Charged domain boundaries stabilized by translational symmetry breaking in the hybrid improper ferroelectric Ca$_{3-x}$Sr$_x$Ti$_2$O$_7$

Charged domain walls and boundaries in ferroelectric materials display distinct phenomena, such as an increased conductivity due to the accumulation of bound charges. Here, we report the electron microscopy observations of atomic-scale arrangements at charged domain boundaries in the hybrid improper ferroelectric Ca$_{2.46}$Sr$_{0.54}$Ti$_2$O$_7$. Like in the prototype improper ferroelectric YMnO$_3$, we find that charged domain boundaries in Ca$_{2.46}$Sr$_{0.54}$Ti$_2$O$_7$ correspond to out-of-phase boundaries, which separate adjacent domains with a fractional translational shift of the unit cell. In addition, our results show that strontium ions are located at charged domain boundaries. The out-of-phase boundary structure may decrease the polarization charge at the boundary because of the ferrielectric nature of Ca$_{2.46}$Sr$_{0.54}$Ti$_2$O$_7$, thereby promoting the stabilization of the charged state. By combining atomic-resolution imaging and density-functional theory calculations, this study proposes an unexplored stabilization mechanism of charged domain boundaries and structural defects accompanying out-of-phase translational shifts.


Introduction
The physical properties of ferroelectric domain walls and boundaries are sometimes different from those of domains themselves [1][2][3] ; examples include ferroelectricity at the antiphase boundary of an antiferroelectric 4 and polar ferroelectric structures at ferroelastic boundaries 5 . Domain walls can have higher or lower conductivity than domains, a feature that could be used in high-density storage and switching devices 2,6,7 . In particular, charged domain walls, in which electric polarization vectors are opposed between adjacent domains, have attracted considerable attention because these walls could induce an enhanced conductivity in ferroelectric materials 8 . Charged domain walls are energetically unfavorable 9 ; however, they are obtained by combining other structural parameters because ferroelectricity is not a primary order parameter in materials known as improper ferroelectrics 10 . For instance, improper ferroelectric hexagonal manganese oxides (RMnO3: R = Sc, Y, or Ho-Lu) exhibit peculiar cloverleaf domains with head-to-head or tail-to-tail domain walls involving polarizationdependent conductivity [11][12][13] , which serves as a bias switching for devices 14 .
Recently, perovskite oxides whose ferroelectricity is induced by more than two nonpolar octahedral tilts (hybrid improper ferroelectricity) have been extensively studied because of the possibility of achieving giant magnetoelectric effects 15 . Through the exploration of hybrid improper ferroelectrics, numerous peculiar charged domain walls were found in Ca3-xSrxTi2O7 (x = 0.54) 16 . This material has a layered perovskite structure with the A21am (No. 36) space group, which comprises a stack of perovskite and rock-salt layers, as shown in Figure 1. The structure is classified as a Ruddlesden-Popper phase (Can+1TinO3n+1) with n = 2. The existence of two nonpolar TiO6 tilts leads to Ca/Sr displacements along the [100] axis, inducing ferroelectricity accompanying charged domain walls. Notably, the number of charged domain walls in the crystal is substantially increased through Sr substitution 16 . However, the microscopic structure of the charged domain walls remains elusive. In particular, the effect of Sr substitution has not been revealed, which motivated us to perform atomic-resolution scanning transmission electron microscopy (STEM) with energy-dispersive X-ray spectroscopy (EDS).
Another motivation for this work is to reveal the role of crystallographic defects in Ca3-xSrxTi2O7.
For RMnO3, recent extensive studies have revealed that antiphase boundaries, which separate adjacent domains with a translational shift, have played an important role in forming charged domain walls [17][18][19][20] .
Translation domains (i.e., domains in which the translational symmetry is broken) are formed due to the trimerization of MnO5, which results in the stabilization of charged ferroelectric domain walls 11 . However, in the improper ferroelectric Ca3-xSrxTi2O7, the existence and effect of translation domains have not been observed at the atomic scale.
Here, we observed atomic-scale local structures to unveil the stabilization mechanism of the numerous charged defects in Ca2.46Sr0.54Ti2O7. The experimental results show that some charged boundaries are out-of-phase boundaries (translational boundaries) that contain Sr ions at the boundaries.
Charged out-of-phase boundaries separate both the ferroelectric polarization directions and the crystallographic domains with phase shifts. Notably, an out-of-phase boundary is defined as a boundary between two regions of a crystal displaced by a fractional translation of a unit cell [21][22][23] . In particular, an out-of-phase boundary is called an antiphase boundary when the displacement phase is π (i.e., half of the lattice constant). Also note that, in this paper, a domain boundary represents a crystallographic defect that separates two ferroelectric domains with opposite polarization directions while a domain wall means a ferroelectric domain boundary without a compositional change. The out-of-phase displacement allows local electric polarization vectors of adjacent domains to align in the same direction at the boundaries, which stabilizes the charged domain boundaries. This study proves that charged domain boundaries are correlated with crystallographic defects in Ca2.46Sr0.54Ti2O7.

