Colossal room-temperature electrocaloric strength aided by hydrostatic pressure in lead-free multiferroic solid solutions

Phase competition in BFCO under pressure. At room temperature and zero pressure, BiFe_1-xCo_xO₃ solid solutions (BFCO) can be stabilized in two different polymorphs, depending on the relative content of Fe/Co atoms, exhibiting rhombohedral (${\cal R}$) and tetragonal (${\cal T}$) symmetries bfco1 ; bfco2 ; menendez20 . For relative cobalt contents of $0\leq x\lesssim 0.25$, the BFCO ground state is the ${\cal R}$ phase, which is analogous to the ground state of bulk BiFeO₃ bfco1 ; bfco2 ; menendez20 ; cazorla13 . This rhombohedral phase presents an electric polarization of $60$–$80$ $\mu$C cm^-2 that is oriented along the pseudocubic direction $[111]$ (Fig. 1a) and G-type antiferromagnetic spin ordering (AFM-G, the net magnetic moment of each transition metal ion is antiparallel to those of its six first nearest neighbours). For larger relative cobalt contents, $0.25<x$, the BFCO ground state corresponds to the ${\cal T}$ phase, which is analogous to the ground state of bulk BiCoO₃ bfco1 ; bfco2 ; menendez20 ; cazorla17 ; cazorla18 . This tetragonal phase presents a giant electric polarization of $160$–$180$ $\mu$C/cm² oriented along the pseudocubic direction $[001]$ (Fig. 1a), hence sometimes it is referred to as “supertetragonal”, and C-type antiferromagnetic spin ordering (AFM-C, the net magnetic moment of each transition metal ion is parallel to those of its two first nearest neighbours located along the polar axis and antiparallel to those of its other four first nearest neighbours).

Under increasing temperature and for relative cobalt contents of $x\lesssim 0.25$, the supertetragonal ${\cal T}$ phase can be stabilized over the ${\cal R}$ phase owing to its larger vibrational entropy bfco1 ; bfco2 ; menendez20 . Such a $T$-induced phase transition clearly is of first-order type (or discontinuous) since the volume change associated to it is huge ($\sim 10$%). To the best of our knowledge, there are not experimental studies on BFCO under pressure. Here, we amend for such a lack of information by carrying out accurate first-principles calculations based on density functional theory (DFT, Methods) menendez20 ; cazorla17d . Figure 1b shows the estimated hydrostatic pressure that is necessary to drive the ${\cal T}\to{\cal R}$ phase transition at low temperatures (i.e., disregarding entropy and also likely quantum nuclear effects) and for compositions in the interval $0.25\leq x\leq 0.50$. This transition pressure is found to steadily, and significantly, decrease under increasing Fe content. For instance, $p_{t}$ amounts to $1.4$ GPa at $x=0.50$ and to $0.3$ GPa at $x=0.25$. As expected, the closer the cobalt content is to the ${\cal T}$–${\cal R}$ morphotropic phase boundary ($x_{c}\approx 0.25$), the easier results to switch from the supertetragonal to the rhombohedral phase with pressure.

Simulating temperature effects in materials with first-principles methods is computationally very intensive and laborious. However, temperature effects are critical for the assessment of possible caloric phenomena hence cannot be neglected in the present study. We employed the quasi-harmonic approximation (QHA) menendez20 ; cazorla17d to calculate ab initio Gibbs free energies for BFCO in the ${\cal T}$ and ${\cal R}$ phases over broad pressure, temperature and electric field conditions, thus allowing for the estimation of barocaloric and electrocaloric effects (Methods).

Figure 1c shows the $p$–$T$ phase diagram calculated for BFCO at a composition of $x=0.50$, hereafter referred to as BFCO_0.5. Therein, it is appreciated that $p_{t}$ consistenly increases under raising temperature, reaching a value of $1.24$ GPa at room temperature. In spite of such a relatively large pressure, in what follows we present multicaloric results obtained for bulk BFCO_0.5 at and near room temperature since from a computational point of view this solid solution is highly affordable (i.e., the size of the corresponding simulation cells are among the smallest thus making the QHA free energy calculations feasible). In practice, much smaller pressures of the order of $0.1$ GPa can be attained by reducing the relative content of Co ions (Fig. 1b) without significantly affecting the main conclusions presented in the next sections.

