# Thermal quench effects on ferroelectric domain walls

###### Abstract

Using piezoresponse force microscopy on epitaxial ferroelectric thin films, we have measured the evolution of domain wall roughening as a result of heat-quench cycles up to 735, with the effective roughness exponent changing from 0.25 to 0.5. We discuss two possible mechanisms for the observed increase: a quench from a thermal 1-dimensional configuration, and from a locally-equilibrated pinned configuration with a crossover from a 2- to 1-dimensional regime. We find that the post-quench spatial structure of the metastable states, qualitatively consistent with the existence of a growing dynamical length scale whose ultra slow evolution is primarily controlled by the defect configuration and heating process parameters, makes the second scenario more plausible. This interpretation suggests that pinning is relevant in a wide range of temperatures, and in particular, that purely thermal domain wall configurations might not be observable in this glassy system. We also demonstrate the crucial effects of oxygen vacancies in stabilizing domain structures.

###### pacs:

77.80.Dj, 68.35.Ct,77.80.Fm## I Introduction

Domain walls separating differently polarized regions in ferroelectric thin films present a powerful model system in which the characteristic roughening and complex dynamics resulting from competing elastic and pinning forces can be readily accessed paruch_dw_review_07 . Understanding this behavior is key to the physics of many diverse disordered elastic systems (DES) agoritsas_physb_12_DES ; yamanouchi_prl_06_creep_ferromagnetic_semiconductor ; ponson_prl_06_fracture ; lemerle_prl_98_FMDW_creep ; blatter_94_4mp_vortex_review ; giamarchi_varenna_wigner_review ; gruner_rmp_88_CDW_review ; kardar_review_lines and of significant technological interest for memory and electromechanical ferroelectric applications scott_memories ; waser_memories ; kumar_apl_04_SAW , and devices using the domain walls as nanoscale functional components bea_natmat_09_domainwalls_NV ; salje_cpc_10_multiferroic_boundaries , in which control over the stability and growth of domain structures is of paramount importance.

Theoretically, although equilibrium DES properties are relatively well understood, much less is know about the out-of-equilibrium behavior of these systems. A major challenge is understanding the non-steady slow dynamics associated with aging cugliandolo_98_aging_review ; cugliandolo_prl_06_keldysh_elastic ; schehr_frg . A specially interesting realization of such out-of-equilibrium phenomena is provided by quenches, in which a parameter is abruptly varied. The study of such quenches is a challenge, not only for classical systems but also for quantum systems cugliandolo_prb_02_dynamics_quantum_aging ; polkovnikov_rmp_11_quantum_nonequilibrium . For classical systems, studies of interfaces subjected to quenches kolton_prl_05_flat_interface ; bustingorry_prb_07_vortex_glass ; iguain_prb_09_aging_line ; kolton_prb_06_slowrelaxatdepinning ; kolton_prl_09_agingatdepinning have shown aging of the interface and a long term memory of the initial configuration.

On the experimental side, studies of ferroelectric domain walls in the DES framework have so far focused mostly on equilibrium properties, with switching current and dielectric measurements showing domain wall pinning in bulk ferroelectrics damjanovic_prb_97_piezo , and random-field disorder inferred for relaxors kleeman_ferro_98_dw_rf , while piezoforce microscopy (PFM) studies of domain wall roughening and creep reported strong dipolar interactions and random bond disorder in epitaxial ferroelectric thin films tybell_prl_02_creep ; paruch_prl_05_dw_roughness_FE ; paruch_jap_06_dynamics_FE . Unlike their theoretical counterparts, these systems also show pinning by individual strong defects rodriguez_apl_08_DW_defect , and local variations of both disorder strength and universality class jesse_natmat_08_SSPFM , further complicating the approach to non-equilibrium behavior. Thermal effects in bulk likodimos_prb_02_thermal_kinetics_TGS and thin film materials paruch_apl_06_stability include increased roughness and higher domain wall mobility upon heating, and transition to domain wall depinning at lower temperatures jo_PRL_09_dw . However, a detailed nanoscale analysis of the complex behavior associated with a thermal quench of these systems remains an open question.

