## Introduction

The development of multiferroic materials aims to realize direct electric-field controlled switching of magnetization at room temperature1,2. However, identifying stable and single-phase multiferroic materials to achieve this objective has been challenging because electrical polarization and magnetization must be strongly coupled together in such systems1,2,3,4,5,6. Recently, a new class of materials, called hybrid improper ferroelectrics, was theoretically proposed as a potential room-temperature multiferroic system2. The principle of the hybrid improper ferroelectric mechanism is to induce ferroelectricity, ferromagnetism, and magnetoelectricity simultaneously with the same set of lattice instabilities2,7. In particular, oxygen octahedron rotations and tilts, which are ubiquitous in perovskite materials and couple strongly to magnetism, combine with a layered crystal structure to induce polarization8,9. Hybrid improper ferroelectricity at room temperature was first experimentally confirmed in the n = 2 Ruddlesden-Popper material Ca3Ti2O710 and was subsequently observed in Sr3Sn2O7 and other layered perovskites11,12,13.

Ca3Mn2O7 (CMO) is another hybrid improper ferroelectric candidate, which was the first theoretically proposed system for achieving room temperature multiferroicity2. Ca3Mn2O7 is an n = 2 member of the Ruddlesden Popper layered perovskite family with the general formula (ABO3)n(AO) or An+1BnO3n+1. The CMO crystal structure consists of double CaMnO3 perovskite blocks stacked along the [001] direction, with an extra CaO rocksalt sheet inserted after each double perovskite block. At room temperature, CMO crystallizes in the polar A21am space group. The condensation of an out-of-phase (aac0 in Glazer notation14) octahedral tilting distortion (Fig. 1a left) and an in-phase (a0a0c+) octahedral rotation distortion (Fig. 1a right) establish the A21am symmetry and induce a polarization2. Figure 1b shows schematic views of the A21am crystal structure viewed along the a ([100] ortho) and b ([010] ortho) projections, with the polarization (P) pointing along [$$\bar{1}$$00]. The polarization arises primarily from a two-against-one displacement of the Ca ions along the a-axis in each perovskite block (Fig. 1b, right). In addition, the aac0 octahedral tilting distortion involves alternating left/right displacements along the b-axis of the Ca ions in each layer (Fig. 1b, left).

Several studies have revealed the complex domain structure and phase transition sequence of CMO. Early transmission electron microscopy (TEM) studies showed twin variants of the A21am phase with an (001) interface15. In addition, Gao et al. observed irregular orthorhombic twins with curved boundaries and further demonstrated the rearrangement of the oxygen octahedral tilts and rotations during a complex phase transition from I4/mmm, to an intermediate nonpolar phase Acaa (a0a0c), and finally to the polar A21am (aac+) phase7. This transition leads to a degeneracy of polarization orientations along either the a-axis ([100]orth) or b-axis ([010]orth) of the CMO crystal and a large number of 90° ferroelectric domain walls perpendicular to the c-axis7. Liu et al. experimentally demonstrated ferroelectric switching by measuring the ferroelectric hysteresis loop, but this could only be accomplished at a very low temperature (T < 28 K) due to the relatively high electrical conductivity of CMO16. On the other hand, the complex domain morphology of CMO leads to many novel physical properties16,17. Most notably, negative thermal expansion behavior was observed in both CMO and Ca3−xSrxMn2O7 (CSMO) crystals and was closely related to the coexistence of competing polar A21am and nonpolar Acaa phases over a large temperature range18,19. It was further demonstrated that negative thermal expansion could be tuned by controlling the phase competition between the polar A21am and nonpolar Acaa phase with different levels of Sr doping19. Although several studies have focused on understanding the macroscale properties of CMO, little has been done to uncover the structure of the polar-nonpolar phase coexistence at the atomic scale.

