Spatial dependent van der Waals energy between graphene and boron-nitride

The small mismatch between the honeycomb lattices of graphene (GE) and boron nitrate (h-BN) leads to long wavelength Moir\'{e} patterns. In order to describe such patterns it will require large size unit cells that are unattainable with \textit{ab-initio} calculations. Earlier density functional theory calculations imposed lattice matching between graphene and h-BN which induces strain and opens a gap of 4 meV [1]. In previous works the Moir\'{e} pattern in GE/h-BN was connected to the van der Waals interaction [2], but a clear theoretical microscopic analysis is still missing. We used atomistic simulations [3] with very large unit cells to investigate quantitative aspects of the connection between the vdW interaction and the Moir\'{e} patterns. The value and symmetry of the spatial dependent vdW energy is obtained which agrees with the recently reported Moir\'{e} patterns. \textit{Acknowledgement}: This work was supported by FWO-Vl, EU-Marie Curie and the Methusalem foundation. [1] B. Sachs \textit{et al}, Phys. Rev. B \textbf{84}, 195414 (2011); M. Zarenia \textit{et al}, Phys. Rev. B \textbf{86}, 085451 (2012). [2] C.R. Dean \textit{et al}, Nat. Nanotech. \textbf{5}, 722 (2010); J.H. Chen \textit{et al}, Nat. Nanotech. \textbf{3}, 206 (2008). [3] M. Neek-Amal and A. Lajevardipour, Comp. Mat. Sc. \textbf{49}, 839 (2010).