Abstract
Moiré patterns formed by stacking atomically thin van der Waals crystals with a relative twist angle can give rise to notable new physical properties1,2. The study of moiré materials has so far been limited to structures comprising no more than a few van der Waals sheets, because a moiré pattern localized to a single two-dimensional interface is generally assumed to be incapable of appreciably modifying the properties of a bulk three-dimensional crystal. Here, we perform transport measurements of dual-gated devices constructed by slightly rotating a monolayer graphene sheet atop a thin bulk graphite crystal. We find that the moiré potential transforms the electronic properties of the entire bulk graphitic thin film. At zero and in small magnetic fields, transport is mediated by a combination of gate-tuneable moiré and graphite surface states, as well as coexisting semimetallic bulk states that do not respond to gating. At high field, the moiré potential hybridizes with the graphitic bulk states due to the unique properties of the two lowest Landau bands of graphite. These Landau bands facilitate the formation of a single quasi-two-dimensional hybrid structure in which the moiré and bulk graphite states are inextricably mixed. Our results establish twisted graphene–graphite as the first in a new class of mixed-dimensional moiré materials.
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Source data are provided with this paper. All other data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
We thank D. Cobden, V. Fal’ko, S. Slizovskiy and M. Rudner for valuable discussions. This work was supported by National Science Foundation (NSF) CAREER award no. DMR-2041972 and NSF MRSEC 1719797. M.Y. acknowledges support from the State of Washington-funded Clean Energy Institute. D.W. was supported by an appointment to the Intelligence Community Postdoctoral Research Fellowship Program at University of Washington administered by Oak Ridge Institute for Science and Education through an interagency agreement between the US Department of Energy and the Office of the Director of National Intelligence. E.T. and E.A.-M. were supported by grant no. NSF GRFP DGE-2140004. Electrical transport calculations were supported by the Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division, Pro-QM Energy Frontier Research Center (grant no. DE-SC0019443). M.F. was supported by JST CREST grant no. JPMJCR20T3 and by JSPS KAKENHI grant nos. JP21J10775 and JP23KJ0339. 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). This research acknowledges usage of the millikelvin optoelectronic quantum material laboratory supported by the M.J. Murdock Charitable Trust.
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D.W., E.T. and E.A.-M. fabricated the devices and performed the measurements. M.F. performed the band structure calculations. Y.R. performed the magnetotransport calculation. T.C. and D.X. supervised the calculations. K.W. and T.T. grew the BN crystals. D.W., E.T., E.A.-M. and M.Y. analysed the data and wrote the paper with input from all authors.
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Extended data figures and tables
Extended Data Fig. 1 Low-field evolution of transport in the t1+10 device.
Longitudinal (top) and Hall (bottom) resistance measurements acquired in steps of B = 50 mT, as indicated in the top left of each column. The zig-zag transport behaviour first becomes evident at fields as low as 50 mT, and becomes more obvious as the field is raised.
Extended Data Fig. 2 High-field transport behavior in a t1+17 device with θ = 1.31°.
a, Landau fan diagram acquired by sweeping Vm at Vgr = 0. The purple (pink) dashed lines denote selected QOs that project to ν = 0 (ν = ± 4). b, Landau fan diagrams acquired by sweeping Vgr at various fixed values of Vm. The black dashed lines denote selected QOs that project to Vgr ≈ 0 at B = 0. The blue lines denote selected QOs that project to a Vgr ≠ 0 that depends on Vm. c, Longitudinal (left) and Hall (right) resistance maps acquired at B = 0.5 T. Zero-field projections of the ν = 0 and ± 4 states from the Vm Landau fans are overlaid on the Rxx map. Zero-field projections of QOs from the Vgr Landau fans are overlaid on the Rxy map. d, Conductance, Gxx, as a function of magnetic field. The blue curve is averaged over all values of Vgr for the Landau fan in (a) acquired at Vm/dm = − 0.11 V/nm. The red curve is averaged over a range of Vm values corresponding to ∣ν∣ < 4 for the Landau fan in (b) acquired at Vgr = 0. Brown-Zak oscillations can be seen upon sweeping either gate. (Inset) Longitudinal resistance map acquired at B = 0 T.
Extended Data Fig. 3 High-field transport in Bernal graphite.
a, Landau fan diagrams from a 24-layer graphite device (different from the one shown in the main text) acquired by sweeping the top gate voltage, Vt, at various fixed values of the bottom gate voltage, Vb, as indicated in each panel. The black dashed lines denote selected QOs that project to Vt ≈ 0 at B = 0, and the pink lines denote QOs that project to a Vt ≠ 0 that depends on the value of Vb. b, Similar Landau fans, but with fixed Vt and sweeping Vb. The QOs projecting to Vb ≠ 0 are denoted in blue. c, Longitudinal (left) and Hall (right) resistance maps acquired at B = 0.5 T. The QOs projecting to approximately zero gate voltage in each Landau fan, corresponding to surface-localized states, are overlaid on the Rxx map and form a cross. The QOs projecting to non-zero gate voltages, corresponding to extended bulk states, are overlaid on the Rxy map and closely track the condition of overall charge neutrality. d, Rxx map acquired at B = 12 T. Dashed black lines denote selected QOs that depend only on a single gate, which arise from localized states on either the top or bottom graphite surfaces. The blue/pink dashed lines denote QOs that depend on both gates, which evolve parallel to the line of overall charge neutrality, arising from the extended bulk states.
