Domain Switching and Energy Harvesting Capabilities in Ferroelectric

Sep 28, 2010 - Appliquées d'El Jadida, B.P 1166, 24000 El Jadida, Morocco, and Laboratoire de Physique de la Matie`re. Condensée, LPMC, Département...
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J. Phys. Chem. C 2010, 114, 20629–20635

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Domain Switching and Energy Harvesting Capabilities in Ferroelectric Materials† Se´bastien Pruvost,*,‡ Abdelowahed Hajjaji,‡,§ Laurent Lebrun,‡ Daniel Guyomar,‡ and Yahia Boughaleb*,| UniVersite´ de Lyon, INSA-Lyon, LGEF Laboratoire de Ge´nie Electrique et Ferroe´lectricite´, EA 682, Baˆtiment GustaVe FERRIE, 8 rue de la Physique, F-69621 Villeurbanne Cedex, France, Ecole Nationale des Sciences Applique´es d’El Jadida, B.P 1166, 24000 El Jadida, Morocco, and Laboratoire de Physique de la Matie`re Condense´e, LPMC, De´partement de Physique, Faculte´ des Sciences, 24000 El Jadida, Morocco ReceiVed: June 8, 2010; ReVised Manuscript ReceiVed: September 10, 2010

This paper deals with the understanding of depolarization mechanisms in ferroelectric materials and how to improve energy harvesting by converting thermal or mechanical energy into electrical energy. Depolarization mechanisms caused by temperature were studied using X-ray diffraction (XRD) to describe the evolution of the domain organization. Polarization variations, constituting the key parameter for energy harvesting, were simulated from the XRD data and explained in terms of domain switching, variation of the dipole moment, and rotation of the dipole. The latter concerned structural transitions induced by an excitation and gave rise to significant nonlinearities in the material properties. Energy harvesting should combine two excitations (e.g., temperature and electric field, or stress and electric field) describing intelligent thermodynamical cycles. Various examples have been given to illustrate the possibilities for harvesting energy. Introduction A large number of applications, including wireless sensors, portable devices, or self-powered electronic systems, need to harvest energy from the environment.1 There are several potential sources of energy including thermal energy, mechanical energy (vibrations, mechanical stress or strain), light energy, electromagnetic energy, and chemical or biological energy. If the energy source is constant, it consequently becomes possible to replace the battery, whereas if the source is intermittent, energy harvesting is used to recharge a battery. Among the various active materials for the conversion, ferroelectric ceramics and single crystals offer the possibility of mechanical-toelectrical conversion using their piezoelectric activity and thermal-to-electrical conversion based on their pyroelectric activity.2-4 To convert energy, the material has to describe a thermodynamic cycle: the Stirling cycle consists of two constant electric induction paths and two isothermal (or constant stress) ones, and the Ericsson cycle corresponds to two constant electric field paths and two isothermal (or constant stress) ones.5,6 The polarization variations due to the excitation (in open circuit for the former cycle and under voltage for the latter one) constitute a key parameter for energy harvesting due to the fact that the larger the polarization variations are, the higher is the harvested energy. It is therefore fundamental to understand the mechanism of polarization/depolarization to improve the efficiency of the thermodynamic cycles. Ferroelectric ceramics or single crystals are organized in domains separated by domain walls for temperatures below the Curie temperature. Some models have been developed to simulate the behavior of the material under excitation (electric †

Part of the “Mark A. Ratner Festschrift”. * Corresponding authors. Se´bastien Pruvost: e-mail: sebastien.pruvost@ insa-lyon.fr. Tel.: +33472436403. Fax: +33472438874. Yahia Boughaleb: e-mail: [email protected]. ‡ Universite´ de Lyon. § Ecole Nationale des Sciences Applique´es d’El Jadida. | Laboratoire de Physique de la Matie`re Condense´e.

