Learning to Love Disorder: Spin-charge Conversion and Other - PowerPoint PPT Presentation

Learning to Love Disorder: Spin-charge Conversion and Other Interesting Effects in 2D Spin-Orbit Coupled Systems Miguel A. Cazalilla National Tsing Hua University & National Center for Theoretical Sciences, Taiwan & Donostia International Physics Center (DIPC) , Spain “Novel Quantum States in CMT NQS2017” 京都大学基礎物理学研究所 （ ２０１７年１１月２日 )

Acknowledgements ChunLi Huang Xianpeng Zhang Junhui Zheng (&Ms. Zheng) Hung-Yu Yang Taiwan NTHU → DIPC (Spain) Taiwan → Frankfurt NTHU → Boston College Mirco Milletari Singapore Hector Ochoa Aires Ferreira Giovanni Vignale Yidong Chong (UCLA) U of Missouri (US) Univ of York (UK) NTU (Singapore)

近藤さん On the importance of disorder

Disorder can be useful… John Bardeen Quantum Hall Plateaux

Outline: Part I Extrinsic Spin Hall Effect and Mott Scattering Indirect Magneto-electric Coupling: Edelstein Effect Direct Magneto-electric Coupling: Anisotropic Spin Precession What about experiments? Non local probes, Hanle, and all that

Outline: Part II Topology = Dimensional reduction? Impurity in a 1D Channel w and w/o interactions: Kane & Fisher Magnetic Impurity near the non-interacting edge of a 2D QSHI Magnetic Impurity near the interacting edge of a 2D QSHI

Outline: Part I Extrinsic Spin Hall Effect and Mott Scattering Indirect Magneto-electric Coupling: Edelstein Effect Direct Magneto-electric Coupling: Anisotropic Spin Precession What about experiments? Non local probes, Hanle, and all that

SHE and Symmetry (2D) 2D rotations J z = L z + S z [ H, J z ] = 0 [U(1) group] Rank-2 Tensor? J = ( J x , J y ) � � J α = J α x , J α y Charge current J α → J z ⊕ ( J x , J y ) (2D vector) (2D vector) Under reflection x ➝ -x and y ➝ y and S z ➝ -S z J z y → J z J y → − J z J x → − J x J y → J y vs . x y Different sign determined by TRS (by Onsager’s reciprocity) J z J x = − θ SHE J z y = θ SHE J x y Symmetry arguments are fine, but what are the mechanisms?

Motivation: Functionalized Graphene Chemisorption: H, F, … Physisorption: Cu, Ag, Au, Th, In,… C Weeks et al Phys Rev X (2012) AH Castro Neto & F Guinea Phys Rev Lett (2009) (a) (c) (a) (b) Graphene h-BN σ(mS) Substrates: TMD V SiO 2 V Si SiO ₂/p-Si WS₂ (kΩ) Z Wang et al Phys. Rev. (2016) (b) Gr aph ene under WS 2 B Wang et al 2D Mater (2016) R Pris tj ne Gr aph ene TMD = WSe 2 , MoS 2 10 WS 2 (V) Ω Ω Ω σ σ σ Ω Ω σ

Extrinsic Mechanisms for SHE Quantum Side Jump Skew Scattering (n imp ⪡ 1) δ elastic channel elastic channel up up − δ down down n imp < 1% 1 σ sH ∼ const σ sH ∼ N imp Nagaosa, Sinova, Onoda, MacDonald, Ong, Rev. Mod. Phys. (2010)

Extrinsic SHE: Mott’s Scattering k Electron-atom scattering θ p 2 x 2 Scattering matrix (to all orders…) T kp = A kp 1 + B kp · s SOC h i cos θ = ˆ B kp = S ( θ ) ( k × p ) k · ˆ p (3D Rotational Sym.) ρ in ( p ) = 1 kp = 1 ρ out ( k ) = T kp ρ in ( p ) T † 2 T kp T † 2 1 kp Unpolarized! ⇥ ⇤ h S z i out = Tr [ s z ρ out ( k )] = Re / sin θ A ∗ kp B z kp Polarization!

