Rashba spin-orbit coupling in the oxide 2D structures: The KTaO 3 (001) Surface

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1 Rashba spin-orbit coupling in the oxide 2D structures: The KTaO 3 (001) Surface Sashi Satpathy Department of Physics University of Missouri, Columbia, USA E Ref: K. V. Shanavas and S. Satpathy, Phys. Rev. Letts. 112, (2014) ; Also: Supplementary section Correlated Oxides 2014, Minneapolis May 3, 2014

2 Gate control of the Rashba SOI in LaAlO 3 /SrTiO 3 interface Measure Rashba coefficient from spin relaxation rates 2DEG Ben Shalom et al, PRL (2010) Caviglia, Gabay, et al., PRL (2010) Ohtomo and Hwang, Nature (2004) Popovic, SS, Martin, PRL (2008) Free-electron result : α R = 2 E 2m 2 c 2 TB result : α R = 2βξ ε, β E

3 Δ SO ΔE = 2α R k Looked for Rashba: Found small

4 The Rashba Effect Momentum-dependent splitting of the spin bands in 2D systems (surfaces, interfaces, heterostructures) Rashba Hamiltonian: H R = α R ( σ k) ẑ = α R (σ x k y σ y k x ) Emmanuel I. Rashba (1927- ) E k x Spin degenerate k y Zeeman splitting B Rashba splitting E Ref: Rashba, Sov. Phys. Solid State (1960)

5 Manifestation of the Rashba effect Theory: band structure Experiment Can drive a wide range of novel phenomena Spin tunneling, Spintronics, spin relaxation rates Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state in superconductors Topological p-wave superconductors Cold atoms Rashba splitting 2 α R k k E = ak 2 ± α R k Expt: spin relaxation rates in STO/LAO: Caviglia et al., PRL (2010)

6 Rashba effect: naïve derivation within the free electron model 1. Apply Electric field E = E ẑ 2. Electron sees the magne4c field in its rest frame: B = ( v E) / c 2 3. B field couples to the magne4c moment of the electron spin: H SO = M B = ( gµ Bσ v E ) ( 2 c 2 ) = α σ ( k E) 4. Rashba Hamiltonian: H R = α R σ. ( k ẑ) = α R (σ x k y σ y k x ) = α R ( k z) σ = H k σ H k = The electric field has been converted into a momentum-dependent magnetic field! 2 E 2m 2 c 2 ( k ẑ)

7 Rashba effect => k-dependent magnetic field Rashba Hamiltonian: H R = α R ( k E) σ = H k σ E = E ẑ H k = 2 E 2m 2 c 2 ( k ẑ) H k Rashba Magnetic field Rashba splitting E = ak 2 ± α R k

8 Broken symmetry and Rashba effect ε Crystal inversion symmetry ε( k ) = ε( k ) k y Time-reversal symmetry (Kramers Degeneracy) ε( k ) = ε( k ) ε( k ) = ε( k ) k x Broken 4me- reversal symmetry (e.g. magne4c field) Zeeman effect Broken inversion symmetry 1. Bulk inversion symmetry broken (zinc blende AlAs): Dresselhaus effect 2. Surface inversion symmetry broken: Rashba effect Spin- Orbit coupling is essen4al to couple the electron mo4on with spin

9 Bulk Inversion symmetry broken: Dresselhaus Effect Dresselhaus Hamiltonian for the X point in zinc-blende crystals: AlAs zinc blende H X = β (σ x k x σ y k y ) ε k = ±βk β = 10.9 mev Å DFT with Spin-orbit interaction Mishra, Thulasi, SS, PRB 72, (2005) G. Dresselhaus, Phys. Rev. 100, 580 (1955)

10 Surface Inversion symmetry broken: Rashba effect The Rashba Hamiltonian: Naïve derivation Electric field Spin-orbit coupling V = E ẑ H SO = 2 2m 2 c 2 ( V k) σ H R = 2 E 2m 2 c 2 (ẑ k) σ ε k = 2 k 2 2m ± α R k α R = 2 E 2m 2 c 2 = (3.8 ev Å2 ) (1 ev / Å) 0.5 MeV ~ 10 5 ev Å 1. Too small by two to three orders of magnitude of observed spin splitting 2. Violates Ehrenfest Theorem (for bound electrons)

11 Tight-Binding model for the Rashba splitting Cubic lattice with a surface and d electrons E sx sin x ; sy sin y See also: Zhong, Toth, Held, PRB 87, (R) (2013); Khalsa, Lee, MacDonald, PRB 88, (R) (2013)

