## Introduction

Many unique properties in functional materials arise from spatially heterogeneous electronic and magnetic states. Examples include the formation of charge and spin stripes in some cuprates and nickelates and phase separation across the Mott transition of VO$${}_{2}$$1,2,3,4,5,6,7. Ferroics also have naturally occurring heterogeneities in the form of domains and domain walls8,9,10,11,12,13,14,15,16,17,18. This is especially true for Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$, a hybrid improper ferroelectric19,20 where the polarization arises from a trilinear coupling mechanism21,22,23 and abundant charged domain walls have been observed19,20,24. Atomic force and piezoforce imaging reveal the different orientations of directional order parameters and domain wall character, providing a physical playground for graph theory. Like other counterpart materials16,17, ferroelectric domain walls in Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$ are anisotropic and more conducting than their surroundings—although the domains themselves are insulating19,20,24. Theoretical modeling and electron diffraction experiments recently revealed that polarization rotates across the ferroelectric walls in a manner that is more Néel than Ising-like25. Ferroelastic walls in Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$ are linear and separate different structural twin domains20. Because the boundaries connect domains with different spontaneous strain states, mechanical compatibility conditions determine their orientation24,26,27. These ferroelastic walls are also 90$${}^{\circ }$$ ferroelectric walls. Within each twin, there also are meandering 180$${}^{\circ }$$ ferroelectric walls20.

There is considerable interest in the dynamics of domain walls. Broadband scanning impedance microscopy of hexagonal ErMnO$${}_{3}$$ reveals a domain wall response dominated by bound charge oscillations rather than free carrier conduction28. Second-harmonic generation spectroscopies also provide insight into local symmetries and chemical structure and, at the same time, are sensitive imaging tools8. X-ray photon correlation spectroscopy has uncovered ferroelectric domain wall dynamics in PbZr$${}_{0.55}$$Ti$${}_{0.45}$$O$${}_{3}$$ as well29. Despite a number of significant discoveries, almost nothing is known about domain wall phonons. Infrared spectroscopy, a powerful tool for probing phonons and their symmetries, has been unable to address the nano-scale heterogeneities of domain walls due to its poor spatial resolution. The long wavelengths of infrared light necessarily mean that, even for a diffraction-limited beam, traditional infrared techniques average over any micro- and nano-structured character of a target material, thereby reducing the sensitivity to these heterogeneities30. The development of near-field spectroscopic techniques has made real space imaging of nano-scale heterogeneity feasible, although most efforts are confined to the middle infrared due to the availability of suitable laser and broadband sources31,32. The extension into the far infrared using an accelerator-based source33 enables broadband measurement of the low-frequency response including phonons. Synchrotron-based near-field infrared spectroscopy34 thus has the potential to unlock the properties of defect states such as domain walls in complex oxides and chalcogenides. Near-field infrared spectroscopy is, of course, quite different than tip-enhanced Raman scattering in that ungerade (odd symmetry) phonon modes, which are crucial to the development of ferroelectricity and other functionalities, can be probed and analyzed.

In order to uncover the behavior of fundamental excitations like phonons at domain walls and to explore the structural distortions that they represent, we performed far infrared synchrotron-based near-field nano-spectroscopy of the local phonon response at domain boundaries in Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$ and compared our findings with theoretical models of how the order parameter evolves across the wall. Analysis of the Ti-O stretching and bending modes across the ferroelastic wall reveals subtle changes in the frequency and matrix element that we relate to the underlying modulation of the crystal structure and to the rotation of the structural order parameters. There is significant width to the residual structural distortion across the walls (60–100 nm) and semiconducting character. This research opens the door to broadband imaging of heterogeneity in ferroics and represents a first step to revealing the rich dynamics of domain walls in these systems. At the same time, it provides crucial information for the development of ultra-low-power devices, switches, polarizers, and computing architectures based upon domain walls35,36. Loss mechanisms involving phonons are also key to controlling decoherence in domain wall-based computing architectures28,37.