Barocaloric performance of BFCO_0.5. We start by analyzing the barocaloric effects induced by hydrostatic pressure in bulk BFCO_0.5 in the absence of electric fields. Figures 2a-b show the compression required to induce the $\cal{T}\to\cal{R}$ phase transition as a function of temperature, $p_{t}$, and the accompanying relative volume change. The estimated phase transition volume change is negative and very large as it amounts to $\sim 8$% in absolute value. Such a huge relative volume change augurs a large phase transition entropy change, as it can be inferred from the Clausius-Clapeyron relation $\Delta S_{t}=\Delta V\cdot\frac{d{p}_{t}}{dT}$. However, after doing the calculations and assuming that $\Delta S_{\rm BC}\approx\Delta S_{t}$ (Methods), it was found that the ensuing barocaloric isothermal entropy shifts were actually quite modest (Fig. 2c). For instance, at room temperature we obtained $|\Delta S_{\rm BC}|=1.7$ J K^-1 mol^-1 ($5.4$ J K^-1 kg^-1), which is about one order of magnitude smaller than the giant barocaloric entropy changes found in superionic and plastic crystals ($\sim 100$ J K^-1 kg^-1) aznar17 ; cazorla16 ; cazorla17b ; cazorla18b ; lloveras19 ; li19 ; cazorla19b ; sau21 ; cazorla19 . Under decreasing temperature, $|\Delta S_{\rm BC}|$ slightly increases (e.g., $2.8$ J K^-1 mol^-1 at $T=200$ K) however the estimated values still are quite reduced. The reason for these outcomes is that $p_{t}$ barely changes with temperature in the explored thermodynamic range (i.e., the temperature derivative of the phase transition pressure amounts only to $\sim 10^{-3}$ GPa K^-1, Fig. 2a).

The revealed minute $T$-induced $p_{t}$ variation, on the other hand, implies sizeable changes in the phase transition temperature, $T_{t}$, induced by small pressure shifts (since $d{T}_{t}/dp=\left[d{p}_{t}/dT\right]^{-1}$), thus suggesting possibly large barocaloric thermal shifts in bulk BFCO_0.5. Figure 2d shows the barocaloric adiabatic temperature changes, $\Delta T_{\rm BC}$, estimated as a function of temperature (Methods). At room temperature ($T=200$ K), $\Delta T_{\rm BC}$ was found to amount to $4.7$ K ($6.5$ K) which, although it cannot rival with the barocaloric adiabatic temperature changes reported for superionic and plastic crystals ($\sim 10$ K) aznar17 ; cazorla16 ; cazorla17b ; cazorla18b ; lloveras19 ; li19 ; cazorla19b ; sau21 ; cazorla19 , it shows promise in the context of electrocaloric effects ($\sim 1$–$10$ K).

The barocaloric results presented above were obtained with the indirect Clausius-Clayperon (CC) method, which is not exact aznar17 . Aimed to assess the extent of the employed approximations, we mimicked with theory quasi-direct barocaloric experiments aznar17 ; lloveras19 in which entropy curves are estimated as a function of pressure and temperature and from which $\Delta S_{\rm BC}$ and $\Delta T_{\rm BC}$ can be straightforwardly determined (Figs. 3a-b) cazorla19 . Moreover, with this quasi-direct estimation approach is also possible to determine for a given pressure shift, $\Delta p$, the temperature span, $T_{\rm span}$, over which barocaloric effects can be operated (Fig. 3b). In view of the huge $d{T}_{t}/dp$ of $\sim 10^{3}$ K GPa^-1 estimated for BFCO_0.5, giant $T_{\rm span}$ values are anticipated li20 .

