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 to or removed from the material. It has received great attention since finding the 12 K temperature change in PbZr_0.95Ti_0.05O₃^[1] and more than 12 K temperature change in P(VDF-TrFE) (55/45 mol%),^[2] and potential applications in chip-scale cooling.^[3–13]

In the study of the EC effect, the indirect measurement has been used based on the Maxwell relations because of its simple testing setup and good consistency.^[14–18] In the method, the adiabatic temperature change ΔT can be evaluated from the following equation:^[19] 1 in which ρ and C_P are the density (in kg/m³) and specific heat capacity (in J/(K⋅kg)), respectively. By using Eq. (1), one must first choose and use appropriate parameters, i.e., specific heat capacity C_P, boundary conditions E₁ and E₂, and more importantly, appropriate temperature step ΔT and electric field step ΔE for precisely deciding (∂P / ∂T)_E. The E₂ is the maximum electric field applied, which is naturally smaller than the breakdown field of material. However, it is interesting to see that the selection principles for other parameters are generally omitted and different authors may adopt different parameters. Moreover, from the view of an experiment, in (E, T) field, it would take lots of time to give a rough answer to the influence of parameter selection on predicted EC ΔT.

Fortunately, the theoretical prediction EC ΔT by phenomenological method is also based on Eq. (1).^[20] Thus, if we use a numerical method rather than an analytical method, the influences of parameter selection on the EC effect can be investigated theoretically. More importantly, the results will help those who work on experiments as a reference to select appropriate parameters to evaluate the EC effect by indirect measurement.

In this work, we investigate the EC ΔT of BaTiO₃ (BTO), a prototype ferroelectric and also an EC material. For a monodomain BTO single crystal with the domain direction along [001] and electric field E along [001], the Landau–Devonshire potential including terms up to the eighth-power term under unconstrained, stress-free state can be written as 2 where α₁, α₁₁, α₁₁₁, and α₁₁₁₁ are the Landau coefficients and g₀ is the Gibbs free energy density without considering the contribution of polarization. In this work, we adopt the Landau coefficients proposed by Wang et al.^[21] The Landau potential for BTO has been developed by different authors, i.e., Mers,^[22] Pertsev et al.,^[23] Bell et al.,^[24] and Wang et al. The advantage of the potential proposed by Wang et al. is its adequate description of cubic-tetragonal phase transition by including either-order term, which exhibits a higher quality in reproducing polar properties. The polarization P under (T, E) field can be evaluated by finding the minimum of Gibbs free energy g, and by differentiating P with respect to T we can calculate EC ΔT according to Eq. (1). It is obvious that the method works in the same manner as experimental indirect measurement.

We first discuss the influence of specific heat capacity C_P on ΔT. It is normally believed that C_P is a function of temperature and electric field. However, almost all the work uses the room temperature zero-field specific heat capacity for calculating ΔT, or casually, the value at phase transition under zero-field.^[18] Is it right to neglect the influence of electric field on C_P?

In the classic ferroelectric book published by Lines and Glass, the variation of specific heat capacity under electric field is generally expressed as^[25] 3 It should be stressed that the entropy S here only counts the contribution of polarization. The contribution of phonons is excluded. For second-order phase transition, ΔC can be evaluated. For first-order phase transition, since there is a jump for entropy, the application for Eq. (2) will produce divergence. Thus, theoretically it is impossible to obtain an accurate value. The phase latent, which can be evaluated and relates to the jump of entropy, is in fact due to the EC effect under electric field. As a result, on the assumption that the entropy relating to phonon contribution is a weak function of electric field, it is reasonable to choose zero-field specific heat capacity. In this work, we choose C_P = 407 and ρ = 6020 to evaluate ΔT.^[19]

For other parameters, almost all experiments choose E₁ = 0 and only very recently it is shown that the initial electric field also has an influence on the EC effect.^[26,27] The choices of temperature step ΔT and electric field step ΔE are arbitrary. Here in this work, we study the influences of parameter choice of E₁, ΔT and ΔE on EC ΔT by a simple scenario, i.e., only one parameter may be arbitrarily changed but the others are fixed.

We first investigate the influence of temperature step ΔT on EC temperature variation ΔT by fixing E₁ = 0 and ΔE = 10⁴ V/m. We choose E₂ = 1.2 MV/m for comparing with the experimental results,^[18,28] which is also a typical value for bulk ferroelectric to switch domain without breakdown. The results are plotted in Fig. 1(a). The predicted ΔT at 1.2 MV/m equals 3.1 K, which is larger than two experimental results, i.e., 0.9 K@1.2 MV/m,^[18] or 1.6 K@1 MV/m,^[29] but smaller than one, i.e., 4.8 K@1 MV/m,^[30] suggesting the validation of prediction. By increasing temperature step ΔT, we can see that the strong EC ΔT zone becomes broader and the EC peak profile changes from asymmetric to symmetric when ΔT > 3 K. In fact, the asymmetric profile is supported by high-resolution direct experimental measurement.^[29] Thus, our conclusion obtained here is that a temperature step of 2 K is reasonable for evaluating the EC effect by the indirect method.

Fig. 1.

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	Fig. 1. (color online) Plots of predicted EC temperature change ΔT versus temperature of BTO under different parameter selections: (a) varying ΔT with ΔE = 104 V/m and E1 = 0; (b) varying ΔE with ΔT = 2 K and E1 = 0; (c) varying E1 with ΔE = 104 V/m and ΔT = 2 K.

Then by fixing the temperature step at 2 K with E₁ = 0, we investigate the influence of electric field step ΔE on ΔT as shown in Fig. 1(b). We can find that when ΔE ≤ 10⁴ V/m, which is the coercive field for BTO at 400 K, an almost identical EC effect is observed. After that, the instability appears. Thus, the result suggests that ΔE should be smaller than coercive field E_C.

Finally, we investigate the influence of E₁ of EC ΔT and we find that there is a big change of EC ΔT if E₁ > E_C. By increasing the initial electric field, the EC zone becomes narrow and the EC peak intensity decreases. Thus, the result suggests that E₁ < E_C is reasonable and E₁ > E_C can underestimate the peak value of EC ΔT. It is common to choose E₁ = 0^[31] or E₁ > E_C.^[20] For saving time and keeping validity, our results suggest that E₀ = E_C is reasonable for evaluating the EC effect.

In conclusion, combining Maxwell equation and phenomenological theory we study the EC ΔT of monodomain BaTiO₃ by emphasizing the influences of parameter selection to serve as a reference for experiments. Our results suggest that the parameters do have significant influence on predicted EC ΔT : the optimized temperature step ΔT is 2 K; the electric field ΔE should be smaller than E_C; and the maximum initial electric field is E_C. In addition, it is reasonable to use zero-field specific heat capacity for predicting the EC effect.

The author thanks professor Z Y Cheng at Auburn University for our meaningful discussion.