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New developments in the AHE: New developments in the AHE: phenomenological regime, unified linear theories, and a new member of the spintronic Hall family Spin Currents 2009 Stanford Sierra Conference Center at Fallen Leaf Lake, South Lake Tahoe April 19 th, 2009 JAIRO SINOVA Texas A&M University Institute of Physics ASCR Research fueled by: Hitachi Cambridge Jorg Wunderlich, A. Irvine, et al Institute of Physics ASCR Tomas Jungwirth, Vít Novák, et al Texas A&M L. Zarbo Stanford University Shoucheng Zhang, Rundong Li, Jin Wang

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OUTLINE Introduction AHE in spin injection Hall effect: – AHE basics – Strong and weak spin-orbit couple contributions of AHE – SIHE observations – AHE in SIHE Spin-charge dynamics of SIHE with magnetic field: – Static magnetic field steady state – Time varying injection AHE general prospective – Phenomenological regimes – New challenges

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Anomalous Hall transport: lots to think about Wunderlich et al SHE Kato et al Fang et al Intrinsic AHE (magnetic monopoles) AHE Taguchi et al AHE in complex spin textures Valenzuela et al Inverse SHE Brune et al Spin Currents 2009

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The family of spintronic Hall effects AHE B=0 polarized charge current gives charge-spin current Electrical detection SHE B=0 charge current gives spin current Optical detection SHE -1 B=0 spin current gives charge current Electrical detection j s ––––––––––– iSHE Spin Currents 2009

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Anomalous Hall effect basics and history Simple electrical measurement of out of plane magnetization Spin dependent “force” deflects like-spin particles I _ F SO _ _ _ majority minority V InMnAs Spin Currents 2009

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Anomalous Hall effect (scaling with ρ) Dyck et al PRB 2005 Kotzler and Gil PRB 2005 Co films Edmonds et al APL 2003 GaMnAs Strong SO coupled regime Weak SO coupled regime Spin Currents 2009

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: Hall discovers the Hall and the anomalous Hall effect The tumultuous history of AHE 1970: Berger reintroduces (and renames) the side-jump: claims that it does not vanish and that it is the dominant contribution, ignores intrinsic contribution. (problem: his side-jump is gauge dependent) Berger 7 Luttinger 1954: Karplus and Luttinger attempt first microscopic theory: they develop (and later Kohn and Luttinger) a microscopic theory of linear response transport based on the equation of motion of the density matrix for non-interacting electrons, ; run into problems interpreting results since some terms are gauge dependent. Lack of easy physical connection. Hall 1970’s: Berger, Smit, and others argue about the existence of side-jump: the field is left in a confused state. Who is right? How can we tell? Three contributions to AHE are floating in the literature of the AHE: anomalous velocity (intrinsic), side-jump, and skew contributions : Smit attempts to create a semi-classical theory using wave- packets formed from Bloch band states: identifies the skew scattering and notices a side-step of the wave-packet upon scattering and accelerating..Speculates, wrongly, that the side-step cancels to zero. The physical interpretation of the cancellation is based on a gauge dependent object!!

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The tumultuous history of AHE: last three decades ’s: Spin-Hall effect is revived by the proposal of intrinsic SHE (from two works working on intrinsic AHE): AHE comes to the masses, many debates are inherited in the discussions of SHE. 1980’s: Ideas of geometric phases introduced by Berry; QHE discoveries 2000’s: Materials with strong spin-orbit coupling show agreement with the anomalous velocity contribution: intrinsic contribution linked to Berry’s curvature of Bloch states. Ignores disorder contributions ’s: Linear theories in simple models treating SO coupling and disorder finally merge: full semi-classical theory developed and microscopic approaches are in agreement among each other in simple models.

