]These authors contributed equally to this work<EMAIL_ADDRESS>
]These authors contributed equally to this work Now at ]DACM, State of Geneva,
Switzerland

Now at ]Federal Institute of Metrology METAS, Switzerland

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
# Dynamic response and roughening of ferroelectric domain walls driven at
planar electrode edges

Guillaume Rapin  [  Department of Quantum Matter Physics, University of
Geneva, 1211, Geneva, Switzerland  Sophia Ehrensperger  [  [  Department of
Quantum Matter Physics, University of Geneva, 1211, Geneva, Switzerland
Cédric Blaser  [  Department of Quantum Matter Physics, University of Geneva,
1211, Geneva, Switzerland  Nirvana Caballero  [  Department of Quantum Matter
Physics, University of Geneva, 1211, Geneva, Switzerland  Patrycja Paruch  [
Department of Quantum Matter Physics, University of Geneva, 1211, Geneva,
Switzerland

###### Abstract

Understanding and controlling the motion, stability, and equilibrium
configuration of ferroelectric domain walls is key for their integration into
potential nanoelectronics applications, such as ferroelectric racetrack
memories. Using piezoresponse force microscopy we analyse the growth and
roughness of ferroelectric domains in epitaxial thin film Pb(Zr0.2Ti0.8)O3,
driven by the electric fields at straight edges of planar electrodes at two
different temperatures. This device relevant geometry allows us to confirm
that the domain walls are well described as 1-dimensional monoaffine elastic
interfaces driven in random-bond disorder. However, we observe a progressive
increase of roughness as initially flat domain walls move through the disorder
landscape, which could prove a significant limiting factor for racetrack-type
memories using ferroelectrics.

††preprint: AIP/123-QED

Ferroelectric materials, in which thin domain walls separate electrically
reconfigurable domains with different polarisation orientation, are widely
used in sensing, actuation, nonlinear optics, and information storage
applications Muralt (2000); Dawber  _et al._ (2005); Qi  _et al._ (2021); Hong
_et al._ (2014). Recent research has focused on emergent structural and
functional properties of these domain walls Cherifi-Hertel  _et al._ (2017);
Meier (2015); Seidel  _et al._ (2009); Guyonnet  _et al._ (2011) and their
potential integration as active nanoelectronic device components Salje (2010);
Catalan  _et al._ (2012); Sharma  _et al._ (2019a). For all such present and
future applications, a detailed understanding of the fundamental physics
underlying domain nucleation, growth, stability, and equilibrium configuration
is essential.

In this context, a useful approach allowing a predictive description of the
motion and geometric properties of domain walls is to model them as elastic
interfaces in a disordered medium Giamarchi  _et al._ (2006); Paruch and
Guyonnet (2013). This general statistical physics framework allows common
features in systems as diverse as flux lines in type II superconductors
Blatter  _et al._ (1994), propagating fractures Santucci  _et al._ , and
proliferating cell fronts Huergo  _et al._ (2010); Chepizhko  _et al._ (2016);
Rapin  _et al._ (2021) to be compared. The behaviour of such systems is
governed by the competition between elasticity, which tends to flatten the
interface, and fluctuations in the potential energy landscape resulting from
the disorder, which allow pinning. This competition leads to a self-affine
equilibrium roughness configuration, whose characteristic power-law scaling is
related to the dimensionality of the system and the type of disorder. When
driven by an external force, elastic disordered systems present a complex and
highly nonlinear dynamic response, with a thermally activated subcritical
creep regime, and depinning when sufficiently high forces are applied.

