The isolation of graphene has triggered an avalanche of studies into the spin-dependent physical properties of this material and of graphene-based spintronic devices. Here, we review the experimental and theoretical state-of-art concerning spin injection and transport, defect-induced magnetic moments, spin-orbit coupling and spin relaxation in graphene. Future research in graphene spintronics will need to address the development of applications such as spin transistors and spin logic devices, as well as exotic physical properties including topological states and proximity-induced phenomena in graphene and other two-dimensional materials.

We present k· p Hamiltonians parametrised by ab initio density functional theory calculations to describe the dispersion of the valence and conduction bands at their extrema (the K, Q, Γ, and M points of the hexagonal Brillouin zone) in atomic crystals of semiconducting monolayer transition metal dichalcogenides. We discuss the parametrisation of the essential parts of the k· p Hamiltonians for MoS 2 , MoSe 2 , MoTe 2 , WS 2 , WSe 2 , and WTe 2 , including the spin-splitting and spin-polarisation of the bands, and we briefly review the vibrational properties of these materials. We then use k· p theory to analyse optical transitions in two-dimensional transition metal dichalcogenides over a broad spectral range that covers the Van Hove singularities in the band structure (the M points). We also discuss the visualisation of scanning tunnelling microscopy maps. PACS numbers:Contents 1 Introduction 2 2 Lattice parameters, band-structure calculations and vibrational properties 43 Band-edge energy differences and spin-splittings 7 4 Valence band width D vb 9 arXiv:1410.6666v3 [cond-mat.mes-hall] 6 Apr 2015 k · p theory for 2D TMDCs to be studied without constructing slabs in three-dimensionally periodic cells and the resulting electronic spectra are free of plane-wave continua. All our fleur calculations were carried out with a cut-off k max of 10.6 eV −1 for the plane-wave basis set and 144 k points corresponding to a 12 × 12 × 1 Monkhorst-Pack grid in the irreducible wedge of the BZ. Muffin-tin radii of 1.0, 1.21, 1.27, 1.27, and 1.27Å were used for S, Se, Te, Mo, and W, respectively. We note that considering local orbitals for Mo (s, p), Se (s, p, d), and W (s, p, f ) to improve the linearised augmented plane-wave basis proved to be crucial for a correct description of the excited states. We used the Perdew-Burke-Ernzerhof (PBE) generalised gradient approximation [83] to the exchange-correlation potential. The structures were relaxed (with the effects of SOC included) until the forces were less than 0.0005 eV/Å. The calculated values of a 0 and d S−S for monolayer TMDCs are shown in Table 1 and compared to measured values for the corresponding bulk materials. The lattice parameters obtained from the first of the DFT approaches described above are shown in the rows labelled by "(HSE)", the ones from the second approach are in the rows labelled by "(PBE)". "(Exp)" indicates experimental results found in the literature. Although there is some scatter in the experimental data, Table 1 suggests that using the HSE06 functional to relax the monolayer crystal structure leads to a good agreement with the room-temperature empirical bulk a 0 values. On the other hand, the PBE functional seems to slightly overestimates a 0 . However, the situation is less clear in the case of d X−X . We note that both the HSE06 and the PBE results are in good agreement with Reference [84].Recent experiments show that the energy of the photoluminescence peak is quite sensitive to the temperature [5,85,86], which can be understood in terms of th...

