Anisotropic Spin Stripe Domains in Bilayer La$_3$Ni$_2$O$_7$
N. K Gupta R. Gong Y. Wu M. Kang C. T. Parzyck B. Z. Gregory N. Costa R. Sutarto S. Sarker A. Singer D. G. Schlom K. M. Shen D. G. Hawthorn

































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Anisotropic Spin Stripe Domains in Bilayer La_3Ni_2O_7
    N. K Gupta,^1,† R. Gong,^1,† Y. Wu,^2,† M. Kang,^2,3,4 C. T. Parzyck,^2 
 
 B. Z. Gregory,^2,3 N. Costa,^1 R. Sutarto,^5 S. Sarker,^6 A. Singer,^3 D. G. Schlom,^3,4,7 

K. M. Shen,^2,4∗ and D. G. Hawthorn^1∗

^1Department of Physics and Astronomy, University of Waterloo,
 
Waterloo, N2L 3G1, Canada

^2Laboratory of Atomic and Solid State Physics, Department of Physics,
 
Cornell University, Ithaca, NY 14853, USA

^3Department of Materials Science and Engineering,
 
Cornell University, Ithaca, NY 14853, USA

^4Kavli Institute at Cornell for Nanoscale Science,Cornell University,

 Ithaca, NY 14853, USA

^5Canadian Light Source, Saskatoon, Saskatchewan, S7N 2V3, Canada

^6Cornell High Energy Synchrotron Source, Cornell University,

Ithaca, NY 14853, USA

^7Leibniz-Institut für Kristallzüchtung, Max-Born-Straße 2, 

12489 Berlin, Germany



^∗To whom correspondence should be addressed;

†These authors contributed equally to this work.

E-mail:  kmshen@cornell.edu and david.hawthorn@uwaterloo.ca.

    
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The discovery of superconductivity in under pressure has motivated the investigation of a parent spin density wave (SDW) state which could provide the underlying pairing interaction. Here, we employ resonant soft x-ray scattering and polarimetry on thin films of bilayer to determine that the magnetic structure of the SDW forms unidirectional diagonal spin stripes with moments lying within the NiO_2 plane and perpendicular to Q_SDW, but without the strong charge disproportionation typically associated with other nickelates. These stripes form anisotropic domains with shorter correlation lengths perpendicular versus parallel to Q_SDW, revealing nanoscale rotational and translational symmetry breaking analogous to the cuprate and Fe-based superconductors, with Bloch-like antiferromagnetic domain walls separating orthogonal domains. 


The discovery of superconductivity with a transition temperature (T_c) above 80 K under hydrostatic pressure has ignited intense interest <cit.> in as a new platform to compare against the Cu- and Fe-based superconductors to understand which common ingredients are essential for realizing high T_c's. In both the cuprates and Fe-based families, parent antiferromagnetic states as well as nematicity are both known to play critical roles in their phase diagrams, the detailed understandings of which are essential for developing microscopic models. Presently, our understanding of the parent state from which the superconductivity condenses in remains incomplete. Recent resonant inelastic x-ray scattering (RIXS) <cit.>, heat capacity and magnetic susceptibility <cit.>, nuclear magnetic resonance (NMR)<cit.> and μSR <cit.> measurements of bulk suggest the presence of a spin density wave (SDW) transition occurring around  T_SDW≈ 150 K, with a wavevector of (H = 1/4, K = 1/4) deduced from RIXS.<cit.> However, essential information regarding the microscopic spin structure and orientation remains unresolved.


Here, we present detailed resonant soft x-ray scattering (RSXS) and polarimetry measurements on epitaxial thin films of bilayer which reveal that the detailed spin structure consists of equal domains of bicollinear spin stripes, with the spins lying in the NiO_2 plane and aligned perpendicular to the SDW wavevector. Temperature-dependent spectroscopic measurements support a scenario with only a single electronic Ni site, with no evidence for charge or bond disproportionation across T_SDW, and where the oxygen ligand holes are strongly hybridized with the Ni d orbitals. The SDW domains exhibit a surprising anisotropy in their correlation lengths parallel versus perpendicular to their wavevector, suggesting that nanoscale rotational symmetry breaking might also play an important role in , reminiscent of the nematic order found in the cuprates and Fe-based superconductors. Finally, a detailed polarization and temperature-dependent analysis of the SDW Bragg peaks suggests the presence of extended, Bloch-like domain walls between orthogonal domains. 

