Anomalous Hall effect

Anomalous Hall effect

Naoto Nagaosa

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

Jairo Sinova

Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USAand Institute of Physics ASCR, Cukrovarnická 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

Published 13 May 2010

The anomalous Hall effect AHE occurs in solids with broken time-reversal symmetry, typically in aferromagnetic phase, as a consequence of spin-orbit coupling. Experimental and theoretical studies ofthe AHE are reviewed, focusing on recent developments that have provided a more completeframework for understanding this subtle phenomenon and have, in many instances, replacedcontroversy by clarity. Synergy between experimental and theoretical works, both playing a crucialrole, has been at the heart of these advances. On the theoretical front, the adoption of the Berry-phaseconcepts has established a link between the AHE and the topological nature of the Hall currents. Onthe experimental front, new experimental studies of the AHE in transition metals, transition-metaloxides, spinels, pyrochlores, and metallic dilute magnetic semiconductors have established systematictrends. These two developments, in concert with first-principles electronic structure calculations,strongly favor the dominance of an intrinsic Berry-phase-related AHE mechanism in metallicferromagnets with moderate conductivity. The intrinsic AHE can be expressed in terms of theBerry-phase curvatures and it is therefore an intrinsic quantum-mechanical property of a perfectcrystal. An extrinsic mechanism, skew scattering from disorder, tends to dominate the AHE in highlyconductive ferromagnets. The full modern semiclassical treatment of the AHE is reviewed whichincorporates an anomalous contribution to wave-packet group velocity due to momentum-space Berrycurvatures and correctly combines the roles of intrinsic and extrinsic skew-scattering and side-jumpscattering-related mechanisms. In addition, more rigorous quantum-mechanical treatments based onthe Kubo and Keldysh formalisms are reviewed, taking into account multiband effects, anddemonstrate the equivalence of all three linear response theories in the metallic regime. Building onresults from recent experiment and theory, a tentative global view of the AHE is proposed whichsummarizes the roles played by intrinsic and extrinsic contributions in the disorder strength versustemperature plane. Finally outstanding issues and avenues for future investigation are discussed.

DOI: 10.1103/RevModPhys.82.1539 PACS numbers: 72.15.Eb, 72.20.Dp, 72.20.My, 72.25.b

CONTENTS

I. Introduction 1540

A. A brief history of the anomalous Hall effect and

new perspectives 1540

B. Parsing the AHE 1543

1. Intrinsic contribution to xyAH 1544

2. Skew-scattering contribution to xyAH 1545

3. Side-jump contribution to xyAH 1545

II. Experimental and Theoretical Studies on Specific

Materials 1546

A. Transition metals 1547

1. Early experiments 1547

2. Recent experiments 1547

a. High conductivity regime 1547

b. Good-metal regime 1548

c. Bad-metal–hopping regime 1548

3. Comparison to theories 1550

B. Complex oxide ferromagnets 1551

1. First-principles calculations and experiments

on SrRuO3 1551

2. Spin-chirality mechanism of the AHE in

manganites 1552

3. Lanthanum cobaltite 1554

4. Spin-chirality mechanism in pyrochlore

ferromagnets 1555

REVIEWS OF MODERN PHYSICS, VOLUME 82, APRIL–JUNE 2010

0034-6861/2010/822/153954 ©2010 The American Physical Society1539

http://dx.doi.org/10.1103/RevModPhys.82.1539

5. Anatase and rutile Ti1−xCoxO2− 1556

C. Ferromagnetic semiconductors 1556

D. Other classes of materials 1559

1. Spinel CuCr2Se4 1559

2. Heusler alloy 1559

3. Fe1−yCoySi 1560

4. MnSi 1560

5. Mn5Ge3 1561

6. Layered dichalcogenides 1562

7. Spin glasses 1562

E. Localization and the AHE 1564

III. Theory of the AHE: Qualitative Considerations 1565

A. Symmetry considerations and analogies between

normal Hall effect and AHE 1565

B. Topological interpretation of the intrinsic

mechanism: Relation between Fermi sea and Fermi

surface properties 1569

IV. Early Theoretical Studies of the AHE 1570

A. Karplus-Luttinger theory and the intrinsic

mechanism 1570

B. Extrinsic mechanisms 1571

1. Skew scattering 1571

2. Kondo theory 1572

3. Resonant skew scattering 1572

4. Side jump 1573

C. Kohn-Luttinger theoretical formalism 1574

1. Luttinger theory 1574

2. Adams-Blount formalism 1575

3. Nozieres-Lewiner theory 1575

V. Linear Transport Theories of the AHE 1576

A. Semiclassical Boltzmann approach 1576

1. Equation of motion of Bloch state wave

packets 1577

2. Scattering and the side jump 1577

3. Kinetic equation for the semiclasscial

Boltzmann distribution 1579

4. Anomalous velocities, anomalous Hall

currents, and anomalous Hall mechanisms 1579

B. Kubo formalism 1580

1. Kubo technique for the AHE 1580

2. Relation between the Kubo and the

semiclassical formalisms 1582

C. Keldysh formalism 1582

D. Two-dimensional ferromagnetic Rashba model—A

minimal model 1584

1. Minimal model 1586

VI. Conclusions: Future Problems and Perspectives on

AHE 1586

A. Recent developments 1586

1. Intrinsic AHE 1587

2. Fully consistent metallic linear response

theories of the AHE 1587

3. Emergence of three empirical AHE regimes 1587

B. Future challenges and perspectives 1588

Acknowledgments 1589

References 1589

I. INTRODUCTION

A. A brief history of the anomalous Hall effect and newperspectives

The anomalous Hall effect has deep roots in the his-tory of electricity and magnetism. In 1879 Edwin H. HallHall, 1879 made the momentous discovery that when acurrent-carrying conductor is placed in a magnetic field,the Lorentz force “presses” its electrons against one sideof the conductor. Later, he reported that his “pressingelectricity” effect was ten times larger in ferromagneticiron Hall, 1881 than in nonmagnetic conductors. Bothdiscoveries were remarkable, given how little wasknown at the time about how charge moves throughconductors. The first discovery provided a simple el-egant tool to measure carrier concentration in nonmag-netic conductors and played a midwife’s role in easingthe birth of semiconductor physics and solid-state elec-tronics in the late 1940s. For this role, the Hall effect wasfrequently called the queen of solid-state transport ex-periments.

The stronger effect that Hall discovered in ferromag-netic conductors came to be known as the anomalousHall effect AHE. The AHE has been an enigmaticproblem that has resisted theoretical and experimentalassaults for almost a century. The main reason seems tobe that, at its core, the AHE problem involves conceptsbased on topology and geometry that have been formu-lated only in recent times. The early investigatorsgrappled with notions that would not become clear andwell defined until much later, such as the concept of theBerry phase Berry, 1984. What is now viewed as theBerry-phase curvature, earlier dubbed “anomalous ve-locity” by Luttinger, arose naturally in the first micro-scopic theory of the AHE by Karplus and Luttinger1954. However, because understanding of these con-cepts, not to mention the odd intrinsic dissipationlessHall current they seemed to imply, would not beachieved for another 40 years, the AHE problem wasquickly mired in a controversy of unusual endurance.Moreover, the AHE seems to be a rare example of apure charge-transport problem whose elucidation hasnot to date benefited from the application of comple-mentary spectroscopic and thermodynamic probes.

Very early on, experimental investigators learned thatthe dependence of the Hall resistivity xy on applied per-pendicular field Hz is qualitatively different in ferromag-netic and nonmagnetic conductors. In the latter, xy in-creases linearly with Hz, as expected from the Lorentzforce. In ferromagnets, however, xy initially increasessteeply in weak Hz but saturates at a large value that isnearly Hz independent Fig. 1. Kundt noted that, in Fe,Co, and Ni, the saturation value is roughly proportionalto the magnetization Mz Kundt, 1893 and has a weakanisotropy when the field z direction is rotated withrespect to the crystal, corresponding to the weak mag-netic anisotropy of Fe, Co, and Ni Webster, 1925.Shortly thereafter, experiments of Pugh 1930 and Pugh

1540 Nagaosa et al.: Anomalous Hall effect

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

and Lippert 1932 established that an empirical relationbetween xy, Hz, and Mz,

xy = R0Hz + RsMz, 1.1

applies to many materials over a broad range of externalmagnetic fields. The second term represents the Hall-effect contribution due to the spontaneous magnetiza-tion. This AHE is the subject of this paper. Unlike R0,which was already understood to depend mainly on thedensity of carriers, Rs was found to depend subtly on avariety of material specific parameters and, in particular,on the longitudinal resistivity xx=.

In 1954, Karplus and Luttinger KL Karplus andLuttinger, 1954 proposed a theory for the AHE that, inhindsight, 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 thereforecould contribute to the Hall effects. In the case of ferro-magnetic conductors, the sum of the anomalous velocityover all occupied band states can be nonzero, implying acontribution to the Hall conductivity xy. Because thiscontribution depends only on the band structure and islargely independent of scattering, it has recently beenreferred to as the intrinsic contribution to the AHE.When the conductivity tensor is inverted, the intrinsicAHE yields a contribution to xyxy /xx

2 and thereforeit is proportional to 2. The anomalous velocity is depen-dent only on the perfect crystal Hamiltonian and can berelated to changes in the phase of Bloch state wavepackets when an electric field causes them to evolve incrystal momentum space Chang and Niu, 1996;Sundaram and Niu, 1999; Bohm et al., 2003; Xiao andNiu, 2009. As mentioned, the KL theory anticipated byseveral decades the modern interest in the Berry phaseand the Berry curvature review here effects, particularlyin momentum space.

Early experiments to measure the relationship be-tween xy and generally assumed to be of the power-law form, i.e., xy, mostly involved plotting xy orRs vs , measured in a single sample over a broad inter-val of T typically 77–300 K. As we explain below, com-

peting theories in metals suggested that either =1 or 2.A compiled set of results was published by Kooi 1954;see Fig. 2. The subsequent consensus was that such plotsdo not settle the debate. At finite T, the carriers arestrongly scattered by phonons and spin waves. These in-elastic 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 from 1to values approaching 2. This was also found by otherinvestigators. A lengthy calculation by Lyo 1973showed that skew scattering at TD the Debye tem-perature leads to xy2+a, with a as a constant. Inan early theory by Kondo considering skew scatteringfrom spin excitations Kondo, 1962, it may be seen thatxy also varies as 2 at finite T.

The proper test of the scaling relation in comparisonwith present theories involves measuring xy and 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 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 semiclassicalAHE theories by Smit and Berger focused instead on

FIG. 1. The Hall effect in Ni data from Smith, 1910. FromPugh and Rostoker, 1953.

FIG. 2. Extraordinary Hall constant as a function of resistivity.The shown fit has the relation Rs1.9. From Kooi, 1954.

1541Nagaosa et al.: Anomalous Hall effect


the influence of disorder scattering in imperfect crystals.Smit argued that the main source of the AHE currentswas asymmetric skew scattering from impurities causedby the spin-orbit interaction SOI Smit, 1955, 1958.This AHE picture predicted that Rsxx =1. Berger,on the other hand, argued that the main source of theAHE current was the side jump experienced by quasi-particles upon scattering from spin-orbit coupled impu-rities. The side-jump mechanism could confusingly beviewed as a consequence of a KL anomalous velocitymechanism acting while a quasiparticle was under theinfluence of the electric field due to an impurity. Theside-jump AHE current was viewed as the product ofthe side jump per scattering event and the scattering rateBerger, 1970. One puzzling aspect of this semiclassicaltheory was that all dependence on the impurity densityand strength seemingly dropped out. As a result, it pre-dicted Rsxx

2 with an exponent identical to that ofthe KL mechanism. The side-jump mechanism thereforeyielded a contribution to the Hall conductivity whichwas seemingly independent of the density or strength ofscatterers. In the decade 1970–1980, a lively AHE de-bate was waged largely between the proponents of thesetwo extrinsic theories. The three main mechanisms con-sidered in this early history are shown schematically inFig. 3.

Some of the confusion in experimental studiesstemmed from a hazy distinction between the KLmechanism and the side-jump mechanism, a poor under-standing of how the effects competed at a microscopiclevel, and a lack of systematic experimental studies in adiverse set of materials.

One aspect of the confusion may be illustrated by con-trasting the case of a high-purity monodomain ferromag-net, which produces a spontaneous AHE current pro-portional to Mz, with the case of a material containingmagnetic impurities e.g., Mn embedded in a nonmag-netic host such as Cu the dilute Kondo system. In afield H, the latter also displays an AHE current propor-

tional to the induced M=H, with as the susceptibilityFert and Jaoul, 1972. However, in zero H, time-reversal invariance TRI is spontaneously broken in theformer but not in the latter. Throughout the period1960–1989, the two Hall effects were often regarded as acommon phenomenon that should be understood micro-scopically on the same terms. It now seems clear thatthis view impeded progress.

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

Since the 1980s, the quantum Hall effect in two-dimensional 2D electron systems in semiconductorheterostructures has become a major field of research inphysics Prange and Girvin, 1987. The accurate quanti-zation of the Hall conductance is the hallmark of thisphenomenon. Both the integer Thouless et al., 1982and fractional quantum Hall effects can be explained interms of the topological properties of the electronicwave functions. For the case of electrons in a two-dimensional crystal, it has been found that the Hall con-ductance is connected to the topological integer Chernnumber defined for the Bloch wave function over thefirst-Brillouin zone Thouless et al., 1982. This way ofthinking about the quantum Hall effect began to have adeep impact on the AHE problem starting around 1998.Theoretical interest in the Berry phase and in its relationto transport phenomena, coupled with many develop-ments in the growth of novel complex magnetic systemswith strong spin-orbit coupling notably the manganites,pyrochlores, and spinels, led to a strong resurgence ofinterest in the AHE and eventually to deeper under-standing.

Since 2003 many systematic studies, both theoreticaland experimental, have led to a better understanding ofthe AHE in the metallic regime and to the recognitionof new unexplored regimes that present challenges tofuture researchers. As it is often the case in condensedmatter physics, attempts to understand this complex andfascinating phenomenon have motivated researchers tocouple fundamental and sophisticated mathematicalconcepts to real-world material issues. The aim of thisreview is to survey recent experimental progress in thefield and to present the theories in a systematic fashion.Researchers are now able to understand the links be-tween different views on the AHE previously thought tobe in conflict. Despite the progress in recent years, un-derstanding is still incomplete. We highlight some in-triguing questions that remain and speculate on the mostpromising avenues for future exploration. In this paper,we focus, in particular, on reports that have contributedsignificantly to the modern view of the AHE. For previ-ous reviews, see Pugh and Rostoker 1953 and Hurd1972. For more recent short overviews focused on thetopological aspects of the AHE, see Sinova, Jungwirth,and Cerne 2004 and Nagaosa 2006. A review of the

a) Intrinsic deflectionInterband coherence induced by ant l l t i fi ld i i t Eexternal electric field gives rise to a

velocity contribution perpendicular tothe field direction. These currents donot sum to zero in ferromagnets.

Electrons have an anomalous velocity perpendicular tothe electric field related to their Berry’s phase curvaturenbEe

kE

dtrd

b) Side jump

The electron velocity is deflected in opposite directions by the oppositeelectric 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 couplingof the electron or the impurity.

FIG. 3. Color online Illustration of the three main mecha-nisms that can give rise to an AHE. In any real material all ofthese mechanisms act to influence electron motion.



modern semiclassical treatment of AHE was recentlywritten by Sinitsyn 2008. The present paper has beeninformed by ideas explained in the earlier works. Read-ers who are not familiar with the Berry-phase conceptsmay find it useful to consult the elementary review byOng and Lee 2006 and the popular commentary byMacDonald and Niu 2004.

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

1 When xyAH is independent of xx, the AHE can often

be understood in terms of the geometric concepts ofthe Berry phase and Berry curvature in momentumspace. This AHE mechanism is responsible for theintrinsic AHE. In this regime, the anomalous Hallcurrent can be thought of as the unquantized ver-sion of the quantum Hall effect. In 2D systems theintrinsic AHE is quantized in units of e2 /h at tem-perature T=0 when the Fermi level lies between theBloch state bands.

2 Three broad regimes have been identified when sur-veying a large body of experimental data for diversematerials: i a high conductivity regime xx106 cm−1 in which a linear contribution toxy

AHxx due to skew scattering dominates xyAH in

this regime the normal Hall conductivity contribu-tion can be significant and even dominate xy; iian intrinsic or scattering-independent regime inwhich xy

AH is roughly independent of xx104 cm−1xx106 cm−1; and iii a bad-metal regime xx104 cm−1 in which xy

AH de-creases with decreasing xx at a rate faster than lin-ear.

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 DMSs. In these systems asystematic nontrivial comparison between the ob-served properties of systems with well controlledmaterial 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 Boltzmannequation taking into account the Berry curvatureand coherent interband mixing effects due to bandstructure and disorder has been formulated. Thistheory provides a clearer physical picture of theAHE than early theories by identifying correctly allthe semiclassically defined mechanisms. This gener-alized semiclassical picture has been verified bycomparison with controlled microscopic linear re-sponse treatments for identical models.

6 The relevance of noncoplanar spin structures withassociated spin chirality and real-space Berry curva-

ture to the AHE has been established both theoreti-cally 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 review asfollows. In the remainder of this section, we provide theminimal theoretical background necessary to understandthe different AHE mechanisms. In particular we explainthe scattering-independent Berry-phase mechanismwhich is more important for the AHE than for any othercommonly measured transport coefficient. In Sec. II wereview recent experimental results on a broad range ofmaterials and compare them with relevant calculationswhere available. In Sec. III we discuss some qualitativeaspects of the AHE theory relating to its topologicalnature and in Sec. IV we discuss AHE theory from ahistorical perspective, explaining links between differentideas which are not always recognized and discussing thephysics behind some of the past confusion. Section Vdiscusses the present understanding of the metallictheory based on a careful comparison between theBoltzmann, Kubo, and Keldysh linear response theorieswhich are now finally consistent. In Sec. VI we present asummary and outlook.

B. Parsing the AHE

The anomalous Hall effect is at its core a quantumphenomenon which originates from quantum coherentband mixing effects by both the external electric fieldand the disorder potential. Like other coherent interfer-ence transport phenomena e.g., weak localization, itcannot be satisfactorily explained using traditional semi-classical Boltzmann transport theory. Therefore, whenparsing the different contributions to the AHE, they canbe defined semiclassically only in a carefully elaboratedtheory.

In this section we identify three distinct contributionswhich sum up to yield the full AHE: intrinsic, skew-scattering, and side-jump contributions. We choose thisnomenclature to reflect the modern literature withoutbreaking completely from the established AHE lexiconsee Sec. IV. However, unlike previous classifications,we base this parsing of the AHE on experimental andmicroscopic transport theoretical considerations ratherthan on the identification of one particular effect whichcould contribute to the AHE. The link to semiclassicallydefined processes is established after developing a fullygeneralized Boltzmann transport theory which takes in-terband coherence effects into account and is fullyequivalent to microscopic theories Sec. V.A. In fact,much of the theoretical effort of the past few years hasbeen expended in understanding this link between semi-classical and microscopic theories which has escaped co-hesion for a long time.



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

AH, which vary as 1 and as 0. In experi-ment a similar separation can sometimes be achieved byplotting xy vs the longitudinal conductivity xx when is varied by altering disorder or varying temperature.More commonly and equivalently the Hall resistivity isseparated into contributions proportional to xx and xx

2 .This partitioning seemingly gives only two contribu-

tions to xyAH, one and the other 0. We define the

first contribution as the skew-scattering contribution,xy

AH-skew. Note that in this parsing of AHE contributionsit is the dependence on or xx which defines it andnot a particular mechanism linked to a microscopic orsemiclassical theory. We further separate the secondcontribution proportional to 0 or independent of xxinto two parts: intrinsic and side jump. Although thesetwo contributions cannot be separated experimentallyby dc measurements, they can be separated experimen-tally as well as theoretically by defining the intrinsiccontribution xy

AH-int as the extrapolation of the ac-interband Hall conductivity to zero frequency in thelimit of →, with 1/→0 faster than →0. This thenleaves a unique definition for the third and last contri-bution, termed side jump, as xy

AH-sjxyAH−xy

AH-skew

−xyAH-int.

We examine these three contributions below still atan introductory level. It is important to note that theabove definitions have not relied on identifications ofsemiclassical processes such as side-jump scatteringBerger, 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. Vthat other contributions are present in the fully general-ized semiclassical theory which were not precisely iden-tified previously and which are necessary to be fully con-sistent with microscopic theories.

The ideas explained in this section are substantiatedin Sec. II by analyses of tendencies in the AHE data ofseveral different material classes and in Secs. III–V byan extensive technical discussion of AHE theory. We as-sume throughout that the ferromagnetic materials of in-terest are accurately described by a Stoner-like mean-field band theory. In applications to real materials weimagine that the band theory is based on spin-density-functional theory Jones and Gunnarsson, 1989 with alocal-spin density or similar approximation for theexchange-correlation energy functional.

1. Intrinsic contribution to xyAH

Among the three contributions, the easiest to evaluateaccurately is the intrinsic contribution. We have definedthe intrinsic contribution microscopically as the dc limitof the interband conductivity, a quantity which is notzero in ferromagnets when SOI is included. There is,however, a direct link to semiclassical theory in whichthe induced interband coherence is captured by amomentum-space Berry-phase related contribution tothe anomalous velocity. We show this equivalence below.

This contribution to the AHE was first derived by KLKarplus and Luttinger, 1954 but its topological naturewas not fully appreciated until recently Jungwirth, Niu,and MacDonald, 2002; Onoda and Nagaosa, 2002. Thework of Jungwirth, Niu, and MacDonald 2002 was mo-tivated by the experimental importance of the AHE inferromagnetic semiconductors and also by the thoroughearlier analysis of the relationship between momentum-space Berry phases and anomalous velocities in semi-classical transport theory by Chang and Niu 1996 andSundaram and Niu 1999. The frequency-dependent in-terband Hall conductivity, which reduces to the intrinsicanomalous Hall conductivity in the dc limit, had beenevaluated earlier for a number of materials by Mainkaret al. 1996 and Guo and Ebert 1995 but the topologi-cal connection was not recognized.

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 from thesimple Kubo formula for the Hall conductivity for anideal lattice, given the eigenstates n ,k and eigenvaluesnk of a Bloch Hamiltonian H,

ijAH-int = e2

nn dk

2d f„nk… − f„nk…

Imn,kvikn,kn,kvjkn,k

nk − nk2 . 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

vk =1

ir,Hk =

1

kHk . 1.3

Note the restriction nn in Eq. 1.2.What makes this contribution quite unique is that, like

the quantum Hall effect in a crystal, it is directly linkedto the topological properties of the Bloch states see Sec.III.B. Specifically it is proportional to the integrationover the Fermi sea of the Berry curvature of each occu-pied band or equivalently Haldane, 2004; Wang et al.,2007 to the integral of the Berry phases over cuts of theFermi-surface FS segments. This result can be derivedby noting that

n,kkn,k =n,kkHkn,knk − nk

. 1.4

Using this expression, Eq. 1.2 reduces to



ijAH-int = − ij

e2

n dk

2df„nk…bnk , 1.5

where ij is the antisymmetric tensor, ank is the Berry-phase connection, ank= in ,kkn ,k, and bnk is theBerry-phase curvature,

bnk = k ank , 1.6

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

conductivity can be obtained from the semiclassicaltheory of wave-packet dynamics Chang and Niu, 1996;Sundaram and Niu, 1999; Marder, 2000. It can be shownthat the wave-packet group velocity has an additionalcontribution in the presence of an electric field: rc=Enk /k− E /bnk see Sec. V.A. The intrinsicHall conductivity formula, Eq. 1.5, is obtained simplyby summing the second anomalous, term over all occu-pied states.

One of the motivations for identifying the intrinsiccontribution xy

AH-int is that it can be evaluated accuratelyeven for relatively complex materials using first-principles electronic structure theoretical techniques. Inmany materials which have strongly spin-orbit coupledbands, the intrinsic contribution seems to dominate theAHE.

2. Skew-scattering contribution to xyAH

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 is the only contribution to the AHE which appearswithin the confines of traditional Boltzmann transporttheory in which interband coherence effects are com-pletely neglected. Skew scattering is due to chiral fea-tures which appear in the disorder scattering of spin-orbit coupled ferromagnets. This mechanism was firstidentified by Smit 1955, 1958.

