Introduction

Ferroelectrics have received much attention in the recent years as potential candidates for solid state cooling applications1,2,3,4. Such applications take advantage of the electrocaloric effect (ECE) that is defined as a reversible change in temperature under the adiabatic application of an electric field. The interest in ferroelectrics as potential electrocaloric materials is largely due to the findings of giant ECE in several ferroics5,6,7,8. Subsequent studies revealed that ferroelectric electrocalorics might have other prospects, thanks to their potential to demonstrate multiple caloric effects9,10,11. Some examples include the coexistence of electrocaloric and piezocaloric effects in the same material10,11 as well as the coexistence of positive and negative ECE in the same ferroelectric below its Curie temperature9. From a practical point of view ferroelectric nanostructures appear to be most attractive as they allow the application of much larger electric fields1,5. Indeed, majority of giant electrocaloric effects were reported for thin film samples5,6,8,12. Interestingly, while ferroelectric thin films received much attention in the caloric effect studies, the research on other low-dimensional structures, such as nanowires and nanodots, is very limited13,14,15. Very large positive and negative ECE was predicted in PbTiO3 nanoparticles with a vortex domain structure13. Ref. 14 predicted giant ECE in ferroelectric nanotubes that originates from an unusual domain transition. Given the very limited number of studies, it is presently unclear what effect the reduced dimensionality has on the electrocaloric properties of ferroelectrics.

The aims of this Paper are: (i) to predict the intrinsic features of ECE in ferroelectric ultrathin nanowires; (ii) to elucidate the effect of reduced dimensionality and size on the electrocaloric properties of nanoscale ferroelectrics; (iii) to explore the potential of ferroelectric nanowires for nanoscale cooling applications.

We simulate three nanowires made of PbTiO3, BaTiO3 and KNbO3 ferroelectric perovskites. These materials are chosen as representatives of ferroelectrics with single (PbTiO3) and multiple (BaTiO3 and KNbO3) phase transitions. Bulk PbTiO3 undergoes a single transition from a paraelectric cubic to a ferroelectric tetragonal phase at 763 K16. Bulk BaTiO3 undergoes a sequence of phase transitions starting from paraelectric cubic to a ferroelectric tetragonal phase transition at 393 K, followed by a transition to a ferroelectric orthorhombic phase at 273 K and a transition to a ferroelectric rhombohedral phase at 183 K17. Bulk KNbO3 exhibits the same sequence of phase transitions as BaTiO3, but at elevated temperatures of 708 K (the Curie point), 498 K and 263 K18. Ferroelectric perovskites have been successfully synthesized in quasi-one-dimensional forms19. These include BaTiO3 nanowires20, PbTiO3 nanorods21 and rodlike nanocrystalline KNbO322. Here we model nanowires with a 12 × 12 unit cells square cross section grown along the <001> pseudocubic direction. Each nanowire is simulated using a 12 × 12 × 20 supercell which is periodic along the nanowire’s axial direction (z− Cartesian axis in our case). Such setup models a nanowire with the aspect ratio D/L ≪ 1, where D and L are the nanowire’s diameter and length, respectively. Note that some of our results were cross-checked using 12 × 12 × 24 supercell. No dependence on the supercell size along the periodic direction was found. The energy of the supercell is given by the first-principles effective Hamiltonian23. The degrees of freedom for the Hamiltonian include local modes, ui, that are proportional to the dipole moment in the unit cell and strain variables tensors ηi (in Voigt notations) that are responsible for mechanical deformations of a unit cell. The energy of the Hamiltonian is23

where EFE is the energy associated with the ferroelectric local modes and includes contributions from the dipole-dipole interactions, short-range interactions and on-site self energy as defined in ref. 23. The on-site self energy gives the energy of an isolated local mode with respect to the perfect cubic structure and contains harmonic as well as anharmonic contributions. The second term, Eelas, is the elastic energy associated with the unit cell deformations. EFE−elas is the energy contribution due to the interactions between the ferroelectric local modes and the strain. The last term, , where Z* is the Born effective charge, gives the interaction energy between the local modes and an external electric field, E. The first-principles parameters for PbTiO3 are those from ref. 24. Parameters for BaTiO3 and KNbO3 were computed from first-principles and and are given in Table 1. The Hamiltonian of Eq.(1) correctly reproduces most of the thermodynamical properties of bulk PbTiO3, BaTiO3 and KNbO3. In particular, the PbTiO3 Hamiltonian predicts a single transition from a paraelectric cubic to a ferroelectric tetragonal phase at 605 K. The present parametrization of BaTiO3 correctly reproduces the sequence of phase transitions in this material with a transition to a ferroelectric tetragonal phase at 405 K, to a ferroelectric orthorhombic phase at 260 K and to a ferroelectric rhombohedral phase at 195 K. Similarly, the present parametrization of KNbO3 predicts the correct sequence of phase transition in bulk KNbO3 with computational transition temperatures of 705, 430 and 345 K for the three phase transitions. Note, that the long-range dipole-dipole interactions in nanowires are computed using the approach of ref. 25.

