Spin Chirality, Berry Phase, and Anomalous Hall Effect in a Frustrated Ferromagnet

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Science  30 Mar 2001:
Vol. 291, Issue 5513, pp. 2573-2576
DOI: 10.1126/science.1058161


An electron hopping on non-coplanar spin sites with spin chirality obtains a complex phase factor (Berry phase) in its quantum mechanical amplitude that acts as an internal magnetic field, and is predicted to manifest itself in the Hall effect when it is not cancelled. The present combined work of transport measurement, neutron scattering, and theoretical calculation provides evidence that the gigantic anomalous Hall effect observed in Nd2Mo2O7, a pyrochlore ferromagnet with geometrically frustrated lattice structure, is mostly due to the spin chirality and the associated Berry phase originating from the Mo spin tilting.

When an electron hops between atoms in solids under magnetic field B, the quantum mechanical amplitude obtains a complex factor with its phase determined by the vector potential A corresponding to B (= × A). In metallic magnets, the analogous complex factor occurs when an electron hops along the non-coplanar spin configurations, and the effective magnetic field is represented by the spin chirality, namely the solid angle subtended by the spins. This internal effective magnetic field is expected to manifest itself in the Hall effect (1, 2)—the voltage drop transverse to the applied current and external magnetic field. The transverse resistivity ρH in ferromagnets consists of two contributions: ρH =R o B + 4πR s M, where B is magnetic induction, M is the magnetization,R o is the ordinary Hall coefficient, andR s is the anomalous Hall coefficient. The second term, which is proportional to the magnetization, is the anomalous Hall effect (AHE). Conventionally, the AHE has been ascribed to spin-orbit interaction and the spin polarization of conduction electrons, which result in asymmetry in terms of orbital angular momentum (3), or to the asymmetric skew scattering of conduction electrons by the fluctuation of localized moments (4). These theories assume the collinear spin structure. Recently, however, the relevance of the non-coplanar spin configuration to the AHE has been discussed in the context of perovskite-type manganites at high temperatures (1, 5,6).

In manganites or related double-exchange ferromagnets, as a result of the strong Hund's-rule coupling between the eg conduction electrons and the localizedt 2g spins, the transfer integral from site i to site j is given bytij =t{cos(θi/2) cos(θj/2) + sin(θi/2) sin(θj/2) exp[ii – φj)]} (7), where θi and φi are the polar coordinates of the spin direction. This transfer integral is a complex number and could produce the gauge flux (8, 9). When we consider an electron hopping along a loop 1 → 2 → 3 → 1, the total phase acquired by the electron is the solid angle subtended by the three spins (Fig. 1A) and is proportional to the spin chirality S 1 ·S 2 × S 3. Such a gauge flux acts as a fictitious magnetic field b and affects the charge dynamics in the same way as a real magnetic field does. At finite temperatures, the spin configuration is disordered by thermal agitation, and the topological excitations (called skyrmions) are activated to produce a finite spin chirality that contributes to the transverse conductivity (1). However, it is rather difficult to specify uniquely the mechanism of AHE at finite temperature: The anomalous Hall term vanishes at low temperatures in the manganites, much as it does in conventional ferromagnets (10), and the temperature dependence may not be contradictory, at least qualitatively, to the existing theories (3, 4). At zero temperature, the periodicity of the crystal causes the uniform component of the gauge flux to vanish. As discussed recently (2), the geometrical and topological properties of the lattice are crucial for the Berry phase mechanism of AHE atT = 0 K, and Kagome and pyrochlore lattice are two of the rare structures that satisfy this condition. In the ground state of a ferromagnet, there is no thermal or quantum fluctuation. Therefore, the connection between spin configuration and transport properties can be studied in an unambiguous way. Here, we present an anomalous experimental observation that the AHE continuously increases down to the ground state in a ferromagnet, Nd2Mo2O7, with a pyrochlore lattice. This behavior is distinct from the prediction of the existing theories (3, 4) and from experimental results for a broad range of ferromagnetic materials (5, 10, 11), but is in accord with the prediction of Berry phase theory (2).

