Friedel oscillation near a van Hove singularity in two-dimensional Dirac materials

We consider Friedel oscillation in the two-dimensional Dirac materials when the Fermi level is near the van Hove singularity. Twisted graphene bilayer and the surface state of topological crystalline insulator are the representative materials which show low-energy saddle points that are feasible to probe by gating. We approximate the Fermi surface near saddle point with a hyperbola and calculate the static Lindhard response function. Employing a theorem of Lighthill, the induced charge density δn due to an impurity is obtained and the algebraic decay of δn is determined by the singularity of the static response function. Although a hyperbolic Fermi surface is rather different from a circular one, the static Lindhard response function in the present case shows a singularity similar with the response function associated with circular Fermi surface, which leads to the δn ∞ R^-2 at large distance R. The dependences of charge density on the Fermi energy are different. Consequently, it is possible to observe in twisted graphene bilayer the evolution that δn ∞ R^-3 near Dirac point changes to δn ∞ R^-2 above the saddle point. Measurements using scanning tunnelling microscopy around the impurity sites could verify the prediction.