# Non-Ising domain walls in c-phase ferroelectric lead titanate thin films

Weymann Christian  Departement of Quantum Matter Physics, University of
Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland  Cherifi-Hertel
Salia  Université de Strasbourg, CNRS, Institut de Physique et Chimie des
Matériaux de Strasbourg, UMR 7504, Strasbourg, 67000, France  Lichtensteiger
Céline  Departement of Quantum Matter Physics, University of Geneva, 24 Quai
Ernest-Ansermet, CH-1211 Geneva 4, Switzerland  Gaponenko Iaroslav
Departement of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-
Ansermet, CH-1211 Geneva 4, Switzerland  Dorkenoo Kokou Dodzi  Université de
Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg,
UMR 7504, Strasbourg, 67000, France  Naden Aaron B  School of Chemistry,
University of St. Andrews, United Kingdom  Paruch Patrycja  Departement of
Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211
Geneva 4, Switzerland

(August 27, 2024)

###### Abstract

Ferroelectrics are technologically important, with wide application in
micromechanical systems, nonlinear optics, and information storage. Recent
discoveries of exotic polarisation textures in these materials, which can
strongly influence their properties, have brought to the forefront questions
about the nature of their domain walls – long believed to be primarily Ising,
with locally null polarisation. Here, combining three complementary techniques
– second harmonic generation microscopy, piezoresponse force microscopy, and
transmission electron microscopy - to cover all the relevant lengthscales, we
reveal the Néel character (non-Ising polarisation oriented perpendicular to
the wall) of $180\text{\,}\mathrm{\SIUnitSymbolDegree}$ domain walls in
c-phase tetragonal ferroelectric lead titanate epitaxial thin films, for both
artificial and intrinsic domains at room temperature. Furthermore, we show
that variations in the domain density – detected both optically and via local
piezoresponse, then quantified by radial autocorrelation analysis – can give
us insight into the underlying defect potential present in these materials.

Ferroelectric materials are characterised by their spontaneous and reversible
electrical polarisation, with a wide variety of technological applications as
microelectromechanical systems (MEMS) and memory devices [1]. The
electromechanical properties of these materials are closely linked to their
domain structure. As such, understanding and controlling the arrangement of
domains with different polarisation orientation remains a key research
objective. In ultrathin films, where strong depolarising fields destabilise
the polarisation and drive the formation of ever smaller domains with
increasing densities of domain walls, strongly modified dielectric and
piezoelectric responses [2, 3, 4] as well as effective negative capacitance
[5, 6] have been observed. In addition, the domain walls themselves can
present emergent properties [7, 8, 9] potentially attractive for domain-wall-
based nanoelectronics [10, 11]. Their complex (de)pinning dynamics when driven
by an electric field [12] have in particular given rise to varied
implementations as nanoscale active device components [13, 14].

While ferroelectric materials were long believed – in contrast to their
ferromagnetic counterparts – to present essentially Ising type
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ domain walls, in which the
polarisation is maintained along the bulk axis, decreasing to zero and
reversing its orientation at the domain wall centre, this is now recognised
not to be the case. Extensive theoretical studies [15, 16, 17, 18, 19, 20, 21]
predicted non-Ising polarisation components oriented either perpendicular to
the wall (Néel type) or in the plane of the wall (Bloch type). Scanning
transmission electron microscopy and second harmonic generation measurement at
room temperature demonstrated the existence of Néel-type domain walls in
Pb(Zr,Ti)O3 [22, 23, 24], and of chiral Bloch walls and Bloch lines, a
particular type of ferroelectric topological structures, in LiTaO3 [24].
Additional exotic polarisation structures, such as arrays of polar vortices
[25], ‘supercrystals’ of highly ordered flux closure domains [26, 27, 28], and
even ferroelectric skyrmions [29, 30] have also recently been reported. In the
canonical tetragonal ferroelectric PbTiO3, hosting many of these emergent
exotic polarisation textures, various theoretical calculations predict either
Néel walls [15, 31] like those observed in Pb(Zr0.2Ti0.8)O3 [23, 24], of which
it is one of the parent compounds, or a low temperature Ising-to-Bloch
transition at the domains walls [20]. Metastable bubble domains observed in
PbTiO3 thin films [32], with no strong delineation between the domain wall and
interior, further suggest that the polarisation in this material can be
extremely ‘soft’ and easy to reorient in either Bloch- or Néel-like textures
under specific electrostatic and strain boundary conditions.

