Introduction

Antiferromagnetic (AFM) materials have attracted considerable interest as promising materials for next-generation spintronics1,2,3,4,5 and novel topological phenomena6,7,8,9,10,11. This interest is largely due to the fact that the AFM spin arrangement produces zero net magnetization, leading to an absence of stray fields and insensitivity to external magnetic fields12,13. Additionally, the spin precession for AFM order is much faster than that in ferromagnets14. The resonance frequencies in antiferromagnets are in the terahertz range15,16,17, whereas those of ferromagnets are in the gigahertz range. In parallel with AFM spintronics, antiferromagnets with topologically protected states have become another focus of recent interest. As AFM spin order breaks the time-reversal symmetry (TRS) or inversion symmetry, various topologically nontrivial states including the Weyl semimetal6,7,8, axion insulator9,10, and Möbius insulator11 can theoretically emerge.

To develop novel AFM spintronics and investigate topological states in AFM materials, understanding the relationship between the spin texture and emergent phenomena in AFM materials is essential. This relationship can be theoretically described by the recently introduced cluster multipole theory (CMT)18. In CMT18, magnetic structures are classified as “cluster multipoles” (dipole and additional high-rank multipoles) according to irreducible representations of the crystallographic point group. These cluster multipoles can be regarded as order parameters that reflect symmetry breaking in AFM materials. In particular, some cluster magnetic octupole (CMO) (i.e., T1 octupole) belongs to the same magnetic point group (−42’m’) as the magnetic dipole18, leading to symmetry breaking and generating nonvanishing Berry curvature. In fact, the analysis using CMOs has been recently extended to understanding the large response of AHE18,19,20,21, magneto-optical signals22, the SHE23,24,25, the anomalous Nernst effect26,27,28, and perpendicular magnetization29,30 in AFM materials. However, due to the absence of net magnetization, experimental identification of the CMOs in AFM materials is very challenging. To date, it has been limited to X-ray magnetic circular dichroism measurements31 and the neutron scattering technique32.

The planar Hall effect (PHE) has been considered a method of probing the physical properties of materials such as magnetism and topology. The PHE corresponds to the development of a Hall voltage when electric and magnetic fields are coplanar, which is different from the usual Hall effect where they are perpendicular to each other. Initially, the PHE was observed in ferromagnetic systems33,34,35,36, detecting the anisotropic magnetization of ferromagnetic materials. Additionally, the PHE has recently been in the spotlight due to its role in detecting topological characteristics such as the chirality arising from Weyl fermions in magnetic Weyl semimetals37,38,39,40. The associated PHE in both ferromagnets and Weyl semimetals exhibit sin (2ϕ) or second harmonic PHE oscillations. In contrast, higher harmonics PHE oscillations beyond second in topological systems41,42,43,44 have been recently reported, in which higher harmonics PHE originated from additional unknown parameters in materials.

Among many kinds of AFM materials, the family of 5d AFM materials R2Ir2O7 (R: rare-earth ions: Eu, Y, Nd, and Pr) has received considerable attention due to a plethora of intriguing properties6,7. The crystal structure of NIO-227 is composed of Ir and Nd tetrahedral sublattices (Fig. 1a), which have a symmetry identical to that of the diamond lattice. Most R2Ir2O7 bulk compounds exhibit an intriguing spin configuration called all-in-all-out (AIAO). According to CMT, the AIAO ordering of bulk R2Ir2O7 is equivalent to the A2-CMO21,45. This A2-CMO breaks the TRS and generates nonvanishing Berry curvature, resulting in topological properties in momentum space such as the magnetic Weyl semimetal6 and Axion insulator6.

Fig. 1: Schematic diagram of strain-induced cluster magnetic multipoles and OM in Nd2Ir2O7 thin films.
figure 1

a Schematic view of the pyrochlore lattice structure of Nd2Ir2O7. b When compressive strain is applied to Nd2Ir2O7 along the [111] direction, the AIAO configuration of the magnetic spin experiences spin canting. Such canting can be represented by three distinctive magnetic multipoles (dipole, A2-octupole, and T1-octupole). Note that the dipole has finite magnetization, whereas the A2-octupole and T1-octupole do not have magnetization. c Schematic diagram of the longitudinal magnetization induced by the dipole. When Hext is applied along the x-y plane, the spin direction is aligned in the direction of Hext. d Schematic diagram of the OM induced by coupling between an external magnetic field (Hext) and magnetic octupoles. The OM can be induced in two directions: normal to the surface and in-plane.