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Boltzmann semiclassical approach: easy physical interpretation of different contributions (used to define them) but very easy to miss terms and make mistakes. MUST BE CONFIRMED MICROSCOPICALLY! How one understands but not necessarily computes the effect. Kubo approach: systematic formalism but not very transparent. Keldysh approach: also a systematic kinetic equation approach (equivalent to Kubo in the linear regime). In the quasi-particle limit it must yield Boltzmann semiclassical treatment. AHE: Microscopic vs. Semiclassical 9

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Intrinsic deflection Electrons have an “anomalous” velocity perpendicular to the electric field related to their Berry’s phase curvature which is nonzero when they have spin-orbit coupling. ~τ 0 or independent of impurity density Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step. They however come out in a different band so this gives rise to an anomalous velocity through scattering rates times side jump. independent of impurity density STRONG SPIN-ORBIT COUPLED REGIME (Δ so >ħ/τ) Side jump scattering V imp (r) Skew scattering Asymmetric scattering due to the spin- orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators. ~1/n i V imp (r) Electrons deflect to the right or to the left as they are accelerated by an electric field ONLY because of the spin- orbit coupling in the periodic potential (electronics structure) E SO coupled quasiparticles Spin Currents 2009

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WEAK SPIN-ORBIT COUPLED REGIME (Δ so <ħ/τ) Side jump scattering from SO disorder Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step. They however come out in a different band so this gives rise to an anomalous velocity through scattering rates times side jump. independent of impurity density λ* V imp (r) Skew scattering from SO disorder Asymmetric scattering due to the spin- orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators. ~1/n i λ* V imp (r) The terms/contributions dominant in the strong SO couple regime are strongly reduced (quasiparticles not well defined due to strong disorder broadening). Other terms, originating from the interaction of the quasiparticles with the SO- coupled part of the disorder potential dominate. Better understood than the strongly SO couple regime Spin Currents 2009

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Consistent theory of anomalous transport in spin- orbit coupled systems Mario F. Borunda, et al, "Absence of skew scattering in two-dimensional systems: Testing the origins of the anomalous Hall Effect", PRL 07 T. S. Nunner, et al“Anomalous Hall effect in a two-dimensional electron gas”, Phys. Rev. B 76, (2007) A. A. Kovalev, et al “Hybrid skew scattering regime of the anomalous Hall effect in Rashba systems”, Phys. Rev. B Rapids 78, (R) (2008). Boltzmann approach Keldysh approach Kubo approach Spin injection Hall device is the perfect testing ground for these theory predictions

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: Spin injection Hall effect: experimental observation n1 ( 4) n2 n3 ( 4) Local Hall voltage changes sign and magnitude along the stripe Spin Currents 2009

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AHE contribution Type (i) contribution much smaller in the weak SO coupled regime where the SO-coupled bands are not resolved, dominant contribution from type (ii) Crepieux et al PRB 01 Nozier et al J. Phys. 79 Two types of contributions: i)S.O. from band structure interacting with the field (external and internal) ii)Bloch electrons interacting with S.O. part of the disorder Lower bound estimate of skew scatt. contribution Spin Currents 2009

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Spin injection Hall effect: Theoretical consideration Local spin polarization calculation of the Hall signal Weak SO coupling regime extrinsic skew-scattering term is dominant Lower bound estimate Spin Currents 2009

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The family of spintronics Hall effects SHE -1 B=0 spin current gives charge current Electrical detection AHE B=0 polarized charge current gives charge-spin current Electrical detection SHE B=0 charge current gives spin current Optical detection SIHE B=0 Optical injected polarized current gives charge current Electrical detection Spin Currents 2009

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OUTLINE Introduction AHE in spin injection Hall effect: – SIHE observations – AHE basics – Strong and weak spin-orbit couple contributions of AHE – AHE in SIHE Spin-charge dynamics of SIHE with magnetic field: – Static magnetic field steady state – Time varying injection AHE prospectives – Phenomenological regimes – New challenges

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Steady state solution for the spin-polarization component if propagating along the [1-10] orientation Steady state spin transport in diffusive regime Spatial variation scale consistent with the one observed in SIHE Spin Currents 2009

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Drift-Diffusion eqs. with magnetic field perpendicular to 110 and time varying spin-injection Spin Currents 2009 σ + (t) B Similar to steady state B=0 case, solve above equations with appropriate boundary conditions: resonant behavior around ω L and small shift of oscillation period Jing Wang, Rundong Li, SC Zhang, et al

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Semiclassical Monte Carlo of SIHE Numerical solution of Boltzmann equation Spin-independent scattering: Spin-dependent scattering: phonons, remote impurities, interface roughness, etc. side-jump, skew scattering. AHE Realistic system sizes ( m). Less computationally intensive than other methods (e.g. NEGF). Spin Currents 2009

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Single Particle Monte Carlo Spin Currents 2009 Spin-Dependent Semiclassical Monte Carlo Temperature effects, disorder, nonlinear effects, transient regimes. Transparent inclusion of relevant microscopic mechanisms affecting spin transport (impurities, phonons, AHE contributions, etc.). Less computationally intensive than other methods(NEGF). Realistic size devices.