Table 1 First-principles parameters for BaTiO3 and KNbO3 in atomic units using the notations of ref. 23.
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The surface of the nanowire creates a boundary at which the polarization discontinuity may occur. Such discontinuity could be eliminated (fully or partially) by an intrinsic or extrinsic free charge - the compensating charge. Here we simulate a realistic situation of a partial surface charge compensation by a limited number of free carriers. Technically this is achieved by compensating only 10% of the surface charge (open-circuit electrical boundary conditions) using the approach of ref. 25. Such setup models experimentally realizable conditions26,27. Under open-circuit electrical boundary conditions nanowires do not develop polarization along their truncated dimensions as it would result in a large residual depolarizing field19,27,28,29,30,31,32,33. The energy given by the effective Hamiltonian is used in the framework of classical Monte Carlo (MC) and Molecular Dynamics (MD) to compute finite temperature properties of the nanowires. Similar computational approach has been previously used to study various properties of ferroelectric nanowires28,34,35.

We begin by investigating the sequence of phase transitions in nanowires using the simulated annealing approach. In such an approach the simulations begin at temperature much above the Curie point and proceeds in steps of 5 K decrements until the simulated temperature reaches 5 K. For each temperature 3*105 MD steps are used. The temperature evolution of the polarization obtained from the annealing simulations is given in Fig. 1. For comparison we also include data for bulk. Note that the bulk material is simulated by applying periodic boundary conditions along all three Cartesian directions. Figure 1(a) shows the data for PbTiO3 nanowire and the bulk. It demonstrates that the reduction in dimensionality leads to a decrease in transition temperature and a smearing of the phase transition36. The decrease in transition temperature is in a qualitative agreement with the predictions from the semi-phenomenological theory37,38,39. For BaTiO3 nanowire (see Fig. 1(b)) we find very little change in the ferroelectric transition temperature as compared to the bulk. However, the phase transition sequence is drastically different from the one in the bulk. We do not find any polarization along the nanowire’s truncated dimensions due to a prohibitively large depolarizing field associated with the chosen electrical boundary conditions. The ferroelectric transition in the nanowire has a slightly smeared character as compared to bulk. KNbO3 nanowire data are given in Fig. 1(c) and exhibit trends similar to the BaTiO3 nanowire. In particular, the ferroelectric phase transition appears to be smeared, while no polarization is developed along the nanowire’s truncated dimensions.

Figure 1
figure 1

Dependence of the polarization components on the temperature in bulk and nanowires (a–c). Dependence of the dielectric constant on the temperature in bulk and nanowire (d–f). The insets give the toroidal moment of polarization.

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To gain further insight into the ferroelectric phases and phase transitions character we computed dielectric constant ε33 for all nanowires and their bulk counterparts using a direct simulation approach. In such an approach the electric field is applied along the z– Cartesian direction in increments of 36 kV/cm, while the polarization is computed for each value of the electric field. For each electric field increment 5*105 MD steps are used. The zero field slope in the polarization versus electric field data is used to calculate the dielectric constant. The data are given in Fig. 1(d–f). For PbTiO3 nanowire we observe a smearing of the dielectric constant. Similar trends were also observed for the ferroelectric thin films40. For the BaTiO3 nanowire the smearing of the dielectric constant is less pronounced. In this case we notice a small peak in the dielectric constant around 120 K. To trace the origin of the peak we turn to the dipole pattern evolution which reveals that at 135 K the nanowire undergoes a transition into a polydomain phase that is best described by a combination of two order parameters - the polarization and the toroidal moment of polarization41, G. The toroidal moment of polarization is defined as , where N is the number of unit cells and ri is the location of the local mode i41. In this case both order parameters align along nanowire’s axial direction. The temperature evolution of the toroidal moment of polarization is given in the inset to Fig. 1(e). A similar transition occurs in KNbO3 nanowire at a higher temperature of 420 K. The dielectric constant of KNbO3 nanowire is also smeared.

Our computational data predict that ferroelectric nanowires with a poorly compensated surface charge develop ferroelectric phases with a polarization along the axial direction. Nanowires may develop polydomain phases with multiple order parameters. The common features are the smearing of the phase transition and the dielectric constant. They can be attributed to the decrease in the correlation length due to the reduced dimensionality. Previously, the correlation effects were found to decrease the transition temperature in nanorods38. Smaller correlation lengths are usually associated with diffuse phase transitions42. Interestingly, ferroics with smeared phase transitions are quite desirable for caloric applications as they usually demonstrate a broadened caloric response43.

