Anomalous Hall effect

Naoto Nagaosa

Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan andCross Correlated Research Materials Group (CMRG), ASI, RIKEN, Wako 351-0198, Saitama, Japan

Jairo Sinova

Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA andInstitute of Physics ASCR, Cukrovarnicka 10, 162 53 Praha 6, Czech Republic

Shigeki Onoda

Condensed Matter Theory Laboratory, ASI, RIKEN, Wako 351-0198, Saitama, Japan

A. H. MacDonald

Department of Physics, University of Texas at Austin, Austin, Texas 78712-1081, USA

N. P. Ong

Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

(Dated: March 23, 2009)

We present a review of experimental and theoretical studies of the anomalous Hall effect (AHE),focusing on recent developments that have provided a more complete framework for understandingthis subtle phenomenon and have, in many instances, replaced controversy by clarity. Synergybetween experimental and theoretical work, both playing a crucial role, has been at the heartof these advances. On the theoretical front, the adoption of Berry-phase concepts has estab-lished a link between the AHE and the topological nature of the Hall currents which originatefrom spin-orbit coupling. On the experimental front, new experimental studies of the AHE intransition metals, transition-metal oxides, spinels, pyrochlores, and metallic dilute magnetic semi-conductors, have more clearly established systematic trends. These two developments in concertwith first-principles electronic structure calculations, strongly favor the dominance of an intrinsicBerry-phase-related AHE mechanism in metallic ferromagnets with moderate conductivity. Theintrinsic AHE can be expressed in terms of Berry-phase curvatures and it is therefore an intrin-sic quantum mechanical property of a perfect cyrstal. An extrinsic mechanisms, skew scatteringfrom disorder, tends to dominate the AHE in highly conductive ferromagnets. We review thefull modern semiclassical treatment of the AHE which incorporates an anomalous contributionto wavepacket group velocity due to momentum-space Berry curvatures and correctly combinesthe roles of intrinsic and extrinsic (skew scattering and side-jump) scattering-related mechanisms.In addition, we review more rigorous quantum-mechanical treatments based on the Kubo andKeldysh formalisms, taking into account multiband effects, and demonstrate the equivalence of allthree linear response theories in the metallic regime. Building on results from recent experimentand theory, we propose a tentative global view of the AHE which summarizes the roles played byintrinsic and extrinsic contributions in the disorder-strength vs. temperature plane. Finally wediscuss outstanding issues and avenues for future investigation.

Contents

I. Introduction 2A. A brief history of the AHE and new perspectives 2B. Parsing the AHE: 5

1. Intrinsic contribution to σAHxy 6

2. Skew scattering contribution to σAHxy 7

3. Side-jump contribution to σAHxy 7

II. Experimental and theoretical studies on specific

materials 8A. Transition-metals 8

1. Early experiments 82. Recent experiments 93. Comparison to theories 11

B. Complex oxide ferromagnets 121. First-principles calculations and experiments on

SrRuO3 122. Spin chirality mechanism of the AHE in manganites14

3. Lanthanum cobaltite 16

4. Spin chirality mechanism in pyrochloreferromagnets 17

5. Anatase and Rutile Ti1−xCoxO2−δ 18

C. Ferromagnetic semiconductors 18D. Other classes of materials 20

1. Spinel CuCr2Se4 20

2. Heusler Alloy 213. Fe1−yCoySi 21

4. MnSi 225. Mn5Ge3 23

6. Layered dichalcogenides 24

E. Localization and AHE 24

III. Theoretical aspects of the AHE and early theories 25A. Symmetry considerations and analogies between

normal Hall effect and AHE 26

B. Topological interpretation of the intrinsic mechanism:relation between Fermi sea and Fermi surface

2

properties 29C. Early theoretical studies of the AHE 30

1. Karplus-Luttinger theory and the intrinsicmechanism 30

2. Extrinsic mechanisms 313. Kohn-Luttinger theory formalism 34

IV. Linear Transport Theories of the AHE 36A. Semiclassical Boltzmann approach 37

1. Equation of motion of Bloch states wave-packets 382. Scattering and the side-jump 383. Kinetic equation for the semiclasscial Boltzmann

distribution 394. Anomalous velocities, anomalous Hall currents, and

anomalous Hall mechanisms 40B. Kubo formalism 41

1. Kubo technique for the AHE 412. Relation between the Kubo and the semiclassical

formalisms 42C. Keldysh formalism 43D. Two-dimensional ferromagnetic Rashba model – a

minimal model 45

V. Conclusions: Future problems and perspectives on

AHE 47

Acknowledgements 50

References 50

I. INTRODUCTION

A. A brief history of the AHE and new perspectives

The anomalous Hall effect has deep roots in the his-tory of electricity and magnetism. In 1879 Edwin H.Hall (Hall, 1879) made the momentous discovery that,when a current-carrying conductor is placed in a mag-netic field, the Lorentz force “presses” its electronsagainst one side of the conductor. One year later, hereported that his “pressing electricity” effect was tentimes larger in ferromagnetic iron (Hall, 1881) than innon-magnetic conductors. Both discoveries were remark-able, given how little was known at the time about howcharge moves through conductors. The first discoveryprovided a simple, elegant tool to measure carrier con-centration more accurately in non-magnetic conductors,and played a midwife’s role in easing the birth of semi-conductor physics and solid-state electronics in the late1940’s. For this role, the Hall effect was frequently calledthe queen of solid-state transport experiments.

The stronger effect that Hall discovered in ferromag-netic conductors came to be known as the anomalous Halleffect (AHE). The AHE has been an enigmatic problemthat has resisted theoretical and experimental assault foralmost a century. The main reason seems to be that,at its core, the AHE problem involves concepts based ontopology and geometry that have been formulated only inrecent times. The early investigators grappled with no-tions that would not become clear and well defined untilmuch later, such as the concept of Berry-phase (Berry,1984). What is now viewed as Berry phase curvature,later dubbed “anomalous velocity” by Luttinger, arose

naturally in the first microscopic theory of the AHE byKarplus and Luttinger (Karplus and Luttinger, 1954).However, because understanding of these concepts, notto mention the odd intrinsic dissipationless Hall currentthey seemed to imply, would not be achieved for another40 years, the AHE problem was quickly mired in a contro-versy of unusual endurance. Moreover, the AHE seemsto be a rare example of a pure, charge-transport problemwhose elucidation has not – to date – benefited from theapplication of complementary spectroscopic and thermo-dynamic probes.

FIG. 1 The Hall effect in Ni [data from A. W. Smith, Phys.Rev. 30, 1 (1910)]. [From Ref. Pugh and Rostoker, 1953.]

Very early on, experimental investigators learned thatthe dependence of the Hall resistivity ρxy on applied per-pendicular field Hz is qualitatively different in ferromag-netic and non-magnetic conductors. In the latter, ρxy

increases linearly with Hz, as expected from the Lorentzforce. In ferromagnets, however, ρxy initially increasessteeply in weak Hz, but saturates at a large value thatis nearly Hz-independent (Fig. 1). Kundt noted that,in Fe, Co, and Ni, the saturation value is roughly pro-portional to the magnetization Mz (Kundt, 1893) andhas a weak anisotropy when the field (z) direction is ro-tated with respect to the cyrstal, corresponding to theweak magnetic anisotropy of Fe, Co, and Ni (Web-ster, 1925). Shortly thereafter, experiments by Pugh andcoworkers (Pugh, 1930; Pugh and Lippert, 1932) estab-lished that an empirical relation between ρxy, Hz, andMz,

ρxy = R0Hz +RsMz, (1.1)

applies to many materials over a broad range of externalmagnetic fields. The second term represents the Hall ef-fect contribution due to the spontaneous magnetization.This AHE is the subject of this review. Unlike R0, whichwas already understood to depend mainly on the densityof carriers, Rs was found to depend subtly on a varietyof material specific parameters and, in particular, on thelongitudinal resistivity ρxx = ρ.

In 1954, Karplus and Luttinger (KL) (Karplus andLuttinger, 1954) proposed a theory for the AHE that, in

3

hindsight, provided a crucial step in unraveling the AHEproblem. KL showed that when an external electric fieldis applied to a solid, electrons acquire an additional con-tribution to their group velocity. KL’s anomalous velocitywas perpendicular to the electric field and therefore couldcontribute to Hall effects. In the case of ferromagneticconductors, the sum of the anomalous velocity over alloccupied band states can be non-zero, implying a contri-bution to the Hall conductivity σxy. Because this contri-bution depends only on the band structure and is largelyindependent of scattering, it has recently been referred toas the the intrinsic contribution to the AHE. When theconductivity tensor is inverted, the intrinsic AHE yieldsa contribution to ρxy ≈ σxy/σ

2xx and therefore it is pro-

portional to ρ2. The anomalous velocity is dependentonly on the perfect crystal Hamiltonian and can be re-lated to changes in the phase of Bloch state wavepacketswhen an electric field causes them to evolve in crystalmomentum space (Bohm et al., 2003; Chang and Niu,1996; Sundaram and Niu, 1999; Xiao and Niu, 2009). Asmentioned, the KL theory anticipated by several decadesthe modern interest in Berry phase and Berry curvatureeffects, particularly in momentum-space.

FIG. 2 Extraordinary Hall constant as a function of resis-tivity. The shown fit has the relation Rs ∼ ρ1.9. [FromRef. Kooi, 1954.]

Early experiments to measure the relationship betweenρxy and ρ generally assumed to be of the power law form,i.e., ρxy ∼ ρβ , mostly involved plotting ρxy (or Rs) vs.ρ, measured in a single sample over a broad interval of T

(typically 77 to 300 K). As we explain below, competingtheories in metals suggested either that β = 1 or β = 2.A compiled set of results was published by Kooi (Kooi,1954) (Fig. 2). The subsequent consensus was that suchplots do not settle the debate. At finite T , the carriersare strongly scattered by phonons and spin waves. Theseinelastic processes – difficult to treat microscopically eventoday – lie far outside the purview of the early theories.Smit suggested that, in the skew-scattering theory (seebelow), phonon scattering increases the value β from 1to values approaching 2. This was also found by otherinvestigators. A lengthy calculation by Lyo (Lyo, 1973)showed that skew-scattering at T ΘD (the Debye tem-perature) leads to the relationship ρxy ∼ (ρ2 + aρ), witha a constant. In an early theory by Kondo consideringskew scattering from spin excitations (Kondo, 1962), itmay be seen that ρxy also varies as ρ2 at finite T .

The proper test of the scaling relation in comparisonwith present theories involves measuring ρxy and ρ in aset of samples at 4 K or lower (where impurity scatteringdominates). By adjusting the impurity concentration ni,one may hope to change both quantities sufficiently todetermine accurately the exponent β and use this iden-tification to tease out the underlying physics.

The main criticism of the KL theory centered on thecomplete absence of scattering from disorder in the de-rived Hall response contribution. The semi-classical AHEtheories by Smit and Berger focused instead on the in-fluence of disorder scattering in imperfect crystals. Smitargued that the main source of the AHE currents wasasymmetric (skew) scattering from impurities caused bythe spin-orbit interaction (SOI) (Smit, 1955, 1958). ThisAHE picture predicted that Rs ∼ ρxx (β = 1). Berger, onthe other hand, argued that the main source of the AHEcurrent was the side-jump experienced by quasiparticlesupon scattering from spin-orbit coupled impurities. Theside-jump mechanism could (confusingly) be viewed as aconsequence of a KL anomalous velocity mechanism act-ing while a quasiparticle was under the influence of theelectric field due to an impurity. The side-jump AHE cur-rent was viewed as the product of the side-jump per scat-tering event and the scattering rate (Berger, 1970). Onepuzzling aspect of this semiclassical theory was that alldependence on the impurity density and strength seem-ingly dropped out. As a result, it predicted Rs ∼ ρ2

xx

with an exponent β identical to that of the KL mech-anism. The side-jump mechanism therefore yielded acontribution to the Hall conductivity which was seem-ingly independent of the density or strength of scatter-ers. In the decade 1970-80, a lively AHE debate waswaged largely between the proponents of these two ex-trinsic theories. The three main mechanisms consideredin this early history are shown schematically in Fig. 3.

Some of the confusion in experimental studies stemmedfrom a hazy distinction between the KL mechanism andthe side-jump mechanism, a poor understanding of howthe effects competed at a microscopic level, and a lack ofsystematic experimental studies in a diverse set of mate-

4

a) Intrinsic deflection

Interband coherence induced by an t l l t i fi ld i i t

Eexternal electric field gives rise to a velocity contribution perpendicular to the field direction. These currents do not sum to zero in ferromagnets.

Electrons have an anomalous velocity perpendicular to the electric field related to their Berry’s phase curvature

nbEe

k

E

dt

rd

b) Side jump

The electron velocity is deflected in opposite directions by the opposite electric fields experienced upon approaching and leaving an impurity.p p pp g g p yThe time-integrated velocity deflection is the side jump.

c) Skew scattering

Asymmetric scattering due toAsymmetric scattering due tothe effective spin-orbit coupling of the electron or the impurity.

FIG. 3 Illustration of the three main mechanisms that cangive rise to an AHE. In any real material all of these mecha-nisms act to influence electron motion.

rials.One aspect of the confusion may be illustrated by con-

trasting the case of a high-purity mono-domain ferromag-net, which produces a spontaneous AHE current propor-tional to Mz, with the case of a material containing mag-netic impurities (e.g. Mn) embedded in a non-magnetichost such as Cu (the dilute Kondo system). In a field H ,the latter also displays an AHE current proportional tothe induced M = χH , with χ the susceptibility (Fertand Jaoul, 1972). However, in zero H , time-reversal in-variance (TRI) is spontaneously broken in the former,but not in the latter. Throughout the period 1960-1989,the two Hall effects were often regarded as a commonphenomenon that should be understood microscopicallyon the same terms. It now seems clear that this viewimpeded progress.

By the mid-1980s, interest in the AHE problem hadwaned significantly. The large body of Hall data gar-nered from experiments on dilute Kondo systems in theprevious two decades showed that ρxy ∼ ρ and thereforeappeared to favor the skew-scattering mechanism. Thepoints of controversy remained unsettled, however, andthe topic was still mired in confusion.

Since the 1980’s, the quantum Hall effect in two-dimensional (2D) electron systems in semiconductor het-erostructures has become a major field of research inphysics (Prange and Girvin, 1987). The accurate quan-tization of the Hall conductance is the hallmark of thisphenomenon. Both the integer (Thouless et al., 1982)and fractional quantum Hall effects can be explained interms of the topological properties of the electronic wave-functions. For the case of electrons in a two-dimensionalcrystal, it has been found that the Hall conductance isconnected to the topological integer (Chern number) de-fined for the Bloch wavefunction over the first Brillouinzone (Thouless et al., 1982). This way of thinking about

the quantum Hall effect began to have a deep impacton the AHE problem starting around 1998. Theoreticalinterest in the Berry phase and in its relation to trans-port phenomena, coupled with many developments in thegrowth of novel complex magnetic systems with strongspin-orbit coupling (notably the manganites, pyrochloresand spinels) led to a strong resurgence of interest in theAHE and eventually to deeper understanding.

Since 2003 many systematics studies, both theoreticaland experimental, have led to a better understanding ofthe AHE in the metallic regime, and to the recognition ofnew unexplored regimes that present challenges to futureresearchers. As it is often the case in condensed matterphysics, attempts to understand this complex and fasci-nating phenomenon have motivated researchers to couplefundamental and sophisticated mathematical concepts toreal-world materials issues. The aim of this review is tosurvey recent experimental progress in the field, and topresent the theories in a systematic fashion. Researchersare now able to understand the links between differentviews on the AHE previously thought to be in conflict.Despite the progress in recent years, understanding isstill incomplete. We highlight some intriguing questionsthat remain and speculate on the most promising avenuesfor future exploration. In this review we focus, in par-ticular, on reports that have contributed significantly tothe modern view of the AHE. For previous reviews, thereader may consult Pugh (Pugh and Rostoker, 1953) andHurd (Hurd, 1972). For more recent short overviews fo-cused on the topological aspects of the AHE, we pointthe reader to the reviews by Nagaosa (Nagaosa, 2006),and by Sinova et al.(Sinova et al., 2004). A review ofthe modern semiclassical treament of AHE was recentlywritten by Sinitsyn (Sinitsyn, 2008). The present reviewhas been informed by ideas explained in the earlier works.Readers who are not familair with Berry phase conceptsmay find it useful to consult the elementary review byOng and Lee (Ong and Lee, 2006) and the popular com-mentary by MacDonald and Niu (MacDonald and Niu,2004).

Some of the recent advances in the understanding ofthe AHE that will be covered in this review are:

1. When σAHxy is independent of σxx, the AHE can

often be understood in terms of the geometric con-cepts of Berry phase and Berry curvature in mo-mentum space. This AHE mechanism is respon-sible for the intrinsic AHE. In this regime, theanomalous Hall current can be thought of as theunquantized version of the quantum Hall effect. In2D systems the intrinsic AHE is quantized in unitsof e2/h at temperature T = 0 when the Fermi levellies between Bloch state bands.

2. Three broad regimes have been identified when sur-veying a large body of experimental data for di-verse materials: (i) A high conductivity regime(σxx > 106 (Ωcm)−1) in which a linear contributionto σAH

xy ∼ σxx due to skew scattering dominates

5

σAHxy . In this regime the normal Hall conductiv-

ity contribution can be significant and even domi-nate σxy, (ii) An intrinsic or scattering-independentregime in which σAH

xy is roughly independent of σxx

(104 (Ωcm)−1 < σxx < 106 (Ωcm)−1), (iii) A bad-metal regime (σxx < 104 (Ωcm)−1) in which σAH

xy

decreases with decreasing σxx at a rate faster thanlinear.

3. The relevance of the intrinsic mechanisms can bestudied in-depth in magnetic materials with strongspin-orbit coupling, such as oxides and diluted mag-netic semiconductors (DMS). In these systems asystematic non-trivial comparison between the ob-served properties of systems with well controlledmaterials properties and theoretical model calcula-tions can be achieved.

4. The role of band (anti)-crossings near the Fermi en-ergy has been identified using first-principles Berrycurvature calculations as a mechanism which canlead to a large intrinsic AHE.

5. Semiclassical treatment by a generalized Boltz-mann equation taking into account the Berry cur-vature and coherent inter-band mixing effects dueto band structure and disorder has been formu-lated. This theory provides a clearer physical pic-ture of the AHE than early theories by identify-ing correctly all the semiclassically defined mech-anisms. This generalized semiclassical picture hasbeen verified by comparison with controlled micro-scopic linear response treatments for identical mod-els.

6. The relevance of non-coplanar spin structures withassociated spin chirality and real-space Berry cur-vature to the AHE has been established both the-oretically and experimentally in several materials.

7. Theoretical frameworks based on the Kubo formal-ism and the Keldysh formalism have been devel-oped which are capable of treating transport phe-nomena in systems with multiple bands.

The review is aimed at experimentalists and theoristsinterested in the AHE. We have structured the reviewas follows. In the remainder of this section, we providethe minimal theoretical background necessary to under-stand the different AHE mechanisms. In particular weexplain the scattering-independent Berry phase mech-anism which is more important for the AHE than forany other commonly measured transport coefficient. InSec. II we review recent experimental results on a broadrange of materials, and compare them with relevant cal-culations where available. In Sec. III we discuss AHEtheory from an historical perspective, explaining linksbetween different ideas which are not always recognized,and discussing the physics behind some of the past con-fusion. The section may be skipped by readers who do

not wish to be burdened by history. Section IV discussesthe present understanding of the metallic theory basedon a careful comparison of the different linear responsetheories which are now finally consistent. In Sec. V wepresent a summary and outlook.

B. Parsing the AHE:

The anomalous Hall effect is at its core a quantum phe-nomena which originates from quantum coherent bandmixing effects by both the external electric field andthe disorder potential. Like other coherent interferencetransport phenomena (e.g. weak localization), it cannotbe satisfactorily explained using traditional semiclassicalBoltzmann transport theory. Therefore, when parsingthe different contributions to the AHE, they can be de-fined semiclassically only in a carefully elaborated theory.

In this section we identify three distinct contributionswhich sum up to yield the full AHE: intrinsic, skewscattering, and side-jump contributions. We choose thisnomenclature to reflect the modern literature withoutbreaking completely from the established AHE lexicon(see Sec. III). However, unlike previous classifications,we base this parsing of the AHE on experimental and mi-croscopic transport theory considerations, rather than onthe identification of one particular effect which could con-tribute to the AHE. The link to semiclassically definedprocesses is established after developing a fully general-ized Boltzmann transport theory which takes inter-bandcoherence effects into account and is fully equivalent tomicroscopic theories (Sec. IV.A). In fact, much of thetheoretical effort of the past few years has been expendedin understanding this link between semiclassical and mi-croscopic theory which has escaped cohesion for a longtime.

A very natural classification of contributions to theAHE, which is guided by experiment and by microscopictheory of metals, is to separate them according to theirdependence on the Bloch state transport lifetime τ . Inthe theory, disorder is treated perturbatively and higherorder terms vary with a higher power of the quasiparti-cle scattering rate τ−1. As we will discuss, it is relativelyeasy to identify contributions to the anomalous Hall con-ductivity, σAH

xy , which vary as τ1 and as τ0. In experi-ment a similar separation can sometimes be achieved byplotting σxy vs. the longitudinal conductivity σxx ∝ τ ,when τ is varied by altering disorder or varying tem-perature. More commonly (and equivalently) the Hallresistivity is separated into contributions proportional toρxx and ρ2

xx.This partitioning seemingly gives only two contribu-

tions to σAHxy , one ∼ τ and the other ∼ τ0. The first

contribution we define as the skew-scattering contribu-tion, σAH−skew

xy . Note that in this parsing of AHE con-tributions it is the dependence on τ (or σxx) which de-fines it, not a particular mechanism linked to a micro-scopic or semiclassical theory. The second contribution

6

proportional to τ0 (or independent of σxx) we furtherseparate into two parts: intrinsic and side-jump. Al-though these two contributions cannot be separated ex-perimentally by dc measurements, they can be sepa-rated experimentally (as well as theoretically) by defin-ing the intrinsic contribution, σAH−int

xy , as the extrapo-lation of the ac-interband Hall conductivity to zero fre-quency in the limit of τ → ∞, with 1/τ → 0 fasterthan ω → 0. This then leaves a unique definition forthe third and last contribution, termed side-jump, asσAH−sj

xy ≡ σAHxy − σAH−skew

xy − σAH−intxy .

We examine these three contributions below ( still atan introductory level). It is important to note thatthe above definitions have not relied on identificationsof semiclassical processes such as side-jump scattering(Berger, 1970) or skew-scattering from asymmetric con-tributions to the semiclassical scattering rates (Smit,1955) identified in earlier theories. Not surprisingly, thecontributions defined above contain these semiclassicalprocesses. However, it is now understood (see Sec. IV),that other contributions are present in the fully general-ized semiclassical theory which were not precisely identi-fied previously and which are necessary to be fully con-sitent with microscopic theories.

The ideas explained briefly in this section are substan-tiated in Sec. II by analyses of tendencies in the AHEdata of several different material classes, and in Sec. IIIand Sec. IV by an extensive technical discussion of AHEtheory. We assume throughout that the ferromagneticmaterials of interest are accurately described by a Stoner-like mean-field band theory. In applications to real ma-terials we imagine that the band theory is based on spin-density-functional theory (Jones and Gunnarsson, 1989)with a local-spin-density or similar approximation for theexchange-correlation energy functional.

1. Intrinsic contribution to σAHxy

Among the three contributions, the easiest to eval-uate accurately is the intrinsic contribution. We havedefined the intrinsic contribution microscopically as thedc limit of the interband conductivity, a quantity whichis not zero in ferromagnets when SOI are included.There is however a direct link to semiclassical theory inwhich the induced interband coherence is captured bya momentum-space Berry-phase related contribution tothe anomalous velocity. We show this equivalence below.

This contribution to the AHE was first derived by KL(Karplus and Luttinger, 1954) but its topological naturewas not fully appreciated until recently (Jungwirth et al.,2002b; Onoda and Nagaosa, 2002). The work of Jung-wirth et al. (Jungwirth et al., 2002b) was motivated bythe experimental importance of the AHE in ferromag-netic semiconductors and also by the thorough earlieranalysis of the relationship between momentum spaceBerry phases and anomalous velocities in semiclassicaltransport theory by Niu et al. (Chang and Niu, 1996;

Sundaram and Niu, 1999). The frequency-dependentinter-band Hall conductivity, which reduces to the intrin-sic anomalous Hall conductivity in the dc limit, had beenevaluated earlier for a number of materials by Mainkaret al. (Mainkar et al., 1996) and Guo and Ebert (Guoand Ebert, 1995) but the topological connection was notrecognized.

The intrinsic contribution to the conductivity is de-pendent only on the band structure of the perfect crys-tal, hence its name. It can be calculated directly fromthe simple Kubo formula for the Hall conductivity for anideal lattice, given the eigenstates |n,k〉 and eigenvaluesεn(k) of a Bloch Hamiltonian H :

σAH−intij = e2~

∑

n6=n′

∫

dk

(2π)3[f(εn(k))− f(εn′(k))]

×Im〈n,k|vi(k)|n′,k〉〈n′,k|vj(k)|n,k〉

(εn(k)− εn′(k))2.

(1.2)

In Eq. (1.2) H is the k-dependent Hamiltonian for theperiodic part of the Bloch functions and the velocity op-erator is defined by

v(k) =1

i~[r, H(k)] =

1

~∇kH(k). (1.3)

Note the restriction n 6= n′ in Eq. (1.2).What makes this contribution quite unique is that,

like the quantum Hall effect in a crystal, it is directlylinked to the topological properties of the Bloch states.(See Sec. III.B.) Specifically it is proportional to theintegration over the Fermi sea of the Berry’s curvatureof each occupied band, or equivalently (Haldane, 2004;Wang et al., 2007) to the integral of Berry phases overcuts of Fermi surface segments. This result can be de-rived by noting that

〈n,k|∇k|n′,k〉 = 〈n,k|∇kH(k)|n′,k〉εn′(k)− εn(k)

. (1.4)

Using this expression, Eq. (1.2) reduces to

σAH−intij = −εij`

e2

~

∑

n

∫

dk

(2π)df(εn(k)) b`n(k), (1.5)

where εij` is the anti-symmetric tensor, an(k) is theBerry-phase connection an(k) = i〈n,k|∇k|n,k〉, andbn(k) the Berry-phase curvature

bn(k) = ∇k × an(k) (1.6)

corresponding to the states |n,k〉.This same linear response contribution to the AHE

conductivity can be obtained from the semiclassical the-ory of wave-packets dynamics (Chang and Niu, 1996;Marder, 2000; Sundaram and Niu, 1999). It can beshown that the wavepacket group velocity has an addi-tional contribution in the presence of an electric field:

7

rc = ∂En(k)/~∂k − (E/~) × bn(k). (See Sec. IV.A.)The intrinsic Hall conductivity formula, Eq. (1.5), is ob-tained simply by summing the second (anomalous) termover all occupied states.

One of the motivations for identifying the intrinsiccontribution σAH−int

xy is that it can be evaluated accu-rately even for relatively complex materials using first-principles electronic structure theory techniques. Inmany materials which have strongly spin-orbit coupledbands, the intrinsic contribution seems to dominates theAHE.

2. Skew scattering contribution to σAHxy

The skew scattering contribution to the AHE can besharply defined; it is simply the contribution which isproportional to the Bloch state transport lifetime. It willtherefore tend to dominate in nearly perfect crystals. It isthe only contribution to the AHE which appears withinthe confines of traditional Boltzmann transport theoryin which interband coherence effects are completely ne-glected. Skew scattering is due to chiral features whichappear in the disorder scattering of spin-orbit coupledferromagnets. This mechanism was first identified bySmit (Smit, 1955, 1958).

Treatments of semi-classical Boltzmann transport the-ory found in textbooks often appeal to the principle ofdetailed balance which states that the transition prob-ability Wn→m from n to m is identical to the transi-tion probability in the opposite direction (Wm→n). Al-though these two transition probabilities are identicalin a Fermi’s golden-rule approximation, since Wn→n′ =(2π/~)|〈n|V |n′〉|2δ(En − En′), where V is the perturba-tion inducing the transition, detailed balance in this mi-croscopic sense is not generic. In the presence of spin-orbit coupling, either in the Hamiltonian of the perfectcrystal or in the disorder Hamiltonian, a transition whichis right-handed with respect to the magnetization di-rection has a different transition probability than thecorresponding left-handed transition. When the transi-tion rates are evaluated perturbatively, asymmetric chi-ral contributions appear first at third order. (See Sec.IV.A). In simple models the asymmetric chiral contribu-tion to the transition probability is often assummed tohave the form (see Sec. III.C.2.a):

WAkk′ = −τ−1

A k × k′ ·Ms. (1.7)

When this asymmetry is inserted into the Boltzmannequation it leads to a current proportional to the lon-gitudinal current driven by E and perpendicular to bothE and Ms. When this mechanism dominates, both theHall conductivity σH and the conductivity σ are propor-tional to the transport lifetime τ and the Hall resistivityρskew

H = σskewH ρ2 is therefore proportional to the longitu-

dinal resistivity ρ.There are several specific mechanisms for skew scat-

tering (see Sec.III.C.2 and Sec. IV.A). Evaluation of

the skew scattering contribution to the Hall conductivityor resistivity requires simply that the conventional lin-earized Boltzmann equation be solved using a collisionterm with accurate transition probabilities, since thesewill generically include a chiral contribution. In prac-tice our ability to accurately estimate the skew scatter-ing contribution to the AHE of a real material is lim-ited only by typically imperfect characterization of itsdisorder. We emphasize that skew scattering contribu-tions to σH are present not only because of spin-orbitcoupling in the disorder Hamiltonian, but also becauseof spin-orbit coupling in the perfect crystal Hamiltonaincombined with purely scalar disorder. Either source ofskew-scattering could dominate σAH−skew

xy depending onthe host material and also on the type of impurities.

We end this subsection with a small note directed tothe reader who is more versed in the latest developmentof the full semiclassical theory of the AHE and in itscomparison to the microscopic theory (see Sec. IV.A andIV.B.2). We have been careful above not to define theskew-scattering contribution to the AHE as the sum ofall the contributions arising from the asymmetric scatter-ing rate present in the collision term of the Boltzmanntransport equation. We know from microscopic theorythat this asymmetry also makes an AHE contribution ororder τ0. There exists a contribution from this asymme-try which is actually present in the microscopic theorytreatment associated with the so called ladder diagramcorrections to the conductivity, and therefore of order τ0.In our experimentally practical parsing of AHE contri-butions we do not associate this contribution with skew-scattering but place it under the umbrella of side-jumpscattering even though it does not physically originatefrom any side-step type of scattering.

3. Side-jump contribution to σAHxy

Given the sharp defintions we have provided for theintrinsic and skew scattering contributions to the AHEconductivity, the equation

σAHxy = σAH−int

xy + σAH−skewxy + σAH−sj

xy (1.8)

defines the side-jump contribution as the difference be-tween the full Hall conductivity and the two-simpler con-tributions. In using the term side-jump for the remainingcontribution, we are appealing to the historically estab-lished taxonomy outlined in the previous section. Estab-lishing this connection mathematically has been the mostcontroversial aspects of AHE theory, and the one whichhas taken the longest to clarify from a theory point ofview. Although this classification of Hall conductivitycontributions is often useful (see below), it is not gener-ically true that the only correction to the intrinsic andskew contributions can be physically identified with theside-jump process defined as in the earlier studies of theAHE (Berger, 1964).

