The quantum nature of electrons’ wavefunction manifests itself by large unconventional Hall responses when coupled to topologically nontrivial spin textures in itinerant frustrated magnets1,2,3,4,5,6. 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 lattices1,3,6,7,8,9,10,11,12,13,14,15, this real-space Berry curvature can effectively produce a giant Hall response, offering a promising route to functionalities for spintronic applications4,5. 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 ferromagnets16,17. 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 temperature18,19,20. 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 Si,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 TN ≈ 37.5 K, yielding a high frustration factor Θ/TN ~ 1314. Below TN, the resulting magnetic phase hosts a noncoplanar 120° spin structure with a \(\sqrt{3}\times \sqrt{3}\) periodicity in the plane21,28 and complex interlayer spin configuration with a possible scalar spin chirality Si (Sj × Sk)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 TN, the highly conducting state of the Pd layers undergoes Fermi surface (FS) reconstruction of the otherwise single hexagonal FS, reflecting the Cr spin texture15,24,25,31,33. Above TN, this single FS is recovered24, while the short-range correlation Cr spins remain significant up to ~200 K21,28. Therefore, PdCrO2 above TN 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.
figure 1

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 TN is shown along q = (1/3, 1/3, 0) (yellow planes). b Cross section of the quasi-2D hexagonal Fermi surface (FS) above TN. 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 TN = 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 TN, 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 (ρxx) and transverse (ρyx) 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 TN = 37.5 K in the temperature-dependent ρxx(T) (Fig. 1e), consistent with the bulk results14,15. A residual resistivity at low temperatures is ρxx ≈ 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 reports15,25. These characters strongly suggest that PdCrO2 is in the ultraclean regime.

Before discussing the Hall response above TN 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 TN reflects the magnetic anomaly observed at low magnetic fields36, above TN, one can expect an H-linear dependence τ/H (χcχa)H, assuming a constant magnetic susceptibility (χa,c) along the a or c axis, described by Ma,c(H) = χa,cH, which differ slightly with each other even above TN14,21,36. However, careful analysis revealed that τ/H curves right above TN 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/HαH 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 TN, 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 (ΔMc) to the field-linear ~χcH dependence. In general, magnetization along a and c axes in the paramagnetic phase is described by Ma,c = χa,cH + ΔMa,c(H), where the deviation ΔMa,c(H) is negligible at low magnetic fields H, but sizable at high H. For a slight tilting angle θ ~ 2°, Ha = \(H\sin \theta\) is much smaller than Hc = \(H\cos \theta \, \approx \, H\), which means that Ma = χaHa + ΔMa(Ha) ≈ χaHa, whereas Mc = χcHc + ΔMc(Hc). 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 ΔMc(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 ρyx(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, ρyx(H) shows a clear concave-shaped non-linearity with magnetic fields, consistent with the previous reports14,15. This concave behavior in ρyx(H) is gradually suppressed with temperature, and the linear field dependence of ρyx(H) is recovered at T ~ 25 K. The corresponding Hall coefficient RH = ρyx/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 oscillations15,25 and angle-resolved photoemission spectroscopy (ARPES)24,31. Interestingly, above T = 25 K, the field-dependent ρyx(H) curves become convex upward15. The convex dependence of ρyx(H), maximized near TN, becomes weaker with increasing temperature and turns into concave behavior. This complex field and temperature dependences are better displayed in the field-dependent dρyx(H)/dH at different temperatures (Fig. 2e and f). Near TN, dρyx/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 ρyx(H). Upon increasing temperature, the dρyx/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 ρyx(H) is also clearly visible in its deviation Δρyx(H) = ρyx(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 TN.
figure 2

a, b Magnetic field dependent Hall resistivity ρyx(H) up to H = 17.5 T at different temperatures. Below TN, ρyx(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 TN, ρyx(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 ρyx(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 Δρxx(H)/ρxx(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 dρyx/dH(H, T) at different magnetic fields and temperatures. The region of the lower dρyx/dH(H, T) is located at lower fields (violet) just above TN, while the opposite behavior is observed with a V-shape feature at high temperatures. f Temperature-dependent dρyx/dH at different magnetic fields. Below T ~ 25 K, the high field value of dρyx/dH matches well with the calculated Hall coefficient of PdCrO2, extracted from quantum oscillations. Non-monotonous temperature dependence of dρyx/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 ρyx(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 studies14,15. In PdCoO2, a conventional linear-field dependent ρyx(H) = R0H was observed at low temperatures, while a nonlinear-field dependence of ρyx(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 K37,38. 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 dρyx/dH larger than R0, similar to the case of PdCoO2, which is opposite to what is observed experimentally in PdCrO2 near TN (Fig. 2f). Moreover, the conventional AHE, proportional to the field-dependent magnetization, cannot explain the observed behavior of ρyx(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 ρyx(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 TN in PdCrO2.

