1. Key Laboratory of Advanced Civil Engineering Materials of Ministry of Education, Functional Materials Research Laboratory, School of Materials Science and Engineering, Tongji University, Shanghai 201804, China
2. Department of Chemistry, City University of Hong Kong, Hong Kong, China
3. Department of Materials Science and Engineering, City University of Hong Kong, Hong Kong, China
ligh@tongji.edu.cn & g.h.li@cityu.edu.hk (Li G.),
apzhai@tongji.edu.cn (Zhai J.),
s.j.zhang@cityu.edu.hk (Zhang S.).
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Received
Accepted
Published Online
2026-04-27
2026-05-27
2026-07-03
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Abstract
Electrocaloric (EC) refrigeration converts electric-field-induced entropy changes into reversible heat exchange and offers a solid-state route for active cooling beyond vapor-compression refrigeration (VCR). Its features, including zero direct global warming potential (GWP), capacitive energy recovery, fast polarization response, and compatibility with miniaturized architectures, make it relevant to thermal management in domestic refrigeration, high-power electronics, and wearable systems. A persistent gap remains, however, between materials-level metrics, such as isothermal entropy change (ΔS) and adiabatic temperature change (ΔT), and device-level metrics, including temperature span (ΔTspan), specific cooling power (SCP), and coefficient of performance (COP). System performance is not set by a single intrinsic materials parameter. It is constrained by the cooling scenario, which defines the device-level thermal, electrical, and mechanical requirements and, in turn, the inverse materials targets. This review examines a scenario-driven systems-engineering framework for EC refrigeration along the sequence of cooling scenarios, device architectures, and materials inverse design. Three representative boundaries are considered: macro-scale refrigeration using multilayer ceramic capacitor (MLCC) arrays and active fluid regenerators, on-chip cooling for localized and transient heat flux, and self-driven wearable systems requiring flexible integration and low-field operation. Regenerative fluid systems, conduction-based contact devices, electro-mechanical actuation systems, and electrostatic self-regenerative topologies are compared in terms of heat-transfer pathways, parasitic thermal mass, charge recovery, contact resistance, and thermodynamic-cycle closure. The review then develops an inverse-design view of EC materials, covering phase-configurational and chemical-configurational high-entropy engineering for smoothing polarization-switching barriers and enlarging reversible entropy changes, together with thermal-impedance management for converting intrinsic entropy gain into usable cooling power. Cross-scale tools, including density functional theory (DFT), phase-field modeling, and machine learning (ML), are assessed for linking atomic-scale dipole physics, mesoscale domain evolution, device-level heat flow, and high-dimensional compositional search. This framework connects intrinsic EC materials optimization with deployable solid-state refrigeration systems.
1.1 Physical limits of vapor-compression cooling across length scales
Thermal management underpins modern industry and infrastructure across building environments, cold-chain logistics, residential applications, and heat dissipation in integrated electronic devices [1,2]. Under the broader transition toward low-carbon energy systems, the energy efficiency of refrigeration has become a key indicator in evaluating industrial sustainability [3]. Data cited from the International Energy Agency (IEA) and Global Cooling Watch 2025 indicate that global demand for cooling continues to rise under the combined influence of extreme heat events, economic growth, and population growth [3–5]. By 2050, the worldwide installed base of cooling equipment is expected to exceed the present level by more than a factor of three [1].
Under the business-as-usual scenario, cooling-related electricity demand is projected to increase from 22 TW in 2022 to 68 TW in 2050, while annual CO2 emissions are expected to approach roughly twice the 2022 level, reaching about 7.2 billion tons [1]. The widespread use of VCR increases peak electrical loads on regional grids. Waste-heat rejection during operation and refrigerant leakage also aggravate urban microclimates and greenhouse-gas control. The development of low-energy, environmentally benign refrigeration technologies is therefore a necessary route toward relieving energy-supply pressure and advancing global climate goals [1].
VCR, which has dominated heat-pump technology for more than a century, has reached a clear thermodynamic limit. After a century of mechanical optimization, the COP of VCR systems has approached the Carnot limit so closely that marginal gains from compressor redesign or expanded heat-exchanger area have entered a regime of strong nonlinear diminishing returns. In parallel, fluorinated working fluids such as hydrofluorocarbons exhibit very high global-warming potential. Under international constraints such as the Kigali Amendment, the phaseout of refrigerants with a high GWP and the search for zero-direct-GWP or low-emission alternatives have become urgent industrial priorities.
A deeper scientific obstacle is the lack of scalability of VCR across geometric length scales. As packaging density rises in the post-Moore era, thermal-management requirements have extended to the micrometer and even nanometer range. VCR relies on a macroscopic cycle composed of compressors, expansion valves, and heat exchangers, and these physical components are difficult to miniaturize without a severe efficiency penalty. Once characteristic dimensions fall below the millimeter scale, viscous drag in microchannels rises sharply, while frictional and thermal losses in micro-pumping components grow rapidly as size decreases. This scale-effect-driven collapse of efficiency limits the suitability of VCR for modern microelectronic systems, flexible wearables, and precision optoelectronic devices for localized, high-precision, highly integrated active cooling.
These physical limits have driven broad interest in solid-state refrigeration. By using external fields to drive entropy changes directly in a solid working medium, solid-state refrigeration eliminates complex moving parts and gas–liquid phase transitions, and thereby opens a new physical route toward efficient and compact thermal-management solutions across a wide range of scales [1,2,5].
1.2 Solid-state refrigeration routes and technical trade-offs
Among candidate replacements for VCR, four active solid-state refrigeration routes involve distinct trade-offs among energy density, efficiency limits, and system integrability.
Thermoelectric cooling (TEC) is the most commercially mature integrated technology. It transfers heat through carrier transport across semiconductor junctions and therefore offers no moving parts, silent operation, and excellent scale-down capability [6]. Its limitation lies in the intrinsic thermoelectric figure of merit ZT. The COP of thermoelectric devices is usually well below 10% of the Carnot limit, and high performance still depends heavily on rare or toxic Bi2Te3-based compounds [7–11]. This low energy-conversion efficiency constrains its use in high-load thermal management and in portable systems that require long endurance.
Caloric effects driven by entropy changes across phase transitions are more attractive from the standpoint of theoretical efficiency [12–14]. Magnetocaloric refrigeration is the most mature branch of this family. It relies on magnetic-field-driven rotation or para-to ferromagnetic phase transitions. Although magnetocaloric systems benefit from long cycle life and negligible hysteretic loss in the driving field [15–17], their system efficiency and power density are restricted by the magnetic-field source. Sufficient magnetic induction usually requires bulky permanent-magnet arrays or complex superconducting magnets. As a result, mass and volume are difficult to reduce, rare-earth costs remain high, and the low operating frequency, typically below 10 Hz, limits deployment in miniaturized and high-power-density applications [18–21].
Mechanocaloric effects (mCE) [22,23] include elastocaloric effect (eCE) [24,25] under uniaxial stress and barocaloric effect (BCE) [26–33] under hydrostatic pressure. Owing to the large latent heat released during structural phase transitions in shape-memory alloys or plastic crystals, mechanocaloric systems exhibit very large intrinsic entropy changes and temperature spans. Their main liabilities are the complexity of the driving system and severe materials fatigue. Elastocaloric devices require large hydraulic or mechanical loading systems, and repeated high-stress cycling readily causes functional degradation or fracture [24,25]. Barocaloric systems face parallel difficulties in high-pressure sealing and in the selection of heat-transfer media. These factors complicate packaging and limit the usefulness of mechanocaloric refrigeration in compact spaces [26–33].
EC refrigeration operates under a different driving mechanism and offers a distinct integration advantage (Fig. 1) [34–50]. Direct electrical driving removes the need for expensive magnets and heavy mechanical actuators, which improves power density and integration flexibility. In addition, the EC working body is capacitive, so charge stored during the field-on step can be partially recovered during field removal. This feature lifts the efficiency potential. Polymers and multilayer ceramics also show extremely fast polarization response, allowing operation at kilohertz frequencies and therefore high-throughput heat pumping within a very small volume [34–43,46].
To provide a device-level basis for comparing solid-state refrigeration routes, representative data for thermoelectric, magnetocaloric (MCE), elastocaloric, barocaloric, and EC technologies (ECE) were compiled in terms of COP, device ΔTspan, cooling/heating power, specific cooling/heating power, and heat flux or power density (Fig. 2A–E and Table 1). The parameter maps place the major solid-state cooling routes in different performance regions, rather than along a single hierarchical line. The power map is the most robust cross-route comparison because it reflects the heat moved by the implemented device or system. The SCP map highlights the advantage of lightweight active bodies and compact heat-transfer paths. EC devices are notable for comparatively high COP values in the available finite-span data, particularly when the reported analysis includes favorable loss accounting or electrical energy recovery. Their specific power is also competitive on a mass-normalized basis, with several polymer-based or self-regenerative configurations extending into the medium-to-high SCP region. At the same time, the absolute cooling or heating power of EC prototypes remains lower than that of the more mature magnetocaloric and elastocaloric platforms, whose representative points continue to dominate the higher-power region. The EC heat-flux data are likewise concentrated in localized or device-level demonstrations, indicating that the presently reported advantages are strongest where short thermal paths and small active volumes are beneficial. Taken together, the present distribution suggests that the current strength of EC devices lies in efficiency-oriented and compact-device operation, whereas their remaining limitations are concentrated in absolute power output, thermal throughput under load, and broader scaling of heat extraction. Further progress should therefore focus on reducing dielectric and leakage losses, improving charge recovery, lowering interfacial thermal resistance, suppressing inactive thermal mass, and integrating multilayer or self-regenerative architectures that can sustain higher heat transfer without eroding COP. By comparison, magnetocaloric and elastocaloric systems already occupy the high-power regime but remain constrained by magnetic-field sources, mechanical loading, fatigue, and auxiliary losses, while barocaloric devices still show considerable potential but are strongly dependent on pressure generation and heat-transfer implementation. Thermoelectric devices retain their value in compact electrical cooling and high local heat-flux operation, but large nominal ΔTspan should not be interpreted alone as cooling capability under practical load conditions.
The ΔTspan plotted here should therefore not be treated as a direct measure of useful cooling capacity. The apparent span can be affected by Joule heating and other thermal factors [87], particularly in thermoelectric devices. In some cases, a large device span may coexist with a stabilized cold-side temperature that remains close to the operating environment, which limits practical cooling. Accordingly, several reports emphasize the active cooling temperature difference relative to the working environment as a more application-relevant descriptor [46,85,88,92]. One TEC report was extracted separately because its values are much higher than the distribution of most devices considered here. Lin et al. [93] reported two operating points, approximately 59 K with COP of about 0.39 and heat flux of about 5700 mW/cm2, and approximately 87 K with COP of about 1 and heat flux of about 6000 mW/cm2. These data were placed at the plot boundary instead of expanding the axis range, so that the main distribution remains visible. More detailed comparisons of maximum cooling power, maximum SCP, and maximum cooling ΔTspan can be found in Ref. [64].
1.3 EC refrigeration as a scenario-driven design problem
1.3.1 From intrinsic materials metrics to system-level cooling performance
Longitudinal monitoring of publication trends over the past two decades underscores the long-standing dominance of the MCE in the field of solid-state refrigeration, with annual academic output approaching 1000 publications by 2024 (Fig. 3A). In contrast, although EC research has expanded significantly since the verification of giant EC responses in 2006, its current annual volume of approximately 400 publications remains less than half of that of MCE. Quantitatively, ECE research occupies a second-tier position alongside mechanocaloric effects, maintaining only a marginal lead over the steadily increasing output of TEC.
The bibliometric distribution shows a marked shift within the ECE literature from isolated materials exploration toward materials-device coupling (Fig. 3B). Historically, EC research was characterized by a severe structural imbalance, where approximately 90% of prior efforts were confined to the isolated exploration of intrinsic material properties. However, statistical data from the most recent decade (2015–2024) indicate a marked shift: the proportion of studies focused exclusively on materials (only materials) has been progressively compressed, while the share of research integrating materials with device architectures (overlap) has increased. By 2024, research involving device-level integration—encompassing both the "overlap" and "only devices" categories—accounted for over 55% of the total annual publications (Fig. 3C). This evolution dictates that the focus of EC research is shifting from the pursuit of idealized intrinsic parameters toward resolving the complexities of system-level cooling capability.
Nevertheless, this proportional redirection has not yet fully bridged a gap between intrinsic material metrics and practical system performance. Intrinsic adiabatic temperature change ΔT or isothermal entropy change ΔS recorded under quasi-static laboratory conditions are often attenuated when translated into functional cooling prototypes, particularly regarding SCP and system-level ΔTspan. This discrepancy is governed by the fact that standard material evaluations typically occur near thermodynamic equilibrium, whereas actual refrigeration cycles constitute non-equilibrium dynamic processes involving polarization kinetics, delayed phonon transport, and interfacial thermal resistance. If the physical feedback constraints imposed by system-level topologies—such as the threshold for lattice thermal conductivity κ and the fatigue boundaries under prolonged alternating electric fields—are neglected during the early stages of inverse material design, single-dimensional optimization of physical parameters is unlikely to fully address the technical barriers between laboratory discovery and industrial deployment.
1.3.2 Energy recovery and thermodynamic cycling in EC devices
High-efficiency EC refrigeration depends on precise control of the thermodynamic cycle. Figure 4 illustrates the canonical EC Brayton cycle, which consists of four steps: adiabatic polarization with heating, isofield heat rejection to the cooling medium, adiabatic depolarization with cooling, and isofield heat absorption at the cold side.
The COP is defined as the ratio of the cooling power delivered by the refrigeration material or device to the total work input consumed by the system.
In contrast to VCR, the EC cycle possesses a distinct physical advantage for efficiency improvement. Because the working body behaves as a nonlinear capacitor, most of the electrical energy stored in the dielectric during polarization can, in principle, be recovered during depolarization through appropriate circuit design rather than being dissipated irreversibly as Joule heat. When this energy-recovery logic is combined with the kilohertz-scale response of EC materials, the system can in principle combine a high COP with a high cycling frequency and thus deliver large heat throughput within a small volume. Realization of this potential, however, depends critically on matching the thermal impedance of the material and the device topology.
1.3.3 Full-link design logic: scenarios, architectures, and materials
To address this gap, the following discussion uses application scenarios as boundary conditions for linking device topologies with materials inverse design. Materials-development targets should not be set by maximizing a single intrinsic parameter in isolation. System performance is defined by the cooling scenario and the associated thermal, electrical, mechanical, and manufacturing constraints. These constraints determine the required device topology and then the inverse materials targets; the same EC material can therefore lead to different useful performance in a regenerator, a contact device, or a wearable self-actuated topology. This review starts from the cooling scenario, identifies the device-level thermal, electrical, mechanical, and manufacturing constraints, and then maps those constraints back to materials targets.
