## Abstract

The recent observation of the anomalous Hall effect (AHE) without notable magnetization in antiferromagnets has suggested that ferromagnetic ordering is not a necessary condition. Thus, recent theoretical studies have proposed that higher-rank magnetic multipoles formed by clusters of spins (cluster multipoles) can generate the AHE without magnetization. Despite such an intriguing proposal, controlling the unconventional AHE by inducing these cluster multipoles has not been investigated. Here, we demonstrate that strain can manipulate the hidden Berry curvature effect by inducing the higher-rank cluster multipoles in spin-orbit–coupled antiferromagnets. Observing the large AHE on fully strained antiferromagnetic Nd_{2}Ir_{2}O_{7} thin films, we prove that strain-induced cluster *T*_{1}-octupoles are the only source of observed AHE. Our results provide a previously unidentified pathway for generating the unconventional AHE via strain-induced magnetic structures and establish a platform for exploring undiscovered topological phenomena via strain in correlated materials.

## INTRODUCTION

The anomalous Hall effect (AHE) is a fundamental transport phenomenon that has been universally observed in time-reversal symmetry broken systems. AHE can arise from two different forms of mechanism (*1*): extrinsic mechanism, such as skew scattering or side jump due to magnetic impurities, and intrinsic mechanism originating from Berry curvature in momentum space. Since the fundamental topological properties of electronic wave functions are encoded in the Berry curvature, AHE is considered as a powerful tool for probing the topological properties of materials (*2*, *3*). In addition to its fundamental interest, AHE can be applied for memory devices (*4*).

Conventionally, AHE has been observed mostly in itinerant ferromagnets. Its magnitude is known to be proportional to the magnetization (*5*), which is a measure of broken time-reversal symmetry. Recently, a large AHE has been unexpectedly found in noncollinear antiferromagnets, such as Mn_{3}X (X = Sn, Ge) (*6*–*8*) and GdPtBi (*9*), which do not exhibit spontaneous magnetization. This unconventional response indicates that ferromagnetism is not a necessary condition for AHE and suggests a possible alternative origin of AHE. A recent theory proposed that higher-rank magnetic multipole (cluster multipole) moments formed from spin clusters in antiferromagnet can induce a nonzero AHE, beyond the conventional dipoles of ferromagnets (*10*). Subsequently, the anomalous Nernst (*11*) and magneto-optical Kerr effects (*12*) in Mn_{3}Sn have also been attributed to its cluster octupoles. However, since antiferromagnets are not easily coupled to both magnetic and electric fields (*13*), it is very difficult to manipulate the higher-rank cluster multipoles. This imposes substantial limitations on controlled experiments on the cluster multipoles and associated AHE.

Here, we demonstrate that the strain can generate the AHE by inducing the higher-rank cluster multipoles, by using antiferromagnetic Nd_{2}Ir_{2}O_{7} (NIO) thin film. Further investigation reveals that biaxial strain on the pyrochlore lattice can modulate the spin structure and induce certain magnetic octupoles. The induced cluster octupoles can generate the net Berry curvature effect hidden in the bulk, leading to a finite AHE. We expect that our method could be widely applied to other spin-orbit–coupled topological magnets (*10*) and antiferromagnetic spintronics (*4*, *14*).

## RESULTS

### Strain-induced cluster multipoles in a pyrochlore lattice

The NIO belongs to the pyrochlore iridates family, *R*_{2}Ir_{2}O_{7} (*R*, rare-earth ions). The members of the family are geometrically frustrated magnets with complex lattice structures. As shown in Fig. 1A, *R*_{2}Ir_{2}O_{7} is composed of linked tetrahedrons with *R* and Ir at each vertex. In *R*_{2}Ir_{2}O_{7}, strong electron correlations and large spin-orbit coupling of Ir *d* electrons result in unique antiferromagnetic spin structures, called all-in-all-out (AIAO) ordering (*15*, *16*). As shown in the circle in Fig. 1B, the spins in one tetrahedron point inward and those in the neighboring tetrahedron point outward. The Néel temperatures of the Ir and Nd sublattices of bulk NIO are *15*) and *17*), respectively. This AIAO ordering breaks the time-reversal symmetry, allowing a nonzero Berry curvature distribution and generating correlated topological phases (*18*, *19*) such as a Weyl semimetal.

