Temperature- and field-driven spin reorientations in triple-layer ruthenate Sr4Ru3O10

Sr4Ru3O10, the n = 3 member of the Ruddlesden-Popper type ruthenate Srn+1RunO3n+1, is known to exhibit a peculiar metamagnetic transition in an in-plane magnetic field. However, the nature of both the temperature- and field-dependent phase transitions remains as a topic of debate. Here, we have investigated the magnetic transitions of Sr4Ru3O10 via single-crystal neutron diffraction measurements. At zero field, we find that the system undergoes a ferromagnetic transition with both in-plane and out-of-plane magnetic components at Tc ≈ 100 K. Below T * = 50 K, the magnetic moments incline continuously toward the out-of-plane direction. At T = 1.5 K, where the spins are nearly aligned along the c axis, a spin reorientation occurs above a critical field Bc, giving rise to a spin component perpendicular to the plane defined by the field direction and the c axis. We suggest that both the temperature- and field-driven spin reorientations are associated with a change in the magnetocrystalline anisotropy, which is strongly coupled to the lattice degrees of freedom. This study elucidates the long-standing puzzles on the zero-field magnetic orders of Sr4Ru3O10 and provides new insights into the nature of the field-induced metamagnetic transition.

Ruddlesden-Popper type perovskite ruthenates (Sr,Ca) n+1 Ru n O 3n+1 are prototypical 4d transition-metal oxides which exhibit a variety of intriguing phenomena. The magnetic and electronic properties of these materials are sensitively dependent on the layer number n and the structural distortions induced by substituting Ca for Sr. For instance, the single-layer Sr 2 RuO 4 (n = 1) shows an unconventional superconducting state 1 , whereas the ground state of the double-layer Sr 3 Ru 2 O 7 (n = 2) is a Fermi liquid and is close to a ferromagnetic instability 2 . The three-dimensional SrRuO 3 (n = ∞) is a ferromagnetic metal with a Curie temperature T c = 160 K 3,4 . On the other hand, Ca 2 RuO 4 (n = 1) is an antiferromagnetic Mott insulator with a Neel temperature T N ~ 110 K 5 , while Ca 3 Ru 2 O 7 (n = 2) exhibits a quasi-two-dimensional metallic behavior and becomes antiferromagnetic below T N ~ 56 K 6,7 , and CaRuO 3 is a paramagnetic metal 8,9 . To date, compared to the intense efforts invested on the single-, double-layer and three-dimensional (n = 1, 2 and ∞) ruthenates, there have been much fewer studies on the triple-layer (n = 3) compounds. Sr 4 Ru 3 O 10 crystallizes in an orthorhombic space group Pbam 10 , as shown in Fig. 1(a), which displays interesting but perplexing magnetic properties. From the magnetic susceptibility measurements, it has been reported that Sr 4 Ru 3 O 10 undergoes a paramagnetic-ferromagnetic transition at T c = 105 K, below which the easy axis is along the c direction 10 . Intriguingly, while an additional transition has been found in the magnetic susceptibility at T * = 50 K 10,11 , no anomaly is revealed in the specific heat measurements 12 . Although several distinct scenarios have been proposed to account for the feature at T * , its intrinsic character remains an open question. On the one hand, below T * a canted antiferromagnetic structure with a ferromagnetic component along the c axis and an antiferromagnetic component in the ab plane has been proposed 11,13 . Nevertheless, no evidence of antiferromagnetism has been observed via neutron diffraction experiments 14,15 . On the other hand, previous neutron diffraction measurements have yielded contradictory results regarding the direction of the ferromagnetic moments in Sr 4 Ru 3 O 10 14,15 . It is initially argued that spins are aligned in the ab plane based on the observation of (0 0 L)-type magnetic Bragg peaks and no anomaly has been found at T * 14 . As a result, the feature at T * in magnetic susceptibility measurements is ascribed to a magnetic domain process 14 . In contrast, a distinct magnetic easy axis has been proposed more recently, where the magnetic moments are determined to be along the c axis since the (0 0 L)-type magnetic Bragg peaks are found absent. In addition, a kink-like feature has been reported in the temperature dependence of the Bragg reflection (2 0 0) at T * , which is also claimed to be observed in all other reflections 15 . Therefore, the easy axis of the magnetic moments and the nature of the anomaly at T * of Sr 4 Ru 3 O 10 remain elusive.