Three representative scenarios therefore serve as the basis for the analysis:
(1) Macro-scale refrigeration based on multilayer ceramics [42,43,94]. The central issue is how large series-parallel architectures can overcome ceramic brittleness while exploiting the long-term reliability of ceramics for residential and industrial service.
(2) Miniaturized and integrated on-chip cooling, with emphasis on ferroelectric polymers such as polyvinylidene fluoride (PVDF) and related copolymers [44–46]. This scenario requires integration flexibility and composite-modification strategies that can satisfy a device-dependent thermal-conductivity threshold for heat extraction within finite cycling periods.
(3) Self-driven wearable systems for human thermal comfort [47,48]. This scenario places stringent limits on driving-field strength and introduces additional design dimensions such as electromechanical energy recovery and self-oscillating device topologies.
These scenario-dependent constraints are tied to the current operating envelopes of solid-state cooling systems (Fig. 2). For instance, the comparatively lower absolute power of current EC prototypes (Fig. 2B) underscores the necessity of scaling active volume and minimizing parasitic loss for macro-scale refrigeration. Conversely, their competitive specific cooling power (Fig. 2C) and high localized heat-flux density (Fig. 2D) support their targeted use in lightweight wearable applications and on-chip thermal management. To support such inverse design, the review further surveys cross-scale theoretical and data-driven tools. At the atomic scale, density functional theory is used to resolve the physical origin of dipolar entropy change. At the mesoscale, phase-field dynamics provide a simulation framework for domain-wall evolution and heat-flow distribution. ML, as an efficient data-driven tool, enables rapid optimization and property prediction across vast compositional spaces. Together, these approaches provide physical criteria and performance guidelines for EC refrigeration systems.
2 Scenario-driven Device Design for EC Refrigeration
2.1 Device topologies beyond intrinsic temperature change
2.1.1 Early cryogenic devices and regenerative cooling concepts
This progression reflects the need to exceed the intrinsic adiabatic temperature change of a single material element through thermodynamic cycling and thermal accumulation.
(1) Cryogenic single-module devices
The engineering exploration of EC systems originated from the thermal management requirements of cryogenic environments. Lawless et al. constructed the first cryogenic EC concepts utilizing SrTiO3 ceramics [95] and KTaO3 single crystals [96]. These studies verified the technical feasibility of solid-state refrigeration for specialized scenarios, such as the cooling of superconducting equipment within the 4 K to 15 K temperature range [97]. Although this work established the concept of system-level cooling, these prototypes functioned as typical single-module refrigerators, where the cooling performance and ΔTspan were strictly limited by the intrinsic ΔT of the material. This single-stage architecture lacked the capacity for cascaded thermal accumulation, resulting in low refrigeration factors—defined as the ratio of actual system cooling capacity to the intrinsic material effect—and failing to achieve temperature gradients of practical engineering value.
(2) Quasi-regenerative dual-module systems
To overcome the physical bottleneck where the ΔTspan of single-module devices is capped by the material's absolute ΔT, the active EC regenerator (AER) topology was introduced. Sinyavsky et al. [98] developed the first prototypical EC refrigerator. The core topological innovation lay in the implementation of a dual-module series system. The structure comprised a parallel-plate matrix of 20 Pb(Sc0.5Ta0.5)O3 (PST) ceramic plates. By operating two independent modules with a 180-degree phase difference in an alternating electric field within a closed-loop fluid circuit, and synchronizing the reversal of fluid pumping, the system realized a quasi-regenerative thermodynamic cycle. Under a driving field of 15 kV/cm, this dual-module prototype achieved a system ΔTspan of 2 K and a SCP of 0.7 W/kg. This advancement established the engineering route for amplifying temperature spans through system-level regenerative design.
(3) Temperature-span amplification by topology
Subsequent deep optimization of the dual-module AER architecture further demonstrated the usefulness of this topology in temperature-lift amplification. Sinyavsky and Brodyansky [99] integrated two PST ceramic modules with a total mass of 35 g (individual plate dimensions: 20 mm × 10 mm × 0.3 mm) into a single housing (Fig. 5A, B). To enhance system energy density, liquid pentane was employed as a high-dielectric-strength heat-transfer medium, enabling the driving field to increase to 60 kV/cm. Experimental results confirmed that through synchronized thermal migration driven by the anti-phase operation of the dual modules, the system achieved a ΔTspan of 5 K, which reached 5 to 6 times the intrinsic temperature change of the working body.
These early devices showed that temperature-span accumulation requires topology-level heat management rather than only larger intrinsic ΔT. This quasi-regenerative logic, based on synchronized phase control, not only addressed the deficiencies of early single-module devices but also provided the physical benchmark for the design of all subsequent complex solid-state regenerative and cascaded refrigeration architectures.
2.1.2 Heat-transfer and actuation pathways in EC devices
The transition from intrinsic materials entropy change to accumulated system-level ΔTspan depends on precise spatiotemporal control of heat flow by device topology. Intrinsic materials parameters represent cooling potential only. The actual system metrics are dictated by the underlying thermodynamic-cycle architecture. On the basis of heat-transfer mechanisms and actuator physics, EC devices can be grouped into three major topological paradigms.
(1) Regenerative fluid systems
Active fluid regeneration architectures use heat-transfer fluids as thermal carriers and rely on phase-synchronized operation to accumulate adiabatic temperature changes. This paradigm originated in the field of MCE, where Brown [100] introduced the active magnetic regenerative (AMR) cycle. By utilizing fluid media to decouple lattice entropy loads from the spin system, the device established a thermodynamic framework for generating spatial temperature gradients through regenerators. Sinyavsky and Brodyansky [99] subsequently mapped this approach to the EC domain, utilizing the dual-module anti-phase operating topology discussed previously to verify the technical feasibility of temperature-span amplification in fluid-based systems.
In subsequent performance optimization, the research focus shifted toward thermal impedance management and flow-field characterization at the fluid-solid interface. Addressing the temperature-span degradation induced by thermal mixing, Plaznik et al. [101] identified that regenerative efficiency is dependent on the physical ratio between the swept volume and the internal stagnant volume (Fig. 5A). By introducing the dimensionless parameter V*, this device quantified the impact of fluid displacement on the integrity of the Brayton cycle. Experimental data demonstrated that within the V* range of 0.8 to 1.0, the system effectively suppresses entropy generation associated with thermal mixing. Under a driving field of 50 kV/cm, the topology achieved a ΔTspan of 3.3 K with a regeneration factor of 3.7, underscoring the necessity of precise hydrodynamic parameter tuning for maximizing heat-exchange potential. Furthermore, numerical modeling projected that substituting the working fluid with water could further elevate the regeneration factor to 9.6.
Building on the numerical framework established by Plaznik et al., Blumenthal and Raatz [102] addressed the mismatch between theoretical designs and engineering reliability (Fig. 5C). While reducing channel spacing facilitates thermal penetration, it simultaneously increases fluidic resistance and risks encapsulation failure in practical applications. The study adopted a balanced strategy by expanding the channel spacing to 0.3 mm and introducing a modular stacked architecture. Although this involved a compromise in ideal heat-exchange efficiency, it mitigated fluidic blockage and electrode short-circuiting. This topological adjustment for engineering tolerance ensured operational robustness over 2000 hours of continuous testing, validating the physical stability of regenerative fluid architectures under long-term service conditions. The regenerative paradigm is also compatible with gaseous media. Annapragada [103] demonstrated an AER prototype using a gas working fluid that achieved a maximum ΔT of 14 K, validating gas-phase thermal transport as a viable alternative for generating system-level temperature gradients.
Expanding upon the modular paradigms of Blumenthal and Raatz [102], Torelló et al. [36,43] showed that reducing inactive structural mass and increasing the active heat-exchange area are central to temperature-span accumulation in parallel-plate AER architectures (bonding 128 PST) (Fig. 5A). This configuration increased the active mass fraction to 60%, expanding the effective heat-exchange area. Coupled with a non-linear feedback control strategy, the system achieved a ΔTspan of 13.0 K at 158 kV/cm. This evolution indicates that minimizing non-active mass to eliminate "thermal trap" effects is essential for enhancing system-level thermal pumping.
Li et al. [42] built on the regenerative framework of Brown et al. and the parallel-plate matrix architecture of Torelló et al., while further shifting the design focus from active-material fraction to flow-path control, terminal placement, and parasitic axial heat leakage, thereby defining a more complete system-level constraint for macro-scale AERs (Fig. 5A, D). The study addressed the degradation caused by electrode terminals by proposing a configuration where terminals are repositioned inside the fluid channels (REG-TI). Concurrently, the system introduced a double-loop asymmetric fluid topology to decouple measurement processes from thermal transport. Under a driving field of 100 kV/cm, this prototype generated a maximum ΔTspan of 20.9 K and a maximum cooling power of 4.2 watts, representing a fifteen-fold increase in cooling power compared to the preceding generation. Under boundary conditions accounting for fluidic pressure drop and energy recovery, the second-law efficiency reached 64% of the Carnot limit. This progression through continuous optimization of channel geometry, active mass fraction, and circuit topology validates the application potential of regenerative fluid architectures within high-efficiency solid-state cooling frameworks. Significantly, the architecture represents a critical pivot toward macroscopic application by breaking the watt-level cooling capacity barrier for the first time. This benchmark provides evidence for the feasibility of large-scale electrocaloric refrigeration, suggesting that solid-state caloric media may be scaled from milliwatt-scale laboratory indicators to high-capacity, engineered thermal management platforms.
(2) Conduction-based contact systems
The transition from intrinsic material response to system-level performance is dictated by the evolution of topological architectures, progressing from micro-scale targeted dissipation to integrated, high-density, and scalable systems.
Addressing the requirements for localized high-heat-flux cooling in microelectronics, Gu et al. [104] demonstrated an EC oscillatory refrigeration (ECOR) architecture (Fig. 6A). This system synchronizes the displacement of EC modules with electric-field excitation to facilitate cyclic contact heat transfer with a solid-state regenerator. The design utilizes graphite-powder/silicone-oil lubricants to enhance interfacial thermal conductance and micro-slitted regenerator geometries to suppress longitudinal parasitic heat conduction, achieving a ΔTspan of 6.6 K at 1 Hz.
Toward practical deployment, Zhang et al. [105] constructed an integrated solid-state rotary prototype leveraging commercial MLCC technology (Fig. 6A, D). By employing dielectric layers with a thickness of 12 µm, the system achieved high-field operation at a domestic-scale driving voltage of 200 V (166 kV/cm). The thermodynamic path follows a Brayton-type cycle through direct solid-solid contact between coaxial counter-rotating MLCC rings, yielding a regeneration factor of 3. This milestone resolved the physical antagonism between the prohibitive driving voltages of bulk ceramics and the insufficient cooling capacity of thin films. Furthermore, by substituting complex fluid-pumping systems with a mechanical rotary topology, the design suppressed parasitic power consumption, verifying the technical viability of low-voltage, miniaturized EC systems. To overcome the limited span of a single unit without an external regenerator, Gu et al. [50] proposed an all-solid-state rotary cascade paradigm (Fig. 6A–C). By arranging multiple discrete EC units on a rotating ring and enforcing stepwise heat exchange, the prototype continuously pumped heat through space, following the heat-accumulation principle used in cascaded magnetic refrigeration. Both experiment and theory confirmed that the total ΔTspan increases significantly with the number of cascaded units Ns. This architecture shows that large temperature-span accumulation can be achieved through device topology alone, without complex fluid loops. Building upon this, Wang et al. [94] developed a scalable all-solid-state system based on a thermal-switch mechanism and physical stacking of multiple Brayton-cycle modules (Fig. 6A, E). Through the synchronized dynamic coupling between PST-MLCC units and high-thermal-conductivity interfaces, the system reached a heat flux of 135 mW/cm2. These studies established the physical feasibility of high-throughput targeted cooling in all-solid-state architectures without fluid assistance, providing a key route for transient thermal management in high-heat-flux electronics.
Addressing the "scaling-effect" challenge—where cooling power in solid-state systems is often offset by parasitic dissipation as structural dimensions increase—He et al. [91] reported an interdigitated all-solid-state device based on lead-free (Ba, SrTiO3) (BST) MLCCs (Fig. 6A, F and G). The topological core features a dual-driving logic: Strategy 1 (parallel) to enhance cooling power and Strategy 2 (cascade) to amplify the ΔTspan. To optimize interfacial heat exchange, the system incorporates thermal pads with a conductivity of 5 W/(m·K), effectively stabilizing contact resistance during mechanical reciprocating motion. The six-stage cascade prototype (REG2) achieved a system ΔTspan of 7.13 K, with the SCP of the active regions reaching 49.9 W/kg and 282 W/L, outperforming existing Pb-based and Ba-based solid-state prototypes. This interdigitated topology indicates its ability to allow cooling power to scale linearly with the mass of the active material, circumventing the bottlenecks of traditional architectures and highlighting the potential for high-density, environment-friendly EC systems in compact thermal management scenarios.
(3) Electro-mechanical actuation systems
To circumvent the parasitic power consumption and volumetric redundancy associated with fluid pumping, electro-mechanical actuation topologies have been proposed. The core thermodynamic pathway involves utilizing reciprocal mechanical motion in physical space to establish alternating thermal contacts, thereby directing heat transfer between a heat source and a heat sink. In the evolution of this topology, the suppression of thermal contact resistance and the internalization of mechanical driving mechanisms constitute the primary physical trajectory.
Early electro-mechanical topologies relied heavily on external mechanical work components (such as a motorized z-stage) to force the up-and-down motion of the EC materials (Fig. 7A, B) [106]. These purely rigid devices exposed a fundamental physical conflict in practice: direct solid–solid contact not only faces severe mechanical wear, but its microscopic surface roughness also leads to excessively high thermal contact resistance, preventing theoretical performance from being replicated in physical entities. To breach this heat-transfer boundary, early experimental prototypes [107] implemented a critical structural trade-off by introducing switchable liquid-based thermal interfaces. This approach verified that introducing a rheological medium into a rigid topology can effectively bridge microscopic roughness and reduce thermal resistance. This exploration of the liquid interface provided direct parametric and structural guidance for resolving contact failure in subsequent self-oscillating topologies.
The external mechanical drivers in early topologies introduced high energy consumption and parasitic volume. Consequently, research shifted toward internalizing the actuation mechanism. Bradeško et al. [108] proposed a self-driven vertical cascade topology that utilizes the coupling of the intrinsic EC and electromechanical piezoelectric properties of the material (Fig. 7A, C). By applying an alternating electric field, this design achieved synchronized heating and bending displacement for an array of such cantilevers without the need for external mechanical motions.
