Giant Hall effect in a highly conductive frustrated magnet GdCu₂

Abstract

The Hall effect is one of the most fundamental but elusive phenomena in condensed matter physics due to the rich variety of underlying mechanisms. Here we report an exceptionally large Hall effect in a frustrated magnet GdCu₂ with high conductivity. The Hall conductivity at the base temperature is as high as the order of 10⁴–10⁵ Ω⁻¹cm⁻¹ and shows abrupt sign changes under magnetic fields. Remarkably, the giant Hall effect is rapidly suppressed as the longitudinal conductivity is lowered upon increasing temperature or introducing tiny amount of quenched disorder. Our systematic transport measurements combined with neutron scattering measurements, ab initio band calculations and spin model calculations indicate that the unusual Hall effect can be understood in terms of spin-splitting induced emergence/disappearance of Fermi pockets as well as skew scattering from spin-chiral cluster fluctuations in a field-polarized state. The present study demonstrates complex interplay among magnetization, spin-dependent electronic structure, and spin fluctuations in producing the giant Hall effect in highly conductive frustrated magnets with a distorted triangular lattice.

Introduction

The Hall effect in magnetic materials has attracted growing attention due to its diverse quantum physical mechanisms^1,2. Traditionally, the microscopic origins of the anomalous Hall effect (AHE) have been classified into two categories: intrinsic and extrinsic ones. The intrinsic mechanism of the AHE is attributed to the Berry curvature associated with the electron bands as affected by magnetic structures. Recently, large intrinsic AHEs due to non-collinear or non-coplanar spin textures have been extensively studied in antiferromagnetic metals^3,4,5,6,7. In addition, large topological Hall effects (THE) arising from magnetic skyrmions, nanometric topological spin textures, have been found in Gd-based intermetallic compounds dominated by magnetic frustration^8,9,10.

The extrinsic AHE, on the other hand, has long been well known to originate from impurity scatterings, where highly conducting electrons are scattered asymmetrically from impurities under the influence of spin-orbit coupling^11,12. Recently, skew scattering from spin clusters was theoretically proposed as a new mechanism of the AHE^13,14. In this spin-cluster scattering, scalar spin chirality (SSC) derived from non-coplanar spin textures induces large Hall conductivities, and hence large Hall angles. This new mechanism of skew scattering explains the giant unconventional Hall effects in the frustrated magnets such as KV₃Sb₅¹⁵, MnGe¹⁶, EuAs¹⁷, SrCo₆O₁₁¹⁸ and PdCrO₂^19,20. The recent progress in these unconventional Hall effects in the high-conductivity regime has stimulated scientific interest in the search for giant Hall effects in magnetic materials with both high conductivity and non-trivial spin textures.

Here, we report a large Hall conductivity and its novel magnetic field response in a highly conductive frustrated magnet GdCu₂, which belongs to the family of RCu₂ (R: rare-earth elements) with the orthorhombic structure (space group: Imma) as shown in Fig. 1a, b. This structure can be viewed as a distorted AlB₂-type hexagonal structure²¹, and Gd atoms form distorted triangular lattice layers perpendicular to the b-axis. GdCu₂ has been known to exhibit antiferromagnetic ordering below T_N = 40 K²². In 2000, using neutron scattering, Rotter et al. reported a non-collinear magnetic structure described by the modulation vector Q₀ = (2/3, 1, 0), which corresponds to an in-plane cycloidal propagation along the a-axis with a pitch angle of 120° and an anti-parallel coupling along the b-axis, as depicted in Fig. 1b²³. The 120° spin texture reflects geometric frustration on the triangular lattice. More recently, incommensurate magnetic order with a slightly modified a-axis propagation Q₀ = (0.678, 1, 0) has been reported²⁴. The magnetization along the a-, b-, and c-axes shows the similar field dependence and exhibits two metamagnetic transitions accompanied by magnetostrictions around 6 and 8 T²⁵⁻²⁷. Despite the observation of such a non-trivial magnetic structure and metamagnetic field response, there have been no experimental reports of Hall effects.

Fig. 1: Crystal structure, and magnetic and transport properties of GdCu₂.

a Schematic of crystal structure of GdCu₂. The black lines show an orthorhombic unit cell. b Schematic of distorted triangular lattices of Gd atoms as viewed from the b-axis for the top layer (y = 0.75) and the bottom layer (y = 0.25). The magnetic moments with 120° cycloid propagation along the a-axis and anti-parallel stacking along the b-axis are drawn with red arrows. c Temperature (T) dependence of magnetic susceptibility χ = M/H at 1 T (blue line). The inverse susceptibility χ⁻¹ (green symbols) is fitted with the Curie-Weiss law above 150 K (θ_p = 10.9 K, μ_eff = 8.59 μ_B/Gd). d Temperature dependence of longitudinal conductivity σ_xx at zero field. e Magnetic field (H) dependence of magnetization M (blue line) and its field derivative dM/dH (green line) at 2 K. f Magnetic field dependence of Hall conductivity σ_xy at 2 K.

