German Physical Society Meeting March 26 th , 2012 Berlin, Germany

1. Theory of the anomalous hall effect: from the metallic fully ab-initio studies to the insulating hopping systems JAIRO SINOVA Texas A&M University Institute of Physics ASCR Institute of Physics ASCR Texas A&M University Xiong-Jun Liu Jülich Forschungszentrum Yuriy Mokrousov, F. Freimuth, H. Zhang, J. Weischenberg, Stefan Blügel UCLA A. Kovalev German Physical Society Meeting March 26th, 2012 Berlin, Germany Research fueled by:

2. Outline • Introduction • Anomalous Hall effect phenomenology: more than meets the eye • AHE in the metallic regime • Anomalous Hall effect (AHE) in the metallic regime • Understanding of the different mechanisms • Full theory of the scattering-independent AHE: beyond intrinsic • ab-initio studies of simple ferromagnets • Scaling of the AHE in the insulating regime • Experiments and phenomenology • phonon-assisted hopping AHE (Holstein) • Percolations theory generalization for the AHE conductivity • Results • Summary

3. V Anomalous Hall Effect Inverse SHE Spin Hall Effect Spin-injection Hall Effect Mesoscopic Spin Hall Effect majority _ _ _ _ _ FSO _ FSO FSO I FSO I minority V V Intrinsic Kato et al Science 03 Wunderlich, Kaestner, Sinova, Jungwirth PRL 04 Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09 Extrinsic Intrinsic Brune,Roth, Hankiewicz, Sinova, Molenkamp, et al Nature Physics 10 Valenzuela et al Nature 06 Anomalous Hall effect: more than meets the eye

4. majority _ _ _ FSO _ FSO I minority InMnAs V Anomalous Hall Effect: the basics Spin dependent “force” deflectslike-spin particles ρH=R0B ┴ +4π RsM┴ Simple electrical measurement of out of plane magnetization

5. Dyck et al PRB 2005 Edmonds et al APL 2003 Material with dominant skew scattering mechanism Material with dominant scattering-independent mechanism Co films GaMnAs Kotzler and Gil PRB 2005 Anomalous Hall effect (scaling with ρ)

6. Skew scattering independent of impurity density Vimp(r) (Δso>ħ/τ)or ∝ λ*∇Vimp(r) (Δso<ħ/τ) A ~σ~1/ni Intrinsic deflection B B Side jump scattering E Vimp(r) (Δso>ħ/τ) or ∝ λ*∇Vimp(r) (Δso<ħ/τ) 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) Asymmetric scattering due to the spin-orbit coupling of the electron or the impurity. Known as Mott scattering. 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. SO coupled quasiparticles 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. Cartoon of the mechanisms contributing to AHE in the metallic regime

7. Anomalous Hall Effect Nagaosa, Sinova, Onoda, MacDonald, Ong 2 1 3 2 1 3 • A high conductivity regime for σxx>106 (Ωcm)-1 in which AHE is skew dominated • A good metal regime for σxx ~104-106 (Ωcm) -1 in which σxyAH~ const • A bad metal/hopping regime for σxx<104 (Ωcm) -1 for which σxyAH~ σxyα with α=1.4~1.7

8. 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 compounds, Lee et al, • Science 303, 1647 (2004) AHE in GaMnAs AHE in Fe Experiment σAH ∼ 1000 (Ω cm)-1 Theroy σAH ∼ 750 (Ω cm)-1 Intrinsic AHE approach in comparing to experiment: phenomenological “proof”

9. Scattering independent regime: towards a theory applicable to real materials Q: is the scattering independent regime dominated by the intrinsic AHE? Challenge: can we formulate a full theory of the ALL the scattering independent contributions that can be coupled to ab-initio techniques?

10. Contributions understood in simple metallic 2D models Kubo microscopic approach: in agreement with semiclassical Borunda, Sinova, et al PRL 07, Nunner, JS, et al PRB 08 Non-Equilibrium Green’s Function (NEGF) microscopic approach Semi-classical approach: Gauge invariant formulation Kovalev, Sinova et al PRB 08, Onoda PRL 06, PRB 08 Sinitsyn, Sinvoa, et al PRB 05, PRL 06, PRB 07

11. Comparing Boltzmann to Kubo (chiral basis) for “Graphene” model Sinitsyn et al 2007 Kubo identifies, without a lot of effort, the order in ni of the diagrams BUT not so much their physical interpretation according to semiclassical theory

12. Generalization to 3D General band structure in the presence of delta-correlated Gaussian disorder N-band projected Hamiltonian expressed via envelope fields In the presence of Gaussian disorder To test our theory we will consider band structures of a 2D Rashba and 3D phenomenological models applicable to DMSs: Kovalev, Sinova, Tserkovnyak PRL 2010

