Anomalous Hall effect

Anomalous Hall effect

Abstract

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

Contents:

I Introduction

i.1 A brief history of the AHE and new perspectives

The anomalous Hall effect has deep roots in the history of electricity and magnetism. In 1879 Edwin H. Hall Hall (1879) made the momentous discovery that, when a current-carrying conductor is placed in a magnetic field, the Lorentz force “presses” its electrons against one side of the conductor. One year later, he reported that his “pressing electricity” effect was ten times larger in ferromagnetic iron Hall (1881) than in non-magnetic conductors. Both discoveries were remarkable, given how little was known at the time about how charge moves through conductors. The first discovery provided a simple, elegant tool to measure carrier concentration more accurately in non-magnetic conductors, and played a midwife’s role in easing the birth of semiconductor physics and solid-state electronics in the late 1940’s. For this role, the Hall effect was frequently called the queen of solid-state transport experiments.

The stronger effect that Hall discovered in ferromagnetic conductors came to be known as the anomalous Hall effect (AHE). The AHE has been an enigmatic problem that has resisted theoretical and experimental assault for almost a century. The main reason seems to be that, at its core, the AHE problem involves concepts based on topology and geometry that have been formulated only in recent times. The early investigators grappled with notions that would not become clear and well defined until much later, such as the concept of Berry-phase Berry (1984). What is now viewed as Berry phase curvature, later dubbed “anomalous velocity” by Luttinger, arose naturally in the first microscopic theory of the AHE by Karplus and Luttinger Karplus and Luttinger (1954). However, because understanding of these concepts, not to mention the odd intrinsic dissipationless Hall current they seemed to imply, would not be achieved for another 40 years, the AHE problem was quickly mired in a controversy of unusual endurance. Moreover, the AHE seems to be a rare example of a pure, charge-transport problem whose elucidation has not – to date – benefited from the application of complementary spectroscopic and thermodynamic probes.

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

Very early on, experimental investigators learned that the dependence of the Hall resistivity on applied perpendicular field is qualitatively different in ferromagnetic and non-magnetic conductors. In the latter, increases linearly with , as expected from the Lorentz force. In ferromagnets, however, initially increases steeply in weak , but saturates at a large value that is nearly -independent (Fig. 1). Kundt noted that, in Fe, Co, and Ni, the saturation value is roughly proportional to the magnetization  Kundt (1893) and has a weak anisotropy when the field () direction is rotated with respect to the cyrstal, corresponding to the weak magnetic anisotropy of Fe, Co, and Ni  Webster (1925). Shortly thereafter, experiments by Pugh and coworkers Pugh (1930); Pugh and Lippert (1932) established that an empirical relation between , , and ,

(1.1)

applies to many materials over a broad range of external magnetic fields. The second term represents the Hall effect contribution due to the spontaneous magnetization. This AHE is the subject of this review. Unlike , which was already understood to depend mainly on the density of carriers, was found to depend subtly on a variety of material specific parameters and, in particular, on the longitudinal resistivity .

In 1954, Karplus and Luttinger (KL) Karplus and Luttinger (1954) proposed a theory for the AHE that, in hindsight, provided a crucial step in unraveling the AHE problem. KL showed that when an external electric field is applied to a solid, electrons acquire an additional contribution to their group velocity. KL’s anomalous velocity was perpendicular to the electric field and therefore could contribute to Hall effects. In the case of ferromagnetic conductors, the sum of the anomalous velocity over all occupied band states can be non-zero, implying a contribution to the Hall conductivity . Because this contribution depends only on the band structure and is largely independent of scattering, it has recently been referred to as the the intrinsic contribution to the AHE. When the conductivity tensor is inverted, the intrinsic AHE yields a contribution to and therefore it is proportional to . The anomalous velocity is dependent only on the perfect crystal Hamiltonian and can be related to changes in the phase of Bloch state wavepackets when an electric field causes them to evolve in crystal momentum space Chang and Niu (1996); Sundaram and Niu (1999); Xiao and Niu (2009); Bohm et al. (2003). As mentioned, the KL theory anticipated by several decades the modern interest in Berry phase and Berry curvature effects, particularly in momentum-space.

Figure 2: Extraordinary Hall constant as a function of resistivity. The shown fit has the relation . [From Ref. Kooi, 1954.]

Early experiments to measure the relationship between and generally assumed to be of the power law form, i.e., , mostly involved plotting (or ) vs. , measured in a single sample over a broad interval of (typically 77 to 300 K). As we explain below, competing theories in metals suggested either that or . A compiled set of results was published by Kooi Kooi (1954) (Fig. 2). The subsequent consensus was that such plots do not settle the debate. At finite , the carriers are strongly scattered by phonons and spin waves. These inelastic processes – difficult to treat microscopically even today – lie far outside the purview of the early theories. Smit suggested that, in the skew-scattering theory (see below), phonon scattering increases the value from 1 to values approaching 2. This was also found by other investigators. A lengthy calculation by Lyo Lyo (1973) showed that skew-scattering at (the Debye temperature) leads to the relationship , with a constant. In an early theory by Kondo considering skew scattering from spin excitations Kondo (1962), it may be seen that also varies as at finite .

The proper test of the scaling relation in comparison with present theories involves measuring and in a set of samples at 4 K or lower (where impurity scattering dominates). By adjusting the impurity concentration , one may hope to change both quantities sufficiently to determine accurately the exponent and use this identification to tease out the underlying physics.

The main criticism of the KL theory centered on the complete absence of scattering from disorder in the derived Hall response contribution. The semi-classical AHE theories by Smit and Berger focused instead on the influence of disorder scattering in imperfect crystals. Smit argued that the main source of the AHE currents was asymmetric (skew) scattering from impurities caused by the spin-orbit interaction (SOI) Smit (1955, 1958). This AHE picture predicted that ( = 1). Berger, on the other hand, argued that the main source of the AHE current was the side-jump experienced by quasiparticles upon scattering from spin-orbit coupled impurities. The side-jump mechanism could (confusingly) be viewed as a consequence of a KL anomalous velocity mechanism acting while a quasiparticle was under the influence of the electric field due to an impurity. The side-jump AHE current was viewed as the product of the side-jump per scattering event and the scattering rate Berger (1970). One puzzling aspect of this semiclassical theory was that all dependence on the impurity density and strength seemingly dropped out. As a result, it predicted with an exponent identical to that of the KL mechanism. The side-jump mechanism therefore yielded a contribution to the Hall conductivity which was seemingly independent of the density or strength of scatterers. In the decade 1970-80, a lively AHE debate was waged largely between the proponents of these two extrinsic theories. The three main mechanisms considered in this early history are shown schematically in Fig. 3.

Figure 3: Illustration of the three main mechanisms that can give rise to an AHE. In any real material all of these mechanisms act to influence electron motion.

Some of the confusion in experimental studies stemmed from a hazy distinction between the KL mechanism and the side-jump mechanism, a poor understanding of how the effects competed at a microscopic level, and a lack of systematic experimental studies in a diverse set of materials.

One aspect of the confusion may be illustrated by contrasting the case of a high-purity mono-domain ferromagnet, which produces a spontaneous AHE current proportional to , with the case of a material containing magnetic impurities (e.g. Mn) embedded in a non-magnetic host such as Cu (the dilute Kondo system). In a field , the latter also displays an AHE current proportional to the induced , with the susceptibility  Fert and Jaoul (1972). However, in zero , time-reversal invariance (TRI) is spontaneously broken in the former, but not in the latter. Throughout the period 1960-1989, the two Hall effects were often regarded as a common phenomenon that should be understood microscopically on the same terms. It now seems clear that this view impeded progress.