In this study, magnetic and electric transport properties of GdCu_2-xAu_x (0 ≤ x ≤ 0.05) were investigated systematically using high-quality single crystals (see Supplementary Fig. 1 for details of sample characterizations). In addition, we conducted neutron scattering measurements and ab initio band calculations to clarify the magnetic and electronic structures under magnetic fields. Furthermore, we calculated spin chirality fluctuations in the field-polarized (FP) state using spin wave approximation. On the basis of those comprehensive data, we discuss the plausible origin of the giant Hall effect in terms of band structure change and cluster skew scattering.

Results

Magnetic and transport properties

First, we show basic magnetic and transport properties of GdCu₂ in Fig. 1c–f. In these measurements, the magnetic field (H) is applied parallel to the b-axis of the crystal (defined as z-axis) and the electric current (J) is applied along the a-axis (defined as x-axis), and thus the Hall voltage is measured along the c-axis (defined as y-axis). Temperature dependence of magnetic susceptibility χ shows a sharp cusp at T_N = 40 K (Fig. 1c), indicating the 120° order of Gd moments. The Curie-Weiss fitting yields the Weiss temperature θ_p = 10.9 K and the effective magnetic moment μ_eff = 8.59 μ_B/Gd, respectively. The positive Weiss temperature may imply the presence of ferromagnetic interaction in the system although the actual magnetic order is antiferromagnetic. As shown in Fig. 1d, the longitudinal conductivity σ_xx shows an abrupt upturn below T_N and reaches as high as 1.6 × 10⁶ Ω⁻¹cm⁻¹ at 2 K (see Supplementary Fig. 3 for the temperature dependence of the longitudinal resistivity ρ_xx). Magnetic field dependence of the magnetization M at 2 K exhibits two metamagnetic transitions around 6 T and 8 T and saturates around 10 T (Fig. 1e), in good agreement with the previous report²⁶. Here, we call the magnetic phases as I (0–6 T), II (6–8 T), III (8–10 T), and FP phase (>10 T). The corresponding Hall conductivity σ_xy at 2 K as a function of the field is shown in Fig. 1f. The Hall conductivity exhibits a large positive peak (σ_xy ~ 4 ×10⁴ Ω⁻¹cm⁻¹) in phase I, followed by a sudden decrease in phase II, and then sharply drops in phase III, exhibiting large negative values (σ_xy ~ −3 ×10⁴ Ω⁻¹cm⁻¹). As the magnetic field is further increased, the Hall conductivity rapidly increases and shows a positive peak (σ_xy ~ 4 ×10⁴ Ω⁻¹cm⁻¹) again in the FP region.

Thermal effect in high fields

To clarify the behavior of the Hall conductivity up to higher magnetic fields and to investigate the influence of thermal fluctuations, we performed high-field resistivity measurements up to 24 T (see Methods for details). Fig. 2a shows the field-swept σ_xy at several elevated temperature, and a contour plot of σ_xy(T, H) on the phase diagram obtained from magnetization measurements (Supplementary Fig. 2) is displayed in Fig. 2b. At 2.3 K, huge positive maxima on the order of 10⁵ Ω⁻¹cm⁻¹ are observed around 5 T (phase I) and 11 T (FP phase), and σ_xy eventually decays with a broad tail as the magnetic field is further increased. Upon increasing temperature, both positive peaks in the phase I and the FP state are suppressed, but their temperature and field dependence are clearly distinct as described below. The large positive Hall conductivity in phase I decreases rapidly with increasing temperature but without changing the peak position, as indicated with the square symbols in Fig. 2a, b, and shows small negative values above 8 K. On the other hand, the decrease in the large positive Hall conductivity in the FP region is more gradual and the peak position shifts to the high field side upon increasing temperature, as indicated with the triangle symbols in Fig. 2a, b, and a broad positive peak is still observed above 10 K in high fields. These differences point to different origins of the large Hall conductivity in the low-field region and the FP region, namely, the ordinary Hall effect (OHE) of high-mobility carriers and the extrinsic AHE due to spin-chirality-induced skew scattering, respectively, as discussed later. Since the large σ_xy in the FP region is ascribed to the extrinsic AHE, the maximum value of σ_xy^max at each temperature from 2–12 K is plotted as a function of σ_xx and compared to the conventional skew scattering of Fe films and the giant AHEs of KV₃Sb₅¹⁵ and MnGe¹⁶ in Fig. 2c. The power law fitting yields a scaling relation of ${\sigma }_{{xy}}\propto {\sigma }_{{xx}}^{1.78}$. This exponent is much larger than that in the conventional skew scattering $({\sigma }_{{xy}}\propto {\sigma }_{{xx}})$. Similar scaling exponents close to two have been reported for KV₃Sb₅¹⁵, SrCo₆O₁₁¹⁸ and PdCrO₂²⁰, in all of which it is argued that SSC fluctuations are the origin of the large AHE. These results suggest that the mechanism of the large AHE in GdCu₂ is related to SSC fluctuations, as discussed in detail later.