Figure 3c shows the results of our quasi-direct barocaloric descriptor estimations. At room temperature and $T=200$ K, we obtained adiabatic temperature changes of $2.0\pm 2.5$ and $4.0\pm 2.5$ K, respectively. Within the numerical uncertainties, these results are compatible with our previous estimations obtained with the CC method; however, it goes without saying that the reported error bars are unacceptably too large. The reasons for the relatively huge numerical uncertainties on $\Delta T_{\rm BC}$ are the small $\Delta S_{t}$ and great $p$-induced $T_{t}$ shifts involved in its quasi-direct estimation (Fig. 3a). Thus, unfortunately, in the present case it is not possible to discern the actual precision of the barocaloric adiabatic temperature changes obtained with the approximate CC method. Nevertheless, the estimation of $T_{\rm span}$ is still possible given its noticeably large size (Figs. 3a-b). By considering an outset compression of $1.18$ GPa, we obtained $T_{\rm span}\approx 60$ K for a small pressure shift of $0.06$ GPa (calculated by adding up all the $\Delta T_{\rm span}$ increments shown in Fig. 3c). This result is very encouraging since it indicates that, in spite of the relative smallness of $\Delta S_{\rm BC}$ and $\Delta T_{\rm BC}$, barocaloric effects in BFCO_0.5 could be operated over unusually ample temperature ranges.

Electrocaloric performance of pressurized BFCO_0.5. The electric polarization, $P$, of BFCO_0.5 in the ${\cal R}$ and ${\cal T}$ phases are significantly different; for instance, $P$ in the supertetragonal phase is more than two times larger than that in the rhombohedral phase menendez20 , adding up to polarization module differences of $>100$ $\mu$C cm^-2 (Fig. 4b). Such a huge electric polarization disparity seems very promising from an electrocaloric (EC) point of view, as it can be inferred from the electric Clausius-Clapeyron relation $\Delta S_{t}=\Delta\bm{P}\cdot\frac{d\bm{{\cal E}_{c}}}{dT}$, where $\Delta S_{t}$ represents the entropy change associated to the field-induced phase transition and ${\cal E}_{c}$ the necessary electric field to switch from the ${\cal R}$ to the ${\cal T}$ phase. Figure 4a shows the ${\cal E}_{c}$ estimated for a fixed pressure of $1.25$ GPa as a function of temperature (Methods), which has been selected to ensure proper stabilization of the ${\cal R}$ phase under conditions $T\leq 300$ K. As clearly appreciated therein, the critical electric field steadily decreases under increasing temperature, ranging from $43$ kV cm^-1 at $200$ K to $\approx 2$ kV cm^-1 at room temperature.

Figures 4c-e show the electrocaloric isothermal entropy and adiabatic temperature changes, $\Delta S_{\rm EC}$ and $\Delta T_{\rm EC}$, estimated for compressed BFCO_0.5 using the indirect CC approach (Methods). In this case, the sign of the EC descriptors indicate that the caloric effect is inverse (i.e., $\Delta T<0$), which follows from the fact that the high-entropy phase ${\cal T}$ presenting the largest electric polarization is stabilized via application of the external electric bias. As expected, the size and temperature dependence of $|\Delta S_{\rm EC}|$ and $|\Delta T_{\rm EC}|$, which are directly related through the temperature and heat capacity (Fig. 4d, Methods), are very much similar to those of $|\Delta S_{\rm BC}|$ and $|\Delta T_{\rm BC}|$ since the underlying phase transitions are equivalent. For instance, at $T=200$ K we estimated an electrocaloric adiabatic temperature change of $-6.9$ K and at room temperature of $-4.8$ K, to be compared with the analogous barocaloric shifts of $+6.5$ and $+4.7$ K. These $\Delta T_{\rm EC}$ values are very much promising, specially when considering the small size of the required driving electric fields (that is, ${\cal E}_{c}\sim 1$–$10$ kV cm^-1).

Figure 4f encloses results for the electrocaloric strength of BFCO_0.5, $\Lambda_{\rm EC}$, expressed as a function of temperature; this quantity is defined like the ratio of $\Delta T_{\rm EC}$ by the corresponding electric bias. At $T=200$ K, the attained adiabatic temperature change is highest however the required switching electric field is also largest, thus the resulting electrocaloric strength is smaller than obtained at higher temperatures. Still, the calculated $\Lambda_{\rm EC}$ amounting to $0.2$ K cm kV^-1 is already comparable to the record experimental values reported for oxide and hybrid organic-inorganic perovskites bto1 ; bto2 ; ec7 . Remarkably, under increasing temperature the electrocaloric strength of BFCO_0.5 noticeably increases reaching a maximum, and colossal, value of $2.2$ K cm kV^-1 at $T=300$ K. These figures will be put into context in a next section; in what follows, we explain how the dual response of BFCO_0.5 to mechanical and electric stimuli may be exploited in practical solid-state cooling cycles.

