1. Spin-orbit coupling in graphene structures D. Kochan, M. Gmitra, J. Fabian Stará Lesná, 25.8.2012

2. Outline Предварительные сведения Bloch vs. Wannier Tight-binding approximation = LCAO Graphene Spin-orbit-interaction in Graphene What we are doing ….

3. Bloch vs. Wannier Periodic structure  Bloch Theorem  Brillouin zone k  set of good quantum numbers Direct lattice Dual lattice

4. Bloch vs. Wannier Bloch states: – delocalized & orthogonal – labeled by the momentum k Wannier states: – localized & orthogonal – labeled by the lattice vector R

5. Tight-binding approximation 1) Wannier states basis = local atomic orbitals 2) Bloch states basis = Bloch sum of local atomic orbitals

6. Tight-binding approximation 3) General solution: 4) Matrix(-secular) equation: How to compute ??-matrix elements?

7. Tight-binding approximation 5) The heart of TB approx: -nearest & next-nearest neighbors only few terms that are lowest in |R|

8. Tight-binding approximation only few terms that are lowest in |R| 5) The heart of TB approx: -nearest & next-nearest neighbors

9. Tight-binding approximation 6) Further simplification – point (local) group symmetries - elements – square lattice non-zero elementszero elements

10. Tight-binding approximation

11. 7) Secular equation + fitting of TB parameters model parameters:

12. Direct lattice Dual lattice Graphene

13. Graphene – basic (orbital) energetics Konschuh, Gmitra, Fabian, PRB 82 245412 (2010) Gmitra, Konschuh, Ertler, Ambrosch-Draxl, Fabian, PRB 80 235431 (2009)

14. 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

15. 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

16. Spin-orbit coupling

17. Spintronics - tunable & strong/week SOC spin relaxation (quantum) spin Hall effect - TI magneto-anisotropy weak (anti-)localization SOC - quintessence of Spin-orbit coupling

18. 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?

19. Graphene - Intrinsic SOC Gmitra et al., PRB 80 235431 (2009) symmetry arguments: Kane, Mele, PRL 95 226801 (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

20. 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

21. 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.

22. Full spin-orbit coupling Hamiltonian Direct lattice Dual lattice Graphene - Intrinsic SOC linearized SOC Hamiltonian at Gmitra et al., PRB 80 235431 (2009)

23. Intrinsic SOC – atomism: - multi-TB perturbation theory Direct lattice Dual lattice Graphene - Intrinsic SOC Konschuh, Gmitra, Fabian, PRB 82 245412 (2010)

24. What will happen if ….??? Direct lattice Dual lattice Graphene – as Topological Insulator Kane, Mele, PRL 95 226801 (2005)

25. Graphene - Extrinsic SOC Graphene – always grown on substrate – background el. field 0 1.0 2.444.0 E [V/nm]

26. 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

27. Full spin-orbit coupling Hamiltonian Graphene - Extrinsic SOC linearized SOC Hamiltonian at

28. Extrinsic SOC – atomism: - multi-TB perturbation theory Direct lattice Dual lattice Graphene - Extrinsic SOC Konschuh, Gmitra, Fabian, PRB 82 245412 (2010)

29. 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