Mapping the complex evolution of ferroelastic/ferroelectric domain patterns in epitaxially strained PbTiO₃ heterostructures

In ferroelectric thin films, the interplay between mechanical and electrostatic boundary conditions allows for the formation of a large variety of domain structures with fascinating properties. This is particularly the case in PbTiO₃, a tetragonal ferroelectric with polarization developing along the c axis mostly due to ionic displacements. In PbTiO₃ thin films, the orientation of the polarization and arrangement into domain structures have been theoretically studied^1–6 and are described as phase diagrams with regions of different domain configurations as a function of epitaxial strain and temperature (see review by Schlom et al.⁷). When the electrostatic boundary conditions are modified and the depolarization field is introduced, the polarization configurations in PbTiO₃ become more complex; for example, in PbTiO₃/SrTiO₃ superlattices, which are periodic repetitions of PbTiO₃ and SrTiO₃ deposited on DyScO₃ substrates, the ferroelectric layers display ordered arrays of polar vortices.⁸ Additionally, the signature of Bloch polarization components was observed using resonant soft x-ray diffraction (RSXD).⁹ A “supercrystal” structure of very ordered flux closure domains was stabilized using ultrafast light pulses,¹⁰ a configuration somewhat similar to the spontaneously ordered phase observed in PbTiO₃/SrRuO₃ superlattices deposited on DyScO₃.^11,12 An incommensurate spin crystal was observed in PbTiO₃ thin films between SrRuO₃ electrodes on DyScO₃.¹³ Ferroelectric skyrmions were predicted in PbTiO₃¹⁴ and subsequently measured in PbTiO₃/SrTiO₃ superlattices on SrTiO₃ substrates,¹⁵ and polar merons were observed in mixed a₁/a₂ − a/c phase PbTiO₃ films under tensile epitaxial strain on a SmScO₃ substrate.¹⁶ All these observations demonstrate that the interplay between elastic and electrostatic energies creates structures, which can be simultaneously controlled using electric fields and light and give rise to novel phenomena, like negative capacitance.¹⁷ The studied systems are usually complicated and are characterized by mixed phases of different domain configurations.¹⁸ 

In this work, a series of samples were grown by off-axis radio frequency (RF) magnetron sputtering on (110)_o-oriented DyScO₃ substrates, with 55 unit cells (u.c.) thick bottom and top SrRuO₃ electrodes and PbTiO₃ film thicknesses ranging from 23 up to 133 u.c. (see Sec. V A for details regarding sample growth). To understand the domain configuration in such samples, both the elastic strain and electrostatic boundary conditions must be considered.

Bulk PbTiO₃ is ferroelectric below a critical temperature of 765 K, with a tetragonal structure and lattice parameters a = b = 3.904 Å and c = 4.152 Å at room temperature.¹⁹ DyScO₃ is orthorhombic with room temperature Pbnm space group lattice parameters²⁰, a_o = 5.443(2) Å, b_o = 5.717(2) Å, and c_o = 7.901(2) Å, corresponding to pseudocubic lattice parameters $apc=cpc=ao2+bo22=3.947$ Å, b_pc = c_o/2 = 3.951 Å, α_pc = γ_pc = 90°, and β_pc = 2 · arctan(a_o/b_o) = 87.187°. For (110)_o-oriented DyScO₃, the out-of-plane [001]_pc direction is equivalent to [110]_o, while the in-plane directions [100]_pc and [010]_pc are equivalent to $[1̄10]o$ and [001]_o, respectively (the subscript “pc” refers to the pseudocubic unit cell, while “o” is used to refer to the orthorhombic unit cell). In our sample series, the in-plane strain imposed by DyScO₃ on PbTiO₃ films at room temperature can thus be calculated as $apc−a0apc=−0.25%$ along [100]_pc and $bpc−a0bpc=−0.16%$ along [010]_pc, where a₀ = 3.957 Å is the equivalent lattice parameter of PbTiO₃ in the room-temperature cubic paraelectric phase. To accommodate this strain, PbTiO₃ thin films on DyScO₃ are expected to be in the a/c-phase, with regions where the c axis points out-of-plane (c-domains) as well as regions where it points in-plane (a-domains), giving rise to a ferroelastic a/c-domain configuration with 90° domain walls, as predicted in Ref. 3 and demonstrated experimentally (see, for example, Refs. 21–23).

