3D Reconstruction of Sawtooth 180° Tail‐to‐Tail Domain Walls in Single Crystal BiFeO3

Previous studies of single crystal BiFeO3 have found a dense domain structure with alternating sawtooth and flat domain walls (DWs). The nature of these domains and their 3D structure has remained elusive to date. Herein, several sections taken at different orientations are used to examine the structure in detail, concentrating here on the sawtooth DWs using diffraction contrast transmission electron microscopy, electron diffraction, and aberration‐corrected scanning transmission electron microscopy (STEM). All DWs are found to be 180° type; the flat walls have head‐to‐head polarity while the sawtooth DWs are tail‐to‐tail with peaks elongated along the polar [111] axis, formed by neutral ( 112¯$11\bar{2}$ ) DW facets and slightly charged facets with orientations close to ( 32¯1$3\bar{2}1$ ) and ( 2¯31$\overline{2}31$ ). The neutral DW facets are Ising type and very abrupt, while the charged DW facets have mixed Néel/Bloch/Ising character with a chiral nature and a width of about 2 nm.


Introduction
Domains form in ferroic materials to minimize the total system energy, consisting of electrostatic, magnetic, and elastic components. [1] At the boundaries between them, regions known as domain walls (DWs), the material has a locally varying structure where the order parameter that characterizes functionality, (e.g., in ferroelectrics the spontaneous polarization P s ), adapts its orientation and/or magnitude over a finite distance. The configurations of ferroelectric domain walls have attracted much attention for the interesting physics they reveal. [2][3][4][5][6] Here, we examine domains in single-crystal BiFeO 3 , one of the most widely studied functional materials, which exhibits simultaneous ferroelectric, antiferromagnetic, and ferroelastic order at room temperature. [7] These BiFeO 3 crystals have been shown to contain a dense 3D network of domain walls [8][9][10] that presents several challenges in characterization and interpretation. Understanding their DOI: 10.1002/adfm.202301171 structure and formation is important for future domain wall applications, in which they may be manipulated, written, erased, and moved to play an active role in future electronic devices. [1,11] Bulk BiFeO 3 has a rhombohedral structure (space groupR3c, No. 161) below its Curie temperature T c = 1100K with lattice parameter of 3.965 Å and rhombohedral angle 89.4°, sufficiently close to 90°f or it to be considered pseudocubic (pc indexing is used throughout this work). Oxygen octahedra in each neighboring unit cell are tilted by about 14°, antiphase about the three-fold [111] axis (a − a − a − in Glazer notation). [12] Its spontaneous ferroelectric polarization P s about 100 μC cm −2 along <111> arises mostly from the displacement of the Bi ions relative to their surrounding FeO 6 cages. [13] Domain walls are classified according to the angle between the different directions of P s on each side, giving three types: 71°, 109°, and 180°. [1,7] The batch of flux-grown single-crystals of BiFeO 3 investigated here have been subject to several previous investigations, [8,9,14] all of which have revealed a dense array of parallel domain walls seen in piezoresponse force microscopy (PFM) and conventional transmission electron microscopy (TEM) as alternating sawtooth and flat bands of contrast. The complex microdomain structure in these crystals is extremely stable, exhibiting no change upon observation even in the thinnest specimens. The first structural study [8] showed the domain walls to be either 180°-or 109°-type, and due to the high predicted energy of 180°-type DWs it was proposed that they were probably 109°-type. A second study [9] of the same batch of crystals using negative C s high resolution TEM imaging found a variety of DW types, including 71°, 109°, and 180°. The most recent study [10] confirmed the sawtooth DWs to be 180°-type and showed that they could be moved with an applied electric field, but proposed that flat DWs were 109°-type. In this article we revisit the domain structure in this same batch of crystals using a combination of PFM, conventional TEM, convergent beam electron diffraction (CBED) and atomic resolution scanning TEM (STEM). We find that there are only two domains in the crystal and all DWs are 180°-type. The difficulties experienced in previous work may be explained due to projection effects when the 3D domain structure is observed in an electron transparent foil, which we overcome here by using focused ion beam (FIB) to prepare multiple sections with different orientations from the same region of crystal. Here we concentrate on the observation and analysis of sawtooth domain walls, while the structure of flat walls is explored elsewhere. [15]

Experimental Section
BiFeO 3 single crystals were obtained from the same batch used in the studies of Marti et al., [14] Berger et al., [8] and Jia et al., [9] the latter describing growth conditions in detail. In brief, crystals were grown from BiFeO 3 reacted powder in a Bi 2 O 3 /B 2 O 3 flux, cooled very slowly from 1170 to 875 K. Much of the growth took place below the paraelectric-ferroelectric phase transition at 1098 K. Only crystals grown in the top of the melt, without any contact with the Pt crucible, were harvested for further investigation. These were octagonal shaped (001) oriented crystals with sizes of half to a few millimeters, with a top surface of four (hhl) facets only a few degrees away from (001). Several well-formed single crystals were selected for this study.
