Giant Direct and Inverse Electrocaloric Effects in Multiferroic Thin Films

TL;DR

This study predicts giant room-temperature electrocaloric effects in multiferroic BiCoO₃ thin films, with tunable direct and inverse responses driven by electric field orientation, promising for environmentally friendly on-chip cooling technologies.

Contribution

It introduces the prediction of large electrocaloric effects in BiCoO₃ thin films at room temperature, highlighting the role of spin-phonon coupling and phase competition, and demonstrates tunability via epitaxial strain.

Findings

Giant electrocaloric effects predicted at room temperature in BiCoO₃ thin films.

Electrocaloric response can be tuned by electric field orientation and epitaxial strain.

Dual (direct and inverse) electrocaloric effects enable advanced cooling cycle designs.

Abstract

Refrigeration systems based on compression of greenhouse gases are environmentally threatening and cannot be scaled down to on-chip dimensions. In the vicinity of a phase transition caloric materials present large thermal responses to external fields, which makes them promising for developing alternative solid-state cooling devices. Electrocaloric effects are particularly well-suited for portable refrigeration applications; however, most electrocaloric materials operate best at non-ambient temperatures or require the application of large electric fields. Here, we predict that modest electric fields can yield giant room-temperature electrocaloric effects in multiferroic BiCoO$_{3}$ (BCO) thin films. Depending on the orientation of the applied field the resulting electrocaloric effect is either direct (heating) or inverse (cooling), which may enable the design of enhanced refrigeration cycles. We show that spin-phonon couplings and phase competition are the underlying causes of the disclosed caloric phenomena. The dual electrocaloric response of BCO thin films can be effectively tuned by means of epitaxial strain and we anticipate that other control strategies like chemical substitution are also possible.

Tables

Table 1. Table 1: Electrocaloric thin film materials. T 𝑇 T represents working temperature, Δ ​ ℰ Δ ℰ \Delta{\cal E} applied electric field (in the BCO case, we also indicate the corresponding orientation in pseudocubic Cartesian notation), C 0 subscript 𝐶 0 C_{0} heat capacity at zero field, Δ ​ S Δ 𝑆 \Delta S isothermal entropy change, and Δ ​ T Δ 𝑇 \Delta T adiabatic temperature change. The chemical composition of the materials are: PbZr 0.95 Ti 0.05 O 3 (PZT), poly(vinylidene fluoride-trifluoroethylene)(55/45 mol%) polymers [P(VDF-TrFE)], multilayer capacitor of doped BaTiO 3 (BTO-MLC), and BiCoO 3 (BCO).

$\mathrm{Material}$	$T$	$\Delta ℰ$	$C_0$	$\Delta S$	$\Delta T$	$\mathrm{Ref}.$
	$\left(K\right)$	$\left({\mathrm{kVcm}}^{-1}\right)$	$\left({\mathrm{JK}}^{-1}{\mathrm{Kg}}^{-1}\right)$	$\left({\mathrm{JK}}^{-1}{\mathrm{Kg}}^{-1}\right)$	$\left(K\right)$
$\mathrm{PZT}$	$495$	$776$	$330$	$-8.0$	$+12.0$	mischenko06
$P\left(\mathrm{VDF}-\mathrm{TrFE}\right)$	$353$	$2000$	$1700$	$-60.0$	$+12.5$	neese08
$\mathrm{BTO}-\mathrm{MLC}$	$350$	$300$	$434$	$-0.7$	$+0.5$	kar-narayan10 ; defay13
		$110\left({\left[111\right]}_{\mathrm{pc}}\right)$		$-11.0$	$+10.0$
$\mathrm{BCO}$	$300$		$345$			This work
		$500\left({\left[001\right]}_{\mathrm{pc}}\right)$		$+60.0$	$-50.0$

