Dissipationless Anomalous Hall Current in the Ferromagnetic Spinel CuCr2Se4-xBrx

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Science  12 Mar 2004:
Vol. 303, Issue 5664, pp. 1647-1649
DOI: 10.1126/science.1094383


In a ferromagnet, an applied electric field E invariably produces an anomalous Hall current JH that flows perpendicular to the plane defined by E and M (the magnetization). For decades, the question of whether JH is dissipationless (independent of the scattering rate) has been debated without experimental resolution. In the ferromagnetic spinel CuCr2Se4–xBrx, the resistivity ρ (at low temperature) may be increased by several decades by varying x (Br) without degrading M. We show that JH/E (normalized per carrier, at 5 kelvin) remains unchanged throughout. In addition to confirming the dissipationless nature of JH, 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 JH appears transverse to E (in practice, a weak H serves to align the magnetic domains) (1, 2). Questions regarding the origin of JH, 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 CuCr2Se4–xBrx. Despite a 70-fold increase in the scattering rate from impurities, we found that JH (per carrier) remains constant, implying that it is indeed dissipationless.

Karplus and Luttinger (3, 4) proposed a purely quantum-mechanical origin for JH. 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: JH remains constant even as the longitudinal current (JE) 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 (79) 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 = λk × 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 Math(1) where n is the carrier density and e is the charge. As noted, JH 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 (1315). 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 ρxy comprises two terms, Math(2) where B is the induction field, R0 is the ordinary Hall coefficient, and Math is the anomalous Hall resistivity. A direct test of the dissipationless nature of JH is to check whether the anomalous Hall conductivity Math (defined as Math) changes as impurities are added to increase 1/τ (and ρ) (3, 7). A dissipationless AHE current implies that Math, 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 αexp = 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, Math 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 CuCr2Se4, a metallic ferromagnet with a Curie temperature TC ∼ 430 K, is particularly well suited for testing the AHE. Substituting Se with Br in CuCr2Se4–xBrx decreases the hole density nh (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 CuCr2Se4–xBrx by chemical vapor transport (24). When x is increased from 0 to 1 in our crystals, nh decreases by a factor of ∼30 while TC decreases from 430 K to 230 K (Fig. 1A). The saturated magnetization Ms at 5 K corresponds to a Cr moment that actually increases from 2.6 to 3 μB (Bohr magneton) (Fig. 1B).

Fig. 1.

(A) Hole density nh (solid circles) in CuCr2Se4–xBrx versus x determined from R0 at 400 K (one hole per formula unit corresponds to nh = 7.2 × 1021 cm–3). The Curie temperature TC is shown as open circles. (B) Curves of the magnetization M versus H at 5 K in three samples (x values indicated). The saturation value Ms = 3.52 × 105 A/m for x = 0, 3.72 × 105 A/m for x = 0.5, and 3.95 × 105 A/m for x = 1.0. (C) Resistivity ρ versus T in 10 samples with Br content x indicated (a and b indicate different samples with the same x). Values of nh in all samples fall in the metallic regime (for x = 1, nh = 1.9 × 1020 cm–3).

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 nh 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 nh = 1.9 × 1020 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 |ρxy| in weak H reflects the rotation of domains into alignment with H. Above the saturation field Hs, when Math is constant, the small ordinary Hall term R0B is visible as a linear background (24). As in standard practice, we used R0 measured above TC to find the nh plotted in Fig. 1A.

Fig. 2.

Curves of the observed Hall resistivity ρxy = R0B + Rsμ0M versus H (at temperatures indicated) in CuCr2Se4–xBrx with x = 0.25 (A) and x = 1.0 (B). In (A), the anomalous Hall coefficient Rs changes sign below 250 K, becomes negative, and saturates to a constant value below 50 K. However, in (B), Rs is always positive. At low T, it rises to very large values (note difference in scale).

By convention, the T dependence of the AHE signal is represented by the anomalous Hall coefficient Rs(T) defined by Math (where μ0 is the vacuum permeability). By scaling the Math-H curve against the M-H curve measured at each T, we have determined (24) Rs versus T in each of the samples studied (Fig. 3). The introduction of Br causes the Rs versus T profiles to change markedly. In the undoped sample (x = 0), Rs 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 Rs and a finite negative value at low T. Increasing the doping to x = 0.25 leads to an Rs 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 Rs increases steeply, but now in the positive direction. At maximum doping (x = 1), the value of Rs at 5 K is very large, corresponding to Math μΩ·cm (Fig. 2B).

Fig. 3.

(A) Values of Rs extracted from the curves of ρxy and M versus H measured at each T in CuCr2Se4–xBrx, with values of x indicated (a and b refer to different crystals with the same x). (B) The corresponding curves for x = 0, 0.1, 0.25, and 0.5 (two crystals a and b). The values of Rs at 5 K are negative at small x (< 0.4), but as x increases, Rs rapidly rises to large positive values.

Our focus is on the low-T values of Math where impurity scattering dominates. At 5 K, Math is too small to be resolved in the sample with x = 0. As x increases to 1, the absolute magnitude Math at 5 K increases by more than three orders of magnitude (hereafter, Math refers to the saturated value measured at 2 T or higher). It is noteworthy that Math 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 Math 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 Math versus ρ, and discuss the change in sign later.

Fig. 4.

The normalized quantity Embedded Image versus ρ (at 5 K) in a log-log plot (Embedded Image is measured at 2 T and 5 K). The 12 samples (with doping x indicated) include ones with negative Embedded Image (open circles) and positive Embedded Image (solid circles). The undoped sample (x = 0) is not shown because Embedded Image at 5 K is unresolved in our experiment (24). The least-squares fit gives Embedded Image with α = 1.95 ± 0.08and A = 2.24 × 10–25 (SI units).

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

Figure 4 shows Math versus ρ in log-log scale for all samples investigated [except x = 0 (24)]. Over several decades, the data fit well to Math with α = 1.95 ± 0.08 (because Ms is nearly insensitive to x, Fig. 4 also gives Rs/nh ∼ ρ2). This immediately implies that the normalized AHE conductivity Math 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 ϵF across the overlap between two narrow bands derived from distinct transition-metal elements. In the alloy Ni1–xFex, the band derived from Fe 3d states lies just above the 3d band of Ni. As ϵF crosses the overlap, Math 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 ϵF moves between overlapping narrow bands. A similar effect is implied in Noziéres and Lewiner's calculation (9). Band-structure calculations (25) on CuCr2Se4 reveal that ϵF lies in a hole-like band of mostly Cu 3d character strongly admixed with Cr 3d states lying just above. We infer that as ϵF rises with increasing Br content, the conduction states acquire more Cr 3d character at the expense of Cu 3d, triggering a sign change in λ. The sign change (negative to positive with increasing x) is consistent with that observed in Ni1–xFex. A change in sign of the AHE conductivity at band crossings is also described in recent theories (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 JH 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 JH ∼ (nn). More important, this also produces a fully polarized spin-Hall current Js 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, JH depends on the direction of the local moment mi on the impurity but not the spin of the incident electron; that is, both JH and Js ∼ (nn).

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.

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