In this context, we set out to characterise the nature of
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ domain walls in PbTiO3. Our work is
based on three 50 nm PbTiO3 thin films deposited by rf-magnetron sputtering on
LaNiO3 electrodes on single crystal (001) SrTiO3 substrates, with an intrinsic
domain configuration governed by the interplay between mechanical and
electrostatic boundary conditions [33, 34] and additionally controlled via
growth temperature defect engineering [35]. X-ray diffraction measurements,
topography and PFM measurements show that all these samples are purely in the
$c$-phase – two samples being in a polydomain state with regions of up- and
down-oriented polarisation, and the other in a monodomain up-oriented state.
To characterise these samples, we used three complementary techniques: second
harmonic generation (SHG), piezoresponse force microscopy (PFM), and
transmission electron microscopy (TEM), which taken together allow us to map
the polarisation at all the relevant lengthscales. All measurements were
carried at room temperature. Using PFM and SHG, we find that Néel-like
behaviour dominates at _artificial_ domain walls introduced by local electric
field switching of polarisation. By combining PFM and careful analysis of TEM
data, we show that the same is true for _naturally occurring_ domain walls in
polydomain samples. Finally, we also show that it is possible to extract
information on the underlying pinning potential by additional SHG and PFM
image treatment and analysis based on auto-correlation.

Figure 1: _Signature of Néel polarisation at written domain walls at room
temperature_ (a) Vertical PFM phase image of the square-in-square domain
structure written in a up-oriented monodomain film. (b) Isotropic SHG image
(without polarisation analysis) acquired at the patterned square-in-square
domain structure, with red corresponding to maximum signal intensity and blue
to minimum signal intensity. We observe a localized second-harmonic emission
at the domain wall regions, which demonstrates their non-Ising polar
character. (c,d) SHG measurements of the domain structure for different
polariser and analyser angle combinations, where the analyser direction is
represented by the white arrow. SHG signal is observed at the domain walls
only when the analyser is perpendicular to the domain wall plane. This
polarimetry analysis shows that the local emission is polarised in the
direction of the white arrows. This demonstrates that the local polarisation
dipole is perpendicular to the domain walls, consistent with a Néel character.
The white scale bar in all panels represents
$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

In the out-of-plane up-oriented monodomain film, domains were patterned by
applying relatively high bias to the scanning probe microscopy tip, first
$+10$ V to switch a $6\times 6$ $\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$
square, then $-10$ V in a $3\times 3$
$\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$ square in the center, as shown in
Fig. 1(a), ensuring a well-spaced square-in-square domain structure whose
walls could be easily individually resolved by SHG. SHG measurements were then
performed to identify the presence and orientation of any in-plane component
of the ferroelectric polarisation at the domain walls. As detailed in Ref.
[24, 36], no SHG signal is expected in the geometry used for this experiment
(see Methods) for tetragonal materials (4mm point group symmetry). From the
anisotropic SHG image (Fig. 1(b)), a localized second-harmonic emission is
observed at the domain wall regions, demonstrating their non-Ising polar
character. Additional polarimetry analysis was performed on this region for
different polariser and analyser angle combinations. The written domain walls
are clearly visible in the SHG image only when the analyser is perpendicular
to the wall plane, as shown in Fig. 1(c-d). We can thus conclude that in this
case the domain walls present an in-plane polarization component with a purely
Néel-type configuration, as similarly observed in the related material
Pb(Zr0.2Ti${}_{0.8})$O3 [24].

