Large anomalous Hall conductivity induced by spin chirality fluctuation in an ultraclean frustrated antiferromagnet PdCrO₂

Abstract

Magnetic frustration, realized in the special geometrical arrangement of localized spins, often promotes topologically nontrivial spin textures in the real space and induces significantly large unconventional Hall responses. This spin Berry curvature effect in itinerant frustrated magnets mainly works with a static spin order, limiting the effective temperature range below the magnetic transition temperature and yielding the typical anomalous Hall conductivity below ~ 10³ Ω⁻¹cm⁻¹. Here we show that an ultraclean triangular-lattice antiferromagnet PdCrO₂ exhibits a large anomalous Hall conductivity up to ~ 10⁶ Ω⁻¹cm⁻¹ in the paramagnetic state, which is maintained far above the Neel temperature (T_N) up to ~ 4T_N. The reported enhancement of anomalous Hall response above T_N is attributed to the skew scattering of highly mobile Pd electrons to fluctuating but locally-correlated Cr spins with a finite spin chirality. Our findings point at an alternative route to realizing high-temperature giant anomalous Hall responses, exploiting magnetic frustration in the ultraclean regime.

Introduction

The quantum nature of electrons’ wavefunction manifests itself by large unconventional Hall responses when coupled to topologically nontrivial spin textures in itinerant frustrated magnets^1,2,3,4,5,6. The resulting spin Berry curvature works as a fictitious magnetic field and induces intriguing transverse motion of charge or spin, dubbed as unconventional anomalous Hall effect (AHE) or topological Hall effect (THE). In various frustrated itinerant magnets containing triangular, Kagome, or pyrochlore lattices^{1,3,6,7,8,9,10,11,12,13,14,15}, this real-space Berry curvature can effectively produce a giant Hall response, offering a promising route to functionalities for spintronic applications^4,5. However, two main limitations are faced while using magnetic frustration for realizing giant AHE. First, the magnetic frustration significantly suppresses the transition temperature of a long-range static spin order, setting the active temperature window of the large AHE at low temperatures. Second, due to the small mean free path in itinerant frustrated magnets, the corresponding Hall conductivity usually shares the limit of the anomalous Hall conductivity (AHC) ~ e²/h per atomic layer by the momentum-space Berry curvature in ferromagnets^16,17. These observations pose a challenge to identify new mechanisms and suitable material candidates for inducing a giant Hall response in a wide range of temperatures.

Recently, multiple spin scattering by fluctuating spin texture has been proposed to induce strong transverse motion of itinerant electrons in frustrated magnets, even above the spin ordering temperature^18,19,20. This spin-cluster skew scattering is proportional to the thermal average of fluctuating spin chirality $\langle {S}_{i}\cdot ({S}_{j}\times {S}_{k}) \rangle$, where S_i,j,k denotes localized spins at the neighboring sites i, j, and k. Thus, the corresponding AHE shares the common origin with THE due to static chiral spin orders. Experimental verification requires a model system satisfying several conditions, including strong magnetic frustration, short-range spin correlation well above the spin ordering temperature, and simple band structure without a complicated multiband effect. In this work, we show that in a triangular-lattice antiferromagnet PdCrO₂, a single Fermi surface of highly mobile electrons is responsible for a giant AHC, two orders of magnitude larger than ~ e²/h per atomic layer. The observed AHC is maintained up to ~150 K, below which short-range correlation of Cr spins exists. Our findings established a concrete example demonstrating spin chirality fluctuation as an effective source of giant anomalous Hall response at high temperatures in ultraclean frustrated antiferromagnets.

Results and discussion

Clean frustrated antiferromagnetic metal

PdCrO₂ is a rare example of antiferromagnetic delafossite metals, consisting of two-dimensional triangular layers of Pd and CrO₂ that provide highly conducting and Mott-insulating layers, respectively (Fig. 1a)^{14,15,21,22,23,24,25,26,27,28,29,30,31,32,33,34}. The localized spins of Cr³⁺ cations (S = 3/2) are antiferromagnetically coupled with a Weiss temperature Θ ~ 500 K, but magnetic frustration in the triangular lattice suppresses the long-range magnetic order with T_N ≈ 37.5 K, yielding a high frustration factor Θ/T_N ~ 13¹⁴. Below T_N, the resulting magnetic phase hosts a noncoplanar 120° spin structure with a $\sqrt{3}\times \sqrt{3}$ periodicity in the plane^21,28 and complex interlayer spin configuration with a possible scalar spin chirality S_i ⋅ (S_j × S_k)^{14,23,25,26,29,30,32}. What makes PdCrO₂ unique is the significant Kondo-type coupling between the localized Cr spins and highly mobile Pd electrons through a characteristic O–Pd–O dumbbell structure (Fig. 1a)^24,31. Below T_N, the highly conducting state of the Pd layers undergoes Fermi surface (FS) reconstruction of the otherwise single hexagonal FS, reflecting the Cr spin texture^{15,24,25,31,33}. Above T_N, this single FS is recovered²⁴, while the short-range correlation Cr spins remain significant up to ~200 K^21,28. Therefore, PdCrO₂ above T_N hosts a single electron band, proximity-coupled with fluctuating localized spins with magnetic frustration. These attributes endow PdCrO₂ with a strong candidate system for the proposed giant AHE due to spin-cluster skew scattering.

