Enhanced Electrocaloric Response in Electrostatically Frustrated Ferroelectrics

The electrocaloric effect (i.e., the temperature change of a material upon the adiabatic application of an electric field) can be large in ferroelectric compounds, to the extent of offering an eco‐friendly alternative to current, polluting and noisy, refrigeration technologies. So far, ferroelectric‐based electrocaloric devices employ materials (from perovskite oxides to polymers) that have been known for decades. At the same time, recent studies have shown that the functional response of a ferroelectric can be drastically enhanced by controlling its mechanical and electric boundary conditions. Here, we use atomistic second‐principles simulations to quantify how much the electrocaloric effect benefits from such an enhancement. Using PbTiO3/SrTiO3 ferroelectric/dielectric superlattices as representative model systems, we predict that temperature changes can indeed be increased significantly (more than two times at room temperature), suggesting electrostatic engineering—to induce very reactive frustrated ferroelectric states—as a promising route for electrocaloric optimization.


Introduction
The state and properties of a ferroelectric material are strongly dependent on the elastic and electric boundary conditions imposed on it. [1] Ferroelectric perovskite oxides like BaTiO 3 or PbTiO 3 provide us with good examples of the many possible effects. For instance, when grown as epitaxial thin films, the strain imposed by the substrate can be used to select the direction of the spontaneous polarization (compression will favor an outof-plane alignment, while expansion will yield in-plane polarity), an effect that reflects perovskites' tendency to stretch along the polar axis. [2] Interestingly, strain clamping by a substrate is known to affect (reduce) the electrocaloric response; [3,4] by contrast, tuning the ferroelectric transition temperature by epitaxial strain can potentially be used to enhance the effect. [3][4][5][6] Even more possibilities can be accessed by controlling the electric boundary conditions the ferroelectric is subject to. For example, PbTiO 3 (PTO) layers in PbTiO 3 =SrTiO 3 superlattices (PTO/STO SLs) are in open-circuit-like conditions and, thus, they adopt a configuration that minimizes the accumulation of bound polarization charge at the ferroelectric/dielectric interfaces. [7][8][9] Typically, these SLs develop a polarization in the plane, with null associated depolarizing field. By contrast, if the SL is subject to an epitaxial compression that forces PTO's polarization to align along the stacking direction, a multidomain configuration forms to minimize the depolarizing field. This possibility has been thoroughly investigated in recent years, as it yields a rich variety of complex ferroelectric textures-including skyrmions and other topological quasi-particles [10] -that open the door to a wealth of novel physical effects and potential applications.
Among the unique features of these electrostatically frustrated ferroelectric states, let us highlight two. On one hand, their confined quasi-two-dimensional nature (the SL layers can be as thin as a few unit cells) gives them properties that are unheard of for bulk materials. For example, the energy barriers for domain wall motion are low, which yields states where the domains may present spontaneous thermally activated diffusion (stochastic dynamics). [11] Such a "domain liquid" state can be further transformed, upon heating, into a gas-like disordered (paraelectric) phase. On the other hand, the existence of many competing orders and strongly fluctuating states makes the materials very sensitive to external perturbations and yields large functional responses. For example, the dielectric response to electric fields has been thoroughly investigated in PTO/STO superlattices (the model system of the field), shedding light on exotic effects such as negative capacitance and voltage amplification. [11][12][13] The electrically frustrated ferroelectric states described above (multidomain, strongly fluctuating) can be expected to present relatively high entropy compared to the corresponding polar state of the bulk material under similar conditions of temperature and pressure. This suggests they will display a comparatively large electrocaloric (EC) response, as the application of an electric field should allow us to freeze the strong dipole fluctuations and thus obtain big changes in temperature (adiabatically) or entropy (isothermally). Here, we explore this possibility by means of DOI: 10.1002/pssr.202300014 The electrocaloric effect (i.e., the temperature change of a material upon the adiabatic application of an electric field) can be large in ferroelectric compounds, to the extent of offering an eco-friendly alternative to current, polluting and noisy, refrigeration technologies. So far, ferroelectric-based electrocaloric devices employ materials (from perovskite oxides to polymers) that have been known for decades. At the same time, recent studies have shown that the functional response of a ferroelectric can be drastically enhanced by controlling its mechanical and electric boundary conditions. Here, we use atomistic second-principles simulations to quantify how much the electrocaloric effect benefits from such an enhancement. Using PbTiO 3 =SrTiO 3 ferroelectric/dielectric superlattices as representative model systems, we predict that temperature changes can indeed be increased significantly (more than two times at room temperature), suggesting electrostatic engineering-to induce very reactive frustrated ferroelectric states-as a promising route for electrocaloric optimization.
atomistic second-principles simulations. [14] We predict very large enhancements (over twofold at room temperature) suggesting electrostatic engineering as a promising route to EC optimization.

