Identification of the defect distribution at ferroelectric domain walls from the evolution of nonlinear dielectric response during aging process

Identification of the defect distribution at ferroelectric domain walls from the evolution of nonlinear dielectric response during aging process

Abstract

Motion of ferroelectric domain walls greatly contributes to the macroscopic dielectric and piezoelectric response of ferroelectric materials. The domain wall motion through the ferroelectric material is however hindered by pinning on crystal defects which substantially reduces these contributions. Here, using thermodynamic models based on the Landau-Ginzburg-Devonshire theory, we find a relation between microscopic reversible motion of non-ferroelastic 180 domain walls interacting with a periodic array of pinning centers and the nonlinear macroscopic permittivity. We show, that the reversible motion of domain walls can be split into two basic modes. First, the bending of a domain wall between pinning centers, and, second, the uniform movement of domain wall plane. We show that their respective contributions may change when the distribution of pinning centers is rearranged during the material aging. We demonstrate that it is possible to indicate which mechanism of the domain wall motion is affected during material aging. This allows to judge whether the defects only homogeneously accumulate at domain walls or prefer to align in certain directions inside the domain wall plane. We suggest that this information can be obtained using simple macroscopic dielectric measurements and a proper analysis of the nonlinear response. Our results may therefore serve as a simple and useful tool to obtain details on domain wall pinning in an aging process.

pacs:
77.80.Dj
1

I Introduction

Domain wall motion has been found responsible for the great contribution to dielectric and piezoelectric response of ferroelectric ceramic Arlt et al. (1985); Li et al. (1991); Xu et al. (2001); Jones et al. (2012), single crystals Braun et al. (2005); Kleemann (2007), and thin films Kim et al. (2003); Gharb and Trolier-McKinstry (2005); Bassiri-Gharb et al. (2007). It is therefore expected that there exists a direct correlation between the fundamental parameters of domain wall motion and the parameters of enhanced macroscopic dielectric response of the studied ferroelectric polydomain systems. In small applied electric fields, the domain walls perform reversible motions, which can be identified by anhysteretic dielectric response with nonlinear field dependence of permittivity. With an increase in the amplitude of applied electric field, the dielectric response becomes hysteretic and follows the so called Rayleigh law Li et al. (1991). In order to explain phenomenological origins of the observed domain wall (extrinsic) contributions, several theoretical concepts of pinning mechanism have been proposed: elastic interaction between domain walls and ceramic grains Arlt and Pertsev (1991), defect orientation Robels and Arlt (1993), presence of electrode-adjacent passive layers Kopal et al. (1999); Bratkovsky and Levanyuk (2001); Mokry et al. (2004), domain wall pinning due to random bonds and random fields in disordered systems Nattermann et al. (1990), or statistical effects of domain wall depinning Boser (1987).

The greatest challenge in characterization of ferroelectric samples is an unambiguous determination of physical mechanism standing behind the macroscopically observed extrinsic enhancement in dielectric response. Techniques frequently used for a dielectric characterization of polydomain ferroelectrics are based on the measurement of frequency dependence of complex permittivity Braun et al. (2005); Kleemann (2007) and on the measurement of Rayleigh-like linear field dependence of permittivity in the sub-switching fields Gharb and Trolier-McKinstry (2005); Zhu et al. (2011); Marincel et al. (2014). Unfortunately, it is known that high-field electric cycling often substantially influences the delicate pinning condition on the ferroelectric domain walls Morozov and Damjanovic (2008); Glaum et al. (2012). It means that Rayleigh-type dielectric measurement in sub-switching regime may not be a suitable dielectric characterization method for studying the aging processes.

Recently, an alternative characterization method Mokry (2008), which is based on the analysis of nonlinear field dependence of macroscopic permittivity in weak electric fields:

(1)

where is the small-signal permittivity and is the dielectric nonlinearity constant, has been used to study the aging of polydomain ferroelectrics Mokry et al. (2009). It was observed that the time-dependent parameters and measured during the aging of [111]-oriented tetragonal lead zirconate titanate films (PZT) follow the relationship

(2)

where is the time-independent value of the permittivity of crystal lattice. This observation brought an evidence that the macroscopic dielectric response in the studied PZT samples was controlled by reversible bending movements of the 180 domain walls. The method based on the analysis of weak-field nonlinear dielectric measurements presented in Ref. Mokry et al. (2009) was however developed with the assumption of the infinitely strong interaction of the domain wall with the pining center. This represents a severe limitation for its applicability as it is evident that the interaction between real domain walls and pinning centers has a finite strength.

The aforementioned issues have motivated the work presented below, where we develop a general thermodynamic model for macroscopic nonlinear dielectric response produced by reversible motion of pinned non-ferroelastic 180 domain walls in ferroelectrics. In contrast to the model presented earlier Mokry et al. (2009), we consider a general situation of finite strength interaction between 180 domain walls and pinning centers. In the first step, we show that the general motion of non-ferroelastic 180 domain walls can be decomposed into two modes: (i) oscillation of planar walls around the pinning center, where the wall keeps its electroneutrality (This mode will be referred to as planar mode, PM.), and (ii) bending movements of domain walls between pinning centers, which are characterized by the appearance of uncompensated bound charge due to the discontinuous normal component of polarization across the domain wall thickness. (This mode will be referred to as bending mode, BM.) We calculate simple formulae for the field dependence of nonlinear extrinsic permittivity controlled by the both considered modes. The both contributions to the extrinsic permittivity are expressed as functions of dielectric parameters of the crystal lattice and representative distances between pinning centers.