One of the challenges of investigating is the recent discovery of multiple structural polymorphs, including the expected n = 2 Ruddlesden-Popper structure comprised of bilayers of NiO_6 octahedra (dubbed “2222”), and a surprising naturally-formed superlattice of alternating single-layer and trilayer blocks of NiO_6 octahedra (dubbed “1313”) <cit.>. It is plausible that both polymorphs could exist within a single macroscopic bulk crystal sample, and both structures have been reported to be the superconducting phase <cit.>. Layer-by-layer reactive oxide molecular beam epitaxy (MBE) synthesis of thin films offers a solution to addressing this polymorphism. Here, we focus our investigation on phase-pure epitaxial thin films of the bilayer “2222” polymorph of . Epitaxial thin films of 16 nm thickness were grown on NdGaO_3 (110) substrates using reactive oxide molecular beam epitaxy with shuttered deposition at a pressure of 8 × 10^-7 torr of 80 % distilled ozone and a temperature of 800^∘ C. Samples were characterized by lab-based x-ray diffraction, electrical transport, and synchrotron-based hard x-ray diffraction (see Supplementary Information). Multiple samples were synthesized and investigated and all exhibited qualitatively similar behaviors; data from two samples are shown in this manuscript.





In Figure <ref>b, we show RSXS measurements on the Ni L_3 edge in π-polarization along the (1, 1, 1.85) direction for a series of temperatures. Here, we employ the pseudo-tetragonal lattice constants of a = b = 3.84 Å and c = 20.44 Å. Below 160 K, a sharp peak emerges at (1/4, 1/4) whose intensity is approximately 45 times stronger in π versus σ polarization at 20 K, suggesting that the scattering originates from magnetic rather than charge ordering, consistent with previous measurements.<cit.> The resonance energy dependence of the peak at (1/4, 1/4, 1.86) is shown in Fig. <ref>c, together with the x-ray absorption spectrum (XAS), which demonstrates that the Bragg peak is peaked only at the Ni L resonances, and not on the La M_4 edge or off-resonance. In Figure <ref>f, we show a scan as a function of L at the in-plane (1/4, 1/4) wavevector on the Ni L_2 edge, corrected for absorption and geometric factors (see Supplementary Information). The data indicate that the SDW Bragg peak is peaked at L = 2, but with only a 14 ± 4 Å correlation length along the c-direction, revealing the highly two-dimensional nature of the magnetic order in . We note that both the resonance energy and L dependence of SDW Bragg peak clearly establish that the SDW intrinsically originates from and not an impurity phase, as has plagued the  identification of ostensible charge or spin density order in other related nickelates.<cit.>

In most related nickelates in the Ruddlesden-Popper sequence, including n = 1 La_2-xSr_xNiO_4 and n = ∞ perovskite RENiO_3 (RE = Nd, Pr, Sm), SDW order coincides with strong charge or bond disproportionation, where the local electronic and orbital environment of the Ni sites is strongly modulated at an atomic scale. Whether such behavior also occurs in remains an important open question that can be addressed by resonant x-ray diffraction and spectroscopy. First, there are no distinct changes in the Ni L edge XAS spectra across  (Figure <ref>e), unlike the situation of RENiO_3 where bond disproportionation results in spectroscopically distinct Ni sites in the Ni L edge XAS across the metal-insulator transition. <cit.> Second, a strongly varying charge density will also modulate the x-ray scattering form factors on the different Ni sites, resulting in the linear dichroism of the SDW peak intensity (σ versus π polarization) becoming energy dependent, as reported in RENiO_3 heterostructures <cit.>. In contrast, we find that the linear dichroism of the peak in does not exhibit any observable energy dependence (Figure <ref>d), suggesting that only a single magnetic Ni site exists in the SDW state. These spectroscopic observations are consistent with transport and thermodynamic measurements which exhibit only weak anomalies across , indicating that the dominant order parameter is magnetic and that unlike many other nickelates, any associated charge modulation is either weak or absent.