In this work, we uncover polar nanoregions (PNRs) in a nonpolar matrix of a layered perovskite at the atomic scale using scanning/transmission electron microscopy (S/TEM) imaging and quantify its structure with picometer precision. We further explore the underlying physics and chemistry of the phase competition transition dynamics as a function of temperature using in-situ high-resolution TEM and monochromated electron energy loss spectroscopy (EELS) techniques. We observe that the formation of Mn antisites drives the phase competition, which is a mechanism that may stabilize similar polar/nonpolar phase competition in other layered perovskite crystals and beyond.

## Results

This study explores Sr-doped CMO because the presence of the Sr cation promotes the coexistence of the A21am and Acaa phases at room temperature. The addition of 3% Sr is expected to decrease the ferroelectric transition temperature of Ca2.9Sr0.1Mn2O7 to below room temperature, and thus at room temperature, we expect a small fraction of the A21am polar phase region to be embedded in a nonpolar matrix with Acaa symmetry, as indicated by previous X-ray measurements19.

Figure 1c, d show the annular dark-field (ADF) and annular bright-field (ABF) STEM micrographs of the CSMO crystal with the nonpolar Acaa space group, respectively. The ADF-STEM shows the contrast from heavier atomic species, Ca, Sr, and Mn. The presence or absence of Ca/Sr displacements can be used to distinguish between the A21am phase, where displacements are expected (Fig. 1a), and the Acaa phase, where such displacements are not allowed by symmetry. The Ca/Sr atoms in the image do not exhibit displacement with regard to the unit cell center, which is expected since the matrix should belong to the nonpolar Acaa space group. On the other hand, we observe many nanometer-sized features along the (001) interface distributed randomly all across the sample on the TEM micrograph taken from the [100]orth zone axis (Fig. 1e). To determine the crystal structure, we obtain the selected area electron diffraction (SAED) patterns from the same sample (Fig. 1f). A row of weak spots at (0, Ɩ, Ɩ) (with Ɩ = 2n + 1) connected with near-continuous streaks is observed in the SAED patterns, suggesting the formation of a superstructure along the (001) interfacial planes, similar to the features in the electron diffraction study reported in an undoped Ca3Mn2O7 crystal with a polar A21am space group15. High-resolution transmission electron microscopy images taken in this region confirm that the weak spots and the streaks in the SAED pattern arise from the linear features within the crystalline matrix (Supplementary Fig. 1).

To directly observe the linear features at the atomic scale, we perform aberration-corrected high-resolution scanning transmission electron microscopy (AC-STEM). Numerous linear features (bright lines) along the (001) interface are observed in the ADF-STEM image in Fig. 2a. The high-resolution ADF-STEM image of the linear features reveals two distinct types of double bilayer polar nanoregions (a-type and b-type db-PNRs), as shown in Fig. 2b, c, respectively. Both types of db-PNRs consist of two adjacent double perovskite blocks (a double bilayer). Importantly, this is the first observation of db-PNRs in a layered perovskite system. In the a-type db-PNRs (Fig. 2b), the Ca/Sr atoms all displace in the same direction in the rocksalt sheet between two double perovskite blocks. In the b-type db-PNRs (Fig. 2c), we observe alternating left/right displacements of the Ca/Sr atoms. The a-type and b-type db-PNRs show displacement patterns similar to those of the CMO A21am polar ground state structure viewed from the [010] and [100] zone axis, respectively (Fig. 1b). To quantify the structural distortions, we use an atom position refinement algorithm to accurately assess the atom positions and measure the center Ca/Sr atomic displacements in the double perovskite blocks20,21. The displacement is measured by comparing the positions of the center Ca/Sr atoms and the average positions of the top and bottom Ca/Sr atoms in the double perovskite blocks (see Supplementary Figs. 2 and 3). The displacement measurement is presented in Fig. 2b, c as colored vector maps superimposed on the ADF-STEM images of the db-PNRs. Figure 2b shows the a-type db-PNRs with a relative displacement of the Ca/Sr atoms of close to 40 pm, and Fig. 2c shows the b-type db-PNRs with an alternating left/right displacement of the Ca/Sr atoms of approximately 20 pm. Additionally, as illustrated in the vector map, the center Ca/Sr atoms exhibit much larger displacements than the matrix of the crystal (close to 0 pm as indicated by purple color, also see Supplementary Fig. 4). The strain mapping from our geometric phase analysis (GPA) overlaid on the ADF-STEM image (Fig. 2d, e) shows a large strain of around 5–10% along the [001] direction. This local strain is likely the cause of the diffraction contrast in the lower magnification ADF-STEM image in Fig. 2a. The schematics of the a-type and b-type db-PNRs are superimposed on the magnified ADF-STEM image and are shown in Fig. 2f, g.