In this paper we report on PFM measurements of domain walls in ferroelectric Pb(ZrTi)O (PZT) thin films, heat-quench cycled up to 735C, in which the value of the roughness exponent , reflecting the system dimensionality and the nature of the disorder, is extracted from an analysis of the domain wall position, and evolves from 0.25 to 0.5. Despite the similarity of the measured exponent with the thermal in one dimension, we show that the observed behavior cannot result from the quench of an initial high temperature thermal configuration, as the observed roughness values are much higher than would be expected in this scenario. Rather, it is clear that disorder pinning continues to dominate the behavior during the quench, with one possibility being that the domain walls cross from a 2- to 1-dimensional regime. We also follow the complex thermal evolution of circular nanoscale ferroelectric domains, in which the combined effects of disorder pinning, line tension and a preferential polarization orientation can be observed, and the crucial effects of oxygen vacancies in stabilizing domain structures are highlighted.

## Ii Materials and Methods

Thermal quench studies were carried out on c-axis oriented PZT films,
epitaxially grown on single crystal (001) Nb:SrTiO substrates at
500C by off-axis radio-frequency magnetron sputtering, in an
Ar-O process gas mixture ^{\theendnote}^{\theendnote}endnote: \theendnoteAr:O = 58:42, 180 mTorr and
Ar:O = 80:20, 225 mTorr were used by P. P. and X. H., respectively,
with high crystal and surface quality confirmed by x-ray diffraction and atomic
force microscopy (AFM). In these films, the polarization vector along the c-axis
can be locally switched by a biased AFM tip, using the substrate as a ground
electrode, and the resulting ferroelectric domains imaged by PFM
kalinin_mm_06_PFM_review . To study the effects of heat-quench cycling on
domain wall roughness we wrote 10 10 m arrays of linear 1
m wide domains with V alternately applied to a scanning AFM tip. To
explore the interaction between heating, line tension and disorder pinning, we
wrote arrays of nanoscale circular domains with V pulses of different
duration in a uniform, oppositely polarized region. The films were heated in air
to progressively higher temperatures for fixed time intervals, then quenched to
room temperature on a large copper block. Ambient temperature PFM images of the
evolving domain structures were acquired after each heating-quench cycle. In
addition, to investigate the role of oxygen vacancies on domain stability, films
grown under the same conditions but cooled in either process gas, Ar, or O
were compared.

As in our previous work paruch_prl_05_dw_roughness_FE , to extract the correlation function of relative displacements for a given PFM phase image of linear domains we first isolated a single domain wall, then binarized the image using a cutoff extracted from the midway point between the phase contrast values for the two polarization orientations. The elastically optimal flat domain wall position was determined from a least-squares linear fit to the binarized image, which finally allowed the extraction of the relative displacements, and their correlation at different length scales. To compensate for possible effects of scanner drift, for each set of walls PFM measurements taken in an “up” vs “down” scan direction were averaged. Although ambient-condition walls after writing were generally well defined, repeated heat-quench cycles, especially close to , led to the appearance of multivalued features such as bubbles or overhangs. In some cases, such features could also be present even immediately after writing at room temperature. Bubbles, being very small, well defined domains of opposite contrast near but not coincident with the position of the primary domain wall, could be removed from the binarized image by hand. For overhangs, the algorithm used to calculate relative displacements would simply result in a renormalized single value “effective position” of the domain wall. Therefore, when such features became prevalent, or when the films switched to the preferred polarization direction upon heating near , quantitative analysis could no longer be carried out, as indicated by the symbol “x” in Table. 1.

