kinetics of polarization reversal in normal and relaxor ... - CiteSeerX

Phase Transitions, V. 65, pp.49-72 (1998)

KINETICS OF POLARIZATION REVERSAL IN NORMAL AND RELAXOR FERROELECTRICS: RELAXATION EFFECTS V.Ya. SHUR

Ural State University Lenin Ave. 51, 620083, Ekaterinburg, Russia The polarization reversal/switching process was investigated in normal and relaxor ferroelectrics and described as a result of field induced evolution of domain/heterophase structure. Hysteresis loops and elastic light scattering data registered under the action of field pulses were measured. The scenario of the evolution peculiarities of heterophase structure in relaxors during "switching" and spontaneous "backswitching" near phase transition point to ferroelectric phase is proposed. Particular attention is given to the non-exponential relaxation behaviour of the bias field value in ferroelectrics during periodical alternating sign switching and of scattered light intensity in relaxors during “switching” and spontaneous "backswitching" near transition point.

KEY WORDS:

Ferroelectric, relaxor, domains, heterophase structure, switching, light scattering.

1 INTRODUCTION In strong electric field the evolution of the domain structure in normal ferroelectrics during switching and the reconstruction of the heterophase structure in relaxor ferroelectrics have much in common. Such similarity has made it possible to use the same methods of investigations and analysis for both of these processes. In ferroelectric single crystals with optically visible domains the most complete information can be obtained by real time registration of the set of instantaneous patterns. The evolution of invisible submicron domains in ceramics and thin films and polar nanoregions in relaxors can be investigated by indirect integral methods only. Using these methods one can obtain the integral data: 1) the angular dependence of elastic scattered light intensity, 2) the transient current pulses and 3) the hysteresis loops. It was shown by us earlier (Shur, Rumyantsev, Makarov and Volegov, 1994; Shur, Rumyantsev and Makarov, 1995). that

2 V.YA. SHUR the original modifications of classic statistic Kolmogorov-Avrami theory for real finite systems can be used for the extraction of the main kinetic parameters, such as nucleation probability and phase growth velocity, from integral data. Application of proposed treatment in model ferroelectric single crystals with simultaneous registration of instantaneous domain patterns revealed its validity. In addition the analysis of set of the instantaneous angular dependencies of the scattered light intensity allow to determine the variation of spatial distribution of the scattering centres in terms of fractal dimensionality. We choose several types of materials for experimental investigation: 1) the single crystals of lead germanate Pb5Ge3O11 (PGO) and gadolinium molybdate Gd2(MoO4)3 (GMO) as a model objects, 2) the dense transparent ceramics of the lanthanum modified lead zirconate titanate solid solutions (PLZT) with La contents between 6 and 8 at. % and with Zr/Ti ratio 65/35 as classic relaxor ferroelectrics. It is well known that the measured characteristics of relaxors and ferroelectrics strongly depend on the prehistory. Extremely long non-exponential relaxation is very typical for such materials. Analysis of two effects demonstrating such behaviour is presented in this paper. First, decreasing of the value of bias field in ferroelectric single crystals during periodical alternating sign switching. Second, relaxation of the intensity of scattered light after removing of electric field, so-called "backswitching", investigated in relaxor PLZT ceramics in wide temperature range. Critical temperature dependence of the "backswitching time" was obtained in the vicinity of the point of phase transition from relaxor phase to ferroelectric one. Influence of bulk screening process on the nonexponential relaxation behaviour is discussed.

2 KINETICS OF THE DOMAIN STRUCTURE IN NORMAL FERROELECTRICS At first we shall review the description of the domain kinetics during polarization reversal in normal ferroelectrics taking into account the role of various screening effects. The specific of switching caused by existence of particular metastable as-grown domain structure with charged domain walls formed after zero field cooling is considered. The simplified idealized scenario of domain kinetics during polarization reversal (Fatuzzo and Merz, 1967) is as follows. The initial state is considered to be completely single domain. In electric field the domains with different direction of spontaneous

