Influence of zigzag edges on the Ruderman-Kittel-Kasuya-Yosida interaction between magnetic impurities in graphene nanoribbons

Abstract

Localized magnetic impurity centres in graphene can interact through the π-electrons, leading to an effective Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. A RKKY-type study is presented for graphene ribbons with zigzag edges. Specifically investigations of how the edges modify the interaction between two localized magnetic moments are made by using a tight-binding Hamiltonian to describe the hopping of the π-electrons between adjacent sites and a contact term for interactions with the localized moments. In terms of a Green's function formalism for the excitation spectrum, which comprises modified bulk modes and two different types of localized edge modes, explicit analytical expressions are obtained for the RKKY interaction for any two magnetic sites on the graphene ribbon. The results enable us to determine the RKKY contributions that arise individually from the bulk-like modes and from the two types of edge modes in the zigzag geometry. The importance of these contributions varies depending on the proximity of the magnetic impurities to each other and to an edge.