However, since AIAO ordering preserves the cubic crystalline symmetry, the net Berry curvature effect vanishes when we integrate over the Brillouin zone (BZ). Unless the cubic crystalline symmetry is broken, AHE cannot be observed in this system. To break the cubic symmetry, a magnetic field was applied to pressured NIO single crystals (*19*) and Pr-doped bulk samples (*20*, *21*). However, the spin structures modulated by the magnetic field are fragile and easily return once the magnetic field is turned off. Thus, a stable method to break the cubic symmetry is highly desirable; here, we choose a strain engineering approach and investigate the associated AHE.

As shown in Fig. 1B, the biaxial strain elongates the unit tetrahedra along the [111] direction. This will naturally break the cubic symmetry of the system. Since the deformation modulates magnetic anisotropy (*22*), the Ir spin directions should be changed. To systematically describe the change of spin direction, we adopted the cluster multipole theory. Since the conduction electrons come from Ir *d* orbitals, we considered Ir sublattice only (*16*). In the cubic pyrochlore lattice, all spin ordering patterns can be classified into five different irreducible representations, carrying 12 distinct cluster multipoles (*18*). Among them, certain cluster multipoles that break the cubic symmetry are responsible for the AHE (see section S1).

In a bulk NIO, the AIAO ordering corresponds to a higher-rank magnetic multipole called the *A*_{2}-octupole (Fig. 1C). Since the *A*_{2}-octupole preserves the cubic symmetry, it cannot generate AHE. However, in a strained NIO (*s*-NIO), the AIAO spin structure becomes modulated under the strain. The resulting spin configuration is denoted by strained AIAO (*s*-AIAO), composed of a superposition of three kinds of cluster multipoles, namely, a dipole, an *A*_{2}-octupole, and a *T*_{1}-octupole (Fig. 1D). Note that the dipole is just the ferromagnetic ordering, while the *T*_{1}-octupole is an antiferromagnetic ordering other than AIAO. Only the *T*_{1}-octupole can induce the AHE without magnetization since it breaks the cubic symmetry as the dipole does.

### Characterizations of relaxed and fully *s*-NIO thin films

To investigate the strain-induced magnetic multipole and associated AHE, we prepared two kinds of NIO thin films on the yttria-stabilized zirconia (YSZ) substrates: relaxed and fully strained films. The biaxial strain can arise from the lattice mismatch between the *R*_{2}Ir_{2}O_{7} film and the YSZ substrate (see Fig. 1B) (*23*, *24*). Since the lattice parameter of YSZ is smaller than those of NIO bulk, the NIO film should be compressively strained. We estimated the strain *a*_{NIO} and *a*_{YSZ} are lattice constants of bulk NIO (10.38 Å) and YSZ (5.14 Å), respectively.

Despite the substantial past efforts (*25*–*27*), the in situ growth of high-quality *R*_{2}Ir_{2}O_{7} thin film is notoriously difficult. Under the proper crystalline growth conditions for pyrochlore oxides (*28*), the corresponding *R*_{2}Ir_{2}O_{7} phase becomes extremely unstable because of the formation of a gaseous IrO_{3} phase (*29*). To avoid this instability, many studies have used the “solid-phase epitaxy (SPE)” (*25*, *27*) method, which involves the initial growth of amorphous *R*_{2}Ir_{2}O_{7} films at a lower temperature (*T*) followed by ex situ thermal annealing in a sealed tube. Although SPE can provide a method for the growth of single-phase *R*_{2}Ir_{2}O_{7} films, it usually produces relaxed films (*25*, *26*). Therefore, we developed a previously unknown in situ film growth method, i.e., repeated rapid high-temperature synthesis epitaxy (RRHSE; see section S2 and Materials and Methods) (*30*).

The RRHSE method made us successfully grow the fully *s*-NIO films on YSZ (111) substrates. Figure 2A shows an x-ray diffraction θ-2θ scan. The NIO (*lll*) and YSZ (*lll*) (*l*: integer) peaks can be seen, indicating epitaxial growth of NIO single phase. Particularly, the satellite peaks near the NIO (222) peaks are observed, which is commonly called “thickness fringes.” These interference peaks indicate the high quality of a sharp interface between film and substrate. Figure 2B shows x-ray reciprocal space mapping around the NIO (662) and YSZ (331) Bragg peaks of a 9-nm-thick NIO film. The lattice parameter of the (662)-plane, *d*_{(662)}, of bulk NIO is 1.19 Å, and the *d*_{(331)} of YSZ is 1.18 Å. Note that the NIO (662) Bragg peak has the same *Q _{x}* value as the YSZ (331) peak, demonstrating that our film becomes fully strained (~1% compressive strain) by the substrate.

