Giant Piezocaloric Effect in PZT Ceramic Film

Transcription

1 Ferroelectrics, 483: , 2015 Copyright Taylor & Francis Group, LLC ISSN: print / online DOI: / Giant Piezocaloric Effect in PZT Ceramic Film ALEXANDER S. STARKOV 1, * AND IVAN A. STARKOV 2 1 Institute of Refrigeration and Biotechnology, University ITMO, St. Petersburg, Russia 2 SIX Research Centre, Brno University of Technology, Brno, Czech Republic 1. Introduction The paper focuses on the analysis of the multicaloric effect in solids. In particular, the presented approach is applicable to thin films as well. For the piezoelectric material, the multicaloric phenomenon includes two well-known effects: electro and elastocaloric; and two new ones - piezoelectocaloric and flexocaloric. These new effects are due to the temperature dependence of the piezoelectric and flexoelectric coefficients. The piezoelectric effect at medium fields is found to be comparable with the electrocaloric one and can be called a giant. The obtained results can be used for the design and optimization of new cooling devices and generators. Keywords Multicaloric effect; piezoelectric film; thermodynamic approach Caloric effects (CEs) are thermal changes in material properties due to the applied driving field. These effects have been named according to the nature of the field. At present there are three well-established CEs: magneto-, electro-, and elastocaloric (MCE, ECE, ElCE) [1 3]. In addition, after applying a thermal field to caloric materials, we could observe an alternation in polarization, magnetization and stresses. This phenomenon is known as the pyroeffect. CEs and pyroeffects are mutually reversible. The former can be used to design solid-state cooling devices [1, 4] while the later is frequently explored in generators and detectors [5]. Multicaloric materials can support more than one type of CE, i.e. demonstrate multicaloric effect (mce) [6]. The interaction between different fields in mce results in a significant increase of the CEs magnitude [7, 8]. Because of this feature, mce has attracted growing interest during the past years [9 11]. The strongest coupling of the fields has been observed in piezoelectric. The piezomagnetic (magnetostrictive) and magnetoelectric materials are rather rare and have weak interactions. Thus, the piezoelectric ceramics with a strong temperature dependence of the physical properties is viewed as the most promising material for practical applications. In this paper we choose lead zirconate titanate (PZT) ceramics as multicaloric material. This material has a giant ECE [12] and its characteristics (such as dielectric susceptibility tensor x ij, stiffness tensor c ijkl, and piezoelectric tensor e ijk ) are highly temperature dependent [13, 14]. Nowadays, the available experimental and theoretical results are rather contradictory attesting to a limited understanding of mce. Moreover, there is a variety of terminology Received October 2, 2014; in final form April 14, *Corresponding author. ferroelectrics@ya.ru Color versions of one or more figures in this article can be found online at 102