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Here, we demonstrate the detection and identification of CMO in fully-strained AFM Nd2Ir2O7 (NIO-227) film via the PHE. Under epitaxial strain, three kinds of cluster multipoles (dipoles and A2- and T1-CMOs) can be induced in the NIO-227 film. By rotating the magnetic field, distinctive harmonics in the PHE oscillation were observed. Specifically, the dipole induces the second harmonic due to longitudinal magnetization (Fig. 1c), whereas the A2- and T1-CMO induce fourth and sixth harmonics in the PHE oscillation, respectively. We demonstrate that the higher harmonics of the PHE oscillation for both CMOs originate from the magnetic response of the CMOs, called orthogonal magnetization (OM) (Fig.1d). Furthermore, cluster multipoles induce nonlinear magnetic field-dependent the planar Hall conductivity. This nonlinear magnetic field behavior of the planar Hall conductivity can be understood in terms of the magnetic properties of the cluster magnetic dipole and CMOs.

Results

To investigate the magnetotransport properties of CMOs, we fabricated 15 nm fully strained NIO-227 films on a (111)-oriented yttria-stabilized zirconia (YSZ) substrate. More details on the growth are provided in the Methods section. Figure 2a shows the X-ray diffraction (XRD) pattern of a (111)-oriented NIO-227 thin film on a YSZ substrate. The strong odd-pair peaks in the XRD pattern exhibit excellent crystallization of the pyrochlore phase in the NIO film. Further analysis of the reciprocal space map (Fig. 2b) shows that the NIO-227 film is fully strained with a compressive strain of ~1%. As mentioned earlier, the presence of compressive strain in the NIO-227 film is crucial, as it modulates the AIAO ordering and develops T1-CMOs. Figure 2c shows the longitudinal resistivity ρxx (T) of the NIO-227 thin film as a function of temperature without a magnetic field. Clearly, with decreasing T, the NIO-227 film exhibits stronger semimetallic behavior than that of bulk NIO-227. At T ~ 2 K, our NIO-227 film has a ρxx of ~2.28 mΩ cm, whereas that of bulk NIO-227 is ~ 2.20 × 103 mΩ cm46. With compressive strain, the valence and conduction bands move in the NIO-227 film. The valence and conduction bands cross near the Fermi level, resulting in the development of electron and hole pockets21. These previous model calculations explain the enhancement of the conductivity in the NIO-227 thin film compared to the bulk.

Fig. 2: Characterization of a Nd2Ir2O7 thin film.
figure 2

a High-resolution XRD pattern of a 15 nm thick Nd2Ir2O7 (NIO-227) thin film. The NIO-227 thin film was grown on a YSZ substrate. b Reciprocal space map of the NIO-227 film in the vicinity of the YSZ (331) reflection, indicating that the NIO-227 film is fully strained. c The plot of the longitudinal resistivity, ρxx, vs. temperature, T, of NIO-227 thin film and single crystal (ref. 46 in the main manuscript). d The AHE is measured by applying Hext along the [111] direction and I along the [1\(\bar{{{{{1}}}}}\)0] direction at 2 K. A clear AHE without a magnetic field is the signature of the presence of the T1-octupole in the NIO-227 thin film.

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Moreover, the AHE (σxyAHE (H)) of the NIO-227 film was measured with Hext applied along the [111] direction below 30 K. As shown in Fig. 2d, σxyAHE (H) at 2 K shows a hysteric feature with a finite value at 0 T, known as the spontaneous Hall effect21. In our strained NIO-227 thin film, the AHE and spontaneous Hall effect developed below 30 and 15 K, respectively. The value of the spontaneous Hall effect approached the maximum value at 2 K. More details are provided in Supplementary Materials (Supplementary Fig. 1 and Note 1). The appearance of AHE below 30 K was previously attributed to Ir spin ordering21. In contrast, the emergence of the spontaneous Hall effect without magnetization below 15 K in the NIO-227 film is known to be the nontrivial contribution of the T1-CMO to the Berry curvature21, which is amplified by Nd ordering through f-d exchange interaction. The effect of Nd ordering can give an additional hysteric component on AHE21 (Supplementary Fig. 2). In our 15 nm NIO-227 film, the spontaneous Hall effect is observed without magnetization, which is induced by T1-CMO in the film (Supplementary Fig. 1c).

