Atomic Structure and Electron Magnetic Circular Dichroism of Individual Rock Salt Structure Antiphase Boundaries in Spinel Ferrites

Spinel ferrites are an important class of materials, whose magnetic properties are of interest for industrial applications. The antiphase boundaries (APBs) that are commonly observed in spinel ferrite films can hinder their applications in spintronic devices and sensors, as a result of their influence on magnetic degradation and magnetoresistance of the materials. However, it is challenging to correlate magnetic properties with atomic structure in individual APBs due to the limited spatial resolution of most magnetic imaging techniques. Here, aberration‐corrected scanning transmission electron microscopy and electron energy‐loss magnetic chiral dichroism are used to measure the atomic structure and electron magnetic circular dichroism (EMCD) of a single APB in NiFe2O4 that takes the form of a rock salt structure interlayer and is associated with a crystal translation of (1/4)a[011]. First principles density functional theory calculations are used to confirm that this specific APB introduces antiferromagnetic coupling and a significant decrease in the magnitude of the magnetic moments, which is consistent with an observed decrease in EMCD signal at the APB. The results provide new insight into the physical origins of magnetic coupling at an individual defect on the atomic scale.


Introduction
Spinel ferrites have a wide spectrum of magnetic properties. They have attracted attention for their utilization in sensors, transformers, antennas, high-frequency inductors, microwave devices, and spintronic devices. [1] Antiphase boundaries (APBs), which are commonly observed in spinel ferrite films, can contribute to magnetoresistance, [2] increase the critical field for magnetic saturation, [3] and cause local magnetic degradation, [4] which can be deleterious for applications. It is therefore important to understand interface configurations and magnetic coupling across individual APBs. Atomic models of APBs have been constructed on the basis of their crystallographic relationships and verified experimentally using dark-field transmission electron microscopy (TEM) and high-resolution scanning TEM (STEM). [5] Domains that are on either side of an APB are typically related to each other by a crystallographic translation of the form (1/4)a [110] or (1/2)a[100], where a is the lattice parameter. Only the cation sublattice is then shifted, with the oxygen sublattice remaining more or less undisturbed. [5b,c] The atomic structure of an APB is sometimes more complicated, for example, if the two shifted domains are not connected directly but are separated by an interlayer. Such configurations have rarely been studied in detail.
The atomic-scale structures and nanoscale magnetic properties of the same APBs or twin boundaries can be studied by using a combination of aberration-corrected (S)TEM with phase contrast techniques such as off-axis electron holography, [6] Lorentz microscopy, [7] and differential phase contrast imaging. [8] However, element-specific magnetic information, which is important for complex oxides such as spinel ferrites, is then missing. In addition, it is difficult to investigate non-conductive materials using phase contrast techniques, as a result of the influence of electron-beam-induced specimen charging. An alternative powerful technique for imaging magnetic microstructure in a (S) TEM that has element specificity is electron energy-loss magnetic chiral dichroism, [9] which is also referred to as electron magnetic circular dichroism (EMCD). This technique is similar to X-ray magnetic circular dichroism (XMCD), [10] in that it permits the quantitative element-selective measurement of spin and orbital magnetic moment in materials from spectra measured at core-loss edges. [11] When it is implemented using convergent electron beam illumination, the spatial resolution of EMCD can reach a few nm, [12] or even the sub-nm scale. [13] Here, we study an APB with a rock salt structure interlayer in inverse spinel NiFe 2 O 4 . In contrast to commonly-observed APBs, high-angle annular dark-field (HAADF) STEM imaging is used to show that the two shifted domains, which have a relative translation of (1/4)a[011], are separated by a rock salt structure interlayer in which only half of the octahedral interstices are occupied. Quantitative analysis of EMCD spectra recorded under optimized dynamical diffraction conditions reveals a significant decrease in experimental EMCD strength at the APB, as compared to that in the adjacent perfect domains. Dynamical diffraction simulations and first-principles density functional theory (DFT) calculations are used to investigate how the atomic structure of the APB affects magnetic coupling within it and to determine the magnetic moments of the constituent Fe and Ni cations.  [14] we refer only to a translation of (1/4)a[011] below. It should be noted that only atomic columns that contain metal atoms are visible in the HAADF STEM image, as the oxygen atomic columns scatter the electron beam too weakly. Unlike the APBs that are more commonly seen in spinel ferrites, the two shifted domains in Figure 1a are not connected to each other directly, but are instead separated by a rock salt structure interlayer, as indicated by orange arrows.

