Anomalous dielectric and thermal properties of Ba-doped PbZrO₃ ceramics

On cooling below the antiferroelectric Curie temperature ${{T}_{\text{c}}}$ in zero applied electric field, antiferroelectrics undergo a phase transition into an ordered antiferroelectric (AFE) phase, characterised by a nonzero value of the order parameter p 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 some anomalous features of the dielectric and thermal behaviour of AFEs. In particular, the electrocaloric temperature change $\Delta T$ of the system has been found to be negative in the AFE phase of pure PbZrO₃ (PZ) in a broad range of temperatures [1–3], as well as in the mixed system (Pb,Ba)ZrO₃ (PBZ) as discussed below. It should be noted that a negative electrocaloric effect was observed before in other systems, for example in the relaxor-based ferroelectric $\langle 0\,1\,1\rangle$-oriented Pb(Mn_1/3Nb_2/3)O₃-0.28PbTiO₃ [4] in a narrow temperature range below 66 °C.

It was shown earlier [3] that the Landau–Kittel model of AFEs provides a relatively simple model system, which enables one to understand the physical mechanism of the negative electrocaloric effect on a semi-phenomenological level. In this paper, we write down the free energy function of a special AFE system, characterised by an intermediate FE phase, within the framework of the Landau–Kittel model [5, 6], which includes a sublattice coupling term introduced earlier by Rae and Dove (RD) [7] for the case of re-entrant phase transitions. Using this model we first calculate numerically the field and temperature dependence of the macroscopic polarisation P(E, T) and the staggered polarisation p(E, T). Finally, the dielectric response and the dipolar entropy change leading to the electrocaloric effect are calculated directly without the use of the familiar Maxwell relation [8].

It has been pointed out [2] that the anomalous electrocaloric effect generally appears when the applied electric field is not collinear with the dielectric polarisation of the material. Thus, the field may decrease the amount of disorder in the dipolar subsytem, leading to a higher value of the dipolar entropy and hence a negative $\Delta T$. An analogous effect exists in antiferromagnets [9]. It is expected that a combination of normal and anomalous electrocaloric materials could be used in practical applications such as in cooling devices [10–12].

The Landau–Kittel model for the free energy density of an AFE system [5, 6, 13–15] can be written as

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}f={{f}_{0}}+\frac{1}{2}a(P_{1}^{2}+P_{2}^{2})+\frac{1}{4}b(P_{1}^{4}+P_{2}^{4})\\ \,\,\,+\,\frac{1}{6}c(P_{1}^{6}+P_{2}^{6})+g{{P}_{1}}{{P}_{2}}-({{P}_{1}}+{{P}_{2}})E,\end{array}\end{eqnarray} \tag{ 1 }$$

where f₀ contains the phonon free energy. As usual, we shall assume that f₀(T) is free of singularities in the temperature range of interest. Also, for simplicity we consider the uniaxial case with P₁ and P₂ representing the effective 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  >  0 is the AFE coupling strength [6]. The first coefficient a is written in the form [5, 6, 13–17]

$$\begin{eqnarray}&&a=g+{{a}_{1}}\left(T-{{T}_{0}}\right).\end{eqnarray} \tag{ 2 }$$

It will be shown below, after transforming to the FE and AFE order parameters P and p, respectively, that T₀ is the antiferroelectric transition temperature in zero field.

The case b  >  0 and $g=\text{const}$ has already been discussed in [3] for a simplified model of AFEs featuring a second-order phase transition in zero field from the paraelectric high-temperature phase to an AFE phase below T₀. In the present work we shall set b  =  −1/3 and c  =  1/5 in order to describe the case of first-order phase transitions.

As originally suggested by Rae and Dove (RD) [7] in a different context, a sequence of phase transitions through an intermediate phase can be described by introducing a quadratic temperature term for the coefficient a(T). It will be discussed in more detail below that in the present problem this is formally equivalent to introducing a temperature-dependent coupling g  =  g(T), namely,

$$\begin{eqnarray}&&g={{a}_{2}}\left(T-{{T}_{0}}\right)\left(T-{{T}_{1}}\right),\end{eqnarray} \tag{ 3 }$$

which is negative in the interval ${{T}_{0}}<T<{{T}_{1}}$. In numerical calculations we shall set ${{a}_{1}}={{a}_{2}}=1$.