As the in-plane lattice parameters of DyScO₃ are different along [100]_pc $(apc=3.974$ Å) and [010]_pc $(bpc=3.951$ Å), the resulting epitaxial strain along these two axes is different too. The larger b_pc lattice parameter favors the a-domains to point with their polarization axis along [010]_pc, so-called a₂-domains, compared to a₁-domains with their polarization axis pointing along [100]_pc. This results in a domain configuration composed of a₁, a₂, and c-domains with different domain population ratios depending on the epitaxial strain and film thickness; for example, Damodaran et al.²⁴ showed that the domain configuration can be modified by the epitaxial strain, going from predominantly c-domains for PbTiO₃ grown on SrTiO₃ to a/c-domain structures on DyScO₃ to a more complex hierarchical domain architecture consisting of a mixture of a/c and a₁/a₂ domain structures when grown on GdScO₃. Langenberg et al.²⁵ observed, in PbTiO₃ grown by molecular beam epitaxy (MBE) onto different (110)_o oriented rare-earth scandates, that the domain configuration can be effectively tuned by altering the strain and film thickness. For PbTiO₃ grown on DyScO₃, they observed the presence of a₂ and c domains for the thinner films, with a₁ domains appearing only for larger thicknesses. The formation of domain walls itself also costs energy. The domain wall nucleation was demonstrated to be very sensitive to the growth mode by Nesterov et al.:²² in PbTiO₃ films grown on DyScO₃ by pulsed laser deposition (PLD), the presence or absence of a₁ domains can be controlled by the growth rate, with a₁, a₂, and c domains obtained for slow-grown films, while a₁ domains were absent for fast-grown films.

Additionally to these ferroelastic domains, the electrostatic boundary conditions also play a role. The polarization charges at the surface of the PbTiO₃ layers are partially screened^26–30 by charges in the SrRuO₃ layers. Bulk SrRuO₃ is a metallic transition-metal oxide and is often used as an electrode in the ferroelectric oxides community.³¹ It is orthorhombic with room temperature Pbnm space group lattice parameters a_o = 5.57 Å, b_o = 5.53 Å, and c_o = 7.85 Å,³² corresponding to the pseudocubic unit cell parameters a_pc = c_pc = 3.924 Å, b_pc = 3.925 Å, α_pc = γ_pc = 90°, and β_pc = 90.413°. The depolarization field arising from the incomplete screening of the surface charges can lead the c-domains to split in alternating “up” (c⁺) and “down” (c⁻) domains with 180° domain walls and plays a role in the domain configuration for the thinner films. The combination of mechanical and electrostatic constraints can then result in flux-closure structures, as observed in tensile-strained PbTiO₃ thin films.^29,33,34

We address the question of domain configuration and evolution as a function of the PbTiO₃ film thickness by using a combination of atomic force microscopy, laboratory and synchrotron x-ray diffraction, and high resolution scanning transmission electron microscopy. We observe the pattern visible at the surface of the SrRuO₃/PbTiO₃/SrRuO₃ heterostructures by atomic force microscopy (Sec. II A), while x-ray diffraction measurements are used to extract the domain periodicity (Sec. II B). We find that the anisotropic strain imposed by the orthorhombic (110)_o-oriented DyScO₃ substrate creates a large asymmetry in the domain configuration, with domain walls macroscopically aligned along one of the two in-plane directions. These measurements not only demonstrate that the evolution of the domain period with film thickness deviates from the Kittel law but also reveal an additional larger period appearing for the thicker films. The origin of this larger period is then investigated using scanning x-ray nanodiffraction microscopy with high spatial resolution, highlighting the arrangement of a/c domains into superdomains (Sec. II C). The evolution of the domain configuration with increasing film thickness from flux-closure-like to a/c domains is also demonstrated by direct imaging using transmission electron microscopy, where the arrangement of the a/c domains into superdomains is further confirmed (Sec. II D). Finally, we show that above a certain critical thickness, the large structural distortions associated with the ferroelastic domains propagate through the top SrRuO₃ layer, creating a modulated structure that extends beyond the ferroelectric layer thickness. The varying length scales of these periodic phenomena reveal the hierarchy of the different energy costs at play within the PbTiO₃ layers.