## Results

### Structure and order parameters across the domain boundary

The structure of Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$ consists of slabs that contain two layers of CaTiO$${}_{3}$$ perovskite separated by CaO rocksalt layers. At room temperature, Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$ crystallizes in the orthorhombic polar space group $$A{2}_{1}am$$38,39. This structure (Fig. 1a) decomposes into three distinct distortions: an out-of-phase octahedral tilt ($${a}^{-}{a}^{-}{c}^{0}$$ in Glazer notation40) that transforms like the $${X}_{3}^{-}$$ irreducible representation of the high-symmetry prototype structure $$I4/mmm$$, an in-phase octahedral rotation ($${a}^{0}{a}^{0}{c}^{+}$$ in Glazer notation) that transforms as $${X}_{2}^{+}$$, and a polar distortion that transforms like $${\Gamma }_{5}^{-}$$. Each distortion is represented by a structural order parameter with amplitude $$Q$$ and phase $$\phi$$24. For the chosen setting of the orthorhombic axes relative to the tetragonal axes, the twin domains are labeled by different settings of the space group symbol: $$A{2}_{1}am$$ (twin A) and $$Bb{2}_{1}m$$ (twin B). Figure 1b shows their structure. In twin A, the $${X}_{3}^{-}$$ tilt axis (and polarization direction) lie along [−110], whereas they lie along [110] in twin B. The $${X}_{2}^{+}$$ rotation axis lies along [001] in both twins, but the adjacent perovskite slabs have different rotation senses: in twin A, the rotations in the two slabs are in-phase (red arrows), whereas in twin B, they are out of phase. The boundaries connect domains with different spontaneous strain states, so mechanical compatibility conditions determine their orientation27. The wall symmetry is obtained by combining the symmetry elements of the two domains and those transforming one domain into the other. To go from twin A to twin B, all three structural order parameters rotate (change phase $$\phi$$) by 90$${}^{\circ }$$ (Fig. 1b, c). Thus the $${X}_{3}^{-}$$ tilt axis rotates by 90$${}^{\circ }$$, whereas the sense of the $${X}_{2}^{+}$$ rotation reverses in every other perovskite slab. At the center of the domain wall, the local structure is $$C2mm$$ (Fig. 1b), where the $${X}_{3}^{-}$$ tilt axes in slabs 1 and 2 are perpendicular to each other (tilts about [100] and [010] axes in slabs 1 and 2, respectively). The $${X}_{2}^{+}$$ rotation is unchanged in slab 1 and zero in slab 2. Thus the amplitude of the $${X}_{3}^{-}$$ order parameter remains relatively constant across the domain wall, whereas the $${X}_{2}^{+}$$ order parameter amplitude is suppressed in the middle of the path (Fig. 1d, e). Whether local $$C2mm$$ structure is realized at the center depends on wall width. If it is atomically thin, the structural change from one domain to the other will be abrupt, but if (as observed here) the walls are wide, a macroscopic region with $$C2mm$$ symmetry may be realized.

### Locating domain walls in different fields of view

Figure 2 summarizes the character of the different ferroelastic and ferroelectric domain walls in Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$ at room temperature. The ferroelastic (90$${}^{\circ }$$ ferroelectric) walls are apparent under cross-polarized light and readily identified by a color change as linear twin boundaries in an optical microscope as illustrated in Supplementary Fig. 1. They appear as faint parallel lines in higher-magnification images. These features are present in a field of view where we have both atomic force microscopic (AFM) topography and near-field infrared spectroscopy. Nano-spectroscopic line scans (where a spectrum is acquired at each pixel) are set up accordingly to cross ferroelastic walls of interest. The line scans discussed here are indicated with arrows in Fig. 2a, b. We also identified candidate 180$${}^{\circ }$$ ferroelectric walls for analysis and near-field line scans using a combination of AFM, piezoforce microscopy, and a careful examination of the ridges and topography of the crystal surface as illustrated in Fig. 2c, d. These walls are much more challenging to locate because they meander between twin boundaries. A detailed explanation of this process and a summary of our spectroscopic findings are available in Supplementary Information.

### Near-field imaging of ferroelastic domain wall phonons

Figure 3a, b displays the near-field scattering amplitude $$A$$($$\omega$$), which encodes the sample dielectric response across two different ferroelastic domain walls in the hybrid improper ferroelectric Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$. Each contour plot shows the line scan distance (the exact position of which is shown in Fig. 2a, b) vs. frequency, with the color scheme indicative of near-field amplitude. We selected these particular scans to illustrate typical wall variations. The features in $$A$$($$\omega$$) are in reasonable agreement with the far-field spectra (Supplementary Information).