Proposed $p$–$E$ multicaloric cycle. Single stimulus solid-state cooling cycles typically consists of four thermodynamic steps, two involving the adiabatic switching on and off of the applied external field and the other two constant-field heat transfer processes with the environment and the system to be refrigerated cazorla19 . In the present work, we propose an original multi-stimuli solid-state cooling cycle consisting of eight thermodynamic steps that has been designed to minimize the applied electric field, thus maximizing $\Lambda_{\rm EC}$, and with a cumulative multicaloric performance of $|\Delta T_{\rm MC}|=|\Delta T_{\rm BC}|+|\Delta T_{\rm EC}|$ and $|\Delta S_{\rm MC}|=|\Delta S_{\rm BC}|+|\Delta S_{\rm EC}|$.

Figure 5 sketches the envisaged multi-stimuli solid-state cooling cycle comprising hydrostatic pressure and electric fields being applied on multiferroic lead-free BFCO solid solutions near room temperature. The cycle starts with multiferroic BFCO in the supertetragonal $\cal{T}$ phase at temperature $T$. Subsequently, hydrostatic pressure is adiabatically applied on BFCO so that it transforms into the $\cal{R}$ phase and experiences a temperature increase of $|\Delta T_{\rm BC}|$. In the third step, heat is released to the ambient, $\delta Q_{\rm BC}$, and the initial temperature of the cycle is restored; compressed BFCO still remains in the $\cal{R}$ phase. Next, an electric field is adiabatically applied on compressed BFCO so that it transforms into the $\cal{T}$ phase, thus experiencing a temperature decrease of $|\Delta T_{\rm EC}|$. In the fifth step, heat is absorbed by the system, $\delta Q_{\rm EC}$, and the initial temperature of the cycle is restored; compressed and electrically biased BFCO remains in the $\cal{T}$ phase. Subsequently, the electric field is adiabatically removed thus BFCO transforms into the $\cal{R}$ phase and experiences a temperature increase of $|\Delta T_{\rm EC}|$. In the seventh step, heat is released to the ambient, $\delta Q_{\rm EC}$, and the initial temperature of the cycle is restored; compressed BFCO remains in the $\cal{R}$ phase. Finally, hydrostatic pressure is adiabatically released so that BFCO transforms back into the $\cal{T}$ phase and experiences a temperature decrease of $|\Delta T_{\rm BC}|$. Heat then is absorbed by the system, $\delta Q_{\rm BC}$, and the initial temperature of the cycle is restored, thus completing an entire multi-stimuli cycle.

Upon completion of a multi-stimuli cycle, multiferroic BFCO is able to remove an amount of heat equal to $|\delta Q_{\rm BC}|+|\delta Q_{\rm EC}|$, or equivalently, $T\cdot\left(|\Delta S_{\rm BC}|+|\Delta S_{\rm EC}|\right)$, from the targeted system to be refrigerated and release it to the ambient (thus cooling it down). The described multi-stimuli cycle lends itself to several useful variations. For instance, the state reached in the seventh step is thermodynamically equivalent to that attained in the third; therefore, one could recursively perform the electrocaloric subcycle consisting of steps (3)–(4)–(5)–(6) which entails application and removal of an electric bias under fixed hydrostatic pressure (dashed lines in Fig. 5). Likewise, if the multi-stimuli cooling cycle starts with multiferroic BFCO in the rhombohedral $\cal{R}$ phase instead of the $\cal{T}$ phase, due to some composition synthesis constraints, for example, then the sequential application of hydrostatic pressure and electric field explained above needs to be swapped.

Table I summarizes some representative materials for which EC effects occurring at or near room temperature have been experimentally measured and reported in the literature. The selected compounds belong to three different families of ferroelectric materials, namely, oxides (e.g., HfO₂ and BaTiO₃), hybrid organic-inorganic perovskites ([(CH₃)₂CHCH₂NH₃]₂PbCl₄) and polymers (Terpolymer). In terms of largest $|\Delta T_{\rm EC}|$, the oxides Y-HfO₂ ec0 and BNBT-BCZT ec6 and the elastomer Terpolymer/PMN-PT ec3 emerge as the most promising since they display colossal values of $20$–$30$ K. Nevertheless, these record materials require of quite large electric fields to realize their full EC potential (${\cal E}_{c}\sim 10^{3}$ kV cm^-1), hence with no exception their associated electrocaloric strengths turn out to be quite mediocre, namely, $\Lambda_{\rm EC}\sim 0.01$ K cm kV^-1.