Treatments of semiclassical Boltzmann transporttheory found in textbooks often appeal to the principleof detailed balance which states that the transition prob-ability Wn→m from n to m is identical to the transitionprobability in the opposite direction Wm→n. Althoughthese two transition probabilities are identical in a Fer-mi’s golden-rule approximation, since Wn→n= 2 /nVn2En−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 direc-tion has a different transition probability than the corre-sponding left-handed transition. When the transitionrates are evaluated perturbatively, asymmetric chiralcontributions appear first at third order see Sec. V.A.In simple models the asymmetric chiral contribution tothe transition probability is often assumed to have theform see Sec. IV.B.1,

WkkA = − A

−1k k · Ms. 1.7

When this asymmetry is inserted into the Boltzmannequation it leads to a current proportional to the longi-tudinal current driven by E and perpendicular to both Eand Ms. When this mechanism dominates, both the Hallconductivity H and the conductivity are approxi-mately proportional to the transport lifetime and theHall resistivity H

skew=Hskew2 is therefore proportional to

the longitudinal resistivity .There are several specific mechanisms for skew scat-

tering see Secs. IV.B and V.A. Evaluation of the skew-scattering contribution to the Hall conductivity or resis-tivity requires simply that the conventional linearizedBoltzmann equation be solved using a collision termwith accurate transition probabilities since these will ge-nerically include a chiral contribution. In practice ourability to accurately estimate the skew-scattering contri-bution to the AHE of a real material is limited only bytypically imperfect characterization of its disorder.

We emphasize that skew-scattering contributions toH are present not only because of spin-orbit coupling inthe disorder Hamiltonian but also because of spin-orbitcoupling in the perfect crystal Hamiltonian combinedwith purely scalar disorder. Either source of skew scat-tering could dominate xy

AH-skew depending on the hostmaterial and also on the type of impurities.

We end this section with a small note directed to thereader who is more versed in the latest development ofthe full semiclassical theory of the AHE and in its com-parison to the microscopic theory see Secs. V.A andV.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 oforder 0. There exists a contribution from this asymme-try which is actually present in the microscopic theoret-ical treatment associated with the so-called ladder dia-gram corrections to the conductivity and therefore oforder 0. In our experimentally practical parsing of AHEcontributions we do not associate this contribution withskew scattering but place it under the umbrella of side-jump scattering even though it does not physically origi-nate from any side-step type of scattering.

3. Side-jump contribution to xyAH

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

xyAH = xy

AH-int + xyAH-skew + xy

AH-sj 1.8

defines the side-jump contribution as the difference be-tween the full Hall conductivity and the two-simplercontributions. In using the term side jump for the re-maining contribution, we are appealing to the histori-cally established taxonomy outlined in Sec. I.B.2. Estab-lishing this connection mathematically has been the



most controversial aspect of AHE theory and the onewhich has taken the longest to clarify from a theoreticalpoint of view. Although this classification of Hall con-ductivity contributions is often useful see below, it isnot generically true that the only correction to the in-trinsic and skew contributions can be physically identi-fied with the side-jump process defined as in the earlierstudies of the AHE Berger, 1964.

The basic semiclassical argument for a side-jumpcontribution can be stated straightforwardly: whenconsidering the scattering of a Gaussian wave packetfrom a spherical impurity with SOI HSO= 1/2m2c2r−1V /rSzLz, a wave packet with incidentwave vector k will suffer a displacement transverse to kequal to 1

6k2 /m2c2. This type of contribution was firstnoticed, but discarded, by Smit 1958 and reintroducedby Berger 1964 who argued that it was the key contri-bution to the AHE. This kind of mechanism clearly liesoutside the bounds of traditional Boltzmann transporttheory in which only the probabilities of transitions be-tween Bloch states appear and not microscopic details ofthe scattering processes. This contribution to the con-ductivity ends up being independent of and thereforecontributes to the AHE at the same order as the intrin-sic contribution in an expansion in powers of scatteringrate. The separation between intrinsic and side-jumpcontributions, which cannot be distinguished by their de-pendence on , has been perhaps the most argued aspectof AHE theory see Sec. IV.B.4.

As explained in a recent review by Sinitsyn 2008,side-jump and intrinsic contributions have quite differ-ent dependences on more specific system parameters,particularly in systems with complex band structures.Some of the initial controversy which surrounded side-jump theories was associated with physical meaning as-cribed to quantities which were plainly gauge depen-dent, like the Berry connection which in early theories istypically identified as the definition of the side step uponscattering. Studies of simple models, for example, mod-els of semiconductor conduction bands, also gave resultsin which the side-jump contribution seemed to be thesame size but opposite in sign compared to the intrinsiccontribution Nozieres and Lewiner, 1973. We now un-derstand Sinitsyn et al., 2007 that these cancellationsare unlikely except in models with a very simple bandstructure, e.g., one with a constant Berry curvature. It isonly through comparison between fully microscopic lin-ear response theoretical calculations, based on equiva-lently valid microscopic formalisms such as the Keldyshnonequilibrium Green’s function or Kubo formalisms,and the systematically developed semiclassical theorythat the specific contribution due to the side-jumpmechanism can be separately identified with confidencesee Sec. V.A.

Having said this, all the calculations comparing theintrinsic and side-jump contributions to the AHE from amicroscopic point of view have been performed for verysimple models not immediately linked to real materials.A practical approach, which is followed at present formaterials in which AH seems to be independent of xx,

is to first calculate the intrinsic contribution to the AHE.If this explains the observation and it appears that itusually does, then it is deemed that the intrinsic mecha-nism dominates. If not, we can take some comfort fromunderstanding on the basis of simple model results thatthere can be other contributions to AH which are alsoindependent of xx and can for the most part be identi-fied with the side-jump mechanism. Unfortunately itseems extremely challenging, if not impossible, to de-velop a predictive theory for these contributions partlybecause they require many higher order terms in theperturbation theory that must be summed but more fun-damentally because they depend sensitively on details ofthe disorder in a particular material which are normallyunknown.

II. EXPERIMENTAL AND THEORETICAL STUDIES ONSPECIFIC MATERIALS

In this section, we review the studies on the AHE ineach specific material, focusing primarily on the experi-mental aspects. The interest here is how the intrinsic andextrinsic contributions described in Sec. I appear in realmaterials. In some cases, the dominance of thescattering-independent contribution, dominated by theintrinsic Berry phase of the material, is reasonably clearby comparison between the experiments and theory, andin some cases the extrinsic skew scattering is identifiedby xy observed in the highly metallic region. How-ever, the situation is not clear yet for many of the mate-rials, and therefore we have tried to provide an unbiaseddescription of the information available at the momentin those cases. The theoretical aspects in this section arefocused on showing the comparison between the theo-retical results and the experimental data rather than go-ing into full detail on the particular theoretical tech-niques which can be obtained directly from thereferences.

The AHE has been studied most intensively in transi-tion metals, and most of the early theories were aimed atunderstanding these materials. Although in early experi-ments there were data supporting both the intrinsic andextrinsic contributions, i.e., xy

2 and xy , respec-tively, recent systematic studies on these systems haverevealed that there occurs a crossover from the extrinsicskew-scattering region to the intrinsic region as the re-sistivity increases. Comparison with first-principles cal-culations indicates that an appreciable part of thescattering-independent AHE contribution comes fromthe Berry-phase curvature in the momentum space,termed as intrinsic contribution in Sec. I.B.3

The transition-metal oxides are the representativestrongly correlated systems, and some of them are me-tallic ferromagnets. For SrRuO3, where the t2g electronsdetermine the conduction properties, detailed compari-son between the experiments and theory seems to indi-cate that the band crossings near the Fermi energy andthe associated enhanced Berry-phase curvature seem tobe the origin of the nonmonotonic temperature depen-dence of the AHE in this material. Manganites are the



typical double-exchange system, where the localizedspins of t2g electrons are coupled to the conducting egelectrons. In these systems, the focus has been the non-coplanar spin configuration with the scalar spin chiralitySi ·SjSk. This spin chirality is produced by the thermalfluctuation of the spins and acts as a fictitious magneticfield in the real space for the conduction electrons. Com-bined with the relativistic spin-orbit interaction, thethermal average of the spin chirality is biased antipar-allel to the spontaneous magnetization, leading to theAHE. In the spin-glass system, the frustration amongthe exchange interactions leads to the noncollinear andnoncoplanar spin configurations providing also thepromising arena for the spin-chirality mechanism ofAHE. Actually, the AHE in metallic spin-glass systemsAuFe, AuMn, and CuMn has been discussed from thisviewpoint. The spiral magnet due to the Dzyaloshinsky-Moriya interaction, e.g., MnSi, is another system wherethe spin chirality is expected to play some role in theAHE. This is because the noncollinear spin configura-tion is already prepared by the Dzyaloshinsky-Moriyainteraction and the noncoplanar configuration is rathereasy to be realized. However, the detailed theoreticalanalysis in this system is still missing.

The spin chirality also exists in the ground state peri-odic spin configuration for the pyrochlore ferromagnets.In this case, the strong single spin anisotropy of rare-earth ions influences the conduction electrons leading tothe periodic distribution of the spin chirality. Then, theBloch wave function is well defined in this case, and theBerry curvature in the momentum space appears to leadto the intrinsic AHE.

Another interesting materials which can serve as abenchmark for the understanding of AHE in metallicferromagnets are DMSs. These materials have relativelyuniform magnetization and have a simple but nontrivialband structure of the carriers. Detailed theoreticalanalysis based on the k ·p method is available, which canbe directly compared with the experiment. In these sys-tems the intrinsic Berry-phase contribution seems to bethe dominant one in the relatively high conductiveDMS.

Note that the spin chirality and Berry curvature pic-tures are real- and momentum-space counterparts. Forthe periodic spin configuration, one can calculate the lat-ter from the former for each example. However, the re-lationship between the two is not yet explored well es-pecially in the temporally and/or spatially nonperiodiccases.

A. Transition metals

1. Early experiments

Four decades after the discovery of the AHE, an em-pirical relation between the magnetization and the Hallresistivity was proposed independently by Smith andPugh Smith and Sears, 1929; Pugh, 1930; Pugh and Lip-pert, 1932; see Sec. I.A. Pugh investigated the AHE inFe, Ni, and Co and the alloys Co-Ni and Ni-Cu in mag-

netic fields up to 17 kG over large intervals in T10–800 K in the case of Ni and found that the Hallresistivity H is comprised of two terms, viz.,

H = R0H + R1MT,H , 2.1

where MT ,H is the magnetization averaged over thesample. Pugh defined R0 and R1 as the ordinary andextraordinary Hall coefficients, respectively. The latterR1=Rs is now called the anomalous Hall coefficient asin Eq. 1.1.

On dividing Eq. 2.1 by 2, we see that it just ex-presses the additivity of the Hall currents: the total Hallconductivity xy

tot equals xyNH+xy

AH, where xyNH is the or-

dinary Hall conductivity and xyAH is the AHE conductiv-

ity. A second implication of Eq. 2.1 emerges when weconsider the role of domains. The anomalous Hall coef-ficient in Eq. 2.1 is proportional to the AHE in a singledomain. As H→0, proliferation of domains rapidly re-duces MT ,H to zero we ignore pinning. Cancella-tions of xy

AH between domains result in a zero net Hallcurrent. Hence the observed AHE term mimics the fieldprofile of MT ,H, as implied by Pugh’s term R1M. Therole of H is simply to align the AHE currents by rotatingthe domains into alignment. The Lorentz-force termxy

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

The most interesting implication of Eq. 2.1 is that, inthe absence of H, a single domain engenders a sponta-neous Hall current transverse to both M and E. Under-standing the origin of this spontaneous off-diagonal cur-rent has been a fundamental problem of chargetransport in solids for the past 60 years. The AHE is alsocalled the spontaneous Hall effect and the extraordinaryHall effect in the older literature.

2. Recent experiments

The resurgence of interest in the AHE motivated bythe Berry-phase approach Sec. I.B 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 xx andcharacterized by the dependence of xy

AH on xx. Thethree regimes are i a high conductivity regime for xx

106 cm−1 in which xyAH-skewxx

1 dominates xyAH

and conductivity contribution dominates xy; ii a good-metal regime for xx

AH104–106 cm−1 in which xy

xx0 ; and iii a bad-metal–hopping regime for xx

104 cm−1 in which xyAHxx

1.6–1.8.We discuss each of these regimes below.

a. High conductivity regime

The Hall conductivity in the high-purity regime, xx0.5106 cm−1, is dominated by the skew-scattering contribution xy

skew. The high-purity regime isone of the least studied experimentally. This regime is



challenging to investigate experimentally because thefield H required for saturating M also yields a very largeordinary Hall effect OHE and R0 tends to be of theorder of Rs Schad et al., 1998. In the limit c1, theOHE conductivity xy

NH may be nonlinear in H c is thecyclotron frequency. Although xy

skew increases as , theOHE term xy

NH increases as 2 and therefore the latterultimately dominates, and the AHE current may be ir-resolvable. Even though the anomalous Hall current cannot always be cleanly separated from the normalLorentz-force Hall effect in the high conductivity re-gime, the total Hall current invariably increases with xxin a way which provides compelling 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 identified a dominant linear relation be-tween xy

AH and xx for xx106 cm−1 Majumdarand Berger, 1973; Shiomi et al., 2009. In an early studyMajumdar and Berger 1973 grew highly pure Fe dopedwith Co. The resulting xy

AH, obtained from Kohler plotextrapolation to zero field, shows a clear dependence ofxy

AHxx Fig. 4a. In a more recent study, a similarfinding linear dependence of xy

AHxx was observedby Shiomi et al. 2009 in Fe doped with Co, Mn, Cr, andSi. In these studies the high-temperature contribution toxy

AH presumed to be intrinsic plus side jump was sub-tracted from xy and a linear dependence of the result-ing xy

AH is observed Figs. 4a and 4b. In this recentstudy the conductivity is intentionally reduced by impu-rity doping to find the linear region and reliably excludethe Lorentz contribution. The results of these authorsshowed, in particular, that the slope of xx vs xy

AH de-pends on the species of the impurities as it is expected inthe regime dominated by skew scattering. It is reassuringto note that the skewness parameters Sskew=AH /xximplied by the older and more recent experiments areconsistent in spite of differences in the conductivityranges studied.

Sskew is independent of xx Majumdar and Berger,1973; Shiomi et al., 2009 as it should be when the skew-scattering mechanism dominates. Further experimentsin this regime are desirable to fully investigate the dif-ferent dependences on doping, temperature, and impu-rity type. Also, new approaches to reliably disentanglethe AHE and OHE currents will be needed to facilitatesuch studies.

b. Good-metal regime

Experiments reexamining the AHE in Fe, Ni, and Cohave been performed by Miyasato et al. 2007. Theseexperiments indicate a regime of xy vs xx in which xyis insensitive to xx in the range xx104–106 cm−1

see Fig. 5. This suggests that the scattering-independent mechanisms intrinsic and side jump domi-nate in this regime. However, in comparing this phenom-enology to the discussion of AHE mechanisms in Sec.I.B, one must keep in mind that the temperature has

been varied in the Hall data on Fe, Ni, and Co in orderto change the resistivity even though it is restricted tothe range well below Tc Fig. 5, upper panel. In themechanisms discussed in Sec. I.B only elastic scatteringwas taken into account. Earlier tests of the 2 depen-dence of xy carried by varying T were treated as suspectin the early AHE period see Fig. 2 because the role ofinelastic scattering was not fully understood. The effectof inelastic scattering from phonons and spin waves re-mains open in AHE theory and is not addressed in thispaper.

c. Bad-metal–hopping regime

Several groups have measured xy in Fe and Fe3O4thin-film ferromagnets Feng et al., 1975; Miyasato et al.,2007; Fernandez-Pacheco et al., 2008; Venkateshvaran etal., 2008; Sangiao et al., 2009; see Fig. 6. Sangiao et al.2009 studied epitaxial thin films of Fe deposited bysputtering on single-crystal MgO001 substrates at pres-sures 510−9 Torr. To vary xx over a broad range,they varied the film thickness t from 1 to 10 nm. The vs T profile for the film with t=1.8 nm displays a resis-

(a)

(b)(b)

FIG. 4. Color online The AHE conductivity in Fe films. axy

AH for pure Fe film doped with Cr, Co, Mn, and Si vs xx atlow temperatures T=4.2 and 5.5 K in. In many of the alloys,particularly in the Co-doped system, the linear scaling in thehigher conductivity sector, xx106 cm−1, implies thatskew scattering dominates xy

AH. After Majumdar and Berger,1973 and Shiomi et al., 2009. The data from Shiomi et al.2009, shown also at a larger scale in b, is obtained by sub-tracting the high-temperature contribution to xy. In the datashown the ordinary Hall contribution has been identified andsubtracted. Panel b from Shiomi et al., 2009.



tance minimum near 50 K, below which shows an up-turn which has been ascribed to localization or electroninteraction effects Fig. 7. The magnetization M isnominally unchanged from the bulk value except possi-bly in the 1.3 nm film. AHE experiments were carriedout from 2 to 300 K on these films and displayed as xyvs xx plots together with previously published resultsFig. 6. In the plot, the AHE data from films with t2nm fall in the weakly localized regime. The combinedplot shows that data of Sangiao et al. are collinear on alogarithmic scale with those measured on 1-m-thick

films by Miyasato et al. 2007. For the three sampleswith t=1.3–2 nm, the inferred exponent in the dirty re-gime is on average 1.66. A concern is that the datafrom the 1.3 nm film were obtained by subtracting a ln Tterm from the subtraction procedure was not de-scribed. How localization affects the scaling plot is anopen issue at present.

In Sec. II.E, we discuss recent AHE measurements indisordered polycrystalline Fe films with t10 nm by Mi-tra et al. 2007. Recent progress in understanding weak-localization corrections to the AHE is also reviewedthere.

In magnetite, Fe3O4, scaling of xyxx with

1.6–1.8 was already apparent in early experiments onpolycrystalline samples Feng et al., 1975. Recently, twogroups have reinvestigated the AHE in epitaxial thinfilms data included in Fig. 6. Fernandez-Pacheco et al.2008 measured a series of thin-film samples of Fe3O4grown by pulse laser deposition on MGO001 sub-strates in ultrahigh vacuum, whereas Venkateshvaran etal. 2008 studied both pure Fe3O4- and Zn-doped mag-netite Fe3−xZnxO4 deposited on MgO and Al2O3 sub-strates grown by laser molecular-beam epitaxy underpure Ar or Ar/O mixture. In both studies, increasesmonotonically by a factor of 10 as T decreases from300 K to the Verwey transition temperature TV=120 K.Below TV, further increases by a factor of 10–100. Theresults for vs T from Venkateshvaran et al. 2008 areshown in Fig. 8.

The large values of 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 xx with 1.6–1.8 when varying T.Fernandez-Pacheco et al. 2008 plotted xy vs xx in therange 150T300 K for several thicknesses t and in-ferred an exponent =1.6. Venkateshvaran et al. 2008

0 50 100 150 200 250 300T [K]

102

103

104

|σxy

|[Ω

-1cm

-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-1cm

-1]

102

103

104

|σxy

|(Ω

-1cm

-1)

FIG. 5. Color online Measurements of the Hall conductivityand resistivity in single-crystal Fe and in thin foils of Fe, Co,and Ni. The top and lower right panels show the T dependenceof xy and xx, respectively. The lower left panel plots xy vsxx. From Miyasato et al., 2007.

100

102

104

σxx [Ω-1cm

-1]

10-5

10-4

10-3

10-2

10-1

100

101

102

103

|σxy

|(Ω

-1cm

-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. Color online Combined plot of the AHE conductiv-ity xy vs the conductivity xx in epitaxial films of Fe grownon MgO with thickness t=2.5, 2.0, 1.8, and 1.3 nm Sangiao etal., 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.

40

45

50

55

60

65

70

75

325

330

335

340

345

350

355

0 50 100 150 200 250 300

MgO(001) // Fe(t) / MgO

(cm)

(cm)

T (K)

t = 1.8 nm

t = 2.5 nm

0

50

100

150

1 1.5 2 2.5 3

T 0(K)

t (nm)

FIG. 7. Color online The T dependence of in epitaxialthin-film MgO001 /Fet /MgO with t=1.8 and 2.5 nm. Theinset shows how T0, the temperature of the resistivity mini-mum, varies with t. From Sangiao et al., 2009.



plotted xy vs xx from 90 to 350 K and obtained power-law fits with =1.69 in both pure and Zn-doped magne-tite data shown in Fig. 6. Significantly, the two groupsfind that is unchanged below TV. There is presently notheory in the poorly conducting regime which predictsthe observed scaling xx10−1 cm−1.

3. Comparison to theories

Detailed first-principles calculations of the intrinsiccontribution to the AHE conductivity have been per-formed for bcc Fe Yao et al., 2004; Wang et al., 2006,fcc Ni, and hcp Co Wang et al., 2007. In Fe and Co, thevalues of xy inferred from the Berry curvature zkare 7.5102 and 4.8102 cm−1, respectively, in rea-sonable agreement with experiment. In Ni, however, thecalculated value of −2.2103 cm−1 is only 30% ofthe experimental value.

These calculations uncovered the crucial role playedby avoided crossings of band dispersions near the Fermienergy F. The Berry curvature bz is always strongly en-hanced near avoided crossings, opposite direction forthe upper and lower bands. A large contribution to xy

int

results when the crossing is at the Fermi energy so thatonly one of the two bands is occupied, e.g., near thepoint H in Fig. 9. A map showing the contributions ofdifferent regions of the Fermi surface to bzk is shownin Fig. 10. The SOI can lift an accidental degeneracy atcertain wave vectors k. These points act as a magneticmonopole for the Berry curvature in k space Fang et al.,2003. In the parameter space of spin-orbit coupling, xyis nonperturbative in nature. The effect of these “parityanomalies” Jackiw, 1984 on the Hall conductivity wasfirst discussed by Haldane 1988. A different conclusionon the role of topological enhancement in the intrinsicAHE was reported for a tight-binding calculation withthe two orbitals dzx and dyz on a square lattice Kontaniet al., 2007.

Motivated by the enhancement of xy at the crossingpoints discussed above, Onoda et al. 2006a, 2008 pro-posed a minimal model that focuses on the topologicaland resonantly enhanced nature of the intrinsic AHEarising from the sharp peak in bzk 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. V.D. In theclean limit, xy is dominated by the extrinsic skew-scattering contribution xy

skew, which almost masks the in-trinsic contribution xy

int. Since xyskew , it is suppressed

by increased impurity scattering, whereas xyint—an inter-

band effect—is unaffected. In the moderately dirty re-gime where the quasiparticle damping is larger than theenergy splitting at the avoided crossing typically theSOI energy but less than the bandwidth, xy

int dominatesxy

skew. As a result, one expects a crossover from the ex-trinsic to the intrinsic regime. When skew scattering isdue to a spin-dependent scattering potential instead ofspin-orbit coupling in the Bloch states, the skew to in-

FIG. 8. Color online Longitudinal resistivity xx vs T forepitaxial Fe3−xZnxO4 films. The 001, 110, and 111 orientedfilms were grown on MgO001, MgO110, and Al2O3 sub-strates. From Venkateshvaran et al., 2008.

FIG. 9. First-principles calculation of the band dispersions andthe Berry-phase curvature summed over occupied bands. FromYao et al., 2004.

FIG. 10. Color online First-principles calculation of the FS inthe 010 plane solid lines and the Berry curvature in atomicunits color map. From Yao et al., 2004.



trinsic crossover could be controlled by a different con-dition. In this minimal model, a plateau in the intrinsicregime ranges over two orders of magnitude in xxwhere xy decays only by a factor of 2 Onoda, 2009.This crossover may be seen clearly when the skew-scattering term shares the same sign as the intrinsic oneKovalev et al., 2009; Onoda, 2009.

With further increase in the scattering strength, spec-tral broadening leads to the scaling relationship xy

xx1.6 as discussed above Onoda et al., 2006b, 2008;

Kovalev et al., 2009; Onoda, 2009. In the strong-disorder regime, xx is no longer linear in the scatteringlifetime . A different scaling, xy xx

2 , attributed tobroadening of the electronic spectrum in the intrinsicregime, has been proposed by Kontani et al. 2007.

As discussed, there is some experimental evidencethat this scaling prevails not only in the dirty metallicregime but also deep into the hopping regime. Sangiaoet al. 2009, obtained the exponent 1.7 in epitaxialthin-film Fe in the dirty regime. In magnetite, the scalingseems to hold, with the same nominal value of evenbelow the Verwey transition where charge transport isdeep in the hopping regime Fernandez-Pacheco et al.,2008; Venkateshvaran et al., 2008. These regimes arewell beyond the purview of either the minimal model,which considers only elastic scattering, or the theoreticalapproximations used to model its properties Onoda etal., 2006b, 2008.

