## Abstract

In a ferromagnet, an applied electric field E invariably produces an anomalous Hall current J_{H} that flows perpendicular to the plane defined by E and M (the magnetization). For decades, the question of whether J_{H} is dissipationless (independent of the scattering rate) has been debated without experimental resolution. In the ferromagnetic spinel CuCr_{2}Se_{4–}* _{x}*Br

*, the resistivity ρ (at low temperature) may be increased by several decades by varying*

_{x}*x*(Br) without degrading M. We show that J

_{H}/E (normalized per carrier, at 5 kelvin) remains unchanged throughout. In addition to confirming the dissipationless nature of J

_{H}, our finding has implications for the generation and study of spin-Hall currents in bulk samples.

A major unsettled question in the study of electron transport in a ferromagnet is whether the anomalous Hall current is dissipationless. In nonmagnetic metals, the familiar Hall current arises when electrons moving in crossed electric (**E**) and magnetic (**H**) fields are deflected by the Lorentz force. However, in a ferromagnet subject to **E** alone, a large, spontaneous (anomalous) Hall current **J**_{H} appears transverse to **E** (in practice, a weak **H** serves to align the magnetic domains) (*1*, *2*). Questions regarding the origin of **J**_{H}, and whether it is dissipationless, have been keenly debated for decades. They have emerged anew because of fresh theoretical insights and strong interest in spin currents for spin-based electronics. We report measurements in the ferromagnet CuCr_{2}Se_{4–}* _{x}*Br

*. Despite a 70-fold increase in the scattering rate from impurities, we found that*

_{x}**J**

_{H}(per carrier) remains constant, implying that it is indeed dissipationless.

Karplus and Luttinger (*3*, *4*) proposed a purely quantum-mechanical origin for **J**_{H}. An electron in the conduction band of a crystal lattice spends part of its time in nearby bands because of admixing caused by the (intracell) position operator **X**. In the process, it acquires a spin-dependent “anomalous velocity” (*5*). Karplus and Luttinger predicted that the Hall current is dissipationless: **J**_{H} remains constant even as the longitudinal current (**J**∥**E**) is degraded by scattering from added impurities. A conventional mechanism was later proposed (*6*) whereby the anomalous Hall effect (AHE) is caused instead by asymmetric scattering of electrons by impurities (skew scattering). Several authors (*7*–*9*) investigated the theoretical ramifications of these competing models. The role of impurities in the anomalous-velocity theory was clarified by Berger's side-jump model (*7*). A careful accounting of various contributions (including side-jump) to the AHE in a semiconductor has been given by Noziéres and Lewiner, who derived **X** = λ**k** × **S**, where λ is the enhanced spin-orbit parameter, **k** is the carrier wave vector, and **S** is its spin (*9*). In the zero-frequency limit, Noziéres and Lewiner obtained the AHE current (1) where *n* is the carrier density and *e* is the charge. As noted, **J**_{H} is linear in **S** but is independent of the electron lifetime τ.

In modern terms, the anomalous-velocity term of Karplus and Luttinger is related to the Berry phase (*10*) and has been applied (*11*) to explain the AHE in Mn-doped GaAs (*12*). The close connection of the AHE to the Berry phase has also been explored in novel ferromagnets in which frustration leads to spin chirality (*13*–*15*). In the field of spintronics, several schemes have been proposed to produce a fully polarized spin current in thin-film structures (*16*) and in bulk *p*-doped GaAs (*17*). The AHE is intimately related to these schemes, and our experimental results are relevant to the spin current problem.

In an AHE experiment (*1*), the observed Hall resistivity ρ* _{xy}* comprises two terms, (2) where

*B*is the induction field,

*R*

_{0}is the ordinary Hall coefficient, and is the anomalous Hall resistivity. A direct test of the dissipationless nature of

**J**

_{H}is to check whether the anomalous Hall conductivity (defined as ) changes as impurities are added to increase 1/τ (and ρ) (

*3*,

*7*). A dissipationless AHE current implies that , with α = 2. By contrast, in the skew scattering model, α = 1.

Tests based on measurements at high temperatures *T* (77 to 300 K) yield exponents in the range α_{e}_{x}_{p} = 1.4 to 2.0 (*18*, *19*). However, it has been argued (*20*) that at high *T*, both models in fact predict α = 2, a view supported by detailed calculations (*21*). To be meaningful, the test must be performed in the impurity-scattering regime over a wide range of ρ. Unfortunately, in most ferromagnets, becomes too small to measure accurately at low *T*. Results on α in the impurity-scattering regime are very limited.