Atomic force microscopy (AFM) topography images obtained on the different samples reveal that as the PbTiO₃ layer thickness increases, trenches develop at the surface of the SrRuO₃ top layer in an organized pattern (Fig. 1). For samples of 45 u.c. and below, this pattern is hardly visible, and the top SrRuO₃ is atomically flat. The pattern gets more pronounced and anisotropic with increasing PbTiO₃ layer thickness, with long and deep trenches parallel to the [010]_pc axis, and smaller trenches parallel to the [100]_pc axis, while the surface roughness stays reasonably low, with root mean square (rms) roughness values ranging from 157 to 393 pm over surfaces of 10 × 10 μm². The pattern that we observe at the surface of the SrRuO₃ top layer is comparable to what has been observed at the surface of PbTiO₃ layers grown on DyScO₃ substrates in Ref. 22 as a result of periodic ferroelastic a/c domains.

To extract the period of the distortions visible on the surface of the samples, we calculate the autocorrelation function (ACF) of the topography image and, subsequently, the fast Fourier transform (FFT) of the autocorrelation image, as shown in Fig. 1(b) for the sample with the 90 u.c. thick PbTiO₃ layer. As elaborated in Ref. 22, this method is more sensitive to the periodic distortions of the surface of the sample than a direct FFT of the topography image. The results for the different samples are shown in Fig. 1(c) and also reported in the concluding Fig. 5. Once visible, the periods along both directions ([100]_pc and [010]_pc) increase as a function of the PbTiO₃ layer thickness, with the period along [100]_pc being always larger than the period along [010]_pc.

To relate this pattern observed at the surface of the SrRuO₃ layer to the domain pattern in the PbTiO₃ layers below, we used in-house x-ray diffraction to measure the reciprocal space maps (RSM) of our different samples (Fig. 2). For the thicker films, the butterfly shape—a signature of the a/c-phase—can be recognized, with the high intensity peak of the substrate overlapping with the peak of the a-domains and the peak of SrRuO₃. The peak of the c domains appears at a lower $Q[001]pc$ value. The butterfly wings arise from the tilts in the a- and c-domains.³⁵ 

Comparing, in Fig. 2, the RSM obtained in the $Q[100]pc$–$Q[001]pc$ plane (top row) and in the $Q[010]pc$–$Q[001]pc$ plane (bottom row), we observe again an anisotropy between the two in-plane crystallographic axes [100]_pc and [010]_pc: periodic peaks are clearly visible along [010]_pc, whereas the peaks along [100]_pc are less well-defined and exhibit lower intensity. The position of these peaks was determined from intensity cuts at $Q[001]pc=1.55$ Å⁻¹, corresponding to the region of the c-domains, reported in Fig. 2(b) (see blue arrows). From the position of these peaks, we extracted the periodicity—plotted in blue in Fig. 2(c) as a function of the PbTiO₃ film thickness. The blue line serves as a guide to the eye and is obtained by fitting the evolution of the period p with film thickness t using an exponential expression, p = a · tⁿ. The best fit was obtained with n = 0.8 ± 0.1, deviating from the Kittel law^36–41 where n should be equal to 1/2 (see Sec. III for a discussion regarding this deviation from the Kittel law).

Upon further analysis of the periodic peaks appearing in the RSM, we note, in Fig. 2(b), that additional peaks appear for the thicker films (see down-facing green triangles). These additional peaks correspond to larger periods, as reported in Fig. 2(c) in green. This time, the fit gives n = 0.4 ± 0.2, much closer to what one would expect from the Kittel law. For comparison, we analyzed a 90 u.c. PbTiO₃ sample without the top SrRuO₃ electrode using piezoresponse force microscopy (PFM) measurements. We observed that in addition to the a/c domains, the out-of-plane polarization arranges into larger regions alternating between up and down polarization, forming c⁺/c⁻ superdomains (see discussion in supplementary material Sec. S1 and in Ref. 42). The period of these superdomains matches with the value obtained from the additional peak in the RSM, confirming that the additional peaks come from the arrangement of the out-of-plane polarization in c⁺/c⁻ superdomains. These additional peaks appear only for samples with a PbTiO₃ layer thicker than ∼45 u.c., which is also the thickness above which the pattern starts to appear in the topography of the surface layer, as observed by AFM.