Figure 2: _Top view of nanoscale domains and domain walls_ (a) Vertical PFM
phase image of the domain structure of a polydomain sample. The white scalebar
represents $400\text{\,}\mathrm{nm}$. The typical thickness of the TEM lamella
prepared from this sample ($\approx 30\text{\,}\mathrm{nm}$) is represented by
the green lines. (b-c) Schematic representation of the polarisation in the
black rectangle in (a), for the case of Néel (b) or Bloch type (c) domain
walls respectively. From this schematic representation, it is clear that
domain walls are highly unlikely to align through the TEM lamella thickness,
but will rather meander, leading to averaging over different polarisation
orientation in the TEM cross-section measurements. For a wall that is not
imaged edge-on, ie orthogonal to the plane of the lamella, the two possible
polarisation configurations at the wall (b) and (c) could lead to similar
projections.

For polydomain samples, vertical PFM imaging shows alternating regions with
up- and down-oriented domains, of the order of a few 10s of nm across, as can
be seen in Fig. 2(a). The high density and small size of these intrinsic
domains make it impossible to individually resolve domain walls and determine
their nature via SHG polarimetry, but non-negligeable contrasts in the
measured intensity point to the presence of in-plane polarisation (as we will
discuss further later). We therefore turn to scanning transmission electron
microscopy (STEM) to visualise the individual atomic displacements
contributing to the macroscopic polarisation. TEM measurements were performed
on a $\approx 30\text{\,}\mathrm{nm}$ lamella prepared with the long axis
parallel to the [100] in-plane axis of the thin film, as schematically
indicated by the green box in Fig. 2(a). Capturing a domain wall perfectly
edge-on in such a lamella is unlikely, and we thus expect to observe a
projection of the wall, in which case the TEM measurements would average over
the two opposing out-of-plane polarisation orientations of the domains as well
as any Néel or Bloch domain wall polarisation, as schematically indicated in
Fig. 2(b) and (c), respectively. Nevertheless, within these constraints we can
preselect regions where the edge-on domain wall alignment in the lamella is at
least somewhat enhanced.

Figure 3: _Transmission electron microscopy and cross-section polarisation
mapping of intrinsic domain walls at room temperature_ (a) Low-magnification
differential phase contrast (DPC) TEM image of the sample, allowing
visualisation of the domain walls positions. We choose to focus on the area
where the domain wall appears thinnest (red box), since this is where it will
be most edge-on. (b) High magnification HAADF-STEM image of the region
indicated by a red box in (a). The domain wall is visible as a shift in the Pb
lattice close to the center of the image (see main text). (c) Cropped version
of the atomic resolution HAADF shown in (b). Here the Ti atom columns are
readily visible, as well as the shift in the Pb lattice, inducing a sharp
change in the $c-a$ angle highlighted by the yellow dotted lines. The domain
wall position is indicated by red arrows in (a-c). (d) $c$-axis and $c-a$
angle extracted from 2D Gaussian fits to the positions of the atomic columns
in (b). The position of the domain wall is clearly visible as an increase in
$c-a$ angle. Concomitant with this increase, one can note a drop in local
$c$-axis (although within the error bars). (e) Ellipticities of the Pb and Ti
columns, caused by the projection of different positions of the atoms
throughout the thickness of the lamella. While both show an increase at the
position of the wall, the Ti ellipticity is higher throughout, consistent with
the averaging over opposite polarisation directions through the thickness of
the lamella. (f) Reconstructed polarisation map based on a three Gaussian fit
to the position of the Ti columns in (c) (see main text). The color scale
indicates the interpolated $c-a$ angle, which we use as a marker for the
position of the domain wall. The arrows represent the polarisation extracted
from the Ti displacement, which follows a typical Néel pattern: the
polarisation rotates through $180\text{\,}\mathrm{\SIUnitSymbolDegree}$
between the two domains on either side, such that it is perpendicular to the
wall plane at the position of the wall.

