INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS D: APPLIED PHYSICS
J. Phys. D: Appl. Phys. 38 (2005) 4145–4152
doi:10.1088/0022-3727/38/23/001
Kinetics of ferroelectric switching in unipolar (CH3NH3)5Bi2Br11 crystal R Z Rogowski1,3 , K Matyjasek1 and R Jakubas2 1
Institute of Physics, Technical University of Szczecin, Al. Piast´ow 48/49, 70-310 Szczecin, Poland 2 Institute of Chemistry, University of Wrocław, F Joliot-Curie 14, 50-383 Wrocław, Poland E-mail:
[email protected]
Received 8 January 2005, in final form 5 September 2005 Published 18 November 2005 Online at stacks.iop.org/JPhysD/38/4145 Abstract Ferroelectric switching in unipolar (CH3 NH3 )5 Bi2 Br11 crystal has been studied by optical observation of the domain structure in low electric fields and switching current registration in high fields. We discuss investigations on the temporal behaviour of the electric polarization in static electric fields as well as polarization decay after removal of the electric field. The backswitching process appears to have a correlation to a defect-induced internal field in the crystal. For the interpretation of experimental data we utilize the classical nucleation and growth model of phase transformation (e.g. Avrami–Kolmogorov theory). An analysis assuming a statistical distribution of the characteristic domain growth times is proposed to explain and to find the functional form of the experimental data.
1. Introduction The mechanism of polarization reversal in ferroelectrics has been intensively discussed in recent years due to potential use of ferroelectric thin films as nonvolatile memory elements [1]. An important area of the study of ferroelectrics is polarization switching, because coercivity, remanence and susceptibility are determined by the switching process. The polarization reversal process, which proceeds by nucleation and growth of domains, has attracted much attention also from the viewpoint of statistical physics related to solid-state transformations. Random nucleation and isotropic growth in an infinite specimen is well described by the Kolmogorov and Avrami model [2, 3]. Ishibashi and Takagi modified the model in such a way as to be convenient for its use in application to ferroelectrics (KAI model) [4]. By analysing switching current data with the KAI model, the authors have been able to extract information concerning the behaviour of the switching kinetics with varying degree of success. The theory describes the switching data as well as the hysteresis loop very well for many single crystals [5–7]. But it has not been completely satisfactory in describing the polarization reversal in ferroelectric thin films [8, 9], 3
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ceramics [10] and defected crystals [11], as it can give physically unclear values of fitting parameters. Moreover, the data analysis has revealed no integer dimensionality of the domain growth, contrary to the Avrami equation restricted to integer values. Recently, several attempts have been made to solve the finite-size problem [12–14]. Contrary to polarization reversal under the influence of an electric field, comparatively less has been reported on the mechanism that controls the spontaneous reversal of polarization, which occurs in the absence of an external field, and is driven by internal field [15–17]. In this report we present the results of investigations of the kinetics of ferroelectric switching in unipolar (CH3 NH3 )5 Bi2 Br11 (MAPBB) single crystal. We discuss the investigations on the temporal behaviour of the electric polarization in dc-electric field as well as polarization decay after removing of the electric field. The switching process was investigated by direct observation of the domain structure and by switching current registration in high electric field ranges. For the interpretation of experimental data we propose a modified version of the KAI model which enables us to describe the switching kinetics in non-uniform crystals. An analysis assuming a statistical distribution of the characteristic domain growth times is proposed to explain and to find the functional form of the experimental data.