Extended Data Fig. 4 High field dual-gate maps for the t1+10, t1+17, and t7+7 devices.
a–c, Dual-gate Rxx maps (top) and corresponding numerical second derivative (bottom) acquired for the (a) t1+10 device at B = 9 T, (b) t1+17 device at B = 12 T, and (c) t7+7 device at B = 12 T. Solid black bars at the top of each map denote regions of gate voltage dominated by vertical QOs, which correspond to surface-localized states. Other regions, though more complicated, contain diagonal QOs which depend on the value of both gate voltages. These features correspond to extended bulk standing wave states. Data in panel a was acquired in a dilution refrigerator with a nominal base temperature of T ≈ 20 mK, all other data was acquired at 1.7 K.
Extended Data Fig. 5 Extracting the zero-field projection points of the QOs.
a, Landau fan diagram of Rxx in the t1+10 device, acquired by sweeping Vgr with Vm = 0. b, The numerical second derivative of the data in a. The color scale is saturated to only show positive values of d2Rxx/dV2, since peaks in d2Rxx/dV2 correspond to minima in Rxx (i.e., QOs). c, Fit results using the unconstrained fitting procedure for QOs that project to Vgr ≈ 0. The green circles are extracted peaks in d2Rxx/dV2 that were used for each fit, and the dashed lines show the result of the fit for each individual QO. d Analogous results using the constrained fitting procedure for the QOs that project to Vgr ≈ 0. The green data points are the same as in c, but now all the states are fit simultaneously and forced to share the same projection point at B = 0 with a defined relationship between their slopes (see Methods for full description). e–f, Similar fitting using the unconstrained and constrained fitting procedures, respectively, but for the sequence of QOs that project to Vgr ≠ 0. g, Projection points determined from the fits shown in c and d. The shaded pink region contains data points and associated error bars from the unconstrained fit, \({V}_{i}^{{\rm{u}}}\pm {{\rm{\sigma }}}_{i}\), corresponding to the dashed lines in c of the same color. The vertical black line and surrounding shaded grey bar denote the weighted average and associated error of the unconstrained fit, \(\bar{{V}_{0}^{{\rm{u}}}}\pm \bar{{{\rm{\sigma }}}^{{\rm{u}}}}\). The shaded blue region contains the result of the constrained fit from d. The single black data point with error bars is the projection point determined by the constrained fit \({V}_{0}^{{\rm{c}}}\pm {{\rm{\sigma }}}^{{\rm{c}}}\). Note that \(\bar{{V}_{0}^{{\rm{u}}}}\pm \bar{{{\rm{\sigma }}}^{{\rm{u}}}}\) and \({V}_{0}^{{\rm{c}}}\pm {{\rm{\sigma }}}^{{\rm{c}}}\) are consistent with one another. h, Similar plot as g, but for the fits shown in e and f. Note that the extracted QO projection points in g and h differ significantly from one another, allowing us to unambiguously identify two distinct sets of QOs corresponding to the surface states and bulk states, respectively.
Extended Data Fig. 6 High-field transport in a device with a buried moiré interface.
a, Rxx (left) and Rxy (right) Landau fans acquired as Vt is swept with Vb = 0 in a device consisting of 7 layers of Bernal graphite stacked and rotated atop another 7 graphite layers with θ = 1.26°. b, Same as (a), but sweeping Vb with Vt = 0. The fans acquired by sweeping each gate are nearly identical to one another owing to the symmetry of the structure. Since the moiré is buried, we do not observe signatures of moiré band filling in either Landau fan. However, both fans show clear horizontal features corresponding to Brown-Zak oscillations arising from the buried moiré. c, Magnetoconductance averaged across all gate voltages for the top (red curve) and bottom (blue curve) gates. These are nearly identical, and both display a clear sequence of Brown-Zak oscillations. The inset shows a cartoon schematic in which separate standing waves couple the top and bottom bulk graphite states to the buried moiré interface. The hybridization of the buried moiré with the bulk states is required to generate the BZ oscillations seen in transport. d, (Left) Band structure calculation of this structure showing a moiré band localized at the center of the twisted graphitic thin film. The color scale is defined as in Fig. 2d of the main text. In this case, the moiré bands are found at ⟨z⟩ ≈ 0, since the moiré is located at the center of the structure. (Right) The density of states integrated over the moiré Brillouin zone. The red filtered curve corresponds to the four central graphene sheets, whereas the black corresponds to the total density of states.