By the mid-1980s, interest in the AHE problem had waned significantly. The large body of Hall data garnered from experiments on dilute Kondo systems in the previous two decades showed that and therefore appeared to favor the skew-scattering mechanism. The points of controversy remained unsettled, however, and the topic was still mired in confusion.

Since the 1980’s, the quantum Hall effect in two-dimensional (2D) electron systems in semiconductor heterostructures has become a major field of research in physics Prange and Girvin (1987). The accurate quantization of the Hall conductance is the hallmark of this phenomenon. Both the integer Thouless et al. (1982) and fractional quantum Hall effects can be explained in terms of the topological properties of the electronic wavefunctions. For the case of electrons in a two-dimensional crystal, it has been found that the Hall conductance is connected to the topological integer (Chern number) defined for the Bloch wavefunction over the first Brillouin zone Thouless et al. (1982). This way of thinking about the quantum Hall effect began to have a deep impact on the AHE problem starting around 1998. Theoretical interest in the Berry phase and in its relation to transport phenomena, coupled with many developments in the growth of novel complex magnetic systems with strong spin-orbit coupling (notably the manganites, pyrochlores and spinels) led to a strong resurgence of interest in the AHE and eventually to deeper understanding.

Since 2003 many systematics studies, both theoretical and experimental, have led to a better understanding of the AHE in the metallic regime, and to the recognition of new unexplored regimes that present challenges to future researchers. As it is often the case in condensed matter physics, attempts to understand this complex and fascinating phenomenon have motivated researchers to couple fundamental and sophisticated mathematical concepts to real-world materials issues. The aim of this review is to survey recent experimental progress in the field, and to present the theories in a systematic fashion. Researchers are now able to understand the links between different views on the AHE previously thought to be in conflict. Despite the progress in recent years, understanding is still incomplete. We highlight some intriguing questions that remain and speculate on the most promising avenues for future exploration. In this review we focus, in particular, on reports that have contributed significantly to the modern view of the AHE. For previous reviews, the reader may consult Pugh Pugh and Rostoker (1953) and Hurd Hurd (1972). For more recent short overviews focused on the topological aspects of the AHE, we point the reader to the reviews by Nagaosa Nagaosa (2006), and by Sinova et al.Sinova et al. (2004). A review of the modern semiclassical treament of AHE was recently written by Sinitsyn Sinitsyn (2008). The present review has been informed by ideas explained in the earlier works. Readers who are not familair with Berry phase concepts may find it useful to consult the elementary review by Ong and Lee Ong and Lee (2006) and the popular commentary by MacDonald and Niu MacDonald and Niu (2004).

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

  1. When is independent of , the AHE can often be understood in terms of the geometric concepts of Berry phase and Berry curvature in momentum space. This AHE mechanism is responsible for the intrinsic AHE. In this regime, the anomalous Hall current can be thought of as the unquantized version of the quantum Hall effect. In 2D systems the intrinsic AHE is quantized in units of at temperature when the Fermi level lies between Bloch state bands.

  2. Three broad regimes have been identified when surveying a large body of experimental data for diverse materials: (i) A high conductivity regime ( ) in which a linear contribution to due to skew scattering dominates . In this regime the normal Hall conductivity contribution can be significant and even dominate , (ii) An intrinsic or scattering-independent regime in which is roughly independent of ( ), (iii) A bad-metal regime ( ) in which decreases with decreasing at a rate faster than linear.

  3. The relevance of the intrinsic mechanisms can be studied in-depth in magnetic materials with strong spin-orbit coupling, such as oxides and diluted magnetic semiconductors (DMS). In these systems a systematic non-trivial comparison between the observed properties of systems with well controlled materials properties and theoretical model calculations can be achieved.

  4. The role of band (anti)-crossings near the Fermi energy has been identified using first-principles Berry curvature calculations as a mechanism which can lead to a large intrinsic AHE.

  5. Semiclassical treatment by a generalized Boltzmann equation taking into account the Berry curvature and coherent inter-band mixing effects due to band structure and disorder has been formulated. This theory provides a clearer physical picture of the AHE than early theories by identifying correctly all the semiclassically defined mechanisms. This generalized semiclassical picture has been verified by comparison with controlled microscopic linear response treatments for identical models.

  6. The relevance of non-coplanar spin structures with associated spin chirality and real-space Berry curvature to the AHE has been established both theoretically and experimentally in several materials.

  7. Theoretical frameworks based on the Kubo formalism and the Keldysh formalism have been developed which are capable of treating transport phenomena in systems with multiple bands.

The review is aimed at experimentalists and theorists interested in the AHE. We have structured the review as follows. In the remainder of this section, we provide the minimal theoretical background necessary to understand the different AHE mechanisms. In particular we explain the scattering-independent Berry phase mechanism which is more important for the AHE than for any other commonly measured transport coefficient. In Sec. II we review recent experimental results on a broad range of materials, and compare them with relevant calculations where available. In Sec. III we discuss AHE theory from an historical perspective, explaining links between different ideas which are not always recognized, and discussing the physics behind some of the past confusion. The section may be skipped by readers who do not wish to be burdened by history. Section IV discusses the present understanding of the metallic theory based on a careful comparison of the different linear response theories which are now finally consistent. In Sec. V we present a summary and outlook.

i.2 Parsing the AHE:

The anomalous Hall effect is at its core a quantum phenomena which originates from quantum coherent band mixing effects by both the external electric field and the disorder potential. Like other coherent interference transport phenomena (e.g. weak localization), it cannot be satisfactorily explained using traditional semiclassical Boltzmann transport theory. Therefore, when parsing the different contributions to the AHE, they can be defined semiclassically only in a carefully elaborated theory.

In this section we identify three distinct contributions which sum up to yield the full AHE: intrinsic, skew scattering, and side-jump contributions. We choose this nomenclature to reflect the modern literature without breaking completely from the established AHE lexicon (see Sec. III). However, unlike previous classifications, we base this parsing of the AHE on experimental and microscopic transport theory considerations, rather than on the identification of one particular effect which could contribute to the AHE. The link to semiclassically defined processes is established after developing a fully generalized Boltzmann transport theory which takes inter-band coherence effects into account and is fully equivalent to microscopic theories (Sec. IV.1). In fact, much of the theoretical effort of the past few years has been expended in understanding this link between semiclassical and microscopic theory which has escaped cohesion for a long time.

A very natural classification of contributions to the AHE, which is guided by experiment and by microscopic theory of metals, is to separate them according to their dependence on the Bloch state transport lifetime . In the theory, disorder is treated perturbatively and higher order terms vary with a higher power of the quasiparticle scattering rate . As we will discuss, it is relatively easy to identify contributions to the anomalous Hall conductivity, , which vary as and as . In experiment a similar separation can sometimes be achieved by plotting vs. the longitudinal conductivity , when is varied by altering disorder or varying temperature. More commonly (and equivalently) the Hall resistivity is separated into contributions proportional to and .

This partitioning seemingly gives only two contributions to , one and the other . The first contribution we define as the skew-scattering contribution, . Note that in this parsing of AHE contributions it is the dependence on (or ) which defines it, not a particular mechanism linked to a microscopic or semiclassical theory. The second contribution proportional to (or independent of ) we further separate into two parts: intrinsic and side-jump. Although these two contributions cannot be separated experimentally by dc measurements, they can be separated experimentally (as well as theoretically) by defining the intrinsic contribution, , as the extrapolation of the ac-interband Hall conductivity to zero frequency in the limit of , with faster than . This then leaves a unique definition for the third and last contribution, termed side-jump, as .