In ferroelectrics, fundamental studies of roughness, memory effects, creep and
depinning dynamics, and their characteristic avalanche statistics Paruch  _et
al._ (2005, 2012); Tybell  _et al._ (2002); Jo  _et al._ (2009); Casals  _et
al._ (2020); Tückmantel  _et al._ (2021) have led to the implementation of a
wide range of prototype nanoelectronic devices based on domain walls, focusing
in particular on racetrack-type memory applications McGilly  _et al._ (2015,
2016); McMillen  _et al._ (2010); McQuaid  _et al._ (2010). However, while in
most device geometries the domain walls are driven by an electric field
applied in a planar capacitor configuration Hong  _et al._ (1999); Gruverman
_et al._ (2008); McQuaid  _et al._ (2017); McConville  _et al._ (2020), in
many fundamental studies the dynamic response is both induced and imaged using
scanning probe microscopy (SPM). In such studies, the extremely local
application of a highly inhomogeneous electric field Blaser and Paruch (2015)
can provoke significant electrochemical effects Kalinin  _et al._ (2011),
strongly dependent on both the polarisation orientation and switching history
of the ferroelectric Domingo  _et al._ (2019). For comparison with theoretical
models of a straight linear interface in equilibrium with the native disorder
present in the sample, and for more direct assessment of what happens in
device-relevant geometries, a useful alternative would be to exploit the
nanoscale resolution of SPM for imaging, but to apply the external field using
the straight edges of patterned macroscale electrodes. Such a protocol would
establish a well defined initial domain wall position, determined by the
electrode geometry, and allow the domain walls to subsequently move away from
the electrode edge and into the pristine region beside it, reflective of the
disorder potential landscape established during sample growth and unperturbed
by electrochemical effects of domain writing using a biased tip. While pairs
of surface electrodes can be used to establish an electric field oriented in
the film plane to drive the switching of in-plane polarisation components
Sharma  _et al._ (2019b); Guy  _et al._ (2021), switching out-of-plane
polarisation components requires an electric field perpendicular to the film
plane.

Here, we report on such a study, following domain wall dynamics and roughness
of initially flat domain wall segments in Pb(Zr0.2Ti0.8)O3 thin films, driven
by the fringing electric fields at the edges of straight patterned electrodes,
as schematically illustrated in Fig. 1(a). We find that the domain walls are
well described as monoaffine 1-dimensional elastic interfaces in random bond
disorder, with measurements at both 23 and 100∘ C demonstrating the importance
of thermal activation in the creep regime, and allowing us to extract the
value of the creep exponent $\mu=0.21\pm 0.02$ with very high precision. We
observe a progressive roughening of the domain walls as they move away from
the electrode edges, with the roughness exponent increasing from $\zeta=0.5--
0.6$ to $\zeta=0.7--0.8$, suggesting a qualitative difference between the
disorder potential landscape of the as-grown sample, and after high electric
field application This picture is compatible with observations in numerical
simulations of interfaces in a Ginzburg-Landau model Caballero (2021). The
observed roughening of the domain walls, which increases their effective
width, may have significant consequences for the use of ferroelectrics in
racetrack memory applications.

All measurements were performed on a $\sim$270 nm Pb(Zr0.2Ti0.8)O3 thin film,
epitaxially grown with a 35 nm SrRuO3 back electrode on (001) oriented single
crystal SrTiO3 by off-axis RF magnetron sputtering Blaser and Paruch (2012).
As detailed in Supplementary Materials, the film shows high crystalline and
surface quality, and presents an out-of-plane up-oriented monodomain
polarisation. 50 nm Au/5 nm Ti top electrodes with extended straight-edged
sections were deposited by e-beam evaporation after photolithographic
patterning. 10 V pulses of varying duration were applied to selected
electrodes using ZN50R-10-BeCu needle contacts in a Lakeshore cryogenic probe
station to induce polarisation switching under the electrode in a standard
parallel plate capacitor geometry, followed by the outward growth of the down-
polarised domains at the electrode edges, driven by the exponentially
decreasing out-of-plane component of the capacitor fringing field. To examine
the effects of thermal activation, this polarisation switching was carried out
at 23∘ C and 100∘ C. The resulting domain configuration was imaged by
piezoresponse force microscopy (PFM) in ambient conditions and analysed with
the Hystorian materials science data analysis package Musy  _et al._ (2021) to
extract the position and follow the evolution of the 180° domain walls.