Fig. 3(a) shows a low magnification differential phase contrast (DPC) STEM
image where the colour scale indicates the direction and relative magnitude of
the deflection of the STEM probe due to diffraction and Lorentz deflection.
Within the PbTiO3 film, elongated linear features between regions of
relatively homogeneous contrast can be identified, appearing dark red and thus
indicating an increased in-plane deflection component. These features appear
narrower and vertically oriented near the film surface, then gradually broaden
and curve towards the interface of the film with the back electrode and
substrate. We therefore focused on the near-surface region, to enhance the
probability of their edge-on alignement. The red box demarcates an area imaged
at atomic resolution in the high angle annular dark field (HAADF) image in
Fig. 3(b). Here, the abrupt feature in the centre can be seen to manifest as a
distinct shift in the Pb atom positions, resulting in apparent ‘kink’ in the
Pb lattice (brightest atom columns). This confirms the prediction of Aguado-
Puente and Junquera for $180\text{\,}\mathrm{\SIUnitSymbolDegree}$ domain
walls in PbTiO3 thin films [37], and is very similar to the Bi-lattice ‘kink’
observed in BiFeO3 related to the polarisation rotation and accumulation of
charged defects at domains walls [38]. Furthermore, it can be seen from Fig.
3(b) and its cropped version shown in Fig. 3(c) that the Ti atom columns sit
closer to the centre of the unit cell on the left whereas they are near the
bottom of the unit cell on the right. In other words, there is evidently a
greater out-of-plane component of polarisation on the right since this is
closely linked to the B-site cation displacement from the centrosymmetric
position within the unit cell [39]. This observation therefore suggests the
presence of a domain wall in the centre of the image.

In order to accurately determine the atom positions and hence polarisation, we
perform 2D Gaussian fitting to the Pb and Ti atom columns which offers
subpixel precision [40, 41, 42]. From this, we can determine a number of
factors that can confirm the presence of a domain wall: $c$-axis and $c-a$
angle (angle between the [100] and [001] directions), as shown in the averaged
line profiles in Fig. 3(d) which go from left to right in Fig. 3(b). In
addition, we can also determine the ellipticity of the atom columns – shown in
Fig. 3(e) – which gives a measure of the disorder through the thickness of the
lamella: an ideally ordered column of atoms would appear perfectly circular.
From the profiles in Fig. 3(d) and (e), it is clear the kink observed in Fig.
3(b) and (c) coincides with an increase in the $c-a$ angle and in both the Pb
and Ti ellipticities. One can also note a concomitant decrease in out-of-plane
lattice parameter $c$ \- although within the error bars. Crucially, all of
this occurs over approximately 2–3 unit cells, and coincides with an increase
in the in-plane component of the polarisation defined by the displacement of
the Ti columns from the centre of the unit cell given by the Pb atoms, as
shown in Fig. 3(e).

To properly interpret the out-of-plane component of the polarisation, careful
analysis has to be performed, bearing in mind the likely scenario of domain
overlap through the thickness of the lamella, as depicted in Fig. 2. Indeed,
Fig. 3(e) shows that the Ti atom columns are significantly more elliptical
than the Pb atom columns, indicating a notable variation in the Ti position
through the lamella thickness, meaning that the observed Ti position is
actually the mean _projected_ position. This results in the initial fitting
shown in Fig. S1(c), where rather than the
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ polarisation rotation that would be
expected on opposing sides of a domain wall, the polarisation goes from
strongly ‘down’ on the right to very weakly ‘down’ on the left.

In order to deconvolute the different domain contributions, we fit three 2D
Gaussians to each of the Ti atom columns: two corresponding to Ti atom shifts
up and down relative to the mean position given by the single Gaussian fitting
and one corresponding to the ‘mean’ position between the first two. This
approach is analogous to the ‘mpfit’ algorithm proposed previously [43]. We
can then reconstruct the dominant polarisation from these three components,
discarding extraneous effects from projection. We note, however, that the
displacement of the Ti atoms only corresponds exactly to the polarisation of
the PbTiO3 if the Born effective charge is constant, as it would be expected
for ferroelectric distortions along the c-axis orientation, with the material
in the P4mm phase in the bulk of the domain, but which may not be the case at
the domain wall. For the sake of brevity, a comprehensive discussion of the
procedure is detailed in the Supporting Information (Section S1). Fig. 3(f)
shows the final result of this reconstruction where the image is an
interpolated map of the $c-a$ angle with the polarisation vectors overlaid.
Moving from left to right on the image, a classic Néel-like behaviour is
observed: the polarisation rotates through
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ from upwards on the left to
downwards on the right with a predominantly in-plane orientation in the 2–3
unit cells at the Pb ‘kink’ in the centre where the $c-a$ angle is maximised
and the tetragonality is minimised.

