Scale-invariant magnetic textures in the strongly correlated oxide NdNiO3

Strongly correlated quantum solids are characterized by an inherently granular electronic fabric, with spatial patterns that can span multiple length scales in proximity to a critical point. Here, we use a resonant magnetic X-ray scattering nanoprobe with sub-100 nm spatial resolution to directly visualize the texture of antiferromagnetic domains in NdNiO3. Surprisingly, our measurements reveal a highly textured magnetic fabric, which we show to be robust and nonvolatile even after thermal erasure across its ordering temperature. The scale-free distribution of antiferromagnetic domains and its non-integral dimensionality point to a hitherto-unobserved magnetic fractal geometry in this system. These scale-invariant textures directly reflect the continuous nature of the magnetic transition and the proximity of this system to a critical point. The present study not only exposes the near-critical behavior in rare earth nickelates but also underscores the potential for X-ray scattering nanoprobes to image the multiscale signatures of criticality near a critical point.

To maximize the scattering intensity, σ polarization was used for all measurements. By raster-scanning the focused X-ray beam across the sample while acquiring the scattering signals, we can record the spatial variations in the magnetic scattering intensity, and consequently map the underlying magnetic texture. The AFM signal was extracted by first subtracting the fluorescence background and then averaging the scattering intensity inside a fixed region of interest (ROI) on the detector. Before acquiring each spatial map, the Cr mask was scanned to ensure all measurements are performed in the same field of view (FOV). The fiducial Cr grid additionally provides a reference frame to correct for the position drift and extract driftcorrected spatial maps over a common spatial range. The position registration enabled by the Cr grid enables to convert the intensity map from the raw pixel grid onto a drift corrected pixel grid by performing an affine transformation followed by point interpolation (Supplementary Fig. 2b).
To recover the signal under the semi-transparent Cr mask, we correct for the photon flux attenuation through the Cr thin layer ( Supplementary Fig. 2c) [the theoretical transmission factor is 12% for a 143 nm thick Cr, at a photon energy of 852 eV and given a mass density of 7.15 g cm -3 ]. The data shown in the main text and presented in Fig. 2 have been obtained after removal of the Cr intensity suppression and position drift correction.
The macroscopic beam measurement is performed by moving the sample away from the focus of the zone plates, where the beam footprint is approximately 200 μm in diameter. The temperature was swept with the rate of 1 K min -1 while recording the sample temperature and CCD image at a constant rate. The AFM signal was extracted by first subtracting the fluorescence background and then averaging the scattering intensity inside a fixed region of interest (ROI) on the detector. While the spot on the sample is large in this out of focus geometry, the photon beam remains strongly divergent (by virtue of passing through the zone plate focusing element), consequently the width of the diffraction peak on the CCD is largely determined by the divergence of the X-ray beam, with the contribution from the intrinsic spatial correlations being minor. As a result, the AFM correlation lengths could not be directly deconvolved from the diffraction peak linewidth.

Supplementary Note 2: Homogeneous insulating phase well below TMIT
To elucidate whether the AFM inhomogeneity is affected by a nanoscale phase coexistence of metallic and insulating domains 1 , we measured the local X-ray absorption spectra (XAS) in the same NdNiO3 film using X-ray PhotoEmission Electron Microscopy (XPEEM) at beamline ESM (21-ID-2) of the National Synchrotron Light Source II. The XAS lineshape at the Ni-L3 edge shows an ostensible transition across the MIT, over an extended spatial region. When below TMIT, the Ni L3 XAS exhibits a 2 eV splitting, corresponding to the presence of two inequivalent Ni sites 2 . Figure S3b shows an overlay of multiple representative local XAS spectra from a given field of view. All spectra show prominent insulating feature as opposed to the metallic profile. To obtain a quantified description of the insulating and metallic property, we performed a principal component analysis (PCA) on the XAS map at 100 K, to separate out the metallic and insulating components. The PCA was done by fitting the local XAS spectra, pixel-by-pixel, using a simple linear combination of two model XAS lineshapes representing the metallic ( , extracted from 300 K absorption spectra) and insulating phase ( , extracted from absorption spectra at 100 K): where the coefficient ( ), constrained within [0,1], represents the local metallicity ( Supplementary Fig.  3c). A typical PCA result is shown in the inset of Fig. S3d. We find that >99% of pixels have a metallic character of less than 0.15 [ Supplementary Fig. 3d], which reveals a spatially-uniform insulating phase across the FOV when the temperature is well below TMIT. We notice that the stripy textures reported in prior studies 1 were also visible in our measurements and were found to correlate with the atomic terraces at the film surface. This result is consistent with other studies reporting that NdNiO3 films are homogeneously insulating at temperature well below TMIT 3,4 . On the basis of the XPEEM spatial maps, we can rule out a coexistence of the metallic and insulating phases. Consequently, we ascribe the AFM inhomogeneity to the inherent distribution of different magnetic twin domains as described in the main text.

Supplementary Note 3: Memory effect
To quantitatively assess the temperature variations in the domain morphology, we define a local correlation metric to monitor the self-similarity of the probed magnetic textures. The local correlation measure ρ(r) is defined as a conventional cross-correlation of the scattering intensity between two maps, calculated over a region of interest centered at coordinate r (see Supplementary Fig. 4a-c). The local spatial filter is a circle of radius of 500 nm, which appears as elliptic because of projection over the sample surface (Fig. 2e). Large and positive values of ρ(r) reflect a higher similarity in the AFM domain morphology across different temperatures ( Supplementary Fig. 4d-f). The average cross-correlation measure over the entire field of view ranges around ̅ ~ 0.6, providing direct statistical evidence of a memory effect during thermal cycling. This apparent resilience against temperature variations suggests the occurrence of domain pinning which is robust across the magnetic ordering transition.

Supplementary Note 4: Is it a disorder effect?
To understand the role of disorders in generating spatial inhomogeneity, we simulated a spatial map with 2d uncorrelated percolation model with the same 25% coverage ( Supplementary Fig. 5). Domains are only organized within a short range, which is unlike the domain organization observed experimentally and shown in the main text. We also summarized the scaling exponents in Supplementary Table 1, where the same exponents based on 2d uncorrelated percolation model are provided for comparison 5 . Most of the exponents disagree with the uncorrelated percolation model. Therefore, the characteristics of the pattern formation observed in the fractal magnetic texture cannot be attributed solely to the effects of material disorder.