Figure 1: _Dynamics of domain walls at electrode edges_ (a) Schematic
representation of the measurement geometry, with the outward growth of a down-
polarised stripe domain (blue, teal, yellow), driven by the out-of-plane
component of the fringing electric fields between the surface Ti-Au electrodes
and the SrRuO3 bottom electrode into the pristine up-polarised as-grown region
of the sample (aubergine) (b) Composite illustration of the displacement
$u(z)$ of the 180° domain wall with respect to the electrode edge, obtained
from PFM phase images taken after the application of 10 V writing pulses for 3
(blue), 30 (teal), and 300 s (yellow) to Au/Ti electrodes patterned on a
Pb(Zr0.2Ti0.8)O3 thin film at 100∘ C. Initially relatively flat, the down-
polarised domains grow outwards from the electrode edge into the up-polarised
as-grown regions of the film (aubergine) with increasing domain wall
roughness, which can be quantified via the correlation of the relative
displacements $\Delta u(r,z)=u(z)-u(z+r)$ measured between pairs of points
separated by a distance $r$. (c) The average width of the growing domains
increases logarithmically with increased switching pulse duration. (d) Domain
wall velocity as a function of the out-of-plane component of the electric
field at the edge of the electrode, showing a highly nonlinear dependence
consistent with a creep process
$v(E)=e^{\frac{-U_{c}}{k_{B}T}(\frac{E_{c}}{E})^{\mu}}$. Comparison between
room temperature (23∘ C, blue–green colour scale) and high temperature (100∘
C, red–yellow colour scale) measurements shows the significant effects of
thermal activation characteristic of creep dynamics. We use the same colour
scale in (c) and (d) to show the correspondence between points in both plots.

For short switching pulses, we initially observe the presence of small
discontinuous down-polarised regions along the electrode edge, which coalesce
into a continuous and progressively wider stripe domains when longer writing
pulses are applied (see Supplementary Materials for PFM phase data) and the
domain walls move further away from the electrode into the up-polarised bulk
of the sample. From the images of the growing domains, we extract the
transverse displacement $u(z,t)$ along the longitudinal coordinate $z$ of the
domain walls with respect to the electrode edge for a given switching pulse
duration $t$, as shown in Fig. 1(b). The average domain width $\langle
u(z,t)\rangle$, obtained by averaging $u(z,t)$ across a minimum of 10 separate
images from at least 4 independent electrodes, increases logarithmically with
the switching pulse duration, as can be seen in Fig. 1(c). While the process
is qualitatively similar, we observe significantly earlier onset and faster
rates of domain growth at the higher temperature. From these measurements, we
extract the domain wall velocity as the increase of the average domain width
$\langle u(z,t_{2})\rangle-\langle u(z,t_{1})\rangle$ over the interval
between two sequential writing times $t_{2}-t_{1}$, and correlate it with the
out-of-plane electric field $E_{z}$ at the edge of the electrode, obtained via
Comsol modelling (as detailed in Supplementary Materials). We find a highly
nonlinear dynamic response, which can be very well described as a creep
process - the very slow, thermally activated dynamics for a subcritical
driving force, characterised by jerky, stochastic jumps of an interface
between different local minima in a highly heterogeneous potential energy
landscape:

$v(E)=e^{\frac{-U_{c}}{k_{B}T}(\frac{E_{c}}{E})^{\mu}}$ (1)

where $U_{c}$ is the characteristic energy barrier, $T$ the temperature,
$k_{B}$ the Boltzmann constant, $E$ the applied electric field driving domain
wall motion, and $E_{c}$ the critical field for depinning. Importantly, since
fitting is performed self-consistently on the full dataset for both
temperatures, a far more precise estimate of the dynamic exponent $\mu=0.21\pm
0.02$ can be obtained (as detailed in Supplementary Materials) than in
previous studies Tybell  _et al._ (2002); Paruch  _et al._ (2005).

