## 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 ~ 10^{3} Ω^{−1}cm^{−1}. Here we show that an ultraclean triangular-lattice antiferromagnet PdCrO_{2} exhibits a large anomalous Hall conductivity up to ~ 10^{6} Ω^{−1}cm^{−1} in the paramagnetic state, which is maintained far above the Neel temperature (*T*_{N}) up to ~ 4*T*_{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.

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## 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) ~ *e*^{2}/*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 PdCrO_{2}, a single Fermi surface of highly mobile electrons is responsible for a giant AHC, two orders of magnitude larger than ~ *e*^{2}/*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

PdCrO_{2} is a rare example of antiferromagnetic delafossite metals, consisting of two-dimensional triangular layers of Pd and CrO_{2} 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 Cr^{3+} 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} ~ 13^{14}. 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 PdCrO_{2} 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 recovered^{24}, while the short-range correlation Cr spins remain significant up to ~200 K^{21,28}. Therefore, PdCrO_{2} above *T*_{N} hosts a single electron band, proximity-coupled with fluctuating localized spins with magnetic frustration. These attributes endow PdCrO_{2} with a strong candidate system for the proposed giant AHE due to spin-cluster skew scattering.

In order to accurately determine the longitudinal (*ρ*_{xx}) and transverse (*ρ*_{yx}) resistivities of highly conducting PdCrO_{2}, we employed microfabrication on a single crystal using the focused ion beam (FIB) technique^{35}. 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 ~10^{4} lattice periods), which is the highest among itinerant frustrated magnets. This highly conducting state of PdCrO_{2} 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 PdCrO_{2} is in the ultraclean regime.

Before discussing the Hall response above *T*_{N} in PdCrO_{2}, 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* = *μ*_{0}*V**M* is determined by the vacuum permeability *μ*_{0}, the sample volume *V*, and magnetization *M*. While *τ*(*H*)/*H* data below *T*_{N} reflects the magnetic anomaly observed at low magnetic fields^{36}, 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,c}*H*, 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 PdCrO_{2} 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* ~ *α**H*, 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*−*α**H* 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 (Δ*M*_{c}) to the field-linear ~*χ*_{c}*H* dependence. In general, magnetization along *a* and *c* axes in the paramagnetic phase is described by *M*_{a,c} = *χ*_{a,c}*H* + Δ*M*_{a,c}(*H*), where the deviation Δ*M*_{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*_{a}*H*_{a} + Δ*M*_{a}(*H*_{a}) ≈ *χ*_{a}*H*_{a}, whereas *M*_{c} = *χ*_{c}*H*_{c} + Δ*M*_{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 Δ*M*_{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} cm^{3}/C agrees well with the expected *R*_{0} = 1/*n**e* 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 upward^{15}. 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 d*ρ*_{yx}(*H*)/d*H* at different temperatures (Fig. 2e and f). Near *T*_{N}, d*ρ*_{yx}/d*H* drops well below *R*_{0} = 1/*n**e* 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 d*ρ*_{yx}/d*H* at the low field limit rises gradually across *R*_{0} 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 Δ*ρ*_{yx}(*H*) = *ρ*_{yx}(*H*)−*α**H*, where the slope *α* is determined by the low field data below ~3 T (Supplementary Fig. S2).

Considering the simple one-band FS of PdCrO_{2} 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, PdCoO_{2} (Supplementary Fig. S3) as recognized in the previous studies^{14,15}. In PdCoO_{2}, a conventional linear-field dependent *ρ*_{yx}(*H*) = *R*_{0}*H* 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 *R*_{0} above 30 K^{37,38}. In PdCrO_{2}, 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 FS^{27}, which has been conjectured to be responsible for unusual field-dependent Hall resistivity at high magnetic fields^{15}. However, since PdCrO_{2} shares the same hot spot structure with PdCoO_{2} and would produce the low field d*ρ*_{yx}/d*H* larger than *R*_{0}, similar to the case of PdCoO_{2}, which is opposite to what is observed experimentally in PdCrO_{2} 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 PdCrO_{2}. 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 PdCrO_{2}.

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 Δ*ρ*_{xx}(*H*)/*ρ*_{xx}(0) follows the Kohler’s scaling rule Δ*ρ*_{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 PdCrO_{2} 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)CoO_{2}, showing clear Kohler’s scaling behaviors at *T* > 150 K^{37}. 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, Δ*ρ*_{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, Δ*ρ*/*ρ*(0) =\(f({\tan }^{2}{\theta }_{{{{{{{{\rm{H}}}}}}}}})\). At high temperatures (*T* > 75 K), Δ*ρ*/*ρ*(0) curves of PdCrO_{2} 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 *R*_{0} = 1/*n**e* 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 PdCrO_{2}^{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 PdCrO_{2}.

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 PdCrO_{2} 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 PdCrO_{2} (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. S8^{48}.

