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Spin-orbit coupling in graphene structures. D. Kochan , M. Gmitra , J. Fabian. Star á Lesná , 25.8.2012. Outline. Предварительные сведения Bloch vs. Wannier Tight-binding approximation = LCAO Graphene Spin-orbit-interaction in Graphene

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## Spin-orbit coupling in graphene structures

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**Spin-orbit coupling in graphene**structures D. Kochan, M. Gmitra, J. Fabian Stará Lesná, 25.8.2012**Outline**• Предварительные сведения • Bloch vs.Wannier • Tight-binding approximation = LCAO • Graphene • Spin-orbit-interaction in Graphene • What we are doing ….**Bloch vs.Wannier**Direct lattice Dual lattice Periodicstructure Bloch Theorem Brillouinzone k set of good quantum numbers**Bloch vs.Wannier**Bloch states: – delocalized & orthogonal – labeled by the momentum k Wannier states: – localized & orthogonal – labeled by the lattice vector R**Tight-binding approximation**1) Wannier states basis = local atomic orbitals 2) Bloch states basis = Bloch sum of local atomic orbitals**Tight-binding approximation**3) General solution: 4) Matrix(-secular) equation: How to compute ??-matrix elements?**Tight-binding approximation**5) The heart of TB approx: -nearest & next-nearest neighbors only few terms that are lowest in |R|**Tight-binding approximation**5) The heart of TB approx: -nearest & next-nearest neighbors only few terms that are lowest in |R|**Tight-binding approximation**6) Further simplification – point (local) group symmetries - elements – square lattice non-zero elements zero elements**Tight-binding approximation**7) Secular equation + fitting of TB parameters model parameters:**Graphene**Direct lattice Dual lattice**Graphene – basic (orbital) energetics**Gmitra, Konschuh, Ertler, Ambrosch-Draxl, Fabian, PRB 80 235431 (2009) Konschuh, Gmitra, Fabian, PRB 82 245412 (2010)**Graphene – basic (orbital) model**Basic TB-model with pz- orbitals structural function of the hexagonal lattice: Direct lattice Dual lattice low energy Hamiltonian: expansion at**Graphene – basic (orbital) model**“relativistic” Hamiltonian - seemingly 2D massless fermions - linear dispersion relation - BUT no-spin degrees of freedom, (when spin ) Direct lattice Dual lattice - acts in pseudospin degrees of freedom – what is that? pseudospin up/down – amplitude to find e-on sublattice A/B**Spin-orbit coupling**Spintronics - tunable & strong/week SOC SOC - quintessence of • spin relaxation • (quantum) spin Hall effect - TI • magneto-anisotropy • weak (anti-)localization**Intra-atomic spin-orbit coupling**Questions: • How does SOC modify in periodically arrayed structures? • Is (and by how much) SOC enhanced in carbon allotropes? • How to further stimulate and control SOC?**Graphene - Intrinsic SOC**Ab-initio Theory next-nearest neighbor interaction symmetry arguments: McClure, Yafet, Proc. of 5th Conf. on Carbon, Pergamon, Vol.1, pp 22-28, 1962 Kane, Mele, PRL 95 226801 (2005) physics behind d-orbitals Gmitra et al., PRB 80 235431 (2009)**Graphene - Intrinsic SOC**How to derive effective SOC? Group theory – invariance: - translations (obvious) - point group D6h – symmetry group of hexagon - time-reversal: k -k, , - Direct lattice Dual lattice**Graphene - Intrinsic SOC**How to compute matrix elements? - go to atomic (Wannier) orbitals Direct lattice Dual lattice - employing all D6h elements + TR one non-zero matr. elem.**Graphene - Intrinsic SOC**Full spin-orbit coupling Hamiltonian Direct lattice Dual lattice linearized SOC Hamiltonian at Gmitra et al., PRB 80 235431 (2009)**Graphene - Intrinsic SOC**Intrinsic SOC – atomism: - multi-TB perturbation theory Direct lattice Dual lattice Konschuh, Gmitra, Fabian, PRB 82 245412 (2010)**Graphene – as Topological Insulator**What will happen if ….??? Kane, Mele, PRL 95 226801 (2005) Direct lattice Dual lattice**Graphene - Extrinsic SOC**Graphene – always grown on substrate – background el. field 1.0 2.44 4.0 0 E [V/nm]**Graphene - Extrinsic SOC**How to derive effective SOC? Group theory – invariance: - translations (obvious) - point group C6v – symmetry group of hexagon without the space inversion - time-reversal Direct lattice Dual lattice**Graphene - Extrinsic SOC**Full spin-orbit coupling Hamiltonian linearized SOC Hamiltonian at**Graphene - Extrinsic SOC**Extrinsic SOC – atomism: - multi-TB perturbation theory Direct lattice Dual lattice Konschuh, Gmitra, Fabian, PRB 82 245412 (2010)**C O N C L U S I O N**• Graphene: • - intrinsic SOC dominated by d-orbitals • - detailed ab-initio and multi-TB-studies • Bilayer graphene: • - symmetry derived SO Hamiltonian • - detailed ab-initio and model studies - band structure & SO-splittings • - SOC comparable with single-layered graphene • Hydrogenized graphene structures: SH & SI • - detailed ab-initio, symmetry and TB-model studies • - substantial SO-splittings compared to single-layered graphene Gmitra et al., PRB 80 235431 (2009) Konschuh et al., PRB 82 245412 (2010) Konschuh et al., PRB 85 1145423 (2012) Gmitra, Kochan, Fabian – work in progress

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