# Nanoscale domain engineering in SrRuO3 thin films

Céline Lichtensteiger<EMAIL_ADDRESS>Department of Quantum
Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva
4, Switzerland  Chia-Ping Su  Université Paris-Saclay, CNRS, Laboratoire de
Physique des Solides, Orsay 91405, France  Iaroslav Gaponenko  Department of
Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211
Geneva 4, Switzerland  Marios Hadjimichael  Department of Quantum Matter
Physics, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva 4,
Switzerland  Ludovica Tovaglieri  Department of Quantum Matter Physics,
University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland
Patrycja Paruch  Department of Quantum Matter Physics, University of Geneva,
24 Quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland  Alexandre Gloter
Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, Orsay
91405, France  Jean-Marc Triscone  Department of Quantum Matter Physics,
University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland

Abstract - We investigate nanoscale domain engineering via epitaxial coupling
in a set of SrRuO3/PbTiO3/SrRuO3 heterostructures epitaxially grown on
(110)o-oriented DyScO3 substrates. The SrRuO3 layer thickness is kept at 55
unit cells, whereas the PbTiO3 layer is grown to thicknesses of 23, 45 and 90
unit cells. Through a combination of atomic force microscopy, x-ray
diffraction and high resolution scanning transmission electron microscopy
studies, we find that above a certain critical thickness of the ferroelectric
layer, the large structural distortions associated with the ferroelastic
domains propagate through the top SrRuO3 layer, locally modifying the
orientation of the orthorhombic SrRuO3 and creating a modulated structure that
extends beyond the ferroelectric layer boundaries.

##  I Introduction

Ferroelectric polarisation can be used to affect the properties of other
materials. This is well know in ferroelectric field-effect transistors for
example, where the polarisation surface charge of the ferroelectric film is
used to reversibly dope the adjacent layer, as demonstrated in epitaxial oxide
thin film heterostructures Ahn-Science-1995 .

In this work, a PbTiO3 layer is sandwiched between two SrRuO3 layers. In bulk,
SrRuO3 is a ferromagnetic metallic transition-metal oxide and is often used as
an electrode in the ferroelectric oxides community Eom-Science-1992 . It is
also an itinerant ferromagnet with a Curie temperature $T_{C}$ = 160 K Cao-
PRB-1997 . In thin films of this material, the formation of complex spin
textures can be induced by the ferroelectric polarisation in an adjacent
ferroelectric layer. These include ferroelectric proximity effect near the
BaTiO3/SrRuO3 interface giving rise to an emergent Dzyaloshinskii–Moriya
interaction, thereby creating robust magnetic skyrmions Wang-NatMat-2018 .
Most recently, in SrRuO3/PbTiO3 heterostructures a ferroelectrically induced
magnetic spin crystal was observed Seddon-NatCom-2021 .

SrRuO3 is not only affected by the polarisation in adjacent layers, but also
by the epitaxial strain imposed by the substrate Koster-RevModPhys-2012 . When
grown on SrTiO3, epitaxial SrRuO3 layers are organized into structural
domains, according to six possible orientations of the orthorhombic unit cell
with respect to the cubic substrate. The orientation of the orthorhombic unit
cell and resulting domains is affected by the steps and terraces at the
surface of the SrTiO3 substrate Jiang-MSEB-1998 . The growth of SrRuO3 onto
vicinal planes of miscut SrTiO3 substrates leads to the privileged development
of a majority single domain orientation in which small domains with different
orientations are embedded Jiang-APL-1998 . Additionally, control of the SrRuO3
can be achieved not only through the choice of substrate Vailionis-APL-2008 ,
but also by modifying the growth temperature Zakharov-JMT-1999 . Further
structural domain engineering has been conducted through control of substrate
miscut direction, demonstrating a one-to-one correspondence between structural
domains and magnetic domains Wang-NPJQuantMats-2020 .

A structural coupling can also be achieved by strain propagation between the
different layers themselves. As was shown in PbTiO3/SrTiO3/PbTiO3
heterostructures on GdScO3 where the structural coupling between the PbTiO3
and SrTiO3 layers resulted in periodic polar waves in the SrTiO3 Tang-
NanoLetters-2021 , in SrRuO3/PbTiO3 superlattices the large local deformations
of the ferroelectric lattice are accommodated by periodic lattice modulations
of the metallic SrRuO3 layers with very large curvatures Hadjimichael-
NatMat-2021 .

At the core of the heterostructure studied here is PbTiO3, a tetragonal
ferroelectric with a polarisation developing along the $c$-axis mostly due to
ionic displacements. In PbTiO3 thin films, the orientation of the polarisation
and arrangement into domain structures have been theoretically studied
Pertsev-PRL-1998 ; Pertsev-PRL-2000 ; Koukhar-PRB-2001 ; Li-APL-2001 ; Jiang-
PRB-2014 ; Chapman-PCCP-2017 , and are described by phase diagrams with
regions of different domain configurations as a function of epitaxial strain
and temperature (see review by Schlom et al. Schlom-AnnuRevMaterRes-2007 ).
The domain pattern is also affected by the film thickness Kittel-PR-1946  and
electrostatic boundary conditions Lichtensteiger-NanoLett-2014 ;
Lichtensteiger-NJP-2016 . Complex polarisation configurations in PbTiO3 have
been reported recently in PbTiO3/SrTiO3 superlattices Yadav2016 ; Stoica2019 ;
Hadjimichael-PhD-2019 ; Hadjimichael-NatMat-2021 ; Goncalves2019 ; Das2019 ;
Wang-NatureMaterials-2020 , with simultaneous control of these configurations
using electric fields and light, giving rise to novel phenomena like negative
capacitance Iniguez-NatureReviewsMaterials-2019 . When grown on DyScO3, PbTiO3
takes the $a/c$ phase, where the polarisation forms ordered ferroelastic
domains ($a/c$ twins), resulting in distortions of the film surface visible by
atomic force microscopy Vlooswijk-APL-2007 ; Catalan-NatMat-2011 ; Nesterov-
APL-2013 ; Lichtensteiger-APLMaterials-2023 .