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2. Theoretical background The Kolmogorov–Avrami statistical model describes the phase transformation kinetics in infinite media and was originally developed to model the crystallization process in metals. For the case of ferroelectric switching, according to the derivation of Ishibashi and Takagi, the model describes a growth of the volume fraction of the aligned domains that can be expressed by the formula n t , (1) Q(t) = 1 − exp − t0 where t is the elapsed time since the application of the electric field, t0 and n are constants. Additionally, the switching current density determined from the derivative of the charge response i = 2P (dQ(t)/dt) can be written as n t 2P n t n−1 , (2) exp − i(t) = t0 t0 t0 where P is the polarization. The characteristic domain growth time t0 is mainly determined by the nucleation rate, the wall mobility and the applied field. The value of n depends on the dimensionality of the domain growth d (which can take integer values of 1, 2, 3) and the mechanism of nucleation. In switching phenomena two mechanisms of nucleation can be distinguished. Heterogeneous nucleation occurs in the vicinity of defects, thus effective dimensionality is n = d. In homogeneous nucleation the nuclei continue to occur at a constant rate during switching, thus n = d +1. The KAI model is a simplification of the real physical conditions assuming a spatial uniform nucleation probability and a constant domain growth velocity. A realistic model describing ferroelectric switching should take into account the initial distribution of the nucleated centres and the dependence of the rate of growth of a particular domain on the state of the crystal (i.e. impurities, defects, etc). In order to find out the functional form of polarization P (t) in non-uniform crystals, we assume that the crystal consists of many regions, which have independent switching kinetics. The complex polarization relaxation P (t) can be expressed as an integral of KAI function equation (1) weighted by a distribution function of characteristic domain growth times g(τ, t0i ): n t dt0i . (3) P (t) = g(τ, t0i ) 1 − exp − t0i From optical observation of the evolution of the domain structure, during polarization reversal, it has been found that the spectra of relaxation times can be described by the Gaussian distribution function C (τ − t0i )2 , (4) exp − g(σ, τ, t0i ) = √ 4σ 2 2πσ with a variance σ . The τ is the averaged characteristic domain growth time and t0i is the characteristic time for the local switching. The averaging over a broad spectrum of relaxation times is widely used in the theory of disordered systems [18]. Our 4146
approach is close to that presented by Tagantsev et al for the description of the switching kinetics in PZT thin films [19]. It was assumed that in thin films the switching is limited by the nucleation of reversed domains rather than by the sidewise motion of domain walls; therefore the switching kinetics was described in terms of the distribution function of the nucleation probabilities. Another approach for investigation of the polarization reversal in inhomogeneous media was developed by the group of Shur [20, 21]. It has been shown that analysis of the switching current data, recorded during quasi-static switching in slow increasing field, enables to extract the distribution function of the local internal bias fields and their evolution during polarization reversal in ferroelectric thin films.
3. Experimental The MAPBB crystal crystallizes from aqueous solution at room temperature in orthorhombic symmetry (space group Pca21 ) with the c-axis being the polar one [22]. The crystal undergoes two second order phase transitions, at Tc = 312 K from paraelectric phase (space group Pcab ) to ferroelectric phase and at 77 K the structural phase transition. The loss of polarity at Tc is related to the motion of the methylammonium cations [23]. To observe the optically indistinguishable 180˚ domains walls, the nematic liquid crystal (NLC) decoration technique was used. For the observation of domains the NLC mixture of p-methoxybenzylidene-p-n-butylaniline (MBBA) and pethoxybenzylidene-p-n-butylaniline (EBBA) was used. For dynamic observation in an electric field a crystal plate (cut perpendicularly to the polar axis) with a thin NLC layer on its upper and lower surface was sandwiched by two glass plates with a tin oxide coating as transparent electrodes. The observation of domains was carried out using a polarization microscope. The switching currents were measured by applying square wave electric pulses, which are amplified with a Kepco bipolar amplifier model BOP 500. The voltage across the resistor 200 , connected in series with the sample, was measured using digital oscilloscope. Hysteresis loop was obtained by means of modified Sawyer–Tower bridge using ac-electric field of 50 Hz. The electrical measurements were carried out with air-drying silver paste as electrodes.
4. Results 4.1. Dynamics of the domain structure on the seconds time scale Here we report the presence of an in-built internal field Eb in crystal which strongly influences the electric field switching of domains. The examined crystal sample reveals an initial static unipolarity that means an initial predominant direction of the spontaneous polarization, which may be due to the characteristic features of the growth of the crystal. Annealing of the crystal above the Curie temperature did not remove the unipolarity. Unipolarity is a stable property, but the value of Eb diminishes in a freshly annealed crystal sample. The results are presented for an aged crystal sample, a few days after the annealing.
Kinetics of ferroelectric switching
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Figure 1. The polarization hysteresis loop of unipolar MAPBB crystal. Vertical scale is 0.57 µC cm−2 per large div., horizontal scale is 100 kV m−1 per large div.
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Figure 2. Sequence of photographs showing domain pattern evolution observed on the ab plate of the crystal (surface area 6.4 mm2 ): (a) switching process in electric field of 100 kV m−1 and (b) backswitching process after removing of the electric field. Polarization reversal time: (a) 4.8 s, (b) 17.4 s.