For PFM a crystal was ground and polished to (001) using diamond lapping film of decreasing sizes to 0.1 μm, finishing with a dilute 0.04 μm colloidal silica solution. PFM measurements were conducted on a Bruker Dimension Icon AFM with a drive frequency around the resonance peak and a drive voltage of 2 V.
To obtain a 3D view of the domains TEM specimens were prepared by lift-out on a Tescan Amber Ga + FIB-SEM from a second crystal from the same batch in its as-grown state, with (110), (010), and (110) orientations. Cutting and thinning was performed using an ion beam energy of 30 kV, with a final low energy polish of 2 kV. The specimens were taken from a compact region on the crystal, and their orientations were verified by their relative position, selected area electron diffraction (SAED) pattern, as well as atomic resolution images. To obtain the thinnest possible TEM specimen for very high-resolution imaging, wedgeshaped lamellae were produced. STEM images were taken with a double-corrected JEOL ARM 200F STEM operating at 200 kV and beam convergence semi-angle of 21 mrad. The annular bright field (ABF) and annular dark field (ADF) detectors covered 11.5-24 and about 70-280 mrad, respectively. [5] Conventional TEM images and selected area/CBED patterns were taken with a JEOL 2100 LaB 6 TEM operating at 200 kV. Atom positions in atomic resolution STEM images were located by fitting 2D Gaussian peaks. [5,16]

Domain Type
Intriguingly, the domains in these BiFeO 3 single crystals always appear with alternating flat DWs and zig-zag sawtooth DWs, irrespective of the plane of section, or which area is chosen. The 3D nature of the domains means that it is essential to obtain views from several different directions to fully understand the structure. Four different views are shown in Figure 1, PFM on a polished (001) face and images from three TEM lamellae cut from this (001) surface, that is, examined with the electron beam along  Figure 1a showing that the domains lie parallel to (112) on the micrometer scale.
We first consider the crystal polarity and the type of domain wall. It is apparent from regions where the periodicity of the domain structure is interrupted that a sawtooth DW can curve round and become a flat DW, completely enclosing a domain (e.g., that marked in Figure 1b,e by x). This observation suggests that the sawtooth and flat DWs therefore simply have opposite polarities. We now consider the PFM and selected area electron diffraction pattern (SADPs) of Figure 1 to deduce the relationship between them.
PFM observations agree with those of Berger, [8] showing a change in the phase of the PFM signal across the domain walls for both out-of-plane (OOP) and in-plane (IP) components ( Figure 15 , all domain orientations produce a PFM response parallel to the scan direction on a (001) surface [17] (Figure S1, Supporting Information) and taking the orientation in Figure 2a as a reference any domain orientation of Figure 2e-h could give similar results, restricting the DW type to 109.5°or 180°type, but giving no further information.
Complementary information can be provided by SADPs at <110> zone axes, which have half odd-odd-odd (½ ooo) spots only when the zone axis is not perpendicular to the [111] polar axis. [18,19] These zone axes are marked in red (with ½ ooo) and blue (without ½ ooo) on Figure 2. With a selected area aperture of several hundred nm in diameter, the diffraction patterns sample many domains and the lack of ½ ooo spots in the [110] SADP, Figure 1h, therefore indicates that the polar axis is perpendicular to the electron beam in both domains. The SADPs of Figure 1f,h are marked in Figure 2 by A and B, respectively, and the condition that B must remain blue for both domains once again limits the possible domain orientations to (a), (d), (e), and (h). Furthermore, the small deviation from cubic symmetry results in splitting of spots in SAED patterns across 70.5°and 109.5°domain walls in BiFeO 3 . [18] The lack of any such splitting thus indicates that the domain walls are of 180°type. From SADP we are therefore sure that the domain walls must be of 180°type, with the same two possibilities as given by PFM, that is, Figure 2(a)+(h) or (d)+(e), but again cannot uniquely distinguish the polar axis. To obtain a definitive answer, a technique that is sensitive to absolute structure is required that can be applied to individual domains. Since Friedel's law is not obeyed by dynamical electron diffraction, [20] the intensities of ±g diffracted beams are different in polar structures, allowing the absolute orientation of the crystal to be determined. However since the SAED patterns average across several domains, this information is superposed and cannot be seen.
