Orbital angular momentum in Rashba, spin Hall and anomalous Hall - PowerPoint PPT Presentation

Orbital angular momentum in Rashba, spin Hall and anomalous Hall effects Changyoung Kim IBS-CCES, SNU, Korea Dept. of Physics and Astronomy, SNU, Korea Center for Correlated Electron Systems

Spin phenomena from orbital degree of freedom (orbital angular momentum) and its connection to Berry curvature Changyoung Kim IBS-CCES, SNU, Korea Dept. of Physics and Astronomy, SNU, Korea Center for Correlated Electron Systems

Spin phenomena Rashba Effect Spin Hall Effect Orbital polarization (angular momentum) + Spin-orbit coupling Center for Correlated Electron Systems

Electric vs magnetic ! ! ˆ • Heisenberg Hamiltonian H H = − J S 1 ⋅ S 2 Ferromagnet Dipole-dipole interaction? è Coulomb interaction in combination with the exclusion principle Factor of ~10 4 too small! Lesson : H electric >> H magnetic Center for Correlated Electron Systems

I. Rashba effect II. Intrinsic spin Hall effect III. Observation of hidden Berry curvature Center for Correlated Electron Systems

Rashba effects E(k) k y k x Rashba Hamiltonian JETP Lett. (1984) ' ( = 𝛽 ( 𝑙×𝑨̂ . 𝜏 𝐼 ⃗ 𝐶 "## 𝐶 "## ∝ 𝑙 k y k x Relativistic effect Center for Correlated Electron Systems

A ‘small’ problem in energy scale Factor of 10 5 ! Center for Correlated Electron Systems

Questions to answer A proper model should explain… • Band splitting & spin degeneracy lifting • Energy scale of the split • Chiral spin structure (including chirality) • The role of atomic SOC parameter a • Asymmetric charge distribution • Chiral orbital angular momentum structure • Conventional interpretation explains only one of them! Center for Correlated Electron Systems

Circular dichroism in ARPES Bi 2 Se 3 surface states Bi 2 Se 3 surface state data with two circular polarizations Right circular Left circular k k Nature Physics 5 , 398 (2009) K x k x Rashba states Binding Energy (eV) Binding Energy (eV) = Chiral spin states S. R. Park et al, PRL 108 , 046805 (2012) Center for Correlated Electron Systems

CD ARPES & Chiral OAM Measures ~OAM Right circular Left circular CD-ARPES <orbital> <spin> J=1/2 S. R. Park et al, PRL 108 , 046805 (2012) Center for Correlated Electron Systems

OAM revives With strong spin-orbit coupling ‘J’ state (a) Atomic (b) H CF (c) H SOC ≫ H CF J 3/2 p z p orbitals p x , p y J 1/2 PRL 101 , 076402 (2008) OAM OAM quenched revives OAM in atomic orbital l x s R Y m Bloch state Center for Correlated Electron Systems

When we have both L & k … l x s R Y m Bloch state in consideration = OAM + linear momentum ⃗ 𝑓 23.4 Two sides look different! Y m l Phase flow What does this to wave ftn? Simulation Center for Correlated Electron Systems

Asymmetric charge distribution Bloch state S L S z y k k k L S L x k = 0.1 π , L x = − 12 0.1 p & -1/2 z k = 0.05 π , L x = 12 k = 0.1 π , L x = 12 high 0.1 p & 1/2 z z z .05 p & 1/2 Surface normal Electric field low y Electron density y y Combination of OAM & k results in an asymmetric charge distribution (electric polarization) Center for Correlated Electron Systems

OAM induced large energy scale Orbital angular Asymmetric charge momentum distribution + Electric energy + Linear momentum Electric field E S z z Bloch state 𝜔 3 𝒔 S z k L y y Electron density x ' 7 = −𝛽 7 𝑀×𝑙 . 𝐹 ; 𝐼 • Interference effect within a Bloch wave function • being complex 𝜔 3 𝒔 Center for Correlated Electron Systems

