We present a detailed theoretical study of effective spin-orbit coupling (SOC) Hamiltonians for graphene based systems, covering global effects such as proximity to substrates and local SOC effects resulting, for example, from dilute adsorbate functionalization. Our approach combines group theory and tight-binding descriptions. We consider structures with global point group symmetries D 6h , D 3d , D 3h , C6v, and C3v that represent, for example, pristine graphene, graphene mini-ripple, planar boron-nitride, graphene on a substrate and free standing graphone, respectively. The presence of certain spin-orbit coupling parameters is correlated with the absence of the specific point group symmetries. Especially in the case of C6v-graphene on a substrate, or transverse electric field-we point out the presence of a third SOC parameter, besides the conventional intrinsic and Rashba contributions, thus far neglected in literature. For all global structures we provide effective SOC Hamiltonians both in the local atomic and Bloch forms. Dilute adsorbate coverage results in the local point group symmetries C6v, C3v, and C2v which represent the stable adsorption at hollow, top and bridge positions, respectively. For each configuration we provide effective SOC Hamiltonians in the atomic orbital basis that respect local symmetries. In addition to giving specific analytic expressions for model SOC Hamiltonians, we also present general (no-go) arguments about the absence of certain SOC terms.

We report on theoretical investigations of the spin-orbit coupling effects in fluorinated graphene. First-principles density functional calculations are performed for the dense and dilute adatom coverage limits. The dense limit is represented by the single-side semifluorinated graphene, which is a metal with spin-orbit splittings of about 10 meV. To simulate the effects of a single adatom, we also calculate the electronic structure of a 10 × 10 supercell, with one fluorine atom in the top position. Since this dilute limit is useful to study spin transport and spin relaxation, we also introduce a realistic effective hopping Hamiltonian, based on symmetry considerations, which describes the supercell bands around the Fermi level. We provide the Hamiltonian parameters which are best fits to the first-principles data. We demonstrate that, unlike for the case of hydrogen adatoms, fluorine's own spin-orbit coupling is the principal cause of the giant induced local spin-orbit coupling in graphene. The sp 3 hybridization induced transfer of spin-orbit coupling from graphene's σ bonds, important for hydrogenated graphene, contributes much less. Furthermore, the magnitude of the induced spin-orbit coupling due to fluorine adatoms is about 1000 times more than that of pristine graphene, and 10 times more than that of hydrogenated graphene. Also unlike hydrogen, the fluorine adatom is not a narrow resonant scatterer at the Dirac point. The resonant peak in the density of states of fluorinated graphene in the dilute limit lies 260 meV below the Dirac point. The peak is rather broad, about 300 meV, making the fluorine adatom only a weakly resonant scatterer.

We propose that the observed spin relaxation in bilayer graphene is due to resonant scattering by magnetic impurities. We analyze a resonant scattering model due to adatoms on both dimer and nondimer sites, finding that only the former give narrow resonances at the charge neutrality point. Opposite to singlelayer graphene, the measured spin-relaxation rate in the graphene bilayer increases with carrier density. Although it has been commonly argued that a different mechanism must be at play for the two structures, our model explains this behavior rather naturally in terms of different broadening scales for the same underlying resonant processes. Not only do our results-using robust and first-principles inspired parameters-agree with experiment, they also predict an experimentally testable sharp decrease of the spin-relaxation rate at high carrier densities. DOI: 10.1103/PhysRevLett.115.196601 PACS numbers: 72.80.Vp, 72.25.Rb Understanding spin relaxation is essential for designing spintronics devices [1,2]. Unfortunately, spin relaxation in graphene structures has been a baffling problem [3]. While experiments in both single layer graphene (SLG) [4][5][6][7][8][9][10][11] and bilayer graphene (BLG) [7,8] yield spin lifetimes on the 100-1000 ps time scale (the highest values achieved in graphene/h-BN structures [12,13]), theories based on realistic spin-orbit coupling and transport parameters predict lifetimes on the order of microseconds [14][15][16][17][18][19][20][21][22][23][24].While the magnitudes of the spin-relaxation rates of SLG and BLG are similar, the dependence of the rates on the electron density is opposite in the two systems. In SLG the spin-relaxation rate decreases with increasing the carrier density [5][6][7][8], in BLG the spin-relaxation rate increases [7,8]. Since the diffusivity in the investigated samples decreases with increasing the electron density, it has been a common practice to assign two different mechanisms to both structures: the Elliott-Yafet mechanism [25,26] to SLG [5,6,9,10] and Dyakonov-Perel mechanism [27] to BLG [7][8][9].The main problem with that assignment is quantitative. Spin-orbit coupling in graphene [28] is too weak to yield such a small spin-relaxation time. An explicit first-principles calculation [23] predicts that one would need 0.1% of adatoms to give a 100 ps spin lifetime. Recently a new mechanism for SLG was proposed [29] (see also Ref.[30]), based on resonant scattering off local magnetic moments. It gives the observed spin-relaxation times with as little as 1 ppm of local magnetic moments and also agrees with the experimental behavior for SLG of decreasing the spinrelaxation rate with increasing electron density. Where do these local moments come from? It was theoretically predicted that adatoms such as hydrogen [31,32], but also chemisorbed organic molecules [33] can be responsible. Experimentally it was demonstrated that hydrogen adatoms indeed induce local moments [34,35], but even untreated graphene flakes were shown to exhibit 20 ppm spin 1=2 para...

