# Ferroelectric texture of individual barium titanate nanocrystals

Athulya Muraleedharan  [  Kevin Co  Maxime Vallet  Abdelali Zaki  Fabienne
Karolak  Christine Bogicevic  [  Karen Perronet  [  Brahim Dkhil  [  Charles
Paillard  [  Céline Fiorini-Debuisschert  [  François Treussart  [
<EMAIL_ADDRESS>
###### Abstract

Ferroelectric materials display exotic polarization textures at the nanoscale
that could be used to improve the energetic efficiency of electronic
components. The vast majority of studies were conducted in two dimensions on
thin films, that can be further nanostructured, but very few studies address
the situation of individual isolated nanocrystals synthesized in solution,
while such structures could open other field of applications. In this work, we
experimentally and theoretically studied the polarization texture of
ferroelectric barium titanate (BaTiO3, BTO) nanocrystals (NC) attached to a
conductive substrate and surrounded by air. We synthesized NC of well defined
quasi-cubic shape and 160 nm average size, that conserve the tetragonal
structure of BTO at room temperature.

We then investigated the inverse piezoelectric properties of such pristine
individual NC by vector piezoresponse force microscopy (PFM), taking
particular care of suppressing electrostatic artifacts. In all the NC studied,
we could not detect any vertical PFM signal, and the maps of the lateral
response all displayed larger displacement amplitude on the edges with
deformations converging toward the center. Using field-phase simulations
dedicated to ferroelectric nanostructures, we were able to predict the
equilibrium polarization texture. These simulations revealed that the NC core
is composed of 180° up and down domains defining the polar axis, that rotate
by 90° in the two facets orthogonal to this axis, eventually lying within
these planes forming a layer of about 10 nm thickness mainly composed of 180°
domains along an edge. From this polarization distribution we predicted the
lateral PFM response, that revealed to be in very good qualitative agreement
with the experimental observations. This work positions PFM as a relevant tool
to evaluate the potential of complex ferroelectric nanostructures to be used
as sensors.

###### keywords:

Ferroelectrics, Barium titanate, Nanocrystal, Piezoresponse force microscopy,
Phase field simulation

LuMIn] Université Paris-Saclay, ENS Paris-Saclay, CNRS, CentraleSupélec,
LuMIn, 91190 Gif-sur-Yvette, France SPMS] Université Paris-Saclay,
CentraleSupélec, CNRS, Laboratoire SPMS, 91190 Gif-sur-Yvette, France LuMIn]
Université Paris-Saclay, ENS Paris-Saclay, CNRS, CentraleSupélec, LuMIn, 91190
Gif-sur-Yvette, France SPMS] Université Paris-Saclay, CentraleSupélec, CNRS,
Laboratoire SPMS, 91190 Gif-sur-Yvette, France SPMS] Université Paris-Saclay,
CentraleSupélec, CNRS, Laboratoire SPMS, 91190 Gif-sur-Yvette, France [UA]
Department of Physics, University of Arkansas, Fayetteville AR 72701, USA
SPEC] Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette, France
LuMIn] Université Paris-Saclay, ENS Paris-Saclay, CNRS, CentraleSupélec,
LuMIn, 91190 Gif-sur-Yvette, France

##  1 Introduction

Ferroelectric materials gather a remarkable set of physical properties,
including a large dielectric permittivity, a piezoelectric response, and
ferroelectric hysteresis that make them vital for several industries 1.
Recently, progress in the synthesis and growth methods of ferroelectrics has
allowed new exotic effects to be discovered at low dimensions. For instance,
new quasiparticles such as (anti)vortices or polar skyrmions have been widely
predicted 2, 3, 4, 5, 6, 7 and subsequently observed 8, 9, 10, 11, 12 due to
electrostatic depolarization in two-dimensional structures such as
superlattices or thin films, or in one-dimensional nanowires embedded in a
dielectric matrix. These topological defects are often endowed with rich
properties, such as negative dielectric capacitance 13, an effect envisioned
to reduce Field Effect Transistor energy consumption. The polarization texture
in 0-dimensional ferroelectric crystals, that is, ferroelectric nanodots and
nanocrystals (NC) have been less studied; yet recent (mainly simulation)
studies have reported exotic polarization textures in NC as well 14, 15, 16,
17.

Therefore, in this work we set out to understand the polarization texture of
ferroelectric BaTiO3 (BTO) NC by an original combination of experiments and
phase field modeling 15. BTO is a prototypical ferroelectric material with a
paraelectric phase above 120∘C in bulk crystal. Below that transition
temperature, and down till about 5∘C, a polar order and a tetragonal
deformation of the unit cell are established. This is the room temperature
ferroelectric phase. Below 5∘C, a succession of ferroelectric orthorhombic and
rhombohedral phases appears 18. The synthesis of BTO nanocrystals with well-
controlled morphology, size, and properties is well established 19, 20.
Compared to bulk, the larger surface-to-volume ratio of BTO NC leads to
starkly different properties, particularly ferroelectric ones, which strongly
depend on boundary conditions. For instance, the phase transition from the
para- to the ferro-electric state is more diffuse in BTO NC 21. Furthermore,
the small dimensions offered in BTO NC, combined with the presence of
electrical polarization, offer competitive advantages for charge separation in
photovoltaic or photocatalytic applications 22, 23, 24, 25, 26, 27, 28.
However, there is little to no report on the nanoscale inhomogeneity of the
polarization distributions, which may be critical for applications such as
photocatalysis.

A frequently used technique of characterization at the nanoscale is the high-
angle annular dark-field (HAADF) imaging mode of scanning transmission
electron microscopy (STEM), harnessing its high sensitivity to ion nuclei
displacements. A polarization structure was observed by HAADF-STEM within 200
nm sized BTO free-standing cuboids milled in a bulk crystal using a focused
ion beam. The observations were interpreted as the results of quadrant
structures of 90∘ polarization domains 29. As the nanocuboids were milled in
bulk, one cannot exclude that the domain structure is partly induced by the
fabrication process, as recently reported 30. Other electron beam-based
techniques, capable of sensing the electric field of the sample via phase
accumulation and its subsequent retrieval, are used to probe the local order
in nanoferroelectrics 31. In particular, Polking et al. harnessed off-axis
electron holography to prove that a 10 nm-sized BTO nanocube retains
ferroelectric properties, as evidenced by the presence of an electric field
around the NC only below Curie temperature 32. Moreover, taking advantage of
the atomic resolution provided by their aberration-corrected STEM, they were
able to reconstruct the atomic displacements in the 10 nm nanocube and
demonstrated that it is made of a single ferroelectric domain. Using Bragg
coherent X-ray diffraction imaging of single BTO nanoparticles of larger size
(160 nm) embedded in a non-ferroelectric polymer matrix, Karpov et al.33
extracted much more complex polarization field (than in 10 nm-sized BTO
nanocube) showing that it arranges in a flux-closure manner (forming a
vortex), that can be consistently reproduced by phase field simulations.
Alternatively to these non-contact methods based on electron or X-ray beams,
near field microscopy in contact mode has been used. For example, the
ferroelectric properties of an array of 60 nm-sized dots nanostructured in a
thin film of bismuth iron oxide grown on a conductive substrate have been
explored by Piezoresponse Force Microscopy (PFM) 14, revealing various
polarization textures in individual dots.

Here, we combine PFM and phase field simulations to study the polarization
texture of isolated BTO nanocubes in their native ferroelectric state, as
produced by solvothermal synthesis. We first present NC synthesis and
structural characterization, then electromechanical mapping on individual NC
using PFM, and finally phase field simulations of such displacement fields. We
show that the simulations are in very good qualitative agreement with the PFM
data, enlarging the domain of application of PFM, mostly used for thin film
characterization, to complex nanostructures, for which phase field simulations
can be demanding.

##  2 Results and discussion

###  2.1 Barium titanate nanocrystals of quasi-cubic shape

Figure 1: Structural characterization of barium titanate nanocubes. (a) X-ray
diffractogram showing all the detected peaks indexed to the tetragonal phase
of BTO. The data points observed are represented as red circles, and the
calculated intensities are shown as a black solid line. The inset graph shows,
near $2\theta=45^{\circ}$, two separate peaks for (002) and (200) planes
evidencing the tetragonal structure. Green stars indicate small peaks
associated to the structure of residual BaCO3 crystallites precursor. The
solid blue line at the bottom represents the difference between the observed
and calculated intensities. (b) TEM image of a small aggregate of
nanocrystals. Scale bar: 200 nm. Inset: electron diffraction pattern on the
encircled region, consistent with a single-crystal nature. Scale bar: 10 nm-1.
(c-e) HR-STEM images of another BTO nanocrystal, in bright-field BF (c), dark-
field DF (d) and high-angle annular dark-field HAADF (e) modes. Scale bar: 20
nm. BF and DF modes display multiple square-shaped spots within the NC, which
may be little pores.

Quasi-cubic shaped barium titanate nanocrystals were synthesized using the
solvothermal method described in Bogicevic et al. 19 (Materials and Methods).
Supporting Information Figure S1a shows a large field-of-view scanning
electron microscopy image of the as-produced powder, confirming the overall
quasi-cubic shape. We first did a structural analysis of this powder, from
which we then prepared the dilute suspension for PFM studies of single
nanocrystals.