8

The basic semiclassical argument for a side-jumpcontribution can be stated straight-forwardly: whenconsidering the scattering of a Gaussian wavepacketfrom a spherical impurity with SOI ( HSO =(1/2m2c2)(r−1∂V/∂r)SzLz), a wavepacket with incidentwave-vector k will suffer a displacement transverse to kequal to 1

6k~2/m2c2. This type of contribution was first

noticed, but discarded, by Smit (Smit, 1958) and reintro-duced by Berger (Berger, 1964) who argued that it wasthe key contribution to the AHE. This kind of mecha-nism clearly lies outside the bounds of traditional Boltz-mann transport theory in which only the probabilities oftransitions between Bloch states appears, and not micro-scopic details of the scattering processes. This contribu-tion to the conductivity ends up being independent of τand therefore contributes to the AHE at the same orderas the intrinsic contribution in an expansion in powersof scattering rate. The separation between intrinsic andside-jump contributions, which cannot be distinquishedby their dependence on τ , has been perhaps the mostargued aspect of AHE theory since they cannot be dis-tinquished by their dependence on scattering rate (seeSec. III.C.2.d).

As explained clearly in a recent review by Sinitsyn(Sinitsyn, 2008), side-jump and intrinsic contributionshave quite different dependences on more specific sys-tem parameters, particularly in systems with complexband structures. Some of the initial controversy whichsurrounded side jump theories was associated with phys-ical meaning ascribed to quantities which were plainlygauge dependent, like the Berry’s connection which inearly theories is typically identified as the definition ofthe side-step upon scattering. Studies of simple models,for example models of semiconductor conduction bands,also gave results in which the side-jump contributionseemed to be the same size but opposite in sign com-pared to the intrinsic contribution (Nozieres and Lewiner,1973). We now understand (Sinitsyn et al., 2007) thatthese cancellations are unlikely, except in models with avery simple band structure, e.g. one with a constantBerry’s curvature. It is only through comparison be-tween fully microscopic linear response theory calcula-tions, based on equivalently valid microscopic formalismssuch as Keldysh (non-equilibrium Grenn’s function) orKubo formalisms, and the systematically developed semi-classical theory that the specific contribution due to theside-jump mechanism can be separately identified withconfidence (see Sec. IV.A).

Having said this, all the calculations comparing theintrinsic and side-jump contibutions to the AHE from amicroscopic point of view have been performed for verysimple models not immediately linked to real materials.A practical approch which is followed at present for ma-terials in which σAH seems to be independent of σxx, isto first calculate the intrisic contribution to the AHE. Ifthis explains the observation (and it appears that it usu-ally does), then it is deemed that the intrinsic mechanismdominates. If not, we can take some comfort from under-

standing on the basis of simple model results, that therecan be other contributions to σAH which are also inde-pendent of σxx and can for the most part be identifiedwith the side jump mechanism. Unfortunately it seemsextremelly challenging, if not impossible, to develop apredictive theory for these contributions, partly becausethey require many higher order terms in the perturbationtheory that be summed, but more fundamentally becausethey depend sensitively on details of the disorder in aparticular material which are normally unknown.

II. EXPERIMENTAL AND THEORETICAL STUDIES ON

SPECIFIC MATERIALS

A. Transition-metals

1. Early experiments

Four decades after the discovery of the AHE, an empir-ical relation between magnetization and Hall resistivitywas proposed independently by A. W. Smith and by E.M. Pugh (Pugh, 1930; Pugh and Lippert, 1932; Smithand Sears, 1929) (see Sec. I.A). Pugh investigated theAHE in Fe, Ni and Co and the alloys Co-Ni and Ni-Cuin magnetic fields up to 17 kG over large intervals in T(10-800 K in the case of Ni), and found that the Hallresistivity ρH is comprised of 2 terms, viz.

ρH = R0H +R1M(T,H), (2.1)

where M(T,H) is the magnetization averaged over thesample. Pugh defined R0 and R1 as the ordinary andextra-ordinary Hall coefficients, respectively. The latterR1 = Rs is now called the anomalous Hall coefficient (asin Eq. 1.1)).

On dividing Eq. (2.1) by ρ2, we see that it just ex-presses the additivity of the Hall currents: the total Hallconductivity σtot

xy equals σNHxy + σAH

xy , where σNHxy is the

ordinary Hall conductivity and σAHxy is the AHE conduc-

tivity. A second implication of Eq. (2.1) emerges whenwe consider the role of domains. The anomalous Hall co-efficient in Eq.(2.1) is proportional to the AHE in a sin-gle domain. As H → 0, proliferation of domains rapidlyreduces M(T,H) to zero (we ignore pinning). Cancella-tions of σAH

xy between domains result in a zero net Hallcurrent. Hence the observed AHE term mimics the fieldprofile ofM(T,H), as implied by Pugh’s term R1M . Therole of H is simply to align the AHE currents by rotat-ing the domains into alignment. The Lorentz force termσNH

xy is a “background” current with no bearing on theAHE problem.

The most interesting implication of Eq. (2.1) is that,in the absence of H , a single domain engenders a spon-taneous Hall current transverse to both M and E. Un-derstanding the origin of this spontaneous off-diagonalcurrent has been a fundamental problem of charge trans-port in solids for the past 60 years. The AHE is alsocalled the spontaneous Hall effect and the extraordinaryHall effect in the older literature.

9

2. Recent experiments

The resurgence of interest in the AHE motivated bythe Berry-phase approach (Sec. I.B.1) has led to manynew Hall experiments on 3d transition metals and theiroxides. Both the recent and the older literature on Feand Fe3O4 are reviewed in this section. An importantfinding of these studies is the emergence of three distinctregimes roughly delimited by the conductivity σxx andcharacterized by the dependence of σAH

xy on σxx. Thethree regimes are:

i) A high conductivity regime for σxx & 106 (Ωcm)−1

in which σAH−skewxy ∼ σ1

xx dominates σAHxy ,

ii) A good metal regime for σAHxx ∼ 104−106 (Ωcm)−1

in which σxy ∼ σ0xx,

iii) A bad metal/hopping regime for σxx < 104

(Ωcm)−1 in which σAHxy ∼ σ1.6−1.8

xx .

We discuss each of these regimes below.

High conductivity regime – The Hall conductivity in thehigh-purity regime, σxx > 0.5 × 106 (Ωcm)−1, is dom-inated by the skew scattering contribution σskew

xy . Thehigh-purity regime is one of the least studied experimen-tally. This regime is very challenging to investigate ex-perimentally because the field H required for saturatingM also yields a very large ordinary Hall effect (OHE)and R0 tends to be of the order of Rs (Schad et al.,1998). In the limit ωcτ 1, the OHE conductivityσNH

xy may be nonlinear in H (ωc is the cyclotron fre-

quency). Although σskewxy increases as τ , the OHE term

σNHxy increases as τ2 and therefore the latter ultimately

dominates, and the AHE current may be unresolvable.Even though the anomalous Hall current can not alwaysbe cleanly separated from the normal Lorentz-force Halleffect in the high conductivity regime, the total Hall cur-rent invariably increases with σxx in a way which providescompelling evidence for a skew-scattering contribution.

In spite of these challenges several studies have man-aged to convincingly separate the competing contribu-tions and have identifed a dominant linear relation be-tween σAH

xy and σxx for σxx & 106 (Ωcm)−1 (Majum-dar and Berger, 1973; Shiomi et al., 2009). In an earlystudy Majumdar et al. (Majumdar and Berger, 1973)grew highly pure Fe doped with Co. The resulting σAH

xy ,obtained from Kohler plots extrapolation to zero field,show a clear dependence of σAH

xy ∼ σxx (Fig. 4 a). In amore recent study, a similar finding (linear dependence ofσAH

xy ∼ σxx) was observed by Shiomi et al. (Shiomi et al.,2009) in Fe doped with Co, Mn, Cr, and Si. In these stud-ies the high temperature contribution to σAH

xy (presumedto be intrinsic plus side jump) was substracted from σxy

and a linear dependence of the resulting σAHxy is observed

(Fig. 4 a and b). In this recent study the conductivityis intentionally reduced by impurity doping to find thelinear region and reliably exclude the Lorernz contribu-tion. The results of these authors show, in particular,

that the slope of σxx vs. σAHxy depends on the species

of the impurities as it is expected in the regime domi-nated by skew scattering. It is reassuring to note thatthe skewness parameters (Sskew = σAH/σxx) implied bythe older and the more recent experiments are consistent,in spite of differences in the conductivity ranges stud-ied. Sskew is independent of σxx (Majumdar and Berger,1973; Shiomi et al., 2009) as it should be when the skewscattering mechanism dominates. Further experimentsin this regime are desirable to fully investiage the dif-ferent dependence on doping, temperature, and impuritytype. Also, new approaches to reliably disentangle theAHE and OHE currents will be needed to faicilitate suchstudies.

Good metal regime – Experiments re-examining the AHEin Fe, Ni, and Co have been performed by Miyasato etal. (Miyasato et al., 2007). These experiments indicate aregime of σxy versus σxx in which σxy is insensitive to σxx

in the range σxx ∼ 104-106 (Ωcm)−1 (see Fig. 5). Thissuggests that the scattering independent mechanisms (in-trinsic and side-jump) dominates in this regime. How-ever, in comparing this phenomenology to the discussionof AHE mechanisms in Sec. I.B, one must keep in mindthat the temperature has been varied in the Hall dataon Fe, Ni, and Co in order to change the resistivity, eventhough it is restricted to the range well below Tc (Fig. 5upper panel). In the mechanisms discussed in Sec. I.Bonly elastic scattering was taken into account. Earliertests of the ρ2 dependence of ρxy carried by varying Twere treated as suspect in the early AHE period (see Fig.2) because the role of inelastic scattering was not fully un-derstood. The effect of inelastic scattering from phononsand spin waves remains open in AHE theory and is notaddressed in this review.

Bad metal/hopping regime– Several groups have mea-sured σxy in Fe and Fe3O4 thin-film ferromagnets (Fenget al., 1975; Fernandez-Pacheco et al., 2008; Miyasatoet al., 2007; Sangiao et al., 2009; Venkateshvaran et al.,2008). (See Fig. 6.) Sangiao et al. (Sangiao et al., 2009)studied epitaxial thin-films of Fe deposited by pulsed-laser deposition (PLD) on single-crystal MgO (001) sub-strates at pressures < 5 × 10−9 Torr. To vary σxx overa broad range, they varied the film thickness t from 1 to10 nm. The ρ vs. T profile for the film with t = 1.8 nmdisplays a resistance minimum near 50 K, below which ρshows an upturn which has been ascribed to localizationor electron interaction effects (Fig. 7). The magnetiza-tion M is nominally unchanged from the bulk value (ex-cept possibly in the 1.3 nm film). AHE experiments werecarried out from 2–300 K on these films and displayed asσxy vs. σxx plots together with previously published re-sults (Fig. 6). In the plot, the AHE data from films witht ≤ 2 nm fall in the weakly localized regime. The com-bined plot shows that Sangio et al.’s data are collinear (ona logarithmic scale) with those measured on 1 µm-thickfilms by Miyasato et al. (Miyasato et al., 2007). For thethree samples with t = 1.3–2 nm, the inferred exponent inthe dirty regime is on average ∼ 1.66. A concern is that

10

0.0 1.0 2.0 3.0 4.0 5.0

σxx [106 Ω -1

cm-1

]

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

σ xy [1

04Ω

-1 c

m-1

]Fe-Co alloy (Majumdar 73)Fe-Co alloy (Shiomi 09)Fe-Cr alloy (Shiomi 09)Fe-Mn alloy (Shiomi 09)Fe-Si alloy (Shiomi 09)

(a)

20

K) (b)

10

xy

A(2

07

Kcm

]

Fe-Cr

Fe-Ni

Fe-Co

(b)

10

(5.5

K)

- [1

03S

/c

Fe-Mn

0 1 2

0

xy

A(

Fe-Si

[ 106]xx(5.5K) [S/cm]

FIG. 4 σAHxy for pure Fe film doped with Cr, Co, Mn and Si

vs. σxx at low temperatures ( T = 4.2 K and T = 5.5 K ) (a).In many of the alloys, particularly in the Co doped system,the linear scaling in the higher conductivy sector, σxx > 106

(Ωcm)−1, implies that skew scattering dominates σAHxy . After

Ref. Majumdar and Berger, 1973 and Ref. Shiomi et al., 2009.The data from (Shiomi et al., 2009), shown also at a largerscale in (b), is obtained by substracting the high temperaturecontribuiton to σxy. In the data shown the ordinary Hallcontribution has been identified and substracted. [Panel (b)From Ref. Shiomi et al., 2009.]

the data from the 1.3 nm film was obtained by subtract-ing a lnT term from ρ (the subtraction procedure wasnot described). How localization affects the scaling plotis an open issue at present. In Sec. II.E, we discuss re-cent AHE measurements in disordered polycrystalline Fefilms with t < 10 nm by Mitra et al. (Mitra et al., 2007).Recent progress in understanding weak-localization cor-rections to the AHE is also reviewed there.

In magnetite, Fe3O4, scaling of σxy ∼ σαxx with

α ∼ 1.6 − 1.8 was already apparent in early experi-ments on polycrystalline samples (Feng et al., 1975).

0 50 100 150 200 250 300T [K]

102

103

104

|σxy

| [Ω

-1 c

m-1

]

Fe crystalsFe filmCo filmNi film

0 50 100 150 200 250 300T [K]

10-7

10-6

10-5

ρ xx [Ω

cm

]

105

106

σxx [Ω-1

cm-1

]

102

103

104

|σxy

| (Ω

-1 c

m-1

)

FIG. 5 Measurements of the Hall conductivity and resistivityin single-crystal Fe and in thin foils of Fe, Co, and Ni. Thetop and lower right panels show the T dependence of σxy andρxx, respectively. The lower left panel plots |σxy | against σxx.After Ref. Miyasato et al., 2007.

100

102

104

σxx [Ω-1

cm-1

]

10-5

10-4

10-3

10-2

10-1

100

101

102

103

|σxy

| (Ω

-1 c

m-1

)

Fe film 1.3 nm (Sangiao 09)Fe film 1.8 nm (Sangiao 09)Fe film 2.0 nm (Sangiao 09) Fe3O4 (001)-oriented (V..09)

Fe3O4 (110)-oriented (V..09)

Fe3O4 (111)-oriented (V..09)

Fe2.9Zn0.1O4 (V..09)

Fe2.5Zn0.5O4 (V..09)

Fe2.5Zn2.5O4 grown in O2/Ar (V..09)

Fe3O4 polycrystalline (Feng 75)

Fe3O4 40 nm (Fernandez 08)

Fe3O4 20 nm (Fernandez o9)

FIG. 6 Combined plot of the AHE conductivity |σxy | versusthe conductivity σxx in epitaxial films of Fe grown on MgOwith thickness t = 2.5, 2.0, 1.8 and 1.3 nm (Sangiao et al.,2009), in polycrystalline Fe3−xZnxO4 (Feng et al., 1975), inthin-film Fe3−xZnxO4 between 90 and 350 K (Venkateshvaranet al., 2008) and above the Verwey transition (Fernandez-Pacheco et al., 2008).

Recently, two groups have re-investigated the AHE inepitaxial thin films (data included in Fig. 6). Fernandez-Pacheco et al. (Fernandez-Pacheco et al., 2008) mea-sured a series of thin-film samples of Fe3O4 grown byPLD on MgO (001) substrates in ultra-high vacuum,whereas Venkateshvaran et al. (Venkateshvaran et al.,2008) studied both pure Fe3O4 and Zn-doped magnetite

11

FIG. 7 The T dependence of ρ in epitaxial thin-filmsMgO(001)/Fe(t)/MgO with t = 1.8 nm and 2.5 nm. Theinset shows how T0, the temperature of the resistivity mini-mum, varies with t.[From Ref. Sangiao et al., 2009.]

Fe3−xZnxO4 deposited on MgO and Al2O3 substratesgrown by laser molecular-beam epitaxy under pure Aror Ar/O mixture. In both studies, ρ increases monotoni-cally by a factor of ∼10 as T decreases from 300 K to theVerwey transition temperature TV = 120 K. Below TV ,ρ further increases by a factor of 10 to 100. The resultsfor ρ vs. T from Venkateshvaran et al. (Venkateshvaranet al., 2008) is shown in Fig. 8.

The large values of ρ and its insulating trend implythat magnetite falls in the strongly localized regime, incontrast to thin-film Fe which lies partly in the weak-localization (or incoherent) regime.

Both groups find good scaling fits extending over sev-eral decades of σxx with α ∼ 1.6 − 1.8 when varyingT . Fernandez-Pacheco et al. (Fernandez-Pacheco et al.,2008) plot σxy vs. σxx in the range 150< T <300 Kfor several thicknesses t and infer an exponent α = 1.6.Venkateshvaran et al. (Venkateshvaran et al., 2008) plotσxy vs. σxx from 90 to 350 K and obtain power-law fitswith α = 1.69, in both pure and Zn-doped magnetite(data shown in Fig. 6). Significantly, the 2 groups findthat α is unchanged below TV . There is presently no the-ory in the poorly conducting regime which predicts theobserved scaling (σxx < 10−1 (Ωcm)−1).

3. Comparison to theories

Detailed first-principles calculations of the intrinsiccontribution to the AHE conductivity have been per-formed for bcc Fe (Wang et al., 2006; Yao et al., 2004),fcc Ni, and hpc Co (Wang et al., 2007). In Fe and Co,the values of σxy inferred from the Berry curvature Ωz(k)are 7.5×102 and 4.8×102 (Ωcm)−1, respectively, in rea-sonable agreement with experiment. In Ni, however, thecalculated value −2.2× 103 (Ωcm)−1 is only 30% of theexperimental value.

These calculations uncovered the crucial role playedby avoided-crossings of band dispersions near the Fermi

FIG. 8 Longitudinal resistivity ρxx vs. T for epitaxialFe3−xZnxO4 films. The (001), (110) and (111) oriented filmswere grown on MgO(001), MgO(110) and Al2O3 substrates.[From Ref. Venkateshvaran et al., 2008.]

FIG. 9 First-principles calculation of the band dispersionsand the Berry-phase curvature summed over occupied bands.[From Ref. Yao et al., 2004.]

energy εF . The Berry curvature bz is always stronglyenhanced near avoided crossings, opposite direction forthe upper and lower bands. A large contribution to σint

xy

when the crossing is at the Fermi energy so that onlyone of the two bands is occupied, e.g. near the point Hin Fig. 9. A map showing the contributions of differentregions of the FS to bz(k) is shown in Fig. 10. The SOIcan lift an accidental degeneracy at certain wavevectorsk. These points act as a magnetic monopole for the Berrycurvature in k-space (Fang et al., 2003). In the parame-ter space of spin-orbit coupling, σxy is nonperturbative innature. The effect of these “parity anomalies” (Jackiw,1984) on the Hall conductivity was first discussed by Hal-dane (Haldane, 1988). A different conclusion on the roleof topological enhancement in the intrinsic AHE was re-ported for a tight-binding calculation with the 2 orbitalsdzx and dyz on a square lattice (Kontani et al., 2007).

Motivated by the enhancement at the crossing pointsdiscussed above, Onoda et al. (Onoda et al., 2006a, 2008)proposed a minimal model that focuses on the topologi-

12

FIG. 10 First-principles calculation of the FS in the (010)plane (solid lines) and the Berry curvature in atomic units(color map). [From Ref. Yao et al., 2004.]

cal and resonantly enhanced nature of the intrinsic AHEarising from the sharp peak in bz(k) near avoided cross-ings. The minimal model is essentially a 2D Rashbamodel (Bychkov and Rashba, 1984) with an exchangefield which breaks symmetries and accounts for the mag-netic order and a random impurity potential to accountfor disorder. The model is discussed in Sec. IV.D. Inthe clean limit, σxy is dominated by the extrinsic skew-scattering contribution σskew

xy , which almost masks the in-

trinsic contribution σintxy . Since σskew

xy ∝ τ , it is suppressed

by increased impurity scattering, whereas σintxy – an in-

terband effect – is unaffected. In the moderately dirtyregime where the quasiparticle damping is larger thanthe energy splitting at the avoided crossing (typically,the SOI energy) but less than the bandwidth, σint

xy domi-

nates σskewxy . As a result, one expects a crossover from the

extrinsic to the intrinsic regime. When skew scatteringis due to a spin-dependent scattering potential instead ofspin-orbit coupling in the Bloch states, the skew to intrin-sic crossover could be controlled by a different condition.In this minimal model, a well-defined plateau is not well-reproduced in the intrinsic regime unless a weak impuritypotential is assumed (Onoda et al., 2008). This crossovermay be seen when the skew-scattering term shares thesame sign as the intrinsic one (Kovalev et al., 2009).

With further increase in the scattering strength, spec-tral broadening leads to the scaling relationship σxy ∝σ1.6

xx , as discussed above (Kovalev et al., 2009; Onodaet al., 2006b, 2008). In the strong-disorder regime, σxx

is no longer linear in the scattering lifetime τ . A dif-ferent scaling, σxy ∝ σ2

xx, attributed to broadening ofthe electronic spectrum in the intrinsic regime, has beenproposed by Kontani et al. (Kontani et al., 2007).

As discussed above, there is some experimental evi-dence that this scaling prevails not only in the dirty

FIG. 11 Anomalous Hall effect in SrRuO3. (A) The magneti-zation M , (B) longitudinal resistivity ρxx, and (C) transverseresistivity ρxy as functions of the temperature T for the sin-gle crystal and thin-film SrRuO3, as well as for Ca-dopedSr0.8Ca0.2RuO3 thin film. µB is the Bohr magneton. [FromRef. Fang et al., 2003.]

metallic regime, but also deep into the hopping regime.Sangio et al. (Sangiao et al., 2009), for e.g., obtained theexponent α '1.7 in epitaxial thin-film Fe in the dirtyregime. In manganite, scaling seems to hold, with thesame nominal value of α, even below the Verwey tran-sition where charge transport is deep in the hoppingregime (Fernandez-Pacheco et al., 2008; Venkateshvaranet al., 2008). These regimes are well beyond the purviewof either the minimal model, which considers only elas-tic scattering, or the theoretical approximations used tomodel its properties (Onoda et al., 2006b, 2008).

Nonetheless, the experimental reports have uncovereda robust scaling relationship with α near 1.6, which ex-tends over a remarkably large range of σxx. The originof this scaling is an open issue at present.

B. Complex oxide ferromagnets

1. First-principles calculations and experiments on SrRuO3

The perovskite oxide SrRuO3 is an itinerant ferromag-net with a critical temperature Tc of 165 K. The electronsoccupying the 4d t2g orbitals in Ru4+ have a SOI en-ergy much larger than that for 3d electrons. Early trans-port investigations of this material were reported in Refs.Allen et al., 1996 and Izumi et al., 1997. The latter au-thors also reported results on thin-film SrTiO3. Recently,the Berry-phase theory has been applied to account for itsAHE (Fang et al., 2003; Mathieu et al., 2004a,b), which

13

FIG. 12 The calculated transverse conductivity σxy as a func-tion of the chemical potential µ for SrRuO3. The chaotic be-havior is the fingerprint of the Berry curvature distributionillustrated in Fig. 13. [From Ref. Fang et al., 2003.]

FIG. 13 The Berry curvature bz(k) for a band as a functionof k⊥ = (kx, ky) with the fixed kz = 0. [From Ref. Fanget al., 2003.]

is strongly T dependent (Fig.11c). Neither the KL the-ory nor the skew-scattering theory seemed adequate forexplaining the T dependence of the inferred AHE con-ductivity σxy (Fang et al., 2003).

The experimental results motivated a detailed first-principles, band-structure calculation that fully incorpo-rated the SOI. The AHE conductivity σxy was calculateddirectly using the Kubo formula Eq. (1.2) (Fang et al.,2003). To handle numerical instabilities which arise nearcertain critical points, a fictitious energy broadening δ= 70 meV was introduced in the energy denominator.Fig. 12 shows the dependence of σxy(µ) on the chemicalpotential µ. In sharp contrast to the diagonal conduc-tivity σxx, σxy(µ) fluctuates strongly, displaying sharppeaks and numerous changes in sign. The fluctuationsmay be understood if we map the momentum dependenceof the Berry curvature bz(k) in the occupied band. Forexample, Fig. 13 displays bz(k) plotted as a function ofk⊥ = (kx, ky), with kz fixed at 0. The prominent peakat k⊥ = 0 corresponds to the avoided crossing of theenergy band dispersions, which are split by the SOI. Asdiscussed in Sec. I.B, variation of the exchange splittingcaused by a change in the spontaneous magnetization Mstrongly affects σxy in a nontrivial way.

FIG. 14 (Upper panel) Combined plots of the Hall conduc-tivity σxy vs. magnetization M in 5 samples of the ruthen-ate Sr1−xCaxRuO3 (0 ≤ x ≤ 0.4). The inset compares dataσxy at x = 0 (triangles) with calculated values (solid curve).(Lower panel) First-principles calculations of σxy vs. M forcubic and orthorhombic structures. The effect of broaden-ing on the curves is shown for the orthorhombic case. [FromRef. Mathieu et al., 2004b.]

From the first-principles calculations, one may esti-mate the temperature dependence of σxy by assumingthat it is due to the temperature-dependence of the Blochstate exchange splitting and that this splitting is propor-tional to the temperature-dependent magnetization. Thenew insight is that the T -dependence of σxy(T ) simplyreflects the M dependence of σxy: at a finite tempera-ture T ′, the magnitude and sign of σxy may be deducedby using the value of M(T ′) in the zero-T curve. Thisproposal was tested against the results on both the purematerial and the Ca-doped material Sr1−xCaxRuO3. Inthe latter, Ca doping suppresses both Tc and M system-atically (Mathieu et al., 2004a,b). As shown in Fig. 14(upper panel), the measured values of M and σxy, ob-tained from 5 samples with Ca content 0.4≤ x ≤ 0, fall on2 continuous curves. In the inset, the curve for the puresample (x = 0) is compared with the calculation. Thelower panel of Fig. 14 compares calculated curves of σxy

for cubic and orthorhombic lattice structures. The sen-sitivity of σxy to the lattice symmetry reflects the dom-inant contribution of the avoided crossing near εF . Thesensitivity to broadening is shown for the orthorhombiccase.

Kats et al. (Kats et al., 2004) also have studied themagnetic field dependence of ρxy in an epitaxial film ofSrRuO3. They have observed sign-changes in ρxy near amagnetic field B = 3 T at T = 130 K and near B = 8T at 134 K. This seems to be qualitative consistent withthe Berry-phase scenario. On the other hand the authorssuggest that the intrinsic-dominated picture is likely in-complete (or incorrect) near Tc (Kats et al., 2004).

14

2. Spin chirality mechanism of the AHE in manganites

In the manganites, e.g. La1−xCaxMnO3 (LCMO), thethree t2g electrons on each Mn ion form a core local mo-ment of spin S = 3

2 . A large Hund energy JH aligns the

core spin S with the s = 12 spin of an itinerant electron

that momentarily occupies the eg orbital. Because thisHund coupling leads to an extraordinary magnetoresis-tance in weak H , the manganites are called colossal mag-netoresistance (CMR) materials (Tokura and Tomioka,1999). The double exchange theory summarized byEq. (2.2) (below) is widely adopted to describe the onsetof ferromagnetism in the CMR manganites.

As T decreases below the Curie temperature TC ' 270K in LCMO, the resistivity ρ falls rapidly from ∼15 mΩcm to metallic values <2 mΩcm. CMR is observed over asignificant interval of temperatures above and below TC ,where charge transport occurs by hopping of electronsbetween adjacent Mn ions (Fig. 15a). At each Mn sitei, the Hund energy tends to align the carrier spin s withthe core spin Si.

Early theories of hopping conductivity (Holstein, 1961)predicted the existence of a Hall current produced by thephase shift (Peierls factor) associated with the magneticflux φ piercing the area defined by 3 non-collinear atoms.However, the hopping Hall current is weak. The observa-tion of a large ρxy in LCMO that attains a broad maxi-mum in modest H (Fig. 15b) led Matl et al. (Matl et al.,1998) to propose that the phase shift is geometric in ori-gin, arising from the solid angle described by s as theelectron visits each Mn site (s||Si at each site i as shownin Fig. 16). To obtain the large ρxy seen, one requiresSi to define a finite solid angle Ω. Since Si graduallyaligns with H with increasing field, this effect should dis-appear along with 〈Ω〉, as observed in the experiment.This appears to be the first application of a geometric-phase mechanism to account for an AHE experiment.

Subsequently, Ye et al. (Ye et al., 1999) consideredthe Berry phase due to the thermal excitations of theSkyrmion (and anti-Skyrmion). They argued that theSOI gives rise to a coupling between the uniform magne-tization M and the gauge field b by the term λM · b. Inthe ferromagnetic state, the spontaneous uniform magne-tization M leads to a finite and uniform b, which acts asa uniform magnetic field. Lyanda-Geller et al. (Lyanda-Geller et al., 2001) also considered the AHE due to thespin chirality fluctuation in the incoherent limit wherethe hopping is treated perturbatively. This approach,applicable to the high-T limit, complements the theoryof Ye et al. (Ye et al., 1999).

The Berry phase associated with non-coplanar spinconfigurations, the scalar spin chirality, was first con-sidered in theories of high-temperature superconductorsin the context of the flux distribution generated by thecomplex order parameter of the resonating valence bond(RVB) correlation defined by χij , which acts as the trans-fer integral of the “spinon” between the sites i and j (Leeet al., 2006). The complex transfer integral also appears

0

4

8

12

16

(m c

m)

250K260K270K280K

290K310K320K

360K330K

(a)

0 2 4 6 8 10 12 14-0.8

-0.6

-0.4

-0.2

0.0

0.2 (b) 100K

200K

oH (T)

H(

cm

) 250K260K270K280K290K

310K320K

360K

330K

FIG. 15 (a) The colossal magnetoresistance ρ vs. H inLa1−xCaxMnO3 (TC = 265 K) at selected T . (b) The Hall re-sistivity ρxy vs. H at temperatures 100 to 360 K. Above TC ,ρxy is strongly influenced by the MR and the susceptibility χ.[From Ref. Matl et al., 1998.]

FIG. 16 Schematic view of spin chirality. When circulatesamong three spins to which it is exchange-coupled, it feels a

fictitious magnetic field ~b with flux given by the half of thesolid angle Ω subtended by the three spins. [From Ref. Leeet al., 2006.]

in the double-exchange model specified by

H = −∑

ij,α

tij(c†iαcjα+h.c.)−JH

∑

i

Si·c†iασαβciβ , (2.2)

where JH is the ferromagnetic Hund’s coupling betweenthe spin σ of the conduction electrons and the localizedspins Si.

