Spin Currents 2009 Stanford Sierra Conference Center at Fallen Leaf Lake, South Lake Tahoe

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

2. 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

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

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

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

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

7. The tumultuous history of AHE • 1880-81: Hall discovers the Hall and the anomalous Hall effect • 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 • 1955-58: 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. Luttinger The physical interpretation of the cancellation is based on a gauge dependent object!! • 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) • 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. Berger

8. The tumultuous history of AHE: last three decades • 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. • 2004’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. • 2004-8’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.

9. 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

10. Intrinsic deflection STRONG SPIN-ORBIT COUPLED REGIME (Δso>ħ/τ) SO coupled quasiparticles E 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) ~τ0 or independent of impurity density 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. Side jump scattering Vimp(r) 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. Skew scattering ~1/ni Vimp(r) 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. Spin Currents 2009

11. WEAK SPIN-ORBIT COUPLED REGIME (Δso<ħ/τ) Better understood than the strongly SO couple regime 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. Side jump scattering from SO disorder  λ*Vimp(r) 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. Skew scattering from SO disorder  λ*Vimp(r) ~1/ni 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. Spin Currents 2009

12. Consistent theory of anomalous transport in spin-orbit coupled systems Boltzmann approach Keldysh approach Mario F. Borunda, et al, "Absence of skew scattering in two-dimensional systems: Testing the origins of the anomalous Hall Effect", PRL 07 A. A. Kovalev, et al “Hybrid skew scattering regime of the anomalous Hall effect in Rashba systems”, Phys. Rev. B Rapids 78, 041305(R) (2008). Kubo approach T. S. Nunner, et al“Anomalous Hall effect in a two-dimensional electron gas”, Phys. Rev. B 76, 235312 (2007) Spin injection Hall device is the perfect testing ground for these theory predictions

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

14. AHE contribution • Two types of contributions: • S.O. from band structure interacting with the field (external and internal) • Bloch electrons interacting with S.O. part of the disorder 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 Lower bound estimate of skew scatt. contribution Spin Currents 2009

15. 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

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

17. 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

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

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

20. 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

21. Single Particle Monte Carlo 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. Spin Currents 2009

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

23. 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

24. Phenomenological regimes of AHE Review of AHE (to appear in RMP 09), Nagaosa, Sinova, Onoda, MacDonald, Ong A high conductivity regime for σxx>106 (cm)-1 in which AHE is skew dominated A good metal regime for σxx ~104-106 (cm) -1 in which σxyAH~const A bad metal/hopping regime for σxx<104 (cm) -1 for which σxyAH~ σxyα with α>1 Skew dominated regime Spin Currents 2009

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

26. intrinsic AHE approach in comparing to experiment: phenomenological “proof” 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 pyrochloreferromagnets[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). • CuCrSeBrcompounts, Lee et al, Science 303, 1647 (2004) AHE in GaMnAs AHE in Fe Experiment sAH  1000 (W cm)-1 Theroy sAH  750 (W cm)-1 Berry’s phase based AHE effect is reasonably successful in many instances Spin Currents 2009

27. Hopping conduction regime: terra incognita • Approximate scaling seen as a function of T • No theory of approximate scaling Spin Currents 2009

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

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

30. CONCLUSION • AHE in SIHE signals in agreement with theory expectations • Predicted resonant behavior for time varying polarization injection with perpendicular magnetic field • New tool to explore the AHE in the strong SO coupled regime • 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 Spin Currents 2009 Sanford Sierra Conference Center at Fallen Leaf Lake, South Lake Tahoe April 19th, 2009 Texas A&MJ. Sinova. L. Zarbo Hitachi Cambridge Jorg Wunderlich, A. Irvine,et al Stanford University Shoucheng Zhang,Rundong Li, Jin Wang Institute of Physics ASCR Tomas Jungwirth,, Vít Novák, et al Research fueled by:

31. Spin Currents 2009

33. What do we mean by gauge dependent? Electrons in a solid (periodic potential) have a wave-function of the form 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 dependent car Gauge invariant car

34. 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 Sx+ and Sz: Spin Currents 2009

35. Why is AHE difficult theoretically in the strong SO couple regime? • AHE conductivity much smaller than σxx : 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.