Heterogeneity of fatigue in bulk lead zirconate titanate

Transcription

1 Acta Materialia 53 (2005) Heterogeneity of fatigue in bulk lead zirconate titanate Yong Zhang a,1, Doru C. Lupascu a, *, Emil Aulbach a, Ivan Baturin b, Andrew Bell c,jürgen Rödel a a Institute of Materials Science, Darmstadt University of Technology, Petersenstr. 23, Darmstadt, Germany b Institute of Physics and Applied Mathematics, Ural State University, Ekaterinburg , Russian Federation c Institute for Materials Research, University of Leeds, LS2 9JT Leeds, UK Received 28 September 2004; received in revised form 19 January 2005; accepted 21 January 2005 Available online 2 April 2005 Abstract The spatial heterogeneity of fatigue in commercial ferroelectric lead zirconate titanate ceramics was experimentally investigated. All parameters measured on fatigued samples are spatially highly heterogeneous including large signal polarisation and strain hysteresis loops (remnant polarisation P r, coercive field E c, bias field E bias, and strain asymmetry c) as well as the small signal parameters determined from field dependent converse piezoelectric measurements (remnant piezoelectric coefficient d r, coercive field E c, offset piezoelectric coefficient d offset, and bias field E bias ). The local strain asymmetry c was found to linearly depend on the local bias field E bias. An analogous relation is established for d offset and E bias. Switching time retardation is similarly heterogeneous across the sample. The previously determined stretched exponential polarisation relaxation equally well occurs locally. A spatial correlation to the observed microcrack densities could not be found. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dielectrics; Ferroelectrics; Functional ceramics 1. Introduction Ferroelectric materials are extensively used in the production of sensors and actuators due to their excellent piezoelectric properties [1,2] and in thin films as ferroelectric memories [3]. Nevertheless, fatigue still poses a problem to the widespread use of this kind of material in certain applications [4,5]. In ferroelectric thin films the suppression of the switchable polarisation after continuous switching makes it difficult to distinguish between the two logical states 0 and 1, and leads to failure * Corresponding author. Tel.: ; fax: address: lupascu@ceramics.tu-darmstadt.de (D.C. Lupascu). 1 Now at: Beijing Fine Ceramics Laboratory, Tsinghua University, Beijing, PR China. of the ferroelectric random access memories (FeRAMs) cell. In multilayer actuators, microcracking and conductive breakdown paths extend under repeated electrical and mechanical loadings and eventually result in catastrophic failure. The detailed origin of fatigue, however, has remained elusive until now. Although numerous efforts have been made to model and elucidate the mechanism, only partly accepted conclusions have been drawn. Some published studies have tended to explain the fatigue as a consequence of domain wall pinning [6,7] which results from the agglomeration of point defects or isolated charged defects. Some studies believed that domain nucleation suppression at electrode ceramics interfaces [8,9] is the cause of fatigue. However, none has been supported by direct experimental evidence, also due to the fact that most studies have been carried out on thin films, where distinct access to the critical sample volumes is difficult /$30 Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi: /j.actamat

2 2204 Y. Zhang et al. / Acta Materialia 53 (2005) The implementation of piezoresponse force microscopy [10 15], a modified form of atomic force microscopy, has been demonstrated to be a powerful tool for the characterisation of the fatigue effect. It has provided for the direct observation of region by region suppression of the switchable polarisation on lead zirconate titanate (PZT) thin films during fatigue [11] and a correlation between frozen polarisation and offsets in strain behaviour [12]. The question driving this study was how far this translates to the macroscopic scale and how far mesoscopic or macroscopic heterogeneities exist in the material after fatigue. If they do, a very local instability must induce the driving force for defect rearrangement which is locally self-stabilizing. In this work, we study bipolar electric fatigue which is of primary concern in thin film devices, but at the scale of macroscopic samples in order to access the spatial distribution of material parameters. A large set of techniques is used in order to capture the different aspects of the heterogeneity effect, local polarisation hysteresis, strain hysteresis, local dielectric constant, piezoelectric hysteresis, measurements of switching dynamics and optical microscopy. In particular strain measurements (s 33 and d 33 ) [16 19] reveal local information about the fatigue of the ceramic without particular treatment of the samples (dicing of electrode). 