field, stress or temperature), for example, micromechanical models,7-11 phase field models,12-14 computational models,15,16 or models based on phenomenological expressions.17-19 The approach depicted in the present paper consists in describing the organization of ferroelectric domains and correlating this organization to macroscopic variables such as the polarization of the sample. In a domain, all of the dipoles are aligned along the same direction. The polarization variation generated by an external excitation can be due to the evolution of the dipole moment (called the intrinsic contribution) or to the motion of domain walls (mobility and/or switching, known as the extrinsic contribution).20 Previous investigations have tried to distinguish between the two contributions by characterizing the material at low temperature to freeze the domain walls. Another option has been based on the frequency dispersion characteristics of the dielectric constant.21-23 Numerous techniques have been used to observe the domain structure of materials such as polarized optical microscopy,24 scanning and transmission electron microscopy (SEM and TEM),25 atomic force microscopy (AFM),26 Raman spectroscopy,27 and X-ray diffraction (XRD).28 XRD provides direct quantitative information regarding the distribution of domains as domain switching provokes a modification of the crystallographic texture of the sample.28 XRD has thus been widely used in previous studies for the investigation of dynamic domain walls under an electric field,29 stress,30 and, more recently, variations in temperature.31 The macroscopic polarization can be simulated from the microscopic description of the domain organization using X-ray data. This article provides a description of polarization/depolarization mechanisms as functions of the electric field, stress, and temperature, with the aim of determining intelligent cycles for energy conversion. For this purpose, X-ray measurements were carried out on Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramics with a rhombohedral structure to quantify domain switching and lattice parameters under the application of a temperature in the range of 20-150 °C. This experiment was used to establish a relation

10.1021/jp105262h  2010 American Chemical Society Published on Web 09/28/2010

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between microscopic mechanisms and the depolarization caused by temperature. Finally, these mechanisms were compared with those described for the same composition but under compressive stress and electric field. Experimental Section A. Sample Preparation. Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramics were prepared using the columbite method as described by Swartz and Shrout.32 The first step of the process consisted in synthesizing MgNb2O6 from Nb2O5 and MgO oxides. These oxides were ground in ethanol, dried, and calcinated at 1100 °C for 6 h. The obtained precursor was milled and mixed with PbO and TiO2. The mixture was treated at 850 °C for 4 h to form the pure perovskite Pb(Mg1/3Nb2/3)0.75Ti0.25O3. At different stages of the synthesis, an excess of MgO and PbO was added to stabilize the perovskite phase and to compensate for PbO loss during heat treatments.33-37 The calcinated powder was mixed with a polyvinyl alcohol binder and cold-pressed into a matrix at 50 MPa. After burning of the organic binder at 600 °C, sintering was performed at 1250 °C in sealed alumina crucibles with a PbZrO3 powder bed to establish a constant PbO vapor pressure during densification. The density of the sintered rods was evaluated from weight and dimension measurements and was equal to 7.7 g · cm-3. Between each stage of the process, X-ray diffractograms were recorded for quality control. Samples were sliced into thin plates with a thickness of 0.5-1 mm. The specimens were annealed at 500 °C to remove texture effects induced by sintering and machining. Subsequently, the ceramics were silver paste electroded on both sides. When necessary, poling was performed in a silicone oil bath at 50 °C for 5 min under an electric field applied along the axial direction. The field value was 3.5 kV · mm-1 for the disk samples. The coercive electric field was determined from bipolar hysteresis cycle realized at 1 Hz for a maximum electric field of 3 kV · mm-1. Its value was equal to 600 V · mm-1 and the remnant polarization equal to 0.21 C · m-2. The temperature at which the macroscopic polarization is equal to zero is around 130 °C. The dielectric constant, K, and dielectric loss factor, tan δ, were determined on polarized samples using an HP 4284 LCR meter at 1 kHz. Moreover, the piezoelectric coefficient, d33, was obtained with a Berlincourt meter 24 h after poling. The dielectric constant at room temperature reached 2000ε0 and the dissipation factor, tan δ, was equal to 2%. The piezoelectric activity was evaluated by measuring the piezoelectric coefficient, d33, which was equal to 260 pC · N-1. B. X-ray Diffraction Measurements. XRD measurements were used to distinguish between intrinsic and extrinsic contributions of the macroscopic polarization. The experiments were performed from room temperature to 150 °C in an X’Pert Pro MPD Panalytical diffractometer equipped with Cu-KR1 radiation (λ ) 1.5406 Å), a focusing incident-beam monochromator, and a real-time multiple strip X’Celerator detector. Diffraction patterns were collected over the angular range 20-90° (2θ) with a step length of 0.0168° (2θ) and a counting time of 200 s step-1 on an Anton Paar HTK 16 high temperature stage. The temperature was stabilized for 15 min prior to recording the pattern. Because of their thickness (several µm), the silver electrodes coated by serigraphy prevented the X-ray analysis on the surface perpendicular to the polarization direction of the ceramic. The silver paste was chosen in such a way that it could be removed with an appropriate solvent after polarization of the sample.