Figure 1a displays the powder X-ray diffractogram at the temperature of 293 K
showing the characteristic peak splitting of tetragonal phase at twice the
diffraction angle $2\theta\approx$45∘, corresponding to the $(hk\ell)$ Miller
indexes $(002)$ and $(200)$, while a purely cubic phase would have a single
peak associated to $(002)$. Rietveld refinement of the diffractogram
(Materials and Methods) reveals a well-known tetragonal structure (space group
symmetry P4mm) for BTO at room temperature (293 K), with
$a=b=4.0072\pm\kern-1.8pt0.0005$ Å and $c=4.0339\pm\kern-1.8pt0.0006$ Å,
leading to a lattice parameter ratio $(c/a)_{\rm
NC}=1.0067\pm\kern-1.8pt0.0002$. However, compared to bulk BTO for which
$(c/a)_{\rm bulk}=1.0101\pm\kern-1.8pt0.0002$ 34, the ratio measured in our NC
is significantly smaller by 0.3% and closer to one, indicating that its
lattice is slightly less tetragonal and more cubic than the bulk one, in
agreement with other reports on BTO nanocrystals 35, 36. We also used
transmission electron microscopy (TEM, Materials and Methods) to investigate
the crystalline structure at the single particle level. Figure 1b shows a TEM
image of a small aggregate accompanied with the diffraction pattern of the
selected area on the central nanocrystal. The regular pattern obtained is
consistent with the monocrystalline nature of the selected NC.

The reduced tetragonality that we observed in BTO nanocrystals compared with
the bulk was reported in several studies and was attributed to different
sources. Firstly, one study evidenced by high-resolution scanning TEM (HR-
STEM) the presence of a thin shell of cubic symmetry at BTO NC surface 37.
This observation was further supported by indirect measurements and models 36.
We examined our BTO NC edge with HR-STEM (see Figure S2) but could not detect
a layer of different symmetry. Secondly, we estimated to 0.19% the
inhomogeneous strain from Figure 1a diffractogram using Williamson-Hall method
with a Cauchy peak profile 38 (see Figure S1b), which is smaller than the
value of 0.26% recently reported 39, but not negligible. This strain is
probably due to hydroxyl groups coming from the barium precursor and remaining
inside the nanocrystals as point structural defects. The latter would migrate
to the surface only at temperatures much higher than those used during
solvothermal synthesis. Other sources of strain are small square-shape pores
that we observed in bright-field (Figure 1c) and dark-field HR-STEM (Figure
1d) of one nanocrystal, but not in high-angle annular dark-field which rather
shows chemical contrast. In some particles with rounder morphology we even
observed much larger pores (see Figure S2c,d). We reported such pores in a
previous study on the same sample 19, and showed evidences that they result
from the merging and restructuration, upon temperature increase above 200∘C,
of nanotori, which are the primary structure formed at low temperature.

As these defects may impact the ferroelectric properties, we further conducted
Raman spectroscopy on BTO powder, at temperatures varying between 80 K and 540
K. In the centro-symmetrical paraelectric cubic phase, BTO is Raman inactive,
contrary to the tetragonal phase, which symmetry leads to height Raman
optical-phonon active modes40. We monitored the area variation of the
narrowest active Raman peak (305 cm-1 wavenumber, at 300 K) with temperature
(see Figure S3a), as in Begg et al. 41 and observed a marked decrease at
$395\pm 2$ K (Figure S3b) in agreement with the commonly accepted BTO Curie
temperature of 120∘C. This observation indicates that, despite a reduced
tetragonality compared with the bulk, the BTO NC we synthesized keeps the
hallmark of the bulk BTO ferroelectric phase transition. Our objective was
then to examine the ferroelectric domain texture of single BTO nanocrystals,
and to this aim we performed vector piezoresponse force microscopy 42.

###  2.2 Piezoresponse displacement measurements

PFM allows nondestructive imaging and control of ferroelectric domains at the
nanoscale via bias-induced converse piezoelectric effect mechanical
deformations. It has gained interest in helping the development of new
nanoferroics 43, 44. The basic principles of PFM are recalled in Supporting
Information Text S1 and for example in ref 45. PFM is a near-field microscopy
operating in contact mode, which makes the study of individual nanocrystals
challenging, as they must be firmly immobilized on a conductive substrate,
while having their upper surface directly exposed to the conductive tip.

For such a study we prepared a sample with isolated NC attached to a
conductive substrate made of indium tin oxide (ITO)-coated coverglass, with a
marking grid on top of the coating, to facilitate the correlation between
scanning electron microscopy prelocalization of isolated NC and their
subsequent study by PFM. The ITO-coated coverglass with the grid is first
covered with a thin layer of a conductive polymer (PEDOT:PSS), on top of which
a dilute aqueous suspension of the NC is spincoated (Materials and Methods),
as displayed on Figure 2a. We optimized the various parameters to achieve a
polymer thickness in the range of 40-50 nm (see Figure S4a). Thanks to a good
wettability of BaTiO3 by water 46, the concave meniscus (visible on Figure
S4b) forming on the edges of the nanoparticles drags them down by surface
tension, and after drying the polymer layer in an oven, about two-thirds of
the particle height emerges and can be directly contacted by the PFM
conductive tip.

Figure 2: Experimental configuration of lateral piezoforce response
measurement on a single BTO nanocrystal. (a) Immobilization of BTO NC onto an
ITO-coated coverglass with a thin film of PEDOT:PSS conductive polymer. (b)
Schematics of lateral PFM measurement. $(x,y,z)$ and $(X_{\rm c},Y_{\rm
c},Z_{\rm c})$ are the laboratory and crystal-attached frames, respectively.
The measurement consists in applying an AC-voltage $V_{\rm AC}(t)=V_{\rm
AC}\cos(2\pi f_{\rm d}t)$ to the tip, with $f_{\rm d}$ being the driving
frequency, set close to the tip contact resonance, and $V_{\rm AC}=1.5$ V. The
PFM tip, aligned along the $y$ axis, is scanned along the $x$ axis, which is
colinear to $X_{\rm c}$ axis initially (left configuration). During scanning,
the nanocrystal in-plane polarization domains (displayed by gray arrows)
induce shear displacements of the tip along the $x$ axis, oscillating in phase
or out of phase with $V_{\rm AC}(t)$, depending on the orientation of the
projection of the polarization along the $X_{\rm c}$ axis. The resulting
displacement amplitude $u_{X_{\rm c}}$ is inferred from the deflection of the
PFM laser as measured by the quadrant photodiode (QPD). To reduce
electrostatic artifacts, we set the laser beam to hit the cantilever at the
electrostatic blind spot (ESBS, see main text). To access the displacement
$u_{Y_{\rm c}}$ along the $Y_{\rm c}$ axis, we rotate the sample by 90∘
counterclockwise around the laboratory $z$ axis, and repeat the scan along the
$x$ axis (right configuration; the laser and QPD are not shown for the sake of
clarity).

Figure S5a shows a typical low magnification SEM image of the sample with an
apparent cell of the numbered grid. To conduct the PFM study, we first
selected and located isolated NC, like the one at the center of Figure S5b,
and characterized their size distribution, that fell in the range of 120-210
nm, as shown on Figure S6a. Moreover, Figure S6b indicates that about 65% of
these NC have an planar aspect ratio larger than 0.95 in the sample plane,
which is a good indication of close to perfect cubic shape.

PFM measurements were conducted according to the experimental configuration
presented in Figure 2b. We first took care of minimizing artifacts that could
stem from electrostatic charge accumulation 47. We evidenced such parasitic
effects when using conventional deflection laser spot setting at the
cantilever tip. These effects include the observation on the conductive and
nonferroelectric surface of the ITO of an apparent contact resonance of the
cantilever (see Figure S7a), an hysteresis loop (Figure S7b), and charge
accumulation in freshly “written” regions (Figure S8a,b). We were able to
strongly reduce these non-electromechanical artifacts by implementing the
recently published electrostatic blind spot (ESBS) settings 47 as detailed in
the Supporting Information Data S2.2 and shown on Figure S7c,d and Figure
S8c-f.

Once these parasitic effects were minimized, we could more reliably map the
lateral PFM two-dimensional displacement field (LPFM) of the top surface of
immobilized NC. As the PFM cantilever scans only along the direction $x$ of
the laboratory frame, the sample must be rotated by 90∘ around the vertical
$z$ laboratory axis, as shown on Figure 2b, to acquire the $Y_{\rm c}$ LPFM
component after the $X_{\rm c}$ one, $(X_{\rm c},Y_{\rm c},Z_{\rm c})$ being a
frame of reference attached to the NC of interest.