Simulation Methods
We use atomistic second-principles potentials [14] that have proved their accuracy for the simulation of the structural and response properties of ferroelectrics and ferroelectric/dielectric SLs, in particular the PTO/STO system. [10][11][12]15] We study n/m superlattices composed of PTO and STO layers with a thickness of n and m perovskite cells, respectively. We consider a periodically repeated simulation supercell composed of 8 Â 8 Â ðn þ mÞ 5-atom perovskite units, which we checked is compatible with the most stable multidomain structures of the PTO layers. [15] As mentioned in Note S1, Supporting Information, we have evidence-from simulations in bulk PbTiO 3 -that the relatively small size of the considered supercell should not have a significant impact on the computed temperature changes. To compute statistical averages-in particular, the temperature (T ) response to the application of an electric field-we employ regular (constant temperature) and microcanonical [16] (constant energy) Monte Carlo methods, in essence adopting the approach of Refs. [17] and [18] (for details see Note S1 and Figure S1 and S2, Supporting Information). We always work in epitaxial strain conditions that force PTO's polarization to be along the vertical direction, which yields a richer phase diagram and, in particular, the liquid-like states we are interested in. The zero of epitaxial strain corresponds to a square STO substrate with lattice constant of 3.905 Å, which is an experimentally relevant reference. Finally, note that we consider relatively large applied fields (a maximum of 4 MV cm À1 , field steps of 1 MV cm À1 ), which we nevertheless deem representative of what can be achieved experimentally in PTO-based materials (see, e.g., Refs. [19,20]). Figure 1 shows representative results of how our PTO/STO superlattices (in particular, the ferroelectric PTO layers) evolve with temperature. At low T (Figure 1a), we have stable stripe ferroelectric domains with alternating up-and down-pointing polarizations; in the following, we will refer to this as the "solid state." Upon heating, the domains become mobile-i.e., their thermal fluctuations yield changes in size, shape, and position-and we enter the liquid state ( Figure 1b). (The "domain liquid" state was first reported in the study by Zubko et al., [11] which provides some illustrative videos as Supporting Information) Upon further heating, the domains progressively break and disorder, entering the gas-like (paraelectric) state ( Figure 1c). In Figure S3, Supporting Information, we show lateral views of these dipole structures, where, for example, one can better appreciate the vortex-like nature of the domain walls in the low-temperature "solid state."

Preliminaries
The transition between solid and liquid is clearly marked by the domains becoming mobile. By contrast, it is challenging to identify a ferroelectric order parameter to characterize the liquidgas transformation. Fortunately, the development of local polar order-even if dynamical or strongly fluctuating-is known to have a definite elastic signature, as it is accompanied by the stretching of the cell along the polar direction. Thus, Figure 1d shows the T-dependence of the c lattice constant of the simulated supercell: at high temperatures, we observe a positive slope (positive thermal expansion), while at low temperatures, the slope is negative as a result of the progressive condensation of the polar order. (This behavior also occurs in bulk PTO. See, e.g., Ref. [21]. Also, see Figure S4, Supporting Information, for the evaluation of the c lattice constants of the individual PTO and STO layers.) We take the minimum of c as the approximate point of the liquid-gas transition. [11] (Alternatively, one might try to identify this transition by computing the spatial decay of dipole-dipole correlations. However, this quantity is hard to obtain with good statistics; hence, here, we rely on the mentioned, much simpler, elastic signature.).
The behavior of these SLs can be readily modified by controlling their design variables: whenever we increase the epitaxial compression or consider thicker PTO layers (or thinner STO layers), we strengthen the polar order, which results in an increase of both the solid-liquid and liquid-gas transition temperatures.  www.advancedsciencenews.com www.pss-rapid.com conditions (0%, À1 %, and À2 %) and at low (100 K) and room (300 K) temperatures. For comparison, the figure also shows the results obtained from simulations of pure PTO in bulk-like (no elastic constraints) and thin-film (epitaxial constraints as for the SLs) forms.

