavli ENSI Philomathia Graduate Student Fellowship. T.S. acknowledges fellowship support from the Masason Foundation.
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C.Z., T.Z. and M.F.C. initiated and conceived the research. C.Z. and T.Z. carried out STM/STS measurements and analyses. M.F.C. supervised STM/STS measurements and analyses. S.K. prepared gate-tuneable devices. A.Z., F.W. and M.F.C. supervised device preparations. T.S. performed theoretical analysis and calculations. M.P.Z. supervised theoretical analysis and calculations. T.T. and K.W. provided the hBN crystals. C.Z., T.Z. and M.F.C. wrote the manuscript with the help of all authors. All authors contributed to the scientific discussion.
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Extended data
Extended Data Fig. 1 Basic STM/STS characterization of gate-tuneable tMBLG.
a, STM topographic image of a tMBLG device surface area (VBias = –200 mV, setpoint I0 = 2 nA). Analysis of the moiré wavelength in all three directions enables the extraction of a local twist angle of θ = 1.251 ± 0.001° and a local hetero-strain of 0.17 ± 0.01% (Methods). b, Gate-dependent dI/dV density plot over the full range –70 V ≤ VG ≤ 70 V (–4.3 ≤ ν ≤ 4.3) obtained for STM tip at the location marked in a (modulation voltage ΔVRMS = 1 mV; setpoint VBias = –100 mV, I0 = 2 nA). Charge gaps emerging at ν = 2, 3 signify the formation of correlated insulating states. All measurements taken at T = 4.7 K.
Extended Data Fig. 2 Topological behaviour of tMBLG QAH states in an out-of-plane B-field.
a–d, Gate-dependent dI/dV density plots near ν = 3 for a B = 0.0 T, b B = 0.4 T, c B = 0.8 T, and d B = 1.2 T (modulation voltage ΔVRMS = 1 mV; setpoint VBias = –75 mV, I0 = 0.2 nA). The white arrow indicates the energy gap for B = 0.0 T, and orange and green arrows indicate C = +2 and C = –2 gaps for finite B. e, dI/dV(VBias = 0) as a function of VG and B-field extracted from a–d (same data as in Fig. 1f but without dashed lines).
Extended Data Fig. 3 Local twist angle and hetero-strain of tMBLG areas with Chern domain interfaces.
a, STM topographic image of a tMBLG area with a Chern domain interface (VBias = –300 mV, setpoint I0 = 0.2 nA; same as Fig. 2a). Local moiré wavelengths in three different directions are shown for locations 1, 2, and 3 from which local twist angle and hetero-strain values are extracted (Methods). b, STM topographic image of another tMBLG area with a Chern domain interface after quantum dot creation (VBias = –1 V, setpoint I0 = 0.01 nA; same as Fig. 4a boxed region). A similar structural analysis is shown.
Extended Data Fig. 4 Experiment–theory comparison for topological phase transition and chiral interface states.
a, dI/dV map of a tMBLG area for B = 0.4 T, VG = 42.0 V, and VBias = 0 mV (same as Fig. 2e). b, dI/dV density plot for B = 0.4 T and VG = 42.0 V obtained along the white dashed line in a (same as Fig. 2f). c, dI/dV spatial line-cuts extracted from b at VBias = 13.5 mV (top), 0 mV (middle), and –16.5 mV (bottom) (corresponding to blue arrows in b). d, Theoretical LDOS map for E = 0 meV calculated for a tMBLG Chern domain wall width of ξ = 125 nm. e, Density plot of theoretical LDOS along the white dashed line in d (same as Fig. 2g). f, Valley-resolved theoretical LDOS line-cuts at E = 15 meV (top), 0 meV (middle), and –15 meV (bottom) (corresponding to blue arrows in e).
Extended Data Fig. 5 Spatially-defined topological phase transition and chiral interface states for a different area.
a, STM topographic image of a different tMBLG area (VBias = –300 mV, setpoint I0 = 0.2 nA). b, dI/dV map of the same area as a for B = 0.4 T, VG = 44.9 V and VBias = 0 mV. c, dI/dV density plot for B = 0.4 T and VG = 44.9 V obtained along the white dashed line in a. d, Gate-dependent dI/dV density plots for B = 0.0 T obtained at locations 1, 2, and 3 marked in a. White arrows indicate the ν = 3 energy gap. e, Same as d, but for B = 0.4 T. Orange and green arrows indicate C = +2 and C = –2 gaps. f, dI/dV spectra for B = 0.4 T and VG = 44.9 V at locations 1, 2, and 3 of a. Spectroscopy parameters: modulation voltage ΔVRMS = 1 mV; setpoint VBias = –60 mV, I0 = 0.5 nA for d, e; setpoint VBias = –300 mV, I0 = 0.2 nA and tip height offset ΔZ = –0.2 nm for b, c, f.