Ferroelectric materials exhibiting moderate or even small $|\Delta T_{\rm EC}|$ but attained under smaller electric fields (${\cal E}_{c}\sim 10$ kV cm^-1), on the other hand, become the clear winners in terms of largest $\Lambda_{\rm EC}$. For instance, the archetypal perovskite oxide BaTiO₃ renders an adiabatic temperature change of roughly $1$ K driven by a minute electric field of $4$ kV cm^-1, thus leading to a huge electrocaloric strength of $0.23$ K cm kV^-1 bto1 ; bto2 . Likewise, the hybrid organic-inorganic perovskite [(CH₃)₂CHCH₂NH₃]₂PbCl₄ holds the record $\Lambda_{\rm EC}$ value of $0.37$ K cm kV^-1, which results from a small electric field of $30$ kV cm^-1 and an adiabatic temperature change of $11.1$ K ec7 . It is worth noting that all these figures correspond to experimental data.

Table I encloses also the EC results that we have theoretically estimated in this study for pressurized BFCO_0.5 at room temperature. According to our QHA-DFT calculations, compressed multiferroic BFCO solid solutions have the potential to surpass all previously known EC materials in terms of largest $\Lambda_{\rm EC}$. In particular, we predict an outstanding electrocaloric strength of $2.18$ K cm kV^-1 that arises from an adiabatic temperature change of $4.8$ K and an electric bias of $\approx 2$ kV cm^-1. This theoretically estimated $\Lambda_{\rm EC}$ value is from one to two orders of magnitude larger than those experimentally measured in uncompressed ferroelectrics. The key mechanism in achieving such a colossal figure is to employ an ancillary field, in our case hydrostatic pressure, to bring the system towards the verge of a ferroelectric phase transition so that it is possible to drive it with a minuscule electric field.

In the specific case considered here, the pressure required to achieve the colossal $\Lambda_{\rm EC}$ value of $2.18$ K cm kV^-1 is higher than $1$ GPa. Obviously, this compression is too large to be considered for practical applications. Nevertheless, as it was argued at the begining of the Results section, it is possible to significantly reduce the size of this ancillary pressure to the order of $0.1$ GPa by decreasing the relative content of cobalt ions down to the critical composition of $\approx 0.25$. Moreover, the $\Lambda_{\rm EC}$ enhancement approach proposed in this study, and theoretically demonstrated for BFCO_0.5, in principle should be generalizable to many other well-known EC materials since most of them are responsive to pressure as well (even though the magnitude of the resulting BC effects may be quite small in comparison to those achieved in state-of-the-art barocaloric materials). Take the archetypal ferroelectric compound BaTiO₃ as an example. The ferro- to paraelectric phase transition temperature in this material can be effectively shifted with pressure, namely, $d{T}_{t}/dp\approx-25$ K GPa^-1 hayward02 , thus its room-temperature EC performance could be potentially improved with our proposed strategy. Finally, to mention that recent developments in the synthesis of ferroelectric membranes and thin films may also allow for the enhancement of the $\Lambda_{\rm EC}$ figure-of-merit by combining electric fields with other types of mechanical stimuli like uniaxial zang22 and biaxial liu16b stress.

In conclusion, we have proposed a new strategy for the enhancement of the electrocaloric strength of ferroelectric materials that consists in concertedly applying pressure and electric fields. We have theoretically proved our new concept on multifunctional BFCO solid solutions, an intriguing family of compounds displaying a discontinuous phase transition between two multiferroic states. In particular, for compressed BFCO_0.5 we estimated a record $\Lambda_{\rm EC}$ parameter of $2.18$ K cm kV^-1 at room temperature resulting from an adiabatic temperature change of $4.8$ K and an electric bias of $\approx 2$ kV cm^-1. This electrocaloric strength turns out to be colossal since it is from one to two orders of magnitude larger than those experimentally measured in uncompressed ferroelectrics. The demonstrated $\Lambda_{\rm EC}$ enhancement strategy can be applied to other types of ferroelectric materials, not necessarily magnetic, and be modified at convenience on the mechanical component. Thus, the combination of multiple stimuli opens new horizons in the field of caloric materials and solid-state refrigeration by expanding the design of possible cooling cycles and boosting current caloric performances. We hope that the present theoretical study will motivate new experimental works on the engineering of original and environmentally friendly solid-state cooling devices.