Results and Discussion
The pure materials present a moderate EC response that peaks at relatively small fields; for example, we obtain a maximum of about 2 K cm MV À1 at 100 K, which amounts to a 2 K temperature increase for an applied field of 1 MV cm À1 . The elastically unconstrained bulk is always more reactive than the films; then, among the films, a stronger epitaxial compression yields a smaller response. Finally, all the pure materials present smaller EC responses as T decreases. To understand all these behaviors, in Figure 3a, we show the evolution of the polarization of the pure compounds under an applied electric field. By comparing Figure 2 and 3a, we note that the stronger EC responses correspond to the pure materials with a smaller spontaneous polarization in the unperturbed state, i.e., the more disordered ones. As we reduce the temperature or increase the epitaxial compression, we enhance the order of the unperturbed state and reduce the ability of the compound to react to the applied field. In these simple cases, where the electric field just contributes to saturate the polarization and no (discontinuous) phase transition is induced, the magnitude of the EC effect correlates strongly with the dielectric response of the compounds, as shown in Figure 3b. Indeed, it is known that, when the action of the field can be treated perturbatively, [22] the EC effect is essentially given by the T-derivatives of polarization (P 0 ) and dielectric susceptibility ( χ 0 ). Further, as shown in Note S2, Supporting Information, in simple cases like these ones, involving pure PTO samples, it can be seen that P 0 ∝ ffiffi ffi χ p and χ 0 ∝ χ 2 ; hence, the susceptibility χ reflects the magnitude of the perturbative EC effect. Figure 2 shows that the EC effect is generally stronger in the SLs than in the pure compounds subject to similar epitaxial constraints. In particular, the maximum enhancements are almost threefold at 100 K and well above twofold at 300 K. This general trend makes physical sense. Because of the electrostatic frustration imposed by the dielectric layers, the SLs are always more disordered; hence, we can expect them to present larger EC responses. However, a second factor is at play. Whenever the PTO volume fraction is relatively small, the EC response of the SL will suffer from the contribution of the comparatively stiff STO layers. This is the likely reason why some of the 3/3 systems Figure 2. Electrocaloric response. Computed electrocaloric response for representative 3/3 and 9/3 superlattices at two initial zero-field temperatures a) 100 K and b) 300 K and for various epitaxial constraints. Results for bulk PTO and PTO thin films are also shown for comparison. www.advancedsciencenews.com www.pss-rapid.com present comparable or smaller responses than the pure PTO compounds at 300 K and small fields. Let us stress that the observed enhancements correspond to the comparison between pure PTO samples and PTO/STO superlattices for initial temperatures of 100 and 300 K, i.e., far from the T C of the pure compounds (e.g., T C % 450 K for our simulated bulk PTO).
Let us now discuss in more detail the response of the SLs. If we consider the maximum responses, we can distinguish two types of systems. In most cases (especially at low T, for stronger epitaxial compressions and for thicker PTO layers), we find that the EC response peaks sharply at relatively small fields of about 1 MV cm À1 . Then, there are also cases (especially at higher T, for weaker epitaxial compressions, and for thinner PTO layers) where the maximum of the EC response is relatively broad and shifts toward higher electric fields. Interestingly, if we recall the discussion of Figure 1, these markedly different behaviors essentially correspond to having an unperturbed system characterized either by a strong polar order (sharp EC peak at relatively small fields) or by very strong dipole fluctuations (broad EC peak at larger fields), respectively.
To verify this connection and gain further insight, in Figure 4, we show snapshots of the dipoles at the center of the PTO layer for a number of representative cases. For example, panels a, b, and c show the 3/3 superlattice at 0% epitaxial strain and 300 K (3/3-0-300 for short) for applied fields of 2, 3, and 4 MV cm À1 , respectively. This is a strongly fluctuating system where, even at such relatively high fields, the polarization is far from having saturated. (Note the pale green tones in Figure 4b,c.) Interestingly, the maximum of the EC response seems to coincide with the disappearance (annihilation) of the regions of minority polarization (magenta areas in Figure 4a).
Let us now consider two cases with slightly stronger polar order: 3/3-2-300 (same SL and T, but now with a À2 % epitaxial compression) and 9/3-0-300. The evolution of the 3/3-2-300 case around its maximum EC response is shown in Figure 4d-f; the situation is similar to that of the 3/3-0-300 case: the EC maximum (at about 2 MV cm À1 ) coincides with the practical annihilation of the minority domains. The 9/3-0-300 case-with an EC maximum also at about 2 MV cm À1 and as shown in Figure 4g-i-is very similar as well.
