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1 Spin-pálya kölcsönhatás grafénben, fotoelektron-spektroszkópia Rakyta Péter EÖTVÖS LORÁND TUDOMÁNYEGYETEM, KOMPLEX RENDSZEREK FIZIKÁJA TANSZÉK 1

2 Introduction to graphene Sp 2 hybridization p z orbitals Low energy excitation around K points with linear dispersion. Spin-degenerated energy bands (p z electrons): Dirac-cones: Conduction Band E k y Valance Band Fermi level 2. oldal

3 Nobel prize in 2010 Andre Geim (University of Manchester in Konstantin Novoselov the UK) The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Physics for 2010 for groundbreaking experiments regarding the two-dimensional material graphene. 3. oldal

4 Spin-orbit coupling in graphene Intrinsic spin-orbit (ISO) originates fromc atoms: SO 12µeV DFT calculations: D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B 74, (2006); M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl, and J. Fabian, Phys. Rev. B 80, (2009)... Rashba spin-orbit coupling from external electric field to the sheet. SARPES (Spin ans Angle Resolved Photoemission Spectroscopy) on Graphene/Au/Ni(111) structure:λ R 4meV Varykhalov at al., PRL 101, (2008) For realistic systems: SO λ R model. We neglect ISO from the 4. oldal

5 Models of Rashba spin-orbit coupling (RSO) in graphene at the K point H 0 R = 3 2 λ R(ˆσ ŝ) z Kane, C. L. and Mele, E. J., PRL 95, (2005). momentum independent RSO based on symmetry considerations H R (k) from tight-binding model with longwave approximatioin: H R = ( vλˆp iλ R 0 0 3iλ R 0 0 v λˆp ) Ψ = ( ) A ˆp± = ˆp B x ±iˆp y A B v λ = 3λ Rr C C 2 v λˆp ± λ R k are missing in HamiltonianH 0 R (λ R γ,k K) Theese terms are responsible for trigonal warping (TW) effect (as in bilayer), and lead to k-dependent bandsplitting. 5. oldal

6 Mapping to Bilayer Graphene including trigonal warping (TW) Bilayer with TW ( 0 vfˆp 0 v 3ˆp + HK B = v Fˆp + 0 γ γ 1 0 v Fˆp v 3ˆp 0 v Fˆp + 0 E B (γ 1,k,ϕ) ) H K = Graphene with RSO ( 0 vfˆp 0 v λˆp + v Fˆp + 0 3λ R i 0 0 3λ R i 0 v Fˆp v λˆp 0 v Fˆp + 0 E(λ R,k,ϕ) = E B (3λ R,k,π +ϕ) ) 6. oldal

7 k-dependent bandsplitting along ΓK E(k) = E 2 (k) E 1 (k), Solid lines: theory including TW E 1,2 (k) are two valance bands Dashed line: theory without TW Errorbars: Spin and Angle Resolved Photoemission Spectroscopy Graphene/Au/Ni(111) structure quasifreestanding graphene (Varykhalov at al., PRL 101, (2008).) 7. oldal

8 Spin structure in the Brillouin zone Energy contours in lower conduction band: P. Rakyta, A. Kormányos, J. Cserti, PRB 82, (2010). The spin structure manifests rotational symmetry (TW leads to higher order correction): ϕ = α+ π 2 cos3α k r C C, 2 s ( /2)2 s = k r C C 2 λ R /γ sin3α k 2 rc C 2 2 λ R /γ 8. oldal

9 Experimental equipment 9. oldal

10 Ionization spectrum from Photoemission Spectroscopy O 2 H 2 O Finestructure: vibration modes of the ionizated molecules Photoemission Spectroscopy of solids: contributions from surface layers ideal for 2D systems. 10. oldal

11 Spin and Angle Resolved Photoemission Spectroscopy (SARPES) F. Meier, J. H. Dill and J. Osterwalder, New Journal of Varykhalov at al., PRL 101, (2008) Physics 11, (2009) 2 detectors detect the spin-polarization in different planes. Restore 3D spin from the projections to the planes. Anisotropic distribution of photoelectrons. Graphene/Au/Ni(111) - Quasifreestanding Graphene 11. oldal

12 SARPES for graphene Calculations based on Fermi s golden rule: Bloch-electron (bandµ) k,µ in graphene+em dipole interactionh int = e m Detected photoelectron with spinσ: p,σ = (H int ) µ σ k p eipr/ σ (H int ) µ σ k p Ap ( Ψ µ Aσ (k)+eigτ Ψ µ Bσ (k) ) i A with momentum and energy conservation. G: reciprocal lattice vector, τ : A B vector Sublattice interference in physical quantities: Mucha-Kruczyński et al., PRB 77, (2008). Ô p = σ={, } p,σ Ô p,σ Measurements on energy-contour with precision E bandstructure k y E = 0 E > 0 E > 0 + A B K 12. oldal

13 Intensity of photoelectrons in the Brillouin Zone 3λ R E K Reciprocal lattice vectors: G= m b + m b [m, m ] λ 2 R = 66 mev = 0 mev 1 E = 650 mev k y [A 1 ] 1 b 2 K: [ 1,0] K: [ 1, 1] K: [0,0] K: [ 1, 1] K: [0,0] 2 3 K: [0, 1] b [A 1 ] InequivalentK andk points (likea B sublattice) 13. oldal

