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 degrees less than or similar to theta less than or similar to 30 degrees. We show that at certain angles theta greater than theta(c) approximate to 1.89 degrees 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 theta behaves nonmonotonously, demonstrating exponentially large jumps for very small variations of theta. This sensitivity to the angle makes it impossible to predict the gap value for a given sample, since in experiment theta is always known with certain error. To establish the connection with experiments, we demonstrate that for a system of finite size (L) over bar 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 (L) over bar. In the regime of small angles theta < theta(c), the system is a metal with a well-defined Fermi surface which is reduced to Fermi points for some values of theta. The density of states in the metallic phase varies smoothly with theta.