In thermodynamic equilibrium, the minimum conditions for the free energy, namely, ${{\left(\partial f/\partial {{P}_{1}}\right)}_{T}}$  =  ${{\left(\partial f/\partial {{P}_{2}}\right)}_{T}}=0$, lead to two coupled equations

$$\begin{eqnarray}&&a{{P}_{1}}+bP_{1}^{3}+cP_{1}^{5}+g{{P}_{2}}-E=0;\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&a{{P}_{2}}+bP_{2}^{3}+cP_{2}^{5}+g{{P}_{1}}-E=0.\end{eqnarray} \tag{ 4b }$$

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 equations (4), but instead calculate P₁(E, T) and P₂(E, T) directly by minimising numerically the free energy (1). It can be checked by evaluating the Hessian of $f\left({{P}_{1}},{{P}_{2}}\right)$ that all solutions ${{P}_{1}},{{P}_{2}}$ obtained in this way are stable.

The physical polarisation P and the staggered polarisation p, which plays the role of an AFE order parameter, are defined by [3]

$$\begin{eqnarray}&&P={{P}_{1}}+{{P}_{2}},\;\;\;p={{P}_{1}}-{{P}_{2}}.\end{eqnarray} \tag{ 5 }$$

The free energy (1) can then be rewritten in terms of the variables P and p, namely [15–17],

$$\begin{eqnarray}&&\begin{array}{*{20}{l}}f={{f}_{0}}+\frac{1}{2}{{a}_{P}}{{P}^{2}}+\frac{1}{2}{{a}_{p}}{{p}^{2}}+\frac{1}{32}b[{{P}^{2}}{{p}^{2}}+6({{P}^{4}}+{{p}^{4}})]\\ \,\,\,+\,\frac{1}{192}c[{{P}^{6}}+{{p}^{6}}+15{{P}^{2}}{{p}^{2}}({{P}^{2}}+{{p}^{2}})]-PE.\end{array}\end{eqnarray} \tag{ 6 }$$

The coefficients a_p and a_P are explicitly given by

$$\begin{eqnarray}&&{{a}_{p}}=\frac{1}{2}{{a}_{1}}\left(T-{{T}_{0}}\right);\;\;{{a}_{P}}={{a}_{p}}+g,\end{eqnarray} \tag{ 7 }$$

where g is defined by equation (3). This shows that the AFE instability temperature T₀ is not affected by the coupling g. On the other hand, the coefficient a_P of the P²-term explicitly contains g  =  g(T), as given by equation (3), which will be responsible for the non-zero value of P in the intermediate region. The coupling between P and p still exists; however, it now appears in higher powers of P and p and is temperature independent. We shall choose ${{P}_{1}}\geqslant 0$ and $-{{P}_{1}}\leqslant {{P}_{2}}\leqslant {{P}_{1}}$, thus $P\geqslant 0$ and $p\geqslant 0$.

It can be shown that for b  >  0 and E  =  0 an intermediate ferroelectric (FE) phase (P  >  0, p  =  0) exists in the interval ${{T}_{\text{c}1}}<T<{{T}_{\text{c}2}}$, where ${{T}_{\text{c}1}}$ and ${{T}_{\text{c}2}}$ are the solutions of the equation a(T)  +  2g(T)  =  0, namely, ${{T}_{\text{c}1}}={{T}_{0}}$ and ${{T}_{\text{c}2}}={{T}_{1}}-{{a}_{1}}/\left(2{{a}_{2}}\right)$.