3 KINETICS OF POLARIZATION REVERSAL polarization are arising, then they are enlarged by forward and sideways motion of domain walls and finally all of them coalesce. The final state is proposed to be also completely single domain. Evolution of domain structure in real ferroelectric samples is more complicated because the initial state is always multidomain and complete switching is never achieved. The domain structure evolution during switching can be considered as an example of the first-order phase transformation, so its kinetics is achieved through nucleation (Shur, and Rumyantsev, 1993; Shur and Rumyantsev, 1994). Any stage of the domain structure evolution can be attributed to the elementary processes of field induced generation of the thermally activated nuclei of various dimensionalities. In particular only 3D nucleation is responsible for the appearance of new domains and the domain growth is due to 1D and 2D nucleation at the domain wall. The nucleation probability depends on the local value of electric field averaged over the volume of the order of nuclei sizes and produced by wide variety of sources. In the simple case of ferroelectric capacitor this local field is determined by the sum of various fields: 1) the external electric field produced by potential difference between the electrodes Eex, 2) the depolarization field of bound charges Edep and 3) the screening fields, which can be divided in two groups: external screening produced by charges situated on electroded surfaces Eex.scr and bulk screening caused by the bulk charges Eb.scr (Shur, 1996; Shur, Popov and Korovina, 1984):

[

El ( r , t ) = E ex − Edep ( r , t ) −

∑ Escr (r, t )]

(1)

where Edep(r,t) - depolarization field produced by all existing bound charges at the point r, where the nucleation occurs at the moment t,

∑ Escr (r , t ) = Eex.scr + Eb.scr

- sum

screening field induced by all screening charges at the surface and in the bulk.

2.1

Depolarization Field

The spatial inhomogeneity of spontaneous polarization is the source of the bound charges. In finite single crystalline ferroelectric sample the bound charges appear at the polar surfaces and at the head to head or tail to tail domain walls in the bulk. In multidomain state arising during switching the spatially nonuniform depolarization field is

4 V.YA. SHUR determined by instantaneous position of the bound charges. Incomplete screening of depolarization field decreases the nucleation probability and slows the switching process therefore the domain growth velocity depends upon its shape and sizes, so as screening kinetics. The relaxation time characterizing the dynamics of compensation of depolarization field tscr is varying sufficiently for different screening mechanisms.

2.2

External Screening

External (surface) screening is the fastest and the most effective way to compensate depolarization field. In the ferroelectric capacitor with the electroded polar surfaces such screening occurs through redistribution of the charges on electrodes by the current in external circuit, which parameters determine the time constant of the process. Its value places the limitation on the switching time. The value of tex.scr for typical experimental situation is about 10-5-10-8 s. The time shorter then 10-10 s was achieved for thin crystals and films (Larsen, Cuppens and Spierings, 1992; Stadler, 1992).

2.3 Bulk Screening One of the essential features of ferroelectrics is the existence of thin surface layers (dielectric gaps) in which spontaneous polarization is absent (Fridkin, 1980; Shur, 1996). Due to these layers the depolarization field in the bulk can not be completely compensated by the surface charges. After finishing of external screening the following residual depolarization field Edr remains in the bulk short-circuited capacitor being in single domain state E dr = E dep − E ex . scr = 2 Ps Ls (ε L ε 0 d )

−1

(2)

where Ps - spontaneous polarization, Ls and d - thickness of the surface layer and the sample, ε L - dielectric permittivity of the surface layer. The bulk screening is the only way to compensate this field. There are several mechanisms of bulk screening, namely redistribution of the space charges or reorientation of the dipolar defects in the bulk (for example, produced by irradiation) (Fridkin, 1980; Lambeck and Jonker, 1978; Shur, Letuchev and Popov, 1982). The bulk

5 KINETICS OF POLARIZATION REVERSAL screening process is of fundamental importance for evolution of the domain structure with charged domain walls.

2.4 Retardation Effect The retardation of the screening of depolarization fields is the most pronounced if the bulk screening prevails. Its time constant is about 103 - 106 s at room temperature (Lambeck and Jonker, 1978). The slow redistribution of the bulk charges results in several effects such as spontaneous backswitching after removing of external field, existing of bias fields in the hysteresis loops and instability of switching characteristics. The bulk screening kinetics determines the motion of the charged domain walls usually existing in real as-grown domain structure.