In the manganese oxides, Si represents the local-ized spin in t2g-orbitals, while c† and c are the opera-tors for eg-electrons. (The mean-field approximations ofHubbard-like theories of magnetism, the localized spin

15

may also be regarded as the molecular field created bythe conduction electrons themselves, in which case JH isreplaced by the on-site Coulomb interaction energy U .)In the limit of large JH , the conduction electron spin s isforced to align with Si at each site. The matrix elementfor hopping from i→ j is then given by

teffij = tij〈χi|χj〉 = tijeiaij cos

(

θij

2

)

, (2.3)

where |χi〉 is the two-component spinor spin wave func-tion with quantization axis ||Si. The phase factor eiaij

acts like a Peierls phase and can be viewed as originat-ing from a fictitious magnetic field which influences theorbital motion of the conduction electrons.

We next discuss how the Peierls phase leads to a gaugefield, i.e. flux, in the presence of non-coplanar spin con-figurations. Let Si, Sj , and Sk be the local spins at sitesi, j, and k, respectively. The product of the three trans-fer integrals corresponding to the loop i → j → k → iis

〈ni|nk〉〈nk|nj〉〈nj |ni〉= (1 + ni · nj + nj · nk + nk · nj) + ini · (nj × nk)

∝ ei(aij+ajk+aki) = eiΩ/2 (2.4)

where |ni〉 is the two-component spinor wavefunction ofthe spin state polarized along ni = Si/|Si|. Its imag-inary part is proportional to Si · (Sj × Sk), which cor-responds to the solid angle Ω subtended by the threespins on the unit sphere, and is called the scalar spinchirality (Fig. 16). The phase acquired by the electron’swave function around the loop is eiΩ/2, which leads tothe Aharonov-Bohm (AB) effect and, as a consequence,to a large Hall response.

In the continuum approximation, this phase factor isgiven by the flux of the “effective” magnetic field b ·dS =∇ × a · dS where dS is the elemental directed surfacearea defined by the three sites. The discussion impliesthat a large Hall current requires the unit vector n(x) =S(x)/|S(x)| to fluctuate strongly as a function of x,the position coordinate in the sample. An insightful wayto quantify this fluctuation is to regard n(x) as a mapfrom the x-y plane to the surface of the unit sphere (wetake a 2D sample for simplicity). An important defectin a ferromagnet – the Skyrmion (Sondhi et al., 1993) –occurs when n(x) points down at a point x′ in a regionA of the x-y plane, but gradually relaxes back to up atthe boundary of A. The map of this spin texture wrapsaround the sphere once as x′ roams over A. The numberof Skyrmions in the sample is given by the topologicalindex

Ns =

∫

A

dxdy n ·(

∂n

∂x× ∂n

∂y

)

=

∫

A

dxdy bz, (2.5)

where the first integrand is the directed area of the imageon the unit sphere. Ns counts the number of times themap covers the sphere as A extends over the sample. The

gauge field b produces a Hall conductivity. Ye et al. (Yeet al., 1999) derived in the continuum approximation thecoupling between the field b and the spontaneuos magne-tization M through the SOI. The SOI coupling producesan excess of thermally excited positive Skyrmions overnegative ones. This imbalance leads to a net uniform“magnetic field” b (anti)parallel to M , and the AHE.In this scenario, ρxy is predicted to attain a maximumslightly below Tc, before falling exponentially to zero asT → 0.

The Hall effect in the hopping regime has been dis-cussed by Holstein (Holstein, 1961) in the context ofimpurity conduction in semiconductors. Since the en-ergies εj and εk of adjacent impurity sites may differ sig-nificantly, charge conduction must proceed by phonon-assisted hopping. To obtain a Hall effect, we consider 3non-collinear sites (labelled as i = 1, 2 and 3). In a fieldH , the magnetic flux φ piercing the area enclosed by the3 sites plays the key role in the Hall response. Accord-ing to Holstein, the Hall current arises from interferencebetween the direct hopping path 1 → 2 and the path1 → 3 → 2 going via 3 as an intermediate step. Takinginto account the changes in the phonon number in eachprocess, we have

(1, Nλ, Nλ′) → (1, Nλ ∓ 1, Nλ′)→ (2, Nλ ∓ 1, Nλ′ ∓ 1),

(1, Nλ, Nλ′) → (3, Nλ ∓ 1, Nλ′)→ (2, Nλ ∓ 1, Nλ′ ∓ 1),

(2.6)

where Nλ, Nλ′ are the phonon numbers for the modesλ, λ′, respectively.

In a field H , the hopping matrix element from Ri to

Rj includes the Peierls phase factor exp[−i(e/c)∫Rj

Ridr ·

A(r)]. When we consider the interference between thetwo processes in Eq. (2.6), the Peierls phase factorscombine to produce the phase shift exp(i2πφ/φ0) whereφ0 = hc/e is the flux quantum. By the Aharonov-Bohmeffect, this leads to a Hall response.

As discussed, this idea was generalized for the mangan-ites by replacing the Peierls phase factor with the Berryphase factor in Eq. (2.4) (Lyanda-Geller et al., 2001).The calculated Hall conductivity is

σH = G(ε) cos(θij/2) cos(θjk/2) cos(θki/2) sin(Ω/2),(2.7)

where ε is the set of the energy levels εa (a = i, j, k),and θij is the angle between ni and nj . When the aver-age of σH over all directions of na is taken, it vanisheseven for finite spontaneous magnetization m. To obtaina finite σxy, it is necessary to incorporate SOI.

Assuming the form of hopping integral with the SOIgiven by

Vjk = V orbjk (1 + iσ · gjk), (2.8)

the Hall conductivity is proportional to the average of

[gjk · (nj × nk)][n1 · n2 × n3]. (2.9)

16

FIG. 17 Comparison between experiment and the theoreti-cal prediction Eq.(2.10). Scaling behavior between the Hallresistivity ρH and the magnetization M is shown. The solidline is a fit to Eq.(2.10); the dashed line is the numerator ofEq.(2.10) only. There are no fitting parameters except theoverall scale. [From Ref. Chun et al., 2000.]

Taking the average of n’s with m = M/Msat where Msat

is the saturated magnetization, we finally obtain

ρxy = ρ0xy

m(1−m2)2

(1 +m2)2. (2.10)

This prediction has been tested by the experiment ofChun et al. (Chun et al., 2000) shown in Fig. 17. Thescaling law for the anomalous ρxy as a function of |M|obtained near TC is in good agreement with the experi-ment.

Similar ideas have been used by Burkov and Ba-lents (Burkov and Balents, 2003) to analyze the vari-able range hopping region in (Ga,Mn)As. The spin-chirality mechanism for the AHE has also been appliedto CrO2 (Yanagihara and Salamon, 2002, 2007), and theelement Gd (Baily and Salamon, 2005). In the formercase, the comparison between ρxy and the specific heatsupports the claim that the critical properties of ρxy aregoverned by the Skyrmion density.

The theories described above assume large Hund cou-pling. In the weak-Hund coupling limit, a perturba-tive treatment in JH has been developed to relate theAHE conductivity to the scalar spin chirality (Tataraand Kawamura, 2002). This theory has been applied tometallic spin-glass systems (Kawamura, 2007).

3. Lanthanum cobaltite

The subtleties and complications involved in analyz-ing the Hall conductivity of tunable ferromagnetic ox-ides are well illustrated by the cobaltites. Samoilov etal. (Samoilov et al., 1998), and Baily and Salamon (Bailyand Salamon, 2003) investigated the AHE in Ca-dopedlanthanum cobaltite La1−xCaxCoO3, which displays anumber of unusual magnetic and transport properties.They found an unusually large AHE near TC as well as

(d)

FIG. 18 Temperature dependence of the magnetization M(a), Hall resistivity ρxy (b), Hall conductivity σxy (c), in fourcrystals of La1−xSrxCoO3 (0.17 ≤ x ≤ 0.30) (all measuredin a field H = 1 T). In Panels a, b and c, the data for x= 0.17 and 0.20 were multiplied by a factor of 200 and 50,respectively. (d) The Hall conductivity σxy at 1 T in the fourcrystals plotted against M . Results for x = 0.17 and 0.20were multiplied by factors 50 and 5, respectively. [From Ref.Onose and Tokura, 2006.]

at low T , and proposed the relevance of spin-orderedclusters and orbital disorder scattering to the AHE inthe low-T limit. Subsequently, a more detailed inves-tigation of La1−xSrxCoO3 was reported by Onose andTokura (Onose and Tokura, 2006). Figs. 18 a, b andc summarize the T dependence of M , ρxy and σxy, re-spectively in four crystals with 0.17 ≤ x ≤ 0.30. Thevariation of ρxx vs. x suggests that a metal-insulatortransition occurs between 0.17 and 0.19. Whereas thesamples with x ≥ 0.2 have a metallic ρxx-T profile, thesample with x = 0.17 is non-metallic (hopping conduc-tion). Moreover, it displays a very large MR at low Tand large hysteresis in curves of M vs. H , features thatare consistent with a ferromagnetic cluster-glass state.

When the Hall conductivity is plotted vs. M (Fig. 18d), σxy shows a linear dependence on M for the mostmetallic sample (x = 0.30). However, for x = 0.17 and0.20, there is a pronounced downturn suggestive of theappearance of a different Hall term that is electron-likein sign. This is most apparent in the trend of the curvesof ρxy vs. T in Fig. 18 b. Onose and Tokura (Onose andTokura, 2006) propose that the negative term may arisefrom hopping of carriers between local moments whichdefine a chirality that is finite, as discussed above.

17

4. Spin chirality mechanism in pyrochlore ferromagnets

In the examples discussed in the previous subsection,the spin-chirality mechanism leads to a large AHE atfinite temperatures. An interesting question is whetheror not there exist ferromagnets in which the spin chiralityis finite in the ground state.

Ohgushi et al. (Ohgushi et al., 2000) considered theground state of the non-coplanar spin configuration in theKagome lattice, which may be obtained as a projectionof the pyrochlore lattice onto the plane normal to (1,1,1)axis. Considering the double exchange model Eq. (2.2),they obtained the band structure of the conduction elec-trons and the Berry phase distribution. Quite similarto the Haldane model (Haldane, 1988) or the model dis-cussed in Eq. (3.21), the Chern number of each band be-comes nonzero, and a quantized Hall effect results whenthe chemical potential is in the energy gap.

Turning to real materials, the pyrochlore ferromag-net Nd2Mo2O7 (NMO) provides a test-bed for exploringthese issues. Its lattice structure consists of two inter-penetrating sublattices comprised of tetrahedrons of Ndand Mo atoms, respectively (the sublattices are shiftedalong the c-axis) (Taguchi et al., 2001; Yoshii et al.,2000). While the exchange between spins on either sub-lattice is ferromagnetic, the exchange coupling Jdf be-tween spins of the conducting d-electrons of Mo and lo-calized f -electron spins on Nd is antiferromagnetic.

FIG. 19 Anomalous Hall effect in Nd2Mo2O7. Magneticfield dependence of (A) the magnetization and (B) the trans-verse resistivity (ρxy) for different temperatures. [From Ref.Taguchi et al., 2001.]

The dependences of the anomalous Hall resistivity ρxy

on H at selected temperatures are shown in Fig. 19. Thespins of Nd begin to align antiparallel to those of Mo be-low the crossover temperature T ∗ ∼= 40 K. Each Nd spinis subject to a strong easy-axis anisotropy along the linefrom a vertex of the Nd tetrahedron to its center. Theresulting noncoplanar spin configuration induces a trans-verse component of the Mo spins. Spin chirality is ex-

pected to be produced by the coupling Jdf , which leads tothe AHE of d-electrons. An analysis of the neutron scat-tering experiment has determined the magnetic structure(Taguchi et al., 2001). The tilt angle of the Nd spins isclose to that expected from the strong limit of the spinanisotropy, and the exchange coupling Jfd is estimatedas Jfd ∼ 5 K. This leads to a tilt angle of the Mo spins of∼5o. From these estimates, a calculation of the anoma-lous Hall conductivity in a tight-binding Hamiltonian oftriply degenerate t2g bands leads to σH ∼ 20 (Ωcm)−1,consistent with the value measured at low T . In a strongH , this tilt angle is expected to be reduced along withρxy. This is in agreement with the traces displayed inFig. 19.

The T dependence of the Hall conductivity has alsobeen analyzed in the spin-chirality scenario by incorpo-rating spin fluctuations (Onoda and Nagaosa, 2003). Theresult is that frustration of the Ising Nd spins leads tolarge fluctuations, which accounts for the large ρ ob-served. The recent observation of a sign-change in σxy ina field H applied in the [1, 1, 1] direction (Taguchi et al.,2003) is consistent with the sign-change of the spin chi-rality.

In the system Gd2Mo2O7 (GMO), in which Gd3+ (d7)has no spin anisotropy, the low-T AHE is an order-of-magnitude smaller than that in NMO. This is con-sistent with the spin-chirality scenario (Taguchi et al.,2004). The effect of the spin chirality mechanism onthe finite-frequency conductivity σH(ω) has been inves-tigated (Kezsmarki et al., 2005).

In another work, Yasui et al. (Yasui et al., 2006,2007) performed neutron scattering experiments over alarge region in the (H,T )-plane with H along the [0, 1, 1]and [0, 0, 1] directions. By fitting the magnetizationMNd(H,T ) of Nd, the magnetic specific heat Cmag(H,T ),and the magnetic scattering intensity Imag(Q,H, T ), theyestimated Jdj

∼= 0.5 K, which was considerably smallerthan estimated previously (Taguchi et al., 2001). Fur-thermore, they calculated the thermal average of the spinchirality 〈Si ·Sj ×Sk〉 and compared its value with thatinferred from the AHE resistivity ρxy. They have empha-sized that, when a 3-Tesla field is applied in the [0, 0, 1]direction (along this direction H cancels the exchangefield from the Mo spins), no appreciable reduction of ρxy

is observed. These recent conclusions have cast doubt onthe spin-chirality scenario for NMO.

A further puzzling feature is that, with H applied inthe [1, 1, 1] direction, one expects a discontinuous transi-tion from the two-in, two-out structure (i.e. 2 of the Ndspins point towards the tetrahedron center while 2 pointaway) to the three-in, one-out structure for the Nd spins.However, no Hall features that might be identified withthis cancellation have been observed down to very low T .This seems to suggest that quantum fluctuations of theNd spins may play an important role, despite the largespin quantum number (S = 3

2 ).

Machida has discussed the possible relevance ofspin chirality to the AHE in the pyrochlore Pr2Ir2O7

18

FIG. 20 Plot of AHE conductivity σAHE vs. conduc-tivity σ for anatase Ti1−xCoxO2−δ (triangles) and rutileTi1−xCoxO2−δ (diamonds). Grey symbols are data taken byother groups. The inset shows the expanded view of data foranatase with x= 0.05 (the open and closed triangles are forT > 150 K and T < 100 K, respectively. [From Ref. Uenoet al., 2007.]

(Machida et al., 2007a,b). In this system, a novel “Kondoeffect” is observed even though the Pr3+ ions with S = 1are subject to a large magnetic anisotropy. The magneticand transport properties of R2Mo2O7 near the phaseboundary between the spin glass Mott insulator and fer-romagnetic metal by changing the rare earth ion R hasbeen studied (Katsufuji et al., 2000).

5. Anatase and Rutile Ti1−xCoxO2−δ

In thin-film samples of the ferromagnetic semiconduc-tor anatase Ti1−xCoxO2−δ, Ueno et al. (Ueno et al.,2008) have reported scaling between the AHE resistanceand the magnetization M . The AHE conductivity σAH

xy

scales with the conductivity σxx as σAHxy ∝ σ1.6

xx (Fig. 20).A similar scaling relation was observed in another poly-morph rutile. See also Ref. Ramaneti et al., 2007 forrelated work on Co-doped TiO2.

C. Ferromagnetic semiconductors

Ferromagnetic semiconductors combine semiconduc-tor tunability and collective ferromagnetic properties ina single material. The most widely studied ferromag-netic semiconductors are diluted magnetic semiconduc-tors (DMS) created by doping a host semiconductor witha transition metal which provides a localized large mo-ment (formed by the d-electrons) and by introducingcarriers which can mediate a ferromagnetic coupling be-tween these local moments. The most extensively studiedare the Mn based (III,Mn)V DMSs, in which substitut-ing Mn for the cations in a (III,V) semiconductor candope the system with hole carriers; (Ga,Mn)As becomes

ferromagnetic beyond a concentration of 1%.

The simplicity of this basic but generally correct modelhides within it a cornucopia of physical and materialsscience effects present in these materials. Among thephenomena which have been studied are metal-insulatortransitions, carrier mediated ferromagnetism, disorderphysics, magneto-resistance effects, magneto-optical ef-fects, coupled magnetization dynamics, post-growth de-pendent properties, etc. A more in-depth discussion ofthese materials, both from the experimental and theoret-ical point of view, can be found in the recent review byJungwirth et al (Jungwirth et al., 2006).

The AHE has been one of the most fundamental char-acterization tools in DMSs, allowing, for example, directelectrical measurement of transition temperatures. Thereliability of electrical measurement of magnetic prop-erties in these materials has been verified by compari-son with remnant magnetization measurements using aSQUID magnetometer (Ohno et al., 1992). The relativesimplicity of the effective band structure of the carriersin metallic DMSs, has made them a playing ground tounderstand AHE of ferromagnetic systems with strongspin-orbit coupling.

Experimentally, it has been established that the AHEin the archetypical DMS system (Ga,Mn)As is in themetallic regime dominated by a scattering-independentmechanism, i.e. ρAH

xy ∝ ρ2xx (Chun et al., 2007; Edmonds

et al., 2002; Pu et al., 2008; Ruzmetov et al., 2004). Thestudies of Edmonds et al., 2002 and Chun et al., 2007have established this relationship in the non-insulatingmaterials by extrapolating the low temperature ρxy(B)to zero field and zero temperature. This is illustrated inFig. 21 where metallic samples, which span a larger rangethan the ones studied by Edmonds et al., 2002, show aclear RS ∼ ρ2

xx dependence.

DMSs grown requires non-equilibrium (low tempera-ture) conditions and the as-grown (often insulating) ma-terials and post-grown annealed metallic materials showtypically different behavior in the AHE response. Asimilar extrapolating procedure performed on insulating(Ga,Mn)As seems to exhibit a somewhat linear depen-dence of RS on ρxx. On the other hand, considerableuncertainty is introduced by the extrapolation to lowtemperatures because ρxx diverges and the complicatedmagetoresistance of ρxx is a priori not known in the lowT range.

A more recent study by Pu et al. (Pu et al., 2008)of(Ga,Mn)As grown on InAs, such that the tensile straincreates a perpendicular anisotropic ferromagnet, has es-tablished the dominance of the intrinsic mechanism inmetallic (Ga,Mn)As samples beyond any doubt. Measur-ing the longitudinal thermo-electric transport coefficients( ρxx, ρxy, αxy, and αxx where J = σE +α(−∇T )), one

can show that given the Mott relation α =π2k2

BT3e ( ∂σ

∂E )EF

and the empirical relation ρxy(B = 0) = λMzρnxx, the re-

lation between the four separately measured transport

19

FIG. 21 (a) Ga1−xMnxAs samples that show insulating andmetallic behavior defined by ∂ρxx/∂T near T = 0. (b) Rs

vs. ρxx extrapolated from ρxy(B) data to zero field and lowtemperatures. [From Ref. Chun et al., 2007.]

FIG. 22 (Top) Zero B field ρxy and ρxx for four samplesgrown on InAs substrates (i.e. perpendicular to plane easyaxis). Annealed samples, which produce perpendicular toplane easy axes, are marked by a ∗. The inset indicates theB dependence of the 7% sample at 10 K. (Bottom) Zero-fieldNerst coefficient αxy for the four samples. The solid red curvesindicate the best fit using Eq. (2.11) and the dashed curvesthe best fit setting n=1. [From Ref. Pu et al., 2008.]

coefficients is:

αxy =ρxy

ρ2xx

(

π2k2BT

3e

λ′

λ− (n− 2)αxxρxx

)

. (2.11)

The fit to the λ′

λ and n parameters are shown in Fig. 22.Fixing n = 1 does not produce any good fit to the dataas indicated by the dashed lines in Fig. 22. These dataexcludes the possibility of a n = 1 type contribution tothe AHE in metallic (Ga,Mn)As and further verify thevalidity the Mott relation in these materials.

Having established that the main contribution to theAHE in metallic (Ga,Mn)As is scattering-independentcontributions, rather than skew-scattering contributions,DMSs are an ideal system to test our understanding ofAHE. In the regime where the largest ferromagnetic crit-ical temperatures are achieved ( for doping levels above1.5% ), semi-phenomenological models that are built onBloch states for the band quasiparticles, rather than lo-calized basis states appropriate for the localized regime(Berciu and Bhatt, 2001), provide the natural startingpoint for a model Hamiltonian which reproduces manyof the observed experimental effects (Jungwirth et al.,2006; Sinova and Jungwirth, 2005). Recognizing thatthe length scales associated with holes in the DMS com-pounds are still long enough, a k · p envelope functiondescription of the semiconductor valence bands is appro-priate. To understand the AHE and magnetic anisotropy,it is necessary to incorporate intrinsic spin-orbit couplingin a realistic way.

A successful model for (Ga,Mn)As is specified by theeffective Hamiltonian

H = HKL + Jpd

∑

I

SI · s(r)δ(r −RI) +Hdis, (2.12)

where HKL is the six-band Kohn-Luttinger (KL) k · pHamiltonian (Dietl et al., 2001), the second term is theshort-range antiferromagnetic kinetic-exchange interac-tion between local spin SI at site RI and the itineranthole spin (a finite range can be incorporated in more real-istic models), and Hdis is the scalar scattering potentialrepresenting the difference between a valence band elec-tron on a host site and a valence band electron on a Mnsite and the screened Coulomb interaction of the itinerantelectrons with the ionized impurities.

Several approximations can be used to vastly simplifythe above model, namely, the virtual crystal approxima-tion (replacing the spatial dependence of the local Mnmoments by a constant average) and mean field theoryin which quantum and thermal fluctuations of the localmoment spin-orientations are ignored (Dietl et al., 2001;Jungwirth et al., 2006). In the metallic regime, disordercan be treated by a Born approximation or by more so-phisticated, exact-diagonalization or Monte-Carlo meth-ods (Jungwirth et al., 2002a; Schliemann and MacDon-ald, 2002; Sinova et al., 2002; Yang et al., 2003).

Given the above simple model Hamiltonian the AHEcan be computed if one assumes that the intrinsic

20

Berry’s phase (or Karplus-Luttinger) contribution willmost likely be dominan, because of the large SOI of thecarriers and the experimental evidence showing the domi-nance of scattering-independent mechanisms. For practi-cal calculations it is useful to use, as in the intrinsic AHEstudies in the oxides (Fang et al., 2003; Mathieu et al.,2004a,b), the Kubo formalism given in Eq. (1.2) with dis-order induced broadening, Γ, of the band-structure (butno side-jump contribution). The broadening is achievedby substituting one of the (εn(k)− εn′(k)) factors in thedenominator by (εn(k) − εn′(k) + iΓ). Applying thistheory to metallic (III,Mn)V materials using both the 4-band and 6-band k · p description of the valence bandelectronic structure one obtains results in quantitativeagreement with experimental data in (Ga,Mn)As and(In,Mn)As DMS (Jungwirth et al., 2002b). In a followup calculation Jungwirth et al. (Jungwirth et al., 2003), amore quantitative comparison of the theory with experi-ments was made in order to account for finite quasiparti-cle lifetime effects in these strongly disorder systems. Theeffective lifetime for transitions between bands n and n′,τn,n′ ≡ 1/Γn,n′ , can be calculated by averaging quasipar-ticle scattering rates obtained from Fermi’s golden ruleincluding both screened Coulomb and exchange poten-tials of randomly distributed substitutional Mn and com-pensating defects as done in the dc Boltzman transportstudies (Jungwirth et al., 2002a; Sinova et al., 2002). Asystematic comparison between theoretical and experi-mental AHE data is shown in Fig. 23 (Jungwirth et al.,2003). The results are plotted vs. nominal Mn concentra-tion x while other parameters of the seven samples stud-ied are listed in the figure legend. The measured σAH

values are indicated by filled squares; triangles are theo-retical results obtained for a disordered system assumingMn-interstitial compensationg defects. The valence bandhole eigenenergies εnk and eigenvectors |nk〉 are obtainedby solving the six-band Kohn-Luttinger Hamiltonian inthe presence of the exchange field, h = NMnSJpdz (Jung-wirth et al., 2006). Here NMn = 4x/a3

DMS is the Mndensity in the MnxGa1−xAs epilayer with a lattice con-stant aDMS , the local Mn spin S = 5/2, and the exchangecoupling constant Jpd = 55 meV nm−3.

In general, when disorder is accounted for, the the-ory is in a good agreement with experimental data overthe full range of Mn densities studied from x = 1.5% tox = 8%. The effect of disorder, especially when assum-ing Mn-interstitial compensation, is particularly strongin the x = 8% sample shifting the theoretical σAH muchcloser to experiment, compared to the clean limit the-ory. The remaining quantitative discrepancies betweentheory and experiment have been attributed to the reso-lution in measuring experimental hole and Mn densities(Jungwirth et al., 2003).

We conclude this section by mentioning the anoma-lous Hall effect in the non-metallic or insulating/hoppingregimes. Experimental studies of (Ga,Mn)As digital fer-romagnetic heterostructures, which consist of submono-layers of MnAs separated by spacer layers of GaAs,

0 1 2 3 4 5 6 7 8 9x (%)

0

10

20

30

40

50

60

70

σ AH

(Ω

-1 c

m-1

)

theory (Mn-interstitials)

experiment

x=1.5 %, p=0.31 nm-3

, e0=-0.065 %

x=2 %, p=0.39 nm-3

, e0=-0.087 %

x=3 %, p=0.42 nm-3

, e0=-0.13 %

x=4 %, p=0.48 nm-3

, e0=-0.17 %

x=5 %, p=0.49 nm-3

, e0=-0.22 %

x=6 %, p=0.46 nm-3

, e0=-0.26 %

x=8 %, p=0.49 nm-3

, e0=-0.35 %

FIG. 23 Comparison between experimental and theoreticalanomalous Hall conductivities. After Ref. Jungwirth et al.,2003.

have shown longitudinal and Hall resistances of the hop-ping conduction type, Rxx ∝ Tα exp[(T0/T )β], and haveshown that the anomalous Hall resistivity is dominatedby hopping with a sublinear dependence of RAH onRxx (Allen et al., 2004; Shen et al., 2008), similar toexperimental observations on other materials. Studies inthis regime are still not as systematic as their metalliccounterparts which clearly indicate a scaling power of 2.Experiments find a sublinear dependence of RAH ∼ Rβ

xx,with β ∼ 0.2−1.0 depending on the sample studies (Shenet al., 2008). A theoretical understanding for this hop-ping regime remains to be worked out still. In previoustheoretical calculations based on the hopping conductionwith the Berry phase (Burkov and Balents, 2003) showedthe insulating behavior RAH → ∞ as Rxx → ∞ butfailed to explain the scaling dependence of RAH on Rxx.

D. Other classes of materials

1. Spinel CuCr2Se4

As mentioned in earlier sections, a key prediction ofthe KL theory (and its modern generalization, based onthe Berry-phase approach) is that the AHE conductivityσAH

yx is independent of the carrier lifetime τ (dissipation-less Hall current) in materials with moderate conductiv-ity. This implies that the anomalous Hall resistivity ρAH

yx

varies as ρ2. Moreover, σAHxy tend to be proportional to

the observed magnetization Mz. We write

σAHxy = SHMz, (2.13)

with SH a constant.Previously, most tests were performed by comparing

ρyx vs. ρ measured on the same sample over an ex-tended temperature range. However, because the skew-scattering model can also lead to the same predictionρyx ∼ ρ2 when inelastic scattering predominates, testsat finite T are inconclusive. The proper test requires a

21

10-7 10-6 10-5 10-4 10-3

10-6

10-5

10-4

10-3

10-2

10-1

= 1.95

0.10

0.250.50

0.60

0.85

1.0

| xy' |

/nh (

10-3

0 m

4 )

Resistivity at 5 K ( m )

FIG. 24 Log-log plot of the quantity |ρ′

xy |/nh vs. ρ in twelvecrystals of Br-doped spinel CuCr2Se4−xBrx with nh the holedensity (ρ is measured at 5 K; ρxy is measured at 2 and 5 K).Samples in which ρyx is electron-like (hole-like) are shown asopen (closed) circles. The straight-line fit implies |ρ′

xy|/nh =Aρα with α = 1.95 ± 0.08 and A = 2.24×10−25 (SI units).[From Ref. Lee et al., 2004.]

system in which ρ at 4 K can be varied over a very largerange without degrading the exchange energy and mag-netization M .

In the spinel CuCr2Se4, the ferromagnetic state is sta-bilized by 90o superexchange between the local momentsof adjacent Cr ions. The charge carriers play only a weakrole in the superexchange energy. The experimental proofof this is that when the carrier density n is varied by afactor of 20 (by substituting Se by Br), the Curie tem-perature TC decreases by only 100 K from 380 K. Signif-icantly, M at 4 K changes negligibly. The resistivity ρ at4 K may be varied by a factor of 103 without weakeningM . Detailed Hall and resistivity measurements were car-ried out by Lee et al. (Lee et al., 2004) on twelve crystalsof CuCr2Se4−xBrx. They found that ρyx measured at 5K changes sign (negative to positive) when x exceeds 0.4.At x = 1, ρyx attains very large values (' 700 µΩcm at5 K).

Lee et al. (Lee et al., 2004) showed that the magni-tude |ρyx|/n varies as ρ2 over 3 decades in ρ (Fig. 24),consistent with the prediction of the KL theory.

The AHE in the related materials CuCr2S4,CuxZnxCr2Se4 (x = 1

2 ) and Cu3Te4 has been investi-gated by Oda et al. (Oda et al., 2001).

2. Heusler Alloy

The full Heusler alloy Co2CrAl has the L21 latticestructure and orders ferromagnetically below 333 K. Sev-

FIG. 25 Combined plots of σxy in the Heusler alloy Co2CrAlversus the magnetization M at selected temperatures from 36to 278 K. The inset shows the inferred values of σAH

xy ≡ σxy/Mat each T . [From Ref. Husmann and Singh, 2006.]

eral groups (Block et al., 2004; Galanakis et al., 2002)have argued that the conduction electrons are fully spinpolarized (“half metal”). The absence of minority carrierspins is expected to simplify the analysis of the Hall con-ductivity. Hence this system is potentially an importantsystem to test theories of the AHE.

The AHE has been investigated on single crystals withstoichiometry Co2.06Cr1.04Al0.90 (Husmann and Singh,2006). Below the Curie temperature TC = 333 K, themagnetization M increases rapidly, eventually saturatingto a low-T value that corresponds to 1.65 µB (Bohr mag-neton) per formula unit. Husmann and Singh show that,

below ∼310 K, M(T ) fits well to the form [1−(T/TC)2]12 .