Figure 1

Schematic magnetic and crystal structures of pyrochlore. (A) Spin chirality, that is, the solid angle subtended by the three spins. (B) “Two-in, two-out” spin structure, in which each spin points along the line that connects the center of the tetrahedron and the vertex. The total fictitious magnetic field is the vector sum of each fictitious magnetic flux that penetrates each plaquette. (C) The B sublattice of pyrochlore structure A2B2O7. The A sublattice is structurally identical with this one, but is displaced by half a lattice constant. (D) Relative position of Nd tetrahedron (green circles) and Mo tetrahedron (blue circles) in Nd2Mo2O7 pyrochlore. (E) The “umbrella” structure observed for Nd2Mo2O7 (A = Nd, B = Mo) by a neutron diffraction study. Magnetic unit cell contains four inequivalent Nd 4f momentsn i and four Mo 4d momentsm i. In the umbrella structure, (m im) ⊥ mand (n in) ⊥n for each m i andn i, where m andn are the average moments of fourm i and fourn i, respectively.

The pyrochlore-type structure (A2B2O7) is well known as a geometrically frustrated lattice (12) and is composed of two sublattices of A and B sites. These sublattices are structurally identical but are displaced by half a lattice constant from each other. In each sublattice, the ions form an infinite network of corner-sharing tetrahedra (Fig. 1C). This connectivity between the magnetic sites gives rise to magnetic frustration, which may lead to intriguing spin states such as spin glass (13). On the other hand, the rare-earth 4f moments are subject to strong spin anisotropy with the easy axis pointing to the center of each tetrahedron. One of the sixfold degenerate “two-in, two-out” spin states in the “spin-ice” model is depicted in Fig. 1B (14). This degeneracy prevents ferromagnetic ordering at finite temperature (14, 15). Another important feature of the spin-ice state is that the total gauge flux for the tetrahedron, which is a vector sum of the flux that penetrates each triangle, is finite and parallel to the total magnetization M.

In R2Mo2O7 materials (where R is a rare-earth ion), the ground state changes from a spin-glass insulator to a ferromagnetic metal with the change of R (16,17). Nd2Mo2O7 (R = Nd) shows the ground state of a ferromagnetic metal (16), in which the Mo spin-polarized delectrons are itinerant while the Nd f electrons are viewed as localized and forming the local moments. The Curie temperature is ∼90 K and the saturated Mo spin moment at low temperature is ∼1.5 μB (where μB is the Bohr magneton). The easy axis of the magnetization is along the (100) direction. Figure 1D shows the relative position of a Nd tetrahedron and a Mo tetrahedron, where another tetrahedron is formed by two Nd ions and two Mo ions. Therefore, the spin chirality of Nd moments is transmitted to the Mo spins via the f-d exchange interactionJfd , and a conduction electron moving in the Mo sublattice would feel a fictitious magnetic field that is parallel toM.

We plot the temperature variation of magnetization and longitudinal conductivity σxx ≈ 1/ρxx, both measured at a field ofH = 0.5 T, in Fig. 2. The magnetization increases rapidly below Curie temperatureT C = 89 K, which corresponds to the ferromagnetic ordering of Mo spins. The Nd moments are then subject to the effective magnetic field through the antiferromagneticJfd , and one of the sixfold degenerate two-in, two-out spin states has more weight than the others. At a crossover temperature ≈ 40 K (which we designate T* hereafter), the magnetization begins to decrease while the neutron scattering intensity of the (200) reflection begins to grow (Fig. 2A). We have determined the magnetic structure from the neutron scattering data of (111) and (200) reflections as well as the low-field magnetization data (Fig. 2A) in the following manner (18): The magnetic unit cell contains four inequivalent Nd 4fmoments n i and four Mo 4dmoments m i. From a general energetic argument that takes into account the nearest-neighbor exchange interaction, we can infer that the “umbrella structure” (Fig. 1E) is stable. This reduces the number of unknown parameters to four, namely, the magnitudes and tilting angles of the Nd and Mo moments. The scattering intensities I 1,I 2, and I 3 corresponding to the (111), (200), and (000) reflections, which are normalized in a unit of μB 2 per magnetic unit cell, can be expressed in terms of the four parameters. In the actual procedure,I 3 data were replaced by the uniform magnetization data with high accuracy, and the magnitude of the Mo moment below T* was extrapolated from the values at higher temperatures. Thus, we could determine three unknown parameters from three experimental constraints. It turns out that the Nd moments grow rapidly below T*, with their total moment pointing antiparallel to the total Mo spin. The tilting angles of the Nd and Mo moments (θn and θm, respectively) are estimated to be θn ≈ 70° to 80° and ∣θm∣ < 10° at 8 K (see below).