To detect CMOs in the NIO-227 film, we performed PHE measurements. As shown in Fig. 3a, the PHE (\(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ, H = ±9 T)) below 30 K was measured across the width of the sample by applying I along the [1\(\bar{1}\)0] direction with the rotation of Hext. The details about the experimental measurement and data process are provided in the “Methods” and Supplementary Materials (Supplementary Fig. 3 and Note 2), respectively. In the range of 15–30 K, the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) curves in the NIO-227 film mostly exhibit sin (2ϕ) oscillations (Fig. 3b). The sin (2ϕ) behavior (second harmonic) of the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation above 15 K can be understood as the magnetization from dipolar order. As shown in Fig. 1b, the NIO-227 film has a cluster dipole, which can be regarded as ferromagnetic ordering. This cluster dipole induces ML to be directed along Hext, which can result in the second harmonic of the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation. Thus, the second harmonic of the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation above 15 K can be understood as the transverse voltage developed by the cluster dipole in the NIO-227 film under Hext.

Fig. 3: Anomalous magnetic oscillation in the \(\Delta {\sigma }_{xy}^{{{{{{\rm{PHE}}}}}}}(\phi,H=\pm 9\,{{{{{\rm{T}}}}}})\) of the NIO-227 thin film.
figure 3

a Schematic diagram of the \({{{{\Delta }}}}\,{{{{{\sigma }}}}}_{{{{{xy}}}}}^{{{{{{{{\rm{PHE}}}}}}}}}\left({{{{\phi }}}}\right)\) measurement geometry. In this geometry, I is applied along the \(\left[1\bar{1}0\right]\) direction and Hext is rotated within the sample plane. b \({{{{\Delta }}}}\,{{{{{\sigma }}}}}_{{{{{xy}}}}}^{{{{{{{{\rm{PHE}}}}}}}}}\left({{{{\phi }}}}\right)\) curves at 20 and 2 K. At 20 K, the \({{{{\Delta }}}}\,{{{{{\sigma }}}}}_{{{{{xy}}}}}^{{{{{{{{\rm{PHE}}}}}}}}}\left({{{{\phi }}}}\right)\) curve has a sin (2ϕ) oscillation. At 2 K, the \({{{{\Delta }}}}\,{{{{{\sigma }}}}}_{{{{{xy}}}}}^{{{{{{{{\rm{PHE}}}}}}}}}\left({{{{\phi }}}}\right)\) curve has an inhomogeneous oscillation, indicating a hidden magnetic origin. c Contour plot of all \({{{{\Delta }}}}\,{{{{{\sigma }}}}}_{{{{{xy}}}}}^{{{{{{{{\rm{PHE}}}}}}}}}\left({{{{\phi }}}}\right)\) curves below 30 K. A gradual appearance of the abovementioned peaks can be seen (red region). While the peaks at ϕ = 120° and 300° (green region) exist above T ~ 15 K, the peaks near ϕ = 60°, 180°, 240°, and 340° (red region) appear below T ~ 15 K. d FFT results of all \({{{{\Delta }}}}\,{{{{{\sigma }}}}}_{{{{{xy}}}}}^{{{{{{{{\rm{PHE}}}}}}}}}\left({{{{\phi }}}}\right)\) curves with different T. The FFT intensities of the second, fourth, and sixth harmonics are related to magnetization from the dipole, OM induced by the A2-CMO, and OM induced by the T1-CMO, respectively. The inset shows the temperature dependence of the sixth harmonic below 30 K. Clearly, the sixth harmonic order emerges at 15 K, which indicates the role of the T1-CMO.

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Below T ~ 15 K, the behavior of the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) curves are complex but periodic, exhibiting oscillations of multiple harmonics. As shown in Fig. 3b, the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation at 2 K cannot be explained only by the typical second harmonic induced by dipolar order. Such complex behavior of the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation can be seen in the contour plot presented in Fig. 3c. Whereas the second harmonic of \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) (green region) exists in all temperature regions below 30 K, additional harmonics (red region) of the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation appears below 15 K. As a result, the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) curve exhibits oscillations of multiple harmonics below 15 K.