## Iii Domain wall roughness

Generally, an elastic manifold in the absence of disorder and thermal fluctuations (0K temperature) would take an ideally flat configuration in order to optimize its energy. Random variations in the potential landscape, whether due to disorder or thermal fluctuations, allow further optimization as the manifold wanders between particularly favorable regions in the potential, giving rise to a characteristic roughening. In the case of single manifold, an interface such as a domain wall, contact line, or fracture, fully in equilibrium with its random potential landscape, the resulting geometrical fluctuations from an elastically optimal flat configuration are present at all length scales, with the system showing mono-affine scaling properties (for a more detailed presentation, see agoritsas_physb_12_DES, and references therein). The key quantity describing this roughening is the correlation function:

(1) |

known as the roughness function. Here are the transverse displacements along the longitudinal coordinate with respect to a flat configuration, measured a distance apart on the interface, and and the thermal and disorder averages, respectively. The value of the roughness exponent depends on the dimensionality of the system, the nature of the disorder potential, and the range of the elastic interactions. Extracting not only yields this important information, which may be used to determine the universality class of the disorder and the dominant pinning defects present in the system, but also allows a full scaling prediction of the complex non-linear creep response of such systems to small driving forces tybell_prl_02_creep . We have previously shown for linear domains created artificially by straight-line scanning with a biased AFM tip that, as a result of this writing process, the domain walls are essentially flat at large length scales, with roughening observed only at short length scales, probably where the stray fields during the writing process allow some small accommodation to the potential landscape paruch_prl_05_dw_roughness_FE . The experimentally obtained thus shows a power-law growth at short length scales, from which the roughness exponent can be extracted, followed by a saturation at large length length scales.

As previously reported in paruch_apl_06_stability , linear domains in process-gas-cooled films show very high stability upon heating, with increased roughness only at small length scales (Fig. 1(a–f)). At 625–785C (depending on the sample), close to the Curie temperature of the films determined by x-ray diffraction gariglio_apl_07_PZT_highTc , extensive polarization switching occurs and the written domains eventually disappear as the region reverts to its as-grown monodomain state. Quasi-identical behaviour was observed in the Ar-cooled samples (Fig. 1(m–o)). In the O-cooled films however, the domain structures showed much lower thermal stability. In these films, expected to show decreased oxygen vacancy densities compared to process-gas or Ar-cooled samples, increased roughening was found even for short (20 min.) heating intervals, with domain disappearance observed already at 250–350C (Fig. 1(j–l)). Nonetheless, x-ray measurements confirmed sample in the 600-700C range, similar to those of the process-gas-cooled samples gariglio_apl_07_PZT_highTc .

To quantitatively examine the roughness evolution, for each sample we extracted
the average roughness function . For the relatively flat, as-written domain walls before thermal cycling, as in our previous studies paruch_prl_05_dw_roughness_FE , we find a power-law growth of at short length
scales followed by saturation at 25–100 nm (comparable to the 40–100 nm film thickness), with 100–1000 nm in the different films. After heating, the power-law-growth region
of extends to 300–700 nm and the saturation value increases (1000–10000
nm), consistent with the observed increase in domain wall roughness
(Fig. 1(h,i)). From the power-law growth region of (Fig. 1(l)), we extract
a value for the roughness exponent , characterizing the static
equilibrium configuration of the domain wall in the random manifold regime,
where . As shown in Table. 1 for domain
walls in different films, heating increases the value of from 0.25
at ambient conditions and lower temperature thermal cycling to 0.5 at
higher temperatures and near domain disappearance ^{\theendnote}^{\theendnote}endnote: \theendnote(A) The 50 nm film
was heated for 3 hour intervals in 100C steps to 485C, then in
50C steps till domain disappearance at 785C. The 52 and 91
nm films were heated for 1 hour intervals in 100C steps to 485C,
then in 25C steps till domain disappearance at 625C. (B)
The 50 nm film was heated for 30 minute intervals in 100C steps to
300C, 50C steps to 450C, then 25C steps till
domain disappearance at 525C. The 40 nm films were heated for 20
minute intervals in 100C steps till 400C, 50C steps till
550C, then 25C steps till domain disappearance at
625C. The 100 nm film was heated for 20 minute intervals in
100C steps till 200C, then 25C steps till domain
disappearance at 325C.. In O-cooled films, higher roughness and increased
values are seen already at relatively low temperatures.

