Hall effects and weak localization in strong SO coupled systems :

1. Hall effects and weak localization in strong SO coupled systems :  merging Keldysh, Kubo, and Boltzmann approaches via the chiral basis. JAIRO SINOVA Texas A&M Univ. and Inst. Phys. ASCR SPIE, San Diego, August 12th 2008 Mario Borunda Texas A&M U. Xin Liu Texas A&M U. Alexey Kovalev Texas A&M U. Nikolai Sinitsyn LANL Ewelina Hankiewicz U. of Missouri Texas A&M U. Other collaborators: Bernd Kästner, Satofumi Souma, Liviu Zarbo, Dimitri Culcer , Qian Niu, S-Q Shen,,Tom Fox, Richard Campton, Artem Abanov Brian Gallagher Laurens Molenkamp Wuerzburg Kentaro Nomura U. Of Texas Branislav Nikolic U. of Delaware Joerg Wunderlich Cambridge-Hitachi Tomas Jungwirth Inst. of Phys. ASCR U. of Nottingham Allan MacDonald U of Texas Research fueled by: SWAN-NRI

2. Anomalous Hall Effect • History, semi-classical mechanism • Microscopic approach, IAHE • Merging the different linear theories • AHE in graphene • AHE in 2DEG+Rashba • Spin Hall Effect • Spin accumulation with strong SO • Weak Localization in GaMnAs • The experimental observations • Theory results

3. majority _ _ _ FSO _ FSO I minority V Anomalous Hall effect: Spin-orbit coupling “force” deflects like-spin particles InMnAs Simple electrical measurement of magnetization controversial theoretically: three contributions to the AHE (intrinsic deflection, skew scattering, side jump scattering)

4. Intrinsic deflection 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) 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 Related to the intrinsic effect: analogy to refraction from an imbedded medium 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. 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.

5. A history of controversy (thanks to P. Bruno– CESAM talk)

6. COLLINEAR MAGNETIZATION AND SPIN-ORBIT COUPLING vs. CHIRAL MAGNET STRUCTURES AHE is present when SO coupling and/or non-trivial spatially varying magnetization (even if zero in average) Spatial dependent magnetization: also can lead to AHE. A local transformation to the magnetization direction leads to a non-abelian gauge field, i.e. effective SO coupling (chiral magnets), which mimics the collinear+SO effective Hamiltonian in the adiabatic approximation SO coupled chiral states: disorder and electric fields lead to AHE/SHE through both intrinsic and extrinsic contributions So far one or the other have been considered but not both together, in the following we consider only collinear magnetization + SO coupling

7. Need to match the Kubo, Boltzmann, and Keldysh Kubo: systematic formalism Boltzmann: easy physical interpretation of different contributions (used to define them) Keldysh approach: also a systematic kinetic equation approach (equivelnt to Kubo in the linear regime). In the quasiparticle limit it must yield Boltzmann eq. Microscopic vs. Semiclassical

8. CONTRIBUTIONS TO THE AHE: MICROSCOPIC KUBO APPROACH n’, k n, q m, p m, p n, q n, q n’n, q Skew σHSkew (skew)-1 2~σ0 S where S = Q(k,p)/Q(p,k) – 1~ V0 Im[<k|q><q|p><p|k>] Skew scattering “side-jump scattering” Vertex Corrections  σIntrinsic Intrinsic AHE: accelerating between scatterings Intrinsic σ0 /εF

9. FOCUS ON INTRINSIC AHE (early 2000’s): semiclassical and Kubo n, q n’n, q STRATEGY: compute this contribution in strongly SO coupled ferromagnets and compare to experimental results, does it work? Kubo: Semiclassical approach in the “clean limit” K. Ohgushi, et al PRB 62, R6065 (2000); T. Jungwirth et al PRL 88, 7208 (2002); T. Jungwirth et al. Appl. Phys. Lett. 83, 320 (2003); M. Onoda et al J. Phys. Soc. Jpn. 71, 19 (2002); Z. Fang, et al, Science 302, 92 (2003).