During the growth of the stripe domain, we also observe a pronounced evolution
of their geometric configuration. As shown in Fig. 1(b), initially relatively
flat domain walls near the electrode edges roughen visibly as they move
progressively further away. This roughening can be quantified by examining the
relative displacements $\Delta u(r,z)=u(z)-u(z+r)$ between two points along
the wall separated by a distance $r$. Extracting the correlation function of
these relative displacements

$B(r,t)=\overline{\left<|\Delta u(r,t)|^{2}\right>}\sim r^{2\zeta}$ (2)

where $\left<...\right>$ and $\overline{...}$ signify the average over
different $z$ values for a single domain wall segment, and the average over
different realizations of disorder, respectively. We find overall a similar
level of roughness for the domain walls driven at 23∘ C (Fig. 2(a)) and 100∘ C
(Fig. 2(b)), comparable to previous reports of roughening at SPM-tip-patterned
ferroelectric domain walls in PZT films subjected to thermal heat–quench
cycles Paruch  _et al._ (2012). At both 23∘ C and 100∘ C, we find that the
domain walls driven by the shortest switching pulses show the expected power-
law scaling growth of $B(r)$ only at short lengthscales $r$ up to 40–50 nm,
followed by an apparent saturation reflecting their essentially flat nature at
large lengthscales and initial times Kolton  _et al._ (2005); Caballero  _et
al._ (2020). For longer writing times $B(r)$ increases and, particularly at
the higher temperature, the power-law scaling region appears to extend
somewhat further to over 200 nm. This behaviour indicates that the electrode
edge does not have a significant effect on the domain wall roughness
exponents.

To further confirm these observations, and compare the scaling exponents
extracted with both approaches, we also carried out a reciprocal space
analysis of the roughness, extracting the structure factor

$S(q,t)=\overline{\langle(\tilde{u}(q,t)\tilde{u}(-q,t))^{n}\rangle},$ (3)

where $\tilde{u}(q,t)$ is the Fourier transform of the displacement field
which defines the domain wall position

$\tilde{u}(q,t)=\frac{1}{L}\int{dz\,u(z,t)e^{-iqz}}.$ (4)

Although $B(r,t)$ and $S(q,t)$ are related, containing the same geometric
information about the domain wall roughness

$B(r,t)=\int{\frac{dz}{\pi}[1-\cos(qr)]S(q,t)},$ (5)

the latter provides more reliable estimates of the roughness exponent value
when sufficient statistics are available Bustingorry  _et al._ (2021). As
shown in Fig. 2(d,e), we observe the expected power-law scaling of the
structure factor $S(q)\propto q^{-(1+2\zeta)}$ at high reciprocal lengthscales
$q$ for the domain walls driven at 23∘ C and 100∘ C, respectively.

Figure 2: _Evolution of domain wall roughness during domain growth_ Averaged
roughness correlation function $B(r)=\left<|\Delta u(r,z)|^{2}\right>$ of
domain walls written with pulse times ranging between 56 and 3000 s for 23∘ C
(a), and between 3 and 10000 s for 100∘ C (b), showing power law scaling
$B(r)\sim r^{2\zeta}$ at short lengthscales $r$. Averaged structure factor
$S(q)=\overline{\langle(\tilde{u}(q,t)\tilde{u}(-q,t))^{n}\rangle}$ of domain
walls written with pulse times ranging between 56 and 3000 s for 23∘ C (c),
and between 3 and 10000 s for 100∘ C (d), showing power law scaling
$S(q)\propto q^{-(1+2\zeta)}$ at large reciprocal lengthscales $q$. Fitting
(black dashed lines) allows the characteristic roughness exponent $\zeta$ to
be extracted from both real space and reciprocal space analysis. Figure 3:
_Evolution of the domain wall roughness exponent during domain growth_ $\zeta$
values extracted from fitting power-law scaling of $B(r)$ (circles) and $S(q)$
(crosses), and followed as a function of (a) the switching pulse duration and
(b) the out-of-plane electric field. While at room temperature (23∘ C, blue-
green colour scale), $\zeta$ values cluster uniformly around 0.5–0.6, at high
temperature (100∘ C, red-yellow colour scale) $\zeta$ values increase to
0.7–0.8 for long switching pulse duration, as the domain walls move further
away from the electrode edges into a region of lower electric field intensity.
The colour scales are used to help the reader to assess the correspondence
between points in (a) and (b).