Whether through ferroelectric polarisation or strain effects, controlling the
structure and morphology of the SrRuO3 thin films is of importance as it will
affect the film electronic resistivity via structural and electronic coupling.
Here, we study the structural coupling between oxide thin film layers on a set
of SrRuO3/PbTiO3/SrRuO3 heterostructures epitaxially grown on (110)o-oriented
DyScO3 substrates. We establish the direct role that the ferroelastic domain
structure in PbTiO3 plays in the determination of the orthorhombic domain
structure in SrRuO3.

##  II Results

A series of samples was grown by off-axis RF magnetron sputtering on
(110)o-oriented DyScO3 substrates, with bottom and top SrRuO3 electrodes of 55
unit cells (u.c.) and PbTiO3 film thickness of 23, 45 and 90 u.c. (see
Section. IV.1 for details of samples growth).

The substrate, DyScO3, is orthorhombic with room temperature lattice
parameters (in $Pbnm$ space group) $a_{o}$ = 5.443(2) Å, $b_{o}$ = 5.717(2) Å
and $c_{o}$ = 7.901(2) Å Velickov-ZKristallogr-2007 . It is often useful to
refer also to the pseudocubic unit cell, where the lattice parameters can be
calculated as $a_{pc}$ = $c_{pc}$ = $\frac{\sqrt{a_{o}^{2}+b_{o}^{2}}}{2}$ =
3.947 Å, $b_{pc}$ = $c_{o}/2$ = 3.951 Å, $\alpha_{pc}$=$\gamma_{pc}$=90∘,
$\beta_{pc}$=2 $\cdot$ $\arctan$(${a_{o}/b_{o}}$)=87.187∘ at room temperature.
Here, “${pc}$” subscript refers to the pseudocubic unit cell, while “$o$” is
used to refer to the orthorhombic unit cell. 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 [$\overline{1}$10]o and
[001]o respectively (see Supplementary Materials, Figure S1).

The bottom and top electrode, SrRuO3, is orthorhombic with bulk room
temperature lattice parameters (in $Pbnm$ space group) $a_{o}$ = 5.57 Å,
$b_{o}$ = 5.53 Å and $c_{o}$ = 7.85 Å Randall-JACS-1959 , corresponding to the
pseudocubic unit cell parameters $a_{pc}$ = $c_{pc}$ = 3.924 Å, $b_{pc}$ =
3.925 Å, $\alpha_{pc}$=$\gamma_{pc}$=90∘, $\beta_{pc}$ = 90.413∘. According to
the Glazer notation, octahedral tilting in orthorhombic SrRuO3 is described by
$a^{-}a^{-}c^{+}$, implying that RuO6 octahedra are rotated in opposite
directions by equivalent magnitude along $[100]_{pc}$ and $[010]_{pc}$ (out-
of-phase) and in the same direction about $[001]_{pc}$ (in-phase) Glazer1972 ;
Woodward-ActaCrystallographicaB-1997 . On (110)o-oriented DyScO3 substrate,
SrRuO3 can grow with different possible orientations Jiang-MSEB-1998 ; Wang-
NPJQuantMats-2020 , as described in Figure 1, and Supplementary Materials
Figure S1 and S2.

Figure 1:  Detailed representation of four of the possible orthorhombic
orientations of the SrRuO3 on (110)o-oriented DyScO3 substrate (in $Pbnm$
space group). Each column corresponds to a different orientation $X$,
$X^{\prime}$, $Y$ and $Y^{\prime}$. First row: perspective view, with arrows
$\vec{a}\parallel[100]_{o}$, $\vec{b}\parallel[010]_{o}$ and
$\vec{c}\parallel[001]_{o}$ corresponding to the axis of the orthorhombic unit
cell. Second row: corresponding pseudocubic representation. Third row:
reciprocal space representation and (hkl) indices corresponding to the
pseudocubic structure. Fourth row: reciprocal space representation and (hkl)
indices corresponding to the orthorhombic structure, highlighting the position
of the “half-order” peaks.