Now, for further understanding, it is important to note that identification of the two types of contributions (with reasonable certainty) is not expected to be possible experimentally from the measurement of an actual value of permittivity, including its nonlinear component, at a given time. It is possible only from the observation of their characteristic changes during processes associated with the rearrangement of pinning centers on the domain wall. Such rearrangements take place during processes like material aging or fatigue. This complication is associated with the necessity to decompose the dielectric response into the lattice (intrinsic) and domain wall (extrinsic) contributions. It was shown in Ref. Mokry et al. (2009) that the decomposition is feasible in polydomain ferroelectric films because the intrinsic contribution to the dielectric response is constant in time while the extrinsic contribution changes substantially when pinning centers rearrange in time. However, the particular procedure of decomposition, which is presented in Ref. Mokry et al. (2009), is applicable only to a special case, when the BM dominates the reversible domain wall motion. In order to eliminate this restraint, a general procedure of decomposition, which is applicable to a general aging process, is introduced in this work.

Therefore, we demonstrate that, once the time changes are observed, it is possible to indicate which mode of the domain wall motion is affected by material aging. This allows to judge how the pinning centers rearrange at the domain walls, namely, it is possible to distinguish whether they homogeneously accumulate at domain walls and/or align in certain directions in the domain wall plane. In order to support our suggestion, we have performed a series of numerical simulations of aging experiments in polydomain BaTiO (BT) and PbTiO (PT) in tetragonal phase. Our results may therefore serve as a simple and useful tool to obtain details on domain wall pinning from conventional dielectric measurements.

Ii Macroscopic dielectric response in films with 180 domain walls

Figure 1: Reversible movements of 180 ferroelectric domain walls pinned by crystal lattice defects. Domain walls in the absence of the external electric field (Position A) are pinned by pinning centers indicated by red spheres. When the external electric field is applied to the system, the equilibrium position of the domain wall (Position C) is given by the superposition of two mechanism: uniform displacement of the wall from the pinning center (Position B) and the spatially dependent displacement due to domain wall bending.

In this section, we analyze the nonlinear macroscopic dielectric response of the polydomain ferroelectric system shown in Fig. 1. We consider a ferroelectric sample with crystallographic axes oriented along the , and axes of the coordinate system. Non-ferroelastic 180 domain walls perpendicular to the axis separate anti-parallel domains with the vectors of polarization oriented along, , and against, , the axis. In the absence of external electric field, the average distance between domain walls is denoted by the symbol . While the domain walls in real materials are pinned by randomly distributed lattice defects (pinning centers), we are limited to assume a regular periodic array of pinning centers in our model. We consider that the largest reduction of the extrinsic dielectric response of the ferroelectric system is produced by the pinning centers, which are localized in representative distances and in the directions of and axes, respectively.

When the external AC electric field with the amplitude is applied in the direction of axis, domain wall displacement produces a change in the volume fractions and of domains with polarization and , respectively:

(3a)
(3b)

Since we have limited our study to 180 ferroelectric domain structure, the symmetry of crystal lattice in the ferroelectric state yields the following expressions for polarizations and in the form of Taylor series with respect to the applied field :

(4a)
(4b)

where is the spontaneous polarization, is the lattice permittivity along the ferroelectric axis, and symbols and stand for bulk dielectric nonlinearity constants, respectively.

The macroscopic polarization response of the polydomain sample can be expressed as the volume-averaged polarization responses of anti-parallel domains:

(5)

It is convenient to introduce the net spontaneous polarization of the polydomain film due to domain wall displacements:

(6)

Using Eq. (6) and considering , the macroscopic polarization of the polydomain film given by Eq. (5) can be expressed in the form:

(7)

Iii Dynamics of pinned 180 domain walls

Objective of this Section is to formulate the equations of state for the net spontaneous polarization and to express extrinsic contributions to the small-signal permittivity and dielectric nonlinearity constant of the ferroelectric system due to the reversible motion of pinned 180 domain walls. The value of at the external field is given by the equilibrium displacement of the domain wall through Eqs. (3) and (6). The equation of state for can be expressed using a standard procedure from the condition for the minimum of the thermodynamic potential , where the function

(8)

is the volume density of the electric enthalpy. The thermodynamic potential is a function of the ferroelectric part of polarization , electric displacement , elastic strain , and electrostatic potential . The indexes with values 1, 2, and 3 denote the directions of the , , and axes of the coordinate system, respectively. The symbol stands for the partial derivative. The electric enthalpy includes the bulk free energy density , the elastic , where is the elastic stiffness tensor, electrostriction , where are the electrostriction coefficients, domain wall , where are the gradient energy coefficients, and electrostatic , where and are permittivity of vacuum and relative background permittivity, respectively. Formulae for particular contributions to the volume density of electric enthalpy are expressed in Appendix A. In this work, we introduce the additional term

(9)

in order to specify the interaction of the domain wall with a pinning center. Symbols , , and stand for the -, -, and -coordinate of the pinning center and symbols and stand for the finite strength and radius of domain wall-defect interaction, respectively. It should be noted here that the term introduces a defect as a nanoscale energy perturbation, but it produces long-range electrostatic and elastic interactions due to its coupling with all remaining energy components [see e.g. Eqs. (43) in Appendix A]. Finally, the last term corresponds to the subtracted work of external electric sources.

