Competing energy scales in correlated electron systems often lead to granular nature (1-7), with substantial spatial variations in charge, spin, and orbital degrees of freedom. Depending on the material and the emergent property, the characteristic length scale of this inhomogeneity can span several orders of magnitude. For instance, charge inhomogeneity ranges from several micrometers in manganites (2) and Hg-based cuprates (3), to a few nanometers in Bi-based cuprates (1) and iridates (4,5). Spin inhomogeneity in correlated quantum solids can exhibit an even more elaborate texture. A ferromagnet can have microscopic domains, each with a different orientation of the local magnetic moment (6), sometimes coexisting with spatially separated nonmagnetic regions (7). In the presence of antisymmetric spin interactions, spins can gradually twist and form complex, long wavelength spin textures such as skyrmion lattices (8)(9)(10). While a larger spatial extent of ferromagnetic domains and skyrmion spin texture facilitates imaging by a variety of probes, antiferromagnetic (AF) ordering has been notoriously difficult to visualize. This is, in large part, due to the cancellation of signals from neighboring spins, which limits the use of probes that are unable to achieve subnanometer resolution. Tremendous progress has recently been made in imaging micron size AF domains using scattering techniques (11). However, to fully understand the nature of the strongly correlated states and quantum fluctuations, it is highly desirable to visualize the formation of AF clusters with atomic resolution, including their size, shape, and distribution in the presence of atomic defects and across different phase transitions.
Over the past decade, Ruddlesden-Popper iridates (12)(13)(14)(15) have emerged as an intriguing family of two-dimensional (2D) AF spinorbit Mott insulators, where the AF Mott ground state can be tuned by different parameters, for example, temperature and chemical composition. The experiments thus far revealed a remarkable nanoscale phase separation in iridates, reflected in the granular fabric of both their electronic and magnetic properties (4,5,16). While, in many correlated oxides, the competition between different phases provides a natural tendency toward phase separation (5,17,18), chemical disorder that is often necessary to induce a new phenomenon can also play a substantial role (5,19,20). Our recent spin-polarized scanning tunneling microscopy (SP-STM) experiments revealed the emergence of AF puddles near insulator-to-metal transition in the Ruddlesden-Popper iridate Sr 2 IrO 4 (16). However, many questions remain regarding the formation of AF clusters in this family of oxides. For example, how sensitive is the domain size and shape to the inevitable presence of chemical disorder? How repeatable is the AF pattern configuration after it is thermally wiped out by increasing the temperature and then cooled back down? In this work, we explore these questions in the bilayer iridate Sr 3 Ir 2 O 7 (Ir-327). Ir-327 is less insulating compared to its single-layer counterpart Sr 2 IrO 4, which facilities tunneling measurements across a wider doping range down to the near "parent" state. Using SP-STM, we find that multiple AF configurations can nucleate at low temperature and reveal the effects of different types of disorder on the local distribution of AF ordering.
Undoped parent Ir-327 exhibits collinear AF ordering below Neel temperature (T N ) ~ 285 K, with local magnetic moment of the Ir atoms pointing out-of-plane (Fig. 1A) (21,22). On the basis of the combination of magnetization and neutron scattering measurements, the long-range AF order can be suppressed by temperature or by chemical substitutions (Fig. 2A) (23,24). To access local spin information related to the AF order in this system, we apply SP-STM imaging (see Materials and Methods and fig. S1) (25). We acquire a pair of STM topographs over an identical area of the sample in magnetic field applied parallel and antiparallel to the c axis (Fig. 1, C andD). The reversal of the magnetic field direction serves to "flip" the spin polarization of the STM tip, without notably affecting the magnetic moment orientation in the sample. By subtracting the two topographs, we can extract the spin-resolved magnetic contrast map M(r), which shows prominent modulations (Fig. 1F). The wave vectors of these modulations, Q a = (0.5, -0.5) and Q b = (0.5, 0.5) (we hereafter define reciprocal lattice vector 2/a 0 = 1), are identical to those attributed to long-range AF order by neutron scattering experiments (23,24).
Using this procedure, we investigate the evolution of spin-resolved modulations in Ir-327 as a function of La substituting for Sr (Fig. 2, A to C) (24,26), while, at the same time, tracking the electronic properties from differential conductance dI/dV spectra (Fig. 2, D andE). While, in the near parent state, we find that the spin-resolved modulations appear spatially uniform (Fig. 2B), in x ~ 3.4% La-substituted sample, we reveal a spatial inhomogeneity in the M(r) maps, with distinct regions where modulations are completely absent (Fig. 2C). This spatial inhomogeneity is also reflected in the larger width of x ≈ 0%
x ≈ 3.4% the Fourier space peaks at Q a and Q b (fig. S1). The inhomogeneous disappearance of the AF order with La substitution in the bilayer iridate Ir-327 is consistent with our observations in La-substituted Sr 2 IrO 4 (16), suggesting that the granular nature of the AF order near the transition is not limited to a single iridate. Moreover, outof-plane spins in Ir-327 allow a complete mapping of all AF domains, in contrast to Sr 2 IrO 4 where in-plane spin orientation may lead to some domains being undetectable by a SP-STM tip (see Materials and Methods).