Another intriguing property of Sr 4 Ru 3 O 10 is the field-induced metamagnetic transition. While a typical magnetic hysteresis loop of ferromagnets is observed as the magnetic field is applied along the c axis, a first-order metamagnetic transition is seen below T * for the field applied in the ab plane 10,11 . Experimental signatures of the electronic phase separation during the phase transition have been observed 16 . Nevertheless, previous neutron diffraction studies reveal controversial features regarding this metamagnetic transition. Across the critical field, while no anomaly has been observed in the field dependence of the magnetic reflection (0 0 8) in ref. 14 , a field-induced transition at the same wave vector is reported in ref. 15 . In addition, it has been reported that the metamagnetic transition is accompanied by a change in the lattice, as evidenced in Raman scattering 13 , neutron diffraction 15 and magnetostriction studies 17 , which suggests the presence of strong spin-lattice coupling in this system. To date, the nature and the origin of this metamagnetic transition are yet to be resolved.
In this paper, we present a single-crystal neutron diffraction study of the magnetic order and the metamagnetic transition in Sr 4 Ru 3 O 10 . We find that the material orders ferromagnetically at T c ≈ 100 K without any signature of antiferromagnetic components, in agreement with the previous studies 14,15 . However, the magnetic moments are found to possess both in-plane and out-of-plane components below T c , in contrast to the previous neutron diffraction studies where the spins are proposed to align either in the ab plane 14 or along the c axis 15 . In addition, we have observed spin reorientations below T * ≈ 50 K, where the magnetic moments incline toward the out-of-plane direction. Furthermore, we show that while the magnetic moments are nearly along the c axis at T = 1.5 K, a magnetic field applied along the in-plane [1 -1 0] direction gives rise to a spin reorientation at a critical field B c , above which there exists a spin component perpendicular to the plane defined by the field direction and the c axis, in accordance to the metamagnetic transition. This observation is distinct from the previous studies, where the metamagnetic transition is ascribed to magnetic domain processes 14 or a coexistence of zero-field c-axis component and the field-induced in-plane spin polarization 15 . Both temperature-and magnetic-field-driven spin reorientations are presumably ascribable to the change in the magnetocrystalline anisotropy that is strongly correlated with the RuO 6 octahedral distortion. Our results reconcile the inconsistency among previous neutron diffraction studies on the magnetic easy axis of Sr 4 Ru 3 O 10 , resolve the puzzling character of the transition at T * , and elucidate the nature of the field-induced metamagnetic transition.

Results
We start with the zero-field neutron diffraction data collected at the HB-3A four-circle diffractometer. Figure 2(a) and (b) present the rocking curves of Bragg reflections (0 0 2) and (1 1 1) at T = 8, 50 and 100 K, respectively. Note that both reflections meet the requirements of nuclear Bragg diffraction. The intensity of (0 0 2) shows a nonmonotonic behavior, which is significantly enhanced at 50 K compared to that measured at 8 K and 100 K; in contrast, the (1 1 1) intensity becomes stronger with decreasing temperature. These observations suggest that there is magnetic intensity superimposed on the nuclear Bragg peaks. To further elucidate the nonmonotonic temperature dependence of the magnetic intensity, Fig. 2(c) and (d) present the peak intensity of (0 0 2) and (1 1 1) Bragg reflections as a function of temperature, respectively. Upon cooling, additional reflection intensity in both peaks emerges below 100 K. Similar enhancement has been observed in other nuclear Bragg peaks including (0 0 6) and (0 0 8), indicating the development of a ferromagnetic order below T c = 100 K which is in line with the magnetic susceptibility measurements 10 . Consistent with Fig. 2(a), the intensity of (0 0 L)-type magnetic reflections, L = 2, 6 and 8, also exhibits nonmonotonic temperature dependence, with a maximum at T * ≈ 50 K that is reminiscent of the anomaly at T * observed in the magnetic susceptibility data 10,11 . In accordance to Fig. 2(b), the magnetic intensity at (1 1 1) emerges at T c and continues to increase below T * , which suggests a spin reorientation at T * . On the one hand, the observation of (0 0 L)-type magnetic reflections is in agreement with the previous neutron diffraction experiments reported in ref. 14 , but is different from that in ref. 15 where they are found to be absent. On the other hand, we have revealed distinct temperature-dependent behaviors of the intensity of (0 0 L) and (1 1 1), which is in sharp contrast to the previous studies where either no anomaly was observed 14 , or a kink feature was claimed to exist in the temperature dependence of all reflections 15 . Note that the HB-3A diffractometer is equipped with a two-dimensional neutron detector, from which the change in the scattering angle 2θ of the diffracted neutrons at different temperatures is found too small to be resolved due to the instrumental resolution limitation. This is in agreement with a change in the lattice constants of only ~0.04% upon cooling observed in the previous studies 15 . Therefore, the variation in the peak intensity of these Bragg peaks cannot be ascribed to the structural change, but is magnetic in origin.