Results and Discussion
In order to determine the atomic density of the metal atoms in this rock salt structure interlayer, we investigated the relationship between the atomic density and the intensities of atomic columns in both the APB interlayer and the perfect domains in the regions of the HAADF STEM image that are marked by colored dots in Figure 2a. The inverse spinel NiFe 2 O 4 crystal structure has Fd3m symmetry, a face centered cubic oxygen array with Fe 3+ ions distributed in one-eighth of the tetrahedral interstices and Ni 2+ and Fe 3+ ions distributed in half of the octahedral interstices. It can be described in the form (Fe 3+ ) tet (Ni 2+ Fe 3+ ) oct O 4 for its nominal stoichiometric ratio. In an atomic model of perfectly-ordered NiFe 2 O 4 viewed along the [110] direction (Figure S1e, Supporting Information), atomic columns along which the octahedral interstices are fully occupied by metal atoms (shown in green) have twice the atomic density of metal atoms compared to atomic columns along which only half of the octahedral interstices are occupied by metal atoms in an ordered manner (shown in blue). Figure S1a, Supporting Information, shows a representative HAADF STEM image of perfectly-ordered NiFe 2 O 4 viewed along the [110] direction, in which atomic columns that have twice the atomic density are brighter than other atomic columns. The intensities of octahedral atomic columns in the HAADF STEM image, which increase monotonically with their atomic density, [15] were analyzed statistically, as shown in Figure 2b. (The intensity difference between Fe and Ni was not taken into account in this analysis, as a result of the close atomic numbers of the two elements). Figure 2b shows that average values of the normalized intensities of fully-occupied atomic columns in the perfect domains (in dimensionless units) are 1.00 ± 0.16 (red) and 0.97 ± 0.20 (green) on the left and right sides, respectively, while average values of the normalized intensities of atomic columns with half of the atomic density in the perfect domains are 0.56 ± 0.07 (blue) and 0.63 ± 0.08 (purple) on the left and right sides, respectively. In contrast, average values of the normalized intensities of atomic columns in the rock salt structure APB interlayer are 0.63 ± 0.15 (orange), that is, close to the intensities of atomic columns with half of the atomic density in the perfect domains. It is therefore inferred that, on average, approximately half of the octahedral interstices in the rock salt structure interlayer are occupied by metal atoms, although the increased standard deviation of the normalized intensities in the rock salt structure APB interlayer suggests a higher degree of randomness of the occupied octahedral interstices when viewed along the [110] direction.
In order to validate the proposed model for this APB, in which half of the octahedral interstices are occupied within the interlayer, DFT calculations were used to fully relax the structure model. Two specific B-site orderings of NiFe 2 O 4 were considered in DFT calculations, with either P4 1 22 (α-type) or Imma (β-type) symmetry instead of Fd m 3 symmetry, for the following reasons: First, short-range B-site ordering has been reported in inverse spinel NiFe 2 O 4 using Raman spectroscopy. [16] Second, the presence of randomly-distributed octahedral Ni 2+ and Fe 3+ in NiFe 2 O 4 with Fd m 3 symmetry could lead to thousands of possible configurations, making theoretical calculations unnecessarily complicated and impractical. In our previous work, calculations have shown that NiFe 2 O 4 with P4 1 22 symmetry has the lowest system energy, whereas NiFe 2 O 4 with Imma symmetry has a system energy that is only 3 meV/atom higher than that for P4 1   The resulting DFT-relaxed atomistic models of the APB in NiFe 2 O 4 are shown in Figure 1c,d for P4 1 22 and Imma symmetry, respectively. Figure 1e,f shows corresponding simulated HAADF STEM images. The positions of atomic columns in both simulated images are consistent with those in the experimental HAADF STEM image (Figure 1a), confirming the consistency of the rock salt structure APB model, with half of the octahedral interstices occupied in the interlayer. Remaining differences in the intensities of atomic columns in the interlayer between the simulated and experimental images are attributed to the use of ordered occupied octahedral interstices in the models, rather than randomly-occupied octahedral interstices.