## Iv Discussion

To understand these data, we first consider the thermal roughening of the linear domain walls within the theoretical framework of DES. Our previous studies have shown that at ambient conditions with no applied electric field, thermal excitations alone are not enough to overcome the energy barriers between different metastable states in the duration of the experiment, although the electric field applied during writing allows domain walls to accommodate at short length scales to the surrounding random potential landscape. A power-law growth of is thus observed only at length scales smaller than the film thickness, where the domain wall behavior agrees well with theoretical predictions for 2-dimensional interfaces pinned by weak, collective random bond disorder and in the presence long-range dipolar forces paruch_prl_05_dw_roughness_FE ; nattermann_jopc_83_dipole_disorder . Given the finite thickness of the sample, much smaller than its lateral dimensions, it is reasonable to expect that in a weak collective pinning scenario, were the walls to fully equilibrate, 1-dimensional behavior would be evident at large length scales. Although it is difficult to estimate the exact length scale for such a 2-dimensional to 1-dimensional crossover in a disordered system, we can expect it to be of the order of the sample thickness. This would be exactly the case for purely thermal roughening. The increased energy provided to the samples on thermal cycling to progressively higher temperatures is one way to promote further equilibration of the domain walls, allowing them to accommodate to the surrounding potential landscape at length scales greater than the film thickness, where the 1-dimensional nature of the domain walls should become evident.

If thermal roughening dominates in the high temperature 1-dimensional regime, the walls should be characterized by a roughness exponent , and slowly age to the equilibrium configuration when quenched. For an increasing time after the quench, a growing length should thus separate the equilibrated short length scales (with exponent ) from the large scales conserving a memory of the roughness at the quench (with exponent ), analogous to a quench from an initially flat configuration kolton_prl_05_flat_interface . We confirmed such a scenario in Langevin dynamics simulations of a one-dimensional interface in a random medium as detailed in Ref.kolton_prl_05_flat_interface, , but this time quenching an initially equilibrated high-temperature configuration. The results indeed show the quasi-freezing of this initial configuration at the largest length scales, with thermally activated dynamic evolution distinguishable only at very small length scales (Fig. 2(a)). This behaviour is also reflected in the evolution of after the quench (Fig. 2(b) ): at large lengths we observe , while at small lengths the line tends to approach the equilibrium roughness exponent (with for an elastic line with short range elasticity (random bond disorder)).

If the barriers are large, as the high stability of ferroelectric domain walls indicates, the length can grow extremely slowly, and most of the observable interface could well be characterized by the exponent . This scenario seems consistent with the exponent observed at length scales up to 10 times greater than the thickness of the film. However, if we consider an elastic 1-dimensional interface with energy , we obtain its thermal roughening at temperature as

(2) |

where the elastic constant in the case of the domain wall is the product of its energy per unit area and the film thickness and is the Boltzmann constant. Taking mJm from ab-initio calculations for PbTiO meyer_prl_02_PTO_DW , we can therefore calculate the gradient , and the roughness expected for the thermal roughening scenario at the length scale corresponding to the experimentally observed saturation length . These values can be directly compared to the experimentally obtained values for the roughness and the gradient in the power-law-growth region where . As shown in Table. 2 for the 50 nm PZT film of Fig. 1, the experimental values are many orders of magnitude higher than those expected for a purely thermal roughening of a 1-dimensional interface. The observed behavior at large length scales therefore cannot be simply a quench of the high temperature configuration. Rather, pinning by disorder in the film continues to be the dominant effect, even during the quench.

