Graphene – basic (orbital) model Basic TB-model with p z - orbitals Direct lattice Dual lattice structural function of the hexagonal lattice: low energy Hamiltonian: expansion at
Graphene – basic (orbital) model “relativistic” Hamiltonian Direct lattice Dual lattice - acts in pseudospin degrees of freedom – what is that? - seemingly 2D massless fermions - linear dispersion relation - BUT no-spin degrees of freedom, (when spin ) pseudospin up/down – amplitude to find e - on sublattice A/B
Spintronics - tunable & strong/week SOC spin relaxation (quantum) spin Hall effect - TI magneto-anisotropy weak (anti-)localization SOC - quintessence of Spin-orbit coupling
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 Gmitra et al., PRB (2009) symmetry arguments: Kane, Mele, PRL (2005) McClure, Yafet, Proc. of 5 th Conf. on Carbon, Pergamon, Vol.1, pp 22-28, 1962 physics behind d-orbitals Ab-initioTheory next-nearest neighbor interaction
How to derive effective SOC? Direct lattice Dual lattice Group theory – invariance: - translations (obvious) - point group D 6h – symmetry group of hexagon - time-reversal: k -k, , - Graphene - Intrinsic SOC
How to compute matrix elements? - go to atomic (Wannier) orbitals Direct lattice Dual lattice Graphene - Intrinsic SOC - employing all D 6h elements + TR one non-zero matr. elem.
Full spin-orbit coupling Hamiltonian Direct lattice Dual lattice Graphene - Intrinsic SOC linearized SOC Hamiltonian at Gmitra et al., PRB (2009)
What will happen if ….??? Direct lattice Dual lattice Graphene – as Topological Insulator Kane, Mele, PRL (2005)
Graphene - Extrinsic SOC Graphene – always grown on substrate – background el. field E [V/nm]
How to derive effective SOC? Direct lattice Dual lattice Group theory – invariance: - translations (obvious) - point group C 6v – symmetry group of hexagon without the space inversion - time-reversal Graphene - Extrinsic SOC
Full spin-orbit coupling Hamiltonian Graphene - Extrinsic SOC linearized SOC Hamiltonian at
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 (2009) Konschuh et al., PRB (2010) Konschuh et al., PRB (2012) Gmitra, Kochan, Fabian – work in progress