To investigate the possibility of oxygen vacancies22,23 as well as to quantify the magnitude of the octahedral tilts, we make use of ABF-STEM. The ABF-STEM mode has been widely used for imaging light elements in a wide range of materials and is especially useful in visualizing oxygen octahedral distortions in oxide materials24,25,26. In this work, our ABF-STEM images uncover the oxygen atom columns in CSMO and show an intact crystal structure at and away from the db-PNRs regions, as shown in Fig. 2h. There is no clear indication of oxygen vacancy at atomic fractions above 40%, as it would significantly modulate the intensity of the oxygen atom columns locally (Supplementary Fig. 5)27. Additionally, the ABF-STEM images clearly illustrate a negligible tilt of the oxygen octahedra in the matrix as expected in the nonpolar Acaa phase, whereas significant octahedral tilting is observed in the vicinity of the db-PNRs. We measure the oxygen octahedra tilting angle to be 158 degrees on average with a standard deviation of 4.7 degrees locally in the vicinity of the db-PNRs, as opposed to 178 degrees on average with a standard deviation of 1.6 degrees in the matrix.

To further understand the dynamics and stabilization of polar nanoregions in CSMO, we perform in-situ heating experiments inside the TEM column. Figure 3 shows the TEM micrographs of the morphology in the CSMO sample at different temperatures during heating. Initial heating of the sample starts from room temperature and goes to 200 °C, as shown in Fig. 3a. As discussed previously, the densely populated, nanometer-sized features are the db-PNRs. During heating from 200 °C to 450 °C, the location and the density of the db-PNRs are relatively stable and do not show much change as shown in Fig. 3a–c. Within this temperature range, the sample undergoes strain relaxation, and the diffraction contrast induced by strain is reduced. As the temperature is further raised to 550 °C, the db-PNRs start to disappear in the nonpolar phase (Fig. 3d). By 600–650 °C, most of the db-PNRs undergo a phase transition into the nonpolar Acaa phase (Fig. 3e, f). However, there are still a small number of db-PNRs remaining at 650 °C. Upon cooling, the db-PNRs start to reappear below 568 °C, and the densely populated db-PNRs are mostly recovered when cooling to room temperature (Supplementary Fig. 6), which shows a reversible phase transition. The oxygen vacancies may emerge during heating in the vacuum, but the reversibility of the formation of db-PNRs during the in-situ heating experiment suggests that the contrast change is less likely to be linked to the oxygen vacancies28. This observation demonstrates a reversible and gradual transition between the polar and nonpolar phases in CSMO, but the mechanism that stabilizes db-PNRs at high temperatures is still unknown.