Electron Magnetic Circular Dichroism of Antiphase Boundary
In order to investigate magnetic coupling at this APB with element specificity and high spatial resolution, EMCD experiments were performed in nanobeam mode using a probe size of 1.8 nm. A so-called two-beam case (2BC) orientation, with the (004) systematic reflection excited, was chosen for recording experimental EMCD spectra, as shown in the insets to Figure 3. The original EMCD spectra, which are shown in blue without smoothing, correspond to differences between normalized experimental electron energy-loss (EEL) spectra S + (black) and S − (red) recorded at the positions "+" and "−", which are  Figure 3. The asymmetry of the non-magnetic contributions to the signal in EMCD experiments in a 2BC orientation can lead to different noise levels in S + and S − spectra recorded using the same exposure time. [17] Figure 3 shows that Fe EMCD spectra recorded from the APB region and the adjacent perfect domains are associated with significant negative signals at the Fe L 3 edge and relatively weak positive signals at the Fe L 2 edge. Figure S2, Supporting Information, shows that Ni EMCD spectra recorded from the APB region and the adjacent perfect domains are associated with very weak but measurable negative signals at the Ni L 3 edge and positive signals at the Ni L 2 edge. The cumulative sums of the Ni EMCD spectra from the APB region and the adjacent perfect domains (shown in green in Figure S2, Supporting Information) decrease under the L 3 edge and rise under the L 2 edge, further confirming the presence of negative and positive signals in the EMCD spectra at the Ni L 3 and L 2 edges, respectively. The L 3 /L 2 white line ratios, which increase with Fe cation valence state, are shown in Figure S3d, Supporting Information [18] in the form of two isotropic spectra that were recorded from a perfect domain and the APB and normalized over the L 3 edge within the energy range from 704 to 717 eV. The resulting integration curve shows that the integral of the step-function-removed isotropic spectrum over the L 2 edge is smaller for the APB than for the perfect domains. The fact that the L 3 /L 2 white line ratio recorded from the APB is higher than that from the perfect region indicates that the Fe ion valence state is higher in the APB area. However, under the current electron beam illumination conditions, shifts in binding energy could not be discerned.
When using convergent beam illumination, the spatial resolution of EMCD depends on the probe size, while the strength of the magnetic signal decreases with decreasing probe size. Because an electron probe with a diameter of only 1.8 nm was used for the detection of local magnetic properties at the present APB, the signal-to-noise ratio (SNR) in the EMCD spectra was relatively poor, albeit still within a reasonable range, especially when considering the relationship between SNR and illumination area in previous works. [13a] Quantitative analysis of the SNR and confidence level in the experimental EMCD spectra was performed using a method that is described by Thersleff et al. [13a] and explained in the Supporting Information. Gaussian curve fitting, as shown in yellow in Figure 3a,b, was used to evaluate the signal strength, which is defined by the peak value of the Gaussian curve. Noise values were calculated as standard deviations of EMCD spectra in the pre-edge and post-edge regions. The SNR was then defined as the ratio between the signal and noise estimates. The SNR values of the EMCD signals at the Fe L 3 edges in the perfect domain and the APB 2.16 and 1.67, respectively, corresponding to confidence levels of 96.92% and 90.50%. The SNR values of the EMCD signals at the Fe L 2 edges in the perfect domain and the APB are 0.38 and 1.50, respectively, corresponding to lower confidence levels of 29.60% and 86.64%. The relatively weak signals at the positions of the Fe L 2 edges may be attributed to a non-zero orbital magnetic moment, plural scattering and shorter core-hole lifetimes of excitations from 2p 1/2 levels, as compared to 2p 3/2 levels. [12c,19] The EMCD intensities at both the L 3 and the L 2 edges were taken into account when quantitatively comparing experimental EMCD signals recorded from the APB and the perfect domains in terms of the normalized strength of the EMCD spectra for Fe was measured to be −0.26 ± 0.04 and −0.14 ± 0.01 in the perfect domains and the APB, respectively, whereas for Ni it was estimated to be −0.91 ± 0.14 and −0.56 ± 0.10 in the perfect domains and the APB, respectively. Considering the relatively low SNR in the Ni EMCD spectra, there is a reduction of approximately 46.8% ± 8.2% and 38.8% ± 14.5% in normalized strength for the Fe and Ni EMCD signals, respectively, in the APB when compared to values measured in the perfect domains. A detailed description of the process of error estimation and confidence level calculation is presented in the Supporting Information. It should be noted that the low signal of the Fe L 2 edge in the perfect domains may lead to an underestimation of the contribution from the Fe L 2 edge to NS emcd for the perfect domains, suggesting that the actual value of NS emcd may be even larger for the perfect domains and may lead to an even greater reduction in experimental EMCD signals.