Figure 1 presents the polarization hysteresis loop obtained by applying ac field of 320 kV m−1 and frequency 50 Hz. The large asymmetry of the hysteresis loop about the electric field axis indicates the presence of internal field Eb , defined as the shift of the hysteresis loop on the E axis. By applying dc-electric field the crystal was switched from its original, monodomain state to the reversed state (switching process) and the domain configuration was recorded as a function of time. Figure 2(a) presents the sequence of photographs showing domain configuration during the switching process in the electric field of 100 kV m−1 . In such an electric field an unstable state of polarization is created, which relaxes towards the original state after removing of the electric field, as illustrated in figure 2(b). The backswitching process, driven by the Eb , occurs very quickly (on the time scale of milliseconds) in the part of the crystal with high spatial density of backswitching domains and more slowly (on the time scale of seconds) in the right-hand side of the images in figure 2(b), where the rearrangements of the domain structure
Figure 3. Distribution of ferroelectric domains in two various electric fields, observed on the small surface area (∼1 mm2 ) of the crystal: (a) 75 kV m−1 , (b) 90 kV m−1 .
occur by domain wall motion. This is a result of spatially non-uniform internal field distribution. Hence, it seems reasonable to assume that the dc field and inhomogeneous Eb add algebraically to determine the effective switching field at a given site of the crystal. It has been found that the intensity of the backswitching process depends on the time the crystal has been left sitting in the electric field as well as the value of the applied field before its removing. This will be discussed in detail in another publication. To study the polarization relaxation phenomena in greater detail, we carried out observations of the domain configurations on a small (∼1 mm2 ) surface area, located in the centre of the image presented in figure 2. A video camera with a 40 ms resolution clock was used to monitor the domain pattern evolution. A typical view of domains nucleated at two different electric field strengths, observed on a small fragment of the ab plate of the crystal, is shown in figure 3. We observed the following qualitative features as the electric field strength was increased. The domains nucleate sooner and in greater number per unit area. The domains nucleate at the same sites for successive electric fields. This means that the latent nuclei are brought about by some defects existing in the crystal. Thus, the domain nucleation is a locally heterogeneous process. The spatially non-uniform distribution of the domains has been observed even at high electric field strength. It means that nucleation probability and consequently the domain growth depend on the local value of internal field. Inspection of the domains, on the opposite polar faces, showed complete penetration of the domains throughout the crystal bulk. Since the velocity of growth of cylindrical domains through the thickness is much larger than along its radius, the problem can be reduced to a two-dimensional one. The dynamics of polarization relaxation was determined by measuring the fraction of the reversed region as a function of time. As shown in figure 4(a), the fraction of the switched area was found out to be well described by KAI function, equation (1) (full curves). However, parameter n which is related to the actual dimensionality of the domain growth, is not consistent with that predicted with the theory (n = 2 for cylindrical domains in the case of heterogeneous nucleation). It seems reasonable to assume that the crystal consists of many regions which have different switching kinetics. The characteristic time t0 of any given small region is a function of the local field in such a region. The distribution of t0 would therefore arise from inhomogeneities in Eb distribution within the crystal sample. The probable spectra of the distribution of characteristic domain growth times can be reconstructed from data, obtained 4147
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Figure 4. Polarization relaxation in various electric fields (a) and corresponding distribution spectra g(σ, τ, t0i ) of characteristic domain growth times (b).
from observation of the domain pattern evolution during the switching and backswitching process. Figure 5 shows the series of video frames, recorded at successive time intervals, illustrating the domain pattern evolution during the switching process at E = 75 kV m−1 and backswitching process after removing of E. To construct the spectra of the domain growth times, each video frame was divided into 60 squareshaped regions and the switching time of each small region was measured until it was covered by the growing domains. Figure 6 shows the distribution function acquired during the switching and backswitching processes. The experimental data (symbols) are best approximated by Gaussian distribution function (broken curve), as given by equation (4). The distribution function can also be obtained by fitting the experimental data P (t) (as given in figure 4(a)) with modified KAI function (equation (3)), taking σ and τ as variable parameters in the function g(σ, τ, t0i ). Figure 4(b) presents the obtained distribution functions for various values of electric fields, assuming the dimensionality of the domain growth n = 2. As shown in figure 6, there is a good correlation between the distribution function obtained by the fitting (full curve) and that obtained from direct measurements of switching times (broken curve). It must be noted that the distribution function obtained by the fitting is centred around the averaged characteristic domain growth time τ , whereas the spectrum obtained by optical observation is centred around the averaged switching time. Figure 6 presents the latter one after transformation on the time axis. In order to compare the results obtained by optical observations with those obtained by switching current 4148
Figure 5. Selected video frames in video recording of domain pattern evolution during (a) switching process in E = 75 kV m−1 and (b) backswitching process after removing of E. Number of regions with a fixed t0i
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Figure 6. Curves of the Gaussian distribution functions g(σ, τ, t0i ) during switching process in E = 75 kV m−1 and backswitching process after removing of E.
