Magnetism in artificial Ruddlesden-Popper iridates leveraged by structural distortions

We report on the tuning of magnetic interactions in superlattices composed of single and bilayer SrIrO$_3$ inter-spaced with SrTiO$_3$. Magnetic scattering shows predominately $c$-axis antiferromagnetic orientation of the magnetic moments for the bilayer justifying these systems as viable artificial analogues of the bulk Ruddlesden-Popper series iridates. Magnon gaps are observed in both superlattices, with the magnitude of the gap in the bilayer being reduced to nearly half that in its bulk structural analogue, Sr$_3$Ir$_2$O$_7$. We assign this to modifications in the anisotropic exchange driven by bending of the $c$-axis Ir-O-Ir bond and subsequent local environment changes, as detected by x-ray diffraction and modeled using spin wave theory. These findings explain how even subtle structural modulations driven by heterostructuring in iridates are leveraged by spin orbit coupling to drive large changes in the magnetic interactions.

We report on the tuning of magnetic interactions in superlattices composed of single and bilayer SrIrO3 inter-spaced with SrTiO3. Magnetic scattering shows predominately c-axis antiferromagnetic orientation of the magnetic moments for the bilayer justifying these systems as viable artificial analogues of the bulk Ruddlesden-Popper series iridates. Magnon gaps are observed in both superlattices, with the magnitude of the gap in the bilayer being reduced to nearly half that in its bulk structural analogue, Sr3Ir2O7. We assign this to modifications in the anisotropic exchange driven by bending of the c-axis Ir-O-Ir bond and subsequent local environment changes, as detected by x-ray diffraction and modeled using spin wave theory. These findings explain how even subtle structural modulations driven by heterostructuring in iridates are leveraged by spin orbit coupling to drive large changes in the magnetic interactions.

I. INTRODUCTION
Recent years have seen iridates, compounds composed of active Ir 5d orbitals in oxygen octahedra, emerge as an important new class of strongly correlated materials [1][2][3][4][5]. The combination of crystal field interactions and strong spin-orbit coupling generates narrow bands leading to insulating antiferromagnetic ground states that arise from modest values of the Coulomb repulsion U [2]. Many appealing structural and electronic analogies between iridates and lighter 3d-electron based cuprates have been identified [1,[6][7][8][9][10]. One crucial difference, however, is that iridates host spin-orbit coupled J eff = 1 2 magnetic moments, which have a more intricate coupling to orbital distortions than pure spin S = 1 2 moments [11]. This is borne out in observations: different magnetic ground states appear in iridates composed of similar Ir-O octahedra when they are subtly distorted or interspaced with different atoms. Sr 2 IrO 4 , hosting isolated IrO 2 layers, forms an ab-plane canted antiferromagnetic state [3,12], while Sr 3 Ir 2 O 7 , hosting isolated IrO 2 bi -layers, has c-axis collinear antiferromagnetic ordering [13,14]. Towards understanding how the structural modulations tailor magnetic ground states, resonant inelastic x-ray scattering (RIXS) has been extremely successful in quantifying the magnetic interactions present in iridate crystals [10,[15][16][17][18][19]. For example, a large (92 meV) spin gap was measured in Sr 3 Ir 2 O 7 , reflecting the substantial interlayer anisotropic coupling that causes a "dimensionality driven spin flop transition" with respect to Sr 2 IrO 4 [16,20].
Recently, artificial layered iridates in analogy to Ruddlesden-Popper iridates were realized via alternating layers of nSrIrO 3 and SrTiO 3 (nSIO/1STO), which showed a metal-insulator transition as a function of n, closely mirroring their bulk analogues [21,22]. However, an ab-plane canted antiferromagnetic state was argued to be maintained for n ≤ 3, suggesting that the spin flop transition is suppressed and breaking the analogy to bulk crystals [21]. To achieve this change in the ground state, anisotropic coupling between the two Ir layers in 2SIO/1STO, which favors the c-axis antiferromagnetic state, would need to be substantially modified compared to Sr 3 Ir 2 O 7 [14]. Further, this behavior is disputed by density functional theory (DFT) predictions that find caxis collinear magnetism in 2SIO/1STO [23,24]. Taken together, these conflicting results point to the need for direct observation of the exchange coupling and the resulting magnetic structure to unravel the impact of heterostructuring on the behavior of these proposed artificial analogues to the Ruddlesden-Popper iridates.
