First-principles-based calculation of the electrocaloric effect in ${\mathrm{BaTiO}}_3$: A comparison of direct and indirect methods

We use molecular dynamics simulations for a first-principles-based effective Hamiltonian to calculate two important quantities characterizing the electrocaloric effect in ${\mathrm{BaTiO}}_3$, the adiabatic temperature change $\mathrm{\Delta }T$ and the isothermal entropy change $\mathrm{\Delta }S$, for different electric field strengths. We compare direct and indirect methods to obtain $\mathrm{\Delta }T$ and $\mathrm{\Delta }S$, and we confirm that both methods indeed lead to an identical result provided that the system does not actually undergo a first order phase transition. We also show that a large electrocaloric response is obtained for electric fields beyond the critical field strength for the first order phase transition. Furthermore, our work fills several gaps regarding the application of the first-principles-based effective Hamiltonian approach, which represents a very attractive and powerful method for the quantitative prediction of electrocaloric properties. In particular, we consider the full temperature and field dependence of the calculated specific heat for the indirect calculation of $\mathrm{\Delta }T$, and we discuss the importance of maintaining thermal equilibrium during the field ramping when calculating $\mathrm{\Delta }T$ using the direct method within a molecular dynamics approach.

Figure 1

(a) Schematic depiction of the simulation cycle (see text). (b) and (c) The total energy as a function of MD steps for instantaneous field application/removal and for slow field ramping, respectively. Here the starting temperature $T_{\text{i}}$ is 530 K and the applied field is 100 kV/cm. In (c) the rate of change of the applied field is equal to $0.05\mathrm{kV}{\mathrm{cm}}^{-1}{\mathrm{fs}}^{-1}$. For clarity, the total energy is plotted only after the system is switched to the microcanonical ensemble.

Figure 2

The change in the total energy on switching off the applied field is plotted as a function of applied electric field for (a) the paraelectric phase at $T_{\text{i}}=530$ K and (b) the ferroelectric phase at $T_{\text{i}}=270$ K. The rate of change of the applied field $d\mathcal{E}/dt$ is equal to $-0.05{\mathrm{kV}\mathrm{cm}}^{-1}{\mathrm{fs}}^{-1}$ for the “ramping” case (red squares). The symbols show the data points, whereas the dashed lines are fits to the corresponding functional forms listed in Table 1.

Figure 3

(a) EC temperature changes for switching the electric field on and off, $\mathrm{\Delta }{T}_{\text{on}}$ and $\mathrm{\Delta }{T}_{\text{off}}$, as function of the inverse rate of change of the applied electric field ${(d\mathcal{E}/dt)}^{-1}$ for a starting temperature $T_{\text{i}}=350$ K. The applied field ${\mathcal{E}}_{\text{app}}$ is equal to 200 kV/cm. The simulation cycle used to obtain $\mathrm{\Delta }{T}_{\text{on/off}}$ is depicted in Fig. 1. (b) The EC temperature change $\mathrm{\Delta }T$ as a function of temperature is plotted for slow ramping and instantaneous ramping of the field. The field is varied from 225 to 0 kV/cm.

Figure 4

Calculated specific heat of the model Hamiltonian as a function of temperature for different applied electric fields. For better comparison, the Dulong-Petit value of $3k_B$ is indicated by the thick horizontal black line and the obtained high temperature limit is indicated by the thin horizontal red line.

Figure 5

(a) Comparison of the EC temperature change as function of the initial temperature obtained using direct and indirect methods. The field is varied from 300 to 75 kV/cm. In the direct calculation slow field ramping with $d\mathcal{E}/dt=-0.002{\mathrm{kV}\mathrm{cm}}^{-1}{\mathrm{fs}}^{-1}$ is used. (b) The EC entropy change as a function of temperature obtained from direct and indirect methods. Here the applied field is varied from 75 to 300 kV/cm. The results of the direct calculations, but with an offset such that it matches the indirect calculation at $T=200$ K (see text), are indicated by the blue dotted line.

Figure 6

EC temperature change $\mathrm{\Delta }T$ as function of temperature for different field intervals with the same total width of 50 kV/cm. Results obtained using the direct (indirect) method are shown as dotted lines with symbols (solid lines without symbols).