A universal Hamiltonian for motion and merging of Dirac points in a two-dimensional crystal
Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-Sud, 91405 Orsay, France
Corresponding author: a firstname.lastname@example.org
Published online: 7 November 2009
We propose a simple Hamiltonian to describe the motion and the merging of Dirac points in the electronic spectrum of two-dimensional electrons. This merging is a topological transition which separates a semi-metallic phase with two Dirac cones from an insulating phase with a gap. We calculate the density of states and the specific heat. The spectrum in a magnetic field B is related to the resolution of a Schrödinger equation in a double well potential. The Landau levels obey the general scaling law ϵn ∝B2/3 fn(Δ/B2/3), and they evolve continuously from a to a linear (n+1/2)B dependence, with a [(n+1/2)B]2/3 dependence at the transition. The spectrum in the vicinity of the topological transition is very well described by a semiclassical quantization rule. This model describes continuously the coupling between valleys associated with the two Dirac points, when approaching the transition. It is applied to the tight-binding model of graphene and its generalization when one hopping parameter is varied. It remarkably reproduces the low field part of the Rammal-Hofstadter spectrum for the honeycomb lattice.
PACS: 73.00.00 – Electronic structure and electrical properties of surfaces, interfaces, thin films, and low-dimensional structures / 71.70.Di – Landau levels / 81.05.Uw – Carbon, diamond, graphite / 67.85.-d – Ultracold gases, trapped gases
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2009