In the following, we shall consider in detail the case b  <  0, which for E  =  0 predicts two first-order FE phase transitions occurring at ${{T}_{\text{c}1}}={{T}_{0}}$ and ${T}_{\text{c}2}^{*}$, where ${T}_{\text{c}2}^{*}>{{T}_{\text{c}2}}$ must be found by minimisation of the free energy. For numerical calculations we shall choose T₀  =  1 and T₁  =  1.6, implying ${{T}_{\text{c}2}}=1.1$ and ${T}_{\text{c}2}^{*}=1.283$. Thus, an intermediate FE phase will exist in the narrow temperature range 1  <  T  <  1.283.

In figure 1(a) we plot the polarisation P(E, T) and in figure 1(b) the AFE order parameter p(E, T). On cooling in zero field (E  =  0), the system undergoes a first-order phase transition from the paraelectric (P) phase (P  =  p  =  0) into the intermediate FE phase (P  >  0, p  =  0) at $T={T}_{\text{c}2}^{*}$, and at T  =  T₀ another first order transition into the AFE or 'antipolar' [17] phase (P  =  0, p  >  0). For E  >  0 the AFE transition is shifted to temperatures below T₀ and occurs at some temperature ${T}_{\text{c}1}^{*}(E)$; however, for fields larger than ${{E}^{*}}\simeq 0.9{{E}_{0}}$, where we choose ${{E}_{0}}=27{{b}^{2}}/(5c)$, the AFE order parameter p is zero at all temperatures and only a field-induced polar phase exists. Thus, for 0  <  E  <  E*and $0<T<{T}_{\text{c}1}^{*}(E)$ the system is in a mixed FE-AFE or 'semipolar' phase [17] (P  >  0, p  >  0).

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An inspection of figure 1(b) reveals that for $0.6\lesssim E/{{E}_{0}}\lesssim 0.9$, the discontinuous jump of the parameter p(E, T) initially reaches an intermediate finite value of p and subsequently drops to zero. An analogous behaviour is found for the parameter P(E, T) in figure 1(a). The corresponding phase diagram is shown in figure 2, where the lines represent first-order transitions. The intermediate mixed FE-AFE phase $\left(\beta \right)$ (shaded area) differs from the remaining FE-AFE phase $\left(\alpha \right)$ in the sense that it is characterised by two distinct non-zero values of the parameters P, p.

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In the case $g=\text{const}$ and b  >  0 discussed in [3] the AFE transition at T₀ is of second order and remains such for electric fields $E<{{E}_{3\text{c}}}$ and $T>{{T}_{3\text{c}}}$, where ${{E}_{3\text{c}}}$ is a tricritical field and ${{T}_{3\text{c}}}$ the tricritical temperature. For $T<{{T}_{3\text{c}}}$ and $E>{{E}_{3\text{c}}}$ the transition becomes first order. In the present model, however, we found no tricritical point even in the case b  >  0, which is not described here.

The static dielectric susceptibility is obtained from the relation $\chi ={{\left(\partial P/\partial E\right)}_{T}}$. By differentiating equations (4) with respect to E we find

$$\begin{eqnarray}&&P_{1}^{\prime}=\frac{{{d}_{2}}-g}{{{d}_{1}}{{d}_{2}}-{{g}^{2}}};\;\;\;P_{2}^{\prime}=\frac{{{d}_{1}}-g}{{{d}_{1}}{{d}_{2}}-{{g}^{2}}}.\end{eqnarray} \tag{ 8 }$$

Here $P_{n}^{\prime}\equiv {{\left(\partial {{P}_{n}}/\partial E\right)}_{T}}$ and ${{d}_{n}}\equiv a+3bP_{n}^{2}+5cP_{n}^{4}$ (n  =  1, 2). Thus, using $\chi =P_{1}^{\prime}+P_{2}^{\prime}$ we obtain the result [3]

$$\begin{eqnarray}&&\chi =\frac{{{d}_{1}}+{{d}_{2}}-2g}{{{d}_{1}}{{d}_{2}}-{{g}^{2}}}.\end{eqnarray} \tag{ 9 }$$

The numerical results for the susceptibility $\chi \left(E,T\,\right)$ are displayed in figure 3 using a logarithmic vertical scale. For E  =  0 the susceptibility shows discontinuous jumps at the boundaries of the intermediate FE phase at ${{T}_{\text{c}1}}={{T}_{0}}$ and ${T}_{\text{c}2}^{*}$, characteristic for first-order transitions. Some double-cusp structure that appears for $T\lesssim 0.16{{T}_{0}}$ and $0.6\lesssim E/{{E}_{0}}\lesssim 0.9$ is related to first-order transitions into an intermediate mixed FE-AFE phase $\left(\beta \right)$ (see figure 2).

