Influence of defects on ferroelectric and electrocaloric properties of ${\mathrm{BaTiO}}_3$

We report modifications of the ferroelectric and electrocaloric properties of ${\mathrm{BaTiO}}_3$ by defects. For this purpose, we have combined ab initio based molecular dynamics simulations with a simple model for defects. We find that different kinds of defects modify the ferroelectric transition temperatures and polarization, reduce the thermal hysteresis of the transition, and are no obstacle for a large caloric response. For a locally reduced polarization, the ferroelectric transition temperature and the adiabatic response are slightly reduced. For polar defects, an intriguing picture emerges. The transition temperature is increased by polar defects and the temperature range of the large caloric response is broadened. Even more remarkable, we find an inverse caloric effect in a broad temperature range.

Figure 1

Schematic sketch of the used defect model. (a) Nonpolar defects with locally frozen soft-mode amplitude, (b) fixed defect dipoles to which parallel or antiparallel external electric field is applied.

Figure 2

Polarization as function of temperature of ${\mathrm{BaTiO}}_3$; red: heating-up simulation; blue: cooling-down simulation; black: the sample has been equilibrated in 100 kV/cm and polarization and temperature have been recorded after field removal (ECE protocol). In all cases, the polarization along [100], [010], and [001] is shown. For the ECE protocol, the data without pressure correction (dashed black lines) is opposed to results for an external pressure correction of $p=-0.005T$, treating the acoustic degrees of freedom explicitly, and shifted by 150 K to lower temperatures (black stars).

Figure 3

Polarization $P_z$ of ${\mathrm{BaTiO}}_3$ as function of temperature $T$ with and without nonpolar defects. Defect concentrations ranges from 0%–3%. Upward and downward arrows indicate cooling-down (blue) and heating-up (red) simulations, respectively. Polarization under an external field of 100 kV/cm along $z$ is given in black. As the external field is larger than the critical field strength, the field-induced transition does not show thermal hysteresis anymore.

Figure 4

Change of polarization for different concentrations of nonpolar defects after an external field of 100 kV/cm has been removed.

Figure 5

Approximate adiabatic temperature change if an external field of 100 kV/cm is removed from samples with different concentration of nonpolar defects. Dotted lines illustrate the linear reduction of peak position and magnitude of the ECE.

Figure 6

Polarization of ${\mathrm{BaTiO}}_3$ with 1% polar defects aligned parallel to the external field of 100 kV/cm and after switching off the field. Black squares: no defects; blue open circles: $50\mu \mathrm{C}/{\mathrm{cm}}^2$; red circles: $100\mu \mathrm{C}/{\mathrm{cm}}^2$. (b) Corresponding approximate adiabatic temperature change.

Figure 7

Influence of antiparallel defects on the polarization of ${\mathrm{BaTiO}}_3$ for different defect densities (4%: black curve at the bottom; 1%: otherwise), strengths [local soft-mode amplitude of 0.4 Å (0.2 Å): stars (squares)] and different external fields (500 kV/cm: magenta; 50 kV/cm: cyan; 100 kV/cm otherwise) for cooling-down simulations. For 1% defects with a strength of 0.4 Å also results for heating-up (red) and initialization from scratch (black) as well as cooling-down simulation for parallel defects (green, with label $P)$ are given.

Figure 8

Polarization of BTO for 1% defects with a soft-mode amplitude of 0.2 Å under cooling down (solid blue lines) and heating up (dashed red lines) for different field strengths. In contrast to Fig. 7, a temperature-dependent expansive pressure has been applied. A $16\times 16\times 16$ supercell only has been used.

Figure 9

Approximate caloric response of BTO with 1% dipole defects if an external field of 100 kV/cm is removed. Black: no defects; cyan: parallel defects $\left(u\right)$; red: 50:50 parallel and antiparallel defects $\left(u/d\right)$; blue: antiparallel defects $\left(d\right)$ with soft-mode amplitudes of 0.4 Å (solid lines) and 0.2 Å (dashed lines). For the 0.4 Å amplitude also the caloric response after field cooling (FC) is shown.

Figure 10

Polarization for 1% polar defects with equal up/down distribution under different simulation protocols (cooling: blue; heating and after field removal: red; within 100 kV/cm: black). For comparison $P\left(T\right)$ under cooling is given for the ideal system (dotted lines). *: after field removal.

Figure 11

Illustration of the temperature-dependent lattice constants along and perpendicular to the tetragonal axis for the ideal system (black), 1% nonpolar defects (blue crosses), and 1% polar defects with 0.4 Å amplitude (dashed lines). Empty squares: no external field; empty circles: 50:50 distribution of parallel and antiparallel dipoles; filled circles: dipoles antiparallel to an external field of 100 kV/cm. Solid lines: cooling-down; dashed lines: after field removal.

Figure 12

Cooling-down and heating-up simulations including a temperature-dependent pressure correction for a $16\times 16\times 16$ simulation cell. 1% defects corresponding to a soft-mode amplitude of 0.2 Å have been taken into account for dipoles pointing randomly along the six cubic crystal directions (blue diamonds) and nonpolar defects (black). The polarization profiles without external field (solid lines) and in an external field of 100 kV/cm along $z$ (dashed lines) are opposed.

Figure 13

(a) Comparison of the adiabatic temperature change found by the direct (dotted lines) and indirect (solid lines) methods for 1% parallel defects with a strength corresponding to a soft-mode amplitude of 0.2 Å. Black: field range: 0–100 kV/cm; red: 0–50 kV/cm. Lines with symbols: ramping down the field with a rate of 0.05 kV/cm/fs, otherwise instantaneous ramping has been used. (b) Corresponding entropy change found by the indirect method.

Figure 14

Dependency of the adiabatic response on the ramping rate for 1% antiparallel defects with a strength corresponding to a soft-mode amplitude of 0.4 Å. The ramping rate is given in kV/cm/fs. Black and red: field removal; blue: field on.