Pruvost et al. Theoretical Calculations A. Domain Switching Quantification. A domain corresponds to a region where the dipoles are aligned in the same direction. Under excitation, the dipole can rotate (or the domain wall can move) to another direction that depends on the crystallographic structure. In the rhombohedral case, the polar axis is parallel to the 〈111〉, direction leading to possible domain reorientations at 71°, 109°, and 180°, whereas in the tetragonal case, only 90°- and 180°-domain switching are possible (the polar axis is parallel to the 〈001〉 direction). XRD renders it possible to separate the intrinsic contribution (change in lattice parameters) from its extrinsic counterpart (domain switching) and to quantitatively describe the domain organization in the sample. The drawback of this technique is the impossibility of distinguishing between two domains when the polarization axes are oriented 180° with regard to one another. Several authors have used XRD for studying the modification of domain organization, and some of them linked the change in peak intensity to the percentage of switched domains.29,38,39 The quantification of domain switching by XRD depends on the configuration of the acquisition (macroscopic polarization with a parallel40,41 or perpendicular31,42 orientation to the analyzed area). The first study using XRD for the evaluation of domain switching was realized by Subbarao et al. on BaTiO3 ceramics under tensile stress.40 These ceramics have a tetragonal structure, and the domain organization was described using (200) and (002) reflections. In the present case, the Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramics had a rhombohedral structure, and their direction of the macroscopic polarization was perpendicular to the analyzed surface. As 71°, 109°, and 180° domain switching could occur, the 222 and 222j diffraction peaks were studied, and the (222) plane depicted domains where the dipole was aligned with the macroscopic polarization. As determined from previous work,30 the percentage of switched domains (PSD) corresponds to the number of switched domains over the total number of domains and can be calculated using the formula:

PSD )

R′ - R (1 + R)(1 + R′)

(1)

Here, R is the intensity ratio (R ) (I222/I222j)) at room temperature, and R′ is the same ratio for a given temperature. The formula takes into account only non-180° domain switching. The PSD could thus be calculated with 111 and 111j peaks, but their deconvolution would be difficult at temperatures near the Curie temperature. The intensity of the 222 and 222j peaks was lower than for the 111 and 111j peaks (compensated by higher time integration); however, the former peaks were more separated. B. Expression of Polarization and Measurement. As mentioned in the introduction, the polarization variations could be divided into two contributions: an intrinsic contribution that corresponds to the variation of the dipole moment, and an extrinsic contribution that corresponds to domain switching. The latter is often associated to nonlinear properties and hystereses. In a previous study, Hajjaji et al. expressed the polarization from microscopic parameters in a tetragonal case31 as

P(θ) ) µ(θ) × N(γ - PSD(θ)) ) a(θ) qδ 1 × N(γ - PSD(θ)) (2) c(θ)