Last, the PbTiO3 is ferroelectric below a bulk critical temperature of 765K
with a tetragonal structure, and lattice parameters $a$ = $b$ = 3.904 Å and
$c$ = 4.152 Å at room temperature. The in-plane strain imposed by DyScO3 on
PbTiO3 films can thus be calculated as $\frac{a_{pc}-a_{0}}{a_{pc}}=-0.25\%$
along $a_{pc}$ and $\frac{b_{pc}-a_{0}}{b_{pc}}=-0.16\%$ along $b_{pc}$, where
$a_{0}$ is the extrapolated lattice parameter of PbTiO3 in the room-
temperature cubic paraelectric phase, $a_{0}$ = 3.957 Å for PbTiO3. To
accommodate this strain111Although the strain with respect to $a_{0}$ is
compressive, the system is usually referred to as being under tensile strain
because the lattice parameter of DyScO3 is larger than the bulk $a$=$b$ axes
of PbTiO3., PbTiO3 thin films on DyScO3 at room temperature 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. The latter are parallel to the $\\{101\\}_{pc}$ crystallographic
planes, and thus are inclined at about 45∘ with respect to the film/substrate
interface, as predicted in Ref. Koukhar-PRB-2001  and demonstrated
experimentally (see for example Ref. Catalan-NatMat-2011 ; Nesterov-APL-2013 ;
Highland-APL-2014 ). Additionally to these ferroelastic domains, the
electrostatic boundary conditions and depolarisation field arising from an
incomplete screening of the surface bound charges can lead the $c$-domains to
alternate between “up” ($c^{+}$) and “down” ($c^{-}$) orientations. Although
the surface bound charges of our PbTiO3 films are screened by the top and
bottom SrRuO3 electrodes, this screening is incomplete Junquera-NAT-2003 ;
Aguado-Puente-PRL-2008 ; Stengel-NatMat-2009 ; Li-APL-2017 ; Hadjimichael-
PRM-2020  and the depolarisation field still plays a role. Such a combination
of mechanical and electrostatic constraints can then result in flux-closure
structures, as observed in strained PbTiO3 thin films Tang-Science-2015 ; Li-
APL-2017 ; Li-ActaMat-2019 ; Lichtensteiger-APLMaterials-2023 .

###  II.1 Topographic modulation observed at the heterostructure surface using
atomic force microscopy

Figure 2(a) shows atomic force microscopy (AFM) images for the three
SrRuO3/PbTiO3/SrRuO3 heterostructures grown on DyScO3. The AFM topography
images reveal that as the PbTiO3 layer thickness increases, trenches develop
at the surface of the SrRuO3 top layer in an organised pattern. For the
samples with 23 and 45 u.c. thick PbTiO3 layers, this pattern is hardly
visible and the top SrRuO3 is smooth. The pattern gets more pronounced and
anisotropic with increasing PbTiO3 layer thickness, with long and deep
trenches parallel to the DyScO3$[001]_{o}$ axis, and smaller trenches parallel
to the DyScO3$[\overline{1}10]_{o}$ axis, while the surface roughness stays
reasonably low (root mean square (RMS) roughness values ranging from 157 to
393 pm over surfaces of 10 $\mu$m $\times$ 10 $\mu$m).

Figure 2:  AFM topography images obtained on the different samples. Color
scale varies between 0 and 2 nm. The sample orientation was fixed with respect
to the substrate pseudocubic axis [100]pc//DyScO3$[\overline{1}10]_{o}$ and
[010]pc//DyScO3$[001]_{o}$. (a-c) 2x2 $\mu$m2 scans for the three samples: (a)
90$\pm$4 u.c., (b) 45$\pm$2 u.c. and (b) 23$\pm$1 u.c. thick PbTiO3 between
top and bottom SrRuO3 electrodes (55$\pm$2 u.c. thick) on DyScO3 substrates,
showing that as the PbTiO3 layer thickness increases, trenches develop at the
surface of the SrRuO3 top layer. (d) A larger 10x10 $\mu$m2 scan for the 90
u.c. thick PbTiO3 sample displays the anisotropic pattern, with long and deep
trenches parallel to the [010]pc axis, and smaller trenches parallel to the
[100]pc axis. In the image obtained from the fast Fourier transform of the
topography measurement (inset), periodic peaks along [100]pc and [010]pc are
visible (see cuts), allowing us to determine the periods. Along [010]pc (red),
two periods are visible, P1 = 77 $\pm$ 1 nm and P2 = 280 $\pm$ 3 nm, while
along [100]pc (blue), a unique period P3 = 335 $\pm$ 4 nm is visible. These
sizes have been drawn on the topography image (a) of the corresponding sample
as yellow and green rectangles.

The pattern that we observe at the surface of the SrRuO3 top layer is
comparable to what has been seen in PbTiO3 layers grown on DyScO3 substrates
in Ref Nesterov-APL-2013 , attributed to the presence of periodic ferroelastic
$a/c$ domains. To extract the period of the distortions visible on the surface
of the samples, we calculate the fast Fourier transform (FFT) of the
autocorrelation image, as shown in Figure 2(d) for the sample with the 90 u.c.
thick PbTiO3 layer. Along DyScO3$[001]_{o}$ (red), two periods are visible, P1
= 77 $\pm$ 1 nm and P2 = 280 $\pm$ 3 nm, while along
DyScO3$[\overline{1}10]_{o}$ (blue), a unique period P3 = 335 $\pm$ 4 nm is
visible. These periods have been illustrated on the topography image of the
corresponding sample as yellow (with dimensions P1 $\times$ P3) and green
(with dimensions P2 $\times$ P3) rectangles. All these values are reported in
Table S1 (Supplementary Materials).

###  II.2 Domain structures observed by scanning transmission electron
microscopy

To better understand the origin of this pattern visible at the surface of the
SrRuO3 top layer, we turned to cross-sectional scanning transmission electron
microscopy (STEM) images. The three samples were cut and prepared for STEM
measurements to obtain slices in the plane defined by the $[001]_{o}$
(horizontal direction) and $[110]_{o}$ (vertical direction) axes of DyScO3.
STEM images were obtained using bright field (BF), annular bright field (ABF),
medium angle annular dark field (MAADF) and high-angle annular dark field
(HAADF) detectors along the DyScO3$[\overline{1}10]_{o}$ zone-axis (see
section IV.3 for more technical details).

The domain walls in the PbTiO3 layers can be directly seen in the STEM images,
as shown in Figure 3 (a-c - HAADF images) and in Figure 6 (a-c - BF images),
while the SrRuO3 layers appear rather homogeneous. The PbTiO3 layers in the
three samples studied have different domain configurations, where the expected
$a/c$ pattern for the thicker PbTiO3 layer transforms into a flux-closure
pattern for the thinner PbTiO3 layers (see Ref. Lichtensteiger-
APLMaterials-2023  for a complete x-ray diffraction based investigation of
ferroelectric domain configuration in an extended series of samples).

