Negative electrocaloric effect in antiferroelectric PbZrO₃

The electrocaloric (EC) effect is defined as the temperature change $\Delta T$ occurring in a system after the application or removal of an external electric field under adiabatic conditions. Recent theoretical [1] and experimental [2–4] studies in various bulk and thin-film systems revealed the co-existence of the normal $(\Delta T>0)$ and anomalous (or negative) $(\Delta T<0)$ EC effect under field application. Earlier, Peräntie et al. [5] observed a negative EC effect in $\langle 011\rangle$-oriented PMN-28PT in a narrow temperature range below $66\ ^{\circ}\text{C}$. It has been argued [3] that the anomalous EC effect appears when the applied electric field is not collinear with the dielectric polarisation of the material. Specifically, a negative EC effect is expected if there is a transition between two states of the system, where the lower-temperature state has a smaller polarisation than the higher-temperature state. Thus the field may increase the amount of disorder in the dipolar subsystem, leading to a lower value of the dipolar entropy and hence a negative $\Delta T$. An analogous effect is known to exist in antiferromagnets [6]. It is expected that a combination of the normal and anomalous EC materials could be used in practical applications such as cooling devices [7,8].

Antiferroelectrics (AFEs) undergo a phase transition in zero electric field into an ordered antiferroelectric phase at the AFE Curie temperature T_c, characterised by the order parameter P_s representing a staggered polarisation. In contrast to normal ferroelectric (FE) systems, the field-induced macroscopic polarisation $P(E,T)$ in the AFE phase may increase with increasing temperature, leading to a negative EC response [2,3]. Thus AFEs represent a relatively simple model system which provides a possibility to understand the physical mechanism of the negative EC effect on a semi-phenomenological level.

In this paper we write down the free energy function of a typical bulk AFE system within the framework of the generic Kittel model [9–11] and calculate numerically the field and temperature dependence of the macroscopic polarisation $P(E,T)$, from which the dipolar entropy change and the EC effect can be derived. To test the predictions of the Kittel model we have carried out direct ECE measurements in PbZrO₃ ceramics, which had been characterised as an AFE system by Shirane et al. already back in 1951 [12]. Our experiments evidence a negative EC effect in a broad temperature range below the AFE transition temperature, which had not been reported earlier in the literature. Thus we feel that PbZrO₃ might be an appropriate system in which to investigate simultaneously the mechanism of antiferroelectricity and a negative EC effect, which are currently of considerable interest both theoretically and experimentally [1–5].

In analogy to the EC effect in ferroelectrics [13] we consider the case where an electric field E is applied at the initial temperature T₁ under adiabatic conditions and subsequently the system reaches the final temperature T₂. The entropy of the system is assumed to be the sum of the phonon contribution S_ph plus a dipolar entropy $S_{\textit{dip}}$, i.e.,

$$\begin{equation} S=S_{ph}+S_{\textit{dip}}. \end{equation} \tag{ 1 }$$

The condition that the change of the total entropy S should be zero leads to the relation [13]

$$\begin{equation} T_2=T_1\exp\left[-\frac{S_{\textit{dip}}(E,T_2)-S_{\textit{dip}}(0,T_1)} {C_{ph}(T_1)}\right]\!. \end{equation} \tag{ 2 }$$

Here C_ph is the heat capacity of the phonon subsystem, which is assumed to have a weak temperature dependence on the interval $(T_1,T_2)$. By contrast, the dipolar specific heat can vary strongly in the same interval, however, it is in principle contained in the change of the dipolar entropy and should thus not be added to C_ph in the denominator.

The EC temperature change is given by ${\Delta T\equiv T_2-T_1}$ and can be calculated by solving the self-consistent equation (2) for T₂. The dipolar entropy $S_{\textit{dip}}(E,T)$ is derived from the Kittel free energy density for an AFE system [9–11,14,15]

$$\begin{eqnarray} f&=f_0+\frac{1}{2}a(P_1^2+P_2^2)+\frac{1}{4}b(P_1^4+P_2^4) \nonumber \\ &+\frac{1}{6}c(P_1^6+P_2^6)+gP_1P_2-(P_1+P_2)E.\qquad \end{eqnarray} \tag{ 3 }$$

For simplicity we consider the uniaxial case with P₁ and P₂ representing the polarisation components for the two sublattices 1 and 2, respectively. Here a, b, c are the Landau-type expansion coefficients for each sublattice, and g is the AFE coupling strength [9]. We write [9–11,15]

$$\begin{equation} a=g+a_1(T-T_0), \end{equation} \tag{ 4 }$$

where T₀ is the paraelectric-to-antiferroelectric transition temperature in zero field. In dimensionless units we shall then set $a_1=1$, $b=c=1/3$, $g=0.5$, and $T_0=1$. In eq. (3) we have included the 6-th order terms in order to facilitate the comparison with the ferroelectric case ${(g=0)}$ discussed earlier [13] and with the case b < 0 describing a first-order transition [9,16].