Fig. 2: Temperature and field dependence of Hall conductivity of GdCu₂.

a Magnetic field dependence of Hall conductivity at various temperatures. The square and triangle open symbols indicate the peak positions. b Contour plot of Hall conductivity on the temperature-field phase diagram. The phase boundaries shown by the closed circle symbols are determined by magnetization measurements (Fig. 1e and Supplementary Fig. 2). The square and triangle open symbols indicate the peak positions of σ_xy(H) in phase I and the FP region, respectively. c Scaling plot of the maximum Hall conductivity σ_xy^max in the FP region versus the longitudinal conductivity σ_xx obtained at the same magnetic field. Blue squares show the data points at 2 K and 5 K obtained from the measurements up to 14 T (Fig. 1f) on sample #1. Red circles show the data points from 2.3 K to 12 K obtained from the measurements up to 24 T (a) on sample #2. The pink line shows the fitting of the scaling law ${\sigma }_{{xy}}\propto {{\sigma }_{{xx}}}^{n}$ (n = 1.78). For comparison, the conventional skew scattering of Fe films and the giant anomalous Hall conductivity of KV₃Sb₅¹⁵ and MnGe films¹⁶ are also plotted.

Influence of quenched disorder

As mentioned above, the large Hall conductivity is observed only at low temperatures, where the longitudinal conductivity is very high on the order of σ_xx ~ 10⁶ Ω⁻¹cm⁻¹. Therefore, next we intentionally polluted the sample by doping a tiny amount of Au to investigate the dependence of the giant Hall effect upon quenched disorder. Au was chosen to be partially substituted for Cu because both are isoelectronic and thus a small amount of Au substitution neither changes the electron filling nor affects the electronic and magnetic structures. The longitudinal and Hall conductivities of GdCu_2-xAu_x (x = 0, 0.02 and 0.05) are summarized in Fig. 3. As expected, σ_xx decreases with increasing the concentration of Au that acts as a scattering center (Fig. 3a), while the magnetic properties are almost unchanged as confirmed by magnetization measurements (see Supplementary Fig. 3 for details). Fig. 3b, c show the magnetic field dependence of σ_xx and σ_xy, respectively, at 2 K for x = 0, 0.02 and 0.05 (see Supplementary Figs. 4 and 5 for the magnetic field dependence of ρ_xx and ρ_yx at all the measurement temperatures). While the magnitude of σ_xx decreases by a factor of 2–3 with increasing x, its field dependence is qualitatively unchanged. On the other hand, σ_xy is more significantly suppressed with increasing x, and the characteristic field dependence with sign reversal completely disappears for x = 0.05. This Au-doping dependence of σ_xy at 2 K bears similarity to the temperature dependence in the non-doped sample. Therefore, the giant Hall conductivity in GdCu₂ is intimately related to the magnitude of the longitudinal conductivity.

Fig. 3: Au-doping effect on the conductivity in GdCu₂.

a Temperature dependence of longitudinal conductivity of GdCu_2-xAu_x (x = 0, 0.02, 0.05) at zero field. Magnetic field dependence of (b) longitudinal and (c) Hall conductivity of GdCu_2-xAu_x (x = 0, 0.02, 0.05) at 2 K.

Magnetic structure under magnetic fields

To search for a possible change in the magnetic structure of GdCu₂ under magnetic fields, we performed single-crystal neutron diffraction measurements (see Methods for details). Fig. 4a shows the reciprocal space intensity mapping on the (h, 1, l) plane obtained at 2 K and 0 T. Magnetic reflections associated with Q₀ = (2/3, 1, 0) are observed clearly as indicated with green circles, which is consistent with the previous study²³. The intensity line cuts along h around (4/3, 1, 0) [= (2, 2, 0) − Q₀] and (8/3, 1, 0) [= (2, 0, 0) + Q₀] at various magnetic fields are plotted in Fig. 4b. These magnetic reflections are found to persist up to 6.5 T above the metamagnetic transition field around 5.5 T. The integrated intensities of these magnetic peaks are plotted as a function of magnetic field in Fig. 4c. As the magnetic field is increased, these intensities decrease above 5.5 T but a half of the intensity remains at 6.5 T. No other magnetic reflections arising from a different finite modulation Q vector are observed at 6.5 T in the detectable range of the reciprocal lattice space (Supplementary Fig. 6). As schematically depicted in Fig. 4d, the Gd 4f local moments lying in the ac-plane tilt towards the b-axis (the canting angle θ from the b-axis decreases) as the magnetic field is increased. The above experimental results, in which the Q vector is invariant up to 6.5 T, indicate that the metamagnetic transition from phase I to II is accompanied by the sudden decrease in θ while the in-plane component maintains the 120° structure and interlayer antiferromagnetic arrangement. Since the diffraction intensity is proportional to the square of the in-plane component of the magnetic moments, if the above scenario is correct, the intensity should be proportional to sin²θ. In fact, as shown in Fig. 4c, the field dependence of the integrated intensity shows a good agreement with the sin²θ calculated from the magnetization curve. Although 6.5 T is insufficient to reach phase III, θ is expected to further decrease from phase II to III around 7.5 T, and finally the Gd moments are forcedly almost aligned to the b-axis above 10 T. As described in the theoretical section later, the conduction bands having large hybridization with Gd 5d orbitals split significantly with increasing magnetization due to strong coupling with the frustrated Gd 4f local moments. As a result, new Fermi surface of spin-polarized conduction electrons emerges, accompanied by the abrupt decrease in the tilt angle θ between the magnetic phases.