Assuming that the ordinary coefficient R0 is negligible,they found that the Hall conductivity σxy = ρyx/ρ

2 isstrictly linear in M (expressed as σxy = σ1

HM) over theT interval 36–278 K (Fig. 25). They interpret the lin-ear variation as consistent with the intrinsic AHE theory.The value of σ1

H = 0.383 G/(4πΩcm) inferred is similarto values derived from measurements on the dilute Nialloys, half Heuslers and silicides.

3. Fe1−yCoySi

The silicide FeSi is a non-magnetic Kondo insulator.Doping with Co leads to a metallic state with a low den-sity p of holes. Over a range of Co doping (0.05 < y <0.8) the ground state is a helical magnetic state witha peak Curie temperature TC ∼ 50 K. The magnetiza-tion corresponds to 1 µB (Bohr magneton) per Co ion.The tunability allows investigation of transport in a mag-netic system with low p. Manyala et al. (Manyala et al.,2004) observe that the Hall resistivity ρH increases to

22

FIG. 26 Hall conductivity σxy of ferromagnetic metals andheavy fermion materials (collected at 1 kG and 5 K unlessotherwise noted). Small solid circles represent Fe1−yCoySifor 5 < T < 75 K and 500 G< H < 50 kG with y = 0.1(filled circles), 0.15 (filled triangles), y = 0.2 (+), and 0.3(filled diamonds). MnSi data (5-35 K) are large, solid squaresconnected by black line. Small asterisks are (GaMn)As datafor 5< T < 120 K. The rising line σxy ∼ M is consistentwith the intrinsic AHE in itinerant ferromagnets while thefalling line σxy ∼ M−3 applies to heavy fermions. [From Ref.Manyala et al., 2004.]

∼1.5 µΩcm (at 5 K) at the doping y= 0.1. By plottingthe observed Hall conductivity σxy against M , they findσxy = SHM with SH ∼ 0.22 G/(4πΩcm). In contrast, inheavy fermion systems (which include FeSi) σxy ∼M−3.

4. MnSi

MnSi grows in the non-centrosymmetric B20 latticestructure which lacks inversion symmetry. Competi-tion between the exchange energy J and Dzyaloshinsky-Moriya term D leads to a helical magnetic state with along pitch λ (∼180 A). At ambient pressure, the heli-cal state forms at the critical temperature TC = 30 K.Under moderate hydrostatic pressure P , TC decreasesmonotonically, reaching zero at the critical pressure Pc

= 14 kbar. Although MnSi has been investigated forseveral decades, interest has been revived recently by aneutron scattering experiment which shows that, abovePc, MnSi displays an unusual magnetic phase in whichthe sharp magnetic Bragg spots at P < Pc are replacedby a Bragg sphere (Pfleiderer et al., 2004). Non-Fermiliquid exponents in the resistivity ρ vs. T are observedabove Pc.

Among ferromagnets, MnSi at 4 K has a low resistivity(ρ ∼ 2-5 µΩcm). The unusually long carrier mean-free-path ` implies that the ordinary term σNH

xy ∼ `2 is greatlyenhanced. In addition, the small ρ renders the total Hall

FIG. 27 (a) Hall resistivity ρyx vs. H in MnSi at selected Tfrom 5 to 200 K. At high T , ρyx is linear in H , but graduallyacquires an anomalous component ρ′

yx = ρyx − R0B with aprominent “knee” feature below TC = 30 K. (b) Matching ofthe field profiles of the anomalous Hall resistivity ρ′

yx to theprofiles of ρ2M , treating SH and R0 as adjustable parameters.Note the positive curvature of the high-field segments. [FromRef. Lee et al., 2007.]

voltage difficult to resolve. Both factors greatly compli-cate the task of separating σNH

xy from the AHE conductiv-ity. However, the long ` in MnSi presents an opportunityto explore the AHE in the high-purity limit of ferromag-nets. Using high-resolution measurements of the Hallresistivity ρyx (Fig. 27 a), Lee et al. (Lee et al., 2007)recently accomplished this separation by exploiting thelarge longitudinal magnetoresistance (MR) ρ(H). FromEq. (2.1) and (2.13), we have ρ′yx(H) = SHρ(H)2M(H),

where ρ′yx = ρAHyx = ρyx − R0B. At each T , the field

profiles of ρ′yx(H) and M(H) are matched by adjustingthe two H-independent parameters SH and R0 (Fig. 27b). The inferred parameters SH and R0 are found to beT independent below TC (Fig. 28 a).

The Hall effect of MnSi under hydrostatic pressure (5–11.4 kbar) was measured recently (Lee et al., 2008). Inaddition to the AHE and OHE terms σAH

xy and σNHxy , Lee

et al. observed a well-defined Hall term σCxy with an un-

usual profile. As shown in Fig. 29 (note that in the figureσAH

xy and σNHxy are labeled σA

xy and σNxy respectively), the

new term appears abruptly at 0.1 T, rapidly rises to a

23

FIG. 28 Comparison of the anomalous Hall conductivity σAxy

and the ordinary term σNxy measured in a 1-Tesla field in

MnSi. σAxy, inferred from the measured M (solid curve) and

Eq. (2.13), is strictly independent of `. Its T dependencereflects that of M(T ). σN

xy ∼ `2 is calculated from R0. Theconductivity at zero H , σ ∼ `, is shown as a dashed curve.[From Ref. Lee et al., 2007.]

large plateau value, and then vanishes at 0.45 T (curvesat 5, 7 and 10 K). From the large magnitude of σC

xy, andits restriction to the field interval in which the cone angleis non-zero, the authors argue that it arises from the cou-pling of the carrier spin to the chiral spin textures in thehelical magnetization, as discussed in Sec. II.B. The au-thors note that MnSi under pressure provides a very rareexample in which the three Hall conductivities co-existat the same T .

5. Mn5Ge3

The AHE in thin-film samples of Mn5Ge3 was investi-gated by Zeng et al. (Zeng et al., 2006). They expressthe AHE resistivity ρAH , which is strongly T depen-dent (Fig. 30a), as the sum of the skew-scattering terma(M)ρxx and the intrinsic term b(M)ρ2

xx, viz.

ρAH = a(M)ρxx + b(M)ρ2xx. (2.14)

The quantity b(M) is the intrinsic AHE conductivityσAH−int.

To separate the 2 terms, Zeng et al. plotted the quan-tity ρAH/(M(T )ρxx) against ρxx with T as a parameter.For T < 0.8TC , the plot falls on a straight line witha small negative intercept (solid squares in Fig. 30b).The intercept yields the skew-scattering term a(M)/Mwhereas the slope gives the intrinsic term b(M)/M . Fromthe constant slope, they derive their main conclusion thatσAH−int varies linearly with M .

To account for the linear-M dependence, Zeng etal. (Zeng et al., 2006) identify the role of long-wavelengthspin waves which cause fluctuations in the local directionof M(x) (and hence of Ω). They calculate the reduc-tion in σAH−int and show that it varies linearly with M(Fig. 30 c).

FIG. 29 (Main panel) −ρyx vs. H in MnSi under hydrostaticpressure P = 11.4 kbar at several T < TC , with H nominallyalong (111). The large Hall anomaly observed (electron-likein sign) arises from a new chiral contribution σC

xy to the totalHall conductivity. In the phase diagram (inset) the shadedregion is where σC

xy is resolved. The non-Fermi liquid regionis shaded blue. [From Ref. Lee et al., 2008.]

FIG. 30 (a) The T dependence of ρAH in Mn5 Ge3 before(solid squares) and after (solid circles) subtraction of theskew-scattering contribution. The thin curve is the resistivityρ. (b) Plots of ρAH/(Mρxx) vs. ρxx before (solid squares)and after (solid circles) subtraction of skew-scattering term.(c) Comparison of calculated σAH−int vs. Mz (open circles)with experimental values before (squares) and after (solid ci-cles) subtraction of skew-scattering term. [From Ref. Zenget al., 2006.]

24

6. Layered dichalcogenides

The layered transition-metal dichalcogenides are com-prised of layers weakly bound by the van der Waals force.Parkin and Friend (Parkin and Friend, 1980) have shownthat a large number of interesting magnetic systems maybe synthesized by intercalating 3d magnetic ions betweenthe layers.

The dichalcogenide FexTaS2 typically displays proper-ties suggestive of a ferromagnetic cluster-glass state atlow T for a range of Fe content x. However, at the com-position x = 1

4 , the magnetic state is homogeneous. Insingle crystals of Fe 1

4TaS2, the easy axis of M is parallel

to c (normal to the TaS2 layers). Morosan et al. (Mo-rosan et al., 2007) observed that the curves of M vs. Hdisplay very sharp switching at the coercive field at allT < TC (160 K). In this system, the large ordinary termσNH

xy complicates the extraction of the AHE term σAHxy .

Converting the ρyx-H curves to σxy-H curves (Fig. 31),Checkelsky et al. (Checkelsky et al., 2008) infer that thejump magnitude ∆σxy equals 2σAH

xy by assuming thatEq. (2.13) is valid. This method provides a direct mea-surement of σAH

xy without knowledge of R0. As shown

in Fig. 31 b, both the inferred σAHxy and measured M

are nearly T -independent below 50 K, but the formerdeviates sharply downwards above 50 K. Checkelsky etal. (Checkelsky et al., 2008) propose that the deviationrepresents a large, negative inelastic-scattering contribu-tion σin

xy that involves scattering from chiral textures of

the spins which increase rapidly as T approaches T−C .

In support, they show that the curve of σinxy/M(T ) vs.

T matches (within the resolution) that of ∆ρ(T )2 with∆ρ(T ) = ρ(T )− ρ(0).

E. Localization and AHE

The role of localization in the anomalous Hall effect isan important issue, and there have been several workson this subject. ( For a review of theoretical works, seeRef. Woelfle and Muttalib, 2006.) The weak localizationeffect on the normal Hall effect due to the external mag-netic field has been studied by Fukuyama (Fukuyama,1980), and the relation δσWL

xy /σxy = 2δσWLxx /σxx has

been obtained where δOWL represents the correction ofthe physical quantity O by the weak localization effect.This means that the Hall coefficient is not subject to thechange due to the weak localization, i.e. δρNH

xy = 0. Anearly experiment by Bergmann and Ye (Bergmann andYe, 1991) on the anomalous Hall effect in the ferromag-netic amorphous metals showed almost no temperaturedependence of the anomalous Hall conductivity, whilethe diagonal conductivity shows the logarithmic temper-ature dependence. Langenfeld and Woelfle (Langenfeldand Wolfle, 1991; Woelfle and Muttalib, 2006) studied

0 50 100 1500.00

0.05

0.10

0.15

0.20

0.00

0.01

0.02

0.03

0.04

-8 -4 0 4 8-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

xy (1

05 -1m

-1)

T(K)

(b)

M (em

u/gram)

(a)

100 K80

6040

25

5 K

xy (1

05-1m

-1)

0H(T)

FIG. 31 (a) Hystersis loops of σxy vs. H in Fe 14TaS2 calcu-

lated from the measured ρ and ρyx curves. The linear portionscorrespond to σNH

xy while the jump magnitude ∆σxy equals

2σAHxy . (b) Comparison of ∆σxy with the magnetization M

measured at 0.1 T. Within the resolution, ∆σxy seems to beproportional to M below 50 K, but deviates sharply from Mat higher temperatures, reflecting the growing dominance ofa negative, inelastic-scattering term σin

xy. [From Ref. Check-elsky et al., 2008.]

theoretically the model

H =∑

k,σ

εkc†k,σck,σ

+∑

i

∑

k,k′,σ

ei(k−k′)·Ri [V + i(k × k′) · Ji]c†k′,σ′ck,σ

(2.15)

where Ji is proportional to the angular momentum of theimpurity at Ri, and the term containing it describes thespin-orbit scattering. They discussed the logarithmic cor-rection to the anomalous Hall conductivity in this model,and found that Coulomb anomaly terms vanished iden-tically, and the weak localization correction was cut-offby the phase-breaking lifetime τφ due to the skew scat-tering, explaining the experiment by Bergmann and Ye(Bergmann and Ye, 1991). Dugaev, Crepieux, and Bruno(Dugaev et al., 2002) found that for the two-dimensionalcase the correction to the Hall conductivity was loga-rithmic in the ratio max(τtr/τSO, τtr/τφ), with τtr beingthe transport lifetime and τSO the lifetime due to thespin-orbit scattering. More recently, Mitra et al. (Mi-tra et al., 2007) first observed the logarithmic tempera-

25

FIG. 32 The resistance (R0)-temperature dependence of thecoefficients AR and AAH defined in Eq. (2.16). Different sym-bols correspond to different methods of sample preparation.[From Ref. Mitra et al., 2007.]

ture dependence of the anomalous Hall conductivity inthe ultrathin film of polycrystalline Fe of sheet resis-tance Rxx less than ∼ 3 kΩ. They defined the quantities∆N (Q) = (1/R0L00)(δQ/Q) for the physical quantity Qwith respect to the reference temperature T = T0 = 5 Kby δQ = Q(T )−Q(T0), R0 = Rxx(T0), and L00 = e2/π~.Defining the coefficients AR and AH by

∆N (σxx) = AR ln

(

T0

T

)

∆N (σxy) = (2AR −AAH) ln

(

T0

T

)

(2.16)

Fig. 32 shows the R0-dependence of the coefficients AR

and AAH defined in the above Eq. (2.16).The change in the interpretation comes from the fact

that the phase-breaking lifetime τφ in the Fe film ismostly from scattering by the magnons and not fromskew scattering, which allows a temperature regimewhere max(1/τs, 1/τSO, ωH) << τφ << 1/τtr ( τs: spin-flip scattering time, ωH : internal magnetic field in the fer-romagnet ). In this region, they found that the weak lo-calization effect can lead to coefficient of the logarithmictemperature dependence of σxy proportional to the factorσSSM

xy /(σSSMxy +σSJM

xy ) where σSSMxy (σSJM

xy ) is the contri-bution from the skew scattering (side jump) mechanism.Assuming that that ratio σSJM

xy /σSSMxy decreases as the

sheet resistance R0 increases, they were able to explainthe sample-dependence of the logarithmic correction tothe anomalous Hall conductivity (Mitra et al., 2007).The absence of the logarithmic term in (Bergmann andYe, 1991) is interpreted as being due to a large ratio ofσSJM

xy /σSSMxy in their sample. It is interesting that the

ratio σSJMxy /σSSM

xy can be estimated from the coefficientof the logarithmic term.

Up to now, we have discussed the weak localizationeffect. When the disorder strength increases, a metal-

insulator transition will occur. For a normal metal underexternal magnetic field, the system belongs to the uni-tary class, and in 2D all the states are localized for anydisorder (Lee and Ramakrishnan, 1985). In the quan-tum Hall system, however, the extended states can sur-vive at discrete energies at the center of the broadeneddensity of states at each Laudau level. This extendedstate carries the quantum Hall current. Field theoret-ical formulation of this localization problem has beendeveloped (Prange and Girvin, 1987). In the presenceof the external magnetic field, a Chern-Simons term ap-pears in the non-linear sigma model whose coefficient isσxy. Therefore, the scaling variables are σxx and σxy,i.e., the two-parameter scaling theory should be appliedinstead of the single-parameter scaling. It has been dis-cussed that the scaling trajectory in the σxy − σxx planehas the fixed point at (σxy, σxx) = ((n + 1/2)(e2/h), σ0)where σ0 is some finite value, and σxy scales to thequantized value n(e2/h) when the initial value (givenby the Boltzmann transport theory) lies in the range

(n− 1/2)(e2/h) < σ(0)xy < (n+ 1/2)(e2/h). In contrast to

this quantum Hall system, there is no external magneticfield or the Landau level formation in the anomalous Hallsystem, and it is not trivial that the same two-parameterscaling theory applies to this case.

Onoda and Nagaosa (Onoda and Nagaosa, 2003) stud-ied this problem using the generalized Haldane model(Haldane, 1988) which shows the quantum Hall effectwithout the external magnetic field. They calculated thescaling function of the localization length in the finite-width stripe sample in terms of the iterative transfermatrix method by MacKinnon (MacKinnon and Kramer,1983), and found that two-parameter scaling holds evenwithout an external magnetic field or Landau level for-mation. For the experimental realization of this quan-

tized anomalous Hall effect, the |σ(0)xy | given by Boltz-

mann transport theory (without the quantum correction) is larger than e2/(2h), and the temperature is lower

than TLoc ∼ εF e−cσ(0)

xx /σ(0)xy , where εF is the Fermi energy

and c is a constant of the order of unity. Therefore, the

Hall angle σ(0)xy /σ

(0)xx should not be so small, hopefully of

the order of 0.1. However, in the usual case, the Hall an-gle is at most 0.01, which makes the quantized anomalousHall effect rather difficult to realize.

III. THEORETICAL ASPECTS OF THE AHE AND

EARLY THEORIES

In this section, we review recent theoretical develop-ments as well as the early theoretical studies of the AHE.First, we give a pedagogical discussion on the differencebetween the normal Hall effect due to the Lorentz forceand the AHE (Sec. III.A). In Sec. III.B we discuss thetopological nature of the AHE. In this Sec. III.C wepresent a wide survey of the early theories from a modernviewpoint in order to bring them into the context of thepresent linear transport theory formalisms now used as

26

a framework for AHE theories.

A. Symmetry considerations and analogies between normal

Hall effect and AHE

Before describing these recent developments, we pro-vide an elementary discussion that may facilitate under-standing of the following sections.

The Hall effect is one of the fundamental transportphenomena in solid-state physics. Its occurrence is adirect consequence of broken time-reversal symmetry inthe ferromagnetic state, T . The charge current J is T -odd,i.e., it changes sign under time reversal. On theother hand, the electric field E is T -even. Therefore,Ohm’s law

J = σE (3.1)

relates two quantities with different T -symmetries, whichimplies that the conductivity σ must be associated withdiispative irreversible processes, and indeed we know thatthe Joule heating Q = σE2/2 always accompanies theconductivity in Eq. (3.1). This irreversibility appearsonly when we consider macroscopic systems with contin-uous energy spectra.

Next, we consider the other aspect of the T -symmetry,i.e., the consequences of the T -symmetry of the Hamil-tonian which governs the microscopic dynamics of thesystem. This important issue has been formulated byOnsager, who showed that the response functions satisfythe following relation (Landau et al., 1984)

Kαβ(ω; , r, r′;B) = εαεβKβα(ω; , r′, r;−B), (3.2)

whereKαβ(ω; , r, r′;B) is the response of a physical quan-tity α at position r to the stimulus conjugate to the quan-tity β at position r′ with frequency ω. Here εα(εβ) = ±1specifies the symmetry property of α (β) with respectto the T -operation. B suggests a magnetic field, butrepresents any time-reversal breaking field. In the caseof a ferromagnet B can be associated with the magne-tization M , the spontaneously generated time-reversalsymmetry breaking field of a ferromagnet. The conduc-tivity tensor σab at a given frequency is proportional tothe current-current response function. We can there-fore make the identification α → Ja, β → Jb, where Ji

(i = x, y, z) are the components of the current operator.Since, εα = εβ = −1, we can conlcude that

σab(ω;B) = σba(ω;−B). (3.3)

Hence, we conclude that σab is symmetric with respectto a and b in systems with T -symmetry. The anti-symmetric part σab(ω) − σba(ω) is finite only if T -symmetry is broken.

Now we turn back to irreversibility for the general formof the conductivity σab, for which the dissipation is given

by (Landau et al., 1984)

Q =∑

ab

1

4(σ∗

ab + σba)EaE∗b

=1

4

∑

ab

[Re(σab + σba)Re(EaE∗b )

+ Im(σab − σba)Im(EaE∗b )]. (3.4)

The real part of the symmetric combination σab + σba

and the imaginary part of the antisymmetric combinationσab − σba contribute to the dissipation, while the imagi-nary part of σab + σba and the real part of σab − σba rep-resent the dispersive (dissipationless) responses. There-fore, Re(σab − σba), which corresponds to the Hall re-sponse, does not produce dissipation. This means thatthe physical processes contributing to this quantity canbe reversible, i.e., dissipationless, even though they arenot necessarily so. This point is directly related to thecontroversy between intrinsic and extrinsic mechanismfor the AHE.

The origin of the ordinary Hall effect is the Lorentzforce due to the magnetic field H :

F = −ecv ×H , (3.5)

which produces an acceleration of the electron perpen-dicular to its velocity v and H . For free electronsthis leads to circular cyclotron motion with frequencyωc = eH/(mc). The Lorentz force leads to charge accu-mulations of opposite signs on the two edges of the sam-ple. In the steady state, the Lorentz force is balanced bythe resultant transverse electric field, which is observed asthe Hall voltage VH . In the standard Boltzmann-theoryapproach, the equation for the electron distribution func-tion f(p,x) is given by (Ziman, 1967):

∂f

∂t+ v · ∂f

∂x+ F · ∂f

∂p=

(

∂f

∂t

)

coll.

, (3.6)

where p and x are the momentum and real space coor-dinates, respectively. Putting F = −e(E + v

c ×H), andusing the relaxatin time approximation for the collisionterm, − 1

τ (f − feq), with feq being the distribution func-tion in thermal equilibrium, one can obtain the steadystate solution to this Boltzmann equation as

f(p) = feq(p) + g(p), (3.7)

with

g(p) = −τe∂feq∂ε

v ·(

E − eτ

mcH ×E

)

, (3.8)

where order H2 terms have been neglected and the mag-netic field Hz is assumed to be small, i.e. ωcτ 1. The

current density J = −e∫

d3p

(2π)3p

mf(p) is obtained from

Eq. (3.8) to order O(E) and O(EH) as

J = σE + σHH ×E, (3.9)

27

where

σ = −1

3

∫

d3p

(2π)3v2e2τ

∂feq∂ε

(3.10)

is the conductivity, while

σH =1

3

∫

d3p

(2π)3v2e2τ

∂feq∂ε

eτ

mc(3.11)

is the ordinary Hall conductivity.Now let us fix the direction of the electric and magnetic

fields as E = Eyey and H = Hzez. Then the Hallcurrent is along the x-direction and we write it as

Jx = σHEy, (3.12)

with the Hall conductivity σH = ne3τ2Hz/m2c with n

being the electron density. The Hall coefficient RH isdefined as the ratio VH/JxHz, and from Eqs. (3.10) and(3.11), one obtains

RH = − 1

nec. (3.13)

This result is useful to determine the electron densityn experimentally, since it does not contain the relax-ation time τ . Note here that for fixed electron densityn, Eq. (3.13) means σH ∝ Hσ2

xx and that ρH ∝ H , in-dependent of the relaxation time τ (or equivalently σxx).In other words, the ratio |σH |/σxx = ωcτ 1. Thisrelation will be compared with the AHE case below.

What happens in the opposite limit ωcτ 1? Inthis limit, the cyclotron motion is completed many timeswithin the lifetime τ . This implies the closed cyclotronmotion is repeated many times before being disruptedby scattering. When treated quantum mechanically theperiodic classical motion leads to kinetic energy quantiza-tion and Landau level formation, which leads in turn tothe celebrated quantum Hall effect. As is well known,in the 2D electron gas realized in semiconductor het-erostructures, Landau-level quantization leads to the cel-ebrated quantum Hall effect (Prange and Girvin, 1987).If we consider the free electrons, i.e. completely neglect-ing the potential (both periodic and random) and theinteraction among electrons, one can show that σH is

given by e2

h ν where ν is the filling factor of the Landaulevels. In combination with electron localization, gapsat integer filling factors lead to quantization of σH . Forelectrons in a two-dimensional cyrstal, the Hall quantiza-tion is still quantized, a property which can be traced tothe topological properties of Bloch state wavefunctionsdiscussed below. (Prange and Girvin, 1987).

Now let us consider the magnetic field effect for elec-trons under the influence of the periodic potential. Forsimplicity we study the tight-binding model

H =∑

ij

tijc†i cj . (3.14)

FIG. 33 Tight-binding model on a square lattice under themagnetic flux φ for each plaquette. We choose the gaugewhere the phase factor of the transfer integral along x-direction is given by e−iφiy with iy being the y-componentof the lattice point position i = (ix, iy).

The magnetic field adds the Peierls phase factor to thetransfer integral tij between sites i and j, viz.

tij → tij exp[iaij ], (3.15)

with

aij =ie

~c

∫ j

i

dr ·A(r). (3.16)

Note that the phase factor is periodic with the period 2πwith respect to its exponent. Since the gauge transfor-mation

ci → ci exp[iθi] (3.17)

together with the redefinition

aij → aij + θi − θj (3.18)

keeps the Hamiltonian invariant, aij itself is not a phys-ical quantity. Instead, the flux φ = aij + ajk + akl + ali

per square plaquette is the key quantity in the problem(see Fig.33 )

For example, one may choose a gauge in which tijalong the directions ±y is a real number t, while tii+x =t exp[−iφiy] to produce a uniform flux distribution φ ineach square plaquette. Here iy is the y-component of thelattice point position i = (ix, iy)). The problem is thatthe iy-dependence breaks the periodicity of the Hamil-tonian along the y-axis, which invalidates the Bloch the-orem. This corresponds to the fact that the vector po-tential A(r) = (−Hy, 0) for the uniform magnetic fieldH is y-dependent, so that momentum component py isno longer conserved. However, when φ/(2π) is a rational

28

number n/m, where n and m are integers that do notshare a common factor, one can enlarge the unit cell bym times along the y-direction to recover the periodicity,because exp[−iφ(iy +m)] = exp[−iφiy]. Therefore, thereappear m sub-bands in the 1st-Brillouin zone, each ofwhich is characterized by a Chern number related to theHall response as will be described in section III.B (Thou-less et al., 1982). The message here is that an externalcommensurate magnetic field leads to a multiband struc-ture with an enlarged unit cell, leading to the quantizedHall response when the chemical potential is within thegap between the sub-bands. When φ/(2π) is an irrationalnumber, on the other hand, one cannot define the Blochwavefunction and the electronic structure is described bythe Hofstadter butterfly (Hofstadter, 1976) .

Since flux 2π is equivalent to zero flux, a commensu-rate magnetic field can be equivalent to a magnetic fielddistribution whose average flux is zero. Namely, the to-tal flux penetrating the m-plaquettes is 2πn, which isequivalent to zero. Interestingly, spatially non-uniformflux distributions can also lead to quantum Hall effects.This possibility was first considered by Haldane (Hal-dane, 1988), who studied a tight-binding model on thehoneycomb lattice. He introduced complex transfer inte-grals between the next-nearest neighbor sites in additionto the real one between the nearest-neighbor sites. Theresultant Hamiltonian has translational symmetry withrespect to the lattice vectors, and the Bloch wavefunctioncan be defined as the function of the crystal momentumk in the first Brillouin zone. The honeycomb lattice hastwo sites in the unit cell, and the tight-binding modelproduces two bands, separated by the band gap. Thewavefunction of each band is characterized by the Berryphase curvature, which acts like a “magnetic field” ink-space, as will be discussed in sections III.B and IV.A.

The quantum Hall effect results when the Fermi en-ergy is within this band gap. Intuitively, this can beinterpreted as follows. Even though the total flux pen-etrating the unit cell is zero, there are loops in the unitcells enclosing a nonzero flux. Each band picks up theflux distribution along the loops with different weightand contributes to the Hall response. The sum of theHall responses from all the bands, however, is zero as ex-pected. Therefore, the “polarization” of Hall responsesbetween bands in a multiband system is a general andfundamental mechanism of the Hall response, which isdistinct from the classical picture of the Lorentz force.

There are several ways to realize a flux distributionwithin a unit cell in momentum space. One is the rela-tivistic spin-orbit interaction given by

HSOI =~e

2m2c2(s×∇V ) · p, (3.19)

where V is the potential, s the spin, and p the momentumof the electron. This Hamiltonian can be written as

HSOI = ASOI · p, (3.20)

with ASOI = ~e2m2c2 (s×∇V ) acting as the effective vector

potential. Therefore, in the magnetically ordered state,

i.e., when s is ordered and can be regarded as a c-number,ASOI plays a role similar to the vector potential of anexternal magnetic field. Note that the Bloch theorem isvalid even in the presence of the spin-orbit interaction,since it preserves the translational symmetry of the lat-tice. However, the unit cell may contain more than twoatoms and each atom may have multiple orbitals. Hence,the situation is very similar to that described above forthe commensurate magnetic field or the Haldane model.

The following simple model is instructive. Let us con-sider the tight-binding model on a square lattice givenby

H = −∑

i,σ,a=x,y

tss†i,σsi+a,σ + h.c.

+∑

i,σ,a=x,y

tpp†i,a,σsi+a,a,σ + h.c.

+∑

i,σ,a=x,y

tsps†iσpi+a,a,σ + h.c.

+ λ∑

i,σ

σ(p†i,x,σ − iσp†i,y,σ)(pi,x,σ + iσpi,y,σ).

(3.21)

On each site, we put the 3 orbitals s, px and py, asso-ciated with the corresponding creation and annihilationoperators. The first three terms represent the transferof electrons between the neighboring sites as shown byFig. 34. The signs in front of t’s are determined by therelative sign of the two orbitals connected by the trans-fer integrals, and all t’s are assumed to be positive. Thelast term is a simplified SOI in which the z-componentof the spin moment is coupled to that of the orbital mo-ment. This 6-band model can be reduced to a 2-bandmodel in the ferromagnetic state when only the σ = +1component is retained. By the spin-orbit interaction, thep-orbitals are split into

|p±〉 = 1√2(|px〉 ± i|py〉) (3.22)

at each site, with the energy separation 2λ. There-fore, when only the lower energy state |p−〉 and thes-orbital |s〉 are considered, the tight-binding Hamilto-nian becomes H =

∑

k ψ†(k)h(k)ψ(k) with ψ(k) =

[sk,σ=1, pk,−,σ=1]T and

h(k) =

[

εs − 2ts(cos kx + cos ky)√

2tsp(i sinkx + sin ky)√2tsp(−i sinkx + sin ky) εp + tp(cos kx + cos ky)

]

.

(3.23)Note that the complex orbital |p−〉 is responsible for

the complex off-diagonal matrix elements of h(k). Thisproduces the Hall response as one can easily see from theformula Eq. (1.5) and Eq. (1.6) in section I.B. When theFermi energy is within the band gap, the Chern numberfor each band is ±1 when εs−4ts < εp+2tp, and they arezero otherwise. This can be understood by consideringthe effective Hamiltonian near k = 0, which is given by

h(k) = ε+mσz +√

2tsp(kyσx − kxσy), (3.24)

29

FIG. 34 Tight-binding model on a square lattice for a 3 bandmodel made from s and px− ipy orbitals with polarized spins.The transfer integrals between s and px− ipy orbitals becomecomplex along the y-direction in a way that is equivalent toeffective magnetic flux

with ε = [(εs − 4ts) + (εp + 2tp)]/2 and m = (εs −4ts) − (εp + 2tp), which is essentially the Dirac Hamil-tonian in (2+1)D. One can calculate the Berry curvatureb±(k) = ∓bz(k⊥)ez defined in Eq. (1.6) for the upper(+)and lower (−) bands, respectively as

bz(k⊥) =m

2[k2⊥ +m2]3/2

. (3.25)

The integral over the two dimensional wavevector k⊥ =(kx, ky) leads to Hall conductance σH = 1

2 sgn(m)e2/h inthis continuum model when only the lower band is fullyoccupied (Jackiw, 1984). The distribution Eq. (3.25) in-dicates that the Berry curvature is enhanced when thegap m is small near this avoided band crossing. Notethat the continuum model Eq. (3.24) cannot describethe value of the Hall conductance in the original tight-binding model since the information is not retained overall the first Brillouin zone. Actually, it has been provedthat the Hall conductance is an integer multiple of e2/hdue to the single-valudeness of the Bloch wavefunctionwithin the first Brillouin zone. However, the change ofthe Hall conductance between positive and negative mcan be described correctly.

