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Chern mosaic and Berry-curvature magnetism in magic-angle graphene

Abstract

Charge carriers in magic-angle graphene come in eight flavours described by a combination of their spin, valley and sublattice polarizations. When inversion and time-reversal symmetries are broken, this ‘flavour’ degeneracy can be lifted, and their corresponding bands can be sequentially filled. Due to their non-trivial band topology and Berry curvature, each band is classified by a topological Chern number C, leading to quantum anomalous Hall and Chern insulator states. Using a scanning superconducting quantum interference device on a tip, we image the nanoscale equilibrium orbital magnetism induced by the Berry curvature, the polarity of which is governed by C, and detect its two constituent components associated with the drift and self-rotation of the electronic wavepackets. At integer filling v = 1, we observe a zero-field Chern insulator, which—rather than being described by a global topologically invariant C—forms a mosaic of microscopic patches of C = −1, 0 or 1. On further filling, we find a first-order phase transition due to the recondensation of electrons from valley K to K′, leading to irreversible flips of the local Chern number and magnetization, as well as to the formation of valley domain walls, giving rise to hysteretic anomalous Hall resistance.

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Fig. 1: Transport characteristics of MATBG.
Fig. 2: Imaging the local magnetic response.
Fig. 3: Imaging local magnetization and equilibrium currents.
Fig. 4: Evolution of orbital magnetism.
Fig. 5: Chern mosaic.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The band structure calculations and magnetization reconstruction codes used in this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We thank M. E. Huber for the SOT readout system. This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 785971) and by the Israel Science Foundation ISF (grant nos. 921/18 and 994/19). A.S. and E.B. acknowledge support from the Israel Science Foundation’s Quantum Science and Technology grant no. 2074/19 and from CRC 183 of the Deutsche Forschungsgemeinschaft (Project C02). E.B. was supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant no. 817799). A.S. acknowledges support from the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 788715 (Project LEGOTOP)). E.Z. acknowledges support of ISF-NRF Singapore grant no. 3518/20, the Andre Deloro Prize for Scientific Research, and the Leona M. and Harry B. Helmsley Charitable Trust (grant no. 2112-04911). D.K.E. acknowledges support from the Ministry of Economy and Competitiveness of Spain through the ‘Severo Ochoa’ program for Centres of Excellence in R&D (SE5-0522), Fundació Privada Cellex, Fundació Privada Mir-Puig, the Generalitat de Catalunya through the CERCA program and funding from the ERC under the European Union’s Horizon 2020 research and innovation programme (grant no. 852927). B.Y. acknowledges financial support by the ERC (Consolidator Grant ‘NonlinearTopo’ no. 815869) and the ISF—Personal Research Grant (no. 2932/21). P.S. acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 754510. G.D.B. acknowledges support from the ‘Presidencia de la Agencia Estatal de Investigación’ (ref. PRE2019-088487). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan (grant no. JPMXP0112101001), and JSPS KAKENHI (grant nos. 19H05790, 20H00354 and 21H05233).

Author information

Authors and Affiliations

Authors

Contributions

S.G., M.B. and E.Z. designed the experiment. S.G. and M.B. performed the measurements. M.B., A.U. and S.G. performed the analysis. P.S., G.D.B. and D.K.E. designed and fabricated the sample and contributed to the analysis of the results. I.R. fabricated the SOTs, and Y.M. fabricated the tuning forks. J.X. and B.Y. performed the band structure calculations. A.Y.M. developed the magnetization reconstruction code. E.B, A.S. and K.P. contributed to the analysis and theoretical modelling. K.W. and T.T. provided the hBN crystals. M.B., A.U., S.G. and E.Z. wrote the manuscript. All the authors participated in the discussions and in writing of the manuscript.

Corresponding author

Correspondence to Eli Zeldov.

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Extended data

Extended Data Fig. 1 Landau fan diagram.

a, Longitudinal resistance Rxx at T= 300 mK reproduced from Fig. 1a. b, Landau level fits. Integer fillings ν of the moiré supercell are indicated by vertical grey lines and the Landau levels and Chern insulating states are indicated by diagonal lines along with their corresponding Chern numbers. Dotted lines are extrapolations.

