Friedel oscillations at the Dirac cone merging point in anisotropic graphene and graphenelike materials

We study the Friedel oscillations induced by a localized impurity in an anisotropic graphenelike structure. We focus on the limit when the two inequivalent Dirac points merge. We find that, in this limit, the Friedel oscillations manifest very peculiar features, such as a strong asymmetry and an atypical inverse square-root decay. Our calculations are performed using both a T-matrix approximation and a tight-binding exact diagonalization technique. They allow us to numerically obtain the local density of states as a function of energy and position as well as an analytical form of the Friedel oscillations in the continuum limit. The two techniques yield results that are in excellent agreement, confirming the accuracy of such methods to approach this problem.