Equations

$${\tilde{E}}_{\mathrm{harm}}(T,a_{\mathrm{in}})$$

$${\tilde{F}}_{\mathrm{vib}}(T,a_{\mathrm{in}})=\frac{1}{N_q}{k}_{B}T\sum _{\mathbf{q}s}\ln \left[2\sinh \left(\frac{\hslash {\tilde{\omega }}_{\mathbf{q}s}}{2k_{\mathrm{B}}T}\right)\right],$$

$${\tilde{F}}_{mag}(T,a_{\mathrm{in}})$$

$${\tilde{F}}_{\mathrm{harm}}(T,a_{\mathrm{in}})$$

$$\Delta S(T,a_{\mathrm{in}})={\int }_0^{\mathcal{E}}{\left(\frac{dP}{dT}\right)}_{{\mathcal{E}}^{\prime }}d{\mathcal{E}}^{\prime },$$

$$\Delta T(T,a_{\mathrm{in}})=-\frac{T}{C_0}\Delta S(T,a_{\mathrm{in}}),$$

$$\Delta S(T,a_{\mathrm{in}})=-\Delta P\frac{d{\mathcal{E}}_c}{dT},$$

$${\mathcal{E}}_c(T,a_{\mathrm{in}})=\frac{\Delta {\tilde{F}}_{\mathrm{harm}}(T,a_{\mathrm{in}})}{\Delta P(T,a_{\mathrm{in}})},$$

$${\tilde{E}}_{\mathrm{harm}}(T,a_{\mathrm{in}})$$

$${\tilde{E}}_0^{\mathcal{O},\mathcal{M}}(a_{\mathrm{in}},T)$$

$${\tilde{\Xi }}_{mn}^{\mathcal{O},\mathcal{M}}(a_{\mathrm{in}},T)$$

$$E^0=\frac{1}{2}\left(E_{\mathrm{eq}}^{\mathrm{FM}}+E_{\mathrm{eq}}^{\mathrm{G}}\right),$$

$${\Phi }_{mn}^0=\frac{1}{2}\left({\Phi }_{mn}^{\mathrm{FM}}+{\Phi }_{mn}^{\mathrm{G}}\right),$$

$$J_a^{(0)}=\frac{1}{6|S{|}^2}\left(E_{\mathrm{eq}}^{\mathrm{FM}}-E_{\mathrm{eq}}^{\mathrm{G}}\right),$$

$$J_{a,mn}^{(2)}=\frac{1}{6|S{|}^2}\left({\Phi }_{mn}^{\mathrm{FM}}-{\Phi }_{mn}^{\mathrm{G}}\right).$$

$${\tilde{E}}_0^{\mathcal{T}}(a_{\mathrm{in}},T)$$

$${\tilde{\Xi }}_{mn}^{\mathcal{T}}(a_{\mathrm{in}},T)$$

$$E^0=\frac{1}{4}\left(E_{\mathrm{eq}}^{\mathrm{FM}}+E_{\mathrm{eq}}^{\mathrm{A}}+E_{\mathrm{eq}}^{\mathrm{C}}+E_{\mathrm{eq}}^{\mathrm{G}}\right),$$

$${\Phi }_{mn}^0=\frac{1}{4}\left({\Phi }_{mn}^{\mathrm{FM}}+{\Phi }_{mn}^{\mathrm{A}}+{\Phi }_{mn}^{\mathrm{C}}+{\Phi }_{mn}^{\mathrm{G}}\right),$$

$$J_a^{(0)}=\frac{1}{8|S{|}^2}\left(E_{\mathrm{eq}}^{\mathrm{FM}}+E_{\mathrm{eq}}^{\mathrm{A}}-E_{\mathrm{eq}}^{\mathrm{C}}-E_{\mathrm{eq}}^{\mathrm{G}}\right),$$

$$J_{a,mn}^{(2)}=\frac{1}{8|S{|}^2}\left({\Phi }_{mn}^{\mathrm{FM}}+{\Phi }_{mn}^{\mathrm{A}}-{\Phi }_{mn}^{\mathrm{C}}-{\Phi }_{mn}^{\mathrm{G}}\right),$$

$$J_c^{(0)}=\frac{1}{4|S{|}^2}\left(E_{\mathrm{eq}}^{\mathrm{FM}}-E_{\mathrm{eq}}^{\mathrm{A}}+E_{\mathrm{eq}}^{\mathrm{C}}-E_{\mathrm{eq}}^{\mathrm{G}}\right),$$

