Chiral Orbital Angular Momentum Perspective on Surface Electronic States of SrTiO 3 and KTaO 3

1. Chiral Orbital Angular Momentum Perspective on Surface Electronic States of SrTiO3 and KTaO3 Kyeong Tae Kang Panjin Kim Jung Hoon Han Department of Physics, SungKyunKwan University, Suwon, South Korea

2. Overview • 1. Introduction • 2D electron gas • STO/KTO materials • Rashba effect/orbital Rashba effect • 2. Solving the TB Hamiltonian • Our model • Checking the OAM chirality • Checking the Rashba splitting • 3. Summary

3. Introduction – Two Dimensional Electron Gas (2DEG) A. Ohtomo and H. Y. Hwang, Nature 427, 423 (2004) LaCrO3 (LCO) YanwaXie et al., Nat. Comm. 2, 494 (2011) Conduction electrons are trapped in the potential well. SrTiO3 (STO) Surface electronic states appear in the LCO/STO interface. Peter V. Sushko, University College London

4. Introduction – 2DEG on the surface of SrTiO3 and KTaO3 ARPES results 2DEG is formed due to the oxygen vacancies at surface of STO. A. F. Santander-Syro et al., Nature, 469, 189 (2011) W. Meevasanaet al., Nat. Mat., 10, 114 (2011) KTaO3, a wide gap 5d insulator presenting a strong spin-orbit coupling, also has 2DEG at their surface. These surface states are believed to originate from t2g-orbitals. Treat them with tight-binding model. A. F. Santander-Syro et al., PRB 86, 121107(R) (2012)

5. Introduction – Rashba effect The Rashba effect is a result of inversion symmetry breaking (ISB). This consideration allows the Rashba-type term, (E : electric field, σ : Pauli matrix) With atomicspin-orbit interaction (SOI), energy splitting appears The Rashba effect also shows chiral spin angular momentum (SAM) structure.

6. Introduction – Multi-orbital character and orbital Rashba effect When the crystal field splitting does not quench the orbital degrees of freedom in a given band, L : orbital angular momentum (OAM) operator for p-orbitals The effect is named orbital Rashba effect in analogy to the similar chiral structure of spins in surface bands. In our case, STO and KTO, the OAM operator L is for t2g-orbitals.

7. Introduction – Multi-orbital character and orbital Rashba effect J.-H. Park et al., PRB 85, 195401 (2012) OAM averages are With SOI Linear Rashba m=0 Chirality corresponds to the linear Rashba effect. m=-1 m=1

8. Introduction – cubic Rashba effect H. Nakamura et al., PRL 108, 206601 (2012) shows the experimental evidence of cubic Rashba, using the magneto resistance (MR) measurement on STO surface Cubic-Rashba-effect-dominant band exists Cubic order of Rashba effect?? m=0 m=-1 m=1

9. Our Work We treat Multi-orbital character (dxy, dyz, dzx) Inversion symmetry breaking (surface effect) Spin-orbit interaction We calculate the OAM averages and the effect of spin-orbit coupling by using the tight-binding Hamiltonian considering cubic Rashba effect to prove that the cubic Rashba effect become dominant in a band that lack the linear orbital Rashba effect.

10. Tight-binding Hamiltonian The Hamiltonian reads in the basis The definition of Hamiltonian is t & t’ : σ- and π-bonding parameters of t2gorbitals γ : Inversion symmetry breaking parameter λso : Spin-orbit interaction parameter δ : On-site energy difference

11. Energy dispersion of the model We obtained values of parameters from Z. Zhong et al., PRB 87, 161102(R) (2012) A. F. Santander-Syro et al., PRB 86, 121107 (2012) Energy dispersions for STO and KTO are E E ky ky STO KTO

12. Analyzing the TB Hamiltonian without SOI The OAM averages for two limiting cases γ≫Δ and γ≪Δ Δ=t-t’-δ/2 : the energy gap at the k=0 point between lowest-energy band and the rest m labels each band , Note that , E 1. γ≫Δ case m=-1 Cubic order is dominant. m=0 2Δ m=1 ky 0

13. Analyzing the TB Hamiltonian without SOI 2. γ≪Δ case Orbital chiralities get mixed between two top and middle bands E Difficult to give a clear assignment of orbital chirality 2Δ ky 0

14. OAM Averages of STO with SOI case We diagonalized the tight-binding Hamiltonian of STO case numerically. 1. γ≫Δ case, γ=0.5, λso=0.01 With the SOI effect, bands are separated by the spin. m=1 Same chiral OAM structure, But counter-direction SAM rotation. The maximum Rashba energy splitting : 21meV. (Can be observed!) m=0 m=-1

15. OAM Averages of STO with SOI case 2. γ≪Δ case, γ=0.02, λso=0.01 Difficult to give a clear assignment of orbital chirality Same chiral OAM structure, But counter-direction SAM rotation. The maximum Rashba energy splitting : 19meV.

16. Summary & Discussion We diagonalized the tight-binding Hamiltonian of STO and KTO. In the large ISB case, bands have chiral OAM structure and this corresponds to the Rashba effect. The band that lack the linear Rashba has cubic Rashba as leading order. Are these realistic? – DFT calculation