Nonetheless, the experimental reports have uncov-ered a robust scaling relationship with near 1.6, whichextends over a remarkably large range of xx. 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 ferro-magnet with a critical temperature Tc of 165 K. Theelectrons occupying the 4d t2g orbitals in Ru4+ have aSOI energy much larger than that for 3d electrons. Earlytransport investigations of this material were reportedby Allen et al. 1996 and Izumi et al. 1997. The latterauthors also reported results on thin-film SrTiO3. Re-cently, the Berry-phase theory has been applied to ac-count for its AHE Fang et al., 2003; Mathieu, Asamitsu,Takahashi, et al., 2004; Mathieu, Asamitsu, Yamada, etal., 2004 which is strongly T dependent Fig. 11c. Nei-ther the KL theory nor the skew-scattering theoryseemed adequate for explaining the T dependence ofthe inferred AHE conductivity xy Fang et al., 2003.

The experimental results motivated a detailed first-principles band-structure calculation that fully incorpo-rated the SOI. The AHE conductivity xy 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 =70 meV was introduced in the energy denominator.Fig. 12 shows the dependence of xy on the chemicalpotential . In sharp contrast to the diagonal conductiv-

ity xx, xy fluctuates strongly, displaying sharp peaksand numerous changes in sign. The fluctuations may beunderstood if we map the momentum dependence of theBerry curvature bzk in the occupied band. For ex-ample, Fig. 13 displays bzk plotted as a function ofk= kx ,ky, with kz fixed at 0. The prominent peak atk=0 corresponds to the avoided crossing of the energyband dispersions, which are split by the SOI. As dis-cussed in Sec. I.B, variation of the exchange splittingcaused by a change in the spontaneous magnetization Mstrongly affects xy in a nontrivial way.

From the first-principles calculations, one may esti-mate the temperature dependence of xy by assumingthat it is due to the temperature dependence of theBloch state exchange splitting and that this splitting isproportional to the temperature-dependent magnetiza-

FIG. 11. Color online Anomalous Hall effect in SrRuO3. aThe magnetization M, b longitudinal resistivity xx, and ctransverse resistivity xy as functions of the temperature T forthe single crystal and thin-film SrRuO3, as well as for Ca-doped Sr0.8Ca0.2RuO3 thin film. B is the Bohr magneton.From Fang et al., 2003.

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



tion. The new insight is that the T dependence of xyTsimply reflects the M dependence of xy: at a finite tem-perature T, the magnitude and sign of xy may be de-duced by using the value of MT in the zero-T curve.This proposal was tested against the results on both thepure material and the Ca-doped material Sr1−xCaxRuO3.In the latter, Ca doping suppresses both Tc and M sys-tematically Mathieu, Asamitsu, Takahashi, et al., 2004;Mathieu, Asamitsu, Yamada, et al., 2004. As shown inFig. 14 upper panel, the measured values of M and xy,obtained from five samples with Ca content 0.4x0,fall on two continuous curves. In the inset, the curve forthe pure sample x=0 is compared with the calculation.The lower panel of Fig. 14 compares calculated curves of

xy for cubic and orthorhombic lattice structures. Thesensitivity of xy to the lattice symmetry reflects thedominant contribution of the avoided crossing near F.The sensitivity to broadening is shown for the ortho-rhombic case.

Kats et al. 2004 also studied the magnetic-field de-pendence of xy in an epitaxial film of SrRuO3. Theyhave observed sign changes in xy near a magnetic fieldB=3 T at T=130 K and near B=8 T at 134 K. Thisseems to be qualitatively consistent with the Berry-phase scenario. On the other hand, they suggested thatthe intrinsic-dominated picture is likely incomplete orincorrect near Tc Kats et al., 2004.

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 localmoment of spin S= 3

2 . A large Hund energy JH aligns thecore spin S with the s= 1

2 spin of an itinerant electronthat momentarily occupies the eg orbital. Because thisHund coupling leads to an extraordinary magnetoresis-tance MR in weak H, the manganites are called colos-sal magnetoresistance CMR materials Tokura and To-mioka, 1999. The double-exchange theory summarizedby Eq. 2.2 is widely adopted to describe the onset offerromagnetism in the CMR manganites.

As T decreases below the Curie temperature TC270 K in LCMO, the resistivity falls rapidly from15 to metallic values 2 m cm. CMR is observedover a significant interval of temperatures above and be-low TC, where charge transport occurs by hopping ofelectrons between adjacent Mn ions Fig. 15a. At eachMn site i, the Hund energy tends to align the carrier spins with the core spin Si.

Early theories of hopping conductivity Holstein,1961 predicted the existence of a Hall current producedby the phase shift Peierls factor associated with themagnetic flux piercing the area defined by three non-collinear atoms. However, the hopping Hall current isweak. The observation of a large xy in LCMO that at-tains a broad maximum in modest H Fig. 15b ledMatl et al. 1998 to propose that the phase shift is geo-metric in origin, arising from the solid angle describedby s as the electron visits each Mn site s"Si at each sitei as shown in Fig. 16. To obtain the large xy seen, onerequires Si to define a finite solid angle . Since Sigradually aligns with H with increasing field, this effectshould disappear along with , as observed in the ex-periment. This appears to be the first application of ageometric-phase mechanism to account for an AHE ex-periment.

Subsequently, Ye et al. 1999 considered the Berryphase due to the thermal excitations of the Skyrmionand anti-Skyrmion. They argued that the SOI gives riseto a coupling between the uniform magnetization M andthe gauge field b by the term M ·b. In the ferromag-netic state, the spontaneous uniform magnetization Mleads to a finite and uniform b, which acts as a uniform

FIG. 13. Color online The Berry curvature bzk for a bandas a function of k= kx ,ky with the fixed kz=0. From Fang etal., 2003.

FIG. 14. Color online The Hall conductivity in Ca-dopedstrontium ruthenate. Upper panel Combined plots of theHall conductivity xy vs magnetization M in five samples of theruthenate Sr1−xCaxRuO3 0x0.4. The inset compares dataxy at x=0 triangles with calculated values solid curve.Lower panel First-principles calculations of xy vs M for cu-bic and orthorhombic structures. The effect of broadening onthe curves is shown for the orthorhombic case. From Mathieu,Asamitsu, Yamada, et al., 2004.



magnetic field. Lyanda-Geller et al. 2001 also consid-ered the AHE due to the spin-chirality fluctuation in theincoherent limit where the hopping is treated perturba-tively. This approach, applicable to the high-T limit,complements the theory of Ye et al. 1999.

The Berry phase associated with noncoplanar spinconfigurations, the scalar spin chirality, was first consid-ered in theories of high-temperature superconductors inthe context of the flux distribution generated by thecomplex order parameter of the resonating valencebond RVB correlation defined by ij, which acts as thetransfer integral of the “spinon” between the sites i andj Lee et al., 2006. The complex transfer integral alsoappears in the double-exchange model specified by

H = − ij,

tijci† cj + H.c. − JH

iSi · ci

† ci, 2.2

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

In the manganese oxides, Si represents the localizedspin in t2g orbitals, while c† and c are the operators for egelectrons. In mean-field approximations of Hubbard-like theories of magnetism, the localized spin may alsobe regarded as the molecular field created by the con-duction electrons themselves, in which case JH is re-placed by the on-site Coulomb interaction energy U. Inthe 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

tijeff = tijij = tije

iaij cosij

2 , 2.3

where i is the two-component spinor spin wave func-tion with quantization axis "Si. The phase factor eiaij actslike a Peierls phase and can be viewed as originatingfrom a fictitious magnetic field which influences the or-bital motion of the conduction electrons.

We next discuss how the Peierls phase leads to agauge field, i.e., flux, in the presence of noncoplanarspin configurations. Let Si, Sj, and Sk be the localspins at sites i, j, and k, respectively. The productof the three transfer integrals corresponding to the loopi→ j→k→ i is

ninknknjnjni = 1 + ni · nj + nj · nk + nk · nj

+ ini · nj nk

eiaij+ajk+aki = ei/2, 2.4

where ni is the two-component spinor wavefunction ofthe spin state polarized along ni=Si / Si. Its imaginarypart is proportional to Si · SjSk, which corresponds tothe solid angle subtended by the three spins on theunit sphere, and is called the scalar spin chirality Fig.16. The phase acquired by the electron’s wave functionaround the loop is ei/2, which leads to the Aharonov-Bohm effect and, as a consequence, to a large Hall re-sponse.

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 nx=Sx / Sx to fluctuate strongly as a function of x, theposition coordinate in the sample. An insightful way toquantify this fluctuation is to regard nx as a map fromthe x-y plane to the surface of the unit sphere we take a2D sample for simplicity. An important defect in a fer-romagnet the Skyrmion Sondhi et al., 1993 occurswhen nx points down at a point x in region A of thex-y plane but gradually relaxes back to up at the bound-ary of A. The map of this spin texture wraps around the

FIG. 15. The field dependence of the resistivity and Hall resis-tivity in the manganite La,CaMnO3. a The colossal magne-toresistance vs H in La1−xCaxMnO3 TC=265 K at selectedT. b The Hall resistivity xy vs H at temperatures of100–360 K. Above TC, xy is strongly influenced by the MRand the susceptibility . From Matl et al., 1998.

FIG. 16. Schematic view of spin chirality. When circulatedamong three spins to which it is exchange coupled, it feels afictitious magnetic field b with flux given by the half of thesolid angle subtended by the three spins. From Lee et al.,2006.



sphere once as x roams over A. The number of Skyrmi-ons in the sample is given by the topological index

Ns = A

dxdyn · nx

ny =

Adxdybz, 2.5

where the first integrand is the directed area of the im-age on the unit sphere. Ns counts the number of timesthe map covers the sphere as A extends over the sample.The gauge field b produces a Hall conductivity. Ye et al.1999 derived in the continuum approximation the cou-pling between the field b and the spontaneous magneti-zation M through the SOI. The SOI coupling producesan excess of thermally excited positive Skyrmions overnegative ones or vice versa. This imbalance leads to anet uniform “magnetic field” b antiparallel to M, andthe AHE. In this scenario, xy is predicted to attain amaximum slightly below Tc before falling exponentiallyto zero as T→0.

The Hall effect in the hopping regime has been dis-cussed by Holstein 1961 in the context of impurity con-duction in semiconductors. Since the energies j and kof adjacent impurity sites may differ significantly, chargeconduction must proceed by phonon-assisted hopping.To obtain a Hall effect, we consider three noncollinearsites labeled as i=1, 2, and 3. In a field H, the magneticflux piercing the area enclosed by the three sites playsthe key role in the Hall response. According to Holstein,the Hall current arises from interference between thedirect hopping path 1→2 and the path 1→3→2 goingvia 3 as an intermediate step. Taking into account thechanges in the phonon number in each process, we have

1,N,N → 1,N 1,N → 2,N 1,N 1 ,2.6

1,N,N → 3,N 1,N → 2,N 1,N 1 ,

where N, N are the phonon numbers for the modes ,, respectively.

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

includes the Peierls phase factor exp−ie /cRi

Rjdr ·Ar.When we consider the interference between the twoprocesses in Eq. 2.6, the Peierls phase factors combineto produce the phase shift expi2 /0, where 0=hc /e is the flux quantum. By the Aharonov-Bohm ef-fect, this leads to a Hall response.

As discussed, this idea was generalized for the manga-nites by replacing the Peierls phase factor with theBerry-phase factor in Eq. 2.4 Lyanda-Geller et al.,2001. The calculated Hall conductivity is

H = G cosij/2cosjk/2coski/2sin/2 ,

2.7

where is the set of the energy levels a a= i , j ,k andij is the angle between ni and nj. When the average ofH over all directions of na is taken, it vanishes even forfinite spontaneous magnetization m. To obtain a finitexy, it is necessary to incorporate SOI.

Assuming the form of hopping integral with the SOIgiven by

Vjk = Vjkorb1 + i · gjk , 2.8

where gjk depends on the spin-orbit coupling for the spe-cific material present in Vjk, the Hall conductivity is pro-portional to the average of

gjk · nj nkn1 · n2 n3 . 2.9

Taking the average of n’s with m=M /Msat, where Msat isthe saturated magnetization, we finally obtain

xy = xy0 m1 − m22

1 + m22 . 2.10

This prediction has been tested by the experiment ofChun et al. 2000 shown in Fig. 17. The scaling law forthe anomalous xy as a function of M obtained near TCis in good agreement with the experiment.

Similar ideas have been used by Burkov and Balents2003 to analyze the variable range hopping region inGa,MnAs. The spin-chirality mechanism for the AHEhas also been applied to CrO2 Yanagihara and Sala-mon, 2002, 2007 and the element Gd Baily and Sala-mon, 2005. In the former case, the comparison betweenxy and the specific heat supports the claim that the criti-cal properties of xy are governed by the Skyrmion den-sity.

The theories described assume large Hund coupling.In the weak-Hund coupling limit, a perturbative treat-ment in JH has been developed to relate the AHE con-ductivity to the scalar spin chirality Tatara and Kawa-mura, 2002. This theory has been applied to metallicspin-glass systems Kawamura, 2007, as reviewed inSec. II.D.7.

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 et al.1998 and Baily and Salamon 2003 investigated the

FIG. 17. Comparison between experiment and theoretical pre-diction Eq. 2.10. Scaling behavior between the Hall resistivityH and the magnetization M is shown for layered manganiteLa1.2Sr1.8Mn2O7. The solid line is a fit to Eq. 2.10; the dashedline is the numerator of Eq. 2.10 only. There are no fittingparameters except the overall scale. From Chun et al., 2000.



AHE in Ca-doped lanthanum cobaltite La1−xCaxCoO3,which displays a number of unusual magnetic and trans-port properties. They found an unusually large AHEnear TC as well as at low T and proposed the relevanceof spin-ordered clusters and orbital disorder scatteringto the AHE in the low-T limit. Subsequently, a moredetailed investigation of La1−xSrxCoO3 was reported byOnose and Tokura 2006. Figures 18a–18c summa-rize the T dependence of M, xy, and xy, respectively, infour crystals with 0.17x0.30. The variation of xx vsx suggests that a metal-insulator transition occurs be-tween 0.17 and 0.19. Whereas the samples with x0.2have a metallic xx-T profile, the sample with x=0.17 isnonmetallic hopping conduction. Moreover, it displaysa very large MR at low T and large hysteresis in curvesof M vs H, features that are consistent with a ferromag-netic cluster-glass state.

When the Hall conductivity is plotted vs M Fig.18d xy shows a linear dependence on M for the mostmetallic sample x=0.30. However, for x=0.17 and 0.20,there is a pronounced downturn suggestive of the ap-pearance of a different Hall term that is electronlike insign. This is most apparent in the trend of the curves ofxy vs T in Fig. 18b. Onose and Tokura 2006 proposedthat the negative term may arise from hopping of carri-ers between local moments which define a chirality thatis finite, as discussed.

4. Spin-chirality mechanism in pyrochlore ferromagnets

In the examples discussed in Sec. II.B.3, the spin-chirality mechanism leads to a large AHE at finite tem-

peratures. An interesting question is whether or notthere exist ferromagnets in which the spin chirality isfinite in the ground state.

Ohgushi et al. 2000 considered the ground state ofthe noncoplanar spin configuration in the Kagome lat-tice, which may be obtained as a projection of the pyro-chlore lattice onto the plane normal to 1,1,1 axis. Con-sidering double-exchange model Eq. 2.2, they obtainedthe band structure of the conduction electrons and theBerry-phase distribution. Quite similar to the Haldanemodel Haldane, 1988 or the model discussed in Eq.3.21, the Chern number of each band becomes non-zero, and a quantized Hall effect results when thechemical potential is in the energy gap.

Turning to real materials, the pyrochlore ferromagnetNd2Mo2O7 NMO provides a test bed for exploringthese issues. Its lattice structure consists of two interpen-etrating sublattices comprised of tetrahedrons of Nd andMo atoms the sublattices are shifted along the c axisYoshii et al., 2000; Taguchi et al., 2001. While the ex-change between spins on either sublattice is ferromag-netic, the exchange coupling Jdf between spins of theconducting d electrons of Mo and localized f-electronspins on Nd is antiferromagnetic.

The dependences of the anomalous Hall resistivity xyon H at selected temperatures are shown in Fig. 19. Thespins of Nd begin to align antiparallel to those of Mobelow the crossover temperature T*40 K. Each Ndspin is subject to a strong easy-axis anisotropy along theline from a vertex of the Nd tetrahedron to its center.The resulting noncoplanar spin configuration induces atransverse component of the Mo spins. Spin chirality isexpected to be produced by the coupling Jdf, which leadsto the AHE of d electrons. An analysis of the neutronscattering experiment has determined the magneticstructure Taguchi et al., 2001. The tilt angle of the Ndspins is close to that expected from the strong limit of

FIG. 18. Temperature dependence of the a magnetization M,b Hall resistivity xy, c Hall conductivity xy in four crystalsof La1−xSrxCoO3 0.17x0.30 all measured in a field H=1 T. In a–c, the data for x=0.17 and 0.20 were multipliedby a factor of 200 and 50, respectively. d The Hall conductiv-ity xy at 1 T in the four crystals plotted against M. Results forx=0.17 and 0.20 were multiplied by factors 50 and 5, respec-tively. From Onose and Tokura, 2006.

FIG. 19. Anomalous Hall effect in Nd2Mo2O7. Magnetic-fielddependence of a the magnetization and b the transverseresistivity xy for different temperatures. From Taguchi et al.,2001.



the spin anisotropy, and the exchange coupling Jfd is es-timated as Jfd5 K. This leads to a tilt angle of the Mospins of 5°. From these estimates, a calculation of theanomalous Hall conductivity in a tight-binding Hamil-tonian of triply degenerate t2g bands leads to H

20 cm−1, consistent with the value measured atlow T. In a strong H, this tilt angle is expected to bereduced along with xy. This is in agreement with thetraces displayed in Fig. 19.

The T dependence of the Hall conductivity has alsobeen analyzed in the spin-chirality scenario by incorpo-rating spin fluctuations S. Onoda and Nagaosa, 2003.The result is that frustration of the Ising Nd spins leadsto large fluctuations, which accounts for the large ob-served. The recent observation of a sign change in xy ina field H applied in the 1,1,1 direction Taguchi et al.,2003 is consistent with the sign change of the spinchirality.

In the system Gd2Mo2O7, in which Gd3+ d7 has nospin anisotropy, the low-T AHE is an order of magni-tude smaller than that in NMO. This is consistent withthe spin-chirality scenario Taguchi et al., 2004. The ef-fect of the spin-chirality mechanism on the finite-frequency conductivity H has been investigatedKezsmarki et al., 2005.

In another work, Yasui et al. 2006, 2007 performedneutron scattering experiments over a large region in

the H ,T plane with H along the 0, 1 ,1 and 0,0,1directions. By fitting the magnetization MNdH ,T ofNd, the magnetic specific heat CmagH ,T, and the mag-netic scattering intensity ImagQ ,H ,T, they estimatedJdj0.5 K, which was considerably smaller than esti-mated previously Taguchi et al., 2001. Furthermore,they calculated the thermal average of the spin chiralitySi ·SjSk and compared its value with that inferredfrom the AHE resistivity xy. They have emphasizedthat, when a 3 T field is applied in the 0,0,1 directionalong this direction H cancels the exchange field fromthe Mo spins, no appreciable reduction of xy is ob-served. These recent conclusions have cast doubt on thespin-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., two of theNd spins point towards the tetrahedron center while twopoint away to the three-in, one-out structure for the Ndspins. However, no Hall features that might be identifiedwith this cancellation have been observed down to verylow T. This seems to suggest that quantum fluctuationsof the Nd spins may play an important role despite thelarge spin quantum number S= 3

2 .Machida discussed the possible relevance of spin

chirality to the AHE in the pyrochlore Pr2Ir2O7Machida, Nakatsuji, Maeno, Tayama, and Sakakibara,2007; Machida, Nakatsuji, Maeno, Tayama, Sakakibara,and Onoda, 2007. In this system, a novel “Kondo ef-fect” is observed even though the Pr3+ ions with S=1 aresubject to a large magnetic anisotropy. The magnetic and

transport properties of R2Mo2O7 near the phase bound-ary between the spin-glass Mott insulator and ferromag-netic metal by changing the rare-earth ion R have beenstudied Katsufuji et al., 2000.

5. Anatase and rutile Ti1−xCoxO2−

In thin-film samples of the ferromagnetic semiconduc-tor anatase Ti1−xCoxO2−, Ueno et al. 2008 reportedscaling between the AHE resistance and the magnetiza-tion M. The AHE conductivity xy

AH scales with the con-ductivity xx as xy

AH xx1.6 Fig. 20. A similar scaling re-

lation was observed in another polymorph rutile seealso Ramaneti et al. 2007 for related work on codopedTiO2.

C. Ferromagnetic semiconductors

Ferromagnetic semiconductors combine semiconduc-tor tunability and collective ferromagnetic properties ina single material. The most widely studied ferromagneticsemiconductors are diluted magnetic semiconductorsDMSs created by doping a host semiconductor with atransition-metal element which provides a localizedlarge moment formed by the d electrons and by intro-ducing carriers which can mediate a ferromagnetic cou-pling between these local moments. The most exten-sively studied are the Mn based III,MnV DMSs, inwhich substituting Mn for the cations in a III,V semi-conductor can dope the system with hole carriers;Ga,MnAs becomes ferromagnetic beyond a concentra-tion of 1%.

The simplicity of this basic but generally correctmodel hides within it a cornucopia of physical and ma-terials science effects present in these materials. Amongthe phenomena which have been studied are metal-insulator transitions, carrier mediated ferromagnetism,disorder physics, magnetoresistance effects, magneto-

FIG. 20. Color online Plot of AHE conductivity AHE vsconductivity for anatase Ti1−xCoxO2− triangles and rutileTi1−xCoxO2− diamonds. Gray 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 for T150 K and T100 K, respectively. From Ueno et al., 2007.



optical effects, coupled magnetization dynamics, post-growth dependent properties, etc. A more in-depth dis-cussion of these materials, from both the experimentaland theoretical points of view, can be found in Jungwirthet 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 proper-ties in these materials has been verified by comparisonwith remnant magnetization measurements using a su-perconducting quantum interference device magneto-meter Ohno et al., 1992. The relative simplicity of theeffective band structure of the carriers in metallic DMSshas made them a playing ground to understand AHE offerromagnetic systems with strong spin-orbit coupling.

Experimentally, it has been established that the AHEin the archetypical DMS system Ga,MnAs is in themetallic regime dominated by a scattering-independentmechanism, i.e., xy

AH xx2 Edmonds et al., 2002; Ruzme-

tov et al., 2004; Chun et al., 2007; Pu et al., 2008. Thestudies of Edmonds et al. 2002 and Chun et al. 2007have established this relationship in the noninsulatingmaterials by extrapolating the low-temperature xyB tozero field and zero temperature. This is shown in Fig. 21where metallic samples, which span a larger range thanthe ones studied by Edmonds et al. 2002, show a clearRsxx

2 dependence.DMS growth process requires nonequilibrium low-

temperature conditions and the as-grown often insulat-ing materials and postgrown annealed metallic materi-als show typically different behaviors in the AHEresponse. A similar extrapolating procedure performedon insulating Ga,MnAs seems to exhibit approximatelylinear dependence of Rs on xx. On the other hand, con-siderable uncertainty is introduced by the extrapolationto low temperatures because xx diverges and the com-plicated magnetoresistance of xx is a priori not under-stood in the low-T range.

A more recent study by Pu et al. 2008 of Ga,MnAsgrown on InAs, such that the tensile strain creates a

perpendicular anisotropic ferromagnet, has establishedthe dominance of the intrinsic mechanism in metallicGa,MnAs samples beyond any doubt. Measuring thelongitudinal thermoelectric transport coefficients xx,xy, xy, and xx where J=E+−T, one can showthat given the Mott relation = 2kB

2 T /3e /EEFand the empirical relation xyB=0=Mzxx

n , the rela-tion between the four separately measured transport co-efficients is

xy =xy

xx2 2kB

2 T

3e

− n − 2xxxx . 2.11

The fits to the / and n parameters are shown in Fig.22. Note that = /EF but it is the ratio / which isthe fitting parameter. Fixing n=1 does not produce anygood fit to the data as indicated by the dashed lines inFig. 22. These data exclude the possibility of a n=1 typecontribution to the AHE in metallic Ga,MnAs and fur-ther verify the validity of the Mott relation in these ma-terials.

