Magnetism without magnetic atoms: The physics of the vacancy in graphene

Abstract

Graphene is a material of considerable current interest owing to its linear band structure and excitations that behave as massless Dirac Fermions. The linear Dirac bands lead to many unique features including high conductivity due to absence of backscattering, Klein tunneling of chiral electrons, new features in the Quantum Hall effect and the RKKY interaction, etc. The vacancy in graphene can be created by ion irradiation and has many novel features whose origin can be traced to the linear band structure. Some of the features including magnetism have been experimentally observed. We have studied the single substitutional vacancy in graphene from density-functional calculations and model studies. The vacancy releases four electrons, to be accommodated by the three sp2σ dangling bond states (Vσ) and a vacancy-induced quasi-localized mid-gap state (Vπ) in the π bands, the linear Dirac bands arising from the carbon pz orbitals. Jahn-Teller distortion of the carbon triangle surrounding the vacancy together with the Hund’s coupling results in the spin occupation of the vacancy states as shown in Fig. 1, resulting in a S=1 configuration. These localized electrons spin-polarize the itinerant π bands in the neighborhood of the vacancy (indicated by π1 in the figure) with an anti-ferromagnetic Kondo-like coupling. The net magnetic moment is 1.7 μB as obtained from the density-functional calculations. Density-functional studies show further that the vacancy acts as a Jahn-Teller center due to the coupling between the vacancy electronic states and the local lattice modes of the carbon triangle surrounding the vacancy. However, the energetics are such that there is only a small potential barrier between the three Jahn-Teller minima, leading to the quantum mechanical tunneling of the nuclei between the minima and resulting in the dynamical Jahn-Teller effect. The Berry phase introduced due to the adiabatic electronic motion leads to observable effects such as the symmetry of the nuclear ground state, which is predicted to be doubly-degenerate, separated by a singly-degenerate excited state removed by an energy of about 65 cm−1. This can be measured by EPR and two-photon scattering experiments. We have also studied using the linear response theory the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic moments, mediated by the linear Dirac band electrons. The RKKY interaction shows many unusual features such as the sign of the interaction, which is ferromagnetic for moments on the same sublattice and antiferromagnetic for moments on opposite sublattices that follow from the particle-hole symmetry for a bipartite lattice. This is different from the standard RKKY interaction in 3D, which shows oscillatory behavior as a function of the distance R between the two moments. Unlike the J∝(2kFR)−2sin(2kFR) behavior of an ordinary 2D metal in the long-distance limit, in graphene the RKKY interaction falls off as R−3 and shows the 1+cos[(K→–K→′)R1] type oscillations with additional phase factors depending on the direction. For graphene doped with electrons or holes by an applied gate voltage, we have obtained analytical results for the linear Dirac bands, assuming that the deviation from linearity for the higher bands may be ignored. The analytical result, expressed in terms of the Meijer G-function, consists of the product of two oscillatory terms, one coming from the interference between the two Dirac cones and the second coming from the finite size of the Fermi surface. These two oscillatory terms sometimes lead to a quantum beating behavior in certain cases, e.g, for the bond-centered magnetic moments as shown in the figure. For large distances, the Meijer G-function behaves as a sinusoidal term, leading to the result J∼R−2kFsin(2kFR){1+cos[(K→–K→′)⋅R1]} for moments located on the same sublattice. The R−2 dependence, which is the same for the standard two-dimensional electron gas, is universal irrespective of the sublattice location and the distance direction of the two moments except when kF = 0 (undoped case), where it reverts to the R−3 dependence as already discussed. We also predict the interesting effect that the RKKY interaction can be switched between ferromagnetic and antiferromagnetic by an applied gate voltage, which controls the concentration of the doped electrons or holes and as a result the Fermi momentum kF.