Extended Data Fig. 7 High-field transport in a t1+6 device with θ = 1.27°.
a, Landau fan diagrams acquired by sweeping Vgr at various fixed values of Vm. The black dashed lines denote selected QOs that project to Vgr ≈ 0 at B = 0. The blue lines denote selected QOs that project to a Vgr ≠ 0 that depends on Vm. b, Landau fan diagrams acquired by sweeping Vm at fixed values of Vgr. The pink dashed lines denote selected QOs from each of the distinct sequences we observe. c, Longitudinal (left) and Hall (right) resistance maps acquired at B = 0.5 T. Zero-field projections of all observed sequences of QOs from the Vm Landau fans are overlaid on the Rxx map. Zero-field projections of QOs from the Vgr Landau fans are overlaid on the Rxy map. We note that, in this device, the observed sequence of QOs in the Vm Landau fans is very complex, and there is not always a clear delineation between QO sequences arising from neutrality (ν = 0) and full filling (ν = ± 4) of the moiré bands. This may be a consequence of the relatively thin nature of the sample, which exhibits features of both atomically-thin graphene and bulk graphite. Supplementary Information Fig. S4 further shows the dimensional crossover from 2D-like to 3D-like behavior as the number of graphene layers in the sample is increased. d, Conductance, Gxx, as a function of magnetic field. The blue curve is averaged over all values of Vgr for the Landau fan acquired at Vm/dm = − 0.05 V/nm. The red curve is averaged over a range of Vm values corresponding to ∣ν∣ < 4 for the Landau fan in b acquired at Vgr = 0. Brown-Zak oscillations case be seen upon sweeping either gate. (Inset) Longitudinal resistance map acquired at B = 0 T.
Extended Data Fig. 8 Representative constrained fit results for Landau fans acquired by sweeping Vm in the t1+10 device.
a, Landau fan diagram of Rxx in the t1+10 device, acquired by sweeping Vm with Vgr/dgr = 0.01 V/nm. b, Numerical second derivative of the data in a. The color scale is saturated to only show positive values, as in Extended Data Fig. 5. c, Results of the constrained fit overlaid on the second derivative data from b, with reduced opacity for clarity. Solid segments denote the range of magnetic field over which the QOs were fit, and the dashed segments denote the projection over the entire range of magnetic field.
Extended Data Fig. 9 Representative constrained fit results for the t1+17 device.
Same plots as in Extended Data Fig. 8, but for the t1+17 device. a–c, Landau fan acquired by sweeping Vgr with Vm/dm = 0.14 V/nm, and associated QO fits. d–e, Landau fan acquired by sweeping Vm with Vgr/dgr = 0, and associated QO fits.
Extended Data Fig. 10 Representative constrained fit results for t1+6 device.
Same plots as in Extended Data Fig. 8, but for the t1+6 device. a–c, Landau fan acquired by sweeping Vgr with Vm/dm = 0.09 V/nm, and associated QO fits. d–e, Landau fan acquired by sweeping Vm with Vgr/dgr = − 0.02 V/nm, and associated QO fits.
Supplementary information
Supplementary Video 1
The left shows the Landau fan diagrams from the t1 + 10 device acquired by sweeping Vgr at the indicated values of Vm. The black and blue dots correspond to the B = 0 projection points of the QO sequences, as described in the main text. In general, the blue dots align with the value of Vgr corresponding to the highest resistance over a wide range of B in the map. The right shows the Rxy map acquired at B = 0.5 T with the corresponding black and blue dots overlaid.
Supplementary Video 2
The left shows Landau fan diagrams from the t1 + 10 device acquired by sweeping Vm at the indicated values of Vgr. The pink and purple dots correspond to the B = 0 projection points of the QO sequences, as described in the main text. The right shows the Rxx map acquired at B = 0.5 T with the corresponding pink (ν = 0) and purple (ν = ± 4) dots overlaid.
Supplementary Video 3
Same as Supplementary Video 1, but for the t1 + 6 device.
Supplementary Video 4
Same as Supplementary Video 2, but for the t1 + 6 device.
Supplementary Video 5
Same as Supplementary Video 1, but for the t1 + 17 device. This video only includes Landau fans with Vgr/dgr ≥ −0.18 V nm−1. The fans acquired for Vgr/dgr ≤ −0.18 V nm−1 were taken only up to B = 5 T due to technical constraints in those particular measurements, and are not included in the video for the sake of continuity. Nevertheless, these lower-field fans still enable unambiguous QO projections, as plotted on the Rxx map.
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Waters, D., Thompson, E., Arreguin-Martinez, E. et al. Mixed-dimensional moiré systems of twisted graphitic thin films. Nature 620, 750–755 (2023). https://doi.org/10.1038/s41586-023-06290-3
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DOI: https://doi.org/10.1038/s41586-023-06290-3
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