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

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

Intrinsic contribution to

Among the three contributions, the easiest to evaluate accurately is the intrinsic contribution. We have defined the intrinsic contribution microscopically as the dc limit of the interband conductivity, a quantity which is not zero in ferromagnets when SOI are included. There is however a direct link to semiclassical theory in which the induced interband coherence is captured by a momentum-space Berry-phase related contribution to the anomalous velocity. We show this equivalence below.

This contribution to the AHE was first derived by KL Karplus and Luttinger (1954) but its topological nature was not fully appreciated until recently Onoda and Nagaosa (2002); Jungwirth et al. (2002b). The work of Jungwirth et al. Jungwirth et al. (2002b) was motivated by the experimental importance of the AHE in ferromagnetic semiconductors and also by the thorough earlier analysis of the relationship between momentum space Berry phases and anomalous velocities in semiclassical transport theory by Niu et al. Chang and Niu (1996); Sundaram and Niu (1999). The frequency-dependent inter-band Hall conductivity, which reduces to the intrinsic anomalous Hall conductivity in the dc limit, had been evaluated earlier for a number of materials by Mainkar et al. Mainkar et al. (1996) and Guo and Ebert Guo and Ebert (1995) but the topological connection was not recognized.

The intrinsic contribution to the conductivity is dependent only on the band structure of the perfect crystal, hence its name. It can be calculated directly from the simple Kubo formula for the Hall conductivity for an ideal lattice, given the eigenstates and eigenvalues of a Bloch Hamiltonian :

In Eq. (LABEL:eq:Kubo) is the -dependent Hamiltonian for the periodic part of the Bloch functions and the velocity operator is defined by

(1.3)

Note the restriction in Eq. (LABEL:eq:Kubo).

What makes this contribution quite unique is that, like the quantum Hall effect in a crystal, it is directly linked to the topological properties of the Bloch states. (See Sec. III.2.) Specifically it is proportional to the integration over the Fermi sea of the Berry’s curvature of each occupied band, or equivalently Haldane (2004); Wang et al. (2007) to the integral of Berry phases over cuts of Fermi surface segments. This result can be derived by noting that

(1.4)

Using this expression, Eq. (LABEL:eq:Kubo) reduces to

(1.5)

where is the anti-symmetric tensor, is the Berry-phase connection , and the Berry-phase curvature

(1.6)

corresponding to the states .

This same linear response contribution to the AHE conductivity can be obtained from the semiclassical theory of wave-packets dynamics Chang and Niu (1996); Sundaram and Niu (1999); Marder (2000). It can be shown that the wavepacket group velocity has an additional contribution in the presence of an electric field: . (See Sec. IV.1.) The intrinsic Hall conductivity formula, Eq. (1.5), is obtained simply by summing the second (anomalous) term over all occupied states.

One of the motivations for identifying the intrinsic contribution is that it can be evaluated accurately even for relatively complex materials using first-principles electronic structure theory techniques. In many materials which have strongly spin-orbit coupled bands, the intrinsic contribution seems to dominates the AHE.

Skew scattering contribution to

The skew scattering contribution to the AHE can be sharply defined; it is simply the contribution which is proportional to the Bloch state transport lifetime. It will therefore tend to dominate in nearly perfect crystals. It is the only contribution to the AHE which appears within the confines of traditional Boltzmann transport theory in which interband coherence effects are completely neglected. Skew scattering is due to chiral features which appear in the disorder scattering of spin-orbit coupled ferromagnets. This mechanism was first identified by Smit Smit (1955, 1958).

Treatments of semi-classical Boltzmann transport theory found in textbooks often appeal to the principle of detailed balance which states that the transition probability from to is identical to the transition probability in the opposite direction (). Although these two transition probabilities are identical in a Fermi’s golden-rule approximation, since where is the perturbation inducing the transition, detailed balance in this microscopic sense is not generic. In the presence of spin-orbit coupling, either in the Hamiltonian of the perfect crystal or in the disorder Hamiltonian, a transition which is right-handed with respect to the magnetization direction has a different transition probability than the corresponding left-handed transition. When the transition rates are evaluated perturbatively, asymmetric chiral contributions appear first at third order. (See Sec. IV.1). In simple models the asymmetric chiral contribution to the transition probability is often assummed to have the form (see Sec. III.3.2):

(1.7)

When this asymmetry is inserted into the Boltzmann equation it leads to a current proportional to the longitudinal current driven by and perpendicular to both and . When this mechanism dominates, both the Hall conductivity and the conductivity are proportional to the transport lifetime and the Hall resistivity is therefore proportional to the longitudinal resistivity .

There are several specific mechanisms for skew scattering (see Sec.III.3.2 and Sec. IV.1). Evaluation of the skew scattering contribution to the Hall conductivity or resistivity requires simply that the conventional linearized Boltzmann equation be solved using a collision term with accurate transition probabilities, since these will generically include a chiral contribution. In practice our ability to accurately estimate the skew scattering contribution to the AHE of a real material is limited only by typically imperfect characterization of its disorder. We emphasize that skew scattering contributions to are present not only because of spin-orbit coupling in the disorder Hamiltonian, but also because of spin-orbit coupling in the perfect crystal Hamiltonain combined with purely scalar disorder. Either source of skew-scattering could dominate depending on the host material and also on the type of impurities.

We end this subsection with a small note directed to the reader who is more versed in the latest development of the full semiclassical theory of the AHE and in its comparison to the microscopic theory (see Sec. IV.1 and IV.2.2). We have been careful above not to define the skew-scattering contribution to the AHE as the sum of all the contributions arising from the asymmetric scattering rate present in the collision term of the Boltzmann transport equation. We know from microscopic theory that this asymmetry also makes an AHE contribution or order . There exists a contribution from this asymmetry which is actually present in the microscopic theory treatment associated with the so called ladder diagram corrections to the conductivity, and therefore of order . In our experimentally practical parsing of AHE contributions we do not associate this contribution with skew-scattering but place it under the umbrella of side-jump scattering even though it does not physically originate from any side-step type of scattering.

Side-jump contribution to

Given the sharp defintions we have provided for the intrinsic and skew scattering contributions to the AHE conductivity, the equation

(1.8)

defines the side-jump contribution as the difference between the full Hall conductivity and the two-simpler contributions. In using the term side-jump for the remaining contribution, we are appealing to the historically established taxonomy outlined in the previous section. Establishing this connection mathematically has been the most controversial aspects of AHE theory, and the one which has taken the longest to clarify from a theory point of view. Although this classification of Hall conductivity contributions is often useful (see below), it is not generically true that the only correction to the intrinsic and skew contributions can be physically identified with the side-jump process defined as in the earlier studies of the AHE Berger (1964).

The basic semiclassical argument for a side-jump contribution can be stated straight-forwardly: when considering the scattering of a Gaussian wavepacket from a spherical impurity with SOI ( ), a wavepacket with incident wave-vector will suffer a displacement transverse to equal to . This type of contribution was first noticed, but discarded, by Smit Smit (1958) and reintroduced by Berger Berger (1964) who argued that it was the key contribution to the AHE. This kind of mechanism clearly lies outside the bounds of traditional Boltzmann transport theory in which only the probabilities of transitions between Bloch states appears, and not microscopic details of the scattering processes. This contribution to the conductivity ends up being independent of and therefore contributes to the AHE at the same order as the intrinsic contribution in an expansion in powers of scattering rate. The separation between intrinsic and side-jump contributions, which cannot be distinquished by their dependence on , has been perhaps the most argued aspect of AHE theory since they cannot be distinquished by their dependence on scattering rate (see Sec. III.3.2).