We find that these broad EC maxima tend to coincide with minimum values of the c lattice constant, as shown in Figure 5. This makes good physical sense. As mentioned above, a minimum of c corresponds roughly to the gas-liquid transition upon zero-field cooling. If we cool down under field, the gas state becomes homogeneously polarized (in essence, as in Figure 4c) and the liquid state presents much reduced areas of minority polarization (similar to the situation depicted in Figure 4a); yet, the minimum of c continues to track (approximately) the point at which spontaneous local polarization appears. Hence, our results suggest that the broad EC maximum observed in some of our superlattices is connected to the  www.advancedsciencenews.com www.pss-rapid.com gas-liquid phase transition. Note that such a transition under field is expected to be diffuse, [23,24] which is consistent with the broad features found. Now let us consider the cases where the EC response presents a sharp peak at low fields. In Figure 4, we show snapshots for two representative situations that correspond to the largest T-changes calculated: 9/3-2-100 (panels j, k, and l) and 9/3-2-300 (panels m, n, and o). In the 9/3-2-100 case, we start from an unperturbed stripe-domain solid-like configuration; under field, the stripes break and we obtain a "bubble solid" of sorts; then, at 2 MV cm À1 the bubbles all but disappear and the polarization is essentially saturated. In the 9/3-2-300 case, we also start from stripes, but now fluctuating in a liquid-like phase; the electric field induces a transition to a "bubble liquid"; finally, at 2 MV cm À1 a wellordered state with polarization essentially saturated is reached. Hence, the maximum EC effects observed correspond to the breaking of the stripes into bubbles, which occurs at fields smaller than 1 MV cm À1 in all the considered cases, and where stripes and bubbles can be either frozen or mobile. The further transformation, from bubbles to a homogeneous monodomain, has a much smaller EC effect associated with it.
Our results thus provide a clear qualitative picture where two different mechanisms lead to strong EC effects, one involving a distinct and sharp transition that occurs at a specific value of the electric field, the other involving a continuous transformation over a wide range of fields. Beyond that basic observation, it is unclear how to rationalize the relative magnitudes of these two effects and explain why one is stronger than the other. Nevertheless, in the following, we propose some factors that we think play a role.
Let us first consider the field-induced breaking of the stripe domains, which we predict leads to the largest EC effects. Our simulations suggest that this transition is discontinuous (as evidenced by the hysteretic behavior of the PðEÞ curves, particularly at low T; see Figure S5, Supporting Information); hence, it must involve a latent heat of the kind that usually constitutes a large fraction of the EC effect in ferroelectrics and antiferroelectrics. [25][26][27] In addition, there are aspects suggesting an important role of the STO layers. First, these sharp EC responses tend to occur in situations (strong epitaxial compression, thick PTO layers, low T ) where the STO layers are about to develop a ferroelectric instability themselves. [13] Second, as shown in Figure S5, Supporting Information, in these SLs the net field-induced polarizations of PTO and STO layers are quantitatively similar so that depolarizing fields are minimized. Hence, a rapid development and saturation of the homogeneous polarization in PTO (as the one associated with our sharp EC peaks) requires a similarly rapid polarization of STO. Therefore, we can expect a significant contribution from STO to the T-change. Unfortunately, we do not see any convincing way to disentangle the PTO and STO contributions to the EC response in our simulations, which complicates a more detailed quantitative analysis.
Our original expectation was to obtain a very large EC effect by the field-driven saturation (freezing) of the very disordered liquid and gas phases of our SLs (e.g., those depicted respectively in Figure 1b,c), as it is intuitively clear that they must be high-entropy states. Let us stress that this expectation is in line with what happens in bulk ferroelectrics, the largest EC effects occur at temperatures around the Curie point, which are the analog of the liquid-gas transition of our SLs. We indeed obtain big T-changes from these states, for example, about 6 K by applying 1 MV cm À1 at T room in the 3/3-2 system. However, the fact that these are not the largest effects computed warrants a comment. Interestingly, these strongly-fluctuating states occur in situations (small epitaxial compression, thin PTO layers, high T ) that also yield dielectrically stiffer STO layers. As mentioned above, a stiff STO makes it hard to induce a homogeneous ferroelectric state with saturated polarization in the PTO layers, which in turn results in a relatively weak EC effect, as indeed observed from our simulations. Hence, these observations suggest that, to maximize the EC performance of these superlattices, it would be convenient to use dielectric layers that are as soft (easily polarizable) as possible, while still forcing the liquid-and gas-like phases to occur in the ferroelectric layers. In practice, this might be achieved by employing a suitably chosen STO-PTO solid solution, instead of pure STO, for the dielectric layers.