Fig. 4: Magnetic structure of GdCu₂.

a Contour map of neutron scattering intensity on the reciprocal lattice (h, 1, l) plane at 2 K and 0 T. The nuclear and magnetic peaks are indicated by light-blue and green circles, respectively. b Neutron scattering intensity profile along the (h, 1, 0) line around the center of h = 4/3 and h = 8/3 at several magnetic fields from 0 to 6.5 T. For clarity, each data was shifted vertically by a constant interval. c Magnetic field dependence of the integrated intensity of the (4/3, 1, 0) magnetic peak (red squares) and (8/3, 1, 0) magnetic peak (green circles). Magnetization normalized by the saturation value at 14 T (M/M_s) and the calculated sin²θ are also plotted with solid orange line and broken blue line, respectively. Here, θ [= arccos(M/M_s)] is defined as the angle of the Gd moments tilted from the b-axis to the ac-plane. d Schematic of the magnetic structure of Gd local moments under magnetic fields.

Origin of the giant Hall effect

Here, we discuss the plausible origin of the giant Hall conductivity from phase I to phase III. Since the upper limit of σ_xy value based on the intrinsic mechanism of AHE is on the order of 10³ Ω⁻¹cm^{−1 1}, the intrinsic mechanism can hardly explain the experimental result of σ_xy ~ 10⁴−10⁵ Ω⁻¹cm⁻¹ and the changes associated with the metamagnetic transitions. In addition, the giant Hall conductivity appears only when the longitudinal conductivity is very high (σ_xx ~ 10⁶ Ω⁻¹cm⁻¹). Therefore, the giant Hall conductivity can be attributed to either extrinsic AHE or OHE. For multiple carriers with high mobility, the OHE often exhibits a nontrivial magnetic field (B) dependence as described by the Drude model,

$${\sigma }_{{xy}}^{\left({\rm{Drude}}\right)}=\mathop{\sum }\limits_{i}{\mu }_{i}{n}_{i}{q}_{i}\cdot \frac{{\mu }_{i}B}{1+{\left({\mu }_{i}B\right)}^{2}}$$

(1)

where μ_i, n_i and q_i are mobility, density and charge of each carrier species, respectively.

To investigate the OHE without the magnetic contribution, we performed similar transport measurements for a non-magnetic isostructural compound YCu₂²⁸ without f moments and observed even higher longitudinal and Hall conductivities (σ_xx ~ 3.6 × 10⁶ Ω⁻¹cm⁻¹ and σ_xy ~ 3 × 10⁵ Ω⁻¹cm⁻¹) at 2 K (Supplementary Fig. 7). The Hall conductivity in YCu₂ shows an initial non-linear slope and a broad single peak as a function of the external field, and thus can be well reproduced with a two-carrier Drude model. Therefore, it is reasonable to assume that the large Hall conductivity in GdCu₂ has also contributions from the OHE of high-mobility carriers arising from a similar band structure. In the case of GdCu₂, as discussed in the following theoretical section, the density as well as the sign of carriers vary significantly at the metamagnetic phase transitions, bringing additional features to the Hall conductivity.

First-principles band structure calculations

First-principles calculations utilizing density functional theory was performed to gain insights into the atypical behavior exhibited by the Hall conductivity concerning the external magnetic field and internal magnetization (see Methods for details). Since the nature of carriers determines the transport property, we plot the band structure near the Fermi level (E_F) and highlight the characteristics of the conduction electrons and holes which have strong hybridization with Gd 5d orbitals. Fig. 5a, b shows band structures with k_z = 0 and k_z = 0.5 along a high-symmetry path within the xy-plane. In order to model the evolution of carrier pockets along with the external field, we present the results of in-plane antiferromagnetic (AFM) states along x-direction with three different spin tilting angles, θ = 0°, 45°, 90°, which correspond to the FM state (red line), intermediate noncolinear state (green line), and AFM state (blue line), respectively (Fig. 5c). Near the Fermi level, we can observe a large energy splitting (~0.5 eV) as the spin orients from the AFM state to the FM state. The large spin splitting clearly demonstrates the strong coupling between the Gd 4f local spins and the conduction electrons in this material. To explicitly show the energy position of the 4f electrons, the band structure expanded over a wider energy scale is presented in Supplementary Fig. 11, indicating that the 4f electrons lie far away from the Fermi level by 3.5 eV for down spin and −8.5 eV for up spin. Given that the observed changes in carrier density at the Fermi level represent a general response to magnetization changes, the subsequent discussion of the transition from AFM to FM phases can be reasonably extended to more complex magnetic structure changes as observed in the experiment.

Fig. 5: First-principles calculations for GdCu₂.

Band structure along the high-symmetry line for (a) k_z = 0 and (b) k_z = 0.5 in AFM state (blue line), intermediate noncollinear state (green line), and FM state (red line). We highlight bands that substantially change the Fermi surface and potentially cause the anomalous behavior of the Hall conductivity. E_F is taken at 0 eV. The corresponding Fermi surface is presented in Supplementary Figs. 9 and 10. c The in-plane AFM spin orientation with different tilting angle adopted in the calculation. θ = 0°,45°, and 90° correspond to the FM state, intermediate state, and AFM state, respectively. d We categorize the bands by their dispersion and their relative level compared to the E_F with/without spin-splitting. In the cases of these four kinds of bands, carrier density has strong magnetization dependence.