2. Experimental 2.1. Samples We used commercially available ceramic discs of 10 mm diameter and 1 mm thickness (PIC 151, PI Ceramic Lederhose, Germany). PIC 151 is a highly compensated soft PZT ceramic of composition Pb 0.99 [Zr 0.45 Ti 0.47 (Ni 0.33 Sb 0.67 ) 8 ]O 3 with a grain size of about 6 lm. The silver electrodes were fired by the manufacturer (9.5 mm in diameter). For the large-signal polarisation and strain measurements, one side of the test samples was divided into 32 segments with parallel grid lines by splitting the top electrode layer to about 50 lm depth using a wire saw (Model 4240, Well Corporation, Germany). The diameter of the wire is 170 lm. This segmentation provides for the local polarisation measurement. The electrodes of the test samples for the small-signal piezoelectric measurements were not segmented, because the strain measurement itself is local Bipolar fatigue cycling Fatigue was induced by subjecting the samples to sinusoidal voltage cycling with amplitude of 2 kv/mm and a frequency of 50 Hz. In the cycling setup, the samples were mounted in between two spherical metal clamps and immersed in silicone oil to avoid arcing. More details on the cycling procedure can be found in Refs. [6,16] Large-signal polarisation and strain measurement Polarisation was measured using a modified Sawyer- Tower bridge with a sensing capacitor of 1 lf. The butterfly hysteresis loop was recorded simultaneously using a linear variable displacement transducer (LVDT) connected to an AC measuring bridge (Hottinger Baldwin Messtechnik, Darmstadt, Germany). The bipolar measuring field of 2 kv/mm at 40 mhz was applied by a high voltage power supply (FUG Elektronik GmbH, Rosenheim, Germany) driven by a frequency generator (HP33120A, CA, USA) Small-signal piezoelectric loop measurement The small ac signal induced strain was measured by a fibre optic displacement sensor (PHILTEC, Inc.). The output signal from the sensor was analyzed by a lock-in amplifier (Stanford Research Systems Model 830). During the measurement, a DC bias field, which was varied step by step, was applied to the samples to trace the hysteresis loop utilizing a large parallel capacitor as charge source. A small AC field was coupled into the circuit using a transformer. The frequency of the AC sinusoidal electric field is 1 khz, 3.36 V RMS. The longitudinal piezoelectric coefficient d 33 was determined from the ratio of strain and applied AC voltage. A two-tip geometry was used for strain measurement to avoid artefacts due to bending and to determine the strain values locally (Fig. 1). The tip of the sensor was placed onto the upper surface of the samples (tip radius 0.5 mm, 0.23 N). The local oscillatory displacement was measured by the optical sensor and delivered to the lock-in amplifier. Data recording and control of the applied voltages was provided by a computer. The samples were mounted on a three-dimensional moving stage. This allowed for measuring the local piezoelectric response on the entire sam- Lock-In in out Computer HV on/off off/on C source ac-amplifier sample Optical Se nso r Fig. 1. Schematic of the measuring arrangement for the small signal converse piezoelectric effect measurement.

3 Y. Zhang et al. / Acta Materialia 53 (2005) Starin [10-3 ] Polarisation [C/m 2 ] (a) E c1 (b) S - P r2 P r1 E c2 S + d 33 [ p m / V] (c) (d) 4 2 / C ] d [ p mv / ] 33 Q [ m eff (e) P [C/m 2 ] (f) -2-3 Electric Field [V/mm] DC Electric Field, E dc [V/mm ] DC Electric Field, E dc [V/mm ] Fig. 2. (a) Polarisation hysteresis (40 MHz). (b) Strain hysteresis (40 MHz). (c) Hysteresis of the piezoelectric constant measured under static bias electric fields. (d) Hysteresis of the dielectric constant under static bias electric fields. (e) Piezoelectric constant vs. polarisation. (f) Q 11 as a function of electric field calculated from the measured polarisation, piezoelectric constant, and dielectric constant. ple surface. Fig. 2 shows a set of data for a non-fatigued sample for reference Switching retardation measurement For the switching experiments after fatigue, the samples were fully poled initially (+2 kv/mm, 5 h, room temperature). Then nearly square pulses of 1, 2, 5, 10, 30, 40 s duration and 2 kv in amplitude with a rise time of about 1 ls in reverse polarity were applied to the samples. All square pulses were supplied using a high current high voltage switch again connected to a set of large high voltage capacitors serving as voltage source and controlled by a function generator (HP33120A). The d 33 value after switching as a function of position was measured using a quasi-static d 33 Berlincourt-meter (Model YE2730 SINOCERA China) with a top spherical probe of 1mm radius. The applied force is kept at 0.23 N (frequency: 110 Hz). Values were measured on a 9 9 point grid (no dicing). After measurements across the whole electrode area another reverse square pulse with longer duration was applied to test the back-switching behaviour. 3. Results 3.1. Large-signal polarisation and strain We will make extensive use of the electrostrictive properties of a ferroelectric to explain the observed heterogeneous fatigue effects. Fundamentally, the strain of a unit cell in any insulator is proportional to the square of the local electric field: SðEÞ ¼qE 2 local with q an electrostrictive coefficient. In a ferroelectric, the local electric field is predominantly determined by the local polarisation. Thus, macroscopically the strain is observed to rather follow S(E)=QP(E) 2 which is the fundamental approximation in Landau Devonshire theory, P being the polarisation and Q an electrostrictive coefficient. For a single crystal, Q would be Q 11, the first order term (tensor of rank 6) in the polynomial expansion. This relation is valid as long as no 90 domain switching occurs. For a macroscopic sample it is found to be only roughly valid (see Fig. 2(f), where the relation was tested by measuring d 33, e 33, and