The physical polarisation P and the staggered polarisation P_s, which plays the role of the order parameter, are given by

$$\begin{equation} P=P_1+P_2, \qquad \qquad P_s=P_1-P_2. \end{equation} \tag{ 5 }$$

In thermodynamic equilibrium, the minimum of the free energy $f(P_1,P_2)$ calls for $(\partial f/\partial P_1)_T=(\partial f/\partial P_2)_T = 0$. This leads to the relations

$$\begin{eqnarray} &aP_1+bP_1^3+cP_1^5+gP_2-E=0; \end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray} &aP_2+bP_2^3+cP_2^5+gP_1-E=0. \end{eqnarray} \tag{ 6b }$$

In general, for a given value of E the solutions of these equations may contain several branches including some metastable states. In order to focus on the stable states only, we shall avoid solving eqs. (6) but instead calculate $P_1(E,T)$ and $P_2(E,T)$ directly by minimising the free energy (3) by means of a suitable numerical routine. We shall select $P_1\ge 0$ and $-P_1\le P_2\le P_1$ so that $P\ge 0$ and ${P_s\ge 0}$. It can be checked by evaluating the Hessian of $f(P_1,P_2)$ that all solutions P₁, P₂ obtained in this way are stable. In fig. 1(a) we plot the order parameter $P_s(E,T)$ and in fig. 1(b) the total polarisation $P(E,T)$. For E = 0 the system undergoes a second-order transition from paraelectric to the AFE phase with non-zero order parameter $P_s(T)$ at the AFE ordering temperature $T_c=T_0$. As E is increased, the transition is shifted to lower temperatures and reaches a tricritical point at the tricritical temperature $T_{3cp}=0.763\ T_0$ and tricritical field $E_{3cp}=0.365\ E_0$, where E₀ is arbitrarily chosen as the critical field of the analogous ferroelectric model with g = 0 and b < 0 [13], i.e., $E_0=(6b^2/25c)\sqrt{3\vert b\vert/10c}$. For fields larger than $\sim 0.564\ E_0$, the order parameter P_s is zero at all temperatures. For E > E_3cp and T < T_3cp, the transition from the paraelectric (or field-induced polar phase [15]) to the AFE (or antipolar [15]) phase is of the first order. The corresponding phase diagram is shown in fig. 2. The solid line represents the line of second-order phase transitions, which is derived from the condition that the derivative $(\partial P_s/\partial T)_E$ should diverge at the AFE transition. From eqs. (6) with $P_2=P_1$ above the line and $P_2=-P_1$ below, we find the equation for the critical value of P_1c at the transition

$$\begin{equation} 5cP_{1c}^4+3bP_{1c}^2+(a-g)=0, \end{equation} \tag{ 7 }$$

which is trivially solved. The critical field is then

$$\begin{equation} E_c=(a+g)P_{1c}+bP_{1c}^3+cP_{1c}^5. \end{equation} \tag{ 8 }$$

A formal solution of this equation exists for all 0 < T < T₀; however, only for T > T_3cp it corresponds to a true phase transition line (solid line in fig. 2), whereas below T_3cp it lies in the metastable regime (dotted line). The line of first-order transitions for $T\le T_{3cp}$ (dashed line) has been determined numerically from the positions of the jumps of $P_s(E,T)$ (see fig. 1(a)).

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The dielectric susceptibility is defined as $\chi=(\partial P/\partial E)_T$ and can be derived from eqs. (5) and (6). The result is

$$\begin{equation} \chi=\frac{d_1+d_2-2g}{d_1d_2-g^2}. \end{equation} \tag{ 9 }$$

Here $d_i\equiv a+3bP_i^2+5cP_i^4\ (i=1,2)$. Numerically, we find that for non-zero fields $\chi(T)$ exhibits a discontinuity on crossing the phase transition line (solid lines in fig. 2) and diverges at the tricritical point from below (see fig. 3). For E = 0, $\chi(T)$ has a cusp at $T=T_0$ (see the inset of fig. 5).