It therefore appears that in our PbTiO3 thin films, at room temperature, both
written and intrinsic domain walls present a Néel character. For the written
domain walls, the microscopic mechanisms underlying this configuration may be
related to the localised high electric field under the tip, necessary to
switch the polarisation, but which can also alter the samples
electrochemically [44] and stabilise the polar discontinuity inherent in the
Néel structure. For intrinsic domain walls, however, first-principles
calculations predict a Bloch character at low temperature and, at higher
temperatures, an Ising character with no in-plane component of polarisation
[20]. Our observations of a Néel component suggest that in real materials such
structures are on the contrary at the very least metastable. The polarisation
discontinuity at domain walls with such a Néel component can be related to
effects of domain wall curvature or inclination [45], in particular at domain
walls reconfigured under high local electric fields. Moreover, even at nominal
straight intrinsic domain walls, distinct few-unit-cell steps with head-to-
head or tail-to-tail polarisation have been observed [46]. Such structures,
with their associated charge, could be potentially important for domain-wall-
based nanoelectronics, as they could contribute to a higher conductivity at
the domain walls [47, 48, 49, 50, 51]. Moreover, in thin films, which
generally present a relatively high density of oxygen vacancies and other
internal charged species with respect to single crystals, such discontinuities
in the polarisation could very easily be stabilised by these defects.

Indeed, defects and disorder are a crucial element of ferroelectric thin
films, acting as nucleation sites during switching [52], pinning domain walls
[12, 53, 54], and segregating along them to provide a path for electrical
conduction [55, 8, 56]. In order to better understand and exploit the
functional properties of these ferroelectric domain walls, it is therefore
crucial to consider them within the defect landscape they inhabit within the
PbTiO3 sample. It was previously shown that variations in Pb and O
stoichiometry, tuned by the temperature and electrostatic boundary conditions
during growth, can lead to strong asymmetry between the two polarisation
orientations and variations in the magnitude of polarisation through the film
thicknesses [35, 57]. In the present study, as can be seen in Fig. 3(a), the
up-oriented domain appear to increase in volume near the surface of the film,
at the expense of the down-oriented domain, with a marked curvature of the
domain walls suggesting components of even stronger head-to-head and tail-to-
tail polar discontinuities.

Further insight into the defect landscape can be obtained from a comparative
analysis of PFM and SHG measurements in another polydomain sample (Fig. 4).
The vertical PFM phase response in Fig. 4(a) reveals intrinsic domains of the
order of $100\text{\,}\mathrm{nm}$, well below the diffraction limit for the
wavelength used in SHG measurements ($\sim$
$300\text{\,}\mathrm{n}\mathrm{m}$). When comparing with the image obtained by
SHG on the same sample and at the same scale (Fig. 4(b)), we nonetheless
observe regions of brighter and darker SHG contrast. This signal confirms the
presence of an in-plane component of polarisation. However, these contrast
variations appear at much larger lengthscales than the individual nanoscale
domains imaged by PFM. Indeed, when we extract the radially averaged
autocorrelation for each image, we find a characteristic pseudo-period of
approximately $800\text{\,}\mathrm{n}\mathrm{m}$ for the SHG contrast
variations, but only about $100\text{\,}\mathrm{nm}$ for the PFM images,
corresponding directly to the domain structure (see Supporting Information
Section S2 for details).

Nonetheless, when the microstructure of domains (Fig. 4(c)) and domain walls
(Fig. 4(d)) in the PFM image is explicitly blurred to mimic the lower
resolution and averaging effect of SHG, longer range variations in density are
clearly revealed. Radially averaged autocorrelation analysis of this resulting
superstructure shows the same approximately $800\text{\,}\mathrm{n}\mathrm{m}$
pseudo-period as that of the intensity variations in the SHG image Fig. 4(e).