To understand the underlying physics of the db-PNR stabilization mechanism, we use monochromated EELS to determine the local chemical environment of the Mn ions in the db-PNRs. The L ionization edges of the transition metal elements (Mn in this case) reflect the electronic transition from 2p to 3d levels, with the L3 and L2 edges showing the transition from 2p3/2 to 3d3/2 3d5/2 levels and from 2p1/2 to 3d3/2 levels respectively29,30,31,32. By performing monochromated EELS, the energy loss near edge structure (ELNES) of Mn L2,3 is resolved for fingerprinting the oxidation states across the db-PNRs in the crystal. The ELNES for Mn with different oxidations states show distinct shapes and varied onset energy33,34. On the other hand, the ELNES for Mn with the same oxidation states and different bonding environment has similar overall shapes and onset energy33. Since the overall shape of the ELNES for the same Mn oxidation state in different compounds is similar, it has been successfully demonstrated that mixed Mn oxidation states can be determined with very high resolution by fitting the experimental Mn ELNES spectra with the existing reference ELNES spectra34,35. Figure 4a is a plot of the line-scan EELS data. In Fig. 4b, the corresponding spatial locations of the electron probe are plotted on the simultaneously acquired ADF-STEM image. Figure 4c shows the averaged EELS Mn L edge spectra from the db-PNRs (red) and nonpolar matrix (blue). We perform linear combination fitting on our line-scan EELS data set; the fitting of the averaged spectra from the polar and nonpolar regions is shown in Fig. 4c. We use the ELNES spectra from MnO (Mn2+) and SrMnO3 (Mn4+) collected by Garvie et al.33 as the reference spectra. The fitting results indicate that the ELNES of the Mn L-edge at the db-PNRs shows more similarity to the Mn2+ reference spectra. The fitting results of the entire line-scan EELS data are plotted (Fig. 4d) and indicate that the Mn2+ percentage is approximately 90% at the center of the db-PNRs.

To confirm this result, we also determine the integrated intensity ratio of the L3/L2 edges of the line-scan dataset acquired from another region near the db-PNRs and obtain a ratio of approximately 2.6 and 2 in polar and nonpolar regions, respectively (Supplementary Fig. 7). The L3/L2 edge intensity ratio confirms that the oxidation state of Mn ions at the db-PNRs changes from the Mn4+ state to either Mn3+ or a mixture of Mn2+/Mn4+ states29. Additionally, we observe a much stronger Mn L edge intensity from the center rocksalt layer of the db-PNRs compared to the perovskite layer (Supplementary Fig. 8). Energy-dispersive X-ray spectroscopy (EDX) mapping of the same region confirms a higher Mn ion and a lower Ca ion content in these rocksalt sheets (Fig. 4e). Our observations indicate that Mn antisite defects form at the rocksalt layer Ca site of the Ruddlesden-Popper structure. The multi-valent Mn ions change oxidation states when forming antisite defects on the Ca sites.

To validate our hypothesis that Mn antisite defects stabilize the db-PNRs in CSMO, we perform density functional theory (DFT) + U calculations on Ca3-xMn2+xO7 for a range of x. We do not include the Sr dopants in our calculations to keep our computational cell to a reasonable size. The main effect of this choice is that A21am is the ground state structure for all Mn dopant concentrations that we consider, however, the energy difference $$\triangle E={E}_{{Acaa}}-{E}_{A{2}_{1}{am}}$$ can still inform us about how the substitution of Mn for some Ca cations impacts the relative energy of the two phases. Figure 5a shows $$\triangle E$$ as a function of Mn-dopant concentration x for Ca3-xMn2+xO7, obtained from DFT + U structural relaxations. As the dopant concentration x increases, the energy difference $$\triangle E$$ grows, indicating that the A21am structure is further stabilized with respect to Acaa. This result supports the interpretation of the experimental data that regions with Mn dopants stabilize the polar structure with A21am-symmetry.