10. Success of intrinsic AHE approach in comparing to experiment: phenomenological “proof” • 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 and Nagaosa, J. Phys. Soc. Jap. 71, 19 (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003), Shindou and Nagaosa, Phys. Rev. Lett. 87, 116801 (2001)] • colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999). • CuCrSeBr compounts, Lee et al, Science 303, 1647 (2004) AHE in GaMnAs AHE in Fe Berry’s phase based AHE effect is quantitative-successful in many instances BUT still not a theory that treats systematically intrinsic and extrinsic contribution in an equal footing Experiment sAH  1000 (W cm)-1 Theroy sAH  750 (W cm)-1

11. INTRINSIC+EXTRINSIC: REACHING THE END OF A 50 YEAR OLD DEBATE AHE in Rashba systems with weak disorder: Dugaev et al (PRB 05) Sinitsyn et al (PRB 05, PRB 07) Inoue et al (PRL 06) Onoda et al (PRL 06, PRB 08) Borunda et al (PRL 07), Nuner et al (PRB 07, PRL 08) Kovalev et al (PRB 08) All are done using same or equivalent linear response formulation–different or not obviously equivalent answers!!! The only way to create consensus is to show (IN DETAIL) agreement between ALL the different equivalent linear response theories both in AHE and SHE and THEN test it experimentally

12. Kubo-Streda formula summary Semiclassical Boltzmann equation Golden rule: In metallic regime: J. Smit (1956): Skew Scattering Calculation done easiest in normal spin basis

13. Semiclassicalapproach II: Golden Rule: Coordinate shift: Modified Boltzmann Equation: Sinitsyn et al PRB 06 Berry curvature: velocity: current:

14. “AHE” in graphene Kubo-Streda formula: In metallic regime: Sinitsyn et al PRB 07

15. Comparing Botlzmann to Kubo in the chiral basis Sinitsyn et al PRB 07

16. A more realistic testAHE in Rashba 2D system n, q n’n, q (differences: spin is a non-conserved quantity, define spin current as the gradient term of the continuity equation. Spin-Hall conductivity: linear response of this operator) Inversion symmetry  no R-SO Broken inversion symmetry  R-SO Bychkov and Rashba (1984)

17. AHE in Rashba 2D system Kubo and semiclassical approach approach: (Nuner et al PRB08, Borunda et al PRL 07) Only when ONE both sub-band there is a significant contribution When both subbands are occupied there is additional higher order vertex corrections that contribute

18. AHE in Rashba 2D system Keldysh and Kubo match analytically in the metallic limit 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 Rapids 08 Numerical Keldysh approach (Onoda et al PRL 07, PRB 08) Solved within the self consistent T-matrix approximation for the self-energy

19. Testing the theory: in progress

20. AHE in Rashba 2D system: “dirty” metal limit? Onoda et al 2008 Is it real? Is it justified? Is it “selective” data chosing? Can the kinetic metal theory be justified when disorder is larger than any other scale?

21. _ _ _ FSO _ non-magnetic FSO I V=0 Spin Hall effect Take now a PARAMAGNET instead of a FERROMAGNET: Spin-orbit coupling “force” deflects like-spin particles Carriers with same charge but opposite spin are deflected by the spin-orbit coupling to opposite sides. Spin-current generation in non-magnetic systems without applying external magnetic fields Spin accumulation without charge accumulation excludes simple electrical detection

22. Spin Hall Effect (Dyaknov and Perel) Interband Coherent Response  (EF) 0 • Occupation # • Response • `Skew Scattering‘ • (e2/h) kF (EF )1 X `Skewness’ [Hirsch, S.F. Zhang] • Intrinsic • `Berry Phase’ • (e2/h) kF [Murakami et al, Sinova et al] Influence of Disorder `Side Jump’’ [Inoue et al, Misckenko et al, Chalaev et al…] Paramagnets