P independently) and we will keep Q as an approximate coefficient. A similar dependency is found in thin films [20]. In the context of this fatigue study this approximation will prove to be sufficient, even though a stringent analysis would necessitate the consideration of higher order terms and of the influence of different variants during 90 switching. Fig. 3 depicts three strain electric field (butterfly) loops determined at different positions of the identical PZT sample. It is obvious that the simple quadratic relation exhibited in non-fatigued samples (Fig. 2(a) and (b)) fails to describe the observed asymmetry. The electrostrictive equation was previously extended [16] in that the total polarisation was divided into a non-switchable contribution p, and a field dependent P(E) assuming constant p and Q. For P(E) the additional complexity may enter that E may also be biased by a local bias field

4 2206 Y. Zhang et al. / Acta Materialia 53 (2005) St ra i n [10-3 ] E [V/mm] St r a in [ 10-3 ] E [V/mm] Stra i n [10-3 ] E [V/mm] After 1.8 x 10 6 cycles Fig. 3. Strain hysteresis loops (butterfly loops) for three different locations on a fatigued PZT disc (checkered electrode). Three different asymmetries are visible. E bias. The charge distribution of free charges around certain volume elements generates a certain field E bias the value of which is not large enough to fully suppress switching, but the effective local field is shifted by the bias value. This was not considered in Ref. [16]. We are thus concerned with the equation SðEÞ ¼Q½PðE þ E bias ÞþpŠ 2 ; ð1þ which is similar to the expression used in Refs. [21 23] for thin or thick films. In order to be able to experimentally disentangle E bias and p, we introduce the following relations: P r ¼ P r2 P r1 2 E bias ¼ E c2 þ E c1 2 and and E c ¼ E c2 E c1 ; ð2þ 2 c ¼ Sþ S S þ þ S : ð3þ Remnant polarisation P r, coercive field E c and bias field E bias, are defined from the polarisation hysteresis and strain asymmetry c from the butterfly hysteresis loops. P r1 and P r2 are remnant polarisations along the vertical axis, E c1 and E c2 are the coercive fields along the field axis in the polarisation hysteresis loop, S + and S are the heights of the two branches in the strain loop at the positive and negative fields. The parameter c is roughly proportional to the offset polarisation and to the saturation value of the switchable polarisation c ¼ Sþ S S þ þ S ¼ 2pP sat P 2 sat þ ð4þ p2 for the saturation strain values S + = Q (p + P sat ) 2 at positive and S = Q (p P sat ) 2 at negative applied electric fields. These relations imply a local measurement of polarisation and strain. For a non-local measurement of polarisation its value is the average over the entire sample surface, while the strain value is determined by the local switchability and offsets only (again different from the assumption in Ref. [16]). Fig. 4 displays colour contour plots of the parameters as defined in Eqs. (2) and (3) as a function of position on a sample fatigued to cycles. The material response is highly heterogeneous at this fatigue stage. Initially at lower cycle numbers, most parts of the sample are still non-fatigued and only small parts already show modified material parameters. Then slowly the heterogeneity increases, finally yielding images as shown in Fig. 4. Fatigue in a ceramic sample thus starts region by region. In order to show the fatigue development in a standard fashion, the development of average mean values of the material parameters are shown in Fig. 5 for 32 segments on one sample as a function of cycle number. The mean polarisation value degrades quickly between 10 5 and 10 6 cycles, after which fatigue enters a relatively steady stage in general agreement with previously reported results for non-segmented samples [16]. The correlation of the strain asymmetry and the offset field from the polarisation hysteresis loops is represented in Fig. 6. Even though significant scatter is observed, bias field and strain asymmetry are nearly linearly coupled. Positive bias fields correspond to negative strain asymmetry and vice versa Small-signal parameters The field dependent measurement of the piezoelectric coefficient directly yields the relation between possible offsets and the small signal parameters of the material. Essentially, the piezoelectric constant is given by d 33 ¼ os oe ¼ 2Q½PðE þ E biasþþpš op oe ¼ 2Q½PðE þ E bias ÞþpŠe 0 e r ; ð5þ

5 Y. Zhang et al. / Acta Materialia 53 (2005) Fig. 4. Contour plots of (a) remnant polarisation, (b) coercive field, (c) bias field, and (d) strain asymmetry as a function of position for a sample fatigued up to cycles. Remnan tp oa l rsa i ti on[ C/m 2 ] Bi a s Fie l d[ V/mm] Cycle Number, N Coercive Field [V/mm] St a r n i Asymme r t y, Fig. 5. The mean absolute value of remnant polarisation, coercive field, bias field, and strain asymmetry vs. switching cycle taken from 32 sample segments on the same sample. Strain Asymmetry Bias Field [V/mm] 1.8 x x x 10 7 Fig. 6. Correlation between the strain asymmetry and the bias field at various segments in samples from three different fatigue states. where an offset piezoelectric coefficient can be defined as d offset =2Qpe 0 e r. The restrictions on the first term approximation in Eq. (1) apply here accordingly. Four parameters can be directly extracted from the piezoelectric hysteresis loops, remnant piezoelectric coefficient d r, coercive