Fig. 1: Electronic and magnetic properties of PdCrO₂ single crystal.

a The crystal structure of PdCrO₂. The blue, red, and white spheres represent Pd, Cr, and O atoms, respectively. The noncollinear and noncoplanar spin structure of PdCrO₂ below T_N is shown along q = (1/3, 1/3, 0) (yellow planes). b Cross section of the quasi-2D hexagonal Fermi surface (FS) above T_N. Significant quasiparticle scattering through a magnetic q-vector produces hot spots at the corners of the hexagonal FS (red-shaded areas). c Schematic illustration of the spin-cluster skew scattering. Asymmetric scattering arises from interference between the scattering processes (the dashed lines) with the localized spins (the red arrow) and conduction spins of electrons (the blue arrows). d False-color SEM images of FIB-fabricated PdCrO₂ single crystals carved in typical Hall bar-type patterns (purple) with gold electrodes (yellow). e Temperature-dependent in-plane resistivity of PdCrO₂ showing a clear anomaly at T_N = 37.5 K. f, g Temperature-dependent torque magnetometry signal τ/H of up to 30 T in the antiferromagnetic phase (f) and the paramagnetic phase (g). The optical image of a single crystal mounted on a piezoresistive cantilever and the observed magnetic quantum oscillations at low temperatures are shown in the insets of (f). Above T_N, a clear deviation from the ideal paramagnetic behavior τ/H ~ H (dashed straight line) is observed in g and the insets of g.

In order to accurately determine the longitudinal (ρ_xx) and transverse (ρ_yx) resistivities of highly conducting PdCrO₂, we employed microfabrication on a single crystal using the focused ion beam (FIB) technique³⁵. The microfabricated crystal with a typical Hall-bar-pattern, in which an electric current flows along the [1,−1,0] axis (Fig. 1d and Supplementary Fig. S1), shows a clear kink at T_N = 37.5 K in the temperature-dependent ρ_xx(T) (Fig. 1e), consistent with the bulk results^14,15. A residual resistivity at low temperatures is ρ_xx ≈ 50 nΩ cm, and the corresponding mean free path is estimated to be ~3.4 μm (or ~10⁴ lattice periods), which is the highest among itinerant frustrated magnets. This highly conducting state of PdCrO₂ is further evidenced by strong de Haas–van Alphen (dHvA) oscillations from the torque magnetometry measurements (Fig. 1f), in good agreement with the previous reports^15,25. These characters strongly suggest that PdCrO₂ is in the ultraclean regime.

Before discussing the Hall response above T_N in PdCrO₂, we focus on its magnetization from the localized Cr spins. The torque magnetometry offers one of the most sensitive probes on magnetic response in the paramagnetic state at high magnetic fields by detecting the torque τ = m × H, where the magnetic moment m = μ₀VM is determined by the vacuum permeability μ₀, the sample volume V, and magnetization M. While τ(H)/H data below T_N reflects the magnetic anomaly observed at low magnetic fields³⁶, above T_N, one can expect an H-linear dependence τ/H ∝ (χ_c−χ_a)H, assuming a constant magnetic susceptibility (χ_a,c) along the a or c axis, described by M_a,c(H) = χ_a,cH, which differ slightly with each other even above T_N^14,21,36. However, careful analysis revealed that τ/H curves right above T_N in PdCrO₂ exhibit a sizable deviation from the expected H-linear dependence. As a representative case, we plotted the τ/H data taken at 40 K together with H-linear line, described by τ/H ~ αH, where the slope α is determined by the low field τ/H data below ~10 T (the inset of Fig. 1g). Here, the field range for the linear fitting is chosen to recover the unconventional linear H-dependence at the low magnetic field. The deviation Δ(τ/H) = τ/H−αH for all measured temperatures, shown in Supplementary Fig. S2, clearly reveals the concave behavior of τ/H data, which becomes weaker with increasing temperature. This captures a signature of short-range correlation of Cr spins above T_N, unlike conventional field-linear dependence of τ/H in the paramagnetic phase.

