Abstract
A real-space approach for the calculation of the moiré lattice parameters for superstructures formed by a set of rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides is presented. Apparent moiré lattices continuously form for all rotation angles, and their lattice parameter to a good approximation follows a hyperbolical angle dependence. Moiré crystals, i.e. moiré lattices decorated with a basis, require more crucial assessment of the commensurabilities and lead to discrete solutions and a non-continuous angle dependence of the moiré-crystal lattice parameter. In particular, this lattice parameter critically depends on the rotation angle, and continuous variation of the angle can lead to apparently erratic changes of the lattice parameter. The solutions form a highly complex pattern, which reflects number-theoretical relations between formation parameters of the moiré crystal. The analysis also provides insight into the special case of a 30° rotation of the constituting lattices, for which a dodecagonal quasicrystalline structure forms.
Highlights
Cao et al (2018) demonstrated that stacked graphene layers with relative rotations can have drastically different properties to their regularly aligned counterparts
In order to fully understand the physical effects in twistronics and to allow precise device design, it is imperative to understand the relation of the relative rotation angle of the constituting layers and the resulting structure of the moirecrystal
In this paper a real-space approach for the calculation of lattice parameters of moirecrystals formed by the relative rotation of two constituting hexagonal crystal layers is worked out
Summary
Cao et al (2018) demonstrated that stacked graphene layers with relative rotations can have drastically different properties to their regularly aligned counterparts. They observed that a relative rotation of about 1.1 leads to superconductivity in double-layer graphene This corresponds to one of the ‘magic angles’ previously predicted by Bistritzer & MacDonald (2011), who calculated the band structure of twisted double-layer graphene and found that the narrowing of bands at small angles is non-continuous, and at 1.05 and other distinct angles the Dirac-point velocity vanishes. In order to fully understand the physical effects in twistronics and to allow precise device design, it is imperative to understand the relation of the relative rotation angle of the constituting layers and the resulting structure of the moirecrystal This is important owing to the fact that both the physical properties – viz. The equations can be used to identify angles of critical angle dependence, or for back-calculation of the rotation angle by measurement of lattice parameters of actual devices in the transmission electron microscope for quality
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