A group theoretical analysis shows 72 zone-center phonons that transform as

$$\Gamma =19{A}_{1}\oplus 17{A}_{2}\oplus 19{B}_{1}\oplus 17{B}_{2}.$$
(1)

The $${A}_{1}$$, $${B}_{1}$$, and $${B}_{2}$$ modes are infrared and Raman active, whereas the $${A}_{2}$$ modes are Raman active only. By considering the atomic displacement patterns arising from the phonon eigenvectors, the broad structures with maxima at 645 and 450 cm$${}^{-1}$$ are assigned as Ti-O stretching and Ti-O-Ti bending modes, respectively. In order to associate these localized vibrations with the rotations and tilts that are crucial for the trilinear coupling and ferroelectric switching, we projected out the character of each calculated phonon. This process is described in detail below. We find that the Ti-O stretching mode has a component that transforms as $${X}_{2}^{+}$$, whereas the Ti-O-Ti bending mode transforms primarily as $${X}_{3}^{-}$$. Thus the symmetry of long-range rotations and tilts are mapped onto the more localized vibrations that are available in our experimental energy window.

Strikingly, the near-field infrared spectrum is sensitive to the ferroelastic (90$${}^{\circ }$$ ferroelectric) domain walls. Focusing first on Fig. 3a, we see that this line scan crosses a twin boundary. Both spectral amplitude and line shape are altered across the wall. This is interesting and important because domain walls in perovskites are traditionally considered to be atomically sharp boundaries41. Turning to the second set of data in Fig. 3b, a near-field scan over an independent ferroelastic domain wall again reveals a significant decrease in phonon amplitude and line width. There are also very slight frequency shifts (toward the blue) that are at the limit of our resolution. We therefore see that, while the height and width of the wall vary somewhat, the general spectral characteristics are similar.

Point spectra taken from the contour data (Fig. 3c) unveil a more traditional spectral view of the ferroelastic wall, which we can compare to that of the surrounding domain. In order to highlight spectral changes between the wall and the surroundings, we calculated a difference spectrum as $$\Delta \overline{A}$$($$\omega$$) = $$\overline{A}$$($$\omega$$)$${}_{{\rm{DW}}2}-\overline{A}{(\omega )}_{{\rm{Domain}}}$$. This quantity, which reveals average changes (indicated by overbars) in the near-field amplitude, is shown in the upper part of Fig. 3c. We find that the wall phonons have reduced amplitude and a slight blue shift. The frequency shift is at the limit of our sensitivity, which unfortunately precludes a more detailed analysis. In any case, the blue shift suggests that there is a spontaneous strain across the wall27. The strain across the wall occurs because a ferroelastic wall connects domains with different spontaneous strain states; this strain together with the modulation of order parameter amplitudes determines the local wall structure. Another property of interest is conductivity. We immediately notice that the twin boundary is semiconducting rather than metallic because there are strong phonons with no hint of a Drude response. Therefore, the ferroelastic walls are not metallic in the conventional sense—although they may be slightly more conducting than their surroundings. To verify this observation, we calculated the band gap for the bulk $$C2mm$$ structure (the hypothetical structure realized at the midpoint of the domain wall). We find that the computed gap is 0.26 eV less than that of the $$A{2}_{1}am$$ structure—not even close to closing the 3.94 eV gap in this system42. This confirms that the ferroelastic walls remain insulating.

Constant frequency cuts in the range of the Ti-O stretching and Ti-O-Ti bending modes uncover another surprising aspect of the walls (Fig. 3d). We take these cuts of the contour data at 460 and 640 cm$${}^{-1}$$, where the change in near-field amplitude between the wall and the surrounding domain as quantified by $$\Delta \overline{A}$$($$\omega$$) in Fig. 3c is strongest. Rather than an atomically sharp boundary, this direct and microscopic probe of the ungerade modes suggests that there is a significant width to the structural distortion. We find widths from 60 to 100 nm for the range of domain walls investigated. This is akin to the length scale of structural relaxations in strained epitaxial thin films. The four different types of ferroelastic domain walls in Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$ (head-to-head, head-to-tail, tail-to-head, and tail-to-tail)20 along with the prediction from Landau theory that charged walls are thicker than their neutral counterparts43 provide a natural explanation for this variation.