§ MEASUREMENT OF SPIN CONFIGURATION


Having demonstrated the absence of strong charge modulations, we now determine the orientation of the staggered moments deep within the SDW state.  For this we make use of the sensitivity of the intensity of resonant scattering to the orientation of the photon polarization relative to the magnetic moments, analogous to polarized neutron scattering.<cit.>. The resonant elastic x-ray scattering cross-section is given by:<cit.>

    I^cr(ϵ_i, ω, Q) ∝|ϵ_out^*·(∑_j F_j(ħω, Q) e^iQ·r_j)·ϵ_in|^2,

where ħω is the photon energy, Q= k_out - k_in is the momentum transfer and ϵ_in and ϵ_out are the incident and scattered x-ray polarization, respectively. F_j is a tensor that encodes the photon energy dependence of the scattering cross-section for site j in the lattice, and its elements depend on the orientation of the magnetic moment at site j.<cit.> (see Supplementary Information) Assuming spherical symmetry of the local valence charge density, consistent with the linear dichroism of the SDW peak being energy-independent (Fig. <ref>d), the scattering depends on the orientation of the staggered Δm⃗, as 

    ∑_j F_j(ħω, Q) e^iQ·r_j∝
    [
    0    Δ m_[[001]    -Δ m_[110]
    
    -Δ m_[001]    0    Δ m_[-110]
    Δ m_[110]    -Δ m_[-110]    0 ],

where Δ m_u are the components of the staggered moments in three orthogonal directions [001], [-110] and [110].  

These components can then be deduced by varying the alignment of ϵ_in relative to the [001], [-110] and [110] directions. In an experiment, this is achieved by rotating the sample azimuthally by an angle ϕ about an axis normal to the [1/4, 1/4, L] set of lattice planes (here L is 1.93), as depicted in Figure <ref>a, which is accomplished by mounting the c-axis normal film on a 43.7^∘ wedge. This approach enables the orientation of the moments to be rotated relative to the incident polarization, which can be set to be π, σ, circular or linear 45^∘ (π - σ), with the scattering wavevector remaining centered on the (1/4, 1/4, 1.94) peak. As shown in Figure <ref>b, the scattering intensity has a strong dependence on the azimuthal angle, ϕ as well as on the polarization of the incident light. In Figure <ref>c, we plot the SDW peak amplitude as a function of azimuthal angle ϕ, for our various incident polarizations. Notably, the peak intensity exhibits a complex, non-monotonic dependence with ϕ and large variations with incident polarization.  This azimuthal dependence can be compared to simulations of the staggered moments oriented along different directions, calculated using Equations <ref> and <ref>, the sample geometry, and a correction for the geometry-dependent absorption of the incident and scattered x-rays (Supplementary Information).  As shown in Figure <ref>c the measurements show remarkable agreement with the moments forming diagonal, bicollinear spin stripes (Figure <ref>d) with the magnetic moments lying entirely within the a-b plane but oriented perpendicular to Q⃗_SDW. We emphasize this model was calculated without any free fitting parameters, apart from an overall scaling factor.