Magnetic field dependence of the longitudinal resistivity ρxx(H) above TN supports the same conclusion. In conventional metals, assuming temperature-independent anisotropy of \({\tau }_{{{{{{{{\rm{tr}}}}}}}}}(k)\) on the FS, the magnetoresistance Δρxx(H)/ρxx(0) follows the Kohler’s scaling rule Δρxx(H)/ρxx(0) = f(H/ρxx(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 TN (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 antiferromagnets39,40,41,42,43, in which fluctuations of local spins dictate quasiparticle scattering near the hot spots. In this case, Δρxx(H)/ρxx(0) is scaled with the Hall angle, \(\tan {\theta }_{{{{{{{{\rm{H}}}}}}}}}={R}_{{{{{{{{\rm{H}}}}}}}}}H/{\rho }_{xx}(0)\), rather than H/ρxx(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 TN (25 K < T < 75 K), in which ρyx(H) shows strong convex-shaped field dependence, which consistently suggests additional scattering channel above TN.

Unconventional anomalous Hall effect above T N

In order to quantify the additional unconventional AHE above TN, we decomposed the field dependent ρyx(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 contribution7,8,9,10,11,12,13,44,44,45,46,47. 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 susceptibilities14,21. 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 scattering16,17. In both cases, a single parameter a or b was determined to reproduce the observed ρyx(H) data above ~ 200 K, in which short-range spin correlation is fully suppressed in PdCrO221,28, 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 ρyx(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 ρyx(H), developed approaching to ~TN, 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 TN due to the rapid decrease of ρxx (Supplementary Fig. S5). Although caution needs to be taken when estimating the magnitude of \({\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)\) below TN, 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 TN.
figure 3

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 TN, \({\rho }_{xy}^{{{{{{{{\rm{T}}}}}}}}}\) 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 TN.

Magnetic field dependent \({\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)\) above TN is highly distinct from \({\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}(H)\) below TN. While the precise magnetic structure of PdCrO2 below TN has remained controversial, a finite scalar spin chirality and the resulting THE well below TN 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 measurements23,29,30,32, 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 TN, we found that additional Hall contribution, opposite in sign, becomes developed and eventually dominant above TN (Fig. 3b). Upon further increasing temperature, \({\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}\) at H = 17.5 T, maximized near TN, is gradually decayed but remains finite up to ~ 150 K (~4 TN). This sign reversal and enhancement of \({\rho }_{yx}^{{{{{{{{\rm{T}}}}}}}}}\) near TN are consistent with the theoretical predictions from the spin-cluster skew scattering model18,19,20. In frustrated magnets, a finite scalar spin chirality induces the THE1,4 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 TN. However, near TN, 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 spins18,19. 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 TN, \({\sigma }_{yx}^{{{{{{{{\rm{THE}}}}}}}}}\) is dominant due to a static spin chiral order, but upon increasing temperature towards TN, \({\sigma }_{yx}^{{{{{{{{\rm{sk}}}}}}}}}\) 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 TN, before \({\sigma }_{yx}^{{{{{{{{\rm{THE}}}}}}}}}\) becomes zero at TN, 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 TN 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 TN 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 σxx= \({\rho }_{xx}/({\rho }_{xx}^{2}+{\rho }_{yx}^{2})\) 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 σxx (Fig. 4b). Consistently, in the PdCrO2 thin films49, of which σxx is reduced by order of magnitude than single crystals, the Hall resistivity ρyx(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 films50,51,52 with σxx ~ 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 σxx ~ 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 TN but different impurity concentrations, where the scattering time and σxx 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.
figure 4

a Anomalous Hall (\({\sigma }_{xy}^{{{{{{{{\rm{A}}}}}}}}\,{{{{{{{\rm{or}}}}}}}}\,{{{{{{{\rm{T}}}}}}}}}\)) and longitudinal (σxx) 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 (Tc or TN).

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 σxx ~ 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 effect16,17. 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 Beff 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 ~TN 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 ~6TN, 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 (~4TN) (Fig. 4c). This unique feature of PdCrO2 is because its itinerant electrons maintain their high mobility well above TN, and the skew scattering with fluctuating spin chirality remains effective at relatively high temperatures.


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 delafossites58,59,60, 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.


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 (ρc/ρab). 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.