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 PdCrO_{2} at ~25 K (Fig. 3a). In fact, a specific heat study on PdCrO_{2} found a small hump observed at ~ 20 K, due to significant fluctuations of frustrated spins^{21}. The onset of spin fluctuation well below *T*_{N} in PdCrO_{2} 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 PdCrO_{2}.

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 PdCrO_{2}. From the data obtained from the seven different PdCrO_{2} crystals, we found that \({\sigma }_{xy}^{{{{{{{{\rm{T}}}}}}}}}\) is strongly enhanced with increasing *σ*_{xx} (Fig. 4b). Consistently, in the PdCrO_{2} thin films^{49}, 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 PdCrO_{2} 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 PdCrO_{2}. 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} ~ 10^{2}–10^{4} S cm^{−1}, is also unlikely to explain our findings on PdCrO_{2}, well inside the clean regime with several orders of magnitude higher *σ*_{xx} ~ 10^{5}-10^{6} 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 intact^{18}. This is indeed the case of our PdCrO_{2} 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 PdCrO_{2}, 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}.

### Comparison with other itinerant magnets

In comparison with other itinerant magnets, including both ferromagnets and frustrated magnets, PdCrO_{2} shows one of the highest values of *σ*_{xx} ~ 10^{6} Ω^{−1} cm^{−1}, placing PdCrO_{2} 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 PdCrO_{2} 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 regime^{19}. 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 ~10^{3} 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 PdCrO_{2}.

Recently, the unconventional AHC, significantly larger than ~*e*^{2}/*h**a*, was reported in a chiral magnet MnGe^{53}, a Kagome metal KV_{3}Sb_{5}^{54} and a triangular magnetic semiconductor EuAs^{55}, 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 temperature^{53}. Such a narrow temperature window below ~*T*_{N} has been similarly observed in other frustrated magnets (Fig. 4c). For a Kagome metal KV_{3}Sb_{5}, the absence of signature of strong localized V spin moments^{56} and the presence of the charge density wave phase in the temperature regime of a large Hall response below ~50 K^{57} 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 EuAs^{55}, an unconventional AHE is observed up to ~6*T*_{N}, which has been attributed to spin cluster scattering by noncoplanar Eu^{2+} 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, PdCrO_{2} exhibits a large AHC far above ~*e*^{2}/*h**a*, persisting up to ~150 K (~4*T*_{N}) (Fig. 4c). This unique feature of PdCrO_{2} 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 PdCrO_{2} 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 PdCrO_{2} were grown using the flux method with a mixture of polycrystalline PdCrO_{2} and NaCl powders. The detailed procedure is described in refs. ^{15,22}. Furthermore, single crystals of PdCoO_{2} were grown using a metathetical reaction method using powders of PdCl_{2} 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 μm^{3}, 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 *a**b* 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 μm^{3} in size, were attached to a Si/SiO_{2} 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 PdCrO_{2} 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 PdCoO_{2} 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.

## Data availability

All data supporting the findings of this study are available within the main text and the Supplementary Information file. The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

The authors thank H.W. Lee and E.G. Moon for the fruitful discussion. We also thank H.G. Kim in Pohang Accelerator Laboratory (PAL) for the technical support. This work was supported by the Basic Science Research Program (NRF-2022R1A2C3009731, RS-2023-00221154, NRF-2019R1A2C1089017), BrainLink program (No. 2022H1D3A3A01077468), the Max Planck POSTECH/Korea Research Initiative (Grant Nos. 2022M3H4A1A04074153 and 2020M3H4A2084417), funded by the Ministry of Science and ICT through the National Research Foundation (NRF) of Korea. This work is also supported by the Institute for Basic Science (IBS) through the Center for Artificial Low-Dimensional Electronic Systems (no. IBS-R014-D1). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement No. DMR-1644779 and the State of Florida. The work at ORNL for film synthesis was supported by the U.S. Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division.

## Author information

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### Contributions

J.S.K., J.M.O. and H.J. conceived the experiments. J.M.O., H.J., J.S., Y.J. E.S.C. and J.S.K. performed the magnetotransport measurements at high magnetic fields. J.M.O., H.J. and H.S. grew high-quality single crystals. J.M.O. and H.N.L. grew high-quality thin films. J.M.O., H.J., Y.H.K. fabricated single-crystal-devices. J.M.O., H.J. and J.S.K. co-wrote the manuscript. All authors discussed the results and commented on the paper.

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### Cite this article

Jeon, H., Seo, H., Seo, J. *et al.* Large anomalous Hall conductivity induced by spin chirality fluctuation in an ultraclean frustrated antiferromagnet PdCrO_{2}.
*Commun Phys* **7**, 162 (2024). https://doi.org/10.1038/s42005-024-01652-3

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DOI: https://doi.org/10.1038/s42005-024-01652-3

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