Domain Switching and Energy Harvesting where N is the total number of dipoles per unit volume, γ is the polarizability of the sample (value ranging between 0 and 1; 0 for a nonpoled sample and 1 for the single domain obtained in a single crystal), q is the total charge (equal to 6e- in a usual perovskite), and δ[1 - (a(θ)/c(θ))]1/2 corresponds to the distance between the negative barycenter and the positive one. As 〈001〉 is the direction of the dipole moment for the tetragonal case, the tetragonality ratio c/a exhibits a similar variation with temperature as the value of the dipole moment. Moreover, δ is a coefficient of proportionality between the distance separating the two barycenters and the square root of the tetragonality ratio. This coefficient is therefore independent of temperature. For the rhombohedral case, the direction of the dipole moment is 〈111〉, and the ratio depicting the variation of the dipole moment with temperature is 1 - (d222j(θ)/d222(θ)), leading to the following expression for the polarization:

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Figure 1. Percentage of switched domains (PSD) versus temperature for a Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramic. The circles correspond to experimental points, and a solid line has been added as a guide to the eye.

P(θ) ) µ(θ) × N(γ - PSD(θ)) ) d222-(θ) qδ 1 × N(γ - PSD(θ)) (3) d222(θ)



The various parameters of the formula can be evaluated from material characteristics. The total number of dipoles (or unit cells), N, per unit volume was calculated as N ) 1/Vunit cell (N ) 1.531 × 1028 m-3). The dipole moment can be expressed as µ(θ) ) qδ[1 - (d222j(θ)/d222(θ))]1/2, the parameters q, δ, and γ are calculated from the value of the remnant polarization and the spontaneous polarization. For the Pb(Mg1/3Nb2/3)0.75Ti0.25O3 composition,30 the spontaneous polarization Ps is equal to 0.39 C · m-2, and the studied ceramics exhibit a remnant polarization Pr of 0.21 C · m-2. The determination of γ is undertaken at the initial state (PSD ) 0) and corresponds to the ratio Pr/Ps representing the disorientation of grains in the ceramic with respect to the macroscopic polar axis. Because of certain vacancies (oxygen or lead) in the material and to the interactions between cations A and B of the perovskite, the total charge would not be exactly equal to 6e-. This uncertainty is compensated by the experimental determination of δ using the microscopic expression of the spontaneous polarization Ps ) q × d(θ) × N ) qδ[1 - (d222jθ)/d222(θ))]1/2 × N. From this formula, the distance between the two barycenters can be calculated and is equal to 0.267 Åsa value similar to that of the displacement of cations in the perovskite structure observed by Grinberg and Rappe.43 The macroscopic polarization was calculated by the integration of the released current measured using a Keithley current amplifier, model 428.

Figure 2. Macroscopic polarization versus temperature and its simulation from X-ray data of a Pb(Mg1/3Nb2/3)0.75Ti0.25O3 ceramic (black circles, experimental polarization; red triangles, simulated polarization).

Moreover, due to the hypothesis (depolarization resulting from the decrease of the dipole moment and the non-180° domain switching), the rhombohedral composition did not exhibit 180° domain switching with temperature, as in the tetragonal case. In a previous study,31 the intrinsic and extrinsic contributions were separated by derivating the polarization (eq 3) with respect to θ, assuming that the number of unit cells per volume unit was almost independent of temperature (variation of approximately 0.3% between room temperature and the highest studied temperature):

dP(θ) ∂µ(θ) ∂PSD(θ) ) N(γ - PSD(θ)) - Nµ(θ) dθ ∂θ ∂θ

(4)

Results and Discussion Figure 1 shows the evolution of the PSD with temperature. Up to Tc - 30° (where Tc is the Curie temperature corresponding to the ferroelectric-to-paraelectric transition), the non-180° domain switching activity remained very low at 107 °C with around 10% of all of the aligned domains. The macroscopic polarization was simulated from eq 3 and is presented with its experimental counterpart in Figure 2. The good agreement between simulation and experiment validated the proposed model. Otherwise, there is a slight difference for the last point at 140 °C, a temperature higher than the temperature at which the sample is well-depolarized (around 130 °C). Near the Curie temperature, as it was specified in the part concerning the domain switching quantification, it is difficult to make the deconvolution of the two peaks. At this temperature, the size of the domains decreases broadening the diffraction peaks.