Figure 2C shows a scanning transmission electron microscopy image that indicates the high quality of our film. The NIO pyrochlore phase is formed with few structural defects or disordered structures. Figure 2 (D and E) shows fast Fourier transform patterns from the selected areas in the film and substrate, respectively, marked in Fig. 2C. As demonstrated by the red dotted lines, the as-grown NIO film has the same inverse lattice constant as the YSZ substrate, which also confirms that our film is fully strained.

### Electronic structures of relaxed and fully *s*-NIO thin films

We compared these fully *s*-NIO films grown by RRHSE with the relaxed NIO (*r*-NIO) films grown by the SPE (see section S3). The resistivity ρ (*T*) curve of a 9-nm-thick *s*-NIO film exhibits a semimetallic behavior at most *T*. As shown in Fig. 3A, the *s*-NIO film has ρ (*T*) an order of magnitude smaller than that of the *r*-NIO film. The ρ (*T*) curve of an 80-nm-thick *r*-NIO film exhibits a metal-insulator transition around ~30 K (black dashed line in Fig. 3A), in agreement with its bulk counterpart (*17*, *31*). The strong upturn of the *r*-NIO film is due to its insulating nature below *17*, *31*). The ρ (*T*) curve of the *r*-NIO film follows the Arrhenius plot (not shown here) in the low *T* region, indicating a bandgap opening. In contrast, the ρ (*T*) curve of the *s*-NIO film has a positive slope for most *T* (orange line in Fig. 3A), suggesting that the film should be in a semimetallic state. Converting the resistivity into conductivity, the *s*-NIO film has σ* _{xx}* ~ 1600 ohms

^{−1}cm

^{−1}at 2 K. The tiny upturn below ~30 K might arise from disorder effects.

To understand the corresponding electronic structure changes, we performed mean-field calculations using the Hubbard model (see section S4). The previous study shows that the most valence and conduction bands near the Fermi energy come mainly from Ir 5*d* electrons (*16*). Our calculated electronic structure of the bulk (the *r*-NIO film in our case) explains its insulating nature. The energy gap opens with a value of about 13 meV (Fig. 3B), which agrees well with the bulk value (*32*). Under 1% compressive strain, the valence and conduction bands move, which slightly increases the direct gap at most ** k** regions. However, some valence and conduction bands become crossed with Fermi level; thus, small electron and hole pockets develop near

*L*

_{1,2,3,4}(Fig. 3C), creating a semimetallic state. These model calculations can explain why the

*s*-NIO film has a much smaller ρ (

*T*) than the

*r*-NIO film.

### Large AHE in fully *s*-NIO thin films

Besides, the *s*-NIO film shows a much larger anomalous Hall conductivity (AHC) compared to the *r-*NIO film. Figure 3D shows the magnetic field (*H*)–dependent AHC *s*- and *r*-NIO films are displayed by the circles and the dashed line, respectively. The *s-* and *r*-NIO films are 2.4 and 0.2 ohms^{−1} cm^{−1}, respectively. The spontaneous Hall conductivity (SHC) *s-*NIO films is 1.04 ohms^{−1} cm^{−1}, which is much larger than that of the *r-*NIO film. Note that the small AHC and SHC in the *r*-NIO film might be induced by the net magnetization of AIAO domain walls (*33*). However, the large AHC and SHC in the *s*-NIO suggest that the net Berry curvature effect can be modulated by the strain.

To cross-check, we compared our magnetotransport property values with those of ferromagnets. For example, (Ga, Mn)As (*34*) and CuCr_{2}Se_{4–x}Br* _{x}* (

*35*) typically exhibit SHC with

*H*= 0 T) ~ 1 to 10 ohms

^{−1}cm

^{−1}and σ

*(*

_{xx}*H*= 0 T) ~ 1000 ohms

^{−1}cm

^{−1}. These ferromagnets follow a scaling relationship

*5*). Since

*values for the*

_{xx}*s*-NIO film also fall on the same scaling curve (see section S5), we confirmed the enhanced AHC and SHC of our fully

*s*-NIO film as the net Berry curvature effect.

Accordingly, we calculated the Berry curvature effect on AHC from the band structure obtained from the mean-field calculations mentioned above (see section S4). AHC can be obtained by integrating the Berry curvature *5*):*H* = 0. Sizable *L*_{1,2,3,4} points in the BZ (Fig. 3F) exists for both the *r*- and *s*-NIO systems. The Berry curvature at each high-symmetry point for the *r*-NIO is somewhat larger than that for the 1% *s*-NIO. However, for the cubic *r*-NIO, *C*_{2} about the *x*, *y*, or *z* axis exist, *r*-NIO, all three *C*_{2} exist, so the net *9*). In contrast, for the trigonal *s*-NIO, the breaking of all *C*_{2} symmetries draw out a finite net *s*-NIO films.

