Transcript Slide 1

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
Direct lattice
Periodic structure
Dual lattice
Bloch Theorem
Brillouin zone
k set of good
quantum numbers
Bloch vs. Wannier
Bloch 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 approximation
3) General solution:
4) Matrix(-secular) equation:
How to compute ??-matrix elements?
Tight-binding approximation
5) The heart of TB approx: -nearest & next-nearest neighbors
only few terms that are lowest in |R|
Tight-binding approximation
5) The heart of TB approx: -nearest & next-nearest neighbors
only few terms that are lowest in |R|
Tight-binding approximation
6) 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:
Graphene
Direct lattice
Dual lattice
Graphene – basic (orbital) energetics
Gmitra, Konschuh, Ertler, Ambrosch-Draxl, Fabian,
PRB 80 235431 (2009)
Konschuh, Gmitra, Fabian, PRB 82 245412 (2010)
Graphene – basic (orbital) model
Basic TB-model with pz- orbitals
Direct lattice
structural function of the hexagonal lattice:
low energy Hamiltonian: expansion at
Dual lattice
Graphene – basic (orbital) model
“relativistic” Hamiltonian
- seemingly 2D massless fermions
Direct lattice
- linear dispersion relation
- BUT no-spin degrees of freedom, (when spin
Dual lattice
)
- acts in pseudospin degrees of freedom – what is that?
pseudospin up/down – amplitude to find e- on sublattice A/B
Spin-orbit coupling
Spin-orbit coupling
Spintronics - tunable & strong/week SOC
SOC - quintessence of
• spin relaxation
• (quantum) spin Hall effect - TI
• magneto-anisotropy
• weak (anti-)localization
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
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)
Gmitra et al., PRB 80 235431 (2009)
physics behind
d-orbitals
Graphene - Intrinsic SOC
How to derive effective SOC?
Group theory – invariance:
Direct lattice
- translations (obvious)
- point group D6h – symmetry group of hexagon
- time-reversal: k
-k,
,
-
Dual lattice
Graphene - Intrinsic SOC
How to compute matrix elements?
- go to atomic (Wannier) orbitals
Direct lattice
- employing all D6h elements + TR
Dual lattice
one non-zero matr. elem.
Graphene - Intrinsic SOC
Full spin-orbit coupling Hamiltonian
Direct lattice
linearized SOC Hamiltonian at
Gmitra et al., PRB 80 235431 (2009)
Dual lattice
Graphene - Intrinsic SOC
Intrinsic SOC – atomism:
- multi-TB perturbation theory
Direct lattice
Konschuh, Gmitra, Fabian, PRB 82 245412 (2010)
Dual lattice
Graphene – as Topological Insulator
What will happen if ….???
Kane, Mele, PRL 95 226801 (2005)
Direct lattice
Dual lattice
Graphene - Extrinsic SOC
Graphene – always grown on substrate – background el. field
0
1.0
2.44
4.0
E [V/nm]
Graphene - Extrinsic SOC
How to derive effective SOC?
Group theory – invariance:
Direct lattice
- translations (obvious)
- point group C6v – symmetry group of hexagon
without the space inversion
- time-reversal
Dual lattice
Graphene - Extrinsic SOC
Full spin-orbit coupling Hamiltonian
linearized SOC Hamiltonian at
Graphene - Extrinsic SOC
Extrinsic SOC – atomism:
- multi-TB perturbation theory
Direct lattice
Konschuh, Gmitra, Fabian, PRB 82 245412 (2010)
Dual lattice
CONCLUSION
• Graphene:
- intrinsic SOC dominated by d-orbitals
- detailed ab-initio and multi-TB-studies
Gmitra et al., PRB 80 235431 (2009)
Konschuh et al., PRB 82 245412 (2010)
• Bilayer graphene:
- symmetry derived SO Hamiltonian
- detailed ab-initio and model studies - band structure & SO-splittings
- SOC comparable with single-layered graphene
Konschuh et al., PRB 85 1145423 (2012)
• Hydrogenized graphene structures: SH & SI
- detailed ab-initio, symmetry and TB-model studies
- substantial SO-splittings compared to single-layered graphene
Gmitra, Kochan, Fabian – work in progress