### Relating near-field amplitude to the order parameters

We now consider how localized phonons map onto the underlying long-range structural order parameters in Fig. 1d, e. We know that the 19 $${A}_{1}$$ phonons maintain $$A{2}_{1}am$$ crystal symmetry across the wall, so we can loosely think of the $${A}_{1}$$ phonons as excitations of the structural order parameters. Since distortions that transform like the $${X}_{3}^{-}$$, $${X}_{2}^{+}$$, $${\Gamma }_{5}^{-}$$, and $${\Gamma }_{1}^{+}$$ irreducible representations of $$I4/mmm$$ contribute to the $$A{2}_{1}am$$ structure, each $${A}_{1}$$ phonon can (in principle) excite a mixture of these four structural order parameters. We therefore performed a change of basis and projected the computed $${A}_{1}$$ phonon eigenvectors $${{\bf{e}}}_{{A}_{1}}$$ onto a basis of symmetry-adapted modes $${{\bf{e}}}_{i\tau }$$ that transform like the irreducible representation $$\tau =\{{\Gamma }_{1}^{+},{\Gamma }_{5}^{-},{X}_{3}^{-},{X}_{2}^{+}\}$$ of the high-symmetry prototype structure $$I4/mmm$$. Figure 4a displays the results of this projection. Details are in Supplementary Information.

Remarkably, certain phonons almost completely overlap with a single symmetry-adapted mode—exciting only one structural order parameter—whereas, others are a mixture. Focusing first on the low-frequency phonons with calculated frequencies 428 and 466 cm$${}^{-1}$$, we find that they transform primarily as $${{\bf{e}}}_{{X}_{3}^{-}}$$. These phonons are Ti-O-Ti bends although they involve different bond angles (Fig. 4b, c). We next consider the phonons calculated to be at 652 and 546 cm$${}^{-1}$$. They primarily overlap with $${{\bf{e}}}_{{\Gamma }_{1}^{+}}$$ and $${{\bf{e}}}_{{X}_{2}^{+}}$$. The atomic displacement patterns (Fig. 4d, e) reveal that the 652 cm$${}^{-1}$$ phonon is a $$c$$-polarized Ti-O stretch, whereas the 546 cm$${}^{-1}$$ mode is an $$ab$$-polarized Ti-O stretch. This motivates our assignment of the broad experimental structure centered at 645 cm$${}^{-1}$$ as bond stretching. The width of the spectral peak suggests that the $$ab$$-plane and $$c$$-directed Ti-O stretching modes overlap.

There are three distinct types of oxygen centers in the layered Ruddlesden–Popper structure. These include the equatorial oxygens O$${}_{{\rm{eq}}}$$, as well as two different types of apical oxygens: those that border the rocksalt layer O$${}_{{\rm{RS}}}$$ and those that lie in the middle of the perovskite slab, O$${}_{{\mathrm{P}}}$$. We find that the 466 cm$${}^{-1}$$ phonon is primarily a Ti-OP-Ti bend, whereas the 428 cm$${}^{-1}$$ mode involves both Ti-O$${}_{{\rm{RS}}}$$-Ti and Ti-O$${}_{{\rm{eq}}}$$-Ti bending motion. From this analysis, we conclude that the broad experimental peak centered near 645 cm$${}^{-1}$$ contains excitations that transform like $${X}_{2}^{+}$$ and $${\Gamma }_{1}^{+}$$, whereas the wide experimental feature centered near 450 cm$${}^{-1}$$ in the near-field spectrum transforms primarily as the $${X}_{3}^{-}$$ irreducible representation.

### Order parameter trends vs. near-field response of ferroelastic domain walls

In order to test the correspondence between these symmetry objects, we plotted the near-field amplitude of the 460 and 640 cm$${}^{-1}$$ phonons as a function of distance across the ferroelastic domain wall and overlaid predictions for how the order parameters change across the wall. Importantly, these two frequencies are most sensitive to the presence of the domain wall, and they are very near the calculated $${A}_{1}$$ mode positions (Fig. 3c). The agreement, while not perfect, has several striking aspects. Our model predicts that the $${X}_{3}^{-}$$ amplitude is relatively constant across the wall, whereas the amplitude of the $${X}_{2}^{+}$$ rotation changes significantly and is suppressed in the center. Domain wall 1 exhibits reasonable overall agreement with these predictions (Fig. 5a, b). The 460 cm$${}^{-1}$$ feature remains relatively constant across the wall, although the anticipated minimum in the fixed frequency scan at 640 cm$${}^{-1}$$ is not well pronounced. Domain wall 2 is different (Fig. 5c, d), illustrating what we have found to be typical variations. The fixed frequency near-field scan at 460 cm$${}^{-1}$$ is relatively flat across the wall—in agreement with the predictions of the $${X}_{3}^{-}$$ order parameter. At the same time, the 640 cm$${}^{-1}$$ feature is suppressed at the center of the wall, as anticipated. Overall, both $${X}_{3}^{-}$$ and $${X}_{2}^{+}$$ track the behavior of the walls fairly well albeit with some deviation. Therefore, we can loosely but not completely think of the long-range rotations and tilts as mapping onto the more localized vibrations that are available in our experimental energy window. Mixing effects (Fig. 4a) and signal-to-noise issues are the primary reasons that the agreement is not better.