In this work, we directly probe the magnetic behavior of nSIO/1STO and extend the sensitivity of Ir L 3 RIXS to quantify the interactions that stabilize this state. We find a c-axis antiferromagnetic ground state in 2SIO/1STO, in contrast with an earlier report [21], demonstrating that the magnetic ground state mimics bulk Sr 3 Ir 2 O 7 . In both 1SIO/1STO and 2SIO/1STO, the magnetic excitation spectrum shows a clear dispersion with a magnon gap of 55 meV in n = 2, substantially reduced to about half that in bulk Sr 3 Ir 2 O 7 [16,20]. Based on modeling the magnetic dispersion, the predom- inately c-axis moments in 2SIO/1STO are stabilized by the anisotropic coupling between Ir-O planes as seen in Sr 3 Ir 2 O 7 . However, the lowering of the magnon gap evidences a significant reduction in the tetragonal distortion of the octahedra, while for 1SIO/1STO the gap size is similar to bulk Sr 2 IrO 4 [25]. The source of the modulation of the tetragonal distortion was determined to be substantial bending of the c-axis Ir-O-Ir bonds alongside changes in the local environment beyond the octahedra. This work establishes these artificial structures as true analogues to the Ruddlesden-Popper iridates, with the caveat that changes in the magnetic ground states are highly susceptible to subtle structural distortions [26]. These distortions push the heterostructure towards a quantum critical point between the ab-plane and c-axis antiferromagnets, exemplifying how magnetic states in iridates can be transformed in a tractable manner owing to their strong spin-orbit coupling. laser deposition using methods described in Ref. [27], as depicted in Fig. 1(a). High sample quality was verified by x-ray diffraction, x-ray magnetic circular dichroism, transport and magnetometry measurements (Fig. S1-S3 of [28]), consistent with previous studies [21,27]. Resonant elastic x-ray scattering (REXS), RIXS, and nonresonant diffraction data were taken at the 6-ID-B, 27-ID-B, and 33-BM-C beamlines of the Advanced Photon Source at Argonne National Laboratory. The RIXS energy resolution was 35 meV, full width half maximum. Further details are available in the Supplemental Material [28].

III. CRYSTAL AND MAGNETIC STRUCTURE
Previous investigations of nSIO/1STO with n = 1, 2, 3 displayed a net ferromagnetic moment for all samples, which was taken as evidence for the stabilization of canted ab-plane magnetic moments as in Sr 2 IrO 4 [21]. Although there is a strong consensus that this is valid for 1SIO/1STO, the result for 2SIO/1STO is more controversial as it breaks the analogy between n = 2 and Sr 3 Ir 2 O 7 , which has purely c-axis collinear antiferromagnetism implying no spontaneous net moment [14,21]. Theory also predicted c-axis moments and posited that the observed net ferromagnetic moment comes from oxygen vacancies [23,24,29]. Establishing the true magnetic ground state is of high importance towards extracting the magnetic exchange parameters that ultimately dictate the overall magnetic behavior of these heterostructures. In view of this controversy, we directly measured the spin ordering direction using azimuthal REXS scans, as was done in Sr 3 Ir 2 O 7 [16,30]. This dependence is shown for 2SIO/1STO in Fig. 1(b), left panel [31]. The calculated azimuthal dependence for c-axis oriented antiferromagnetic moments, shown as the grey line, matches the data well and establishes predominately c-axis moments [28]. To further emphasize this distinction, we also show the magnetic Bragg peaks where the maximum intensity for both cases is expected (-90 • ), and also where no intensity for in-plane moments is expected (7 • ), Fig. 1 (b), right panel. Clearly, a magnetic peak persists with integrated intensity that matches that expected for caxis orient moments (∼ 30%). These results then agree with theoretical predictions, showing the 2SIO/1STO SL maintains the same magnetic ground state as Sr 3 Ir 2 O 7 , strengthening the analogy to Ruddlesden-Popper series iridates [23]. Armed with the correct ground state, the magnetic excitation spectrum can be correctly interpreted, allowing the probing of the magnetic exchange couplings to unravel the true impact of artificial heterostructuring.