This deviation of τ/H from the H-linear dependence, obtained under magnetic fields nearly along the c-axis, probes additional contribution (ΔM_c) to the field-linear ~χ_cH dependence. In general, magnetization along a and c axes in the paramagnetic phase is described by M_a,c = χ_a,cH + ΔM_a,c(H), where the deviation ΔM_a,c(H) is negligible at low magnetic fields H, but sizable at high H. For a slight tilting angle θ ~ 2°, H_a = $H\sin \theta$ is much smaller than H_c = $H\cos \theta \, \approx \, H$, which means that M_a = χ_aH_a + ΔM_a(H_a) ≈ χ_aH_a, whereas M_c = χ_cH_c + ΔM_c(H_c). Then, the resulting torque signal $\tau (H)\propto {M}_{c}{H}_{a}-{M}_{a}{H}_{c} \, \approx \, ({\chi }_{c}-{\chi }_{a}){H}^{2}\sin 2\theta +\Delta {M}_{c}(H)H\sin \theta$. Thus, for the τ/H data, the H-linear dependence reflects the susceptibility anisotropy, whereas its deviation from the H-linear behavior probes additional contribution to the c-axis magnetization ΔM_c(H), which can be compared with the field-dependent Hall resistivity as discussed below.

Complex magnetotransport behaviors above T _N

This short-range correlation of localized spins affects the highly mobile electrons in the paramagnetic state. The transverse resistivity ρ_yx(H) under magnetic fields up to 17.5 T (Fig. 2a and b) exhibits complex evolution in a wide range of temperatures (5–260 K). At low temperatures, ρ_yx(H) shows a clear concave-shaped non-linearity with magnetic fields, consistent with the previous reports^14,15. This concave behavior in ρ_yx(H) is gradually suppressed with temperature, and the linear field dependence of ρ_yx(H) is recovered at T ~ 25 K. The corresponding Hall coefficient R_H = ρ_yx/H = −2.7 × 10⁻⁴ cm³/C agrees well with the expected R₀ = 1/ne from the total carrier density (n) obtained by the dHvA oscillations^15,25 and angle-resolved photoemission spectroscopy (ARPES)^24,31. Interestingly, above T = 25 K, the field-dependent ρ_yx(H) curves become convex upward¹⁵. The convex dependence of ρ_yx(H), maximized near T_N, becomes weaker with increasing temperature and turns into concave behavior. This complex field and temperature dependences are better displayed in the field-dependent dρ_yx(H)/dH at different temperatures (Fig. 2e and f). Near T_N, dρ_yx/dH drops well below R₀ = 1/ne at low magnetic fields and then grows slowly with the magnetic field, consistent with the convex field dependence of ρ_yx(H). Upon increasing temperature, the dρ_yx/dH at the low field limit rises gradually across R₀ and becomes saturated at high temperatures above ~200 K. The evolution of convex-to-concave type field dependence of ρ_yx(H) is also clearly visible in its deviation Δρ_yx(H) = ρ_yx(H)−αH, where the slope α is determined by the low field data below ~3 T (Supplementary Fig. S2).

Fig. 2: Magnetotransport properties of PdCrO₂ above T_N.

a, b Magnetic field dependent Hall resistivity ρ_yx(H) up to H = 17.5 T at different temperatures. Below T_N, ρ_yx(H) shows a clear hump at H ~ 5 T (a), which disappears at T ~ 25 K, recovering the linear field dependence from a single Fermi surface (FS) (dashed line). Above T_N, ρ_yx(H) changes from the concave to convex behaviors, manifested by opposite deviation from the low-field linear behavior (dot lines) at T = 45 and 260 K. c Deviation of ρ_yx(H) from the conventional Hall effect (solid line), determined by the ordinary Hall (${\rho }_{xy}^{{{{{{{{\rm{O}}}}}}}}}(H)$) and the conventional anomalous Hall (${\rho }_{yx}^{{{{{{{{\rm{A}}}}}}}}}(H)$) contributions. The resulting unconventional anomalous Hall resistivity (${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$) is indicated by the shaded area. d The modified Kohler’s plot of Δρ_xx(H)/ρ_xx(0) as a function of the Hall angle, $\tan {\theta }_{H}$. Strong violation from the scaling behavior is observed below ~75 K. e Contour plot of dρ_yx/dH(H, T) at different magnetic fields and temperatures. The region of the lower dρ_yx/dH(H, T) is located at lower fields (violet) just above T_N, while the opposite behavior is observed with a V-shape feature at high temperatures. f Temperature-dependent dρ_yx/dH at different magnetic fields. Below T ~ 25 K, the high field value of dρ_yx/dH matches well with the calculated Hall coefficient of PdCrO₂, extracted from quantum oscillations. Non-monotonous temperature dependence of dρ_yx/dH becomes saturated at high temperatures T ~ 150 K.