To obtain further insight into the anomalous behavior of \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) curves below 15 K, we performed a fast Fourier transform (FFT) of the measured \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) curves. The detailed FFT procedure is provided in Supplementary Materials (Supplementary Fig. 4). The harmonics (n) dependency of the FFT intensity in the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation is shown in Fig. 3d. At 30 K, the second harmonic always exists, which is induced by the cluster dipole. The additional higher harmonics appear below 30 K. The appearance of the fourth harmonic in the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation has been attributed to the A2-CMO-induced chiral anomaly in Pr2Ir2O7 thin films41. Considering that both Pr2Ir2O7 and NIO-227 films have the A2-CMO, the observed fourth harmonic of the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation in the NIO-227 film should originate from the A2-CMO. However, the appearance of the sixth harmonic below 15 K (inset in Fig. 3d) cannot be explained by either the dipole or A2-CMO.

To elucidate the appearance of the sixth harmonic in the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation, we calculate \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) from the dipole and A2- and T1-CMOs using the phenomenological model. Notably, the A2-CMO induces an unusual magnetic response perpendicular to Hext. This magnetic response was initially observed in a Eu2Ir2O7 single crystal by torque magnetometry and defined as OM30. As mentioned above, the T1-CMO is also expected to exhibit OM. Employing the expression30 of the OM in Eu2Ir2O7, we obtained the \(\phi\) dependence of \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) for our system, expressed as follows:

$$\Delta {\sigma }_{xy}^{{{{{{\rm{PHE}}}}}}}(\phi )={\sigma }_{xy}^{{{{{{\rm{Dipole}}}}}}}+{\sigma }_{xy}^{A2}+{\sigma }_{xy}^{T1}=A{{{{{\rm{sin}}}}}}(2\phi )+B{{{{{\rm{sin}}}}}}(4\phi )+C\,{{{{{\rm{sin}}}}}}(6\phi )$$
(1)

where A is the PHE coefficient of the dipole. B and C are the PHE coefficients of the OM induced by A2- and T1-CMOs, respectively. The detailed theoretical calculation is discussed in Supplementary Materials (Supplementary Figs. 5 and 6, and Note 3): Then, Eq. (1) can explain the appearance of peaks of the second, fourth, and sixth harmonics (Fig. 3d) by the ML induced by the dipole, OM induced by the A2-CMO, and OM induced by the T1-CMO, respectively. In the range of 15–30 K, the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation can be fitted with the ML and OM induced by the dipole and A2-CMO, respectively. The fitting of the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation at 20 K without the sin (6ϕ) term in Eq. (1) is shown in Fig. 3b. This indicates that the cluster dipole and A2-CMO of Ir spin ordering induce second and fourth harmonics of the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation, respectively, while the contribution of the T1-CMO is small in this temperature region.

In contrast, the \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) oscillation below 15 K can only be fitted by including the T1-CMO contribution. As shown in Fig. 3b, Eq. (1), by including the sin (6ϕ) term from the T1-CMO (red), well fits \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) at 2 K. The fitting results of \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) for measured temperature using Eq. (1) is provided in the Supplementary Materials (Supplementary Fig. 7). The temperature dependence of the fitting result is consistent with the appearance of the sixth harmonic peak below 15 K in the NIO-227 film. Furthermore, a similar temperature characteristic of T1-CMOs is observed in not only AHE but also in anisotropic magnetoconductivity and out-of-plane rotation magnetoconductivity (Supplementary Figs. 8 and 9). The multiple harmonics of anisotropic magnetoconductivity and out-of-plane rotation magnetoconductivity are developed below 15 K. These complex behaviors can be explained by including the contribution of T1-CMOs, which is amplified by Nd ordering through f-d exchange interaction21. Therefore, we can confirm that the higher harmonics of \(\Delta {\sigma }_{{xy}}^{{{{{{\mathrm{PHE}}}}}}}\) (ϕ) originate from A2- and T1-CMOs via OM.