In the following, we categorize the bands by their spin-resolved electronic states at the E_F. There are four types, and we summarize them in Fig. 5d. As in the standard approach, since the upward parabolic bands form an electron pocket when they cross the E_F, we call upward parabolic bands the “electron type” (Type-e); since the downward parabolic bands contribute as holes, we call downward parabolic bands the “hole type” (Type-h). Besides, for both electron and hole types, the bands that cross the E_F in the AFM state without spin splitting, but only for the majority spin band in the FM state with spin splitting, we call them type 1; for those that do not touch the E_F in the AFM state but come across it in the FM state, we call them type 2. With this categorization, we can discuss their contribution to the Hall conductivity as we increase the external magnetic field, while the intermediate state helps to track the evolution between the two states.

According to the Boltzmann transport equation, the electron (q_i = −e) contributes negatively to the Hall conductivity while the hole (q_i= +e) contributes positively. Therefore, for the band of Type-e1(h2), the spin-down band stops (starts) contributing as electrons (holes) to the transport property as the spin-splitting is enhanced by the external magnetic field. Therefore, the Hall conductivity increases as a response to the changing of carrier density for each carrier species. On the other hand, for the band of Type-e2(h1), the spin-up band starts (stops) contributing as electron (hole) to the transport as the spin-splitting is increased, resulting in a decrease in the Hall conductivity. In Fig. 5a, b, we highlight some notable bands that belong to the four introduced types and attribute the magnetization dependence of Hall conductivity to the change of band structure near the E_F. Therefore, the sharp decrease in the Hall conductivity in phases II and III can be explained by the significant change in the Fermi surface, which is plausibly dominated by Type-e2.

Skew scattering by spin clusters

Based on the above band calculations, we fitted the observed large Hall conductivity of GdCu₂ with a multi-carrier Drude model incorporating the sudden appearance of electron carriers in phases II and III (see Supplementary Note 6 for details). As shown in Supplementary Fig. 8, the multi-carrier model well explains the observed Hall conductivity from phase I to phase III. To further explain the sharp increase in σ_xy from negative values in phase III to positive ones in the FP phase by OHE alone, the sudden appearance of another hole-type Fermi pocket is necessary. On the other hand, as magnetization smoothly connects from phase III to the FP region, it is unlikely that such a significant change in the Fermi surface occurs around the saturation field. Therefore, we attribute this part to the extrinsic AHE. As shown in the scaling plot of AHE in Fig. 2c, the large magnitude of σ_xy ~ 10⁴–10⁵ Ω⁻¹cm⁻¹ and its strong dependence on σ_xx cannot be explained by the intrinsic mechanism. In the extrinsic mechanisms, i.e., conventional skew scattering and side jump, on the other hand, the anomalous Hall conductivity is proportional to the magnetization at a fixed temperature, and the anomalous Hall angle ratio (tanθ_H = σ_xy/σ_xx) is typically less than 1%²⁹. Therefore, these conventional impurity scattering mechanisms can never explain the exceptionally large anomalous Hall angle (tanθ_H ~ 12% at 2 K) and unusual temperature-field dependence of the Hall conductivity in the FP state as shown in Fig. 2a, b. Thus, we can finally conclude that skew scattering by spin clusters with spin chirality fluctuations is the most plausible scenario. The neutron scattering measurement indicates that the b-axis magnetization process is caused by the decrease in the canting angle of Gd moments from the b-axis while keeping the in-plane 120° modulation (Fig. 4d). Thus, spin chirality fluctuations near the phase boundary (local spin canting from the collinear FP state) may induce skew scattering.

While the total spin chirality cancels out in a uniform regular triangle lattice, complete cancellation does not occur in a distorted triangular lattice as in GdCu₂, as discussed quantitatively in the following theoretical section.

Model calculation of spin chirality

To investigate how the scalar spin chirality of fluctuating spins occurs in the FP phase, we consider an effective triangular lattice model

$$\begin{array}{c}H=-{J}_{1}\mathop{\sum}\limits_{{\left\langle i{\alpha },j{\alpha }\right\rangle }_{1}}{\vec{S}}_{i{\alpha }}\,\cdot\, {\vec{S}}_{j{\alpha }}-{J}_{2}\mathop{\sum}\limits _{{\left\langle i{\alpha },j\beta \right\rangle }_{2}}{\vec{S}}_{i{\alpha }}\,\cdot\, {\vec{S}}_{j\beta }-{J}_{2}^{{\prime} }\mathop{\sum}\limits _{{\left\langle i{\alpha },j\beta \right\rangle }_{2}^{{\prime} }}{\vec{S}}_{i{\alpha }}\,\cdot\, {\vec{S}}_{j\beta }\\ +\mathop{\sum}\limits _{{\left\langle i{\alpha },j{\alpha }\right\rangle }_{1}}{D}_{{\alpha }}\hat{z}\,\cdot\, {\vec{S}}_{i{\alpha }}\times {\vec{S}}_{j{\alpha }}-h\mathop{\sum}\limits _{i{\alpha }}{S}_{i{\alpha }}^{z},\end{array}$$

(2)

where ${\vec{S}}_{i{\alpha }}=({S}_{i{\alpha }}^{x},{S}_{i{\alpha }}^{y},{S}_{i{\alpha }}^{z})$ is the Gd spin on α(= A, B)th sublattice in ith unit cell, and the summations are for 1, 2 and 2′ bonds in Fig. 6a; the first three terms are antiferromagnetic Heisenberg interaction (J₁, J₂, J₂’ < 0) between the Gd spins and the fourth term is the Dzyaloshinskii-Moriya (DM) interaction allowed by the symmetry. Unlike the uniform triangular lattice, the DM interaction is allowed in GdCu₂ by the lattice distortion, which breaks the inversion symmetry on bond 1. The DM interaction favors a spin configuration with opposite spin chirality for blue and red triangles. In addition, the inequivalence of blue and red triangles violates the cancellation of net spin chirality, which often prohibits a spin-chirality-related AHE in the uniform triangular lattice. Hence, the lattice distortion in GdCu₂ can give rise to a non-zero spin chirality and chirality-related AHE.