From eq 4, the polarization variations can be expressed as:

∆P ) ∆Pint + ∆Pext

(5)

with

∆Pint )

dθ ∫ N(γ - PSD(θ)) ∂µ(θ) ∂θ

(6)

and

∆Pext ) -

dθ ∫ Nµ(θ) ∂PSD(θ) ∂θ

(7)

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Figure 3. Simulation of the polarization (black circles) from X-ray data and its separation into an intrinsic (green triangles) and extrinsic (red squares) contribution. A solid line has been added as a guide to the eye.

As pointed out in Figure 3, the intrinsic contribution reached 80% of the complete depolarization at 75 °C and 70% at 107 °C and decreased to 50% at the Curie temperature. The sudden increase of the domain switching activity in the vicinity of the Curie temperature implied an irreversible depolarization of the sample, but larger polarization variations were obtained. This point is explicated below with regards to establishing good conditions for energy harvesting. The first conclusion of this study concerns the description of depolarization mechanisms owing to the excitation. Equations 2 and 3 rendered it possible to simulate the macroscopic polarization from microscopic parameters describing the domain configuration for respectively a tetragonal and a rhombohedral structure. From this simulation, depolarization mechanisms can be defined for materials with tetragonal or rhombohedral structures, regardless of the excitation. Using the results of the study concerning depolarization due to temperature and previous ones regarding depolarization due to stress30,41,44 and electric field,45 the mechanisms could be described and utilized to depict intelligent thermodynamic cycles for energy harvesting. The reversible or irreversible nature of polarization variations generated by an excitation is an important parameter to improve the harvested energy. Depolarization mechanisms versus temperature, stress, and electric field determined in this work and previous ones will be detailed below and used to define after optimal conditions for thermal to electrical energy conversion and mechanical to electrical energy conversion. From the results presented above, the temperature depolarization had two origins. The first was the decrease of the dipole moment for temperatures lower than Tc - 30°. Near the ferroelectric to paraelectric transition, the non-180° domain switching became important and could be associated to the energetic threshold necessary for depinning domain walls. The first contribution was reversible except after crossing the Curie temperature, whereas the second was irreversible. The ferroelectric-to-paraelectric transition was observed to be due to both domain switching and to a decrease of the dipole moment. The energy for nucleating a domain with its polar axis at 180° from the macroscopic polarization was too high as compared to the thermal energy brought to the sample. Above the transition, the macroscopic polarization was equal to zero, and when the temperature was decreased below the Curie temperature, the minimization of electrostatic interactions between domains generated 180° domains, causing the macroscopic polarization to be null. For the compressive stress measurements, the depolarization was caused only by the 71° and 109° domain switching (or 90°