One possible scenario is that of a dimensional crossover dominated by disorder: in equilibrium with a random bond disorder potential, below a typical length scale , comparable to the film thickness, the domain walls should act as 2-dimensional elastic sheets with –0.3, and as 1-dimensional elastic strings with for higher length scales (see Fig.3(d)). If this crossover is smeared out, it is not improbable that an effective is observed experimentally. According to the theory of DES kolton_prl_05_flat_interface , the locally equilibrated regime just after the quench should extend up to a length-scale that grows with the heating-time and the heating temperature , as the dynamical length , with a positive exponent, a characteristic temperature, and a characteristic microscopic time. We then expect to observe equilibrium power law behavior (possibly including the equilibrium dimensional crossover) below and a memory of the flat initial condition, imposed by the linearly scanning AFM tip during writing, above it. The theory thus predicts a temperature dependence of the effective roughness exponent, from at low temperatures, (see Fig.3(b)), to at higher temperatures (see Fig.3(c)) where the power law behavior is effectively dominated by the equilibrium 1-dimensional regime. The results displayed Fig.1(e,l) qualitatively agree with this theoretically expected scenario, and in particular, the results in table 1 are consistent with the expected heating-temperature dependence of the effective roughness exponent. The theory also predicts, at a fixed temperature, a slow (logarithmic) time dependence of that should translate into a heating-time dependence of , from at short heating times , to at large times, , as summarized in Fig.3 . In this respect, the results shown in Table 1 for the 50 nm PG-cooled samples in (A) and (B) seem to be also qualitatively consistent with the theory.

Moreover, the values in table 2 show no clear linear temperature dependence, a fact that is also qualitatively consistent with large length scales dominated by disorder, and which we can further analyze through the following estimates. Phenomenologically, in the case of disorder-dominated roughening of an elastic interface larkin_70 ; agoritsas_physb_12_DES , we can consider in the 2-dimensional regime at short length scales that , where is the correlation length of the disorder potential and the Larkin length (minimum pinning length, below which the interface would behave purely elastically). In the 1-dimensional regime at large length scales, this relation would become , where the characteristic length in this case would be the crossover length scale and . Taking nm from paruch_jap_06_dynamics_FE, , , , and for the 50 and 91 nm samples, we find that the experimentally obtained values would correspond to nm in the 2-dimensional regime (ambient conditions) and –8 nm in the 1-dimensional regime (thermally cycled). For reference, the –cm impurity densities extracted from current-voltage measurements in PZT films zubko_jap_06_PZT correspond to an inter-defect spacing of –10 nm, and would thus qualitatively agree with the extracted disorder correlation lengths.

An additional important consideration, especially at increasing temperatures, is the exact nature of the disorder. Although surface deterioration is seen upon heating, especially at the highest temperatures, x-ray characterization of the samples demonstrates continued high crystalline quality paruch_apl_06_stability . However, the mobility of point defects, such as oxygen vacancies, increases strongly with temperature. From the comparison of PG-, Ar- and O-cooled films, it appears that the primary defects responsible for the domain wall pinning are indeed oxygen vacancies. These defects can appreciably lower their energy by associating with a domain wall he_prb_03_DW_ovac , so the probability of a migration and segregation of defects at the domain walls upon thermal cycling, effectively increasing the strength of the pinning potential and stabilizing the domain structures against large-scale thermal roughening, is very high. In this respect it is worth noting that in the theory of DES an analogous “slow structural relaxation” mechanism was very recently shown jagla_jgr_10_pinning_relaxation, to provide a minimal model for predicting pinning enhancement or aging of the static friction force, and the phenomenon of “aftershocks” or static-relaxation-induced instabilities in the interface. This is interesting as these phenomena are both absent in the standard DES models with truly quenched disorder, but they are for instance observed in solid friction experiments.

Another possibility is that the thermal cycling steps, by removing the water layer normally present on the film surface in air at ambient temperature, significantly affect the electrostatic boundary conditions, and thus the effective potential landscape experienced by the domain walls. In this case, the higher roughness and increased values could reflect the configuration adapted by the walls under these new conditions, and frozen-in after a quench from high temperature, possibly in parallel to a dimensional crossover. Further investigation would be needed to asses the potential importance of this effect and also to quantitatively test the scenario described above, summarized in Fig.3.