In CBED the electron probe can be made small enough (<10 nm diameter) to be placed inside individual domains, avoiding the averaging effect of SAED. The patterns obtained from such a measurement at the [110] zone axis are shown in   Figure 1g), while in the [110] SADP the streak is strongly inclined to the Ewald sphere, producing a subsidiary spot visible on reflections far away from the direct beam aligned with [001] (inset, Figure 1f). As described in the partner article, [15] these flat head-to-head walls form around a negatively charged Fe-rich, Bipoor monolayer that is formed on the (112) plane during crystal growth.
The tail-to-tail sawtooth DW is difficult to characterize due to its 3D nature and varying appearance when seen from different viewpoints. Its straight DW segments show that it is faceted, although in most projections the facets are seen obliquely and DWs are not edge-on. In the thinnest part of the (110) TEM specimen (top, Figures 1b and 3a) the DW has a W shape, with peaks that have a spacing of 30-40 nm and symmetrical vertices (consistent with the mirror symmetry of the diffraction pattern). However, in thicker parts of the same sample multiple peaks are seen in projection, and peaks appear in arrays that are aligned in some direction offset to the point of view (bottom Figure 1b). In the (110) TEM specimen, the structure is asymmetric, with the peaks pointing toward the polar direction. Since the average orientation of the tail-to-tail DW is dictated by the adjacent (112) flat head-tohead walls it must be, on average, a charged domain wall (CDW). However, as can be observed in Figure 1d and Figure S2 (Supporting Information), re-entrant (112) facets are clearly visible edgeon in the (110) section and since this plane contains the [111] polar axis, these are neutral domain wall facets (NDWs). Based on the stereology of these TEM observations we propose that the tailto-tail DWs form a crinkled 3D structure shown schematically in Figure 4, comprised of three-faceted peaks consisting of a (112) NDW and two CDWs with orientations close to (321) and (231).
While the energy of any given DW configuration requires the calculation of short-and long-range electrostatic and polarization/screening components for all DW facets and surrounding material, [21,22] it is not immediately obvious that a crinkled DW that has re-entrant facets (and thus a larger CDW area than a flat DW) is the lowest energy configuration. The presence of these facets therefore requires some consideration. Importantly, the 180°ferroelectric DWs are inherently more flexible than 71°or 109°ferroelectric-ferroelastic DWs, since the latter have two constraints, that is, matching of lattice planes to minimize strain [23] and continuity of oxygen octahedral rotations across the DW. [24,25] These constraints favor certain orientations and control the geometry of ferroelastic DW configurations. Conversely, both octahedral rotations and lattice strain are unaffected in principle by a change in polarization magnitude (including reversal), allowing 180°DWs to take any orientation. The formation of re-entrant facets from an initially flat tail-to-tail (112) CDW may be understood using the principle that the local energy per unit area of a DW increases with its charge density, proportional to P s ·n, where n is the unit normal. Thus, NDWs have very low energy per unit area, and CDWs have an energy per unit area that increases as n is more parallel to P s , that is, the [111] polar axis. A flat CDW is unstable if the local reduction in energy, produced by a change in CDW orientation that gives lower P s ·n, is larger than the increase in energy arising from the increased DW area (that must take place, if the average orientation of the DW remains unchanged). To understand how the crinkled surface develops, it is instructive to first consider a corrugated surface consisting of just two facets as shown in Figure 4a. Due to the angle between P s and the DW, an "up" step A rotates the local DW normal away from P s while a "down" step Z does the opposite. Therefore, energy is lowered at the A step and there is a driving force for it to expand into a re-entrant NDW facet A'A″. Conversely, local energy is increased at the Z step, which suppresses the formation of non-re-entrant NDW facets. This means that re-entrant facets readily form to reduce local energy-even though a lower total CDW wall area, and perhaps lower total energy, could be achieved with facets that form a surface that is not re-entrant.
