Electronic spectrum of twisted bilayer graphene

We study the electronic properties of twisted bilayer graphene in the tight-binding approximation. The interlayer hopping amplitude is modeled by a function which depends not only on the distance between two carbon atoms, but also on the positions of neighboring atoms as well. Using the Lanczos algorithm for the numerical evaluation of eigenvalues of large sparse matrices, we calculate the bilayer single-electron spectrum for commensurate twist angles in the range ${1}^{\ensuremath{\circ}}\ensuremath{\lesssim}\ensuremath{\theta}\ensuremath{\lesssim}{30}^{\ensuremath{\circ}}$. We show that at certain angles $\ensuremath{\theta}$ greater than ${\ensuremath{\theta}}_{c}\ensuremath{\approx}1.{89}^{\ensuremath{\circ}}$ the electronic spectrum acquires a finite gap, whose value could be as large as 80 meV. However, in an infinitely large and perfectly clean sample the gap as a function of $\ensuremath{\theta}$ behaves nonmonotonously, demonstrating exponentially large jumps for very small variations of $\ensuremath{\theta}$. This sensitivity to the angle makes it impossible to predict the gap value for a given sample, since in experiment $\ensuremath{\theta}$ is always known with certain error. To establish the connection with experiments, we demonstrate that for a system of finite size $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{L}$ the gap becomes a smooth function of the twist angle. If the sample is infinite, but disorder is present, we expect that the electron mean-free path plays the same role as $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{L}$. In the regime of small angles $\ensuremath{\theta}<{\ensuremath{\theta}}_{c}$, the system is a metal with a well-defined Fermi surface which is reduced to Fermi points for some values of $\ensuremath{\theta}$. The density of states in the metallic phase varies smoothly with $\ensuremath{\theta}$.