However, when this cascade topology was materialized as rigid multifunctional cantilevers [109], the material rigidity and bending deflection reintroduced the issue of poor solid-to-solid heat transfer (Fig. 7A, D). To overcome the high thermal contact resistance caused by the sharply reduced contact area, the researchers directly adopted the liquid interface paradigm established by early prototypes, precisely introducing a droplet of ethylene glycol at the cantilever contact interfaces. During mechanical deflection, the droplet spreads into a continuous liquid film, instantaneously forming a low-thermal-resistance contact zone. This physical strategy of interface softening elevated the effective operating frequency to 8 Hz and achieved a 2.4-fold enhancement in temperature drop at 45 kV/cm, validating the high efficiency of liquid-film-assisted contacts in rigid cascade topologies.
Although introducing fluid interfaces can patch the contact defects of rigid ceramics, Han et al. [46] attempted to eliminate this physical contradiction at the material root through a heterogeneous fusion topology (Fig. 7A, E and F). This device discarded purely inorganic ceramic actuators and constructed flat panel EC cooling devices, wherein the core working body was replaced by a flexible 30 µm thick P(VDF-TrFE-CFE) polymer film. Leveraging ceramic systems, early investigations pioneered the physical blueprint for integrated "actuation-cooling" by partitioning electrical energy between the EC effect and the converse piezoelectric (or electrostrictive) response [108–110]. By employing asymmetric laminated architectures, these studies converted microscopic lattice strains into macroscopic out-of-plane bending to trigger intrinsic thermal switching under antiphase control. While constrained by the limited strain and high interfacial thermal resistance of early rigid ceramics, this "system subtraction" paradigm established the foundational topological lineage for subsequent high-performance, flexible self-driven EC heat pump systems.
The device reported by Han et al. is based on a coupled EC and electrostrictive response: under an alternating electric field of 667 kV/cm, the polymer film generates an EC effect while simultaneously exhibiting a 1.9% in-plane electrostriction, thereby spontaneously inducing vertical deflection. Benefiting from the high mechanical compliance of the polymer, the film achieves good contact with the external surface directly. Without any liquid medium assistance, the system obtained a steady-state thermal contact resistance as low as 0.6 K·cm−2·W−1 under a minimal contact pressure of 160 Pa. This topological innovation not only realized an external-driver-free, self-cycling, soft refrigerator, but also delivered a system ΔTspan of 8 K at 0.5 Hz. After 70,800 testing cycles, the cooling power was maintained at 99.8% of its initial state. This demonstrates that fusing self-oscillating mechanisms with the intrinsic deformation capabilities of flexible polymers is an effective pathway to reduce the solid-solid thermal impedance mismatch and mechanical redundancy in electro-mechanical actuation systems. Importantly, the electro-thermo-mechanical synergistic device [46] establishes a critical milestone in system integration. By introducing a novel design model that seamlessly couples electro-induced volume strain (electrostriction) with entropy change, this architecture reduces the mechanical redundancy of earlier prototypes. This advancement elevates electrocaloric refrigeration to a new stage of full integration, moving closer to practical application. Its cooling performance surpasses that of previous self-actuated devices in the same class, providing an essential structural blueprint for the development of future single-module and cascaded electrocaloric heat pumps.
(4) Electrostatic self-actuation systems
Electrostatic actuation topologies utilize Maxwell stress to drive flexible EC thin films in rapid reciprocal motion between a heat source and a heat sink. Compared to active fluid regeneration or mechanical drives, this topology reduces system volume, mass, and parasitic heat generated by frictional forces by eliminating external electric motors or pumps, providing a physical pathway for constructing compact solid-state cooling systems.
The paradigm of electrostatic-driven EC cooling was established by performing a "subtraction" of mechanical redundancy. Ma et al. [45] proposed a prototype based on unipolar electrostatic actuation. The core of this topology is an S-shaped film actuator with low bending stiffness (Fig. 8A, B), utilizing P(VDF-TrFE-CFE) as the active material and single-walled carbon nanotubes (CNTs) as ultra-lightweight electrodes to minimize mechanical constraints. By executing a six-step thermal pumping cycle under an EC field of 667 kV/cm and an actuation field of 610 kV/cm, the device achieved a system ΔTspan of 1.4 K and a COP of 13 at 0.8 Hz. The device demonstrated the physical feasibility of using reversible electrostatic forces to simultaneously achieve film actuation and tight thermal contact at the interface.
To overcome the physical limits of intrinsic temperature changes in single-layer films, the topological structure evolved toward spatial cascading. Meng et al. [44] constructed a cascade EC cooling device that integrates multiple EC units via thermal tandem junctions (Fig. 8A, C). The key to this topology lies in the antiphase mode of operation, where adjacent film stages maintain synchronized but opposite phases in both spatial displacement and EC effects, allowing heat flow to be transferred consecutively through the films. Experiments showed that the ΔTspan increases linearly with the number of stages, with a four-layer cascaded device achieving a zero-load span of 8.7 K at 600 kV/cm and 1.0 Hz. Furthermore, the system introduced an internal charge recovery mechanism using an energy-recovery circuit to transfer charges between conjugated stacks, recovering 65% to 70% of the input electrical energy and enhancing the overall exergy efficiency of the cascaded system.
As application requirements shifted from global cooling to localized precision thermal control, electrostatic actuation topologies incorporated matrix control logic. Bai et al. [90] developed an active pixel-matrix device by pre-depositing pixelated CNT electrode pairs on modified polymer films, enabling spatial decoupling of the cooling regions (Fig. 8A, D). The main advantage of this topology is targeted and differential thermal management, where individual pixel units can be independently activated based on local heat loads. At the material level, the study introduced double bonds (DB) via triethylamine dehydrochlorination and doped the matrix with Ba0.6Sr0.4TiO3 nanoparticles (BST NPs) to construct a DB 7.1@BST-15 nanocomposite, enhancing both thermal conductivity and Young's modulus. Experiments demonstrated that a single pixel (6.5 mm × 6.5 mm) of this matrix device could achieve a surface temperature reduction of 33.2 K within 120 s in tests targeting central processing unit (CPU) hotspots, validating the flexibility of this topology in complex thermal environments.
The most integrated form of electrostatic actuation topologies is the elimination of all inactive supporting components to achieve functional integration at the material level. Drawing upon the topological lineage of early self-driven ceramic prototypes [46,108–110], Wu et al. [88] further refined this self-driven paradigm and proposed a self-regenerative heat pump (SRHP), where area expansion (electrostriction) triggered by internal dipole reorientation in P(VDF-TrFE-CFE) occurs synchronously with the EC effect (Fig. 8A, E). This topology utilizes polyimide (PI) tape as an asymmetric backing layer to convert in-plane strain into regular out-of-plane deformation, generating a blocking force of 0.39 N at 800 kV/cm.
In the SRHP topology, six film stacks execute a self-regenerative cycle through precise spatiotemporal coordination, with adjacent stages alternately generating heat and bending into contact under an antiphase logic. This design removes external actuator(s) and redundant interfacial materials, reducing the thermal mass of inactive components to a minimum. To ensure extreme rigor, the polymer underwent deep purification via Soxhlet extraction, and the infrared thermography data were validated through emissivity calibration based on Stefan-Boltzmann's law. This integrated device achieved a temperature lift of 14.2 K and a SCP of 1.52 W/g at 800 kV/cm and 1 Hz. Combined with an energy recovery circuit reaching a 73% recovery rate, the COP reached 10.1 (at a 7.4 K span). This stage of research marks the integrated form of electrostatic actuation topologies from complex mechanical-assisted systems into integrated, low-thermal-inertia material-level heat pump units.
After establishing these fundamental heat-transfer and actuation pathways, the subsequent sections will evaluate how these structural topologies behave under specific application constraints, including macro-scale refrigeration, localized on-chip cooling, and flexible wearable thermal management.
2.2 Macro-scale refrigeration and high-capacity heat pumping
2.2.1 Active regenerators and thermal accumulation
When EC technology scales up to replace traditional VCR systems—such as household air conditioners or industrial refrigerators—the core engineering challenge shifts from merely generating intrinsic cooling capacity to efficiently extracting and transporting macroscopic thermal energy without severe self-consumption. In these large-scale scenarios, the AER topology, which utilizes heat-transfer fluids to transport massive thermal loads, has emerged as a primary paradigm. Because the main device classes have been introduced above, this section focuses on how these topologies behave under macro-scale refrigeration constraints.
In macroscopic AER systems, the physical stacking of large-mass active materials inherently introduces significant non-active components, such as insulating frames, spacers, and electrode terminals. These components act as "thermal traps," absorbing the generated cooling energy and severely limiting the system-level ΔTspan. To overcome this bottleneck, modern macro-topologies execute extreme structural de-redundancy (Fig. 9A–C). By constructing dense parallel-plate matrices and repositioning inward or stripping away non-active support structures, these systems maximize the active heat-transfer area and minimize the parasitic thermal mass that restricts macroscopic cooling potential [42,43,111].
Furthermore, macroscopic fluid flow inevitably introduces hydraulic pressure drops and pumping power consumption, which directly degrades the overall COP. Therefore, a critical engineering trade-off lies in balancing the channel porosity to minimize pumping resistance while maintaining sufficient convective heat transfer. This balancing act also highlights the fluid-selection dilemma in macroscopic devices: while dielectric oils prevent electrical breakdown, their high viscosity and limited thermal capacity induce frictional heating and restrict heat transfer efficiency. Future transitions toward water-based fluids—contingent upon advances in perfect macroscopic electrode insulation—are poised to reduce pressure drops and improve the feasibility of macroscopic cooling power.
Finally, macroscopic fluid channels are susceptible to parasitic axial heat conduction and thermal mixing, which can easily destroy the established temperature gradients. To reduce this heat leakage, advanced macro-topologies introduce unidirectional double-loop architectures equipped with check valves, or physically segment the active working bodies along the fluid flow direction. These sophisticated flow-engineering mechanisms help maintain the spatial temperature gradient and prevent the failure of the thermodynamic cycle caused by the mixing of hot and cold fluids, thereby enabling the stable accumulation of massive temperature spans across the regenerator (Fig. 9A–C) [42,99,111]. Notably, although existing large-scale prototypes establish substantial temperature spans, extending these spans below room temperature remains challenging. Conventional EC working bodies operate optimally above their room-temperature Curie temperatures; at sub-ambient conditions, their EC effects either degrade sharply or change sign, rendering continuous cooling through room temperature thermodynamically impossible. Addressing this fundamental limitation, Guo et al. [112] developed unannealed MLCCs based on (1-x) PbSc0.5Ta0.5O3-xPbMg0.5W0.5O3 (PST-PMW) solid solutions. By diluting PST with PMW, they deliberately disrupted long-range dipolar order to suppress the Curie temperature down to 230 K. Crucially, by exploiting valence and size mismatch, the strategy preserves crystallographic B-site order without the necessity of an energetically expensive anneal. This high degree of structural ordering maintains the substantial latent heat associated with the material's strongly first-order ferroelectric phase transition. Consequently, supercritically driving this first-order phase transition yields highly reversible EC temperature changes of approximately 3 K across and well below room temperature. For current cooling device research, this represents a pivotal materials breakthrough: it provides a macroscopic working body capable of true sub-ambient refrigeration, establishing a viable pathway to replace conventional PST in EC prototypes and actualize solid-state cooling through room temperature.
For application scenarios demanding zero tolerance for fluid leakage and complex plumbing (e.g., aerospace environments, large-scale data centers, or precision high-power electronics), macroscopic cooling relies on pure solid-state topologies. The fundamental challenge in this domain shifts from fluid dynamics to overcoming the "scaling effect"—specifically, how to physically enlarge the active volume without being overwhelmed by mechanical friction and macroscopic contact thermal resistance.
To achieve an approximately linear scaling in cooling power proportional to material mass, modern solid-state architectures abandon low-thermal-conductivity polymers in favor of large-thermal-mass MLCCs. By constructing modular linear-sliding or interdigitated arrays, these systems employ a dual-dimensional "scaling-up" strategy: parallel integration is utilized to multiply the absolute cooling capacity, while cascaded integration is enforced to step-wise amplify the overall system ΔTspan [91,94].
However, repeated solid–solid contact inevitably introduces severe interfacial thermal resistance and mechanical wear. To circumvent this, macroscopic solid-state systems mandate the implementation of physical "thermal switches." Engineering innovations such as anisotropic thermal conductive (ATC) plates—which exhibit high through-plane thermal conductivity but extreme in-plane thermal insulation—paired with ultra-thin fluid lubricants, or conductive flexible thermal pads, are heavily utilized. These macro-interfaces not only enable precise thermal routing (switching "on" and "off" seamlessly) during relative macroscopic displacement but also effectively absorb mechanical shocks and eliminate frictional heating during continuous reciprocating motions.
Beyond structural and thermal mechanics, evaluating macroscopic EC systems against traditional VCR necessitates a rigorous accounting of electrical energy. In macro-scale operations, the polarization of massive capacitive arrays consumes substantial electrical work. If this stored electrical energy is merely dissipated as Joule heat during the depolarization phase, the system's COP will remain uncompetitive regardless of the material's intrinsic caloric response.
Consequently, macroscopic charge transfer mechanisms—such as LC oscillation circuits or diode-assisted charge recovery—are no longer treated as mere "bonus features" for efficiency enhancement (Fig. 10A–D) (proposed by Defay et al. [113]). Instead, they have been elevated to an important engineering requirement for large-scale deployment (Fig. 11A–D) [42,44–46,88]. By recovering a substantial majority of the driving electrical work, these macro-scale energy recovery circuits reduce the active power requirement. This critical technological integration reduces part of the efficiency gap associated with electrical work input, establishing large-scale solid-state EC heat pumps as a viable, zero-direct-GWP alternative to traditional refrigerant-based cooling systems. While the capacitive nature of EC systems allows for energy recuperation, charge recovery improves the device COP only when dielectric loss, current leakage, driving circuit efficiency, and parasitic thermal mass are concurrently minimized.
2.2.2 Parasitic heat loads and positive-negative EC coupling
Although AER topologies are effective in amplifying ΔTspan, further increases in COP are still limited by thermodynamic losses. The sensible-heat load of the external heat-transfer fluid, the mechanical work required for pumping and switching, and the lattice heat capacity of the dielectric itself all contribute to a substantial parasitic thermal burden. Large MLCC or capacitive arrays in the fluid regenerators add a different loss set, including dielectric loss, leakage current, incomplete charge recovery, inactive thermal mass, and interfacial thermal resistance.
At the materials-response level, one promising route is positive-negative EC synergy. In contrast to the positive EC effect, in which conventional ferroelectrics release heat as field increases, some ferroelectric or antiferroelectric systems show anomalous negative EC responses under field because of field-induced antiferroelectric-ferroelectric (AFE-FE) phase transitions or field-driven disordering of polar nanoregions [114]. Representative cases include PbZrO3 (PZO)-based systems from room temperature to elevated temperature, BiFeO3-BaTiO3 (BF-BT) systems at high temperature [115], and some relaxors such as PWM near room temperature [116]. Rotary architectures or stacked configurations incorporating thermal insulation layers offer a structural means to deconstruct these positive and negative elements. Such coupling may reduce the sensible-heat burden handled by the external heat-transfer path, although its device-level benefit depends on phase matching, thermal insulation, and cycle synchronization.