23. First experimentalobservations at the end of 2004 Wunderlich, Kästner, Sinova, Jungwirth, cond-mat/0410295 PRL 05 Experimental observation of the spin-Hall effect in a two dimensional spin-orbit coupled semiconductor system Co-planar spin LED in GaAs 2D hole gas: ~1% polarization CP [%] 1.505 1.52 Light frequency (eV) Kato, Myars, Gossard, Awschalom, Science Nov 04 Observation of the spin Hall effect bulk in semiconductors Local Kerr effect in n-type GaAs and InGaAs: ~0.03% polarization (weaker SO-coupling, stronger disorder)

24. OTHER RECENT EXPERIMENTS Transport observation of the SHE by spin injection!! Valenzuela and Tinkham cond-mat/0605423, Nature 06 Saitoh et al APL 06 Sih et al, Nature 05, PRL 05 “demonstrate that the observed spin accumulation is due to a transverse bulk electron spin current” Room temperature SHE in ZnSe ??? Stern et al 06 (signal same as GaAs but SO smaller????)

25. The challenge: understanding spin accumulation in strongly spin-orbit coupled systems Spin is not conserved; analogy with e-h system Spin Accumulation – Weak SO Quasi-equilibrium Parallel conduction Spin diffusion length Burkov et al. PRB 70 (2004)

26. Spin Accumulation – Strong SO ? Mean Free Path? Spin Precession Length

27. SPIN ACCUMULATION IN 2DHG: EXACT DIAGONALIZATION STUDIES so>>ħ/ Width>>mean free path Nomura, Wundrelich et al PRB 06 Key length: spin precession length!! Independent of  !!

28. n LED 1 p y m 1.5 m x n channel LED 2 z SHE experiment in GaAs/AlGaAs 2DHG Wunderlich, Kaestner, Sinova, Jungwirth, Phys. Rev. Lett. '05 10m channel - shows the basic SHE symmetries - edge polarizations can be separated over large distances with no significant effect on the magnitude - 1-2% polarization over detection length of ~100nm consistent with theory prediction (8% over 10nm accumulation length) Nomura, Wunderlich, Sinova, Kaestner, MacDonald, Jungwirth, Phys. Rev. B '05

29. H-bar for detection of Spin-Hall-Effect Charge based measurements of ISHE (electrical detection through inverse SHE) (Numerical Keldysh calculation: no SO in leads) E.M. Hankiewicz et al ., PRB 70, R241301 (2004)

30. Mesoscopic electron SHE L/2 L/6 L calculated voltage signal for electrons (Hankiewicz and Sinova)

31. Mesoscopic hole SHE L/2 L/6 L calculated voltage signal (Hankiweicz, Sinova, & Molenkamp) L more than 10 time larger!

32. New (smaller) sample sample layout 200 nm 1 mm Experiments by Laruens Molenkamp group

33. SHE-Measurement

34. SUMMARY (AHE AND SHE) • All linear theories treating disorder and non-trivial band structure have been merged in agreement • Clear identification of semi-clasical contributions from the microscopic theory • Many strongly spin-orbit coupled systems are dominated by the intrinsic contribution: old side-jump+intrinsic cancellations were an artifact of simple band structure (e.g. constant Berry curvature) • Intrinsic SHE can also be observed in strongly spin-orbit coupled system with the induced spin-accumulation length scale in agreement with theory • Charge based detection of intrinsic SHE seen in inverted semiconductor systems SWAN-NRI

35. Weak Localization in GaMnAs Quantum driven localization of time reversed paths interference. Each spin channel adds to the localization. e.g. 2D In the presence of spin-orbit coupling one decouples channels in total angular momentum states. Singlet (zero total spin) is the one not affected BUT contributes with a negative sign to diffusion, i.e. Weak Antilocalization.