Having established that the main contribution to theAHE in metallic Ga,MnAs is given by scattering-independent contributions rather than skew-scattering

FIG. 21. Color online The resistivity and anomalous Hallcoefficient in Ga, MnAs. a Ga1−xMnxAs samples that showinsulating and metallic behavior defined by xx /T near T=0. b Rs vs xx extrapolated from xyB data to zero fieldand low temperatures. From Chun et al., 2007.

FIG. 22. Color online The Hall resistivity and Nernst effectin Ga,MnAs grown on InAs substrate. Top Zero B field xyand xx for four samples grown on InAs substrates i.e., per-pendicular to plane easy axis. Annealed samples, which pro-duce perpendicular to plane easy axes, are marked by an .The inset indicates the B dependence of the 7% sample at10 K. Bottom Zero-field Nernst coefficient xy for the foursamples. The solid curves indicate the best fit using Eq. 2.11and the dashed curves indicate the best fit setting N=1. FromPu et al., 2008.



contributions, DMSs are an ideal system to test our un-derstanding of AHE. In the regime where the largestferromagnetic critical temperatures are achieved fordoping levels above 1.5%, semiphenomenological mod-els that are built on the Bloch states for the band quasi-particles rather than localized basis states appropriatefor the localized regime Berciu and Bhatt, 2001 pro-vide the natural starting point for a model Hamiltonianwhich reproduces many of the observed experimentaleffects Sinova and Jungwirth, 2005; Jungwirth et al.,2006. Recognizing that the length scales associated withholes in the DMS compounds are still long enough, ak ·p envelope function description of the semiconductorvalence bands is appropriate. To understand the AHEand magnetic anisotropy, it is necessary to incorporateintrinsic spin-orbit coupling in a realistic way.

A successful model for Ga,MnAs is specified by theeffective Hamiltonian

H = HK-L + JpdI

SI · srr − RI + Hdis, 2.12

where HK-L is the six-band Kohn-Luttinger k ·p Hamil-tonian Dietl et al., 2001, the second term is the short-range antiferromagnetic kinetic-exchange interactionbetween local spin SI at site RI and the itinerant holespin a finite range can be incorporated in more realisticmodels, and Hdis is the scalar scattering potential repre-senting the difference between a valence band electronon a host site and a valence band electron on a Mn siteand 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 moresophisticated exact-diagonalization or Monte Carlomethods Jungwirth, Abolfath, et al., 2002; Schliemannand MacDonald, 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 Berryphase or Karplus-Luttinger contribution will mostlikely be dominant because of the large SOI of the car-riers and the experimental evidence showing the domi-nance of scattering-independent mechanisms. For prac-tical calculations it is useful to use, as in the intrinsicAHE studies in the oxides Fang et al., 2003; Mathieu,Asamitsu, Takahashi, et al., 2004; Mathieu, Asamitsu,Yamada, et al., 2004, the Kubo formalism given in Eq.1.2 with disorder induced broadening of the bandstructure but no side-jump contribution. The broaden-ing is achieved by substituting one of the nk−nkfactors in the denominator by nk−nk+ i. Ap-plying this theory to metallic III,MnV materials usingboth the four-band and six-band k ·p descriptions of thevalence band electronic structure one obtains results in

quantitative agreement with experimental data inGa,MnAs and In,MnAs DMSs Jungwirth, Niu, andMacDonald, 2002. In a follow up calculation by Jung-wirth et al. 2003, a more quantitative comparison of thetheory with experiments was made in order to accountfor finite quasiparticle lifetime effects in these stronglydisordered systems. The effective lifetime for transitionsbetween bands n and n, n,n1/n,n, can be calculatedby averaging quasiparticle scattering rates obtainedfrom Fermi’s golden rule including both screened Cou-lomb and exchange potentials of randomly distributedsubstitutional Mn and compensating defects as done inthe dc Boltzmann transport studies Jungwirth, Abol-fath, et al., 2002; Sinova et al., 2002. A systematic com-parison between theoretical and experimental AHEdata is shown in Fig. 23 Jungwirth et al., 2003. Theresults are plotted versus nominal Mn concentration xwhile other parameters of the seven samples studied arelisted in the figure legend. The measured AH values areindicated by filled squares; triangles are theoretical re-sults obtained for a disordered system assuming Mn-interstitial compensation defects. The valence band holeeigenenergies nk and eigenvectors nk are obtained bysolving the six-band Kohn-Luttinger Hamiltonian in thepresence of the exchange field, h=NMnSJpdz Jungwirthet al., 2006. Here NMn=4x /aDMS

3 is the Mn density in theMnxGa1−xAs epilayer with a lattice constant aDMS, thelocal Mn spin S=5/2, and the exchange coupling con-stant Jpd=55 meV nm−3.

In general, when disorder is accounted for, the theoryis in a good agreement with experimental data over thefull range of Mn densities studied from x=1.5% to 8%.The effect of disorder, especially when assuming Mn-interstitial compensation, is particularly strong in the x=8% sample shifting the theoretical AH much closer toexperiment compared to the clean limit theory. The re-maining quantitative discrepancies between theory andexperiment have been attributed to the resolution inmeasuring experimental hole and Mn densities Jung-wirth et al., 2003.

We conclude this section by mentioning the anoma-lous Hall effect in the nonmetallic or insulating-hoppingregimes. Experimental studies of Ga,MnAs digital fer-

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

0

10

20

30

40

50

60

70

σ AH

(Ω-1

cm-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. From Jungwirth et al., 2003.



romagnetic heterostructures, which consist of submono-layers of MnAs separated by spacer layers of GaAs,have shown longitudinal and Hall resistances of the hop-ping conduction type, Rxx T expT0 /T, and haveshown that the anomalous Hall resistivity is dominatedby hopping with a sublinear dependence of RAH on RxxAllen et al., 2004; Shen et al., 2008 similar to experi-mental observations on other materials. Studies in thisregime are still not as systematic as their metallic coun-terparts which clearly indicate a scaling power of 2. Ex-periments find a sublinear dependence of RAHRxx

,with 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. Previous the-oretical calculations based on the hopping conductionwith the Berry phase Burkov and Balents, 2003showed the 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, a key prediction of the KL theory andits modern generalization based on the Berry-phase ap-proach is that the AHE conductivity yx

AH is indepen-dent of the carrier lifetime dissipationless Hall cur-rent in materials with moderate conductivity. Thisimplies that the anomalous Hall resistivity yx

AH varies as2. Moreover, xy

AH tends to be proportional to the ob-served magnetization Mz. We write

xyAH = SHMz, 2.13

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

yx vs measured on the same sample over an extendedtemperature range. However, because the skew-scattering model can also lead to the same predictionyx2 when inelastic scattering predominates, tests atfinite T are inconclusive. The proper test requires a sys-tem in which 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 90° superexchange between the local mo-ments of adjacent Cr ions. The charge carriers play onlya weak role in the superexchange energy. The experi-mental proof of this is that when the carrier density n isvaried by a factor of 20 by substituting Se by Br, theCurie temperature TC decreases by only 100 K from380 K. Significantly, M at 4 K changes negligibly. Theresistivity at 4 K may be varied by a factor of 103

without weakening M. Detailed Hall and resistivity mea-surements were carried out by Lee et al. 2004 on 12crystals of CuCr2Se4−xBrx. They found that yx measuredat 5 K changes sign negative to positive when x ex-ceeds 0.4. At x=1, yx attains very large values700 cm at 5 K.

Lee et al. 2004 showed that the magnitude yx /nvaries as 2 over three decades in Fig. 24, consistentwith the prediction of the KL theory.

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

2 , and Cu3Te4, has been investigatedby Oda et al. 2001. These materials however are be-lieved to have nontrivial spin structures below a tem-perature Tm. For CuCr2S4 and Cu3Te4, the Hall resistiv-ities were observed to deviate from H=R0H+4RsM below Tm through sign changes in Rs. Theyproposed adding a third term aH3 to this standardequation to understand the strong T dependence of theHall resistivity and argued that the nontrivial spin struc-ture plays an important role.

The origin of this nontrivial behavior is believed tohinge on the nontrivial magnetic structures although fur-ther theoretical studies will be needed to establish thisconnection.

2. Heusler alloy

The full Heusler alloy Co2CrAl has the L21 latticestructure and orders ferromagnetically below 333 K.Several groups Galanakis et al., 2002; Block et al., 2004have 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 crystalswith stoichiometry Co2.06Cr1.04Al0.90 Husmann andSingh, 2006. Below the Curie temperature TC=333 K,

FIG. 24. Color online Log-log plot of xy /nh vs in 12 crys-tals of Br-doped spinel CuCr2Se4−xBrx with nh as the hole den-sity is measured at 5 K; xy is measured at 2 and 5 K.Samples in which yx is electronlike holelike are shown asopen closed circles. The straight-line fit implies xy /nh=A, with =1.95±0.08 and A=2.2410−25 SI units. FromLee et al., 2004.



the magnetization M increases rapidly, eventually satu-rating to a low-T value that corresponds to 1.65B Bohrmagneton/f.u. Husmann and Singh showed that, below310 K, MT fits well to the form 1− T /TC21/2. As-suming that the ordinary Hall coefficient R0 is negligible,they found that the Hall conductivity xy=yx /2 isstrictly linear in M expressed as xy=SH

1 M over the Tinterval 36–278 K Fig. 25. They interpret the linearvariation as consistent with the intrinsic AHE theory.The value of S

1 =0.383 G/ 4 cm inferred is similarto values derived from measurements on the dilute Nialloys, half Heuslers, and silicides.

3. Fe1−yCoySi

The silicide FeSi is a nonmagnetic Kondo insulator.Doping with Co leads to a metallic state with a low den-sity p of holes. Over a range of Co doping 0.05y0.8 the ground state is a helical magnetic state with apeak Curie temperature TC50 K. The magnetizationcorresponds to 1B Bohr magneton per Co ion. Thetunability allows investigation of transport in a magneticsystem with low p. Manyala et al. 2004 observed thatthe Hall resistivity H increases to 1.5 cm at 5 Kat the doping y=0.1. By plotting the observed Hall con-ductivity xy against M, they found xy=SHM with SH0.22 G/ 4 cm. In contrast, in heavy fermion sys-tems which include FeSi xyM−3 Fig. 26.

4. MnSi

MnSi grows in the noncentrosymmetric B20 latticestructure which lacks inversion symmetry. Competitionbetween the exchange energy J and Dzyaloshinsky-Moriya term D leads to a helical magnetic state with a

long pitch 180 Å. At ambient pressure, the helicalstate forms at the critical temperature TC=30 K. Undermoderate hydrostatic pressure P, TC decreases mono-tonically, reaching zero at the critical pressure Pc=14 kbar. Although MnSi has been investigated for sev-eral 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 PPc are replaced bya Bragg sphere Pfleiderer et al., 2004. Non-Fermi liquidexponents in the resistivity vs T are observed abovePc.

Among ferromagnets, MnSi at 4 K has a low resistiv-ity 2–5 cm. The unusually long carrier meanfree path implies that the ordinary term xy

NH2 isgreatly enhanced. In addition, the small renders thetotal Hall voltage difficult to resolve. Both factorsgreatly complicate the task of separating xy

NH from theAHE conductivity. However, the long in MnSi pre-sents an opportunity to explore the AHE in the high-purity limit of ferromagnets. Using high-resolution mea-surements of the Hall resistivity yx Fig. 27a, Lee etal. 2007 recently accomplished this separation by ex-ploiting the large longitudinal magnetoresistance MRH. From Eqs. 2.1 and 2.13, they have yx H=SHH2MH, where yx =yx

AH=yx−R0B. At each T,the field profiles of yx H and MH are matched byadjusting the two H-independent parameters SH and R0Fig. 27b. The inferred parameters SH and R0 arefound to be T independent below TC Fig. 28.

FIG. 25. Color online Combined plots of xy in the Heusleralloy Co2CrAl vs the magnetization M at selected tempera-tures from 36 to 278 K. The inset shows the inferred values ofxy

AHxy /M at each T. From Husmann and Singh, 2006.

FIG. 26. Color online Hall conductivity xy of ferromagneticmetals and heavy fermion materials collected at 1 kG and 5 Kunless otherwise noted. Small solid circles representFe1−yCoySi for 5T75 K and 500 GH50 kG with y=0.1 filled circles, 0.15 filled triangles, y=0.2 , and 0.3filled diamonds. MnSi data 5–35 K are large, solid squaresconnected by black line. Small asterisks are GaMnAs datafor 5T120 K. The rising line xyM is consistent with theintrinsic AHE in itinerant ferromagnets while the falling linexyM−3 applies to heavy fermions. From Manyala et al.,2004.



The Hall effect of MnSi under hydrostatic pressure5–11.4 kbar was measured recently Lee et al., 2009.In addition to the AHE and OHE terms xy

AH and xyNH,

Lee et al. observed a well-defined Hall term xyC with an

unusual profile. As shown in Fig. 29 note that in thefigure xy

AH and xyNH are labeled xy

A and xyN , respec-

tively, the new term appears abruptly at 0.1 T, rapidlyrises to a large plateau value, and then vanishes at0.45 T curves at 5, 7, and 10 K. From the large magni-tude of xy

C and its restriction to the field interval inwhich the cone angle is nonzero, they argued that itarises from the coupling of the carrier spin to the chiralspin textures in the helical magnetization, as discussed inSec. II.B. They noted that MnSi under pressure providesa rare example in which the three Hall conductivitiescoexist at the same T.

In the A phase of MnSi, which occurs at ambient pres-sure in a narrow range of temperature 28T29 Kand field 0.1H0.2 T, Neubauer et al. 2009 ob-served a distinct contribution to the Hall effect whichthey identified with the existence of a Skyrmion lattice.Diffraction evidence for the Skyrmion lattice in the Aphase has been obtained by small-angle neutron scatter-ing.

5. Mn5Ge3

The AHE in thin-film samples of Mn5Ge3 was inves-tigated by Zeng et al. 2006. They expressed the AHEresistivity AH, which is strongly T dependent Fig.30a, as the sum of the skew-scattering term aMxx

and the intrinsic term bMxx2 , viz.,

AH = aMxx + bMxx2 . 2.14

The quantity bM is the intrinsic AHE conductivityAH-int.

FIG. 27. Color online The anomalous Hall effect in MnSi. aHall resistivity yx vs H in MnSi at selected T from 5 to 200 K.At high T, yx is linear in H but gradually acquires an anoma-lous component yx =yx−R0B with a prominent “knee” fea-ture below TC=30 K. b Matching of the field profiles of theanomalous Hall resistivity yx to the profiles of 2M, treatingSH and R0 as adjustable parameters. Note the positive curva-ture of the high-field segments. From Lee et al., 2007.

FIG. 28. Color online Comparison of the anomalous Hallconductivity xy

A and the ordinary term xyN measured in a 1 T

field in MnSi. xyA , inferred from the measured M solid curve

and Eq. 2.13, is strictly independent of . Its T dependencereflects that of MT. xy

N 2 is calculated from R0. The con-ductivity at zero H, , is shown as a dashed curve. FromLee et al., 2007.

FIG. 29. Color online Weak-field peak anomaly in the Halleffect of MnSi under pressure. −yx vs H in MnSi under hydro-static pressure P=11.4 kbar at several TTC, with H nomi-nally along 111. The large Hall anomaly observed electron-like in sign arises from a new chiral contribution xy

C to thetotal Hall conductivity. In the phase diagram inset the shadedregion is where xy

C is resolved. The non-Fermi-liquid region isshaded. From Lee et al., 2009.



To separate the two terms, Zeng et al. plotted thequantity AH /MTxx against xx with T as a parameter.For T0.8TC, the plot falls on a straight line with asmall negative intercept solid squares in Fig. 30b. Theintercept yields the skew-scattering term aM /M,whereas the slope gives the intrinsic term bM /M.From the constant slope, they derive their main conclu-sion that AH-int varies linearly with M.

To account for the linear-M dependence, Zeng et al.2006 identified the role of long-wavelength spin waveswhich cause fluctuations in the local direction of Mxand hence of . They calculated the reduction inAH-int and showed that it varies linearly with M Fig.30c.

6. Layered dichalcogenides

The layered transition-metal dichalcogenides are com-prised of layers weakly bound by the van der Waalsforce. Parkin and Friend 1980 showed that a largenumber of interesting magnetic systems may be synthe-sized by intercalating 3d magnetic ions between the lay-ers.

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 Fe1/4TaS2, the easy axis of M is parallel

to c normal to the TaS2 layers. Morosan et al. 2007observed that the curves of M vs H display very sharpswitching at the coercive field at all TTC 160 K. Inthis system, the large ordinary term xy

NH complicates theextraction of the AHE term xy

AH. Converting the yx-Hcurves to xy-H curves Fig. 31, Checkelsky et al. 2008inferred that the jump magnitude xy equals 2xy

AH byassuming that Eq. 2.13 is valid. This method provided adirect measurement of xy

AH without knowledge of R0. Asshown in Fig. 31b, both the inferred xy

AH and measuredM are nearly T independent below 50 K, but the formerdeviates sharply downward above 50 K. Checkelsky etal. 2008 proposed that the deviation represents a largenegative inelastic-scattering contribution xy

in that in-volves scattering from chiral textures of the spins whichincrease rapidly as T approaches TC

−. In support, theyshowed that the curve of xy

in /MT vs T matches withinthe resolution that of T2 with T=T−0.

7. Spin glasses

In canonical spin glasses such as dilute magnetic alloysAuFe, AgMn, and CuMn, localized magnetic momentsof randomly distributed Fe or Mn ions interact with eachother through the Ruderman-Kittel-Kasuya-Yoshida in-teraction mediated by conduction electrons Binder and

FIG. 30. Color online The anomalous Hall effect in Mn5Ge3.a The T dependence of AH in Mn5Ge3 before solid squaresand after solid circles subtraction of the skew-scattering con-tribution. The thin curve is the resistivity . b Plots ofAH / Mxx vs xx before solid squares and after solidcircles subtraction of skew-scattering term. c Comparison ofcalculated AH-int vs Mz open circles with experimental val-ues before squares and after solid circles subtraction ofskew-scattering term. From Zeng et al., 2006.

FIG. 31. Color online Jumps in the AHE conductivity in theFe-doped chalcogenide TaS2. a Hysteresis loops of xy vs Hin Fe1/4TaS2 calculated from the measured and yx curves.The linear portions correspond to xy

NH, while the jump magni-tude xy equals 2xy

AH. b Comparison of xy with the mag-netization M measured at 0.1 T. Within the resolution, xyseems to be proportional to M below 50 K but deviates sharplyfrom M at higher temperatures, reflecting the growing domi-nance of a negative inelastic-scattering term xy

in . From Check-elsky et al., 2008.



Young, 1986; Campbell and Senoussi, 1992. Then, uponcooling, there occurs an equilibrium phase transition tothe spin glass state, for instance, at Tg=28 and 24 K forAuFe 8 at. % Fe and AuMn 8 at. % Mn, respectivelyPureur et al., 2004. The theoretical understanding ofthe spin-glass transition remains controversial. In fact,from the experimental viewpoint, the spin-glass transi-tion appears to be almost independent of the strength ofthe spin anisotropy due to the Dzyaloshinsky-Moriya in-teraction, though there is no finite-temperature spin-glass transition in three-dimensional 3D random iso-tropic Heisenberg ferromagnets. A chirality-drivenmechanism for the spin-glass transition has been sug-gested by Kawamura 1992, who pointed out that forsmall spin anisotropy the nature of the spin-glass transi-tion is dominated by that of a transition to a chiral glassKawamura, 1992. He also argued that the AHE due tothe spin chirality, explained in Sec. II.B.4, should appearin this class of materials and be useful as an experimen-tal test of the mechanism Kawamura, 2003.

The Hall transport experiments in these systemsstarted with pioneering works of McAlister and Hurd1976 and McAlister 1978, who measured the Hall re-sistivity of zero-field-cooled samples at a low field andfound a shoulder in the Hall coefficient around the spin-glass transition. Pureur et al. performed more detailedmeasurements across the spin-glass transition Fig. 32.Below Tg, it clearly showed a bifurcation of the anoma-lous Hall coefficient Rs depending on the condition ofcooling the sample, namely, whether the field cooling orthe zero-field cooling is adopted Pureur et al., 2004.Taniguchi et al. also performed the Hall transport mea-surements on the canonical spin-glass AuFe, reproduced

the above result, and showed that xy basically scaleswith the magnetic susceptibility at least at the field H=0.05 T, as shown in Fig. 33. They also analyzed thedata of xy by assuming

Rs = Rs0 + CX + X

nlM2 , 2.15

which was introduced by Tatara and Kawamura 2002and Kawamura 2003 for the anomalous Hall coefficientRs in the empirical relation xyR0H+RsM. Here X

and X nl are the linear susceptibility and the nonlinear

susceptibility of the uniform spin chirality. Their experi-mental results of Rs are shown in Fig. 34, which indicatethat after removing the bifurcation of M by dividing xyby M, the data contain a bifurcation irrespective of theapplied field strength and thus the value of M. Since theresistivity does not contain the hysteresis, this suggests

FIG. 32. The anomalous Hall coefficient Rs for AuMn 8 at. %Mn and AuFe 8 at. % Fe. under field-cooled and zero-field-cooled protocols. From Pureur et al., 2004.

FIG. 33. The Hall resistivity xy and magnetization M simulta-neously measured for AuFe 8 at. % Fe at a field H=500 G.From Taniguchi et al., 2004.

FIG. 34. The temperature dependence of Rs=xy /M for AuFe8 at. % at several magnetic fields The arrows indicate thefreezing temperature at each field strength. From Taniguchi etal., 2004.



that the linear chirality susceptibility X bifurcates be-low the freezing temperature Tf. This is consistent withthe chirality hypothesis for the spin-glass transition pro-posed by Kawamura 1992 in that the chirality shows acusp at the transition temperature.

The chirality-induced contribution to the AHE in re-entrant alloys of AuFe with higher Fe concentrationswas also found Pureur et al., 2004; Taniguchi et al., 2004;Fabris et al., 2006, where a cusp appears in at the reen-trant spin-glass transition at Tk=75 T from the collinearferromagnetic phase TC165 K for AuFe 18 at. % FePureur et al., 2004. In this low-temperature phase be-low Tk, the transverse components of spins freeze andhence the material becomes a ferromagnet having a qua-sistatic spin canting. Then, they attributed it to an onsetof this noncoplanarity and the associated spin chiralityat Tk.

Direct experimental estimates on the noncoplanarchiral spin configuration or the spin chirality togetherwith microscopic arguments are required and will needfor further studies.

E. Localization and the AHE

The role of localization in the anomalous Hall effect isan important issue. For a review of theoretical works,see Woelfle and Muttalib 2006.

The weak-localization effect on the normal Hall effectdue to the external magnetic field has been studied byAltshuler et al. 1980 and Fukuyama 1980, and therelation xy

WL /xy=2xxWL /xx has been obtained,

where OWL represents the correction of the physicalquantity O by the weak-localization effect. This meansthat the Hall coefficient is not subject to the change dueto the weak localization, i.e., xy

NH=0. An early experi-ment by Bergmann and Ye 1991 on the anomalousHall effect in the ferromagnetic amorphous metalsshowed almost no temperature dependence of theanomalous Hall conductivity, while the diagonal conduc-tivity shows the logarithmic temperature dependence.Langenfeld and Wolfle 1991, Woelfle and Muttalib2006, Muttalib and Woelfle 2007, and Misra et al.2009 studied theoretically the model

H = k,

kck,† ck, +

i

k,k,

eik−k·Ri

V + ik k · Jick,† ck,, 2.16

where Ji is proportional to the angular momentum ofthe impurity at Ri and the term containing it describesthe spin-orbit scattering. They discussed the logarithmiccorrection to the anomalous Hall conductivity in thismodel and found that Coulomb anomaly terms vanishedidentically; the weak-localization correction was cut offby the phase-breaking lifetime due to the skew scat-tering, explaining the experiment of Bergmann and Ye1991. Dugaev et al. 2001 found that for the two-dimensional case the correction to the Hall conductivitywas logarithmic in the ratio maxtr /SO,tr /, with tr

being the transport lifetime and SO being the lifetimedue to the spin-orbit scattering. More recently, Mitra etal. 2007 first observed the logarithmic temperature de-pendence of the anomalous Hall conductivity in the ul-trathin film of polycrystalline Fe of sheet resistance Rxxless than 3 k. They defined the quantities NQ= 1/R0L00Q /Q for the physical quantity Q with re-spect to the reference temperature T=T0=5 K by Q=QT−QT0, R0=RxxT0, and L00=e2 /22. Definingthe coefficients AR and AH by

Nxx = AR lnT0

T ,

2.17

Nxy = 2AR − AAHlnT0

T .