Figure 3: Lateral piezoresponse displacement mapping on a single BTO NC. (a)
SEM scan of the investigated particle, with an approximately square shape and
size of 125 nm. The laboratory frame is denoted $(x,y,z)$, with (O$z$)
perpendicular to the substrate. We scan the cantilever along the $x$ axis. (b)
AFM height measurement (contact mode) of the same BTO NC oriented as shown in
(a). (c,d) Amplitudes $u_{X_{\rm c}}$ and $u_{Y_{\rm c}}$ of lateral
piezoresponses along the $X_{\rm c}$ and $Y_{\rm c}$ axis respectively of the
frame $(X_{\rm c},Y_{\rm c},Z_{\rm c})$ attached to the nanocrystal and
confounded with $(x,y,z)$ before sample rotation. To record $u_{Y_{\rm c}}$
the sample was rotated 90∘ counterclockwise around the $z$ axis. The amplitude
of the electric field modulation applied was 1.5 V. (e,f) Lateral
piezoresponse phase associated to $X_{\rm c}$\- and $Y_{\rm c}$ axis
respectively. (g,h) Cross sections of the (e) and (f) phase maps. (i) Mixed
lateral piezoresponse field $\boldsymbol{u}^{\rm m}(u_{X_{\rm
c}}\cos(\varphi_{X_{\rm c}}-\varphi_{\rm offset}),u_{Y_{\rm
c}}\cos(\varphi_{Y_{\rm c}}-\varphi_{\rm offset}))$, displayed by a
superimposed norm (color code) and direction maps (array of fixed-length
arrows at the nodes of a 7.8 nm pitch grid). Note that the lateral
displacement values are not absolute, as we did not perform any calibration.
Scale bars (all scans): 50 nm.

We implemented this procedure to map the LPFM of four nanocrystals, setting
the voltage modulation amplitude applied to the tip at $V_{\rm AC}=1.5$ V for
all of them. Figure S9 shows a typical cantilever contact resonance curve,
used to set the driving frequency close to its maximum. Figure 3 exemplifies
LPFM measurements in one of the four NC, whose lateral dimensions measured
using SEM (Figure 3a) are $125\text{~{}nm}\times 125\text{~{}nm}$. The height
of the portion of the NC emerging from the PEDOT:PSS layer, as measured by
AFM, is $\approx 72$ nm (Figure 3b). Assuming a cubic shape, the depth of
penetration of the particle in the polymer layer is then $\approx 53$ nm,
which is consistent with Figure S4a thickness estimate.

Figure 3c-h then display LPFM measurements carried out on this NC, for which
the electric field applied by the tip (along the laboratory $z$ axis) has an
amplitude $E^{\rm exp}_{z}=120$ kV/cm across the supposedly 125 nm-sized
particle. Figure 3c shows a map of $u_{X_{\rm c}}$, which is the amplitude of
the voltage-induced local lateral deformation along the $X_{\rm c}$ direction.
Figure 3e shows the map of the phase difference $\varphi_{X_{\rm c}}$ between
$V_{\rm AC}$ applied to the tip and the NC piezoresponse along the $X_{\rm c}$
direction. To access LPFM measurement along the $Y_{\rm c}$ direction, the
sample was rotated counterclockwise by $90\pm\kern-1.8pt2$∘, around the
laboratory $z$ axis. Figure 3d and Figure 3f show the amplitude $u_{Y_{\rm
c}}$ and phase $(\varphi_{Y_{\rm c}}+180\text{${}^{\circ}$})$ maps
respectively, along the $Y_{\rm c}$-direction. Note that the phase actually
measured in the rotated-sample orientation is $\varphi_{Y_{\rm c}}+180$∘ to
account for the fact that the $Y_{\rm c}$ axis, after rotation, points in the
opposite direction of the original $X_{\rm c}$ axis. We observed a small
instrumental phase offset, $\varphi_{\rm offset}=-10\pm\kern-1.8pt2$∘, for the
in-phase piezoresponse (refer to Materials and Methods for its determination),
which we arbitrarily chose to be the phase of the in-phase domain (with a
positive $X_{\rm c}$ component). The cross-sections of Figure 3e,f evidence a
phase difference of $\approx 180^{\circ}$ between the red and green regions,
revealing the presence of in-plane electric polarization domains with opposite
orientations along the scanning direction.

We finally inferred the mixed lateral piezoresponse displacement field
$\boldsymbol{u}^{\rm m}$ components, defined as $u^{\rm m}_{X_{\rm c}}\equiv
u_{X_{\rm c}}\cos(\varphi_{X_{\rm c}}-\varphi_{\rm offset})$ and $u^{\rm
m}_{Y_{\rm c}}\equiv u_{Y_{\rm c}}\cos(\varphi_{Y_{\rm c}}-\varphi_{\rm
offset})$, displayed in Figure 3i. We observe that the displacement directions
close to the nanocrystal edges are either parallel or orthogonal to these
edges, and that these orientations propagate from the surface to the center.

The same behavior was observed for the lateral PFM applied to the three other
nanocrystals investigated (see Figures S10 to S12.) The measured lateral
responses display non-homogeneous converging deformation fields that reflect
complex underlying ferroelectric textures, and that we further investigated by
simulations, as discussed later. Note that similar converging deformation
fields were reported in single bismuth iron oxide dots of cylindrical shape,
nanostructured as an array in a thin layer 14.

Figure 4: Vertical piezoresponse of the same nanocrystals as the one of Figure
3, which footprint is marked by the white dashed line square. (a,b) PFM
amplitude (a) and phase (b) maps showing no significant contrast. (c) Cross
section of the phase image along the solid white line. Scale bars (all
images): 50 nm.

We also measured the vertical piezoresponse (VPFM) that is shown in Figure 4,
and detected no significant vertical piezoresponse amplitude compared to the
lateral one as $u_{Z_{\rm c}}^{\rm max}/u_{X_{\rm c}}^{\rm max}\approx 4\times
10^{-3}$ (Figure 4a). We observe a phase difference of $\approx-80$∘ relative
to the surrounding, in the exact same region where there is no displacement
amplitude at all. We therefore suspect that this phase value has no physical
meaning.

In order to interpret these PFM data, we carried out ferroelectric phase field
simulations, first without any applied external electric field, to get the
equilibrium ferroelectric domain distribution, and then in condition of PFM
measurement, with an external applied electric field which amplitude is of the
order of magnitude of the one used in the experiment.

###  2.3 Phase field simulations

We considered a cubic-shape BTO nanocrystal of 100 nm size, embedded in air
and used the FERRET module 15 of the Multiphysics Object Oriented Simulation
Environment (MOOSE), an open-source software maintained by Idaho National
Laboratory 48. FERRET is a finite element implementation of a ferroelectric
phase field model, which we utilise here to capture the polar and elastic
equilibrium state of BTO nanocrystals, as well as their response to static
electric fields. In the experiment, one facet of the particle is in contact
with a conductive medium. However, as shown in Prosandeev and Bellaiche 49,
asymmetric screening with a metal electrode does not destroy the domain
structure in ferroelectric thin films, and the simulations conducted probably
extend to the experimental case of the bottom facet surface attached to a
conductive layer with the rest of the nanocube exposed to air.

##### Equilibrium domain structure.

Figure 5: Polarization components distribution at equilibrium in the middle
plane $X_{\rm c}=50$ nm of a 100 nm sized cube of BTO. (a) Planar distribution
of the $P_{Z_{\rm c}}$ component defining the main (O$Z_{\rm c}$) polar axis
([001] direction). The alternation of red and blue shadings associated with
positive and negative $P_{Z_{\rm c}}$ components, respectively, illustrates
the multiple 180∘ domains. The black arrow glyphs map the polarization field
3D orientation every 2 nm. Note the small thickness of the surface domains
(top and bottom surfaces) with $P^{\rm eq}_{Z_{\rm c}}\approx 0$. (b) Profiles
of the polarization components along the line L1 ($X_{\rm c}=50$ nm, $Y_{\rm
c}=50$ nm, $Z_{\rm c}$) showing the presence of surface domains with
polarization perpendicular to the main (O$Z_{\rm c}$) polar axis. (c) Profiles
of the polarization components along the line L2 ($X_{\rm c}=50$ nm, $Y_{\rm
c}$, $Z_{\rm c}=50$ nm) showing the partial Bloch and Néel nature of some
domain walls. (d) 3-dimensional representation of the polarization vector
along the same line as in (c), with labeled types of domain walls (B stands
for Bloch, N for Néel and I for Ising; I+B, I+N and I+B+N indicate a mixed
Ising-Bloch, mixed Ising-Néel and mixed Ising-Bloch-Néel character
respectively).

We first determined the equilibrium polarization and displacement fields,
$\boldsymbol{P}^{\rm eq}(\boldsymbol{r})$ and $\boldsymbol{u}^{\rm
eq}(\boldsymbol{r})$ by minimising the energy of a BTO nanocube with random
and small initial polarization and mechanical displacement field components.
Figure 5a shows that minimization of the free energy of the system leads to a
multidomain structure consisting of domains polarized upward and downward
along the (O$Z_{\rm c}$) axis, which we now refer to as the polar axis [001].
We estimate that the core of the BTO NC made of upward and downward tetragonal
domains along (O$Z_{\rm c}$) represents 60% of the fraction of the volume of
the NC. From the polarization profiles along lines L1 and L2 (Figure 5b,c), we
inferred a polarization component amplitude $P_{Z_{\rm c}}=\pm 0.265$ C/m2 in
these up and down domains, which is consistent with what has been measured 50
or calculated 51, 52 in the tetragonal ferroelectric phase of barium titanate.