### Antihysteresis of AHC

Another notable feature of *s*-NIO film is that its *H*) curve shows an intriguing antihysteresis-like behavior, displayed in Fig. 3D. When we sweep the *H*-field from −9 to +9 T, a sign change occurs at an *H* value of about −1 T (circles in Fig. 3D). Similar behavior is also observed when we reverse the *H-*field sweep from +9 to −9 T. This *H*-dependent sign change of the AHC differs from a typical hysteretic response of most ferroic materials, where the sign change occurs during the domain switch to the opposite direction. Although a similar antihysteresis-like behavior has been also reported in an earlier Hall conductivity study of an NIO single crystal under hydrostatic high pressure (*21*), its origin has not fully investigated yet.

To understand our antihysteresis-like *H*) curve, we used a phenomenological model (see section S6). The model is composed of two tangent hyperbolic functions; one is hysteretic (blue line) and the other is nonhysteretic (green line) (see Fig. 4A). Since the experimental data (orange circles) agree with the sum of two tangent hyperbolic functions (black solid line), the antihysteretic curve can be explained by the two different origins of *T*-dependent *H*) curves of *s*-NIO film below 40 K. As shown in Fig. 4B, *T* decreases, *T* < 30 K, *T* decreases further below 15 K, *T*-dependent *17*), suggesting that the hysteretic and nonhysteretic responses are developed because of the magnetic orderings of Nd and Ir spins, respectively.

Figure 4C summarizes the results of the AHC fitting at *H* = −9 T with the *d* electrons near the Fermi level. This carrier transport can be affected by the spin ordering at the Ir and Nd sublattices. In Fig. 4C, the nonhysteretic component (green circles) starts to emerge below *f-d* exchange with either Nd-3O1I or Nd-3I1O, which can be formed by large Nd moments under a [111] magnetic field (see section S6). This hysteretic behavior of

### AHE from strain-induced T_{1}-octupoles

To reveal the relation of AHE and cluster multipoles under strain, we should compare *M* and *H*-dependent *M* and *5*), i.e., *M* (0 T), does not hold for the *s*-NIO film. Although the *s*-NIO film has a large SHC signal (orange circles) shown in Fig. 5A, it has no spontaneous *M* at 3 K with *H* = 0 (purple squares) within the measurement error (± 0.01 μ_{B}/NdIrO_{3.5}). As shown in Fig. 1D, the biaxial deformation of pyrochlore lattice can generate three kinds of multipoles, i.e., a dipole, an *A*_{2}-octupole, and a *T*_{1}-octupole. The dipole is crossed out because of the zero magnetization of our data, and the *A*_{2}-octupole is crossed out because of its zero contribution to AHC. Therefore, the strain-induced *T*_{1}-octupole should play important roles in generating the AHC without magnetization.

To elucidate how *T*_{1}-octupole emerges under the strain, we calculated the spin structure from the spin model. Since both Nd and Ir spins play important roles, we included the Heisenberg, Dzyaloshinskii-Moriya, anisotropic spin-exchange interactions between Ir spins (*36*), the *f-d* exchange interaction between the Ir and Nd spins (*17*), and the Zeeman energy for both the Ir and Nd spins (for details, see section S7). On the basis of the calculated spin structure, we obtained the cluster multipoles (table S1 in section S1). Figure 5C shows the calculated dipole (*M*, green circles) and *T*_{1}-octupole (ω, blue circles) as a function of the effective Zeeman energy *h* in the *r*-NIO. According to our calculation, *r*-NIO does not have a finite *M* or ω value for the Ir sublattice at *h* = 0. The zero values of *M* and ω can explain the negligible SHC of the *r*-NIO film (see Fig. 3D). Figure 5D shows the calculated *M* and ω of *s*-NIO, which are finite even for *h* = 0. Particularly, the hysteresis curve of ω looks similar to the *s*-NIO film.

## DISCUSSION

Our work demonstrates that the strain-engineering of an antiferromagnet can generate the net Berry curvature effects by modulating its cluster multipoles. In particular, our findings highlight that the strain-induced *T*_{1}-octupole is closely connected with the topological properties of NIO. We can further extend this strain-engineering approach to search for the other novel topological phenomena in correlated magnets. For example, the strain engineering approach on numerous series of *R*_{2}Ir_{2}O_{7} can realize novel correlated topological phases, such as Weyl semimetal, axion insulator (*16*), a strong topological insulator (*18*), and line-node semimetal (*19*, *21*) by properly modifying their magnetic structure. In this perspective, we believe that our strain study on NIO could provide a cornerstone to discover strain-engineered novel topological phenomena in oxides and to understand their fundamental mechanisms.