If such negative-EC elements are arranged alternately with conventional positive-EC materials such as PST, BST, or Ba(Zr, Ti)O3 (BZT), the resulting array can form an internal thermal push-pull mechanism. Upon synchronized application or removal of the field, adjacent arrays with opposite thermal responses can exchange heat internally through solid-state thermal conduction across their interfaces. The phase-transition heat generated in the positive unit can then be partially compensated by the heat absorbed in the negative unit, creating an internal self-thermal-compensation loop. In principle, this mechanism lowers the sensible-heat burden that must be handled by the external fluid circuit, can increase system COP under suitable cycle and loss conditions, and shortens the response time required to establish a large macro-scale ΔTspan. It therefore represents a forward-looking strategy for breaking the efficiency bottleneck of current AER architectures.
2.2.3 Thermal-transfer impedance and device-dependent design boundaries
(1) Multiscale heat-flux boundaries
To circumvent the intrinsic thermal impedance bottleneck of polymer-based EC materials, structural inverse design dictates the multiscale reconstruction of continuous phonon pathways. At the mesoscopic and microscopic scales, enforcing geometric confinement via rigid inorganic templates (Fig. 12A) [117] or embedding continuous three-dimensional ferroelectric networks [118] systematically bypasses the insulating polymer bulk (Fig. 12B). These topological configurations construct directional heat dissipation pathways that actively suppress interfacial phonon scattering. Enforcing a cross-scale physical bridge, these continuous thermal networks dually function to dictate local polarization dynamics. By acting as predefined nucleation sites and geometrically unclamping molecular chains, these architectures lower the thermodynamic energy barrier for polar domain expansion. This structural coupling directly aligns enhanced heat transport with amplified EC entropy changes under applied electric fields.
However, thermodynamic audits of dynamic macroscopic architectures reveal that interfacial contact thermal resistance, rather than intrinsic material thermal conductivity, dominates the cyclic heat flux [46]. Translating this impedance suppression mechanism into system-level design, the deployment of compliant, conductive buffer layers at solid–solid contact interfaces effectively reduces insulating air gaps and microscopic surface roughness [91]. This macroscopic interfacial reconstruction ensures continuous heat flux and minimizes structural thermal conduction losses, thereby dictating the physical upper bounds of the device's operational frequency.
Despite these multiscale impedance optimizations, the physical integration of rigid heat-conducting inorganic templates or macroscopic compliant buffers inevitably injects parasitic sensible mass into the active core. This architectural compromise establishes a strict physical limit on the maximum achievable system COP and exposes the heterostructures to severe interfacial mechanical fatigue under continuous high-frequency operational stress.
(2) Perspective on thermal-transport thresholds in regenerative fluid devices
The steady-state and transient performance of macro-scale active regenerators should be governed jointly by internal heat-transfer impedance and materials thermodynamic parameters. The relevant boundary should be treated as device-dependent rather than as a universal material constant. It is expected to vary with regenerator geometry, active-material dimensions, cycle period, fluid velocity, volumetric heat capacity, and interfacial thermal resistance.
A device-dependent thermal-conductivity threshold is expected to exist in regenerative EC devices. This threshold should depend on regenerator geometry, cycle period, fluid velocity, active-material dimensions, volumetric heat capacity, and interfacial thermal resistance. Above this threshold, heat generated or absorbed during field cycling can be extracted within the available cycle period, and the regenerator may operate close to the intended thermal-accumulation mode. Below this boundary, incomplete heat extraction is expected to reduce the effective ΔTspan and prolong the approach to steady operation. This threshold should therefore be treated as a design parameter of the device architecture, not as an intrinsic material constant.
Component count, channel geometry, and flow velocity define a second boundary. Increasing the number of MLCC elements can increase thermal accumulation, but it also enlarges dead volume, flow resistance, and parasitic thermal mass. A practical inverse-design rule is therefore to match the thermal-transport threshold of the EC medium with the cycle period and the regenerator geometry, while keeping the hot- and cold-end components at low thermal mass. This perspective reframes thermal conductivity as a coupled system parameter rather than a standalone materials target.
2.3 Miniaturized and integrated cooling for electronics
With the continued rise of chip integration density in the post-Moore era, thermal-management systems for high-performance electronics must cope with extreme heat flux. Other integrated cooling technologies, particularly embedded microfluidics, have already demonstrated the ability to dissipate local heat flux exceeding 1.7 kW/cm2 within a chip through monolithic integration [119]. This value serves as an external benchmark for on-chip cooling rather than a demonstrated EC heat flux. Under such loads, which also exhibit strong transience, remote heat rejection faces a fundamental resistance bottleneck. EC refrigeration must therefore achieve very high SCP through inverse design if it is to satisfy on-chip targeted cooling without relying on complex fluid loops.
2.3.1 Scaling rules for dielectric cooling elements
As demonstrated by the fully integrated architecture of Han et al. [46], the electro-thermo-mechanical coupling model effectively bypasses macroscopic contact limitations, suggesting that integrated material/device design can be important for scalable thermal-impedance management. Physically shortening the heat-transfer path is the primary engineering route to achieving faster heat extraction and higher SCP in EC systems. The reduction of dielectric-layer thickness minimizes intrinsic phonon-transport resistance, allowing the material to reach thermal equilibrium within microsecond-scale relaxation times. While academic research on EC MLCCs often utilizes layers around 1.4–3 µm [120,121], industrial manufacturing (e.g., by Taiyo Yuden, Murata, and Samsung Electro-Mechanics) has pushed these boundaries further to meet high-density integration demands [122–125]. Leading manufacturers such as Taiyo Yuden have reportedly reduced the effective dielectric-layer thickness in commercial MLCCs to approximately 0.6 μm (estimated) [124,125]. Furthermore, advanced deconstruction research at the Technical University of Darmstadt suggests that dielectric-layer thicknesses as low as 0.3 μm (at least below 0.5 μm) can be achieved in an EIA 0402 case-size 22 μF MLCC components from Murata [122,123,126]. This extreme miniaturization not only increases the heat-transfer area per unit volume but also defines the power-density boundaries of miniature refrigeration elements in high-power-density electronics.
2.3.2 Ceramic stacking for high-density cooling architectures
The core advantage of ceramic EC devices in board-level integration lies in their native compatibility with existing electronic industrial manufacturing systems. Unlike discrete heat sinks, the design logic of these devices focuses on modular deployment to achieve a "scaling-up" effect where cooling power increases linearly with active mass. The work of Zhang et al. [105] demonstrated that mature commercial MLCC packaging can be directly transformed into compact refrigeration units. It uses layer miniaturization (single layer 12 µm) to achieve high-field operation directly using standard board-level voltages without step-up circuits, resolving the driving voltage bottleneck for bulk ceramics in integration (Fig. 6A, B). For local transient thermal loads in microelectronics, the modular topology proposed by Wang et al. [94] utilizes mechanical contact "thermal switches" to achieve a high-throughput heat flux of 135 mW/cm2, providing a dynamic cooling path for high-power components such as laser diodes (Fig. 6A, E). Furthermore, the interdigitated topology by He et al.[91] stabilizes contact resistance via high-conductivity thermal pads [5 W/(m·K)] (Fig. 13E), verifying that lead-free ceramics can maintain high SCPin miniaturized packaging through multi-stage parallelization.
2.3.3 Polymer-based topologies for on-chip heat management
Despite the high thermal conductivity of ceramics, thin-film devices based on PVDF copolymers serve as the mainstay of miniature scenarios due to their ultra-low profile and functional integration potential. To address the local heat accumulation bottleneck caused by the extremely low thermal conductivity of polymer EC materials [approximately 0.2 W/(m·K)], integrated design focuses on constructing efficient "heat routing". The AAO hybrid thin-film topology that utilizes vertically conducting paths [6 W/(m·K)] established by AAO channels was developed by Zhang et al. [117] to shorten heat exchange time by two orders of magnitude, achieving a record cooling power density of 225 W/cm3 at 100 Hz (Fig. 11A). Additionally, the ECOR architecture proposed by Gu et al. [104] focuses on resolving interfacial challenges during integration. By combining lubricated interfaces with micro-slitted structures, this topology achieves directional heat transport at the chip scale (Fig. 5A, B). This precise regulation of microscopic interfacial thermal resistance is a prerequisite for solid-state cooling devices to enter the interior of millimeter-scale chip packaging.
In the progression toward miniaturization and chip-level integration, traditional external motors or fluid pumps often act as major parasitic sources that occupy system volume, introduce inactive thermal mass, and hinder high-density packaging. Integrated devices centered on PVDF copolymers and terpolymers achieve "system subtraction" through innovations in physical mechanisms and topological structures, compressing system volume to the extreme via actuator-free or one-dimensional integrated topologies. Early electrostatic actuation mechanisms eliminated noisy and bulky mechanical components [45,89]. This mechanism utilizes unipolar electrostatic forces to drive S-shaped films with low bending stiffness. The reversible electrostatic force serves not only as the driving source for displacement within micro-spaces but also directly acts as an interfacial clamping force, ensuring tight thermal contact between the film and the heat source or sink. In more complex cascaded structures, the system further achieves efficient transfer of heat and charge within an extremely compact physical stacking space through the antiphase operation of conjugated films and internal charge recovery mechanisms [44].
When addressing complex heat sources in micro-packaging, the thermal distribution of integrated circuits (such as high-computing-power CPUs) often exhibits extreme non-uniformity. Traditional global cooling methods are inefficient in miniaturized scenarios and must rely on topologies with high spatial resolution to achieve "precise targeting". Consequently, matrix topologies and local hotspot management constitute another core innovative mechanism for integrated devices (Fig. 13C) [90]. By pre-depositing pixelated carbon nanotube electrode pairs onto the polymer film, this topology constructs an active pixel-matrix. It enables targeted and differential thermal management: the system can spatially decouple cooling regions based on the distribution of local hotspots on the chip surface, independently waking up cooling pixels at corresponding locations or activating them in specific array combinations. This spatial decoupling design thoroughly breaks the morphological limits of traditional bulk cooling devices, endowing solid-state cooling modules with the capability for dynamic, high-precision addressable cooling of complex transient heat sources in micro-packaging.
In extremely compact multi-layer cascaded packaging, electrical insulation, lightweighting, and structural stability are the fundamental engineering boundaries that polymer materials must address. To avoid adding rigid layers that cause mechanical constraints on soft polymer films, ultra-lightweight electrodes (such as sprayed CNT percolation networks) are widely applied, ensuring unhindered, high-efficiency actuation. Meanwhile, in stacked structures subjected to frequent micro-deformations and high-frequency electric field switching, active heating elements and electrodes are susceptible to physical interference. By introducing ultra-thin insulating barriers [such as micrometer-scale thick polymethylpentene (TPX) insulating films] between key functional layers, the system constructs clear physical interface boundaries, effectively preventing short-circuit faults that are prone to occur in miniaturized devices [47]. To support such high-density packaging, the interior of the polymer is typically doped with nanoparticles to construct a percolation network, which not only enhances local heat routing but also increases the Young's modulus of the film to accommodate the dual rigorous demands of micro-scale stacking.
2.3.4 Multiphysics coupling for driver-free integrated devices
Furthermore, functional integration and self-regenerative architectures establish an integrated route for high-density, actuator-free packaging (Fig. 13B) [88]. This topology achieves a coupling of thermodynamic cooling and mechanical actuation functions within a single flexible film: it utilizes the EC effect and electrostriction synchronously triggered by internal polarization conformation transitions of the polymer, combined with an asymmetric backing layer structure to convert disordered in-plane expansion into regular out-of-plane deformation, allowing the film itself to act directly as a micro-actuator. This self-regenerative mechanism executes precise spatiotemporal heat relays through direct physical contact between adjacent film layers, removing external displacement actuators and reducing inactive thermal mass, thereby reducing parasitic components along the heat-transfer path to a minimum. Its logic of triggering thermal cycles through intrinsic material deformation eliminates dependence on external displacement controllers in traditional cascade systems. This self-regenerative architecture further marks the transition of integrated devices from unidirectional heat dissipation to self-sustaining cycles. In parallel, Bai et al. [127] developed a cascaded EC cooling tube for integrated microfluidics. By implementing spatial segmentation within microscale channels, this topology compensates for the limited intrinsic span of individual polymer units and offers a high-SCP response route for integrated components. A limiting target of integrated design is the elimination of all inactive volumes. The self-oscillating topology [46] defines a new efficiency benchmark for chip-level integration (Fig. 13F). Its main feature is the integration of driving and cooling—utilizing the intrinsic electrostriction of the polymer film to generate displacement, eliminating external actuators such as motors and pumps. For high-density packaging (such as 3D stacking chips), this driver-free topology represents an extremely low volume penalty (zero-footprint), and its self-oscillating logic exhibits millisecond-scale responses that can synchronize with the power fluctuations of computing chips. In the context of integration, it serves not only as a cooler but as a "thermal load smoothing layer", achieving an additional 17.5 K temperature suppression on continuously operating high-performance chips. Under the ideal limit of zero initial temperature span, the system COP reached 58. Even at a maintained system ΔTspan of 4 K, the COP remained 24. Although establishing a finite temperature span inevitably increases heat-pumping work, these combined metrics confirm the advantage of electro-mechanical co-design for reducing parasitic power consumption and increasing power density. The resulting performance serves as a new energy-efficiency benchmark for integrated cooling. These methods of simplifying system structures through multiphysics coupling represent the inevitable direction for the miniaturization and efficiency of future board-level thermal management systems.
2.4 Wearable EC cooling and personal thermal management
2.4.1 Low-field, flexible, and skin-compatible device requirements
Wearable EC coolers aim to extend the human thermal comfort zone by providing precise heat pumping within the microenvironment. Unlike industrial or desktop-scale integrated cooling, wearable applications impose complex physical constraints on the EC topology: first, the human safety boundary requires a low driving voltage, ensuring effective entropy change within safe thresholds; second, extreme mechanical compliance is necessary for the device to maintain structural and thermodynamic integrity under repeated stretching and bending caused by body movement; finally, the system must be lightweight with low thermal inertia, necessitating the removal of redundant mechanical actuators to maximize SCP. Beyond simple flexibility, wearable EC systems must satisfy strict operational boundary conditions, including skin-safe temperature variation, soft mechanical contact, structural breathability, sweat and humidity resistance, and long-cycle user comfort.