Looking at the domain walls separating up and down domains in Figure 5c
polarization profile, we observe that they are mostly of the Ising-type, i.e.
the transition is accomplished by modulating the $P_{Z_{\rm c}}$ component. We
do see, however, that some walls retain partial Bloch characteristics
(rotation of the polarization in planes parallel to the domain wall, i.e.
showing non-vanishing $P_{X_{\rm c}}$ component, for example at $Y_{\rm
c}=$11, 35, 55 and 90 nm) or Néel characteristics (rotation of the
polarization in a plane orthogonal to the domain wall, i.e. with non-zero
$P_{Y_{\rm c}}$, for example at $Y_{\rm c}=$45 and 55 nm), as schematized on
the polarization profile line L2 in Figure 5d.

Note that, in surface planes along the polar axis like $X_{\rm c}=0$ nm, the
polarization aligns up and down along (O$Z_{\rm c}$) like in the bulk (see
Figure S13a) and the domain walls are also of Ising type with partial Néel
characteristics (Figure S13b).

Let us now focus our attention on the facets of the NC perpendicular to the
polar axis, $Z_{c}=0$ and $Z_{c}=100$ nm. Interestingly, the $P_{Z_{\rm c}}$
component of the polarization field vanishes near the surface of these facets
(see Figure 5a,b). Instead, the polarization lies in the plane of these
facets, as shown in Figure 6a and Figure S13c (facet $Z_{c}=100$ nm), with
mostly tetragonal domains polarized along the (O$X_{\rm c}$) direction
(representing about 16% of the total NC volume), as well as a lower number of
domains polarized along the (O$Y_{\rm c}$) direction (about 7% of the NC
volume). These surface domains extend at an average depth of 8 nm (see Figure
5a,b) from the surface of the $Z_{\rm c}=0$ and $Z_{\rm c}=100$ nm facets, and
they also display Ising domain walls with Bloch and/or Néel characteristics
(Figure S13d) like in the core of the nanocrystal (Figure 5d).

Rotation of the polarization from (O$Z_{\rm c}$) direction to either (O$X_{\rm
c}$) or (O$Y_{\rm c}$) one allows the system to prevent a strong electrostatic
depolarizing field. Indeed, the averaged planar electrostatic energy density
in the equilibrium structure is mostly zero (see Figure S14). Depolarizing
fields, which appear when polarization-induced bound charges are imperfectly
screened at an interface, are well known in ferroelectric thin films 53, 54.
They are responsible for the emergence of flux closure patterns 55, 49 in
which the polarization field rotates from the out-of-plane direction to lie in
the plane of the thin film surface, similarly to our NCs, as well as for the
appearance of more exotic polar textures such as polar vortices 56, which we
do not observe here. Rotation of polarization on surfaces normal to (O$Z_{\rm
c}$) thus allows the NC to sustain uniformly polarized domains, at the cost of
an increase in electromechanical energy near the surfaces. Nonetheless, this
is counteracted by the release of elastic energy by the $90^{\circ}$ domain
walls near the surface (see Figure S14).

##### Lateral piezoresponse force response simulation.

Figure 6: Simulated in-plane polarization texture and piezoresponce of 100 nm
sized BTO cube, with polar axis along (O$Z_{\rm c}$). (a) Polarization
gradient in the $Z_{\rm c}=100$ nm (001) plane at equilibrium (at the
temperature of 293 K), with light gray indicating domains and dark grey/black
indicating domain walls. The solid arrow glyphs (green) represent the
orientation of the polarization vector. (b) Polarization gradient in the
$Y_{\rm c}=100$ nm (010) plane. (c) In-plane mixed piezoresponse in the
$Z_{\rm c}=100$ nm plane, for a crystal oriented with the $Z_{\rm c}$ axis
parallel to the laboratory $z$ axis, in response to a field of amplitude
$E_{z}=150$ kV/cm. The solid arrow glyphs indicate the piezoresponse vector
orientation at the surface. The color map corresponds to a gaussian resampling
of the simulation values of the piezoresponse displacement to better match
experiments (kernel standard deviation of $\approx 20$ nm, corresponding to
the experimental imaging resolution). The largest calculated displacement,
located at the bottom-right corner ($Y_{\rm c}=0$ nm and $X_{\rm c}=100$ nm)
is 75 pm, and the displacement in the center of the cube is 4 pm. (d) In-plane
mixed piezoresponse in the $Y_{\rm c}=100$ nm plane, for a crystal oriented
with $Z_{\rm c}$ axis parallel to laboratory $x$ axis, in response to a field
applied along $z$ axis (amplitude $E_{z}=150$ kV/cm, very close to the field
amplitude applied in the PFM measurements).

Starting from the equilibrium solution described in the previous paragraph, we
now apply a finite electric field to simulate PFM experimental conditions. The
BTO nanocrystal is then relaxed under this applied electric field, leading to
new equilibrium values of the polarization and mechanical displacement fields.

Let us first consider the case when an electric field of amplitude $E^{\rm
sim}_{z}=150$ kV/cm is applied along the polar axis (O$Z_{\rm c}$) direction
of the BTO nanocrystal. The difference between the zero-field and finite-field
mechanical displacement fields gives us an estimate of the electromechanical
response of the nanocube. We show, in Figure 6c, the resulting electric-field-
induced mixed piezoresponse in the “top” facet perpendicular to the polar axis
($Z_{\rm c}=100$ nm facet). We observe qualitatively similar LPFM data of
Figure 3i, Figure S10f, S11f and S12f, with local regions of pronounced in-
plane displacement at the corners and edges of the facet. We also observe a
low to vanishing in-plane piezoelectric displacement in the inner region of
the facet, as evidenced experimentally as well.

To investigate the other experimental configuration of the nanocube having its
polar axis (O$Z_{\rm c}$) lying in the substrate plane, parallel to the
laboratory plane $(xy)$, we also simulated the application of an electric
field along the (O$Y_{\rm c}$) axis using the same equilibrium condition as
depicted in Figure 5. The resulting mixed piezoresponse simulation, presented
on Figure 6d, exhibits lower (by one order of magnitude in the center of the
facet and by a factor of 2 to 5 at the edges) and contrast than the simulation
for the field applied along (O$Z_{\rm c}$). It is well known that the presence
of 90∘ twin domain walls contribute to the piezoelectric response of BTO 57,
58, 59 and can strongly enhance it. Indeed, contrary to the (O$Y_{\rm c}$)
facet, the (O$Z_{\rm c}$) possesses multiple 90∘ domain walls, at a distance
of about 8 nm from the surface, due to a 90∘ rotation of the polarization from
a $Z_{\rm c}$ axis aligned orientation (as displayed for $X_{\rm c}=0$ nm
facet in Figure S13a) to an in-$(X_{\rm c},Y_{\rm c})$ plane lying one. This
may explain the larger simulated in-plane electromechanical response when the
electric field is applied along the $Z_{\rm c}$ axis (Figure 6c).

We note that the magnitude of the maximum in-plane displacement observed
experimentally (Figure 3i) is one order of magnitude larger than that obtained
from the simulation (Figure 6b). Apart from the fact that we did not calibrate
PFM lateral displacement, two other factors may contribute to this difference.
First, in the experiment the amplitude of the electric field effectively
applied to the nanocrystal is not known precisely because the PFM geometry
results in a pronounced inhomogeneous radial distribution with a larger field
valued compared to the one of the uniform field applied in the simulations
(considering the same applied voltage). Second, in the simulations the applied
electric field is static, while the PFM tip applies an AC voltage oscillating
at a frequency set close to the electromechanical resonance, hence enhancing
the displacement.

##  3 Conclusion

It is now well established theoretically and experimentally, that
ferroelectric materials at the nanoscale exhibit exotic polarization texture,
leading to unconventional properties. The case of 0-dimension ferroelectric
nanocrystals, in particular isolated ones, has been far less studied
experimentally, despite the opportunity it offers to bring novel properties
due to their large surface to volume ratio. In this work we considered
isolated barium titanate nanocrystals immobilized on a conductive surface and
exposed to air. Using a solvothermal approach 19 we were able to synthesize NC
of a well-defined cubic shape and 160 nm average size, that conserve the
tetragonal structure of BTO at room temperature (293 K), are individual
monocrystallites (as observed by TEM) and exhibit the expected transition to a
cubic symmetry phase above 120∘C, the known Curie temperature of the ferro- to
para-electric phases. We then investigated, at room temperature, the inverse
piezoelectric properties of such pristine individual NC by piezoresponse force
microscopy. To this end, we developed a methodology that combines the firm
attachment of the NC to the substrate with a conductive polymer and the fine
adjustment of the deflection laser in a position along the PFM cantilever that
strongly reduces electrostatic artifacts 47. This strategy allowed repeated
measurements on the same particle without risk of being dragged with the PFM
tip and provided a high confidence in the crystal deformation fields recorded.
We observed that none of the NC showed any vertical PFM displacement, while
all showed lateral displacements. We reconstructed mixed piezoresponse fields
from the 2D LPFM recording of the same NC for two orthogonal orientations
relative to the cantilever (attached to the laboratory frame). All the NC
studied showed an inhomogeneous LPFM field of larger amplitude along the
edges, a minimum at the center with deformations that tend to be oriented
towards this lower-amplitude central region.

