Continuum Model of the twisted graphene bilayer

The electronic structure of the twisted bilayer was first considered [1] in the context of a continuum description of the two layers, coupled by a spatially modulated hopping. The model's predictions were subsequently confirmed by experiments [2,3], including a scanning tunneling spectroscopy finding of two low energy Van-Hove peaks in the density of states [4], and by band structure calculations [5,6]. We discuss the extension of the model in several directions: the two families of commensurate structures discovered by Mele [7], will be characterized by elementary geometrical arguments; it will be shown that it is possible to calculate analytically \emph{all} Fourier components of the hopping amplitudes for any kind of commensurate structure with large period; the calculations will be extended beyond the perturbative regime in the interlayer coupling to address the electronic structure and local density of states in the very small angle limit. \\[4pt] [1] J.~M.~B. Lopes dos Santos, \emph{et. al}, Phys. Rev. Lett. \textbf{99}, 256802, (2007).\\[0pt] [2] Z.~Ni, \emph{et. al.}, Phys. Rev. B \textbf{77}, 235403, 2008.\\[0pt] [3] A.~Luica, \emph{et. al}, Phys. Rev. Lett. \textbf{106}, 126802 (2011).\\[0pt] [4] G.~Li, \emph{et. al}, Nature Physics \textbf{6}, 109 (2010).\\[0pt] [5] S. Shallcross, \emph{et. al}, Phys. Rev. B \textbf{81}, 165105 (2010).\\[0pt] [6] G.~de Laissardiere, \emph{et. al}, Nano Letters \textbf{10},804 (2010).\\[0pt] [7] E.~J. Mele, {Phys. Rev. B} \textbf{81}, 161405 (2010).