As explained clearly in a recent review by Sinitsyn Sinitsyn (2008), side-jump and intrinsic contributions have quite different 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 ascribed to quantities which were plainly gauge dependent, like the Berry’s connection which in early theories is typically identified as the definition of the side-step upon scattering. Studies of simple models, for example models of semiconductor conduction bands, also gave results in which the side-jump contribution seemed to be the same size but opposite in sign compared to the intrinsic contribution Nozieres and Lewiner (1973). We now understand Sinitsyn et al. (2007) that these cancellations are unlikely, except in models with a very simple band structure, e.g. one with a constant Berry’s curvature. It is only through comparison between fully microscopic linear response theory calculations, based on equivalently valid microscopic formalisms such as Keldysh (non-equilibrium Grenn’s function) or Kubo formalisms, and the systematically developed semi-classical theory that the specific contribution due to the side-jump mechanism can be separately identified with confidence (see Sec.  IV.1).

Having said this, all the calculations comparing the intrinsic and side-jump contibutions to the AHE from a microscopic point of view have been performed for very simple models not immediately linked to real materials. A practical approch which is followed at present for materials in which seems to be independent of , is to first calculate the intrisic contribution to the AHE. If this explains the observation (and it appears that it usually does), then it is deemed that the intrinsic mechanism dominates. If not, we can take some comfort from understanding on the basis of simple model results, that there can be other contributions to which are also independent of and can for the most part be identified with the side jump mechanism. Unfortunately it seems extremelly challenging, if not impossible, to develop a predictive theory for these contributions, partly because they require many higher order terms in the perturbation theory that be summed, but more fundamentally because they depend sensitively on details of the disorder in a particular material which are normally unknown.

Ii Experimental and theoretical studies on specific materials

ii.1 Transition-metals

Early experiments

Four decades after the discovery of the AHE, an empirical relation between magnetization and Hall resistivity was proposed independently by A. W. Smith and by E. M. Pugh Smith and Sears (1929); Pugh (1930); Pugh and Lippert (1932) (see Sec. I.1). Pugh investigated the AHE in Fe, Ni and Co and the alloys Co-Ni and Ni-Cu in magnetic fields up to 17 kG over large intervals in (10-800 K in the case of Ni), and found that the Hall resistivity is comprised of 2 terms, viz.

(2.1)

where is the magnetization averaged over the sample. Pugh defined and as the ordinary and extra-ordinary Hall coefficients, respectively. The latter is now called the anomalous Hall coefficient (as in Eq. 1.1)).

On dividing Eq. (2.1) by , we see that it just expresses the additivity of the Hall currents: the total Hall conductivity equals , where is the ordinary Hall conductivity and is the AHE conductivity. A second implication of Eq. (2.1) emerges when we consider the role of domains. The anomalous Hall coefficient in Eq.(2.1) is proportional to the AHE in a single domain. As , proliferation of domains rapidly reduces to zero (we ignore pinning). Cancellations of between domains result in a zero net Hall current. Hence the observed AHE term mimics the field profile of , as implied by Pugh’s term . The role of is simply to align the AHE currents by rotating the domains into alignment. The Lorentz force term is a “background” current with no bearing on the AHE problem.

The most interesting implication of Eq. (2.1) is that, in the absence of , a single domain engenders a spontaneous Hall current transverse to both and . Understanding the origin of this spontaneous off-diagonal current has been a fundamental problem of charge transport in solids for the past 60 years. The AHE is also called the spontaneous Hall effect and the extraordinary Hall effect in the older literature.

Recent experiments

The resurgence of interest in the AHE motivated by the Berry-phase approach (Sec. I.2.1) has led to many new Hall experiments on 3 transition metals and their oxides. Both the recent and the older literature on Fe and FeO are reviewed in this section. An important finding of these studies is the emergence of three distinct regimes roughly delimited by the conductivity and characterized by the dependence of on . The three regimes are:

  • A high conductivity regime for in which dominates ,

  • A good metal regime for in which ,

  • A bad metal/hopping regime for in which .

We discuss each of these regimes below.

High conductivity regime – The Hall conductivity in the high-purity regime, , is dominated by the skew scattering contribution . The high-purity regime is one of the least studied experimentally. This regime is very challenging to investigate experimentally because the field required for saturating also yields a very large ordinary Hall effect (OHE) and tends to be of the order of  Schad et al. (1998). In the limit , the OHE conductivity may be nonlinear in ( is the cyclotron frequency). Although increases as , the OHE term increases as and therefore the latter ultimately dominates, and the AHE current may be unresolvable. Even though the anomalous Hall current can not always be cleanly separated from the normal Lorentz-force Hall effect in the high conductivity regime, the total Hall current invariably increases with in a way which provides compelling evidence for a skew-scattering contribution.

In spite of these challenges several studies have managed to convincingly separate the competing contributions and have identifed a dominant linear relation between and for Majumdar and Berger (1973); Shiomi et al. (2009). In an early study Majumdar et al. Majumdar and Berger (1973) grew highly pure Fe doped with Co. The resulting , obtained from Kohler plots extrapolation to zero field, show a clear dependence of (Fig. 4 a). In a more recent study, a similar finding (linear dependence of ) was observed by Shiomi et al. Shiomi et al. (2009) in Fe doped with Co, Mn, Cr, and Si. In these studies the high temperature contribution to (presumed to be intrinsic plus side jump) was substracted from and a linear dependence of the resulting is observed (Fig. 4 a and b). In this recent study the conductivity is intentionally reduced by impurity doping to find the linear region and reliably exclude the Lorernz contribution. The results of these authors show, in particular, that the slope of vs. depends on the species of the impurities as it is expected in the regime dominated by skew scattering. It is reassuring to note that the skewness parameters () implied by the older and the more recent experiments are consistent, in spite of differences in the conductivity ranges studied. is independent of Majumdar and Berger (1973); Shiomi et al. (2009) as it should be when the skew scattering mechanism dominates. Further experiments in this regime are desirable to fully investiage the different dependence on doping, temperature, and impurity type. Also, new approaches to reliably disentangle the AHE and OHE currents will be needed to faicilitate such studies.

Figure 4: for pure Fe film doped with Cr, Co, Mn and Si vs. at low temperatures ( K and K ) (a). In many of the alloys, particularly in the Co doped system, the linear scaling in the higher conductivy sector, , implies that skew scattering dominates . After Ref. Majumdar and Berger, 1973 and Ref. Shiomi et al., 2009. The data from Shiomi et al. (2009), shown also at a larger scale in (b), is obtained by substracting the high temperature contribuiton to . In the data shown the ordinary Hall contribution has been identified and substracted. [Panel (b) From Ref. Shiomi et al., 2009.]

Good metal regime – Experiments re-examining the AHE in Fe, Ni, and Co have been performed by Miyasato et al. Miyasato et al. (2007). These experiments indicate a regime of versus in which is insensitive to in the range - (see Fig. 5). This suggests that the scattering independent mechanisms (intrinsic and side-jump) dominates in this regime. However, in comparing this phenomenology to the discussion of AHE mechanisms in Sec. I.2, one must keep in mind that the temperature has been varied in the Hall data on Fe, Ni, and Co in order to change the resistivity, even though it is restricted to the range well below (Fig. 5 upper panel). In the mechanisms discussed in Sec. I.2 only elastic scattering was taken into account. Earlier tests of the dependence of carried by varying were treated as suspect in the early AHE period (see Fig. 2) because the role of inelastic scattering was not fully understood. The effect of inelastic scattering from phonons and spin waves remains open in AHE theory and is not addressed in this review.

Figure 5: Measurements of the Hall conductivity and resistivity in single-crystal Fe and in thin foils of Fe, Co, and Ni. The top and lower right panels show the dependence of and , respectively. The lower left panel plots against . After Ref. Miyasato et al., 2007.