####  II.2.1 Geometric Phase Analysis

To study the local strain induced by these different domain configurations, we
turn to Geometric Phase Analysis (GPA) Hytch-Ultramicroscopy-1998 . This is
done by taking the FFT of the HAADF-STEM images in Figure 3, selecting two
peaks (here $(01\overline{1})_{pc}$ and $(011)_{pc}$) corresponding to two
reciprocal lattice vectors defining the lattice, and getting the inverse
Fourier transform containing information about local displacements of the
atomic planes along these two vectors. The local strain components are
calculated from the derivative of the obtained displacement field: in-plane
strain $\varepsilon_{yy}$ (along DyScO3$[001]_{o}$), out-of-plane strain
$\varepsilon_{zz}$ (along DyScO3$[110]_{o}$), shear strain $\varepsilon_{yz}$
and rotation $\omega_{yz}$.

Figure 3:  HAADF images and strain maps for the three samples: (top row)
90$\pm$4 u.c., (center row) 45$\pm$2 u.c., and (bottom row) 23$\pm$1 u.c.
thick PbTiO3 between top and bottom SrRuO3 electrodes (55$\pm$2 u.c. thick) on
DyScO3 substrates. (a-c) HAADF images, with the inset showing the FFT and the
two vectors used for the GPA analysis indicated by the blue and purple arrows.
(d-f) Out-of-plane strain $\varepsilon_{zz}$. (g-i) In-plane strain
$\varepsilon_{yy}$. (j-l) Shear strain $\varepsilon_{yz}$. (m-o) Rotation
$\omega_{yz}$. The strain and rotation values are calculated with respect to a
reference lattice here chosen as the substrate. The same intensity scales from
-8% to 8% (strain) and from -5∘ to 5∘ (rotation) are used for the three
samples to allow for comparison. Different contrasts are observed in the
PbTiO3 layers, corresponding to the different ferroelastic domain
configurations, from $a/c$ for the thickest to flux-closure for the thinner
one. The images interestingly reveal an additional contrast appearing only in
the top SrRuO3 layers, for the samples with the 45 u.c. and the 90 u.c. thick
PbTiO3 layers, and propagating all the way to the top surface.

Looking first at the results within the PbTiO3 layers, we see from the HAADF
images and from the GPA maps that the domain patterns vary with PbTiO3
thickness. For the 90 u.c. thick PbTiO3 in Figure 3 (top row), we see large
regions with a high out-of-plane strain, but low in-plane strain, shear and
rotation, corresponding to $c$-domains, i.e. regions where the polarisation is
out-of-plane. These regions are separated by narrower features, with high in-
plane strain and rotation, but low out-of-plane strain and shear,
corresponding to $a$-domains, i.e. regions where the polarisation is in-plane.
These results confirm a typical well developed $a/c$-phase. For the 45 u.c.
thick PbTiO3 layer in Figure 3 (center row), the strain map is more complex,
with alternating regions with large out-of-plane strain or large in-plane
strain close to each interface, and reduced strain at the center of the PbTiO3
layer, clearly different from an $a/c$-phase (see Supplementary Materials
Figure S4). This pattern is more comparable to the flux closure configuration
observed for PbTiO3 with similar thickness grown without electrodes Tang-
Science-2015 . Finally, for the 23 u.c. thick PbTiO3 layer in Figure 3 (bottom
row), the PbTiO3 strain maps are more homogeneous compared to the results
obtained for the two other samples. This indicates that for this sample, the
distortions related to the ferroelectric/ferroelastic domain configuration in
the PbTiO3 layer are small with respect to the homogeneous strain induced by
the substrate. The most pronounced contrast is visible in the rotation map and
corresponds to a pattern with a period of $\sim$ 16 nm, in agreement with the
value found by XRD (see Supplementary Materials Figure S3 and Table S1 for
comparison).

Concentrating now on the SrRuO3 layers, we see that the bottom ones are
predominantly homogeneous in all three samples. However, this is not the case
for the top SrRuO3 layers, where different contrasts appear for the three
different PbTiO3 thicknesses. While the SrRuO3 layer on top of the 23 u.c.
PbTiO3 looks rather homogeneous, regions with different shear strain and
rotation values alternate in the SrRuO3 layers grown on top of the 45 u.c. and
90 u.c. PbTiO3 with boundaries propagating along the [001]pc growth direction.

For the sample with 90 u.c. PbTiO3, at the interface with each $a$-domain and
above the obtuse angle formed by the $a/c$ domain wall, the rotation is
positive (red), while it is negative (green) above the acute angle (Figure
3(m)). The rotation then propagates directly to the top surface along the
growth direction DyScO3$[110]_{o}$. A similar modulation of the rotation is
also observed in the SrRuO3 bottom electrode, with positive rotation below the
obtuse angle of the $a/c$ domain wall, and negative rotation below the acute
angle. However, for the bottom electrode, this modulation is limited to the
vicinity of the interface and does not propagate through the whole SrRuO3
bottom electrode thickness, most likely due to substrate clamping.

For the sample with 45 u.c. PbTiO3, regions with a positive rotation (red)
alternate with regions with a negative rotation (green) (Figure 3(n)), with a
reduced rotation amplitude compared to the sample with 90 u.c. thick PbTiO3,
but with sharper boundaries. The period of this pattern follows the period of
the ferroelectric domains underneath.

