Band-structure topologies of graphene: Spin-orbit coupling effects from first principles

Abstract

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...

Structure of graphene and its disorders: a review

“…The asymmetry between the conduction band () and the valence band (), which is especially pronounced in the vicinity of the point, which is attributed to the non-zero overlap parameter . However, the electronic band structure of graphene can be simply altered by applying electric field [6,57–61] or providing substrates [62,63], and precisely engineered by introducing disorders into the hexagonal lattice [64–68], which will be discussed in detail in later sections.…”

Edge states in graphene-like systems

“…Second order interband scattering gives the first non-zero contribution but it is not obvious how to include it in the Hamiltonian. Further work, 49 showed that inclusion of d − channels lead to a first order contribution in SOC. In their seminal work, based on symmetry considerations, Kane and Mele postulated that the effective Hamiltonian for SOC in the subspace of the π orbitals would be given by a spin dependent second neighbor hopping, eq.…”

“…The original atomic spin-orbit term, λ S · L, has a vanishing value on the p z . However, higher order processes involving orbitals from the p x and p y manifold, or even from the d manifold, 49 will add an effective SOC term to the Hamiltonian.…”

Spin-dependent Klein tunneling in graphene: Role of Rashba spin-orbit coupling

“…The latter is induced by the electric field perpendicular to the graphene plane, which can be externally controlled, and resembles the Rashba model 12,13 for the two-dimensional electron gas. Agreement has been achieved, based on first-principles calculations, 9,10 that the intrinsic SOC term opens a gap of the order of 2λ I ≈ 24 μeV, while the Rashba SOC removes the spin degeneracy and creates a spin-splitting 2λ R at the K and K points that has a linear dependence on an external electric field E with the slope of about 100 μeV per V/Å of E. Under a strong gate voltage, the Rashba coupling may in principle dominate the intrinsic SOC in MLG. 9,10 The low-energy spectrum of MLG plus the Rashba coupling (MLG + R) was derived by Rashba, 14 based on the Kane-Mele model 7 (i.e., an effective Dirac Hamiltonian).…”

“…6 The question about the role of SOC effects in graphene then naturally emerged, including the proposal of graphene as a topological insulator, 7 which attracted the attention of various first-principles-based studies. [8][9][10] SOC in MLG includes an intrinsic and an extrinsic term. The former reflects the inherent asymmetry of electron hopping between next nearest neighbors 7 (i.e., a generalization of Haldane's model 11 ).…”