The roughness exponent $\zeta$, whose value depends on the dimensionality of
the system, the nature of the disorder potential, and the range of the elastic
interactions, can then be obtained by fitting this characteristic power-law
scaling of $B(r)$ and $S(q)$. We find that the roughness exponents obtained
from both the real space and reciprocal space analysis are in excellent
agreement, although the latter consistently gives slightly lower values. As
can be seen in Fig. 3(a), at room temperature $\zeta$ values cluster uniformly
around 0.5–0.6, but increase to 0.7–0.8 for long switching pulse duration at
the higher temperature.

Multiscaling analysis of the probability distribution function of relative
displacements following Santucci  _et al._ ; Guyonnet  _et al._ (2012)
(detailed in Supplementary Materials) confirms that the observed scaling
appears to be monoaffine, as expected for elastic interfaces subject to weak
collective pinning by random disorder Halpin-Healy (1991); Mézard and Parisi
(1991); Rosso  _et al._ (2005), rather than multiaffine Nattermann  _et al._
(1990); Barabasi  _et al._ (1992); Kolton  _et al._ (2005).

These observations of ferroelectric domain wall roughness and dynamics in our
specific measurement geometry reveal a number of important features. First, by
comparing domain walls driven at 23∘ C and 100∘ C we explicitly confirm the
very significant role of thermal activation during creep motion, with speeds
up to 2 orders of magnitude higher observed for comparable electric fields at
the higher temperatures, an effect that is particularly important at low
fields. This thermal activation allows a much larger stripe domain to grow for
a given switching pulse duration, in spite of the exponentially decreasing
magnitude of the out-of-plane component of the electric field as one moves
further away from the electrode edge. Second, the very different aspect
presented by the domain walls in the immediate vicinity of and furthest away
from the electrode edges, with over twofold increase in roughness $B(r)$ and
an evolution of the roughness exponent $\zeta$ from $\sim 0.5$ to $\sim 0.8$
points towards qualitative differences in the disorder potential landscape in
the two cases.

Indeed, past studies of the effects of high intensity electric fields applied
via an SPM tip demonstrated that polarisation reversal can be accompanied by
the injection and reorganisation of charged defects, long-lasting
electrochemical changes on the ferroelectric surface, and even significant
material degradation for the highest field intensity Kalinin  _et al._ (2011);
Ievlev  _et al._ (2016); Domingo  _et al._ (2019). The resulting patterned
domains commonly show marked pinning of domain walls at their initial
position, even after subsequent field application or heating Tückmantel  _et
al._ (2021); Paruch  _et al._ (2012). We believe the strong electric field
right under the electrode and at its edge, comparable to that applied under a
biased SPM tip Blaser and Paruch (2012, 2015), has similar effects in our
sample.

We note, moreover, that the relatively flat geometry of the domain walls in
the immediate vicinity of the electrode edges, in spite of the higher electric
field intensity, suggests the system is not simply in a depinning regime,
where in fact higher growth of domain wall roughness may be expected López
_et al._ (1997); Purrello  _et al._ (2019). However, as the domain walls
progressively move away from the electrodes into the region of decreasing
electric field intensity, their growing roughness, increasing roughness
exponent values, and the extension of the power-law scaling region to higher
$r$ suggest that here the system is more effectively relaxing toward
equilibrium with the pristine disorder potential landscape of the as-grown
sample.

Values of the roughness exponent $\zeta$ in the 0.6–0.7 range had previously
been observed in for both artificial and naturally occurring domain walls in
Pb(Zr0.2Ti0.8)O3 Pertsev  _et al._ (2011); Guyonnet  _et al._ (2012), and
together with dynamic exponent $\mu$ of 0.21 would be compatible with the
theoretical predictions for 1-dimensional elastic interfaces in random bond
disorder Nattermann (1987); Agoritsas  _et al._ (2012). The random bond
disorder at the origin of the observed domain wall roughening can be
interpreted as site-to-site fluctuations in the energy barrier that the
polarisation has to overcome during 180° switching Caballero  _et al._ (2020).
These fluctuations can be related to uncharged defects in the crystalline
order or locally varying surface morphology Paruch and Guyonnet (2013); Lu
_et al._ (2019).