Figure 35 shows the R0 dependence of the coefficientsAR and AAH defined in Eq. 2.17.

The change in the interpretation comes from the factthat the phase-breaking lifetime in the Fe film ismostly from scattering by the magnons and not fromskew scattering, which allows a temperature regimewhere max1/s ,1 /SO,H!1/!1/tr s—spin-flipscattering time; H—internal magnetic field in the ferro-magnet. In this region, they found that the weak-localization effect can lead to coefficient of the logarith-mic temperature dependence of xy proportional to thefactor xy

SSM / xySSM+xy

SJM, where xySSM xy

SJM is the con-tribution from the skew-scattering side-jump mecha-nism. Assuming that ratio xy

SJM /xySSM 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. Theabsence of the logarithmic term of Bergmann and Ye1991 is interpreted as being due to a large ratio ofxy

SJM /xySSM in their sample. It is interesting that the ratio

xySJM /xy

SSM can be estimated from the coefficient of thelogarithmic term. Another recent study on weak local-

FIG. 35. Color online The resistance R0 temperature de-pendence of the coefficients AR and AAH defined in Eq. 2.17.Different symbols correspond to different methods of samplepreparation. From Mitra et al., 2007.



ization was done by Meier et al. 2009 on granular fer-romagnetic materials which were found to be in agree-ment with normal metals.

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 two dimensions all the states are local-ized for any disorder Lee and Ramakrishnan, 1985. Inthe quantum Hall system, however, the extended statescan survive at discrete energies at the center of thebroadened density of states at each Landau level. Thisextended state carries the quantum Hall current. Fieldtheoretical formulation of this localization problem hasbeen developed Prange and Girvin, 1987. In the pres-ence of the external magnetic field, a Chern-Simonsterm appears in the nonlinear sigma model whose coef-ficient is xy. Therefore, the scaling variables are xx andxy, i.e., the two-parameter scaling theory should be ap-plied instead of the single-parameter scaling. It has beendiscussed that the scaling trajectory in the xy-xx planehas the fixed point at xy ,xx= „n+1/2e2 /h ,0…,where 0 is some finite value and xy scales to the quan-tized value ne2 /h when the initial value given by theBoltzmann transport theory lies in the range n−1/2e2 /hxy

0 n+1/2e2 /h. In contrast to thisquantum Hall system, there is no external magnetic fieldor the Landau-level formation in the anomalous Hallsystem, and it is not trivial that the same two-parameterscaling theory applies to this case.

M. Onoda and Nagaosa 2003 studied this problemusing the generalized Haldane model Haldane, 1988which shows the quantum Hall effect without the exter-nal magnetic field. They calculated the scaling functionof the localization length in the finite-width stripesample in terms of the iterative transfer matrix methodby MacKinnon and Kramer 1983 and found that two-parameter scaling holds even without an external mag-netic field or Landau-level formation. For the experi-mental realization of this quantized anomalous Halleffect, xy

0 given by Boltzmann transport theory with-out the quantum correction is larger than e2 / 2h, and

the temperature is lower than TLocFe−cxx0/xy

0, where

F is the Fermi energy and c is a constant of the order ofunity. Therefore, the Hall angle xy

0 /xx0 should not be

so small, hopefully of the order of 0.1. However, in theusual case, the Hall angle is at most 0.01, which makesthe quantized anomalous Hall effect rather difficult torealize.

III. THEORY OF THE AHE: QUALITATIVECONSIDERATIONS

In this section, we review recent theoretical develop-ments as well as the early theoretical studies of theAHE. First, we give a pedagogical discussion on the dif-ference between the normal Hall effect due to the Lor-entz force and the AHE Sec. III.A. In Sec. III.B, wediscuss the topological nature of the AHE. In Sec. IV,

we present a wide survey of the early theories from amodern viewpoint in order to bring them into the con-text of the present linear transport theoretical formal-isms now used as a framework for AHE theories.

A. Symmetry considerations and analogies between normalHall effect and AHE

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

The AHE is one of the fundamental transport phe-nomena in solid-state physics. Its occurrence is a directconsequence of broken time-reversal symmetry in theferromagnetic state, T. The charge current J is T odd,i.e., it changes sign under time reversal. On the otherhand, the electric field E is T even. Therefore, Ohm’s law

J = E 3.1

relates two quantities with different T symmetries, whichimplies that the conductivity must be associated withdissipative irreversible processes, and indeed we knowthat the Joule heating Q=E2 /2 always accompaniesthe conductivity in Eq. 3.1. This irreversibility appearsonly when we consider macroscopic systems with con-tinuous 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., 2008:

K ;r,r;B = K ;r,r ;− B , 3.2

where K ;r ,r ;B is the response of a physical quan-tity at position r to the stimulus conjugate to the quan-tity at position r with frequency . Here = ±1specifies the symmetry property of with respect tothe T operation. B suggests a magnetic field but repre-sents any time-reversal breaking field. In the case of aferromagnet B can be associated with the magnetizationM, the spontaneously generated time-reversal symmetrybreaking field of a ferromagnet. The conductivity tensorab at a given frequency is proportional to the current-current response function. We can therefore make theidentification →Ja, →Jb, where Ji i=x ,y ,z are thecomponents of the current operator. Since ==−1,we can conclude that

ab ;B = ba ;− B . 3.3

Hence, we conclude that ab is symmetric with respectto a and b in systems with T symmetry. The antisymmet-ric part ab−ba is finite only if T symmetry isbroken.

Now we turn back to irreversibility for the generalform of the conductivity ab, for which the dissipation isgiven by Landau et al. 2008,



Q =14

abab

* + baEaEb*

=14

abReab + baReEaEb

*

+ Imab − baImEaEb* . 3.4

The real part of the symmetric combination ab+ba andthe imaginary part of the antisymmetric combinationab−ba contribute to the dissipation, while the imagi-nary part of ab+ba and the real part of ab−ba rep-resent the dispersive dissipationless responses. There-fore, Reab−ba, which corresponds to the Hallresponse, 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 mechanismsfor the AHE.

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

F = −e

cv H , 3.5

which produces an acceleration of the electron perpen-dicular to its velocity v and H. For free electrons thisleads to circular cyclotron motion with frequency c=eH /mc. The Lorentz force leads to charge accumula-tions of opposite signs on the two edges of the sample.In the steady state, the Lorentz force is balanced by theresultant transverse electric field, which is observed asthe Hall voltage VH. In the standard Boltzmann-theoretical approach, the equation for the electron dis-tribution function fp ,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. Setting F=−eE+ v

c H and usingthe relaxation time approximation for the collision term,−1f− feq, with feq being the distribution function inthermal equilibrium, one can obtain the steady-state so-lution to this Boltzmann equation for a uniform systemas

fp = feqp + gp , 3.7

with

gp = − efeq

v · E −

e

mcH E , 3.8

where order H2 terms have been neglected and the mag-netic field Hz is assumed to be small, i.e., c!1. Thecurrent density J=−ed3p / 23 p

m fp is obtainedfrom Eq. 3.8 to order OE and OEH as

J = E + HH E , 3.9

where

= −23 d3p

23v2e2feq

3.10

is the conductivity, while

H =23 d3p

23v2e2feq

eHz

mc3.11

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

fields as E=Eyey and H=Hzez. Then the Hall current isalong the x direction and we write it as

Jx = HEy, 3.12

with the Hall conductivity H=ne32Hz /m2c with n be-ing the electron density. The Hall coefficient RH is de-fined as the ratio VH /JxHz, and from Eqs. 3.10 and3.11 one obtains

RH = −1

nec. 3.13

This result is useful to determine the electron density nexperimentally since it does not contain the relaxationtime . Note here that for fixed electron density n, Eq.3.13 means H Hxx

2 and that H H, independent ofthe relaxation time or equivalently xx. In otherwords, the ratio H /xx=c!1. This relation will becompared with the AHE case below.

What happens in the opposite limit c1? In thislimit, the cyclotron motion is completed many timeswithin the lifetime . This implies that the closed cyclo-tron motion is repeated many times before being dis-rupted by scattering. When treated quantum mechani-cally the periodic classical motion leads to kinetic-energy quantization and Landau-level formation. As iswell known, in the 2D electron gas realized in semicon-ductor heterostructures, Landau-level quantization leadsto the celebrated quantum Hall effect Prange andGirvin, 1987. If we consider the free electrons, i.e., com-pletely neglecting the potential both periodic and ran-dom and the interaction among electrons, one can showthat H is given by e2 /h", where " is the filling factor ofthe Landau levels. In combination with electron local-ization, gaps at integer filling factors lead to quantizationof H. For electrons in a two-dimensional crystal, theHall conductance is still quantized, a property which canbe traced to the topological properties of Bloch statewavefunctions discussed below Prange and Girvin,1987.

Now consider the magnetic-field effect for electronsunder the influence of the periodic potential. For sim-plicity we study the tight-binding model,

H = ij

tijci†cj. 3.14

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

tij → tij expiaij , 3.15

with



aij =e

c

i

j

dr · Ar . 3.16

Note that the phase factor is periodic with the period 2with respect to its exponent. Since the gauge transforma-tion

ci → ci expii 3.17

together with the redefinition

aij → aij + i − j 3.18

keeps the Hamiltonian invariant, aij itself is not a physi-cal quantity. Instead, the flux =aij+ajk+akl+ali persquare plaquette is the key quantity in the problem seeFig. 36

One may choose, for example, a gauge in which tij

along the directions ±y is a real number t, while tii+x= t exp−iiy to produce a uniform flux distribution ineach square plaquette. Here iy is the y component of thelattice point position i= ix , iy. The problem is that the iydependence breaks the periodicity of the Hamiltonianalong the y axis, which invalidates the Bloch theorem.This corresponds to the fact that the vector potentialAr= −Hy ,0 for the uniform magnetic field H is y de-pendent, so that momentum component py is no longerconserved. However, when / 2 is a rational numbern /m, where n and m are integers that do not share acommon factor, one can enlarge the unit cell by m timesalong the y direction to recover the periodicity becauseexp−iiy+m=exp−iiy. Therefore, there appear msubbands in the first-Brillouin zone, each of which ischaracterized by a Chern number related to the Hallresponse as described in Sec. III.B Thouless et al.,1982. The message here is that an external commensu-rate magnetic field leads to a multiband structure with

an enlarged unit cell, leading to the quantized Hall re-sponse when the chemical potential is within the gapbetween the subbands. When / 2 is an irrationalnumber, on the other hand, one cannot define the Blochwavefunction and the electronic structure is describedby the Hofstadter butterfly Hofstadter, 1976.

Since flux 2 is equivalent to zero flux, a commensu-rate magnetic field can be equivalent to a magnetic-fielddistribution whose average flux is zero. Namely, the totalflux penetrating the m plaquettes is 2n, which isequivalent to zero. Interestingly, spatially nonuniformflux distributions can also lead to quantum Hall effects.This possibility was first considered by Haldane 1988,who studied a tight-binding model on the honeycomblattice. He introduced complex transfer integrals be-tween the next-nearest-neighbor sites in addition to thereal one between the nearest-neighbor sites. The result-ant Hamiltonian has translational symmetry with respectto the lattice vectors, and the Bloch wavefunction can bedefined as the function of the crystal momentum k in thefirst-Brillouin zone. The honeycomb lattice has two sitesin the unit cell, and the tight-binding model producestwo bands separated by the band gap. The wavefunctionof each band is characterized by the Berry-phase curva-ture, which acts like a magnetic field in k space, as willbe discussed in Secs. III.B and V.A.

The quantum Hall effect results when the Fermi en-ergy is within this band gap. Intuitively, this can be in-terpreted 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 weightsand contributes to the Hall response. The sum of theHall responses from all the bands, however, is zero asexpected. Therefore, the “polarization” of Hall re-sponses between bands in a multiband system is a gen-eral and fundamental mechanism of the Hall response,which is distinct from the classical picture of the Lorentzforce.

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 is the spin, and p is the mo-mentum of the electron. This Hamiltonian can be writ-ten as

HSOI = ASOI · p , 3.20

with ASOI= e /2m2c2sV acting as the effectivevector potential. Therefore, in the magnetically orderedstate, i.e., when s is ordered and can be regarded as a cnumber, ASOI plays a role similar to the vector potentialof an external magnetic field. Note that the Bloch theo-rem is valid even in the presence of the spin-orbit inter-action since it preserves the translational symmetry ofthe lattice. However, the unit cell may contain morethan two atoms and each atom may have multiple orbit-

i + x

i + y

φ

φ

φ

φ

φ

φi = ( i x , i y )

yii

t e⋅φ

t

i + x

i + y

φ

φ

φ

φφ

φ

φi = ( i x , i y )

yii

t e⋅φ

t

FIG. 36. Tight-binding model on a square lattice under themagnetic flux for each plaquette. We choose the gaugewhere the phase factor of the transfer integral along x direc-tion is given by e−iiy, with iy being the y component of thelattice point position i= ix , iy.



als. Hence, the situation is similar to that describedabove for the commensurate magnetic field or theHaldane model.

The following simple model is instructive. Considerthe tight-binding model on a square lattice given by

H = − i,,a=x,y

tssi,† si+a, + H.c. +

i,,a=x,ytppi,a,

† pi+a,a,

+ H.c. + i,,a=x,y

tspsi† pi+a,a, + H.c.

+ i,

pi,x,† − ipi,y,

† pi,x, + ipi,y, . 3.21

On each site, we put the three orbitals s, px, and pyassociated with the corresponding creation and annihila-tion operators. The first three terms represent the trans-fer of electrons between the neighboring sites as shownby Fig. 37. 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 component ofthe spin moment is coupled to that of the orbital mo-ment. This six-band model can be reduced to a two-bandmodel in the ferromagnetic state when only the =+1component is retained. By the spin-orbit interaction, thep orbitals are split into

p ± =12

px ± ipy 3.22

at each site, with the energy separation 2. Therefore,when only the lower energy state p− and the s orbitals are considered, the tight-binding Hamiltonian be-comes H=k#

†khk#k, with #k= sk,=1 ,pk,−,=1T

and

hk

= s − 2tscos kx + cos ky 2tspi sin kx + sin ky2tsp− i sin kx + sin ky p + tpcos kx + cos ky .

3.23

Note that the complex orbital p− is responsible for thecomplex off-diagonal matrix elements of hk. This pro-duces the Hall response as one can easily see from for-mulas Eqs. 1.5 and 1.6 in Sec. I.B. When the Fermienergy is within the band gap, the Chern number foreach band is ±1 when s−4tsp+2tp and they are zerootherwise. This can be understood by considering theeffective Hamiltonian near k=0, which is given by

hk = + mz + 2tspkyx − kxy , 3.24

with = s−4ts+ p+2tp /2 and m= s−4ts− p+2tp, which is essentially the Dirac Hamiltonian in2+1 dimensions. One can calculate the Berry curva-ture b±k=bzkez defined in Eq. 1.6 for the upper and lower bands as

bzk =m

2k2 + m23/2 . 3.25

The integral over the two-dimensional wave vector k

= kx ,ky leads to Hall conductance H= 12 sgnme2 /h in

this continuum model when only the lower band is fullyoccupied Jackiw, 1984. The distribution Eq. 3.25 indi-cates that the Berry curvature is enhanced when the gapm is small near this avoided band crossing. Note that thecontinuum model Eq. 3.24 cannot describe the value ofthe Hall conductance in the original tight-binding modelsince the information is not retained over all the first-Brillouin zone. Actually, it has been proven that the Hallconductance is an integer multiple of e2 /h due to thesingle valudeness of the Bloch wave function within thefirst-Brillouin zone. However, the change of the Hallconductance between positive and negative m can bedescribed correctly.

Onoda and Nagaosa 2002 demonstrated that a simi-lar scenario also emerges for a six-band tight-bindingmodel of t2g orbitals with SOI. They showed that theChern number of each band can be nonzero leading to anonzero Hall response arising from the spontaneousmagnetization. This suggests that the anomalous Hall ef-fect can be of topological origin, a reinterpretation ofthe intrinsic contribution found by Karplus and Lut-tinger 1954. Another mechanism which can produce aflux is a noncoplanar spin structure with an associatedspin chirality as discussed in Sec. II.B.

The considerations explained here have totally ne-glected the finite lifetime due to the impurity and/orphonon scattering, a generalization of the parameter cwhich appeared in the case of the external magneticfield. A full understanding of disorder effects usually re-quires the full power of the detailed microscopic theo-ries reviewed in Sec. V. The ideas explained in this sec-tion though are sufficient to understand why the intrinsicBerry-phase contribution to the Hall effect can be so

t p

t s

t s p

p x - i p y

s

FIG. 37. Color online Tight-binding model on a square lat-tice for a three-band model made from s and px− ipy orbitalswith polarized spins. The transfer integrals between s and px− ipy orbitals become complex along the y direction in a waythat is equivalent to effective magnetic flux.



important. 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 c re-lated to this flux is that of the spin-orbit interaction forEq. 3.20 and hence of the order of 30 meV in 3dtransition-metal elements. This corresponds to a mag-netic field of the order of 300 T. Therefore, one can ex-pect that the intrinsic AHE is easier to observe com-pared with the quantum 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 consider-ations and KL theory. A more elementary treatment isgiven by Ong and Lee 2006.

The topological expression given in Eq. 1.5 was firstobtained by Thouless, Kohmoto, Nightingale, and NijsTKNN Thouless et al., 1982 for Bloch electrons in a2D insulating crystal lattice immersed in a strong mag-netic field H. They showed that each band is character-ized by a topological integer called the Chern number

Cn − dkxdky

22 bnzk . 3.26

According to Eq. 1.5, the Chern number Cn gives thequantized Hall conductance of an ideal 2D insulator,viz., xy=e2Cn /h. As mentioned, these topological argu-ments were subsequently applied to semiclassical trans-port theory Sundaram and Niu, 1999 and the AHEproblem in itinerant ferromagnets Jungwirth, Niu, andMacDonald, 2002; Onoda and Nagaosa, 2002, and theequivalence to the KL expression Karplus and Lut-tinger, 1954 for H was established.

Although the intrinsic Berry-phase effect involves theentire Fermi sea, as is clear from Eq. 1.5, it is usuallybelieved that transport properties of Fermi liquids atlow temperatures relative to the Fermi energy should bedependent only on Fermi-surface properties. This is asimple consequence of the observation that at thesetemperatures only states near the Fermi surface can beexcited to produce nonequilibrium transport.

The apparent contradiction between conventionalFermi liquid theoretical ideas and the Berry-phasetheory of the AHE was resolved by Haldane 2004, whoshowed that the Berry-phase contribution to the AHEcould be viewed in an alternate way. Because of its to-pological nature, the intrinsic AHE, which is most natu-rally expressed as an integral over the occupied Fermi

sea, can be rewritten as an integral over the Fermi sur-face. We start with the topological properties of theBerry phase. The Berry-phase curvature bnk is gauge in-variant and divergence-free except at quantized mono-pole and antimonopole sources with the quantum ±2,i.e.,

k · bnk = i

qni3k − kni, qni = ± 2 . 3.27

These monopoles and antimonopoles appear at isolatedk points in three dimensions. This is because in complexHermitian eigenvalue problems accidental degeneracycan 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

ij = ije2

h

K

2, 3.28

where

K = n

Kn, 3.29

Kn = −1

2

F.B.Z.d3kfnkbnk. 3.30

We note that if the band dispersion nk does not cross F,Kn

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 independentprimitive reciprocal lattice vectors Kohmoto, 1985.

We next discuss the nonquantized part of the anoma-lous Hall conductivity. In a real material with multipleFermi surface FS sheets indexed by , we can describeeach sheet by kF

s, where s= s1 ,s2 is a parametriza-tion of the surface. It is convenient to redefine theBerry-phase connection and curvature as

ais = a„ks… · siks , 3.32

bs = ijsiajs , 3.33

with ij as the rank-2 antisymmetric tensor.By integrating Eq. 3.30 by parts, eliminating the in-

tegration over the Brillouin zone boundary for eachband, and dropping the band indices, we may write atT=0 the vector K in Eq. 3.29 as

K = a

CaGa +

K, 3.34



K =1

2

S

ds1 ∧ ds2bskFs

+1

4i

GiS

ids · as . 3.35

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

i is theintersection of the Fermi surface S with the Brillouinzone boundary i where kF

s jumps by Gi. Note thatthe quantity

1

2

S

ds1 ∧ ds2bs 3.36

gives an integer Chern number. Hence gauge invariancerequires that

1

2

S

ds1 ∧ ds2bs = 0. 3.37

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

IV. EARLY THEORETICAL STUDIES OF THE AHE

A. Karplus-Luttinger theory and the intrinsic mechanism

The pioneering work of Karplus and Luttinger KLKarplus and Luttinger, 1954 was the first theory of theAHE fully based on Bloch states #nk. As a matter ofcourse, their calculations uncovered the important roleplayed by the mere existence of bands and the associ-ated overlap of Bloch states. The KL theory neglects alllattice disorder, so that the Hall effect it predicts is basedon an intrinsic 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, asmentioned, is described by the Hamiltonian given byEq. 3.19, where Vr 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 Hamil-tonian 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, 4.1

where H0 is the Hamiltonian in the absence of SOI andHE is the perturbation due to the applied electric field E.In simple models the consequences of magnetic ordercan 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 satu-rated moment. The matrix element of HE=−eEbxb canbe written as

nkHEnk = − eEbin,n

kbk,k + ik,kJb

nnk ,

4.2

where the “overlap” integral

Jbnnk =

drunk* r

kbunkr 4.3

is regular function of k. In hindsight, Jbnnk may be rec-

ognized as the Berry-phase connection ank discussedin Sec. I.B.1.

We next divide HE into HEr +HE

a , where HEa HE

r cor-responds to the first second term on the right-hand siderhs of Eq. 4.2. Absorbing HE

r into the unperturbedHamiltonian, i.e., Hp=H0+HSOI+HE

r , KL treated the re-maining term HE

a as a perturbation. Accordingly, thedensity matrix was written as

= 0Hp + 1, 4.4

where 0Hp is the finite-temperature equilibrium den-sity matrix and 1 is the correction. KL assumed that 1gives only the ordinary conductivity, whereas the AHEarises solely from 0Hp. Evaluating the average veloc-ity va as va=Tr0va, they found that

va = − ieEbn,k

0Enkp Ja

nnk , 4.5

where 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 exter-nal electric field, i.e., HE

r . The AHE contribution arisesfrom the interband coherence induced by an electronicfield and not from the more complicated rearrangementsof states within the partially occupied bands.

A second assumption of KL is that, in 3d metals, theSOI energy HSO!F the Fermi energy and W thebandwidth, so that it suffices to consider HSO to leadingorder. Using Eq. 3.19, the AHE response is then pro-portional to Ms, consistent with empirical relationship2.1. More explicitly, to first order in HSOI, KL obtained

v = −e

m2k,n

0nkE · vnkFnk, 4.6

where nk is the energy of the Bloch state for H0, isthe averaged value of interband energy separation, andFnk is the force inkHSOI,pnk. This gives the anoma-lous Hall coefficient

Rs 2e2HSO

m2 m

m*2, 4.7

where Fe /cHSOv, m* is the effective mass, is thenumber of incompletely filled d orbitals, and is theresistivity.

Note that the 2 dependence implies that the off-diagonal Hall conductivity H is independent of thetransport lifetime , i.e., it is well defined even in theabsence of disorder, in striking contrast with the diago-



nal conductivity . The implication that H=H2 varies

as the square of was immediately subjected to exten-sive 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=n,k /k.

In the Bloch basis

r#nk = eik·rrunk , 4.8

the N-orbital Hamiltonian for a given k may be de-composed into the 2N2N matrix hk, where spin hasbeen included, viz.,

hk = n,m

nkhkmkan†kamk . 4.9

The corresponding current operator for a given k is

Jt,k = n,m

nkhkk

mkan†kamk . 4.10

According to the Hellman-Feynman theorem,

nkhkk

nk =nk

k

, 4.11

the diagonal contribution to the current is with n=m

Jintrat = − e

nkvgkNnk , 4.12

where Nnk=an†kank is the occupation number in the

state nk. Equation 4.12 corresponds to the expressionin Boltzmann transport theory mentioned before. Theinterband matrix element nkhk /kmk with nm corresponds to interband transitions. As shown byKL, the interband matrix elements have profound con-sequences for the Hall current, as has become apparentfrom the Berry-phase approach.