Figure 5b–g show the Mn-O-Mn bond angles for several Mn-dopant concentrations, obtained from our DFT + U structural relaxations. The structures in Fig. 5(b–g) depict the lowest energy dopant configuration for each doping level. Other Mn-dopant configurations are shown in Supplementary Figs. 10 and 11. The n = 2 Ruddlesden-Popper A3B2O7 structure contains two symmetry-distinct A-sites: a larger A-site situated in the center of the perovskite bilayer, and a smaller A-site located in the rock salt layer. Our DFT calculations show that it is always energetically favorable to place the smaller Mn2+ dopants in the rocksalt A-sites, which agrees with prior experimental observations. Interestingly, when more than one dopant is present in the supercell (Fig. 5c–f), we find that it is energetically favorable for the Mn2+ dopants to cluster in a single rocksalt layer. We also find that as the Mn-dopant concentration increases, the bond angles bend further from 180o. This effect can be rationalized by considering the tolerance factor: a decrease in the A-site cation ionic radius decreases the tolerance factor, which leads to increased octahedral rotation angles36,37. The Shannon radii of Ca2+ and Mn2+ (in 8-fold coordination) are 1.12 and 0.96 respectively. Thus, as the Mn2+ dopant concentration increases, the tolerance factor decreases, and the octahedral tilting increases (more bending of the Mn-O-Mn bond). We further notice that the bond angles closest to the Mn2+ dopants change the most compared to their values in the undoped compound in Fig. 5b, whereas the bond angles further away from the dopant change a smaller amount.

We also use our DFT + U calculations to probe the charge state of the Mn dopants substituted on the Ca sites. In Ca3Mn2O7, the Mn cations have a formal charge of 4+, whereas we expect a formal charge of 2+ if Mn substitutes onto a Ca site. Mn2+ can adopt high- or low-spin states, which would correspond to magnetic moments of 5 $${{{{{\rm{\mu }}}}}}{{{{{\rm{B}}}}}}$$ and 1 $${{{{{\rm{\mu }}}}}}{{{{{\rm{B}}}}}},$$ respectively. Table 1 reports the magnetic moments of the Mn-dopants obtained from our DFT + U calculations. We find that for all dopant concentrations, the moment of the Mn dopants is 4.3–4.4 $${{{{{\rm{\mu }}}}}}{{{{{\rm{B}}}}}}$$, which is consistent with the Mn2+ charge state in the high-spin configuration. This agrees with the experimental observation of the existence of the Mn2+ oxidation state in the polar region. Overall, the DFT calculations exhibit excellent agreement with our experimental observations from atomic resolution STEM combined with EELS.

In conclusion, our study uncovers that the coexistence of polar A21am and nonpolar Acaa phases in CSMO leads to the formation of db-PNRs. PNRs are widely viewed as the embryo of the ferroelectric phase and are critical to the relaxor properties of ferroelectrics in ultrasonic applications due to their superior piezoelectric properties38, and the discovery of db-PNRs in a hybrid improper ferroelectric system is rare. The interpretation of the origin of the polar nanoregions in the relaxor ferroelectric materials has been challenging because of the heterogeneity over different time and length scales39,40,41. By employing atomic-resolution STEM imaging in combination with EELS and DFT calculations, this study demonstrates the polar A21am phase stabilization mechanism of db-PNRs in CSMO to be the formation of the Mn antisites on the Ca sites that increases the octahedral tilting amplitudes. This study utilizes an in-situ heating experiment to further explore the polar/nonpolar phase transition as a function of temperature and observed the presence of the db-PNRs at temperatures as high as 650 °C. Both previous studies19 and our in-situ heating experiment show that the polar A21am and nonpolar Acaa phases in CSMO exhibit a competition over a large temperature range, but the stabilization mechanism has not been clear until now. The stabilization mechanism of db-PNRs in CSMO due to Mn antisites is similar to Ti antisites producing PNRs and a switchable polarization in SrTiO342,43, as well as the recent report of Y antisites leading to room temperature ferroelectricity in yttrium orthoferrite YFeO344. Our work shows that antisite defects play an important role in stabilizing polar nanoregions in the family of layered perovskite crystals and beyond. This study provides a path toward engineering polar nanoregions and designing novel lead-free relaxor ferroelectrics in hybrid improper ferroelectric materials by tuning the stoichiometry during growth for sophisticated defect engineering.