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It should be noted that a similar phase diagram and the corresponding structure of the susceptibility was described by Okada [17], who investigated the case without an intermediate FE phase ($g=\text{const}$) but with a first-order AFE phase transition, i.e. b  <  0 in the present notation.

The electrocaloric effect in an AFE system with constant coupling g and b  >  0 has recently been studied in [3]. As usual, the electrocaloric temperature change $\Delta T$ is observed by applying and/or removing an electric field E to the system under adiabatic conditions. Following the approach used in the case of relaxor ferroelectrics [18] we consider the case where the field is applied at some initial temperature ${{T}_{\text{i}}}$ and the system reaches the final temperature ${{T}_{\text{f}}}$. The electrocaloric temperature change $\Delta T$ can be estimated by formally integrating the thermodynamic Maxwell relation ${{\left(\partial S/\partial E\right)}_{T}}={{\left(\partial P/\partial T\right)}_{E}}$, leading to an integral equation for $\Delta T(E)$, which is usually solved by iteration [19]. The application of this method is, however, far from trivial in the case of two order parameters P and p. Alternatively, in cases where the Landau-type free energy is known, one can obtain a simple expression for ${{T}_{\text{f}}}$ by assuming that the entropy of the system can be written as the sum of the phonon contribution ${{S}_{\text{ph}}}=-{{\left(\partial {{f}_{0}}/\partial T\right)}_{E}}$ and a dipolar entropy ${{S}_{\text{dip}}}$, namely [18, 19],

$$\begin{eqnarray}&&S={{S}_{\text{ph}}}+{{S}_{\text{dip}}}.\end{eqnarray} \tag{ 10 }$$

The condition that the total entropy S must remain constant, together with the assumption that the function ${{S}_{\text{ph}}}\left(E,T\,\right)$ is free of singularities, leads to the relation [18, 19]

$$\begin{eqnarray}&&{{T}_{\text{f}}}\cong {{T}_{\text{i}}}\exp \left[-\frac{{{S}_{\text{dip}}}\left(E,{{T}_{\text{f}}}\right)-{{S}_{\text{dip}}}\left(0,{{T}_{\text{i}}}\right)}{{{C}_{\text{ph}}}\left({{T}_{\text{i}}}\right)}\right].\end{eqnarray} \tag{ 11 }$$

Here ${{C}_{\text{ph}}}=T{{\left(\partial {{S}_{\text{ph}}}/\partial T\right)}_{E}}$ is the heat capacity of the phonon subsystem, which is assumed to have a weak temperature dependence on the interval $\left({{T}_{\text{i}}},{{T}_{\text{f}}}\right)$. By contrast, the dipolar specific heat ${{C}_{\text{dip}}}$ may have strong anomalies in the same interval; however, ${{C}_{\text{dip}}}$ is already implicitly contained in the change of the dipolar entropy, which can be derived from the free energy. Thus, ${{C}_{\text{dip}}}$ should not be added to ${{C}_{\text{ph}}}$ in the denominator.