B. Extrinsic mechanisms

1. Skew scattering

In a series of reports, Smit 1955, 1958 mounted aserious challenge to the basic findings of KL. In thelinear-response transport regime, the steady-state cur-rent balances the acceleration of electrons by E againstmomentum relaxation by scattering from impuritiesand/or phonons. Smit pointed out that this balancingwas entirely absent from the KL theory and purportedto show that the KL term vanishes exactly. His reasoningwas that the anomalous velocity central to KL’s theory is

proportional to the acceleration k which must vanish onaverage at steady state because the force from E cancelsthat 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 ofSOI, the matrix element of the impurity scattering po-tential reads

ksVk,s = Vk,ks,s +i2

4m2c2 ss k · k .

4.13

Microscopic detailed balance would require that thetransition probability Wn→m between states n and m isidentical to that proceeding in the opposite directionWm→n. It holds, for example, in Fermi’s golden-ruleapproximation,

Wn→m =2

nVm2En − Em , 4.14

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

WkkA = − A

−1k k · Ms. 4.15

When the asymmetric scattering processes is includedcalled skew scattering, the scattering probability Wk→k is distinct from Wk→k.

Physically, scattering of a carrier from an impurity in-troduces a momentum perpendicular to both the inci-dent momentum k and the magnetization M. This leadsto a transverse current proportional to the longitudinalcurrent driven by E. Consequently, the Hall conductivityH and the conductivity are both proportional to thetransport lifetime . Equivalently, H=H

2 is propor-tional to the resistivity .

As mentioned in Secs. I and 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 nonmagnetichosts Au or Cu. At higher concentrations, these sys-tems become spin glasses. Although these systems donot exhibit magnetic ordering even at T as low as 0.1 K,their Hall profiles H vs H display an anomalous compo-nent derived from polarization of the magnetic local mo-ments Fig. 38a.

Empirically, the Hall coefficient RH has the form

RH = R0 + A/T , 4.16

where R0 is the nominally T-independent OHE coeffi-cient and A is a constant see Fig. 38b. Identifying thesecond term with the paramagnetic magnetization M=H, where T 1/T is the Curie susceptibility of thelocal moments, we see that Eq. 4.16 is of the Pughform Eq. 2.1.

Many groups have explored the case in which andH can be tuned over a large range by changing themagnetic-impurity concentration ci. Hall experiments on



these systems in the period 1970–1985 confirmed theskew-scattering prediction H . This led to the conclu-sion that the KL theory had been “experimentally dis-proved.” As mentioned in Sec. I, this invalid conclusionignores the singular role of TRI breaking in ferromag-nets. While Eq. 4.16 indeed describes skew scattering,the dilute Kondo system respects TRI in zero H. Thisessential qualitative difference between ferromagnetsand systems with easily aligned local moments impliesessential differences in the physics of their Hall effects.

2. Kondo theory

Kondo proposed a finite-temperature skew-scatteringmodel in which spin waves of local moments at finite Tlead to asymmetric scattering Kondo, 1962. In the KLtheory, the moments of the ferromagnetic state are itin-erant: the electrons carrying the transport current alsoproduce the magnetization. Kondo has considered theopposite limit in which nonmagnetic s electrons scatterfrom spin-wave excitations of the ordered d-band localmoments with the interaction term JSn ·s. Retainingterms linear in the spin-orbit coupling and cubic in J,Kondo derived an AHE current that arises from transi-tion probabilities containing the skew-scattering termkk. The T dependence of H matches well espe-cially near TC the H vs T profiles measured in Ni Jan,1952; Jan and Gijsman, 1952; Lavine, 1961 and Fe Jan,1952 Fig. 39.

Kondo has noted that a problem with his model is thatit predicts that H should vanish when the d orbital an-gular momentum is quenched as in Gd whereas H isobserved to be large. A second problem is that theoverall scale for H fixed by the exchange energies F0,F1, and F2 is too small by a factor of 100 compared withexperiment. Kondo’s model with the s-d spin-spin inter-action replaced by a d spin-s orbital interaction hasalso been applied to antiferromagnets Maranzana,1967.

3. Resonant skew scattering

Resonant skew scattering arises from scattering of car-riers from virtual bound states in magnetic ions dis-solved in a metallic host. Examples are XM, where X=Cu, Ag, and Au and M=Mn, Cr, and Fe. The proto-typical model is a 3d magnetic ion embedded in a broads band, as described by the Anderson model Hewson,1993. As shown in Fig. 40, a spin-up s electron tran-siently occupies the spin-up bound state which liesslightly below F. However, a second spin-down s elec-tron cannot be captured because the on-site repulsionenergy U raises its energy above F. As seen in Fig. 40,the SOI causes the energies Ed

m of the d orbitals la-belled by m to be individually resolved. Here =± isthe spin of the bound electron. The applied H merelyserves to align spin =+ at each impurity.

The scattering of an incident wave eik·r is expressed bythe phase shifts d

m E in the partial-wave expansion ofthe scattered wave. The phase shift is given by

FIG. 38. The dependences of the Hall resistivity in AuFe ona magnetic field and b temperature. From Hurd, 1972.

FIG. 39. Comparison of the curve of H vs T measured in Feby Jan with Kondo’s calculation. From Kondo, 1962.

FIG. 40. Sketch of the broadened virtual bound states of a 3dmagnetic impurity dissolved in a nonmagnetic metallic host.The SOI lifts the degeneracy of the d levels indexed by m, themagnetic quantum number. From Fert and Jaoul, 1972.



cot dm E =

Edm − E

, 4.17

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

m E is independent of m andthere is no Hall current. The splitting caused by SOIresults in a larger density of states at F for the orbital mcompared with −m in the case where the spin-up elec-tron is more than half filled as shown in Fig. 40. Becausethe phase shift is sensitive to occupancy of the impuritystate Friedel sum rule, we have d

m d−m. This leads to

a right-left asymmetry in the scattering, and a large Hallcurrent ensues. Physically, a conduction electron inci-dent with positive z component of angular momentumm hybridizes more strongly with the virtual bound statethan one with −m. This results in more electrons beingscattered to the left than the right. When the spin-updensity of states is filled less than half, i.e., Ed+

0 0, thedirection becomes the opposite, leading to a sign changeof the AHE.

Explicitly, the splitting of Edm is given by

Ed±m = Ed±

0 ± 12m±, 4.18

where is the SOI energy for spin . Using Eq. 4.18in Eq. 4.17, we have, to order /2,

2m = 2

0 m

2sin2 2

0 +

2m2

42 sin3 20 cos 2

0 .

4.19

Inserting the phase shifts into the Boltzmann transportequation, we find that the Hall angle $xy

/xx for

spin is, to leading order in /,

$ = 35

sin22

0 − 1sin 1, 4.20

where 1 is the phase shift of the p-wave channel whichis assumed to be independent of m and s Fert andJaoul, 1972; Fert et al., 1981. After thermal averaging

over , we can obtain an estimate of the observed Hallangle $. Its sign is given by the position of the energylevel relative to F. With the rough estimate /0.1and sin 10.1, we have $10−2. Without the resonantscattering enhancement, the typical value of $ is 10−3.

Heavy-electron systems are characterized by a verylarge resistivity above a coherence temperature Tcohcaused 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. 41, the Hall coefficient RH increases to a broadmaximum at Tcoh before decreasing sharply to a smallvalue in the low-T coherent-band regime. Resonantskew scattering has also been applied to account for thestrong T dependence of the Hall coefficient in CeCu2Si2,UBe13, and UPt3 Fert et al., 1981; Coleman et al., 1985;Fert and Levy, 1987. In addition, the theory of Kondowas tested in the context of resonant skew scattering in aseries of Ag thin films doped with rare-earth impurities.The interplay of the spin-orbit coupling of the d5 statesof the impurities and the pure spin s-f exchange wasfound to be proportional to g−2 and therefore vanishedfor the Gd impurity s-state ion, g=2 but not for theother impurities in the series Fert and Friederich, 1976.

4. Side jump

An extrinsic mechanism distinct from skew scatteringis side jump Berger, 1970 Fig. 3. Berger consideredthe scattering of a Gaussian wave packet from a spheri-cal potential well of radius R given by

Vr = 2

2mk2 − k1

2 r R

0 r R 4.21

in the presence of the SOI term HSO= 1/2m2c2r−1V /rSzLz, where Sz Lz is the z com-

FIG. 41. The Hall coefficient RH in heavy electron systems. Left panel: The T dependence of RH in CeAl3 open triangle, UPt3solid square, UAl2 solid circle, and a single crystal of CeRu2Si2 circle dot symbol. The field H is along the c axis. Right panel:Interpretation of the RH-T curve based on skew scattering from local moments. After Hadzicleroux et al., 1986 and Lapierre et al.,1987. From Fert and Levy, 1987.



ponent of the spin orbital angular momentum. For awave-packet incident with wave vector k, Berger foundthat the wave packet suffers a displacement y trans-verse to k given by

y = 16kc

2, 4.22

with c= /mc as the Compton wavelength. For kkF1010 m−1 in typical metals, y310−16 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*q 3.4 104, 4.23

with q= m* /32mn%2 /Enmmqpn2. HereEnm0.5 eV is the gap between adjacent d bands,0.3 is the overlap integral, 2.510−10 m is

the nearest-neighbor distance, and %=−2 /2m2c2r−1V /r0.1 eV is the atomic SOI energy.The factor in Eq. 4.23 is essentially the ratio of theelectron rest mass energy mc2 to the energy gap Enm.With this enhancement, the transverse displacement isy0.810−11 m, which renders the contribution rel-evant to the AHE.

Because the side-jump contribution to H is indepen-dent of , it is experimentally difficult to distinguish fromthe KL mechanism. We return to this issue in Sec. V.A.

C. Kohn-Luttinger theoretical formalism

1. Luttinger theory

Partly motivated by the objections raised by Smit,Luttinger 1958 revisited the AHE problem using theKohn-Luttinger formalism of transport theory Kohnand Luttinger, 1957; Luttinger and Kohn, 1958. Em-ploying a systematic expansion in terms of the impuritypotential &, he solved the transport equation for thedensity matrix and listed several contributions to theAHE current, including the intrinsic KL term, the skew-scattering term, and other contributions. He found thatto zeroth order in &, the average velocity is obtained asthe sum of the three terms v

11, u, and vb defined by

Luttinger 1958,

v11 = − ieE J

k

−J

k

k=0 F

3ni& , 4.24

u = − ieE J

k

−J

k

k=0, 4.25

and

vb = ieE J

k

−J

kC − FT

0 , 4.26

with ni as the impurity concentration. For the definitionof T

0, see Eq. 4.27 of Luttinger 1958. The first termv

11 is the skew-scattering contribution, while the sec-

ond u is the velocity obtained by KL Karplus and Lut-tinger, 1954. The third term v

b is another term of thesame order as u.

The issues raised by comparing intrinsic versus extrin-sic AHE mechanisms involve several fundamental issuesin the theory of transport and quantum systems awayfrom equilibrium. Following Smit 1955, 1958 and Lut-tinger 1958, we consider the wavefunction expanded interms of Bloch waves #’s, viz.,

#t =

at#, 4.27

where = n ,k represents the band index n and wavevector k. The corresponding expectation value of theposition operator x is given by

x = i

a

k

a* + i

,

aa*J

,, 4.28

where J, was defined in Eq. 4.3. Taking the quantum-

statistical average using the density matrix T,

= aa*, the expectation value can be written as

x = i T,

k

k=k+ i

,

TJ,. 4.29

It may be seen that the second term in Eq. 4.29 doesnot contribute to the current because it is a regular func-tion of k, and the expectation value of its time derivativeis zero. Finally, one obtains

v = i T,

k

k=k4.30

by taking the time derivative of Eq. 4.29.Smit 1955, 1958 assumed at= ae−inkt and ob-

tained the diagonal part as

iT, =nk

k

a2 , 4.31

while the off-diagonal part averages to zero because ofthe oscillatory factor e−i−t. Inserting Eq. 4.31 intoEq. 4.30 and using T= a2, one obtains the usualexpression for the velocity, i.e.,

v =

T,nk

k

, 4.32

which involves the group velocity but not the anomalousvelocity. A subtlety in expression Eq. 4.30 is clarifiedby writing =0+ fest, where est is the adiabatic factorand T=sfest. This leads to

v = is fnk,nk

k

k=k

, 4.33

which approaches a finite value in the limit s→0. Thismeans thatfnk,nk /kk=k1/s, i.e., a singular func-tion of s. This is in sharp contrast to f itself, which is aregular function as s→0, which KL estimated. They con-sidered the current or velocity operator instead of the



position operator, the latter of which is unbounded andeven ill-defined for periodic boundary conditions.

Luttinger 1958 argued that the ratio v11 /u equals

F /3ni&, which may become less than 1 for large impu-rity concentration ni. However, if we assume & is com-parable to F, the expansion parameter is F /, i.e.,v

11 /u=F /.The “metallicity” parameter F / plays a key role in

modern quantum-transport theory, especially in theweak localization and interaction theory Lee and Ra-makrishnan, 1985. Metallic conduction corresponds toF /1. More generally, if one assumes that theanomalous Hall conductivity H is first order in the spin-orbit energy , it can be written in a scaling form as

H =e2

ha

Ff

F , 4.34

where the scaling function can be expanded as

fx = n=−1

cnxn. 4.35

Here cn’s are constants of the order of unity. The leadingorder term c−1x−1 corresponds to the skew-scatteringcontribution , while the second constant term c0 is thecontribution found by KL. If this expansion is valid, theintrinsic contribution by KL is always smaller than theskew-scattering contribution in the metallic region with /F!1. One should note, however, that the right-handside of this inequality should really have a smaller valueto account for the weakness of the skew-scattering am-plitude.

When both are comparable, i.e., /F1, one needsto worry about the localization effect and the system isnearly insulating, with a conduction that is of the hop-ping type. This issue is discussed in more detail in Secs.II.A, II.E, and V.D.

2. Adams-Blount formalism

Adams and Blount 1959 expressed the KL theory ina way that anticipates the modern Berry-phase treat-ment by introducing the concept of “field-modified en-ergy bands” and the “intracell” coordinate. They consid-ered the diagonal part of the coordinate matrix in theband n as

xc = i

p

+ Xnnp , 4.36

which is analogous to Eq. 4.2 obtained by KL. The firstterm is the Wannier coordinate identifying the latticesite, while the second term the intracell coordinate lo-cates the wave-packet centroid inside a unit cell. Signifi-cantly, the intracell nature of X

nn implies that it involvesvirtual interband transitions. Although the motion of thewave packet is confined to the conduction band, its po-sition inside a unit cell involves virtual occupation ofhigher bands whose effects appear as a geometric phaseOng and Lee, 2006. They found that the curl kXnn

acted like an effective magnetic field that exists in kspace.

Application of an electric field E leads to an anoma-lous Luttinger velocity, which gives a Hall current thatis manifestly dissipationless. This is seen by evaluatingthe commutation relationship of xc

and xc". We have

xc,xc

" = i X"nn

p

−X

nn

p" = i"Bnp, 4.37

where we have defined the field Bnp=pXnnp. Thisis analogous to the commutation relationship among thecomponents of =p+ e /cA in the presence of the vec-tor potential, i.e., ,"= i"rAr= iBr.The anomalous velocity arises from the fictitious mag-netic field Bnp which exists in momentum space andnoncommutation of the gauge-covariant coordinatesxc’s. This insight anticipated the modern idea of Berry-

phase curvature see Secs. III.B and V.A for details.Taking the commutator between xc

and the Hamil-tonian Hnn=Enp−F"xc

", we obtain

vnn = − ixc,Hnn =Enp

p− F Bnp . 4.38

The second term on the right-hand side is called theanomalous velocity. Adams and Blount reproduced theresults of Luttinger 1958 and demonstrated it for thesimple case of a uniform field Bnp=D. As recognizedby Smit 1955, 1958, the currents associated with theanomalous velocities driven by E and by the impuritypotential mutually cancel in the steady state. However,D introduces corrections to the “driving term” and the“scattering term” in the transport equation. As a conse-quence, the current J is

J = ne2k−

23

Edf0

dEF

m−

F D

+ 2k2

3mAF D . 4.39

Note that the current arises entirely from the average ofthe “normal current.” However, the second term in Eq.4.39 is similar to the anomalous velocity and is consis-tent with the conclusions of KL Karplus and Luttinger,1954 and Luttinger 1958. The consensus now is thatthe KL contribution exists. However, in the clean limit→, the leading contribution to the AHE conductivitycomes from the skew-scattering term. A more completediscussion of the semiclassical treatment is given in Sec.V.A.

3. Nozieres-Lewiner theory

Adopting the premises of Luttinger’s theory Lut-tinger, 1958, Nozieres and Lewiner 1973 investigated asimplified model comprised of one conduction and onevalence band to derive all possible contributions to theAHE. The Fermi level F was assumed to lie near thebottom of the conduction band. Integrating over the va-lence band states, Nozieres and Lewiner derived an ef-



fective Hamiltonian for a state k in the conduction band.The derived position operator is reff=r+, where thenew term which involves the SOI parameter is givenby

= − k S . 4.40

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

JAHE = 2Ne2E S , 4.41

where N is the carrier concentration and S is the aver-aged spin polarization. The sign is opposite to that of the

intrinsic term Jintrinsic=−2Ne2E S obtained from theSOI in an ideal lattice. They identified all the terms asarising 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 cancella-tions occur among the various contributions and suchcancellations can be traced back to the fact that theBerry curvature is independent of k. In particular, it isoften the case that the intrinsic contribution surviveseven when lies inside an energy gap, i.e., N=0 theside-jump contribution is zero in this case. This issuelies at the heart of the discussion of the topological as-pect of the AHE. For a recent consideration on the side-jump mechanism, see Rashba 2008.

An experimental test of the theoretical expectationsin nonmagnetic semiconductors, induced by magneticfield and using spin-dependent optical excitation of thecarriers, was done by Chazalviel 1975 with reasonableagreement with theory.

V. LINEAR TRANSPORT THEORIES OF THE AHE

In this section, we describe the three linear responsetheories now used to describe the AHE. All three theo-ries are formally equivalent in the F limit. Thethree theories make nearly identical predictions; thesmall differences between the revised semiclassicaltheory and the two formalism of the quantum theory arethoroughly understood at least for several different toymodel systems. The three different approaches haverelative advantages and disadvantages. Considering allthree provides a more nuanced picture of AHE physics.Their relative correspondence has been shown analyti-cally in several simple models Sinitsyn et al., 2007; Sin-itsyn, 2008. The generalized semiclassical Boltzmanntransport theory which takes the Berry phase into ac-count is reviewed in Sec. V.A; this theory has the advan-tage of greater physical transparency, but it lacks thesystematic character of the microscopic quantum-mechanical theories whose machinery deals automati-cally with the problems of interband coherence. An-other limitation of the semiclassical Boltzmannapproach is that the “quantum correction” to the con-ductivities, i.e., the higher order terms in /F, which

lead to the Anderson localization and other interestingphenomena Lee and Ramakrishnan, 1985, cannot betreated systematically.

The microscopic quantum-mechanical linear responsetheories based on the Kubo formalism and the Keldyshnonequilibrium Green’s function formalism are re-viewed in Secs. V.B.1 and V.C, respectively. These for-mulations of transport theory are organized differentlybut are essentially equivalent in the linear regime. In theKubo formalism, which is formulated in terms of theequilibrium Green’s functions, the intrinsic contributionis more readily calculated, especially when combinedwith first-principles electronic structure theory in appli-cations to complex materials. The Keldysh formalismcan more easily account for finite lifetime quantum scat-tering effects and take account of the broadened quasi-particle spectral features due to the self-energy while atthe same time maintaining a structure more similar tothat of semiclassical transport theory Onoda et al.,2006a. We contrast these techniques by comparing theirapplications to a common model, the ferromagneticRashba model in two dimensions, which has been stud-ied intensively in recent years Sec. V.D.

A. Semiclassical Boltzmann approach

As should be clear from the complex phenomenologyof the AHE in various material classes discussed in Sec.II, it is not easy to establish a one-size-fits-all theory forthis phenomenon. 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 the Keldysh and Kubo linear re-sponse theoretical formalisms and are organized aroundan expansion in disorder strength, characterized by thedimensionless quantity /F. A small value for this pa-rameter may be taken as a definition of the good metal.We start with semiclassical theory, however, because ofits greater physical transparency. In this section, we out-line the modern version Sinitsyn et al., 2007; Sinitsyn,2008 of the semiclassical transport theory of the AHE.This theoretical augments standard semiclassical trans-port theory by accounting for coherent band mixing bythe external electric field which leads to the anomalousvelocity contribution and by a random disorder poten-tial which leads to side jump. For simple models inwhich a comparison is possible, the two theories i.e., ithe Boltzmann theory with all side-jump effects formu-lated now in a proper gauge-invariant way and theanomalous velocity contribution included and ii themetallic limit of the more systematic Keldysh and Kuboformalism treatments give identical results for theAHE. This section provides a more compact version ofthe material presented by Sinitsyn 2008, to which werefer readers interested in further detail. Below we take=1 to simplify notation.

In semiclassical transport theory one retreats from amicroscopic description in terms of delocalized Blochstates to a formulation of transport in terms of the dy-namics of wave packets of Bloch states with a well-



defined band momentum and position n ,kc ,rc andscattering between Bloch states due to disorder. The dy-namics of the wave packets between collisions can betreated by an effective Lagrangian formalism. The wave-packet distribution function is assumed to obey a classi-cal Boltzmann equation which in a spatially uniform sys-tem takes the form Sundaram and Niu, 1999

fl

t+ kc ·

fl

kc= −

l

l,lfl − l,lfl . 5.1

Here the label l is a composition of band and momentan ,kc labels and l,l, the disorder averaged scatteringrate between wave packets defined by states l and l, isto be evaluated fully quantum mechanically. The disor-der potential and fields should vary slowly 10 nm forthe semiclassical approximation to be valid. The semi-classical description is useful in clarifying the physicalmeaning and origin of the different mechanisms contrib-uting to the AHE. However, as explained in this reviewcontributions to the AHE which are important in a rela-tive sense often arise from interband coherence effectswhich are neglected in conventional transport theory.The traditional Boltzmann equation therefore requireselaboration in order to achieve a successful descriptionof the AHE.

There is a substantial literature Smit, 1955; Berger,1970; Jungwirth, Niu, and MacDonald, 2002 on the ap-plication of Boltzmann equation concepts to AHEtheory see Sec. IV. However, stress was often placedonly on one of the several possible mechanisms, creatingmuch confusion. A cohesive picture has been lackinguntil recently Sinitsyn et al., 2005, 2006; Sinitsyn, 2008.In particular, a key problem with some prior theory wasthat it incorrectly ascribed physical meaning to gaugedependent quantities. In order to build the correctgauge-invariant semiclassical theory of AHE we musttake the following steps Sinitsyn, 2008: i obtain theequations of motion for a wave packet constructed fromspin-orbit coupled Bloch electrons; ii derive the effectof scattering of a wave packet from a smooth impurity,yielding the correct gauge-invariant expression for thecorresponding side jump; iii use the equations of mo-tion and the scattering rates in Eq. 5.1 and solve for thenonequilibrium distribution function, carefully account-ing for the points at which modifications are required toaccount for side jump; and iv utilize the nonequilib-rium distribution function to calculate the dc anomalousHall currents, again accounting for the contribution ofside jump to the macroscopic current.

The validity of this approach is partially established inthe following sections by direct comparison with fullymicroscopic calculations for simple model systems inwhich we are able to identify each semiclassically de-fined mechanisms with a specific part of the microscopiccalculations.

1. Equation of motion of Bloch state wave packets

We begin by defining a wave packet in band n cen-tered at position rc with average momentum kc,

'n,kc,rcr,t =

1V

k

wkc,rckeik·r−rcunkr . 5.2

A key aspect of this wave packet is that the complexfunction, sharply peaked around kc, must have a specificphase factor in order to have the wave packet centeredaround rc. This can be shown to be Sundaram and Niu,1999; Marder, 2000

wkc,rck = wkc,rc

kexpik − kc · an , 5.3

where anunk ikunk is the Berry connection of theBloch state see Sec. I.B. We can generate dynamics forthe wave packet parameters kc and rc by constructing asemiclassical Lagrangian from the quantum wave func-tions

L = 'n,kc,rci

t− H0 + eV'n,kc,rc

= kc · rc + kc · ankc − Ekc + eVrc . 5.4

All the terms in the above Lagrangian are common toconventional semiclassical theory except for the secondterm, which is a geometric term in phase space depend-ing only on the path of the trajectory in this space. Thisterm is the origin of the momentum-space Berry-phaseBerry, 1984 effects in anomalous transport in the semi-classical formalism. The corresponding Euler-Lagrangeequations of motion are

kc = − eE , 5.5

rc =Enkc

kc− kc bnkc , 5.6

where bnkc=an is the Berry curvature of the Blochstate. Compared to the usual dynamic equations forwave packets formed by free electrons, a new termemerges due to the nonzero Berry curvature of theBloch states. This term, which is already linear in elec-tric field E, is of the Hall type and as such will give risein the linear transport regime to a Hall current contri-bution from the entire Fermi sea, i.e., jHall

int =−e2EV−1kf0Enkbnk.

