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

2 Outline 1.Introduction Anomalous Hall effect phenomenology: more than meets the eye 2.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 3.Scaling of the AHE in the insulating regime Experiments and phenomenology phonon-assisted hopping AHE (Holstein) Percolations theory generalization for the AHE conductivity Results 4.Summary

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

4 Simple electrical measurement of out of plane magnetization InMnAs Spin dependent “force” deflects like-spin particles ρ H =R 0 B ┴ +4π R s M ┴ Anomalous Hall Effect: the basics I _ F SO _ _ _ majority minority V

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

6 Cartoon of the mechanisms contributing to AHE in the metallic regime 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. 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 Intrinsic deflection B 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 Side jump scattering V imp (r) (Δso>ħ/τ)  ∝ λ* ∇ V imp (r) (Δso<ħ/τ) B Skew scattering Asymmetric scattering due to the spin-orbit coupling of the electron or the impurity. Known as Mott scattering. ~σ~1/n i V imp (r) (Δso>ħ/τ)  ∝ λ* ∇ V imp (r) (Δso<ħ/τ) A

7 Nagaosa, Sinova, Onoda, MacDonald, Ong Anomalous Hall Effect 1.A high conductivity regime for σ xx >10 6 (Ωcm) -1 in which AHE is skew dominated 2.A good metal regime for σ xx ~ (Ωcm) -1 in which σ xy AH ~ const 3.A bad metal/hopping regime for σ xx <10 4 (Ωcm) -1 for which σ xy AH ~ σ xy α with α=1.4~

8 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 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) Experiment  AH  1000 (  cm) -1 Theroy  AH  750 (  cm) -1 AHE in Fe AHE in GaMnAs

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 Semi-classical approach: Gauge invariant formulation Sinitsyn, Sinvoa, et al PRB 05, PRL 06, PRB 07 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 Kovalev, Sinova et al PRB 08, Onoda PRL 06, PRB 08

11 Comparing Boltzmann to Kubo (chiral basis) for “Graphene” model 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

12 Generalization to 3D 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 General band structure in the presence of delta-correlated Gaussian disorder

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

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20 Outline 1.Introduction Anomalous Hall effect phenomenology: more than meets the eye 2.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 3.Scaling of the AHE in the insulating regime Experiments and phenomenology phonon-assisted hopping AHE (Holstein) Percolations theory generalization for the AHE conductivity Results 4.Summary

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

22 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? S. H. Chun et al., PRL 2000; Lyanda-Geller et al., PRB 2001 (theory for manganites; ) A. A. Burkov and L. Balents, PRL 2003;

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

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

25 Percolation theory for AHE: the resistor network connected disconnected Cut-off 1.Map the random impurity system to a random resistor network based on direct conductance: Treated as perturbation 2. Introduce the cut-off to redefine the connectivity (Ambegaokar etal., 1971):

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

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

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: The approach

Let: Extreme situation (I): In each triad of the whole percolation cluster, it is the two bonds with larger conductance (, ) that dominate the charge transport. The lower limit of the AHC. Extreme situation (II): In each triad of the whole percolation cluster, it is the two bonds with smaller conductance (, ) that dominate the charge transport. The upper 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 where Limits of distributions: final result Depends weakly on the type of hopping! Generic to hopping conductivity Xiong-Jun Liu, Sinova PRB 2011 Direct numerical calculation gives 1.6

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