Fig. 6: Effective model and spin chirality in the FP phase.

a Schematic of the spin model. The circles with dots and crosses represent the direction of Dzyaloshinskii-Moriya vector and the plus and minus signs represent the signs of spin chirality when D < 0. b Magnetic field (h) dependence of scalar spin chirality χ₁ for J₁ = J₂ = J₂’ = − 1 and D = − 0.1. Each line is for T/|J₁| = 0.1 (purple), 0.3 (blue), 0.5 (yellow), 0.7 (green), and 1.0 (red). The plots are for when the thermal average of transverse spin components is $\langle {S}_{{ix}}^{2}+{S}_{{iy}}^{2}\rangle \le 0.1$.

We calculated the spin chirality of the above model in the FP phase using linear spin-wave theory. This theory is reliable in the high field region where the tilting of spins by the thermal fluctuation is small; as a threshold, we used $\langle {S}_{{ix}}^{2}+{S}_{{iy}}^{2}\rangle \le 0.1$, where $\langle {S}_{{ix}}^{2}+{S}_{{iy}}^{2}\rangle$ is the thermal average of transverse spin components. The magnetic field dependence of spin chirality for the blue triangles is shown in Fig. 6b, which shows non-zero chirality at a finite temperature. Note that the spin chirality is induced by the fluctuating spins, not by a chiral magnetic order. In the presence of such chiral spin fluctuation, it induces a skew-scattering AHE proportional to the spin chirality, ${\sigma }_{{xy}}\propto \chi (T,h)$¹³. The spin chirality increases as the magnetic field decreases, which is qualitatively consistent with the AHE curve in experiment (Fig. 2a). Furthermore, in the high-field limit, the spin chirality is the largest at finite temperatures (Supplementary Fig. 12), reproducing the temperature dependence of the observed Hall conductivity at 24 T (Supplementary Fig. 13). Hence, the temperature and magnetic field dependence of AHE in the FP phase is qualitatively reproduced by the skew-scattering AHE.

Similar large AHEs from thermally-induced spin chirality fluctuations have been reported in frustrated magnets¹⁷⁻²⁰, where the maximum of σ_xy is observed around T_N. On the other hand, in the high-field limit of the FP state as in the present study, the maximum of σ_xy is determined by the magnon excitation gap. Furthermore, in GdCu₂, σ_xy strongly depends on σ_xx, as shown in Fig. 2c, as well as on spin chirality. Since the suppression of σ_xx at high temperatures is dominant compared to the increase in spin chirality fluctuations, σ_xy reaches a maximum at low temperatures in the FP region (Fig. 2a, b), as in the case of MnGe¹⁶.

Finally, we discuss the role of geometric effects in the formation of spin chirality. As shown in Supplementary Fig. 14, the sharp positive peak of σ_xy in the FP region is suppressed when a magnetic field is applied parallel to the ac-plane, namely, the magnetic easy plane. These experimental results support the chirality-induced skew scattering scenario. Triangular lattice magnets with 120° spin ordering have been less investigated than kagome lattice magnets in the studies of anomalous Hall effects because of the spin-chirality cancelation problem. Here we demonstrate that giant spin-cluster skew scattering occurs in triangular lattice magnets and that symmetry breaking due to the lattice distortion is important for the non-zero spin chirality. As shown in Fig. 2c, the maximum value of the anomalous Hall conductivity in GdCu₂ is even larger than those in the kagome magnet KV₃Sb₅¹⁵ and the chiral magnet MnGe¹⁶ with similar mechanisms, exhibiting the largest value ever reported.

Discussion

We performed comprehensive magnetic and transport measurements for the highly conductive frustrated magnet GdCu₂ and discovered the giant Hall conductivity on the order of 10⁴–10⁵ Ω⁻¹cm⁻¹, which shows an atypical response to the increase in net magnetization and decreases significantly as the longitudinal conductivity is lowered. The large Hall conductivity in the intermediate field region from phase I to phase III is governed by the ordinary Hall effect of high mobility carriers where carrier densities vary significantly with increasing magnetization due to spin-splitting induced change in the Fermi surface. Similar large Hall conductivity with complex magnetic field dependence has been reported in other high-mobility f-electron systems such as CeAgBi₂³⁰ and (Tb, Dy)₃Sn₇³¹. In the present GdCu₂, however, another characteristic positive peak of the Hall conductivity emerges in the field-polarized phase, and is attributed to the extrinsic anomalous Hall effect due to skew scattering from spin clusters accompanying scalar spin chirality, as proposed by the recent theory¹³. Therefore, this study not only reveals the important role of band structures for the significant modulation of the ordinary Hall effect under magnetic fields, but also demonstrates the substantial contribution of spin chirality fluctuations to the extrinsic anomalous Hall effect. These results advance understanding of the diverse mechanisms of the complex Hall response in frustrated magnets with high mobility carriers.