Onoda and Nagaosa (Onoda and Nagaosa, 2002)demonstrated that a similar scenario emerges also fora 6-band tight binding model of t2g-orbitals with SOI.They showed that the Chern number of each band canbe nonzero leading to a nonzero Hall response arisingfrom the spontaneous magnetization. This suggests thatthe anomalous Hall effect can be of topological origin,a reinterpretation of the intrinsic contribution found byKarplus-Luttinger (Karplus and Luttinger, 1954). An-other mechanism which can produce a flux is a non-coplanar spin structure with an associated spin chiralityas has been discussed in section II.B.

The considerations explained here have totally ne-glected the finite lifetime τ due to the impurity and/or

phonon scattering, a generalization of the parameter ωcτwhich appeared in the case of the external magnetic field.A full understanding of disorder effects usually requiresthe full power of the detailed microscopic theories re-viewed in Sec. IV. The ideas explained in this sectionthough are sufficient to understand why the intrinsicBerry phase contribution to the Hall effect can be soimportant. The SOI produces an effective vector poten-tial and a “magnetic flux distribution” within the unitcell. The strength of this flux density is usually muchlarger than the typical magnetic field strength availablein the laboratory. In particular, the energy scale “~ωc”related to this flux is that of the spin-orbit interaction forEq. (3.20), and hence of the order of 30 meV in 3d tran-sition metal elements. This corresponds to a magneticfield of the order of 300 T. Therefore, one can expect thatthe intrinsic AHE is easier to observe compared with thequantum Hall effect.

B. Topological interpretation of the intrinsic mechanism:

relation between Fermi sea and Fermi surface properties

The discovery and subsequent investigation of thequantum Hall effect led to numerous important concep-tual advances (Prange and Girvin, 1987). In particular,the utility of topological considerations in understandingelectronic transport in solids was first appreciated. Inthis section, we provide an introduction to the applica-tion of topological notions such as the Berry phase andthe Chern number to the AHE problem, and explain thelong-unsuspected connections between these considera-tions and KL theory. A more elementary treatment isgiven in Ref. Ong and Lee, 2006.

The topological expression given in Eq. (1.5) was firstobtained by Thouless, Kohmoto, Nightingale and Nijs(TKNN) (Thouless et al., 1982) for Bloch electrons in a2D insulating crystal lattice immersed in a strong mag-netic field H . These authors showed that each band ischaracterized by a topological integer called the Chernnumber

Cn ≡ −∫

dkxdky

(2π)2bzn(k). (3.26)

According to Eq. (1.5), the Chern number Cn gives thequantized Hall conductance of an ideal 2D insulator, viz.σxy = e2Cn/h. As mentioned in prior sections, thesetopological arguments were subsequently applied to semi-classical transport theory (Sundaram and Niu, 1999) andthe AHE problem in itinerant ferromagnets (Jungwirthet al., 2002b; Onoda and Nagaosa, 2002), and the equiva-lence to the KL expression (Karplus and Luttinger, 1954)for σH was established. (Note however, that in multibandsystems, induced interband coherence among states withthe same crystal momentum is a possibility.)

Although the intrinsic Berry-phase effect involves theentire Fermi sea, as is clear from Eq. (1.5), it is usu-ally believed that transport properties of Fermi liquids

30

at low temperatures relative to the Fermi energy shouldbe dependent only on Fermi-sruface properties. This isa simple consequence of the observation that at thesetemperatures only states near the Fermi surface can beexcited to produce non-equillibirium transport.

The apparent contradiction between conventionalFermi liquid theory ideas and the Berry phase theoryof the AHE was resolved by Haldane (Haldane, 2004),showed that the Berry phase contribution to the AHEcould be viewed in an alternate way. Because of its topo-logical nature, the intrinsic AHE, which is most naturallyexpressed as an integral over the occupied Fermi sea, canbe rewritten as an integral over the Fermi surface. Let usstart with the topological properties of the Berry phase.The Berry-phase curvature bnk is gauge-invariant anddivergence-free except at quantized monopole and anti-monopole sources with the quantum ±2π, i.e.,

∇k · bnk =∑

i

qniδ3(k − kni), qni = ±2π. (3.27)

These monopoles and anti-monopoles appear at isolatedk points in three dimensions. This is because in com-plex Hermitian eigenvalue problems, accidental degener-acy can occur by tuning the three parameters of k.

To gain further insight on the topological nature of 3Dsystems, it is useful to rewrite Eq. (1.5) as

σij = εij`e2

h

K`

2π, (3.28)

where

K =∑

n

Kn, (3.29)

Kn = − 1

2π

∫

F.B.Z.

d3k f(εnk) bnk. (3.30)

We note that if the band dispersion εnk does not cross εF ,K`

n is quantized in integer multiples (Cna) of a primitivereciprocal lattice vector Ga at T = 0. We have

Kn =∑

a

CnaGa, (3.31)

where the index a runs over the three independent prim-itive reciprocal lattice vectors (Kohmoto, 1985).

We next discuss the non-quantized part of the anoma-lous Hall conductivity. In a real material with multipleFermi surface (FS) sheets indexed by α, we can describe

each sheet by k(α)F (s), where s = (s1, s2) is a parame-

terization of the surface. It is convenient to redefine theBerry-phase connection and curvature as

ai(s) = a(k(s)) · ∂sik(s), (3.32)

b(s) = εij∂si aj(s), (3.33)

with εij the rank-2 antisymmetric tensor.By integrating Eq. (3.30) by parts, eliminating the in-

tegration over the Brillouin zone boundary for each band,

and dropping the band indices, we may write (at T = 0)the vector K in Eq. (3.29) as

K =∑

a

CaGa +∑

α

Kα, (3.34)

Kα =1

2π

∫

Sα

ds1 ∧ ds2 b(s)k(α)F (s)

+1

4π

∑

i

Gαi

∫

∂Siα

ds · a(s). (3.35)

Here, Ca is the sum of Cna over the fully occupied bands,α labels sheets of the Fermi surface Sα, and δSi

α is theintersection of the Fermi surface Sα with the Brillouin

zone boundary i where k(α)F (s) jumps by Gαi. Note that

the quantity

1

2π

∫

Sα

ds1 ∧ ds2 b(s) (3.36)

gives an integer Chern number. Hence gauge invariancerequires that

∑

α

1

2π

∫

Sα

ds1 ∧ ds2 b(s) = 0. (3.37)

The second term in Eq. (3.35) guarantees that Kα is un-changed by any continuous deformation of the Brillouinzone into another primitive reciprocal cell.

C. Early theoretical studies of the AHE

1. Karplus-Luttinger theory and the intrinsic mechanism

The pioneering work of Karplus and Luttinger (KL)(Karplus and Luttinger, 1954) was the first theory of theAHE fully based on Bloch states ψnk. As a matter ofcourse, their calculations uncovered the important roleplayed by the mere existence of bands and the associatedoverlap of Bloch states. The KL theory neglects all latticedisorder, so that the Hall effect it predicts is based on anintrinsic mechanism.

In a ferromagnet, the orbital motion of the itinerantelectrons couples to spin ordering via the SOI. Hence alltheories of the AHE invoke the SOI term, which, as wehave mentioned, is described by the Hamiltonian givenby Eq. 3.19, where V (r) is the lattice potential. The SOIterm preserves lattice translation symmetry and we cantherefore define spinors which satisfy Bloch’s theorem.When SO interactions are included the Bloch Hamilto-nian acts in a direct product of orbital and spin-space.The total Hamiltonian HT can be separated into contri-butions as follows:

HT = H0 +HSOI +HE (3.38)

where H0 is the Hamiltonian in the absence of SOI andHE the perturbation due to the applied electric field E.In simple models the consequences of magnetic order can

31

be represented by replacing the spin operator in Eq.(3.19)by the ordered magnetic moment Ms as s→ ~

2Ms

M0, where

M0 is the magnitude of the saturated moment. The ma-trix element of HE = −eEbxb can be written as

〈nk|HE |n′k′〉 = −eEb

(

iδn,n′

∂

∂kbδk,k′ + iδk,k′Jnn′

b (k)

)

,

(3.39)where the “overlap” integral

Jnn′

b (k) =

∫

Ω

dr u∗nk(r)∂

∂kbun′k(r) (3.40)

are regular functions of k. In hindsight, Jnn′

b (k) maybe recognized as the Berry-phase connection an(k) dis-cussed in Sec. I.B.1.

We next divide HE into HrE + Ha

E , where HaE (Hr

E)corresponds to the first (second) term on the right handside of Eq. (3.39). Absorbing Hr

E into the unperturbedHamiltonian, i.e., Hp = H0 + HSOI + Hr

E , KL treatedthe remaining term Ha

E as a perturbation. Accordingly,the density matrix ρ was written as

ρ = ρ0(Hp) + ρ1 (3.41)

where ρ0(Hp) is the finite-temperature equilibrium den-sity matrix and ρ1 is the correction. KL assumed that ρ1

gives only the ordinary conductivity, whereas the AHEarises solely from ρ0(Hp). Evaluating the average veloc-ity va as va = Tr[ρ0va], they found that

va = −ieEb

∑

n,k

ρ′0(Epnk

)Jnna (k), (3.42)

where ρ′0 is the derivative with respect to the energy. Asit is clear here, this current is the dissipationless currentin thermal equilibrium under the influence of the externalelectric field, i.e., Hr

E . The AHE contribution arises fromthe interband coherence induced by an electronic field,and not from the more complicated rearrangements ofstates within the partially occupied bands.

A second assumption of KL is that, in 3d metals, theSOI energy HSO εF (the Fermi energy) and W (thebandwidth), so that it suffices to consider HSO to lead-ing order. Using Eq. (3.19), the AHE response is thenproportional to |Ms|, consistent with the empirical rela-tionship Eq. (2.1). More explicitly, to first-order inHSOI ,KL obtained

v = − e

m∆2

∑

k,n

ρ0(εnk)[E · vnk]Fnk, (3.43)

where εnk is the energy of the Bloch state for H0, ∆ is theaveraged value of interband energy separation, and Fnk

is the force i〈nk|[HSOI ,p]|nk〉. This gives the anomalousHall coefficient

Rs∼= 2e2HSO

m∆2δ⟨ m

m∗

⟩

ρ2, (3.44)

where |F | ∼= (e/c)HSOv, m∗ is the effective mass, δ is

the number of incompletely filled d-orbitals, and ρ is theresistivity.

Note that the ρ2-dependence implies that the off-diagonal Hall conductivity σH is independent of thetransport lifetime τ , i.e., it is well-defined even in theabsence of disorder, in striking contrast with the diag-onal conductivity σ. The implication that ρH = σHρ

2

varies as the square of ρ was immediately subjected toextensive experimental tests, as described in Sec. I.A.

An important finding in the KL theory is that inter-band matrix elements of the current operator contributesignificantly to the transport currents. This contrastswith conventional Boltzmann transport theory, wherethe current arises solely from the group velocity vn,k =∂εn,k/∂k.

In the Bloch basis

〈r|ψnk〉 = eik·r〈r|unk〉, (3.45)

the N -orbital Hamiltonian (for a given k) may be decom-posed into the 2N × 2N matrix h(k), viz.

h(k) =∑

n,m

〈nk|h(k)|mk〉 a†n(k)am(k). (3.46)

The corresponding current operator for a given k is

Jµ(t,k) =∑

n,m

〈nk|∂h(k)

∂~kµ|mk〉 a†n(k)am(k). (3.47)

According to Hellman-Feynman’s theorem

〈nk|∂h(k)

∂kµ|nk〉 = ∂εn(k)

∂k, (3.48)

the diagonal contribution to the current is (with n = m)

J intraµ (t) = −e

∑

nk

vg(k)Nn(k), (3.49)

where Nn(k) = a†n(k)an(k) is the occupation numberin the state |nk〉. Eq. (3.49) corresponds to the ex-pression in Boltzmann transport theory mentioned be-

fore. The inter-band matrix element 〈nk|∂h(k)∂kµ|mk〉 with

n 6= m corresponds to inter-band transitions. As shownby KL, the inter-band matrix elements have profoundconsequences for the Hall current, as has become appar-ent from the Berry-phase approach.

2. Extrinsic mechanisms

a. Skew scattering

In a series of reports, Smit (Smit, 1955, 1958) mounteda serious challenge to the basic findings of KL. Inthe linear-response transport regime, the steady-statecurrent balances the acceleration of electrons by E

32

against momentum relaxation by scattering from impu-rities and/or phonons. Smit pointed out that this bal-ancing was entirely absent from the KL theory, and pur-ported to show that the KL term vanishes exactly. Hisreasoning wasa that the anomalous velocity central toKL’s theory is proportional to the acceleration k whichmust vanish on average at steady state because the forcefrom E cancels that from the impurity potential.

More significantly, Smit proposed the skew-scatteringmechanism (Fig. 3) as the source of the AHE (Smit, 1955,1958). As discussed in Sec. I.B.2, in the presence of SOI,the matrix element of the impurity scattering potentialreads

〈k′s|V |k, s〉 = Vk,k′

(

δs,s′ +i~2

4m2c2(〈s′|σ|s〉 × k′) · k)

)

.

(3.50)Microscopic detailed balance would require that the tran-sition probability Wn→m between states n and m isidentical to that proceeding in the opposite direction(Wm→n). It holds, for example, in the Fermi’s golden-rule approximation

Wn→m =2π

~|〈n|V |m〉|2δ(En − Em), (3.51)

where V is the perturbation inducing the transition.However, microscopic detailed balance is not generic. Incalculations of the Hall conductivity, which involve thesecond Born approximation (third order in V ), detailedbalance already fails. In a simple N=1 model, skew scat-tering can be represented by an asymmetric part of thetransition probability

WAkk′ = −τ−1

A k × k′ ·Ms. (3.52)

When the asymmetric scattering processes is included(dubbed skew scattering), the scattering probabilityW (k→ k′) is distinct from W (k′ → k).

Physically, scattering of a carrier from an impurity in-troduces a momentum perpendicular to both the incidentmomentum k and the magnetization M . This leads to atransverse current proportional to the longitudinal cur-rent driven by E. Consequently, the Hall conductivityσH and the conductivity σ are both proportional to thetransport lifetime τ . Equivalently, ρH = σHρ

2 is propor-tional to the resistivity ρ.

As mentioned in Sec. I and in Sec. II, this prediction –qualitatively distinct from that in the KL theory – is con-sistent with experiments, especially on dilute Kondo sys-tems. These are systems which are realized by dissolvingmagnetic impurities (Fe, Mn or Cr) in the non-magnetichosts Au or Cu. (At higher concentrations, these systemsbecome spin glasses.) Although these systems do not ex-hibit magnetic ordering even at T as low as 0.1 K, theirHall profiles ρH vs. H display an anomalous componentderived from polarization of the magnetic local moments(Fig. 35 a).

Empirically, the Hall coefficient RH has the form

RH = R0 +A/T (3.53)

FIG. 35 The dependences of the Hall resistivity in AuFe onmagnetic field (Panel a) and temperature (Panel b). [AfterRef. Hurd, 1972.]

where R0 is the nominally T -independent OHE coeffi-cient and A is a constant (see Fig. 35 b). Identifyingthe second term with the paramagnetic magnetizationM = χH , where χ(T ) ∝ 1/T is the Curie susceptibilityof the local moments, we see that Eq. (3.53) is of thePugh form Eq. (2.1).

Many groups have explored the case in which ρ andρH can be tuned over a large range by changing themagnetic-impurity concentration ci. Hall experimentson these systems in the period 1970-1985 by and largeconfirmed the skew-scattering prediction ρH ∝ ρ. Thisled to the conclusion – often repeated in reviews – thatthe KL theory had been “experimentally disproved”. Asmentioned in Sec. I, this invalid conclusion ignores thesingular role of TRI-breaking in ferromagnets. WhileEq. (3.53) indeed describes skew scattering, the diluteKondo system respects TRI in zero H . This essentialqualitative differences between ferromagnets and systemswith easily aligned local moments implies essential differ-ences in the physics of their Hall effects.

b. Kondo theory

Kondo has proposed a finite-temperature skew-scattering model in which spin waves of local moments atfinite T lead to asymmetric scattering (Kondo, 1962). Inthe KL theory, the moments of the ferromagnetic stateare itinerant: the electrons carrying the transport currentalso produce the magnetization. Kondo has consideredthe opposite limit in which nonmagnetic s electrons scat-ter from spin-wave excitations of the ordered d-band localmoments (with the interaction term JSn · s). Retainingterms linear in the spin-orbit coupling λ and cubic in J ,Kondo derived an AHE current that arises from tran-sition probabilities containing the skew-scattering term

33

∼ k × k′. The T dependence of ρH matches well (espe-cially near TC) the ρH vs. T profiles measured in Ni (Jan,1952; Jan and Gijsman, 1952; Lavine, 1961) and Fe (Jan,1952) (Fig. 36).

FIG. 36 Comparison of the curve of ρH vs. T measured in Feby Jan with Kondo’s calculation [From Ref. Kondo, 1962.]

Kondo has noted that a problem with his model isthat it predicts that ρH should vanish when the d orbitalangular momentum is quenched as in Gd (whereas ρH

is observed to be large). A second problem is that theoverall scale for ρH (fixed by the exchange energies F0,F1 and F2) is too small by a factor of 100 comparedwith experiment. Kondo’s model with the s-d spin-spininteraction replaced by a d(spin)-s(orbital) interactionhas also been applied to antiferromagnets (Maranzana,1967).

c. Resonant skew scattering

Resonant skew scattering arises from scattering of car-riers from virtual bound states in magnetic ions dissolvedin a metallic host. Examples are XM, where X = Cu,Ag and Au and M = Mn, Cr and Fe. The prototypicalmodel is a 3d magnetic ion embedded in a broad s band,as described by the Anderson model (Hewson, 1993). Asshown in Fig. 37, a spin-up s electron transiently occu-pies the spin-up bound state which lies slightly belowεF . However, a second (spin-down) s electron cannot becaptured because the on-site repulsion energy U raisesits energy above εF . As seen in Fig. 37, the SOI causesthe energies Em

dσ of the d orbitals (labelled by m) to beindividually resolved. Here, σ = ± is the spin of thebound electron. The applied H merely serves to alignspin σ = + at each impurity.

The scattering of an incident wave eik·r is expressedby the phase shifts δm

ls (E) in the partial-wave expansionof the scattered wave. The phase shift is given by

cot δmls (E) =

(Emdσ − E)

∆, (3.54)

where ∆ is the half-width of each orbital.In the absence of SOI, δm

ls (E) is independent of m andthere is no Hall current. The splitting caused by SOI

FIG. 37 Sketch of the broadened virtual bound states of a 3dmagnetic impurity dissolved in a non-magnetic metallic host.The SOI lifts the degeneracy of the d levels indexed by m,the magnetic quantum number. [From Ref. Fert and Jaoul,1972.]

results in a larger density of states at εF for the orbitalm compared with −m in the case where the up spin elec-tron is more than half-filled as shown in Fig. 37. Becausethe phase shift is sensitive to occupancy of the impuritystate (Friedel sum-rule), we have δm

ls 6= δ−mls . This leads

to a right-left asymmetry in the scattering, and a largeHall current ensues. Physically, a conduction electron in-cident with positive z-component of angular momentumm hybridizes more strongly with the virtual bound statethan one with negative −m. This results in more elec-trons being scattered to the left than the right. Whenthe up spin density of states are filled less than half, i.e.,E0

d+ > 0, the direction becomes the opposite, leading asign change of the AHE.

Explicitly, the splitting of Emdσ is given by

Emd± = E0

d± ±1

2mλ±, (3.55)

where λσ is the SOI energy for spin σ. Using Eq. (3.55)in (3.54), we have, to order (λσ/∆)2,

δm2σ = δ02σσ

λσm

2∆sin2 δ02σ +

λ2σm

2

4∆2sin3 δ02σ cos δ02σ. (3.56)

Inserting the phase shifts into the Boltzmann transportequation, we find that the Hall angle γσ ' ρσ

xy/ρσxx for

spin σ is, to leading order in λσ/∆,

γσ = σ3

5

λσ

∆sin(2δ02σ − δ1) sin δ1, (3.57)

where δ1 is the phase shift of the p-wave channel which isassumed to be independent of m and s (Fert et al., 1981;Fert and Jaoul, 1972). Thermal averaging over σ, we canobtain an estimate of the observed Hall angle γ. Its signis given by the position of the energy level relative to εF .With the rough estimate λσ/∆ ' 0.1, and sin δ1 ∼ 0.1,we have |γ| ' 10−2. Without the resonant scatteringenhancement, the typical value of γ is ∼ 10−3.

34

FIG. 38 The Hall coefficient RH in heavy electron systems.Left panel: The T dependence of RH in CeAl3 (open triangle),UPt3 (solid square), UAl2 (solid circle), and a single crystalof CeRu2Si2 (circle dot symbol). The field H is along the c-axis. Right panel: Interpretation of the RH -T curve based onskew scattering from local moments. After Ref. Hadziclerouxet al., 1986 and Lapierre et al., 1987. [From Ref. Fert andLevy, 1987.]

Heavy-electron systems are characterized by a verylarge resistivity ρ above a coherence temperature Tcoh

caused by scattering of carriers from strong fluctuationsof the local moments formed by f -electrons at each lat-tice site. Below Tcoh, the local-moment fluctuations de-crease rapidly with incipient band-formation involvingthe f -electrons. At low T , the electrons form a Fermiliquid with a greatly enhanced effective mass. As shownin Fig. 38, the Hall coefficient RH increases to a broadmaximum at Tcoh before decreasing sharply to a smallvalue in the low-T coherent-band regime. Resonant skewscattering has also been applied to account for the strongT dependence of the Hall coefficient in CeCu2Si2, UBe13,and UPt3 (Coleman et al., 1985; Fert et al., 1981; Fertand Levy, 1987).

d. Side-jump

An extrinsic mechanism distinct from skew scatteringis side-jump (Berger, 1970) (Fig. 3). Berger consideredthe scattering of a Gaussian wavepacket from a sphericalpotential well of radius R given by

V (r) =~

2

2m(k2 − k2

1) (r < R)

V (r) = 0 (r > R), (3.58)

in the presence of the SOI term HSO =(1/2m2c2)(r−1∂V/∂r)SzLz, where Sz (Lz) is thez-component of the spin (orbital) angular momentum.For a wavepacket incident with wavevector k, Bergerfound that the wavepacket suffers a displacement ∆ytransverse to k given by

∆y =1

6kλ2

c , (3.59)

with λc = ~/mc the Compton wavelength. For k ∼= kF∼=

1010m−1 (in typical metals), ∆y ∼= 3×10−16 m is far toosmall to be observed.

In solids, however, the effective SOI is enhanced byband-structure effects by the factor (Fivaz, 1969)

2m2c2

m∗~τq ∼= 3.4× 104, (3.60)

with τq = (m∗/3~2)∑

m 6=n(χξρ2/∆Enm)|〈m|q × p|n〉|2.Here, ∆Enm

∼= 0.5 eV is the gap between adjacentd-bands, χ ∼= 0.3 is the overlap integral, ρ ∼= 2.5 ×10−10 m is the nearest-neighbor distance, and ξ =−(~2/2m2c2)〈(r−1∂V/∂r)〉 ∼= 0.1 eV is the atomic SOIenergy. The factor in Eq. (3.60) is essentially the ra-tio of the electron rest mass energy mc2 to the energygap ∆Enm. With this enhancement, the transverse dis-placement is ∆y ' 0.8 × 10−11 m, which renders thecontribution relevant to the AHE.

However, because the side-jump contribution to σH isindependent of τ , it is experimentally difficult to distin-guish from the KL mechanism. We return to this issuein Sec. IV.A.

3. Kohn-Luttinger theory formalism

a. Luttinger theory

Partly motivated by the objections raised by Smit,Luttinger (Luttinger, 1958) revisited the AHE problemusing the Kohn-Luttinger formalism of transport the-ory (Kohn and Luttinger, 1957; Luttinger and Kohn,1958). Employing a systematic expansion in terms of theimpurity potential ϕ, he solved the transport equation forthe density matrix and listed several contributions to theAHE current, including the intrinsic KL term, the skew-scattering term, and other contributions. They foundthat to zero-th order in ϕ, the average velocity is ob-

tained as the sum of the 3 terms v(11)β , uβ and v

(b)β defined

by (Luttinger, 1958)

v(11)β = −ieEα

(

∂J`β

∂kα− ∂J`

α

∂kβ

)

k=0

(

εF

3niϕ

)

, (3.61)

uβ = −ieEα

(

∂J`β

∂kα− ∂J`

α

∂kβ

)

k=0

, (3.62)

and

v(b)β = ieEα

[(

∂J`β

∂kα− ∂J`

α

∂kβ

)

C − εFT(0)αβ

]

, (3.63)

with ni the impurity concentration. (For the definition

of T(0)αβ , see Eq. (4.27) of Ref. Luttinger, 1958).

The first term v(11)β is the skew-scattering contribu-

tion, while the second uβ is the velocity obtained by KL

(Karplus and Luttinger, 1954). The third term v(b)β is

another term of the same order as uβ.The issues raised by comparing intrinsic vs. extrinsic

AHE mechanisms involve several fundamental issues in

35

the theory of transport and quantum systems away fromequilibrium. Following Smit (Smit, 1955, 1958) and Lut-tinger (Luttinger, 1958), we consider the wavefunctionexpanded in terms of Bloch waves ψ`’s, viz.

ψ(t) =∑

`

a`(t)ψ`, (3.64)

where ` = (n,k) stands for the band index n andwavevector k. The corresponding expectation value ofthe position operator xβ is given by

xβ = i∑

`

∂a`

∂kβa∗` + i

∑

`,`′

a`′a∗`J

`,`′

β (3.65)

where J`,`′

β was defined in Eq. (3.40). Taking thequantum-statistical average using the density matrix(ρT )`′,` = 〈a`′a

∗` 〉, the expectation value can be written

as

〈xβ〉 = i∑

`

∂(ρT )`,`

∂kβ|k=k′ + i

∑

`,`′

(ρT )`′`J`,`′

β . (3.66)

It may be seen that the second term in Eq. (3.66) does notcontribute to the current because it is a regular functionof k, and the expectation value of its time-derivative iszero. Finally, one obtains the expression

〈vβ〉 = i∑

`

∂(ρT )`,`

∂kβ|k=k′ (3.67)

by taking the time-derivative of Eq.(3.66).Smit (Smit, 1955, 1958) assumed a`(t) = |a`|e−iεnkt

and obtained the diagonal part as

i(ρT )`,` =∂εnk

∂kβ〈|a`|2〉, (3.68)

while the off-diagonal part averages to zero because ofthe oscillatory factor e−i(ε`−ε`′)t. Inserting Eq. (3.68) inEq. (3.67) and using (ρT )`` = 〈|a`|2〉, one obtains theusual expression for the velocity, i.e.,

〈vβ〉 =∑

`

(ρT )`,`∂εnk

∂kβ(3.69)

which involves the group velocity, but not the anomalousvelocity. A subtlety in the expression Eq. (3.67) is clar-ified by writing ρ = ρ0 + fest where est is the adiabaticfactor, and ρT = sfest. This leads to the expression

〈vβ〉 = is∑

`

∂(f)nk,nk′

∂kβ|k=k′ , (3.70)

which approaches a finite value in the limit s→ 0. This

means that∂(f)nk,nk′

∂kβ|k=k′ ∼ 1/s, i.e., a singular function

of s. This is in sharp contrast to f itself, which is aregular function as s → 0, which KL estimated. They

considered the current or velocity operator instead of theposition operator, the latter of which is unbounded andeven ill-defined for periodic boundary conditions.

Luttinger (Luttinger, 1958) argued that the ratio

v(11)β /uβ equals εF /(3niϕ), which may become less than

1 for large impurity concentration ni. However, if we as-sume ϕ is comparable to εF , the expansion parameter is

εF τ/~, i.e., v(11)β /uβ = εF τ/~.

The “metallicity” parameter εF τ/~ plays a key rolein modern quantum-transport theory, especially in theweak localization and interaction theory (Lee and Ra-makrishnan, 1985). Metallic conduction corresponds toεF τ/~ 1. More generally, if one assumes that theanomalous Hall conductivity σH is first order in the spin-orbit energy ∆, it can be written in a scaling form as

σH =e2

ha· ∆

εFf

(

~

εF τ

)

, (3.71)

where the scaling function can be expanded as

f(x) =

∞∑

n=−1

cnxn. (3.72)

Here, cn’s are constants of the order of unity. The leadingorder term c−1x

−1 corresponds to the skew-scatteringcontribution ∝ τ , while the second constant term c0 isthe contribution found by KL. If this expansion is valid,the intrinsic contribution by KL is always smaller thanthe skew scattering contribution in the metallic regionwith ~/εF τ 1. One should note however that the righthand side of this inequality should really have a smallervalue to account for the weakness of the skew scatteringamplitude. When both are comparable, i.e., ~/εF τ ∼ 1,one needs to worry about the localization effect and thesystem is nearly insulating, with a conduction that is ofthe hopping type. This issue is discussed in more detailin Secs. II.A, II.E, and IV.D.

b. Adams-Blount formalism

Adams and Blount (Adams and Blount, 1959) ex-pressed the KL theory in a way that anticipates the mod-ern Berry-phase treatment by introducing the concept of“field-modified energy bands” and the “intracell” coordi-nate. They considered the diagonal part of the coordi-nate matrix in the band n as

xµc = i~

∂

∂pµ+Xnn

µ (p), (3.73)

which is analogous to Eq. (3.39) obtained by KL. Thefirst term is the Wannier coordinate identifying the lat-tice site, while the second term – the “intracell” coor-dinate – locates the wavepacket centroid inside a unitcell. Significantly, the intracell nature of Xnn

µ impliesthat it involves virtual interband transitions. Althoughthe motion of the wavepacket is confined to the conduc-tion band, its position inside a unit cell involves virtual

36

occupation of higher bands whose effects appear as a ge-ometric phase (Ong and Lee, 2006). They found that thecurl ∇k×Xnn acted like an effective magnetic field thatlives in k space.

The application of an electric field E leads to ananomalous (Luttinger) velocity, which gives a Hall cur-rent that is manifestly dissipationless. This is seen byevaluating the commutation relationship of xµ

c and xνc .

We have

[xµc , x

νc ] = i~

[

∂Xnnν

∂pµ−∂Xnn

µ

∂pν

]

= i~εµνλBn(p)λ, (3.74)

where we have defined the field Bn(p) = ∇p ×Xnn(p).This is analogous to the commutation relationship amongthe components of π = p+(e/c)A in the presence of thevector potential,i.e., [πµ, πν ] = i~εµνλ[∇r × A(r)]λ =i~Bλ(r). The anomalous velocity arises from the ficti-tious “magnetic field” Bn(p) (which lives in momentumspace), and non-commutation of the gauge-covariant co-ordinates xµ

c ’s. This insight anticipated the modern ideaof Berry phase curvature (see Secs. III.B and IV.A fordetails).