We present a combined DFT and model Hamiltonian analysis of spin-orbit coupling in graphene induced by copper adatoms in the bridge and top positions, representing isolated atoms in the dilute limit. The orbital physics in both systems is found to be surprisingly similar, given the fundamental difference in the local symmetry. In both systems the Cu p and d contributions at the Fermi level are very similar. Based on the knowledge of orbital effects we identify that the main cause of the locally induced spin-orbit couplings are Cu p and d orbitals. By employing the DFT+U formalism as an analysis tool we find that both the p and d orbital contributions are equally important to spin-orbit coupling, although p contributions to the density of states are much higher. We fit the DFT data with phenomenological tight-binding models developed separately for the top and bridge positions. Our model Hamiltonians describe the low-energy electronic band structure in the whole Brillouin zone and allow us to extract the size of the spin-orbit interaction induced by the local Cu adatom to be in the tens of meV. By application of the phenomenological models to Green's function techniques, we find that copper atoms act as resonant impurities in graphene with large lifetimes of 50 and 100 fs for top and bridge, respectively.

We present first-principles calculations of the electronic band structure and spin-orbit effects in graphene functionalized with methyl molecules in dense and dilute limits. The dense limit is represented by a 2 × 2 graphene supercell functionalized with one methyl admolecule. The calculated spin-orbit splittings are up to 0.6 meV. The dilute limit is deduced by investigating a large, 7 × 7, supercell with one methyl admolecule. The electronic band structure of this supercell is fitted to a symmetry-derived effective Hamiltonian, allowing us to extract specific hopping parameters including intrinsic, Rashba, and pseudospin inversion asymmetry spin-orbit terms. These proximity-induced spin-orbit parameters have magnitudes of about 1 meV, giant compared to pristine graphene whose intrinsic spin-orbit coupling is about 10 μeV. We find that the origin of this giant local enhancement is the sp 3 corrugation and the breaking of local pseudospin inversion symmetry, as in the case of hydrogen adatoms. Similarly to hydrogen, also methyl acts as a resonant scatterer, with a narrow resonance peak near the charge neutrality point. We also calculate STM-like images showing the local charge densities at different energies around methyl on graphene.

We present a theoretical study of resonance characteristics in graphene from adatoms with s or pz character binding in top, bridge, and hollow positions. The adatoms are described by two tightbinding parameters: on-site energy and hybridization strength. We explore a wide range of different magnitudes of these parameters by employing T -matrix calculations in the single adatom limit and by tight-binding supercell calculations for dilute adatom coverage. We calculate the density of states and the momentum relaxation rate and extract the resonance level and resonance width. The top position with a large hybridization strength or, equivalently, small on-site energy, induces resonances close to zero energy. The bridge position, compared to top, is more sensitive to variation in the orbital tight-binding parameters. Resonances within the experimentally relevant energy window are found mainly for bridge adatoms with negative on-site energies. The effect of resonances from the top and bridge positions on the density of states and momentum relaxation rate is comparable and both positions give rise to a power-law decay of the resonant state in graphene. The hollow position with s orbital character is affected from destructive interference, which is seen from the very narrow resonance peaks in the density of states and momentum relaxation rate. The resonant state shows no clear tendency to a power-law decay around the impurity and its magnitude decreases strongly with lowering the adatom content in the supercell calculations. This is in contrast to the top and bridge positions. We conclude our study with a comparison to models of pointlike vacancies and strong midgap scatterers. The latter model gives rise to significantly higher momentum relaxation rates than caused by single adatoms.