2. Scattering and the side jump

From the theory of elastic scattering we know that thetransition rate l,l in Eq. 5.1 is given by the T-matrixelement of the disorder potential

ll 2Tll2l − l . 5.7

The scattering T matrix is defined by Tll= lV#l,where V is the impurity potential operator and #l is the

eigenstate of the full Hamiltonian H=H0+ V that satis-

fies the Lippman-Schwinger equation #l= l+ l−H0

+ i(−1V#l#l is the state which evolves adiabaticallyfrom l when the disorder potential is turned on slowly.For weak disorder one can approximate the scattering



state #l by a truncated series in powers of Vll= lVl,

#l l + l

Vll

l − l + i(l + ¯ . 5.8

Using this expression in the above definition of the Tmatrix and substituting it into Eq. 5.7, one can expandthe scattering rate in powers of the disorder strength

ll = ll2 + ll

3 + ll4 + ¯ , 5.9

where ll2=2Vll

2disl−l,

ll3 = 2

l

VllVllVlldis

l − l − i(+ c.c.l − l , 5.10

and so on.We can always decompose the scattering rate into

components that are symmetric and antisymmetric inthe state indices: ll

s/all±ll /2. In conventionalBoltzmann theory the AHE is due solely to the antisym-metric 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 tothe Bloch state lifetime . Since ll

2 is symmetric, theleading contribution to ll

a appears at order V3. Partlyfor this reason the skew-scattering AHE conductivitycontributions is always much smaller than the longitudi-nal conductivity. It is this property which motivates theidentification below of additional transport mechanismwhich contributes to the AHE and can be analyzed insemiclassical terms. The symmetric part of ll

3 is notessential since it only renormalizes the second-order re-sult for ll

2 and the antisymmetric part is given by

ll3a = − 22

l

l − l

ImVllVllVlldisl − l . 5.11

This term is proportional to the density of scatterers, ni.Skew scattering has usually been associated directly with3a, neglecting in particular the higher order term ll

4a

which is proportional to ni2 and should not be disre-

garded because it gives a contribution to the AHE whichis of the same order as the side-jump contribution con-sidered below. This is a common mistake in the semiclas-sical analyses of the anomalous Hall effect Sinitsyn,2008.

Now we come to the interesting side-jump story.Because the skew-scattering conductivity is small, wehave to include effects which are absent in conventionalBoltzmann transport theory. Remarkably it is possible toprovide a successful analysis of one of the main addi-tional effects, the side-jump correction, by means of acareful semiclassical analysis. As mentioned previouslyside jump refers to the microscopic displacement rl,lexperienced by a wave packets formed from a spin-orbitcoupled Bloch states when scattering from state l to

state l under the influence of a disorder potential. In thepresence of an external electric field side jump leads toan energy shift Ul,l=−eE ·rl,l. Since we are assumingonly elastic scattering, an upward shift in potential en-ergy requires a downward shift in band energy and viceversa. We therefore need to adjust Eq. 5.1 by addingll,l

s f0l /leE ·rl,l to the rhs.But what about the side-jump itself? An expression

for the side-jump rl,l associated with a particular tran-sition can be derived by integrating rc through a transi-tion Sinitsyn, 2008. We can write

rn,c =d

dt'n,kc,rc

r,tr'n,kc,rcr,t

=d

dt dr

V dk dkwkw*ke−ik·rreik·r

unk* runkreiEk−Ekt/

=dEnkc

dkc+

d

dtcelldr dkw*kunkr

i

kwkunkr . 5.12

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. 5.12 it appears that the scattering shift will go as

rl,l uli

kul − uli

kul . 5.13

This quantity has usually been associated with the sidejump, although it is gauge dependent and therefore ar-bitrary in value. The correct expression for the sidejump is similar to this one, at least for the smooth impu-rity potentials situation, but was derived only recently bySinitsyn et al. 2005, 2006,

rll = uli

kul − uli

kul

− Dk,k argulul , 5.14

where arga 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 detailsof the impurity potential or of the scattering process. Asthis discussion shows, the side-jump contribution to mo-tion during a scattering event is analogous to the anoma-lous velocity contribution to wave-packet evolution be-tween collisions, with the role of the disorder potentialin the former case taken over in the latter case by theexternal electric field.



3. Kinetic equation for the semiclasscial Boltzmanndistribution

Equations 5.7 and 5.14 contain the quantum-mechanical information necessary to write down a semi-classcial Boltzmann equation that takes into accountboth the change of momentum and the coordinate shiftduring scattering in the presence of a driving electricfield E. Keeping only terms up to linear order in theelectric field the Boltzmann equation reads Sinitsyn etal., 2006

fl

t+ eE · v0l

f0ll

= − l

llfl − fl −f0l

leE · rll , 5.15

where v0l is the usual group velocity v0l=l /k. Notethat for elastic scattering we do not need to take accountof the Pauli blocking which yields factors like fl1− flon the rhs of Eq. 5.15 and that the collision terms arelinear in fl as a consequence. For further discussion ofthis point see Appendix B of Luttinger and Kohn1955. This Boltzmann equation has the standard formexcept for the coordinate shift contribution to thecollision integral explained above. Because of theside-jump effect, the collision term does not vanishwhen the occupation probabilities fl are replaced bytheir thermal equilibrium values when an external elec-tric field is present: f0l− f0l−eE ·rll−f0l /leE ·rll0. Note that the term containing l,lfl

should be written as l,ls fl−l,l

afl. In making this simpli-fication we are imagining a typical simple model inwhich the scattering rate depends only on the angle be-tween k and k. In that case, ll,l

a =0 and we can ig-nore a complication which is primarily notational.

The next step in the Boltzmann theory is to solve forthe nonequilibrium distribution function fl to leading or-der in the external electric field. We linearize by writingfl as the sum of the equilibrium distribution f0l andnonequilibrium corrections,

fl = f0l + gl + gladist, 5.16

where we split the nonequilibrium contribution into twoterms, gl and gadist, in order to capture the skew-scattering effect. gl and gadist solve independent self-consistent time-independent equations Sinitsyn et al.,2006,

eE · v0lf0l

l= −

l

llgl − gl 5.17

and

l

llgladist − gl

adist −f0l

leE · rll = 0. 5.18

In Eq. 5.14 we have noted that rll=rll. To solve Eq.5.17 we further decompose gl=gl

s+gla1+gl

a2 so that

l

ll3agl

s − gls +

l

ll2gl

a1 − gla1 = 0, 5.19

l

ll4agl

s − gls +

l

ll2gl

a2 − gla2 = 0. 5.20

Here gls is the usual diagonal nonequilibrium distribu-

tion function which can be shown to be proportional toni

−1. From Eq. 5.11 and ll3ani, it follows that gl

3a

ni−1. Finally, from ll

4ani2 and Eq. 5.20, it follows

that gl4ani

0 illustrating the dangers of ignoring the ll4a

contribution to l,l. One can also show from Eq. 5.18that gl

adistni0.

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

We are now at the final stage where we use the non-equilibrium distribution function derived from Eqs.5.17–5.20 to compute the anomalous Hall current. Todo so we need first to account for all contributions to thevelocity of semiclassical particles that are consistent withthis generalized semiclassical Boltzmann analysis.

In addition to the band state group velocity v0l=l /k, we must also take into account the velocitycontribution due to the accumulations of coordinateshifts after many scattering events, another way in whichthe side-jump effect enters the theory, and the velocitycontribution from coherent band mixing by the electricfield the anomalous velocity effect Nozieres andLewiner, 1973; Sinitsyn et al., 2006; Sinitsyn, 2008,

vl =l

k+ bl eE +

l

llrll. 5.21

Combining Eqs. 5.16 and 5.21 we obtain the total cur-rent

j = el

flvl = el

f0l + gls + gl

a1 + gla2 + gl

adist

l

k+ l eE +

l

llrll . 5.22

This gives five nonzero contributions to the AHE up tolinear order in E,

xytot = xy

int + xyadist + xy

sj + xysk1 + xy

sk2-sj. 5.23

The first term is the intrinsic contribution,

xyint = − e2

lf0lbz,l. 5.24

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

xyadist = e

lgl

adist/Eyv0lx 5.25

follows from the distribution function correction due toside jumps, while



xysj = e

lgl/Ey

l

llrllx 5.26

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 xy

adist and for xysj , there is, unsurprisingly, an intimate

relationship between those two contributions. In most ofthe literature, xy

adist is usually considered to be part ofthe side-jump contribution, i.e., xy

adist+xysj →xy

sj . We dis-tinguish between the two because they are physicallydistinct and appear as separate contributions in the mi-croscopic formulation of the AHE theory.

Finally, xysk1 and xy

sk2-sj are contributions arising fromthe asymmetric part of the collision integral Sinitsyn,2008,

yxsk1 = − e

lgl

a1/Exv0ly ni−1, 5.27

yxsk2-sj = − e

lgl

a2/Exv0ly ni0. 5.28

According to the old definition of skew scatteringboth could be viewed as skew-scattering contributionsbecause they originate from the asymmetric part of thecollision term Smit, 1955. However, if instead we defineskew scattering as the contribution proportional to ni

−1,i.e., linear in , as in Sec. I.B, it is only the first contri-bution Eq. 5.27 which is the skew scattering Lut-tinger, 1958; Leroux-Hugon and Ghazali, 1972. The sec-ond contribution Eq. 5.28 was generally discarded inprior semiclassical theories, although it is parametricallyof the same size as the side-jump conductivity. Explicitquantitative estimates of yx

sk2-sj so far exist only for themassive 2D Dirac band Sinitsyn et al., 2007. In parsingthis AHE contribution we incorporate it within the fam-ily of side-jump effects due to the fact that it is propor-tional to ni

0, i.e., independent of xx. However, it is im-portant to note that this side-jump contribution has nophysical link to the side step experienced by a semiclass-cial quasiparticle upon scattering. An alternative termi-nology for this contribution is intrinsic skew scattering todistinguish its physical origin from side-jump deflectionsSinitsyn, 2008, but, to avoid further confusion, we sim-ply include it as a contribution to the Hall conductivitywhich is of the order of xx

0 and originates from scatter-ing.

As we will see below, when connecting the micro-scopic formalism to the semiclasscial one, xy

int can bedirectly identified with the single bubble Kubo formal-ism contribution to the conductivity, xy

adist+xysj +xy

sk2

constitute the usually termed ladder-diagram vertex cor-rections to the conductivity due to scattering and there-fore it is natural to group them together although theirphysical origins are distinct. xy

sk1 is identified directlywith the three-scattering diagram used in the literature.This comparison has been made specifically for twosimple models, the massive 2D Dirac band Sinitsyn et

al., 2007 and the 2D Rashba with exchange model Bo-runda et al., 2007.

B. Kubo formalism

1. Kubo technique for the AHE

The Kubo formalism relates the conductivity to theequilibrium 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 it can be found inmany textbooks. Instead, we emphasize the key issues instudying the AHE within this formalism and how it re-lates to the semiclassical formalism described in Sec.V.A.4.

For the purpose of studying the AHE it is best toreformulate the current-current Kubo formula for theconductivity in the form of the Bastin formula see Ap-pendix A of Crépieux and Bruno 2001 which can bemanipulated into the more familiar form for the conduc-tivity of the Kubo-Streda formula for the T=0 Hall con-ductivity xy=xy

Ia+xyIb+xy

II , where

xyIa =

e2

2VTrvxGRFvyGAFc, 5.29

xyIb = −

e2

4VTrvxGRFvyGRF

+ vxGAFvyGAFc, 5.30

xyII =

e2

4V

−

+

dfTrvxGRvyGR

d

− vxGR

dvyGR + c.c.GRF − GAF .

5.31

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

II was first derived byStreda in the context of studying the quantum Hall ef-fect Streda, 1982. In these equations GR/AF= F−H± i−1 are the retarded and advanced Green’s func-tions evaluated at the Fermi energy of the total Hamil-tonian.

Looking more closely at xyII we notice that every term

depends on products of the retarded Green’s functionsonly or products of the advanced Green’s functions only.It can be shown that only the disorder free part of xy

II isimportant in the weak disorder limit, i.e., this contribu-tion is zeroth order in the parameter 1/kFlsc. The onlyeffect 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. Itcan therefore be shown by a similar argument that ingeneral xy

Ib is of order 1/kFlsc and can be neglected inthe weak scattering limit Mahan, 1990. Thus, impor-tant disorder effects beyond simple quasiparticle life-



time broadening are contained only in xyIa. For these

reasons, it is standard within the Kubo formalism to ne-glect xy

Ib and evaluate the xyII contribution with a

simple lifetime broadening approximation to theGreen’s function.

Within this formalism the effect of disorder on thedisorder-configuration averaged Green’s function is cap-tured by the use of the T matrix defined by the integralequation T=W+WG0T, where W=iV0r−ri is adelta-scatterer potential and G0 are the Green’s functionof the pure lattice. From this one obtains

G = G0 + G0TG0 = G0 + G0)G . 5.32

Upon disorder averaging we obtain

) = Wc + WG0Wc + WG0WG0Wc + ¯ . 5.33

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

)z,k = niVk,k +ni

Vk

Vk,kG0k,zVk,k + ¯ ,

5.34

with Vk,k=Vk−k being the Fourier transform of thesingle impurity potential, which in the case of delta scat-terers is simply V0 see Fig. 44 for a graphical represen-

tation. Note that G and G0 are diagonal in momentumbut, due to the presence of spin-orbit coupling, nondi-agonal in spin index in the Pauli spin basis. Hence, the

lines depicted in Fig. 42 represent G and are in generalmatrices 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 xy

Ia and xyII Eqs. 5.29 and 5.31. This step

captures the intrinsic contribution to the AHE and the

effect of disorder on it, which is generally weak in me-tallic systems. This contribution is separately identifiedin Fig. 42a.

The so-called ladder diagram vertex corrections, alsoseparately identified in Fig. 42, contribute to the AHE atthe same order in 1/kFl as the intrinsic contribution. It isuseful to define a ladder-diagram corrected velocity ver-tex vFv+vF, where

vF =niV0

2

V k

GRFv + vFGAF ,

5.35

as depicted in Fig. 42b. Note again that vF and v=H0 /k 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. 42.

As may seem obvious from the above machinery, cal-culating the intrinsic contribution is not very difficult,while calculating the full effects of the disorder in a sys-tematic way beyond calculating a few diagrams is chal-lenging for any disorder model beyond the simple delta-scattering model.

Next we illustrate the full use of this formalism for thesimplest nontrivial model, massive 2D Dirac fermions,with the goals of illustrating the complexities present ineach contribution to the AHE and the equivalence ofquantum and semiclassical approaches. This model is ofcourse not directly linked to any real material reviewedin Sec. II and its main merit is the possibility of obtain-ing full simple analytical expressions for each of the con-tributions. A more realistic model of 2D fermions withRashba spin-orbit coupling will be discussed in Sec. V.C.The ferromagnetic Rashba model has been used to pro-pose a minimal model of AHE for materials in whichband crossing near the Fermi surface dominates theAHE physics Onoda et al., 2008.

The massive 2D Dirac fermion model is specified by

H0 = vkxx + kyy + z + Vdis, 5.36

where Vdis=iV0r−Ri, x and y are Pauli matrices,and the impurity free spectrum is k

±= ±2+ vk2, withk= k and the labels * distinguish bands with positiveand negative energies. We ignore in this simple modelspin-orbit coupled disordered contributions which canbe directly incorporated through similar calculations asin Crépieux and Bruno 2001. Within this model thedisorder averaged Green’s function is

GR =1

1/G0R − )R

=F + i + vkxx + kyy + − i1z

F − + + i+F − − + i−, 5.37

where =niV02 /4v2, 1= cos, $±=0±1 cos,

and cos = /vk2+2. Note that within this disordermodel 1/ni. Using the result in Eq. 5.37 one can

IIxyxy x x x

(a)xyxy

intrinsic “side-jump”

x x

j p

“skew scattering”

(b)

yv x xx x x

FIG. 42. Color online Graphical representation of the Kuboformalism application to the AHE. The solid lines are the dis-order averaged Green’s function G, the circles are the barevelocity vertex v=H0 /k, and the dashed lines withcrosses represent disorder scattering niV0

2 for the delta-scattermodel. b vy is the velocity vertex renormalized by vertexcorrections.



calculate the ladder diagram correction to the bare ve-locity vertex given by Eq. 5.35,

vy = 8v cos 1 + cos2

1 + 3 cos2 2x

+ v + vsin2

1 + 3 cos2 y, 5.38

where is evaluated at the Fermi energy. The details ofthe calculation of this vertex correction are described inAppendix A of Sinitsyn et al. 2007. Incorporating thisresult into xy

1a we obtain the intrinsic and side-jumpcontributions to the conductivity for F,

xyint =

e2

2Vk

TrvxGvyG = −e2 cos

4, 5.39

xysj =

e2

2Vk

TrvxGvyG

= −e2 cos

4 3 sin2

1 + 3 cos2 +

4 sin2

1 + 3 cos2 2 .

5.40

The direct calculation of the skew-scattering diagramsof Fig. 42 is xy

sk =−e2 /2nV0 vkF4 / 42+ vkF22and the final total result is given by

xy = −e2

4vkF2 + 21 +4vkF2

42 + vkF2

+3vkF4

42 + vkF22 −e2

2nV0

vkF4

42 + vkF22 .

5.41

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 thesemiclassical formalism the natural basis is the one thatdiagonalizes 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, inthe 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. V.A and obtain the fol-lowing results for the five distinct contributions: xy

int,xy

adist, xysj , xy

sk2-sj, and xysk1. Below we quote the results

for the complicated semiclassical calculation of eachterm see Sec. IV of Sinitsyn et al. 2007 for details,

xyint = −

e2

42 + vkF2, 5.42

xysj = xy

adist = −e2kF

2

2kF2 + 2kF

2 + 42, 5.43

xysk-sj = −

e23vkF4

4vkF2 + 242 + vkF22, 5.44

xysk = −

e2

2niV0

vkF4

42 + vkF22 . 5.45

The correspondence with the Kubo formalism resultscan be seen after a few algebra steps. The contributionsxy

int and xysk1 are equal in both cases Eqs. 5.39 and

5.41. As expected, the intrinsic contribution xyint is in-

dependent of disorder in the weak scattering limit andthe skew-scattering contribution is inversely propor-tional to the density of scatterers. However, recall that inSec. I.B we have defined the side-jump contribution asthe disorder contributions of zeroth order in ni, i.e., 0,as opposed to being directly linked to a side step in thescattering process in the semiclassical theory. Hence, it isthe sum of the three physically distinct processes xy

adist

+xysj +xy

sk2-sj which can be shown to be identical to Eq.5.40 after some algebraic manipulation and showngraphically in Fig. 43. Therefore, the old notion of asso-ciating the skew scattering directly with the asymmetricpart of the collision integral and the side jump with theside-step scattering alone leads to contradictions withtheir usual association with respect to the dependenceon or equivalently 1/ni. We also note that unlikewhat happens in simple models where the Berry curva-ture is a constant in momentum space, e.g., the standardmodel for electrons in a 3D semiconductor conductionband Nozieres and Lewiner, 1973, the dependences ofthe intrinsic and side-jump contributions are quite differ-ent with respect to parameters such as Fermi energy,exchange splitting, etc.

C. Keldysh formalism

Keldysh developed a Green’s function formalism ap-plicable even to the nonequilibrium quantum states, forwhich the diagram techniques based on Wick’s theoremcan be used Baym and Kadanoff, 1961; Kadanoff andBaym, 1962; Keldysh, 1964; Rammer and Smith, 1986;Mahan, 1990. Unlike with the thermal MatsubaraGreen’s functions, the Keldysh Green’s functions are de-fined for any quantum state. The price for this flexibilityis that one needs to introduce the path-ordered productfor the contour from t=−→ t= and back again fromt=→−. Correspondingly, four kinds of the Green’sfunctions, GR, GA, G, and G, need to be considered,although only three are independent Baym andKadanoff, 1961; Kadanoff and Baym, 1962; Keldysh,1964; Rammer and Smith, 1986; Mahan, 1990. 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 ther-mal Green’s function and Kubo formalism. Since ap-proximations are normally required in treating disor-dered systems, it is important to make them in a waywhich at least satisfies gauge invariance. In both formal-isms this is an important theoretical requirement which



requires some care. Roughly speaking, in the Keldyshformalism, GR and GA describe the single particle states,while G represents the nonequilibrium particle occupa-tion distribution and contains vertex corrections. There-fore, the self-energy and vertex corrections can betreated in a unified way by solving the matrix Dysonequation. This facilitates the analysis of some modelsespecially when multiple bands are involved.

Another and more essential advantage of Keldysh for-malism over the semiclassical formalism is that one cango beyond a finite order perturbative treatment of im-purity scattering strength by solving a self-consistentequation, as will be discussed in Sec. V.D. In essence, weare assigning a finite spectral width to the semiclassicalwave packet to account for an important consequence ofquantum scattering effects. In the Keldysh formalism,the semiclassical limit corresponds to ignoring the his-tory of scattering particles by keeping the two time la-bels in the Green’s functions identical.

We restrict ourselves below to the steady and uniformsolution. For more generic cases of electromagneticfields, see Sugimoto et al. 2007. Let x= t ,x be thetime-space coordinate. The Green’s functions depend ontwo space-time points x1 and x2, and the matrix Dysonequation for the translationally invariant system readsRammer and Smith, 1986

− Hp − ) ,p G ,p = 1,5.46

G ,p − Hp − ) ,p = 1,

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 dxeit−p·x/

¯ ,

5.47

with p= ,p. In this Dyson equation, the product isreserved for matrix products in band indices like thosethat also appear in the Kubo formalism.

In the presence of the external electromagnetic fieldA, we must introduce the mechanical or kinetic-energy-momentum variable,

X ;p = p + eAX , 5.48

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

GX ; /2i is the quantum mechanical generalizationof the semiclassical distribution function. When an ex-ternal electric field E is present, the equation of motionor, equivalently the Dyson equation retains the sameform as Eq. 5.46 when the product is replaced by theso-called Moyal product Moyal, 1949; Onoda et al.,2006b given by

= exp i− e2

E · p − p . 5.49

Henceforth, and denote the derivatives operating onthe right- and left-hand sides, respectively, and the sym-bol p= ,p is used to represent the mechanical energy-momentum . In this formalism, only gauge-invariantquantities appear. For example, the electric field E ap-pears instead of the vector potential A.

Expanding Eq. 5.49 in E and inserting the result intoEq. 5.46, one obtains the Dyson equation to linear or-der in E, corresponding to linear response theory. The

linear order terms GE and )E

in E are decomposed intotwo parts as

GE = GE,I

f + GEA − GE

Rf , 5.50

FIG. 43. Color online Graphical representation of the AHE conductivity in the chiral band eigenstate basis. The two bands ofthe two-dimensional Dirac model are labeled *. The subsets of diagrams that correspond to specific terms in the semiclassicalBoltzmann formalism are indicated. From Sinitsyn et al., 2007.



)E = )E,I

f + )EA − )E

Rf . 5.51

Here f represents the Fermi distribution function. Inthese decompositions, the first term on the rhs corre-sponds to the nonequilibrium deviation of the distribu-tion 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 ij=ij

I +ijII with

ijI = e22 dd+1p

2d+1iTrvipGEj,I

pf , 5.52

ijII = e22 dd+1p

2d+1iTr vipGEj

A p − GEj

R pf .

5.53

This is in the same spirit as the Streda version Streda,1982 of the Kubo-Bastin formula Kubo, 1957; Bastin etal., 1971. 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. 44 corresponding tothe T-matrix approximation. In this approximation, theself-consistent integral equation for the self-energy andGreen’s function can be solved and the solution can beused to evaluate the first and second terms in Eqs. 5.50and 5.51.

In general, Eqs. 5.52 and 5.53, together with theself-consistent equations for GR, GA, and G Onoda etal., 2008, define a systematic diagrammatic method forcalculating ij in the Streda decomposition Streda,1982 of the Kubo-Bastin formula Kubo, 1957; Bastin etal., 1971.