Pruvost et al. domain switching for the material with a tetragonal structure), and no 180° domain switching was observed. Nor was there any change in the value of the dipole moment (the intrinsic contribution was null). When comparing the results obtained on the tetragonal composition (Pb(Mg1/3Nb2/3)0.60Ti0.40O3) to those on the rhombohedral one (Pb(Mg1/3Nb2/3)0.75Ti0.25O3), it appeared that the domain switching was not governed by the macroscopic polarization but rather by long-range interactions. Two types of local interactions could be considered: electrostatic ones due to dipole-dipole interactions and mechanic ones due to internal stress generated by domain switching. It can be concluded that dipole-dipole interactions seemed to be small compared to the strain induced by domain switching. Indeed, domain switching caused by a compressive stress was controlled by a mechanical parameter. The mechanisms of depolarization under an electric field (applied in the opposite direction with respect to the macroscopic polarization) were more complicated since they depended on three possible types of domain switching: 0-90°, 0-180°, and 90°-180° (or similar domain switching with 71° and 109° for the rhombohedral case). As mentioned for the compressive stress, the value of the dipole moment was not modified for the electric field that a ceramic could withstand. The depolarization process can be explained by detailing only the extrinsic contribution. When comparing the strain for two frequencies with a difference of five decades, 0-90° domain switching remained the same regardless of the frequency. This leads to the conclusion that 0-90° domain switching is kinetically fast. A similar result has also been observed on a rhombohedral composition in a study on Pb(Zn1/3Nb2/3)0.955Ti0.045O3 single crystals.46 At the same time, 0-180° domain switching occurred, as demonstrated in the simulation of polarization. The 0-180° domain switching is kinetically slow and controlled the value of the coercive electric field. For this reason, at the highest frequency, 0-90° domain switching accumulated, converging on a higher strain. The 90-180° domain switching was observed only near the coercive electric field where the internal electric field was the lowest. The 90-180° domain switching was therefore said to be energetically controlled. To summarize, on the basis of the results obtained under an electric field, the intrinsic contribution was negligible for depolarization. Depolarization mechanisms were more intricate under an electric field as opposed to under compression: 0-90° and 0-180° were kinetically controlled, and 0-180° domain switching was slower than its 0-90° counterpart. Moreover, it was responsible for the coercive electric field value. For 90-180° domain switching, the key parameter is energetic. A transition is not always associated with irreversible polarization variation. In a study on Pb(Zn1/3Nb2/3)0.955Ti0.045O3 single crystals, Benayad et al. observed that the variation of polarization due to rhombohedral-to-tetragonal transition induced by the temperature was reversible.47 The rotation of the polarization axis during the phase transition can be viewed as a reversible domain wall motion and can be attributed to a reversible extrinsic contribution, as also described by Davis et al.48 For electric field excitation, the polarization variation due to the ferroelectric-to-ferroelectric transition is also reversible as it has been observed for rhombohedral Pb(Mn1/3Nb2/3)1-xTixO349,50 and Pb(Zn1/3Nb2/3)1-xTixO351,52 single crystals. For both cases, the crossing of the transition, and thereby increasing and decreasing the excitation, marked a hysteresis whose origin may be kinetic. Furthermore, the corresponding phase was metastable on this gap. The reversible aspect of domain switching due to rotation of the polarization with

Domain Switching and Energy Harvesting

Figure 4. Scheme of an Ericsson cycle for energy harvesting. The hachured area corresponds to the harvested energy.

respect to the ferroelectric-to-ferroelectric transition was due to the thermodynamical stability of the phase. For example, for a 〈110〉 Pb(Mn1/3Nb2/3)0.7Ti0.3O3 single crystal under 6 MPa compression stress along the 〈001〉 direction, the electric field necessary for the transition decreased from 0.7 kV · mm-1 to 0.2 kV · mm-1.50 When approaching the paraelectric phase, polarization variations resulted from non-180° domain switching and were irreversible. In conclusion, large polarization variations observed within small excitation ranges resulted in extrinsic contributions whose reversibility can be associated to ferroelectric-to-ferroelectric transitions. The irreversible polarization variations originated from non-180° domain switching, including the ferroelectric-to-paraelectric transitions. Lastly, to adapt the material to the working temperature for energy harvesting, the value of the electric field inducing the transition can be tuned with the chemical composition and/or a prestress on the sample. Energy harvesting consists of using the energy conversion ability of materials to transform waste energy (vibration, thermal) into electrical energy. Figure 4 illustrates this principle with a graph of the polarization versus the electric field. The material produced electrical energy if the closed cycle followed the clockwise direction, whereas electrical energy was dissipated for the counter clockwise direction. The area of the cycle corresponded to the harvested energy. Some thermodynamic cycles, like the Stirling cycle (i.e., two isothermal processes and two constant polarization paths) or the Ericsson cycle (i.e., two isothermal paths and two constant electric field paths), were described by Olsen and Evans for thermal-to-electrical energy conversion.53 These cycles can also be used for the mechanical-to-electrical conversion. The Ericsson cycle presented in Figure 4 points out two criteria required by a material for it to be a good candidate for energy harvesting: a low injected energy in the material during the AB path (for a good Eharvested/Einjected ratio) and a large polarization variation during the BC path (for a large harvested energy) within a narrow range of excitation. The latter criterion can be improved by understanding the mechanisms at the origin of the polarization variations. Harvesting energy from thermodynamic cycles needs two types of excitation (i.e., electric field and stress, or electric field and temperature). The BC path uses polarization variations under an electric field, and the mechanisms of the polarization variations presented above for compressive stress and temperature were at a zero electric field. Coupled excitations had to be taken into account. Energy harvesting from temperature has been developed using ferroelectric materials.2,5,53-56 In Figure 4, the polarization