In order to explore the possible magnetic configurations in Sr 4 Ru 3 O 10 , we carried out the magnetic representational analysis using the program SARAh 18 . The space group of the crystal structure is No. 55 Pbam and the propagation vector is k = (0 0 0). There are four inequivalent Ru atoms in a chemical unit cell which are located at Ru1 (0 0 0), Ru2 (0 0 0.1402), Ru3 (0.5 0 0.3598) and Ru4 (0.5 0 0.5). The analysis shows that an in-plane antiferromagnetic component would give rise to strong magnetic reflections at (1 0 0), (1 0 1) or (0 1 1), which, nevertheless, are all absent in our measurements. Therefore, the feature at T * in the magnetic susceptibility cannot be ascribed to the formation of antiferromagnetic components in the ab plane that was claimed previously 11,13 . As a result, the only symmetry-allowed magnetic configurations in Sr 4 Ru 3 O 10 is the ferromagnetic structure with the magnetic moments either along the c axis or in the ab plane. Since neutrons couple to the magnetic moment perpendicular to the momentum transfer q, the observation of magnetic intensities at (0 0 L), L = 2, 6 and 8, at T * < T < T c suggests the development of a magnetic order with spins aligned in the ab plane. This, combined with the magnetic susceptibility measurements which show an easy axis closer to the c axis, suggests that the magnetic moments have both in-plane and out-of-plane components in this temperature regime. Note that at T * < T < T c , the intensity of (0 0 2) increases more rapidly than that of (1 1 1), which implies that the magnetic moments incline toward the ab plane, though the easy axis is always closer to the c axis. Below T * , the decrease in the intensity of (0 0 L)-type magnetic peaks suggests that the in-plane ferromagnetic moment reduces and the spins incline continuously toward the out-of-plane direction upon cooling. This speculation is supported by the observation of an increase in the reflection intensity of the (1 1 1) Bragg peak below T * , as the direction of the (1 1 1) wave vector is almost perpendicular to the c axis (note that a * ≈ b * ≈ 1.137 Å −1 , c * ≈ 0.219 Å −1 ). We have collected the nuclear and magnetic Bragg peaks at 8 K, 50 K and 100 K and attempted to perform Rietveld analysis using the program FullProf 19 . Note that the magnetic moments of the outer-layer Ru ions (Ru1 and Ru4) and the central-layer ones (Ru2 and Ru3) can be different, owing to the difference in the direction and magnitude of the RuO 6 octahedral rotation 10 . At T = 8 K, due to the very weak intensity of the (0 0 L) peaks, the spins are nearly aligned along the c axis. Since only the (1 1 1) peak is associated with this spin component, the magnitude of the magnetic moments of outer-and central-layer Ru ions cannot be exclusively determined at this temperature. Similarly, more magnetic reflections are required in order to determine the moment sizes and the canting angles quantitatively at T * . The schematic of the magnetic structure at zero field is plotted in Fig. 1(b).