In order to develop an analytical thermodynamic model for the finite-strength interaction of the non-ferroelastic 180 domain wall with a pinning center, we consider a simplified situation, where the polarization is oriented along axis and that the component of the polarization changes its value linearly from to across the domain wall of thickness . The points with zero polarization in the middle of the domain wall comprise a curved surface, which is given by the equation . This situation can be described mathematically by considering the ansatz of the domain wall profile in the form of piecewise linear function:

(10a)
(10b)

where symbol stands for the distance from the mid-point of the domain wall. Then, we consider that the spatial distribution of polarization within the ferroelectric can be approximated by:

(11)

The assumption of the piecewise linear profile of the polarization at domain walls will be later proven as an adequate model by the correspondence between the analytical and phase field simulation results.

An essential concept in our analysis, which is indicated in Fig. 1, is the consideration that the reversible domain wall displacement in the -direction is given by the sum of two contributions:

(12)

where is the uniform displacement of the domain wall due to its PM motion (indicated by B in Fig. 1), which is controlled by the local interaction of 180 domain wall with the pinning center, and is the spatially dependent displacement of the domain wall due to its BM motion (indicated by C in Fig. 1). When Eq. (12) is substituted into Eqs. (3) and when the linearity of the operation of integration in Eqs. (3) is considered, it is convenient to express the net spontaneous polarization given by Eq. (6) as a sum of three contributions:

(13)

where is the net spontaneous polarization at zero applied field. Symbols and stand for the field dependent contributions due to the uniform reversible displacement of domain wall (PM) and due to the reversible domain wall bending (BM), respectively. In accordance with the definition of symbols and , it is noted here that these two contributions are produced by and , respectively, as it is indicated in Fig. 1.

In order to express the equations of state for the quantities and , the ansatz for the distribution of polarization within the domain wall given by Eq. (10) is substituted into the volume density of electric enthalpy given by Eq. (8) and integrated over the volume of the sample. Using a detailed analysis of particular contributions to the electric enthalpy given by Eq. (8) it is possible to show that, in the first approximation, the is the only contribution to , which depends solely on . The reason for this is that is the only contribution to , which is sensitive to the uniform displacement of the domain wall from the pinning center at . On the other hand, it is possible to show that the remaining contributions depend solely on . The reason for this is that spatially non-uniform displacement of the domain wall, which is proportional to , is associated with an increase in the domain wall area and with the production of depolarizing field due to the discontinuity of the normal component of polarization across the domain wall thickness. As a result of the above arguments, it is clear that, in the first approximation, the thermodynamic function can be expressed in the form:

(14)

where

(15)

and

(16)

are the contributions to the macroscopic volume density of electric enthalpy due to PM and BM motions of the domain wall, respectively. Symbol is the constant term in the expansion of the thermodynamic potential , which is independent on the reversible displacements of the domain wall.

The value of the first contribution is controlled by the additional term in the volume density of electric enthalpy, which describes the interaction of domain wall with a pinning center. The most significant terms of with respect to are calculated in Appendix B and equal

(17a)
where
(17b)
(17c)

are numerical functions that depend on the spatial distribution of polarization within the domain wall.

The second contribution to the macroscopic electric enthalpy is due to the BM motion of 180 domain wall. Its value is controlled by energy of the electric field produced by uncompensated bound charges due to the discontinuity in the normal component of polarization across the domain wall thickness and due to the increase in the area of the bent domain wall. The leading terms of with respect to are calculated in Appendix C and equal

(18)

The equilibrium response of the net spontaneous polarization is found from the minimum of the thermodynamic function considering the principle of superposition for the net spontaneous polarization given by Eq. (13). Substituting Eq. (13) into Eq. (14), the equilibrium values of and are found by solving the system of equations of state:

(19a)
(19b)

Solution of Eq. (19a) can be expressed in the form of Taylor series (see Appendix B):

(20a)
where
(20b)
(20c)

stand for the small-signal permittivity and dielectric nonlinearity constant of the extrinsic dielectric response due to the movement of planar pinned domain walls, respectively. Symbols and are dimensionless numerical refinement factors, which are computed in Appendix D.

Similarly, solution of Eq. (19b) yield the Taylor expansion (see Appendix C):

(21a)
where
(21b)
(21c)

stand for the small-signal permittivity and dielectric nonlinearity constant of the extrinsic dielectric response due to the bending movement of domain walls, respectively. Symbols and are dimensionless numerical refinement factors, which are computed in Appendix D.