To check whether these observations were consistent with the underlying
texture of the volume polarization, we conducted phase field simulations 15,
17. These simulations revealed that in the equilibrium state at the
temperature of 293 K, a polar axis (O$Z_{\rm c}$) with up and down 180∘
domains emerges in the core, accompanied with 90∘ domains on the two surfaces
normal to this axis in a layer of an average thickness of 8 nm. In this layer,
there is a dominant polarization orientation in the plane with mostly 180∘ and
a few 90∘ domains (on the edges). Hence, the phase field simulations predict
that none of the cube facets possess an out-of-plane polarization component,
which is consistent with the PFM observations. We investigated the nature of
the domain walls, as predicted by the simulation, and found that they are
mostly of the Ising type with Bloch and/or Néel characteristics. We then
simulated the LPFM field by calculating the difference between the equilibrium
position of atomic displacement in the zero electric field and in the presence
of an amplitude-similar electric field as applied during the PFM measurement.
The resulting simulated LPFM 2D-maps strongly resemble the experimental ones,
with maximum displacement amplitudes along the edges and in the corners, and
displacement vectors mostly following the gradient of amplitude.

The results of our study challenge the interpretation of second harmonic
generation (SHG) by $\approx 100$ nm-sized BTO nanocrystals, implicitly
considered composed of a single static polarization domain in all reported
studies 60, eventually bearing a cubic symmetry shell61. As the 180∘ domains
dominate the core and surface of the NC in a balanced up- and down-fraction,
we expect that it yields a destructively interfering SHG field, leaving open
the origin of the observed nonlinear contrast. Correlative SHG and PFM
investigations may contribute to resolve this issue. Finally, ferroelectric
matrices doped with luminescent reporting ionic species have also been
considered for sensing applications, including temperature 62 and electric
fields 63, 64. The complex polarization structure of BTO nanocrystals that we
revealed in our study should be considered to design optimal ferroelectric-
based nanosensors.

##  4 Materials and Methods

###  4.1 Sample preparation

##### BTO nanocube synthesis and characterization.

Barium titanate nanocrystals were synthesized using the solvothermal method
described in Bogicevic et al. 19. Briefly, in a solvent made of volumic
proportions of 60% ethanol and 40% deionized water, we mixed, in a nitrogen
environment, solid Na2Ti3O7 titanium and soluble Ba(OH)${}_{2}\cdot 8$H2O
barium precursors. We then heated the mixture at 250∘C in an autoclave under
autogeneous pressure for 24 hours. We selected a Ba/Ti atomic ratio of 1.6 as
it leads to BTO nanocrystals of mostly cubic shape and relatively homogeneous
size, as can be observed on Figure S1a SEM image, in the range of 100-350 nm.

XRD measurement (Figure 1b) was performed at 293 K using the Cu K$\alpha$1
($\lambda$= 1.5418 Å) and Cu K$\alpha$2 radiations ($\lambda$= 1.5444 Å) of a
diffractometer (D2 Phaser from Bruker AXS GmbH, Germany) and collecting data
in steps of 0.02∘. We then extracted the structural model with DIFFRAC.EVA
(Bruker) software, and performed Rietveld refinement of the structural model
using the WinPlotr/FullProf package 65 with peak shape described by a pseudo-
Voigt function and background level modelled using a polynomial function. The
refined free parameters were background coefficients, scale factor, lattice
constants, zero shift, peak profile, and asymmetry parameters.

For single BTO NC experiments, we prepared an aqueous colloidal suspension of
BTO NC at the stock concentration of 1 mg/mL.

##### BTO NC immobilization.

To enable contact mode PFM imaging on individual BTO NC, we immobilize them on
a 170 $\mu$m thick glass coverslip with a conductive 80 nm-thick layer of
indium-tin-oxide (ITO) deposited on it. Furthermore, a numbered grid made by
gold-deposition and with the smallest cells of 30 $\mu$m size was deposited on
the ITO-coated coverslip. This allows us to address the same individual BTO NC
after 90∘ rotation and also with other analytical methods. For immobilization
while maintaining electrical contact, we first spin-coated a 40-80 nm thick
layer of an aqueous suspension (at 1% weight concentration) of a conductive
polymer PEDOT:PSS (Ref. 768618, Sigma Aldrich) onto the marked ITO-coated
coverslip. Subsequently, the stock solution of BTO NCs was first diluted by a
factor of 50 in water, then spin-coated onto the coated coverslip and the
sample was finally annealed at 95∘C for 30 minutes. While the literature
suggests annealing PEDOT:PSS at temperatures ranging from 80∘C to 160∘C, we
deliberately maintained the annealing temperature below 100∘C to avoid any
potential phase transition of BTO, which typically occurs at a Curie
temperature of 120∘C.

###  4.2 PFM measurements

For PFM we used a commercial apparatus (Dimension ICON head system and
Nanoscope V controller, Bruker Inc.). The measurements were conducted using
SCM-PIT-V2 (Bruker) probes, which have conductive platinum-iridium coated tips
with a radius of curvature of 25 nm and a nominal spring constant of 3 N/m. We
acquired the piezoresponse phase and amplitude signal while scanning the tip
along the laboratory frame $x$ axis direction (see Figure 2b), with the
cantilever along $y$ axis.

##### Minimization of electrostatic effects.

It is widely recognized that PFM measurements can exhibit artifacts, leading
to possible misinterpretations of the data. The signal artifacts are mainly
caused by non-electromechanical effects such as electrostatic force that
arises due to the electrostatic potential difference between the cantilever
and the sample66. This force generates a bending response in the cantilever of
the AFM that is not distinguishable from the true PFM signal. The strength of
this force is highest near the tip-sample junction and decreases with
increasing distance along the cantilever. To address this issue, we employed
the concept of electrostatic blind spot position (ESBS) for the laser, as
recently introduced by Killgore et al. 47. In ESBS, the laser spot, which is
usually placed just above the tip, is positioned along the cantilever where
the bending induced by the electrostatic force is minimised, in order to get a
piezoresponse amplitude dominated by the electromecanichal contribution. The
signal thus obtained will be predominantly dominated by the true PFM signal
caused by the bending of the cantilever due to inverse piezoelectric effect.
The ESBS position was determined to be at a distance of 0.45 times the length
of the cantilever.

PFM measurements were done on single isolated BTO NCs preidentified using
scanning electron microscopy (SEM). The lateral PFM phase and amplitude images
were taken using a drive voltage of 1.5 V and a drive frequency of 727.856 kHz
(near in-plane contact resonance of the cantilever). We used a scan speed of
0.35 Hz and utilized 192 scan lines across a full scan size of 300 nm
resulting in 1.56 nm pixel size.

##### LPFM phase offset $\varphi_{\rm offset}$ estimation.

To estimate $\varphi_{\rm offset}$, we first selected one domain with the
smallest phase values, close to 0, as the reference, and computed a first
phase offset as the phase average value within this domain. In parallel, we
identified another domain with phase values close to 180∘. Subtracting 180∘
from the phase of this second domain led to a second phase offset. By
averaging these offsets, we obtained a final offset value of
$-12\pm\kern-1.8pt2$∘ for the PFM scan along the (O$X_{\rm c}$) direction. The
same process was repeated for the PFM scan along the (O$Y_{\rm c}$) direction,
producing an average offset value of $-9\pm\kern-1.8pt4$∘. As this offset is
attributed to the instrumental response and should be the same for both scans,
regardless of particle orientation, we took the average of these two values,
resulting in $\varphi_{\rm offset}=-10\pm 2^{\circ}$.

##### Vector lateral PFM mapping.

To build the $\boldsymbol{u}^{\rm m}$ lateral displacement field of probed-NC
facets, we first had to register $u_{Y_{\rm c}}$ component map with $u_{X_{\rm
c}}$ map, as they were acquired separately. We only considered translations
shifts between the two scans, as rotation mismatch could not be assessed with
a precision better than the uncertainty on the physical 90∘ sample rotation.
The translation vector shift between $X_{\rm c}$ and $Y_{\rm c}$ components
was determined using Fiji-ImageJ software 67, based on the largest overlap of
the raw PFM amplitude images. We then represented the mixed displacement
amplitude and vector map.

###  4.3 Piezoresponse simulation

Simulations of an embedded BTO nanoparticle have been performed using
FERRET15, a module for simulating multiferroic phenomena in the MOOSE finite
element framework48. We consider the equilibrium domain structure and
piezoresponse of a stress-free BTO nano-cube (100 nm size) embedded in a
dielectric infinite medium (here assumed to be air), by evolving a time-
dependent Landau-Ginzburg-Devonshire equation,
$\displaystyle\frac{\delta\boldsymbol{P}}{\delta t}=-\frac{\delta
F}{\delta\boldsymbol{P}}$, with $F$ representing the free energy of the
system.

