Emerging Diluted Ferromagnetism in High‐T c Superconductors Driven by Point Defect Clusters

Defects in ceramic materials are generally seen as detrimental to their functionality and applicability. Yet, in some complex oxides, defects present an opportunity to enhance some of their properties or even lead to the discovery of exciting physics, particularly in the presence of strong correlations. A paradigmatic case is the high‐temperature superconductor YBa2Cu3O7‐δ (Y123), in which nanoscale defects play an important role as they can immobilize quantized magnetic flux vortices. Here previously unforeseen point defects buried in Y123 thin films that lead to the formation of ferromagnetic clusters embedded within the superconductor are unveiled. Aberration‐corrected scanning transmission microscopy has been used for exploring, on a single unit‐cell level, the structure and chemistry resulting from these complex point defects, along with density functional theory calculations, for providing new insights about their nature including an unexpected defect‐driven ferromagnetism, and X‐ray magnetic circular dichroism for bearing evidence of Cu magnetic moments that align ferromagnetically even below the superconducting critical temperature to form a dilute system of magnetic clusters associated with the point defects.


S1. Structural analysis
Microstructural analysis. Figure S1a and S1b are low magnification HAADF images showing, respectively, the global microstructure of a standard Y123 thin film and a Y123 nanocomposite.
These Z-contrast Scanning Transmission Electron Microscopy (STEM) images were acquired with a FEI Titan (60-300 kV) equipped with a probe-aberration corrector. In the HAADF Zcontrast image Y124 intergrowths appear as horizontal black stripes. The Y124 planar defects discussed in the manuscript are common structural defects in both films; numerous intergrowths are observed close to the surface in the two samples. However, the introduction of secondary phase nanoparticles into the Y123 matrix dramatically increases the density of Y124 planar defects in the central bulk part of the nanocomposite thin film.  HAADF Image simulation. The HAADF-STEM simulations were performed using the STEM_CELL software [1] , a simulation package based on Kirkland routines [2] . This software is based on the multislice technique in the 'frozen-lattice' approximation.
Y123 and Y124 supercells, with approximately 3.5nm thickness, were created in order to simulate the contrast modulations due to the presence of Cu vacancies within the double CuO chain ( Figure S5). Experimental HAADF images in Figure 2 and Figure

S2. Computational details
Defect formation energies and chemical potentials. We use density functional theory (DFT) to determine the most stable Cu vacancies by calculating their formation energies. This requires the knowledge of the chemical potential of individual copper (µ Cu ) and oxygen (µ O ) atoms within the compound, which is not well defined. Instead, we vary µ Cu and µ O over a range of values bound by any two phases of Cu and O that are considered to be under thermodynamic equilibrium [3] . For "O-deficient" (equivalent to "Cu-rich") conditions corresponding to lower oxygen chemical potential, we use the energy of a Cu atom in bulk Cu (with face centered cubic lattice) as µ Cu Table S1.
Electronic Structure. The effect of the defects on the electronic structure was studied within the approximations described above. In order to understand the effect of the stacking faults in the electronic structure, we compared the linear charge density of the YBa 2 Cu 4 O 8 (Y-248) a supercell with and without 2V Cu +3V O . The linear charge density is obtained by numerically integrating the charge density in the planes parallel to the CuO 2 plains where superconductivity takes place. The electronic densities for these two cases are shown in Fig. S6. Although the combinations of Cu and O vacancies that appear in the double chain do affect the charge density around the double chain that contains the vacancy, the charge density around the superconducting planes is very similar for both cases. The main difference is actually due to the slightly smaller c parameter when the vacancies are considered. Also, note that the magnetic density in Fig. 3b of the main text appears only when the vacancies within the double chain are considered (it is zero for Y248), despite the overall similar electronic density.

S3. XMCD analysis
XMCD sum rules. XAS&XMCD measurements as a function of the temperature (at 6T) and as a function of the magnetic field (at 1.6 K) were performed in normal beam incidence (θ=0 o ).
The sum rules [4,5] were applied to the background subtracted XAS spectra to evaluate the Cu orbital and effective spin moment: where n h is the number of holes in the 3d shell, which we assumed to be n h =1 for Cu 2+ sites Finally, the orbital to effective-spin ratio is determined as: and thus is independent of the number of holes n h and from the factor r(θ). Figure S7a shows the effective spin and orbital moment field dependence, and , obtained from XMCD(B) measurements at 1.6 K, for the two studied samples. Note that the effective spin moment increases almost linearly with the field but tends towards saturation for very high fields (>4T). The orbital-to-effective spin moment ratio, at high fields, where the error is smaller, is close to ≈0.22, the value predicted from atomic model calculations for Cu 2+ with a x 2 -y 2 ground state [6] .
Effective spin and spin moments. As extensively discussed in Ref. [6,7] , for Cu 2+ sites with high anisotropy, the effective spin moment  [6,7] . The spin-dipole moment expresses the inhomogeneous spatial distribution of the spin density over the atomic unit cell, due to the anisotropic charge distribution arising from strongly directional bonds or crystal field [8] .
We adopted a methodology [9] , previously applied to highly-anisotropic atoms on surfaces [10,11] , to experimentally determine the isotropic moment (m s ) from XAS&XMCD measurements performed in grazing incidence, under the so-called "magic-angle" (θ*=54.7 o ). The measured XMCD with B parallel to the beam direction and incidence angle θ gives the projection of the Cu magnetic moment along the direction of the applied field. The anisotropic effective spin moment determined in grazing incidence can be written as:  [8] . f θ θ = + . As pointed out by Stepanow [7] , this correction needs to be done for very anisotropic atoms to account for the angular dependence of the absorption intensity due to the spatial distribution of the 3d-orbitals. XAS&XMCD measurements in grazing incidence at the "magic angle" (θ * =54.  The      Note that there are significant differences in the charge density around the double chain due to the vacancies, but in the CuO 2 planes the charge density is very similar. Figure S7. a) Magnetic field dependence of the (bottom) effective spin moment, (middle) orbital moment and (top) orbital-to-effective spin moment ratio for the standard YBCO sample at 1.6K, in normal incidence (θ * =54.7 o ); b) Cu L 2,3 edge background-subtracted XAS and XMCD spectra measured on standard YBCO standard at 6 T, 1.6 K in grazing incidence, at the magic angle (θ * =54.7 o ).