Results and discussion
Dark-field images of ferroelectric domains. To explore charged structural defects in the hybrid improper ferroelectric Ca2.46Sr0.54Ti2O7, its (11 ̅ 0) plane was observed. Observations from the [11 ̅ 0] direction are expected to be crucial because numerous charged defects run along this direction, which is at 45° to the macroscopic polarization directions 16 . Figure 2a,b shows dark-field images in the (11 ̅ 0) plane, which depict the characteristic defects and the ferroelectric domains with polarization pointing toward the left-and right-hand side of the image, as denoted by the blue and red arrows, respectively.
The image was obtained under a two-beam condition by exciting a Bragg reflection, which causes the breakdown of Friedel's law [24][25][26] . The imaging method identifies ferroelectric domains as bright and dark areas when the ferroelectric polarization is parallel and antiparallel to the Bragg reflection used in the dark-field imaging, respectively. Thus, the images demonstrate that the specimen has numerous neutral 180° and charged head-to-head and tail-to-tail ferroelectric domains. Curved charged structural defects exist, as reported in Ref 27 . However, sharp straight charged domain boundaries are frequently formed, as indicated by the squares labeled 3a and 3b. Moreover, broad domain boundaries are also visualized (labeled S6a). Atomic-resolution observations were performed to reveal these characteristic defects. Figure 3a shows the high-angle annular dark-field STEM (HAADF-STEM) image of the corresponding area indicated in Figure 2. The lattice structure can be described using five ions between the rock-salt layers, as depicted in Figure 1:

Atomic-resolution observations in charged domain boundaries.
The Ca, Ti, Sr, Ti, and Ca ions are arranged along the [001] axis. The dark lines correspond to the rocksalt layer, since the rock-salt layer has less atomic density than the perovskite layer. Noticeably, the left region is misaligned compared with the right region. The boundary is pinched at an intergrowth with seven layers (yellow arrowhead, Figure 3b), which agrees with the contrast of the region labeled 3b in From the microscopy study, it was also found that another out-of-phase boundary with one-layer shift coincides with a head-to-head charged boundary involving Sr segregation (Supplementary Figure 1 It is important to explain why charged domain boundaries are formed at out-of-phase boundaries. Ferroelectric domains are formed as a consequence of the interplay between the wall energy and the depolarization fields induced by the bound charges due to the electric polarization 28 . Charged domain walls (boundaries) are energetically unfavorable compared with neutral domain walls because they create bound charges, which increase the electrostatic energy of the wall. Bulk Ca2.46Sr0.54Ti2O7 is a ferrielectric material whose polarization is caused by the combination of the two left displacements and one right displacement in perovskite and rock-salt layers, respectively ( Figure 1). When the structure is Furthermore, the difference between Ca2.46Sr0.54Ti2O7 and YMnO3 can also be explained. In YMnO3, the trimerization of MnO5 causes cloverleaf charged ferroelectric domain walls due to the structural phase transition 11 . The cloverleaf domains comprise ferroelectric domain walls that coincide with antiphase boundaries because ferroelectric walls with antiphase boundaries have lower energy than only ferroelectric walls or antiphase boundaries 11,17 . Although YMnO3 and Ca2.46Sr0.54Ti2O7 share several properties, such as the formation of charged defects, translation domains, and improper ferroelectricity due to structural order parameters, the origin of the charged domain boundaries accompanying the translation domains in Ca2.46Sr0.54Ti2O7 differs from that in YMnO3. The out-of-phase boundaries in Ca2.46Sr0.54Ti2O7 are not induced by the phase transition from the I4/mmm to the A21am space group because the shear structure of the layer observed in Figure 3 is not related to the symmetry of the phase transition. This is significantly different from the antiphase boundary due to the transition in YMnO3.
The result obtained for Ca2.46Sr0.54Ti2O7 is also supported by the presence of Sr ions at the boundaries because the phase transition will not move Sr, which would take their positions after the material has become crystalline. Thus, the out-of-phase boundaries should exist in the I4/mmm paraelectric phase. Therefore, as the temperature is decreased, charged boundaries are selectively formed at the out-ofphase boundary positions because of the above-explained energy gain at the ferroelectric transition.
Notably, in planar defects with dislocations, one domain is shifted with respect to the adjacent domain, and this shift is characterized by a displacement vector 30,31 . If the displacement vector is a translation vector of the disordered structure, the boundary is known as an antiphase boundary. By contrast, if the displacement vector is not a lattice translation vector, the boundary is known as a stacking fault. As shown in Figure 3, the domains are shifted by fractions of the unit cell, demonstrating that the displacement vector is not a lattice translation vector. Thus, the out-of-phase boundary observed in Figure 3 is also defined as a stacking fault. This study thus reveals charged stacking faults in ferroelectrics, which have not been reported in layered perovskite structures. The displacement vector of the stacking fault could not be determined here because of the diffraction contrast due to the ferroelectricity and the low symmetry of the structure via dark-field imaging. However, Figure 3 shows that the displacement in the (11 ̅ 0) plane occurs only along the [001] direction. Besides, the HAADF-STEM images shown in Supplementary Figure 4 indicate that no displacement can be seen in boundaries for which the dark-field image and the STEM images display boundary contrast when the (001) plane is investigated. These images also show that this type of boundary runs along the [11 ̅ 0] direction, and the boundary is a planar defect.

Charged domain boundaries with rotated out-of-phase domains.
Another structural defect is observed in Figure 4a, in which a head-to-head boundary is formed. Figure 4b-f shows magnified HAADF-STEM images and corresponding EDS maps around the defect. Similar to Figure 3, the rock- axis in the HAADF-STEM image, whose contrast is characteristic of the rock-salt layer. The elemental maps in Figure 4c demonstrate that these edges comprise Ca ions parallel to the [001] axis. Since this region has Ca layers that correspond to the rock-salt layer along the [001] direction, the region is considered a 90°-rotated domain whose [001] axis is perpendicular to the crystal [001] axis. The darkfield image in Figure 4 indicates that this boundary has a head-to-head charged configuration, as illustrated by the arrows in Figure 4a. These SrTiO3 regions seemingly stabilize the charged domain boundary. Contrary to Ca3Ti2O7, the perovskite structure of simple cubic SrTiO3 is nonpolar. Thus, it is energetically more favorable to form the SrTiO3 perovskite structure at the charged domain boundary.
The presence of a nonpolar region increases the stability of the charged domain boundary because these regions separate oppositely polarized domains. Thus, these results reveal that the local structures and origins of the charged domain boundaries in this case are different from those illustrated in Figure 3, although they are formed in the same crystal.
Finally, the effects of Sr substitution in Ca2.46Sr0.54Ti2O7 are explained. Previous polarizationhysteresis measurements have revealed that the spontaneous polarization magnitude decreases as the Sr content increases 16 . Such a reduction occurs because the structure of Sr-substituted Ca3-xSrxTi2O7 is nonpolar (space group P42/mnm for 0.915 < x < 1 and I4/mmm for x > 1) 32 . Accordingly, the increased Sr content reduces the octahedral tilt in the crystal structure, which decreases the polarization.
Additionally, Sr substitution results in a decrease in the polarization through the formation of SrTiO3 regions in a crystal. Our observations show out-of-phase boundaries that comprise the SrTiO3 perovskite structure. Furthermore, a large perovskite region with higher Sr content than the matrix region was observed as shown in Figure 2a and Supplementary Figure 6. The areas with the SrTiO3 perovskite structure have no polarization because of the nonpolar simple cubic structure of SrTiO3. Hence, the extended perovskite region reduces the spontaneous polarization, which is defined as the electric polarization per volume.