Forces from collisions θ “Orbital” pseudo magnetic field from collisions kp = S ( θ ) ( k × p ) · ˆ z ∝ sin θ B z Lorentz-like force Z h � ∗ i � d θ sin θ Re ( J = J x ˆ x ) F s ∝ n imp ˆ y s z A kp B z kp Forces “emerge” from collisions (akin to the Bernoulli principle)

Graphene: Resonant Enhancement Graphene is very prone Graphene to resonant scattering γ = spin Hall angle (T = 0) Linear Density of States ⇢ ( ✏ ) ∼ | ✏ | A Ferreira, T Rappoport, MAC, Yang et al. Science (2013) AH Castro Neto Phys Rev Lett (2014)

SHE in CVD Graphene Spin Hall Angle Electric Conductivity J Balakrishnan et al Nat. Comm. (2014) Model: Kane-Mele SOC impurities plus scalar resonant scatterers

Outline: Part I Extrinsic Spin Hall Effect and Mott Scattering Indirect Magneto-electric Coupling: Edelstein Effect Direct Magneto-electric Coupling: Anisotropic Spin Precession What about experiments? Non local probes, Hanle, and all that

2D Magnetoelectric Effect & Symmetry provided INVERSION SYMMETRY IS BROKEN J M must be related to M = ( M x , M y , M z ) = M z ⊕ M k =( M x , M y ) (pseudo-vector under 3D rotation) J = ( J x , J y ) Charge current (2D vector) Under reflection x ➝ -x and y ➝ y and S z ➝ -S z M y = α CISP J x α CISP M y J x = ˜

CISP from Edelstein Effect k y M y 6 = 0 E k x H R = B R ( k ) · s B R ( k ) = α R ( ˆ z × k )

Indirect DMC: Eldestein Effect Two-step process J SHE y − δ S y z : δ S y ¼ − 2 m α J z (a) τ EY k y δ S z y x J z M y k k x z (b) Rashba SOC y δ S z x H R = B R ( k ) · s K Shen, G Vignale & R Raimondi PRL (2014) B R ( k ) = α R ( ˆ z × k ) R Raimondi, P Schwab, C Gorinni, and G Vignale Ann Phys (Berlin) 2012

Outline: Part I Extrinsic Spin Hall Effect and Mott Scattering Indirect Magneto-electric Coupling: Edelstein Effect Direct Magneto-electric Coupling: Anisotropic Spin Precession What about experiments? Non local probes, Hanle, and all that

Revisiting Mott: Anisotropic Spin Precession Understanding disorder from one impurity k T kp = A kp 1 + B kp · s p Unpolarized ρ in ( p ) = 1 kp = 1 ρ out ( k ) = T kp ρ in ( p ) T † 2 T kp T † 2 1 kp Mott’s scattering ⇥ ⇤ � � ASP h S i = Tr [ s ρ out ( p )] = Re + i B ∗ kp ⇥ B kp A ∗ kp B kp Isotropic TR-invariant B kp = S ( θ ) ( k × p ) ∝ B ∗ kp ⇒ ASP = 0 No ASP scattering!

ASP in 2D Materials C Huang, Y Chong, and MAC, Phys Rev B (2016) T kp = A kp 1 + B kp · s k ˆ z p 3D Rotational Invariance broken to 2D rotation B kp = B k z B z kp + ˆ kp “Zeeman-like” Orbital-like (Rashba disorder fluctuations) 2D Mott’s scattering MM Glazov, E Ya Sherman, VK Dugaev Physica E (2010) kp = S ( θ ) ( k × p ) · ˆ B z z JP Binder et al Nature Physics (2016)

Anisotropic Precession Scattering (ASP) C Huang, Y Chong, and MAC, Phys Rev B (2016) in-plane Scattering θ B k angle kp out of plane B z kp ∝ sin θ � ⇤ ⇥ ˆ / iB k B kp ⇥ B ⇤ � � � z / h S y i out ASP rate = i B z kp kp kp Electron current → spin polarization perp. to E (i . e . M y )