$$J_{c,mn}^{(2)}=\frac{1}{4|S{|}^2}\left({\Phi }_{mn}^{\mathrm{FM}}-{\Phi }_{mn}^{\mathrm{A}}+{\Phi }_{mn}^{\mathrm{C}}-{\Phi }_{mn}^{\mathrm{G}}\right),$$

$$J_{ac}^{(0)}=\frac{1}{16|S{|}^2}\left(E_{\mathrm{eq}}^{\mathrm{FM}}-E_{\mathrm{eq}}^{\mathrm{A}}-E_{\mathrm{eq}}^{\mathrm{C}}+E_{\mathrm{eq}}^{\mathrm{G}}\right),$$

$$J_{ac,mn}^{(2)}=\frac{1}{16|S{|}^2}\left({\Phi }_{mn}^{\mathrm{FM}}-{\Phi }_{mn}^{\mathrm{A}}-{\Phi }_{mn}^{\mathrm{C}}+{\Phi }_{mn}^{\mathrm{G}}\right).$$

Full text

I Introduction

Caloric materials react thermally to external fields as a result of induced transformations involving sizeable entropy changes. Their compactness and fast response to external stimuli raise high hopes to surpass the performance, environmental compliance, and portability of current refrigeration technologies based on compression cycles of greenhouse gases. A major breakthrough occurred in the late nineties with the discovery of giant magnetocaloric effects in Gd5(Si2Ge2) pecharsky97 . Yet, most magnetocaloric materials are based on scarce rare-earth elements and require the generation of high magnetic fields. Meanwhile, mechanocaloric effects are attracting much attention due to their large latent heat and adiabatic temperature changes bonnot08 ; cazorla16 ; cazorla17a ; cazorla17b ; cazorla18 . The practical implementation of mechanocaloric effects, however, becomes difficult when scaling down towards on-chip dimensions.

In the context of microelectronic cooling, electrocaloric (EC) materials emerge as particularly promising owing to their high energy density, natural integration in circuitry scott11 ; kar-narayan10 ; defay13 ; jiang17 ; geng15 ; mangeri16 ; zhang11 ; tong14 ; pandya17 , and the possibility of implementing charge-recovery strategies to increase efficiency defay18 . Unfortunately, the largest EC effects observed to date occur at high temperatures mischenko06 or require large electric fields neese08 , which may lead to the appearance of impeding leakage current and dielectric loss problems defay18 ; asbani17 . Here, we demonstrate giant room-temperature EC effects ($|\Delta T|\sim 10$ K) in epitaxially grown BiCoO3 (BCO) thin films induced by modest electric fields ($\sim 100$ kVcm*-1*). We use first-principles methods cazorla17 ; cazorla13 to resolve the phase diagram of BCO thin films and show that it is possible to obtain field-driven transitions from a non-polar antiferromagnetic (AFM) phase to either a high-entropy polar paramagnetic (PM) phase (inverse effect, $\Delta T<0$) or a low-entropy polar AFM phase (direct effect, $\Delta T>0$), depending on the orientation of the applied electric field. We show that the causes of the disclosed EC phenomena are strong phase competition and spin-phonon couplings.

II Methods

II.1 First-principles calculations

In our first-principles calculations, we use the GGA-PBE approximation to density functional theory (DFT) pbe96 ; vasp because this functional provides an accurate description of the relative stability of bulk BCO polymorphs at zero temperature cazorla17 . A “Hubbard-U” scheme with $U=6$ eV is employed for a better treatment of Co’s $3d$ electrons cazorla17 . We use the projector-augmented wave method to represent the ionic cores bloch94 , considering the following electrons as valence states: Co’s $3p$, $3d$, and $4s$; Bi’s $5d$, $6s$, and $6p$; and O’s $2s$ and $2p$. Wave functions are represented in a plane-wave basis truncated at $650$ eV and for integrations within the first Brillouin zone (BZ) we employ a dense $\Gamma$-centered ${\rm q}$-point grid of $12\times 12\times 12$ for a $2\times\sqrt{2}\times\sqrt{2}$ supercell containing $20$ atoms (i.e., $4$ formula units) cazorla15 ; cazorla17c . By using these parameters we obtain zero-temperature energies converged to within $0.5$ meV per formula unit. Our strained-bulk geometry relaxations are performed with a conjugate-gradient algorithm that allows for volume variations while imposes the structural constraints defining $(100)$-oriented perovskite thin films cazorla15 ; cazorla17c . The relaxations are halted when the forces in the atoms fall all below $0.01$ eV$\cdot$Å*-1*. The calculation of phonon frequencies is performed with the small displacement method kresse95 ; alfe09 . We find the following settings to provide quasi-harmonic free energies converged to within $5$ meV per formula unit: 160–atom supercells, atomic displacements of $0.02$ Å, and ${\rm q}$-point grids of $16\times 16\times 16$ for BZ integrations.

