View
216
Download
0
Category
Preview:
Citation preview
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
Recommended