Low Dimensional Systems Workshop KITP, Santa Barbara, CA May 14 th , 2009

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 Low Dimensional Systems Workshop KITP, Santa Barbara, CA May 14th , 2009 Stanford University Shoucheng Zhang,Rundong Li, Jin Wang Research fueled by:

2. 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

3. OUTLINE • Introduction • SIHE experiment • Making the device • Basic observation • Analogy to AHE • Photovoltaic and high T operation • The effective Hamiltonian • Spin-charge Dyanmcis • 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

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

5. Towards a spin-based non-magnetic FET device: can we electrically measure the spin-polarization? Can we achieve direct spin polarization detection through an electrical measurement in an all paramagnetic semiconductor system? Long standing paradigm: Datta-Das FET • Unfortunately it has not worked: • no reliable detection of spin-polarization in a diagonal transport configuration • No long spin-coherence in a Rashba SO coupled system

6. Spin-detection in semiconductors • Magneto-optical imaging non-destructive  lacks nano-scale resolution and only an optical lab tool Crooker et al. JAP’07, others • MR Ferromagnet  electrical  destructive and requires semiconductor/magnet hybrid design & B-field to orient the FM Ohno et al. Nature’99, others • spin-LED  all-semiconductor  destructive and requires further conversion of emitted light to electrical signal

7. Spin-injection Hall effect  non-destructive  electrical  100-10nm resolution with current lithography in situ directly along the SmC channel (all-SmC requiring no magnetic elements in the structure or B-field) Wunderlich et al. arXives:0811.3486

8. J. Wunderlich, B. Kaestner, J. Sinova and T. Jungwirth, Phys. Rev. Lett. 94 047204 (2005) Spin-Hall Effect Utilize technology developed to detect SHE in 2DHG and measure polarization via Hall probes B. Kaestner, et al, JPL 02; B. Kaestner, et al Microelec. J. 03; XiulaiXu, et al APL 04, Wunderlich et al PRL 05 Proposed experiment/device: Coplanar photocell in reverse bias with Hall probes along the 2DEG channel Borunda, Wunderlich, Jungwirth, Sinova et al PRL 07

9. Device schematic - material p 2DHG i n

10. Device schematic - trench - p 2DHG i n

11. p i 2DHG n 2DEG Device schematic – n-etch

12. Vd VH Vs Device schematic – Hall measurement 2DHG 2DEG

13. Vd VH h h h h h h Vs e e e e e e Device schematic – SIHE measurement 2DHG 2DEG

14. Reverse- or zero-biased: Photovoltaic Cell • Red-shift of confined 2D hole  free electron trans. • due to built in field and reverse bias • light excitation with  = 850nm • (well below bulk band-gap energy) Band bending: stark effect -1/2 +1/2 bulk -1/2 +1/2 -3/2 +3/2 -1/2 +1/2 -3/2 -1/2 +1/2 +3/2 Transitions allowed for ħω>Eg Transitions allowed for ħω<Eg Transitions allowed for ħω<Eg trans. signal σo σ- σ+ σo VL

15. Spin injection Hall effect: experimental observation - + n2 n2 n2 n3 (4) n1 (4) n1 (4) Local Hall voltage changes sign and magnitude along the stripe

16. Spin injection Hall effect  Anomalous Hall effect

17. Persistent Spin injection Hall effect and high temperature operation Zero bias- - + - +

18. THEORY CONSIDERATIONS Spin transport in a 2DEG with Rashba+Dresselhaus SO The 2DEG is well described by the effective Hamiltonian: For our 2DEG system: Hence

19. What is special about ? Ignoring the term for now • spin along the [110] direction is conserved • long lived precessing spin wave for spin perpendicular to [110] • The nesting property of the Fermi surface:

20. An exact SU(2) symmetry The long lived spin-excitation: “spin-helix” • Finite wave-vector spin components • Shifting property essential Only Sz, zero wavevector U(1) symmetry previously known: J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90,146801 (2003). K. C. Hall et. al., Appl. Phys. Lett 83, 2937 (2003).

21. Physical Picture: Persistent Spin Helix • Spin configurations do not depend on the particle initial momenta. • For the same x+ distance traveled, the spin precesses by exactly the same angle. • After a length xP=h/4mα all the spins return exactly to the original configuration. Thanks to SC Zhang, Stanford University

22. Persistent state spin helix verified by pump-probe experiments Similar wafer parameters to ours

23. 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: For Dresselhauss = 0, the equations reduce to Burkov, Nunez and MacDonald, PRB 70, 155308 (2004); Mishchenko, Shytov, Halperin, PRL 93, 226602 (2004)

24. 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

25. majority _ _ _ FSO _ FSO I minority V Understanding the Hall signal of the SIHE: Anomalous Hall effect Spin dependent “force” deflects like-spin particles Simple electrical measurement of out of plane magnetization InMnAs

26. 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

27. 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

28. 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.

29. 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

30. 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

33. 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

34. 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

35. 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

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

37. 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

38. SIHE: a new tool to explore spintronics • nondestructive electric probing tool of spin propagation without magnetic elements • all electrical spin-polarimeter in the optical range • Gating (tunes α/β ratio) allows for FET type devices (high T operation) • New tool to explore the AHE in the strong SO coupled regime

40. AHE in the strong SO regime

41. 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

42. 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.

43. 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.

44. 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

45. 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. Microscopic vs. Semiclassical AHE in the strongly SO couple regime

46. n, q n, q = + Kubo microscopic approach to transport: diagrammatic perturbation theory Need to perform disorder average (effects of scattering) Real Eigenstates Bloch Electron = 1/0 = 0 Averaging procedures: Perturbation Theory: conductivity Drude Conductivity σ = ne2  /m*~1/ni Vertex Corrections  1-cos(θ)

47. 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 Berry’s phase based AHE effect is reasonably successful in many instances BUT still not a theory that treats systematically intrinsic and ext rinsic contribution in an equal footing Experiment sAH  1000 (W cm)-1 Theroy sAH  750 (W cm)-1

48. n’, k n, q m, p m, p n, q n, q n’n, q Kubo microscopic approach to AHE Early identifications of the contributions “Skew scattering” n’, k m’, k’ m, p n, q m, p n, q “Side-jump scattering” matrix in band index Vertex Corrections  σIntrinsic ~ 0or n0i Intrinsic AHE: accelerating between scatterings Intrinsic σ0 /εF~ 0or n0i

49. “AHE” in graphene: linking microscopic and semiclassical theories EF Armchair edge Zigzag edge

50. Kubo-Streda calculation of AHE in graphene Single K-band with spin up Don’t be afraid of the equations, formalism can be tedious but is systematic (slowly but steady does it) Kubo-Streda formula: A. Crépieux and P. Bruno (2001) In metallic regime: Sinitsyn, JS, et al PRB 07