####  II.2.2 Discriminating between $X/X^{\prime}$ and $Y/Y^{\prime}$ using
fast Fourier transforms

To better understand the origin of this contrast, one can use the FFT and
deduce the orientation of the SrRuO3 layers from the obtained Bragg peak
positions. In the FFT images in Figure 4, the bright peaks of the pseudo-cubic
lattice are clearly visible, corresponding to {0 k l}pc with $k$ and $l$
integer indices. Additionally to these peaks, weaker peaks also appear at
positions corresponding to half-integer Miller indices {0 1/2 1/2}pc
highlighted in blue and {0 1/2 1}pc highlighted in yellow. These peaks come
from the orthorhombic unit cell, which is composed of 4 pseudo-cubic unit
cells, as described in Figure 1. The position of the additional peaks is a
clear indication of the orientation of the orthorhombic unit cell. The {0 1/2
1}pc peaks in the FFT appear when the orthorhombic long axis [001]o is
oriented in-plane, parallel to the [010]pc axis, and correspond to the $X$ or
$X^{\prime}$ orientation (note that it is not possible to discriminate between
$X$ or $X^{\prime}$ in this measurement geometry). The {0 1/2 1/2}pc peaks on
the other hand are the signature of the orthorhombic long axis [001]o being
oriented in-plane, parallel to the [100]pc axis, corresponding to $Y$ or
$Y^{\prime}$ orientation.

Figure 4:  FFT analysis of the SrRuO3 electrodes orientation in the three
samples with (a) 90 u.c., (b) 45 u.c. and (c) 23 u.c. thick PbTiO3 layers
showing the presence of bright peaks corresponding to the pseudo-cubic lattice
at {0 k l}pc with $k$ and $l$ integer indices, and weaker peaks at half-
integer Miller indices {0 1/2 1}pc, corresponding to $X/X^{\prime}$
(highlighted in yellow), and {0 1/2 1/2}pc, corresponding to $Y/Y^{\prime}$
orientation (highlighted in blue). The color maps are obtained by FFT
filtering the orthorhombic superstructures, demonstrating the $X/X^{\prime}$
orientation for the bottom SrRuO3 for the three samples, and the
$Y/Y^{\prime}$-orientation for the top SrRuO3 for the two samples with the
thinner PbTiO3 layers (b,c) and a mixed $X/X^{\prime}$-$Y/Y^{\prime}$ for the
sample with the thickest PbTiO3 layer (a).

By selecting the different half order peaks and reconstructing the images in
Figure 4, we find that the {0 1/2 1}pc peaks corresponding to $X/X^{\prime}$
orientation originate from the substrate and the bottom SrRuO3 electrode for
all three samples, while the {0 1/2 1/2}pc peaks corresponding to
$Y/Y^{\prime}$ originate from the top SrRuO3 electrode. We also note that for
the sample with the thickest PbTiO3 layer, the top SrRuO3 shows a mixed
$X/X^{\prime}$ and $Y/Y^{\prime}$ character. Although already highlighting
differences in the top SrRuO3 layers for the different samples, this is not
enough yet to explain the contrast observed in the strain and rotation maps in
the GPA analysis in the SrRuO3 top layers for the samples with 45 u.c. and 90
u.c. thick PbTiO3 layers. This will be further investigated below, where we
show that it is possible to discriminate between $Y$ and $Y^{\prime}$
222Although the orientation of the lamella does not allow us to discriminate
between $X$ and $X^{\prime}$, we can go one step further by discriminating
between $Y$ and $Y^{\prime}$, the two orientations corresponding to very small
horizontal shift in the peak positions..

####  II.2.3 Discriminating between $Y$ and $Y^{\prime}$ using fast Fourier
transforms

Figure 5:  FFT of $Y/Y^{\prime}$ oriented top SrRuO3 regions of the samples
with (a) 90 u.c. PbTiO3 (b) 45 u.c. PbTiO3 and (c) 23 u.c. PbTiO3 (c) compared
to the simulation (g) with $a_{o}=5.62$ and $b_{o}=5.48$ parameters (in
arbitrary units). The colored diffraction patterns of the top SrRuO3 in
(a,b,c) combine the FFT Bragg peaks of the $Y$ domain in red and
$Y\textquoteright$ domain in green, where the regions of interest were
selected based on the regions of largest contrast in the GPA rotation maps in
(d,e,f).

FFT was performed locally on selected regions corresponding to different shear
strain and rotation values in the top SrRuO3 layers of the three samples, as
shown in Figure 5. The regions of interest are selected based on the largest
contrast in the GPA rotation maps in Figure 5(d-f), and give the colored
diffraction patterns in (a-c) for the sample with 90 u.c. (a), 45 u.c. (b) and
23 u.c. (c) PbTiO3 , respectively, combining the FFT Bragg peaks in red and in
green from the two regions. For the samples with 90 u.c. and 45 u.c. PbTiO3 ,
the two diffraction patterns do not overlap perfectly, and a very small
horizontal shift can be observed (the $(02\overline{2})_{pc}$ peak is enlarged
for clarity). For comparison, the diffraction pattern has been simulated for
the $Y$ and $Y^{\prime}$ orientations by arbitrarily increasing the difference
between the $a_{o}$ and $b_{o}$ parameters ($a_{o}=5.62$ and $b_{o}=5.48$ in
arbitrary units), making the difference in the Bragg peak positions more
visible: the difference between $Y$ and $Y^{\prime}$ results in horizontal
shifts of peaks, as measured experimentally. No shift is observed for the
SrRuO3 top electrode above 23 u.c. PbTiO3. From this we can conclude that the
contrast seen in the GPA strain and rotation maps for the two samples with the
thickest PbTiO3 layers arises from alternating $Y$ and $Y^{\prime}$ domains.
Moreover, Figure 5(d) shows that the transition from $Y$ to $Y^{\prime}$
domains for the sample with 90 u.c. PbTiO3 is gradual with a modulation of the
rotation following a sinusoidal behavior between 0.3∘ and -0.3∘ with a period
of approximately 30 nm. In comparison, this transition is much sharper for the
sample with the 45 u.c. PbTiO3 layer, with the modulation of the rotation
following a step-like function between 0.15∘ to -0.2∘ with a period of
approximately 20 nm as seen in Figure 5(e), hinting at the presence of proper
twin boundaries between $Y$ and $Y^{\prime}$ domains. Further discussion and
high resolution TEM image of such a twin boundary is shown in Supplementary
Materials, Section S4.