The 90 u.c. thick PbTiO₃ layer sample was also analyzed using scanning x-ray nanodiffraction microscopy (SXDM). This technique allows us to measure raster scans of local 3D-RSMs, with a spatial resolution determined by the focused x-ray beam size,⁴² here $∼30×30nm2$ full width at half maximum (FWHM). The output of an SXDM map is thus the diffracted intensity I as a function of three reciprocal space coordinates (⁠$Q[100]pc$⁠, $Q[010]pc$⁠, and $Q[001]pc$⁠), and two direct space coordinates (⁠$R[100]pc$ and $R[010]pc$⁠) (Fig. 3).

The average 002 3D-RSM obtained by summing I over all sample positions $(R[100]pc,R[010]pc)$ is shown in Fig. 3(a). The projections along $Q[010]pc$ and $Q[101]pc$ are shown in Figs. 3(b) and 3(c). This RSM displays the “butterfly” shape characteristic of the a/c phase already evident in Fig. 2. Compared to Fig. 2, however, in Figs. 3(a)–3(c), no satellites are present: because the domain period is comparable to the nanofocused beam size in a local 3D-RSM, the number of coherently illuminated periods is insufficient to give rise to the constructive interference that generates the satellites. The colored boxes in Figs. 3(b) and 3(c) define a region of interest (ROI), corresponding to an a or c domain tilted in one of the four ⟨100⟩_pc directions. Along $Q[010]pc$ [Fig. 3(b)], we can identify the peaks of the a₂-domains with different tilts: a_I and a_II, and the correspondingly tilted c-domains c_I and c_II. Similarly, along $Q[100]pc$ [Fig. 3(c)], we can identify the peaks of the a₁-domains with different tilts: a_III and a_IV, and the correspondingly tilted c-domains c_III and c_IV.

The sum of the intensity scattered within each ROI is computed for each local RSM and plotted as a function of $(R[100]pc,R[010]pc)$ in Figs. 3(e) and 3(f). Here, each plot is labeled with the domain type and color of the respective ROI, allowing us to map the presence or absence of the different domains. Note that due to their small size, it is clusters, or “bundles” of domains rather than individual ones that are visible. By comparing the intensity maps to the topography of the sample displayed at the same scale in Fig. 3(d), one sees that the contrast is comparable, showing that the pattern observed in the topography of the SrRuO₃ top layer is related to the arrangement of the a/c domains in the PbTiO₃ layer below.

Comparing the a and c domain maps in Figs. 3(e) and 3(f), we see a clear correlation between the spatial distribution of different tilts: the spatial distribution of a_I domains matches that of the c_I domains (pink), the a_II of the c_II (green), the a_III of the c_III (yellow), and the a_IV of the c_IV (blue). Such pairing is to be expected given the crystallography of a/c twins.⁴⁴ More interestingly, a_I/c_I and a_II/c_II pairs appear to further aggregate at a larger scale into homogeneous areas, forming superdomains compatible with the observations in Figs. 2 and S1 (supplementary material).

A more direct way to image the domain structure is the use of cross-sectional scanning electron microscopy (STEM) (Fig. 4). Atomically resolved high angle annular dark field (HAADF)-STEM imaging allows us to observe the lattice and atomic displacements at the atomic level. The samples were cut and prepared for the STEM measurement so as to obtain slices in the plane defined by the [010]_pc (horizontal direction) and [001]_pc (vertical direction) axes of DyScO₃, imaging along the [100]_pc zone-axis. In the 90 u.c. thick PbTiO₃ in Fig. 4, the a-domains are clearly visible as very narrow domains separated from the larger c-domains by domain walls in the (011)_pc and $(01̄1)pc$ planes.

Focusing on a region with the domain walls in the (011)_pc planes [Fig. 4(c)], we observe the tilted a and c-domains, corresponding to an a_I-domain surrounded by c_I-domains. In the case of domain walls in the $(01̄1)pc$ planes [Fig. 4(d)], we observe an a_II-domain surrounded by c_II-domains. These correspond to the a_I/c_I and a_II/c_II pairing observed in the nanodiffraction data (Fig. 3).

