1. New developments in the AHE: New developments in the AHE: phenomenological regime, unified linear theories, and a new member of the spintronic Hall family Low Dimensional Systems Workshop KITP, Santa Barbara, CA May 14 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

2. 2 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

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. The family of spintronic Hall effects 4 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

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

7.  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 Spin-injection Hall effect

8. Utilize technology developed to detect SHE in 2DHG and measure polarization via Hall probes J. Wunderlich, B. Kaestner, J. Sinova and T. Jungwirth, Phys. Rev. Lett. 94 047204 (2005) Spin-Hall Effect 8 B. Kaestner, et al, JPL 02; B. Kaestner, et al Microelec. J. 03; Xiulai Xu, 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. i p n 2DHG material Device schematic - material

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

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

12. VdVd VHVH 2DHG 2DEG VsVs 12 Hall measurement Device schematic – Hall measurement

13. 2DHG 2DEG e h e e ee e h h h h h VsVs VdVd VHVH 13 SIHE measurement Device schematic – SIHE measurement

14. Photovoltaic Cell Reverse- or zero-biased: Photovoltaic Cell trans. signal 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) σoσoσoσo σ+σ+σ+σ+ σ-σ-σ-σ- σoσoσoσo VLVL 14 -1/2 +1/2 +3/2 -3/2 bulk Transitions allowed for ħω>E g Transitions allowed for ħω<E g -1/2 +1/2 +3/2-3/2 Band bending: stark effect Transitions allowed for ħω<E g

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

16. Spin injection Hall effect  Anomalous Hall effect 16

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

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

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

20. The long lived spin-excitation: “spin-helix” An exact SU(2) symmetry 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). Finite wave-vector spin components Shifting property essential 20

21. 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 x P =h/4mα all the spins return exactly to the original configuration. Physical Picture: Persistent Spin Helix Thanks to SC Zhang, Stanford University 21

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

24. Steady state solution for the spin-polarization component if propagating along the [1-10] orientation 24 Steady state spin transport in diffusive regime Spatial variation scale consistent with the one observed in SIHE

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

26. 26 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

27. 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

28. 28 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

29. 29 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

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 30 Lower bound estimate

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33. 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

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 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.

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

37. 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 37 SIHE B=0 Optical injected polarized current gives charge current Electrical detection

38. 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

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40. AHE in the strong SO regime 40

41. 1880-81: 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 41 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. 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. The physical interpretation of the cancellation is based on a gauge dependent object!!

42. The tumultuous history of AHE: last three decades 42 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. 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. 2004-8’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 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. 43

44. 44 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

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 45

46. Kubo microscopic approach to transport: diagrammatic perturbation theory Averaging procedures: = 1/  0 = 0  = + Bloch Electron Real Eigenstates Need to perform disorder average (effects of scattering) n, q Drude Conductivity σ = ne 2  /m * ~1/n i Vertex Corrections  1-cos(θ) Perturbation Theory: conductivity n, q 46

47. 47 intrinsic AHE approach in comparing to experiment: phenomenological “proof” 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 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

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

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

50. Single K-band with spin up In metallic regime: Sinitsyn, JS, et al PRB 07 50 Kubo-Streda calculation of AHE in graphene 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)

51. 51 Semiclassical transport of spin-orbit coupled Bloch electrons: Boltzmann Eq. and Hall current As before we do this in two steps: first calculate steady state non- equilibrium distribution function and then use it to compute the current. Set to 0 for steady state solution Only the normal velocity term, since we are looking for linear in E equation order of the disorder potential strength and symmetric and anti-symmetric components 1 st Born approximation 2 nd Born approximation (usual skew scattering contribution) To solve this equation we write the non-equilibrium component in various components that correspond to solving parts of the equation the corresponding order of disorder

52. 52 Semiclassical transport of spin-orbit coupled Bloch electrons: Boltzmann Eq. and Hall current ~V 0 ~V ~V 2 2 nd step: (after solving them) we put them into the equation for the current and identify from there the different contributions to the AHE using the full expression for the velocity

53. Comparing Boltzmann to Kubo (chiral basis) 53 Kubo identifies, without a lot of effort, the order in n i of the diagrams BUT not so much their physical interpretation according to semiclassical theory Sinitsyn et al 2007

54. Intrinsic deflection 54 Popular believe: ~τ 1 or ~1/n i WRONG E ~n i 0 or independent of impurity density Skew scattering (2 contributions) term missed by many people using semiclassical approach Side jump scattering (2 contributions) Popular believe: ~n i 0 or independent of impurity density Origin is on its effect on the distribution function

55. 55 Recent progress: full understanding of simple models in each approach Semi-classical approach: Gauge invariant formulation; shown to match microscopic approach in 2DEG+Rashba, Graphene Sinitsyn et al PRB 05, PRL 06, PRB 07 Borunda et al PRL 07, Nunner et al PRB 08 Sinitsyn JP:C-M 08 Kubo microscopic approach: Results in agreement with semiclassical calculations 2DEG+Rashba, Graphene Sinitsyn et al PRL 06, PRB 07, Nunner PRB 08, Inoue PRL 06, Dugaev PRB 05 NEGF/Keldysh microscopic approach: Numerical/analytical results in agreement in the metallic regime with semiclassical calculations 2DEG+Rashba, Graphene Kovalev et al PRB 08, Onoda PRL 06, PRB 08 – – – – – – – – – – – + + + + + jqsjqs nonmagnetic Spin-polarizer current injected optically Spin injection Hall effect (SIHE) Up to now no 2DEG+R ferromagnetis: SIHE offers this possibility

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57. 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 ~10 4 -10 6 (  cm) -1 in which σ xy AH ~ const 3.A bad metal/hopping regime for σ xx 1 Skew dominated regime

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

59. 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

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

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

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

63. EXTRA SLIDES 63

64. A2070 Kelvin Nanotechnology, University of Glasgow thicknesscompositiondopingfunction 5nmGaAsp=1E19 (Be)cap 2MLGaAsun 50nmAl x Ga 1-x As, x=0.5p=8E18 (Be) 3nmAl x Ga 1-x As, x=0.3un 90nmGaAsunchannel 5nmAl x Ga 1-x As, x=0.3unspacer 2MLGaAsun n=5E12 delta (Si)delta-doping 2MLGaAsun 300nmAl x Ga 1-x As, x=0.3un 50 period(9ML GaAs: 9ML AlGaAs, x=0.3)superlattice 1000nmGaAsun GaAsSIsubstrate 64

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67. How does side-jump affect transport? 67 Side jump scattering The side-jump comes into play through an additional current and influencing the Boltzmann equation and through it the non-equilibrium distribution function VERY STRANGE THING: for spin-independent scatterers side-jump is independent of scatterers!! 1 st -It creates a side-jump current: 2 nd -An extra term has to be added to the collision term of the Boltzmann eq. to account because upon elastic scattering some kinetic energy is transferred to potential energy. full ω ll’ does not assume KE conserved, T-matrix approximation of ω ll’ (ω T ll’ ) does.

68. AHE in Rashba 2D system When both subbands are occupied the skew scattering is only obtained at higher Born approximation order AND the extrinsic contribution is unique (a hybrid between skew and side-jump) Kovalev et al PRB 08 Keldysh and Kubo match analytically in the metallic limit Numerical Keldysh approach (Onoda et al PRL 07, PRB 08) 68