Having established the equilibrium phases and electric properties of nanowires, we turn to the modeling of the ECE. To simulate the ECE we apply an electric field along the nanowire’s axial direction under adiabatic conditions using the computational approach of refs 9,10. In bulk the electric field is applied along the polarization direction. Technically, the electric field was first applied and then removed very slowly at a rate of 100 V/m per one MC sweep to ensure reversibility. The electrocaloric temperature was computed as a function of the applied field. The electrocaloric change in temperature as a function of the initial temperature is given in Fig. 2 for a few different values of the applied electric field. We have also included data for the linear electrocaloric response, dT/dE, at low fields which were computed by taking the zero field slope of temperature versus electric field data. It should be noted that our computational data for BaTiO3 are in excellent agreement with the direct experimental measurements on BaTiO3 multilayer thick film (see Fig. 2(b)). We begin the discussion with the PbTiO3 nanowire. Here we find a reduction in the maximum electrocaloric response as compared to the bulk. Such a reduction could be elucidated with the help of Maxwell relation for the electrocaloric change in temperature , where CE is the volumetric heat capacity, P is the polarization, T and E are the temperature and the electric field, respectively. For low fields the relationship can be approximated as , where Pspon is the spontaneous polarization and ε is the dielectric constant. The latter expression suggests that the linear ECE is proportional to the pyroelectric coefficient and the derivative of the dielectric constant with respect to temperature. Both the spontaneous polarization and the dielectric constant are reduced in the nanowire as evident from Fig. 1 due to the reduced correlation length. This explains the observed decrease in the maximum ECE in the nanowire as compared to bulk. Similarly, we observe a reduced maximum electrocaloric response in both BaTiO3 and KNbO3 nanowires as compared to their bulk counterparts. Both BaTiO3 and KNbO3 mostly loose their electrocaloric properties in polydomain phases. To elucidate the reason for this we first notice that below the ferroelectric transition temperature the ECE is dominated by the pyroelectric coefficient. Indeed in a ferroelectric phase the two terms in the Maxwell relation have the opposite sign (as could be seen from inspection of Fig. 1) and compete with each other. The overall sign of the ECE is determined by the largest of the two terms and is positive in this case (see Fig. 2). The positive sign of ECE implies that the pyroelectric coefficient makes a dominant contribution to the ECE. In the polydomain phase the spontaneous polarization depends on the temperature only weakly (see Fig. 1) resulting in very small values of the pyroelectric coefficient and the associated ECE. More intuitively, the reduction of ECE in a polydomain phase could be understood by recalling that inside each domain the polarization is at an angle with the applied electric field. As a result the application of a relatively low electric field does not significantly affect the configurational disorder and the entropy associated with it. The lack of the entropy change results in a negligible ECE.

Figure 2
figure 2

The electrocaloric change in temperature as a function of initial temperature for a few values of the electric field [(a–c)]. In (b) the triangles give the experimental data from the direct measurements on thick films45. The linear ECE as a function of the initial temperature. Note, that for bulk the electric field is applied along the polarization direction.

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To investigate the potential of the nanowires for cooling applications we computed the relative cooling power44 RCP = ΔTmax × δTFWHM, where ΔTmax is the maximum of the electrocaloric temperature change and, δTFWHM is the full width at half maximum. The RCP values for different electric fields are given in Table 2. In all cases we find that the RCP is reduced in nanowires, on an average by 20%.

Table 2 RCP values in K2 for different electric fields.
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Interestingly, our computational data predict that, while the maximum electrocaloric ΔT as well as RCP are reduced in nanowires, their room temperature response could be significantly enhanced. Indeed all nanowires exhibit a decrease in the ferroelectric transition temperature and associated shift in both dielectric and electrocaloric response (see Figs 1(c,d) and 2(c,d)). As a result the room temperature ΔT of BaTiO3 nanowire remains comparable to ΔT in bulk, while PbTiO3 nanowire exhibits a 30% enhancement in the room temperature ΔT, which is very attractive for near room temperature refrigeration.

In summary, we studied the ECE in poorly compensated ferroelectric ultrathin nanowires and compared the findings to the ECE in bulk ferroelectrics. The computational data demonstrate a reduction in the ECE in nanowires as compared to bulk which is attributed to the reduced correlation length. In nanowires with polydomains the ECE is nearly negligible. The electrocaloric change in temperature exhibits a strong correlation with the dielectric susceptibility. Among the three ferroelectric perovskites studied in this work we find the largest ECE and RCP in PbTiO3 followed by KNbO3 and BaTiO3. This behavior is well correlated with the trends in the dielectric susceptibility and the spontaneous polarization data where we find the largest values in PbTiO3, followed by KNbO3 and BaTiO3. While nanowires exhibit reduction in maximum ECE, they offer the opportunity to tune the electrocaloric temperature change through variation of the transition temperature. For example, in case of PbTiO3 nanowires we find nearly 30% increase in the room temperature ΔT, which makes such nanowires attractive for room temperature cooling applications.

Additional Information

How to cite this article: Herchig, R. et al. Electrocaloric effect in ferroelectric nanowires from atomistic simulations. Sci. Rep. 5, 17294; doi: 10.1038/srep17294 (2015).