Materials science

Graphene moiré mystery solved?


  • A Correction to this article was published on 13 July 2011

In systems consisting of just a few layers of graphene, the relative orientation of adjacent layers depends on the material's preparation method. Light has now been shed on the relationship between stacking arrangement and electronic properties.

Moiré patterns appear whenever two regular templates are overlaid at an angle — commonly as artful designs in textiles, and unfortunately also as annoying distractions in digital images. More than 20 years ago, periodic modulations (superlattices) were discovered1 in scanning tunnelling microscopy (STM) studies of graphite surfaces and correctly interpreted as moiré patterns caused by misorientations between subsurface graphene layers. (Graphene is a two-dimensional, pure carbon material that forms three-dimensional graphite when stacked in layers.)

The graphitic moiré pattern phenomenon has recently reappeared in studies of materials consisting of only a few layers of graphene, especially in systems prepared by the technique of epitaxial growth on a silicon carbide (SiC) substrate2 or by the process of chemical-vapour-decomposition growth on metallic substrates. But as in the graphite case3, the relationship between stacking rotations and electronic properties has been difficult to sort out. Writing in Physical Review Letters, Luican et al.4 provide insight into this relationship by combining advanced microscopy and spectroscopy techniques.

The problem of identifying the relationship between stacking rotations and electronic properties is particularly intriguing in the few-layer case, both because it should be tractable and because it is known5 that stacking arrangements can radically alter a material's electronic properties. For few-layer graphene systems, the main sticking point has been that studies6,7,8 of epitaxial graphene layers grown on SiC have given the impression that adjacent layers that are misoriented by only a few degrees could, in stark contradiction to theory9, be very weakly coupled electronically.

By combining high-magnetic-field STM and Landau-level spectroscopy on samples consisting of a few layers of graphene grown by chemical-vapour deposition, Luican and colleagues4 now convincingly establish that electronic decoupling occurs between layers misoriented by more than about 20° and that strong coupling occurs between layers rotated by less than about 2°. Moreover, the authors point to hints in previous epitaxial graphene data8 which suggest that the layer pairs identified as weakly coupled may not have been adjacent. If confirmed by additional STM studies, this conjecture would allow for a consistent interpretation of all current data.

The authors' findings4 provide an attractive jumping-off point for systematic studies of interlayer coupling in misoriented few-layer graphene systems. Many important and subtle mysteries remain, both at large and small rotation angles. For example, what does effective decoupling at rotation angles larger than about 20° mean quantitatively? Complete decoupling at the typical separation between adjacent graphene sheets of only approximately 0.3 nanometres would be truly unexpected.

When two graphene sheets, each with a honeycomb lattice structure, are overlaid at an angle (Fig. 1), the result is almost always not a crystal. A crystal forms only at a discrete set of commensurate rotation angles9,10,11,12. In graphene, most observable properties depend on electrons close to the Dirac point, the point at which the gap between the material's conduction and valence energy bands vanishes. These electrons have an unusual ultrarelativistic, massless behaviour, which is characterized by a velocity that does not vanish with momentum. And in the presence of a magnetic field they show a characteristic pattern of discrete energy states, known as Landau levels, that is disturbed when layers couple.

Figure 1: Two examples of misoriented honeycomb lattices.

a, The two lattices are overlaid at an angle of about 27.8°. The resulting atomic arrangement is precisely periodic, but has more atoms per period than has perfectly oriented (no rotation) Bernal stacking. b, Lattices rotated by 9°. Although the atomic arrangement never precisely repeats, there is a periodic pattern of points in space at which atoms from the two layers are nearly on top of each other, making the structure appear more open in this top view. When the local stacking arrangement varies slowly in space, electronic properties are insensitive to the atomic details that distinguish commensurate and nearby incommensurate rotation angles.

Because Dirac-point electronic states at crystalline misorientations are equal-weight sums of components in the two layers9,10,11,12, it is not immediately obvious in what sense the layers can be considered to be decoupled. Theory answers this question in the first place in terms of energy gaps induced by energy-level repulsion between Dirac-point states in different layers. Numerical ab initio calculations find, however, that these gaps rapidly decrease with increases in the unit-cell area of commensurately rotated bilayers. In fact, for all but the commensurate structures with the smallest crystalline unit cells9,10,11,12 (which occur at rotation angles near 21.8°, 27.8°, 32.2° and 38.2°), the gaps are smaller than the energy uncertainty implied by Heisenberg's uncertainty principle and the typical time between electron–sample-impurity collisions (the quantum lifetime), making them unobservable. A sample's quantum lifetime can be extracted13 from STM Landau-level spectroscopy measurements. So far, gaps due to coupling between rotated layers have never been observed, suggesting either that they are smaller than generally expected14 or that the rotation angles found experimentally do not correspond to short-period commensurate structures.