Next, we discuss the neutron diffraction measurements done on the HB-1A triple-axis spectrometer with the magnetic field applied along the in-plane [1 -1 0] direction, where a first-order metamagnetic transition is observed in the magnetic susceptibility measurements 10 . Figure 3(a) and (b) display the peak intensity of (0 0 2) and (1 1 1) Bragg peaks as a function of the magnetic field at T = 1.5 K. It is remarkable that the (0 0 2) Bragg peak intensity increases significantly at B c = 1.5 T, whereas that of the (1 1 1) reflection decreases. These observations are distinct from the previous studies where no anomaly has been observed in the intensity of magnetic reflections as a function of magnetic field 14 . It is worth noting that in another neutron diffraction study a similar trend to (0 0 2) [ Fig. 3(a)] has been observed in the field dependence of the intensity of (0 0 8) Bragg peak 15 . However, a slight reduction in the scattering intensity of (H K 0) reflections was claimed and suggested to be associated with the magnetic moment along the c axis, which led the authors to argue that the field-induced transition is due to a coexistence of the initial zero-field ferromagnetic order with a field-induced in-plane spin polarization 15 . If this were the case, one would expect a slight increase in the magnetic intensity of (1 1 1) reflection above the critical field B c , as (1 1 1) is perpendicular to [1 -1 0] but is about ~82.2° with respect to the [0 0 1] direction. This is not in line with our observation of the field-induced decrease in the intensity of (1 1 1) reflection, which suggests that the applied magnetic field gives rise to a spin component perpendicular to the plane defined by the field direction and the c axis. This is supported by a recent study which, by measuring the magnetic moment vectors, reports the existence of a magnetic moment component that is perpendicular to the field rotation plane 20 . Note that the slight difference in the critical field compared with the magnetic susceptibility data may arise from different applied magnetic field directions in the ab plane. To further support the variation of the magnetic scattering intensity below and above B c , Fig. 3(c) and (d) plot the θ-2θ scans over (0 0 2) and (1 1 1) Bragg peaks measured at T = 1.5 K with B = 0 and 3.5 T, respectively. We can clearly see that above B c the intensity of (0 0 2) peak becomes enhanced while that of the (1 1 1) peak is suppressed. Furthermore, there is no noticeable change in the 2θ values of both peak positions across the field-induced magnetic transition, which is consistent with the previous study that the lattice constants a and c change only by ~0.04% at the critical field that is beyond the instrumental resolution in our study 15 . For comparison, in the inset of Fig. 3(d), we present the θ-2θ scan of a high-|Q| nuclear peak (0 0 16), where the magnetic intensity is expected to be very weak due to the small magnetic form factor. The very little difference in the intensity of (0 0 16) at 0 and 3.5 T magnetic field further substantiates that the field-induced intensity variation at low-|Q| peaks (0 0 2) and (1 1 1) are magnetic in origin, rather than a structural change. Figure 4(a) and (b) show the θ-2θ scans of (0 0 2) and (1 1 1) Bragg peaks measured at T = 50 K with B = 0 and 3.5 T, respectively. Both (1 1 1) and (0 0 2) scattering intensity displays negligible changes upon applying the magnetic field. These results, combined with the features observed at low temperature shown in Fig. 3, are in agreement with the magnetic susceptibility measurements that the metamagnetic transition only occurs below T * = 50 K.

Discussions
The observation of a spin reorientation at T * in Sr 4 Ru 3 O 10 at zero field has elucidated the long-standing puzzle on the anomaly in magnetic susceptibility measurements 10 . This spin reorientation suggests a change in the magnetic easy axis as a function of temperature. This feature is reminiscent of the widely studied spin reorientation transitions in rare earth magnets and orthoferrites 21 , which are ascribed to the change in the magnetic anisotropy constants as a function of external parameters, such as temperature, magnetic field and pressure, etc 22 . Our finding raises an intriguing question: What is the underlying mechanism responsible for the change in the magnetic easy axis in Sr 4 Ru 3 O 10 ? It is known that in Ruddlesden-Popper type ruthenates, the magnetic anisotropy is strongly coupled to the structural distortions of the RuO 6 octahedra. For instance, magnetocrystalline anisotropy has been extensively investigated in SrRuO 3 thin films, where the magnetic anisotropy and the saturated magnetic moments can be readily tuned by controlling the RuO 6 octahedral distortion via the epitaxial strain imposed by substrates 23 . Considering the fact that the ferromagnetic state with T c ≈ 100 K in triple-layer (n = 3) Sr 4 Ru 3 O 10 bridges the physics of the double-layer (n = 2) Sr 3 Ru 2 O 7 (Fermi liquid, ferromagnetic instability) 2 and the three-dimensional (n = ∞) SrRuO 3 (ferromagnetic, T c = 160 K) 24 , the change in the magnetocrystalline anisotropy in Sr 4 Ru 3 O 10 across T * may arise from the corresponding changes in lattice structures. Indeed, anomalies  in the lattice constants with c expanding while a contracting below T * at zero field have been observed previously 15,17 . These changes in lattice parameters are expected to alter the occupancy of the Ru t 2g orbitals in this multiband system, which consequently affects the magnetic anisotropy through spin-orbit coupling 25 .