We should also note that while we have discussed the caloric effect caused by an increasing electric field (heating), for applications one is more interested in the response upon removal of the applied field (cooling). In materials as complex as the ones here studied-which undergo several kinds of field-induced transitions-and for the relatively large electric field steps we have used (1 MV cm À1 ), we cannot expect the computed caloric effect to be reversible. [28] Indeed, as shown in Figure S6 and S7 and discussed in Note S3, Supporting Information, preliminary calculations show a significant asymmetry between heating and cooling, even in cases for which isothermal PðεÞ loops present no hysteresis at all. Hence, our results for heating cannot be taken as accurate quantitative predictions for the related cooling. Nevertheless, as shown in the study by Marathe et al., [28] a (large) heating effect associated with a field-induced transition is accompanied by a similarly large cooling response, associated with the reversed transformation but occurring at slightly different conditions. Hence, our present findings for electrocaloric heating can be expected to reflect similar trends and effects (even in magnitude) for electrocaloric cooling. A detailed analysis of (ir)reversibility in these superlattices requires a dedicated investigation and remains for future work.
Before we conclude, let us note that the EC response of PTO/STO superlattices has recently been investigated theoretically, using a phase-field approach, by Ji et al. [29] . It is not clear to us whether the phase-field methods employed by these authors can be expected to capture the dynamical liquid-and gas-like states that occur in these systems. [11] By contrast, we think they should describe the behavior of the solid-like state in a qualitatively correct way, and Ji et al. [29] report some interesting results in that regime. More precisely, it is shown that some superlattices present a negative EC response for applied fields never exceeding 0.2 MV cm À1 . The negative effect is found to be associated with a stripes-to-monodomain ("vortex-to-polar") transformation. By contrast, we find no indication of a negative EC response in our calculations. Admittedly, we employ a very different physical model and computational (statistical) approach; yet, we think the main difference with respect to the study by Ji et al. [29] may be that we consider much stronger applied fields (a minimum of 1 MV cm À1 ), which takes us well into a nonperturbative regime with sizeable latent-heat contributions. Let us also note that, in absence of a net spontaneous polarization, the EC response of a material is controlled by the T-derivative of its dielectric susceptibility, χ 0 ; in particular, if χ 0 > 0, general perturbative arguments guarantee a negative EC effect at small fields. [22] We think that the main findings in the study by Ji et al. [29] can be interpreted in this way, which suggests a similarity with the negative EC effect in antiferroelectrics. Interestingly, some of us have recently proposed ferroelectric/ dielectric superlattices as artificial antiferroelectrics that can be optimized for energy-storage applications. [15] Studying further their anomalous EC response at small fields indeed appears as an interesting direction for future work. As a final note, let us point out that phase-field calculations can treat simulation boxes that are much larger than those considered in our atomistic calculations. However, the multidomain states studied by Ji et al. [29] are qualitatively similar to the ones discussed here. Hence, we believe that size effects play no significant role when comparing our results with those of in the study by Ji et al. [29]

Summary
We have used atomistic second-principles simulations to investigate the electrocaloric response of model ferroelectric/dielectric PbTiO 3 =SrTiO 3 superlattices. We find that the electrostatic frustration imposed by the paraelectric layers results in strongly responsive states yielding caloric effects up to twice bigger than the response of pure PbTiO 3 compounds at room temperature. Our results reveal two distinct situations (physical origins) leading to large temperature changes: one related to the field-driven transition between relatively well-ordered stripe-domain and bubbledomain states, and a second one associated with the field-driven saturation of the polarization in (liquid-and gas-like) phases of strongly fluctuating dipoles. Our findings evidence the key role of the paraelectric layers, and suggest strategies to enhance the caloric effects by increasing their polarizability (e.g., by applying an epitaxial compression). Hence, given the notable enhancements in electrocaloric response that we obtain, the variety of physical mechanisms that can potentially be exploited, and the wealth optimization possibilities these materials offer, ferroelectric/dielectric superlattices seem a suitable platform to further optimize electrocaloric effects. More generally, electrostatic engineering appears as a promising route to enhance the electrocaloric performance of any ferroelectric material.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.