field E c, offset piezoelectric coefficient d offset, and bias electric field E bias [24].Asd offset and E bias are directly read from experiment, their correlation is better defined than in the butterfly loops, where fitting of Eq. (1) to the experimental data is necessary [24]. Fig. 7 displays d 33 hysteresis loops from a single location on one sample for different fatigue stages along with the dielectric constant averaged across the intact initial electrode. The test results for d 33 on a 6 6 matrix with an interval distance of 1 mm over the centre part of the fatigued sample ( cycles) are shown in Fig. 8 as colour contour plots of position. Highly fatigued regions display low remnant and large offset piezoelectric coefficients, large coercive and large bias electric fields. In less fatigued regions this is not as pronounced.

6 2208 Y. Zhang et al. / Acta Materialia 53 (2005) d 3 3 [ p mv / ] / 3 3 o x x x x 10 7 DC Electric Field [ V/mm ] Fig. 7. Evolution of the piezoelectric coefficient d 33 and dielectric constant e 33 hysteresis loops. Cycle numbers are labelled. The comparison of Fig. 8(a) and (c) demonstrates that the decrease of the remnant piezoelectric coefficient (switchable polarisation) is accompanied by an increase of the offset piezoelectric coefficient (non-switchable polarisation). The results presented in Figs. 8(a) and (b) indicate that less switchable polarisation is coupled to a higher coercive field which is in general agreement with the results measured across the entire sample. Furthermore, the linear relation between offset piezoelectric coefficient and bias electric field [24] can be directly mapped onto the data of Fig. 6 where the biggest scatter results from the error in measured local electric displacement at finite grid size. This error comes in, because the grid pads cannot be reduced more to infinitely small size. They already represent some local average values. An experimental detail to be pointed out is that measurements on segmented samples like in Section 3.1 induce a mechanical constraint due to the neighbouring segments not subject to an applied field. Fig. 9 illustrates the effect with respect to the dielectric constant. The coercive field is smeared out and reduced to lower values. The maximum in dielectric constant occurs at essentially zero applied field. For the d 33 measurements, this is not a concern, because d offset is directly determined from experiment independent of a dielectric constant measurement. Furthermore, the sample surface is electroded throughout circumventing any artificial constraint. We will see in the discussion that fatigue itself induces additional constraints. Independent of the mechanical constraint, Fig. 9 furthermore demonstrates that the dielectric constant also becomes highly heterogeneous and the measured values vary strongly with location. Fig. 8. Contour plots of (a) remnant piezoelectric coefficient, (b) coercive field, (c) offset piezoelectric coefficient, and (d) bias electric field as a function of position for a fatigued sample after cycles.

7 Y. Zhang et al. / Acta Materialia 53 (2005) / 3 3 o Switching retardation segment in virgin sample three segments in fatigued sample DC Electric Field [V/mm] Fig. 9. Field dependence of the dielectric constant in a sample with segmented electrodes. The mechanical clamping by the surrounding material suppresses the two distinct peaks at each coercive field which unite into a single peak. Four data sets are shown, one for a fresh sample and three for different segments in a fatigued sample showing the additional heterogeneity ( cycles). Fig. 10 displays the heterogeneous distribution of d 33 across the sample surface for switching and reverse switching experiments of different durations (Fig. 10(a) (c), for 2, 5, and 30 s) which fatigue is known to alter drastically [25]. Fig. 10(d) illustrates the d 33 values after the sample was subjected to square pulses of opposite polarity (40 s width). Like the other properties of the sample, the retardation in switching is highly heterogeneous. The regions which are retarded in one polarity are also retarded in the opposite polarity. A second technique for testing switching rates was to apply increasing maximum electric fields to the sample during bipolar strain hysteresis measurement. Fig. 11 displays the dependence of strain asymmetry on the applied maximum electric field at constant measuring frequency. The line perfectly follows the same functional relation of the relaxation time constant on applied electric field for switching in ferroelectrics [26]: s = s 0 exp(e A /E) with an activation electric field E A. For increasing maximum applied field, the rate of relaxation becomes faster than the threshold rate value given by the frequency of the applied measuring field. Local switching is thus activated in a classical fashion. Strain As ym mtry e S S E S E Electric Field [ kv/mm ] Fig. 11. Strain asymmetry as a function of the applied maximum electric field during bipolar measurement (25 s per cycle). The fit shows c = c 0 exp(e A /E applied ). The insets show the corresponding strain hysteresis loops. E Fig. 10. Contour plots of the d 33 [pc/n] values from the direct piezoelectric effect determined at E dc = 0 for a sample after cycles. After poling (5 h at 2 kv/mm) reverse switching pulses were applied with different durations: (a) 2 s; (b) 5 s; (c) 30 s. Subsequent backswitching is shown after 40 s (d).