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Effects of B field: current set-up Spin Currents 2009 In-Plane magnetic field Out-of plane magnetic field

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OUTLINE Introduction AHE in spin injection Hall effect: – SIHE observations – AHE basics – Strong and weak spin-orbit couple contributions of AHE – AHE in SIHE Spin-charge dynamics of SIHE with magnetic field: – Static magnetic field steady state – Time varying injection AHE general prospective – Phenomenological regimes – New challenges

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Phenomenological regimes of AHE Spin Currents 2009 Review of AHE (to appear in RMP 09), Nagaosa, Sinova, Onoda, MacDonald, Ong 1.A high conductivity regime for σ xx >10 6 ( cm) -1 in which AHE is skew dominated 2.A good metal regime for σ xx ~ ( cm) -1 in which σ xy AH ~ const 3.A bad metal/hopping regime for σ xx 1 Skew dominated regime

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Scattering independent regime Spin Currents 2009 Q: is the scattering independent regime dominated by the intrinsic AHE?

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intrinsic AHE approach in comparing to experiment: phenomenological “proof” Berry’s phase based AHE effect is reasonably successful in many instances n, q n’ n, q DMS systems (Jungwirth et al PRL 2002, APL 03) Fe (Yao et al PRL 04) layered 2D ferromagnets such as SrRuO3 and pyrochlore ferromagnets [Onoda et al (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003) colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999). CuCrSeBr compounts, Lee et al, Science 303, 1647 (2004) Experiment AH 1000 ( cm) -1 Theroy AH 750 ( cm) -1 AHE in Fe AHE in GaMnAs Spin Currents 2009

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Hopping conduction regime: terra incognita Approximate scaling seen as a function of T No theory of approximate scaling

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Spin Currents 2009 Nagaosa et al RMP 09 Tentative phase diagram of AHE

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Spin Currents 2009 AHE Review, RMP 09, Nagaosa, Sinova, Onoda, MacDonald, Ong

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Spin Currents 2009 Sanford Sierra Conference Center at Fallen Leaf Lake, South Lake Tahoe April 19 th, 2009 Research fueled by: Hitachi Cambridge Jorg Wunderlich, A. Irvine, et al Institute of Physics ASCR Tomas Jungwirth,, Vít Novák, et al Texas A&M J. Sinova. L. Zarbo Stanford University Shoucheng Zhang, Rundong Li, Jin Wang CONCLUSION Predicted resonant behavior for time varying polarization injection with perpendicular magnetic field New tool to explore the AHE in the strong SO coupled regime AHE in SIHE signals in agreement with theory expectations Three rough AHE regimes appear when reviewing a large range of materials AHE linear response regimes in metallic regime are now well understood in simple models

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Spin Currents 2009

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33 What do we mean by gauge dependent? Electrons in a solid (periodic potential) have a wave-function of the form Gauge dependent car BUT is also a solution for any a(k) Any physical object/observable must be independent of any a(k) we choose to put Gauge wand (puts an exp(ia(k)) on the Bloch electrons) Gauge invariant car

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The Spin-Charge Drift-Diffusion Transport Equations For arbitrary α,β spin-charge transport equation is obtained for diffusive regime For propagation on [1-10], the equations decouple in two blocks. Focus on the one coupling S x+ and S z : Spin Currents 2009

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Why is AHE difficult theoretically in the strong SO couple regime? AHE conductivity much smaller than σ xx : many usual approximations fail Microscopic approaches: systematic but cumbersome; what do they mean; use non-gauge invariant quantities (final result gauge invariant) Multiband nature of band-structure (SO coupling) is VERY important; hard to see these effects in semi-classical description (where other bands are usually ignored). Simple semi-classical derivations give anomalous terms that are gauge dependent but are given physical meaning (dangerous and wrong) Usual “believes” on semi-classically defined terms do not match the full semi-classical theory (in agreement with microscopic theory) What happens near the scattering center does not stay near the scattering centers (not like Las Vegas) T-matrix approximation (Kinetic energy conserved); no longer the case, adjustments have to be made to the collision integral term Be VERY careful counting orders of contributions, easy mistakes can be made. 35

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