II.2 Estimation of free energies and EC effects

We calculate the Helmholtz free energy of competitive polymorphs in $(001)$-oriented BCO thin films as a function of temperature ($T$) and in-plane lattice parameter ($a_{\rm in}$) with first-principles methods (see works cazorla17 ; cazorla13 and Appendix). Our approach takes into account the couplings between magnetic spin disorder and lattice phonons, which strongly depend on temperature and epitaxial strain. We start by expressing the internal energy of the thin film as:

[TABLE]

where $\tilde{E}_{0}$ represents an effective static energy, $\tilde{\Xi}_{mn}$ an effective force-constant matrix (see Appendix), $u$’s atomic displacements, and the dependences of the various terms on $T$ and $a_{\rm in}$ are explicitly noted. The Helmoltz free energy associated to the lattice vibrations, $\tilde{F}_{\rm vib}$, is calculated by finding the phonon eigenfrequencies of the dynamical matrix obtained from $\tilde{\Xi}_{mn}$, namely, $\tilde{\omega}_{{\bf q}s}$, and subsequently using the formula cazorla17c ; cazorla17d :

[TABLE]

where $N_{q}$ is the total number of wave vectors used for integration in the BZ, and the summation runs over all wave vectors ${\bf q}$ and phonon branches $s$. The magnetic free energy stemming exclusively from spin fluctuations is estimated through the analytical mean-field solution to the spin $\frac{1}{2}$-Ising model at zero magnetic field strecka15 , which reads:

[TABLE]

where $q$ represents the number of nearest-neighboring spins, $J\left(a_{\rm in}\right)$ the exchange constant between nearest-neighboring spins, and $m\left(T,a_{\rm in}\right)$ the magnetization of the system. In practice, we estimate the value of the magnetic exchange constants from zero-temperature DFT calculations (see Appendix) and the magnetization of the system through DFT-based spin-model Monte Carlo simulations (see next section). The total Helmholtz free energy of the system then is estimated as:

[TABLE]

Temperature–induced phase transitions are determined via the condition $\tilde{F}_{\rm harm}^{A}(T_{c},a_{\rm in})=\tilde{F}_{\rm harm}^{B}(T_{c},a_{\rm in})$, where $A$ and $B$ represent two different phases.

The isothermal entropy change that a polar material undergoes under the action of a varying external electric field is moya14 :

[TABLE]

where $P$ represents the polarization of the system and ${\cal E}$ the applied electric field. Likewise, the corresponding adiabatic temperature change can be estimated as manosa17 :

[TABLE]

where $C_{0}(T,a_{\rm in})$ is the heat capacity of the system at zero electric field. $\Delta S$ and $\Delta T$ become large when the system undergoes a ${\cal E}$–induced phase transition; if such a phase transition is of first-order type, as it occurs in BCO belik06 ; oka10 , the isothermal entropy change can be estimated with the Clausius-Clapeyron method as moya14 :

[TABLE]

where $\Delta P(T,a_{\rm in})$ is the change in polarization along the electric field direction, and ${\cal E}_{c}(T,a_{\rm in})$ the critical electric field inducing the phase transformation.

In the presence of an electric field the thermodynamic potential that appropriately describes the stability of a particular phase is the Gibbs free energy, defined as $\tilde{G}_{\rm harm}=\tilde{F}_{\rm harm}-\bm{{\cal E}}\cdot\bm{P}$, where $\tilde{F}_{\rm harm}$ corresponds to the Helmholtz free energy in Eq.(4). In this case, the thermodynamic condition that determines a ${\cal E}$–induced phase transition is $\tilde{G}_{\rm harm}^{A}(T,a_{\rm in},{\cal E}_{c})=\tilde{G}_{\rm harm}^{B}(T,a_{\rm in},{\cal E}_{c})$. The value of the corresponding critical electric field then is estimated as:

[TABLE]

where $\Delta\tilde{F}_{\rm harm}$ 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. By using the formulas (6)–(8) and knowing $\Delta\tilde{F}_{\rm harm}$, one can calculate $\Delta S$ and $\Delta T$ as a function of temperature and in-plane lattice parameter.