Taking the commutator between xµc and the Hamilto-

nian Hnn = En(p)− F νxνc , we obtain

vnn = −i[xc, Hnn] =∂En(p)

∂p− F ×Bn(p). (3.75)

The second term on the right hand side is called theanomalous velocity. Adams and Blount reproduced theresults of Luttinger (Luttinger, 1958) and demonstratedit for the simple case of a uniform field Bn(p) = D.As recognized by Smit (Smit, 1955, 1958), the currentsassociated with the anomalous velocities driven by E andby the impurity potential mutually cancel in steady state.However, D introduces corrections to the “driving term”and the “scattering term” in the transport equation. Asa consequence, the current J is

J = ne2∑

k

(

−2

3Edf (0)

dE

)

×[

F τ

m− F ×D

~+

(

τ2k2

3mτA

)

(F ×D)

]

.(3.76)

Note that the current arises entirely from the averageof the “normal current”. However, the second term inEq. (3.76) is similar to the anomalous velocity and isconsistent with the conclusions of KL (Karplus and Lut-tinger, 1954) and Luttinger (Luttinger, 1958). The con-sensus now is that the KL contribution exists. However,in the clean limit τ → ∞, the leading contribution tothe AHE conductivity comes from the skew-scatteringterm. A more complete discussion of the semiclassicaltreatment is in Sec. IV.A.

c. Nozieres-Lewiner theory

Adopting the premises of Luttinger’s theory (Lut-tinger, 1958), Nozieres and Lewiner (Nozieres andLewiner, 1973) investigated a simplified model comprisedof one conduction and one valence band to derive all thepossible contributions to the AHE. The Fermi level εF

was assumed to lie near the bottom of the conductionband. Integrating over the valence band states, Nozieresand Lewiner derived an effective Hamiltonian for a statek in the conduction band. The derived position operatoris reff = r + ρ, where the new term ρ which involves theSOI parameter λ is given by

ρ = −λk × S. (3.77)

This polarization or effective shift modifies the transportequation to produce several contributions to σH . In theirsimple model, the anomalous Hall current JAHE (besidesthat from the skew scattering) is

JAHE = 2Ne2λE × S, (3.78)

where N is the carrier concentration, and S is the aver-aged spin polarization. The sign is opposite to that of theintrinsic term Jintrinsic = −2Ne2λE × S obtained fromthe SOI in an ideal lattice. They identified all the termsas arising from the spin-orbit correction to the scatteringpotential, i.e., in their theory there is no contributionfrom the intrinsic mechanism. However, one should becautious about this statement because these cancellationsoccur among the various contributions and such cancel-lations can be traced back to the fact that the Berry cur-vature is independent of k. In particular, it is often thecase that the intrinsic contribution survives even whenµ lies inside an energy gap, i.e., N = 0 (the side-jumpcontribution is zero in this case). This issue lies at theheart of the discussion of the topological aspect of theAHE.

IV. LINEAR TRANSPORT THEORIES OF THE AHE

In this section we describe the three linear responsetheories now used to describe the AHE. All three theoriesare formally equivalent in the εF τ 1 limit. The threetheories make nearly identical predictions; the small dif-ferences between the revised semiclassical theory and thetwo formalism of the quantum theory are thoroughly un-derstood at least for several different toy model systems.The three different approaches have relative advantagesand disadvantages. Considering all three provides a morenuanced picture of AHE physics. Their relative corre-spondence has been shown analytically in several simplemodels (Sinitsyn, 2008; Sinitsyn et al., 2007). The gen-eralized semiclassical Boltzmann transport theory whichtakes the Berry phase into account is reviewed in Sec.IV.A; this theory has the advantage of greater physi-cal transparency, but it lacks the systematic character ofthe microscopic quantum-mechanical theories whose ma-chinery deals automatically with the problems of inter-band coherence. Another limitation of the semiclassical

37

Boltzmann approach is that the “quantum correction” tothe conductivities, i.e., the higher order terms in ~/εF τ ,which lead to the Anderson localization and other inter-esting phenomena (Lee and Ramakrishnan, 1985) cannotbe treated systematically.

The microscopic quantum-mechanical linear responsetheories based on the Kubo formalism and the Keldysh(non-equiliribrum Green’s function) formalism are re-viewed in Sec. IV.B.1 and Sec. IV.C, respectively. Theseformulations of transport theory are organized differentlybut are essentially equivalent in the linear regime. Inthe Kubo formalism, which is formulated in terms of theequilibrium Green’s functions, the intrinsic contributionis more readily calculated, especially when combined withfirst-principles electronic structure theory in applicationsto complex materials. The Keldysh formalism can moreeasily account for finite lifetime quantum scattering ef-fects and take account of the broadened quasiparticlespectral features due to the self-energy, while at the sametime maintaining a structure more similar to that of semi-classical transport theory (Onoda et al., 2006a). AWecontrast these techniques by comparing their applicationsto a common model, the ferromagnetic Rashba model intwo-dimensions (2D) which has been studied intensivelyin recent years (Sec.IV.D).

A. Semiclassical Boltzmann approach

As should be clear from the complex phenomenology ofthe AHE in various materials classes discussed in Sec. II,it is not easy to establish a one-size-fits-all theory forthis phenomenom. From a microscopic point of view theAHE is a formidable beast. In subsequent sections weoutline systematic theories of the AHE in metallic sys-tems which employ Keldysh and Kubo linear responsetheory formalisms and are organized around an expan-sion in disorder strength, characterized by the dimen-sionless quantity ~/εF τ . A small value for this parame-ter may be taken as a definition of the good metal. Westart with semiclassical theory, however, because of itsgreater physical transparency. In this section we out-line the modern version (Sinitsyn, 2008; Sinitsyn et al.,2007) of the semiclassical transport theory of the AHE.This theory augments standard semiclassical transporttheory by accounting for coherent band-mixing by theexternal electric field (which leads to the anomalous ve-locity contribution) and by a random disorder potential(which leads to side jump). For simple models in whicha comparison is possible, the two theories (i.e. (i) theBoltzmann theory with all side-jump effects (formulatednow in a proper gauge-invariant way) and the anoma-lous velocity contribution included and (ii) the metalliclimit of the more systematic Keldysh and Kubo formal-ism treatments) give identical results for the AHE. Thissection provides a more compact version of the materialpresented in the excellent review by Sinitsyn (Sinitsyn,2008), to which we refer readers interested in further de-

tail. Below we take ~ = 1 to simplify notation.In semiclassical transport theory one retreats from a

microscopic description in terms of delocalized Blochstates to a formulation of transport in terms of the dy-namics of wavepackets of Bloch states with a well definedband momentum and position (n,kc, rc) and scatteringbetween Bloch states due to disorder. The dynamics ofthe wavepackets between collisions can be treated by aneffective Lagrangian formalism. The wavepacket distri-bution function is assummed to obey a classical Boltz-mann equation which in a spatially uniform system takesthe form (Sundaram and Niu, 1999):

∂fl

∂t+ ~kc ·

∂fl

∂kc= −

∑

l′

(ωl′,lfl − ωl,l′fl′). (4.1)

Here the label l is a composition of band and momenta(n,kc) labels and Ωl′,l, the disorder averaged scatteringrate between wavepackets defined by states l and l′, is tobe evaluated fully quantum mechanically. The semiclas-sical description is useful in clarifying the physical mean-ing and origin of the different mechanism contributing tothe AHE. However, as explained in this review, contribu-tions to the AHE which are important in a relative senseoften arise from inter-band coherence effects which areneglected in conventional transport theory. The tradi-tional Boltzmann equation therefore requires elaborationin order to achieve a successful description of the AHE.

There is a substantial literature (Berger, 1970; Jung-wirth et al., 2002b; Smit, 1955) on the application ofBoltzmann equation concepts to AHE theory (see Sec.III.C). However stress was often placed only on one ofthe several possible mechanisms, creating a lot of confu-sion. A cohesive picture has been lacking until recently(Sinitsyn, 2008; Sinitsyn et al., 2006, 2005). In particu-lar, a key problem with some prior theory was that it in-correctly ascribed physical meaning to gauge dependentquantities. In order to build the correct gauge-invariantsemiclassical theory of AHE we must take the followingsteps (Sinitsyn, 2008):

i) obtain the equations of motion for a wavepacketconstructed from spin-orbit coupled Bloch elec-trons,

ii) derive the effect of scattering of a wave-packetfrom a smooth impurity, yielding the correct gauge-invariant expression for the corresponding side-jump,

iii) use the equations of motion and the scatteringrates in Eq. (4.1) and solve for the non-equillibirumdistribution function, carefully accounting for thepoints at which modifications are required to ac-count for side-jump,

iv) utilize the non-equilibrium distribution function tocalculate the dc anomalous Hall currents, again ac-counting for the contribution of side-jump to themacroscopic current.

38

The validity of this approach is partially established inthe following sections by direct comparison with fully mi-croscopic calculations for simple model systems in whichwe are able to identify each semiclassically defined mech-anisms with a specific part of the microscopic calcula-tions.

1. Equation of motion of Bloch states wave-packets

We begin by defining a wavepacket centered at positionrc with average momentum kc:

Ψkc,rc(r, t) =

1√V

∑

k

wkc,rc(k)eik·(r−rc)unk(r). (4.2)

A key aspect of this wavepacket is that the complex func-tion, sharply peaked around kc, must have a very specificphase factor in order to have the wavepacket centeredaround rc. This can be shown to be (Marder, 2000; Sun-daram and Niu, 1999):

wkc,rc(k) = |wkc,rc

(k)| exp[i(k − kc) · an], (4.3)

where an ≡ 〈unk|i∂kunk〉 is the Berry’s connection of theBloch state (see Sec. I.B). We can generate dynamicsfor the wavepacket parameters kc and rc by constructinga semiclassical Lagrangian from the quantum wavefunc-tions:

L = 〈Ψkc,rc|i ∂∂t−H0 + eV |Ψkc,rc

〉

= ~kc · rc + ~kc · an(kc)− E(kc) + eV (rc).(4.4)

All the terms in the above Lagrangian are common toconventional semiclassical theory except for the secondterm, which is a geometric term in phase space dependingonly on the path of the trajectory in this space. This termis the origin of the momentum-space Berry phase (Berry,1984) effects in anomalous transport in the semiclassicalformalism. The corresponding Euler-Lagrange equationsof motion are:

~kc = −eE (4.5)

rc =∂En(kc)

∂kc− ~kc × bn(kc) (4.6)

where bn(kc) = ∇ × an is the Berry’s curvature ofthe Bloch state. Compared to the usual dynamic equa-tions for wave-packets formed by free electrons, a newterm emerges due to the non-zero Berry’s curvature ofthe Bloch states. This term, which is already linearin electric field E, is of the Hall type and as such willgive rise in the linear transport regime to a Hall cur-rent contribution from the entire Fermi sea, i.e., jint

Hall =−e2E × 1

V

∑

k f0(Enk)bn(k).

2. Scattering and the side-jump

From the theory of elastic scattering we know that thetransition rate ωl,l′ in Eq. (4.1) is given by the T -matrix

element of the disorder potential:

ωl′l ≡ 2π|Tl′l|2δ(εl′ − εl). (4.7)

The scattering T -matrix is defined by Tl′l = 〈l′|V |ψl〉,where V is the impurity potential operator and |ψl〉 is

the eigenstate of the full Hamiltonian H = H0 + Vthat satisfies the Lippman-Schwinger equation |ψl〉 =

|l〉+(εl − H0 + iη)−1V |ψl〉. |ψl〉 is the state which evolvesadiabatically from |l〉 when the disorder potential isturned on slowly. For weak disorder one can approxi-mate the scattering state |ψl〉 by a truncated series in

powers of Vll′ = 〈l|V |l′〉:

|ψl〉 ≈ |l〉+∑

l′′

Vl′′l

εl − εl′′ + iη|l′′〉+ . . . (4.8)

Using this expression in the above definition of the T-matrix and substituting it into Eq. (4.7), one can expandthe scattering rate in powers of the disorder strength

ωll′ = ω(2)ll′ + ω

(3)ll′ + ω

(4)ll′ · · · , (4.9)

where ω(2)ll′ = 2π〈|Vll′ |2〉disδ(εl − εl′),

ω(3)ll′ = 2π

(

∑

l′′

〈Vll′Vl′l′′Vl′′l〉dis

εl − εl′′ − iη+ c.c.

)

δ(εl − εl′),

(4.10)and so on.

We can always decompose the scattering rate intocomponents that are symmetric and anti-symmetric in

the state indices: ω(s/a)l′l ≡ (ωll′ ± ωl′l)/2. In conven-

tional Boltzmann theory the AHE is due solely to theanti-symmetric contribution to the scattering rate (Smit,1955).

The physics of this contribution to the AHE is quitesimilar to that of the longitudinal conductivity. In par-ticular, the Hall conductivity it leads to is proportional

to the Bloch state lifetime τ . Since ω(2)l′l is symmetric, the

leading contribution to ω(a)l′l appears at order V 3. Partly

for this reason the skew scattering AHE conductivity con-tributions is always much smaller than the longitudinalconductivity. (It is this property which motivates theidentification below of additional transport mechanismwhich contribute to the AHE and can be analyzed in

semiclassical terms.) The symmetric part of ω(3)ll′ is not

essential since it only renormalizes the second order re-

sult for ω(2)l′l and the antisymmetric is given by

ω(3a)ll′ = −(2π)2

∑

l′′δ(εl − εl′′)Im〈Vll′Vl′l′′Vl′′l〉disδ(εl − εl′).

(4.11)This term is proportional to the density of scatterers, ni.Skew scattering has usually been associated directly with

ω(3a), neglecting in particular the higher order term ω(4a)ll′

which is proportional to n2i and should not be disregarded

because it gives a contribution to the AHE which is of

39

the same order as the side-jump contribution consideredbelow. This is a common mistake in the semiclassicalanalyses of the anomalous Hall effect (Sinitsyn, 2008).

Now we come to the interesting side-jump story. Be-cause the skew scattering conductivity is small, we haveto include effects which are absent in conventional Boltz-mann transport theory. Remarkably it is possible to pro-vide a successful analysis of one of the main additionaleffects, the side-jump correction, by means of a carefulsemiclassical analysis. As we have mentioned previouslyside jump refers to the microscopic displacement δrl,l′

experienced by a wave-packets formed from a spin-orbitcoupled Bloch states, when scattering from state l tostate l′ under the influence of a disorder potential. Inthe presence of an external electric field side jump leadsto an energy shift ∆Ul,l′ = −eE · δrl,l′ . Since we areassuming only elastic scattering, an upward shift in po-tential energy requires a downward shift in band energyand vice-versa. We therefore need to adjust Eq. (4.1) by

adding∑

l′ ωsl,l′

∂f0(εl)εl

eE · δrl,l′ to the r.h.s.But what about the side-jump itself? An expression for

the side-jump δrl,l′ associated with a particular transi-tion can be derived by integrating rc through a transition(Sinitsyn, 2008). We can write

rc =d

dt〈Ψkc,rc

(r, t)|r|Ψkc,rc(r, t)〉 = d

dt

∫

dr

V

∫

dk

∫

dk′w(k)w∗(k′)e−ik′·r(

reik·r)

u∗nk′(r)

unk(r)ei(E(k′)−E(k))t/~

=dEn(kc)

dkc+ (4.12)

d

dt

∫

cell

dr

∫

dkw∗(k)unk(r)

(

i∂

∂k

w(k)unk(r)

)

This expression is equivalent to the equations of motionderived within the Lagrangian formalism; this form hasthe advantage of making it apparent that in scatteringfrom state l to a state l′ a shift in the center of masscoordinate will accompany the velocity deflection. FromEq. (4.12) it appears that the scattering shift will go ap-proximately as:

δrl′,l ≈ 〈ul′ |i∂

∂k′ul′〉 − 〈ul|i

∂

∂kul〉. (4.13)

This quantity has usually been associated with the side-jump, although it is gauge-dependent and therefore arbi-trary in value. The correct expression for the side jumpis similar to this one, at least for the smooth impurity po-tentials situation, but was derived only recently by Sinit-syn et al. (Sinitsyn et al., 2006, 2005):

δrl′l = 〈ul′ |i∂

∂k′ul′〉 − 〈ul|i

∂

∂kul〉 − Dk′,karg[〈ul′ |ul〉],

(4.14)where arg[a] is the phase of the complex number a and

Dk′,k = ∂∂k′

+ ∂∂k

. The last term is essential and makesthe expression for the resulting side-jump gauge invari-

ant. Note that the side-jump is independent of the de-tails of the impurity potential or of the scattering pro-cess. As this discussion shows, the side jump contribu-tion to motion during a scattering event is analogous tothe anomalous velocity contribution to wave-packet evo-lution between collisions, with the role of the disorderpotential in the former case taken over in the latter caseby the external electric field.

3. Kinetic equation for the semiclasscial Boltzmann distribution

Equations (4.7) and (4.14) contain the quantum me-chanical information necessary to write down a semiclass-cial Boltzmann equation that takes into account boththe change of momentum and the coordinate shift dur-ing scattering in the presence of a driving electric field E.Keeping only terms up to linear order in the electric fieldthe Boltzmann equation reads (Sinitsyn et al., 2006):

∂fl

∂t + eE · v0l∂f0(εl)

∂εl= −∑

l′ωll′ [fl − fl′ − ∂f0(εl)

∂εleE · δrl′l],

(4.15)where v0l is the usual group velocity v0l = ∂εl/∂k. Notethat for elastic scattering we do not need to take accountof the Pauli blocking which yields factors like fl(1 − fl′)on the r.h.s. of Eq. (4.15), and that the collision termsare linear in fl as a consequence. (For further discus-sion of this point see Appendix B in Ref. Luttinger andKohn, 1955.) This Boltzmann equation has the standardform except for the coordinate shift contribution to thecollision integral explained above. Because of the side-jump effect, the collision term does not vanish when theoccupation probabilities fl are replaced by their ther-mal equilibrium values when an external electric field is

present: f0(εl)−f0(εl−eE ·δrll′) ≈ −∂f0(εl)∂εl

eE ·δrl′l 6= 0.Note that the term containing ωl,l′fl should be written

as ω(s)l,l′fl − ω(a)

l,l′ fl. In making this simplification we areimagining a typical simple model in which the scatteringrate depends only on the angle between k and k′. In

that case,∑

l′ ω(a)l,l′ = 0 and we can ignore a complication

which is primarily notational.

The next step in the Boltzmann theory is to solve forthe non-equilibrium distribution function fl to leadingorder in the external electric field. We linearize by writ-ing fl as the sum of the equilibrium distribution f0(εl)and non-equilibrium corrections:

fl = f0(εl) + gl + gadistl , (4.16)

where we split the non-equilibrium contribution into twoterms gl and gadist in order to capture the skew scatter-ing effect. gl and gadist solve independent self-consistenttime-independent equations (Sinitsyn et al., 2006):

eE · v0l∂f0(εl)

∂εl= −

∑

l′

ωll′(gl − gl′) (4.17)

40

and

∑

l′

ωll′

(

gadistl − gadist

l′ − ∂f0(εl)

∂εleE · δrl′l

)

= 0.

(4.18)In Eq. (4.14) we have noted that δrl′l = δrll′ . To solveEq. (4.17) we further decompose gl = gs

l + ga1l + ga2

l sothat∑

l′

ω(3a)ll′ (gs

l − gsl′) +

∑

l′

ω(2)ll′ (ga1

l − ga1l′ ) = 0, (4.19)

∑

l′

ω(4a)ll′ (gs

l − gsl′) +

∑

l′

ω(2)ll′ (ga2

l − ga2l′ ) = 0. (4.20)

Here gsl is the usual diagonal non-equilibrium distribu-

tions function which can be shown to be proportional to

n−1i . From Eq. (4.11) and ω

(3a)ll′ ∼ ni, it follows that

g3al ∼ n−1

i . Finally, from ω(4a)ll′ ∼ n2

i and Eq. (4.20), itfollows that g4a

l ∼ n0i ; illustrating the dangers of ignoring

the ω(4a)ll′ contribution to ωl,l′ . One can also show from

Eq. (4.18) that gadistl ∼ n0

i .

4. Anomalous velocities, anomalous Hall currents, andanomalous Hall mechanisms

We are now at the final stage where we usethe non-equilibrium distribution function derived fromEqs. (4.17)-(4.20) to compute the anomalous Hall cur-rent. To do so we need first to account for all contri-butions to the velocity of semiclassical particles that areconsistent with this generalized semiclassical Boltzmannanalysis. In addition to the band state group velocityv0l = ∂εl/∂k, we must also take into account the veloc-ity contribution due to the accumulations of coordinateshifts after many scattering events, another way in whichthe side-jump effect enters the theory, and the veloc-ity contribution from coherent band mixing by the elec-tric field (the anomalous velocity effect) (Nozieres andLewiner, 1973; Sinitsyn, 2008; Sinitsyn et al., 2006):

vl =∂εl∂k

+ bl × eE +∑

l′

ωl′lδrl′l. (4.21)

Combining Eqs. (4.16) and (4.21) we obtain the totalcurrent

j = e∑

l

flvl = e∑

l

(f0(εl) + gsl + ga1

l + ga2l + gadist

l )

×(∂εl∂k

+ ~Ωl × eE +∑

l′

ωl′lδrl′l) (4.22)

This gives five non-zero contributions to the AHE up tolinear order in E:

σtotalxy = σint

xy + σadistxy + σsj

xy + σsk1xy + σsk2−sj

xy . (4.23)

The first term is the intrinsic contribution which shouldbe by now familiar to the reader:

σintxy = −e2

∑

l

f0(εl)bz,l. (4.24)

Next are the effects due to coordinate shifts during scat-tering events (for E along the y-axis):

σadistxy = e

∑

l

(gadistl /Ey)(v0l)x (4.25)

follows from the distribution function correction due toside jumps while

σsjxy = e

∑

l

(gl/Ey)∑

l′

ωl′l(δrl′l)x (4.26)

is the current due to the side-jump velocity, i.e., due tothe accumulation of coordinate shifts after many scatter-ing events. Since coordinate shifts are responsible bothfor σadist

xy and for σsjxy, there is, unsurprisingly, an inti-

mate relationship between those two contributions. Inmost of the literature, σadist

xy is usually considered to be

part of the side-jump contribution, i.e., σadistxy + σsj

xy →σsj

xy. We distinguish between the two because they arephysically distinct and appear as separate contributionsin the microscopic formulation of the AHE theory.

Finally, σsk1xy and σsk2−sj

xy are contributions arising fromthe asymmetric part of the collision integral (Sinitsyn,2008):

σsk1yx = −e

∑

l

(ga1l /Ex)(v0l)y ∼ n−1

i , (4.27)

σsk2−sjyx = −e

∑

l

(ga2l /Ex)(v0l)y ∼ n0

i . (4.28)

According to the old definition of skew scattering bothcould be viewed as skew scattering contributions becausethey originate from the asymmetric part of the collisionterm (Smit, 1955). However, if instead we define skew-scattering as the contribution proportional to n−1

i , i.e.linear in τ , as in Sec. I.B, it is only the first contribu-tion, Eq. (4.27), which is the skew scattering (Leroux-Hugon and Ghazali, 1972; Luttinger, 1958). The secondcontribution, Eq. (4.28), was generally discarded in priorsemiclassical theories, although it is parametrically of thesame size as the side-jump conductivity. Explicit quanti-tative estimates of σsk2−sj

yx so far exist only for the mas-sive 2D Dirac band (Sinitsyn et al., 2007). In parsingthis AHE contribution we will incorporate it within thefamily of side-jump effects due to the fact that it pro-portional to n0

i , i.e. independent of σxx. However, it isimportant to note that this ”side-jump” contribution hasno phyiscal link to the side-step experienced by a semi-classcial quasiparticle upon scattering. An alternativeterminology for this contribution is intrinsic skew scat-tering to distinguish its physical origin from side-jump

41

deflections (Sinitsyn, 2008), but, to avoid further confu-sion, we simply include it as a contribution to the Hallconductivity which is of the order of σ0

xx and originatesfrom scattering.

As we will see below, when connecting the microscopicformalism to the semiclasscial one, σint

xy , can be directlyidentified with the single bubble (Kubo formalism) con-tribution to the conductivity, σadist

xy + σsjxy + +σsk2

xy con-stitute the usually termed ladder-diagram vertex correc-tions to the conductivity due to scattering and there-fore it is natural to group them together although theirphysical origins are distinct. σsk1

xy is identified directlywith the three-scattering diagram used in the literature.This comparison has been made specifically for two sim-ple models, the massive 2D Dirac band (Sinitsyn et al.,2007) and the 2D Rashba with exchange model (Borundaet al., 2007).

B. Kubo formalism

1. Kubo technique for the AHE

The Kubo formalism relates the conductivity to thequilibrium current-current correlation function (Kubo,1957). It provides a fully quantum mechanical formallyexact expression for the conductivity in linear responsetheory (Mahan, 1990). We do not review the formal ma-chinery for this approach here since can be found in manytextbooks. Instead, emphasize the key issues in studyingthe AHE within this formalism and how it relates to thesemiclassical formalism described in the previous section.

For the purpose of studying the AHE it is best to re-formulate the current-current Kubo formula for the con-ductivity in the form of the Bastin formula (see appendixA in (Crepieux and Bruno, 2001)) which can be manipu-lated into the more familiar form for the conductivity ofthe Kubo-Streda formula for the T = 0 Hall conductivity

σxy = σI(a)xy + σ

I(b)xy + σII

xy where:

σI(a)xy =

e2

2πVTr〈vxG

R(εF )vyGA(εF )〉c, (4.29)

σI(b)xy = − e2

4πVTr〈vxG

R(εF )vyGR(εF )+viG

A(εF )vjGA(εF )〉c,

(4.30)

σIIxy =

e2

4πV

∫ +∞

−∞

dεf(ε)Tr[vxGR(ε)vy

GR(ε)

dε

−vxGR(ε)

dεvyG

R(ε) + c.c.]. (4.31)

Here the subscript c indicates a disorder configurationaverage. The last contribution, σII

xy, was first derived byStreda in the context of studying the quantum Hall effect(Streda, 1982). In these equations GR/A(εF ) = (εF−H±iδ)−1 are the retarded and advanced Green’s functionsevaluated at the Fermi energy of the total Hamiltonian.

Looking more closely σIIxy we notice that every term

depends on products of retarded Green’s functions onlyor products of advanced Green’s functions only. It canbe shown that only the disorder free part of σII

xy is impor-tant in the weak disorder limit, i.e. , this contributionis zeroth order in the parameter 1/kF lsc. The only ef-fect of disorder on this contribution (for metals) is tobroaden the Green’s functions (see below) through theintroduction of a finite lifetime (Sinitsyn et al., 2007).It can therefore be shown by a similar argument that ingeneral σIb

xy, is of order 1/kF lsc and can be neglected inthe weak scattering limit (Mahan, 1990). Thus, impor-tant disorder effects beyond simple quasiparticle lifetimebroadening are contained only in σIa

xy. For these reasons,

it is standard within the Kubo formalism to neglect σIbxy

and evaluate the σIIxy contribution with a simple lifetime

broadening approximation to the Green’s function.Within this formalism the effect of disorder on the

disorder-configuration averaged Green’s function is cap-tured by the use of the T-matrix, defined by the integralequation T = W +WG0T , where W =

∑

i V0δ(r− ri) isa delta-scatterers potential and G0 are the Green’s func-tion of the pure lattice. From this one obtains

G = G0 +G0TG0 = G0 +G0ΣG. (4.32)

Upon disorder averaging we obtain

Σ = 〈W 〉c + 〈WG0W 〉c + 〈WG0WG0W 〉c + ... (4.33)

To linear order in the impurity concentration, ni, thistranslates to

Σ(z,k) = niVk,k +ni

V

∑

k

Vk,k′G0(k′, z)Vk′,k + · · · ,

(4.34)with Vk,k′ = V (k−k′) being the Fourier transform of thesingle impurity potential, which in the case of delta scat-terers is simply V0 (see Fig. 41 for a graphical reprenta-tion). Note that G and G0 are diagonal in momentumbut, due to the presence of spin-orbit coupling, non-diagonal in spin-index in the Pauli spin-basis. Hence,the blue lines depicted in Fig. 39 represent G and are ingeneral matrices in band levels.

One effect of disorder on the anomalous Hall conduc-tivity is taken into account by inserting the disorder av-eraged Green’s function, GR/A, directly into the expres-sions for σIa

xy and σIIxy, Eq. (4.29) and Eq. (4.31). This

step captures the intrinsic contribution to the AHE andthe effect of disorder on it, which is generally weak inmetallic systems. This contribution is separately identi-fied in Fig. 39 a.

The so-called ladder diagram vertex corrections, alsoseparately identified in Fig. 39, contribute to the AHEat the same order in 1/kF l as the intrinsic contribution.It is useful to define a ladder-diagram corrected veloctyvertex vα(εF ) ≡ vα + δvα(εF ), where

δvα(εF ) =niV

20

V

∑

k

GR(εF )(vα + δvα(εF ))GA(εF ),

(4.35)

42

IIII x x xII x x xx x xxyxy xyxy xyxy yy

intrinsic “side jump”intrinsic side-jump”intrinsic side-jumpj p

x xx xx xx

“ k tt i ”“skew scattering”skew-scatteringskew scatteringg

x xx x xx xx x xx xx x xxxyyy

FIG. 39 Graphical representation of the Kubo formalism ap-plication to the AHE. The solid blue lines are the disorderaveraged Green’s function, G, the red circles the bare velocityvertex vα = ∂H0/∂~kα, and the dashed lines with crosses rep-resent disorder scattering (niV

20 for the delta-scatter model).

(b) δvy is the velocity vertex renormalized by vertex correc-tions.

as depicted in Fig. 39 b. Note again that vα(εF ) and vα =

∂H0/∂~kα are matrices in the spin-orbit coupled bandbasis. The skew scattering contributions are obtained byevaluating, without doing an infinite partial sum as inthe case of the ladder diagrams, third order processes inthe disorder scattering shown in Fig. 39.

As may seem obvious from the above machinery, calcu-lating the intrinsic contribution is not very difficult, whilecalculating the full effects of the disorder in a systematicway (beyond calculating a few diagrams) is challengingfor any disorder model beyond the simple delta-scatteringmodel.

Next we illustrate the full use of this formalism for thesimplest nontrivial model, massive 2D Dirac fermions,with the goals of illustrating the complexities presentin each contribution to the AHE and the equivalenceof quantum and semiclassical approaches. This modelis of course not directly linked to any real material re-viewed in Sec. II and its main merit is the possibility ofobtaining full simple analytical expressions for each ofthe contributions. A more realistic model of 2D fermionswith Rashba spin-orbit coupling will be discussed in Sec.IV.C. The ferromagnetic Rashba model has been used topropose a minimal model of AHE for materials in which

band crossing near the Fermi surface dominate the AHEphysics (Onoda et al., 2008).