Figure 2

Temperature evolution of magnetic and transport properties. (A) Temperature dependence of magnetization at H = 0.5 T and the background-subtracted and properly normalized neutron scattering intensity I of the (200) reflection, which represents the ordering of the transverse component. (B) Temperature dependence of the longitudinal conductivity σxx at H = 0.5 T and the Hall conductivity σxy atH = 0.5 T. The value of σxy atH = 0.5 T is derived from the values of ρH and ρxx at H= 0.5 T through the relation σxy = ρH/(ρxx 2 + ρH 2). Single crystals of Nd2Mo2O7 were grown by the floating-zone method in Ar atmosphere. Typical size is 4 mm in diameter and 30 mm in length. The Curie temperature of the crystal used for the transport measurements is 89 K. The Hall effect and resistivity measurements were done by the conventional method. Neutron diffraction measurements were carried out on a general-purpose triple-axis spectrometer in the JRR-3M of the Japan Atomic Energy Research Institute, Tokai, Japan. An incident neutron momentumk i = 2.559 Å−1 was selected with a pyrolytic graphite filter. The single-crystal sample was placed in an Al can with helium gas, and it was attached at the cold head of a closed-cycle helium gas refrigerator.

Figure 3A shows the magnetic field dependence of magnetization of a single crystal of Nd2Mo2O7 at several temperatures when the magnetic field is applied along the (100) direction. The magnetization monotonically increases with decreasing temperature toT* ≈ 40 K at both low and high fields. BelowT*, in contrast, the magnetization at a low field (e.g., 0.5 T) decreases, reflecting the rapid growth of Nd moments with their total moment pointing antiparallel to that of the Mo spin. However, the Nd moment is very susceptible to the magnetic field at low temperatures (for free Nd3+ ion at H = 10 T and T = 2 K,gJ μBH/kBT= 11, where gJ is the Landé gfactor and kB is the Boltzmann constant). Therefore, the total magnetization shows a substantial increase with application of high magnetic field at 2 K.

Figure 3

Magnetic field dependence of magnetization and Hall resistivity. (A) Magnetic field dependence of magnetization at various temperatures. The field is applied along the (100) direction. (B) Magnetic field dependence of the Hall resistivity ρH with H ∥ (100) at several temperatures. The inset shows the temperature dependence of ρH at 0.5 T, which is a measure of the anomalous Hall term. The ρH at a low field (<0.3 T) is finite at 2 K, whereas it tends to zero above 10 K, in accord with the presence or absence of remnant magnetization at the respective temperatures.

In plotting the magnetic field dependence of the Hall resistivity ρH at various temperatures (Fig. 3B) with the magnetic field H also applied along the (100) direction, the anomalous term is dominant over the ordinary term at all the temperatures. Because the ordinary term is negligible at a weak magnetic field, the value of ρH at 0.5 T can be a good measure of the anomalous term. This quantity continues to increase down to the lowest temperature, 2 K (Fig. 3B, inset), in marked contrast to the prediction of the existing theories (3, 4) and to experimental results (5, 10, 11). In a plot of the Hall conductivity σxy at H = 0.5 T, together with the results for cases where a magnetic field is applied along high-symmetric directions (110) and (111) (Fig. 2B), the σxy value for every direction continues to increase below T C all the way down to the ground state. This is true even below 20 K ≈ T C/5, where σxx changes very little (Fig. 2B). The magnitude of σxy is almost isotropic aboveT* but shows large anisotropy belowT*, clearly indicating an intimate relation with the spin configuration.

The steep decrease of ρH in a high-field region at low temperatures (Fig. 3B) is quite remarkable and should be attributed to the anomalous Hall term. If it were due to the ordinary Hall coefficient R o, the value ofR o would increase by a factor of 5 asT decreases from 40 K to 2 K, which is obviously incompatible with the minimal change in σxx (Fig. 2B).

As we have argued above, both the temperature and field dependencies of ρH are anomalous within the conventional view of AHE. Furthermore, these features strongly suggest that the relevant variable is the directional degree of freedom, rather than the magnitude, of the spin moment: The direction of the spin can vary with the change of temperature or magnetic field even at temperatures well belowT C, whereas its magnitude cannot.

