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

Periodic structure Bloch Theorem Brillouin zone

k set of goodquantum numbers

Direct lattice Dual lattice

Bloch vs. WannierBloch 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 approximation3) General solution:

4) Matrix(-secular) equation:

How to compute ??-matrix elements?

Tight-binding approximation5) The heart of TB approx: -nearest & next-nearest neighbors

only few terms that are lowest in |R|

Tight-binding approximation

only few terms that are lowest in |R|

5) The heart of TB approx: -nearest & next-nearest neighbors

Tight-binding approximation6) Further simplification – point (local) group symmetries

- elements – square lattice

non-zero elements zero elements

Tight-binding approximation

Tight-binding approximation

7) Secular equation + fitting of TB parameters

model parameters:

Direct lattice Dual lattice

Graphene

Graphene – basic (orbital) energetics

Konschuh, Gmitra, Fabian, PRB 82 245412 (2010)

Gmitra, Konschuh, Ertler, Ambrosch-Draxl, Fabian, PRB 80 235431 (2009)

Graphene – basic (orbital) model

Basic TB-model with pz- 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

Spin-orbit coupling

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 80 235431 (2009)

symmetry arguments:

Kane, Mele, PRL 95 226801 (2005)

McClure, Yafet, Proc. of 5th Conf. on Carbon, Pergamon, Vol.1, pp 22-28, 1962

physics behind d-orbitals

Ab-initio Theory

next-nearest neighbor interaction

How to derive effective SOC?

Direct lattice Dual lattice Group theory – invariance:

- translations (obvious)

- point group D6h – 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 D6h 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 80 235431 (2009)

Intrinsic SOC – atomism:

- multi-TB perturbation theory

Direct lattice Dual lattice

Graphene - Intrinsic SOC

Konschuh, Gmitra, Fabian, PRB 82 245412 (2010)

What will happen if ….???

Direct lattice Dual lattice

Graphene – as Topological Insulator

Kane, Mele, PRL 95 226801 (2005)

Graphene - Extrinsic SOC

Graphene – always grown on substrate – background el. field

0 1.0 2.44 4.0

E [V/nm]

How to derive effective SOC?

Direct lattice Dual lattice Group theory – invariance:

- translations (obvious)

- point group C6v – 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

Extrinsic SOC – atomism:

- multi-TB perturbation theory

Direct lattice Dual lattice

Graphene - Extrinsic SOC

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