2 Giant Piezocaloric Effect in PZT Ceramic Film 103 and methods used in various papers, which obscures the links between different results. Therefore, the goal of our work is to accurately describe mce in the PZT ceramic film. It is important to note that our study includes the flexoelectric effect. That is the response of electric polarization to an elastic strain gradient [15]. It may be explained as a secondary effect with respect to the piezoelectricity, which is the response of polarization to strain itself. The flexoelectric effect in PZT ceramic was revealed in 2003 [16]. In turn, the temperature dependence of the flexocoupling tensor f ijkl leads to the flexocaloric effect [11], which is a part of mce. Typically, CEs are described on the basis of the thermodynamic approach. The first thermodynamic theory of mce has been presented by I.N. Flerov [6]. A significant extension of this work was reported in [11] where the gradient effects were taken into account. 2. The Thermodynamics of Piezocaloric Effect Let a generic piezoelectric system in thermodynamic equilibrium be described in terms of temperature T, vectors of polarization P i and electric field E i, tensors of strain u ij and stress s ij. For the study of the piezoelectric, flexoelectric and CEs we will use the following expansion of the free energy [15] F D a ij 2 P ip j C c ijkl 2 u iju kl u ijk P i u jk f ijkl.p k u ij;l u ij P k;l / P i E i ; (1) where a ij is the tensor inverse to x ij and u ijk is the tensor linking energy with polarization and strain. Hereafter, the Einstein summation notation is used, i.e., repeated indexes are summed over the Cartesian components; the subscript after the comma denotes differentiation with respect to the corresponding variable P k /@x l. The strain tensor is defined in a standard manner u ij D.u i;j C u j;i //2, with u i as the displacement vector. The expansion (1) does not contain nonlinear terms, i.e. the electrostriction is omitted. It may be worthwhile to point out that the nonlinear thermodynamics was considered in [11,17]. The presence of the flexoelectric terms in (1) indicates that the electric, elastic, and temperature fields are inhomogeneous. Hence, it is necessary to consider derivatives of the polarization and temperature in (1). Apparently, these additions lead to a more complicated differential equation [15, 17]. Moreover, in this case the electric field E i should be replaced on the potential gradient [18]. At the same time, it is not difficult to verify that we can neglect the above effects for small f ijkl [15] and for films with a thickness greater than 20nm. The variation of (1) with respect top i and u i results in where the stress tensor is defined as E i D a ij P j u ijk u jk f klij u kl;j ; s ij;j D 0; (2) s ij D c ijkl u kl u ijk P k C f ijkl P k;l : (3) It should be noted that, because of the presence of the flexoelectric term in (1) (3), the thermodynamical Maxwell s relations are not fulfilled [11, 19]. Exactly these equations are the basis for obtaining experimental data on CEs [12]. Equation (3) describes the generalized Hook s law and converse piezoelectric and flexoelectric effects. Despite

3 104 A. S. Starkov and I. A. Starkov the fact that the equations (2), (3) completely govern the behavior of flexoelectric, they not suitable for analyzing the caloric effects. For the description of CEs it is necessary to exclude polarization and strain from (2), (3) [11,20]. In general, this is a quite difficult problem because these equations are differential ones. However, the following approximate relations are fulfilled for a small flexoelectric effect P i D x ij E i C d ijk s jk C m klij u kl;j ; u ij D s ijkl s kl C d kij E k n ijkl P k;l : (4) Here, d ijk are piezoelectric modules, m klij flexoelectric tensor, s ijkl elastic compliance tensor, n ijkl flexoelastic tensor. It should be emphasized that the last tensor is introduced for the first time in this paper. All the coefficients in (4), with the exception of n ijkl, have been measured experimentally for PZT ceramics [13,14,16]. The system (4) provides the opportunity to derive expressions for the pyroelectric p i i =@T and pyroelastic p ij ij =@T coefficients. After differentiation of (4) with respect to T, we come to the relations p i E i s ik u kl;j; p ij s kl E k P k;l: (5) The right-hand side of the first of equations (5) consists of three terms. The first one describes the primary pyroelectric effect that exists due to the temperature dependence of the dielectric susceptibility. The second links with the secondary pyroelectric effect. This effect is explained by the dependence of the polarization on elastic stresses. The last term (the tertiary effect, according to the terminology of [20]) is caused by the polarization dependence on the strain gradient. Identical senses have three terms in the right-hand side of the second equation (5). The general formula for CE in piezoelectric has the form [11, 20] ds D p i de i C p ij ds ij ; dt D T C.p ide i C p ij ds ij /: (6) where S is the entropy and C is the heat capacity. The first equation in (6) describes the isothermal entropy changes, while the second the adiabatic temperature change. For the modeling of the hard ferroelectric [21], the above formulas should be supplemented by the spontaneous polarization P Si and spontaneous strain u sij. This approach is the most appropriate for the description of mce. The proposed model does not take into account hysteresis phenomenon [22]. Nevertheless, hysteresis represents a drawback because it reduces the refrigerant efficiency and should be avoided. In our case the resulting formulas have the form ds E j Si de i ijkl s kl d.e is jk / ds ij u kl;jde i P k;lds ij : (7) The expressions for the adiabatic temperature change differ from (7) by the multiplier T/C.