The field-dependent net spontaneous polarization can be expressed in the form:

(22a)
where
(22b)
(22c)

are the small-signal contribution to the extrinsic permittivity and the nonlinearity constant of the anhysteretic field dependence of nonlinear extrinsic permittivity, respectively.

In the final step of our analysis, it is convenient to introduce the field-dependent extrinsic contribution to permittivity by the formula:

(23)

It follows from Eq. (22) that can be expressed in the form of Taylor series

(24)

Since in dielectric experiments it is possible to directly measure only the field dependence of macroscopic permittivity , which is defined as

(25)

it follows from Eq. (7) that

(26)

where

(27a)
(27b)

and symbols and stand for the lattice values of permittivity and dielectric nonlinearity constant [see Eqs. (4)].

Analytical results obtained above describe the general nonlinear macroscopic dielectric response of ferroelectric polydomain system controlled by the reversible motion of non-ferroelastic 180 domain walls interacting with pinning centers. It should be noted here that the continuous spatial distribution of the polarization vector was considered in the analytical solution of the thermodynamic model. This may seem to be inconsistent with the well-known concept of domain wall pinning at atomic planes by Miller and Weinreich Miller and Weinreich (1960). However, the recent results based on the atomistic molecular dynamic simulations by Shin et al. Shin et al. (2007) indicated that the classical model by Miller and Weinreich is outdated and that the critical domain nuclei responsible for the side-wise domain wall motion have actually much lower activation energy. This should reasonably justify our use of continuous thermodynamic model even the phenomena we analyse are on the atomistic level. In the next Section, we will show that these results can be used for the identification of microscopic processes responsible for aging or fatigue in ferroelectric polydomain films.

Iv Nonlinear dielectric characterization during aging

It was shown in Sec. II that the macroscopic dielectric response of polydomain ferroelectric systems is given by the superposition of intrinsic (lattice) and extrinsic (domain wall) contributions and that these contributions can be, in principle, decomposed in the nonlinear dielectric response of these systems Mokry et al. (2009). Such a decomposition is possible because the intrinsic parameters and are virtually time-independent while the extrinsic parameters and undergo the process of aging, which are frequently explained by defect migration. Several types of defects that may act as pinning centers have been discussed in the works on lead titanate Morozov and Damjanovic (2008); Morozov and Damjanovic (2010) and bismuth ferrite Rojac et al. (2010, 2014). In addition, several mechanisms that represent a driving force for defect formation and migration have been suggested Chandrasekaran et al. (2013); Morozov and Damjanovic (2008); Genenko et al. (2015). In the terms of our model, some of these mechanisms may result in two different processes that may be responsible for the dielectric aging in polydomain ferroelectric: (i) uniform accumulation of pinning centers at the domain wall, which is expressed by an increase in the value of pinning center concentration , and, (ii) redistribution and alignment of pinning centers along certain directions on the domain wall, which is expressed by a change in the value of the anisotropy of the pinning center distribution . The key objective of this Section is to show that the nonlinear analysis of time changes in the correlation between the small-signal permittivity and dielectric nonlinearity constant allows to obtain information about the preferred modes of domain wall motion (PM or BM) and, consequently, also some details about the microscopic process responsible for the dielectric aging.

An important result presented in Sec. II is that the dielectric parameters of extrinsic dielectric response due to the PM motion of domain walls, i.e. and , depend only on the concentration of pinning centers on the domain wall , but do not depend on the anisotropy of the pinning centers distribution [see Eqs. (20)]. On the contrary, the dielectric parameters due to BM motion of domain walls, i.e. and , depend on the both parameters and [see Eqs. (21)]. Let us consider for a moment that the anisotropy parameter is constant and the aging process is driven by uniform accumulation of pinning centers on the domain wall. Under this assumption, the both contributions to the extrinsic small-signal permittivity due to PM and BM domain wall motion and change proportionally to . On the contrary, there exist a difference in scaling the dielectric nonlinearity constants. In the case of PM domain wall motion, it is , whereas it is in the case of BM domain wall motion.

The above statement indicates that the nonlinear extrinsic dielectric response due to PM motion of domain walls is characterized by a typical correlation between and , which can be expressed as:

(28)

This result suggests that when the correlation is identified in the extrinsic dielectric response during aging process, it is a rather unambiguous indication that the aging process is dominated by PM domain wall motion. Since the dielectric response in the PM is insensitive to the anisotropy parameter , it can be clearly concluded that the experimentally observed aging process is caused by the accumulation of the pinning centers on the domain wall.

Similarly, a correlation between and , which satisfies:

(29)

represents a clear indication that the nonlinear dielectric response is dominated by the BM motion of domain walls. Identification of the microscopic origin of the aging process, however, is more difficult and requires more detailed analysis. First it should be noted that there exist a critical defect concentration which represents a threshold between the dominant modes in the dielectric response. For smaller values of pinning center concentration than , the contribution of PM to extrinsic permittivity becomes negligible compared to BM and makes further analysis of the microscopic origin of the aging process impossible, since the aging in the BM can be caused by the both aforementioned processes. However, above this critical concentration the effects of PM and BM modes on the nonlinear dielectric response are comparable. It means that when the is identified in the extrinsic dielectric response during the aging process in samples with supercritical concentration of pinning centers at the domain wall, it is an indication that the aging process is caused by the redistribution and alignment of pinning centers along certain directions in the domain wall plane.