8 2210 Y. Zhang et al. / Acta Materialia 53 (2005) Fig. 12. Optical photographs of different samples polished to a 1 lm finish using diamond paste: (a) a non-fatigued sample; (b) a sample fatigued to cycles, (c) corresponding to Fig. 4 and (d) to Fig. 10. The feathery structures are microcracks Microstructural effects The strong modifications of the microstructure are illustrated in Fig. 12. As previously reported [6], all samples show some change in colour after fatigue. Most samples show a certain density of very heterogeneously distributed cracks. They form feather-like structures sometimes along the electrode edge, sometimes in certain arc-like distributions only underneath the electrodes, sometimes across the entire sample area excluding the not electroded rim outside the near electrode edge region (>50 lm off the edge). The local distribution of the heterogeneous material properties does not correlate with the observed spatial crack distribution. Even samples without any cracks exhibit heterogeneous fatigue. 4. Discussion 4.1. Switching and clamping Fatigue in perovskite ferroelectrics has been interpreted to result from many different sources, but none of them has been conclusively identified to be the fundamental origin of the process. A purely mechanical source is microcracking [27,28]. A second source is the reduction of domain wall mobility in the bulk of the material finally clamping the entire dynamics of switching [11]. The third is the inhibition of seed formation from the surface of the ferroelectric [4,29]. This latter scenario involves the blocking of polarisation reversal by inhibiting the formation of new domains right at the interface to the electrode material. The differentiation of these two scenarios is difficult and still lacking in rigour. Independent of the mechanism in discussion, the amount of switchable polarisation at a certain frequency, namely the shape and width/height of a polarisation hysteresis loop, has always been the reference property of the material. In thin films it has been known that the small signal properties also change significantly due to fatigue [30] and ageing [31]. Furthermore, heterogeneity of the fatigue effect has been observed for domain switching in thin films of single grain thickness using scanning force microscopy [14]. From the present results it is apparent that similar effects are encountered in a bulk material but on a completely different length scale. Furthermore, it is clear that microcracking is a secondary effect only following the initial clamping of domain switching. In thin films, switching is strongly heterogeneous throughout individual grains and thus on a very local length scale [11]. After fatigue of films, certain regions completely stop switching. In films, switching occurs predominantly by 180 domain wall motion, because 90 domain wall motion is mostly clamped by the mechanical constraints imposed by the substrate. The 180 domain walls themselves are not smooth lines but highly bent. Their propagation is locally hindered either at the electrode or in the bulk, while propagation of the switching information across grain boundaries seems not to be a source of switching inhibition [14]. The polarisation mismatch [32] introduced at the grain boundary upon switching of a neighbouring grain directly translates into a high driving force in this grain and domain walls appear to propagate straight through the grain boundary [10]. Thin films are also known to show strong imprint [10,33], meaning that certain volumes in the film are already frozen due to the processing and do not even necessitate any further fatigue treatment to loose their switching ability. Imprint is also highly heterogeneous [10].