Next, we set to explore whether and how chemical disorder affects the AF texture observed after introducing a dilute concentration of La substitutions. We focus on the three types of defects that we most commonly observe in this system: La substitution for Sr in the topmost SrO plane seen as bright squares in STM topographs (La 1 ), La substitution for Sr in the second SrO layer seen as bright features in dI/dV(r, V) maps at low energy (La 2 ) ( 27), and dark features in STM topographs (D 3 ) (Figs. 1A and3, A to C and H, and figs. S3 and S4). To visualize how the AF order arranges itself relative to local disorder, we superimpose the positions of each type of defect on top of the local AF amplitude map |M(r)| (Fig. 3,E to G). Visually, the AF order appears to be the strongest away from La 2 substitutions, while La 1 and D 3 defects are nearly randomly distributed with respect to the AF order. This conclusion is supported by the cross-correlation of the |M(r)| maps and dopant density maps (r) (Fig. 3D and fig. S4), which shows a cross-correlation coefficient ~ -0.3 for La 2 , while coefficients for La 1 and D 3 defects are substantially weaker. Bulk measurements of La-doped Ir-327 found that long-range AF order is gradually suppressed with the increase in La concentration (24). Our SP-STM experiments are consistent with this global picture and further reveal that the inhomogeneous AF order near the transition point has a tendency to weaken in proximity to a particular type of La disorder, located in the middle SrO plane.
In general, strong disorder pinning could suggest that a spatial pattern associated with the AF order will be repeatable after cycling through the magnetic transition and back. To investigate this, we image the evolution of spin-resolved AF modulations as a function of temperature, tracking the same area of the sample (Fig. 4, A to D). We find that the average |M(r)| signal quickly subsides by ~9 K (Fig. 4F). This provides further evidence that our sample is near the transition point at this La composition. We also note that the spectral gapmaps at 4.5 and 9 K are nearly identical (Fig. 4, G andH), thus confirming a lack of direct dependence of the most prominent spectral gap to the local AF order (16). After increasing the temperature to ~10 K, we cool the sample back down to ~4.5 K and repeat the measurement. Although the cross-correlation between the AF amplitude maps before (|M 1 (5 K)|) and after thermal cycling (|M 2 (5 K)|) is relatively high (Fig. 4F, inset), we observe distinct changes in the patterns (Fig. 4E), indicating a change in the lowtemperature AF configuration. In both cases, however, the AF patterns form away from La 2 substitutions, while other defects play a minor role (Fig. 3,E to G).
The ability to visualize the AF clusters in real space allows us to investigate their size, shape, and distribution in more detail. We apply 2D cluster analysis theory, a powerful statistical analysis tool that can be used to probe the strength of electronic correlations and near-critical behavior from the geometric metrics of the clusters (11,(28)(29)(30). We first binarize the AF amplitude map |M 1 (5 K)| based on an intensity cutoff (Fig. 5A and section S2). Then, using logarithmic binning, a standard technique for power law distribution analysis (31), we plot the relationship between several geometrical descriptors of the clusters: area A, perimeter P, gyration radius R g [defined
, where r i(j) is the position of the i(j)th site in the cluster (32) and s is the size of the cluster], number of clusters of given area D(A), and pair connectivity PC(r) (defined as the probability that two sites separated by a distance r belong to the same connected finite cluster). In the analysis, we exclude the clusters touching the boundaries, as we do not have a complete set of geometric metrics for these. We find that the domain area distribution histogram D(A) versus A shows a linear dependence on a logarithmic scale spanning over 2.5 decades. By fitting D(A) = A - , we determine the critical exponent = 0.76 ± 0.18 (Fig. 5B). Next, we study the relationship between the area A (perimeter P) and the gyration radius R g across all domains, as shown in Fig. 5C (Fig. 5D). We find that both A and P scale as a power of R g (P = R g dh * and A = R g dv *) for exponents dv* (dh*) often referred to as effective volume (hull) dimensions (32). From the fitting analysis, we determine these to be dv* = 1.95 ± 0.18 and dh* = 1.12 ± 0.04. To put these values in perspective, we note that coefficients for an uncorrelated percolation should be dv* ≈ 1.90 and dh* = 1.75 (32). The difference between these coefficients and those extracted from our data suggests that the formation of AF patterns in our sample may be influenced by electronic correlations in the presence of disorder (section S2). To further support the validity of our cluster analysis, we point to the following. First, the exponent ratio dh*/dv* determined directly from the slope in Fig. 5E (3/5 ± 0.04) is nearly identical to the ratio of dh* and dv* calculated individually from Fig. 5 (C andD). Second, we note that the hyperscaling relation, d -2 + = 2 (d -dv*), expected to hold near the transition point within this model, is satisfied (section S2). Here, d is the dimensionality of the system taken to be 2 given the quasi-2D nature of Ir-327, and = 0.05 ± 0.02 is the coefficient obtained from fitting pair connectivity function PC(r) ~ r - •e -r/ (Fig. 5F).