Our observations of progressively increasing roughness as initially flat
domain walls move through the disorder landscape are particularly pertinent
for potential ferroelectric-based racetrack memory applications McGilly  _et
al._ (2015, 2016); McMillen  _et al._ (2010); McQuaid  _et al._ (2010), where
the resulting broadening of the information-carrying region could prove a
significant limiting factor. A possible solution would be to focus on
ferroelastic ferroelectric domain walls Scott  _et al._ (2017); Salje (2020);
Barzilay and Ivry (2020), where additional strain terms could help maintain
straighter walls.

## Author contributions statement

P.P. designed and supervised the work. S.E. performed the PFM switching
measurements, and carried out preliminary analysis with assistance from C.B.
G.R carried out the full multiscaling data analysis. G.R., N.C., and P.P.
wrote the manuscript. All authors contributed to the scientific discussion and
manuscript revisions.

The authors declare no competing interests.

###### Acknowledgements.

This work was partially supported by Division II of the Swiss National Science
Foundation under project 200021_178782. N.C. acknowledges support from the
Federal Commission for Scholarships for Foreign Students for the Swiss
Government Excellence Scholarship (ESKAS No. 2018.0636).

## Data Availability Statement

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

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Supplementary material for
“Dynamic response and roughening of ferroelectric domain walls driven at
planar electrode edges”

###  .1 PFM measurements of domain growth at electrode edges

To follow the growth of the down-polarised domains into the up-polarised as-
grown regions of the sample, vertical piezoresponse force microscopy (PFM)
imaging was carried out in a Bruker Dimension V AFM system at ambient
conditions, using Bruker MESP tips, with a 20 kHz drive frequency, 3000 mV
drive amplitdue, and 3 $\upmu$m/s tip scanning velocity). After the
application of a 10 V switching pulse for the chosen duration, concurrent PFM
phase and amplitude 5 $\times$ 5 $\upmu$m2 images were taken along the
electrode edge with a 1024 $\times$ 1024 pixel resolution, as can be seen in
Figs. 4– 7 for domains switched at 23∘ C and 100∘ C, respectively. Given the
intrinsically almost perfectly binary nature of phase imaging (180° phase
difference with relatively low noise) with a steep change between the two
phase states corresponding to up- vs. down-polarised domains, convoluted
essentially only by the tip size, vertical PFM phase provides the most
reliable and least noisy way to precisely determine the domain wall position.
The amplitude channel can be used, tracking its minimum as the position of the
domain wall in each scan line, and gives comparable domain wall geometry, but
with far more noise. This is because the amplitude channel is more prone to
variability via electrostatic and electrochemical effects, and as a result of
contact resonance variations giving rise to crosstalk with the sample
topography.

Figure 4: _PFM phase images of domain growth at electrode edges at 23 ∘C._
Domains written with 10 V pulses applied to the top electrode for the
indicated switching pulse duration. The white bar represents 1 $\upmu$m, and
all images are shown at the same PFM phase scale, with the white and black
regions on the scale representing the approximate value of the down and up
polarised domains, respectively. Figure 5: _PFM amplitude images of domain
growth at electrode edges at 23 ∘ C._ Acquired concurrently with the
measurements shown in Fig. 4, the amplitude images recapitulate the same
information, with the 180° domain walls visible as a narrow dark line to the
right of the electrode, itself corresponding to a region of minimum amplitude
at the left of each image, since it blocks our PFM signal. The white bar
represents 1 $\upmu$m. Figure 6: _PFM phase images of domain growth at
electrode edges at 100 ∘ C._ Domains written with 10 V pulses applied to the
top electrode for the indicated switching pulse duration. The white bar
represents 1 $\upmu$m, and all images are shown at the same PFM phase scale.
with the white and black regions on the scale representing the approximate
value of the down and up polarised domains, respectively. Figure 7: _PFM
amplitude images of domain growth at electrode edges at 100 ∘ C._ Acquired
concurrently with the measurements shown in Fig. 6, the amplitude images
recapitulate the same information, with the 180° domain walls visible as a
narrow dark line to the right of the electrode, itself corresponding to a
region of minimum amplitude at the left of each image, since it blocks our PFM
signal. The white bar represents 1 $\upmu$m.