12 Tight-Binding model for the Rashba splitting (focus on d electrons) Use Löwdin downfolding to get an effective 2 x 2 Hamiltonian (Rashba) for each state such as Γ 6 h = Δ + α R σ. ( k ẑ)

13 TB model: Rashba splitting is orbital dependent Pseudo spins Δ = t 2g e g splitting H R = α R ( k ẑ) σ ξ = SOC strength α,β,γ = electric field induced hoppings ε = ε xy ε xz/yz = surface asymmetry

14 Rashba Hamiltonian from 2 nd order Perturbation Theory E field SO For the conduction band state h V = i V i Δ = ( 2βikx )( 3ξ / 2) Δ i = α R (k y + ik x ) + ( 2βik y )(iξ) Δ α R = 2 3βξ / Δ h = Δ + α R σ. ( k ẑ) Both the broken inversion symmetry (β) and the SOC term (ξ) must be present for the Rashba effect.

15 Rashba form of the Hamiltonian not always good H SO = ξ L S d Δ e g ξ Δ SO + Γ 8 + Γ 7 ξ / 2 δ Γ 7 Γ 6 Γ 7 If level separation is not large enough, a 2 x 2 Rashba Hamiltonian not possible. t 2g ξ / 2 Γ 8 + ξ / 2 Γ 6 Cubic crystal field O h Spin-orbit ε Γ 7 Symmetry-breaking surface field (C 4v )

16 Rashba splitting in nearly-degenerate case h = Δ + α R σ. ( k ẑ) Case : ε ~ α Rashba 2 x 2 Hamiltonian With Rashba No Rashba ξ = SOC strength ε = ε xy ε xz/yz Rashba 4 x 4 Hamiltonian α = E field induced hopping

17 Looked for Rashba: Found small

18 Polar catastrophe and the origin of the 2DEG at the KTaO 3 surface Ta O Polar Catastrophe 1/2 e Unit cell K Chemical valence K + Ta 5+ O 3 6- => TaO 2 + and KO - layers. This again leads to diverging potential. The polar catastrophe is satisfied by inserting ½ e - at the surface. TaO 2 KO TaO 2 KO TaO 2 KO W. Harrison (1978) Origin of 2DEG at LAO/STO oxide interfaces: Ohtomo and Hwang, Nature (2004) Popovic, SS, Martin, PRL 101, (2008)

19 2DEG at the KTO surface: DFT results unrelaxed relaxed surface surface

20 Rashba Splitting of the KTO surface bands Add SO Δ SO ΔE = 2α R k k ε k = 2 k 2 2m ± α R k

21 Where does the Rashba effect come from? : The nuclear region in the surface layers H SO = 1 m 2 c 2 r V r L S = ξ L S Retain SO coupling only on a specific surface layer or only within radius r from the nucleus Δ SO ΔE = 2α R k Origin of Rashba Splitting (a) From electric fields near the nucleus and (b) Broken inversion symmetry in the first few surface layers

22 2DEG profile due to surface relaxation Unrelaxed Relaxed 2DEG 2DEG Relaxed structure: Surface electric field becomes weaker due to ionic screening and the 2DEG spreads deeper into the bulk surface surface

23 Tuning of the Rashba effect by E field: DFT results

24 DFT Calculations of the Rashba Effect with Electric field 17 KTO layers slab Two identical surface

25 Control of Rashba Effect by an electric field Violet: increase in electron density blue: decrease 2DEG at the surface can be moved in and out by an applied electric field Rashba effect comes from the first couple of surface layer electrons

26 Tuning of the Rashba effect by E field: DFT results

27 Tuning of the Rashba effect by the electric field Expt: STO/LAO: Caviglia et al., PRL (2010) Measured α from electron spin relaxation rates

28 So far: TaO 2 -terminated Surface Strong Rashba (2DEG confined to surface) Weak Rashba (weak SOI, O (p) states delocalized into the bulk)

29 KO-terminated Surface (2D Hole gas) KO TaO 2 KO TaO 2 KO 1/2 h Rashba splitting much smaller, as conduction states not confined to the surface. Rashba splitting much stronger, as the O(p) holes are confined to the surface DFT bands: Ideal surface

30 Summary Rashba spin splitting: TB model serves as a framework for undersanding of the Rashba effect. Origin of the effect: First few surface layers and mostly from spin-orbit coupling in the nuclear region Controlled by several factors spin-orbit coupling strength Strength of the broken-inversion symmetry Gate control achieved via (i) the 2DEG moving in and out of the surface region (ii) Reorganization of the xy/yz/zx energies at the surface. DFT results for KTaO 3 surface => Oxides are rich systems for manipulating the Rashba effect with the electric field