Moire Bloch Bands in Twisted Bilayer Graphene

A moir\'{e} superlattice pattern is formed when two copies of a periodic lattice are overlaid with a relative twist. I will address the electronic structure of a twisted two-layer graphene system by generalizing the Dirac equation continuum models that are used to describe single-layer graphene and untwisted bilayers. In the Dirac model electrons in graphene have a pseudospin degree-of-freedom corresponding to the honeycomb sublattice dependence of wavefunction amplitudes. The continuum model of twisted bilayers can be derived systematically [1] by assuming that interlayer tunneling amplitudes are non-nocal with a range that is large compared to the honeycomb lattice constant, and leads to an appealing picture in which the tunneling operator has a position-dependent pseudospin dependence that simply reflects the local registry between the two honeycombs. The continuum model twisted bilayer Hamiltonian is therefore periodic, with the periodicity of the moir\'{e} pattern, and insensitive to the incommensurability of the microscopic Hamiltonian. I will discuss the properties of the Bloch bands of this periodic Hamiltonian, which become highly non-trivial at small twist angles. In particular the Dirac velocity crosses zero several times as the twist angle is reduced and vanishes at a discrete set of magic angles. I will also briefly discuss the Hofstadter butterfly spectral patterns [2] created by incommensurability between the moir\'{e} pattern and magnetic lengths when a twisted bilayer is placed in an external magnetic field, and the electronic structure [3] of a single graphene layer that is twisted with respect to a boron nitride substrate.\\[4pt] [1] R. Bistritzer and A.H. MacDonald, PNAS {\bf 108}, 12233 (2011). \hfill \newline [2] R. Bistritzer and A.H. MacDonald, Phys. Rev. B {\bf 84}, 035440 (2011). \hfill \newline [3] A. Raoux and A.H. MacDonald, arXiv:1112.nnnn.