Extended Data Fig. 2 Orbital magnetization calculations.

a, The evolution of the integrated magnetization Mz (dashed) and its two components MSR and MC vs. the filling factor v in the compressible region and vs. μ in the gap region (cyan) in the KA flat band for twist angle θ = 1.08°, staggered potential δ = 17 meV, and tunnelling ratio w = 0.8. b, The corresponding evolution of the differential magnetization mz and its mSR and mC components. c, The self-rotation magnetization \(m_{SR}({{{\boldsymbol{k}}}})\) in the first mini-Brillouin zone in the KA flat band. d, The calculated peak value of the differential magnetization \(m_z^{peak}\) near the top of the band as a function of δ and w for θ = 1.08°.

Extended Data Fig. 3 Evolution of mz through recondensation phase transition.

a, Numerically calculated mz in the vicinity of v = 1, similar to Fig. 4d, but including the first-order recondensation transition marked by the red dot, upon which the mz flips its sign discontinuously. The value of the Chern gap is taken to be 3.5 meV and \(m_z = dM_z/dn\) is calculated taking into account the finite vac = 0.083 modulation. b, The measured mz along the vertical yellow dashed line in Fig. 4g. The black dots are the experimental data points and the solid curve is guide to the eye. The first sign change of mz at \(\nu \cong\) 0.9 marks the continuous crossover from mSR to mC dominated magnetization. The second sign change at \(\nu \cong\) 1 is discontinuous (and hysteretic) and reflects the first-order K to K’ recondensation transition.

Extended Data Fig. 4 Moiré commensurability conditions.

Calculated pairs of graphene-graphene θGG and graphene-hBN θGBN twist angles that result in commensurability of the two moiré lattices following Ref. 20. Each solution is labelled by its (n, p, q) integer triplet. Shown are only solutions for n = 1,2 as the super moiré cell scales like n2, but numerous additional solutions exist for larger values of n with proportionally larger super moiré cells. The blue error bars show deviations from the exact commensurate angles where a Chern mosaic structure is still expected to form.

Extended Data Fig. 5 Local twist angle.

a, \(B_z^{ac}(x)\) measured along the dotted line in Fig. 2b vs. the global filling factor v in the vicinity of v = 4 (top) and v = −4 (bottom). The black lines mark \(\nu _s^ + (x)\) and \(\nu _s^ - \left( x \right)\) corresponding the local electron and hole dispersive band edges, \(n_s^ + (x)\) and \(n_s^ - \left( x \right)\). b, The derived local twist angle\(\theta \left( x \right) = \sqrt {\frac{{\sqrt 3 }}{8}a^2\frac{{n_s^ + - n_s^ - }}{2}}\).

Extended Data Fig. 6 Comparison of magnetization and transport hysteresis.

a, Percentage of sample area that has locally hysteretic mz represented as a histogram as a function of filling factor v. b, The hysteresis in transport, \({{\Delta }}R_{yx}\left( \nu \right) = R_{yx}\left( {\nu _ \uparrow } \right) - R_{yx}(\nu _ \uparrow )\), at Ba = 47 mT from data presented in Fig. 1d.

Extended Data Fig. 7 Optical and AFM images of the device.

a–d, Optical images of graphene (a), top hBN (b), bottom hBN (c) and the final stack (d). The white (graphene), yellow (top hBN), and red (bottom hBN) dashed lines show that the top (bottom) hBN layer is at an angle of 68° (6°) with respect to the graphene. e, Zoomed-in AFM image of the device. The two large dark spots are due to damage caused to the device after the measurements presented in this paper.

Extended Data Fig. 8 Sublattice polarization.

a–b, Single particle band structure of the K valley with colour indicating the degree of sublattice polarization P in the bottom (a) and top (b) graphene layers. Red (blue) colour shows polarization of A (B) sublattice with P = 0.2 corresponding to 60% occupation weight on A and 40% on B sublattices. The sign of P determines the sign of C. Staggered potential δ = 17 meV is applied to the bottom graphene, θ = 1.08° and w = 0.95.