## Methods

### Computational details

We perform density functional theory (DFT) calculations using the Vienna Ab initio Simulation Package (VASP)45,46. We employ the PBEsol exchange-correlation functional and a plane-wave basis with an energy cutoff of 600 eV47. All calculations are performed in a 48-atom computational cell with a $$6\times 6\times 2$$ Monkhorst-Pack k-point mesh to sample the Brillouin zone. For structural relaxations, a force convergence tolerance of 2 meV/$$\mathring{\rm A}$$ is used, and both lattice parameters and atomic positions are optimized. We make use of the Liechtenstein formulation of the DFT + U method48 to treat the Mn on-site Coulomb interaction and set the Coulomb and exchange parameters to be U = 4.5 eV and J = 1.0 eV, respectively, in agreement with previous work on Ca3Mn2O72. We have checked that our results are robust against reasonable variations of the U parameter. In addition, we note that in principle describing the A-site Mn2+ dopant atoms may require a different value of U compared to the Mn4+ atoms. We have checked that our results are robust against varying the U on the Mn2+ dopant atoms (while keeping U = 4.5 eV on the Mn4+ atoms). We impose G-type antiferromagnetic order in the perovskite bilayers, which is the known ground state magnetic structure of Ca3Mn2O72. For compositions with Mn dopants substituted on some Ca sites, the Mn dopant spins are placed in a ferromagnetic order. We have checked that selecting an antiferromagnetic order for the Mn dopant spins does not produce a qualitative change to our results. The DFT + U-relaxed lattice parameters of all Ca3-xMn2+xO7 compositions discussed in the main text are reported in Supplementary Table 1. We note that due to the ordered nature of the Mn dopant atoms in the periodic computational supercell, for most dopant concentrations the symmetry of the crystal structure is lower than the A21am or Acaa symmetry of Ca3Mn2O7. However, for simplicity we choose to refer to these Mn-doped structures by the symmetry of the corresponding Ca3Mn2O7 structure (A21am or Acaa) throughout the text. We use the ISOTROPY Software Suite49 for group-theoretic analysis and VESTA50 to visualize crystal structures.

### Sample preparation

The high quality polycrystalline CSMO crystals were first prepared with the solid state reaction method by mixing the powders of CaCO3 (99.99%, Alfa Aesar), SrCO3 (99.99%, Alfa Aesar), and MnO2 (99.997%, Alfa Aesar) in the stoichiometric ratio, and then pelletized and heated at 1350–1650 °C for 200 h in oxygen flow. Single-crystalline CSMO were grown by using optical floating zone growth methods (also see refs. 7, 19). The TEM samples were prepared using the Thermo Scientific Helios NanoLab Dual-Beam Focused Ion Beam (FIB) system. Before mounting the CSMO crystal into the FIB/SEM column, a thin layer of carbon was deposited for better electron conductivity and preventing charging. The regular TEM samples were prepared by first lifting out the 1–2 μm lamella from the crystal surface and mounting the lamella onto the copper V-post TEM grid. The mounted lamella was then thinned down using the 30 kV Ga ion beam to around 400 nm and subsequently to around 100 nm (electron transparent) using the 5 kV Ga ion beam. Later, 2 kV and 1 kV ion beams were applied to gently reduce the amorphization and Ga ion implantation caused by Ga ion bombardment.

The in-situ TEM samples were prepared by an ex-situ lift-out method. First, we created a trench at the crystal surface with the lamella sitting vertically in the center. Second, the lamella was thinned down to around 100 nm using the 30 kV and 5 kV ion beam. Third, we used the 2 kV ion beam to mill the sample surface for final cleaning. Then, the arms of the lamella that attaches to the trench were cut off and this step makes the electron transparent sample left free-standing in the trench. Subsequently, the crystal was carefully taken out of the SEM column and transferred to the EXpressLO workstation. Lastly, we used a glass needle with a very fine tip to lift out the sample using electrostatic force and placed the sample flat onto the windows in the SiC membrane of the Protochips heating E-chip.