When the energy uncertainty is larger than the Dirac-point gap, the strength of interlayer coupling is more appropriately characterized by the interlayer conductance. This conductance is proportional to the area of the sample, increases without limit with quantum lifetime at commensurate rotation angles, and vanishes with quantum lifetime for incommensurate rotation angles. When the conductance is small, it can be characterized by the RC time, the time it takes for a voltage difference between the layers to relax to zero. The RC time is expected14 to be very sensitive to quantum lifetimes in the individual graphene sheets, and long enough to be experimentally accessible — as long as about 10−8 seconds — when sample disorder is weak.

In their study, Luican et al.4 find that, at small rotation angles, the local density of electronic states develops a dependence on position within the moiré-pattern unit cell and no longer exhibits the Dirac-like, decoupled-layer, Landau-level pattern. Layer coupling becomes strong in this sense for rotation angles less than about 2°, corresponding to moiré-pattern periods longer than about 10 nanometres. Here it is tempting to conjecture — from the spatial dependence of the density of electronic states — that bilayer wavefunctions have become localized, so that an STM measurement at one position reflects the stacking arrangement only at that position.

But continuum models, which ignore local atomic arrangements and are expected to be more accurate at small rotation angles, suggest otherwise. We have separately used15 continuum models to argue that Dirac-point states in the small-rotation-angle limit are characterized by narrow moiré energy bands whose width oscillates as a function of the reciprocal of the rotation angle. Greater understanding of these small-angle states is an interesting theoretical challenge, which may be informed by future experiments using samples that have weaker disorder than has so far been available, or by improved resolution of STM spectroscopy. Continuum models capture the moiré-pattern periodicity but ignore some details of the atomic arrangement. It is interesting that Bloch's theorem, which allows the electronic structure to be calculated relatively simply for periodic structures, can also be applied to these systems despite the fact that, crystallographically, they are quasi-periodic rather than periodic15.

It has long been recognized16 that commensurability between a unit-cell area in a periodic system and the area that encloses one quantum of magnetic flux should theoretically lead to complex, fractal electronic structures. This physics has, however, been difficult to address in the real world. On the one hand, typical unit-cell areas for crystalline periodicity are about 0.1 nm2, dramatically smaller than the magnetic-flux quantum areas achievable experimentally (about 3,000 nm2 divided by the magnetic field measured in tesla). On the other hand, any artificial periodicity fabricated lithographically tends to be too large. The moiré-pattern periods seem to fall nicely in the middle ground. This could allow17 the physics of periodic electronic systems in external magnetic fields to be studied more fully than has previously been possible.

The extraordinary sensitivity of the electronic properties of few-layer graphene systems to the relative orientations of their layers could prove useful in various applications, for example in ultra-sensitive strain gauges, pressure sensors or ultra-thin capacitors. Further progress requires an improved understanding of both large and small rotation-angle limits, and also improved experimental control of rotation angles.

Change history

  • 13 July 2011

    The figure in this article has been replaced.


  1. 1

    Kuwabara, M., Clarke, D. R. & Smith, D. A. Appl. Phys. Lett. 56, 2396–2398 (1990).

  2. 2

    First, P. N. et al. MRS Bull. 35, 296–305 (2010).

  3. 3

    Pong, W.-T. & Durkan, C. J. Phys. D 38, R329–R355 (2005).

  4. 4

    Luican, A. et al. Phys. Rev. Lett. 106, 126802 (2011).

  5. 5

    McCann, E. & Falko, V. I. Phys. Rev. Lett. 96, 086805 (2006).

  6. 6

    Sadowski, M. L. et al. Phys. Rev. Lett. 97, 266405 (2006).

  7. 7

    Sprinkle, M. et al. Phys. Rev. Lett. 103, 226803 (2009).

  8. 8

    Miller, D. L. et al. Science 324, 924–927 (2009).

  9. 9

    Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Phys. Rev. Lett. 99, 256802 (2007).

  10. 10

    Shallcross, S. et al. Phys. Rev. B 81, 165105 (2010).

  11. 11

    Trambly de Laissardière, G., Mayou, D. & Magaud, L. Nano Lett. 10, 804–808 (2010).

  12. 12

    Mele, E. J. Phys. Rev. B 81, 161405 (2010).

  13. 13

    Song, Y. J. et al. Nature 467, 185–189 (2010).

  14. 14

    Bistritzer, R. & MacDonald, A. H. Phys. Rev. B 81, 245412 (2010).

  15. 15

    Bistritzer, R. & MacDonald, A. H. Preprint at arXiv:1009.4203 (2010).

  16. 16

    Hofstadter, D. R. Phys. Rev. B 14, 2239–2249 (1976).

  17. 17

    Bistritzer, R. & MacDonald, A. H. Preprint at arXiv:1101.2606 (2011).

Download references

Author information



Corresponding author

Correspondence to Allan H. MacDonald.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

MacDonald, A., Bistritzer, R. Graphene moiré mystery solved?. Nature 474, 453–454 (2011).

Download citation

Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.