The electrocaloric (EC) effect refers to the isothermal entropy change ΔS or adiabatic temperature change ΔT of a dielectric material when an electric field is applied or removed. It has attracted great attention after the findings of 12-K temperature change in PbZr_0.95Ti_0.05O₃^[1] and more than 12-K temperature variation in P(VDF-TrFE) (55 mol%/45 mol%).^[2] It has been particularly used for applications in cooling.^[3–6] Lead-based electrocaloric materials show a large EC effect. However, the toxicity of lead has limited their applications. Therefore, lead-free ferroelectrics have been considered as better candidates for EC refrigeration, such as Bi_0.5Na_0.5TiO₃-based ceramics,^[7–9] and BaTiO₃-based materials.^[10]

The EC ΔT is closely related to the applied electric field. The electrocaloric strength ΔT/ΔE is also evaluated in the studies of EC effect. For instance, the large EC effects in PbZr_0.95Ti_0.05O₃ were induced with a field of 77.6 MV/m,^[1] which is huge in comparison with the dielectric strengths of common inorganic ferroelectric materials.

To gain an insight into the effect of the applied electric field, we conducted a theoretical study of the EC effect of the model ferroelectric BaTiO₃ (BTO). A variety of experimental results has been reported on the EC effect of this material. In single crystal, the reported ΔT varies from 0.9 K@1.2 MV/m,^[11] ∼1.2 K@1.2 MV/m,^[12] to 1.6 K@1 MV/m.^[13] In multilayer films, the ΔT varies from 1.8 K@17.6 MV/m,^[14] to 7.1 K@80 MV/m.^[15] These results indicate that the electric field strength is crucial for the EC response.

Based on the Maxwell equation, the EC adiabatic temperature change ΔT can be evaluated by the following equation^[16,17] 1 in which ρ and C_p are density and specific heat capacity, respectively, here ρ = 6020 kg · m⁻³,^[16] and C_p = 407 J · kg⁻¹ · K⁻¹.^[11] For a monodomain BTO single crystal with both the polarization P and the applied electric field along [001] axis, the Landau–Devonshire potential including up to eighth-power term can be written as

The EC ΔT of BTO at the different electric field strengths are plotted in Fig. 1. For the case of an applied electric field less than E_CP, there exists a sharp peak of EC ΔT versus T; as shown in Fig. 1(a).

Fig. 1.

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	Fig. 1. (color online) Calculated EC ΔT of BTO at different electric field strengths by the phenomenological method.

This profile of ΔT–T is consistent with the experimental observations.^[11–13] Both the intensity and the width of the peak increase slightly with the increase of the applied electric field strength. The starting temperature of the ΔT peak is T_C, and the stopping temperature is T_CP. The predicted ΔT at 1 MV/m is 1.2 K, and the EC strength ΔT/ΔE is 1.2 × 10⁻⁶ K · m · V⁻¹. These values agree with the experimental observations in the crystals of BTO as listed in Table 1.

Table 1.

Table 1.

Table 1. Data of EC properties of BTO and doped BTO in the literature. . Materials Form ΔE/(MV/m) ΔT/K ΔT/ΔE/10−6 K · m · V−1 Ref. BaTiO3 single crystal 1 1.6 1.6 [13] BaTiO3 single crystal 1.2 0.9 0.75 [11] BaTiO3 single crystal 1.2 ∼ 1.2 1 [12] BaZr0.5Ti0.95O3 ceramic 3 2.4 0.8 [19] BaHf0.5Ti0.95O3 ceramic 5 1.64 0.33 [20] BaZr0.21Ti0.79O3 ceramic 9.9 4.7 0.46 [21] BaZr0.20Ti0.80O3 ceramic 14.5 4.5 0.31 [10] BaTiO3 multilayer film 80 7.1 0.09 [15]

Table 1. Data of EC properties of BTO and doped BTO in the literature. .

In contrast, the EC ΔT–T profiles become broad and flat at the electric field strengths approaching and higher than E_CP; as shown in Fig. 1(b). These profiles are similar to the experimental results in the thin films of BTO,^[15] PbZr_0.95Ti_0.05O₃,^[1] and P(VDF-TrFE) (55mol%/45 mol%).^[2] At 100 MV/m, ΔT is 3 K and ΔT/ΔE is 0.03 × 10⁻⁶ K · m · V⁻¹, which are comparable with those obtained with BTO multilayer film (see Table 1).

To disclose these two types of EC responses, we calculated P–E loops at selected temperatures and P at selected electric field strengths. The typical P–E loops are shown in Fig. 2(a). These temperatures satisfy the following relations: 400 K < T_C, T_C < 41 K < T_CP, and T_CP < 420 K. Thus, the BTO shows ferroelectric P–E loop at T < T_C. However, the coercive field E_C decreases with the increasing temperature. It becomes zero at T_C. The double loop presents at T_C < T < T_CP. Finally, linear P–E relation is observed at T_CP < T. Since a small electric field can induce a large variation of polarization at T_C < T < T_CP, a sharp EC ΔT peak with large ΔT/ΔE is observed; as shown in Fig. 2(a).

Fig. 2.

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	Fig. 2. (color online) The P–E loops at selected temperatures (a) and the polarization as a function of temperature at selected electric field strengths (b).

The electric field effect on the evolution of polarization is depicted in Fig. 2(b) at selected electric field strengths. Note that the critical point E_CP is 1.5 MV/m for BTO. The discontinuity of polarization versus temperature becomes faint after 1.2 MV/m. When the electric field strength is higher than 10 MV/m, the polarization is a weak function of temperature and the polarization variation decreases with the increasing electric field. Thus, a giant electric field can induce large ΔT but the ΔT/ΔE is small, and a flat ΔT–T profile is observed; as shown in Fig. 2(b).

Finally, we suggest that the finding in BTO is universal for ferroelectrics with first-order phase transition. At an electric field approaching E_CP, ferroelectric materials have small ΔT about 1 K but high EC strength, and a sharp EC ΔT–T peak. Under a strong electric field, they produce giant ΔT but low EC strength, and a flat ΔT–T profile is observed. Since ferroelectric ceramics in bulk have low dielectric strengths around E_CP, a sharp ΔT–T peak and tiny ΔT are normally observed. Because the dielectric strength is significantly enhanced in the thin films of ferroelectrics, a broad ΔT–T profile and large ΔT are observed. Table 1 lists some recent experimental observations on EC effect of BTO-based materials, which agree quantitatively with the theoretical prediction.

In conclusion, the present study indicates that there are two distinguishing EC responses for ferroelectrics with first-order phase transition, which are mainly decided by the applied electric field. When the applied electric field strength is comparable with E_CP, there exists a sharp ΔT–T peak with large ΔT/ΔE between [T_C, T_CP]. When the applied electric field strength is much higher than E_CP, because of the vanishing of the first-order phase transition, the polarization is a weak function of temperature. The large EC ΔT can be induced by a large electric field but the EC ΔT/ΔE is quite small. A flat EC ΔT–T profile is observed commonly. This result explains the experimental observations on BTO-based materials from bulks to thin films. It also inspires researchers to find alternative ways to achieve both large EC ΔT and ΔT/ΔE simultaneously.