Moiré Patterns and Commensurability in Rotated Graphene Bilayers

This Demonstration shows the appearance of Moiré patterns in hexagonal lattice systems as you rotate a honeycomb layer on top of another identical layer by an angle . The rotation angle can be freely chosen or, as done in this Demonstration, specified by a couple of integer indices following [1]. These integers also give the number of atoms and the unit cell basis vectors of the supercell of the bilayer system. Some example systems where Moiré patterns on hexagonal lattices that have been recently experimentally observed are bilayer graphene systems.
The controls in this Demonstration let you explore the variety of patterns and supercells by choosing the size needed for building the graphene layers and the integers giving the mutual rotation angle . After checking the "show supercell" checkbox, the locator control allows you to translate the supercell in order to cover the periodic pattern of the resulting commensurate system.
Two limit cases are possible: if , the honeycomb layers perfectly match () and the so-called AA stacking is obtained (this means that atoms of the same type in the graphene unit cell usually denoted in graphene literature as A- or B-type atoms do overlay as shown in Snapshot 1); when A-type atoms are superimposed on B-type atoms (or vice versa), one gets the so-called AB stacking, which can be obtained by a full rotation by or locally within the Moiré pattern for a generic angle . Large commensurate supercells can be obtained for either by choosing large and values, for which is small, or for angles with large . In order to get an angle close to the perfect AB stacking one can choose (or ) and large (or large ).


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Snapshot 1: perfect AA stacking () with
Snapshot 2: quasi-perfect AB stacking limit () with large
Snapshot 3: example of an extremely large supercell ( limit) with small
Snapshot 4: example of a commensurate bilayer cell , symmetric with respect to the case in the thumbnail snapshot with
[1] G. T. de Laissardière, D. Mayou, and L. Magaud, "Localization of Dirac Electrons in Rotated Graphene Bilayers," Nano Letters, 10(3), 2010 pp. 804–808. pubs.acs.org/doi/abs/10.1021/nl902948m.
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