## MATERIALS AND METHODS

### Film growth and structural characterization

Fully *s*-NIO films were in situ grown on insulating YSZ substrates using the RRHSE method. This film growth method is a modified form of pulsed laser deposition, based on repeated short-term thermal annealing processes using an infrared laser. RRHSE consists of two key steps in one thermal cycle. During the first step, amorphous stoichiometric NIO and IrO_{2} layers were deposited by a KrF excimer laser (λ = 248 nm, 5 Hz) at *T* ~ 600°C with *P*_{O2} ~ 50 mtorr. The additional IrO_{2} layer was deposited to compensate for the Ir loss that would unavoidably occur later during the synthesis process. During the second step, the pyrochlore phase is formed by rapidly raising *T* to 800°C (up to ~400°C min^{−1}). We must expose the sample to the high *T* for a period that is sufficiently long to synthesize the pyrochlore phase but short enough to minimize the formation of IrO_{3}. Last, we repeated these deposition and thermal synthesis processes until the desired film thickness was obtained. During the growth, the reflection high-energy electron diffraction pattern was monitored and the intensity of the oscillation was recorded. After growth, NIO films were characterized by an x-ray diffractometer (Bruker Corp.) and an atomic-resolution high-angle annular dark-field scanning transmission electron microscope (JEM-ARM200F; JEOL) equipped with an energy-dispersive x-ray spectrometer.

### 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), which has a base *T* of 2 K and a maximum magnetic field of 9 T. During the measurements, the current was applied along the [1-10] direction, and *H* was applied along the [111] direction. Magnetization data were obtained using a commercial superconducting quantum interference device magnetometer (MPMS, Quantum Design) with the magnetic field applied normal to the film.

The AHC value * _{xx}* is longitudinal resistivity. To exclude the longitudinal contribution from the raw Hall resistivity data

*8*,

*9*,

*13*). We separated the positive and negative field sweep branches and then antisymmetrized ρ

*using*

_{xy}### Self-consistent mean-field Hubbard model

We developed the Hubbard model for the *s*-NIO thin film under the magnetic field and acquired the ground state and electronic structure by the self-consistent mean-field method. We adopted 24 × 24 × 24 and 32 × 32 × 32 *k*-point mesh and found that the results are consistent. We calculated the AHC by integrating the Berry curvature, adopting a 48 × 48 × 48 *k*-point mesh of the entire BZ. Details of the calculation are provided in section S4.

### Spin model calculation

We developed the spin model including Heisenberg exchange, Dzyaloshinskii-Moriya interaction, anisotropic exchange, the *f-d* exchange between Nd and Ir electrons, and the Zeeman effect by applying second-order perturbation theory to the Hubbard model and referring to previous works. We calculated the ground state by the iterative minimization method, which repeatedly aligns spins to the effective field direction until each spin is fixed. Details of the calculation are provided in section S7.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/29/eabb1539/DC1

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**We acknowledge the invaluable comments and suggestions from D. Lee, S. H. Chang, T. H. Kim, and C. H. Sohn.

**Funding:**This work was supported by the Research Center Program of the Institute for Basic Science in Korea (grants no. IBS-R009-D1 and no. IBS-R009-G1). T.O. and B.-J.Y. acknowledge the support by the Institute for Basic Science in Korea (Grant No. IBS-R009-D1), Basic Science Research Program through the National Research Foundation of Korea (NRF) (Grant No. 0426-20200003), and the U.S. Army Research Office under Grant Number W911NF-18-1-0137. STEM measurement was supported by the National Center for Inter-University Research Facilities (NCIRF) at Seoul National University in Korea.

**Author contributions:**W.J.K., T.O., B.-J.Y., and T.W.N. conceived the original idea. W.J.K. and T.W.N designed the experiments. T.O. performed the tight-binding model calculations and spin model calculations under the supervision of B.-J.Y. J.M. performed the STEM measurements under the supervision of M.K., W.J.K., J.Song, and E.K.K. grew and characterized the structure of the samples. W.J.K., J.Song, Y.L., Z.Y., and Y.K. performed the magnetotransport measurements. W.J.K., T.O., J.Song, B.-J.Y., and T.W.N. analyzed the results and wrote the manuscript with contribution from all authors. All authors participated in the discussion during the manuscript preparation.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).