Chiral OAM & Rashba E s + _ z + + <orbital> _ _ k x k y <spin> k + L _    • Asymmetric charge (‘electric polarization’) determined by p ~ L × k   p ⋅  k ) ⋅  U = −  o o • Energy from E s ~ ( L × E s ~ ( eA ) × ( V / A ) ~ eV • Chiral structure determined by OAM • Spin chirality follows from SOC Center for Correlated Electron Systems

LDA on single layer of Bi w/ external field Single layer of Bi with 3 V/A M G K M J. S. Hong, et al., Scientific Reports 5, 13488 (2015) LDA results reveal asymmetric charge distribution for Rashba states Center for Correlated Electron Systems

OAM based Hamiltonian for Rashba effect Conventional Rashba (spin) New Hamiltonian (orbital) z ×  ˆ ' 7 = −𝑞 ) ⋅ ˆ 𝐼 ⃗ . 𝐹 ; = −𝛽 7 𝑀×𝑙 . 𝐹 ; H Rashba = α R ˆ ( p σ spin OAM ' "## = 𝜁 3 + 𝛽𝑀 . 𝑇 ⃗ − 𝛽 7 𝑀×𝑙 . 𝐹 ; 𝐼 Crystal field + atomic SOC + Electrostatic PRL 108, 046805 (2012); PRB 85, 195402 (2012); PRB 88, 205408 (2013) Center for Correlated Electron Systems

Summary on Rashba • Orbital angular momentum induces asymmetric charge distribution which can result in a large energy term • Chiral OAM structure exists in Rashba states resulting from the energy term • Spin chirality follows the OAM chirality through SOC • OAM plays the essential role in Rashba effect. Effective Hamiltonian ' ( = 𝛽 ( 𝜏 𝐼 ⃗×𝑙 . 𝑨̂ <orbital> J=1/2 <spin> ' 7 = −𝛽 7 𝑀×𝑙 . 𝐹 @ + 𝜇𝑀 . 𝜏 ⃗ 𝐼 PRL 107 , 156803 (2011); PRL 108 , 046805 (2012); PRB 85 , 195402 (2012); PRB 88 , 205408 (2013) Sci. Rep. 5, 13488 (2015); J. Electr. Spectr. Rel. Phenom, 201 , 6 (2015) Center for Correlated Electron Systems

I. Rashba effect II. Intrinsic spin Hall effect Is OAM important in other phenomena? III. Observation of hidden Berry curvature Center for Correlated Electron Systems

Anomalous and spin Hall effects 𝐶 ↑ 𝐶 ↓ 𝐶 ↑ 𝐶 ↓ • Ferromagnetic system • Non-magnetic metallic system • Hall effect without external B-field • Spin accumulation Center for Correlated Electron Systems

Anomalous Hall effect • Evaluation of current operator • Very general formula • AHE in terms of Berry curvature • No true microscopic picture Center for Correlated Electron Systems

Issues in spin Hall effect • Issues 1. Role of SOC? 2. Sign reversal issue? (Pt vs Ta) è Need a more intuitive picture è OAM Hamiltonian can help Center for Correlated Electron Systems

Rashba vs Spin Hall • Rashba case • Spin Hall case E s L z k x k y k k E L ' 7 = −𝛽 7 𝑀×𝑙 . 𝐹 ; Band is spin degenerate 𝐼 Degeneracy lifted Inversion symmetry breaking Inversion symmetry breaking from intrinsic field applied field Center for Correlated Electron Systems

Spin Hall effect from the new Hamiltonian J = ½ case OAM z y w SAM k y H 0 = ! 2 k 2 / 2 m OAM x d ! ! SAM L ⋅ S + α k x 𝐼 D = −𝛽 7 𝑀×𝑙 . 𝐹 E L W k y E −𝛽 7 𝑀×𝑙 . 𝐹 E E k y • causes OAM dependent transverse motion k x • behaves like an effective magnetic field • should be related to Berry curvature Δ E = H 1 * Spin Hall current is by-product due to SOC Center for Correlated Electron Systems