2.5

As-Grown Domain Structure

The specific initial state (so-called virgin or as-grown domain structure) is forming during zero field cooling below Tc. Its parameters and geometry are determined by the evolution of heterophase state existing in the vicinity of Tc in real samples (Kamzina and Korzhenevskii, 1992; Shur, Negashev, Rumyantsev, Subbotin, Demina and Naumova, 1997). In such structure the choice between possible orientations of spontaneous polarization in the isolated volumes of polar phase is determined by the local electric fields and mechanical stresses produced by the structural defects. As a result of zero field cooling the microscaled as-grown domain structure with charged domain walls evolves. For example, the periodical macrodefects (growth layers) existing in ferroelectric single crystals leads to formation of the regular domain structure with charged domain walls (Shur, Rumyantsev and Subbotin, 1993). Being metastable this structure nevertheless can have extremely long decay time (Shur, Guriev, Bunina, Subbotin and Popov, 1988) due to bulk screening charges which partly compensate the depolarization fields. Existence of the “freezed in” charged domain walls hinders the polarization reversal, moreover it leads to pronounced memory effect and even to complete backswitching after removing of external electric field.

6 V.YA. SHUR In the relaxor ferroelectrics with diffused phase transitions these effects are even more pronounced as the heterophase (relaxor phase) is existing in extremely wide temperature range. Examination of this phase by electron microscope revealed a multitude of isolated polar nanoregions contained in non-polar matrix (Cross, 1994; Schmidt, 1990). The local defects are responsible for arising of such structure (Dai, Xu, and Viehland, 1994). The evolution of heterophase during zero field cooling to ferroelectric phase leads to formation of nanoscale “freezed in” as-grown domain structure with charged domain walls. Despite the fact that the scale of as-grown structure in normal and relaxor ferroelectrics differs on several orders of magnitude the main details of the evolution scenario during switching will be shown to be similar.

3 SWITCHING PROCESS IN FINITE MEDIA Usually the fast switching is investigated by registration of integral characteristics such as switching current or scattered light intensity (Shur, Negashev, Rumyantsev, Subbotin, and Makarov, 1995). For interpretation of experimental data the statistical approach developed by Kolmogorov-Avrami (K-A) can be used (Kolmogorov, 1937; Avrami, 1939; Duiker and Beale, 1990; Ishibashi and Orihara, 1992). In our recent papers we proposed the modification of the K-A formulas for transformation in finite media (Shur, Rumyantsev, Makarov and Volegov, 1994; Shur, Rumyantsev and Makarov, 1995). Two limiting situations are considered usually: (Kolmogorov, 1937; Shur et al., 1994) α model, in which arising of new domains exists during whole process (α α(t) - nucleation probability per volume) and β model, when all domains are arising instantaneously at the very beginning with the density β per volume. In K-A theory the main equations used for the description of the time dependence of the fraction of the nonswitched volume q(t) runs as follows (for simplicity the phase growth rate v and α(t) are taken to be constant):

[

q( t ) = exp − ( n + 1)−1 C ⋅ α ⋅ v n ⋅ t n +1

(

q(t ) = exp − C ⋅ β ⋅ v n ⋅ t n

)

]

for α model

(3)

for β model

(4)

where n - dimensionality of domain growth, C - shape constant.

7 KINETICS OF POLARIZATION REVERSAL For interpretation of the switching in real finite objects (such as ceramics and thin films) we take into account the fact that at the moment, when the growing domains touch the boundaries of the media (grain), change of their shape constants occurs. It was shown that the influence of media boundaries leads to the time variation of the shape constant averaged over the whole media at given moment (Shur et al., 1994; Shur et al., 1995). As a result the following formula was written for β model :

[

]

q(t ) = exp − c ⋅ β ⋅ v n ⋅ t n (1 − t t m )