The experimental EMCD strengths are determined by the intrinsic magnetic moments and dynamical diffraction coefficients of each atom in the material. In order to quantitatively evaluate the influence of the dynamical diffraction coefficients, beam-position-dependent dynamical diffraction calculations based on the DFT-relaxed APB model were performed to predict experimental EMCD strengths on the assumption that the Fe and Ni atoms in the octahedral interstices do not change their electronic structure, magnetic moment direction, or strength inside the APB interlayer, when compared to the Fe and Ni atoms in the octahedral interstices in the perfectly-ordered region of NiFe 2 O 4 . Under this assumption, the normalized strengths of the EMCD spectra are proportional to the calculated strengths of the relative EMCD (REMCD) coefficients (in dimensionless units), as shown in the Supporting Information. The REMCD strengths of Fe and Ni in perfectly-ordered NiFe 2 O 4 were calculated to be −8.4% and 7.8%, respectively, as indicated by dashed green and blue lines in Figure 3c. Absolute numbers for REMCD coefficients in the APB region are taken at the position marked by a red arrow on the horizontal axis. The calculated REMCD strengths of Fe and Ni in the APB region are found to be −7.9% and 7.7%, respectively, as indicated by green triangles and blue diamonds in Figure 3c. The standard deviations of all of the REMCD strengths for Fe (green triangles) and Ni (blue diamonds), as shown in Figure 3c, are only ≈0.3%, indicating that both the rock salt structure in the Considering that the theoretically-predicted NS emcd is proportional to the calculated REMCD strength, the NS emcd is predicted to be approximately the same in the APB as in perfectly-ordered NiFe 2 O 4 , in contradiction to the experimental results. Therefore, the initial assumption that was made in the simulations is necessarily incorrect, suggesting that the Fe and Ni atoms in the APB region have different electronic structures, or different magnitudes and/or directions of magnetic moment, when compared to atoms in perfectly-ordered NiFe 2 O 4 .

Magnetic Coupling at Antiphase Boundary
In order to understand the physical origin of the reduced EMCD strength in the APB region, first-principles DFT calculations [19,20] were performed. Both α-type (P4 1 22) and β-type (Imma) B-site ordering structures were considered in DFT + U calculations. The plausibility of these crystallographic structures was discussed above with reference to HAADF STEM simulations. Further support for these structures, based on DFT calculations, is provided below (for P4 1

symmetry) and in the Supporting Information (for Imma symmetry).
