Dimensional reduction, quantum Hall effect and layer parity in graphite films


The quantum Hall effect (QHE) originates from discrete Landau levels forming in a two-dimensional electron system in a magnetic field1. In three dimensions, the QHE is forbidden because the third dimension spreads Landau levels into overlapping bands, destroying the quantization. Here we report the QHE in graphite crystals that are up to hundreds of atomic layers thick, a thickness at which graphite was believed to behave as a normal, bulk semimetal2. We attribute this observation to a dimensional reduction of electron dynamics in high magnetic fields, such that the electron spectrum remains continuous only in the field direction, and only the last two quasi-one-dimensional Landau bands cross the Fermi level3,4. Under these conditions, the formation of standing waves in sufficiently thin graphite films leads to a discrete spectrum allowing the QHE. Despite the large thickness, we observe differences between crystals with even and odd numbers of graphene layers. Films with odd layer numbers show reduced QHE gaps, as compared to films of similar thicknesses but with even numbers because the latter retain the inversion symmetry characteristic of bilayer graphene5,6. We also observe clear signatures of electron–electron interactions including the fractional QHE below 0.5 K.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: QHE in 3D graphite.
Fig. 2: Landau levels in thin graphite films.
Fig. 3: Thickness and layer-parity dependence of the 2.5D QHE.
Fig. 4: Hierarchy of QHE gaps and level crossings in the UQR.

Data availability

All relevant data are available from the corresponding authors on reasonable request.


  1. 1.

    Yoshioka, D. The Quantum Hall Effect (Springer, Berlin, 1998).

  2. 2.

    McClure, J. W. Band structure of graphite and de Haas–van Alphen effect. Phys. Rev. 108, 612–618 (1957).

  3. 3.

    Shovkovy, I. A. Magnetic catalysis: a review. Lecture Notes Phys. 871, 13–49 (2013).

  4. 4.

    McClure, J. W. & Spry, W. J. Linear magnetoresistance in the quantum limit in graphite. Phys. Rev. 165, 809–815 (1968).

  5. 5.

    McCann, E. & Fal’ko, V. I. Landau-level degeneracy and quantum Hall effect in a graphite bilayer. Phys. Rev. Lett. 96, 086805 (2006).

  6. 6.

    Koshino, M. & McCann, E. Landau level spectra and the quantum Hall effect of multilayer graphene. Phys. Rev. B 83, 165443 (2011).

  7. 7.

    Landau, L. D. & Lifshitz, E. M. Quantum Mechanics 3rd edn (Pergamon, Oxford, 1977).

  8. 8.

    Slonczewski, J. C. & Weiss, P. R. Band structure of graphite. Phys. Rev. 109, 272–279 (1958).

  9. 9.

    Brandt, N. B., Kapustin, G. A., Karavaev, V. G., Kotosonov, A. S. & Svistova, E. A. Investigation of galvanomagnetic properties of graphite in magnetic-fields up to 500 kOe at low temperatures. Zh. Eksp. Teor. Fiz. 40, 564–569 (1974).

  10. 10.

    Kopelevich, Y. et al. Reentrant metallic behavior of graphite in the quantum limit. Phys. Rev. Lett. 90, 156402 (2003).

  11. 11.

    Luk’yanchuk, I. A. & Kopelevich, Y. Phase analysis of quantum oscillations in graphite. Phys. Rev. Lett. 93, 166402 (2004).

  12. 12.

    Morozov, S. V. et al. Two-dimensional electron and hole gases at the surface of graphite. Phys. Rev. B 72, 201401 (2005).

  13. 13.

    Zhang, Y. B., Small, J. P., Pontius, W. V. & Kim, P. Fabrication and electric-field-dependent transport measurements of mesoscopic graphite devices. Appl. Phys. Lett. 86, 073104 (2005).

  14. 14.

    Zhu, Z. et al. Magnetic field tuning of an excitonic insulator between the weak and strong coupling regimes in quantum limit graphite. Sci. Rep. 7, 1733 (2017).

  15. 15.

    Arnold, F. et al. Charge density waves in graphite: towards the magnetic ultraquantum limit. Phys. Rev. Lett. 119, 136601 (2017).

  16. 16.

    Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).

  17. 17.

    Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).

  18. 18.

    Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419–425 (2013).

  19. 19.

    Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures. Science 353, aac9439 (2016).

  20. 20.