(5)

where tm accounts for the interaction of growing domains with boundaries of the switched volume. It was shown that for anisotropic growth the whole process must be divided into two stages. The final stage must be described by the same formula with reduced integer value of the growth dimensionality n. In other words geometrical transformation (catastrophe) occurs (Figure 1, in inset). For 2D growth in rectangular shaped volume with area S = A . L2 (A - anisotropy of sizes, L - shortest size) q(t) is as follows:

( )

exp − t t 2 ⋅ 1 + t / t , for 0 < t < t ( o1 m1 ) c     q( t ) =   exp − t t o2 ⋅ (1 + t / t m 2 ) , for t c < t < t s

[( )

]

(6)

where: tc = L / v - time of geometrical catastrophe, v - velocity of sideways domain wall motion, t o1 = ( C1 ⋅ N 0 A)

− 12

⋅ t c , t o2 = ( C2 ⋅ N 0 A)

−1

⋅ t c , C 1 ,C 2 - the shape

constants of the first and second stages, N 0 - number of remnant and backswitched −1 −1 −1 domains, t m 1 = t m + t m2 ,

−1 t m2 = ( dN / dt ) ⋅ N 0−1 , dN / dt - derivative of number of

arising domains. The anisotropy can be induced not only by non-equivalent geometrical sizes of media but also by the anisotropy of the growth rate. For β model time of catastrophe tc corresponds to the moment, when all growing domains have reached both opposite boundaries of the media.

8 V.YA. SHUR The results of computer simulation of anisotropic 2D growth in finite media are presented in Figure 1. Well fitting by Equation (6) demonstrate that proposed approach adequately describes the whole process from the beginning up to the end. 1-q(t) 1,0

β(2D)

β(1D)

2D 0,5

1D

2D 0,0 0,0

1D

0,4

0,8

1,2

TIME

Figure 1 Time dependence of the new phase fraction obtained as a result of computer simulation of phase growth in 2D finite media. Results of computer experiments (circles) are fitted by Equation (6). In inset - scheme of anisotropic phase growth in finite media.

4 EVOLUTION OF THE BIAS FIELD DURING PERIODICAL SWITCHING The measurements of hysteresis loops in ferroelectrics always demonstrate the unipolarity of switching process which can be measured in terms of the bias of the loop.

(

Eb = 0.5 ⋅ Ec+ + Ec−

)

(7)

It was shown by us earlier that during long time periodical switching the bias field in PGO single crystals slowly decreases. The fact that the relaxation demonstrate nonexponential behaviour was pointed out. The investigated single crystalline plates of the uniaxial ferroelectric lead germanate PGO (thickness about 0.5 - 1.0 mm) and improper ferroelectric-ferroelastic gadolinium molybdate GMO (thickness 0.4 mm) were cut perpendicular to polar axes and polished. Measurements of the hysteresis loops were done in traditional manner at 50 Hz. The chosen amplitude of external ac field was enough for the polarization reversal of the whole volume of the sample at chosen temperature. Before the investigation of the bias

9 KINETICS OF POLARIZATION REVERSAL field relaxation the sample was staying in the single domain state for 20 hours at the proper temperature. The non-exponential behaviour of Eb decay in PGO and GMO is shown in Figure 2. Eb GMO

600

τ = 38000 s

400

Eb PGO

200

τ = 8000 s 100

0

2000

4000

6000

TIME (s) Figure 2

Relaxation of bias field during ac switching in single crystalline plates of gadolinium

molybdate GMO and lead germanate PGO. Experimental points are fitted by Equation (10)

It was proposed that the bias field appears a result of bulk screening of residual depolarization field. The bound trapped charges are the sources of the field. During periodical switching the depolarization field averaged on time is about zero in the whole sample. In this situation decay of bias field take place due to spatial redistribution of the bound charges, in other words the current in the bulk. dEb / dt − j ( Eb ) = 0

(8)

For calculation of the time dependence of Eb one must take into account the nonlinearity of the current voltage dependence in ferroelectrics. j ( E ) = aE + bE 2 The following time dependence of Eb will be obtained

(9)

10 V.YA. SHUR

[

]−1 + Eun

Eb ( t ) = A ⋅ B ⋅ exp(− t / τ ) − 1

[

where A = a / b , B = 1 + ( a / b) ⋅ Eb (0) − Eun

]−1 , τ = 1 / a , E

un

(10)

- the static unipolar field.