J. Phys. Chem. C, Vol. 114, No. 48, 2010 20633 variation during the BC path was governed by the pyroelectric activity of the material under a DC electric field. For the Pb(Mn1/3Nb2/3)0.9Ti0.1O3 ceramics, the Ericsson cycle with an electric field of 3 kV · mm-1 and a temperature variation of 30° (between 35 and 65 °C) rendered it possible to harvest 107 mJ · cm-3.55 For the same electric field, 〈110〉 Pb(Zn1/3Nb2/3)0.955Ti0.045O3 single crystals enabled the harvesting of 112 mJ · cm-3 for a temperature variation of 35° (between 100 and 135 °C).2 In both cases, electric field-induced structural transitions were associated with the process of energy extraction. In the vicinity of a transition, dipoles are highly mobile, thus generating large polarization variations. With the exception of the working temperature, the difference between the two investigated materials lays in the injected energy. The Curie temperature of the Pb(Mn1/3Nb2/3)0.9Ti0.1O3 ceramic was close to room temperature. During the AB path, the electric field applied to the sample tended to align dipoles in the same direction by domain switching. The paraelectric-to-ferroelectric transition induced by the electric field in each grain was diffuse as both non-180° and 180° domain switching occurred, and the injected energy was equal to 100 mJ · cm-3. The 〈110〉 Pb(Zn1/3Nb2/3)0.955Ti0.045O3 single crystals exhibited an electric field-induced rhombohedral-to-orthorhombic transition near 80-100 °C and a temperature-induced rhombohedralto-tetragonal transition at 110 °C.57 Contrary to the previous case, the polarization variation was not due to domain switching but to the rotation of the polarization axis. This mechanism explains why the polarization variation occurred abruptly and the injected energy was only 41 mJ · cm-3. For both cases, the high pyroelectric activity was justified by the closeness of the transition and the associated high dipole mobility. This transition could be tuned by applying a prestress to the sample so as to obtain the temperature range associated with the application as presented in the previous paragraph. As described above, nonlinear energy harvesting can also be carried out by exciting the material with an electric field and a compressive stress. For the case shown in Figure 4, the electric field would be applied and withdrawn during the AB and CD paths, respectively. The polarization variation during the BC path would obtained by compressing the sample. The compressive stress is withdrawn at a zero electric field (CD path). Unruan et al. studied the effect of a compressive stress on cycles of polarization versus electric field (at 60 Hz) for Pb(Mn1/3Nb2/3)1-xTixO3 ceramics.58 The harvested energy can be evaluated from these cycles. For example, a 20 MPa compressive stress applied to Pb(Mn1/3Nb2/3)0.7Ti0.3O3 ceramics reduced the remnant polarization from 0.10 to 0.08 C · m-2, and at 1.5 kV · mm-1, the polarization variation was around 0.025 C · m-2. Hence, the harvested energy was equal to 33.7 mJ · cm-3. The same cycle for Pb(Mn1/3Nb2/3)0.9Ti0.1O3 ceramics, for which the remnant polarization was nearly null and the polarization variation under electric field equal to 0.008 C · m-2, rendered it possible to harvest only 6 mJ · cm-3. The second case is worse due to the lower polarization variation when the compressive stress is applied. The mobility of domain walls is increased near the MPB region due to the smaller size of the domains. Applying a compressive stress in the same direction as the polarization axis led to non-180° domain switching since the previous polarization step of the sample had eliminated all 180° domain switching.59