##  III Discussion

Figure 6:  STEM-BF images (a-c) on three different samples, together with the
sketches (d-f) representing the different polarisation patterns in the PbTiO3
layers and the induced crystallographic domains in the top SrRuO3 layers :
(top row) 90$\pm$4 u.c., (center row) 45$\pm$2 u.c. and (bottom row) 23$\pm$1
u.c. thick PbTiO3 between top and bottom SrRuO3 electrodes (55$\pm$2 u.c.
thick) on DyScO3 substrates. From these images, the domain walls are visible
in the PbTiO3 layers, forming the expected $a/c$ pattern for the thicker
PbTiO3 layer, transforming into a flux-closure pattern for the thinner PbTiO3
layers. In the PbTiO3 layers, domains with up polarisation are shown in red,
down in blue, left in yellow and right in green. In the SrRuO3 layers, the $X$
(or $X^{\prime}$) orientation is shown in brown, $Y$ in red and $Y^{\prime}$
in green.

We show that the complex tilt pattern of the PbTiO3 layer is responsible for
the deformation of the SrRuO3 layer deposited on top, resulting in the
periodic pattern visible in the topography by AFM. Our STEM measurements
highlight how the domain pattern in the PbTiO3 layer affects the strain state
and crystal orientation in the SrRuO3 top layer, with clear differences for
the three samples with different PbTiO3 layer thickness.

We find that the top SrRuO3 layers show a different behaviour for each of the
three samples, summarized in Figure 6, while the bottom SrRuO3 layers
systematically have the same orientation as the DyScO3 substrate
($X$-orthorhombic orientation), probably pinned by the interfacial continuity
of the oxygen octahedral rotation imposed by the substrate.

For the sample with a 23 u.c. PbTiO3 layer, where no pattern is visible in the
topography, we observe a flux-closure type PbTiO3 domain structure and
correspondingly a null or weakly distorted SrRuO3 structure, homogeneously in
the $Y$-orthorhombic orientation as shown in Figure 6 (bottom row).

For the sample with a 45 u.c. PbTiO3 layer, where a pattern is only weakly
observed in the topography, we again have a flux-closure type PbTiO3 domain
structure. However, in Figure 3 if we compare the out-of-plane (e) and in-
plane (h) strain maps close to the interfaces, the strain pattern locally
resembles that of the $a/c$ phase (Figure 3 (g-h)). This points at a nascent
$a/c$ phase near the top interface, allowing a more pronounced deformation
that translates into a clearly distorted SrRuO3 structure that shares the same
periodicity as the PbTiO3. The top SrRuO3 thus alternates between $Y$\- and
$Y^{\prime}$-orthorhombic orientations - with a period corresponding to the
period of the domain pattern in PbTiO3. The mechanism that drives the rotation
of the top SrRuO3 is strain-induced: due to the small shear and rotation in
the top layer of PbTiO3, the PbTiO3/top SrRuO3 interface is not totally flat
(left and right inclination). The formation of $Y/Y^{\prime}$ domains is a
good way to minimize the interface strain (Figure 6(center row)).

Finally, in the sample with the 90 u.c. thick PbTiO3 layer, where the
topography displays noticeable tilts and trenches, the $a/c$-phase is fully
developed, with strong PbTiO3 lattice heterogeneity that partially imprints on
the top SrRuO3 electrode. Here the top SrRuO3 exhibits domains with
alternating primary $Y$\- and $Y^{\prime}$\- orthorhombic orientations, and
some $X$ interstitial regions. The positive and negative rotation on the
SrRuO3 correspond to $Y$ and $Y\textquoteright$ domains as evidenced by the
GPA shear strain and rotation maps. In between these $Y$ and
$Y\textquoteright$ domains, a very tiny region with a typical $X$-orientation
is often observed in the FFT. This is typically restrained to a 10 nm width
domain at the interface between $Y$ and $Y\textquoteright$ domains where no
rotation was observed on the GPA map. It is different from the case with the
45 u.c. thick PbTiO3 layer, where sharp interfaces and no $X$ oriented domain
were observed between $Y$ and $Y\textquoteright$ domains.

Our work demonstrates that the large structural distortions associated with
ferroelastic domains propagate through the top SrRuO3 layer, creating a
modulated structure that extends beyond the ferroelectric layer thickness,
allowing domain engineering in the top SrRuO3 electrode. Since there exists a
one-to-one correspondence between the structural and magnetic domains Wang-
NPJQuantMats-2020 , our approach allows magnetic domain engineering in SrRuO3
thin films through structural domain engineering to be realized. This work
paves a new path towards control of magnetic domains via structural coupling
to ferroelastic domains.