## V Domain switching and growth

The crucial role of disorder in stabilizing the domain walls in a given configuration is further highlighted by the behavior of nanoscale circular domains followed in parallel measurements during the same heat-quench cycles. In these measurements, we note the strong polarization switching asymmetry: domains written with positive tip voltage in a negatively pre-polarized area grow significantly larger for a given pulse duration (Fig. 4(a-f)), even in films polarized as-grown. We relate this feature to the highly asymmetric device configuration maksymovych_nl_11_BFO_DW_conductivity ; guyonnet_am_11_DW_conduction . In PG- and Ar-cooled films at ambient temperature, both tip voltage polarities give uniform, circular domains, which become irregular upon heating, with apparent bowing of the domain walls. In as-grown films, domains decrease in size after thermal cycling, while in as-grown films a slight increase in domain size was observed. Some of the smaller domains (20–50 nm radius) of both polarities collapse even after low-temperature heating intervals (Fig. 4(e–f)). In as-grown films, significant displacements and an “amoeboid-like” growth are observed for domains near , with the structures eventually coalescing (Fig. 4(g–i)). In the O-cooled films, even at ambient conditions domains were unstable and domains were stable only for domain radii greater .

This complex thermal evolution can be understood by considering the forces on
the domain wall due to its curvature and the polarization orientation as well as
disorder and thermal effects. A preferred monodomain state indicates an
asymmetry of the ferroelectric double well potential, which promotes growth or
collapse of domains depending on their polarization orientation. In addition,
the “line tension” or elastic energy cost of circular domain walls promotes domain collapse, its effects
diminishing with increasing domain size. Finally, disorder allows domain wall
pinning, and therefore deformations from the elastically optimal circular
geometry. The relative strength of these contributions during thermal cycling
can be gauged from the degree of domain stability. Although polarization
orientation and device asymmetry ^{\theendnote}^{\theendnote}endnote: \theendnoteThe significant asymmetry in domain
switching and stability as a function of the polarity of the tip bias is a
feature already noted in previous studies guyonnet_am_11_DW_conduction
and can be related to the particular device geometry used. Essentially, the
metallic electrodes on either side of the ferroelectric thin films can be
considered as two back-to-back Schottky diodes, with the barrier height
determined by the work functions of the respective electrode materials
zubko_jap_06_PZT ; maksymovych_nanotech_11_PZT_conduction . In addition,
further asymmetry in current-voltage and switching characteristics can result
from strain and other boundary-specific conditions, and, in thin films, the
direction of polarization. Finally, the device architecture itself is strongly
asymmetric: one electrode is planar, while the other is the nanoscale AFM tip,
with a nominal radius at apex form metal-coated tips given as 25-50 nm. In such
devices, the nucleation and growth of domains is therefore strongly dependent on
the polarity of the voltage applied to the AFM tip. determine the domain size
for a given writing time, for domains of comparable size, line tension determines
stability. Larger and domains remain, while 20–50
nm radius domains readily collapse on heating. Domain wall roughening is
observed for all domain sizes, although the extent of the deformations increases
with domain size. Disorder appears to be crucial in stabilizing the domain
structures, in particular the smallest domains, and protects them against
thermal effects.

## Vi Conclusion

In conclusion, our studies show that in epitaxial ferroelectric thin films, domain wall roughening is governed primarily by disorder pinning, even upon heating to progressively higher temperatures, followed by a quench. However, the increased energy provided by the thermal cycling does allow accommodation of the artificially straight AFM-written domain walls to the surrounding disorder potential at progressively larger length scales, with the roughness exponent evolution possibly related to a 2-to-1 dimensional crossover in the behavior of the system. From a comparison of O-poor vs O-rich PZT films, it appears that oxygen vacancies are particularly important for pinning domain walls, and allow even relatively small domains to be stabilized at high temperatures, close to . This observation is especially interesting for potential applications, for which, in fact, it may be more useful to have higher oxygen vacancies and thus more stable domain structures.

###### Acknowledgements.

The authors thank J.-M. Triscone for useful discussions, and M. Lopes for technical support. Work at UniGE supported by the Swiss National Science Foundation through the NCCR MaNEP and Division II. Work at Yale supported by the National Science Foundation under MRSEC DMR-1119826, DMR-1006256, and FENA. ABK acknowledges the Universidad de Barcelona, Ministerio de Ciencia e Innovación (Spain) and Generalitat de Catalunya for partial support through the I3 program.## References

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