D. Two-dimensional ferromagnetic Rashba model—A minimalmodel

A useful model to study fundamental aspects of theAHE is the ferromagnetic two-dimensional 2D Rashbamodel Bychkov and Rashba, 1984,

Hptot =p2

2m− p m · ez − 0

z + Vx . 5.54

Here m is the electron mass, is the Rashba spin-orbitinteraction strength, 0 is the mean-field exchange split-

ting, = x , y , z and 0 are the Pauli and identity ma-

trices, ez is the unit vector in the z direction, and Vx=V0ir−ri is a -scatterer impurity potential with im-purity density ni. Quantum transport properties of thissimple but nontrivial model have been intensively stud-ied in order to understand fundamental properties of theAHE in itinerant metallic ferromagnets. The metallicRashba model is simple, but its AHE has both intrinsicand extrinsic contributions and has both minority andmajority spin Fermi surfaces. It therefore captures mostof the features that are important in real materials witha minimum of complicating detail. The model has there-fore received a lot of attention Dugaev et al., 2005; In-oue et al., 2006; Onoda et al., 2006b, 2008; Borunda et al.,2007; Kato et al., 2007; Nunner et al., 2007; Kovalev etal., 2008, 2009; Onoda, 2009.

The bare Hamiltonian has the band dispersion,

p =p2

2m− p, p = 2p2 + 0

2, 5.55

illustrated in Fig. 45a, and Berry-phase curvature,

bzp = 2p ip,bmpp,z =

220

2p3 ,

5.56

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

FIG. 44. Color online Diagrammatic representation of theself-energy in the self-consistent T-matrix approximation inthe Keldysh space, which is composed of the infinite series ofmultiple Born scattering amplitudes. From Onoda et al., 2008.

(b)

(a)

FIG. 45. Color online The intrinsic AHE conductivity in the2D Rashba model. a Band dispersion of the ferromagnetic2D Rashba model in the clean limit. b The intrinsic anoma-lous Hall conductivity of the Hamiltonian in Eq. 5.54 as afunction of the Fermi level F measured from the bottom ofthe majority band and /mnimpV0

2 /2, which is the Bornscattering amplitude for =0=0 without the vertex correction)E. For the parameter set 0=0.1, 2m=3.59, and 2mV0 /2

=0.6, with energy unit has been taken as 0,−. From Onoda etal., 2006b.



We can then obtain the intrinsic contribution to theAHE by integrating over occupied states at zero tem-perature Culcer et al., 2003; Dugaev et al., 2005,

xyAH-int =

e2

2h

1 −0

p„ − p… , 5.57

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

An important feature of xyAH-int is its enhancement in

the interval 0,+0,−, where it approaches a maxi-mum value close to e2 /2h no matter how small 0 isprovided that it is larger than /. Near p=0 the Berrycurvatures of the two bands are large and opposite insign. The large Berry curvatures translate into large in-trinsic Hall conductivities only when the chemicalpotential lies between the local maximum of oneband and the local minimum of the other. When Fis far away from the resonance energy, the contributionsfrom the momentum near p=0 cancel each otherout or does not appear, leading to a suppression ofxy

AH-inte2 /h0 /F, where the perturbation expan-sion in 0 is justified. This enhancement of the intrinsicAHE near avoided band crossings is illustrated in Fig.45b, where it is seen to survive moderate disorderbroadening of several times 0. This peaked featurearises from the topological nature of xy

AH-int. As a conse-quence it is important to note that the result is nonper-turbative in SOI; only a perturbative expansion on /Fis justified.

Culcer et al. 2003 were the first to study this model,obtaining Eq. 5.57. They were followed by Dugaev etal. 2005 where the intrinsic contribution was calculatedwithin the Kubo formalism. Although these studiesfound a nonzero xy

AH-int, they did not calculate all con-tributions arising from disorder some aspects of the dis-order treatment by Dugaev et al. 2005 were correctedby Sinitsyn et al. 2007. The intrinsic AHE xy

AH-int

comes form both xyI and xy

II . The first part xyI contains

the intraband contribution xyIa which is sensitive to the

impurity scattering vertex correction.The calculation of xy

Ia incorporating the effects ofdisorder using the Kubo formalism, i.e., incorporatingthe ladder vertex corrections side jump and the leadingOV0

3 skew-scattering contributions Sec. V.B.1, yieldsa vanishing xy

AH for the case where F is above the gap atp=0 i.e., both subbands are occupied irrespective ofthe strength of the spin-independent scattering ampli-tude Inoue et al., 2006; Borunda et al., 2007; Nunner etal., 2008. On the other hand, when only the majorityband =+ is occupied, xy

Ia is given by the skew-scattering contribution Borunda et al., 2007,

xyIa −

e2

h

1

nimpuimp

2p+4D+0p+

302 + p+

2 2 , 5.58

in the leading order in 1/nimp.Some of the above leading order results from the fer-

romagnetic Rashba model appear unphysical in the limit

→: xy vanishes discontinuously as the chemical po-tential crosses the edge 0,− of the minority bandwhich leads to a diverging anomalous Nernst effect at=0,−, irrespective of the scattering strength, if one as-sumes the Mott relation Smrcka and Streda, 1977 to bevalid for anomalous transport Onoda et al., 2008. How-ever, this unphysical property does not really holdOnoda et al., 2008, in fact, xy does not vanish evenwhen both subbands are occupied, as shown by includ-ing all higher order Born scattering amplitudes as it isdone automatically in the Keldysh approach Onoda etal., 2006a, 2008. In particular, the skew-scattering con-tribution arises from the odd-order Born scattering i.e.,even order in the impurity potential beyond the con-ventional level of approximation, OV0

3, which gives riseto the normal skew-scattering contributions Kovalev etal., 2008. This yields the unconventional behaviorxy

AH-skew 1/nimp independent of V0 Kovalev et al.,2008. The possible appearance of the AHE in the casewhere both subbands are occupied was also suggested inthe numerical diagonalization calculation of the Kuboformula Kato et al., 2007. The influence of spin-dependent impurities has also been analyzed Nunner etal., 2008.

A numerical calculation of xyAH based on the Keldysh

formalism using the self-consistent T-matrix approxima-tion, shown in Fig. 46, suggests three distinct regimes forthe AHE as a function of xx at low temperaturesOnoda et al., 2006a, 2008. In particular, it shows acrossover from the predominant skew-scattering regionin the clean limit xy xx to an intrinsic-dominatedmetallic region xyconst. In this simple model nowell-defined plateau is observed. These results also sug-gest another crossover to a regime, referred to as theincoherent regime by Onoda et al. 2008, where xy de-cays with the disorder following the scaling relationxy xx

n with n1.6. This scaling arises in the calcula-

FIG. 46. Color online Total anomalous Hall conductivity vsxx for the Hamiltonian in Eq. 5.54 obtained in the self-consistent T-matrix approximation to the Keldysh approachOnoda et al., 2006b, 2008; Kovalev et al., 2009. Curves are fora variety of disorder strengths. The same parameter valueshave been taken as in Fig. 45 with the chemical potential beinglocated at the center of the two subbands. The dashed curvesrepresent the corresponding semiclassical results. From Kova-lev et al., 2009.



tion due to the influence of finite-lifetime disorderbroadening on xy

AH-int, while the skew-scattering contri-bution is quickly diminished by disorder as expected.Kovalev et al. 2009 revisited the Keldysh calculationsfor this model, studying them numerically and analyti-cally. In particular, their study extended the calculationsto include the dependence of the skew-scattering contri-bution on the chemical potential both for 0,− andfor 0,− as shown in Fig. 46. They demonstratedthat changing the sign of the impurity potential changesthe sign of the skew-scattering contribution. The datacollapse illustrated in Fig. 46 then fails, especially nearthe intrinsic-extrinsic crossover. Data collapse in xy vsxx plots is therefore not a general property of 2DRashba models. There is, however, a sufficient tendencyin this direction to motivate analyzing experiments byplotting data in this way. It is also noted that in the elas-tic scattering regime, a plateau expected in the intrinsicregime is not clearly observed in the above results ob-tained with the self-consistent T-matrix approximationOnoda et al., 2006a, 2008; Kovalev et al., 2009, whichoverestimates the impurity scattering amplitude andthus the extrinsic skew-scattering contribution. This issignificantly modified by adopting a better approxima-tion, the coherent potential approximation Yonezawaand Morigaki, 1973, to the same model Onoda, 2009.This can cover a wider range of impurity concentrationsand reveals a clear plateau ranging over two orders ofmagnitude in xx while xy decays only by a factor of 2Onoda, 2009.

1. Minimal model

The above results have the following implications forthe generic nature of the AHE Onoda et al., 2006b,2008. The 2D ferromagnetic Rashba model can beviewed as a minimal model that takes into account boththe “parity anomaly” Jackiw, 1984 associated with anavoided crossing of dispersing bands, as well as impurityscattering in a system with two Fermi-surface sheets.Consider then a general 3D ferromagnet. When the SOIis neglected, majority and minority spin Fermi surfaceswill intersect along lines in three dimensions. For a par-ticular projection kx of Bloch momentum along the mag-netization 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 xy

AH-int will be a maximumand therefore the region around this point should ac-count for most of xy

AH, as the first-principles calculationsseem to indicate Fang et al., 2003; Yao et al., 2004, 2007;Wang et al., 2006, 2007. We emphasize that the Berry-phase contributions from the two bands are nearly op-posite, so that a large contribution from this region of kspace accrues only if the Fermi level lies between thesplit bands. Expanding the Hamiltonian at this particu-lar k point, one can then hope to obtain an effectiveHamiltonian of the form of the ferromagnetic Rashbamodel. Note that the gap 0 in this effective modelHamiltonian Eq. 5.54 comes physically from the SOI

splitting at this particular k point, while the “RashbaSOI” is proportional to the Fermi velocity near thiscrossing point. In three dimensions, the anomalous Hallconductivity is then given by the 2D contribution inte-grated over pz near this minimum gap region and re-mains of order of e2 /ha a is the lattice constant if noaccidental cancellation 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 it is0 and not the SOI which opens the gap, so this is al-ready one key difference. The SO interactions in anyeffective Hamiltonian of this type should in general con-tain at least Rashba-like and Dresselhaus-like contribu-tions. Further studies examining the crossing pointsmore closely near these minimum gap regions will shedfurther light on the relationship between the AHE inreal materials and the AHE in simple models for whichdetailed 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 in-trinsic one, ii the intrinsic metallic regime where xybecomes more or less insensitive to the scatteringstrength and xx, and iii the dirty regime with kF=2F /+1 exhibiting a sublinear dependence of xy

xx2−n or equivalently xy xx

n with n1.6. It is impor-tant to note that this minimal model is based on elasticscattering and cannot explain the scaling observed in thelocalized hopping conduction regime as xx is tuned bychanging T. Nevertheless, if we multiply the 2D anoma-lous Hall conductivity by a−1 with the lattice constant a5 Å for comparison to the experimental results onthree-dimensional bulk samples, then an enhanced xy

AH

of the order of the quantized value e2 /h in the intrinsicregime should be interpreted as e2 /ha103 −1 cm−1.This value compares well with the empirically observedcross over seen in the experimental findings on Fe andCo, as discussed in Sec. II.A.

VI. CONCLUSIONS: FUTURE PROBLEMS ANDPERSPECTIVES ON AHE

In this concluding section, we summarize what hasbeen achieved by the recent studies of the AHE andwhat is not yet understood, pointing out possible direc-tions for future research on this fascinating phenom-enon. To keep this section brief, we exclude the histori-cal summary presented in Secs. I.A and IV which outlinethe early debate on the origin of the AHE. We avoidrepeating all the points highlighted already in Sec. I.Aand focus on the most salient ones.

A. Recent developments

The renewed interest in the AHE, which has lead to aricher and more cohesive understanding of the problem,began in 1998 and was fueled by other connected devel-



opments in solid-state physics. These were i the devel-opment of geometrical and topological concepts usefulin understanding electronic properties such as quantumphase interference and the quantum Hall effect Leeand Ramakrishnan, 1985; Prange and Girvin, 1987, iithe demonstration of the close relation between the Hallconductance and the topological Chern number re-vealed by the TKNN formula Thouless et al., 1982, iiithe development of accurate first-principles band-structure calculation which accounts realistically forSOI, and iv the association of the Berry-phase conceptBerry, 1984 with the noncoplanar spin configurationproposed in the context of the RVB theory of cupratehigh-temperature superconductors Lee et al., 2006.

1. Intrinsic AHE

The concept of an intrinsic AHE, debated for a longtime, was brought back to the forefront of the AHEproblem because of studies which successfully connectedthe topological properties of the quantum states of mat-ter and the transport Hall response of a system. In Sec.I.B.1 we have defined xy

AH-int both experimentally andtheoretically. From the latter, it is rather straightforwardto write xy

AH-int in terms of the Berry curvature in the kspace, from which the topological nature of the intrinsicAHE can be easily recognized immediately. The topo-logical nonperturbative quality of xy

AH-int is highlightedby the finding that, for simple models with spontaneousmagnetization and SOI, bands can have nonzero Chernnumbers even without an external magnetic fieldpresent. This means that expansion with respect to SOIstrength is sometimes dangerous since it lifts the degen-eracy between the spin-up and -down 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 thatxy

AH-int can be the dominant contribution to xyAH in cer-

tain regimes has been strengthened by recent compari-sons of experiment and theory. The intrinsic AHE canbe calculated from first-principles calculations or, in thecase of semiconductors, using k ·p theory. These calcula-tions have been compared to recent experimental mea-surements for several materials such as Sr1−xCaxRuO3Sec. II.B, Fe Sec. II.A, CuCr2Se4−xBrx Sec. II.D,and dilute magnetic semiconductors Sec. II.C. The cal-culations and experiments show semi-quantitative agree-ment. More importantly, however, violations of the em-pirical relation H M have been established boththeoretically and experimentally. This suggests that theintrinsic contribution has some relevance to the ob-served AHE. On the other hand, these studies do notalways provide a compelling explanation for dominanceof the intrinsic mechanism

2. Fully consistent metallic linear response theories of theAHE

Important progress has been achieved in AHE theory.The semiclassical theory, appropriately modified to ac-

count for interband coherence effects, has been shownto be consistent with fully microscopic theories based onthe Kubo and Keldysh formalisms. All three theorieshave been shown to be equivalent in the F limit,with each having their advantages and disadvantagesSec. V. Much of the debate and confusion in earlyAHE literature originated from discrepancies and far-raginous results from earlier inconsistent application ofthese linear response theories.

A semiclassical treatment based on the Boltzmanntransport equation, but taking into account the Berrycurvature and interband coherence effects, has been for-mulated Sec. V.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 themultiband nature of the Green’s functions in terms ofthe Kubo and Keldysh formalism have been fully devel-oped Sec. V.C. These have been applied to a particularmodel, 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 anomalousHall conductivity stays more or less constant as a func-tion of xx, corresponding to the intrinsic-dominated re-gime. The emergence of this regime has been linked tothe topological nature of the intrinsic contribution,analogous to the topologically protected quantized Halleffect.

3. Emergence of three empirical AHE regimes

Based on the large collections of experimental resultsand indications from some theoretical calculations, it isnow becoming clear that there are at least three differ-ent regimes for the behavior of AHE as a function ofxx. i xx106 cm−1 a high conductivity regime inwhich xy

AHxx, skew scattering dominates xyAH, and the

anomalous Hall angle H /xx is constant. In this regime,however, the normal Hall conductivity from the Lorentzforce, proportional to xx

2 H, is large even for the smallmagnetic field H used to align ferromagnetic domainsand separating xy

AH and xyNH is therefore challenging. ii

104 cm−1xx106 cm−1 An intrinsic orscattering-independent regime in which xy

AH is roughlyindependent of xx. In this intermediate metallic region,where the comparison between the experiments andband-structure calculations have been discussed, the in-trinsic mechanism is assumed to be dominant as men-tioned. The dominance of the intrinsic mechanism overside jump is hinted in some model calculations, but thereis no firm understanding of the limits of this simplifyingassumption. iii xx104 cm−1 A bad-metal re-gime in which xy

AH decreases with decreasing xx at arate faster than linear. In this strong disorder region, ascaling H xx

n with 1.6n1.7 has been reported ex-perimentally for a variety of materials discussed in Sec.II. This scaling is primarily observed in insulating mate-



rials exhibiting variable range hopping transport andwhere xx is tuned by varying T. The origin of this scal-ing is not yet understood and is a major challenge forAHE theory in the future. For metallic ultrathin thinfilms exhibiting this approximate scaling, it is naturalthat H is suppressed by the strong disorder excludingweak-localization corrections. Simple considerationfrom the Kubo formula where the energy denominatorincludes a /2 is that H

−2 when this broadening islarger than the energy splitting between bands due tothe SOI. Since in this large broadening regime xx isusually no longer linear in , an upper limit of =2 forthe scaling relation H xx

is expected. The numericalKeldysh studies of the ferromagnetic 2D Rashba modelindicate that this power is 1.6, close to what is ob-served in the limited dirty-metallic range considered inthe experiments. It is a surprising feature that this scal-ing seems to hold for both the metallic and insulatingsamples.

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-electron and/or electron-phonon interac-tions give additional contributions to the resistivity whilesuppressing localization effects through a reduction ofthe phase coherence length. In addition one needs toconsider quantum correction due to the electron-electron interaction in the presence of the disorder.These issues, revealed in the 1980s, must be consideredto scrutinize the microscopic mechanism of xx or xxbefore studying the AHE. This means that it seems un-likely that only xxT characterizes the AHE at eachtemperature. We can define the Boltzmann transport xx

B

only when the residual resistivity is well defined at lowtemperature before the weak-localization effect sets in.Therefore, we need to understand first the microscopicorigin of the resistivity. Separating the resistivity intoelastic and inelastic contributions via Mathhiessen’s ruleis the first step in this direction.

To advance understanding in this important issue, oneneeds to develop the theoretical understanding for theeffect of inelastic scattering on AHE at finite tempera-ture. This issue has been partly treated in the hoppingtheory of the AHE described in Sec. II.B where phonon-assisted hopping was assumed. However, the effects ofinelastic scattering on the intrinsic and extrinsic mecha-nisms are not clear at the moment. Especially, spin fluc-tuation at finite T remains the most essential and diffi-cult problem in the theory of magnetism, and usually themean-field approximation breaks down there. The ap-proximate treatment in terms of the temperature depen-dent exchange splitting, e.g., for SrRuO3 Sec. II.B,

needs to be reexamined by more elaborated methodsuch as the dynamical mean-field theory, taking into ac-count the quantum or thermal fluctuation of the ferro-magnetic moments. These types of studies of the AHEmay shed some light on the nature of the spin fluctua-tion in ferromagnets. Also the interplay between local-ization and the AHE should be pursued further in theintermediate and strong disorder regimes. These are allvital issues for quantum transport phenomena in solidsin general, as well as for AHE specifically. In addition,the finite-temperature effect tends to suppress the finite-temperature skew scattering and make the scattering-independent contributions dominate Tian et al., 2009.

Admitting that more work needs to be done, in Fig. 47we propose a speculative and schematic crossover dia-gram in the plane of diagonal conductivity xx of theBoltzmann transport theory corresponding to the disor-der strength and the temperature T. Note that a realsystem should move along the y axis as temperature ischanged, although the observed xx changes with T. Thisphase diagram reflects the empirical fact that inelasticscattering removes the extrinsic skew-scattering contri-bution more effectively, leaving the intrinsic and side-jump contributions as dominant at finite temperature.We want to stress that the aim of this figure is to pro-mote further studies of the AHE and to identify thelocation of each region or system of interest. Of course,the generality of this diagram is not guaranteed and it ispossible that the crossover boundaries and even the to-pology 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 materialsand should be extended to many other ferromagnets.Especially, the heavy fermion systems are an importantclass of materials to be studied in detail. Concerning thispoint, a more economical numerical technique is nowavailable Marzari and Vanderbilt, 1997. This methodemploys the maximally localized Wannier functions,which can give the best tight-binding model parametersin an energy window of several tens of eV near the

Ferromagnetic critical region

Incoherent region

T

Localized

T

Intrinsic metallic region

Mott-Anderson

Localized-hoppingconductionregion

Skewscattering

critical regiong

region

[-1cm-1]103 106

xx [ 1cm 1]

FIG. 47. Color online A speculative and schematic phasediagram for the anomalous Hall effect in the plane of the di-agonal conductivity xx and the temperature T.



Fermi level. The algorithm for the calculation of xy inthe Wannier interpolation scheme has also been devel-oped Wang et al., 2006. Application of these newly de-veloped methods to a large class of materials should be ahigh priority in the future.

AHE in the dynamical regime is a related interestingproblem. The magneto-optical effects such as the Kerror Faraday rotation have been the standard experimen-tal methods to detect the ferromagnetism. These tech-niques usually focus, however, on the high energy regionsuch as the visible light. In this case, an atomic or localpicture is usually sufficient to interpret the data, and thespectra are not directly connected to the dc AHE. Re-cent studies have revealed that the small energy scalescomparable to the spin-orbit interaction are relevant toAHE which are typically 10 meV for 3d transitionmetal and 100 meV in DMSs Sinova et al., 2003. Thismeans that the dynamical response, i.e., xy, in theTHz and infrared region will provide important informa-tion on the AHE.

A major challenge for experiments is to find examplesof a quantized anomalous Hall effect. There are twocandidates at present: i a ferromagnetic insulator witha band gap Liu et al., 2008 and ii a disorder inducedAnderson insulator with a quantized Hall conductanceM. Onoda and Nagaosa, 2003. Although theoreticallyexpected, it is an important issue to establish experimen-tally that the quantized Hall effect can be realized evenwithout an external magnetic field. Such a finding wouldbe the ultimate achievement in identifying an intrinsicAHE. The dissipationless nature of the anomalous Hallcurrent will manifest itself in this quantized AHE; engi-neering systems using quantum wells or field effect tran-sistors are a promising direction to realize this novel ef-fect.

There are many promising directions for extensions ofideas developed through studies of the AHE. For ex-ample, one can consider several kinds of “current” in-stead of the charge current. An example is the thermalor heat current, which can also be induced in a similarfashion by the anomalous velocity. The thermoelectriceffect has been discussed in Sec. II.C, where combiningall measured thermoelectric transport coefficientshelped settle the issue of the scaling relation xy

AHxx2

in metallic DMSs Pu et al., 2008. Recent studies in Fealloys doped with Si and Co discussed in Sec. II.A fol-lowed a similar strategy Shiomi et al., 2009. From thetemperature dependence of the Lorentz number, theyidentified the crossover between the intrinsic and extrin-sic mechanisms. Further studies of thermal transport willshed some light on the essence of AHE from a differentside.

Spin current is also a quantity of recent great interest.A direct generalization of AHE to the spin current is thespin Hall effect Murakami et al., 2003; Kato et al., 2004;Sinova, Culcer, et al., 2004; Wunderlich et al., 2005,which can be regarded as the two copies of AHE forspins up and down with the opposite sign of xy. In thiseffect, a spin current is produced perpendicular to thecharge current. A recent development in spin Hall effect

is that the quantum spin Hall effect and topological in-sulators have been theoretically predicted and experi-mentally confirmed. We did not include this new and stilldeveloping topic in this review article. Interested read-ers are referred to the original papers and referencestherein Kane and Mele, 2005; Bernevig et al., 2006;Koenig et al., 2007.

ACKNOWLEDGMENTS

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. Ka-wasaki, S. Murakami, Q. Niu, T. S. Nunner, K. Ohgushi,M. Onoda, Y. Onose, J. Shi, R. Shindou, N. A. Sinitsyn,N. Sugimoto, Y. Tokura, and J. Wunderlich. This workwas supported by Grant-in-Aids Grants No. 15104006,No. 16076205, No. 17105002, No. 19048015, and No.19048008 and NAREGI Nanoscience Project from theMinistry of Education, Culture, Sports, Science, andTechnology of Japan. N.N. was supported by FundingProgram for World-Leading Innovative R&D on Sci-ence and Technology First Program. S.O. acknowl-edges partial support from Grants-in-Aid for ScientificResearch under Grants No. 19840053 and No. 21740275from the Japan Society of the Promotion of Science andunder Grants No. 20029006 and No. 20046016 from theMEXT of Japan. J.S. acknowledges partial support fromONR under Grant No. ONR-N000140610122, by NSFunder Grant No. DMR-0547875, by SWAN-NRI, and byCottrell Scholar grant of the Research Corporation.N.P.O. acknowledges partial support from a MRSECgrant from the U.S. National Science Foundation GrantNo. DMR-0819860. A.H.M. acknowledges supportfrom the Welch Foundation and from the DOE.

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