Bad metal/hopping regime– Several groups have measured in Fe and FeO thin-film ferromagnets Miyasato et al. (2007); Fernandez-Pacheco et al. (2008); Venkateshvaran et al. (2008); Sangiao et al. (2009); Feng et al. (1975). (See Fig. 6.) Sangiao et al. Sangiao et al. (2009) studied epitaxial thin-films of Fe deposited by pulsed-laser deposition (PLD) on single-crystal MgO (001) substrates at pressures Torr. To vary over a broad range, they varied the film thickness from 1 to 10 nm. The vs. profile for the film with = 1.8 nm displays a resistance minimum near 50 K, below which shows an upturn which has been ascribed to localization or electron interaction effects (Fig. 7). The magnetization is nominally unchanged from the bulk value (except possibly in the 1.3 nm film). AHE experiments were carried out from 2–300 K on these films and displayed as vs. plots together with previously published results (Fig. 6). In the plot, the AHE data from films with 2 nm fall in the weakly localized regime. The combined plot shows that Sangio et al.’s data are collinear (on a logarithmic scale) with those measured on m-thick films by Miyasato et al. Miyasato et al. (2007). For the three samples with = 1.3–2 nm, the inferred exponent in the dirty regime is on average . A concern is that the data from the 1.3 nm film was obtained by subtracting a term from (the subtraction procedure was not described). How localization affects the scaling plot is an open issue at present.

Figure 6: Combined plot of the AHE conductivity versus the conductivity in epitaxial films of Fe grown on MgO with thickness = 2.5, 2.0, 1.8 and 1.3 nm  Sangiao et al. (2009), in polycrystalline FeZnO Feng et al. (1975), in thin-film FeZnO between 90 and 350 K Venkateshvaran et al. (2008) and above the Verwey transition Fernandez-Pacheco et al. (2008).

In Sec. II.5, we discuss recent AHE measurements in disordered polycrystalline Fe films with 10 nm by Mitra et alMitra et al. (2007). Recent progress in understanding weak-localization corrections to the AHE is also reviewed there.

Figure 7: The dependence of in epitaxial thin-films MgO(001)/Fe(t)/MgO with nm and 2.5 nm. The inset shows how , the temperature of the resistivity minimum, varies with .[From Ref. Sangiao et al., 2009.]

In magnetite, FeO, scaling of with was already apparent in early experiments on polycrystalline samples  Feng et al. (1975). Recently, two groups have re-investigated the AHE in epitaxial thin films (data included in Fig. 6). Fernandez-Pacheco et al. Fernandez-Pacheco et al. (2008) measured a series of thin-film samples of FeO grown by PLD on MgO (001) substrates in ultra-high vacuum, whereas Venkateshvaran et al. Venkateshvaran et al. (2008) studied both pure FeO and Zn-doped magnetite FeZnO deposited on MgO and AlO substrates grown by laser molecular-beam epitaxy under pure Ar or Ar/O mixture. In both studies, increases monotonically by a factor of 10 as decreases from 300 K to the Verwey transition temperature = 120 K. Below , further increases by a factor of 10 to 100. The results for vs. from Venkateshvaran et al. Venkateshvaran et al. (2008) is shown in Fig. 8.

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

Both groups find good scaling fits extending over several decades of with when varying . Fernandez-Pacheco et alFernandez-Pacheco et al. (2008) plot vs. in the range 150300 K for several thicknesses and infer an exponent . Venkateshvaran et al. Venkateshvaran et al. (2008) plot vs. from 90 to 350 K and obtain power-law fits with = 1.69, in both pure and Zn-doped magnetite (data shown in Fig. 6). Significantly, the 2 groups find that is unchanged below . There is presently no theory in the poorly conducting regime which predicts the observed scaling ( ).

Figure 8: Longitudinal resistivity vs. for epitaxial FeZnO films. The (001), (110) and (111) oriented films were grown on MgO(001), MgO(110) and AlO substrates. [From Ref. Venkateshvaran et al., 2008.]

Comparison to theories

Detailed first-principles calculations of the intrinsic contribution to the AHE conductivity have been performed for bcc Fe Yao et al. (2004); Wang et al. (2006), fcc Ni, and hpc Co Wang et al. (2007). In Fe and Co, the values of inferred from the Berry curvature are and , respectively, in reasonable agreement with experiment. In Ni, however, the calculated value is only of the experimental value.

These calculations uncovered the crucial role played by avoided-crossings of band dispersions near the Fermi energy . The Berry curvature is always strongly enhanced near avoided crossings, opposite direction for the upper and lower bands. A large contribution to when the crossing is at the Fermi energy so that only one of the two bands is occupied, e.g. near the point H in Fig. 9. A map showing the contributions of different regions of the FS to is shown in Fig. 10. The SOI can lift an accidental degeneracy at certain wavevectors . These points act as a magnetic monopole for the Berry curvature in -space Fang et al. (2003). In the parameter space of spin-orbit coupling, is nonperturbative in nature. The effect of these “parity anomalies” Jackiw (1984) on the Hall conductivity was first discussed by Haldane Haldane (1988). A different conclusion on the role of topological enhancement in the intrinsic AHE was reported for a tight-binding calculation with the 2 orbitals and on a square lattice Kontani et al. (2007).

Figure 9: First-principles calculation of the band dispersions and the Berry-phase curvature summed over occupied bands. [From Ref. Yao et al., 2004.]
Figure 10: First-principles calculation of the FS in the (010) plane (solid lines) and the Berry curvature in atomic units (color map). [From Ref. Yao et al., 2004.]

Motivated by the enhancement at the crossing points discussed above, Onoda et al. Onoda et al. (2006a); Onoda et al. (2008) proposed a minimal model that focuses on the topological and resonantly enhanced nature of the intrinsic AHE arising from the sharp peak in near avoided crossings. The minimal model is essentially a 2D Rashba model Bychkov and Rashba (1984) with an exchange field which breaks symmetries and accounts for the magnetic order and a random impurity potential to account for disorder. The model is discussed in Sec. IV.4. In the clean limit, is dominated by the extrinsic skew-scattering contribution , which almost masks the intrinsic contribution . Since , it is suppressed by increased impurity scattering, whereas – an interband effect – is unaffected. In the moderately dirty regime where the quasiparticle damping is larger than the energy splitting at the avoided crossing (typically, the SOI energy) but less than the bandwidth, dominates . As a result, one expects a crossover from the extrinsic to the intrinsic regime. When skew scattering is due to a spin-dependent scattering potential instead of spin-orbit coupling in the Bloch states, the skew to intrinsic crossover could be controlled by a different condition. In this minimal model, a well-defined plateau is not well-reproduced in the intrinsic regime unless a weak impurity potential is assumed Onoda et al. (2008). This crossover may be seen when the skew-scattering term shares the same sign as the intrinsic one Kovalev et al. (2009).

With further increase in the scattering strength, spectral broadening leads to the scaling relationship , as discussed above Onoda et al. (2006b); Onoda et al. (2008); Kovalev et al. (2009). In the strong-disorder regime, is no longer linear in the scattering lifetime . A different scaling, , attributed to broadening of the electronic spectrum in the intrinsic regime, has been proposed by Kontani et al.  Kontani et al. (2007).

As discussed above, there is some experimental evidence that this scaling prevails not only in the dirty metallic regime, but also deep into the hopping regime. Sangio et alSangiao et al. (2009), for e.g., obtained the exponent 1.7 in epitaxial thin-film Fe in the dirty regime. In manganite, scaling seems to hold, with the same nominal value of , even below the Verwey transition where charge transport is deep in the hopping regime Fernandez-Pacheco et al. (2008); Venkateshvaran et al. (2008). These regimes are well beyond the purview of either the minimal model, which considers only elastic scattering, or the theoretical approximations used to model its properties Onoda et al. (2006b); Onoda et al. (2008).