Considering the simple one-band FS of PdCrO₂ in the paramagnetic state, these complex temperature and field dependence of ρ_yx(H, T) are highly unusual and clearly distinct from those observed in the nonmagnetic delafossite metal, PdCoO₂ (Supplementary Fig. S3) as recognized in the previous studies^14,15. In PdCoO₂, a conventional linear-field dependent ρ_yx(H) = R₀H was observed at low temperatures, while a nonlinear-field dependence of ρ_yx(H) appears in the intermediate temperatures due to the momentum (k) dependent scattering time ${\tau }_{{{{{{{{\rm{tr}}}}}}}}}(k)$³⁷. On the quasi-2D hexagonal FS (Fig. 1b), quasiparticle scattering occurs more strongly near the highly curved corner of the hexagonal FS, called hot spots, than near the flat part of the FS. Consequently, the Hall coefficient at low magnetic fields becomes larger than R₀ above 30 K^37,38. In PdCrO₂, the fluctuating Cr spins introduce magnetic scattering with q = (1/3,1/3,0), resulting in a significant scattering rate near the corners of the hexagonal FS²⁷, which has been conjectured to be responsible for unusual field-dependent Hall resistivity at high magnetic fields¹⁵. However, since PdCrO₂ shares the same hot spot structure with PdCoO₂ and would produce the low field dρ_yx/dH larger than R₀, similar to the case of PdCoO₂, which is opposite to what is observed experimentally in PdCrO₂ near T_N (Fig. 2f). Moreover, the conventional AHE, proportional to the field-dependent magnetization, cannot explain the observed behavior of ρ_yx(H, T) in PdCrO₂. One of the key findings in this work is that the evolution of the concave-to-convex-type field dependence of ρ_yx(H) with increasing temperature (Fig. 2b and Supplementary Fig. S2) is inconsistent with a monotonous convex-type field dependence of magnetization, obtained from the τ(H)/H data (Fig. 1g). These observations strongly suggest that an additional mechanism due to fluctuating spins is needed to understand the transverse motion of quasiparticles above T_N in PdCrO₂.

Magnetic field dependence of the longitudinal resistivity ρ_xx(H) above T_N supports the same conclusion. In conventional metals, assuming temperature-independent anisotropy of ${\tau }_{{{{{{{{\rm{tr}}}}}}}}}(k)$ on the FS, the magnetoresistance Δρ_xx(H)/ρ_xx(0) follows the Kohler’s scaling rule Δρ_xx(H)/ρ_xx(0) = f(H/ρ_xx(0)), where f denotes a temperature-independent scaling function. However, we found that Kohler’s rule is strongly violated in PdCrO₂ for the entire temperature range up to 260 K above T_N (Supplementary Fig. S4). This behavior contrasts with the case of nonmagnetic (Pd,Pt)CoO₂, showing clear Kohler’s scaling behaviors at T > 150 K³⁷. Similar violation of Kohler’s scaling was reported in various itinerant antiferromagnets^{39,40,41,42,43}, in which fluctuations of local spins dictate quasiparticle scattering near the hot spots. In this case, Δρ_xx(H)/ρ_xx(0) is scaled with the Hall angle, $\tan {\theta }_{{{{{{{{\rm{H}}}}}}}}}={R}_{{{{{{{{\rm{H}}}}}}}}}H/{\rho }_{xx}(0)$, rather than H/ρ_xx(0), following the modified Kohler’s rule, Δρ/ρ(0) =$f({\tan }^{2}{\theta }_{{{{{{{{\rm{H}}}}}}}}})$. At high temperatures (T > 75 K), Δρ/ρ(0) curves of PdCrO₂ collapse onto a single curve, nicely following the modified Kohler’s rule (Fig. 2d). However, a strong violation from the modified Kohler’s rule occurs near T_N (25 K < T < 75 K), in which ρ_yx(H) shows strong convex-shaped field dependence, which consistently suggests additional scattering channel above T_N.