Our experiments provide a new insight into the inhomogeneous distribution of spin degrees of freedom relative to chemical disorder, complementary to the reports of nanoscale separation in the charge degrees of freedom in the same family of materials (4,23,27). We find that chemical disorder can put some constraints on the morphology of the AF texture. At the same time, thermal cycling experiments reveal substantial spatial fluctuations and rearrangement of the AF puddles around this disorder. While the AF distribution maps at different temperatures appear markedly different (Fig. 4, C andD), spectral gapmaps look virtually identical (Fig. 4, G and H), thus confirming that short-range AF order is not the cause behind the spatially inhomogeneous gap closing (16). While our work points toward La 2 substitutions in the middle SrO layer having the strongest correlation with the regions where the AF order is absent, it remains to be seen what is the physical effect driving this behavior. The site-dependent effect of La substitutions on the local distribution of the AF order is particularly puzzling and warrants further experimental and theoretical work.
Revealing scale invariance of the AF order in real space near the doping induced transition is an important step toward understanding the correlation physics in this spin orbit-coupled Mott system. The statistical analysis of the cluster geometric metrics suggests that the disappearance of the AF order exhibits signatures of a continuous phase transition near criticality. However, we caution that the comparison of the exponents to the oversimplified model of uncorrelated classical percolation provides only a simple benchmark for the possible emerging continuous transition. The deviation of the exponents from the percolation model also points toward the role of electronic correlations, which can influence the magnetic correlation length and, as such, the size of the clusters. Note that even for a percolative transition, electronic correlations can still play a role since they may control the low-energy excitations at the boundary of the clusters where the electrons can quantum tunnel at low temperature to other clusters or to the nonmagnetic region. While, in this work, we reveal spatial variations of the AF order parameter, future experiments could attempt to detect AF fluctuations in time domain by measuring step-like jumps in tunneling current in constant height STM mode, in analogy to sudden changes in the optical response in 2D magnets (33). By exploring the evolution of the coherence length with temperature in the statistical analysis of fractal patterns, future experiments focusing on large sample areas to gain higher statistics could shed more light on the underlying criticality in this family of iridates.
Single crystals of (Sr were dried, and stoichiometric amounts were measured out, in a 15:1 molar ratio between SrCl 2 flux and the target composition.
Powders were loaded into a platinum crucible and further contained inside alumina crucibles to limit volatility. Mixtures were heated to 1300°C, slowly cooled to 850°C at a rate of 3.5°C hour -1 , and then furnace-cooled to room temperature. The resulting boule was etched with deionized water, revealing black plate-like crystals with typical dimensions of ~1 mm by 1 mm by 0.1 mm. More detailed chemical, structural, and electronic characterization can be found in (24).
(Sr 1-x La x ) 3 Ir 2 O 7 samples were cleaved in ultrahigh vacuum at ~80 K and immediately inserted into the STM head. All STM data were acquired using a customized Unisoku USM1300 system at the base temperature of ~5 K, unless otherwise noted. A typical STM topograph of reveals a square lattice of Sr atoms in the topmost SrO plane, with the lattice constant a 0 ~ 0.39 Å (Fig. 1, C and D). To produce the spin-resolved magnetic contrast M(r) maps, the Lawler-Fujita drift-correction algorithm (34) was applied to eliminate the inevitable effects of piezo hysteresis and thermal drift and align the topographic images acquired at two different fields using various defects as position markers. Spectroscopic measurements were taken using a standard lock-in technique at 915-Hz frequency and varying bias excitation as detailed in the figure captions. Spin-polarized tips were electrochemically etched from a Cr wire in 2 M NaOH solution, similar to the process described in our previous work (16). The tip was trained to be ferromagnetic on the surface of ultrahigh vacuum-cleaved single crystals of antiferromagnet Fe 1+y Te by fast scanning and bias pulsing to produce the desired sharpness and the spin-resolved contrast. An advantage of a spin-polarized tip prepared in this method is low "stray" magnetic field because the bulk of the tip remains AF. Last, we note that, in the single-layer iridate Sr 2 IrO 4 , spins are oriented in-plane, which can lead to AF domains with spins rotated by 90° in-plane (16). This can, in principle, lead to some AF domains that are undetectable by an STM tip, if tip polarization is exactly orthogonal to the spin orientation of the sample [see more details in supplementary information of ( 16)]. This is not the case in Ir-327 because spins point out-of-plane and no orthogonally oriented domains should exist.