Spin-polarized DFT calculations were performed with the generalized gradient approximation proposed by Perdew, Burke and Ernzerhof (PBE) as it is implemented in the VASP package vasp ; pbe96 . The “Hubbard-$U$” scheme due to Dudarev et al. was employed in the PBE calculations for treating better the Co (Fe) $3d$ electrons, adopting a $U$ value of $6$ ($4$) eV menendez20 ; menendez20b ; cazorla17 ; cazorla18 ; cazorla13 . The “projected augmented wave” method bloch94 was used to represent the ionic cores considering the following electronic states as valence: Co $4s^{1}3d^{8}$, Fe $3p^{6}4s^{1}3d^{7}$, Bi $6s^{2}5d^{10}6p^{3}$, and O $2s^{2}2p^{4}$. An energy cut-off of $800$ eV and a $\Gamma$-centered ${\bf k}$-point grid of $4\times 6\times 6$ were employed for a $2\times\sqrt{2}\times\sqrt{2}$ simulation cell containing $20$ atoms cazorla15 , thus obtaining zero-temperature energies converged to within $0.5$ meV/f.u. Geometry relaxations were performed for an atomic force threshold of $0.005$ eV$\cdot$Å^-1. Electric polarizations were accurately estimated with the hybrid HSE06 functional hse06 and the Berry phase formalism king93 ; resta94 ; bellaiche99 .

Ab initio free energies were estimated within the quasi-harmonic approximation (QHA) cazorla13 ; cazorla17d ; phonopy as a function of $p$ and $T$. Phonon frequencies were calculated with the small displacement method phonopy . The following technical parameters provided QHA free energies converged to within $5$ meV per formula unit: $160$-atom supercells, atomic displacements of $0.01$ Å, and ${\rm q}$-point grids of $16\times 16\times 16$ for integration within the first Brillouin zone. The effects of chemical disorder were addressed by generating all possible atomic Co–Fe and magnetic spin arrangements (ferromagnetic –FM– and antiferromagnetic –AFM– of type A, C, and G) for a $2\times 2\sqrt{2}\times\sqrt{2}$ supercell containing $40$ atoms. Quasi-harmonic free energies were calculated only for the lowest-energy configurations. Our spin-polarized DFT calculations were performed for bulk BiFe_0.5Co_0.5O₃.

Within the QHA cazorla13 ; cazorla17d ; phonopy , the Gibbs free energy of a given crystal phase, $G_{\rm harm}$, is expressed as:

$$G_{\rm harm}(p,T)=E(p)+pV(p,T)+F_{\rm harm}(p,T)~{},$$

(1)

where $E$ is the static energy of the system (i.e., as directly obtained from zero-temperature DFT calculations), $p$ the pressure, $V$ the volume, and $F_{\rm harm}$ the lattice Helmholtz free energy. (The dependence of the different energy terms on $p$ and $T$ have been explicitly noted.) For given $V$ and $T$, $F_{\rm harm}$ can be determined with the formula:

	$\displaystyle F_{\rm harm}(V,T)$	$\displaystyle=$	$\displaystyle\frac{1}{N_{q}}~{}k_{B}T\times$		(2)
			$\displaystyle\sum_{{\bf q}s}\ln\left[2\sinh\left(\frac{\hbar\omega_{{\bf q}s}}{2k_{B}T}\right)\right]~{},$

where $\omega_{{\bf q}s}(V)$ are the phonon frequencies obtained at the reciprocal lattice vector ${\bf q}$ and phonon branch $s$, $N_{q}$ the total number of wave vectors used for integration in the Brillouin zone, and $k_{B}$ the Boltzmann constant. Meanwhile, the hydrostatic pressure $p$ is calculated via the expression:

$$p(V,T)=-\frac{\partial\left[E(V)+F_{\rm harm}(V,T)\right]}{\partial V}~{},$$

(3)

which numerically allows to determine $V(p,T)$. Thus, by performing $E$ and $\omega_{{\bf q}s}$ DFT calculations for a set of $V$ points, over which interpolation is applied to describe $F_{\rm harm}$ and $p$ continuously, and by using Eqs. (1)–(3), it is possible to estimate $G_{\rm harm}(p,T)$. To quantify the temperature at which the $\cal{T}\leftrightarrow\cal{R}$ phase transition occurs at a given pressure, $T_{t}$, the condition $\Delta G_{\rm harm}(p,T_{t})\equiv G_{\rm harm}^{\cal{T}}(p,T_{t})-G_{\rm harm}^{\cal{R}}(p,T_{t})=0$ was employed.