Nonetheless, the experimental reports have uncovered a robust scaling relationship with near 1.6, which extends over a remarkably large range of . The origin of this scaling is an open issue at present.

ii.2 Complex oxide ferromagnets

First-principles calculations and experiments on SrRuO

Figure 11: Anomalous Hall effect in SrRuO. (A) The magnetization , (B) longitudinal resistivity , and (C) transverse resistivity as functions of the temperature for the single crystal and thin-film SrRuO, as well as for Ca-doped SrCaRuO thin film. is the Bohr magneton. [From Ref. Fang et al., 2003.]
Figure 12: The calculated transverse conductivity as a function of the chemical potential for SrRuO. The chaotic behavior is the fingerprint of the Berry curvature distribution illustrated in Fig. 13. [From Ref. Fang et al., 2003.]

The perovskite oxide SrRuO is an itinerant ferromagnet with a critical temperature of 165 K. The electrons occupying the 4 orbitals in have a SOI energy much larger than that for 3 electrons. Early transport investigations of this material were reported in Refs. Allen et al., 1996 and Izumi et al., 1997. The latter authors also reported results on thin-film SrTiO. Recently, the Berry-phase theory has been applied to account for its AHE Fang et al. (2003); Mathieu et al. (2004a, b), which is strongly dependent (Fig.11c). Neither the KL theory nor the skew-scattering theory seemed adequate for explaining the dependence of the inferred AHE conductivity  Fang et al. (2003).


Figure 13: The Berry curvature for a band as a function of with the fixed . [From Ref. Fang et al., 2003.]
Figure 14: (Upper panel) Combined plots of the Hall conductivity vs. magnetization in 5 samples of the ruthenate (). The inset compares data at (triangles) with calculated values (solid curve). (Lower panel) First-principles calculations of vs. for cubic and orthorhombic structures. The effect of broadening on the curves is shown for the orthorhombic case. [From Ref. Mathieu et al., 2004b.]

The experimental results motivated a detailed first-principles, band-structure calculation that fully incorporated the SOI. The AHE conductivity was calculated directly using the Kubo formula Eq. (LABEL:eq:KuboFang et al. (2003). To handle numerical instabilities which arise near certain critical points, a fictitious energy broadening = 70 meV was introduced in the energy denominator. Fig. 12 shows the dependence of on the chemical potential . In sharp contrast to the diagonal conductivity , fluctuates strongly, displaying sharp peaks and numerous changes in sign. The fluctuations may be understood if we map the momentum dependence of the Berry curvature in the occupied band. For example, Fig. 13 displays plotted as a function of , with fixed at 0. The prominent peak at corresponds to the avoided crossing of the energy band dispersions, which are split by the SOI. As discussed in Sec. I.2, variation of the exchange splitting caused by a change in the spontaneous magnetization strongly affects in a nontrivial way.

From the first-principles calculations, one may estimate the temperature dependence of by assuming that it is due to the temperature-dependence of the Bloch state exchange splitting and that this splitting is proportional to the temperature-dependent magnetization. The new insight is that the -dependence of simply reflects the dependence of : at a finite temperature , the magnitude and sign of may be deduced by using the value of in the zero- curve. This proposal was tested against the results on both the pure material and the Ca-doped material SrCaRuO. In the latter, Ca doping suppresses both and systematically Mathieu et al. (2004a, b). As shown in Fig. 14 (upper panel), the measured values of and , obtained from 5 samples with Ca content 0.4 0, fall on 2 continuous curves. In the inset, the curve for the pure sample () is compared with the calculation. The lower panel of Fig. 14 compares calculated curves of for cubic and orthorhombic lattice structures. The sensitivity of to the lattice symmetry reflects the dominant contribution of the avoided crossing near . The sensitivity to broadening is shown for the orthorhombic case.

Kats et alKats et al. (2004) also have studied the magnetic field dependence of in an epitaxial film of SrRuO. They have observed sign-changes in near a magnetic field T at = 130 K and near T at 134 K. This seems to be qualitative consistent with the Berry-phase scenario. On the other hand the authors suggest that the intrinsic-dominated picture is likely incomplete (or incorrect) near Tc Kats et al. (2004).

Spin chirality mechanism of the AHE in manganites

In the manganites, e.g. (LCMO), the three electrons on each Mn ion form a core local moment of spin . A large Hund energy aligns the core spin with the spin of an itinerant electron that momentarily occupies the orbital. Because this Hund coupling leads to an extraordinary magnetoresistance in weak , the manganites are called colossal magnetoresistance (CMR) materials Tokura and Tomioka (1999). The double exchange theory summarized by Eq. (2.2) (below) is widely adopted to describe the onset of ferromagnetism in the CMR manganites.

As decreases below the Curie temperature 270 K in LCMO, the resistivity falls rapidly from 15 m cm to metallic values 2 mcm. CMR is observed over a significant interval of temperatures above and below , where charge transport occurs by hopping of electrons between adjacent Mn ions (Fig. 15a). At each Mn site , the Hund energy tends to align the carrier spin with the core spin .

Early theories of hopping conductivity Holstein (1961) predicted the existence of a Hall current produced by the phase shift (Peierls factor) associated with the magnetic flux piercing the area defined by 3 non-collinear atoms. However, the hopping Hall current is weak. The observation of a large in LCMO that attains a broad maximum in modest (Fig. 15b) led Matl et alMatl et al. (1998) to propose that the phase shift is geometric in origin, arising from the solid angle described by as the electron visits each Mn site ( at each site as shown in Fig. 16). To obtain the large seen, one requires to define a finite solid angle . Since gradually aligns with with increasing field, this effect should disappear along with , as observed in the experiment. This appears to be the first application of a geometric-phase mechanism to account for an AHE experiment.

Figure 15: (a) The colossal magnetoresistance vs. in ( = 265 K) at selected . (b) The Hall resistivity vs. at temperatures 100 to 360 K. Above , is strongly influenced by the MR and the susceptibility . [From Ref. Matl et al., 1998.]

Figure 16: Schematic view of spin chirality. When circulates among three spins to which it is exchange-coupled, it feels a fictitious magnetic field with flux given by the half of the solid angle subtended by the three spins. [From Ref. Lee et al., 2006.]

Subsequently, Ye et al. Ye et al. (1999) considered the Berry phase due to the thermal excitations of the Skyrmion (and anti-Skyrmion). They argued that the SOI gives rise to a coupling between the uniform magnetization and the gauge field by the term . In the ferromagnetic state, the spontaneous uniform magnetization leads to a finite and uniform , which acts as a uniform magnetic field. Lyanda-Geller et alLyanda-Geller et al. (2001) also considered the AHE due to the spin chirality fluctuation in the incoherent limit where the hopping is treated perturbatively. This approach, applicable to the high- limit, complements the theory of Ye et al. Ye et al. (1999).

The Berry phase associated with non-coplanar spin configurations, the scalar spin chirality, was first considered in theories of high-temperature superconductors in the context of the flux distribution generated by the complex order parameter of the resonating valence bond (RVB) correlation defined by , which acts as the transfer integral of the “spinon” between the sites and  Lee et al. (2006). The complex transfer integral also appears in the double-exchange model specified by

(2.2)

where is the ferromagnetic Hund’s coupling between the spin of the conduction electrons and the localized spins .