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The dipolar entropy is derived from eq. (3), namely, $S_{\textit{dip}}=-[\partial(f-f_0)/\partial T]_E$, and is given by

$$\begin{equation} S_{\textit{dip}}=-\frac{1}{2}\,a_1\left(P_1^2+P_2^2\right). \end{equation} \tag{ 10 }$$

To calculate the electrocaloric temperature change $\Delta T$ we insert the above expression for $S_{\textit{dip}}$ into eq. (2) with $a_1=1$ and find

$$\begin{equation} \frac{\Delta T}{T}=\exp{\left\{\frac{\sum_{i=1}^2\left[P_i^2(0,T)-P_i^2(E,T+\Delta T)\right]}{2C_{ph}(T)}\right\}}-1. \end{equation} \tag{ 11 }$$

This is a self-consistent equation for $\Delta T$ as a function of E and T, which can be solved by iteration or by some other suitable method [13]. The results are shown in fig. 4 for $C_{ph}=15$ and a set of field values in the range $0.06\ E_0\le E \le 0.6\ E_0$. For non-zero fields and T < T₀ the EC effect is negative, i.e., $\Delta T<0$ in a broad range of temperatures and fields. The maximum value of $\vert\Delta T\vert$ occurs at the tricritical point T_3cp, E_3cp. For T > T₀ the EC effect is always positive and it decreases with the temperature in a manner similar to normal ferroelectrics [13]. From the experimental point of view, it is of special interest to investigate the EC in antiferroelectric systems in the range of temperatures and fields where a negative EC effect occurs.

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The main prediction of the above analysis based on the Kittel model for AFE systems is the occurrence of a negative EC effect in a broad temperature range below the AFE Curie temperature T₀ (fig. 4). It should be noted, however, that the Kittel model is now believed to be rather trivial and thus unable to describe in detail all the properties of real AFE systems such as single crystal PbZrO₃ (PZ) [17,18]. For example, it has been reported that the AFE phase of PZ below $T_0=506\ \text{K}$ shows weak transient ferroelectricity in a temperature range of 20 K [18]; however, it has further been suggested that the FE phase might be defect induced and thus not present in pure crystals [19]. On the other hand, it has long been known that a FE phase between the paraelectric and the AFE phase appears in PbZrO₃ ceramics doped with a small amount of Ba [20].

In order to test the predictions of the above model, we have investigated experimentally the EC effect in PbZrO₃ ceramics (PZ). The ceramic powder was prepared by solid-state synthesis using PbO (Sigma, 99.9% purity, 211907) and ZrO₂ (TZ-0, Tosoh) as precursos. The powders were mixed in a stochiometric ratio, homogenized in a planetary mill and calcined twice in alumina crucibles at $850\ ^{\circ}\text{C}$ for 2 hours with intermediate milling. The sample was then sintered for 2 hours at $950\ ^{\circ}\text{C}$ using the pressure-assisted sintering technique and a uniaxial pressure of 24.5 MPa. During sintering the sample was embedded into a packing powder of the same composition in order to prevent the evaporation of PbO. The density after the sintering was $7.83\ \text{g/cm}^3$, as determined by the Archimedes method. The phase purity of the sample was confirmed by X-ray diffraction, whereby the structure was identified as orthorhombic, as previously reported by others [21]. The sintered sample was cut, ground to final thickness of about $70\text{-}100\ \mu \text{m}$, and sputter-coated with gold electrodes. Finally, the sample was annealed to relax any internal stresses, possibly induced during the preparation procedure.

From the dielectric measurements, the AFE transition temperature in PZ was found at $T_0=507\ \text{K}$ and no FE phase was observed on heating the system starting from temperatures far below T₀ (see fig. 5). It seems that any FE phase that might exist in the individual grains is smeared out in the ceramic sample, whereas the AFE transition temperature T₀ is not affected by averaging over the grain orientations [12].

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The EC temperature changes were measured by a specially made high-resolution calorimeter [22,23], allowing a precise temperature stabilisation (within 0.1 K) as well as high-resolution measurements of the EC temperature changes induced by an external field E [24]. In fig. 6 a typical EC response of an AFE material, i.e., the negative EC effect is shown. The results for the EC temperature change $\Delta T(T)$ in PZ ceramics are plotted in fig. 7 for a discrete set of applied fields E in the temperature range $\sim0.6~T_0<T<0.87~T_0$, and agree qualitatively with the calculated behaviour of $\Delta T$ displayed in fig. 4. Since the largest observed value of $\vert\Delta T\vert$ in PZ is found for $E=80\ \text{kV/cm}$, we can tentatively assume that this corresponds to the tricritical field $E_{3cp}=0.365~E_0$, giving an estimate for the parameter E₀ in PZ, i.e., $E_0\gtrsim 220\ \text{kV/cm}$.

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The predictions of the generic Kittel model of antiferroelectrics are in qualitative agreement with the experimental data for the negative electrocaloric effect in PbZrO₃ ceramics, obtained from direct measurements of the EC temperature change in this system. Another prediction is the occurrence of a tricritical point at some special values of the electric field and temperature, which has not been investigated in detail so far. Obviously, in spite of its apparent simplicity, the model seems capable of describing the basic mechanism of antiferroelectricity in PZ and related perovskites, and could probably be extended to include an intermediate ferroelectric phase occurring, for example, in barium-doped PZ.

This work was supported by the Slovenian Research Agency under programmes P1-0044, P1-0125, and P2-0105, and by the NAMASTE Centre of Excellence.