On the basis of the spin chirality mechanism of the AHE, we have calculated σxy as a function of the tilting angle of the Mo spins using the tight-binding Hamiltonian for triply degenerate t 2g bands. In Fig. 4, the calculated results for σxy are shown as a function of the tilting angle θm in the two-in, two-out spin state under the applied field along the (100) direction. We could reproduce the correct sign (σxy > 0) and obtained the relation σxy ∝ θm 2, as is expected from ∣b∣ ∝ θm 2. The magnitude of σxy is also consistent with the above experimental results; that is, the observed σxy can be reproduced with θm ≈ 4°. The conventional on-site spin-orbit interaction λ ≈ 400 cm−1 gives comparable or slightly larger values of σxy, and the value is rather sensitive to the details of the band structure such as the crystal field splitting. However, this mechanism cannot explain the rapid increase of σxy accompanied by the tilting, because it gives only a weak dependence on θm (∝ cos θm).

Figure 4

Theoretical and experimental results of Hall conductivity. The theoretically calculated Hall conductivity σxy for the two-in, two-out spin configuration for H ∥ (100) (solid circles) is plotted against the tilting angle of the spins. Note the scale change at a tilting angle of 9°. The tight-binding Hamiltonian parameters oft dpσ = –0.5 eV,t dpπ = 0.1 eV are used for the triply degenerate t 2g bands. The transverse conductivity σxy is calculated in terms of the Kubo formula, where the k-integral over the first Brillouin zone is replaced by the discrete sum over the 40 × 40 × 40 mesh. The open circles represent 1/Embedded Image times the respective solid-circle values, which corresponds to the case withH ∥ (111) assuming that the two-in, two-out spin configuration remains stable against the magnetic field. The inset shows the experimental results of σxy plotted against applied magnetic field. The data below and above T* are shown for H ∥ (100) and (111) configurations. The experimental zero-field value at 2 K crudely corresponds to the calculated tilting angle of ∼4° of the Mo spins, which is in accord with the estimate at 8 K as judged by neutron scattering [(18); see also text] and the following molecular-field consideration: Using the moments of Nd and Mo at 50 K derived from the neutron data, we can estimate the molecular field at the Nd site and hence the value of Jfd (= 1.5 K). GivenJfd , together with the magnitude and tilting angle of the Nd moments (θn = 70° to 80°) deduced from neutron data at 8 K, the effective transverse field at the Mo site at 8 K produced by the tilted Nd moments can be evaluated. Together with the estimated value ofJdd (= 11 K), this allows us to estimate the tilting angle of Mo to be 2.4°.

We also plotted the calculated σxy for the two-in, two-out structure for the (111) direction. These values are simply 1/3 times the values for the (100) direction, because the “easy axis” of the spin chirality is the (100) direction and only 1/3 of the fictitious field can affect σxy. Indeed, the observed anisotropy at 2 K at a low field, σxy (100)xy (111)≈ 1.6, nearly coincides with the predicted value (= 3) (Fig. 4, inset).

The magnetic field dependence of σxy at low temperatures (Fig. 4, inset) can also be explained in terms of Berry phase theory as a field suppression of the spin chirality. In a low-field region, the tilting angle of the spins is relatively large, which gives rise to the large spin chirality and hence the large anomalous Hall term. Once a high field is applied and the spins are aligned along the field direction, the spin chirality or the fictitious magnetic field is reduced, resulting in the diminished AHE.

At 40 K, σxy is almost isotropic, in accord with the absence or minute residue of the long-range ordering of the transverse component. The magnetic field dependence of σxy at 40 K (Fig. 4, inset) is weaker than at 2 K because the Zeeman energy is small compared with the temperature, and the thermal fluctuation of moments cannot be suppressed by the magnetic field.

We have found anomalous temperature and field dependence of the AHE. Below T* ≈ 40 K, where the magnetic structure changes, the Hall conductivity shows clear anisotropy. Such behaviors of the AHE can be successfully ascribed to the Berry phase produced by the spin chirality on the pyrochlore lattice. Our results show that the control of spin texture reflecting lattice topology provides a new way to tailor electronic properties in a variety of magnetic materials in bulk and at interfaces.


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