Supplementary information

Supplementary Video 1

Minor hysteresis loops of Ryx around v = 1 at Ba = 46 mT. Top: the system is initialized at filling factor νlow = 0.55 and Ryx is measured and sweeping v up to νmax (magenta curves) and decreasing back to νlow (cyan curves). The procedure is then repeated on gradually incrementing νmax up to νhigh = 1.3. Bottom: minor loops performed in the opposite direction. The system is initialized at νhigh = 1.3 and v is decreased down to νmin (cyan curves) and then increased back to νhigh (magenta curves), and the procedure is repeated on decrementing νmin down to νlow. When the system is initialized at νlow, reversible Ryx is observed on sweeping the filling factor up to νmax 1, above which a progressively increasing hysteresis develops with sharp jumps. A similar behaviour is observed on initializing the system at νhigh, with reversible Ryx down to νmin 0.95 and appearance of increasing hysteresis on lowering νmin. The global Ryx dynamics is consistent with the observation of the flipping of magnetization domains in the local measurements (their hysteresis is shown in Fig. 5e,f).

Supplementary Video 2

Evolution of \({{{{B}}}}_{{{{z}}}}^{{{{{\mathrm{a.c.}}}}}}\) with filling factor. Top: the measured local a.c. magnetic field \(B_z^{\mathrm{a.c.}}(x,y,\nu _ \uparrow )\) induced by small a.c. modulation of the filling factor νa.c. = 0.083 on sweeping the d.c. filling factor from ν = 0.737 to 1.174. The system is initialized at ν = 0 before the measurement. Middle: same as the top panel after initializing the system at ν = 2 and measuring \(B_z^{\mathrm{a.c.}}\left( {x,y,\nu _ \downarrow } \right)\) on sweeping v down (from 1.174 to 0.737). Bottom: numerical difference, \({{\Delta }}B_z^{\mathrm{a.c.}}\left( {x,y,\nu } \right)\), of the sweep-up and sweep-down data revealing the local hysteresis in \(B_z^{\mathrm{a.c.}}\). A complex pattern of positive and negative \(B_z^{\mathrm{a.c.}}\) is shown, where different areas of the sample develop hysteresis at different filling factors, with some areas showing completely reversible behaviour in the full range of v. The left side of the sample shows local hysteresis over a larger range of v.

Supplementary Video 3

Evolution of magnetization and equilibrium currents with filling factor. Numerical reconstruction of the magnitude of the equilibrium current Ja.c.(x, y, v) (top) and local magnetization mz(x, y, v) = dMz(x, y, v)/dn (bottom) obtained by the inversion of \(B_z^{\mathrm{a.c.}}(x,y,\nu _ \uparrow )\) for v = 0.737–1.174. In two dimensions, magnetization M (magnetic dipole moment per unit area) is given in units of current (A), which—for the case of a uniformly magnetized domain—describes the equilibrium current that circulates along the edges of the domain. Thus, mz(x, y) and Ja.c.(x, y) present equivalent descriptions of the change in magnetization and in equilibrium currents, respectively, due to a change in filling factor. At low v, isolated patches of positive mz(x, y, v) are present along with areas of negative response, but at higher v, the system develops into an intricate pattern of positive and negative patches and an equivalently intricate current network.

Supplementary Video 4

Tomographic rendering of magnetization. Evolution of mz(x, y, v), mz(x, y, v) and Δmz(x, y, v) as a function of the spatial axes x and y and filling factor v. Different two-dimensional slices in the xv, yv and xy planes are sequentially presented. The bottom panel reveals a wider hysteresis in v in the left part of the sample and very narrow or no hysteresis in the right part.

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Grover, S., Bocarsly, M., Uri, A. et al. Chern mosaic and Berry-curvature magnetism in magic-angle graphene. Nat. Phys. 18, 885–892 (2022). https://doi.org/10.1038/s41567-022-01635-7

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