Thus, the results above indicate that the microscopic process, which is responsible for the dielectric aging, can be determined by means of the identification of the dominant mode of domain wall motion in the extrinsic dielectric response. The last task that has to be resolved is the decomposition of the intrinsic and extrinsic responses from the macroscopic dielectric response of the ferroelectric system.

In order to identify the aging mechanism in a real ferroelectric system, we suggest the following procedure. Let us introduce a general aging mechanism, which is characterized by a general correlation between the small-signal extrinsic permittivity and the extrinsic nonlinearity constant in the form

(30)

It follows from Eqs. (27) that the coupling between the macroscopic small-signal permittivity and the macroscopic nonlinearity constant satisfies the relation:

(31)

It is considered that during aging, the lattice parameters and remains constant and the evolution of the macroscopic dielectric response (i.e. parameters and ) is produced due to change in the extrinsic dielectric response parameters and .

Let us conduct an aging experiment and measure the time dependence of macroscopic values and . Let us define the symbols and for the minimum measured values of and , respectively. Let us define the parameter

(32)

Substituting Eq. (31) into Eq. (32), it can be calculated with the use of elementary calculus rules that

(33)

It means that using the estimation of the limiting value of parameter for very large values of macroscopic small-signal permittivity from measured experimental data, it is possible to determine the dominant aging mechanism in the ferroelectric system.

Unfortunately, it is rather difficult to achieve the condition given by Eq. (33) in real experimental conditions. The reason is that using dielectric measurements, it is possible to study only intermediate stages of the aging process. Now, the key objective is to find a procedure for the determination of the extrapolated value of for very large values of macroscopic small-signal permittivity .

One possibility is offered by expressing the Taylor expansion of given by Eq. (32) with respect to the inverse macroscopic small-signal permittivity normalized to its minimal value measured during aging experiment. For integer numbers of in Eq. (31), one yields:

(34)

In order to estimate (i) the value , which identifies the mechanism of the aging, (ii) lattice permittivity , and (iii) parameter , which contains information on the pinning condition on the domain wall, we suggests to compute the parameter from the values of and measured during the aging experiment and to fit the values of parameter to the function of the form:

(35)

where are the fitting parameters. Comparing the coefficients in the Taylor expansions in Eqs. (34) and (35), it is possible to roughly estimate the unknown parameters of the ferroelectric system:

(36a)
(36b)
(36c)

Thus, it is seen that it is possible to decompose the intrinsic and extrinsic contributions to the dielectric response of ferroelectric polydomain samples and that it is possible to identify the dominant aging mechanism by proper analysis of the nonlinear dielectric response.

V Phase-field model simulations of aging

The key objective of this Section is to provide independent arguments supporting the idea that results presented in the above Section can be used to study processes of aging in real ferroelectric systems. We have performed numerical simulations of two aging mechanisms using the standard thermodynamic model Zhirnov (1959); Cao and Cross (1991); Hu and Chen (1998). The process of aging has been modeled by means of a series of numerical simulations performed on a representative volume element (RVE) in a periodic structure of a ferroelectric material with a 180 domain wall. The RVE is considered as a cuboid with dimensions , , and along , , and axes of the coordinate system as it is indicated by a dashed line in Fig. 1. In accordance with two aging mechanisms considered in the above Section, we have performed two types of numerical simulations. The PM aging mechanism can be modeled in the system, where the parameter equals a constant value , which is smaller than the value of parameter . The BM aging mechanism, where the dielectric response is controlled by BM movements of 180 domain walls, can be modeled in the system with the constant value of parameter and decreasing value of parameter .

We have performed numerical simulations of two aforementioned aging mechanisms using the numerical method, which is described in Appendix A. In the numerical simulations, the parameter was set to be equal to 4 nm. The effect of aging is introduced into the numerical simulations by means of a decrease in the representative distance between pinning centers in the particular direction. The parameter in the RVE, which is subjected to change during the simulation of aging, is denoted by a symbol and its value has been decreasing from 25 nm to 4 nm.

(a) Bending mode (BM) aging
(b) Planar mode (PM) aging
Figure 2: Result of the phase-field model simulation of the 180 domain wall interacting with a pinning center in BaTiO for  nm and  nm during aging. Two aging mechanisms were modeled: \subreffig:PFMres:a bending mode (BM) aging, where the dielectric response is controlled by bending movements of 180 domain walls, and \subreffig:PFMres:b planar mode (PM) aging, where the planar 180 domain walls oscillate around the pinning centers. Magnitude of the external field is kV/m. Color on the surface indicates the value of the component of the polarization vector. Blue and red colors indicate the values of equal to -0.221 C/m and 0.221 C/m, respectively.