In three dimensions, further reduction of local energy can be obtained by CDW orientations that further minimize P s ·n, that is, forming peaks rather than corrugations. Such a structure can be obtained from an initial flat CDW by an array of nodes of alternating type, labeled A and B (Figure 4b). The nodes A act as nucleation sites for NDWs that expand to form diamond-shaped re-entrant NDW facets with vertices A′, B, A″, and B as shown in Figure 4b. The 3D shape is illustrated in Figure 4c where it can be seen that A′ vertices move downward while A″ vertices move up. Although the real structure appears much less regular than the illustration of Figure 4, this model satisfies the requirement that the domain wall must be continuous and agrees with all the observations of Figure 1 (and indeed all other investigations of these crystals). [8,9]

Domain Wall Structure
Most atomic resolution studies of DW structure in BiFeO 3 have been performed using thin films, [11,[26][27][28][29][30][31][32] which displays a variety of domain structures and DWs that are very dependent on the substrate material, the misfit strain it induces in the BiFeO 3 layer and the deposition methods and conditions. In these studies 180°DWs are relatively rare, and when they are present, they are constrained by the film geometry and the other DWs with which they interact. The large area of 180°DW in the single crystals examined here, and their ability to take on complex shapes, is therefore very unusual and provides a unique opportunity for their investigation. An atomic resolution study of the sawtooth DW structure in these crystals in response to applied fields has recently been presented by Condurache et al., [10] showing that the sidewalls could be moved in response to an applied electric field while the peaks remained pinned and immobile. The flat DWs remained pinned by the charged defects at their center. Their use of <100> sections, in which DWs do not naturally appear edgeon in this crinkled structure, meant that their width and atomic structure was not readily determined.
Here, with the use of different sections, we may view DWs without projection effects, in particular in the (110) orientation the (112) NDW facets can be captured cleanly. An example is shown in the atomic resolution images of Figure 5. In the BF-STEM image Figure 5b, the (321) and (231) CDW facets are almost parallel to the plane of section, perpendicular to the electron beam, and appear as diffuse darker bands. The (112) NDW facets are seen exactly edge-on and appear as sharp dark lines along [111]. On a unit-cell level, polarization P s is commonly taken to be proportional to the negative of the displacement of the Fe atom column relative to its surrounding Bi atom columns, namely − FB . [26,[33][34][35] Atom positions were extracted from aligned image stacks using 2D Gaussian fitting. [5,15] − FB vector is calculated as the displacement of Fe atom to the geometric center of two nearest Bi atoms (Figure 5e,g), with the minus sign indicating a reversal of direction. [13] As illustrated in Figure 5e, in the (110) projection this displacement is relative to the mid-point of the line joining Bi atom columns along [001], and because the [111] polar axis lies in the image plane the full shift is observed. Figure 5c shows a map of − FB magnitudes obtained from on the ADF-STEM image of Figure 5a using 2D Gaussian fitting. [5,16] The result confirms that − FB reverses direction across the DW and has roughly the same magnitude of about 40 pm in the two domains, in good accordance with the theoretical value of 41 pm. The correlation between the CDW bands in Figure 5b,c is quite poor; while they appear very diffuse in the ABF image, they present as sharp, but irregular, DWs in the − FB map. Condurache et al. [10] found that the BF-STEM contrast of these DWs did not follow their movement under an applied field, and suggested that the contrast was due to a concentration of oxygen vacancies that had accumulated at the original position of the DW. This may also be the case here, although it is certain that the DW is seen in projection through the thickness of the TEM specimen, with varying polarization along the electron beam. (The apparent sharp location of the inclined DW segments in Figure 5c is simply due to the absence of information on − FB magnitude.) For these reasons, the − FB measurements cannot be taken as reliable at the CDWs in this projection, nor give useful information about the local polarization at the CDW. In comparison, the (112) NDW appears sharp and straight in Figure 5c and is concurrent with the dark line in Figure 5b, giving confidence that − FB measurements have meaning. Figure 5d shows a quiver plot, and Figure 5f a line plot, of − FB for the NDW highlighted in Figure 5c. No rotation of polarization is apparent at the DW; rather, FB drops to zero for a single unit cell at the DW. This neutral domain wall is thus of Ising type. The magnitude of polarization is also maintained up to the DW. This agrees both with predictions that NDWs are in general much sharper than CDWs, [1,36,37] and contrasts with observations of other CDWs in BiFeO 3 , including 180°DWs [2,30,38] and 71°/109°DWs, [30,33,39,40] as well as the tail-to-tail CDW facets described below.