Unconventional anomalous Hall effect above T _N

In order to quantify the additional unconventional AHE above T_N, we decomposed the field dependent ρ_yx(H) into the following three terms ${\rho }_{yx}(H)={\rho }_{xy}^{{{{{{{{\rm{O}}}}}}}}}(H)+{\rho }_{yx}^{{{{{{{{\rm{A}}}}}}}}}(H)+{\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$, where ${\rho }_{xy}^{{{{{{{{\rm{O}}}}}}}}}(H)={R}_{0}H$ denotes the H-linear ordinary Hall contribution, reflecting a single Fermi surface in the paramagnetic state, ${\rho }_{yx}^{{{{{{{{\rm{A}}}}}}}}}(H)\propto M(H)$ corresponds to the conventional anomalous Hall contribution proportional to finite magnetization M(H) along the c axis, and ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$ represents the unconventional anomalous Hall contribution^{7,8,9,10,11,12,13,44,44,45,46,47}. Here, we precisely determined R₀ = 1/ne from both ARPES and dHvA oscillations (Fig. 1f)^15,24,31 and M(H) from measurements of the magnetic susceptibilities^14,21. For conventional anomalous Hall contribution ${\rho }_{yx}^{{{{{{{{\rm{A}}}}}}}}}$, we considered two possible cases based on the dominant sources of AHE, ${\rho }_{yx}^{{{{{{{{\rm{A}}}}}}}}}=aM{\rho }_{xx}^{2}$ for the k-space Berry curvature or the side-jump impurity scattering, or ${\rho }_{yx}^{{{{{{{{\rm{A}}}}}}}}}=bM{\rho }_{xx}$ for impurity skew impurity scattering^16,17. In both cases, a single parameter a or b was determined to reproduce the observed ρ_yx(H) data above ~ 200 K, in which short-range spin correlation is fully suppressed in PdCrO₂^21,28, and assumed to be temperature independent (Supplementary Fig. S5). Then, the unconventional contribution ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$ was obtained by ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)={\rho }_{yx}(H)-{\rho }_{xy}^{{{{{{{{\rm{O}}}}}}}}}(H)-{\rho }_{yx}^{{{{{{{{\rm{A}}}}}}}}}(H)$ down to low temperatures (Fig. 3a). We note that ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$ data in the two cases mentioned above were qualitatively similar to each other (Supplementary Fig. S6). Each contribution ${\rho }_{yx}^{{{{{{{{\rm{O}}}}}}}}}(H)$, ${\rho }_{yx}^{{{{{{{{\rm{A}}}}}}}}}(H)$ and ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$ are plotted in Fig. 2c for T = 45 K as a representative case and in Supplementary Fig. S7 for all measured temperatures. We found that the weak convex behaviors of ρ_yx(H) at high temperatures above ~150 K are well explained by the contributions of ${\rho }_{yx}^{{{{{{{{\rm{O}}}}}}}}}(H)$ and ${\rho }_{yx}^{{{{{{{{\rm{A}}}}}}}}}(H)$ only, while the concave behaviors of ρ_yx(H), developed approaching to ~T_N, are due to significant contribution of ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$. We note that the conventional AHE contribution ${\rho }_{yx}^{{{{{{{{\rm{A}}}}}}}}}$ becomes negligible near T_N due to the rapid decrease of ρ_xx (Supplementary Fig. S5). Although caution needs to be taken when estimating the magnitude of ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$ below T_N, such qualitative changes suggest that the field- and temperature-dependent ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H,T)$ capture of the intrinsic transport properties due to fluctuating spins in PdCrO₂.

Fig. 3: Unconventional anomalous Hall effect above T_N.

a Magnetic field-dependent unconventional anomalous Hall resistivity (${\rho }_{xy}^{{{{{{{{\rm{T}}}}}}}}}$) at different temperatures. Below T = 25 K, ${\rho }_{xy}^{{{{{{{{\rm{T}}}}}}}}}(H)$ shows a positive hump, which disappears completely at T = 25 K. Near T_N, ${\rho }_{xy}^{{{{{{{{\rm{T}}}}}}}}}$ at high magnetic fields grows rapidly, followed by its suppression at high temperatures. b Temperature-dependent ${\rho }_{xy}^{{{{{{{{\rm{T}}}}}}}}}$ at H = 17.5 T. At T ~ 25 K, ${\rho }_{xy}^{{{{{{{{\rm{T}}}}}}}}}$ shows a sign-change from the positive (blue shaded) to the negative (red shaded) and a strong deep immediately above T_N.