The electrocaloric temperature change is given by $\Delta T\equiv {{T}_{\text{f}}}-{{T}_{\text{i}}}$ and can be calculated by solving the self-consistent equation (11) for ${{T}_{\text{f}}}$. The dipolar entropy ${{S}_{\text{dip}}}(E,T)$ is derived from equation (6), namely, ${{S}_{\text{dip}}}=-{{\left[\partial \left(\,f-{{f}_{0}}\right)/\partial T\right]}_{E}}$, and is given by

$$\begin{eqnarray}&&{{S}_{\text{dip}}}=-\frac{1}{2}{{\overset{\centerdot}{\mathop{a}}}_{P}}{{P}^{2}}-\frac{1}{2}{{\overset{\centerdot}{\mathop{a}}}_{p}}{{p}^{2}},\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&{{\overset{\centerdot}{\mathop{a}}}_{p}}\equiv \text{d}{{a}_{p}}/\text{d}T=\frac{1}{2}{{a}_{1}};\;\;\;\;\;{{\overset{\centerdot}{\mathop{a}}}_{P}}=\frac{1}{2}{{a}_{1}}+\overset{\centerdot}{\mathop{g}},\end{eqnarray} \tag{ 13 }$$

with $\overset{\centerdot}{\mathop{g}}={{a}_{2}}\left[2T-\left({{T}_{0}}+{{T}_{1}}\right)\right]$.

To calculate the electrocaloric temperature change $\Delta T$ we insert the above expression for ${{S}_{\text{dip}}}$ into equation (11) and find

$$\begin{eqnarray}&&{{T}_{\text{f}}}={{T}_{\text{i}}}\exp \left\{\frac{\Phi\left(E,{{T}_{\text{f}}}\right)-\Phi\left(0,{{T}_{\text{i}}}\right)}{2{{C}_{\text{ph}}}\left({{T}_{\text{i}}}\right)}\right\},\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}&&\Phi(E,T)={{\overset{\centerdot}{\mathop{a}}}_{p}}(T){{p}^{2}}(E,T)+{{\overset{\centerdot}{\mathop{a}}}_{P}}{{P}^{2}}(E,T).\end{eqnarray} \tag{ 15 }$$

Equation (14) is a self-consistent equation for ${{T}_{\text{f}}}$ as a function of E and T, which can be solved by iteration or by any other suitable method [18].

We have evaluated $\Delta T(E,T)$ for the present model of AFEs with g given by equation (3) and b  <  0, and assuming a₁  =  1 and ${{C}_{\text{ph}}}=20$. The results are displayed in figure 4 for a set of field values in the range $0\leqslant E\leqslant 3{{E}_{0}}$. For non-zero fields and T  <  1.0 T₀ the electrocaloric effect is negative, i.e. $\Delta T<0$ in a range of temperatures, the width of which is field dependent and increases with the field value.

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For T  >  1.05 T₀, the electrocaloric effect is always positive, reaching a maximum that increases with the field, and then decreases with the temperature at higher values of T. Clearly, for E  =  0 one has $\Delta T=0$ at all temperatures; however, $\Delta T$ is also zero for all field values at a temperature ${{T}_{z}}>{{T}_{0}}$, where the exponent in equation (14) vanishes, i.e. ${{T}_{z}}=\left({{T}_{0}}+{{T}_{1}}\right)/2-{{a}_{1}}/\left(4{{a}_{2}}\right)$. Using the parameter values T₀  =  1, T₁  =  1.6, and ${{a}_{1}}={{a}_{2}}=1$, we obtain T_z  =  1.05, in agreement with the numerical results shown in figure 4. It should also be noted that a discontinuous field-dependent jump of $\Delta T$ occurs at the first-order FE phase transition at ${T}_{\text{c}2}^{*}=1.283$.

The heat capacity of the system at constant field and pressure σ follows from equation (10), namely,

$$\begin{eqnarray}&&{{C}_{E,\sigma}}={{C}_{\text{ph}}}+{{C}_{\text{dip}}},\end{eqnarray} \tag{ 16 }$$

where ${{C}_{\text{ph}}}=T{{\left(\partial {{S}_{\text{ph}}}/\partial T\right)}_{\sigma ,E}}$ and the dipolar contribution to the heat capacity is given by ${{C}_{\text{dip}}}=T{{\left(\partial {{S}_{\text{dip}}}/\partial T\right)}_{\sigma ,E}}$. Ignoring the pressure dependence of the coefficients a(T) and g(T) we obtain from equation (12)