Figure 2 shows results of the numerical simulations of the two considered aging mechanisms in BT considering  nm,  nm, and  kV/m. It indicates the clear differences in the reversible 180 domain wall motion during PM and BM aging mechanisms. Spatially dependent bending displacement of the curved domain wall, which dominates the BM aging is clearly presented in Fig. 2(a). On the other hand, Fig. 2(b) shows the spatially uniform displacement of the planar 180 domain wall is characteristic for the PM aging. Numerical results make it possible to estimate the domain wall displacements. The displacement of the domain wall in the PM mode is about  nm in BT at 104 C and considering the conditions specified in Fig. 2. The displacement of the apex of the domain wall during the BM mode is about 1.82 nm under the same external conditions. This gives the maximum deflection of the bent domain wall  nm [See Eq. (62)]. In the PT single crystal, the displacement of the domain wall is 0.15 nm in the PM at external electric field 441 kV/m, temperature 25 C, and  nm and  nm. The displacement of the apex of the domain wall during the BM mode is about 0.38 nm under the same external conditions. The value of the maximum deflection due to domain wall bending is  nm. It should be noted that the simulations were computed under the conditions of relatively high pinning centers concentrations, which yields to the small domain wall displacements (compared to the lattice constant) at the given magnitude of the external field.

The result of the numerical simulations of aging gives the spatial distribution of polarization at a given external field . The field dependence of the lattice polarization and and the field dependence of the macroscopic polarization in the whole ferroelectric RVE were computed according to the following formulae:

(37a)
(37b)
(37c)

where is the distance at which the averaged dielectric response of the crystal lattice is not affected by the movement of the domain wall. Such a condition is usually satisfied for . Thickness of the domain wall was numerically estimated from the numerical simulations according to the formula:

(38)
Material
[C] [C/m] [1] [10 C/V] [10 m C/V] [nm]
PbTiO 25 0.61 53.5 1.57 9.5 2.00
50 0.59 58.4 1.97 10.5 2.09
75 0.57 64.6 2.63 12.0 2.20
BaTiO 104 0.22 273.1 98.4 404.8 1.44
106 0.22 289.3 117.8 486.3 1.48
108 0.22 289.3 114.2 450.3 1.48
Table 1: Lattice values fitted from the phase field model simulations.

The computed values of and were used for the identification of parameters that control the nonlinear dielectric response of the crystal lattice, i.e. , , , and , from Eqs. (4) using the method of least squares. The resulting numerical values are presented in Tab. 1.

(a)  PbTiO (BM aging)
(b)  PbTiO (PM aging)
(c)  BaTiO (BM aging)
(d)  BaTiO (PM aging)
Figure 3: Evolution of the electric field dependence of extrinsic permittivity computed from numerical phase-field model simulations of BM aging \subreffig:EpsW-E:a and \subreffig:EpsW-E:c; and PM aging \subreffig:EpsW-E:b and \subreffig:EpsW-E:d. Numerical simulations computed for PbTiO are presented in graphs \subreffig:EpsW-E:a and \subreffig:EpsW-E:b. Graphs \subreffig:EpsW-E:c and \subreffig:EpsW-E:d show results for BaTiO.

In the next step, the field dependence of the net spontaneous polarization can be expressed from Eq. (7) in the form:

(39)

where is the macroscopic polarization at given external field computed using Eq. (37c) from the results of numerical simulations. It follows from Eq. (23) that the field dependence of the extrinsic permittivity can be expressed as:

(40)

The values of are then fitted to Eq. (24) and numerical values of the small-signal extrinsic permittivity and the nonlinearity constant of the field dependence of extrinsic permittivity at given stage of aging are computed using the method of least squares. The evolution of electric field dependence of extrinsic permittivity computed from numerical simulations of the considered PM and BM aging mechanisms is presented in Fig. 3.

(a)  PbTiO
(b)  BaTiO
Figure 4: Dielectric nonlinearity constant versus small-signal extrinsic permittivity during PM (filled markers) and BM (open markers) aging. Numerical computations were carried out for PbTiO \subreffig:GammaW-EpsL:a and BaTiO \subreffig:GammaW-EpsL:b. Dashed and solid lines indicate fitting to the model of PM domain wall motion [see Eq. (28)] and BM domain wall motion [see Eq. (29)], respectively.

Figure 4 shows plots of the dielectric nonlinearity constant versus small-signal extrinsic permittivity in the logarithmic scales for PT [Fig. 4(a)] and BT [Fig. 4(b)], respectively. The values of and correspond to the best fit of the field dependence of the nonlinear extrinsic permittivity at different stages during the considered PM and BM aging mechanisms. Filled and empty markers correspond to the PM and BM aging mechanisms, respectively. Both graphs in Fig. 4 clearly indicate that the curves presented by filed and empty markers differ in their slope. Curves, which correspond to the PM aging mechanism have the slope equal approximately to 3. On the other hand, the slope of curves corresponding to the BM aging mechanism equals to 2. It should be however noted that graphs shown in Fig. 4 were displayed for the parameters of extrinsic nonlinear permittivity, which were extracted using the numerical procedure described above. The essential point of this numerical procedure is the knowledge of dielectric response of the crystal lattice, which is possible to compute from the numerical simulations with a high accuracy. This is a principal difference from the analysis of experimental data where the dielectric response of the crystal lattice is sometimes difficult to obtain.

