Spin-orbit coupling in graphene structures

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 Direct lattice Dual lattice Periodicstructure Bloch Theorem  Brillouinzone k set of good quantum numbers

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

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

10. Tight-binding approximation

11. Tight-binding approximation 7) Secular equation + fitting of TB parameters model parameters:

12. Graphene Direct lattice Dual lattice

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

14. 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

15. 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

16. Spin-orbit coupling

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

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 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)

20. 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

21. 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.

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

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

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

25. Graphene - Extrinsic SOC Graphene – always grown on substrate – background el. field 1.0 2.44 4.0 0 E [V/nm]

26. 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

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

28. Graphene - Extrinsic SOC Extrinsic SOC – atomism: - multi-TB perturbation theory Direct lattice Dual lattice 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