Abstract
Self-similarity and fractals have fascinated researchers across various disciplines. In graphene placed on boron nitride and subjected to a magnetic field, self-similarity appears in the form of numerous replicas of the original Dirac spectrum, and their quantization gives rise to a fractal pattern of Landau levels, referred to as the Hofstadter butterfly. Here we employ capacitance spectroscopy to probe directly the density of states (DoS) and energy gaps in this spectrum. Without a magnetic field, replica spectra are seen as pronounced DoS minima surrounded by van Hove singularities. The Hofstadter butterfly shows up as recurring Landau fan diagrams in high fields. Electron–electron interactions add another twist to the self-similar behaviour. We observe suppression of quantum Hall ferromagnetism, a reverse Stoner transition at commensurable fluxes and additional ferromagnetism within replica spectra. The strength and variety of the interaction effects indicate a large playground to study many-body physics in fractal Dirac systems.
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References
Xue, J. M. et al. Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride. Nature Mater. 10, 282–285 (2011).
Decker, R. et al. Local electronic properties of graphene on a BN substrate via scanning tunneling microscopy. Nano Lett. 11, 2291–2295 (2011).
Yankowitz, M. et al. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nature Phys. 8, 382–386 (2012).
Ponomarenko, L. A. et al. Cloning of Dirac fermions in graphene superlattices. Nature 497, 594–597 (2013).
Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598–602 (2013).
Hunt, B. et al. Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427–1430 (2013).
Park, C. H., Yang, L., Son, Y. W., Cohen, M. L. & Louie, S. G. New generation of massless Dirac fermions in graphene under external periodic potentials. Phys. Rev. Lett. 101, 126804 (2008).
Barbier, M., Vasilopoulos, P. & Peeters, F. M. Extra Dirac points in the energy spectrum for superlattices on single-layer graphene. Phys. Rev. B 81, 075438 (2010).
Wu, S., Killi, M. & Paramekanti, A. Graphene under spatially varying external potentials: Landau levels, magnetotransport, and topological modes. Phys. Rev. B 85, 195404 (2012).
Ortix, C., Yang, L. & van den Brink, J. Graphene on incommensurate substrates: Trigonal warping and emerging Dirac cone replicas with halved group velocity. Phys. Rev. B 86, 081405 (2012).
Wallbank, J. R., Patel, A. A., Mucha-Kruczynski, M., Geim, A. K. & Fal’ko, V. I. Generic miniband structure of graphene on a hexagonal substrate. Phys. Rev. B 87, 245408 (2013).
Chen, X. et al. Dirac edges of fractal magnetic minibands in graphene with hexagonal moiré superlattices. Phys. Rev. B 89, 075401 (2014).
Brown, E. Bloch electrons in a uniform magnetic field. Phys. Rev. 133, A1038–A1044 (1964).
Zak, J. Magnetic translation group. Phys. Rev. 134, A1602–A1611 (1964).
Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).
Pfannkuche, D. & Gerhardts, R. R. Theory of magnetotransport in two-dimensional electron systems subjected to weak two-dimensional superlattice potentials. Phys. Rev. B 46, 12606–12626 (1992).
Albrecht, C. et al. Evidence of Hofstadter’s fractal energy spectrum in the quantized Hall conductance. Phys. Rev. Lett. 86, 147–150 (2001).
Geisler, M. C. et al. Detection of a Landau band-coupling-induced rearrangement of the Hofstadter butterfly. Phys. Rev. Lett. 92, 256801 (2004).
Bistritzer, R. & MacDonald, A. H. Moire butterflies. Phys. Rev. B 84, 035440 (2011).
Yang, K., Das Sarma, S. & MacDonald, A. H. Collective modes and skyrmion excitations in graphene SU(4) quantum Hall ferromagnets. Phys. Rev. B 74, 075423 (2006).
Goerbig, M. O. & Regnault, N. Analysis of a SU(4) generalization of Halperin’s wave function as an approach towards a SU(4) fractional quantum Hall effect in graphene sheets. Phys. Rev. B 75, 241405 (2007).
Jiang, Z., Zhang, Y., Stormer, H. L. & Kim., P. Quantum Hall states near the charge-neutral Dirac point in graphene. Phys. Rev. Lett. 99, 106802 (2007).
Young, A. F. et al. Spin and valley quantum Hall ferromagnetism in graphene. Nature Phys. 8, 550–556 (2012).
Yu, G. L. et al. Interaction phenomena in graphene seen through quantum capacitance. Proc. Natl Acad. Sci. USA 110, 3282–3286 (2013).
Kretinin, A. et al. Accurate determination of electron-hole asymmetry and next-nearest neighbor hopping in graphene. Phys. Rev. B 88, 165427 (2013).
Woods, C. R. et al. Commensurate–incommensurate transition for graphene on hexagonal boron nitride. Nature Phys. 10, 451–456 (2014).
Fang, T., Konar, A., Xing, H. & Jena, D. Carrier statistics and quantum capacitance of graphene sheets and nanoribbons. Appl. Phys. Lett. 91, 092109 (2007).
Wannier, G. H. A result not dependent on rationality for Bloch electrons in a magnetic field. Phys. Status Solidi b 88, 757–765 (1978).
Sondhi, S. L., Karlhede, A., Kivelson, S. A. & Rezayi, E. H. Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies. Phys. Rev. B 47, 16419–16426 (1993).
Fertig, H. A., Brey, L., Côté, R. & MacDonald, A. H. Charged spin-texture excitations and the Hartree–Fock approximation in the quantum Hall effect. Phys. Rev. B 50, 11018–11021 (1994).
Barrett, S. E., Dabbagh, G., Pfeiffer, L. N., West, K. W. & Tycko, R. Optically pumped NMR evidence for finite-size skyrmions in GaAs quantum wells near Landau level filling ν = 1. Phys. Rev. Lett. 74, 5112–5115 (1995).
Apalkov, V. M. & Chakraborty, T. Gap structure of the Hofstadter system of interacting Dirac fermions in graphene. Phys. Rev. Lett. 112, 176401 (2014).
Acknowledgements
This work was supported by the European Research Council, the Royal Society, Graphene Flagship, Science and Innovation Award from the EPSRC (UK) and EuroMagNET II (EU Contract 228043).
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J.S.T., A.V.K., Y.C., R.J. and F.W. designed and fabricated the devices. A.M. and G.L.Y. carried out the measurements. B.A.P. and M.P. helped with high-field experiments. X.C. and V.I.F. provided theoretical support. K.W. and T.T. provided hBN crystals. R.V.G. devised the fabrication technology for graphene capacitors. A.M. developed the on-chip capacitance bridge. A.M., V.I.F. and A.K.G. analysed the results. A.K.G. together with V.I.F. wrote the manuscript. K.S.N., I.V.G., L.A.P. and D.C.E. helped with experiments and/or writing the paper. Section 4 of the Supplementary Information was written by V.I.F. All authors contributed to discussions.
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Yu, G., Gorbachev, R., Tu, J. et al. Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices. Nature Phys 10, 525–529 (2014). https://doi.org/10.1038/nphys2979
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DOI: https://doi.org/10.1038/nphys2979
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