SHE current (intuitive) • J = ½ case k y S z = +1 Spin current within dk x : 2𝑓ℏ𝑙 G 2𝑓ℏ𝑙 G 𝑜 𝑜 2𝑙 T 𝑒𝑙 E = 𝑜𝑓ℏ𝑙 T Δ𝑙 G 𝑒𝑙 E = 𝑙 G 𝑒𝑙 E k x 4𝜌 O 4𝜌 O 𝜌 O 𝑛 " 𝑛 " 𝑛 " S z = -1 Total spin current : 3 W @J2K = 𝑜𝑓ℏ𝑙 T O 2𝜌𝑛 " ⁄ 𝑘 G V 𝑙 G 𝑒𝑙 E = 𝑜𝑓ℏ𝑙 T 𝑙 # ← 𝑙 T = 𝛽 7 𝑛 " 𝐹 E /ℏ 𝜌 O 𝑛 " X3 W O 2𝜌 ⁄ = 𝑜𝑓𝛽 7 𝐹 E 𝑙 # Spin Hall voltage : O 𝜍 GG 𝑋 O 𝑘 E 𝜍 EE 𝜍 GG 𝑋 𝑜𝑓𝛽 7 𝐹 E 𝑙 # 𝑜𝑓𝛽 7 𝑙 # ;H = 𝑘 G @J2K 𝜍 GG 𝑋 = 𝑊 = G 2𝜌 2𝜌 O 𝑋𝜍 O 𝑘 E ∝ 𝜍 O ≈ 𝑜𝑓𝛽 7 𝑙 # ç well known result from AHE Center for Correlated Electron Systems

Connection to Berry phase - I Spin Hall effect OAM driven intrinsic spin Hall effect ! ↑ ! ↓ w k y OAM H 0 = ! 2 k 2 / 2 m ! ! SAM L ⋅ S + α k x 𝐼 D = −𝛽 7 𝑀×𝑙 . 𝐹 E Spin dependent effective B-field • Equation of motion d k Anomalous velocity d r 1 ( ) ( ) - dt ´ dt = Ñ k c B k c E k n c n c ! Berry curvature B.Z. d k e E dt = - c i k r r e u r ( ) ⋅ ( ) ψ = ! n k n k B n related to OAM? This contains 𝑀 Center for Correlated Electron Systems

� � � � Connection to Berry phase - II Spin Hall effect Hint Berry curvature ! ! ! ! ! ↑ B ( k ) ! × A ( k ) ! ↓ = ∇ n n k PRB 47, 1651 (1993) Berry connection ~ ! ! ! ∂ A ( k ) i n k n k = − = ni k ∂ i ⃗(𝑙 ) 𝐵 Spin dependent effective B-field Dipole moment of asymmetric charge distribution (momentum dependent) 𝑞 ⃗~𝛽 7 𝑀×𝑙 𝑒 e 𝑙~ V 𝛽 7 𝑀×𝑙 𝑒 e 𝑙 Polarization 𝑄 = V 𝑞 ⃗ ⃗ 𝑙 ~𝛽 7 𝑀×𝑙 ? 𝐵 Center for Correlated Electron Systems

Rigorous theory Daegeun Jo, 1 *To appear in PRL, Aug 2018 ⃗(𝑙) ≈ 𝜇 J @ 𝑀×𝑙 𝐵 @ 𝑀 𝐶 𝑙 ≈ 2𝜇 J Center for Correlated Electron Systems

Summary on SHE • OAM plays the key role in intrinsic SHE • OHE is generated even when SOC=0 • OHE is more fundamental than SHE (SHE is a concomitant effect of OHE through SOC) Berry connection and curvature are directly related to 𝑀 • Center for Correlated Electron Systems