(a)  PbTiO
(b)  BaTiO
Figure 5: Parameter computed according to Eq. (32) as a function of the inverse macroscopic small-signal permittivity normalized to its minimal value measured during aging experiment for the PM (filled markers) and BM (open markers) aging mechanisms. It is seen that values of the parameter can be used for the identification of aging mechanism. Numerical computations were carried out for PbTiO \subreffig:CoefA-EpsF:a and BaTiO \subreffig:CoefA-EpsF:b.

In the next step, it is necessary to verify whether it is possible to determine the dominant aging mechanism from measurements of the macroscopic dielectric response, which is given by the superposition of the time-constant intrinsic and time-dependent extrinsic contributions. Filled and open markers in Fig. 5 show values of the parameter computed from the numerical simulations of the PM and BM aging experiments, respectively. In order to estimate the correlation between the dielectric nonlinearity constant and small-signal permittivity , which is expressed by the parameter [see Eq. (31)], the numerically computed values of the parameter have been fitted to the function given by Eq. (35).

Material [C] [1] [1] [1] [1] [1]
Tab. 1 BM Aging PM Aging
PbTiO 25 53.5 1.8 82.6 2.8 36.0
50 58.4 1.8 87.3 2.8 38.2
75 64.6 1.9 88.2 2.8 41.8
BaTiO 104 273.1 1.7 218.7 2.5 331.6
106 289.3 1.7 220.4 2.5 347.2
108 289.3 1.7 224.7 2.5 347.9
Table 2: Results of the numerical simulation of BM and PM aging mechanisms. The value of parameter was extrapolated using Eqs. (35) and (36a). It is seen that the extrapolated values of correspond to the theoretical values 2 and 3 for BM and PM aging mechanisms. The values of lattice permittivity were calculated from simulations of aging using Eqs. (35) and (36b). It is seen that the values of calculated from aging simulations are in the agreement with values computed using Eqs. (4) and (37a) and presented in Tab. 1.

Result of this fitting procedure are presented in Tab. 2. The value of parameter was extrapolated using Eqs. (35) and (36a). It is seen that the extrapolated values of correspond to the theoretical values 2 and 3 for BM and PM aging mechanisms, respectively. The values of lattice permittivity were calculated from the simulations of aging using Eqs. (35) and (36b). It is seen that the values of calculated from aging simulations are in the good agreement with values computed using Eqs. (4) and (37a) and presented in Tab. 1.

Therefore, it was concluded that the measurement of nonlinear dielectric response may serve as a tool for identification of the mechanism of reversible domain wall motion, which controls the dielectric response during the processes of aging in the polydomain ferroelectric samples. In addition, it is possible to estimate important material parameters and microstructure of the pinning centers of the ferroelectric system by proper analysis of nonlinear dielectric data.

Vi Conclusions

In this Article, we presented the main results of our detailed theoretical analysis of the so-called extrinsic contributions to macroscopic dielectric constant due to reversible 180 domain wall movements in polydomain ferroelectric samples. It was considered that the reversible movement of 180 domain walls is hindered by their interaction with crystal lattice defects, which behave as pinning centers. It was demonstrated that the general reversible motion of 180 domain walls can be decomposed into two modes: (i) oscillation of planar walls around a pinning center (PM mode) and (ii) bending movements of domain walls between pinning centers (BM mode). We have expressed simple formulae for the extrinsic contributions to the small-signal permittivity and dielectric nonlinearity constant due to the PM and BM modes as functions of dielectric parameters of the crystal lattice and the parameters of the pinning center distribution on the domain wall.

Domain wall motion Pinning centers concentration
“Small” () “High” ()
Bending mode
() Aging can originate from the both mechanisms. Aging originates from redistribution of pinning centers along certain directions inside the domain wall planes.
Planar mode
() Not available Aging originates from uniform accumulation of pinning centers at domain walls.
Table 3: The table concludes the main results of this work, i. e. identification of the microscopic mechanism that controls the reversible domain wall motion at small and high concentration of pinning centers in polydomain ferroelectrics. In this work, two microscopic mechanisms responsible for dielectric aging are considered: (i) redistribution of pinning centers along certain directions and (ii) uniform accumulation of pinning centers on the domain wall. Figure 1 shows two modes of the reversible domain wall motion considered in this work: (a) uniform motion of the domain wall as a plane (PM) and (b) bending movement of free segments of the domain wall between pinning centers (BM). Identification of the mode of the reversible domain wall motion using the measurements of nonlinear permittivity it is demonstrated in Sec. IV. Table briefly reviews the origins of the PM and BM domain wall motion at small and high pinning centers concentrations.