In contrast, the calculated polarization and ϕ dependence for other possible SDW scenarios (Figure <ref>f and g), including Δ m out of the plane, Δ m ∥Q⃗_SDW, or a non-collinear configuration, all bear no qualitative resemblance to the experimental data (Figure <ref>c). Having constrained the orientation of moments within the NiO_2 planes, we can now consider the full 3D magnetic structure, which is informed by the L dependence of the SDW scattering cross-section. The fact that the SDW order is peaked at L = 2 and is minimal at L = 1.5, (Fig. <ref>f) indicates that with the coupling within the bilayer (intra-bilayer coupling) and between bilayers (inter-bilayer coupling) are either both ferromagnetic or both antiferromagnetic (see Supplemental Information). Given that the experimental magnon dispersion is well described by a model with a large AF bilayer coupling,<cit.> this suggests the magnetic structure is consistent with the one depicted in Figure <ref>e, with both inter- and intra-bilayer AF coupling. We note that this bicollinear double spin-stripe  configuration is notably different from other nickelates, including the non-collinear spin-spiral magnetic order in bulk RENiO_3 <cit.>, the collinear order with moments parallel to (1/4, 1/4, 1/4) in ultrathin RENiO_3, or the reported magnetic order, with moments along c, in La_2Ni_2O_5 <cit.> It is, however, remarkably similar to the bicollinear double spin stripe observed in FeTe <cit.>.



§ ANISOTROPIC, UNIDIRECTIONAL MAGNETIC DOMAINS


The unidirectional, stripe-like character of the antiferromagnetic order is also manifest in the shape of the SDW domains which break the rotational symmetry of the lattice. We measured the shapes and intensities of the SDW Bragg peaks in the H-K plane for both sets of SDW domains with orthogonal Q⃗ vectors, around (1/4, 1/4, L) (in red) and (1/4, -1/4, L) (in blue) using a two-dimensional microchannel plate detector, shown in Fig. <ref>a. The Bragg peaks around (1/4, 1/4, L) and (1/4, -1/4, L) have equal intensities to within experimental uncertainty (± 3 %), indicating an equal population of domains.  In both sets of domains, the SDW peaks have highly anisotropic shapes, with a 2D Lorentzian fit giving correlation lengths that are much longer parallel to Q⃗_SDW (ξ_∥ = 292 Å) versus perpendicular to Q⃗_SDW (ξ_⊥ = 134 Å).



This peak shape is consistent with anisotropic domains of unidirectional SDW order, with each domain characterized by either (1/4, 1/4) or (-1/4, 1/4) order, as depicted in Figure <ref>d. This is in contrast to anisotropic domains of bidirectional order or isotropic domains of unidirectional order, which would exhibit x-shaped or isotropic peak shapes respectively.  This anisotropy could be associated with orthorhombic structural twin domains or may be orthogonal domains of unidirectional order occurring within a single structural domain.  Intriguingly, this latter scenario is reminiscent of the anisotropic charge ordering reported in underdoped YBa_2Cu_3O_7-δ and Bi_2Sr_2-xLa_xCuO_6+δ, where the CDW Bragg peaks likewise exhibit an anisotropy, with longer correlations lengths parallel to the CDW wavevector, indicative of orthogonal anisotropic unidirectional CDW domains within a single orthorhombic structural domain.<cit.>



§ PROBING MAGNETIC DOMAIN WALLS


In Figure <ref>, the orientation of the staggered moment at 20 K was determined to be almost entirely within the a-b plane and perpendicular to Q⃗_SDW. Insights may be gleaned from the small discrepancies between the model and measurements. In particular, measurements using σ polarization at ϕ = 0 are at a minimum in intensity for this magnetic orientation, with deviations of the staggered moment parallel to Q⃗_SDW or along [001] required to provide a finite scattering intensity. We now leverage this sensitivity of the polarization to the staggered moment orientation to investigate how the spin configuration evolves with temperature. In Figure <ref>b, we show a comparison between the temperature dependence of the SDW peak when measured with π versus σ polarization. At low temperatures (T < 50 K), the intensity of the SDW peak measured in π polarization, I_π, is more than an order of magnitude stronger than in σ polarization, I_σ. Upon raising the temperature to T_SDW, I_π, smoothly decreases whereas I_σ exhibits a non-monotonic temperature dependence, growing in intensity before peaking around 130 K, and then falling rapidly to zero at T_SDW. On the other hand, the ratio of I_σ / I_π, shown in figure <ref>c grows smoothly and monotonically with increasing temperature all the way up to T_SDW when the peak vanishes, indicating that the component of the spins oriented away from the low T configuration (Figure <ref>c) grows with increasing temperature, with I_σ/I_π > 0.6 near T_SDW. 