##  IV Experimental techniques

###  IV.1 Sample growth

The three samples were deposited using our in-house constructed off-axis
radio-frequency magnetron sputtering system, equipped with three different
guns allowing the deposition of heterostructures and solid-solutions of high
crystalline quality. PbTiO3 thin films were deposited at 560∘C, in 180 mTorr
of a 20:29 O2/Ar mixture, at a power of 60 W, and using a Pb1.1TiO3 target
with 10% excess of Pb to compensate for its volatility. SrRuO3 layers were
deposited from a stoichiometric target in 100 mTorr of O2/Ar mixture of ratio
3:60, at a power of 80 W. The bottom layer was grown at 640∘C, while for the
top layer the temperature was kept at the growth temperature used for PbTiO3,
i.e. 560∘C, to avoid possible damage of the PbTiO3 layer. 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.

###  IV.2 Atomic force microscopy

Topography measurements were performed using a Digital Instrument Nanoscope
Multimode DI4 with a Nanonis controller.

###  IV.3 Scanning transmission electron microscopy

Cross-sectional slices prepared by focus ion beam allow the imaging of domain
structures by scanning transmission electron microscopy. 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, annular bright field or medium
angle annular dark field, and high angle annular dark field images. For ABF-
MAADF imaging, the inner-outer angles can be continuously adjusted between
10-20 to 60-120 mrad. Most ABF images were collected with 15-30 mrad and MAADF
images with 40-80 mrad angular ranges.

For the high-resolution HAADF images used to extract GPA, to minimize the
influence of the sample drift and environmental noise, a series of fast-scan
(low exposure time) HAADF images was taken in the same region; afterward, a
script based on Gatan DigitalMicrograph software aligned and summed them
together. This technique typically used twenty 4kx4k images with 1$\mu$s
exposure time per pixel.

GPA is an algorithm that reconstructs the displacement field
$\vec{u}(\vec{r})$ from HAADF images by measuring the displacement of lattice
fringes with respect to a reference lattice here chosen as the substrate. GPA
thus allows the local strain present in the different layers to be revealed:
in-plane strain $\varepsilon_{yy}=\frac{\partial u_{y}}{\partial y}$ (along
[010]pc, i.e. perpendicular to the growth direction), out-of-plane strain
$\varepsilon_{zz}=\frac{\partial u_{z}}{\partial z}$ (along [001]pc i.e. along
the growth direction), shear strain
$\varepsilon_{yz}=\frac{1}{2}\left(\frac{\partial u_{z}}{\partial
y}+\frac{\partial u_{y}}{\partial z}\right)$ and rotation
$\omega_{yz}=\frac{1}{2}\left(\frac{\partial u_{z}}{\partial y}-\frac{\partial
u_{y}}{\partial z}\right)$. This is particularly useful for the study of the
domain configuration in PbTiO3, as the polarisation is related to the strain
from the strong strain-polarisation coupling Cohen-Ferroelectrics-1992 . At
room temperature, PbTiO3 is tetragonal with the polarisation pointing along
the long tetragonal axis. The strain orientation and amplitude therefore
indicates the orientation and magnitude of the polarisation.

We determine the periodicity of the superstructures in the PbTiO3 and SrRuO3
layers 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.

###  IV.4 X-ray diffraction

In-house XRD measurements were performed using a Panalytical X’Pert
diffractometer with Cu K$\alpha_{1}$ radiation (1.5405980 Å) equipped with a
2-bounce Ge(220) monochromator and a triple axis detector in our laboratory in
Geneva. The $\theta$-2$\theta$ scans were analysed using the InteractiveXRDFit
software Lichtensteiger-JApllCryst-2018 . This XRD system is also equipped
with a PIXcel 1D detector, used for faster acquisition of the reciprocal space
maps.

##  V Data availability

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

##  VI Acknowlegements

The authors thank Lukas Korosec and Christian Weymann for support and
discussions.

This work was supported by Division II of the Swiss National Science
Foundation under projects 200021_178782 and 200021_200636. STEM experiments
were supported by the EU Horizon research and innovation program under grant
agreement ID 823717-ESTEEM3. C.-P. Su acknowledges Taiwan Paris-Saclay
doctoral scholarship, which is cost-shared by the Ministry of Education,
Taiwan, and the Université Paris-Saclay, France. M.H. acknowledges funding
from the SNSF Scientific Exchanges Scheme (Grant Number IZSEZ0_212990).

##  VII Author contributions

C.L, M.H., 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. C.-P.S.
and A.G. conducted the STEM measurements. I.G., C.-P.S. and A.G. performed
advanced STEM analysis. C.L. 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.

##  VIII Bibliography

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## Supplementary Materials

##  S1 Orthorhombic orientation

Figure S1:  Top - Schematic diagram showing the pseudocubic representation of
the (110)o-oriented DyScO3 substrate and the equivalence between the
pseudocubic and orthorhombic axis (in $Pbnm$ notation). Bottom - The six
possible orthorhombic orientations of the SrRuO3 on (110)o-oriented DyScO3
substrate (in $Pbnm$ space group). Figure adapted from Ref. Jiang-MSEB-1998 .
The blue arrow points in the direction of the orthorhombic long axis $c_{o}$.
The yellow and black arrows point along the two diagonals in the
($a_{o}$,$b_{o}$) plane, along $[110]_{o}$ (yellow) and $[\overline{1}10]_{o}$
(black). The 6 possible orthorhombic orientations are obtained by
$\pm\frac{\pi}{2}$ rotations around each of the three pseudo-cubic axes.
Figure S2:  Detailed representation of the six possible orthorhombic
orientations of the SrRuO3 on (110)o-oriented DyScO3 substrate (in $Pbnm$
space group).