The electronic band structure of graphene in the presence of spin-orbit coupling and transverse electric field is investigated from first principles using the linearized augmented plane-wave method. The spin-orbit coupling opens a gap of 24 eV ͑0.28 K͒ at the K͑KЈ͒ point. It is shown that the previously accepted value of 1 eV, coming from the -mixing, is incorrect due to the neglect of d and higher orbitals whose contribution is dominant due to symmetry reasons. The transverse electric field induces an additional ͑extrinsic͒ Bychkov-Rashba-type splitting of 10 eV ͑0.11 K͒ per V/nm, coming from the -mixing. A "miniripple" configuration with every other atom shifted out of the sheet by less than 1% differs little from the intrinsic case.The fascination with graphene, 1 the one-atom-thick allotrope of carbon, comes from its two-dimensional structure as well as from its unique electronic properties. 2-7 The latter originate from the specific electronic band structure at the Fermi level: electrons move with a constant velocity, apparently without mass and a spectral gap. Analogy with massless Dirac fermions is often drawn, presenting graphene as a solid-state toy for relativistic quantum mechanics. Ironically, this nice analogy is broken by the relativistic effects themselves. In particular, the interaction of the orbital and spin degrees of freedom, spin-orbit coupling, gives the electrons in graphene a finite mass and induces a gap in the spectrum. How large is the gap and which orbital states contribute to it? This question is crucial for knowing graphene's bandstructure topology, understanding its spin transport and spin relaxation properties, 8,9 or for assessing prospects of graphene for spin-based quantum computing. 10 By performing comprehensive first principles calculations we predict the spectral gap and establish the relevant electronic spectrum of graphene in the presence of external transverse electric field. We find that realistic electric fields can tune among different band structure topologies with important ramifications for the physics of graphene.Carbon atoms in graphene are arranged in a honeycomb lattice which comprises two triangular Bravais lattices; the unit cell has two atoms. The corresponding reciprocal lattice is again honeycomb, with two nonequivalent vertices K and KЈ which are the Fermi momenta of a neutral graphene. The states relevant for transport are concentrated in two touching cones with the tips at K͑KЈ͒-the Dirac points-as illustrated in Fig. 1. The corresponding Bloch states are formed mainly by the carbon valence p z orbitals ͑the z axis is perpendicular to the graphene plane͒ forming the two bands ͑cones͒. The other three occupied valence states of carbon form the deep-lying bands by sp 2 hybridization; these are responsible for the robustness of graphene's structure. The states in the lower cones are holelike or valencelike, while the upper cone states are electronlike or conductionlike, borrowing from semiconductor terminology. These essentials of the electronic band struct...

The spin-orbit coupling in graphene induces spectral gaps at the high-symmetry points. The relevant gap at the ⌫ point is similar to the splitting of the p orbitals in the carbon atom, being roughly 8.5 meV. The splitting at the K point is orders of magnitude smaller. Earlier tight-binding theories indicated the value of this intrinsic gap of 1 eV, based on the -coupling. All-electron first-principles calculations give much higher values, between 25 and 50 eV, due to the presence of the orbitals of the d symmetry in the Bloch states at K. A realistic multiband tight-binding model is presented to explain the effects the d orbitals play in the spin-orbit coupling at K. The -coupling is found irrelevant to the value of the intrinsic spin-orbit-induced gap. On the other hand, the extrinsic spin-orbit coupling ͑of the Bychkov-Rashba type͒, appearing in the presence of a transverse electric field, is dominated by the -hybridization, in agreement with previous theories. Tightbinding parameters are obtained by fitting to first-principles calculations, which also provide qualitative support for the model when considering the trends in the spin-orbit-induced gap in graphene under strain. Finally, an effective single-orbital next-nearest-neighbor hopping model accounting for the spin-orbit effects is derived.

Proximity orbital and spin-orbital effects of graphene on monolayer transition-metal dichalcogenides (TMDCs) are investigated from first-principles. The Dirac band structure of graphene is found to lie within the semiconducting gap of TMDCs for sulfides and selenides, while it merges with the valence band for tellurides. In the former case, the proximity-induced staggered potential gaps and spin-orbit couplings (all on the meV scale) of the Dirac electrons are established by fitting to a phenomenological effective Hamiltonian. While graphene on MoS 2 , MoSe 2 , and WS 2 has a topologically trivial band structure, graphene on WSe 2 exhibits inverted bands. Using a realistic tight-binding model we find topologically protected helical edge states for graphene zigzag nanoribbons on WSe 2 , demonstrating the quantum spin Hall effect. This model also features "half-topological states," which are protected against time-reversal disorder on one edge only.