In the manganese oxides, represents the localized spin in -orbitals, while and are the operators for -electrons. (The mean-field approximations of Hubbard-like theories of magnetism, the localized spin may also be regarded as the molecular field created by the conduction electrons themselves, in which case is replaced by the on-site Coulomb interaction energy .) In the limit of large , the conduction electron spin is forced to align with at each site. The matrix element for hopping from is then given by

(2.3)

where is the two-component spinor spin wave function with quantization axis . The phase factor acts like a Peierls phase and can be viewed as originating from a fictitious magnetic field which influences the orbital motion of the conduction electrons.

We next discuss how the Peierls phase leads to a gauge field, i.e. flux, in the presence of non-coplanar spin configurations. Let , , and be the local spins at sites , , and , respectively. The product of the three transfer integrals corresponding to the loop is

(2.4)

where is the two-component spinor wavefunction of the spin state polarized along . Its imaginary part is proportional to , which corresponds to the solid angle subtended by the three spins on the unit sphere, and is called the scalar spin chirality (Fig. 16). The phase acquired by the electron’s wave function around the loop is , which leads to the Aharonov-Bohm (AB) effect and, as a consequence, to a large Hall response.

In the continuum approximation, this phase factor is given by the flux of the “effective” magnetic field where is the elemental directed surface area defined by the three sites. The discussion implies that a large Hall current requires the unit vector to fluctuate strongly as a function of , the position coordinate in the sample. An insightful way to quantify this fluctuation is to regard as a map from the - plane to the surface of the unit sphere (we take a 2D sample for simplicity). An important defect in a ferromagnet – the Skyrmion Sondhi et al. (1993) – occurs when points down at a point in a region of the - plane, but gradually relaxes back to up at the boundary of . The map of this spin texture wraps around the sphere once as roams over . The number of Skyrmions in the sample is given by the topological index

(2.5)

where the first integrand is the directed area of the image on the unit sphere. counts the number of times the map covers the sphere as extends over the sample. The gauge field produces a Hall conductivity. Ye et al. Ye et al. (1999) derived in the continuum approximation the coupling between the field and the spontaneuos magnetization through the SOI. The SOI coupling produces an excess of thermally excited positive Skyrmions over negative ones. This imbalance leads to a net uniform “magnetic field” (anti)parallel to , and the AHE. In this scenario, is predicted to attain a maximum slightly below , before falling exponentially to zero as .

The Hall effect in the hopping regime has been discussed by Holstein Holstein (1961) in the context of impurity conduction in semiconductors. Since the energies and of adjacent impurity sites may differ significantly, charge conduction must proceed by phonon-assisted hopping. To obtain a Hall effect, we consider 3 non-collinear sites (labelled as = 1, 2 and 3). In a field , the magnetic flux piercing the area enclosed by the 3 sites plays the key role in the Hall response. According to Holstein, the Hall current arises from interference between the direct hopping path and the path going via 3 as an intermediate step. Taking into account the changes in the phonon number in each process, we have

where are the phonon numbers for the modes , respectively.

In a field , the hopping matrix element from to includes the Peierls phase factor . When we consider the interference between the two processes in Eq. (LABEL:eq:TwoProc), the Peierls phase factors combine to produce the phase shift where is the flux quantum. By the Aharonov-Bohm effect, this leads to a Hall response.

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

(2.7)

where is the set of the energy levels , and is the angle between and . When the average of over all directions of is taken, it vanishes even for finite spontaneous magnetization . To obtain a finite , it is necessary to incorporate SOI.

Assuming the form of hopping integral with the SOI given by

(2.8)

the Hall conductivity is proportional to the average of

(2.9)

Taking the average of ’s with where is the saturated magnetization, we finally obtain

(2.10)

This prediction has been tested by the experiment of Chun et al. Chun et al. (2000) shown in Fig. 17. The scaling law for the anomalous as a function of obtained near is in good agreement with the experiment.

Figure 17: Comparison between experiment and the theoretical prediction Eq.(2.10). Scaling behavior between the Hall resistivity and the magnetization is shown. The solid line is a fit to Eq.(2.10); the dashed line is the numerator of Eq.(2.10) only. There are no fitting parameters except the overall scale. [From Ref. Chun et al., 2000.]

Similar ideas have been used by Burkov and Balents Burkov and Balents (2003) to analyze the variable range hopping region in (Ga,Mn)As. The spin-chirality mechanism for the AHE has also been applied to CrO Yanagihara and Salamon (2002, 2007), and the element Gd Baily and Salamon (2005). In the former case, the comparison between and the specific heat supports the claim that the critical properties of are governed by the Skyrmion density.

The theories described above assume large Hund coupling. In the weak-Hund coupling limit, a perturbative treatment in has been developed to relate the AHE conductivity to the scalar spin chirality Tatara and Kawamura (2002). This theory has been applied to metallic spin-glass systems Kawamura (2007).

Lanthanum cobaltite

The subtleties and complications involved in analyzing the Hall conductivity of tunable ferromagnetic oxides are well illustrated by the cobaltites. Samoilov et alSamoilov et al. (1998), and Baily and Salamon Baily and Salamon (2003) investigated the AHE in Ca-doped lanthanum cobaltite , which displays a number of unusual magnetic and transport properties. They found an unusually large AHE near as well as at low , and proposed the relevance of spin-ordered clusters and orbital disorder scattering to the AHE in the low- limit. Subsequently, a more detailed investigation of was reported by Onose and Tokura Onose and Tokura (2006). Figs. 18 a, b and c summarize the dependence of , and , respectively in four crystals with . The variation of vs. suggests that a metal-insulator transition occurs between 0.17 and 0.19. Whereas the samples with have a metallic - profile, the sample with = 0.17 is non-metallic (hopping conduction). Moreover, it displays a very large MR at low and large hysteresis in curves of vs. , features that are consistent with a ferromagnetic cluster-glass state.

When the Hall conductivity is plotted vs. (Fig. 18 d), shows a linear dependence on for the most metallic sample ( = 0.30). However, for = 0.17 and 0.20, there is a pronounced downturn suggestive of the appearance of a different Hall term that is electron-like in sign. This is most apparent in the trend of the curves of vs. in Fig. 18 b. Onose and Tokura Onose and Tokura (2006) propose that the negative term may arise from hopping of carriers between local moments which define a chirality that is finite, as discussed above.

Figure 18: Temperature dependence of the magnetization (a), Hall resistivity (b), Hall conductivity (c), in four crystals of () (all measured in a field = 1 T). In Panels a, b and c, the data for = 0.17 and 0.20 were multiplied by a factor of 200 and 50, respectively. (d) The Hall conductivity at 1 T in the four crystals plotted against . Results for = 0.17 and 0.20 were multiplied by factors 50 and 5, respectively. [From Ref. Onose and Tokura, 2006.]

Spin chirality mechanism in pyrochlore ferromagnets

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

Ohgushi et al. Ohgushi et al. (2000) considered the ground state of the non-coplanar spin configuration in the Kagome lattice, which may be obtained as a projection of the pyrochlore lattice onto the plane normal to (1,1,1) axis. Considering the double exchange model Eq. (2.2), they obtained the band structure of the conduction electrons and the Berry phase distribution. Quite similar to the Haldane model Haldane (1988) or the model discussed in Eq. (LABEL:eq:spmodel), the Chern number of each band becomes nonzero, and a quantized Hall effect results when the chemical potential is in the energy gap.

Turning to real materials, the pyrochlore ferromagnet NdMoO (NMO) provides a test-bed for exploring these issues. Its lattice structure consists of two interpenetrating sublattices comprised of tetrahedrons of Nd and Mo atoms, respectively (the sublattices are shifted along the -axis) Taguchi et al. (2001); Yoshii et al. (2000). While the exchange between spins on either sublattice is ferromagnetic, the exchange coupling between spins of the conducting -electrons of Mo and localized -electron spins on Nd is antiferromagnetic.