registration (on the milliseconds time scale) the measured polarization versus time (P –t) were converted to the switching current density versus time (i–t), as shown in figure 7. The experimental data were fitted by the KAI function equation (1) (broken curves), with exponent n = 1.73 and characteristic time t0 = 3.56 s for switching process in the electric field (figure 7(a)) and n = 1.45, t0 = 0.44 s for spontaneous backswitching process (figure 7(b)). The full curves are the best fittings, in the whole time range, by the function given by equation (3) and taking the value of exponent n = 2, which is related to the actual dimensionality of the domain growth (see figure 5). The distribution function g(σ, τ, t0i ) obtained by fitting with equation (3) is centred around τ = 3.46 s for switching and τ = 0.99 s for backswitching process (see figure 6). Thus, the value of τ is close to the characteristic
Kinetics of ferroelectric switching
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Figure 7. Time dependence of the switching current density: (a) in electric field of 75 kV m−1 and (b) for the decay of the polarization at E = 0.
time t0 in the case of the switching process and differs considerably from t0 (0.44 s) for the backswitching process. As is seen in figure 7 the switching current curve is more symmetrical than the backswitching one. It has been discerned by optical observations that τ reflects the time in which the majority of domains begin to coalesce. Figure 4(b)) characterizes the changes in the distribution function with external electric field. The increase of the electric field strength diminishes the broadening of the distribution function. The spectra shift as a whole together with their maximum at τ to smaller values of time on increase of the electric field strength. 4.2. Dynamics of the polarization relaxation on the milliseconds time scale In this section validity of the approach will be verified for high speed switching process, involving a large number of domains. The switching current registration was made for the same unipolar crystal sample, with the electrode area of 9 mm2 and a thickness of 1 mm. The existence of Eb in the crystal sample is manifested by switching current registration, by applying a train of bipolar pulse sequence to the sample as shown in figure 8(a). Figure 8(b) presents the trace of the corresponding switching current. One can see that after the removal of the applied positive voltage pulse, spontaneous depoling process occurs, indicated by negative current response. Figure 9 illustrates the switching current density versus time at E = 260 kV m−1 . The experimental data were fitted by the KAI function as given by equation (2) (broken curve)
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Figure 9. Time dependence of the switching current density at E = 260 kV m−1 . The broken curve shows the KAI function (equation (2)) with n = 2.29; full curve shows modified KAI function with n = 2. The inset shows corresponding Gaussian distribution function acquired by computer simulation.
with the exponent n = 2.29. The full curve is the best fit in the whole time regime, using representation determined from the derivative of P (t) given by equation (3), and the exponent n = 2. It means that the dimensionality, which represents the behaviour of the domain growth, is unchanged as compared to its value for slow polarization reversal. The inset of figure 9 shows the Gaussian distribution deduced from the parameters obtained by applying the fitting procedure to the switching current curve. The results for the backswitching current density (under no external field) are shown in figure 10. The quality of the fit is not good, especially in the case of the backswitching process. The backswitching current curve with its maximum centred on the time axis in the vicinity of zero cannot be represented by symmetrical Gaussian function (equation (4)). In the last stage of this process, the slowly decreasing backswitching current density is a result of the highly inhomogeneous distribution of domain nuclei. As shown in figure 2(b), some regions of the sample are switched very rapidly, whereas others (where the nucleation is restrained) are switched slowly. The slow sidewise motion of domains cannot play a role on milliseconds time scale. 4149
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Figure 10. Time dependence of the backswitching current density after removing of the electric field of E = 240 kV m−1 . The broken curve shows the KAI function (equation (2)) with n = 1.35; full curve shows modified KAI function with n = 2. The inset shows corresponding Gaussian distribution function acquired by computer simulation.