36. Matsukura et al Physica E 2004 Weak Localization at high magnetic fields Low field MR dominated by complicated AMR effects Kawabata A., Solid State Commun. 34 (1980) 432

38. Focus is on low magnetic field region where AMR dominates • Rely on subtracting e-e interaction contribution which they attribute to the 1-D theory proportional to T-1/2 . However they ignore that e-e contribution depends on the conductivity and strong AMR contributions will influence it. • 1-D dimensionality is not quite justified given the length scales at the temperatures considered • Lso seems too large to have real meaning. For a strongly spin-orbit coupled system is should be lower. • High field contribution ignored

39. B2?

40. Rokhinson et al observed a ~1% negative MR at low temperature in a GaMnAs film which saturates at ~20mT and “states” that it is isotropic in field (ignoring the clear AMR in the data). The magnitude of the WL is stronger than the largest expected from the simples theory. One expects saturation at very large fields, not present in their experiment ? ? ? ? ? ? Success of metallic disorder valence band theory seems unimportant • Ferromagnetic transition temperatures  •  Magneto-crystalline anisotropy and coercively  •  Domain structure  •  Anisotropic magneto-resistance  •  Anomalous Hall effect  •  MO in the visible range  •  Non-Drude peak in longitudinal ac-conductivity  • Ferromagnetic resonance  • Domain wall resistance  • TAMR  But they still ascribe this feature to weak localisation and furthermore argue that the presence of weak localisation is incompatible with the Fermi level being in strongly spin orbit coupled valence band ??!! But is their main basis even right?

41. Theory of WL in GaMnAs Unlike the case of time-reversal symmetric systems there are no obvious Invariant representation when the energy scales are similar (exchange field, disorder, spin-orbit coupling, etc.)

42. Key result: for typical doping values and disorder WL is present!!! The main point is b/c disorder affects most the inter-band correlations which in the case of GaMnAs dominates the WAL contribution so the cross over from WAL to WL occurs before Eso is of the order of exchange energy.

43. SUMMARY (Weak Localization in GaMnAs) • Interpretation of low magnetic field MR effects do not support a clear signature of WL (or WAL). Complicated AMR effects need to be taken into account more carefully • For moderate Mn doping GaMnAs should show WL due to the large disorder scattering which limits the WAL corrections coming from interband correlations • Interpretation of WL-> impurity band has no basis since the presence of SO coupling in the model does not create a WAL regime for moderate Mn doping!!! • Effects of e-e interactions at low fields should incorporate AMR effects to correctly analyze the data

45. Spin-orbit coupling interaction(one of the few echoes of relativistic physics in the solid state) Produces an electric field Ingredients: -“Impurity” potential V(r) - Motion of an electron In the rest frame of an electron the electric field generates and effective magnetic field This gives an effective interaction with the electron’s magnetic moment • CONSEQUENCES • If part of the full Hamiltonian quantization axis of the spin now depends on the momentum of the electron !! • If treated as scattering the electron gets scattered to the left or to the right depending on its spin!!

46. 3. Charge based measurements of SHE Non-equilibrium Green’s function formalism (Keldysh-LB) • Advantages: • No worries about spin-current definition. Defined in leads where SO=0 • Well established formalism valid in linear and nonlinear regime • Easy to see what is going on locally • Fermi surface transport

47. PRL 05

48. INTRINSIC SPIN-HALL EFFECT: Murakami et al Science 2003 (cond-mat/0308167)Sinova et al PRL 2004 (cont-mat/0307663) n, q n’n, q as there is an intrinsic AHE (e.g. Diluted magnetic semiconductors), there should be an intrinsic spin-Hall effect!!! (differences: spin is a non-conserved quantity, define spin current as the gradient term of the continuity equation. Spin-Hall conductivity: linear response of this operator) Inversion symmetry  no R-SO Broken inversion symmetry  R-SO Bychkov and Rashba (1984)

49. ‘Universal’ spin-Hall conductivity n, q n’n, q Color plot of spin-Hall conductivity: yellow=e/8π and red=0

50. Disorder effects: beyond the finite lifetime approximation for Rashba 2DEG n, q n’n, q Question: Are there any other major effects beyond the finite life time broadening? Does side jump contribute significantly? +…=0 + For the Rashba example the side jump contribution cancels the intrinsic contribution!! Inoue et al PRB 04 Dimitrova et al PRB 05 Raimondi et al PRB 04 Mishchenko et al PRL 04 Loss et al, PRB 05 Ladder partial sum vertex correction: the vertex corrections are zero for 3D hole systems (Murakami 04) and 2DHG (Bernevig and Zhang 05)