Therefore, it is natural to expect that the extrinsic contributions may change when the distribution of pinning centers is rearranged during dielectric aging of polydomain ferroelectric. We have find out that the PM mode of domain wall motion is controlled solely by the concentration of pinning center on the domain wall and that it is insensitive to the anisotropy of the pinning centers distribution on the domain wall. On the other hand, the BM mode is controlled by the anisotropy parameter in addition to the concentration . It can be concluded from the above statements that, when the PM domain wall motion is identified as a dominant contribution to the macroscopic dielectric response during aging, the microscopic evolution of the pinning centers is dominated by the uniform accumulation of the pinning centers on the domain wall and not by the redistribution and alignment of pinning centers along certain directions on the domain wall. On the contrary, identification of the microscopic detail in the evolution of pinning centers on the domain wall is more difficult when the BM domain wall motion is identified as a dominant in the macroscopic dielectric response. The reason is that PM mode becomes negligible compared to BM mode in the extrinsic dielectric response in systems with low concentration of pinning centers and the identification of the mechanism of the pinning centers evolution is impossible in this situation. However, BM mode may suggest that the evolution of pinning centers is dominated by the alignment of pinning centers along certain directions in systems with a high concentration of the pinning centers. These results are briefly concluded in Tab. 3.

In order to identify the dominant mode of the reversible domain wall motion, we have suggested a method based on the nonlinear analysis of macroscopic dielectric response of polydomain films. The identification can be obtained by plotting the nonlinearity constant of the macroscopic dielectric response versus the small-signal macroscopic permittivity in the logarithmic scale (see Fig. 4). When the condition is valid, it is an indication that the dominant mode is the oscillation of planar 180 domain walls around pinning centers. Bending movements of 180 domain walls are indicated by the condition . In addition, we have developed a refinement of this simple method, which is applicable for the analysis of nonlinear dielectric data measured in the intermediate stages of the aging process and in systems with small extrinsic contributions compared to intrinsic ones [see Eq. (32), (35) and (36)].

In order to support our statements, we have performed several phase-field model simulations of two aging mechanisms in polydomain ferroelectric: planar mode aging, where the macroscopic dielectric response is controlled by the oscillation of planar 180 domain walls around pinning centers, and bending mode aging, where the dielectric response is controlled by the bending movements of 180 domain walls. We have processed numerically computed data according to our suggested method and demonstrated that it is possible to identify which mechanism of the domain wall motion is dominant in a particular aging mechanism using simple macroscopic dielectric measurements and proper analysis of the nonlinear response. We have further demonstrated that using the proper analysis of nonlinear dielectric data, it is possible to obtain information on the domain pattern microstructure and further material parameters of the ferroelectric polydomain system.

Acknowledgment

The support from the Czech Science Foundation, Project No. GA14-32228S is acknowledged. The research leading to these results has received also funding from the European Research Council under the EU 7th Framework Program (FP7/2007–2013)/ERC grant agreement no 268058, MOBILE-W. Authors would like to thank Alexander K. Tagantsev and Dragan Damjanovic of Ceramics Laboratory, Swiss Federal Institute of Technology, Lausanne, Switzerland for reading the manuscript and many useful discussions.

Appendix A Thermodynamic model of pinned 180 domain wall

The thermodynamic model of the interaction of 180 domain walls with pinning centers is based on the standard thermodynamic approach Zhirnov (1959); Cao and Cross (1991); Hu and Chen (1998). Introduction of the model will be carried out in three steps. The definition of state quantities and thermodynamic potentials is presented in Subsection A.1. Subsection A.2 presents the equations of state governing the evolution of domain pattern. Finally, geometry of the representative volume element (RVE) model of the polydomain ferroelectric with a pinned 180 domain wall and the boundary conditions are specified in Subsection A.3.

For the sake of convenience of notation in this Appendix, the , , and axes of the coordinate system are denoted by symbols , , and , respectively. Einstein summation is considered for indexes if not indicated otherwise.

a.1 State quantities and thermodynamic potentials

In the thermodynamic model, we consider seven independent state quantities: scalar electrostatic potential , three components of polarization , and three components of mechanical displacement . For the purpose of clearer definition of thermodynamic potentials, it is convenient to introduce following physical quantities: electric field

(41a)
where the symbol stands for the partial derivative, elastic strain
(41b)
and the dielectric displacement
(41c)

where symbols and stand for the permittivity of vacuum and the background permittivity, respectively.

In order to calculate the dielectric response, it is necessary to introduce the Helmholtz free energy (per unite volume), which is given by the sum of following contributions:

(42)

where

(43a)
is the bulk free energy density, where zero-strain coefficients can be expressed in terms of the stress-free coefficient as follows:
Symbol
(43b)
is the elastic energy density,
(43c)
is the domain wall energy density,
(43d)
is the electrostriction energy density,
(43e)
is the electrostatic energy density. Symbol stands for .
Interaction of the domain wall with a pinning center has been introduced by an additional term in the Helmholtz free energy (referred to as “pinning energy density” hereafter):
(43f)

where symbols , , and stand for the -, -, and -coordinate of the pinning center, symbols and stand for the strength and radius of domain wall-defect interaction.

a.2 Equations of state

In order to express the equations of state that govern the evolution of the domain pattern at given boundary conditions, it is convenient to introduce the volume density of the electric enthalpy :

(44)

Evolution of the domain pattern is then governed by the following equations:

(45a)
(45b)
(45c)

which express the mechanical equilibrium, the Poisson equation for the electric displacement , and the time-dependent Landau-Khalatnikov equation. Symbol stands for the kinetic constant.

Parameter PbTiO BaTiO
(J m C)