II.3 Spin-model Monte Carlo simulations

To simulate the effects of thermal excitations on the magnetic order of energetically competitive polymorphs in $(100)$-oriented BCO thin films, we construct several spin Heisenberg models of the form $\hat{H}=\frac{1}{2}\sum_{ij}J^{(0)}_{ij}S_{i}S_{j}$ based on density functional calculations (see Appendix). We use those models to perform Monte Carlo (MC) simulations in a periodically-repeated simulation box of $20\times 20\times 20$ spins; thermal averages are computed from runs of $50,000$ MC sweeps after equilibration. These simulations allow us to monitor the $T$-dependence of the magnetic order via the computation of the AFM-C and AFM-G order parameters (see Fig.1d), defined as: $S^{\rm C}\equiv\frac{1}{N}\sum_{i}(-1)^{n_{ix}+n_{iy}}S_{iz}$ and $S^{\rm G}\equiv\frac{1}{N}\sum_{i}(-1)^{n_{ix}+n_{iy}+n_{iz}}S_{iz}$. Here, $n_{ix}$, $n_{iy}$, and $n_{iz}$ are the three integers locating the $i$-th lattice cell, and $N$ is the total number of spins in the simulation cell. For the calculation of $S^{\rm C}$ and $S^{\rm G}$ we consider only the $z$ component of the spins since a small symmetry-breaking magnetic anisotropy is introduced in the system Hamiltonian to facilitate the numerical analysis.

III Results and Discussion

III.1 Phase transitions in BCO thin films

Figure 1 shows the relevant polymorphs of bulk BCO cazorla17 . At ambient conditions, bulk BCO presents the ferroelectric (FE) tetragonal ${\cal T}$ phase shown in Fig.1a. This structure has a relatively small in-plane lattice constant ($c/a\approx 1.3$), hence perovskite substrates with small in-plane cell parameters $a_{\rm in}$ will be required for stabilizing it in thin-film form. The competing structures are the paraelectric (PE) orthorhombic ${\cal O}$ phase (Fig.1b) and the FE monoclinic ${\cal M}$ phase (Fig.1c); both phases have cells that are slightly distorted versions of the ideal cubic perovskite structure, with $c/a\approx 1$. Note that the mentioned FE phases present spontaneous polarizations along quite different crystallographic directions – i.e., pseudocubic $[001]_{\rm pc}$ for ${\cal T}$ and $\sim[111]_{\rm pc}$ for ${\cal M}$. As regards magnetism (Fig.1d), both the ${\cal O}$ and ${\cal M}$ phases display G-type AFM order with a quite high Néel temperature $T_{\rm N}\sim 500$ K. In contrast, the ${\cal T}$ phase presents C-type AFM order with a relatively low $T_{\rm N}\sim 310$ K.

We thus expect the competition between BCO polymorphs to be strongly affected by epitaxially growing thin films on perovskite substrates, as this imposes an in-plane lattice constant $a_{\rm in}$ in the system. Our zero-temperature first-principles calculations confirm this conjecture, yielding the results summarized in Fig.2. We predict a zero-temperature strain-driven ${\cal T}\to{\cal M}$ phase transformation at $a_{\rm in}=3.925$ Å  (Fig.2a) that involves rotation of the polarization (Figs.2c,e) and the appearance of anti-phase oxygen octahedral rotations along the three pseudocubic directions (Figs.2d,f). The ${\cal O}$ phase remains close in energy to the ${\cal M}$ polymorph over the whole $a_{\rm in}$ interval; however, it never becomes stable at zero temperature, in contrast to what occurs in compressed bulk BCO cazorla17 . Meanwhile, the Néel temperature of epitaxially grown BCO thin films decreases mildly with increasing $a_{\rm in}$ (Fig.2b).