### Transmission electron microscopy

The aberration-corrected scanning transmission electron microscopy (STEM) images and the electron energy loss spectroscopy (EELS) spectra were both collected using the Thermo Scientific Titan3 S/TEM equipped with a spherical aberration corrector, a monochromator, and a Gatan imaging filter (GIF). The STEM imaging was operated at the acceleration voltage of 300 kV with the probe convergence angle of 30 mrad. The ADF/ABF STEM images are acquired using HAADF and ABF detectors, respectively. The collection angle for the ADF detector is 44–244 mrad while for the ABF detector is 9–51 mrad. During the STEM-EELS data acquisition, the acceleration voltage is reduced to 80 kV for alleviating the beam damage. The EELS data was acquired in dual-EELS mode, in which each EELS data set consists of the low-loss spectra containing the zero-loss peak (ZLP) and the high-loss spectra with the core-loss edges. The collection angle for EELS data acquisition is 9.2 mrad. After activating the monochromator, the aberration correction is performed to enhance spatial resolution. The energy resolution is approximately 0.15 eV by measuring the FWHM of the ZLP. The exposure time for each probe position is 4 s.

The in-situ heating experiment was performed inside the Thermo Scientific Talos F200X TEM. The Protochips heating E-chip was first mounted onto the Protochips Fusion Select holder with the double tilt capability. After the holder was inserted into the TEM column, we first used the double tilt function to find the correct zone axis for the observation, and then start the heating process at the rate of 10 °C per minute until the temperature reached set goals. Several TEM images were captured during the heating process. After the sample reached a certain temperature, the sample position and tilt angles were adjusted to ensure the image to be acquired from relatively the same region, and then the temperature was held for around 10–15 min for the potential dynamics to take place. Multiple TEM images were taken at different magnifications every one to two minutes. The TEM images in Figs. 1 and 3 are acquired without inserting the objective aperture and at a slight defocus to enhance the contrast.

### STEM and EELS analysis

Each STEM image analyzed and presented in this work is drift corrected with a non-linear drift correction algorithm, with the input being a pair of sequentially acquired STEM images with perpendicular scanning directions51. The atom position analysis and displacement measurement were performed using the customized Matlab code21,52. The atom positions in ADF/ABF STEM images were determined by fitting a 2D-gaussian over the atomic columns and we defined the refined atomic positions as the center of the Gaussian peak. The displacement is measured by comparing the positions of the center Ca/Sr atoms and the average positions of the top and bottom Ca/Sr atoms in the double perovskite blocks (see Figs. S2 and S3). The GPA strain measurement was performed using Strain ++  software53,54. Prior to the analysis, a Hann window was applied on the STEM image input to reduce the fast Fourier transform (FFT) artifact. Then we applied FFT to the input image and obtained the power spectrum. Afterward, the Bragg-filtered images were generated by first applying two gaussian masks over the first Brillouin zone of two reciprocal lattice vectors and then perform an inverse Fourier transform. Later the phase images can be extracted from the Bragg-filtered images and the strain field can be determined from the phase images.

Before analyzing the EELS line-scan data, all energy loss axes in the high-loss spectra were first corrected according to the shift in the ZLP positions measured in the simultaneously collected low-loss spectra. The corrected high-loss EELS spectra were then denoised by employing the principal component analysis method and select the first five principal components to reconstruct the line-scan EELS data55. For the EELS white-line ratio calculation, we first subtract the continuum contribution using a Hartree–Slater cross-section and determined L3/L2 intensity by fitting Gaussian peaks and integrating the intensity within the fitted Gaussian peaks (Supplementary Fig. 12)29. The ELNES fitting was performed on monochromated EELS spectra and the oxidation state was obtained by fitting the Mn L edges as a linear combination of two reference spectra of different oxidation states. The spectra are adjusted to accommodate the different calibration of the drift tube in different TEM instrument.