In perfectly-ordered NiFe 2 O 4 that has P4 1 22 symmetry, there are five kinds of interactions between pairs of nearest neighbor magnetic ions that are connected by oxygen anions: 90°-Fe oct OFe oct , 90°-Fe oct ONi oct , 90°-Ni oct ONi oct , 120°-Fe tet OFe oct and 120°-Fe tet ONi oct (Figure 4, Table 1), where Fe tet , Fe oct , and Ni oct represent Fe ions in tetrahedral interstices, and Fe and Ni ions in octahedral interstices, respectively. 90° interactions are formed between magnetic ions on octahedral sites, whereas 120° interactions are formed between magnetic ions on tetrahedral sites and magnetic ions on octahedral  The sequence number corresponds to that indicated in Figure 4. sites. Interactions of the five magnetic pairs were analyzed using a four-state method. [21] We adopted the Heisenberg spin Hamil- where J ij are exchange coupling parameters and S  i represents a spin moment vector at site i. In our work, a negative value of J corresponds to ferromagnetic coupling. Calculated exchange parameters (J) are given in Table 1. Strong antiferromagnetic coupling is present for both of the 120° interactions, with magnitudes of 0.0352 and 0.0178 eV for 120°-Fe tet OFe oct and 120°-Fe tet ONi oct , respectively. 90° interactions with Ni 2 + on the octahedral sites tend to couple ferromagnetically, with magnitudes of −0.0014 and −0.0012 eV for 90°-Fe oct ONi oct and 90°-Ni oct ONi oct , respectively. In contrast, 90° interactions between Fe 3 + tend to be weakly antiferromagnetic (0.0012 eV), in agreement with the Goodenough-Kanamori rule. [22] All 90° interactions between octahedral sites are much weaker than 120° interactions, by approximately an order of magnitude. The overall magnetic coupling in the P4 1 22 NiFe 2 O 4 structure therefore remains the same as that in Fd3m NiFe 2 O 4 , in which magnetic ion on octahedral sites couple antiferromagnetically to magnetic ion on tetrahedral sites. Figure 4c shows a histogram of the magnetic moments of Fe atoms and Ni atoms in the P4 1 22 − APB model based on a statistical analysis of their magnitudes and directions, as indicated by red arrows in Figure 4b. In perfect domains in the P4 1 22 − APB model, Fe ions in tetrahedral interstices are antiferromagnetically coupled with Fe and Ni atoms in octahedral interstices, as expected, while the magnitudes of the magnetic moments of the Fe atoms in both the octahedral and the tetrahedral interstices are almost identical. Some Ni atoms in the perfect domains adjacent to the APB have lower magnitudes of their magnetic moments, when compared with other Ni atoms in the perfect domains. Furthermore, the magnetic moments of the Ni ions decrease from 1.6 μ B in the perfectly-ordered domains to approximately −0.1 μ B in the rock salt structure APB interlayer, which partly explains the decreased Ni EMCD signals at the APB. The vanishingly small local magnetic moment and the smaller radii of the Ni ions near the APB implies that these Ni ions take the form of low-spin Ni 4+ (d 6 ) ions. Coupling between Fe atoms in the rock salt structure interlayer is antiferromagnetic. When compared to Fe atoms in perfect domains of the P4 1 22 − APB model (Figure 4b,c), most of the Fe atoms in the APB have nearly unchanged magnitudes of their magnetic moments, while some Fe atoms have slightly lower values as a result of the presence of Fe 4+ ions in the rock salt structure interlayer, as confirmed theoretically in projected densities of states (PDOS). Plots of the PDOS for Fe and Ni ions in the perfect domain and APB interlayer can be found in Figure S4, Supporting Information. For Fe ions, the spin-down e g orbitals are partly occupied, with some e g lying above the Fermi level, corresponding to a reduced number of valence electrons at the Fe atoms in the APB interlayer and resulting in an Fe 4+ valence state (the local magnetic moment is 3.1 μ B ). For Ni ions, the t 2g orbitals are fully occupied and the e g orbitals are empty, meaning that Ni ions in the APB interlayer have very small magnetic moments.
At the APB, two types of 90° magnetic interactions were considered in the two interlayers with rock salt structures. When the magnetic ions are located on the same {111} plane, the 90°-Fe oct OFe oct interactions are weakly ferromagnetic (−0.0014 eV). However, when the magnetic ions come from different {111} planes, the 90°-Fe oct OFe oct interactions are strongly antiferromagnetic (0.0326 eV). In a region of the perfect domain that is closest to the APB, 90°-Fe oct OFe oct from different {111} planes remain weakly antiferromagnetically coupled (0.0067 and 0.0040 eV), yet 120°-Fe tet OFe oct tend to be coupled antiferromagnetically even more strongly (0.0565 eV) when compared to the perfect domain. More importantly, cations on octahedral sites in the two rock salt structure {111} layers also couple antiferromagnetically to cations on octahedral sites in the (111) layers in the adjacent perfect domains (Figure 4b).