We have performed first-principles Helmholtz free-energy calculations of the three relevant BCO polymorphs within the quasi-harmonic approximation (see Sec. II and Appendix), to determine their relative stability as a function of $T$ and $a_{\rm in}$. Our predicted $T$–$a_{\rm in}$ phase diagram for epitaxially grown BCO thin films is shown in Fig.3a. For relatively small $a_{\rm in}$’s, we find that the ${\cal T}$ phase dominates and extends its stability region to temperatures much higher than observed in bulk BCO (we recall that bulk BCO presents a ${\cal T}\rightarrow{\cal O}$ transition at $T\approx 900$ K cazorla17 ). The reason for this ${\cal T}$ stability enhancement is that the competing ${\cal O}$ polymorph remains highly strained at such $a_{\rm in}$ conditions (Fig.2a), hence its free energy increases considerably as compared to the bulk case. As $a_{\rm in}$ is increased, the ${\cal T}$ phase eventually is replaced by the ${\cal O}$ and ${\cal M}$ polymorphs, yielding a very rich phase diagram that exhibits concurrent structural and spin-ordering transformations.

Figures 3b–d show the calculated Helmholtz free-energy differences $\Delta\tilde{F}_{\rm harm}$ between the three relevant BCO polymorphs expressed as a function of $T$ and $a_{\rm in}$. As noted previously, our calculations take into account all possible sources of entropy, namely, magnetic and vibrational, and the interplay between spin disorder and lattice vibrations (see Sec. II and Appendix). We find that at high temperatures ($T\gtrsim 600$ K) the vibrational contributions to $\tilde{F}_{\rm harm}$ always favor the ${\cal O}$ and ${\cal M}$ phases over the ${\cal T}$ phase. Nonetheless, whenever a polymorph becomes magnetically disordered, the corresponding Helmholtz free energy decreases significantly as a consequence of $T$–induced lattice phonon softenings (as is appreciated, for instance, in the densities of vibrational states calculated at $T<T_{\rm N}$ and $T>T_{\rm N}$, not shown here). Accordingly, abrupt $\Delta\tilde{F}_{\rm harm}$ changes appear in Figs.3b–d at the corresponding AFM $\to$ PM magnetic transition temperatures. The strong spin-phonon couplings in epitaxially grown BCO thin films are responsible for the stabilization of the ${\cal O}$–AFM(G) phase at temperatures near ambient and $3.88$ Å $\leq a_{\rm in}\leq 3.96$ Å. Indeed, when spin-phonon couplings are disregarded (i.e., quasi-harmonic free energies are calculated in the standard manner by considering “frozen” spin arrangements) the ${\cal O}$ phase becomes stable only after reaching the paramagnetic state at high temperatures (Figs.3b–d and Fig.2b).

III.2 Electrocaloric effects in epitaxially grown BCO thin films

We now focus on the $a_{\rm in}$ region $3.89$ Å $\leq a_{\rm in}\leq 3.93$ Å, which is particularly convenient from a practical perspective since many perovskite substrates present lattice constants in this range (where all the phases remain strained by $+4$–$5$% –${\cal T}$– and $\pm 0.1$–$1$% –${\cal O}$ and ${\cal M}$– as compared to their corresponding equilibrium in-plane lattice parameters). Interestingly, we find a $T$–driven reentrant behavior that is reminiscent of bulk BCO under compression cazorla17 : upon heating, the BCO film transforms first from a FE (${\cal T}$–AFM(C) or ${\cal M}$–AMF(G)) phase to a PE (${\cal O}$–AFM(G)) state, then back to a FE (${\cal T}$–PM) phase, and finally to a PE (${\cal O}$–AFM(G)) state. Of particular interest is the PE ${\cal O}$–AFM(G) region appearing near room temperature $T_{\rm room}$, which is surrounded by two FE phase domains presenting markedly different features. In particular, the phase diagram in Fig.3a suggests that the ${\cal O}$ phase may be transformed into the ${\cal T}$ or ${\cal M}$ states by applying an electric field ${\cal E}$ along the [001]pc or [111]pc directions, respectively. Such ${\cal E}$–driven phase transformations involve drastic structural changes as well as magnetic transitions, hence big entropy shifts are likely to occur as a consequence.