The nature of the in-plane field-induced metamagnetic transition in Sr 4 Ru 3 O 10 below T * , which has been an unresolved puzzle, can now be readily understood. It is also due to a spin reorientation where the magnetic moments reorient from the nearly c axis toward the ab plane. Intriguingly, we find that there exists a magnetic moment component perpendicular to the plane defined by the field direction and the c axis above the critical field, which is consistent with the observation of a recent study where it has been ascribed to the Dzyaloshinskii-Moriya interaction assuming that the easy axis is along the c axis 20 . On the other hand, it is worth noting that across the field-induced metamagnetic transition below T * the lattice parameter a expands while c shrinks 15 , a trend which is opposite to that observed upon cooling through T * at zero field discussed above 15,17 . This suggests that the magnetic anisotropy is altered above the critical field due to the strong spin-lattice coupling, which needs to be taken into account in the future theoretical modeling.
In summary, the magnetic structure and the metamagnetic transition in the triple-layer ruthentate Sr 4 Ru 3 O 10 (n = 3) are revisited by neutron diffraction measurements. The magnetic order below T c is found to be ferromagnetic with both in-plane and out-of-plane spin components, while it evolves toward the c axis below T * = 50 K. This finding elucidates the long-standing puzzle of the nature of the magnetic transition at T * in Sr 4 Ru 3 O 10 . Below T * , upon the metamagnetic transition with a magnetic field applied along the in-plane [1 -1 0] direction, the magnetic moments undergo a spin reorientation from the c axis toward the ab plane with a spin component perpendicular to the plane defined by the magnetic field direction and the c axis. Both temperature-and field-induced spin reorientations can be attributed to the change of magnetocrystalline anisotropy via spin-orbit coupling due to the change in the RuO 6 octahedral distortion.

Methods
The single-crystal Sr 4 Ru 3 O 10 was grown by the floating zone method 26 . The lattice constants are a = 5.5280 Å, b = 5.5260 Å and c = 28.651 Å. The phase and quality of the crystal have been verified by x-ray diffraction measurements. The magnetization as a function of both temperature and magnetic field has been measured using SQUID and the results are in agreement with previous studies 10,11 . A very small amount of intergrowth of SrRuO 3 has been revealed in the magnetic susceptibility measurements, which is easy to be separated in the neutron diffraction measurements due to its very different lattice parameter c (c = 7.8446 Å in orthorhombic unit cell). Zero-field and field-dependent neutron diffraction measurements were performed using the HB-3A four-circle diffractometer (λ = 1.5426 Å) and the HB-1A triple-axis spectrometer (λ = 2.36 Å) respectively at High Flux Isotope Reactor in Oak Ridge National Laboratory. At HB-3A, the sample was mounted on an aluminum stick and loaded into a closed-cycle Helium displex. At HB-1A, the sample was oriented in the horizontal (H H L) scattering plane and loaded into a vertical-field cryomagnet such that the magnetic field is applied along the [1 −1 0] direction. The Bragg reflections (H K L) were in the reciprocal lattice units 2π/a, 2π/b and 2π/c of the orthorhombic space group Pbam (No. 55) 10 .

Data availability.
The data that support the findings of this study are available from the corresponding author upon reasonable request.