Conclusions
The dark-field and HAADF-STEM images prove that the crystal has distinct structural boundaries and defects. The elemental maps show the presence of out-of-phase boundaries, which contain Sr ions.
Furthermore, it was demonstrated that charged domain boundaries correspond to out-of-phase boundaries because the ferrielectric structure compensates for the polarization charge at the boundaries by means of translational shifts. Notably, this study reveals that some charged boundaries considered as charged domain walls in a previous work 16 could actually be charged defects. The structural differences between charged walls and defects are evidenced in their macroscopic behaviors. For example, charged domains walls can be moved via an external electric field, whereas domains separated by charged defects remain stationary under an electric field higher than the coercive field because they are pinned by Sr atoms. Besides, in charged defects of ferroelectrics, the valence states of cations are found to be different from those of domains 33 . Thus, the charged out-of-phase boundaries presented in this study may have different valence states of Sr and Ti ions at the boundaries. These properties will be explored in future works. Our results provide mechanistic insights into the microscopic structure of boundaries and the Sr substitution effect on the properties of the layered perovskite oxide Ca3-xSrxTi2O7. Moreover, this study shows that atomic-resolution elemental mapping is important for understanding structural defects and macroscopic properties in ferroelectric materials.

Scanning transmission electron microscopy
Atomic-resolution HAADF-STEM and EDS-STEM mapping were performed using a transmission electron microscope (JEM-ARM200F, JEOL Co., Ltd., Japan) equipped with a spherical aberration corrector and double silicon drift detectors. The acceleration voltage, probe semiangle, and current were 200 kV, 22 mrad, and 60 pA, respectively. The solid angle for the whole collection system was ~1.96 Sr.
in the EDS mapping. The angular detection range of the HAADF detector for scattered electrons was 90-170 mrad. EDS-STEM images were processed using Fourier filtering for noise reduction 34 . The single crystal was grown via the floating zone method. Thin specimens along the (11 ̅ 0) plane were prepared via focused ion beam at 30 kV up to a thickness <100 nm. Subsequent Ar-ion milling at 4 kV and an incident angle of 8° was used to remove the damage from the specimen and reduce the specimen thickness.
The polarization direction vector Ps of each ferroelectric domain was determined from the contrast of the dark-field images, which were captured under two-beam conditions 24,25 . In these conditions, the intensity of a Bragg reflection is different from that of a reflection with space inversion ̅̅ ̅ , and the breakdown of Friedel's law occurs. Consequently, the contrast in the ferroelectric domain can be expressed as Ps • : The dark-field image indicates ferroelectric domains as bright areas when the Bragg reflection is parallel to the polarization. Conversely, ferroelectric domains are depicted as dark areas when the reflection is antiparallel to the polarization. The contrast disappears when the polarization is perpendicular to the reflection. The disappearance of the contrast was confirmed by using the 008 reflection, as shown in Supplementary Figure 7.

Density-functional theory calculations
All calculations were conducted based on the spin-polarized density-functional theory using the Vienna ab initio simulation package (VASP 6.1.0) 35 . The core-valence electron interaction was described using the projector augmented wave method 36 , and the exchange-correlation part was treated within the Perdew-Burke-Ernzerhof generalized gradient approximation 37