This order parameter framework provides appealing insight into the spatial extent of the ferroelastic domain walls in Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$. Twin boundaries are traditionally considered to be quite narrow41,44, although recent work suggests that they may be wider than previously supposed45. In systems with octahedral rotations, the width and energy of a ferroelastic wall depends on the orientation of the octahedral rotation axes with respect to the domain wall plane41,46. In particular, symmetry constraints that require the octahedral rotation amplitude to go to zero at the center of the wall can increase the wall’s width. In Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$, as the $${X}_{2}^{+}$$ order parameter rotates by 90$${}^{\circ }$$ across the ferroelastic wall, symmetry dictates that the $${a}^{0}$$$${a}^{0}$$$${c}^{+}$$ rotation turns off in every other perovskite slab (Fig. 1c). The spatial extent of the ferroelastic walls—ranging from 60 to 100 nm in our measurements—may therefore originate from frustration of the $${X}_{2}^{+}$$ rotation. In addition to the octahedral rotation order parameters discussed here, the spontaneous strain also changes across the ferroelastic wall. This means that the elastic properties also play a role in determining the domain wall width and energy27,47. Based on experimentally reported room temperature lattice parameters38, Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$ has a small orthorhombic distortion with spontaneous strain $$\eta =(b-a)/(b+a)=8\times 1{0}^{-4}$$. The elastic energy per area of a ferroelastic wall can be estimated as $${\eta }^{2}Cd$$, where $$C$$ is the elastic stiffness coefficient and $$d$$ is the domain wall width. Taking $$C$$ from density functional theory (DFT) calculations48 and estimated wall widths from our experiments, we find the elastic energy per area to be roughly 10 mJ/m$${}^{2}$$. This relatively small elastic energy contribution may make wide walls a favorable situation.

## Discussion

In this work, we combine synchrotron-based infrared nano-spectroscopy and theoretical modeling to unlock a nano-scale view of ferroelastic domain walls in hybrid improper ferroelectric Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$. While domain walls have long been known to play a key role in determining the functionality of ferroic materials, exploration of the atomic- and nano-scale structure and properties of these walls is in its early stages, enabled by new high-resolution imaging techniques and complementary theoretical methods. In particular, we elucidate how the phonon response evolves across a 90$${}^{\circ }$$ ferroelastic wall and relate it to the underlying structural changes that occur within the wall. We find that these twin boundaries have a surprisingly large spatial extent, suggesting that ferroelastic walls are not always narrow objects.

## Methods

### Crystal growth and scanning techniques to locate domain walls

High-quality single crystals of Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$ were synthesized by optical floating zone techniques as described previously20. Surface topography was scanned using AFM, and ferroelastic domain walls were identified using a combination of visual inspection of orthorhombic twin boundaries and cross-polarized light. We also carried out piezoforce microscopy to identify ferroelectric domains and walls. Identification of common step edges and defects allow these images to be overlaid—even though they are not in the same field of view. In this manner, we are able to navigate around the surface and examine the lattice dynamics across different ferroelastic and ferroelectric domain walls.