Interestingly, in these two different regions, the polarization in the c-domains is always oriented down (in the [001]_pc direction), while the polarization in the a-domains is always oriented to the right (in the [010]_pc direction).

At a larger scale, the periodic pattern is directly visible in real space, and the period can be estimated by taking the FFT of the image, as shown in Fig. 4(f). Looking at the (001)_pc Bragg spot in the obtained reciprocal space map [Fig. 4(g)], one can see the superstructure with a periodicity corresponding to 40–45 nm, in perfect agreement with the values obtained from the RSM (see Fig. 5 for the comparison between values obtained by the different techniques).

In Fig. 4(g), one also recognizes the butterfly shape as observed by XRD and nanodiffraction. By selecting different regions of interest, it is then possible to reconstruct the direct space images in Figs. 4(h) and 4(i), revealing the mapping of the a and c-domains (a_I and c_I in pink, a_II and c_II in green). Although the reconstructed image is sharper for the a-domains [Fig. 4(h)] than for the c-domains [Fig. 4(i)], it allows us to confirm at a local scale and with direct imaging the result obtained by nanodiffraction: the a and c-domains organize themselves, not only in pairs but also in larger regions composed of several a_I/c_I pairs alternating with larger regions composed of several a_II/c_II pairs, forming superdomain structures.

The different periods observed in this work along [010]_pc from the AFM topography, the RSM, and the STEM images are summarized in Fig. 5(d). From this Figure, we see that two different regions can be defined: region I where no pattern is visible in the topography for the thinner PbTiO₃ films (45 u.c. and lower), and region II where the topography displays tilts and trenches, and where additional peaks appear in the RSM for the thicker PbTiO₃ films (50 u.c. and higher).

In the STEM images [Figs. 5(a)–5(c)], the domain walls are visible in the PbTiO₃ layers. The sample in region II has a clear a/c-phase domain structure [Fig. 5(a)], while the two samples measured in region I show a more complex domain pattern [Figs. 5(b) and 5(c)], sharing similarities with the flux-closure like pattern observed in PbTiO₃/SrTiO₃ heterostructures on GdScO₃.³³ We extracted the periodicity of these different domain configurations by performing FFT filtering and found 40–45 nm for the 90 u.c. thick PbTiO₃, 25–30 nm for the 45 u.c. thick PbTiO₃, and 16 nm for the 23 u.c. thick PbTiO₃, in perfect agreement with the values obtained from the RSM [see Fig. 5(d) - black dots].

Looking at the evolution of the periods as a function of film thickness, we observe that the period of the ferroelastic domain structure in the PbTiO₃ layers (blue triangles and black dots) decreases with decreasing film thickness. When considering the energetics at play in ferroelastic domains in epitaxial ferroelectric and ferroelastic films, Pertsev and Zembilgotov⁴¹ showed that the dependence of the equilibrium domain period on film thickness varies from linear to the usual square root law at very large film thickness, as the domain wall energy density varies with film thickness. At very low thicknesses, the domain size can even increase as the film thickness is reduced.^41,45,46 In our work, we did not observe such a turnover of domain size for our samples within the studied thickness range, but rather a continuous decrease of periodicity p with decreasing film thickness t through both regions as p ∼ tⁿ with n = 0.8 ± 0.1, following an intermediate behavior between Kittel’s square-root law^36–41 and Pertsev’s linear law.⁴¹ This is very close to the value obtained by Nesterov et al.²² (n = 0.68 ± 0.3). It also compares well with the results obtained in different works on PbTiO₃ epitaxially strained to DyScO₃,^8,9,12,13,18 as reported in the supplementary material (Sec. S2). Interestingly, this scaling is also observed in other materials: in K_0.9Na_0.1NbO₃ thin films epitaxially strained to (110)-NdScO₃, the domain period scales with an exponent n = 0.8 ± 0.1 across a range of film thicknesses and domain structures, from in-plane a₁/a₂ domains for very thin films, to the onset of a ferroelectric monoclinic M_C phase at larger film thickness, accompanied by a transformation of the domain configuration from stripe domains to a herringbone pattern and then eventually to a checkerboard pattern as the film thickness increases from 10 to 60 nm.⁴⁷ 