The Helmoltz free energy volume density $f$ includes several terms, introduced
in Hlinka and Marton 51 and which analytical expressions are given in the
Supporting Information Text S2:

$f_{\rm H}\equiv f_{\rm bulk}+f_{\nabla{P}}+f_{\rm elastic}+f_{\rm
electrostr}+f_{\rm elec},$ (1)

where $f_{\rm bulk}$ is the contribution to the energy stemming from the local
polarization field $\boldsymbol{P}(\boldsymbol{r})$. In a uniaxial
ferroelectric, this typically has two degenerate minima and a double-well
shape as a function of the polarization value. This is typically the case in
BTO and indicates the desire of the system to form a uniformly polarized
ferroelectric domain. The term $f_{\nabla{P}}$ depends on the gradient of the
polarization field and tends to oppose variations of the polarization field.
The term $f_{\rm elastic}$ depends on the derivative of the local atomic
displacement field $\boldsymbol{u}(\boldsymbol{r})$ (defined relative to a
cubic parent lattice structure) and describes the linear elastic response of
the system. The electrostrictive term $f_{\rm electrostr}$ gathers the
electromechanical energy couplings between the polarization field
$\boldsymbol{P}(\boldsymbol{r})$ and the displacement field
$\boldsymbol{u}(\boldsymbol{r})$. Finally, $f_{\rm elec}$ represents the
energy density caused by internal/external electric fields. As stated earlier,
as we consider a stress-free condition, the strain energy term
$-\boldsymbol{\sigma.\varepsilon}$, where $\boldsymbol{\sigma}$ is the stress
and $\boldsymbol{\varepsilon}$ the strain tensor, resumes at zero. To realize
our simulations of ferroelectric domains and displacements on the BTO NC at
the ambient temperature of 298 K, we adopted numerical coefficients from ref.
51 for the numerical values of the coefficients involved in each term of
energy density. The displacement field is obtained by solving the stress-
divergence equation $\nabla.\boldsymbol{\sigma}=0$ (neglecting gravity) with
mechanical boundary conditions without stress at each time step.

The computational domain comprises the BTO nanocrystal, with 100 nm long sides
embedded at the center of a dielectric matrix, with 200 nm sides in order to
capture the depolarization field around the inclusion. The relative
permittivity of the dielectric matrix is set to 1 and the NC is considered
stress-free, simulating an open air environment. Dirichlet boundary conditions
on ($u,\Phi$), are applied to ensure that displacement and electrostatic
fields disappear far from the particle. Partial clamping and dielectric
screening by PEDOT:PSS could lead to a slight alteration of the ground state
and the piezoelectric response through modulation of the surface layer
thickness and potential Poisson effects which could lead to enhanced out-of-
plane mechanical response.

The equilibrium structure at zero applied electric field is obtained by
relaxing the system starting from a reasonable approximation of the
paraelectric state, realized by initializing random, small component
($<10^{-4}$C/m2) values to the polarization and local displacement fields.
Through the gradient flow approach, the system variables are evolved through a
dynamic time stepper until a local minimum is reached, defined by a
convergence criterion of within $10^{-6}\%$ of the total energy of the
previous time step.

The average thickness of the surface layers are estimated by the ratio of the
volume having $P_{Z_{\rm c}}\approx 0$ to the volume of the nanocrystal,
multiplied by the nanoparticle size. The obtained averaged thickness of 8 nm
correlates well with typical thicknesses which can be estimated from Figure
5b.

Coupling to internal/external electrical fields is accomplished through the
Poisson equation, $\displaystyle\nabla^{2}\Phi=\frac{1}{\varepsilon_{\rm
r}\varepsilon_{0}}\nabla.\boldsymbol{P}$, where $\Phi$ is the electric
potential, related to the electric field $\boldsymbol{E}$ through
$\boldsymbol{E}=-\nabla\Phi$, and $\varepsilon_{\rm r}$ is the surrounding
medium relative permittivity, which in our case is air for most of the
particle volume, hence $\varepsilon_{\rm r}=1$. Hence, to mimic PFM
experiments, we use the following procedure. First, we start with the zero-
field equilibrium structure. We then relax the polarization and local
displacement fields by solving the Landau-Ginzburg-Devonshire equation and the
stress-divergence equation at each time step, under an applied external
electric field of 150 kV/cm, until the new equilibrium is reached. The
difference in equilibrium local displacement fields between the configuration
with and without applied electric field gives us the electric-field induced
displacement in the BTO nanocube, which we report in Figure 6c,d and compare
to lateral PFM experiments. Note that to compare with the experimental
observations, a Gaussian blur with 20 nm diameter was applied to the local
displacement field.

## Author contributions

Conceptualization: BD, CFD, CP, and FT; Methodology: AM, KC, KP, CFD, CP and
FT; Software: KC and CP; Validation: AM, KC, CFD, CP and FT; Formal analysis:
KC and CP; Investigation: AM, AZ, MV, and KC; Resources: CB and FK; Data
curation: CB, MV, AM and KC; Writing – Original draft: AM, KC, CP and FT;
Writing – Review and Editing: CFD, KP, CP and FT; Visualization: AM, KC, MV,
AZ, KP, CP, and FT; Supervision: CFD and FT ; Project administration: CFD, CP
and FT; Funding acquisition: BD, CP, CFD and FT. All authors have read and
agreed to the published version of the manuscript.

We thank Pascale Gemeiner for the measurements of the temperature dependence
of the BTO Raman spectrum. The authors thank Nataliya Alyabyeva for her
initial support in PFM imaging and Dana Stanescu and Cindy Rountree for their
training on the Bruker Icon PFM of the CEA/SPEC Interdisciplinary Multiscale
Atomic Force Microscope Platform. The authors thank Pierre Audebert, Guy
Deniau, and Noelle Gogneau for their suggestions to bind the nanocrystals to
the substrate; Laureen Moreau for her guidance in SEM imaging; and Simon
Vassant for the realization of the marking grids enabling correlative SEM-PFM
imaging. C.P. and K.C. acknowledge support from GENCI-TGCC computing resources
through grant AD010913519 and computational resources from the “Mésocentre”
computing center of Université Paris-Saclay, CentraleSupélec and École Normale
Supérieure Paris-Saclay supported by CNRS and Région Île-de-France
(https://mesocentre.universite-paris-saclay.fr/). This work has received
financial support to B.D., C.F. and F.T. from the CNRS through the MITI
interdisciplinary program and from the French National Research Agency (ANR,
grant numbers ANR-21-CE09-0028 and ANR-21-CE09-0033).

Two supporting texts: PFM basics; analytical expressions of the contributions
to the free energy volume density used in phase field simulations. Supporting
data figures: Additional characterizations of the BTO sample (morphology and
strain, HR-STEM images, temperature-dependent Raman spectroscopy and Curie
temperature determination, PEDOT:PSS layer thickness measurement, size
distribution of BTO NC inferred from SEM measurements); Evidence of non-
electromechanical effects and use of ESBS setting to suppress them; LPFM
resonance curve of a cantilever; Lateral PFM mapping of three additional BTO
NC; Additional data on phase field simulations (distribution of polarization
in facets orthogonal or parallel to the polar axis, distribution of the
different components of the energy density across a NC).

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Supporting information for:
Ferroelectric texture of individual barium titanate nanocrystals
A. Muraleedharan, K. Co, M. Vallet, A. Zaki, F. Karolak, C. Bogicevic, K.
Perronet, B. Dkhil, C. Paillard, C. Fiorini, and F. Treussart

## 1\. Supporting text

###  4.4 S1. Basic principles of piezoresponse force microscopy

In piezoresponse force microscopy, a conducting tip to which an oscillating
voltage $V_{\rm tip}=V_{\rm AC}\cos(2\pi f_{\rm d}t)$ is applied is brought
into contact with the grounded sample, causing a surface deformation due to
the inverse piezoelectric effect. From the resulting oscillatory response of
the material, the device extracts the first harmonic component of the
deflection of the AFM tip $D=A\cos(2\pi f_{\rm d}t+\varphi)$, where $A$ is the
deflection amplitude to which the magnitude of the local piezoresponse
displacement $u$ is proportional, and $\varphi$ is the phase change between
the driving voltage and the voltage-induced deformation that depends on the
local polarization of the material 45. The deflection is measured using a
laser beam that is reflected off the cantilever onto a position-sensitive
quadrant photodiode. PFM cantilever undergoes two types of deformations:
vertical deformation due to the out-of-plane/vertical domains (VPFM) or/and
torsion due to the shear deformation induced by the in-plane/lateral domains
(LPFM). PFM measurement consists of mapping the piezoresponse amplitude and
phase signals in both the VPFM and LPFM modes. The electromechanical response
of a ferroelectric material of arbitrary crystallographic orientation to an
applied voltage is a vector having three independent components of
piezoresponse: one VPFM measurement and two LPFM measurements in two
independent directions. The third component of the displacement vector can be
determined by imaging the sample region of the sample after a 90∘ rotation
around the $z$ axis. When all three components of the piezoresponse are
acquired, it is possible to reconstruct a 3-dimensional local
electromechanical mixed response vector map. This approach is called a vector
PFM, described in details in Kalinin et al.42.