The orientation of the CDW facets means that no low-index zone axis is available that would allow them to be imaged at atomic resolution and edge-on. However, they are inclined by only about 10°from the point of view in a (110) section, and by choosing the very thinnest part of the specimen we may hope to minimize projection effects and obtain a reliable measurement of − FB at the unit cell level, as shown in Figure 6. Here, the tips of two peaks are shown, corresponding to vertices of type  Figure 4, bounded by CDWs. In this projection, the (112) NDW facets lie at about 20°to the plane of section. If one of these facets is captured in the TEM lamella, there will be a 180°change of polarisation at some point in the specimen, leading to unreliable results as observed for the CDWs in the (110) section. However, with a distance between NDW facets of about 40 nm and an estimated specimen thickness of 7 nm it is unlikely that one has been captured in the region of Figure 6, meaning that the polarization should not change significantly through its thickness. Confidence in the results is bolstered by the coincidence between the DW location in the ABF image Figure 6b and − FB map Figure 6c. The average value of FB measured away from the DW is 23 pm, as expected for this (110) projection-the component of 34 pm parallel to the electron beam is not observed-and antiphase tilting of oxygen octahedra is clearly visible as a curvature of O-Fe-O-Fe-O chains in the ABF images (Figure 6g). Figure 6d-f shows enlarged parts of the quiver plot at the tip of the domain and on the sidewalls, respectively. At the tip of the domain where the DW is perpendicular to the polar axis, Figure 6d, there is no obvious trend in the orientation of P s and its magnitude drops close to zero, that is, the DW has Ising character. Figure 6f shows a region away from the tip where the 180°change in P s is uniform and the DW is flat, with a plot of the orientation and magnitude of P s in the marked region given in Figure 6h. Here, P s rotates clockwise as the DW is crossed from right (P s down) to left (P s up) and in contrast to the very abrupt change in P s seen at the NDW facet in Figure 5, this Néel-type rotation oc-curs over a width of almost 2 nm. There is also a decrease in the magnitude of P s , over the same width, dropping almost to zero at the DW center. Even though the P s component parallel to the point of view is not seen, this reduction indicates that the DW also has Ising-and Bloch-type character, with a reduced magnitude of P s and a chiral rotation. This is perhaps to be expected since the mixed Néel/Bloch/Ising-type of CDWs in ferroic materials is well established, [41] and both 71°and 109°CDWs in thin-film BiFeO 3 have been confirmed to have a chiral nature. [42,43] Interestingly, the clockwise rotation of P s seen in Figure 6f is not found in all places along the sawtooth DW, and regions can also be found with anticlockwise rotation. Between the two chiralities, an interfacial line defect forms with vortex or skyrmionic character as shown in Figure 6e (see also Figure S3, Supporting Information).

Conclusion
We have re-examined the domain structure in BiFeO 3 single crystals, using TEM imaging, electron diffraction, and atomic resolution STEM in FIB-prepared sections taken in several orientations. They have a dense 180°domain structure that is dictated by the formation of charged non-stoichiometric monolayers on (112) planes, which form flat, immobile head-to-head 180°CDWs with a spacing of about100 nm, characterized in detail elsewhere. [15] Between them, tail-to-tail 180°walls form a crinkled, sawtooth DW with peaks elongated along the [111] polar axis, bounded by facets with orientations close to (321), (231), and (112). Their www.advancedsciencenews.com www.afm-journal.de formation is driven by the reduction in the local charge per unit area for DW facets that maximise the angle between their normal n and the polar axis P s . The 70°angle between P s and the average (112) plane of the DW favors the formation of re-entrant (112) NDW facets, even though this results in a larger total CDW area. These NDWs are found to be Ising-type and very abrupt, with an unpolarised centre only a single unit cell in width. The CDW facets are chiral, mixed Néel/Bloch/Ising-type with a width of about 2 nm. These observations show the complexity that 180°d omain structures can attain in three dimensions and highlights the competing driving forces that drive their formation.

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