Download presentation

Presentation is loading. Please wait.

Published byAmber Singleton Modified over 2 years ago

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

Similar presentations

OK

Michael S. Fuhrer University of Maryland Graphene: Scratching the Surface Michael S. Fuhrer Professor, Department of Physics and Director, Center for Nanophysics.

Michael S. Fuhrer University of Maryland Graphene: Scratching the Surface Michael S. Fuhrer Professor, Department of Physics and Director, Center for Nanophysics.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on c language functions Ppt on preservation of public property definition Ppt on maternal health community Ppt on emotional intelligence by daniel goleman Ppt on electrical power transmission Ppt on grade of concrete Ppt on energy giving food proteins Ppt on tata trucks new Ppt on surface water flow Ppt on ancient indian mathematicians and their contributions