In Figure S2, the six possible orthorhombic orientations of the SrRuO3 on
(110)o-oriented DyScO3 substrate (in $Pbnm$ space group) are represented. Each
column corresponds to a different orientation $X$, $X^{\prime}$, $Y$,
$Y^{\prime}$, $Z$ and $Z^{\prime}$. First row: projection along [100]pc. The
oxygen octahedra rotations are represented as blue (in-phase along the [001]o
axis), yellow (out of phase along the [110]o axis) and white (out of phase
along the $[1\overline{1}0]_{o}$ axis) rotating arrows, and the antipolar
motion of the A-cations are represented by green straight arrows. Second row:
perspective view. $\vec{a}\parallel[100]_{o}$, $\vec{b}\parallel[010]_{o}$ and
$\vec{c}\parallel[001]_{o}$ correspond to the axis of the orthorhombic unit
cell. Third row: corresponding pseudocubic representation. Forth row:
reciprocal space representation and (hkl)pc indices corresponding to the
pseudocubic structure. Fifth row: reciprocal space representation and (hkl)o
indices corresponding to the orthorhombic structure structure, highlighting
the position of the “half-order” peaks.

##  S2 X-ray diffraction data for the three samples

Figure S3:  Reciprocal space maps showing the high crystalline quality of the
samples studied here - Reciprocal space maps around the (001)pc peak of the
substrate in the $Q_{[001]_{pc}}-Q_{[100]_{pc}}$ plane and in the
$Q_{[001]_{pc}}-Q_{[010]_{pc}}$ plane. Below each map is the corresponding
intensity obtained from a cut at $Q_{[001]_{pc}}$=1.55 Å-1 ($c$-domains),
displaying the periodic peaks of the domain pattern. This demonstrates that
the periodic pattern is heterogeneous, with periodicity detected only with the
incident XRD beam pointing along the DyScO3$[001]_{o}$-axis. It also shows the
different values reported in Table S1, with the additional periodic peak
visible for the sample with the thickest PbTiO3 layer only, marked as a green
star.

##  S3 Table summarizing the different periods

PbTiO3 | Period in PbTiO3 | Period in PbTiO3 | Period in SrRuO3 | Period
---|---|---|---|---
thickness | from XRD | from STEM | from STEM | from topo
(u.c.) | (nm) | (nm) | (nm) | (nm)
23 $\pm$ 1 | 14.3 $\pm$ 0.1 | 13 $\pm$ 1 | n.a. | n.a.
45 $\pm$ 2 | 23.4 $\pm$ 0.2 | 26 $\pm$ 1 | 25 $\pm$ 1 | n.a.
90 $\pm$ 4 | 44 $\pm$ 3 (146.4 $\pm$ 0.3) | 46 $\pm$ 3 | 80 $\pm$ 30 | 77 $\pm$ 1 (280 $\pm$ 3)
Table S1: Table summarizing the different periods along DyScO3$[001]_{o}$.

##  S4 Twin boundary observed by GPA and HAADF in the SrRuO3 top electrode
above the 45 u.c. thick PbTiO3 layer

Figure S4:  (a) High-resolution HAADF image of the 45 u.c. PbTiO3 sample. The
red lines indicate the atomic layers, showing a small angular difference
between the $Y$ and $Y^{\prime}$ domains. overlaying the corresponding
rotation map of GPA. (b) The overlapping of the crystal structure of the $Y$
and $Y\textquoteright$ domains. It reveals the continuity of cation along
[110]o and a slight clockwise or counterclockwise rotation.

As mentioned in the paper, the strain map for the 45 u.c. thick PbTiO3 layer
in Figure 3 (center row) is complex, with regions alternating with large out-
of-plane strain or large in-plane strain close to each interface, and reduced
strain at the center of the PbTiO3 layer. The succession of sequences close to
the interfaces resembles with local $a/c$-domains, but with an offset of about
half a period between the sequences at the top and at the bottom interface. In
the center of the layer the out-of-plane extension is rather weak, while
periodic strain contrasts are present and alternating rotations are visible in
the GPA maps. It is clearly different from the expanded $a/c$-phase of the 90
u.c. thick PbTiO3. Horizontal flux-closure quadrants as observed for PbTiO3
with similar thickness grown without electrodes Tang-Science-2015  will give
resembling features, but does not explain the local $a/c$-domains at the
interface, hinting at a more complicated three-dimensional flux-closure
domain. For instance, combination of “horizontal” and “vertical” flux closures
have already been reported in metal-ferroelectric superlattices to minimize
the macroscopic polarisation for imperfect electrode while still competing
with $a/c$-pattern at longer scale Hadjimichael-NatMat-2021 . Such combination
can explain the strong flux closure contrast from the HAADF and MAADF STEM
images and the presence of nascent local $a/c$-phases at the interface. The
complex shear strains and rotations in the PbTiO3 layer results in a distorted
top interface, exhibiting staggered left or right inclination confirmed by the
atomic-resolved HAADF image shown in Figure S4. The interface has a zigzag
behaviour due to the presence of the small $a$-domains, inducing SrRuO3
twinning. This image demonstrates how the growing $a/c$ domains near the top
of the PbTiO3 film imposes the crystallographic domain distribution in the top
SrRuO3 layer. The corresponding rotational and shear patterns of the top
PbTiO3 part are thus corresponding to the $Y$ and $Y^{\prime}$ domains of the
SrRuO3, as shown in Figure S4, both having the same periodicity (ca. 26 nm),
sharing interface connectivity.