Methods

Sample preparations

Bulk single crystals of GdCu₂ and GdCu_2-xAu_x (x = 0.02 and 0.05) were prepared by the Bridgman method. First, polycrystalline samples were synthesized from pure Gd and Cu metals in the stoichiometric ratio by arc melting in an Ar atmosphere. The product was crushed into small pieces and loaded in an yttria crucible and sealed in an evacuated quartz tube. Then, the quartz tube was set in a vertical Bridgman furnace and slowly lowered in the region with a temperature gradient between the upper heater (910 °C) and the lower heater (800 °C) at a rate of 1.4 mm/h. The obtained single-crystal ingot was cut into rectangular pieces after determining the crystal orientation with an X-ray Laue camera (RASCO-BL II, Rigaku). The RCu₂-type orthorhombic crystal structure and phase purity were confirmed by powder X-ray diffraction with Cu Kα radiation (RINT-TTR III, Rigaku) as shown in Supplementary Fig. 1. These samples were used in the magnetization and transport measurements up to 14 T and the neutron diffraction measurement. For high-field transport measurements up to 24 T, we used different single crystal of GdCu₂ growth by the Czochralski pulling method in an induction furnace as detailed in ref. ²².

Magnetization measurements

Temperature-swept magnetization measurements were performed by a superconducting quantum interference device magnetometer (MPMS3, Quantum Design), while field-swept magnetization was measured up to 14 T using a physical properties measurement system (PPMS, Quantum Design) equipped with a vibrating-sample magnetometer (VSM) option. In all measurements, the field direction is H || b-axis.

Electric transport measurements

Longitudinal resistivity (ρ_xx) and Hall resistivity (ρ_yx) were measured up to 14 T using a conventional four-terminal method in a PPMS (Quantum Design) equipped with an AC transport option (40 mA, 197 Hz). The current direction is J || a-axis, and the field direction is H || b-axis. To eliminate the influence of voltage probe misalignment, the raw data of field-swept longitudinal and Hall resistivity were processed as ρ_xx(H) = [ρ_xx(+H) + ρ_xx(−H)]/2 and ρ_yx(H) = [ρ_yx(+H) − ρ_yx(−H)]/2, respectively. The field-swept longitudinal and Hall conductivities were calculated with the formulae σ_xx(H) = ρ_xx(H)/[ρ_xx²(H) + ρ_yx²(H)] and σ_xy(H) = ρ_yx(H)/[ρ_xx²(H) + ρ_yx²(H)], respectively, where we assume ρ_yy = ρ_xx and ρ_xy = −ρ_yx Temperature-swept longitudinal conductivity was obtained from σ_xx(T) ~ 1/ρ_xx(T).

High-field transport measurements

The high-field transport measurements of GdCu₂ up to 24 T were performed using a 25 T cryogen-free superconducting magnet (25T-CSM) in Institute for Materials Research (IMR), Tohoku University. The longitudinal and Hall resistivity values ρ_xx and ρ_yx were measured using lock-in amplifiers with AC electric current (35.4 mA). To eliminate the voltage probe misalignment, the resistivity values were averaged with the positive field sweep and the negative field sweep (measured after rotating the sample stage by 180°) as described above.

Neutron scattering measurements

Neutron scattering measurements on GdCu₂ up to 6.5 T were performed using a time-of-flight (TOF) single-crystal neutron diffractometer (SENJU, BL-18) at the Materials and Life Science Experimental Facility (MLF) of the Japan Proton Accelerator Research Complex (J-PARC)³². The neutron wavelength is 0.4 − 4.4 Å. The incident neutron beam direction is k_i || ac-plane and the field direction is H || b-axis. The neutron diffraction data were accumulated for 3–4 h for each rocking angle at ω = 12°, 30°, 47° and 102°, where ω is defined as the angle between k_i and c-axis. The data were processed and the intensity mapping on the reciprocal lattice space was obtained by the software STARGazer³³.

First-principles calculations

In the first-principles calculations for GdCu₂, the experimental lattice constants are set to a = 4.320 Å, b = 6.858 Å, and c = 7.33 Ų¹, with optimized atomic position with QUANTUM ESPRESSO code³⁴. We use pseudopotentials with PBEsol exchange-correlation functional generated from the Standard Solid-State Pseudopotentials library (SSSP)^35,36. To take account of the strong correlation among electrons within the Gd atom, we add an empirical Hubbard U parameter of 7.0 eV on the 4f-orbitals. Using a converged energy cutoff of 70 Ry with a 6 × 4 × 4 k-grid, we obtained a magnetic ground state with 7.12 μ_B/atom for Gd and −0.01 μ_B/atom for Cu, with negligible variation for different spin-tilting angle.