Equation (10) was used successively for approximation of experimental data in PGO and GMO from the beginning to the end of decay process (Figure 2).

5 “SWITCHING” AND “BACKSWITCHING” IN RELAXOR CERAMICS Elastic light scattering technique was used for the investigation of switching behaviour in transparent ceramics because it is extremely sensitive to spatial inhomogeneities of polarization (Ivey and Bolie, 1991). Having high time resolution it allows to obtain the statistical information about heterophase patterns at any moment from instantaneous angular dependence of scattered light. Moreover the light scattering characteristics of relaxors drastically change under the action of electric field. Direct observation using high resolution TEM shows that the typical sizes of the nanoregions in relaxor phase are about 10 - 30 nm (Cross, 1994; Kleemann, Bianchi, Bürgel, Prasse and Dec, 1995), so the scattering of visible light at the individual regions can not be registered (Ivey and Bolie, 1991). It was proposed that spontaneous polarization in the individual nanoregions is randomly oriented along the allowable crystallographic directions and value of averaged “macropolarization” is near zero. Under the action of electric field during “switching” the spontaneous polarization in nanoregions is aligned.

In relaxor phase as compare with ferroelectric one the

backswitching after removing of electric field to the initial disorder state is more pronounced (Cross, 1994; Kleemann et al., 1995). It was proposed that stability of disorder state is due to the quenched random fields of defect nature acting onto the local polar order parameter (compare it with the role of trapped bulk charges in normal ferroelectrics) (Kleemann, 1993). It is known that birefringent scattering on the optically anisotropic regions is the main light scattering mechanism in the dense ceramics (Ivey and Bolie, 1991). The thermally depoled (zero field cooled) PLZT ceramics is practically transparent in ferroelectric phase due to the existence of as-grown nanoscale domain structure with random

11 KINETICS OF POLARIZATION REVERSAL orientation of spontaneous polarization. The visible light scattering which depends on the scattering particle size to wavelength ratio is negligible in this case. After applying of strong enough electric field the birefringent scattering strongly increases due to irreversible realignment of domain structure. I (arb.units) I max 20

PLZT 8/65/35 T = 60 o C

15 10

I min 5 E = 0 E = 13 kV/cm 0

1

E=0

2 TIME (ms)

3

Figure 3 Time dependence of scattered light intensity during application of electric field pulse.

In this case the scattering is determined by the difference in the orientation of refractive index ellipsoid depending on the relative orientation of polarization in the neighbouring domains/grains (Ivey and Bolie, 1991; Krumins, Shiosaki and Koizumi, 1994). The same consideration can be applied for the explanation of the drastic reversible change of the light scattering characteristics under the action of electric field in relaxor phase having the random orientation of polarization in polar nanoregions in initial state. The microvolumes combining the number of neighbouring nanoregions with aligned polarization arising during switching (Figure 4) are equivalent to the ferroelectric domains. Therefore their kinetics can be analyzed within the above mentioned approach. The thin plates (thickness about 100 mm) of coarse grained hot-pressed optically transparent ceramics of PLZT X/65/35 (X = 6 - 8 at.% La) covered by the transparent

12 V.YA. SHUR (a)

(b)

E=0

in field ( t < ts)

(c)

in field ( t > ts)

Figure 4 Scheme of evolution of heterophase structure existing in relaxor phase far above Tf under the action of electric field.

Black and white points represents the polar nanoregions with different

orientation of polarization in non-polar matrix. The “cluster” is marked on (b). ts - time of “switching”.

electrodes of indium oxide were investigated in wide range of fields and temperatures. The time dependence of scattered laser beam intensity was measured for fixed angle during application of the sequence of rectangular field pulses. Successive measurements in the angle range from 1 to 20 degrees allows to obtain the instantaneous angular dependence of scattered light due to detail reproducibility of data at the different cycles. Under the application of field pulse one can observe two processes: 1) “switching” increasing of scattered light intensity in the field and 2) “backswitching” - the gradual diminishing of scattered light to initial value after removing of the field (Figure 3). The reproducibility of the process during periodical pulse switching far above the point of transition to ferroelectric phase demonstrate that initial relaxor state is quite stable during the time of measurement.