Since the OM is the coupling effect between CMOs and Hext and is known to have an H-field dependence30, the \({\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) of NIO-227 should exhibit behavior different from that of a ferromagnet. We measured \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(\phi={45}^{^\circ },H)\) below 30 K by fixing ϕ = 45° between I and Hext. (Fig. 4a). Rxx and Rxy were simultaneously measured and symmetrized, and \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) was obtained (Supplementary Fig. 10). The \({\Delta \sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) curves with respect to Hext below 20 K are shown in Fig. 4b. In the range of 15–20 K, the values of the \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) curves remain near zero at Hext = +9 and −9T. However, below 15 K, the values of the \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) curves have finite values with increasing Hext and reach \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)=+ 9{{{{{\rm{T}}}}}}\)~ −1.103 Ω−1 cm−1 at 2 K.

Fig. 4: Nonlinear magnetic behavior of the \(\Delta {\sigma }_{xy}^{{{{{{\rm{PHE}}}}}}}(\phi=45^\circ,H)\) of the NIO-227 thin film.
figure 4

a Schematic diagram of the \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) measurement geometry. In this geometry, the angle between the current and magnetic field (\({\pm} \! {{{{9}}}}\,{{{{T}}}}\)) is 45°. Surprisingly, \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) shows Hext dependency below T ~ 15 K due to OM. b \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) curves as a function of Hext below T ~ 20 K. In the region of 15 K < T < 20 K, the value of \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) is nearly zero. The slight deviation of \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) near Hext ~ \({{{{\pm }}}}\!{{{{9}}}}\,{{{{T}}}}\) at T ~ 15 K is possibly due to dipolar order. c Logarithmic plot of the results of fitting \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) at T ~ 4 K with \({{{{\Delta }}}}{{{{{\sigma }}}}}_{{{{{xy}}}}}^{{{{{{{{\rm{PHE}}}}}}}}}\left({{{{H}}}}\right){{=}}\,{{{{{a}}}}}_{{{{{2}}}}}{{{{{H}}}}}^{{{{{2}}}}}{{{{+}}}}{{{{{a}}}}}_{{{{{3}}}}}{{{{{H}}}}}^{{{{{3}}}}}{{{{+}}}}{{{{{a}}}}}_{{{{{4}}}}}{{{{{H}}}}}^{{{{{4}}}}}\) (red line). The a2, a3, and a4 terms correspond to longitudinal magnetization, OM from the A2-CMO, and OM from the T1-CMO, respectively. Note that without the a3 and a4 contributions (black dashed line), the experimental data (blue dots) do not fit well, indicating the role of OM induced by both the A2- and T1-octupoles. d The plot of extracted fitting parameters a2, a3, and a4 and \({{{{\Delta }}}}{{{{{\sigma }}}}}_{{{{{xy}}}}}^{{{{{{{{\rm{PHE}}}}}}}}}\left({{{{H}}}}\,{{=}}\,{{9\,{{{{T}}}}}}\right)\) for the measured T.

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By considering the OM induced by the A2-CMO and T1-CMO, we calculated \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) using the phenomenological model. Based on our calculations, the relation of \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) to Hext can be expressed as follows:

$$\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)={a}_{2}{H}^{2}+{a}_{3}{H}^{3}+{a}_{4}{H}^{4}$$
(2)

where a2 is the coefficient of longitudinal magnetization induced by the dipole. Both a3 and a4 are the coefficients of OM induced by A2- and T1-CMOs. The details about the data process and theoretical calculation are provided in Supplementary Materials (Supplementary Note 4). Figure 4c shows the fitting results for \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) at a selected temperature of ~ 4 K with (red line) and without (black dashed line) the H3 and H4 terms. Note that the exact proportions of dipole and CMOs cannot be determined by experimental PHE due to the complexity behind transport phenomena41, but we can get it theoretically (Supplementary Fig. 11). The fitting with the OM contribution explains the experimental results better than that with the dipole, demonstrating the development of OM from Hext. Additionally, all measured \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) values below 20 K are well fitted by Eq. (2) (Supplementary Fig. 12). The contribution of the PHE coefficients of a2, a3, and a4 to \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) below 30 K is shown in Fig. 4d. Below 30 K, the dipole induces an H2 dependence of \(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}(H)\) due to the longitudinal magnetization, as shown in Fig. 1c. In contrast, below 15 K, the contributions of a2, a3, and a4 coexist. While dipolar order still affects \(\Delta {{{{{{\rm{\sigma }}}}}}}_{{xy}}^{{{{{{\rm{PHE}}}}}}}(H)\), a3 and a4 may appear due to the OM induced by both the A2 and T1-CMOs. Through theoretical analysis and the PHE results, we confirm that the anomalous behavior of the PHE indeed originates from A2- and T1-CMOs. Our results summarizing the contributions to the AHE and PHE of the cluster magnetic dipole, A2-CMO, and T1-CMO are shown in Table 1 in the Supplementary Materials.