###  4.5 S2. Phase field simulations: analytical expressions of the
contributions to the free energy volume density

In this section, $(x,y,z)$-coordinates are attached to the material. Unless
otherwise specified we use in an indiscriminate manner the letter $(x,y,z)$ or
number (1,2,3) indices. Below are the analytical expressions of the different
terms of Helmoltz free energy density $f_{\rm H}\equiv f_{\rm
bulk}+f_{\nabla{P}}+f_{\rm elastic}+f_{\rm electrostr}+f_{\rm elec}$, as
stated in refs.51, 15

$\displaystyle f_{\rm bulk}$ $\displaystyle=$
$\displaystyle\alpha_{1}(P_{x}^{2}+P_{y}^{2}+P_{z}^{2})+\alpha_{11}^{(e)}(P_{x}^{4}+P_{y}^{4}+P_{z}^{4})$
(2) $\displaystyle+$
$\displaystyle\alpha_{12}^{(e)}(P_{x}^{2}P_{y}^{2}+P_{x}^{2}P_{z}^{2}+P_{y}^{2}P_{z}^{2})+\alpha_{111}(P_{x}^{6}+P_{y}^{6}+P_{z}^{6})$
$\displaystyle+$
$\displaystyle\alpha_{112}(P_{x}^{4}(P_{y}^{2}+P_{z}^{2})+P_{y}^{4}(P_{x}^{2}+P_{z}^{2})+P_{z}^{4}(P_{y}^{2}+P_{x}^{2}))$
$\displaystyle+$ $\displaystyle\alpha_{123}P_{x}^{2}P_{y}^{2}P_{z}^{2}$

The gradient energy density, $f_{\nabla P}$, contains the first-order Lifshitz
invariant

$\displaystyle f_{\nabla P}$ $\displaystyle=$
$\displaystyle\frac{G_{11}}{2}\left(\frac{\partial P_{x}}{\partial
x}+\frac{\partial P_{y}}{\partial y}+\frac{\partial P_{z}}{\partial z}\right)$
(3) $\displaystyle+$ $\displaystyle G_{14}\left(\frac{\partial P_{x}}{\partial
x}\frac{\partial P_{y}}{\partial y}+\frac{\partial P_{x}}{\partial
x}\frac{\partial P_{z}}{\partial z}+\frac{\partial P_{z}}{\partial
z}\frac{\partial P_{y}}{\partial y}\right)$ $\displaystyle+$
$\displaystyle\frac{G_{44}}{2}\left(\left(\frac{\partial P_{x}}{\partial
y}\right)^{2}+\left(\frac{\partial P_{y}}{\partial
x}\right)^{2}+\left(\frac{\partial P_{x}}{\partial
z}\right)^{2}+\left(\frac{\partial P_{z}}{\partial
x}\right)^{2}+\left(\frac{\partial P_{z}}{\partial
y}\right)^{2}+\left(\frac{\partial P_{y}}{\partial z}\right)^{2}\right)$

Contributions to the elastic volume energy density comes from the strain
tensor $\varepsilon_{\alpha\beta}$ defined in terms of the local displacement
field $\boldsymbol{u}(\boldsymbol{r})$:

$\varepsilon_{\alpha\beta}=\frac{1}{2}\left(\frac{\partial
u_{\alpha}}{\partial\beta}+\frac{\partial
u_{\beta}}{\partial\alpha}\right),\alpha,\beta=x,y\text{ or }z.$ (4)

The elastic energy density is then written in terms of the elastic constants
$C_{ij}$, and using Voigt’s notation
($\varepsilon_{1}=\varepsilon_{11},\varepsilon_{2}=\varepsilon_{22},\varepsilon_{3}=\varepsilon_{33},\varepsilon_{4}=2\varepsilon_{23},\varepsilon_{5}=2\varepsilon_{13}$,
and $\varepsilon_{6}=2\varepsilon_{12}$) as

$\displaystyle f_{\rm elastic}$ $\displaystyle=$
$\displaystyle\frac{1}{2}C_{11}\left(\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\varepsilon_{3}^{2}\right)$
(5) $\displaystyle+$ $\displaystyle
C_{12}\left(\varepsilon_{1}\varepsilon_{2}+\varepsilon_{1}\varepsilon_{3}+\varepsilon_{2}\varepsilon_{3}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}C_{44}\left(\varepsilon_{4}^{2}+\varepsilon_{5}^{2}+\varepsilon_{6}^{2}\right)$

The electrostrain energy density is written

$\displaystyle f_{\rm electrostr}$ $\displaystyle=$ $\displaystyle-
q_{11}\left[\varepsilon_{1}P_{x}^{2}+\varepsilon_{2}P_{y}^{2}+\varepsilon_{3}P_{z}^{2}\right]$
(6) $\displaystyle-$ $\displaystyle
q_{12}\left[\varepsilon_{1}\left(P_{y}^{2}+P_{z}^{2}\right)+\varepsilon_{2}\left(P_{x}^{2}+P_{z}^{2}\right)+\varepsilon_{3}\left(P_{y}^{2}+P_{x}^{2}\right)\right]$
$\displaystyle-$ $\displaystyle
q_{44}\left[\varepsilon_{6}P_{x}P_{y}+\varepsilon_{5}P_{x}P_{z}+\varepsilon_{4}P_{y}P_{z}\right]$

At last, the electric energy $f_{{\rm
elec}}=-\boldsymbol{E}(\boldsymbol{r}).\boldsymbol{P}(\boldsymbol{r})$, where
$\boldsymbol{E}(r)$ is the electric field obtained from solving the Poisson
equation.

All numerical coefficients involved here were taken from Ref. 51.

## 2\. Supporting data

###  4.6 2.1. BTO sample additional characterizations

####  4.6.1 BTO nanocrystals morphology and strain

Figure S1: Morphology and strain of as-produced BTO nanocrystals. (a) SEM
image of BTO NC powder deposited on a carbon layer, showing mostly
parallelepipedic/cubic shape particles. SEM model: XL30S FEG from FEI/Philips
(The Netherlands). Scale bar: 400 nm. (b) Estimate of the strain from the
diffractogram Figure 1a, using Williamson-Hall method. The experimental points
are the diffraction peaks positions, and the solid line a fit according to
equation (7).

Figure S1a shows a “large” SEM field of view of the as-produced BTO
nanocrystal powder. This powder was the starting material to prepare the BTO
NC aqueous suspension from which individual particles were isolated. We also
estimate the strain from Figure 1a diffractogram of this powder using the
Williamson-Hall equation for a Cauchy peak shape:

$\beta_{1/2}\cos\theta=\frac{K\lambda}{D}+4e\sin\theta,$ (7)

where $\beta_{1/2}$ is the full-width at half-maximum of each diffraction
peak, $\lambda$ is the X-ray radiation wavelength (here $\lambda=1.5418$ Å,
$D$ and $e$ the grain size and inhomogeneous strain, respectively, and $K$ is
a NC form factor that we took equal to 1. Figure S1b displays the experimental
points for each diffraction peak and the linear fit according to equation 7,
leading to a strain $e=0.19\pm 0.04\%$ and a crystallographic grain size
$D=81$ nm.

####  4.6.2 HR-STEM Image

Figure S2: Additional STEM images of single BTO nanocrystals. (a) HAADF STEM
image of one BTO NC. Scale bar: 20 nm. (b) Atomic resolution image of the
region delineated by the dashed line square in (a), with no evidence of a thin
cubic layer at the NC surface. Scale bar: 5 nm. (c) Bright-field and (d) HAADF
images of a different, much rounder NC, showing a $\approx 20$ nm-sized pore
defect within the particle. Scale bars: 20 nm.

Figure S2 shows HR-STEM images of an individual BTO NC oriented relative to
the electron beam to yield atomic column resolution (Figure S2b). We do not
observe that the NC has any layer of different crystallographic symmetry than
the one at its center. Note that the line crossing the NC from top to bottom
on Figure S2a may be either a structural defect or an in-plane domain wall.

####  4.6.3 Evidence of a phase transition from temperature dependence Raman
spectroscopy of BTO nanopowder

Figure S3: Temperature dependence of BTO Raman spectrum and Curie temperature
estimate. (a) Raman spectrum of BTO nanopowder at temperatures varying from 80
K to 540 K. In the tetragonal phase (space group P4mm) BTO symmetry belongs to
the point group $C^{1}_{4v}$ which irreducible representation is $\Gamma^{\rm
optic}_{C_{4v}}=3(A_{1}+E)+E+B_{1}$. Due to long-range electrostatic forces,
the $A_{1}$ and $E$ modes are further divided into transverse and longitudinal
optical modes (LO and TO, respectively 40). On top of the graph, we identified
by their group nomenclature a series of four groups of spectrally not-all-
resolved peaks. All spectra were normalized to the $\approx 250$ cm-1 peak
intensity (to account for temperature depopulation), and the spectra were
shifted vertically for clarity. (b) Area of the $\approx 305$ cm-1 normalized
peak versus temperature. The shaded zone indicates the range of temperature in
which the peak area change most, and from which we infer a phase transition
Curie temperature of $395\pm 2$ K.

####  4.6.4 Characterization of BTO nanocrystals bound to the substrate

The NC are attached to the numbered grid with a thin layer of PEDOT:PSS
conductive polymer. A layer thickness of $\approx 45$ nm was measured by AFM,
as shown on Figure S4a,b, after having removed a tape that had been placed on
an edge of the substrate before spin-coating PEDOT:PSS. Similarly, we
estimated how much the NC sinks into the PEDOT:PSS before its annealing, by
measuring the depth of the hole that remains in the event of the PFM tip drags
the NC (Figure S4c,d). We estimate this depth to be $\approx 30$ nm. The
heightened contour encircling the hole is likely to be the concave meniscus
that holds the particle, confirming the wetting of the BTO by PEDOT:PSS.

Figure S4: PEDOT:PSS layer thickness and depth of penetration of NC in the
layer. (a) PEDOT:PSS layer near an edge of the substrate after tape removal.
(b) Height profile along the dashed white line in panel (a). (c) Hole in the
PEDOT:PSS layer after a NC was dragged by the PFM tip during scanning. (d)
Height cross-section along the dashed line of panel (c) showing that the
particle was immersed partially in the polymer layer.