13. Idea behind this calculation 1. Use Kubo-Streda formalism or linearized version of Keldysh formalism to obtain where 2. Integrate out sharply peaked Green’s functions which leads to integrals over the Fermi sphere and no dependence on disorder 3. In order to identify the relevant terms in the strength of disorder it is convenient to use diagrams (Synistin et al PRB 2008) Kovalev, Sinova, Tserkovnyak PRL 2010

14. Scattering independent AHE conductivity expressed through band structure Well known intrinsic contribution Side jump contribution related to Berry curvature (arises from unusual disorder broadening term usually missed) Remaining side jump contribution (usual ladder diagrams) Kovalev, Sinova, Tserkovnyak PRL 2010

15. Application to simple ferromagnets

20. Outline • Introduction • Anomalous Hall effect phenomenology: more than meets the eye • AHE in the metallic regime • Anomalous Hall effect (AHE) in the metallic regime • Understanding of the different mechanisms • Full theory of the scattering-independent AHE • ab-initio studies of simple ferromagnets • Scaling of the AHE in the insulating regime • Experiments and phenomenology • phonon-assisted hopping AHE (Holstein) • Percolations theory generalization for the AHE conductivity • Results • Summary

21. This scaling has been confirmed in many experiments. Below are some examples: S. Shen etal (2008) H.Toyosaki etal (2004) AHE in hopping conduction regime

22. S. H. Chun et al., PRL 2000; Lyanda-Geller et al., PRB 2001 (theory for manganites; ) A. A. Burkov and L. Balents, PRL 2003; A. Fernández-Pacheco etal (2008) Deepak Venkateshvaran etal (2008) In magnetite thin films: • Microscopic mechanisms? • Why is irrespective of material? • Why doesn’t it depend on type of conduction?

23. : responsible for longitudinal conductance. represents the local on-site total angular-momentum state. phonon 1. Two-sitedirect hopping with one-phononprocess : localization i j k i j k Phonon-assisted hopping The minimal Hamiltonian for the AHE in insulating regime: Longitudinal hopping charge transport

24. Including these triads the electric current between two sites is: 2. How to capture the Hall effect? Three-site hopping (Holstein, 1961) Two-sitedirect hopping preserves the TR symmetry. Hall transition rate The transition must break the time-reversal (TR) symmetry Need three site hopping : direct conductance due totwo-sitehopping. Interference term : off-diagonal conductance due to hopping via three-sites. Challenging: Macroscopicanomalous Hall conductivity/resistivity? Geometric phase: break TR symmetry m: the number of real phonons included in the whole transition. Phonon-assisted hopping: Hall charge transport

25. connected disconnected Cut-off Treated as perturbation Percolation theory for AHE: the resistor network • Map the random impurity system to a random resistor network based on direct conductance: 2. Introduce the cut-off to redefine the connectivity (Ambegaokar etal., 1971):

26. For a site with energy , the average number of sites connecting to it under the condition (Pollak, 1972) : Percolating path/cluster appears when the averaging connectivity (G.E. Pike, etal 1974): Percolation path/cluster the probability that the n-th smallest resistor connected to the i site has the conductance larger than . Percolation theory for AHE: the resistor network 3. Percolation path/cluster 4. Configuration averaging of general m-site function along the critical path/cluster, with the i-th site has at least sites connected to it: Transverse resistivity/conductivity:each site has at least three sites connected to it?

27. Macroscopic anomalous Hall conductivity/resistivity Percolation path/cluster appears when (G.E. Pike, etal 1974): The averaging transverse voltage is given by: In the thermodynamic limit, we get the AHC:

28. The approach However, instead of an exact calculation, one may find the upper and lower limits of AHC by imposing different restrictions for the configuration integrals in it. Once we obtain the two limits of the AHC, we can determine the range of the scaling relation between the AHC and longitudinal conductivity. Note:

29. Extreme situation (I): Extreme situation (II): Let: In each triad of the whole percolation cluster, it is the two bonds with smaller conductance ( , ) that dominate the charge transport. In each triad of the whole percolation cluster, it is the two bonds with larger conductance ( , ) that dominate the charge transport. The upper limit of the AHC. The lower limit of the AHC. What bonds in a triad play the major role for the charge current flowing through it is determined by the optimization on both the resistance magnitudes and spatial configuration of the three bonds.

30. Limits of distributions: final result where Direct numerical calculation gives 1.6 • Depends weakly on the type of hopping! • Generic to hopping conductivity Xiong-Jun Liu, Sinova PRB 2011

31. SUMMARY • AHE general theory for metallic multi-band systems which contains all scattering-independent contributions developed: useful for ab-initio studies • (Kovalev, Sinova, Tserkovnyak PRL 2010) • AHE ab-initio theory of of simple ferromagnetic metals of the scattering independent contribution (Weischenberg, Freimuth, Sniova, Blügel, Mokrousov, PRL 2011) • AHE hopping regime approximate scaling arises directly from a generalization of the Holstein theory to AHE (Xiong-Jun Liu, Sinova, PRB 2011) • AHE hopping regime scaling remains even when crossing to different types of insulating hopping regimes, only algebraic pre-factor changes