Figure 4: _Characteristic lengthscale of domain superstructure_ (a) Vertical
PFM phase image of the domain structure of the polydomain sample in blue
(ranging from 0 to $180\text{\,}\mathrm{\SIUnitSymbolDegree}$), with the edges
detected using the Canny edge detection algorithm in purple. (b) SHG image of
the same sample. The dark and bright contrast cannot be directly related to
the domain pattern or to the domain edges. (c,d) Local density variations
revealed by blurring applied on the vertical PFM phase image of the domains in
(c) and on the domain domain wall structure (d). These domain density
variations appear to closely match the variation observed in SHG intensity.
All images are shown at the same scale, indicated by the
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ white scale bar in (a). (e) Radially
averaged autocorrelation for the SHG image and both the domain and domain wall
distributions obtained from the PFM phase image, and their superstructure
obtained after blurring. While only very small structures are revealed in the
raw PFM images, the superstructure in the density has the same pseudoperiod as
the contrast variations found in the SHG image.

This superstructure in the domain density appears to be directly linked to the
underlying defect landscape. To demonstrate this, we are applying this
radially averaged autocorrelation analysis to a simple Ising model simulation
shown in Fig. 5(a), in which a disorder potential with a characteristic
lengthscale $\xi$ is added to the Ising Hamiltonian, and the resulting domain
density fluctuations are blurred to reveal their superstructure, as for the
experimental analysis (see Supporting Information Section S3 for details). As
can be seen in Fig. 5(b), we find that the typical feature size extracted from
radially averaged autocorrelation analysis exactly matches the characteristic
lengthscale $\xi$ of the disorder potential.

Figure 5: _Simulations of domain density variations demonstrating the
importance of the disorder correlation length._ Experimentally observed
fluctuations of domain density were reproduced using an Ising system with a
random disorder potential of characteristic lengthscale $\xi$. (a) Relaxed
domain structure after randomly distributed up and down polarisation regions,
initially in equal proportion, were subjected to a random field disorder
potential. (b) The radially averaged autocorrelation function of the domain
superstructure, whose first minima, corresponding to the typical feature size,
exactly match the disorder correlation length.

While the effect of strong, localised defects can be seen in, for example, the
changes in roughness at domain walls [58], our observation of a characteristic
lengthscale of 800 nm seems more compatible with variations of a collective
background defect potential. This potential has been previously mapped out
using spectroscopic techniques [52], recently further assisted by machine
learning [59]. We note, moreover, that the key contributions to the disorder
landscape do not appear to be variations of the surface morphology (see
Supporting Information Fig. S10 for details). Our analysis therefore
demonstrates that quantitative information on this defect potential – its
characteristic lengthscale – can be obtained using non-invasive and non-
destructive optical measurements, without the need to switch the intrinsic
domains and thus modify the defect landscape of the film [44, 60]. This method
can be applied more generally, for example to samples where the defect density
has been engineered, to reveal the resulting change in domain density
superstructure.

In summary, our complementary PFM, SHG, and TEM measurements demonstrate that
both artificial and naturally occurring domain walls in PbTiO3 thin films
present components of Néel polarisation at room temperature, possibly
stabilised over the predicted Ising structure by defect segregation. Indeed,
radially averaged autocorrelation analysis of domain density fluctuations
reveals characteristic lengthscales which can be related to variations in the
underlying disorder potential landscape. While it remains to be explored if
and how such domain walls transition to a Bloch configuration at low
temperature, our observations suggest potential utility in novel
nanoelectronic applications, as Néel domain walls would be locally more likely
to show increased electrical conductivity than their Ising counterparts (see
Supporting Information Section 4 for details).

## Methods

### Thin film growth

Three different samples have been used in this study:

  * •

M: monodomain sample studied by SHG

  * •

P1: polydomain sample studied by TEM

  * •

P2: polydomain sample studied by SHG

The samples were grown epitaxially using off-axis radio frequency magnetron
sputtering on TiO2 terminated, (001)-oriented, undoped SrTiO3 with a LaNiO3
back electrode. LaNiO3 was deposited at a temperature of
$510\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for samples M and P1 and
$555\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for sample P2, in 180 mTorr of a
10:35 O2/Ar mixture using a power of 50 W. The PbTiO3 thin films were
deposited at temperatures ranging from
$540\text{\,}\mathrm{\SIUnitSymbolCelsius}555\text{\,}\mathrm{\SIUnitSymbolCelsius}$
($540\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$545\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$555\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for sample M, P1 and P2
respectively) in 180 mTorr of a 20:29 O2/Ar mixture using a power of 60 W.
Additionally, the PbTiO3 film in sample P2 was grown using the SKIS technique
(Slow Kinetics Intermittent Sputtering, see [61])), while standard
(continuous) RF growth was used for all the others. A Pb1.1TiO3 target with a
10 % excess of Pb was used to compensate for Pb volatility, while
stoichiometric targets were used for all other materials.