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The compressive stress controls non-180° domain switching and therefore reduces the switchable part of the domains through the application of an electric field. Furthermore, the harvested energy using an Ericsson cycle can be improved by using ceramics that are softer than Pb(Mn1/3Nb2/3)0.7Ti0.3O3 or single crystals. Some single crystals exhibit large polarization variations (around 0.10-0.15 C · m-2) for the same compressive stress and can be repoled at room temperature if the electrodes do not reach the sides of the sample. This avoids possible arcing. Compared with the previous case, the harvested energy can be multiplied by 5-8. The best case would be to employ compressive stressinduced transitions in single crystals by a rotation of the polarization axis to find a compressive stress-temperature diagram similar to the electric field-temperature diagram defined by Renault et al. for Pb(Zn1/3Nb2/3)0.955Ti0.045O3 single crystals.57 Contrary to ceramics studied before, the polarization would be reversible, thus decreasing the injected energy for following cycles. Energy harvesting using compressive stress and electric fields can for instance be found in the heel of a shoe or on a floor (ticket gates of the subway in Tokyo, dance floors, or roads). The force developed under the heel of a man (weighing 70 kg) can be evaluated to 5 kg · cm-2, and the active area is equal to 15 cm2. The corresponding compressive stress is nearly equal to 8 MPa. On the other hand, a tensile stress on a ceramic plate also reduces the perpendicular polarization at lower values than the compressive stress (due to the Poisson ratio). In this case, the ceramic is placed on the sole of the shoe. Conclusions This work has established a link between microscopic parameters and the macroscopic polarization of rhombohedral ferroelectric ceramics regardless of the excitation. Depolarization mechanisms due to temperature have been determined and compared to those induced by compressive stress and electric field. It was deemed interesting to compare macroscopic models with these mechanisms to quantify the energy associated with depolarization mechanisms. Such a method can be used to obtain a precise understanding of the role of dopants. Subsequently, depolarization mechanisms were used to explain and improve thermodynamic cycles for energy harvesting from temperature and stress. The vicinity of a transition increased the mobility of the domain walls and thus the polarization variation for a small excitation interval. Transitions associated with the rotation of the polarization axis (i.e., ferroelectric-to-ferroelectric transitions) generated reversible polarization variations, whereas other transitions such as ferroelectric-to-paraeletric transitions were found to be associated with irreversible domain switching. The reversible or irreversible part of extrinsic or intrinsic contributions revealed its importance for energy harvesting, especially for decreasing the energy injected in the material during the first step of the cycle. References and Notes (1) Roundy, S.; Leland, E. S.; Baker, J.; Carleton, J.; Reilly, E.; Lai, E.; Otis, B.; Rabaey, J. M.; Wright, P. K.; Sundararajan, V. IEEE PerVasiVe Comput. 2005, 4, 28. (2) Guyomar, D.; Pruvost, S.; Sebald, G. IEEE Trans. Ultrason., Ferroelect., Freq. Control 2008, 55, 279. (3) Khodayari, A.; Pruvost, S.; Sebald, G.; Guyomar, D. IEEE Trans. Ultrason., Ferroelect., Freq. Control 2009, 56, 693. (4) Guyomar, D.; Sebald, G.; Pruvost, S.; Lallart, M.; Khodayari, A.; Richard, C. J. Intell. Mater. Syst. Struct. 2009, 20, 609. (5) Olsen, R. B.; Bruno, D. A.; Briscoe, J. M.; Jacobs, E. W. J. Appl. Phys. 1985, 57, 5036.

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