Magnetic field dependent ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$ above T_N is highly distinct from ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$ below T_N. While the precise magnetic structure of PdCrO₂ below T_N has remained controversial, a finite scalar spin chirality and the resulting THE well below T_N was proposed by H. Takatsu et al. to explain to the nonlinear field dependence of ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$¹⁴. This has been supported by several subsequent studies, including single crystal neutron diffraction and nonreciprocal magnetotransport measurements^23,29,30,32, although detailed field dependence of ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$ such as plateau behaviors above ~ 10 T (Fig. 3a) remains to be understood. As the static magnetic order melts down near T_N, we found that additional Hall contribution, opposite in sign, becomes developed and eventually dominant above T_N (Fig. 3b). Upon further increasing temperature, ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}$ at H = 17.5 T, maximized near T_N, is gradually decayed but remains finite up to ~ 150 K (~4 T_N). This sign reversal and enhancement of ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}$ near T_N are consistent with the theoretical predictions from the spin-cluster skew scattering model^18,19,20. In frustrated magnets, a finite scalar spin chirality induces the THE^1,4 and the resulting Hall conductivity ${\sigma }_{yx}^{{{{{{{{\rm{THE}}}}}}}}}$ are determined by $\langle {S}_{i} \rangle \cdot \langle {S}_{j} \rangle \times \langle {S}_{k} \rangle$, where $\langle \rangle$ denotes thermal average. Thus ${\sigma }_{yx}^{{{{{{{{\rm{THE}}}}}}}}}$ is reduced to zero as temperature approaches to T_N. However, near T_N, fluctuating but short-range-correlated spins with a scalar spin chirality can produce asymmetric scattering due to interference between the one- and two-spin scattering processes (Fig. 1c), named as spin-cluster skew scattering. Boltzmann calculations reveal that the resulting Hall conductivity is proportional to ${\sigma }_{yx}^{{{{{{{{\rm{sk}}}}}}}}} \sim {J}^{3} \langle {S}_{i}\cdot {S}_{j}\times {S}_{k} \rangle$, where J is Kondo coupling between itinerant electrons and localized spins^18,19. Therefore, it can persist at high temperatures until the short-range spin correlation is fully suppressed. Moreover, for ferromagnetic Kondo coupling (J < 0), ${\sigma }_{yx}^{{{{{{{{\rm{THE}}}}}}}}}$ is opposite in sign and comparable in size with ${\sigma }_{yx}^{{{{{{{{\rm{sk}}}}}}}}}$. All of such hallmarks of spin-cluster skew scattering are well reproduced in ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(T)$ of PdCrO₂ (Fig. 3b). Furthermore, the resulting ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)$ exhibits a characteristic non-monotonic field dependence (Fig. 3a), resembling the typical topological Hall resistivity ${\rho }_{xy}^{{{{{{{{\rm{T}}}}}}}}}(H)$ from a static scalar spin chirality, as discussed in Supplementary Fig. S8⁴⁸.

In the spin cluster skew scattering model, the sign crossover temperature is determined by competition between ${\sigma }_{yx}^{{{{{{{{\rm{THE}}}}}}}}}$ and ${\sigma }_{yx}^{{{{{{{{\rm{sk}}}}}}}}}$. Well below T_N, ${\sigma }_{yx}^{{{{{{{{\rm{THE}}}}}}}}}$ is dominant due to a static spin chiral order, but upon increasing temperature towards T_N, ${\sigma }_{yx}^{{{{{{{{\rm{sk}}}}}}}}}$ with an opposite sign becomes significant and eventually dominates over ${\sigma }_{yx}^{{{{{{{{\rm{THE}}}}}}}}}$, leading to sign crossover of the resulting Hall conductivity. Therefore the sign crossover is expected to occur below T_N, before ${\sigma }_{yx}^{{{{{{{{\rm{THE}}}}}}}}}$ becomes zero at T_N, in PdCrO₂ at ~25 K (Fig. 3a). In fact, a specific heat study on PdCrO₂ found a small hump observed at ~ 20 K, due to significant fluctuations of frustrated spins²¹. The onset of spin fluctuation well below T_N in PdCrO₂ can suppress static spin chirality for ${\sigma }_{yx}^{{{{{{{{\rm{THE}}}}}}}}}$ and enhance skew scattering contribution ${\sigma }_{yx}^{{{{{{{{\rm{sk}}}}}}}}}$, leading to the observed sign crossover in the Hall conductivity at ~25 K. The observed sign reversal, active temperature window, and magnetic field dependence of ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H,T)$ are in good agreement with the theory, suggesting that the spin-cluster skew scattering is the primary origin of ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H,T)$ above T_N in PdCrO₂.

The scaling properties of additional Hall conductivity ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}}$= ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}/({\rho }_{xx}^{2}+{\rho }_{yx}^{2})$ with the longitudinal conductivity σ_xx= ${\rho }_{xx}/({\rho }_{xx}^{2}+{\rho }_{yx}^{2})$ further support the spin-cluster skew scattering model in PdCrO₂. From the data obtained from the seven different PdCrO₂ crystals, we found that ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}}$ is strongly enhanced with increasing σ_xx (Fig. 4b). Consistently, in the PdCrO₂ thin films⁴⁹, of which σ_xx is reduced by order of magnitude than single crystals, the Hall resistivity ρ_yx(H) recovers the completely linear H dependence up to 30 T without any detectable contribution of ${\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}$ (Supplementary Fig. S9). The scaling behavior of ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}} \sim {\sigma }_{xx}^{2}$, observed in PdCrO₂ single crystals, differs clearly from the scaling behaviors expected for the impurity skew scattering, ${\sigma }_{xy}^{{{{{{{{\rm{A}}}}}}}}}\propto {\sigma }_{xx}$, or for the side jump scattering or the intrinsic k-space Berry curvature, ${\sigma }_{xy}^{{{{{{{{\rm{A}}}}}}}}}\propto const$^16,17. The THE with static scalar chirality and the AHE in the strong dirty limit are known to produce the same scaling relationship of ${\sigma }_{xy}^{{{{{{{{\rm{A}}}}}}}}\,{{{{{{{\rm{or}}}}}}}}\,{{{{{{{\rm{T}}}}}}}}} \sim {\sigma }_{xx}^{2}$, but they cannot be applied to the case of PdCrO₂. For the THE, it requires a long-range spin ordering, which cannot explain the significant ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}}$ observed in the paramagnetic state. The strong dirty limit behavior, often found in the ferromagnetic thin films^50,51,52 with σ_xx ~ 10²–10⁴ S cm⁻¹, is also unlikely to explain our findings on PdCrO₂, well inside the clean regime with several orders of magnitude higher σ_xx ~ 10⁵-10⁶ Ω⁻¹ cm⁻¹. Instead, the spin-cluster skew scattering model predicts the scaling, ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}}\propto {\sigma }_{xx}^{2}$, when dominant impurity scattering introduces variation in the scattering time while the scalar spin chirality $\langle {S}_{i}\cdot ({S}_{j}\times {S}_{k}) \rangle$ remains nearly intact¹⁸. This is indeed the case of our PdCrO₂ crystals with identical T_N but different impurity concentrations, where the scattering time and σ_xx change without spoiling the magnetism. The clear ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}}\propto {\sigma }_{xx}^{2}$ behavior in PdCrO₂, consistent with the theory, contrasts with the recent case of chiral magnet films of MnGe, in which both scattering time and magnetic anisotropy are tuned by thickness reduction, introducing the scaling of ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}} \sim {\sigma }_{xx}$⁵³.