9 Y. Zhang et al. / Acta Materialia 53 (2005) The dominant question arising for bulk material is why a similar heterogeneity is observed on a much larger length scale. One would expect that if heterogeneity occurs in individual grains there should be a sufficient number of such grains that their effect cancels out for a macroscopic sample. In particular, a large number of grains exists in thickness direction and the fatigue offsets should average out. Furthermore, many 90 domain walls are moved upon switching of a macroscopic bulk ceramic [34]. Thus the overall switchability should decrease, but this would be expected to occur rather homogeneously. Let us first consider the evolution of the fatigue effect with cycle number. For fatigue cycles the degradation process just starts. A large part of the sample is still in an unfatigued state and switching occurs homogeneously on large parts of the sample. Only isolated locations across the sample surface exhibit lower polarisation, higher coercive field, larger bias field, and strain asymmetry. For cycles, which is in the rapid fatigue range, a major part of the sample has fatigued, only small parts remain almost unfatigued. Some regions exhibiting large offset polarisation have occurred. For cycles, most of the sample surface shows remnant (switchable) polarisation lower than 0.20 C/m 2. The regions with larger strain asymmetry (up to 1) have coalesced. This implies that totally clamped polarisation was generated in these regions. There are thus two effects. Firstly, the switchable polarisation independently freezes in certain locations. Secondly, these centres then grow until large regions of non-switchable polarisation unite. The first effect is rather elucidating already. Arlt and Calderwood recently discussed the implications of the continuity of electric displacement in space [32]. Within individual columns of grains in thickness direction, switching is highly correlated, meaning that the switching of one grain at the surface highly enhances the driving force on the next grain in thickness direction and so on. We previously used this explanation to interpret part of the switching retardation due to fatigue [25]. Thus, if the blocking mechanism is determined by the properties of the interface between the electrode and the ferroelectric, entire columns in thickness direction quit switching. As the effective piezoelectric effect is given by the integral over the thickness direction, the effect on the d 33 measurements is immediate. The measurement tip in our measurements is testing a contact area of around 400 lm 2 when assuming the Hertz solution for a rigid spherical indenter on an infinite half-space [35]. Several tens of grains directly touch the strain measurement tip (6 lm grain size) and the stress field of a Hertzian indenter is known to further spread sideways. Many columns of grains are thus tested simultaneously. The suppression of polarisation switching is enhanced at the rim of the electrode. These regions also exhibit the lowest remnant polarisation, largest coercive field and offset field, and biggest strain asymmetry (Fig. 3 should not be misread in that e.g. left asymmetry is correlated to a particular side of the sample, it is completely arbitrary). This reduction of switchable polarisation could be attributed to the observed microcracks which start at the boundary between electroded and non-electroded regions due to strain incompatibility and propagate perpendicular to the electrode edge [36]. The close relationship between offset field and strain asymmetry of Fig. 6 shows a similar trend to the ones between bias fields and offset piezoelectric coefficients from piezoelectric hysteresis loops [24], despite a larger scatter of the data. This suggests that the generation of offset polarisation and bias field is a simultaneous process, in that the value and the direction of bias field and strain asymmetry are linearly coupled. The relatively large data scatter could be explained by the discrepancy of local strain and large area of polarisation averaging on each electrode pad Small-signal piezoelectric hysteresis loop The fatigue heterogeneity is represented in the position-dependent piezoelectric properties [24]. Figs. 8(c) and (d) display that the offset piezoelectric coefficient and the bias electric field are closely related. As we showed before [24], this relation is very reliable and more so than the one given from the fits of strain butterfly loops. In the latter, bias electric field and offset polarisation cannot be well separated. Some averaging on finite electrode surfaces always obscures the true offset electric field, which needs to be obtained from an electrical measurement for fitting of the strain loop. The colour contours of Fig. 8 illustrate the close relation between E bias and d offset and to a lesser degree to the remnant (switchable) polarisation. At the same time hardly any correlation to the coercive field is obtained. A further important observation is that the transitions between regions of opposite or highly dissimilar offsets are smooth. Thus, even though the offsets are highly heterogeneous, they can obviously not change arbitrarily from area to area. It is also directly visible, that positive offset dielectric constants correspond to negative bias fields. We previously analysed this relation by assigning neighbourhood relations between grains that are still switchable and those that have frozen [24]. The latter highly determine the fields in their immediate neighbourhood and the inverse relation becomes evident. Generally speaking, the piezoelectric hysteresis loops proved to be the most reliable measurement for determining the heterogeneity effect Switching retardation In a previous study we found that switching as determined from polarisation measurements becomes highly