The single NC that we studied by PFM were first identified using SEM and their
position in the numbered gold grid, as shown on Figure S5a, allowed to
retrieve them in PFM.

Figure S5: BTO nanocrystals bound to the numbered gold-made grid. (a) Large
field of view that allows one to read the grid number (here B06). Scale bar: 5
$\mu$m. (b) Smaller field of view with an isolated NC candidate in the middle.
Scale bar: 1 $\mu$m.

For the PFM study, we considered only isolated nanocrystals of close to cubic
shape, such as the one in the middle of Figure S5b. Figure S6 shows the size
and shape distributions of the isolated NC subsequently studied by PFM.

Figure S6: Size distribution and morphology of the individual BTO nanocrystals
spincoated on ITO-coated coverslip. (a) Distribution of NC size (defined as
the mean value of the side length), with an average and standard deviation of
$160\pm 28$ nm. (b) Aspect ratio of the particles (defined as the size of the
short side on the one of the long side), revealing a majority of particles
with ratio 1, indicating a close-to-cubic shape. Inset: SEM images of single
nanoparticles with aspect ratio 1 and 0.8. Scale bar: 100 nm.

###  4.7 2.2. Evidence of non-electromechanical effects and use of
electrostatic blind spot setting to suppress them

We evidenced the existence of non-electromechanical effects in PFM
measurements and demonstrated their suppression when the laser spot used to
measure the cantilever deflection is placed at an electrostatic blind spot
(ESBS) position. This approach was first described in Killgore et al. 47.

Figure S7: PFM artifacts and their suppression by finding the ESBS position.
(a,b) When the deflection laser spot is positioned above the tip at the
extremity of the cantilever, it induces a contact resonance on ITO in the
220-280 kHz range (a) and an hysteresis loop when ramping a DC bias between +2
V and -2 V (b). (c,d) By moving the laser spot away from the cantilever
extremity to the ESBS position, the contact resonance can be suppressed (c) as
well as the hysteresis loop (d). Inset of panel (c): picture of ESBS position
of the red laser on the cantilever (length $L=225~{}\mu$m) located at 100
$\mu$m from the cantilever tip (corresponding to $0.45L$). The conventional
laser position is indicated by the white circle. Scale bar: 50 $\mu$m.

We first confirmed non-electromechanical effects leading to a PFM signal on an
ITO coated substrate alone, with no ferroelectric material deposited on it.
These effects were observed through different signatures that were also useful
to determine ESBS position by suppressing them in situations where they should
not exist, typically on non-ferroelectric materials like in our case the ITO
layer. The artifacts observed included the presence of a cantilever contact
resonance (Supporting Figure S7a), or a hysteresis loop (Figure S7b), or of a
PFM contrast in freshly “written” regions, when the deflection laser spot was
adjusted at the extremity of the cantilever, as conventionally done in atomic
force microscopy.

As we moved the deflection laser spot further from the cantilever extremity,
as shown on the inset picture of Figure S7c, we managed to find a position,
the ESBS position, for which the contact resonance disappeared (Figure S7c)
and the hysteresis cycle flattened (Figure S7d), removing the previous
artifacts.

Figure S8: Suppression of electrostatic cross-talk after adjusting AFM laser
spot at ESBS location (a,b) PFM displacement amplitude and phase maps after
writing on ITO with +10 V and -10 V with the laser spot positioned at the
cantilever tip. (c,d) Same region with the laser positioned halfway between
the ESBS position and the tip. (e,f) Amplitude and phase images of the same
region with laser at ESBS positioned approximately 100 $\mu$m away from the
cantilever tip showing absence of contrast in amplitude and phase. Scale bar:
1 $\mu$m.

As mentioned, another type of electrostatic artifact could be detected in PFM
phase and amplitude images of ITO surface after having written on a square of
1 $\mu$m side with +10 and -10 V potentials. Supporting Figure S8a,b shows PFM
images with the deflection laser positioned at the extremity of the cantilever
just above the tip. Figure S8c,d shows the same region when the laser was
positioned between the tip and the ESBS position, showing a decrease in
contrast in phase and amplitude signals. Finally, Figure S8e,f shows PFM
amplitude and phase images of the same region when we placed the laser at ESBS
position (identified by the disappearance of the contact resonance, like in
Figure S7c). The absence of phase and amplitude contrast confirms the
suppression of electrostatic parasitic effects in the PFM signal.

###  4.8 2.3. Lateral PFM resonance curve of a cantilever

Figure S9: In-plane resonance curve at 730 kHz at 1.5 V drive voltage. PFM
imaging was done near the resonance peak at 727.8 kHz, indicated by the black
dashed line.

###  4.9 2.4. Lateral PFM mapping of three additional BTO NC

Figure S10: Lateral piezoresponse displacement mapping on the single BTO NC#2.
(a,b) Lateral piezoresponse phase $\varphi_{x}$ and amplitude $u_{x}$ along
$x$\- axis. (c,d) Lateral piezoresponse phase $\varphi_{y}$ and amplitude
$u_{y}$ along the $y$\- axis (sample rotated counterclockwise). (e) Topography
image of the particle. (f) Mixed lateral piezoresponse field
$\boldsymbol{u}^{\rm m}(u_{X_{\rm c}}\cos(\varphi_{X_{\rm c}}-\varphi_{\rm
offset}),u_{Y_{\rm c}}\cos(\varphi_{Y_{\rm c}}-\varphi_{\rm offset}))$, with
$\varphi_{\rm offset}=34^{\circ}$, displayed by superimposed norm maps (color
code) and direction (array of fixed-length arrows at nodes of a 7.8 nm pitch
grid). Scale bars (all scans): 50 nm. Figure S11: Lateral piezoresponse
displacement mapping on the single BTO NC#3. (a,b) Lateral piezoresponse phase
$\varphi_{x}$ and amplitude $u_{x}$ along $x$\- axis. (c,d) Lateral
piezoresponse phase $\varphi_{y}$ and amplitude $u_{y}$ along $y$\- axis
(Sample rotated counterclockwise. (e) Topography image of the particle. (f)
Mixed lateral piezoresponse field $\boldsymbol{u}^{\rm m}(u_{X_{\rm
c}}\cos(\varphi_{X_{\rm c}}-\varphi_{\rm offset}),u_{Y_{\rm
c}}\cos(\varphi_{Y_{\rm c}}-\varphi_{\rm offset}))$ with $\varphi_{\rm
offset}=20^{\circ}$, displayed by superimposed maps of norm (color code) and
direction (array of fixed-length arrows at nodes of a 7.8 nm pitch grid).
Scale bars (all scans): 50 nm. Figure S12: Lateral piezoresponse displacement
mapping on the single BTO NC#4. (a,b) Lateral piezoresponse phase
$\varphi_{x}$ and amplitude $u_{x}$ along $x$\- axis. (c,d) Lateral
piezoresponse phase $\varphi_{y}$ and amplitude $u_{y}$ along $y$\- axis
(Sample rotated counterclockwise. (e) Topography image of the particle. (f)
Mixed lateral piezoresponse field $\boldsymbol{u}^{\rm m}(u_{X_{\rm
c}}\cos(\varphi_{X_{\rm c}}-\varphi_{\rm offset}),u_{Y_{\rm
c}}\cos(\varphi_{Y_{\rm c}}-\varphi_{\rm offset}))$, with $\varphi_{\rm
offset}=15^{\circ}$, displayed by superimposed maps of norm (color code) and
direction (array of fixed-length arrows at nodes of a 7.8 nm pitch grid).
Scale bars (all scans): 50 nm.

###  4.10 2.5. Additional data on phase field simulations

Figure S13: Distribution of polarization in facets parallel or orthogonal to
the polar axis, and illustration of the type of domain walls they host. (a)
The distribution of polarization on facet $(Y_{\rm c}OZ_{\rm c})$ at $X_{\rm
c}=0$ nm shows the presence of $\pm Z_{\rm c}$-oriented domains with $\pm
Y_{\rm c}$ domains near the surfaces. (b) Three-dimensional representation of
the direction of the polarization vector $\boldsymbol{P}/|\boldsymbol{P}|$
showing the partial Néel nature of the walls of the predominantly Ising domain
along the line $(X_{\rm c}=0\text{ nm},Y_{\rm c},Z_{\rm c}=-71\text{ nm})$.
(c) The distribution of polarization on the $Z_{\rm c}=100$ nm top facet shows
a predominantly $\pm X_{c}$ oriented polar domain (red and blue) with some
$\pm Y_{c}$ oriented domains (gray). (d) Three-dimensional representation of
the direction of the polarization vector showing that the domain walls along
the line ($X_{\rm c}=50\text{ nm},Y_{\rm c},Z_{\rm c}=100\text{ nm})$ present
the partial Néel or/and Bloch character of the main Ising domain walls. Figure
S14: Distribution of the different components of the energy density across a
nanocrystal. Planar averaged energy density contributions showing the bulk
energy density (blue), the gradient of polarization energy density (orange),
the elastic energy density (green), the electrostatic energy density (purple)
and electromechanical energy density (red). Energy densities are averaged in
planes perpendicular to the main polar axis $Z_{\rm c}$.