In addition to this anomalous temperature dependence, the width of the SDW Bragg peak is surprisingly broader when measured with σ versus π incident polarization (Fig. <ref>a and d). This indicates that the out-of-plane moments have a shorter correlation length than the in-plane moments, which would be inconsistent with a uniform, temperature-dependent canting of all the spins. Instead, this could be consistent with the existence of real-space defects of the magnetic order, such as Bloch-like domain walls or antiferromagnetic skyrmion-like topological defects. Indeed, Bloch-like domain walls should naturally exist between two orthogonal unidirectional domains, where the transition between domains would necessitate a reorientation of the spins by 90^∘ from one domain to its orthogonal counterpart, which could occur over an extended region, depending on the relative magnitude of the magnetic anisotropy and exchange terms. 

In the Bloch domain wall scenario, the staggered moments would rotate away from [110] or [-110] and out of the a-b plane, as illustrated in Figure <ref>e. Here, I_π would probe the bicollinear spin stripe regions within the core of each domain, while I_σ would be sensitive to the rotated spin component in the domain walls. The relative volume fraction of domain walls would grow with increasing temperature until SDW order is lost at T_SDW. This interpretation could explain the broader peak for I_σ when compared to I_π, as shown in Fig. <ref>d. Finally, the influence of the domain walls may also be the source of the small deviations from a perfectly in-plane moment depicted in Fig. <ref>c. 

This analysis represents a new approach to detecting defects in antiferromagnetic order.  Unlike ferromagnetic domain walls, antiferromagnetic domain walls are often difficult to detect.<cit.> Further analysis of the polarization and temperature dependence of the (1/4 1/4) SDW peaks in may provide key insights into the width, density and detailed magnetic configuration of the domain walls. Such investigations may be of key importance as the magnetic configuration within the domain walls may represent the melting of long-range SDW order as the material enters its superconducting phase, since a similar phenomenology also exists in the cuprates. 



§ DISCUSSION


A comparison between these experiments with those on bulk crystals <cit.> reveal a number of commonalities, including similar T_SDW's, temperature dependences, photon polarization dependence, and peak widths along the (H H L) direction, suggesting that the results reported here are universal to both bulk and thin film samples.

The structure of the bicollinear spin stripe order in bears a strong resemblance to the magnetically ordered state in FeTe, but with a major distinction: in FeTe, the spin stripe ordering is accompanied by a monoclinic structural transition that stabilizes the magnetic order.<cit.> In contrast, temperature-dependent, synchrotron-based hard x-ray diffraction measurements of our thin films do not reveal any lowering of structural symmetry or the appearance of superlattice peaks upon entering the SDW state (see Supplementary Information). Furthermore, the existence of domain walls or topological defects indicate that both orthogonal magnetic domains occur within a single NiO_2 plane. This may imply an inherent instability in to rotational symmetry breaking, possibly accompanied by disordered or frustrated nematic order,<cit.> analogous to the Fe-based and cuprate superconductors where unidirectional density wave and nematic orders are pervasive and closely intertwined with superconductivity. <cit.> These close analogies with the cuprates and Fe-based superconductors suggest that the melting of this SDW state, via the likely formation of Bloch-like domain walls, is directly related to the formation of the superconducting state under high pressure. 