5.1 Temperature dependence of scattered light intensity During “switching” the intensity of scattered light changes from minimum (residual) value Imin to maximum Imax with amplitude ∆ I = Imin - Imax (Figure 3). The temperature dependence of all these quantities is presented on Figure 5. It is seen that pronounced increase of field induced light scattering ∆ I is observed during cooling in the temperature range from 120 to 45 OC. At T = 45OC field induced change ∆ I abruptly decreases and residual value of scattered light Imin increases (Figure 5). The temperature hysteresis of about 1.5 K is obtained (Figure 6).

13 KINETICS OF POLARIZATION REVERSAL I (arb.units) PLZT 15

COOLING

8/65/35

I min I max

10

∆I

5 0

40

60

80

100 o

TEMPERATURE ( C ) Figure 5 Temperature dependence of scattered light integral intensity under application of the train of unipolar pulses (E = 10 kV/cm).

∆I PLZT 8/65/35 100

E = 8.8 kV/cm field induced

50

I max in field

160 140

COOLING HEATING

Imin

background

100 50

42

44

46

48 o

TEMPERATURE, C Figure 6 Temperature hysteresis of scattered light integral intensity during heating and cooling in the vicinity of transition to ferroelectric phase.

14 V.YA. SHUR

dI/dT 0 PLZT 8/ 65/ 35 d = 90 µm

-1 .

ε 10

(a)

-3

16

tgδ HEATING 0,20

12

0,15

tgδ ε

8

(b) 30

Tf

60

0,10

90

120 T c TEMPERATURE ( C) o

Figure 7 Temperature dependence of (a) derivative on temperature of scattered light intensity and (b) dielectric permittivity and losses measured at 30 kHz.

Comparison of the temperature dependencies of light scattering and dielectric data (Figure 7) demonstrate that first critical temperature coincides with temperature of dielectric permittivity maximum (Tc ~ 120OC), which is commonly attributed to the transition from paraelectric phase to relaxor one. Low critical temperature is due to transition to ferroelectric phase (Tf = 45OC). It is confirmed by singularities on the temperature dependencies of dielectric permittivity and losses which are observed only during heating of the sample after its polarization in electric field (Figure 7b). This critical point is the most pronounced on the derivative of scattered light intensities on temperature (Figure 7a). For explanation of obtained results we propose that light scattering mechanism in relaxor phase ( Tf < T < Tc) is also due to the birefringence as in ferroelectric phase (Ivey and Bolie, 1991; Krumins, Shiosaki and Koizumi, 1994). Without external field there exist the polar nanoregions with randomly oriented polarization (Figure 4a). The value of polarization averaged over the volume of grain is negligible. Under the action of external field the process of aligning of the polarization directions in different regions of one grain occurs.

15 KINETICS OF POLARIZATION REVERSAL During “switching” the macroscopic “clusters” of neighbouring polar nanoregions with similar direction of polarization (microregions with correlated polarization) are organizing (Figure 4b). The scattering on the boundaries of such “clusters” arises, when its sizes exceeds the threshold value for the elastic light scattering. Finally, for sufficiently strong field and long enough time (more then “switching” time), the complete “switching” of all polar regions occurs (Figure 4c), leading to the maximum value of scattered light Imax at given temperature. The mechanism of scattering on the grain boundaries is dominated in this case. The increase of Imax during cooling is due to increasing not only of the number and volume of polar regions, but also of the value of spontaneous polarization in given polar region.