Figure 19: Anomalous Hall effect in NdMoO. Magnetic field dependence of (A) the magnetization and (B) the transverse resistivity () for different temperatures. [From Ref. Taguchi et al., 2001.]

The dependences of the anomalous Hall resistivity on at selected temperatures are shown in Fig. 19. The spins of Nd begin to align antiparallel to those of Mo below the crossover temperature K. Each Nd spin is subject to a strong easy-axis anisotropy along the line from a vertex of the Nd tetrahedron to its center. The resulting noncoplanar spin configuration induces a transverse component of the Mo spins. Spin chirality is expected to be produced by the coupling , which leads to the AHE of -electrons. An analysis of the neutron scattering experiment has determined the magnetic structure Taguchi et al. (2001). The tilt angle of the Nd spins is close to that expected from the strong limit of the spin anisotropy, and the exchange coupling is estimated as K. This leads to a tilt angle of the Mo spins of 5. From these estimates, a calculation of the anomalous Hall conductivity in a tight-binding Hamiltonian of triply degenerate bands leads to , consistent with the value measured at low . In a strong , this tilt angle is expected to be reduced along with . This is in agreement with the traces displayed in Fig. 19.

The dependence of the Hall conductivity has also been analyzed in the spin-chirality scenario by incorporating spin fluctuations Onoda and Nagaosa (2003). The result is that frustration of the Ising Nd spins leads to large fluctuations, which accounts for the large observed. The recent observation of a sign-change in in a field applied in the direction Taguchi et al. (2003) is consistent with the sign-change of the spin chirality.

In the system GdMoO (GMO), in which Gd () has no spin anisotropy, the low- AHE is an order-of-magnitude smaller than that in NMO. This is consistent with the spin-chirality scenario Taguchi et al. (2004). The effect of the spin chirality mechanism on the finite-frequency conductivity has been investigated Kezsmarki et al. (2005).

In another work, Yasui et al. Yasui et al. (2006, 2007) performed neutron scattering experiments over a large region in the -plane with along the and directions. By fitting the magnetization of Nd, the magnetic specific heat , and the magnetic scattering intensity , they estimated K, which was considerably smaller than estimated previously Taguchi et al. (2001). Furthermore, they calculated the thermal average of the spin chirality and compared its value with that inferred from the AHE resistivity . They have emphasized that, when a 3-Tesla field is applied in the direction (along this direction cancels the exchange field from the Mo spins), no appreciable reduction of is observed. These recent conclusions have cast doubt on the spin-chirality scenario for NMO.

A further puzzling feature is that, with applied in the direction, one expects a discontinuous transition from the two-in, two-out structure (i.e. 2 of the Nd spins point towards the tetrahedron center while 2 point away) to the three-in, one-out structure for the Nd spins. However, no Hall features that might be identified with this cancellation have been observed down to very low . This seems to suggest that quantum fluctuations of the Nd spins may play an important role, despite the large spin quantum number ().

Machida has discussed the possible relevance of spin chirality to the AHE in the pyrochlore PrIrO  Machida et al. (2007a, b). In this system, a novel “Kondo effect” is observed even though the Pr ions with are subject to a large magnetic anisotropy. The magnetic and transport properties of MoO near the phase boundary between the spin glass Mott insulator and ferromagnetic metal by changing the rare earth ion has been studied Katsufuji et al. (2000).

Anatase and Rutile

In thin-film samples of the ferromagnetic semiconductor anatase , Ueno et al. Ueno et al. (2008) have reported scaling between the AHE resistance and the magnetization . The AHE conductivity scales with the conductivity as (Fig. 20). A similar scaling relation was observed in another polymorph rutile. See also Ref. Ramaneti et al., 2007 for related work on Co-doped TiO.

Figure 20: Plot of AHE conductivity vs. conductivity for anatase (triangles) and rutile (diamonds). Grey symbols are data taken by other groups. The inset shows the expanded view of data for anatase with = 0.05 (the open and closed triangles are for 150 K and 100 K, respectively. [From Ref. Ueno et al., 2007.]

ii.3 Ferromagnetic semiconductors

Ferromagnetic semiconductors combine semiconductor tunability and collective ferromagnetic properties in a single material. The most widely studied ferromagnetic semiconductors are diluted magnetic semiconductors (DMS) created by doping a host semiconductor with a transition metal which provides a localized large moment (formed by the d-electrons) and by introducing carriers which can mediate a ferromagnetic coupling between these local moments. The most extensively studied are the Mn based (III,Mn)V DMSs, in which substituting Mn for the cations in a (III,V) semiconductor can dope the system with hole carriers; (Ga,Mn)As becomes ferromagnetic beyond a concentration of 1%.

The simplicity of this basic but generally correct model hides within it a cornucopia of physical and materials science effects present in these materials. Among the phenomena which have been studied are metal-insulator transitions, carrier mediated ferromagnetism, disorder physics, magneto-resistance effects, magneto-optical effects, coupled magnetization dynamics, post-growth dependent properties, etc. A more in-depth discussion of these materials, both from the experimental and theoretical point of view, can be found in the recent review by Jungwirth et al Jungwirth et al. (2006).

The AHE has been one of the most fundamental characterization tools in DMSs, allowing, for example, direct electrical measurement of transition temperatures. The reliability of electrical measurement of magnetic properties in these materials has been verified by comparison with remnant magnetization measurements using a SQUID magnetometer Ohno et al. (1992). The relative simplicity of the effective band structure of the carriers in metallic DMSs, has made them a playing ground to understand AHE of ferromagnetic systems with strong spin-orbit coupling.

Experimentally, it has been established that the AHE in the archetypical DMS system (Ga,Mn)As is in the metallic regime dominated by a scattering-independent mechanism, i.e. Edmonds et al. (2002); Ruzmetov et al. (2004); Chun et al. (2007); Pu et al. (2008). The studies of Edmonds et al., 2002 and Chun et al., 2007 have established this relationship in the non-insulating materials by extrapolating the low temperature to zero field and zero temperature. This is illustrated in Fig. 21 where metallic samples, which span a larger range than the ones studied by Edmonds et al., 2002, show a clear dependence.

DMSs grown requires non-equilibrium (low temperature) conditions and the as-grown (often insulating) materials and post-grown annealed metallic materials show typically different behavior in the AHE response. A similar extrapolating procedure performed on insulating (Ga,Mn)As seems to exhibit a somewhat linear dependence of on . On the other hand, considerable uncertainty is introduced by the extrapolation to low temperatures because diverges and the complicated magetoresistance of is a priori not known in the low range.

Figure 21: (a) GaMnAs samples that show insulating and metallic behavior defined by near . (b) R vs. extrapolated from data to zero field and low temperatures. [From Ref. Chun et al., 2007.]
Figure 22: (Top) Zero field and for four samples grown on InAs substrates (i.e. perpendicular to plane easy axis). Annealed samples, which produce perpendicular to plane easy axes, are marked by a . The inset indicates the dependence of the 7% sample at 10 K. (Bottom) Zero-field Nerst coefficient for the four samples. The solid red curves indicate the best fit using Eq. (2.11) and the dashed curves the best fit setting n=1. [From Ref. Pu et al., 2008.]

A more recent study by Pu et alPu et al. (2008)of (Ga,Mn)As grown on InAs, such that the tensile strain creates a perpendicular anisotropic ferromagnet, has established the dominance of the intrinsic mechanism in metallic (Ga,Mn)As samples beyond any doubt. Measuring the longitudinal thermo-electric transport coefficients ( , ,