The different growth conditions have been used to optimize the sample quality
and achieve the desired domain configuration (monodomain or polydomain) needed
for this study. Note that this explains the slightly different vertical PFM
phase images obtained on sample P1 (figure 2(a): mix of up and down regions)
and on sample P2 (figure 4(a): uniform background with well defined
nanoregions of the opposite polarization).

### SPM Characterisation

All SPM characterisation was performed using an Asylum Research Cypher atomic
force microscope. Surface topographies were obtained in tapping mode using
Bruker TESPA tips. Piezoresponse force microscopy vertical phase and amplitude
were recorded in dual resonance tracking (DART) mode [62] with excitation
amplitudes typically less than 500 mV, using conductive Mikromasch
HQ:NSC18/Cr-Au coated tips.

### SHG

Local SHG measurements were conducted by means of an inverted optical
microscope. The fundamental wave is provided by a laser source emitting pulses
of 100 fs duration at a repetition rate of $80$ MHz, centered at a wavelength
$\lambda=800$ nm. The sample was illuminated at normal incidence with a time-
averaged power of $10-14$ mW. The SHG images are obtained by scanning the
sample with respect to the focused laser beam (objective x$40$, $0.66$
numerical aperture) using computer-controlled stepping motors. The output
intensity was spectrally filtered and collected into a photomultiplier.
Polarimetry measurements are performed by recording the images at different
polarizer and analyzer angles (P and A, respectively). Ref. [24, 36] provide
detailed information on the measurement geometry, and on how the internal
domain wall structure is determined using SHG polarimetry.

### TEM

Cross-sections for STEM were prepared via FIB milling at accelerating voltages
of 30, 16, 8, 5 and 2 kV on an FEI Scios DualBeam instrument. STEM
measurements were performed on a probe-corrected FEI Titan Themis operated at
200 kV and with a probe convergence angle of 21.2 mrad. High angle annular
dark field (HAADF) imaging was performed with inner/outer collection angles of
56.3 and 200 mrad, respectively. DPC imaging was performed with a 4 segment
annular detector with inner/outer collection angles of 13.2 and 73.6 mrad,
respectively. The specimen thickness was estimated from the $t/\lambda$
(thickness/inelastic mean free path) from a zero loss electron energy loss
(EEL) spectrum acquired using a Gatan Enfinium spectrometer. Atomic resolution
images were acquired as series of 15 frames with pixel dwell time of 100 ns
before being summed and corrected for drift in Velox by Thermo Fisher
Scientific. Gaussian fitting was performed using Atomap [63] and Matlab.

## Data availability

The data that support the findings of this study are available at Yareta
(DOI).

## Acknowledgements

This work was supported by Division II of the Swiss National Science
Foundation under project 200021_178782. A.B.N. greatfully acknowledges support
from the EPSRC (EP/R023751/1 and EP/L017008/1). S.C.-H. acknowledges the
support of the French Agence Nationale pour la Recherche (ANR) through grant
ANR-18-CE92-0052 and the Programme d’Investissement d’Avenir
ANR-10-IDEX-0002-02 of the University of Strasbourg under contract
ANR-11-LABX-0058_NIE. C.L. acknowledges the support from the Division II of
the Swiss National Science Foundation under project 200021_200636. The authors
thank S. Mohapatra for his help for the domain poling.

## Author contributions

P.P. and S.C.-H. designed the experiment. C.W. and C.L. grew the samples and
conducted the piezoresponse force microscopy. S.C.-H. carried out the second
harmonic generation microscopy in collaboration with K.D.D. A.B.N. performed
the transmission electron microscopy and the atomic position fitting. C.W.
carried out the auto-correlation analysis. I.G. performed the Monte Carlo
simulations of the perturbed Ising Model. C.W., C.L., A.B.N., 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.

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