The insertion of rock salt structure interlayers also leads to a new type of magnetic coupling (termed 170°-Fe oct OFe oct / Ni oct , as the Fe oct OFe oct /Ni oct angle is ≈170°). The DFT calculations suggest rather strong antiferromagnetic coupling for the 170° magnetic interaction, which agrees with the Goodenough-Kanamori rule (whereby the 180°-Fe oct OFe oct spin exchange is predicted to be strongly antiferromagnetic). [22] It results in antiferromagnetic coupling between cations that are located on two neighboring rock salt structure {111} planes. The antiferromagnetic coupling between the Fe ions on the octahedral sites leads to a significant decrease in Fe EMCD signal strength recorded from the APB region. The magnetic structure of the P4 1 22 − APB model deduced from calculated exchange parameters (J) in Figure 4b is the most stable structure with the lowest energy, when compared with other magnetic structures considered in Figure S5, Supporting Information.
The magnetic interactions ( Figure S8 and Table S1, Supporting Information) and the lowest magnetic states ( Figure S9, Supporting Information) of NiFe 2 O 4 in the Imma perfect domain and the Imma − APB model are discussed in detail in the Supporting Information. Similarly, the local magnetic moments of the Ni ions in the APB interlayer decrease by almost 50% as a result of their change to low-spin state Ni 4+ (d 6 ) ions. The insertion of rock salt structure layers here also leads to the introduction of strongly antiferromagnetic 170°-Fe oct ONi oct interactions, which result in antiferromagnetic coupling between Fe ions in the inserted {111} planes and therefore a decreased Fe EMCD signal from the APB region. We also analyzed the PDOS of the Imma-APB structure ( Figure S4, Supporting Information) and obtained the same conclusions.
It is worth noting that an APB with a rock salt structure potentially has a Néel temperature that is very close to or higher than room temperature, given that FeO and NiO have Néel temperatures of 198 and 523 K, respectively. In AFM/FM or FM/AFM/FM heterostructures, the ordering temperature of the AFM interlayer can be enhanced significantly or influenced by the neighboring FM layer due to the magnetic proximity effect. [23] In the present case, NiFe 2 O 4 has a Curie temperature of 865 K, which is much higher than room temperature. [24] Taking magnetic proximity effects into account, an APB with a rock salt structure buried between adjacent ferrimagnetic NiFe 2 O 4 grains is likely to have a Néel temperature above room temperature. Studies of the Néel temperature of the APB and its effects on the local magnetization are planned for future temperature-dependent experiments and theoretical calculations.

Conclusions
High-spatial-resolution EMCD and HAADF STEM imaging have been used to experimentally demonstrate reductions of 46.8% ± 8.2% and 38.8% ± 14.5% in the EMCD strengths of Fe and Ni, respectively, at a new type of APB in NiFe 2 O 4 that has a rock salt structure and a relative translation of (1/4)a[011], compared to perfectly-ordered NiFe 2 O 4 . DFT calculations and dynamical diffraction calculations suggest that the reduced EMCD strengths result from the fact that Fe ions at the APB interlayer are antiferromagnetically coupled with each other, whereas Ni ions show a significant decrease in magnetic moment as a result of the formation of low-spin state Ni 4+ (d 6 ) ions. Intensity analysis of octahedral atomic columns in HAADF STEM images suggests that only half of the octahedral interstices in the rock salt structure interlayer are occupied. Such local magnetic information with element specificity is crucial for understanding magnetic coupling within the APB. Our combined approach of using high-spatial-resolution EMCD and first-principles calculations to resolve and identify the atomic structure and magnetic coupling of an individual APB in spinel ferrite is applicable to studies of a broad spectrum of other defects in magnetic materials.

Experimental Section
NiFe 2 O 4 thin films were fabricated on MgAl 2 O 4 substrates using reactive radio frequency magnetron sputtering. The MgAl 2 O 4 substrates were first annealed at 400 °C for 30 min under vacuum prior to deposition. After 10 min of pre-sputtering, the nickel ferrite films were deposited at a substrate temperature of 300 °C. [25] Cross-sectional TEM samples were prepared using an FEI Helios NanoLab 400S FIB SEM. [26] Atomic-resolution HAADF STEM images were recorded at 200 kV in an FEI Titan G2 80-200 ChemiSTEM microscope [27] with a beam convergence semi-angle of ≈17.7 mrad and collection semi-angles ranging from 70 to 200 mrad. These experimental parameters were also used to perform HAADF STEM image simulations using Dr Probe software [28] for 188 projected potential slices and 47 unit cells in the [1 10] direction, without introducing any aberrations and with 20 variants per slice in frozen-lattice configurations. The image simulations were convoluted by a Gaussian function to take into account partial spatial coherence using a probe with a half width at half-maximum of 0.04 nm. Quantitative HAADF STEM intensity profiles were extracted with an integration width of 10 pixels, corresponding to 0.09 nm in real space, using Digital Micrograph software. Background subtraction and Gaussian function fitting were employed before integration over 0.05 nm of the intensity profiles for each atomic column. [29] All of the intensities were normalized to the average intensities of fully-occupied atomic columns in the perfect grain on the left side of the APB.