2.4.2 Materials design for wearable EC systems
To meet the specific demands of wearable scenarios, material-level research has shifted from merely increasing entropy change toward a comprehensive optimization of driving energy barriers, thermal conductivity, and ambient temperature matching. Since human skin is sensitive to voltage, developing low-field high-response materials is a primary technical priority. Zhang et al. [128] reshaped phase-transition thermodynamics by constructing vertically aligned ferroelectric nanowire arrays (Ba0.67Sr0.33TiO3 NW array) on flexible substrates (Fig. 14A). This topology utilizes the size effect on dipole orientation to increase surface tension, suppressing the switching of non-working direction domains and allowing dipoles to align effectively at an ultra-low electric field of 120–150 kV/cm. The study demonstrated that the device could generate a ΔT of 3 K and a ΔS of 4.9 J/(kg·K) under a human-safe driving voltage of only 36 V. Furthermore, by adjusting the Ba/Sr ratio, the Curie temperature (Tc) was precisely tuned to 20 °C, perfectly covering the cooling requirements of human skin.
To further enhance the conformability of devices during dynamic body movement and eliminate interfacial thermal resistance, EC topologies evolved from planar films to one-dimensional fiber-based forms. Wang et al. [129] developed a self-driven EC fiber based on P(VDF-TrFE-CFE), featuring a co-axial layers structure with diameters refined to 70–160 µm (Fig. 14B). The core of this topology lies in utilizing the electrostriction effect to induce asymmetric bending of the fiber under an electric field, thereby achieving a self-actuation mechanism. This physical deformation allows the fiber to actively shuttle between the heat source and heat sink, acting as a micro-actuator to automatically switch thermal contact interfaces. Experiments showed a measured ΔT of 0.7 K at a driving field of 1000 kV/cm. This approach removes the volumetric burden of external actuators and maintains performance without degradation after 2000 cycles at a 2.5 mm bending radius.
2.4.3 Wearable device architectures and heat-exchange modes
In addition to nano-size effects, regulating molecular chain dynamics to enhance the EC strength is an effective path to lowering driving barriers. Bo et al. [89] reported a modification of P(VDF-TrFE-CFE) by introducing 0.5 wt.% dioctyl phthalate (DOP) as an anti-plasticizer (Fig. 13D). The introduction of DOP increased spontaneous polarization and material crystallinity, resulting in an additional 1 K temperature change at 1300 kV/cm compared to the pure polymer. The study further employed a double-unit system topology, achieving a COP as high as 8.3 at an electric field of 417 kV/cm. This method of regulating molecular chain rotation barriers to maintain high performance under low fields provides a critical reference for resolving the physical conflict between "safety and efficiency" in the wearable field.
Regarding the evolution of device topologies, wearable EC technology has progressed from basic actuation verification to complex system integration. As mentioned previously in Section 2.1.2(3), the electrostatic actuation topologies developed by Pei’s research group [44,45] established the intrinsic advantages of solid-state cooling in reducing system volume and power consumption by eliminating bulky electromagnetic motors and pumps (Fig. 15A). Subsequently, the active pixel-matrix topology proposed by Bai et al. [90] provided a physical paradigm for spatial differential thermal management of human skin hotspots by pre-depositing pixelated CNT electrodes on flexible polymer films. These works mark the evolution of wearable topologies from single-function energy conversion units to thermal management systems with logical control capabilities.
For specialized wearable thermal management needs, such as the dynamic concealment of human thermal signatures, EC topologies have further integrated ultrafast thermal response characteristics. Bo et al. [47] utilized the very high entropy change rate of EC polymers (theoretical limit of 10−8 s/K) to develop a flexible ultrafast thermal camouflage skin (Fig. 15B). To address wearable safety challenges, a 6 µm TPX insulation layer was introduced between the Ag NWs heater and the CNT electrode. The key to its topological design lies in the modulation of square-wave temperature: utilizing the EC effect for instantaneous temperature changes combined with a flexible heater for balance, achieving a macroscopic temperature change rate of 3.8 × 10−3 s/K. This flexible camouflage device maintains stable performance at bending angles from 0° to 120°, enabling instantaneous changes in its own radiant temperature to match complex backgrounds, demonstrating the potential of EC topologies in the field of intelligent electronic skins (e-skin).
At a high level of wearable system integration, achieving a logical closed loop between energy supply and thermal management functions is essential for practical application. Wang et al. [48] developed an all-day self-sustaining wearable thermoregulatory clothing (OETC) (Fig. 15C). The topological innovation of this system involves the parallel integration of flexible OPV modules with multiple bidirectional EC devices, creating a compliant system only 180 µm thick. Its operation relies on efficient energy management: under 100 mW/cm2 sunlight, the OPV modules generate 298.58 mW of electrical energy—sufficient to drive the EC array, which has a power consumption of only 1.91 mW/cm2, while storing redundant energy in an integrated storage system (ESS). Leveraging the high-performance response of P(VDF-TrFE-CFE), this clothing achieved an ultrafast cooling rate of 14.0 K/min and a heating rate of 15.6 K/min within the first 5 s. Experimental verification showed that only 12 hours of sunlight energy input is required to support 24 hours of self-sustaining operation, extending the human thermal comfort zone from 22–28 °C to 12.5–37.6 °C (a span of 19.1 K). This deep coupling of flexible photovoltaic collection and EC heat pumping resolves the energy anxiety of wearable devices in mobile scenarios and establishes a system-level paradigm for future smart clothing.
The evolution of EC topologies in wearable scenarios clearly illustrates a developmental trajectory from single-material modification to complex system functional integration. From the early establishment of electrostatic actuation paradigms to the work of Zhang et al. [128] in reducing the driving voltage to the human safety threshold (36 V) through nanowire size effects, and the utilization of fiber-level self-actuation by Wang et al. [129] to eliminate interfacial thermal contact resistance, each evolutionary step has addressed the fundamental physical conflicts of wearable devices. Recent advances by Ma’s research group [47,90] in pixel-matrix control and energy-sustaining clothing have further pushed EC technology to expanded device autonomy.
More recently, as demonstrated by the latest research from Wu et al. [88], the self-regenerative topology—which strips away inactive supporting structures—allows the EC system to maintain a high power density of 1.52 W/g while achieving a significant temperature lift of 14.2 K. This series of studies indicates that a low driving energy barrier (safety voltage), high thermal conductivity interfaces, mechanical flexibility, and system-level energy self-sufficiency are the core parameters for constructing future personal thermal management systems. As topological structures are further refined and material efficiency is continuously optimized, flexible cooling technology based on the EC effect is expected to see large-scale application in smart wearables, medical thermotherapy, and specialized protective equipment.
3 Materials Inverse Design for System-level EC Performance
EC materials design must be constrained by device-level thermodynamic targets. Macro-scale regenerators, on-chip cooling units, and self-actuated wearable systems impose different requirements on polarization entropy, switching field, heat extraction, mechanical compliance, and cycling losses. Materials metrics such as ΔS and ΔT therefore become useful only when they are evaluated together with driving-field burden, thermal-transfer impedance, and energy-recovery conditions.
In device terms, barrier reduction is not only a materials-level objective. High-polarization-active ions, defect regulation, and phase-configurational entropy can reduce the field required for reversible polarization switching, thereby lowering electrical work input and providing a direct route to higher effective COP when heat extraction and charge recovery are maintained. The following sections therefore treat entropy enlargement, barrier smoothing, defect-mediated reversibility, and thermal-transport control as coupled design variables rather than independent materials descriptors.
3.1 High-entropy engineering for reversible entropy release
The EC entropy change ΔSECE is bounded by the intrinsic configurational-entropy ceiling of the material itself. In recent years, high-entropy design has emerged as a key route toward breaking the correlation constraints of conventional ferroelectrics and achieving a step change in entropy by introducing crystallographic asymmetry and disorder in local polar states. In this review, configurational entropy is discussed in two distinct dimensions: phase-configurational entropy (PCE) [130,131] and chemical configurational entropy (CCE) [132–139]. Correspondingly, the associated high-entropy design strategies in EC materials are referred to as phase-configurational high entropy (PCHE) and chemical configurational high entropy (CCHE), respectively. PCE, and thus PCHE, are associated primarily with phase degeneracy and orientational disorder at the mesoscale, whereas CCE, and thus CCHE, originate from atomic-scale chemical disorder induced by the random occupation of crystallographically equivalent lattice sites. At the device level, barrier reduction reduces the field required for reversible polarization reconfiguration and thereby lowers electrical work input when charge recovery is maintained. A design that enlarges configurational entropy but requires a high irreversible work input will not necessarily increase COP; the useful route is to enlarge reversible entropy release while lowering the field required for polarization reconfiguration.
3.1.1 PCHE and polarization degeneracy
In the search for routes beyond the intrinsic polarization-correlation limits of conventional ferroelectrics, Du et al. [40] proposed a strategy based on polar disorder and introduced the concept of polar entropy as a statistical metric for evaluating and predicting the theoretical upper bound of giant EC effect in perovskite oxides. Their analysis showed that multielement atomic distortion in a conventional ferroelectric lattice can generate a disordered yet field-responsive local polar state. This state relaxes the restriction imposed by strongly ordered polarization correlation on the available configurational fluctuation space and therefore provides a much larger phase space for EC response on the microscopic scale.
According to the statistical-mechanics model established in the Supporting Information of their work, polar entropy can be written in Boltzmann form as
where Ωi denotes the statistical distribution density of polar-disordered states on the local scale. This framework provides a direct theoretical tool for quantifying the statistical configurational gain of local dipoles and, more importantly, reveals the physical evolution from long-range polar order to polar disorder. By maximizing the local spatial degeneracy of the polarization vector, the model establishes a priori design criteria for evaluating the potential of giant EC response and therefore sets a meaningful physical boundary for the development of high-entropy EC dielectrics.
The macroscopic theoretical framework of PCE was established primarily by Pirc et al. [140], which was subsequently developed by Liu et al. (2012) [130]; both studies originated from the group of Q. M. Zhang. Through detailed thermodynamic analysis, they showed that exceptionally large electrically driven response and entropy gain can be induced near invariant critical points (ICPs) in ferroelectric phase diagrams by maximizing the number of coexisting phases through compositional tuning or stress-field design. This theory showed strong predictive power in constructing the phase diagram of PMN-PT relaxor ferroelectrics and first identified a six-phase coexistence point composed of rhombohedral (R), orthorhombic (O), tetragonal (T), cubic (C), and two monoclinic phases (MB and MC) (Fig. 16A, B). Such an extreme state of physical degeneracy greatly enlarges the configurational fluctuation space and state density in a statistical sense.
Building on this framework, Chen et al. [131] from the group of Q. M. Zhang later refined the molar quantification model of dipolar entropy (i.e., PCE in this review) as
where R is the molar gas constant and Ω is the number of equivalent polarization axes, that is, the number of degenerate states, in a given phase region. Near an invariant critical point, multiphase coexistence compresses the free-energy difference among competing structures to a very small value and thereby produces an isotropic and flattened multidimensional free-energy landscape. This barrier-smoothing mechanism drastically lowers the driving barrier required for polarization switching. Because anisotropy is weakened so strongly, the polarization vector can rotate rapidly among multiple equivalent crystallographic directions under a very small external field with almost no dissipative penalty. This mechanism explains the origin of giant EC response and provides a core methodological basis for triggering large entropy change under low field and thus for high-efficiency solid-state refrigeration.
Lead-free ferroelectric systems based on BaTiO3 (BT) and (K, Na)NbO3 (KNN) provide the most representative experimental platforms for validating and implementing the theory of PCHE [141]. Although many early studies focused on compositional modification to enhance macroscopic polarization response, their underlying performance gains can be traced back to precise control of the invariant critical point and the number of associated degenerate phase states. In BT-based ceramics, inverse design has centered on chemical modification of the A and B sublattices and the associated reconstruction of phase-diagram topology [142–150]. B-site substitution with ions of strongly mismatched radius but stable chemistry, as in classic Ba(Bx, Ti1-x)O3 (B = Zr, Hf, Sn) systems [151–155], causes a clear convergence of phase boundaries. In BaZrxTi1-xO3, for example (Fig. 16C-F), when the Zr4+ content is tuned to a critical composition, typically x = 0.15–0.20, the transition lines of the R, O, and T phases converge rapidly toward lower temperature and finally meet the cubic phase near room temperature, creating a physically important R-O-T-C four-phase coexistence point [151,156]. According to the PCE model, formation of this ICP raises the number of equivalent polarization axes Ω from 6 or 8 in a single-phase region to 26, thereby providing a major statistical gain in PCE.
The same design logic has been deepened further through co-doping on both the A and B sites. In systems such as (Ba, Sr, Ca)(Ti, Zr, Hf, Sn)O3 or doped system, the Curie-point suppression induced by Sr2+ and Ca2+ and B-site ions generate compositions near a morphotropic phase boundary, where giant EC entropy change has been observed and where the flattened free-energy surface associated with multiphase coexistence has been revealed directly (Fig. 17A–E) [40,143–145,150,157–160].
It is worth noting that while the intensifying focus on environmental sustainability in the new century has directed increasing scholarly attention toward Pb-free materials, early pursuits of maximized EC performance relied heavily on Pb-based systems. In these classical materials, such as PST, PMN-PT and (Pb, La)(Zr, Ti)O3 (PLZT), the deliberate construction of multiphase regions—often by tuning composition toward a morphotropic phase boundary or introducing complex ionic doping—serves as a powerful mechanism to increase phase-configurational entropy. For instance, single crystals and ceramics of PMN-PT and its solid solutions (e.g., PMN-PT/BZT) have demonstrated that varying the coexistence of R, O, T, and cubic (C) phases can produce distinct, and sometimes massive, positive or negative EC responses, a phenomenon further corroborated by studies across different device architectures like MLCCs and thick films [112,161–168]. Similarly, in PLZT systems, strategic doping (e.g., with Gd3+, Sn4+, or tuning La/Zr/Ti ratios) orchestrates transitions among monoclinic, O, T, and R phases. Depending on crystallographic orientation and the applied electric field, these multiphase configurations can induce a high EC strength (e.g., 1.52 K·mm/kV) and ultrahigh refrigeration efficiency, alongside substantial temperature changes (ΔT) around the depolarization temperature [169–175]. Furthermore, recent advancements in related fields, such as the discovery of a quadruple phase point in Pb(Zr, Ti)O3 (PZT) for enhanced piezoelectricity [176], suggest that applying these rich, multi-pathway phase transition designs to the EC domain holds significant probability for achieving multi-fold enhancements in cooling response. This historical perspective on Pb-based systems underscores the fundamental thermodynamic utility of manipulating phase coexistence—a strategy that remains structurally relevant even as the field transitions toward environmentally benign alternatives.