10 2212 Y. Zhang et al. / Acta Materialia 53 (2005) retarded by fatigue [25]. One of the possible explanations was the cascading of the switching process across the sample thickness, the other a lateral heterogeneity across the electrode surface. As switching retardation of highly fatigued samples is in the range of hours, a set of measurements of partial switching could easily be performed. A sample was tested using the local d 33 technique after switching different times. A strong position-dependence was observed for the switching process from the contour plots in Fig. 10. Certain regions reproducibly switch faster than others. Thus, retardation occurs in each column of material to a different degree. Overall, both effects contribute to switching retardation, laterally distributed regions of different switching speeds, but each being retarded in its own switching process. The switching speed is associated with the fatigue state of the corresponding regions. The local distribution of switching speed, d 33, and the offset polarisation determined from butterfly hysteresis loops show the same lateral distribution on the sample indicating their strong correlation High field rejuvenation An evidence for changes in offset polarisation in certain regions stems from an observed recovery of strain asymmetry by higher external driving force [37]. This has been denoted as high field rejuvenation. The strain asymmetry decreases in an almost exponential fashion with the amplitude of the applied electric field as is verified in Fig. 11. The experimental data could be described using an exponential relationship: E A c ¼ A exp ; ð6þ E applied where E A is an activation field (E A 12 kv/mm, A 017). This relation is very similar to the relaxation time dependence on applied electric field known in the polarisation switching process of single crystals [26]. This provides a fairly straightforward evidence of attributing the fatigue mechanism to a distribution of defect sizes representing increasing barrier heights to domain wall motion. Along with increasing applied field values, the number of clamped domains thus diminishes. If the applied field is high enough, the offset polarisation fully disappears and switching recovers. This result should be a stimulus to reconsider the old and well established empirical exponential relationship and afford a forthcoming modelling of possible microscopic origins Microscopic origins Switching retardation (Fig. 10) as well as the optical images of samples after fatigue in Fig. 12 indicate certain spots in the centre of the sample that behave significantly different from the rest of the electroded area. These were easily identified to have resided underneath the stamps providing electrical contact during the fatigue treatment. These locations are thus subject to different mechanical boundary conditions, because the stamps exert a clamping pressure onto the sample. Their fatigue behaviour is obviously different, which is accompanied by a different colouring in the ceramic. The remainder of the sample drastically changes colour, indicating certain defects or colour centres to be partly responsible for the fatigue effect. Even though a large heterogeneity was seen, the colouring is actually rather homogenous except for the spot underneath the contacting stamps already discussed. If so, fatigue may induce the colour changes, but in turn these do not reflect the spatial heterogeneity. The heterogeneity most likely stems from regions underneath the electrodes. This particular point is further supported by other measurements attempting to extend the spatial investigations into the axial dimension. Using again the arguments by Arlt and Calderwood [32], it is not surprising that, once near electrode regions become blocking areas for domain wall switching, entire columns of grains fail to follow the externally applied electric field and remain clamped. On the other hand, each column is simply only retarded in its switching behaviour and will eventually also switch. Thus, whatever blocking mechanisms prevail, it can be modified by an extended application of electric fields. The heterogeneity data also confirm a certain degree of agglomeration or cluster formation of defects. These are most likely the origin of the non-switchable areas. On the other hand, the formation of such clusters across areas as large as tens or hundreds of grains is nevertheless unlikely. The high degree of coherence of the domain system, particularly in thickness direction then provides the clamping of extended columns of grains as seen by the d 33 measurements. Unfortunately, the present set of data is also still not able to give an answer as to whether the defect clusters are formed by electronic or ionic defects. The time dependence of switching in each column assures that it cannot be simple charge injection, because this should be much faster. If charge injection is relevant, it must be mediated through localised states of low mobility or in other words high activation energies. Trapping of electronic carriers in clusters of ionic vacancies or already existing clusters of electronic defects underneath the electrodes is equally possible. Details of such kind will have to be answered by techniques identifying electronic defect states Cracking Once it is clear that polarisation clamping is heterogeneous, it is rather straightforward to explain the observed cracking. The previous observation that cracking predominantly occurs in the near electrode region [38] is reproduced here. If one column can switch, but a