EEL spectra and EMCD experiments were performed under nanobeam illumination at 300 kV using an FEI Titan 80-300 STEM [30] with an energy resolution of ≈1.2 eV, a beam full width at half maximum (FWHM) of ≈1.8 nm, [12f] a collection semi-angle of ≈3.72 mrad and a convergence semi-angle of ≈0.65 mrad. In order to track the same APB using atomicscale imaging and in nanobeam mode, four auxiliary holes were drilled on the nanometer scale using the electron beam. EMCD signals from the APB were recorded at the intersection of connecting lines between holes 1 and 3 and holes 2 and 4 ( Figure S10, Supporting Information). Background subtraction, deconvolution of plural scattering, and postedge normalization were performed during EELS data processing. The term, after subtracting a step function, [31] was used for normalizing the spectra between the ordered area and the APB. The integration ranges used were 704-717 eV for the Fe L 3 edge, 717-731 eV for the Fe L 2 edge, 849-861 eV for the Ni L 3 edge, and 866-878 eV for the Ni L 2 edge. The sample thickness was measured using low-loss EEL spectra. Structural relaxation and electronic structure calculations were carried using DFT methods with the Vienna ab initio simulation package (VASP) [32] with the projector-augmented-wave (PAW) method. [33] The Perdew-Burke-Ernzerhof functional of the generalized gradient approximation (GGA-PBE function) [34] was used to describe the exchange-correlation energy. In order to better describe strong electron-electron correlations, the GGA with on-site Coulomb repulsion U (GGA+U) method was adopted. [35] The value of the effective on-site repulsion U was set to be 4 and 4.5 eV for Ni and Fe ions, respectively, in agreement with previous work. [36] The spin exchange interaction parameters were computed using U eff = 4.0 and 3.0 eV for Ni and Fe ions, respectively, [37] and similar results were obtained (see Table S2, Supporting Information). The energy cutoff and convergence criteria for total energy and forces were set to be 400, 10 -6 , and 0.01 eV Å −1 , respectively. Further computational details are given in the Supporting Information. For the structural model with APBs, a supercell containing two APBs in the [110] direction was constructed to fulfill periodic boundary conditions. With the available computational resources, only a distance of 15 Å between adjacent APBs could be taken into consideration. In the present structure model, the distances between periodic images of individual magnetic atoms were more than 10 Å, which should lead to negligible interactions between them. The supercell contained 640 atoms and its cell parameters were a = 35.58 Å, b = 20.47 Å, c = 11.84 Å. All of the atom positions in the structure models were fully relaxed. For the purpose of more intuitive visualization, only the original P4 1 22 and Imma structures containing 28 atoms, and P4 1 22 − APB and Imma − APB supercells containing 160 atoms, are shown in Figure 4 and Figure S8 EMCD simulations of the beam-position-dependent strength of EMCD and the non-magnetic part of the inelastic scattering crosssection were performed using MATS.v2 software [38] separately for the tetrahedral and octahedral sites and for both the Fe and the Ni 2p corelevel excitations. The DFT-relaxed atomistic APB model with a distance of 29 Å between adjacent APBs for NiFe 2 O 4 with P4 1 22 symmetry was used for the construction of a large orthogonal supercell consisting of 3936 atoms, with dimensions of 7.1 nm × 10.1 nm × 0.6 nm. The simulation parameters were chosen to match the experimental parameters, including accelerating voltage, convergence semi-angle, and probe size. The MATS.v2 summation convergence parameter was set to 2 × 10 −5 . The beam scanning step across the APB was set to 0.5 nm.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.