5.2 Critical slow-down of “backswitching” process The pronounced slowing of “backswitching” process is observed during cooling in the vicinity of the transition to ferroelectric phase (Figure 8). I (arb. units)

COOLING

PLZT 8/65/35 E = 8.8 kV/cm ∆ tpulse= 2 ms

150

6

6

5 100

5

4 50

4 3 2

1,2,3

1

0 field on

50 off

100

TIME (ms)

Figure 8 Time dependence of the scattered light integral intensity during application of the set of electric field pulses at various temperatures just above point of transition to ferroelectric phase. T,oC: 1 - 47.0, 2 - 46.0, 3 - 45.5, 4 - 45.0, 5 - 44.5, 6 - 42.0

16 V.YA. SHUR The temperature dependence of the time constant of “backswitching” process demonstrate the critical slow-down in the interval about 2 K above Tf (Figure 9).

τ 10000

PLZT 8/65/35 E = 8.8

∆ tpulse= 2 ms

1000

100

COOLING 10 45

46

47

TEMPERATURE ( o C)

Figure 9 Temperature dependence of backswitching time constant just above Tf

For explanation of this effect we propose the following temperature dependent mechanism of stabilization of the polarized (scattering) state induced by external electric field. Let us analize the nature of “backswitching field” in the whole temperature range of existence of the relaxor phase. Far above Tf every grain contains the number of randomly oriented isolated polar nanoregions (Figure 4). Their sizes are too small to be screened by any mechanism. The growth of their sizes during cooling leads to the (ms)

formation of microregions (clusters) of polar phase. The value of spontaneous polarization averaged over the volume of any cluster is still negligible. This clusters can kV/cm be viewed as polydomain regions being made of the nanodomains with random

orientation of the spontaneous polarization (Figure 10a). The clusters can be switched into single domain state in strong enough electric field. The bound charges existing on the boundaries of these field induced single domain clusters produce depolarization field. The partial compensation of this field during application of external field is achieved through bulk screening (Shur et al., 1984; Fridkin, 1980), as their sizes are large enough to be screened by bulk charges.

17 KINETICS OF POLARIZATION REVERSAL

(a)

E=0

in field (t > t s)

E=0

in field (t > t s)

(b)

Figure 10 Scheme of evolution of heterophase structure in the relaxor phase under the action of electric field at different temperatures: a - in the vicinity of Tf, b - just above Tf. Black and white points represent the polar nanoregions with different orientation of spontaneous polarization, black and white squares - different domains, grey area - non-polar matrix.

This is very slow process because its kinetics is limited by bulk conductivity. Nevertheless under application of long enough pulses of electric field the bulk screening field can be achieved. As a result the complete “backswitching” to the initial random state after removing of external field is hindered due to pronounced compensation of depolarization field. Near the transition to ferroelectric phase the sizes of polydomain microclusters are of the order of the grains (Figure 10b) and fast external screening due to the current along the grain boundaries allows to obtain the stabilization of single domain state even in short pulses. The role of external screening rapidly increases while approaching the transition to ferroelectric phase. After complete transition to ferroelectric phase only irreversible “switching” from transparent to scattering state takes place.

5.3 Kinetics of the “switching” and “backswitching” processes The proposed approach was applied for the treatment of the scattering data (Shur et al., 1995) registered during the “switching” and “backswitching” processes (Figure 11). We

18 V.YA. SHUR assume that the instantaneous value of the total scattered light intensity is proportional to the fraction of the grain volume occupied by the “clusters” consisting of nanoregions with correlated polarization. The fitting shows that evolution of heterophase during “switching” is divided in two stages with pronounced geometrical (topological) catastrophe β (2D) → β (1D) (Shur et al., 1994; Shur et al., 1995). I (arb.units) PLZT 8/65/35 d = 90 µm

80 60

field on 40

2D 1D

0,0

0,3

0,6

0,9

1,2

TIME (ms) Figure 11 Field induced growth of scattered light intensity during “switching” (E = 6.1 kV/cm, T = 60oC). Experimental points are fitted by Equation (6). V (arb.units)

100

PLZT 8/65/35

β (1D) growth

10

5

6 7 8 FIELD (kV/cm)

Figure 12 Field dependence of the 1D growth velocity of “clusters“ at the final stage of switching (T = 60oC). Experimental points are fitted by Equation (11).