The massive 2D Dirac fermion model is specified by:

H0 = v(kxσx + kyσy) + ∆σz + Vdis, (4.36)

where Vdis =∑

i V0δ(r − Ri), σx and σy are Pauli

matrices and the impurity free spectrum is ε±k

=±√

∆2 + (vk)2 where k = |k| and the labels ± distin-guish bands with positive and negative energies. Weignore in this simple model spin-orbit coupled disordercontributions which can be directly incorporated throughsimilar calculations as in Crepieux et al (Crepieux andBruno, 2001). Within this model the disordered aver-aged Green’s function is

GR = 11/GR

0 −ΣR =εF +iΓ+v(kxσx+kyσy)+(∆−iΓ1)σz

(εF −ε++iΓ+)(εF −ε−+iΓ−) ,

(4.37)where Γ = πniV

20 /(4v

2), Γ1 = Γ cos(θ), γ± = Γ0 ±Γ1 cos(θ), and cos θ = ∆/

√

(vk)2 + ∆2. Note that withinthis disorder model τ ∝ 1/ni. Using the result inEq. (4.37) one can calculate the ladder diagram correc-tion to the bare velocity vertex given by Eq. (4.35):

vy = 8vΓ cos θ(1 + cos2 θ)

λ(1 + 3 cos2 θ)2σx+

(

v + vsin2 θ

(1 + 3 cos2 θ)

)

σy,

(4.38)where θ is evaluated at the Fermi energy. The details ofthe calculation of this vertex correction is described inAppendix A of Ref. Sinitsyn et al., 2007. Incorporatingthis result in σIa

xy we obtain the intrinsic and side-jumpcontributions to the conductivity for εF > ∆

σintxy = e2

2π~V

∑

k Tr [vσxGvσyG] = − e2 cos θ4~π

, (4.39)

σsjxy = e2

2π~V

∑

k Tr [vσxGδvyG]

= − e2 cos θ4π~

(

3 sin2 θ(1+3 cos2 θ) + 4 sin2 θ

(1+3 cos2 θ)2

)

. (4.40)

The direct calculation of the skew-scattering diagrams of

Fig. 39 is σskxy = − e2

2π~nV0

(vkF )4∆(4∆2+(vkF )2)2 and the final total

result is given by

σxy = − e2∆

4π~

√(vkF )2+∆2

[1 + 4(vkF )2

4∆2+(vkF )2

+ 3(vkF )4

(4∆2+(vkF )2)2 ]− e2

2π~nV0

(vkF )4∆(4∆2+(vkF )2)2 . (4.41)

2. Relation between the Kubo and the semiclassical formalisms

When comparing the semiclassical formalism to theKubo formalism one has to keep in mind that in the

semiclassical formalism the natural basis is the one thatdiagonalize the spin-orbit coupled Hamiltonian. In thecase of the 2D massive Dirac model this is sometimescalled chiral basis in the literature. On the other hand

43

js

1sk

xy

xy

int dint

xyadis

xysjsk

xy

2

FIG. 40 Graphical representation of the AHE conductivity in the chiral (band eigenstate) basis. The two bands of the two-dimensional Dirac model are labeled ”±”. The subset of diagrams that correspond to specific terms in the semiclassicalBoltzmann formalism are indicated.

in the application of the Kubo formalism it is simplest tocompute the different Green’s functions and vertex cor-rections in the Pauli basis and take the trace at the endof the calculation. In the case of the above model onecan apply the formalism of Sec. IV.A and obtain thefollowing results for the five distinct contributions: σint

xy ,

σadistxy , σsj

xy, σsk2−sjxy and σsk1

xy . Below we quote the resultsfor the complicated semiclassical calculation of each term(see Sec. IV of Sinitsyn et al. (Sinitsyn et al., 2007) fordetails):

σintrxy = − e2∆

4π~√

∆2 + (vkF )2; (4.42)

σsjxy = σadist

xy = − e2∆k2F

2π~√

k2F + ∆2(k2

F + 4∆2); (4.43)

σsk−sjxy = − e23∆(vkF )4

4π~√

(vkF )2 + ∆2[4∆2 + (vkF )2]2; (4.44)

σskxy = − e2

2π~niV0

(vkF )4∆

(4∆2 + (vkF )2)2. (4.45)

The correspondance with the Kubo formalism resultscan be seen after a few algebra steps. The contribu-tions σint

xy and σsk1xy are equal in both cases (Eq. (4.39)

and Eq. (4.41)). As expected, the intrinsic contribution,σintr

xy , is independent of disorder in the weak scatteringlimit and the skew scattering contribution is inversely

proportional to the density of scatterers. However, recallthat in Sec. I.B we have defined the side-jump contri-bution as the disorder contributions of zeroth order inni, i.e., τ0, as opposed to being directly linked to a side-step in the scattering process in the semiclassical theory.Hence, it is the sum of the three physically distinct pro-cesses σadist

xy + σsjxy + σsk2−sj

xy which can be shown to beidentical to Eq. (4.40) after some algebraic manipula-tion. Therefore, the old notion of associating the skewscattering directly with the asymmetric part of the col-lision integral and the side-jump with the side-step scat-tering alone leads to contradictions with their usual as-sociation with respect to the dependence on τ (or equiv-alenty 1/ni). We also note, that unlike what happens insimple models where the Berry’s curvature is a constantin momentum space, e.g., the standard model for elec-trons in a 3D semiconductor conduction band (Nozieresand Lewiner, 1973), the dependence of the intrinsic andside-jump contributions are quite different with respectto parameters such as Fermi energy, exchange splitting,etc.

C. Keldysh formalism

Keldysh has developed a Green’s function formalismapplicable even to the nonequilibrium quantum states,for which the diagram techniques based on Wick’s the-orem can be used (Baym and Kadanoff, 1961; Kadanoffand Baym, 1962; Keldysh, 1965; Mahan, 1990; Rammerand Smith, 1986). Unlike with the thermal (Matsubara)Green’s functions, the Keldysh Green’s functions are de-

44

fined for any quantum state. The price for this flexibilityis that one needs to introduce the path-ordered prod-uct for the contour from t = −∞ → t = ∞ and backagain from t = ∞ → −∞. Correspondingly, four kindsof the Green’s functions GR, GA, G< and G> need to beconsidered, although only three are independent (Baymand Kadanoff, 1961; Kadanoff and Baym, 1962; Keldysh,1965; Mahan, 1990; Rammer and Smith, 1986). There-fore, the diagram technique and the Dyson equation forthe Green’s function have a matrix form.

In linear response theory, one can use the usual thermalGreen’s function and Kubo formalism. Since approxima-tions are normally required in treating disordered sys-tems, it is important to make them in a way which atleast satisfies gauge invariance. In both formalisms thisis an important theoretical requirement which requiressome care. Roughly speaking, in the Keldysh formal-ism, GR and GA describe the single particle states, whileG< represents the non-equilibrium particle occupationdistribution and contains vertex corrections. Therefore,the self-energy and vertex corrections can be treated in aunified way by solving the matrix Dyson equation. Thisfacilitates the analysis of some models especially whenmultiple bands are involved.

Another and more essential advantage of Keldysh for-malism over the semiclassical formalism is that one can gobeyond a finite order perturbative treatment of impurityscattering strength by solving a self-consistent equation,as will be discussed in the next subsection. In essence,we are assigning a finite spectral width to the semiclassi-cal wave packet to account for an important consequenceof quantum scattering effects. In the Keldysh formalism,the semiclassical limit corresponds to ignoring the historyof scattering particles by keeping the two time labels inthe Greens functions identical.

We restrict ourselves below to the steady and uniformsolution. For more generic cases of electromagnetic fields,see Ref. Sugimoto et al., 2007. Let x = (t,x) be thetime-space coordinate. Green’s functions depend on twospace-time points x1 and x2, and the matrix Dyson equa-tion for the translationally invariant system reads (Ram-mer and Smith, 1986):

(ε− H(p)− Σ(ε,p))⊗ G(ε,p) = 1,

G(ε,p)⊗ (ε− H(p)− Σ(ε,p)) = 1, (4.46)

where we have changed the set of variables (x1;x2) tothe center-of-mass and the relative coordinates, and thenproceeded to the Wigner representation (X ; p) by meansof the Fourier transformation of the relative coordinate;

(X,x) ≡(

x1 + x2

2, x1 − x2

)

→∫

dt

∫

dx ei(εt−p·x)/~ · · · ,(4.47)

with p = (ε,p). In this Dyson equation, the product ⊗ iisreserved for matrix products in band indices, like thosethat also appear in the Kubo formalism.

In the presence of the external electromagnetic field

Aµ, we must introduce the mechanical or kinetic energy-momentum variable

πµ(X ; p) = pµ + eAµ(X). (4.48)

replacing p as the argument of the Green’s function, asshown by Onoda et al (Onoda et al., 2006b). In this rep-

resentation, G<(X ;π)/2πi is the quantum-mechanicalgeneralization of the semi-classical distribution function.When an external electric field E is present, the equa-tion of motion, or equivalently, the Dyson equation, re-tains the same form as Eq. (4.46) when the product ⊗is replaced by the so-called Moyal product (Moyal, 1949;Onoda et al., 2006b) given by

⊗ = exp

[

i~(−e)2

E · (←−∂ ε−→∇p −

←−∇p−→∂ ε)

]

. (4.49)

Henceforth,−→∂ and

←−∂ denote the derivatives operating

on the right-hand and left-hand sides, respectively, andthe symbol p = (ε,p) is used to represent the mechanicalenergy-momentum π. In this formalism, only gauge in-variant quantities appear. For example, the electric fieldE appears instead of the vector potential A.

Expanding Eq. (4.49) in E and inserting the result inEq. (4.46), one obtains the Dyson equation to linear orderin E, corresponding to linear response theory. The linearorder terms Gα

E and ΣαE in E are decomposed into two

parts as

G<E = G<

E,I∂εf(ε) +(

GAE − GR

E

)

f(ε), (4.50)

Σ<E = Σ<

E,I∂εf(ε) +(

ΣAE − ΣR

E

)

f(ε). (4.51)

Here, f(ε) represents the Fermi distribution function. Inthese decompositions, the first term on the r.h.s. cor-responds to the nonequilibrium deviation of the distri-bution function due to the electric field E. The secondterm, on the other hand, represents the change in quan-tum mechanical wavefunctions due to E, and arises dueto the multiband effect (Haug and Jauho, 1996) throughthe noncommutative nature of the matrices.

The corresponding separation of conductivity contri-butions is: σij = σI

ij + σIIij with

σIij = e2~

2

∫

dd+1p

(2π~)d+1iTr[

vi(p)G<Ej ,I(p)

]

∂εf(ε), (4.52)

σIIij = e2~

2

∫

dd+1p

(2π~)d+1iTr[

vi(p)(

GAEj

(p)− GREj

(p))]

f(ε).

(4.53)

This is in the same spirit as the Streda version (Streda,1982) of the Kubo-Bastin formula (Bastin et al., 1971;Kubo, 1957). The advantage here is that we can use thediagrammatic technique to connect the self-energy andthe Green’s function. For dilute impurities, one can takethe series of diagrams shown in Fig. 41 correspondingto the T -matrix approximation. In this approximation,

45

the self-consistent integral equation for the self-energyand Green’s function can be solved and the solution usedto evaluate the first and second terms in Eq. (4.50) andEq. (4.51).

FIG. 41 Diagrammatic representation of the self-energy in theself-consistent T -matrix approximation in the Keldysh space,which is composed of the infinite series of multiple Born scat-tering amplitudes. [From Ref. Onoda et al., 2008.]

In general, Eqs. (4.52) and (4.53), together with theself-consistent equations for GR,GA, and G< (Onodaet al., 2008) defines a systematic diagrammatic methodfor calculating σij in the Streda decomposition (Streda,1982) of the Kubo-Bastin formula (Bastin et al., 1971;Kubo, 1957).

D. Two-dimensional ferromagnetic Rashba model – a

minimal model

A very useful model to study fundamental aspectsof the AHE is the ferromagnetic two-dimensional (2D)Rashba model (Bychkov and Rashba, 1984):

H(p)tot =p2

2m− λp× σ · ez −∆0σ

z + V (x). (4.54)

Here m is the electron mass, λ is the Rashba spin-orbitinteraction strength, ∆0 is the mean field exchange split-ting, σ = (σx, σy, σz) and σ0 are the Pauli and identitymatrices, ez is the unit vector in the z direction, andV (x) = V0

∑

i δ(r−ri) is a δ-scatterer impurity potentialwith impurity density ni. Quantum transport propertiesof this simple but non-trivial model have been intensivelystudied in order to understand fundamental properties ofthe AHE in itinerant metallic ferromagnets. The metal-lic Rashba is a simple, but its AHE has both intrinsicand extrinsic contributions and both minority and ma-jority spin Fermi surfaces. It therefore captures most ofthe features that are important in real materials with aminimum of complicating detail. The model has there-fore received a lot of attention (Borunda et al., 2007;Dugaev et al., 2005; Inoue et al., 2006; Kato et al., 2007;Kovalev et al., 2009, 2008; Nunner et al., 2007; Onodaet al., 2006b, 2008).

The bare Hamiltonian has the band dispersion:

εσ(p) =p2

2m− σ∆p, ∆p =

√

λ2p2 + ∆20, (4.55)

illustrated in Fig. 42 a, and Berry-phase curvature

bzσ(p) = ~2 [∇p × (i〈p, σ|∇p|p, σ〉)]z =

λ2~

2∆0σ

2∆3p

,

(4.56)

where σ = ± labels the two eigenstates |p, σ〉 at momen-tum p.

With this we can then obtain the intrisic contributionto the AHE by integrating over occupied states at zerotemperature (Culcer et al., 2003; Dugaev et al., 2005):

σAH−intxy =

e2

2h

∑

σ

σ

[

1− ∆0

∆pσ

]

θ(µ− εσ(pσ)), (4.57)

where p± denotes the Fermi momentum for the band σ =±.

An important feature of σAH−intxy is its enhancement

in the interval of ε0,+ < µ < ε0,−, where it approaches amaximum value close to e2/2h, no matter how small ∆0

is (provided that it is larger than ~/τ). Near p = 0 theBerry curvatures of the two bands are large and oppositein sign. The large Berry curvatures translate into largeintrinsic Hall conductivities only when the chemical po-tential lies between the local maximum of one band andthe local minimum of the other. This enhancement ofthe intrinsic AHE near avoided band crossings is illus-trated in Fig. 42 b, where it is seen to survive moderatedisorder broadening of several times ∆0. This peaked fea-ture arises from the topological nature of σAH−int

xy . Asa consequence it is important to note that the result isnon-perturbative in SOI; only a perturbative expansionon ~/εF τ is justified.

Cucler et al. (Culcer et al., 2003) were the first to studythis model, obtaining Eq. (4.57). They were was followedby Dugaev et al. (Dugaev et al., 2005) where the intrinsiccontribution was calculated within the Kubo formalism.Although these studies found a non-zero σAH−int

xy , theydid not calculate all the contributions arising from disor-der (some aspects of the disorder treatment in (Dugaevet al., 2005) where corrected by these authors in (Sinit-syn et al., 2007)). The intrinsic AHE σAH−int

xy comes

form both σIxy and σII

xy. The first part σIxy contains the

intra-band contribution σI(a)xy which is sensitive to the

impurity scattering vertex correction.

The calculation of σI(a)xy incorporating the effects of dis-

order using the Kubo formalism, i.e., incorporating theladder vertex corrections (”side-jump”) and the leadingO(V 3

0 ) skew-scattering contributions (Sec. IV.B.1), yieldsa vanishing σAH

xy for the case where εF is above the gapat p = 0 (i.e. both subbands are occupied), irrespec-tive of the strength of the spin-independent scatteringamplitude (Borunda et al., 2007; Inoue et al., 2006; Nun-ner et al., 2008). On the other hand, when only the

majority band σ = + is occupied, σI(a)xy is given by the

skew-scattering contribution (Borunda et al., 2007),

σI(a)xy ≈ −e

2

h

1

nimpuimp

λ2p4+D+(µ)∆0∆p+

(3∆20 + ∆2

p+)2

, (4.58)

in the leading-order in (1/nimp).Some properties of the ferromagnetic Rashba model

appear unphyiscal in the limit τ → ∞ (Onoda et al.,

46

(a)

FIG. 42 (a) Band dispersion of the ferromagnetic 2D Rashbamodel in the clean limit. (b) The intrinsic anomalous Hallconductivity of the Hamiltonian Eq. (4.54) as functions ofthe Fermi level εF measured from the bottom of the majorityband and ~/τ ≡ mnimpV 2

0 /~2, which is the Born scattering

amplitude for λ = ∆0 = 0. This figure is for the parameterset ∆0 = 0.1, 2mλ = 3.59, and 2mV0/~

2 = 0.6, with energyunit has been taken as ε0,−. [From Ref. Onoda et al., 2006b.]

2008): σxy vanishes discontinuously as the chemical po-tential µ crosses the edge ε0,− of the minority bandwhich leads to a diverging anomalous Nernst effect atµ = ε0,−, irrespective of the scattering strength, if one as-sumes the Mott relation to be valid for anomalous trans-port (Smrcka and Streda, 1977). However, this unphys-ical property does not really hold, in fact, σxy does notvanish even when both subbands are occupied, as shownby including all higher-order Born scattering amplitudesas it is done automatically in the numerical Keldysh ap-proach (Onoda et al., 2006a). In particular, the skew-scattering contribution arises from the odd-order Bornscattering (i.e., even order in the impurity potential) be-yond the conventional level of approximation, O(V 3

0 ),that gives rise to the normal skew scattering contribu-tions (Kovalev et al., 2008). This yields the uncon-ventional behavior σAH−skew

xy ∝ 1/nimp independent ofV0 (Kovalev et al., 2008). The possible appearance ofthe AHE in the case where both subbands are occupiedwas also suggested in the numerical diagonalization cal-culation of the Kubo formula (Kato et al., 2007). Theinfluence of spin-dependent impurities has also been an-alyzed (Nunner et al., 2008).

A numerical calculation of σAHxy based on the Keldysh

formalism using the self-consistent T -matrix approxima-tion, shown in Fig. 43, suggests three distinct regimes for

FIG. 43 Total anomalous Hall conductivity vs. σxx forthe Hamiltonian Eq. (4.54) obtained in the self-consistentT -matrix approximation to the Keldysh approach (Kovalevet al., 2009; Onoda et al., 2006b, 2008). Curves are for avariety of disorder strengths. The same parameter valueshave been taken as in Fig. 42 with the chemical potentialbeing located at the center of the two subbands. The dashedcurves represent the corresponding semiclassical results. [Af-ter Ref. Kovalev et al., 2009.]

the AHE as a function of σxx at low temperatures (Onodaet al., 2006a, 2008). In particular, it shows a crossoverfrom the predominant skew-scattering region in the cleanlimit (σxy ∝ σxx) to an intrinsic-dominated metallic re-gion (σxy ∼ constant). In this simple model no welldefined plateau is observed. These results also suggestsanother crossover to a regime, referred to as the inco-herent regime by (Onoda et al., 2008), where σxy de-cays with the disorder following the scaling relation ofσxy ∝ σn

xx with n ≈ 1.6. This scaling arises in the cal-culation due to the influence of finite-lifetime disorderbroadening on σAH−int

xy , while the skew-scattering contri-bution is quickly dimished by disorder as expected. Ko-valev et al. (Kovalev et al., 2009) revisited the Keldyshcalculations for this model, studying them numericallyand analytically. In particular, their study extendedthe calculations to include the dependence of the skew-scattering contribution on the chemical potential µ bothfor µ < ε0,− and for µ > ε0,− (as shown in Fig. 43). Theseauthors demonstrated that changing the sign of the im-purity potential changes the sign of the skew-scatteringcontribution. The data collapse illustrated in Fig. 43then fails, especially near the intrinsic-extrinsic crossover.Data collapse in σxy vs. σxx plots is therefore not a gen-eral property of 2D Rashba models and should not beexpected in real materials. There is, however, a suffi-cient tendency in this direction to motivate analyzingexperiments by plotting data in this way.

Minimal model– The above results have the followingimplications for the generic nature of the AHE (Onodaet al., 2006b, 2008). The 2D ferromagnetic Rashba modelcan be viewed as a minimal model that takes into ac-

47

count both the “parity anomaly” (Jackiw, 1984) asso-ciated with an avoided-crossing of dispersing bands, aswell as impurity scattering in a system with two Fermisurface sheets. Consider then a general 3D ferromag-net. When the SOI is neglected, majority and minorityspin Fermi surfaces will intersect along lines in 3D. Fora particular projection kx of Bloch momentum along themagnetization direction, the Fermi surfaces will touch atpoints. SOI will generically lift the band degeneracy atthese points. At the k-point where the energy gap is aminimum the contribution to σAH−int

xy will be a maxi-mum and therefore the region around this point shouldaccount for most of σAH

xy , as the first-principles calcula-tions seem to indicate (Fang et al., 2003; Wang et al.,2007, 2006; Yao et al., 2004, 2007). We emphasize thatthe Berry phase contributions from the two bands arenearly opposite, so that a large contribution from thisregion of k-space accrues only if the Fermi level lies be-tween the split bands. Expanding the Hamiltonian atthis particular k-point, one can then hope to obtain aneffective Hamiltonian of the form of the ferromagneticRashba model. Note that the gap ∆0 in this effectivemodel Hamiltonian, Eq. (4.54), comes physically fromthe SOI splitting at this particular k-point, while the”Rashba SOI” λ is proportional to the Fermi velocitynear this crossing point. In 3D, the anomalous Hall con-ductivity is then given by the 2D contribution integratedover pz near this minimum gap region, and remains oforder of e2/ha (a the lattice constant) if no accidentalcancellation occurs.

We point out that a key assumption made in the abovereasoning is that the effective Hamiltonian obtained inthis expansion is sufficiently similar to the ferromagneticRashba model. We note that in the Rashba model itis ∆0 and not the SOI which opens the gap, so thisis already one key difference. The SO interactions inany effective Hamiltonian of this type should in generalcontain at least Rashba-like and Dresselhaus-like contri-butions. Further studies examining the crossing pointsmore closely near these minimum gap regions will shedfurther light on the relationship between the AHE in realmaterials and the AHE in simple models for which de-tailed perturbative studies are feasible.

The minimal model outlined above suggests the pres-ence of three regimes: (i) the superclean regime domi-nated by the skew-scattering contribution over the intrin-sic one, (ii) the intrinsic metallic regime where σxy be-comes more or less insensitive to the scattering strengthand σxx, and (iii) the dirty regime with kF ` = 2εF τ/~ .1 exhibiting a sublinear dependence of ρxy ∝ ρ2−n

xx , orequivalently σxy ∝ σn

xx with n ≈ 1.6. It is impor-tant to note that this minimal model is based on elas-tic scattering and cannot explain the scaling observedin the localized hopping conduction regime as σxx istuned by changing T . Nevertheless, if we multiply the2D anomalous Hall conductivity by a−1 with the lat-tice constant a ∼ 5 A for comparison to the experi-mental results on three-dimensional bulk samples, then

an enhanced σAHxy of the order of the quantized value

e2/h in the intrinsic regime should be interpreted as∼ e2/ha ∼ 103 Ω−1 cm−1. This value compares well withthe empirically observed cross over seen in the experimen-tal findings on Fe and Co, as discussed in Sec. II.A.

V. CONCLUSIONS: FUTURE PROBLEMS AND

PERSPECTIVES ON AHE

In this concluding section, we summarize what hasbeen achieved by the recent studies of the AHE and whatis not yet understood, pointing out possible directions forfuture research on this fascinating phenomenon. To keepthis section brief, we exclude the historical summary pre-sented in Sec. I.A and Sec. III.C which outline the earlydebate on the origin of the AHE. We avoid repeating allthel points highlighted already in Sec. I.A and focus onthe most salient ones. Citations are kept to a minimumas well and we refer mostly to the sections in which thematerial was presented.

a. Recent developments

The renewed interest in the AHE, which has lead toa richer and more cohesive understanding of the prob-lem, began in 1998 and was fueled by other connecteddevelopments in solid state physics. These were: i) thedevelopment of geometrical and topological concepts use-ful in understanding electronic properties such as quan-tum phase interference and the quantum Hall effect. (Leeand Ramakrishnan, 1985; Prange and Girvin, 1987), ii)the demostration of the close relation between the Hallconductance and the toplogical Chern number revealedby the TKNN formula (Thouless et al., 1982), iii) thedevelopment of accurate first-principles band structurecalculation which account realistically for SOI, and iv)the association of the Berry-phase concept (Berry, 1984)with the noncoplanar spin configuration proposed in thecontext of the resonating valence bond (RVB) theory ofcuprate high temperature superconductors (Lee et al.,2006).Intrinsic AHE– The concept of an intrinsic AHE, debatedfor a long time, was brought back to the forefront of theAHE problem because of studies which successfully con-nected the topological properties of the quantum statesof matter and the transport Hall response of a system. InSec. I.B.1 we have defined σAH−int

xy both experimentallyand theoretically. From the latter, it is rather straight-forward to write σAH−int

xy in terms of the Berry curvaturein the k-space, from which the topological nature of theintrinsic AHE can be easily recognized immediately. Thetopological non-perturbative quality of σAH−int

xy is high-lighted by the finding that for simple models with spon-taneous magnetization and SOI, bands can have nonzeroChern numbers even without an external magnetic fieldpresent. This means that expansion with respect to SOIstrength is sometimes dangerous since it lifts the degen-

48

eracy between the up- and down-spin bands, leading toavoided band crossings which can invalidate such expan-sion.

Even though the interpretations of the AHE in realsystems are still subtle and complicated, the view thatσAH−int

xy can be the dominant contribution to σAHxy in

certain regimes has been strengthened by recent compar-isons of experiment and theory. The intrinsic AHE canbe calculated from first-principles calculations or, in thecase of semiconductors, using k · p theory. These calcu-lations have been compared to recent experimental mea-surements for several materials such as Sr1−xCaxRuO3

(section II.B), Fe (section II.A), CuCr2Se4−xBrx (sectionII.D), and dilute magnetic semiconductors (section II.C).The calculations and experiments show semi-quantitativeagreement. More importantly however, violations of theempirical relation σH ∝ M have been established boththeoretically and experimentally. This suggests that theintrinsic contribution has some relevance to the observedAHE. On the other hand, these studies do not alwaysprovide a compelling explanation for dominance of theintrinsic mechanism

Fully consistent metallic linear response theories of theAHE – Important progress has been achieved in AHEtheory. The semiclassical theory, appropriately modi-fied to account for interband coherence effects, has beenshown to be consistent with fully microscopic theoriesbased on Kubo and Keldysh formalisms. All three the-ories have been shown to be equivalent in the εF τ 1limit, with each having their advantages and disadvan-tages (Sec. IV). Much of the debate and confusion inearly AHE literature originated from discrepancies andfarraginous results from earlier inconsistent applicationof these linear response theories.

A semiclassical treatment based on the Boltzmanntransport equation, but taking into account the Berrycurvature and inter-band coherence effects, has been for-mulated (Sec. IV.A). The physical picture for each pro-cess of AHE is now understood reasonably well in thecase of elastic impurity scattering.

More rigorous treatments taking into account themulti-band nature of the Green’s functions in terms ofKubo and Keldysh formalism have been developed fully(section IV.C). These have been applied to a particu-lar model, i.e., the ferromagnetic Rashba model, with astatic impurity potential which produces elastic scatter-ing. The ferromagnetic Rashba model has an avoidedcrossing which has been identified as a key player in theAHE of any material. These calculations have shown aregion of disorder strength over which the anomalous Hallconductivity stays more or less constant as a functionof σxx, corresponding to the intrinsic-dominated regime.The emergence of this regime has been linked to the topo-logical nature of the intrinsic contribution, analogous tothe topologically protexted quantized Hall effect.

Emergence of three empirical AHE regimes— Based onthe large collections of experimental results and indi-cations from some theoretical calculations, it is now

becoming clear that there are at least three differentregimes for the behavior of AHE as a function of σxx:(i) (σxx > 106 (Ωcm)−1) A high conductivity regime inwhich σAH

xy ∼ σxx, skew scattering dominates σAHxy , and

the anomalous Hall angle σH/σxx is constant. In thisregime however the normal Hall conductivity from theLorentz force, proportional to σ2

xxH , is large even forthe small magnetic field H used to align ferromagneticdomains and separating σAH

xy and σNHxy is therefore chal-

lenging. (ii) (104 (Ωcm)−1 < σxx < 106 (Ωcm)−1) An in-trinsic or scattering-independent regime in which σAH

xy isroughly independent of σxx. In this intermediate metallicregion, where the comparison between the experimentsand band structure calculations have been discussed, theintrinsic mechanism is assumed to be dominant as men-tioned above. The dominance of the intrinsic mechanismover side-jump is hinted at in some model calculations,but there is no firm understanding of the limits of thissimplifying assumption. (iii) (σxx < 104 (Ωcm)−1) Abad-metal regime in which σAH

xy decreases with decreas-ing σxx at a rate faster than linear. In this strong disor-der region, a scaling σH ∝ σn

xx with 1.6 < n < 1.7 hasbeen reported experimentally for a variety of materialsdiscussed in Sec.II. This scaling is primarily observedin insulating materials exhibiting variable range hoppingtransport and where σxx is tuned by varying T . The ori-gin of this scaling is not yet understood and is a majorchallenge for AHE theory in the future. For metallic ul-trathin thin films exhibiting this approximate scaling, itis natural that σH is suppressed by the strong disorder(excluding weak localization corrections). Simple con-siderations from the Kubo formula where the energy de-nominator includes a (~/τ)2 is that σH ∝ τ−2 when thisbroadening is larger than the energy splitting betweenbands due to the SOI. Since in this large broadeningregime σxx is usually no longer linear in τ , an upper limitof β = 2 for the scaling relation σH ∝ σβ

xx is expected.The numerical Keldysh studies of the ferromagnetic 2DRashba model indicates that this power is β ∼ 1.6, closeto what is observed in the limited dirty-metallic rangeconsidered in the experiments. It is a surprising featurethat this scaling seems to hold for both the metallic andinsulating samples.

b. Future challenges and perspectives

In the classical Boltzmann transport theory, the resis-tivity or conductivity at the lowest temperature is simplyrelated to the strength of the disorder. However, quan-tum interference of the scattered waves gives rise to aquantum correction to the conductivity and eventuallyleads to the Anderson localization depending on the di-mensionality. At finite temperature, inelastic scatteringby electron-electon and/or electron-phonon interactionsgive additional contributions to the resistivity, while sup-pressing localization effects through a reduction of thephase coherence length. In addition one needs to con-sider quantum correction due to the electron-electon in-

49

teraction in the presence of the disorder. These issues,revealed in the 80’s, must be considered to scrutinize themicroscopic mechanism of σxx or ρxx before studying theAHE. This means that it seems unlikely that only σxx(T )characterizes the AHE at each temperature. We can de-fine the Boltzmann transport σB

xx only when the residualresistivity is well defined at low temperature before theweak localization effect sets in. Therefore, we need tounderstand first the microscopic origin of the resistivity.Separating the resistivity into elastic and inelastic con-tributions via Mathhiessen’s rule is the first step in thisdirection.