###  .2 Image processing

Each of the 1024 $\times$ 1024 px vertical PFM phase images, with phase values
normalised on a 0–360∘ range, were offset to obtain comparable minimum values
(centered around $180^{\circ}$, crimson colour in Figs. 4,6) for the as-grown
up-polarised domains, and maximum values (centered around $360^{\circ}$,
yellow colour in Figs. 4,6) for the down-polarised domains growing outwards
from the electrode edges. The images were then binarised using in-house
developed algorithms within the Hystorian materials science data analysis
Python package Musy  _et al._ (2021), allowing the positions of the electrode
edges and domain walls to be identified, and their $x,y$ coordinates
extracted. Finally, the relative front displacement $u(z)$ was obtained as the
difference in the $x$ coordinates of the electrode edge and domain wall for
each $y$ coordinate value.

###  .3 Modelling electric field intensity at electrode edges

When voltage pulses are applied across a ferroelectric film sandwiched between
macroscopic electrodes in a planar capacitor geometry, as schematically
illustrated in Fig. 8(a–c), previous studies have shown that polarisation
switching proceeds by the nucleation, growth, and coalescence of multiple
domains Hong  _et al._ (1999); Gruverman  _et al._ (2008) in the intense,
homogeneous and purely out-of-plane electric field within the capacitor. Once
the ferroelectric volume within the capacitor is fully switched into the
polarisation state orientated parallel to the applied electric field, much
slower outward domain growth is possible, driven by the fringing fields
extending beyond the edges of the electrodes.

Figure 8: _Schematic representation of polarisation reversal in a planar
capacitor geometry_ (a) The initially up-polarised PZT thin film, with a
SrRuO3 epitaxial bottom electrode and Au/Ti patterned top electrodes, between
which 10 V pulses are applied, giving rise to uniform, out-of-plane, high
intensity electric field under the top electrode, and fringing fields
extending from its edges (field lines represented in red). As a result,
initial domain nucleation and growth occurs directly under the top electrode
(b), followed by propagation of domain walls away from the electrode edges
(c).

Using Comsol finite element 2D (slab) simulations of a 270 nm thick, 10
$\upmu$m long PZT film with its lower boundary fixed at ground, and a 55 nm
thick, 5 $\upmu$m wide perfectly conducting top electrode to which 10 V
potential could be applied, we numerically simulated the fringing electric
field to extract its out-of-plane component, shown in Fig. 9(a). As can be
seen in Fig. 9(b), the intensity of the fringing fields decreases very rapidly
as a function of distance from the electrode edge. We therefore expect
progressively slower domain wall motion and less significant effects of
surface and bulk charge dynamics as the domain walls are driven further from
the electrodes.

Figure 9: _Electric field driving domain wall dynamics at electrode edges_ (a)
Finite element simulation of the out-of-plane component of the electric field
$E_{z}$ in the planar capacitor geometry of the measurements, and (b) the
corresponding values of $E_{z}$ extracted at the ferroelectric surface, as a
function of the distance from the electrode edge. The inset shows the $E_{z}$
values at length scales corresponding to the width of the domains imaged in
our experiments.

###  .4 Extracting the creep exponent $\mu$

The domain wall velocity during creep depends on both the temperature $T$ and
the vertical component of the electric field $E_{z}$ which drives the motion.
We therefore carried out a self-consistent two-dimensional surface fitting of
the creep equation (Eq. 1 of the main text) to the cloud of points combining
data obtained from switching at both 23∘ C and 100∘ C, as shown in Fig. 9(a).
Seed values for the two-dimensional fitting parameters were obtained from fits
carried out separately on the datasets obtained at the two different
temperatures, shown for comparison in Fig. 9(b). Based on these preliminary
values, the two-dimensional fits were processed using the Python _Scipy_
library and the _curve_fit_ function.

