Electrocaloric effects in the lead-free $\mathrm{Ba}(\mathrm{Zr},\mathrm{Ti}){\mathrm{O}}_3$ relaxor ferroelectric from atomistic simulations

Atomistic effective Hamiltonian simulations are used to investigate electrocaloric (EC) effects in the lead-free $\mathrm{Ba}\left({\mathrm{Zr}}_{0.5}{\mathrm{Ti}}_{0.5}\right){\mathrm{O}}_3$ (BZT) relaxor ferroelectric. We find that the EC coefficient varies nonmonotonically with the field at any temperature, presenting a maximum that can be traced back to the behavior of BZT's polar nanoregions. We also introduce a simple Landau-based model that reproduces the EC behavior of BZT as a function of field and temperature, and which is directly applicable to other compounds. Finally, we confirm that, for low temperatures (i.e., in nonergodic conditions), the usual indirect approach to measure the EC response provides an estimate that differs quantitatively from a direct evaluation of the field-induced temperature change.

Figure 1

Physical properties associated with the MC-1 method. (a) shows the temperature dependency of the polarization in BZT systems subject to different dc electric fields, all applied along the pseudocubic [001] direction but varying from $2.0\times {10}^7$ to $3.0\times {10}^8$ V/m in magnitude in steps of $2.0\times {10}^7$ V/m. (b) shows the resulting change in temperature as a function of $\mathrm{\Delta }\mathcal{E}={\mathcal{E}}_2-{\mathcal{E}}_1$ for four selected initial temperatures, as computed from Eq. (1) and choosing ${\mathcal{E}}_1=2.0\times {10}^7$ V/m. Note that (b) also further reports the direct change in temperature at 100 K as a function of ${\mathcal{E}}_f$.

Figure 2

Electrocaloric coefficient $\alpha$ as a function of the applied dc electric field $\mathcal{E}$, as predicted for the different indirect and direct approaches at (a) 100, (b) 200, (c) 300, and (d) 500 K. The solid green line represent the fit of the MC-1 and MC-2 results by the second line of Eq. (9), i.e., $\alpha =\beta T{\left.\frac{\partial P^2}{\partial \mathcal{E}}\right|}_T$, where $\beta$ is a constant and ${\left.\frac{\partial P^2}{\partial \mathcal{E}}\right|}_T$ is obtained from the data of Fig. 3. Error bars (resulting from the use of 20 different disordered alloy configurations) are also shown for the MC-2 data.

Figure 3

The square of the macroscopic polarization as a function of the applied dc electric field, at (a) 100, (b) 200, (c) 300, and (d) 500 K. The red line represents a fit by seventh degree polynomials, which were then used to calculate the derivative $d{P}^2/d\mathcal{E}$ that is shown in the corresponding central inset of each panel. The other insets of (a) show the dipolar configurations in a given $(x,z)$ plane at 100 K, as obtained from MC simulations for different dc electric fields (0, $1.2\times {10}^8$, and $3.0\times {10}^8$ V/m) applied along the pseudocubic [001] direction. In these latter insets, the blue and green colors indicate that the local dipoles are centered on Ti and Zr ions, respectively, and the red solid lines delimit the PNRs.

Figure 4

The ratio of dipoles that are pointing along $〈111〉$ directions having a positive $z$ component, as a function of the magnitude of the electric field applied along the pseudocubic [001] direction at 100 K. Note that these $〈111〉$ directions are thus away from the [001] field's direction.