    Partoens, B. & Peeters, F. M. From graphene to graphite: electronic structure around the K point. Phys. Rev. B 74, 075404 (2006).

  21. 21.

    Spain, I. L., Ubbelohde, A. R. & Young, D. A. Electronic properties of well oriented graphite. Phil. Trans. R. Soc. A 262, 345–386 (1967).

  22. 22.

    Hedley, J. A. & Ashworth, D. R. Imperfections in natural graphite. J. Nucl. Mater. 4, 70–78 (1961).

  23. 23.

    Arovas, D. P. & Guinea, F. Stacking faults, bound states, and quantum Hall plateaus in crystalline graphite. Phys. Rev. B 78, 245416 (2008).

  24. 24.

    Soule, D. E. Magnetic field dependence of the Hall effect and magnetoresistance in graphite single crystals. Phys. Rev. 112, 698–707 (1958).

  25. 25.

    Ono, S. & Sugihara, K. Theory of the transport properties in graphite. J. Phys. Soc. Jpn 21, 861–868 (1966).

  26. 26.

    Brandt, N. B., Chudinov, S. M. & Ponomarev, Y. G. Semimetals: 1. Graphite and its Compounds (North-Holland, Amsterdam, 1988).

  27. 27.

    Guinea, F. Charge distribution and screening in layered graphene systems. Phys. Rev. B 75, 235433 (2007).

  28. 28.

    Mikitik, G. P. & Sharlai, Y. V. Band-contact lines in the electron energy spectrum of graphite. Phys. Rev. B 73, 235112 (2006).

  29. 29.

    McClure, J. W. Theory of diamagnetism of graphite. Phys. Rev. 119, 606–613 (1960).

  30. 30.

    De Poortere, E. P., Tutuc, E., Papadakis, S. J. & Shayegan, M. Resistance spikes at transitions between quantum Hall ferromagnets. Science 290, 1546–1549 (2000).

  31. 31.

    Halperin, B. I. Possible states for a three-dimensional electron gas in a strong magnetic field. Jpn. J. Appl. Phys. 26, 1913 (1987).

  32. 32.

    Yoshioka, D. & Fukuyama, H. Electronic phase-transition of graphite in a strong magnetic-field. J. Phys. Soc. Jpn 50, 725–726 (1981).

  33. 33.

    Mishchenko, A. et al. Twist-controlled resonant tunnelling in graphene/boron nitride/graphene heterostructures. Nat. Nanotechnol. 9, 808–813 (2014).

  34. 34.

    Henni, Y. et al. Rhombohedral multilayer graphene: a magneto-Raman scattering study. Nano Lett. 16, 3710–3716 (2016).

  35. 35.

    Yu, G. L. et al. Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices. Nat. Phys. 10, 525–529 (2014).

  36. 36.

    Nakao, K. Landau level structure and magnetic breakthrough in graphite. J. Phys. Soc. Jpn 40, 761–768 (1976).

  37. 37.

    Inoue, M. Landau levels and cyclotron resonance in graphite. J. Phys. Soc. Jpn 17, 808–819 (1962).

  38. 38.

    Koshino, M., Sugisawa, K. & McCann, E. Interaction-induced insulating states in multilayer graphenes. Phys. Rev. B 95, 235311 (2017).

Download references


This work was supported by the EU Graphene Flagship Program, the European Research Council, the Royal Society and the Engineering and Physical Sciences Research Council. J.Y. and A.M. acknowledges the support of EPSRC Early Career Fellowship EP/N007131/1.

Author information

A.M., A.K.G. and J.Y. conceived the experiments. J.Y., I.L., S.O. and B.P. conducted the transport measurements. J.Y., Y.C., S.H., Y.Y. and S.-K.S. prepared the samples. A.M. and J.Y. performed data analysis. S.S. and V.F. developed theory and S.S., V.F. and F.G. interpreted the data and performed the tight-binding simulations. T.T. and K.W. provided hBN crystals. A.M. wrote the manuscript with input from V.F., A.K.G., J.Y., S.S., K.S.N and F.G.

Correspondence to A. K. Geim or Vladimir Fal’ko or Artem Mishchenko.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figures 1–9.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yin, J., Slizovskiy, S., Cao, Y. et al. Dimensional reduction, quantum Hall effect and layer parity in graphite films. Nat. Phys. 15, 437–442 (2019) doi:10.1038/s41567-019-0427-6

Download citation

Further reading