The copper-chromium selenide spinel CuCr_{2}Se_{4}, a metallic ferromagnet with a Curie temperature *T*_{C} ∼ 430 K, is particularly well suited for testing the AHE. Substituting Se with Br in CuCr_{2}Se_{4–}* _{x}*Br

*decreases the hole density*

_{x}*n*

_{h}(

*22*). However, because the coupling between local moments on Cr is primarily from 90° superexchange along the Cr-Se-Cr bonds (

*23*), this does not destroy the magnetization. We have grown crystals of CuCr

_{2}Se

_{4–}

*Br*

_{x}*by chemical vapor transport (*

_{x}*24*). When

*x*is increased from 0 to 1 in our crystals,

*n*

_{h}decreases by a factor of ∼30 while

*T*

_{C}decreases from 430 K to 230 K (Fig. 1A). The saturated magnetization

*M*

_{s}at 5 K corresponds to a Cr moment that actually increases from 2.6 to 3 μ

_{B}(Bohr magneton) (Fig. 1B).

As shown in Fig. 1C, all samples except the ones with *x* = 1.0 lie outside the localization regime. In the “metallic” regime, the low-*T* resistivity increases by a factor of ∼270 as *x* increases from 0 to 0.85; this increase is predominantly due to a decrease in τ by a factor of 70. The hole density *n*_{h} decreases by only a factor of 4. In the localization regime (*x* = 1.0), strong disorder causes ρ to rise gradually with decreasing *T*. We emphasize, however, that these samples are not semiconductors (ρ is not thermally activated, and *n*_{h} = 1.9 × 10^{20} cm^{–3} is degenerate).

The field dependence of the total Hall resistivity (Eq. 2) is shown in Fig. 2 for *x* = 0.25 and *x* = 1.0 [see (*24*) for measurement details]. The steep increase in |ρ* _{xy}*| in weak

*H*reflects the rotation of domains into alignment with

**H**. Above the saturation field

*H*

_{s}, when is constant, the small ordinary Hall term

*R*

_{0}

*B*is visible as a linear background (

*24*). As in standard practice, we used

*R*

_{0}measured above

*T*

_{C}to find the

*n*

_{h}plotted in Fig. 1A.

By convention, the *T* dependence of the AHE signal is represented by the anomalous Hall coefficient *R*_{s}(*T*) defined by (where μ_{0} is the vacuum permeability). By scaling the -*H* curve against the *M-H* curve measured at each *T*, we have determined (*24*) *R*_{s} versus *T* in each of the samples studied (Fig. 3). The introduction of Br causes the *R*_{s} versus *T* profiles to change markedly. In the undoped sample (*x* = 0), *R*_{s} is positive and monotonically decreasing below 360 K, as is typical in high-purity ferromagnets (Fig. 3B). Weak doping (*x* = 0.1) produces a negative shift in *R*_{s} and a finite negative value at low *T*. Increasing the doping to *x* = 0.25 leads to an *R*_{s} profile that is large, negative, and nearly independent of *T* below 50 K (Fig. 3B). At mid-range doping and higher (*x* ≥ 0.5), the magnitude of *R*_{s} increases steeply, but now in the positive direction. At maximum doping (*x* = 1), the value of *R*_{s} at 5 K is very large, corresponding to μΩ·cm (Fig. 2B).

Our focus is on the low-*T* values of where impurity scattering dominates. At 5 K, is too small to be resolved in the sample with *x* = 0. As *x* increases to 1, the absolute magnitude at 5 K increases by more than three orders of magnitude (hereafter, refers to the saturated value measured at 2 T or higher). It is noteworthy that is negative at low doping (0 < *x* < 0.4) but becomes positive for *x* > 0.5. Initially, the sign change seemed to suggest to us that there might exist two distinct mechanisms for the AHE in this system. As more samples were studied, however, it became apparent that irrespective of the sign, the magnitude versus ρ falls on the same curve over several decades (Fig. 4), providing strong evidence that the same AHE mechanism occurs in both sign regimes. We focus first on the magnitude versus ρ, and discuss the change in sign later.