Likewise, the entropy of the crystal can be obtained through the expression:

$$S(V,T)=-\left(\frac{\partial F_{\rm harm}}{\partial T}\right)_{V}~{},$$

(4)

and the heat capacity like:

	$\displaystyle C(V,T)$	$\displaystyle=$	$\displaystyle k_{B}\sum_{{\bf q}s}\left(\frac{\hbar\omega_{{\bf q}s}}{k_{B}T}\right)^{2}\times$		(5)
			$\displaystyle\frac{\exp{\left(\hbar\omega_{{\bf q}s}/k_{B}T\right)}}{\left[\exp{\left(\hbar\omega_{{\bf q}s}/k_{B}T\right)}-1\right]^{2}}~{}.$

Through the knowledge of $V(p,T)$ and Eqs. (2)–(5), then it is possible to determine $S(p,T)$ and $C(p,T)$.

In the absence of electric fields, the isothermal entropy change associated to barocaloric effects was approximately estimated with the Clausius-Clapeyron (CC) method like sau21 :

$$\Delta S_{\rm BC}(p,T)=\Delta V\cdot\frac{d{p}_{t}}{dT}~{},$$

(6)

where $\Delta V$ is the change in volume occurring during the phase transition and $p_{t}(T)$ the critical pressure. Likewise, the corresponding adiabatic temperature change can be approximated with the expression manosa17 :

$$\Delta T_{\rm BC}(p,T)=-\frac{T}{C}\cdot\Delta S_{\rm BC}(p,T)~{},$$

(7)

where $C(T)$ is the heat capacity of the system at zero pressure.

In the presence of electric fields, and assuming zero pressure, the thermodynamic potential that describes a particular phase is the Gibbs free energy defined as $G_{\rm harm}=E-\bm{{\cal E}}\cdot\bm{P}+F_{\rm harm}$, where $E$ and $F_{\rm harm}$ are the same terms that appear in Eq. (1), $\bm{P}$ the electric polarization and $\bm{{\cal E}}$ the applied electric field. In this case, the thermodynamic condition that determines a ${\cal E}$–induced phase transition is $G_{\rm harm}^{\cal T}(T,{\cal E}_{c})=G_{\rm harm}^{\cal R}(T,{\cal E}_{c})$. The value of the corresponding critical electric field then can be estimated like:

$${\cal E}_{c}(T)=\frac{\Delta\left(E+F_{\rm harm}(T)\right)}{\Delta P(T)}~{},$$

(8)

where $\Delta\left(E+F_{\rm harm}\right)$ is the Helmholtz free energy difference between the two phases, and $\Delta P$ the resulting change in the electric polarization along the electric field direction. For $p\neq 0$ conditions, an additional $p\Delta V$ term should appear in the right-hand side of Eq. (8).

Once the value of ${\cal E}_{c}$ and its dependence on temperature are determined through Eq. (8), the isothermal entropy change associated to electrocaloric effects can be approximately estimated with the CC method like cazorla18 :

$$\Delta S_{\rm EC}({\cal E},T)=-\Delta P\cdot\frac{d{\cal E}_{c}}{dT}~{}.$$

(9)

Likewise, the corresponding adiabatic temperature change was approximated with the expression manosa17 :

$$\Delta T_{\rm EC}({\cal E},T)=-\frac{T}{C}\cdot\Delta S_{\rm EC}({\cal E},T)~{},$$

(10)

where $C(T)$ is the heat capacity of the system at zero electric field.

The authors thank Riccardo Rurali for insightful discussions on the calculation of phonon properties of BFCO solid solutions. C.C. acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities under the “Ramón y Cajal” fellowship RYC2018-024947-I and TED2021-130265B-C22 project. Computational support was provided by the Red Española de Supercomputación (RES) under the grants FI-2022-1-0006, FI-2022-2-0003 and FI-2022-3-0014.