4 Giant Piezocaloric Effect in PZT Ceramic Film 105 Let us discuss the obtained result. The first two terms in (7) describe the wellknown ECE and ElCE. The third term is responsible for the piezoelectrocaloric effect. This effect exists only in the case when the both (electric and elastic) fields are applied. The last two terms describe the flexocaloric effect. They are proportional to the strain and polarization gradients. The equation (7) establishes the change of entropy for the multicaloric effect in piezoelectric solids. In order to describe the mce in films, it is necessary to apply appropriate boundary conditions to equations (4). We employed the uniaxial boundary conditions for the electric field and the elastic stress (E! D.0; 0; E 3 /, s 33 D const, s 13 D s 23 D 0). In addition, the approximate boundary conditions were imposed for the flexoelectric effect. That is, the same conditions as used in the experiment [16]. The exact boundary conditions for the flexoelectric effect are given in [23]. The solution of the piezoelectric problem for the film structure is possible to find in [24]. The results of calculations based on this solution and equation (7) are summarized in Figs.1, 2. The analysis has been performed for the PZT ceramic film. More precisely, the following calculation steps have been applied. As a starting point, we can use the expressions for finding the electric and elastic fields for the film presented in [24]. By substituting these equations in (7), we get an expression containing derivatives of the material parameters with respect to the temperature and material parameters themselves (piezoelectric modules, flexoelectric tensor, elastic compliance tensor, flexoelastic tensor). The values of these parameters can be found in [13,14,16]. The evaluation of the derived equation and determination of the temperature derivatives is quite cumbersome and, therefore, we do not include it here. As a result, we obtain expressions for the elasto/electro/piezoelectro-caloric (Fig. 1) and flexocaloric (Fig. 2) effects. Note that each of the graphs corresponds to one of the terms in the equation (7), or rather its analogue for the film. We can see that the piezoelectric effect under an electric field of 48 MV/m is comparable with the electrocaloric one and can be called a giant. Figure 1. The multicaloric effect and its components: the electrocaloric, piezocaloric and elastocaloric effects. The calculations have been performed for the PZT ceramic film under an electric field of 48 MV/m and pressure of 0.3 GPa [12 14].

5 106 A. S. Starkov and I. A. Starkov Figure 2. The entropy change as a function of temperature for the flexocaloric effect in PZT ceramic film. The strain gradient is taken to be equal to 1 m 1 [16]. 3. Conclusion We have predicted a new giant multicaloric effect in the PZT ceramic film, which exceeds either electrocaloric or elastocaloric effects alone. This effect owes its origin to the temperature dependencies of the dielectric susceptibility tensor as well as the stiffness and piezoelectric tensors. We have demonstrated that the account of the flexoelectricity leads to inapplicability of the Maxwell s relations. The main advantages of the flexocaloric effect are the existence in all types of crystals (regardless of their symmetry) and increasing with decreasing sample size. The calculated values for mce at moderate fields are large enough to envisage future applications of PZT ceramics in efficient refrigerators and energy generators. Funding This work was financially supported by the Government of Russian Federation, Grant 074-U01. References 1. A. Tishin and Y. Spichkin: The Magnetocaloric Effect and Its Applications. Bristol: Institute of Physics Publishing; P. Kobeco and I. V. Kurtchatov: Dielectric properties of Rochelle salt crystal. Z. Phys. 66, (1930). 3. E. Bonnot, R. Romero, L. Manosa, E. Vives, and A. Planes: Elastocaloric effect associated with the martensitic transition in shape-memory alloys. Phys Rev Lett. 100, (2008). 4. Y. V. Sinyavsky and V. M. Brodyansky: Experimental testing of electrocaloric cooling with transparent ferroelectric ceramic as a working body. Ferroelectrics. 131, (1992). 5. G. Sebald, D. Guyomar and A. Agbossou: On thermoelectric and pyroelectric energy harvesting. Smart Mater. Struct. 18, p (2009).

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