Kohn & Luttinger: Strong disorder C Huang, Y Chong, and MAC, Phys Rev B (2016) δ n k = n k − n 0 k Spin δ n k = ρ k 1 + m k · s Charge Yidong Chong C-L Huang @ t ⇢ k + e E · r k n 0 k = I 1 [ ⇢ k , m k ] , ~ ~ ⇣ ⌘ ⇣ � ~ H � n imp ⌘ @ t m k + Re B kk ⇥ m k = I 2 [ ⇢ k , m k ] . ~ Collision integral k k Impurity (Rashba) I 1 , I 2 ∝ n imp self-energy

Collision Integral SHE / ISHE CISP Conductivity  � Z I 1 = n imp d 2 p Charge c 1 ( ⇢ p − ⇢ k ) + c 2 · ( m p − m k ) − c 3 · ( m p + m k ) � ( ✏ p − ✏ k ) 2 ⇡ ~  � Z I 2 = n imp Spin d 2 p c 1 ( m p − m k ) + c 2 ( ⇢ p − ⇢ k ) + c 3 ( ⇢ p − ⇢ k ) + K � ( ✏ p − ✏ k ) 2 ⇡ ~ Anisotropic spin precession (ASP) Skew scattering rate Drude relaxation rate scattering rate c 3 c 1 c 2 Scattering rate ∝ Probability = (Amplitude) × (Amplitude) ∗ 2 n imp n imp n imp = T + T − kp × kp T + kp k k p p k p C Huang, Y Chong, and MAC, Phys Rev B (2016)

Keldysh: Smooth weak disorder C Huang, M Milletari, and MAC, arxiv:17061316 (2017) (accepted in PRB)

Linear Response & Direct Magnetoelectric Coupling (DMC) C Huang, Y Chong, and MAC, Phys Rev B (2016)         O θ sH τ D α asp J x J x E x 0 E  =  + σ D J z J z − θ sH τ D α R 0 0       y y O τ EY α asp − τ EY α R M y M y 0 0 Onsager R ij = ✏ i ✏ j R ji (TR parity ✏ i = ± 1) a ) J Charge current J z y θ sH α asp M y E x J M Magnetization Spin current (Non-equilibrium spin polarization) α R

Extrinsic CISP? Yes! Two mechanisms a ) J 50 Charge current 40 θ sH α asp 30 20 10 J M Magnetization Spin current (Non-equilibrium 0 spin polarization) α R Eq. (1) to firs 50 M y = σ cisp E x , w 40 30 σ cisp = σ D ( θ sH α R τ s + α asp τ s ) . 20 10 ASP Edelstein effect: 0 SHE × Rashba Scattering - 0.4 - 0.2 0.0 0.2 0.4 Controlled by gating C Huang, Y Chong, and MAC, Phys Rev B (2016)

Outline: Part I Extrinsic Spin Hall Effect and Mott Scattering Indirect Magneto-electric Coupling: Edelstein Effect Direct Magneto-electric Coupling: Anisotropic Spin Precession What about experiments? Non local probes, Hanle, and all that

Nonlocal Resistance (or Mott’s double scattering) Nonlocal Electric J c Voltage field C2 C4 Direct Inverse E m J s C3 C1 Direct Magnetoelectric SHE Coupling (ASP) J Hirsch PRL (1999) C Huang, Y Chong, and MAC, E M Hankiewiz et al PRB (2004) arXiv:1702.04955 (2017) D Abanin et al PRB (2007)

Anomalous Hanle spin precession C Huang, Y Chong, and MAC, Phys. Rev. Lett. (2017) Hanle effect is qualitatively different for different spin- charge conversion mechanism. It may become asymmetric. 3 3 3 SHE~ASP SHE>>ASP 2 2 2 R nl ê r xx H 10 - 3 L R nl ê r xx H 10 - 3 L R nl ê r xx H 10 - 3 L τ J B k 1 1 1 0 0 0 - 1 - 1 - 1 ASP>>SHE - 2 - 2 - 2 τ J B k - 3 - 3 - 3 - 10 - 10 - 5 - 5 0 0 5 5 10 10 - 10 - 5 0 5 10 w L t s w L t s ∝ B k w L t s