Environmentally benign KNN-based architectures dictate a similarly powerful PCE platform, governed by their intrinsically rich phase evolution. The targeted substitution of modifier elements into the baseline lattice triggers the convergence of crystallographic phase boundaries, systematically inducing R-O-T or even broader multiphase coexistence regions [177–179]. This engineered structural convergence drastically enriches the local spatial degeneracy of the polarization vector. By flattening the thermodynamic free-energy surface, the multiphase topology underlies substantial entropy fluctuations under low driving fields. In bulk ceramics, the diffuse orthorhombic-tetragonal (O-T) phase transition governs a broad EC response peak, which remains precisely tunable through compositional modification [180]. Direct calorimetric assessments in these systems not only validate the magnitude of the response but also accurately depict the evolution of the EC effect relative to the phase diagram boundaries, demonstrating high consistency with indirect thermodynamic methodologies [180]. This robust thermodynamic foundation is further corroborated across a diverse range of KNN-based architectures, including confined thin-film devices and optimized bulk systems [181–187].
Viewed from the perspective of bringing theory back to experiment, both the four-phase convergence in BZT and the multiple phase-boundary evolution in KNN show the same underlying microscopic dynamics. Relative to the six-phase coexistence point predicted earlier for PMN-PT [130], the exact number of coexisting phases in existing experiments may differ slightly, but the physical essence is unchanged: maximization of PCHE produces strong smoothing of the polarization-switching barrier. Once free-energy differences among competing phases are minimized near the ICP, the polarization vector can rotate quickly across phase boundaries among multiple equivalent axes under a very weak external field. This result confirms experimentally that PCHE is a genuine engine of materials-performance enhancement and provides a firm physical basis for achieving large EC output and high system-level efficiency under low driving field.
3.1.2 CCHE and local polar disorder
(1) CCE versus PCE
Unlike PCE, which reflects phase degeneracy and the orientational disorder of polarization vectors at the mesoscale, CCE is rooted in atomic-scale chemical disorder [132–138]. In perovskite EC systems, this contribution is manifested specifically as ionic-site configurational entropy. Highly diverse elemental substitution on crystallographically equivalent sublattice sites reconstructs the local potential-energy landscape and provides a configurational fluctuation space broader than that generated by multiphase coexistence alone. From an inverse-design perspective, CCE governs phase stability and relaxor behavior, while also acting as an entropy reservoir for enhanced EC effect. Its contribution to the total entropy change can therefore couple with mesoscopic polarization processes and produce cross-scale amplification.
(2) Sublattice contributions to CCHE
According to the statistical definition of high-entropy oxides [188,189], CCE arises from random occupation of crystallographically equivalent sites by multiple components (Fig. 18A). For EC dielectrics with the perovskite ABO3 structure, a rigorous CCE model must go beyond a single sublattice and instead treat the A- and B-site sublattices jointly. For multicomponent systems on a molar basis, the CCE is
where xi and yj are the molar fractions of species on the A and B sites, respectively. Under the standard thermodynamic criterion, when Sconf_chem exceeds the high-entropy threshold of 1.5R, the material is expected to cross from the conventional doped regime into a high-entropy-stabilized phase. In this strong disorder state, entropy-driven stabilization suppresses secondary-phase precipitation while preserving the overall high-symmetry perovskite framework. At the same time, the strong mismatch in ionic radius and valence among the constituent species generates severe local lattice distortion. Materials design therefore shifts from compositional fine tuning toward entropy-driven structural reconstruction.
(3) Disorder-driven polarization response
CCHE systems exhibit what may be called an entropy-reservoir effect. Strong chemical disorder creates a network of randomly distributed barriers on the microscopic scale, disrupts long-range ferroelectric coupling, and pushes the system toward a relaxor-like state with high dynamical activity. This disordered background stores enormous configurational potential. Under zero field, dipoles occupy an extremely disordered statistical distribution. Once a strong external field is applied, these ions and dipolar clusters undergo a large cooperative reorientation from a disordered state into an ordered polarized state.
Because the configurational space of a high-entropy system is much broader than that of a conventional low-entropy system, the polarization response and the associated phase-transition entropy change generated by the transition from strong disorder to high order can exceed the theoretical ceiling of conventional ferroelectric or relaxor materials. By weakening long-range coupling and providing a large configurational fluctuation margin for field-induced reversible polarization, CCHE design resolves the contradiction between high cooling output and the intrinsic limitation of polarization response. It therefore acts as a central kinetic engine for activating giant EC response.
(4) System-specific design rules
In the inverse-design map for CCHE EC materials, BT-based systems constitute the primary platform for exploring excess polarization response driven by strong chemical disorder because the perovskite framework tolerates compositional complexity well and provides broad flexibility for design.
BT-based and KNN-based (relaxor) ferroelectric ceramics are well suited for exploring CCHE because their lattices tolerate equiatomic multicomponent substitution on both the A and B sublattices while maintaining relatively homogeneous distribution. Detailed studies of (Ba, A)(Ti, B)O3 and (K, Na, A)(Nb, B)O3 systems, where A can be Sr, Ca, La, Bi, Na, or Sm and B can be Zr, Hf, Sn, Ta, Sb, Zn and related species, reveal how multidimensional atomic distortion under extreme chemical disorder reconstructs microscopic polarization behavior (Fig. 18B, C). From a theoretical standpoint, the local-field fluctuations induced by high-entropy design can break the long-range polarization-correlation constraints of conventional ferroelectrics at an intrinsic level. In an idealized limit, specifically within the KNN system, maximizing chemical disorder on sublattice sites is predicted to raise the EC ceiling (Fig. 18D). The synergy between PCHE and CCHE is projected to trigger a high model-predicted upper limit of isothermal entropy change, reaching 73 J/(kg·K) at a polarization level of 100 µC/cm2 [139]. This value, representing the coupled contribution of phase-configurational degeneracy and atomic-scale chemical disorder, is far above that of present commercial EC dielectrics and suggests, within the thermodynamic model, that the CCHE route, when integrated with phase-configurational control, is effective for achieving excess entropy output.
Converting this theoretical ceiling into experimental reality, however, requires overcoming the kinetic bottleneck introduced by high entropy itself. Because high CCE systems are intrinsically strongly relaxor-like, their dipoles are often bound and hidden within the random energy landscape generated by strong chemical disorder. Under conventional driving fields, these locked configurational degrees of freedom cannot be activated efficiently because reversible polarization remains severely limited. Existing experiments reveal this competition clearly: in BT-based high-entropy multiphase-coexisting systems, even a field as high as 140 kV/cm yields an EC temperature change of only 2.58 K [133].
Closer examination shows that the polarization response of such high-entropy materials remains far from saturation at this field, which suggests that the observed performance accesses only a small fraction of the stored CCHE potential. Because high-entropy design raises the switching field required for polarization reversal, dielectric breakdown often occurs before full polarization saturation can be reached. To release the large entropy reservoir hidden in this complex chemical network, MLCC architectures with high density and high breakdown strength therefore become a natural next step. Through the integration advantages of MLCC technology, raising the sustainable excitation field toward 300 kV/cm or higher is a possible route for further activating the hidden EC response of high-entropy systems and for converting configurational entropy efficiently into macroscopic cooling capacity. This progression from compositional disorder to field-driven polarization order defines the core logic by which BT-based high-entropy EC materials can move from theoretical assessment to experimental advance.
Within the design landscape of CCHE EC materials, KNN-based systems form a second major pillar. Their intrinsically rich phase-boundary evolution and strong lattice tolerance define a physical route different from, yet complementary to, that of BT-based systems.
Because of the particular ionic radii on the A site (K+, Na+) and B site (Nb5+), KNN provides broad compositional freedom for equiatomic multicomponent substitution and thus acts as a natural testbed for CCHE design. In an inverse-design workflow, the absolute value of Sconf_chem can be used directly as a central measure of disorder, which clarifies the mapping between chemical disorder and macroscopic electrical response. Although dedicated studies of genuinely high-entropy KNN-based EC behavior are still at an early stage, advances in adjacent areas already provide a strong logical basis for this direction.
For instance, Chen et al. [191] obtained very high polarization response in a high-entropy KNN system. Although that study focused on piezoelectric and dielectric properties rather than EC response, the underlying design logic, namely enhancement of polarization activity through CCHE, is instructive for EC materials. It confirms that within an extremely disordered CCHE environment, cooperative multielement substitution can break long-range ferroelectric correlation and induce polar states with high dynamical activity, thereby providing the physical basis for excess field-driven polarization reversal.
Experimental studies of EC behavior in KNN were initially concentrated in the low-and medium-entropy range by Yang et al. [192,193]. Even before the high-entropy threshold is crossed, KNN-based ceramics already show considerable EC temperature and entropy changes in multilayer architectures under strong fields. These low- and medium-entropy performance baselines provide a firm experimental foundation for extension into the high-entropy regime.
Theory and preliminary experiments both suggest that if these mature low- and medium-entropy compositions are used as parent systems and are then pushed across 1.5R threshold by introducing additional polarization-active constituents, the upper bound of EC performance should expand. Under this evolution logic, the large entropy-reservoir effect associated with CCHE would no longer remain a purely theoretical quantity. Instead, when coupled with the intrinsic R-O-T phase-boundary evolution of KNN, it compensates for the enhanced relaxor character typical of PCHE disorder while maintaining high polarization magnitude. This design strategy draws significant methodological inspiration from the piezoelectric domain, where chemical disorder-based medium-entropy design by Lin et al. [194,195] has been utilized to stabilize R-O-T-C multiphase coexistence and enhance ferroelectric and piezoelectric activity. The translation of this entropy-driven structural heterogeneity—specifically the induction of phase-boundary convergence via chemical configurational disorder—into the EC field is expected to enable the simultaneous achievement of high polarization response and giant configurational entropy change across an expanded compositional landscape. However, CCHE should be treated as a stability-constrained design variable rather than as a monotonic descriptor of EC enhancement. When the number or mismatch of A- and B-site species exceeds the solid-solution window, phase segregation, secondary phases, and non-equilibrium processing can reduce reproducibility and introduce leakage pathways; these constraints are discussed again in Section 5.2.
3.1.3 Negative EC response in disordered polar states
Unlike the positive EC effect, in which dipoles become more ordered with increasing external field, the negative EC effect (NECE) originates from field-driven evolution toward disorder or toward noncollinear dipolar states. Within the framework of high-configurational-entropy engineering, the central issue is how extreme compositional disorder can tune the energetic stability of antiferroelectric phases precisely enough to optimize negative EC response and shift it into more useful temperature windows.
(1) PZO-based negative EC response
PZO is a prototypical antiferroelectric system and exhibits a pronounced negative EC effect near its intrinsic AFE-paraelectric or AFE-FE transition, around 220 °C [114] (Fig. 19A). The drawback is that the intrinsic transition is usually accompanied by an abrupt energetic discontinuity, which yields a narrow response window and a working temperature well above room temperature. High-entropy modification offers a route to suppress this limitation. By introducing multicomponent substitution on the sublattices of PZO, the sharp transition barrier can be smoothed through the increase in configurational entropy. The more complex local chemical environment flattens the steep free-energy curve and reduces the temperature sensitivity of the transition. The resulting disordered high-entropy state helps preserve a relatively large negative entropy change while creating the physical basis for stabilizing the antiferroelectric phase over a wider temperature range.
(2) A- and B-site control in PST and PMW
In complex oxides, high-entropy design provides multiple handles for NECE regulation. In PST, recent results suggest that Ca2+ introduction on the A site can tune the stability of polar states and the dipole-reversal pathway, thereby inducing a tunable negative EC response [196]. In PMW, the characteristic chessboard ordering of heterovalent B-site ions provides a valuable platform for examining the connection between ionic order and EC behavior (Fig. 19B) [116,197]. If multicomponent design is imposed on the B site so as to increase CCE and disrupt long-range order through strong chemical disorder, the probability of noncollinear dipole reversal should increase. This combined route, based on A-site ion introduction and enhanced CCE on the B site, provides a sound basis for identifying new high-entropy negative-EC candidate systems.
(3) High-entropy response in BFO-BT
In 0.7BF-0.3BT, increasing CCE is expected to be a key lever for a step change in NECE performance. The available evidence suggests that simultaneous multicomponent high-entropy design on both the A and B sites can generate a large order-disorder phase response than that found in conventional doped systems. The gain in CCE should not only increase the absolute magnitude of the negative entropy change, but also smooth the lattice-energy barriers globally and thereby shift the phase-transition temperature window. On this basis, high-entropy design is expected to move the optimum NECE response of BFO-BT from elevated temperature toward room temperature, potentially overcoming the long-standing difficulty of obtaining practically useful room-temperature NECE.
(4) Configurational fluctuations and negative entropy change
From the standpoint of statistical physics, the central contribution of high-entropy disorder to NECE is the enlargement of the configurational fluctuation space available for field-induced reversible polarization. This space contains many metastable pathways and allows dipoles to reconfigure and rotate under the field in a much more flexible nonlinear fashion. Once the local transition from order to disorder is strengthened by a high-entropy background, the material shows a larger macroscopic negative entropy change. In this way, the cooling potential and energy-conversion efficiency associated with NECE are improved at a fundamental level.
3.2 Barrier regulation for low-field EC operation
3.2.1 Polarization-switching barriers and driving-field requirements
In perovskite ferroelectrics, the dynamics of polarization-vector reversal are governed by the anisotropy of the free-energy surface and the associated phase-transition barrier. Phenomenological theory shows that the critical switching field is set directly by the highest free-energy barrier along the switching pathway. In conventional ferroelectrics, this barrier is mainly determined by intrinsic lattice anisotropy. In systems with CCHE, however, strong chemical disorder introduces strong local random-field fluctuations.
On the macroscopic scale, this microscopic randomness appears as a rugged energy landscape. Random site occupation by different elements makes the local free-energy curves inhomogeneous and therefore generates barrier pinning. As a result, the giant configurational entropy change potentially available in high-entropy ceramics is often bound and hidden in local potential wells of varying depth, so that the system exhibits a very high switching-field limit. Under low field, dipoles cannot easily cross these randomly distributed barriers, reversible polarization is suppressed, and giant EC response is usually released only near the dielectric-breakdown limit. Clarifying the intrinsic connection between switching barrier and driving field is therefore the first physical prerequisite for obtaining giant entropy change under low-field excitation.
3.2.2 Free-energy landscape flattening by local structural heterogeneity
The realization of giant EC response requires not only a high entropy ceiling but also the ability to trigger these states under realistic operating conditions. In conventional ferroelectrics, the deep energy wells associated with long-range order lock the configurational entropy, necessitating extreme fields for transition. To circumvent this, materials inverse design must focus on manipulating the free energy landscape through two synergistic routes: intrinsic "softening" of the switching barrier and extrinsic regulation of the polarization kinetics.
(1) Symmetry breaking and barrier flattening
The primary objective is to transform the rugged energy landscape of a long-range ordered ferroelectric into a "slush-like" or flattened landscape where the polarization vector can rotate with minimal energetic penalty.