$$\begin{eqnarray}&&{{\left(\frac{\partial {{S}_{\text{dip}}}}{\partial T}\right)}_{\sigma ,E}}=-\frac{1}{2}{{\ddot{\mathop{a}}}_{P}}{{P}^{2}}-{{\overset{\centerdot}{\mathop{a}}}_{P}}P\overset{\centerdot}{\mathop{P}}-{{\overset{\centerdot}{\mathop{a}}}_{p}}p\overset{\centerdot}{\mathop{p}},\end{eqnarray} \tag{ 17 }$$

where ${{\ddot{\mathop{a}}}_{P}}\equiv {{\text{d}}^{2}}{{a}_{P}}/\text{d}{{T}^{2}}=2$, and we have used ${{\ddot{\mathop{a}}}_{p}}=0$. The temperature derivatives $\overset{\centerdot}{\mathop{P}}={{\overset{\centerdot}{\mathop{P}}}_{1}}+{{\overset{\centerdot}{\mathop{P}}}_{2}}$ etc are calculated from equations (4) by differentiating over T. From equation (17) we thus obtain the result

$$\begin{eqnarray}&&{{C}_{\text{dip}}}=T\,\frac{{{d}_{2}}P_{1}^{2}+{{d}_{1}}P_{2}^{2}-g{{P}_{1}}{{P}_{2}}}{{{d}_{1}}{{d}_{2}}-{{g}^{2}}}.\end{eqnarray} \tag{ 18 }$$

In figure 5 the heat capacity from equation (18) is plotted as a function of temperature for several values of the electric field. Surprisingly, in zero field (E  =  0) the dipolar heat capacity is negative within the intermediate FE phase. The main reason for this is the first term in equation (17) arising from the second derivative of the coupling g(T). An analogous result was obtained some time ago by Dove et al [20], who studied the re-entrant phase transitions in malononitrile by applying the RD model [7]. As E increases, the width of the negative ${{C}_{\text{dip}}}$ region broadens because P²(E) increases with E (see figure 1). It should be noted that ${{C}_{\text{ph}}}$ in equation (16) is normally expected to be a monotonically increasing function of temperature. Therefore, the negative dipolar contribution could be observed as either a weaker increase or a decrease in the total specific heat capacity at a given temperature interval [20].

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Until now, no monocrystalline systems possessing a low temperature AFE phase followed by an intermediate FE phase have been reported. However, it is known that a polycrystalline system with these properties can be obtained by doping a pure AFE system such as PbZrO₃ (PZ) with ions similar to Pb²⁺ , for example Ba²⁺ , as first described by Shirane [21]. In this work we investigated the ceramic material Pb_0.94Ba_0.06ZrO₃ (PBZ06). The ceramic powder was prepared by solid-state synthesis using PbO (Sigma, 99.9$\%$ purity, 211907), ZrO₂ (TZ-0, Tosoh), and BaCO₃ (Alfa, 99.8$\%$ purity, 014341) as precursors. The powders were mixed in a stochiometric ratio, homogenised in a planetary mill, and calcined twice in alumina crucibles at 850 °C for 2 h with intermediate milling. The PBZ06 was sintered by conventional pressure-less sintering for 2 h at 1250 °C using a tube furnace. The samples were embedded into a packing powder of the same composition in order to prevent the evaporation of PbO. The density after the sintering was 7.79 g cm⁻³, as determined by the Archimedes method. The phase purity of the samples was confirmed by x-ray diffraction, where the samples were identified as orthorhombic, as previously reported by others [22, 23]. The sintered samples were cut, ground to a final thickness of about 70–100 μm, and sputter-coated with gold electrodes.

The temperature dependence of the dielectric constant was measured upon heating in the temperature range between 350 K and 575 K at the frequency of 1 kHz by using an HP4284A LCR meter. The dielectric constant shown in figure 6 exhibits two distinctive first-order anomalies. The low temperature weakly-first-order anomaly at $T\sim 422$ K is due to the transition from the AFE phase to the intermediate FE phase, while the strong first-order jump at $T\sim 490$ K marks the transition from the FE phase to the paraelectric phase. This behaviour agrees qualitatively with the E  =  0 line of figure 3 (plotted on a logarithmic vertical scale), thus confirming the existence of an intermediate FE phase in PBZ06.