Fig. 4: Anomalous Hall conductivities for various itinerant magnets.

a Anomalous Hall (${\sigma }_{xy}^{{{{{{{{\rm{A}}}}}}}}\,{{{{{{{\rm{or}}}}}}}}\,{{{{{{{\rm{T}}}}}}}}}$) and longitudinal (σ_xx) conductivities for various ferromagnets (open symbols)^16,17 and frustrated magnets (solid symbols)^{4,7,8,9,10,11,12,13,45,46,47,48,54,55,56}. For conventional ferromagnets, scaling behaviors of ${\sigma }_{xy}^{{{{{{{{\rm{A}}}}}}}}} \sim {\sigma }_{xx}^{\alpha }$ were observed in the hopping regime (α =  1.6, green dashed line) and in the ultraclean regime (α = 1, orange dashed line), except in the intermediate regime showing a nearly constant ${\sigma }_{xy}^{{{{{{{{\rm{A}}}}}}}}}$ due to the momentum-space Berry curvature effect. For frustrated magnets, the topological Hall effect follows the trend ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}} \sim {\sigma }_{xx}^{2}$ (red dashed line). b The detailed scaling behavior of the candidate magnets showing a spin cluster skew scattering effect, including PdCrO₂ (red circles), MnGe film (navy circles), and KV₃Sb₅ (green circles). For comparison, ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}} \sim {\sigma }_{xx}^{\alpha }$ with different exponents, α is shown with the quantum conductivity e²/ha, where a corresponds to the lattice parameter (solid cyan line). c, the anomalous Hall conductivity ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}}$ for frustrated magnets as a function of the normalized temperature with the magnetic transition temperatures (T_c or T_N).

Comparison with other itinerant magnets

In comparison with other itinerant magnets, including both ferromagnets and frustrated magnets, PdCrO₂ shows one of the highest values of σ_xx ~ 10⁶ Ω⁻¹ cm⁻¹, placing PdCrO₂ in the ultraclean regime (Fig. 4a). In the ultraclean regime, the extrinsic impurity scattering mechanism is responsible for large AHC in ferromagnets, surpassing an upper limit of ${\sigma }_{yx}^{{{{{{{{\rm{A}}}}}}}}}={e}^{2}/ha \sim 1{0}^{3}\ {\Omega }^{-1}{{{{{{{{\rm{cm}}}}}}}}}^{-1}$ (a is a lattice parameter) from the intrinsic Berry curvature effect^16,17. However, ${\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}}$ of PdCrO₂ is nearly two orders of magnitude higher than the AHC by the impurity skew scattering (Fig. 4a), indicating that magnetic fluctuations with scalar spin chirality can induce skew scattering much more effectively than backward scattering in the ultraclean regime¹⁹. Moreover, this is also much larger than the AHC induced by the THE from static chiral spin textures found in frustrated magnets (Fig. 4a). For the THE, the scalar spin chirality or skyrmion density works as an effective fictitious magnetic field B_eff in the real space, leading to ${\sigma }_{yx}^{{{{{{{{\rm{T}}}}}}}}} \sim {B}_{{{{{{{{\rm{eff}}}}}}}}}{\sigma }_{xx}^{2}$. However, the observed ${\sigma }_{yx}^{{{{{{{{\rm{T}}}}}}}}}$ of frustrated magnets is far smaller than ~10³ Ω⁻¹ cm⁻¹, due to their low conductivity. Therefore, highly mobile electrons with a large scattering time are essential for intensifying the spin cluster skew scattering effect in PdCrO₂.