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The switching current curves obtained for three values of electric field strength are presented in figure 11(a). The backswitching current curves obtained after removing of E are presented in figure 11(b). One can see that the amount of the switched charge, Q = i dt, increases distinctly with the electric field strength. The increase of E from 150 to 260 kV m−1 causes nearly twice the increase of imax , whereas switching time ts , changes insignificantly. This effect, typical in the case of not full switching process, is the result of nonuniformity of the domain nuclei distribution, taking place even in high electric fields. It must be noted, that when the complete switching process occurs imax ∼ 1/ts [24]. The total charge switched during a complete reversal of the polarization is 4150
switching backswitching
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Figure 12. Electric field dependence of the polarization in unipolar MAPBB crystal sample.
expressed by the formula Q = i dt = 2Ps S, where Ps is the spontaneous polarization and S is the electrode area. As can be seen in figure 12 the polarization increases distinctly with the electric field strength and levels off to a constant value for high fields, at E ∼ 210 kV m−1 , comparable to coercive field determined from the hysteresis loop. The measured value of Ps ∼ 1.75 µC cm−2 is comparable to the spontaneous polarization value 1.4 µC cm−2 obtained from pyroelectric measurements. The switched charge calculated under the backswitching current curves increases with increasing E until E ∼ 220 kV m−1 and then decreases. It means that the role of the internal field diminishes as E values are strong enough to involve in reversal most of the quantity of domains with stabilized Ps . Thus, the spontaneous backswitching process is less intensive and did not occur completely. It means that the portion of the crystal that was switched does not return to its original polarization state, after the high electric field is removed. It may be due to the fact that at high electric fields defect dipoles are switched as well, thus preventing complete backswitching.
5. Discussion It is well known that nucleation events are proportional to the defect density, because the activation energy for nucleation and domain wall motion can be lowered by the presence of impurities in the crystal [24]. Differences in the local defect density distribution in real crystal should yield a distribution of Eb and thus a distribution of activation energies. According to the Miller and Weinreich theory [25], the sidewise motion of the domains can be treated as a continuous nucleation process of domain nuclei at the existent domain wall. Thus, it is reasonable to describe the polarization relaxation process in non-uniform crystal, by a set of relaxation times. To analyse the switching process in non-uniform media in the framework of classical theory of nucleation and growth of opposite domains, one should perform the averaging over the broad spectrum of the characteristic domain growth times t0i with the smooth distribution function centred around the time τ , which signifies the range where the most of the single relaxation times lie. We have shown that this approach leads
Kinetics of ferroelectric switching
to correct assumptions about dimensionality of the domain growth in the case of a slow switching process and the index n is related to the dimension d which characterizes the shape of the reversed domains. This approach is also a good description for the time dependence of the polarization and the switching current for non-uniform crystals in the case of a slow switching process (switching time of the order of milliseconds or seconds). In this case the switching speed is determined by the low nucleation rate of steps at the domain walls. During the fast polarization reversal process the switching speed is determined by the high nucleation rate of forward growing domains (along the polarization direction). It seems that the regions with a high density of nucleated domains switches completely without stimulation of domain wall motion, when a high electric field is applied. In such a case the index n increases from two for low speed switching to three at high speed switching, as was observed in the experiments in strong electric fields and at higher temperatures (as Tc is approached) [11]. As the index n is not constant with respect to temperature and electric field, one can conclude (based on optical observation of the domain structure) that the real nucleation in MAPBB crystal is a mixed nucleation process—locally one step nucleation at the beginning of the switching process (heterogeneous nucleation), but it is apparent continuous nucleation throughout the polarization reversal, if observations are made on a large surface area of the crystal. These results may be relevant to the recent investigations of the switching kinetics in ferroelectric thin films, which have been explained in terms of a wide spectrum of waiting times for nucleation [19]. It has been discerned only by optical observations that the components of the relaxation response can be related to the size distribution of relaxing domains, as shown in figures 3 and 5. This effect is more clearly seen during the relaxation of the domains and the domain walls when the external field is removed, as presented in figure 2(b). We can distinguish two distinct time regimes of polarization relaxation: the fast one decays on the time scale of milliseconds (in the region with a large density of the nucleated backswitching domains) while the slow one in longer times (on the seconds time scale). Long time relaxation process was observed in the region of the sample with a small number of domains involved in the backswitching process. Our results pointed out that the domain structure of unipolar MAPBB crystal can resemble relaxor type behaviour, observed in disordered systems; as such crystals have a great number of metastable states, separated by energy barriers [26]. It should be noted that the domain patterns revealed by the NLC method may not thoroughly reflect the domain structure evolution in electrical switching at the same pulse amplitudes. The results of the reported works show a strong influence of the interface (electrode, ferroelectric surface) conductivity on the kinetics of polarization reversals [24]. In a metal contact, electrical charges can freely move and can effectively screen the depolarization field accompanying the ferroelectric polarization, which decreases the switching process [16]. There is little direct evidence concerning the exact nature of the internal field, caused by structural disorder, which accounts for the broad spectrum of potential barriers for
domain nucleation and growth. It has been known from the literature [27–30] that defects (e.g. dopes, impurities, non-stoichiometric defects) can occupy energetically preferred sites in the lattice and form anisotropic centres which locally favour a certain direction of the spontaneous polarization. At this moment, it is hard to explain what kind of crystal imperfections or non-uniformities are responsible for such stabilization effects in MAPBB crystal. The MAPBB crystals were grown in ferroelectric phase, under some excess of HBr and Br2 molecules may form defects. Polar properties of the crystal depend upon methylammonium cations ordering [23]. The switching phenomena should then be ascribed to the cations movement due to their free orientation ability and to their limitations due to Br2 defects. A number of regions in MAPBB crystal may have different defect concentrations than the average, as the spatially non-uniform distribution of domain nuclei has been observed during the backswitching process, governed by internal field.