To advance understanding in this important issue, oneneeds to develop the theoretical understanding for the ef-fect of inelastic scattering on AHE at finite temperature.This issue has been partly treated in the hopping the-ory of the AHE described in section II.B where phononassisted hopping was assumed. However, the effects ofinelastic scattering on the intrinsic and extrinsic mech-anisms are not clear at the moment. Especially, spinfluctuation at finite T remains the most essential and dif-ficult problem in the theory of magnetism, and usuallythe mean field approximation breaks down there. Theapproximate treatment in terms of the temperature de-pendent exchange splitting, e.g., for SrRuO3 (Sec. II.B),needs to be reexamined by more elaborated method suchas the dynamical mean field theory, taking into accountthe quantum/thermal fluctuation of the ferromagneticmoments. These type of studies of the AHE may shedsome light on the nature of the spin fluctuation in fer-romagnets. Also the interplay between localization andthe AHE should be pursued further in the intermediateand strong disorder regimes. These are all vital issuesfor quantum transport phenomena in solids in general,as well as for AHE specifically.

Admitting that more work needs to be done, in Fig.44 we propose a speculative and schematic crossover di-agram in the plane of diagonal conductivity σxx of theBoltzmann transport theory (corresponding to the dis-order strength) and the temperature T . Note that areal system should move along the y-axis as tempera-ture is changed, although the observed σxx changes withT. This phase diagram reflects the empirical fact thatinelastic scattering kills off the extrinsic skew scatteringcontribution more effectively, leaving the intrinsic andside-jump contributions as dominant at finite tempera-ture. We want to help stress that the aim of this figureis to promote further studies of the AHE and to iden-tify the location of each region/system of interest. Ofcourse, the generality of this diagram is not guaranteedand it is possible that the crossover boundaries and eventhe topology of the phase diagram might depend on thestrength of the spin-orbit interaction and other details ofthe system.

There still remain many other issues to be studied inthe future. First-principles band structure calculationsfor AHE are still limited to a few number of materi-als, and should be extended to many other ferromag-

Ferromagnetic critical region

Incoherent region

T

Localized

T

Intrinsic metallic region

Mott-Anderson

Localized-

hopping

conduction

regionSkew

scatteringcritical region

g

region

[ -1cm-1]103 106

xx [ 1cm 1]

FIG. 44 A speculative and schematic phase diagram for theanomalous Hall effect in the plane of the diagonal conductivityσxx and the temperature T .

nets. Especially, the heavy fermion systems are an im-portant class of materials to be studied in detail. Con-cerning this point, a more economical numerical tech-nique is now available (Marzari and Vanderbilt, 1997).This method employs the maximally localized Wannierfunctions, which can give the best tight-binding modelparameters in an energy window of several tens of eV’snear the Fermi level. The algorithm for the calculation ofσxy in the Wannier interpolation scheme has also been de-veloped (Wang et al., 2006). Application of these newlydeveloped methods to a large class of materials shouldbe a high priority in the future.

AHE in the dynamical regime is a related interest-ing problem. The magneto-optical effects such as theKerr/Faraday rotation have been the standard experi-mental methods to detect the ferromagnetism. Thesetechniques usually focus, however, on the high energy re-gion such as the visible light. In this case, an atomic orlocal picture is usually sufficient to interpret the data,and the spectra are not directly connected to the d.c.AHE. Recent studies have revealed that the small energyscale comparable to the spin-orbit interaction is relevantto AHE which is typically ∼ 10 meV for 3d transitionmetal and ∼ 100 meV in DMSs (Sinova et al., 2003).This means that the dynamical response, i.e., σxy(ω),in the THz and infrared region will provide importantinformation on the AHE.

A major challenge for experiments is to find exam-ples of a quantized anomalous Hall effect. There aretwo candidates at present: (i) a ferromagnetic insula-tor with a band gap (Liu et al., 2008), and (ii) a disorderinduced Anderson insulator with a quantized Hall con-ductance (Onoda and Nagaosa, 2003). Although theo-retically expected, it is an important issue to establishexperimentally that the quantized Hall effect can be re-alized even without an external magentic field. Such a

50

finding would be the ultimate achievement in identify-ing an intrinsic AHE. The dissipationless nature of theanomalous Hall current will manifest itself in this quan-tized AHE; engineering systems using quantum wells orfield effect transistors is a promising direction to realizethis novel effect.

There are many promising directions for extensions ofideas developed through studies of the AHE. For exam-ple, one can consider several kinds of “current” insteadof the charge current. An example is the thermal/heatcurrent, which can be also induced in a similar fashionby the anomalous velocity. The thermoelectric effect hasbeen discussed briefly in Sec. II.C, where combining allthe measured thermoelectric transport coefficients helpedsettle the issue of the scaling relation σAH

xy ∼ σ2xx in

metallic DMSs (Pu et al., 2008). Recent studies in Fe al-loys doped with Si and Co discussed in Sec. II.A followeda similar strategy (Shiomi et al., 2009). From the temper-ature dependence of the Lorentz number, they identifiedthe crossover between the intrinsic and extrinsic mech-anisms. Further studies of thermal transport will shedsome light on the essence of AHE from a different side.

Spin current is also a quantity of recent great inter-est. A direct generalization of AHE to the spin currentis the spin Hall effect (Kato et al., 2004; Murakami et al.,2003; Sinova et al., 2004; Wunderlich et al., 2005), whichcan be regarded as the two copies of AHE for up anddown spins with the opposite sign of σxy. In this effect,a spin current is produced perpendicular to the chargecurrent. An interesting recent development in spin Halleffect is that the quantum spin Hall effect and topologicalinsulators have been theoretically predicted and experi-mentally confirmed. We did not include this excitingnew and still developing topic in this review article. In-terested readers are referred to the original papers andreferences therein (Bernevig et al., 2006; Kane and Mele,2005; Koenig et al., 2007).

Acknowledgements

We thank the following people for collaboration anduseful discussion. A. Asamitsu, P. Bruno, D. Culcer,V. K. Dugaev, Z. Fang, J. Inoue, T. Jungwirth, M.Kawasaki, S. Murakami, Q. Niu, T. S. Nunner, K. Oh-gushi, M. Onoda, Y. Onose, J. Shi, R. Shindou, N.A. Sinitsyn, N. Sugimoto, Y. Tokura, and J. Wunder-lich. The work was supported by Grant-in-Aids (No.15104006, No. 16076205, No. 17105002, No. 19048015,N0. 19048008) and NAREGI Nanoscience Project fromthe Ministry of Education, Culture, Sports, Science, andTechnology of Japan. S. O. acknowledges partial sup-port from Grants-in-Aid for Scientific Research underGrant No. 19840053 from the Japan Society of the Pro-motion of Science, and under Grant No. 20029006 andNo. 20046016 from the MEXT of Japan. J. S. ac-knoledges partial support from ONR under Grant No.ONR-N000140610122, by NSF under Grant No. DMR-

0547875, by SWAN-NRI, by the Cottrell Scholar grantof the Research Corporation. N. P. O. acknowledges par-tial support from a MRSEC grant from the U. S. Na-tional Science Foundation (DMR-0819860). AHM ac-knowledges support from the Welch Foundation and fromthe DOE.

References

Adams, E., and E. Blount, 1959, J. Phys. Chem. Solids 10,286.

Allen, P. B., H. Berger, O. Chauvet, L. Forro, T. Jarlborg,A. Junod, B. Revaz, and G. Santi, 1996, Phys. Rev. B53(8), 4393.

Allen, W., E. G. Gwinn, T. C. Kreutz, and A. C. Gossard,2004, Phys. Rev. B 70, 125320.

Baily, S. A., and M. B. Salamon, 2003, J. Appl. Phys. 93,8316.

Baily, S. A., and M. B. Salamon, 2005, Phys. Rev. B 71(10),104407.

Bastin, A., C. Lewinner, O. Betbeder-Matibet, andP. Nozieres, 1971, J. Phys. Chem. Solids 32, 1811.

Baym, G., and L. P. Kadanoff, 1961, Phys. Rev. 124, 287.Berciu, M., and R. N. Bhatt, 2001, Phys. Rev. Lett. 87,

107203.Berger, L., 1964, Physica 30, 1141.Berger, L., 1970, Phys. Rev. B 2, 4559.Bergmann, G., and F. Ye, 1991, Phys. Rev. Lett. 67, 735.Bernevig, B. A., T. L. Hughes, and S.-C. Zhang, 2006, Science

314, 1757.Berry, M. V., 1984, Proc. Roy. Soc. of London 392, 45.Block, T., M. J. Carey, B. A. Gurney, and O. Jepsen, 2004,

Phys. Rev. B 70, 205114.Bohm, A., A. Mostafazadeh, H. Koizumi, Q. Niu, and

J. Zwanziger, 2003, The Geometric Phase in Quantum Sys-tems (Springer, Berlin, 2003).

Borunda, M., T. Nunner, T. Luck, N. Sinitsyn, C. Timm,J. Wunderlich, T. Jungwirth, A. H. MacDonald, andJ. Sinova, 2007, Phys. Rev. Lett. 99, 066604.

Burkov, A. A., and L. Balents, 2003, Phys. Rev. Lett. 91,057202.

Bychkov, Y. A., and E. I. Rashba, 1984, J. Phys. C 17, 6039.Chang, M., and Q. Niu, 1996, Phys. Rev. B. 53, 7010.Checkelsky, J. G., M. Lee, E. Morosan, R. J. Cava, and N. P.

Ong, 2008, Phys. Rev. B 77, 014433.Chun, S. H., Y. S. Kim, H. K. Choi, I. T. Jeong, W. O. Lee,

K. S. Suh, Y. S. Oh, K. H. Kim, Z. G. Khim, J. C. Woo,and Y. D. Park, 2007, Phys. Rev. Lett. 98, 026601.

Chun, S. H., M. B. Salamon, M. Jaime, R. Suryanarayanan,G. Dhalenne, and A. Revcolevschi, 2000, Physica B 284,1442.

Coleman, P., P. W. Anderson, and T. V. Ramakrishnan, 1985,Phys. Rev. Lett. 55, 414.

Crepieux, A., and P. Bruno, 2001, Phys. Rev. B 64, 014416.Culcer, D., A. H. MacDonald, and Q. Niu, 2003, Phys. Rev.

B 68, 045327.Dietl, T., H. Ohno, and F. Matsukura, 2001, Phys. Rev. B

63, 195205.Dugaev, V. K., P. Bruno, M. Taillefumier, B. Canals, and

C. Lacroix, 2005, Phys. Rev. B 71, 224423.Dugaev, V. K., A. Crepieux, and P. Bruno, 2002, J. Mag. and

Mag. Mat. 240, 159.

51

Edmonds, K. W., K. Y. Wang, R. P. Campion, A. C. Neu-mann, C. T. Foxon, B. L. Gallagher, and P. C. Main, 2002,Appl. Phys. Lett 81, 3010.

Fang, Z., N. Nagaosa, K. S. Tahakashi, A. Asamitsu, R. Math-ieu, T. Ogasawara, H. Yamada, M. Kawasaki, Y. Tokura,and K. Terakura, 2003, Science 302, 92.

Feng, J. S.-Y., R. D. Pahsley, and M.-A. Nicolet, 1975, J.Phys. C 8, 1010.

Fernandez-Pacheco, A., J. M. De Teresa, J. Orna, L. Morel-lon, P. A. Algarabel, J. A. Pardo, and M. R. Ibarra, 2008,Phys. Rev. B 77, 004.

Fert, A., A. Friederich, and A. Hamzic, 1981, J. Mag. andMag. Mat. 24(3), 231.

Fert, A., and O. Jaoul, 1972, Phys. Rev. Lett. 28(5), 303.Fert, A., and P. M. Levy, 1987, Phys. Rev. B 36(4), 1907.Fivaz, R. C., 1969, Phys. Rev. 183, 586.Fukuyama, H., 1980, J. Phys. Soc. Jap. 49(2), 644.Galanakis, I., P. H. Dederichs, and N. Papanikolaou, 2002,

Phys. Rev. B 66, 174429.Guo, G. Y., and H. Ebert, 1995, Phys. Rev. B. 51, 12633.Hadzicleroux, M., A. Hamzic, A. Fert, P. Haen, F. Lapierre,

and O. Laborde, 1986, Europhysics Lett. 1(11), 579.Haldane, F. D. M., 1988, Phys. Rev. Lett. 61, 2015.Haldane, F. D. M., 2004, Phys. Rev. Lett. 93, 206602.Hall, E., 1881, Phil. Mag. 12, 157.Hall, E. H., 1879, Amer. J. Math. 2, 287.Haug, H., and A.-P. Jauho, 1996, Quantum Kinetics in Trans-

port and Optics of Semiconductors (Springer, Berlin).Hewson, A. C., 1993, The Kondo Problem to Heavy Fermions.

(Cambridge University Press, Cambridge).Hofstadter, D. R., 1976, Phys. Rev. B 14(6), 2239.Holstein, T., 1961, Phys. Rev. 124(5), 1329.Hurd, C. M., 1972, The Hall Effect in Metals and Alloys

(Plenum Press, New York).Husmann, A., and L. J. Singh, 2006, Phys. Rev. B 73, 724.Inoue, J., T. Kato, Y. Ishikawa, H. Itoh, G. E. W. Bauer, and

L. W. Molenkamp, 2006, Phys. Rev. Lett. 97, 046604.Izumi, M., K. Nakazawa, Y. Bando, Y. Yoneda, and H. Ter-

auchi, 1997, J. Phys. Soc. Jap. 66, 3893.Jackiw, R., 1984, Phys. Rev. D 29, 2375.Jan, J.-P., 1952, Helv. Phys. Acta 25, 677.Jan, J.-P., and J. M. Gijsman, 1952, Physica 18, 339.Jones, R. O., and O. Gunnarsson, 1989, Rev. Mod. Phys. 61,

689.Jungwirth, T., M. Abolfath, J. Sinova, J. Kucera, and A. H.

MacDonald, 2002a, Appl. Phys. Lett. 81, 4029.Jungwirth, T., Q. Niu, and A. H. MacDonald, 2002b, Phys.

Rev. Lett. 88, 207208.Jungwirth, T., J. Sinova, J. Masek, J. Kucera, and A. H.

MacDonald, 2006, Rev. Mod. Phys. 78, 809.Jungwirth, T., J. Sinova, K. Y. Wang, K. W. Edmonds, R. P.

Campion, B. L. Gallagher, C. T. Foxon, Q. Niu, and A. H.MacDonald, 2003, Appl. Phys. Lett. 83, 320.

Kadanoff, L. P., and G. Baym, 1962, Quantum Statistical Me-chanics (Benjamin, Menlo Park).

Kane, C. L., and E. J. Mele, 2005, Phys. Rev. Lett. 95,226801.

Karplus, R., and J. M. Luttinger, 1954, Phys. Rev. 95, 1154.Kato, T., Y. Ishikawa, H. Itoh, and J. Inoue, 2007, New J.

Phys. 9, 50.Kato, Y. K., R. C. Myers, A. C. Gossard, and D. D.

Awschalom, 2004, Science 306, 1910.Kats, Y., I. Genish, L. Klein, J. W. Reiner, and M. R. Beasley,

2004, Phys. Rev. B 70, 8040.

Katsufuji, T., H. Y. Hwang, and S. W. Cheong, 2000, Phys.Rev. Lett. 84, 1998.

Kawamura, H., 2007, J. Mag. and Mag. Mat. 310, 1487.Keldysh, L. V., 1965, Sov. Phys. JETP 20, 1018, [Zh. Eksp.

Teor. Fiz. 47, 1515 (1964).].Kezsmarki, I., S. Onoda, Y. Taguchi, T. Ogasawara, M. Mat-

subara, S. Iguchi, N. Hanasaki, N. Nagaosa, and Y. Tokura,2005, Phys. Rev. B 72, 094427.

Koenig, M., S. Wiedmann, C. Bruene, A. Roth, H. Buhmann,L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, 2007, Science318, 766.

Kohmoto, M., 1985, Ann. Phys. 160, 343.Kohn, W., and J. M. Luttinger, 1957, Phys. Rev. 108, 590.Kondo, J., 1962, Prog. Theor. Phys. 27, 772.Kontani, H., T. Tanaka, and K. Yamada, 2007, Phys. Rev. B

75, 184416.Kooi, C., 1954, Phys. Rev. 95, 843.Kovalev, A. A., Y. Tserkovnyak, K. Vyborny, and J. Sinova,

2009, unpublished.Kovalev, A. A., K. Vyborny, and J. Sinova, 2008, Phys. Rev.

B 78, 41305.Kubo, R., 1957, J. Phys. Soc. Jpn 12, 570.Kundt, A., 1893, Wied. Ann. 49, 257.Landau, L. D., E. M. Lifshitz, and L. P. Pitaevskii, 1984,

Electrodynamics of Continuous Media (Oxford), 2nd edi-tion.

Langenfeld, A., and P. Wolfle, 1991, Phys. Rev. Lett. 67(6),739.

Lapierre, F., P. Haen, R. Briggs, A. Hamzic, A. Fert, andJ. P. Kappler, 1987, J. Mag. and Mag. Mat. 63-4, 338.

Lavine, J. M., 1961, Phys. Rev. 123, 1273.Lee, M., W. Kang, Y. Onose, Y. Tokura, and N. P. Ong, 2008,

eprint arXiv:0811.3146.Lee, M., Y. Onose, Y. Tokura, and N. Ong, 2007, Phys. Rev.

B 75, 172403.Lee, P. A., N. Nagaosa, and X. G. Wen, 2006, Reviews of

Modern Physics 78(1), 17.Lee, P. A., and T. V. Ramakrishnan, 1985, Reviews of Modern

Physics 57(2), 287.Lee, W.-L., S. Watauchi, V. L. Miller, R. J. Cava, and N. P.

Ong, 2004, Science 303, 1647.Leroux-Hugon, P., and A. Ghazali, 1972, J. Phys. C: Solid

State Phys. 5, 1072.Liu, C.-X., X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, 2008,

Phys. Rev. Lett. 101, 468.Luttinger, J. M., 1958, Phys. Rev. 112, 739.Luttinger, J. M., and W. Kohn, 1955, Phys. Rev. 97, 869.Luttinger, J. M., and W. Kohn, 1958, Phys. Rev. 109, 1892.Lyanda-Geller, Y., S. Chun, M. Salamon, P. Goldbart,

P. Han, Y. Tomioka, A. Asamitsu, and Y. Tokura, 2001,Phys. Rev. B 63, 184426.

Lyo, S. K., 1973, Phys. Rev. B 8, 1185.MacDonald, A. H., and Q. Niu, 2004, Physics World 53, 18.Machida, Y., S. Nakatsuji, Y. Maeno, T. Tayama, and

T. Sakakibara, 2007a, J. Mag. and Mag. Mat. 310, 1079.Machida, Y., S. Nakatsuji, Y. Maeno, T. Tayama, T. Sakak-

ibara, and S. Onoda, 2007b, Phys. Rev. Lett. 98, 5720.MacKinnon, A., and B. Kramer, 1983, Z. Physik. B. 53, 1.Mahan, G. D., 1990, Many-Particle Physics (Plenum Press,

New York), pp. 671–686.Mainkar, N., D. A. Browne, and J. Callaway, 1996, Phys. Rev.

B. 53, 3692.Majumdar, A., and L. Berger, 1973, Phys. Rev. B 7, 4203.Manyala, N., Y. Sidis, J. F. Ditusa, G. Aeppli, D. P. Young,

52

and Z. Fisk, 2004, Nature Materials 3, 255.Maranzana, F. E., 1967, Phys. Rev. 160, 421.Marder, M. P., 2000, Condensed Matter Physics (Wiley, New

York), eprint Supplementary material by author.Marzari, N., and D. Vanderbilt, 1997, Phys. Rev. B 56, 12847.Mathieu, R., A. Asamitsu, K. Takahashi, H. Yamada,

M. Kawasaki, Z. Fang, N. Nagaosa, and Y. Tokura, 2004a,J. Mag. and Mag. Mat. 272, E785.

Mathieu, R., A. Asamitsu, H. Yamada, K. S. Takahashi,M. Kawasaki, Z. Fang, N. Nagaosa, and Y. Tokura, 2004b,Phys. Rev. Lett. 93, 1660.

Matl, P., N. P. Ong, Y. F. Yan, Y. Q. Li, D. Studebaker,T. Baum, and G. Doubinina, 1998, Phys. Rev. B 57, 10248.

Mitra, P., R. Mitra, A. F. Hebard, K. A. Muttalib, andP. Wolfle, 2007, Phys. Rev. Lett. 99, 046804.

Miyasato, T., N. Abe, T. Fujii, A. Asamitsu, S. Onoda,Y. Onose, N. Nagaosa, and Y. Tokura, 2007, Phys. Rev.Lett. 99, 086602.

Morosan, E., H. W. Zandbergen, L. Li, M. Lee, J. G. Check-elsky, M. Heinrich, T. Siegrist, N. P. Ong, and R. J. Cava,2007, Phys. Rev. B 75, 104401.

Moyal, J. E., 1949, Proc. Cambridge Philos. Soc. 45, 99.Murakami, S., N. Nagaosa, and S. C. Zhang, 2003, Science

301, 1348.Nagaosa, N., 2006, J. Phys. Soc. Jap. 75, 420.Nozieres, P., and C. Lewiner, 1973, J. Phys. (Paris) 34, 901.Nunner, T. S., N. A. Sinitsyn, M. F. Borunda, V. K. Dugaev,

A. A. Kovalev, A. Abanov, C. Timm, T. Jungwirth, J.-i.Inoue, A. H. MacDonald, and J. Sinova, 2007, Phys. Rev.B 76, 235312.

Nunner, T. S., G. Zarand, and F. von Oppen, 2008, Phys.Rev. Lett. 100, 36602.

Oda, K., S. Yoshii, Y. Yasui, M. Ito, T. Ido, Y. Ohno,Y. Kobayashi, and M. Sato, 2001, J. Phys. Soc. Jap. 70,2999.

Ohgushi, K., S. Murakami, and N. Nagaosa, 2000, Phys. Rev.B 62, 6065.

Ohno, H., H. Munekata, T. Penney, S. von Molmar, and L. L.Chang, 1992, Phys. Rev. Lett. 68, 2664.

Ong, N. P., and W.-L. Lee, 2006, in Foundations of quan-tum mechanics in the light of new technology (ISQM Tokyo2005), edited by S. Ishioka and K. Fujikawa (World Scien-tific), p. 121, eprint arXiv:cond-mat/0508236.

Onoda, M., and N. Nagaosa, 2002, J. Phys. Soc. Jpn. 71, 19.Onoda, M., and N. Nagaosa, 2003, Phys. Rev. Lett. 90, 0660.Onoda, S., and N. Nagaosa, 2003, Phys. Rev. Lett. 90,

196602.Onoda, S., N. Sugimoto, and N. Nagaosa, 2006a, Phys. Rev.

Lett. 97, 126602.Onoda, S., N. Sugimoto, and N. Nagaosa, 2006b, Prog. Theor.

Phys. 116, 61.Onoda, S., N. Sugimoto, and N. Nagaosa, 2008, Phys. Rev.

B 77, 165103.Onose, Y., and Y. Tokura, 2006, Phys. Rev. B 73, 7442.Parkin, S. S. P., and R. H. Friend, 1980, Philos. Mag. 41, 65.Pfleiderer, C., D. Reznik, L. Pintschovius, H. v. Lohneysen,

M. Garst, and A. Rosch, 2004, Nature 427, 227.Prange, R. E., and S. M. Girvin (eds.), 1987, The Quantum

Hall Effect (Springer, Berlin).Pu, Y., D. Chiba, F. Matsukura, H. Ohno, and J. Shi, 2008,

Phys. Rev. Lett. 101, 172.Pugh, E. M., 1930, Phys. Rev. 36, 1503.Pugh, E. M., and T. W. Lippert, 1932, Phys. Rev. 42, 709.Pugh, E. M., and N. Rostoker, 1953, Rev. Mod. Phys. 25,

151.Ramaneti, R., J. C. Lodder, and R. Jansen, 2007, Appl. Phys.

Lett. 91, 012502.Rammer, J., and H. Smith, 1986, Rev. Mod. Phys. 58, 323.Ruzmetov, D., J. Scherschligt, D. V. Baxter, T. Wojtow-

icz, X. Liu, Y. Sasaki, J. K. Furdyna, K. M. Yu, andW. Walukiewicz, 2004, Phys. Rev. B 69, 155207.

Samoilov, A. V., G. Beach, C. C. Fu, N. C. Yeh, and R. P.Vasquez, 1998, Phys. Rev. B 57(22), 14032.

Sangiao, S., L. Morellon, G. Simon, J. M. D. Teresa, J. A.Pardo, J. Arbiol, and M. R. Ibarra, 2009, Phys. Rev. B 79,014431.

Schad, R., P. Belien, G. Verbanck, V. V. Moshchalkov, andY. Bruynseraede, 1998, J. Phys.-Cond. Matt. 10, 6643.

Schliemann, J., and A. H. MacDonald, 2002, Phys. Rev. Lett.88, 137201.

Shen, S., X. Liu, Z. Ge, J. K. Furdyna, M. Dobrowolska, andJ. Jaroszynski, 2008, J. Appl. Phys. 103, 7D13.

Shiomi, Y., Y. Onose, and Y. Tokura, 2009, Phys. Rev. B 79,100404, and unpublished data.

Sinitsyn, N. A., 2008, J. Phys.: Condens. Matter 20, 023201.Sinitsyn, N. A., A. H. MacDonald, T. Jungwirth, V. K.

Dugaev, and J. Sinova, 2007, Phys. Rev. B 75, 045315.Sinitsyn, N. A., Q. Niu, and A. H. MacDonald, 2006, Phys.

Rev. B 73, 075318.Sinitsyn, N. A., Q. Niu, J. Sinova, and K. Nomura, 2005,

Phys. Rev. B 72, 045346.Sinova, J., D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth,

and A. H. MacDonald, 2004, Phys. Rev. Lett. 92, 126603.Sinova, J., and T. Jungwirth, 2005, in Frontiers in mag-

netic materials, edited by A. V. Narlikar (Springer-VerlagBerlin), p. 185.

Sinova, J., T. Jungwirth, and J. Cerne, 2004, Int. J. Mod.Phys. B 18, 1083.

Sinova, J., T. Jungwirth, J. Kucera, and A. H. MacDonald,2003, Phys. Rev. B 67, 352.

Sinova, J., T. Jungwirth, S. Yang, J. Kucera, and A. H. Mac-Donald, 2002, Phys. Rev. B 66, 412.

Smit, J., 1955, Physica 21, 877.Smit, J., 1958, Physica 24, 39.Smith, A. W., and R. W. Sears, 1929, Phys. Rev. 34, 1466.Smrcka, L., and P. Streda, 1977, J. Phys. C: Solid State Phys.

10, 2153.Sondhi, S. L., A. Karlhede, S. A. Kivelson, and E. H. Rezayi,

1993, Phys. Rev. B 47, 16419.Streda, P., 1982, J. Phys. C: Solid State Phys. 15, L717.Sugimoto, N., S. Onoda, and N. Nagaosa, 2007, Prog. Theor.

Phys. 117, 415.Sundaram, G., and Q. Niu, 1999, Phys. Rev. B 59, 14915.Taguchi, Y., Y. Oohara, H. Yoshizawa, N. Nagaosa, T. Sasaki,

S. Awaji, Y. Iwasa, T. Tayama, T. Sakakibara, S. Iguchi,K. Ohgushi, T. Ito, et al., 2004, J. Phys.-Cond. Matt. 16,S599.

Taguchi, Y., Y. Oohara, H. Yoshizawa, N. Nagaosa, andY. Tokura, 2001, Science 291, 2573.

Taguchi, Y., T. Sasaki, S. Awaji, Y. Iwasa, T. Tayama,T. Sakakibara, S. Iguchi, T. Ito, and Y. Tokura, 2003, Phys.Rev. Lett. 90, 257202.

Tatara, G., and H. Kawamura, 2002, J. Phys. Soc. Jpn. 71,2613.

Thouless, D. J., M. Kohmoto, M. P. Nightingale, and M. denNijs, 1982, Phys. Rev. Lett. 49, 405.

Tokura, Y., and T. Tomioka, 1999, J. Mag. and Mag. Mat.200(1-3), 1.

53

Ueno, K., T. Fukumura, H. Toyosaki, M. Nakano, andM. Kawasaki, 2007, Appl. Phys. Lett. 90, 072103.

Ueno, K., T. Fukumura, H. Toyosaki, M. Nakano, T. Ya-masaki, Y. Yamada, and M. Kawasaki, 2008, J. Appl. Phys.103, 7D1.

Venkateshvaran, D., W. Kaiser, A. Boger, M. Althammer,M. S. R. Rao, S. T. B. Goennenwein, M. Opel, andR. Gross, 2008, Phys. Rev. B 78, 092405.

Wang, X., D. Vanderbilt, J. R. Yates, and I. Souza, 2007,Phys. Rev. B 76, 195109.

Wang, X., J. R. Yates, I. Souza, and D. Vanderbilt, 2006,Phys. Rev. B 74, 195118.

Webster, W. L., 1925, Proc. Cambridge Phil. Soc. 23, 800.Woelfle, P., and K. A. Muttalib, 2006, Ann. Der Phys. 15,

508.Wunderlich, J., B. Kaestner, J. Sinova, and T. Jungwirth,

2005, Phys. Rev. Lett. 94, 472.Xiao, D., and Q. Niu, 2009, review in progress to be submitted

to Rev. Mod. Phys.Yanagihara, H., and M. B. Salamon, 2002, Phys. Rev. Lett.

89, 8720.

Yanagihara, H., and M. B. Salamon, 2007, J. Phys.-Cond.Matt. 19, 1520.

Yang, S. R. E., J. Sinova, T. Jungwirth, Y. P. Shim, and A. H.MacDonald, 2003, Phys. Rev. B 67, 045205.

Yao, Y., L. Kleinman, A. H. MacDonald, J. Sinova, T. Jung-wirth, D.-S. Wang, E. Wang, and Q. Niu, 2004, Phys. Rev.Lett. 92, 037204.

Yao, Y. G., Y. Liang, D. Xiao, Q. Niu, S. Q. Shen, X. Dai,and Z. Fang, 2007, Phys. Rev. B 75, 020401(R).

Yasui, Y., T. Kageyama, T. Moyoshi, M. Soda, M. Sato, andK. Kakurai, 2006, J. Phys. Soc. Jpn. 75, 084711.

Yasui, Y., T. Kageyama, T. Moyoshi, M. Soda, M. Sato, andK. Kakurai, 2007, J. Mag. and Mag. Mat. 310, E544.

Ye, J., Y. B. Kim, A. J. Millis, B. I. Shraiman, P. Majumdar,and Z. Tesanovic, 1999, Phys. Rev. Lett. 83, 3737.

Yoshii, S., S. Iikubo, T. Kageyama, K. Oda, Y. Kondo,K. Murata, and M. Sato, 2000, J. Phys. Soc. Jpn. 69, 3777.

Zeng, C., Y. Yao, Q. Niu, and H. H. Weitering, 2006, Phys.Rev. Lett. 96, 037204.

Ziman, J. M., 1967, Adv. in Phys. 16, 551.