Recently, the unconventional AHC, significantly larger than ~e²/ha, was reported in a chiral magnet MnGe⁵³, a Kagome metal KV₃Sb₅⁵⁴ and a triangular magnetic semiconductor EuAs⁵⁵, which has been attributed to spin-cluster skew scattering. In the case of MnGe films, magnetic-field-induced melting of the static chiral spin order leads to a large ${\sigma }_{yx}^{{{{{{{{\rm{T}}}}}}}}}$ at high magnetic fields, probably due to significant skew scattering. However, because of a relatively short mean free path, the temperature window of the large ${\sigma }_{yx}^{{{{{{{{\rm{T}}}}}}}}}$ is limited to below ~50 K, close to the spin ordering temperature⁵³. Such a narrow temperature window below ~T_N has been similarly observed in other frustrated magnets (Fig. 4c). For a Kagome metal KV₃Sb₅, the absence of signature of strong localized V spin moments⁵⁶ and the presence of the charge density wave phase in the temperature regime of a large Hall response below ~50 K⁵⁷ have raised questions on the validity of spin-cluster skew scattering model, without ruling out the possibility of the conventional multiband effect. In the triangular-lattice magnetic semiconductor EuAs⁵⁵, an unconventional AHE is observed up to ~6T_N, which has been attributed to spin cluster scattering by noncoplanar Eu²⁺ spins on the triangular lattice. However, the semiconducting character of EuAs results in a small AHC in the hopping regime. In contrast to these materials, PdCrO₂ exhibits a large AHC far above ~e²/ha, persisting up to ~150 K (~4T_N) (Fig. 4c). This unique feature of PdCrO₂ is because its itinerant electrons maintain their high mobility well above T_N, and the skew scattering with fluctuating spin chirality remains effective at relatively high temperatures.

Conclusion

Our findings clearly demonstrate that thermally excited spin clusters with scalar spin chirality can be an effective source for large anomalous Hall responses in itinerant frustrated magnets, particularly at the ultraclean limit. Unlike the conventional intrinsic mechanisms by the static chiral spin structures, the spin cluster skew scattering mechanism works in a wide temperature range well above the magnetic-ordering temperature. These properties found in PdCrO₂ are mainly due to the unique layered structure where the metallic layers host highly mobile electrons coupled with the spatially separated layers of frustrated local spins. Therefore, similar to magnetic delafossites^58,59,60, we envision that heterostructures with clean metallic layers and frustrated magnetic layers can provide a promising material platform for even larger anomalous Hall responses at high temperatures due to proximity-coupling with the underlying spin textures and their excitations.

Methods

Single crystal growths

Single crystals of PdCrO₂ were grown using the flux method with a mixture of polycrystalline PdCrO₂ and NaCl powders. The detailed procedure is described in refs. ^15,22. Furthermore, single crystals of PdCoO₂ were grown using a metathetical reaction method using powders of PdCl₂ and CoO in sealed quartz tubes, following the recipe described in refs. ^61,62. X-ray diffraction and energy-dispersive spectroscopy were used to verify confirming high crystallinity and stoichiometry of the crystals.

Torque magnetometry

A small single crystal, typically ~50 × 50 × 10 μm³, was used in torque measurements and mounted onto a miniature Seiko piezoresistive cantilever as described in ref. ¹⁵. Magnetic field and temperature were controlled in a 31 T bitter magnet at the National High Magnetic Field Laboratory, Tallahassee, FL, USA.

Device preparations

The well-defined geometry of the specimen is important for precise measurements of its electronic transport, especially the Hall resistivity, because metallic delafossites have a very low resistivity with a long mean free path (especially at in-plane transport) and high anisotropy in resistivities along the ab plane and c-axis (ρ_c/ρ_ab). We employed the focused-ion-beam technique to prepare the devices by following the procedures described in ref. ³⁵. Single crystals, ~100 × 100 × 10 μm³ in size, were attached to a Si/SiO₂ substrate. Metal deposition of Cr(10 nm)/Au(150 nm) was performed through a shadow mask. The direction of the current path was defined with respect to the well-defined hexagonal facets of the crystals. We used the conditions of the beam current 10 and 1 nA for rough and find structuring, respectively.

Electric transport measurements

Transport measurements for PdCrO₂ were performed using standard a.c. technique at a measurement frequency and current of 17.77 Hz 1 mA, respectively. We used preamplifiers in Hall measurements for accurate resistivity measurements and noise reduction. The measurements for PdCoO₂ were performed using d.c. technique at a current of 10 mA in a physical properties measurement system. Hall measurements were performed using standard a.c. technique at a measurement frequency and current of 17.77 Hz and 5 mA, respectively. Magnetic field and temperature control in both measurements were obtained using an Oxford variable temperature insert and an 18 T superconducting magnet.