To directly probe the magnetic interactions, we utilize RIXS to map the magnetic dispersion of the established magnetic ground state. Although Ir L 3 -edge RIXS has been applied extensively to iridate crystals and thin films, a full 2D magnetic dispersion curve has never been characterized on SL heterostructures due to the relatively large (5 µm) x-ray penetration depth at the Ir L 3 edge [10,15,16,20,25,[32][33][34][35]. This challenge was overcome by growing relatively thick SLs (60 IrO 2 planes) and working near grazing incidence (1 • ) [28]. Raw RIXS spectra for the SLs are displayed in Fig. 2. Each spec-tra displays a high energy feature around 0.75 eV energy loss, corresponding to both an intra-t 2g orbital excitation and the e − h continuum [19]. A sharp peak arises at zero energy due to elastic scattering, along with a small phonon feature at around 40 meV. Finally, a dispersive feature from 50 to 140 meV is seen in all spectra, and is identified as the magnon excitation [36], with the higher energy tail including multimagnon excitations [10,15,16,32,34]. The spectra were fit using a combination of peaks in a similar approach to that used previously (see supplemental materials) [16,19,25,28,32]. Examples of fits along the nodal direction for each sample are displayed in Fig. 3(a). From these one can extract the energy, width, and integrated intensity of the magnetic excitation, Figs. 3(b). The intensity peaks at the magnetic ordering wave vector (0.5, 0.5) and the energy loss is within the bandwidth seen for Sr 2 IrO 4 and Sr 3 Ir 2 O 7 , corroborating our assignment of the feature as a magnetic type excitation [15,16].
From the extracted magnon dispersion, some important observations are immediately clear: (i) both SLs have nearly identical dispersion around the (0.25, 0.25) and (0.5, 0) points with maxima of ∼ 120 and 150 meV, respectively, (ii) both samples show magnon gaps. In the case of the 1SIO/1STO, the size of the gap is not well defined, being ∼ 11 − 36 meV, due to the worse reciprocal space resolution, 0.46Å −1 (0.073 r.l.u.) [28]. For 2SIO/1STO, a mask was used to improve the resolution to 0.12Å −1 (0.018 r.l.u.) for the (0.5, 0.5) and (0, 0) Q-points. Here, a larger gap is much better defined as between 50 and 60 meV at (0.5, 0.5) [28]. Compared with the dispersion for Sr 3 Ir 2 O 7 , Fig. 3(b), the overlap is very robust everywhere except at these minima [16]. Phase boundaries between the canted in-plane |xy and collinear out-of-plane |z orderings are displayed for single layer (blue) and bilayer iridates (red). The position for Sr3Ir2O7 (green triangle) was taken from [16]. Error bars are the statistical error (magnon gap range) of θ for 2SIO/1STO (1SIO/1STO).
We analyzed the origin of this anomalous behavior using linear spin wave theory [37] applied to the Hamiltonian described in Ref. [16], with ab-plane canted and c-axis collinear ground states for the single and bilayer SLs, respectively [16,28]. Importantly, the canted nature of the moments in 1SIO/1STO gives a dispersion relation that fundamentally differs from the out-of-plane Néel state [16,28]. As was the case for Sr 3 Ir 2 O 7 , the nine magnetic couplings can be parameterized in terms of: (i) the tetragonal distortion θ, defined by tan 2θ = 2 √ 2λ λ−2∆ , with spin-orbit coupling λ and tetragonal splitting ∆, (ii) η = J H U , with Hund's coupling J H and Coulomb repulsion U , (iii) the octahedral rotation angle α. The form of the exchange couplings in terms of these parameters is described in the supplemental (J ab , J ab , and J c are treated as free fit parameters) [14,16,28]. Concerning (ii), η = 0.24 was established for Sr 3 Ir 2 O 7 and is unlikely to change significantly, leaving the tetragonal distortion and rotation angles to explain the observed dispersion and magnon gaps [38].
Regarding α, Sr 2 IrO 4 and Sr 3 Ir 2 O 7 both feature large in-plane rotations (α = 12 • and 11 • respectively), but no tilts (rotations about a/b axes that bend the c-axis bond) [1,13,39]. Bulk-like SIO films, on the other hand, show substantial tilts and rotations implying that similar effects may be present in SLs [40]. We consequently tested for the presence of octehedral tilts and rotations by scanning the half order Bragg peaks locations. While an exact structural solution of the SLs is unfeasible due to the complex orthorhombic structure of SIO [40], pub- lished methods allow us to associate different half order reflections with different antiphase distortions [41][42][43].
We measured several reflections for the n = 1 and 2 samples and illustrate the important behavior in Fig. 4(a) [28]. The ( 1 2 , 3 2 , 3 2 ) reflection (left panel) arises from a combination of rotations and tilts; whereas the ( 1 2 , 1 2 , 3 2 ) reflection (right panel) comes from only tilts. Both peaks are of similar magnitude for n = 2, but the tilt-peak is suppressed by an order of magnitude in n = 1. This data suggests that both SLs have similar rotations of ∼ 8 • .
In contrast with the nearly straight c-axis bonds seen in Sr 3 Ir 2 O 7 [13], n = 2 likely hosts tilting of a similar size (of order 8 • ), while n = 1 has small, but finite tilts [44]. Most importantly, the presence of tilting generates the ab-plane ferromagnetic moment in 2SIO/1STO, observed experimentally [21,27], through canting the c-axis antiferromagnetic moments, resolving the conflict between previous experimental interpretations and theory [23,24,45].

