RKKY interaction in heavily vacant graphene

Dirac electrons in clean graphene can mediate the interactions between two localized magnetic moments. The functional form of the RKKY interaction in pristine graphene is specified by two main features: (i) an atomic-scale oscillatory part determined by a wavevector Q connecting the two valleys; with doping another longer range oscillation appears which arises from the existence of an extended Fermi surface characterized by a momentum scale kF; (ii) an algebraic R(α) decay in large distances where the exponent α =- 3 is a distinct feature of undoped Dirac sea in two dimensions. In this work, we investigate the effect of a few per cent vacancies on the above properties. Depending on the doping level, if the chemical potential lies on the linear part of the density of states, the exponent α remains at -3 even in vacant graphene. Otherwise α reduces towards more negative values. Presence of vacancies washes out both atomic-scale and Friedel oscillations of the RKKY interaction. The absence of atomic-scale oscillations indicates the destruction of two-valley structure of the parent graphene material. However, the absence of Friedel oscillations upon 'alloying' with vacancies indicates that a quantum ground state of heavily vacant doped graphene is not given by a unique kF momentum scale.