11 Y. Zhang et al. / Acta Materialia 53 (2005) neighbouring one cannot, the column subject to switching will attempt to ferroelastically contract at the coercive field while the neighbouring volume only performs piezoelectric contraction. A considerable strain mismatch develops. If the shear strength of the electrode is high, the tensile load exerted on the non-switching volume will increase even more. The highest tensile stress occurs at the edge of the non-switching volume. If these areas contain a pore or microcrack of sufficient size, a crack will start growing from here (Fig. 12). 5. Conclusion In this work, experimental studies of the fatigue behaviour of commercial soft piezoelectric PZT ceramics were conducted using large-signal polarisation and strain hysteresis loops and small-signal piezoelectric measurements based on the direct and converse piezoelectric effects. Experimental results demonstrate that apparent fatigue heterogeneity exists down to the submillimetre scale across disc shaped samples 10 mm in diameter. The distribution and evolution during fatigue of the four parameters, remnant polarisation P r, coercive field E c, bias field E bias, and strain asymmetry c from the polarisation and butterfly loops were discussed in terms of clamped polarisation. The variation of strain asymmetry was found to depend on the bias field. Likewise, the distribution of the four parameters from the piezoelectric hysteresis loops, remnant piezoelectric coefficient d r, coercive field E c, offset piezoelectric coefficient d offset, and bias electric field E bias point to a strongly heterogeneous material behaviour. The switching heterogeneity based on the direct piezoelectric effect measurement exemplifies region by region suppression of switching ability of local polarisation. The exponential dependence of fatigue recovery on rising electric fields correlates to classical activated domain wall motion for distributions of defects of different energetic barrier heights. The observed fatigue heterogeneity is accompanied by different degrees of increase of the relaxation time in each column of grains in thickness direction. Acknowledgements This research was supported by the Deutsche Forschungsgemeinschaft (SFB 595). Y.Z. wishes to acknowledge the support by the Alexander-von-Humboldt Foundation. We gratefully acknowledge the use of the switching equipment of Heinz von Seggern and discussions with Vladimir Ya. Shur and Alexei Gruverman. References [1] Cross LE. Jpn J Appl Phys 1995;34:2525. [2] Setter N, Waser R. Acta Mater 2000;48:151. [3] Scott JF. Ferroelectric memories. Heidelberg: Springer; [4] Tagantsev AK, Stolichnov I, Colla EL, Setter N. J Appl Phys 2001;90:1387. [5] Yang W. Mechatronic reliability. Beijing: Tsinghua University Press; [6] Nuffer J, Lupascu DC, Glazounov A, Kleebe HJ, Rödel J. J Eur Ceram Soc 2002;22:2133. [7] Lupascu DC, Rabe U. Phys Rev Lett 2002;89: [8] Lee JJ, Thio CL, Desu SB. J Appl Phys 1995;78:5073. [9] Pöykkö S, Chadi DJ. Phys Rev Lett 1999;83:1231. [10] Gruverman A, Auciello O, Tokumoto H. Appl Phys Lett 1996;69:3191. [11] Colla EL, Hong S, Taylor DV, Tagantsev AK, Setter N, No K. Appl Phys Lett 1998;72:2763. [12] Dunn S. J Appl Phys 2003;94:5964. [13] Bdikin IK, Shvartsman VV, Kholkin AL. Appl Phys Lett 2003;83:4232. [14] Rodriguez BJ, Gruverman A, Kingon AI, Nemanich RJ, Cross JS. J Appl Phys 2004;95:1958. [15] Abplanalp M, Barosova D, Bridenbaugh P, Erhart J, Fousek J, Gunter P, et al. J Appl Phys 2002;91:3797. [16] Nuffer J, Lupascu DC, Rödel J. Acta Mater 2000;48:3783. [17] Weitzing H, Schneider GA, Steffens J, Hammer M, Hoffmann MJ. J Eur Ceram Soc 1999;19:1333. [18] Verdier C, Lupascu DC, Rödel J. J Eur Ceram Soc 2003;23: [19] Chaplya P, Carman GP. J Appl Phys 2001;90:5278. [20] Kholkin AL, Akdogan EK, Safari A, Chauvy PF, Setter N. J Appl Phys 2001;89:8066. [21] Kholkin AL, Colla EL, Tagantsev AK, Taylor DV, Setter N. Appl Phys Lett 1996;68:2577. [22] Ricinschi D, Okuyama M. Appl Phys Lett 2002;81:4040. [23] Bobnar V, Kutnjak Z, Levstik A, Holc J, Kosec M, et al. J Appl Phys 1999;85:622. [24] Zhang Y, Baturin IS, Lupascu DC, Kholkin AL, Shur VY, Rödel J. Appl Phys Lett 2005;86: [25] Lupascu DC, Fedosov S, Verdier C, Rödel J, von Seggern H. J Appl Phys 2004;95:1386. [26] Lines ME, Glass AM. Principles and applications of ferroelectrics and related materials. Oxford: Clarendon Press; [27] Jiang Q, Cao W, Cross LE. J Am Ceram Soc 1994;77:211. [28] Hill MD, White GS, Hwang C-S, Lloyd IK. J Am Ceram Soc 1996;79:1915. [29] Taylor DV, Damjanovic D, Colla E, Setter N. Ferroelectrics 1999;225:91. [30] Colla EL, Kholkin AL, Taylor D, Tagantsev AK, Brooks KG, Setter N. Microelectron Eng 1995;29:145. [31] Kholkin A, Colla E, Brooks K, Muralt P, Kohli M, Maeder T, et al. Microelectron Eng 1995;29:261. [32] Arlt G, Calderwood JH. Appl Phys Lett 2002;81:2605. [33] Warren WL, Dimos D, Pike GE, Tuttle BA, Raymond MV, Ramesh R, et al. Appl Phys Lett 1995;67:866. [34] Hoffmann MJ, Hammer M, Endriss A, Lupascu DC. Acta Mater 2001;49:1301. [35] Ling FF, Lai WM, Lucca DA. Fundamentals of surface mechanics with applications. New York: Springer; [36] Lucato SL, Lupascu DC, Kamlah M, Rödel J, Lynch CS. Acta Mater 2001;49:2751. [37] Nuffer J, Lupascu DC, Rödel J. Appl Phys Lett 2002;80:1049. [38] Nuffer J, Lupascu DC, Rödel J. J Eur Ceram Soc 2001;21:1421.