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In order to obtain the electrocaloric temperature changes, a specially-made high resolution calorimeter [24, 25] was utilised, allowing a precise temperature stabilisation (within 0.1 K) and high resolution measurements of the sample temperature variations. In the electrocaloric experiments, step-like voltage pulses were applied always starting from zero. A typical electrocaloric signal as a function of time, measured by a small glass bead thermistor in the AFE phase of bulk PBZ06, is shown in figure 7. Here the negative electrocaloric response is observed when the electric field is switched on at time zero, i.e. instead of heating, a cooling effect occurs.

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The electrocaloric response of PBZ06 was then measured at different temperatures under various electric field amplitudes. In figure 8, the temperature variation of the electrocaloric temperature change $\Delta {{T}_{\text{EC}}}$ in the AFE phase is shown for two values of the amplitude of the electric field. It should be noted that $\Delta {{T}_{\text{EC}}}$ was found to be negative in the entire experimental temperature range. In view of the breakdown field, the maximum amplitude achieved was 40 kV cm⁻¹. For this relatively modest electric field amplitude, $\Delta {{T}_{\text{EC}}}$ reaches nearly  −1 K.

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The heat capacity of PBZ06 ceramics has not been measured so far.

We have shown that the Landau–Kittel model of antiferroelectricity can be generalised to the case of an intermediate ferroelectric phase occurring between the low-temperature antiferroelectric (AFE) phase and the high-temperature paraelectric (P) phase. This has been accomplished by adopting a temperature dependent coupling between the two sublattices of the AFE model, which has a quadratic form, as proposed by Rae and Dove (RD) for the re-entrant phase transitions in malononitrile [7]. By numerically minimising the free energy, the temperature and electric field dependence of the FE polarisation P and the AFE order parameter p have been calculated for the case of first-order AFE-FE and FE-P phase transitions. The corresponding phase diagram and the dielectric susceptibility have been calculated. It has been shown that the electrocaloric temperature change $\Delta T$ is negative in the AFE phase, similar to the case of pure AFE material without an intermediate FE phase [3], but becomes positive in the FE phase and above. In zero applied field the dipolar heat capacity ${{C}_{\text{dip}}}$ is also negative in the intermediate FE phase, implying that the total specific heat may be reduced in this temperature range. In a non-zero field the temperature region of negative ${{C}_{\text{dip}}}$ broadens and its magnitude increases.

Preliminary experiments on the mixed ceramic system Pb_0.94Ba_0.06ZrO₃ (PBZ06 ceramics) reveal two first-order jumps of the dielectric susceptibility occurring at $T\sim 422$ K and $T\sim 490$ K. By comparison with the predictions of the Landau–Kittel model containing the RD-type sublattice coupling term, we conclude that these anomalies occur at the AFE-FE and FE-P boundaries of an intermediate FE phase, in analogy with earlier experiments at other concentrations [21, 23]. The main conclusion of this work is that the electrocaloric temperature change $\Delta {{T}_{\text{EC}}}$, measured directly in PBZ06, is negative in the AFE phase, in agreement with the model. Thus, PBZ06 ceramics appears to be a possible physical realisation of the Landau–Kittel model with a quadratic sublattice coupling term of the RD type. The present experiments do not allow a quantitative comparison with the model predictions and further work in a broader temperature range, including the heat capacity measurements, is clearly needed.

This work was supported by the Slovenian Research Agency under programs P1-0125, P2-0105 and P1-0044, and by the NAMASTE Centre of Excellence. GC acknowledges the support of the project PE3-1535 in the frame of the Action 'Supporting Postdoctoral Researchers' of the Operational Program 'Education and Lifelong Learning', co-financed by the European Social Fund and the Greek State.