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§ ACKNOWLEDGMENTS


This work was supported by the the Air Force Office of Scientific Research (Grant No. FA9550-21-1-0168, FA9550-23-1-0161), the National Science Foundation through Grant No. DMR-2104427 and the Platform for the Accelerated Realization, Analysis and Discovery of Interface Materials (PARADIM) under Cooperative Agreement No. DMR-2039380, the U.S. Department of Energy, Office of Basic Energy Sciences, under contract no. DE-SC0019414, and the Natural Sciences and Engineering Research Council (NSERC) of Canada. Additional support for materials synthesis was provided by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant Nos. GBMF3850 and GBMF9073. N.K. Gupta acknowledges support from the Waterloo Institute of Nanotechnology (WIN). This work is based on research conducted at the Center for High-Energy X-ray Sciences (CHEXS), which is supported by the National Science Foundation (BIO, ENG and MPS Directorates) under award DMR-1829070. Part of the research described in this paper was performed at the Canadian Light Source, a national research facility of the University of Saskatchewan, which is supported by the Canada Foundation for Innovation (CFI), the Natural Sciences and Engineering Research Council (NSERC), the National Research Council Canada (NRC), the Canadian Institutes of Health Research (CIHR), the Government of Saskatchewan, and the University of Saskatchewan. Substrate preparation was performed in part at the Cornell NanoScale Facility, a member of the National Nanotechnology Coordinated Infrastructure, which is supported by the NSF (Grant No. NNCI-2025233); the authors would like to thank Sean Palmer and Steven Button for their assistance in substrate preparation and Michel Gingras for helpful discussions.



§ AUTHOR CONTRIBUTIONS

D.G.H. and K.M.S. conceived of, designed, and supervised the experiment. N.K.G. conceived of the experiment and identified anisotropy in the SDW peak shape. The resonant scattering experiments were performed by N.K.G., R.G., M.K., C.T.P., N.C., R.S., and D.G.H. Samples were synthesized and characterized by Y.W. with assistance from C.T.P, D.G.S., and K.M.S. Hard x-ray scattering measurements were conducted and analyzed by B.Z.G., S.S. and A.S. The data were analyzed by N.K.G., R.G., C.T.P., K.M.S. and D.G.H. Model calculations were performed by D.G.H. The manuscript was written by D.G.H. and K.M.S. with input from all authors. 



§ COMPETING INTERESTS

The authors declare that they ave no competing interests.



§ DATA AND MATERIALS AVAILABILITY

All data needed to evaluate the conclusions in the paper are present in the manuscript and supplementary information.  Additional data available upon request.



§ SUPPLEMENTARY MATERIALS

Materials and Methods

Supplementary Text

Figs. S1 to S9


References





§ MATERIALS AND METHODS


Two samples A and B were measured for this study.  A more extensive set of data was measured for sample A. However, sample B was found to have comparable peak intensity, peak width, and temperature dependence as sample A, as well as reproducing the anisotropy in the width parallel and perpendicular to Q (see supplementary information).  

Resonant soft x-ray scattering measurements were performed at the Canadian Light Source REIXS beamline.<cit.>  The majority of the measurements were made using an energy resolved silicon drift detector, with measurements of the peak shape utilizing a 2D micro-channel plate detector.

The samples were oriented using the (001) Bragg peak of the NGO substrate at 2500 eV. The in-plane orientation was performed using (1/41/4 L) SDW Bragg peaks at 852 eV. The samples were found to have a c axis lattice constant of 20.44 Å from lab-based x-ray diffraction. From the location of the (1/41/4 L) peaks, √(a_T^2 + b_T^2) = 5.426(3) Å for sample A.

The scattering intensities, I_S, shown in Figures 1b (inset), 1c, 1d, 2b, 3c and 4, are deduced by subtracting the background intensity measured above the SDW transition (at 200 K or 220 K) from the measured intensity at lower temperature.  

For measurements with σ incident polarization, shown in Figure 4, the incident beam includes a small ≃ 1.5%π contribution due to synchrotron light from bending chicane magnets that is in addition to the primary σ polarized light from the EPU. In <ref>d, the full width at half maximum  (FWHM) with σ polarization is determined by fitting the data in Fig.<ref>a to a narrow peak from π polarized light, with FWHM equal to that of a pure π polarization (blue curve in Fig. <ref>d), as well as broader peak with σ polarization (red curve in Fig. <ref>d).






