Figure 10: _Fitting domain wall motion as a creep process
$v(E)=e^{\frac{-U_{c}}{k_{B}T}(\frac{E_{c}}{E})^{\mu}}$ _ Domain wall velocity
as a function of the out-of-plane electric field and temperature, self-
consistently fitting the data obtained at both 23∘ C and 100∘ C as a two-
dimensional surface. The inset shows domain wall velocity as a function of the
out-of-plane component of the electric field at the edge of the electrode,
where solid lines are extracted from the two-dimensional fit, and the dashed
lines from fits done separately on the data obtained at the two different
temperatures. 23∘ C data are shown on the blue–green colour scale, and 100∘ C
data on the red–yellow colour scale.

###  .5 Multiscaling analysis of the probability distribution function of
relative displacements

The statistical distribution of the fluctuation of the domain wall position
can be quantified by defining the probability distribution function (PDF) of
the relative displacements $\Delta u(r,z)$ at a given length scale $r$

$P[\Delta u(r,z)]=\frac{1}{N}\int{dz\cdot\Delta u(r,z)}$ (6)

where the factor $N$ ensures normalisation. The central moments of this PDF,
reflecting its characteristic scaling properties Agoritsas  _et al._ (2012),
are the real-space displacement autocorrelation functions

$\sigma_{n}(r)=\overline{\left<|\Delta u(r)|^{n}\right>}\sim r^{n\zeta_{n}},$
(7)

where $\zeta_{n}$ are the associated scaling exponents for the $n$th moment.

For monoaffine systems, such as 1-dimensional equilibrated interfaces at zero
temperature in weak collective pinning, the PDF is well approximated by a
Gaussian function Halpin-Healy (1991); Mézard and Parisi (1991); Rosso  _et
al._ (2005), and the second moment or roughness $B(r)\equiv\sigma_{2}(r)\sim
r^{2\zeta}$ is sufficient to fully characterise the scaling, with a single-
valued exponent $\zeta_{n}=\zeta$ $\forall n$. For multiaffine systems, such
as out-of-equilibrium, correlated disorder, or strong pinning scenarios
Nattermann  _et al._ (1990); Barabasi  _et al._ (1992); Kolton  _et al._
(2005) the full set of higher order scaling exponents
$\zeta_{n}\neq(n/2)\zeta_{2}$ are necessary to characterise the interface
roughening.

To investigate the nature of the domain walls and the symmetry of their
relative displacements, we therefore carried out a multiscaling analysis
Santucci  _et al._ ; Guyonnet  _et al._ (2012) to evaluate their PDF and its
central moments. As shown in Figs. 11 (a–d) and 12(a–d), the PDFs obtained for
different $r$ values and switching pulse durations for domain walls written at
23∘ C and 100∘ C, respectively, are generally quite symmetric. The
corresponding renormalised central moments, shown in Figs. 11 (e–h) and
12(e–h)

$C_{n}(r,z)=[\sigma_{n}(r,z)]^{1/n}$ (8)

show some fanning at the highest and lowest $r$ values, but collapse to within
15 % in the intermediate range.

Figure 11: _Multiscaling analysis for domain walls written at 23 ∘ C_ (a–d)
Probability distribution function of relative displacements $\Delta u(r,z)$
for $r\in{6,40,60,400}$ nm and (e–h) power law scaling of its averaged
renormalised central moments $C_{2}$–$C_{8}$, for domain walls written with
switching pulse duration of 56, 178, 562, and 1778 s. Figure 12: _Multiscaling
analysis for domain walls written at 100 ∘ C_ (a–d) Probability distribution
function of relative displacements $\Delta u(r,z)$ for $r\in{6,40,60,400}$ nm
and (e–h) power law scaling of its averaged renormalised central moments
$C_{2}$–$C_{8}$, for domain walls written with switching pulse duration of 3,
30, 300, and 3000 s.