Drawing crucial inspiration from relaxor dielectrics designed for capacitive energy storage, the introduction of cations with stereochemically active 6 s2 lone pairs, most notably Bi3+, serves as a fundamental mechanism for intrinsic lattice destabilization. In BT-based systems, Bi3+ strongly hybridizes with O 2p orbitals, inducing large-scale local polar displacements that break A-B sublattice matching [198]. When combined with small-radius modifiers like Li+ on the A-site, the resulting short-range chemical ordering triggers the formation of polar nanoclusters (PNCs) at the 1–3 nm scale (Fig. 20A, B) [199]. These clusters act as dynamic polarization reservoirs; their disordered orientation and high local polarizability flatten the Gibbs free energy distribution, enabling a massive maximum polarization while effectively decoupling it from hysteresis loss [198,199].
The strategic introduction of heterovalent ion pairs (e.g., Zn2+/Nb5+ or La3+/Mg2+) with significant size mismatch creates intense local random electric and strain fields (Fig. 20D) [200,201]. These fields prevent the formation of macroscopic domains, forcing the system into a state of multidimensional phase coexistence. Reverse Monte Carlo (RMC) modeling of such systems reveals intense polarization fluctuations (ranging from 10 to 90 μC/cm2), suggesting the presence of local morphotropic phase boundaries at the unit-cell level [200]. This mechanism is further exemplified in Sm-doped PMN-PT. Borrowing the concept of local structural heterogeneity utilized to boost piezoelectric or ferroelectric activities, this engineered heterogeneity reduces the anisotropy energy and brings the polarization switching barrier to markedly reduced, thereby providing a template for the low-field release of configurational entropy in EC materials [202–206].
(2) Defect dipoles and restoring fields
While intrinsic softening lowers the driving field, extrinsic defect engineering is required to govern the reversibility and directionality of the entropy change, particularly in relaxor-ferroelectric transitions and negative EC systems. Defect regulation is therefore relevant not only to polarization reversibility but also to leakage suppression, breakdown strength, and field-compatible device operation.
The formation of defect dipoles—such as the () pair in BT (Fig. 20C) or composite Li+/Nb5+ clusters in Bi0.5Na0.5TiO3 (BNT)—introduces a localized internal bias field that acts as a deterministic restoring force [207,208]. Unlike the random fields of chemical disorder, these structured defect fields provide a nucleation template for polarization reversal. Upon field removal, the internal field accelerates the destabilization of field-induced macrodomains, forcing them to disperse rapidly back into a high-entropy polar nanoregion. This mechanism is critical for maximizing reversible entropy change and has enabled giant negative EC temperature changes of 1.49 K under moderate fields of 60 kV/cm [208].
In BNT-BT systems, the deliberate introduction of A-site vacancies serves to fine-tune the ergodic-to-nonergodic transition temperature. By hovering the material's operational state precisely at the P-WP (Paraelectric-Weak Polar) phase boundary, ADE minimizes the thermodynamic penalty for field-induced phase transitions [209–213]. This allows the material to exploit the massive entropy change associated with the relaxor-to-ferroelectric transition at room temperature, while ensuring rapid, low-hysteresis recovery [214].
(3) Electronic barriers and operating windows
System-level deployment of EC materials is limited by the trade-off between polarization magnitude and dielectric breakdown strength. Advanced defect engineering addresses this by modulating the electronic energy levels of the dielectric.
Adapting strategies originally developed to stabilize high-temperature piezoelectric ceramics, the incorporation of specific modifiers (e.g., Ga3+ in BF-BT or Sm-Li co-doping in BT) introduces deep-level trap states within the bandgap [207,215]. These traps increase the carrier activation energy (Ea) to the 1.33–1.98 eV range, effectively immobilizing oxygen vacancies and charge carriers under high fields [208,216]. By suppressing the non-linear leakage current and managing transport modes, this electronic barrier engineering extends the dielectric breakdown strength (Eb)—reaching 152 kV/cm in optimized BT systems—thereby allowing the material to reach its theoretical entropy ceiling without premature failure [198,207].
3.2.3 Grain orientation and texturing for anisotropic barrier control
In conventional polycrystalline high-entropy ceramics, random grain orientation introduces severe intergranular constraint and mechanical clamping. This stress mismatch broadens the phase transition and imposes a large elastic-energy penalty during polarization switching, thereby creating a macroscopic anisotropic barrier against low-field entropy release. Crystallographic orientation engineering, through design along selected directions such as <011>, <111>, or <100>c, offers a direct route toward choosing the lowest-barrier switching path [217–221]. Highly ordered single crystals provide the ideal physical model for eliminating grain-boundary clamping and exploring barrier-free switching. Once microscopic interfaces are removed, the switching barrier is governed by the chosen crystallographic orientation. In large BNT-6BT-3KNN single crystals prepared by molten-salt methods, for example, the <100>c orientation locks onto a nonergodic/ergodic relaxor phase boundary and permits smooth polarization evolution along <001>c in an interface-free lattice. A giant polarization jump can then be triggered under a field as low as 16.5 kV/cm, from which a substantial EC response may be inferred, although the EC effect itself was not directly measured in the original study [217].
The deeper physical mechanism is even clearer in BT single crystals. Under the conventional <001> orientation, the applied field is collinear with the spontaneous polarization axis, so that the transition must proceed by polarization extension, which is accompanied by strong electrothermal hysteresis and elastic-energy dissipation. The crystal exhibits a positive adiabatic temperature change of 0.90 K (reported data but uncalibrated, actual value might be higher) and a negative temperature change of 0.68 K (evaluated data, uncalibrated, actual value might be higher) under 12 kV/cm (Fig. 21A) [222]. In contrast, the <011> design introduces a noncollinear field that forms a defined angle with the spontaneous polarization direction. Physically, this geometric design converts a high-barrier polarization-extension process into a lower-dissipation dipole-rotation process. The elastic-energy penalty associated with the transition is then reduced strongly, allowing <011>-oriented BT to produce an adiabatic temperature change of 1.33 K near 288 K under only 15 kV/cm (Fig. 21B), with an EC strength of 88.7 mK·cm·kV−1. This crystallographic design therefore defines an upper benchmark for overcoming intrinsic barrier pinning [218].
For integrated microsystems, the most effective engineering route toward low-barrier behavior analogous to that of a single crystal is to build highly textured polycrystalline ceramics through template grain growth. The core rule of texture engineering is crystallographic symmetry matching between the texture direction and the intrinsic spontaneous polarization vector Ps. Without such matching, anisotropic barriers cannot be removed on the macroscopic scale.
PMN-10PT and BaHf0.11Ti0.89O3 provide representative examples of materials whose intrinsic Ps lies strictly along <111> [220,221]. If <001> texture is imposed, the field and the polarization vector differ by 54.7° (Fig. 21C) [220]. The system is forced into a high-energy 4R domain configuration, the projection of polarization is reduced by , and nonpolar-direction driving produces strong structural distortion. As a result, the coercive field can rise to 4.53 kV/cm. A targeted <111> texture, by contrast, reproduces the single-crystal-like 1R equivalent polarization configuration on the macroscopic scale. Because the applied field is coaxial with the intrinsic polar easy axis, geometric resistance caused by grain misalignment is removed. In addition, the perovskite lattice shows intrinsically weak electrostriction along <111>, so that the strain accompanying polarization reversal is strongly compressed. This geometrically declamped state lowers the switching barrier sharply and reduces Ec to 2.07 kV/cm. Under this low-barrier condition, local dipoles reverse cooperatively with little obstruction, enabling PMN-10PT to produce 1.30 K under 50 kV/cm [221] and BaHfTiO3 to deliver 1.48 K [220]. Targeted construction of the crystallographic plane thus unlocks giant polarization potential and large thermodynamic entropy change under much lower field. Recent work on <111>c-textured BT-based ceramics confirmed the validity of this strategy: through the synergy between PCE and orientational response, the material showed an adiabatic temperature change of 3.9 K across a broad temperature range, with fluctuations below ± 10% (Fig. 21D) [223]. Although these reported values were obtained in different laboratories and were not subjected to a unified calibration protocol, they consistently support the same EC mechanism governed by targeted texture engineering.
3.2.4 Dynamic barrier smoothing and energy-conversion efficiency
Dynamic smoothing of polarization-switching barriers through the synergy of intrinsic active ions and extrinsic defect engineering not only enables low-field response on the microscopic scale, but also carries decisive implications for the COP and reliability of EC cycles on the macroscopic scale.
(1) Hysteresis suppression by barrier smoothing
In EC refrigeration, hysteresis loss generated during polarization and depolarization is converted into irreversible Joule heat. This parasitic heat source directly offsets the cooling produced during depolarization and is therefore a key thermodynamic penalty. In conventional high-entropy systems with strong barrier pinning, polarization reversal is usually accompanied by large hysteresis loops. By contrast, barrier-smoothing strategies open low-energy polarization tunnels through local fluctuation fields and defect dipoles, allowing the polarization vector to rotate reversibly under very low field through near-barrier-free nonlinear pathways. On the macroscopic scale, this appears as extreme slimming of the P-E loop, which minimizes irreversible energy dissipation and ensures that the large configurational entropy stored in high-entropy materials is converted into net cooling with minimal loss.
(2) Driving-field reduction and COP gain
COP is the core metric for engineering usefulness because it measures the ratio of effective cooling output to electrical work input. In high-configurational-entropy systems, maximizing the absolute entropy change alone cannot guarantee a practical COP if the required switching field remains very high. Once the polarization barrier is lowered, both the field-induced transition temperature and the saturation field are reduced. The same or better giant EC response can then be triggered at much lower voltage input. This physical decoupling between entropy release and ultrahigh field not only reduces input power sharply, but also acts together with the near-elimination of hysteresis loss to improve device COP when reduced switching work is accompanied by controlled dielectric, leakage, and thermal losses.
(3) Reliability through reduced switching barriers
Operation near the dielectric-breakdown field is a fundamental cause of electrical fatigue and dielectric failure in EC ceramics. Barrier smoothing raises the carrier activation energy and lowers the critical switching field required for polarization reversal. In effect, it alleviates the intrinsic trade-off between high dielectric strength and high polarization sensitivity. Deep polarization cycling can be achieved without driving the material close to its dielectric limit, which widens the safety margin and improves fatigue resistance and cycling reliability under broad temperature windows and high-frequency alternating fields. This system-level optimization, extending from microscopic barrier control to macroscopic operating stability, closes a key gap between theoretical design and microsystem integration.
Notably, barrier flattening lowers the field needed for reversible polarization reconfiguration, but it does not by itself ensure a higher device COP. A COP benefit is expected only when leakage current, dielectric loss, heat-transfer loss, circuit efficiency, charge recovery, and parasitic thermal mass are controlled within the same cycle.
3.3 Decoupling entropy change from heat-transport bottlenecks
This section treats heat transport as a device-level constraint on materials design. Thermal conductivity, interfacial resistance, and domain-scale phonon scattering determine whether the generated entropy change can be extracted within the operating period, and therefore affect SCP and ΔTspan.
As discussed in the preceding section, although Han et al. [46] showed that material thermal conductivity had only a minor effect on the self-oscillating device examined in their study, thermal conductivity and thermal impedance still constitute key limiting factors for most EC refrigeration devices. This conclusion is topology-dependent: in the self-oscillating polymer device, contact resistance dominates the measured heat flux, whereas intrinsic thermal conductivity can still become limiting in thicker films, composite stacks, and regenerative architectures with finite cycle periods.
3.3.1 Semiconductor heterointegration and lattice-resonance effects
A major challenge in high-entropy EC materials is strong phonon scattering induced by severe chemical disorder. One effective solution is to introduce high-thermal-conductivity semiconductor phases, such as GaN, AlN and AlGaN, into a ferroelectric matrix through heterointegration, thereby creating high-speed channels for phonon transport and increasing the effective thermal conductivity of the composite heat path [224]. Both theory and experiment indicate that in lead-free matrices such as BNT-BZT, precise addition of high-thermal-conductivity GaN can bypass part of the thermal bottleneck imposed by the disordered host (Fig. 22A). Physically, the added Ga atoms induce lattice resonance with Ti and O in the matrix. Analysis of the phonon density of states shows substantial overlap between the Ga phonon distribution in the 10–375 cm−1 range and the density of states (DOS) associated with Na, Ti, and O in the host lattice. This strong lattice resonance increases the phonon group velocity and thereby strengthens lattice thermal conductivity intrinsically. At 300 cm−1 the group velocity in BNT-BZT-0.1 wt.% GaN rises from 651.0 to 684.1 m/s. The same thermo EC coupling also enhances the vibration frequency of TiO6 octahedra and the spontaneous polarization, which yields simultaneous optimization of heat conduction and polarization behavior. At the optimum GaN concentration of 0.1 wt.%, the thermal conductivity increases from 1.48 to 1.61 W/(m·K).
3.3.2 Polar disorder, lattice order, and thermal-transport decoupling
Increasing chemical disorder provides a large entropy reservoir but at the price of limited thermal conductivity because of strong anharmonic phonon scattering. Breaking this contradiction requires a topological design logic in which disorder is used where entropy gain is needed, whereas order is retained where heat conduction is needed. Through nonuniform structural design, thermal impedance and polarization response can be decoupled across scales. This paradigm targets three coupled contradictions simultaneously: the conflict between the poor thermal conductivity of high-entropy disorder and the higher thermal conductivity of low-entropy order; the tradeoff between the high excitation field of high-entropy systems and the limited entropy ceiling of low-entropy systems; and the need to understand how macroscopic PCE and local CCE can cooperate under structural regulation. Cross-scale spatial ordering allows the system to maintain giant entropy output while increasing heat-transfer rate and energy-conversion efficiency intrinsically. Thermal conductivity and interfacial resistance determine whether the generated entropy change can be extracted within the operating period, thereby affecting SCP and temperature-span accumulation.
(1) High-entropy clusters in ordered matrices
To circumvent the intrinsic thermal impedance bottleneck of polymer-based EC materials, inverse design within the systems-engineering paradigm must dictate the reconstruction of phonon transport pathways at the micro-topological level. An interpenetrating heterogeneous composite engineered by embedding a continuous three-dimensional (3D) thermally conductive network within the polymer matrix provides a structural solution to this constraint [118]. This topological configuration establishes high-speed, directional pathways for phonon transport, effectively suppressing the interfacial thermal scattering and high contact resistance typically induced by randomly distributed fillers. Consequently, the continuous phonon network governs a macroscopic absolute thermal conductivity of 0.84 W/(m·K) (Fig. 12B), achieving a 300% enhancement over the intrinsic 0.21 W/(m·K) baseline of the neat polymer. This architecture ensures the rapid and directional dissipation of thermal energy upon the cyclic application of an electric field. Additionally, the inherent organic structural homology between the embedded covalent organic framework nanosheets (COF-NS) and the polymer matrix facilitates optimal phonon matching, which effectively minimizes interfacial thermal resistance and sustains robust in-plane heat dissipation pathways to elevate the macroscopic thermal conductivity of the composite [225].