In region II, where the samples are in the a/c-phase, an additional periodicity appears, which is observed in XRD as additional periodic peaks. The evolution of this superdomain period is closer to the Kittel square-root law [Fig. 5(d)]. Nanodiffraction measurements and STEM allowed us to spatially resolve the a/c pattern in the sample with 90 u.c. thick PbTiO₃ and to demonstrate the coupling of the tilts of the a- and c-domains, revealing the organization of these domains at a larger scale: the a/c-domains organize in “superdomains,” i.e., regions composed of either a_I/c_I, a_II/c_II, a_III/c_III, or a_IV/c_IV domains. This gives rise to an additional periodicity that can be detected as additional peaks in the RSM. From the PFM measurements on a sample without the top SrRuO₃ electrode, we relate these additional periodic peaks to the 180° ferroelectric domains (c⁺/c⁻), where the out-of-plane polarization alternates between up and down, in a configuration determined by the screening of the depolarization field and electrostatic boundary conditions. Although we could not confirm it directly, one possibility is that these c⁺/c⁻ domains correspond to the superdomains observed with different a/c tilts.

In region I, the domains form a complex polarization texture, similar to the flux-closure pattern, with a period following the behavior of the a/c-domain pattern at a larger thickness and without any additional peaks or superdomain structure. The different polarization patterns appearing at the different thicknesses are the result of the best compromise to accommodate the cost of strain and depolarization field at the same time. This explains why, in this regime, it is possible to observe very complex patterns, such as flux-closure,^12,33 vortices,^8,9,18 supercrystals,¹² and incommensurate spin (polarization) structures,¹³ all with the period determined by the strain from the substrate and the layer thickness.

Our work demonstrates the presence of two regimes with different ferroelectric/ferroelastic domain configurations. For the smaller PbTiO₃ thicknesses, the combination of the effect of strain and electrostatic boundary conditions gives rise to a complex domain configuration where flux-closure, vortices, or supercrystal configurations can develop. For larger PbTiO₃ thicknesses, the a/c-phase induced by epitaxial strain is recovered, with large superdomains appearing to screen the depolarization field. The large structural distortions associated with the ferroelastic domains propagate through the top SrRuO₃ layer, creating a modulated structure that extends beyond the ferroelectric layer thickness and allowing for domain engineering in the top SrRuO₃ electrode.⁴⁸ These domain structures not only change the properties of the ferroelectric itself but can also be used to change the properties of other materials through electrostatic and structural coupling.

All the samples were deposited using our in-house constructed off-axis radio-frequency magnetron sputtering system, which was equipped with three different guns allowing the deposition of heterostructures and solid-solutions of high crystalline quality composed of up to three different materials: PbTiO₃, SrTiO₃, and SrRuO₃. PbTiO₃ thin films are typically deposited around 560 and 580 °C, in 180 mTorr of a 20:29 O₂/Ar mixture, at a power of 60 W, and using a Pb_1.1TiO₃ target with 10% excess of Pb to compensate for its volatility. SrRuO₃ electrodes were deposited in situ from a stoichiometric target at 640 °C in 100 mTorr of O₂/Ar mixture of ratio 3:60 at a power of 80 W. Huettinger PFG 300 RF power supplies are used in power control mode. The sample holder is grounded during deposition, but the sample surface is left floating.

Topography measurements were performed using a Digital Instrument Nanoscope Multimode DI4 with a Nanonis controller. Piezoresponse force microscopy measurements under ambient conditions to image the intrinsic domain patterns were performed on an Asylum Research Cypher or MFP-3D atomic force microscopes.

In-house XRD measurements were performed using a Panalytical X’Pert diffractometer with Cu Kα₁ radiation (1.540 598 0 Å) equipped with a 2-bounce Ge(220) monochromator and a triple axis detector in our laboratory in Geneva. The θ–2θ scans were analyzed using the InteractiveXRDFit software.⁴⁸ This XRD system is also equipped with a PIXcel1D detector, which is used for faster acquisition of the reciprocal space maps.