Figure 4 shows the direct EC effect associated to the field-induced ${\cal O}\to{\cal M}$ transformation (Fig.4a), in which the entropy of the system decreases ($\Delta S<0$, Fig.4c). For the smallest $a_{\rm in}$ values, a maximum adiabatic temperature change $\Delta T$ of $+10$ K (Fig.4d) is estimated for a maximum critical electric field of $110$ kVcm*-1* (Fig.4b). The magnitude of this effect and of the accompanying critical electric field decrease with increasing $a_{\rm in}$, as the region of ${\cal M}$ stability is approached. Similarly, Fig.4 shows the inverse EC effect associated to the field-induced ${\cal O}\to{\cal T}$ transformation (Fig.4a), in which the entropy of the film increases ($\Delta S>0$, Fig.4c). A maximum $\Delta T$ of $-50$ K (Fig.4d) is estimated for a maximum critical electric field of $500$ kVcm*-1* at $a_{\rm in}=3.93$ Å  (Fig.4b). The magnitude of this effect and of the corresponding ${\cal E}_{c}$ decrease with decreasing $a_{\rm in}$, as the region of ${\cal T}$ stability is approached. We note in passing that application of electric fields of the order of $100$ kVcm*-1* in oxide perovskite thin films appears to be non-problematic in practice (see, for instance, Ref.defay13 and experimental references therein).

The predicted giant $\Delta T$ and $\Delta S$ values, which can be achieved with relatively small driving fields, turn epitaxially grown BCO thin films into very attractive EC materials. Table 1 offers a comparison with other promising EC compounds. Note that BCO thin films would operate at $T_{\rm room}$ and display significantly larger adiabatic $T$-spans than observed in other oxides. Meanwhile, the required electric fields are much smaller than reported for FE polymers while similar $\Delta T$ values should be attained in both cases. (The smaller heat capacity of perovskite oxides contributes favorably to this latter outcome, see Table 1 and Eq.(6).) Hence, BCO films somehow unite the best of the FE ceramics and FE polymers worlds. On the down side, the reported ${\cal E}$–driven BCO phase transitions are strongly discontinuous and thus irreversibility issues are likely to appear in practical cooling applications moya14 .

Let us stress that the coexistence of giant direct and inverse EC effects at room temperature offers the possibility of designing improved EC refrigeration cycles (as originally proposed by Ponomareva and Lisenkov for Ba0.5Sr0.5TiO3 alloys ponomareva12 and subsequently discussed by Marathe et al. in BaTiO3 marathe17 ). In the present case, coexistence of direct and inverse EC effects suggests a refrigeration cycle based on the direct transformation between the high-entropy FE ${\cal T}$ and low-entropy FE ${\cal M}$ phases as induced by ${\cal E}$ rotation, with a cooling performance equal to the sum of the individual ${\cal O}\leftrightarrow{\cal T}$ and ${\cal O}\leftrightarrow{\cal M}$ cycles. In addition, we foresee alternative strategies for tuning and optimizing BCO’s EC behavior beyond epitaxial strain. Chemical substitution, for instance, represents an obvious option for stabilizing phases that are similar to the ${\cal M}$ (alike to the ground state of BiFeO3 cazorla15 ) and ${\cal O}$ (most common among perovskites peng18 ) polymorphs discussed here, and to control the corresponding magnetic transition temperatures. One particular example is provided by BiCo1-xFexO3 solid solutions, where a morphotropic transition between a ${\cal T}$–like and a ${\cal M}$–like phase is observed to occur at room temperature azuma08 ; azuma18 . Likewise, bulk Bi1-xLaxCoO3 appears to be a good candidate where to realize field-driven ${\cal O}\to{\cal T}$ transformations cazorla17 . Hence BCO offers a variety of experimental possibilities to achieve giant EC, bringing new exciting prospects to the field of solid-state cooling.