6. Conclusions In the present work we have demonstrated that there are no qualitative differences in switching properties resulting from having a small number of domains and that involving a large number of domains. The KAI model has been modified to allow more reasonably the description of the switching process in real crystals, by including information concerning the dependence of the rate of growth of particular domains on the state of the crystal (i.e. impurities, defects). The proposed approach describes the relaxation of the domains with a continuous distribution of relaxation times. It has been showed, by optical observation of the domain structure, that the broadened response reflects the distribution of the domain sizes.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Scott J F and Paz de Araujo C A 1989 Science 246 1400 Kolmogorov A N 1937 Izv. Akad. Nauk 3 355 Avrami M 1939 J. Chem. Phys. 8 212 Ishibashi Y and Takagi Y 1971 J. Phys. Soc. Japan 31 506 Usher T D, Poole C P and Farach H A 1991 Ferroelectrics 120 201 Matyjasek K 2000 J. Phys. D: Appl. Phys. 33 2553 Hashimoto S, Orihara H and Ishibashi Y 1994 J. Phys. Soc. Japan 63 1601 Dimmler K, Parris M, Butter D, Eaton S, Pouligny B, Scott J F and Ishibashi Y 1987 J. Appl. Phys. 61 5467 Scott J F, Kammerdiner L, Parris M, Traynor S, Ottenbacher V, Shavabkeh A and Oliver W F 1988 J. Appl. Phys. 64 787 Levstik A, Kosec M, Bobnar V, Filipic C and Holc J 1997 Japan. J. Appl. Phys. 36 2744 Matyjasek K 2001 J. Phys. D: Appl. Phys. 34 2211 Shur V Ya, Rumyantsev E L and Makarov S D 1998 J. Appl. Phys. 84 445 Duiker M and Beale P D 1990 Phys. Rev. 41 490 Orihara H and Ishibashi Y 1992 J. Phys. Soc. Japan 61 1919 Taylor G W 1966 J. Appl. Phys. 37 593 Shur V Ya, Gruvermann A L, Kuminov V P and Tonkahova N A 1990 Ferroelectrics 111 197 Matyjasek K 2001 J. Appl. Phys. 90 5328 Courtens E 1986 Phys. Rev. B 33 2975 Tagantsev A K, Stolichnov J, Setter N, Cross S J and Tsukada M 2002 Phys. Rev. B 66 214109
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[20] Shur V Ya, Baturin I S, Shishkin E I and Belousova M V 2003 Ferroelectrics 291 27 [21] Shur V Ya, Baturin I S, Shishkin E I and Belousova M V 2003 Integr. Ferroelectr. 53 379 [22] Jakubas R 1989 Solid State Commun. 69 267 [23] Jakubas R and Sobczyk L 1989 Ferroelectrics 92 365 [24] Lines M E and Glass A M 1977 Principles and Applications of Ferroelectrics and Related Materials (Oxford: Clarendon)
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[25] Miller R C and Weinreich G 1960 Phys. Rev. 117 1460 [26] Gladkii V V, Kirikov V A, Volk T R and Ivleva L J 2003 Ferroelectrics 285 275 [27] Keve E T, Bye K L, Whipps P W and Annis A D 1971 Ferroelectrics 3 39 [28] Hilczer B 1974 Ferroelectrics 7 319 [29] Arlt G and Neumann H 1988 Ferroelectrics 87 109 [30] Gopalan V, Mitchell T E, Furukawa Y and Kitamura K 1998 Appl. Phys. Lett. 72 1981