Hybrids of graphene and two-dimensional transition-metal dichalcogenides (TMDCs) have the potential to bring graphene spintronics to the next level. As we show here by performing first-principles calculations of graphene on monolayer MoS 2 , there are several advantages of such hybrids over pristine graphene. First, Dirac electrons in graphene exhibit a giant global proximity spin-orbit coupling, without compromising the semimetallic character of the whole system at zero field. Remarkably, these spin-orbit effects can be very accurately described by a simple effective Hamiltonian. Second, the Fermi level can be tuned by a transverse electric field to cross the MoS 2 conduction band, creating a system of coupled massive and massless electron gases. Both charge and spin transport in such systems should be unique. Finally, we propose to use graphene/TMDC structures as a platform for optospintronics, in particular, for optical spin injection into graphene and for studying spin transfer between TMDCs and graphene.

First-principles calculations of the spin-orbit coupling in graphene with hydrogen adatoms in dense and dilute limits are presented. The chemisorbed hydrogen induces a giant local enhancement of spin-orbit coupling due to sp(3) hybridization which depends strongly on the local lattice distortion. Guided by the reduced symmetry and the local structure of the induced dipole moments, we use group theory to propose realistic minimal Hamiltonians that reproduce the relevant spin-orbit effects for both single-side semihydrogenated graphene (graphone) and for a single hydrogen adatom in a large supercell. The principal linear spin-orbit band splittings are driven by the breaking of the local pseudospin inversion symmetry and the emergence of spin flips on the same sublattice.

Theory of the electron spin relaxation in graphene on the SiO2 substrate is developed. Charged impurities and polar optical surface phonons in the substrate induce an effective random BychkovRashba-like spin-orbit coupling field which leads to spin relaxation by the D'yakonov-Perel' mechanism. Analytical estimates and Monte Carlo simulations show that the corresponding spin relaxation times are between micro-to milliseconds, being only weakly temperature dependent. It is also argued that the presence of adatoms on graphene can lead to spin lifetimes shorter than nanoseconds.PACS numbers: 72.25. Rb, 73.61.Wp, 73.50.Bk Since the experimental realization of graphene, a single stable 2D-monolayer of carbon atoms arranged in a honeycomb lattice, considerable research has been done to enlighten its peculiar electronic transport properties originating from the Dirac-like band structure at the K and K ′ points in the momentum space [1]. Long spin relaxation times and phase coherence lengths in graphene are expected based on the weak atomic spin-orbit coupling in carbon (Z = 6). However, recent spin injection measurements based on a non-local spin valve geometry [2, 3, 4] revealed surprisingly short spin relaxation times of only about 100-200 ps, being only weakly dependent on the charge density and temperature. These results appear puzzling, although the low mobilities of the samples (about 2000 cm 2 /Vs) suggest that the measured spin relaxation times are likely due to extrinsic effects [2].Very recent experiments on the charge transport in graphene affirmed the importance of the underlying substrate [5,6,7]. At low temperatures the transport properties have been shown to be dominated by scattering from the charged impurities residing in the substrate [6,8]. The conductivity of graphene placed on a SiO 2 substrate starts to decrease above 200 K. The observed temperature and density dependence of the resistivity are most likely explained by remote phonon scattering due to occurrence of polar optical surface modes in the substrate [9, 10, 11].These findings naturally raise the question if (i) charged impurities and (ii) remote surface phonons are also relevant for the spin relaxation in graphene. As argued here both mechanisms provide a temperaturedependent, random spin-orbit coupling field, which limits the spin relaxation via the D'yakonov-Perel' (DP) mechanism [12,13,14]. The calculated spin relaxation times are micro to milliseconds. In addition, we give estimates for the spin relaxation times due to the possible presence of adatoms on graphene. For reasonable adatom densities the spin lifetimes can be lower than nanoseconds.Several other mechanisms have already been investigated theoretically, such as the spin relaxation due to the corrugations (ripples) of graphene and due to exchange interaction with local magnetic moments [15], or spin-