Extended Data Fig. 6 Fitting the spatial profile of chiral interface states.
a, Experimental dI/dV map showing 1D chiral interface states obtained in area #1 for B = 0.4 T, VG = 42.0 V and VBias = 0 mV (same as Fig. 2e). b, 2D fit to image in a with a Gaussian peak and a constant background (see Methods). c, Experimental dI/dV map showing 1D chiral interface states obtained in area #2 for B = 0.4 T, VG = 44.9 V and VBias = 0 mV (Same as Extended Data Fig. 5b). d, 2D fit to image in c with a Gaussian peak and a constant background (see Methods). e, Experimental dI/dV map showing 1D chiral interface states obtained in area #3 for B = 0.3 T, VG = 42.0 V and VBias = 0 mV. f, 2D fit to image in e with a Gaussian peak and a constant background (see Methods). g, Experimental dI/dV map showing 1D chiral interface states obtained in area #4 for B = 0.3 T, VG = 42.1 V and VBias = 0 mV. h, 2D fit to image in g with a Gaussian peak and a constant background (see Methods). i, Summary of 2D Gaussian fitting for datasets from areas #1–4. inc. = increasing gate voltage; dec. = decreasing gate voltage.
Extended Data Fig. 7 Reversible displacement of chiral interface states via back-gating.
a–f, dI/dV maps of the same area as Fig. 2a at B = 0.4 T and VBias = 0 mV for a VG = 42.5 V, b VG = 42.3 V, c VG = 42.1 V, d VG = 41.9 V, e VG = 41.7 V, and f VG = 41.5 V. Modulation voltage ΔVRMS = 1 mV; setpoint VBias = –300 mV, I0 = 0.2 nA; tip height offset ΔZ = –0.075 nm.
Extended Data Fig. 8 Local electron density profile of an n-type quantum dot.
a, STM topographic image of a tMBLG area (VBias = –1 V, setpoint I0 = 0.01 nA; same as Fig. 4a). b, Representative gate-dependent dI/dV density plots for B = 0.0 T obtained at different locations indicated on white dashed line of a. The ν = 3 gap (white arrows) always appears near VG = 44.3 V, indicating a low variation of local electron density over the entire area in a (Δν < 0.02). c, Representative gate-dependent dI/dV density plots for B = 0.0 T obtained at the same locations as in b after the formation of an n-type quantum dot. A significant electron density gradient occurs in the rightmost region (highlighted by the red dashed box in a) where a Chern domain interface hosting 1D chiral states can be realized (Fig. 4c). Spectroscopy parameters: modulation voltage ΔVRMS = 1 mV; setpoint VBias = –60 mV, I0 = 0.5 nA.
Extended Data Table. 9 Spatially-defined topological phase transition and 1D chiral states at quantum-dot-generated Chern domain interfaces.
a, Zoom-in STM topographic image of the boxed region shown in Fig. 4a (VBias = –1 V, setpoint I0 = 0.01 nA). b, dI/dV density plot for B = 0.4 T and VG = 43.0 V obtained along the white dashed line in a after formation of an n-type quantum dot (modulation voltage ΔVRMS = 1 mV; setpoint VBias = –300 mV, I0 = 0.2 nA; tip height offset ΔZ = –0.075 nm). Charge gaps in the left and right regions are attributed to the C = +2 and C = –2 insulating states. The absence of a gap in the middle signifies the presence of chiral interface states. c, dI/dV spatial line-cut at VBias = 0 mV extracted from b. d–f, Same as a–c, but after formation of a p-type quantum dot. Charge gaps in the left and right regions are attributed to the C = –2 and C = +2 insulating states. The absence of a gap in the middle signifies the presence of chiral interface states.
Extended Data Table. 10 1D band structure and chiral state image for a tMBLG Chern domain interface.
a, Energy eigenvalues as a function of momentum along the interface (k||) calculated for a Chern domain wall width of ξ = 10 nm (the minimum value allowed in our model). Only electronic states for spin up are shown (corresponding to the sub-bands in Fig. 2h that shift upward/downward). Two branches of chiral interface states (marked by red and blue) emerge for each valley that connect the occupied and unoccupied bulk bands (thus spanning the bulk gap) and disperse unidirectionally in momentum. The total number of chiral modes (four) is equal to the Chern number difference between neighbouring domains. b, Theoretical LDOS map for E = 0 meV showing a FWHM of w = 9 nm for the 1D chiral interface modes arising from a Chern domain wall width of ξ = 10 nm. c, Energy eigenvalues calculated for a larger ξ = 50 nm. Here more in-gap eigenstates appear at a given momentum, but the number of chiral branches crossing EF (dashed lines) remains the same (two per valley, marked by red and blue). This is because the group velocity of interface modes becomes smaller (see Supplementary Note 4 for details) so they now extend over more than one Brillouin zone in momentum. d, Theoretical LDOS map for E = 0 meV showing a FWHM of w = 17 nm for the 1D chiral interface modes arising from a Chern domain wall width of ξ = 50 nm.
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Zhang, C., Zhu, T., Kahn, S. et al. Manipulation of chiral interface states in a moiré quantum anomalous Hall insulator. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02444-w
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DOI: https://doi.org/10.1038/s41567-024-02444-w