IV Conclusions

We have disclosed a number of multiferroic phase transitions in epitaxially grown BCO thin films that render giant and dual (that is, direct and inverse) room-temperature EC effects ($|\Delta T|\sim 10$ K) driven by moderate electric fields ($\sim 100$ kVcm*-1*). The multiferroic phase transitions and accompanying EC effects are originated by strong spin-phonon couplings and polymorph competition, which can be efficiently tuned over a wide operating-temperature interval by means of epitaxial strain. Our findings in BCO thin films should stimulate the development of cooling devices based on multiferroic EC materials, whose energy efficiency and refrigerant performance are good as compared to magnetocaloric and mechanocaloric materials. Unlike mechanocaloric materials driven by uniaxial stress or hydrostatic pressure, scalability down to transistor sizes and implementation on circuitry are possible without compromising cooling performance. Unlike magnetocaloric compounds, the generation of large external fields at great expense is not necessary. Thus, we hope that our results in multiferroic thin films will bring new prospects to the field of solid-state cooling.

Acknowledgments

This research was supported under the Australian Research Council’s Future Fellowship funding scheme (project number FT140100135) and the Luxembourg National Research Fund through the PEARL (Grant P12/4853155 COFERMAT) and CORE (Grant C15/MS/10458889 NEWALLS) programs. Computational resources and technical assistance were provided by the Australian Government and the Government of Western Australia through Magnus under the National Computational Merit Allocation Scheme and The Pawsey Supercomputing Centre.

Appendix: Inclusion of spin-phonon couplings in quasi-harmonic free energy calculations

To calculate the Helmholtz free energy of the competitive polymorphs in $(001)$-oriented BCO thin films as a function of temperature ($T$) and in-plane lattice parameter ($a_{\rm in}$), we employ a variant of the approaches described in works cazorla17 ; cazorla13 . Our approach takes into consideration the couplings between magnetic spin disorder and lattice phonons, which strongly depend on the temperature and epitaxial strain conditions. Specifically, we start by expressing the internal energy of the thin film as:

[TABLE]

where $\tilde{E}_{0}$ represents an effective static energy, $\tilde{\Xi}_{mn}$ an effective force-constant matrix, $u$’s atomic displacements, and the dependences of the various terms on $T$ and $a_{\rm in}$ are explicitly noted.

For the ${\cal O}$ and ${\cal M}$ phases ($c\approx a$), the quantities entering Eq. (9) can be calculated as:

[TABLE]

where $\gamma_{a}(a_{\rm in},T)\equiv\langle S_{i}S_{j}\rangle/|S|^{2}$ represents the correlation function between neighboring spins (see Fig.1d) and $\langle...\rangle$ the thermal average as obtained from our Monte Carlo simulations of the corresponding spin Heisenberg hamiltonian (see Sec. II.3). The rest of parameters in $\tilde{E}_{0}^{{\cal O},{\cal M}}$ and $\tilde{\Xi}_{mn}^{{\cal O},{\cal M}}$ correspond to:

[TABLE]

In the equations above, superscripts “FM” and “G” represent perfect ferromagnetic and antiferromagnetic G-type spin arrangements, respectively. The $J_{a}^{(0)}$ parameter describes the magnetic interactions when the atoms remain frozen at their equilibrium positions; typically, this captures the bulk of the exchange couplings. Meanwhile, the $J_{a,mn}^{(2)}$ parameter captures the dependence of the phonon spectrum on the spin configuration (i.e., spin-phonon coupling effects).

For the ${\cal T}$ phase ($c>a$), we express the corresponding static energy and force constant matrix as:

[TABLE]

and

[TABLE]

where $\gamma_{\alpha}(a_{\rm in},T)\equiv\langle S_{i}S_{j}\rangle/|S|^{2}$, with $\alpha=a,b,ac$, represent the correlation functions between in-plane and out-of-plane neighboring spins (see Fig.1d). (We note that in this case we need to differentiate among in-plane and out-of-plane neighboring spins due to the highly anisotropic nature of the tetragonal phase.) The rest of parameters in $\tilde{E}_{0}^{\cal T}$ and $\tilde{\Xi}_{mn}^{\cal T}$ can be obtained as:

[TABLE]

In the equations above, superscripts “FM”, “G”, “A”, and “C” mean perfect ferromagnetic, antiferromagnetic G-type, antiferromagnetic A-type, and antiferromagnetic C-type spin arrangements, respectively. We note that the explained first-principles quasi-harmonic approach could also be employed to estimate pyroelectric coefficients liu16 ; liu18 in multiferroic materials.