The 90 u.c. thick PbTiO₃ layer sample was further analyzed using scanning x-ray diffraction microscopy (SXDM)⁴² on the ID01 beamline at the European Synchrotron Radiation Facility (ESRF).⁵⁰ An incident x-ray energy of 9.5 keV was selected using a Si(111) double crystal monochromator with resolution Δλ/λ = 10⁻⁴. A 0.7 μm-thick tungsten Fresnel zone plate with a 300 μm diameter and a 20 nm outer zone width was used to focus the x-ray beam down to a spot size of ∼30 × 30 nm² FWHM, as measured via a ptychography scan of a known reference object.⁵⁰ The sample was placed at the 002 diffraction condition and raster scanned relative to the focused beam in 25 nm steps over a 4 × 4 μm² area using a commercial piezo scanner. A two-dimensional MAXIPIX detector positioned ∼0.4 m downstream of the sample stage was exposed for 12 ms at each sample position. The procedure was repeated at different incidence angles about the 002 condition to obtain a 5-dimensional dataset describing the diffracted intensity as a function of three reciprocal and two direct space dimensions, i.e., $I(Q[100]pc,Q[010]pc,Q[001]pc,R[100]pc,R[010]pc)$⁠. We note that reaching the 002 diffraction condition entails forming a $∼19°$ angle with the (00L) planes; since the measurements were performed with an x-ray beam parallel to [100]_pc, the direct space resolution is degraded along this direction, and features appear elongated parallel to it.

Cross-sectional lamella prepared by a focus ion beam allows the imaging of domain structures by scanning transmission electron microscopy (STEM). Experiments were acquired on Nion Cs-corrected UltraSTEM200 at 100 kV operating voltage. A convergence angle of 30 mrad was used to allow high-resolution atomic imaging with a typical spatial resolution of 1 Å. Three imaging detectors in the STEM are used to simultaneously obtain bright field (BF), annular bright field (ABF) or medium angle annular dark field (MAADF), and high angle annular dark field (HAADF) images.

For ABF-MAADF imaging, the inner-outer angles can be continuously adjusted between 10 and 20 to 60–120 mrad. Most ABF images were collected with 15–30 mrad and MAADF images with 40–80 mrad angular ranges.

We determine the periodicity of the superstructures in the PbTiO₃ layer by measuring the distances between the additional reciprocal space spots obtained after FFT. The accuracy of the measurement was estimated by considering the diffraction spot extension as the lower and upper limit for the superstructure length estimation.

The supplementary material includes a discussion about the observation of c+/c− superdomains in a sample without the top SrRuO3 electrode (Sec. S1) and a discussion about deviation from Kittel law, with the comparison with ferroelastic and flux-closure domains in other PbTiO3 thin films and superlattices on DyScO3 (Sec. S2).

The authors thank Kumara Cordero-Edwards and Christian Weymann for their support and discussions. This work was supported by the Division II of the Swiss National Science Foundation under Project No. 200021_200636. STEM experiments were supported by the EU Horizon research and innovation program under Grant Agreement ID 823717-ESTEEM3. M.H. acknowledges funding from the SNSF Scientific Exchanges Scheme (Grant No. IZSEZ0_212990).

The authors have no conflicts to disclose.

C.L., M.H., P.P., A.G., and J.-M.T. designed the experiment. C.L., M.H., and L.T. grew the samples and conducted the AFM and XRD measurements and analysis. E.Z. conducted the synchrotron x-ray nanodiffraction measurements and analysis. C.-P.S. and A.G. conducted the STEM measurements and analysis. I.G. performed additional STEM analysis. C.L., M.H., I.G., and P.P. wrote the manuscript with contributions from all authors. All authors discussed the experimental results and models, commented on the manuscript, and agreed on its final version.

Céline Lichtensteiger: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Marios Hadjimichael: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Edoardo Zatterin: Data curation (equal); Formal analysis (equal); Investigation (equal); Resources (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Chia-Ping Su: Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Iaroslav Gaponenko: Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Ludovica Tovaglieri: Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Patrycja Paruch: Conceptualization (equal); Funding acquisition (equal); Writing – original draft (equal); Writing – review & editing (equal). Alexandre Gloter: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Jean-Marc Triscone: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are openly available in Yareta at https://doi.org/10.26037/yareta:7eof5tfyjrdpblws5vsqrvf3ny.