### Synchrotron-based near-field infrared spectroscopy

Near-field infrared spectroscopy was performed using the set-ups at beamlines 5.4 and 2.4 at the Advanced Light Source, Lawrence Berkeley National Laboratory34. The apparatus is configured as an asymmetric Michelson interferometer, in which one arm consists of an AFM (Bruker Innova or Neaspec neaSNOM) and the other arm is a moving mirror. Synchrotron infrared light is focused and scattered off an AFM tip in close proximity to the sample surface. The scattered light is combined with the reference beam from the moving mirror on a silicon or KRS-5 beamsplitter and detected by a liquid helium-cooled Ge:Cu detector. Scanning the mirror of the reference arm creates an interferogram, which is Fourier-transformed to obtain both amplitude and phase information, which is related to the real and imaginary parts of the optical dielectric function of the material. The incident light is $$p$$-polarized with respect to the sample such that the polarization is parallel to the tip axis. This configuration enables the most efficient coupling to the antenna modes of the tip, but there is also a component of polarization in the plane of the sample due to the focusing angle of the off-axis parabolic mirror. However, the strong enhancement of the metal tip localizes and enhances the optical field linearly polarized parallel to the tip axis, such that the technique is most sensitive to phonon and vibrational modes perpendicular to the sample surface49. To discriminate the near-field signal from the far-field background signal, the AFM is operated in non-contact (tapping) mode at a typical frequency of 250–300 kHz, and the corresponding detector signal is demodulated at twice the tapping frequency. Our measurements were performed with typical free oscillation tip amplitudes in the 80–100-nm range with a setpoint between 70% and 78% of the free-tapping amplitude, such that the engaged tapping amplitude values were in the 55–75-nm range. Routine approach curves with free oscillation tip amplitudes in the 80–100-nm range taken on gold indicate a near-field tip enhancement within $$<$$30 nm of the surface with little-to-no second-harmonic signal when the sample is withdrawn further. Different setpoints within the 70–78% range have minimal-to-no effects on the sample amplitude or shape. The setpoint is typically chosen to have the least damping (i.e., closer to 78%) while still maintaining good AFM feedback and reproducibility between the forward and backward traces. This approach minimizes tip wear as well.

Our near-field infrared measurements were carried out at room temperature over a frequency range between 330 and 800 cm$${}^{-1}$$. The near-field spectra are corrected for the limited transmissivity of the entire set-up by normalizing the power spectrum of the sample to that of a gold reference mirror. Both second- and third-harmonic signals were analyzed. The spatial resolution of this technique is limited by the AFM tip radius, which is typically $$<$$25 nm for the tips used (Nanosensor PtSi-NCH). Our line scans employed a step size of 20 nm. Multiple line scans across the same domain wall are very similar in character with variances on the order of the noise level. Scans across different domain walls reveal some variations, which is why we show the results for two different walls in this work.

### Far-field infrared spectroscopy

We also carried out traditional, far-field spectroscopy using a suite of Fourier transform and grating spectrometers covering the spectral range from 15 to 65,000 cm$${}^{-1}$$ for comparison with the near-field infrared response. Data were collected in reflectance mode in the $$ab$$-plane and along the $$c$$-direction. A summary of mode positions and assignments is available in Supplementary Information.

### Symmetry analysis and lattice dynamics calculations

DFT calculations were performed using projector augmented wave pseudopotentials50 and the PBEsol functional, as implemented in VASP51. We used the (Ca_sv, Ti_sv, O) VASP pseudopotentials with electronic configurations 3$${s}^{2}$$3$${p}^{6}$$4$${s}^{2}$$, 3$${s}^{2}$$3$${p}^{6}$$4$${s}^{2}$$3$${d}^{2}$$, and 2$${s}^{2}$$2$${p}^{4}$$, respectively. Calculations were performed in a 48-atom supercell commensurate with both orthorhombic twins of Ca$${}_{3}$$Ti$${}_{2}$$O$${}_{7}$$ with a $$6\times 6\times 2$$ Monkhorst–Pack $$k$$-point mesh and a 600 eV plane wave cutoff. All structural relaxations were performed with a force convergence tolerance of 2 meV/$$\mathring{\rm{A}}$$. The theoretical curves across domain walls were obtained by considering a path through the bulk energy surface connecting the two domains and calculating a sequence of bulk structures along that path. This was done using nudged elastic band calculations, allowing the ions to relax at each image along the path—between the $$A{2}_{1}am$$ domain to the midpoint structure $$C2mm$$ and into the $$Bb{2}_{1}m$$ domain. Note that we are considering a sequence of bulk structures along the path through the bulk energy landscape that connects the two domains; we are not taking into account order parameter gradients and strains, which are known to be important at the domain wall. While this level of theory does not permit quantitative predictions of, e.g., domain wall widths, we find that it places a simple focus on the evolution of the structural order parameters across the domain wall. Phonon frequencies and eigenvectors were calculated using density functional perturbation theory52. We performed group theoretic analysis with the aid of ISOTROPY53 and the Bilbao Crystallographic Server54,56,57,57 and visualized crystal structures using VESTA58.