Spin wave approximation in the field-polarized (FP) phase

We computed the scalar spin chirality using classical spin wave approximation. In the FP phase, at a low temperature, the Gd spins points along the z axis, i.e., ${\vec{S}}_{i\alpha } \sim (\mathrm{0,0,1})$. Hence, by assuming ${S}_{i\alpha }^{x,y}\,\ll\, 1$ and ${S}_{i\alpha }^{z}=\sqrt{1-{{S}_{i\alpha }^{x}}^{2}-{{S}_{i\alpha }^{y}}^{2}} \sim 1-\frac{1}{2}({{S}_{i\alpha }^{x}}^{2}+{{S}_{i\alpha }^{y}}^{2})$, the Hamiltonian reads $H=\frac{1}{2}\sum _{\vec{k}}{\vec{S}}_{-\vec{k}}{h}_{\vec{k}}{\vec{S}}_{\vec{k}}$, where

$$\begin{array}{c}{h}_{\vec{k}}=\left(-2{J}_{1}\cos \left({k}_{x}\right)+2{J}_{1}+2{J}_{2}+2{J}_{2}^{{\prime} }+{h}_{z}\right){\sigma }_{0}\otimes {\sigma }_{0}-i2{D}_{z}\sin \left({k}_{x}\right){\sigma }_{3}\otimes {\sigma }_{2}\\ -2\left({J}_{2}+{J}_{2}^{{\prime} }\right)\cos \left(\frac{\sqrt{3}{k}_{y}}{2}\right){\sigma }_{1}\otimes {\sigma }_{0}+2i\left({J}_{2}-{J}_{2}^{{\prime} }\right)\sin \left(\frac{\sqrt{3}{k}_{y}}{2}\right){\sigma }_{2}\otimes {\sigma }_{0}\end{array}$$

(3)

with ${\sigma }_{1,2,3}$ being the Pauli matrices and σ₀ is the 2 × 2 unit matrix, ${\vec{S}}_{\vec{k}}=({S}_{\vec{k}A}^{x},{S}_{\vec{k}A}^{y},{S}_{\vec{k}B}^{x},{S}_{\vec{k}B}^{y})$, ${S}_{\vec{k}\alpha }^{a}=\frac{1}{\sqrt{N}}\sum _{i}{S}_{i\alpha }^{a}{e}^{-i\vec{k}\cdot {\vec{r}}_{i\alpha }}$, and ${\vec{r}}_{i\alpha }$ is the position of spin ${\vec{S}}_{i\alpha }$. Similarly, the spin chirality for the blue triangles reads

$$\begin{array}{c}{\chi }_{1}=\frac{1}{N}\mathop{\sum}\limits _{\vec{r}}\hat{z}\cdot \left({\vec{S}}_{B,\vec{r}}\times {\vec{S}}_{A,\vec{r}}+{\vec{S}}_{A,\vec{r}}\times {\vec{S}}_{A,\vec{r}+\vec{x}}+{\vec{S}}_{A,\vec{r}+\vec{x}}{\times \vec{S}}_{B,\vec{r}}\right)\\ +\hat{z}\cdot \left({\vec{S}}_{B,\vec{r}}\times {\vec{S}}_{A,\vec{r}+\vec{x}}+{\vec{S}}_{A,\vec{r}+\vec{x}}\times {\vec{S}}_{B,\vec{r}+\vec{x}}+{\vec{S}}_{B,\vec{r}+\vec{x}}{\times \vec{S}}_{B,\vec{r}}\right)\\ =\frac{1}{N}\mathop{\sum}\limits _{\vec{r}}i\sin \left({k}_{x}\right)\left[{S}_{A-\vec{k}}^{x}{S}_{A\vec{k}}^{y}-{S}_{A-\vec{k}}^{y}{S}_{A\vec{k}}^{x}-{S}_{B-\vec{k}}^{x}{S}_{B\vec{k}}^{y}+{S}_{B-\vec{k}}^{y}{S}_{B\vec{k}}^{x}\right],\end{array}$$

(4)

and the magnetization is

$${M}^{z} \sim 1-\frac{1}{4N}\sum _{\vec{r},\alpha }\left[{\left({S}_{\alpha \vec{r}}^{x}\right)}^{2}+{\left({S}_{\alpha \vec{r}}^{y}\right)}^{2}\right]=1-\frac{1}{4N}\sum _{\vec{k},\alpha }\left[{S}_{\alpha -\vec{k}}^{x}{S}_{\alpha \vec{k}}^{x}+{S}_{\alpha -\vec{k}}^{y}{S}_{\alpha \vec{k}}^{y}\right].$$

(5)

Using these formulas, the thermal average of a physical observable $O=\sum _{k}{\vec{S}}_{-k}{o}_{\vec{k}}{\vec{S}}_{k}$, including chirality and magnetization, is given by

$$\left\langle O\right\rangle \sim \frac{1}{Z}{2}^{\displaystyle{{N}_{{asym}}}/{2}}\int D\,{\vec{S}}_{\vec{k}}\frac{1}{N}\sum _{k}{\vec{S}}_{-k}{o}_{\vec{k}}{\vec{S}}_{k}\exp \left(-\frac{\beta }{2}\sum _{k}{\vec{S}}_{-\vec{k}}{h}_{k}{\vec{S}}_{\vec{k}}\right),$$

(6)

where

$$Z={tr}\left({e}^{-\beta H}\right) \sim {2}^{{N}_{{asym}}}\int D\,{\vec{S}}_{\vec{k}}\exp \left(-\frac{\displaystyle{\beta }}{2}\sum _{k}{\vec{S}}_{-\vec{k}}{h}_{k}{\vec{S}}_{\vec{k}}\right)$$

(7)

is the partition function.