It means that kinetics of “switching” is controlled by anisotropic growth of “clusters” arising in finite grain at the very beginning of the process (β model).

19 KINETICS OF POLARIZATION REVERSAL The field dependence of cluster growth velocity can be obtained from the voltage dependence of time constant to presented in Figure 12, which is well fitted by formula for field activated process:

[

]

V ( E ) ~ 1 t o ( E ) ~ exp a( E − Eth )

(11)

We attribute the threshold field Eth to the quenched bulk screening field.

I (arb.units) 16

PLZT 8/65/35

8 4

∆t = 3.1 ms

2

D 1 = 1.55

1

D 2 = 2.52

0,5

1

2

ANGLE (degree)

Figure 13

Instantaneous angular dependence of scattered light intensity during “backswitching”

(E = 6.3 kV/cm, T = 75OC). Experimental points are fitted by Equations (12) and (13).

5.4 Fractal aspects of the growth kinetics The most interesting information about the evolution of heterophase structure was obtained from the measurement of the sequence of instantaneous angular dependencies of scattered light. It is known that for light scattering on fractal objects the following dependence occurs (Feder, 1988): I sc ~ q - D ,

for bulk fractals

(12)

I sc ~ q D - 6 ,

for surface fractals

(13)

where q = (4π π /λ λ ) sin(Θ Θ /2) - wave vector, D - fractal dimensionality, Θ - scattering angle. Such treatment of the sequence of experimental instantaneous angular dependence data registered at different angles (Figure 13) allows to obtain the time dependence of the

20 V.YA. SHUR fractal dimensionality of clusters. It shows that heterophase kinetics during “switching” can be analysed as evolution of the fractal objects. The tendency of the changes of fractal dimensionality during “switching” (Figure 14) is similar to the behaviour observed while investigating the growth of the fractal objects in the other kinetic processes (Feder, 1988; Russ, 1994).

D 1,6 PLZT 1,2

8/65/35 T = 60 oC

0,8

E = 0 E = 13kV/cm

0

1

E=0

2 3 TIME (ms)

4

Figure 14 Evolution of fractal dimensionality under application of electric field pulse, extracted from light scattering data at small angles.

6

SUMMARY

In this paper we describe the switching process in normal and relaxor ferroelectrics under the unified approach. Within this approach we analyze the light scattering experimental data and

propose the following scenario of heterophase evolution in relaxor

ferroelectrics during cooling and under action of the rectangular field pulses. 1. During cooling the fraction of polar phase increases in relaxor phase ( Tf < T < Tc ) by growth of polar regions and reaches unit in ferroelectric phase ( T < Tf ). Formation of the multidomain microclusters of polar phase occurs near Tf, when the fraction of polar phase is large enough. Below Tf the nanoscale multidomain structure with chaotically oriented polarization in different domains occurs.

21 KINETICS OF POLARIZATION REVERSAL 2. In relaxor phase far above Tf the field induced “switching” to final “scattering stage” is due to alignment of spontaneous polarization of all polar nanoregions in individual grain. During “switching” the total intensity of scattered light is proportional to fraction of the grain volume occupied by the growing “clusters” consisting of the neighbouring polar nanoregions with similar orientation of polarization. The angular dependence of scattered light intensity reveals their fractal geometry. 3. Spontaneous “backswitching” to random oriented (transparent) stage after removing of external field is due to the action of the depolarization field produced by bound charges of polarized clusters in the vicinity of Tf. Far above Tf the role of the frozen-in fields of defects prevails. The redistribution of screening charges is very slow as compare with the time of measurement. This leads to the re-establishing of the initial random relaxor state after removing of external field. 4. Charged domain walls in multidomain polar clusters disappear under the action of strong electric field. So only partial “backswitching” occurs. The role of external screening due to the current along the grain boundaries rapidly increases while approaching to Tf that allows to obtain stabilization of the microdomain structure. In ferroelectric phase (below Tf) the domain structure without charged domain walls can be formed in electric field by irreversible switching. 5. The non-exponential behaviour of bulk screening effects is caused by nonlinearity of the current voltage dependence in ferroelectrics.

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