14 Graphene/Y/Ni(111) structures with different Y atoms Y = Au band structure of graphene is unaffected within experimental precision: ideal graphene + RSO coupling ( Quasifreestanding graphene, Rashba spin splitting 13 mev) Y = (Cu,Ag) measured gap ( 200 mev) at the Dirac-point. The ideal model is not enough for description. (Rashba spin splitting 100 mev) sublattice asymmetry: differenta B on-site energy E H AB = 2 (σ z Î 2 ) not coupled to the spin gap opens at the Dirac-points 14. oldal

15 Sublattice asymmetry in graphene with RSO ( ) s z (k 0) = ±(1 δ 0, ) 1 γ2 2λRk 2 r 2 2 C C s z (k ) = ± λ R 3γ 2 k 2 rc C 2 = 0 mev = 200 mev λ = 66 mev R 0.05 Spin structure around the K point: Out of plane spin polarization around the center (K point) λ=30mev, =20meV Dark Corridor E E = 120 mev E E = 120 mev = 0 mev = 200 mev λ R = 66 mev r C C Sublattice interference in the spin structure Dark Corridor 0 k y r C C E E 0 k y r C C E = 660 mev E = 660 mev r C C 0 = 0 case: F. Kuemmeth, E. I. Rashba, PRB 80, (R), (2009). 15. oldal

16 Spin polarization in SARPES experiment Polarization: Z Polarization: Y Intensity k [A 1 ] Polarization: X y z x K E K x E = 50meV, = 40meV,λ R = 33meV, E = 8meV 16. oldal

17 Spin polarization in SARPES experiment Polarization: Z Polarization: Y Intensity k [A 1 ] Polarization: X y x K E z K x E = 50meV, = 40meV,λ R = 33meV, E = 42meV 17. oldal

18 Summary Anisotropic bandsplitting due to corrections of first order in Rashba spin-orbit coupling. SARPES experiments are heavily affected by sublatice interference: The intesity of the photoelectrons is minimal in the region called the dark corridor. Sublattice asymmetry: Gap opens at the Dirac-point. Finite out-of-plane spin polarization can be measured in SARPES experiments over the dark corridor. 18. oldal

19 Rashba spin-orbit coupling in tight-binding model (TB) Lattice representation: H TB R = iλ R i,j,µ,ν [ a iµ ) (ŝ µν d i,j d z ] b jν h.c. λ R E z ; i,j nearest neighbours; d i,j points from sitej toi. Momentum representation: D(q) = 3 j=1 e iqa j d j d D ± (q) = ±D x (q) id y (q) H TB R (q) = ( λ R D + (q) 0 0 λ R D (q) 0 0 λ R D (q) 0 0 λ R D+ (q) ) Ψ = ( A ) B A B 19. oldal

20 The role of intrinsic SO: continous model H = H K +H SO, H SO = SOˆσ z ŝ z = 0 Realistic parameters: SO 12µeV (theory: first principles, DFT, Gmitra, M. at al., Phys. Rev. B 80, (2009).) λ R 4meV (measurment: SARPES, Graphene/Au/Ni(111) Varykhalov at al., PRL 101, (2008). ) E L λ2 R 4γ, E L 1.6µeV The pockets are smaller than the ISO energy scale. (E L < SO ) 20. oldal

21 The role of intrinsic SO: continous model = 0 Relevant case: SO 12µeV λ R 4meV The pocket structure survives despite of finite ISO coupling. The dynamics of the electrons is affected by the TW: minimal conductivity:σ λr = 3σ λr =0 (as in bilayer: J. Cserti at al., PRL. 99, ) 21. oldal

22 Spin splitting along ΓM Solid lines: theory including TW Experimental data Dedkov at al., PRL. 100, (2008). Angle Resolved Photoemission Spectroscopy Measurements: Graphene/Ni(111) structure (λ R 80meV) Rader at al.: The origin of the splitting is not clear PRL. 102, (2009). 22. oldal

23 Minimal Conductivity (MC) MC: conductivity ate F = 0. The MC is 3 larger as the MC of the bulk without TW: σ = 3σ 0 For realistic systems the interference between pockets leads to the anisotropy of the MC: σ = κ(ϕ)σ 0 σ: Landauer formalism in Continous (W ) and tightbinding (TB: finitew ) model. Characteristic length of RSO coupling:l = π/k SO 23. oldal

24 Minimal conductivity for Zig-Zag orientation (ϕ = 30 0 ) Solid line: continous model (W ). Dashed lines: TB model for aspect ratio R 1 = W/L = 4.71 andr 2 = Destructive interference between P 3 andk σ 2(σ P1 +σ P2 ) = 7/3σ 0. Good agreement between Continous and TB results. 24. oldal

25 Minimal conductivity for Armchair orientation (ϕ = 0 0 ) Solid line: continous model (W ) Dashed lines: TB model for aspect ratio R 1 = W/L = 3.12 andr 2 = 5.80 Lower number of propagating modes in armchair nanoribbon due to boundary conditions. σ TB max = σ P1 +σ P2 +2(σ P3 +σ K ) = 2.5σ oldal

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