Discussion

In summary, the PHE of the AFM NIO-227 film exhibits unique features different from those of ferromagnetic materials. The CMOs without magnetization affect the PHE, resulting in higher harmonics of the PHE oscillation beyond the second harmonic. Notably, A2- and T1-CMOs produce fourth and sixth harmonics of PHE oscillations, respectively. This feature of CMOs can be well explained by considering the magnetic characteristic of CMOs called OM in our theoretical calculations. Moreover, the OM of CMOs shows an intriguing Hext dependency, which induces the nonlinear PHE when the angle between Hext and I is 45°.

We would like to make some remarks about the CMOs in AFM materials. According to recent theoretical studies and experiments on antiferromagnets, the magnetic octupole is crucial for stabilizing Weyl fermions near the Fermi energy18,27,47,48,49,50,51 and inventing magnetic field-free switching spintronic devices23,24,25,48,49,50. Moreover, AHE, SHE, and Nernst effects have been experimentally observed in kagome antiferromagnets19,20,25,26,29, in which magnetic structure can also be classified with CMO18. Therefore, understanding and manipulating the CMOs are thus important to achieve novel topological phases and physics in AFM systems. However, due to the absence of magnetization, detection and characterization of CMOs and multipoles beyond dipoles have been difficult. In this context, our strategy to distinguish A2- and T1-CMOs could be extensively used to detect and identify cluster multipoles18,45 via the PHE. Thus, our work paves the way for detecting and identifying AFM order, which is expected to facilitate the development of novel functionalities using AFM materials.

Methods

Film fabrication and structural characterization

Fully strained NIO-227 films were grown in situ on insulating YSZ substrates using the “repeated rapid high-temperature synthesis epitaxy” (RRHSE) method21. This film growth method is a modified form of pulsed laser deposition, in which repetitive short-term thermal annealing processes are conducted using an infrared laser. The RRHSE consists of two key steps completed in one thermal cycle. In the first step, amorphous stoichiometric NIO-227 and compensational IrO2 layers are deposited with a KrF excimer laser (λ = 248 nm, 5 Hz) at T ~ 600 °C and PO2 ~ 50 mTorr. During the second step, the pyrochlore phase is formed by rapidly raising T to 850 °C at a rate of ~400 °C min−1, and the sample is exposed to the high T for a sufficient period. We repeated the deposition and thermal synthesis processes until the desired film thickness was obtained.

Transport and magnetic properties

Magnetotransport properties were measured via a standard four-point probe method using a commercial physical property measurement system (PPMS; Quantum Design Inc., San Diego, CA, USA), which has a base T of 2 K and a maximum magnetic field of 9 T. Au layer (thickness: ~50 nm) was deposited by an e-beam evaporator on the NIO-227 film and used as an electrode. For the AHE measurement, the current was applied along the \(\left[1\bar{1}0\right]\) direction, while H was applied in the [111] direction. For the PHE measurement, the current was applied along the \(\left[1\bar{1}0\right]\) direction and H was rotated between the \(\left[1\bar{1}0\right]\) and \(\left[11\bar{2}\right]\) directions. The Rxy and longitudinal magnetoresistance Rxx below 30 K were simultaneously measured while varying ϕ and fixing Hext at \(\pm \,\) T. The out-of-plane contribution of Hext caused by a slight misalignment of the electrode can be removed by symmetrizing the Rxy acquired at +9 and –9 T. Then, the planar Hall conductivity \({\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}\) is calculated by \({\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}=\frac{-{\rho }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}}{{{\rho }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}}^{2}+{{\rho }_{{xx}}}^{2}}\) and normalized (\(\Delta {\sigma }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}\)(ϕ)); here, \(-{\rho }_{{xy}}^{{{{{{{\rm{PHE}}}}}}}}\) and ρxx are the PHE resistivity and longitudinal resistivity, respectively. The detailed symmetrization process of the PHE is provided in Supplementary Materials (Supplementary Fig. 3).