It is worth emphasizing that is proportional to the carrier density *n*_{h} (see Eq. 1). For our goal of determining whether the AHE current is dissipationless, it is clearly necessary to factor out *n*_{h} before comparing against ρ. Hence, we divide by *n*_{h}. We refer to as the normalized AHE conductivity (*24*).

Figure 4 shows versus ρ in log-log scale for all samples investigated [except *x* = 0 (*24*)]. Over several decades, the data fit well to with α = 1.95 ± 0.08 (because *M*_{s} is nearly insensitive to *x*, Fig. 4 also gives *R*_{s}/*n*_{h} ∼ ρ^{2}). This immediately implies that the normalized AHE conductivity at 5 K is dissipationless. Increasing ρ by a factor of ∼100 leaves the AHE current per carrier unchanged to our measurement accuracy [see (*24*) for a discussion of our resolution]. As noted, the two samples with *x* = 1 are in the localization regime. The fact that their points also fall on the line implies that the dissipationless nature of the normalized AHE current extends beyond the Bloch-state regime (where most AHE theories apply) into the weak localization regime, where much less has been done. This supports recent theories (*10*, *11*, *17*) that the anomalous-velocity origin is topological in nature and is equally valid in the Bloch and localization regimes.

The sign change at *x* ∼ 0.4 is reminiscent of sign changes observed in ferromagnetic alloys (versus composition). The common feature is that doping drives the Fermi energy ϵ_{F} across the overlap between two narrow bands derived from distinct transition-metal elements. In the alloy Ni_{1–}* _{x}*Fe

*, the band derived from Fe 3*

_{x}*d*states lies just above the 3

*d*band of Ni. As ϵ

_{F}crosses the overlap, changes from negative to positive. Similar sign changes are observed in Au-Fe and Au-Ni alloys. It has been pointed out (

*2*) that the effective spin-orbit parameter λ in Eq. 1 changes sign whenever ϵ

_{F}moves between overlapping narrow bands. A similar effect is implied in Noziéres and Lewiner's calculation (

*9*). Band-structure calculations (

*25*) on CuCr

_{2}Se

_{4}reveal that ϵ

_{F}lies in a hole-like band of mostly Cu 3

*d*character strongly admixed with Cr 3

*d*states lying just above. We infer that as ϵ

_{F}rises with increasing Br content, the conduction states acquire more Cr 3

*d*character at the expense of Cu 3

*d*, triggering a sign change in λ. The sign change (negative to positive with increasing

*x*) is consistent with that observed in Ni

_{1–}

*Fe*

_{x}*. A change in sign of the AHE conductivity at band crossings is also described in recent theories (*

_{x}*10*).

We now discuss the relevance of our findings to spin current production. To produce fully polarized spin currents, it is ideal to use “half metals” (ferromagnets in which all conduction electrons are, say, spin-up). However, only a few examples are known (*26*). Alternate schemes based on elemental ferromagnets have been proposed (*16*). As is evident from Eq. 1, anomalous-velocity theories predict that **J**_{H} depends on the carrier spin **S**. If a beam of electrons with spin populations *n*_{↑} and *n*_{↓} enters a region of fixed **M**, the spin-up and spin-down electrons are deflected in opposite directions transverse to **E**, just as in the classic Stern-Gerlach experiment. This results in a Hall charge current proportional to the difference between the spin populations, namely **J**_{H} ∼ (*n*_{↑} – *n*_{↓}). More important, this also produces a fully polarized spin-Hall current **J**_{s} proportional to the sum (*n*_{↑} + *n*_{↓}). Hence, in a ferromagnet that is not a half metal, the spin-Hall current is fully polarized according to these theories. By contrast, in skew-scattering theories, **J**_{H} depends on the direction of the local moment **m**_{i} on the impurity but not the spin of the incident electron; that is, both **J**_{H} and **J**_{s} ∼ (*n*_{↑} – *n*_{↓}).

In confirming that the normalized AHE current is dissipationless over a multidecade change in ρ, we verify a specific prediction of the anomalous-velocity theories and resolve a key controversy in ferromagnets. The implication is that fully polarized spin-Hall currents are readily generated in ferromagnets (at low *T*) by simply applying **E**. Although this realization does not solve the conductivity-mismatch problem at interfaces (*27*), it may greatly expand the scope of experiments on the properties of spin currents.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/303/5664/1647/DC1

Materials and Methods

Figs. S1 and S2

References