IV. TUNING MAGNETIC GROUND STATES
Having established the approximate rotation angles of both samples as α = 8 • , we can fit the dispersion of the SLs, as discussed above, displayed for 2SIO/1STO in Fig. 3(b) left panel [28]. For 1SIO/1STO with gap ∼ 11 − 36 meV (compared to ∼25 meV for Sr 2 IrO 4 ), we find θ = 0.221-0.247π. Within this range, spin wave theory can model well the dispersion throughout reciprocal space [25]. For 2SIO/1STO, fitting with θ = 0.225 ± 0.009π adequately reproduces the gaps and dispersion. For the bilayer, optical and acoustic modes are present, but only one mode is observed due to the Qdependence of their intensities, discussed in the supplemental [28]. The similar theta values, which do not reflect the differences seen between bulk Sr 2 IrO 4 and Sr 3 Ir 2 O 7 , are initially surprising due to the less distorted octahedra observed for both SIO and Sr 3 Ir 2 O 7 (≤ 2%, ∼ 8% for Sr 2 IrO 4 ) [13,16,26,39,40]. However, the Ir 5d orbitals are rather extended spatially and couple strongly to next nearest neighbors, breaking the local symmetry [26]. The similar θ of the SLs further corroborates this, pointing to the SL structure as the dominant determinant of θ.
Extracted values of the exchange couplings for each of the SLs are shown in Table I [28]. For 1SIO/1STO, the values are similar to those found in both doped and undoped Sr 2 IrO 4 , owing to the relatively small change in θ between the SL and Sr 2 IrO 4 (∼ 0.01π) [25]. This in-dicates the 1SIO/1STO provides another magnetic analogue to cuprates, similar to that found in Sr 2 IrO 4 , but with the higher tunability afforded by heterostructuring [4,[46][47][48]. Comparing 2SIO/1STO with Sr 3 Ir 2 O 7 , on the other hand, the changes are quite substantial, owing to the much more significant shift of θ (0.035π) [16]. Here, the ratio J ab /J c is half that found in Sr 3 Ir 2 O 7 , a reasonable change in light of the more uniform octahedra expected in the SL (due to the ability of tilting to circumvent distortion) [49]. Finally, from fitting the smaller magnon gap, a 28% decrease in the pseudodipolar anisotropic coupling Γ c is observed, which is chiefly responsible for stabilizing the c-axis magnetic ground state.
To investigate the stability of the observed magnetic phases, we map the SLs on the classical (θ, η) magnetic phase diagram in Fig. 4(b) alongside Sr 2 IrO 4 and Sr 3 Ir 2 O 7 [14]. Intriguingly, both SLs lie close to their respective phase transitions. This is especially significant for 2SIO/1STO, where the material lies closer to the phase transition than its bulk analogue, within 0.02π or ∼ 10 meV tetragonal splitting change. The fact that such a shift happens despite the similar structures of 2SIO/1STO and Sr 3 Ir 2 O 7 shows how relatively small distortions can strongly modify the interactions of these SLs. Further bending of the c-axis bond can then be expected to drive the system closer to, and eventually through, a quantum critical point between the abplane canted and c-axis collinear antiferromagnets. This could be accomplished through applying epitaxial tensile strain, by changing the substrate, or by substituting Sr with Ca. Based on the calculated change in the crystal field of strained films of Sr 2 IrO 4 , applied strain of only a few percent could be enough to drive 2SIO/1SIO to the quantum critical point [23,33]. In this way, strong spinorbit coupling provides a means to exploit small structural distortions to stabilize large changes in the magnetic ground state.

V. CONCLUSION
In conclusion, we demonstrate that 2SIO/1STO has predominantly c-axis antiferromagnetic moments, establishing the nSIO/1STO SL series as viable artificial ana-logues to the Ruddlesden-Popper crystals. We furthermore reconcile previous contradictory reports by identifying finite octahedral tilting that generates a net canted moment [21,23]. Hard x-ray RIXS is shown to be sufficiently sensitive to probe the magnetic interactions that stabilize the observed ground state for the first time. For the bilayer 2SIO/1STO, the magnon gap is significantly smaller than that observed in Sr 3 Ir 2 O 7 and spin-wave based modelling shows that this material is closer to a phase transition between different ground states. Heterostructuring iridates and probing their magnetic interactions with RIXS thus shows how spin orbit coupling can leverage small structural distortions to alter magnetic interactions with potential to realize quantum critical artificial Ruddlesden-Popper phases.