German Physical Society MeetingTU Berlin

March 26th, 2012

Research fueled by:

JAIRO SINOVATexas A&M University

Institute of Physics ASCR

Institute of Physics ASCR

UCLA A. Kovalev

Y. Tserkovnyak

Texas A&M UniversityXiong-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

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

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⇢xy

= � �xy

�2xx

+ �2xy

⇡ ��xy

�2xx

⇡ ��xy

⇢2xx

⇡ �A⇢xx

� B⇢2xx

Simple electrical measurement of out of plane magnetization

InMnAs

Spin dependent “force” deflects like-spin particles

ρH=R0B ┴ +4π RsM┴

Anomalous Hall Effect: the basics

I

_ FSO

FSO

_ __majority

minority

V

�AHxy

⇡ B + A�xx

3

⇢xy

= �A⇢xx

� B⇢2xx

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 mechanismMaterial with dominant scattering-independent mechanism

�AHxy

⇡ B + A�xx

4

Valenzuela et al Nature 06

Inverse SHE

5

Anomalous Hall effect: more than meets the eye

Wunderlich, Kaestner, Sinova, Jungwirth PRL 04

Kato et al Science 03

IntrinsicExtrinsic

V

Mesoscopic Spin Hall Effect

Intrinsic

Brune,Roth, Hankiewicz, Sinova, Molenkamp, et al

Nature Physics 2010

Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09

Spin-injection Hall Effect

Anomalous Hall Effect

I

_ FSO

FSO

_ _majority

minority

V

Spin Hall Effect

I

_ FSO

FSO

_ _

V

Topological Insulators

Kane and Mele PRL 05

Nagaosa, Sinova, Onoda, MacDonald, Ong

Anomalous Hall effect phase diagram

1. A high conductivity regime for σxx>106 (Ωcm)-1 in which AHE is skew dominated2. A good metal regime for σxx ~103-106 (Ωcm) -1 in which σxy

AH~ const3. A bad metal/hopping regime for σxx<103 (Ωcm) -1 for which σxy

AH~ σxyα with α=1.4~1.7

123

12

3

6

Cartoon of the mechanisms contributing to AHE in the metallic regime �AHxy

⇡ B + A�xx

independent of impurity density

xc =@"(~k)~@

~

k

+e

~~

E ⇥ ~⌦

⌦z

(~k, n) = 2Im⌧

@

@y

n~k

����@

@x

n~k

�

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 scatteringVimp(r) (Δso>ħ/τ) or ∝ λ*∇Vimp(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/niVimp(r) (Δso>ħ/τ) or ∝ λ*∇Vimp(r) (Δso<ħ/τ) A

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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 σAH ∼ 1000 (Ω cm)-1

TheroyσAH ∼ 750 (Ω cm)-1

AHE in Fe

AHE in GaMnAs

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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?

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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

�AHxy

⇡ B + A�xx

10

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

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

Sinitsyn et al 2007

11

1. Use Kubo-Streda formalism or linearized version of Keldysh formalism to obtain

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 2007)

Kovalev, Sinova, Tserkovnyak PRL 2010

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Generalization to 3D

Scattering independent AHE conductivity expressed through band structure

Well known intrinsic contributionSide 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 201013

Application to simple ferromagnets

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Evaluating “intrinsic” vs. “side-jump”DIRECTLY from the band structure alone

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Some further challenges

1. A key assumption so far: short range isotropic disorder

2. Is CPA applicable?• CPA is useful for self-averaging quantities, e.g. diagonal

conductivity. What about AHE which is so dependent on inter-band coherence scattering?

• How are the different length scales coming into play

3. How sensitive are each of the scattering independent contributions to disorder?• Are both contributions topological?• How do magnetization spatial variation (topological AHE) play a

role in some of the systems?

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

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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

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A. Fernández-Pacheco etal (2008)

Deepak Venkateshvaran etal (2008)

In magnetite thin films:

•Microscopic mechanisms?•Why is it 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;

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The minimal Hamiltonian for the AHE in insulating regime: Phonon-assisted hopping

i j k

phononi 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

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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?

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The transition must break the time-reversal (TR) symmetryTwo-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

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):

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3. Percolation path/clusterFor 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

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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:

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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.

where

Limits of distributions: final result

•Depends weakly on the type of hopping!•Generic to hopping conductivity

Xiong-Jun Liu, Sinova PRB 201127

Direct numerical calculation gives 1.6

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

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