Enforcing a cross-scale physical bridge, this thermal transport augmentation directly couples with local polarization dynamics. The continuous 3D network simultaneously functions as nucleation sites for ordered dipoles under an applied electric field. By lowering the energy barrier for polar domain growth compared to initial nucleation, these predefined interfaces trigger a 240% enhancement in the EC cooling performance. However, the integration of a rigid 3D continuous ceramic framework into a soft polymer matrix inevitably introduces severe mechanical stress concentration at the phase boundaries, establishing a physical limit on the long-term electromechanical fatigue endurance and mechanical flexibility required for scalable macroscopic device operation.
Recent work in dielectric energy storage and high-performance piezoelectrics has shown that local disordered regions embedded in a long-range ordered matrix (Fig. 22B, C) provide an effective route toward combining strong polarization response with efficient phonon transport [226–230]. Through multicomponent ionic introduction or selective elemental enrichment, these designs create strong local random fields on the atomic scale and divide an ordered structure into local high-entropy clusters or multiphase nanoclusters roughly 1–2 nm in size.
Transferred to EC materials, these local high-entropy clusters can be regarded as thermodynamically active islands or polarization-fluctuation centers. Their local potential surfaces are extremely flat and display multiphase degeneracy, so under an external field they can undergo sensitive polarization rotation and configurational transformation, thereby contributing a large local CCE.
The deeper innovation is that this design also decouples local functional response from global phonon transport. Severe lattice distortion induced by high entropy is confined to discrete nanoclusters rather than being distributed randomly throughout the whole material. The majority ordered matrix therefore remains continuous and highly symmetric, which creates fast channels for phonon propagation. This ordered-framework-with-local-disorder architecture discretizes and localizes phonon scattering centers, minimizes the decay of phonon mean free path across the entire sample, and breaks the conventional assumption that all-scale high-entropy design must necessarily lead to a sharp drop in thermal conductivity.
(2) Phonon highways within a disordered matrix
Complementary to disorder embedded at the local scale is a second engineering strategy: the creation of macroscopically ordered phonon-transport tracks in a disordered relaxor or high-entropy matrix through oriented growth. Inspired by highly oriented piezoelectric ceramics, this design offers a geometric route toward addressing strong phonon scattering while preserving a large EC entropy reservoir.
Recent textured systems produced by template grain growth establish a distinctive physical balance between macroscopic order and microscopic disorder [223]. Even though the high-entropy matrix retains typical relaxor features, including dynamically evolving polar nanoclusters and strong dielectric dispersion, the high Lotgering factor indicates that macroscopic lattice alignment is strong enough to reduce internal stress and structural defects associated with randomly oriented grains.
If applied to EC refrigeration, the main contribution of this architecture is that geometric declamping opens low-scattering channels for phonon transport on the macroscopic scale. Along selected directions such as <111>c or <001>c, the structure can not only lock in the preferred low-barrier switching pathway, but also increase the phonon mean free path because of the long-range order of the aligned lattice. In this way, high entropy at the microscopic scale can coexist with improved thermal conductivity and higher breakdown field at the macroscopic scale.
(3) Coupled PCE–CCE gains
The deeper significance of the ordered-disordered structural game lies in the fact that it uses ordered intervention at both microscopic and macroscopic scales to couple macroscopic PCE and local CCE under different field regimes. This design directly addresses three central tensions in the development of high-entropy EC materials.
First, the conflict between poor heat conduction in disordered materials and good heat conduction in ordered materials is resolved through spatial decoupling. Whether by embedding local high-entropy islands in an ordered framework or by opening textured phonon highways in a disordered matrix, the system separates the entropy-generating and heat-conducting functions hierarchically while preserving continuous channels for phonon transport.
Second, the tradeoff between the high driving field of high-entropy systems and the limited entropy ceiling of low-entropy systems is eased through barrier reconstruction. Ordered regions act as low-barrier nuclei for field-induced polarization reversal and thus lower the macroscopic driving field, whereas the local high-entropy zones provide a very large number of microscopic states and therefore preserve a high entropy ceiling. The result is multilevel cooperative response from the unit-cell scale of CCE to the domain scale of PCE under comparatively low field.
Third, this ordered-disordered design improves thermodynamic efficiency by enabling PCE and CCE to act not as isolated contributions, but as a dynamically correlated polarization network coupled through local random fields and global lattice orientation. Such multiscale synergy yields giant EC ΔT while also suppressing hysteresis loss and improving cycle reliability. High-entropy EC design thus progresses from simple compositional disorder toward a more advanced stage of structural cooperation.
3.3.3 Domain simplification and phonon phase-space reconstruction
In addition to building high-speed channels through heterointegration, thermal-transport optimization in high-entropy materials can also be pursued by reducing intrinsic phonon scattering at the level of crystal-structure topology. Work on the heat-transport evolution of the antiferroelectric prototype PZO demonstrates that thermal transport can be regulated through two coupled dimensions: simplification of the number of atoms in the unit cell n and active management of domain dynamics [231,232].
(1) Unit-cell complexity and phonon scattering
According to the Slack model, lattice thermal conductivity scales negatively with the number of atoms per unit cell as κ proportional to n−2/3. In complex antiferroelectrics, the zero-field AFE phase, with space group Pbam, contains as many as 40 atoms in the unit cell. This large n causes strong Brillouin-zone folding and produces an extremely crowded phase space for phonon-phonon scattering, thereby shortening the phonon mean free path and keeping κ in a low-conductivity OFF state.
If this logic is translated to high-entropy EC design, the key innovation is to drive the material by an electric field from a disordered and structurally complex ground state toward a metastable state of smaller n and higher symmetry. As shown in PZO, the field-induced AFE-to-FE transition can reduce the number of atoms in the unit cell from 40 to 10, while the thermally induced AFE-to-PE transition can reduce the number of formula units even further. This dramatic unit-cell contraction causes the weighted phonon-phonon scattering phase space to collapse across the full spectrum and reduces optical-acoustic scattering. Experimentally observed switching ratios above 2.2 and sub-150 ns response suggests a viable route for transient enhancement of thermal conductivity during an EC cycle (Fig. 23A–C) [231].
(2) Domain control and interfacial scattering
Besides the intrinsic contribution of unit-cell topology, the dynamic evolution of domains also exerts strong control over extrinsic thermal resistance. During field-induced phase transitions in PZO, a bidirectional switching feedback emerges. Unit-cell simplification tends to increase κ, but the transition is often accompanied by volume expansion and domain restructuring, which can transiently generate high-density ferroelastic domains. The resulting increase in domain-wall density strengthens phonon-interface scattering and can lower thermal conductivity temporarily by about 10% [232] (Fig. 23D, E).
The implication for EC design is that an ideal high-entropy microstructure should not only permit field-induced simplification of the unit cell, but should also minimize domain complexity. Compositional strategies that reduce domain-wall density during polarization switching, or pseudocubic structures with intrinsically sparse domain walls, can cut parasitic thermal resistance arising from interface scattering. Combined with unit-cell compression, such domain simplification ensures that the material enters a phonon-transparent, high-thermal-conductivity state at the same moment that it exhibits strong field-driven EC response.
(3) Coupled entropy release and heat transport
In future high-entropy EC materials, the desired design should synchronize thermodynamic entropy change with dynamic heat transport. During field application, the electric field would drive the structure toward a more symmetric state with lower n and fewer domain walls, so that thermal conductivity rises at the same time as a large entropy change is released, thereby shortening isothermal heat rejection. During field removal, the structure would return to a more disordered and more strongly scattering low-conductivity state, physically suppressing backflow of heat. If dynamic thermal-conductivity control of this kind can be integrated into a solid-state cycle, then low-voltage drive below 10 V and nanosecond-scale heat switching could raise the operating frequency of EC refrigeration to a new level. In addition, the regulation of thermal conductivity, phonon transport, and lattice vibration can, to some extent, be coupled to lattice heat capacity. This coupling provides a further opportunity to tune heat capacity indirectly and thereby reinforce EC performance within the same dynamic thermal cycle.
3.4 Heterogeneous integration for broad-temperature EC response
A single-phase EC material usually exhibits giant response only within a narrow temperature interval near the phase-transition temperature Tc. Macro-heterogeneous integration offers a route around this intrinsic limitation. By constructing compositional gradients from dielectric layers with different phase-transition characteristics under a multiphysics framework, one can build a continuous and stable EC plateau on the macroscopic scale.
3.4.1 Multistage phase transitions and performance-envelope reconstruction
The first logic of macro-heterogeneous integration is the physical superposition of units with different transition temperatures in order to compensate for the low response of a single-phase material away from its transition point. In BT-based systems, stacking units with different B-site dopants and different four-phase coexistence points, such as BSnT11, BZT15, and BHT11, creates strong responses because the free-energy profiles remain extremely flat near tricritical points (Fig. 24A) [233]. In such layered heterostructures, interfacial coupling makes the phase transitions of neighboring layers no longer independent. Once one functional layer undergoes a field-induced transition, its polarization jump and lattice strain can help destabilize neighboring layers through interfacial stress, generating an emergent enhancement beyond the simple sum of the individual layers.
A parallel strategy has been demonstrated in BNT-based systems by stacking gradient layers with different ferroelectric-relaxor transition temperatures (Fig. 24B, C) [234]. In that case, discrete EC peaks can be merged into a continuous performance envelope. Whenever the ambient temperature approaches the transition temperature of a particular layer, the large EC contribution of that layer compensates for the lower contribution of layers away from transition. The result is diffuse variation of the polarization current with temperature and a significant broadening of the useful temperature window.
3.4.2 Compositional gradients and interfacial barrier suppression
The fundamental bottleneck of solid-state cooling is the inherently narrow operational window associated with a single critical transition point. To bypass this, materials design must shift toward macro-heterogeneous integration. To smooth abrupt performance changes in layered heterostructures further, parabolic compositional gradients provide an effective route toward lowering interfacial barriers. Through tape casting followed by high-temperature sintering and trace diffusion, continuous composition profiles can be formed. Phase-field simulations show that this parabolic design creates a continuous and smooth potential-energy landscape and thereby suppresses the interfacial barrier between adjacent heterogeneous layers (Fig. 24D) [235]. Such continuous gradients improve both mechanical and physical integrity across the stack while also enabling dynamic coupling between long-range ferroelectric domains and local polarization clusters. The polarization response is then no longer limited to abrupt jumps at discrete layer boundaries but instead evolves continuously across a broad temperature range. The smoothing of microscopic barriers together with macroscopic gradients therefore provides a key physical basis for broad-temperature EC plateaus with high stability.
By geometrically stacking compositionally graded multilayers or directly blending granules with distinct structural boundaries (e.g., R-O, O-T, and T-C transitions), discrete Landau energy wells are smeared into a continuous thermodynamic trough. This macroscopic chemical heterogeneity introduces pervasive interfacial energy that competes directly with the bulk matrix energy. By systematically lowering local polarization rotation barriers, the discrete EC and piezoelectric response peaks merge into a flexible, broad-temperature performance plateau [142,236,237].
3.4.3 Dielectric gradients and adaptive field redistribution
A further advantage of macro-heterogeneous structures arises from differences in relative permittivity among the constituent layers. Under the series-capacitor model, the local field borne by each layer under a fixed total voltage is inversely proportional to its relative permittivity. When temperature approaches the transition of a particular layer, the relative permittivity of that layer rises sharply and the local field borne by it decreases spontaneously. Layers outside their transition regions, which retain lower permittivity, then bear a larger fraction of the field. This self-driven redistribution of the field ensures that the largest field is directed preferentially toward the low-response region rather than driving excessively rapid switching near the phase-transition point. The resulting adaptive compensation smooths the macroscopic EC response, broadens the ΔTspan, and reduces the risk of local dielectric failure [234].
The response of a multilayer or gradient EC stack is not the algebraic sum of the single-layer ΔT values. Dielectric mismatch, local field redistribution, interfacial thermal resistance, mechanical constraint, and nonuniform heat extraction can reshape both the local EC event and the net device heat flow.
3.4.4 Topological constraints and entropy relay across layers
While macroscopic gradients provide the thermodynamic basis for broad-temperature operation, their physical realization requires strict microstructural topologies to prevent complete homogenization during high-temperature processing. Compositionally sandwiched core-shell architectures—such as an ordered core@double-shell hierarchy—serve as an ideal topological constraint (Fig. 25A–C). This architecture induces controlled, nanoscale gradient diffusion and structurally anchors the relaxor behavior of the constituent layers. As the ambient temperature fluctuates, the nested composition gradients orchestrate an uninterrupted "entropy relay": as the polar transition of the core wanes, the adjacent shell is thermodynamically activated. This continuous compensation mechanism neutralizes the narrow operational window of conventional relaxors, establishing a robust microstructural blueprint for sustaining massive configurational entropy release across complex thermal environments [238].
3.4.5 Microscopic disorder and macroscopic integration under low fields
Broad-temperature EC materials often suffer from a high driving-field requirement. This bottleneck can be alleviated through a cooperative design in which microscopic barrier softening is combined with macroscopic gradient integration. On the microscopic scale, local polarization fluctuations, for example those induced by A-site Sr2+ substitution, or grain coarsening to average sizes near 4 µm, can lower the activation energy for polarization reversal. Coarser grains reduce grain-boundary density and provide spatial continuity for reversible long-range domain motion.
When such microscopic barrier softening is combined with a macroscopic gradient in transition temperature, local disordered polarization regions act as cores for low-field phase change. Experiments show that heterogeneously integrated ceramics designed under this logic can still exhibit strong EC intensity under fields as low as 20 kV/cm, while maintaining broad-temperature response and improving COP [235].
Macroscopic EC systems require efficient electrical energy recovery, but any polarization hysteresis within the active material generates intrinsic entropy and parasitic heat that external circuits cannot recapture [239]. Consequently, achieving zero or extremely narrow hysteresis loops is a prerequisite for material optimization, as these unrecoverable microscopic losses will otherwise multiply during high-frequency operation and paralyze the system's overall efficiency.
By drawing on mature co-firing methods used in MLCC technology, macro-heterogeneous integration is moving toward micrometer-scale and integrated device forms. Alternating stacks of gradient dielectric layers only tens of micrometers thick strengthen interfacial coupling, reduce macroscopic defects, and enable joint optimization of electric-field distribution and heat flow. In future device-level design, this interlayer heterogeneity can be extended further to anisotropic thermal-conductivity engineering. By adjusting the contact interfaces between layers, the rates of heat absorption and release can be controlled precisely, which opens a path toward all-solid-state refrigeration systems with large cooling capacity and stable interlayer heat exchange over a broad operating temperature range.
While ordered bulk ceramics, such as PST, exhibit extraordinary intrinsic material efficiency (ηmat up to 128) driven by their strong first-order phase transitions, this peak thermodynamic performance is strictly confined to an extremely narrow operational temperature window (typically around 2.5 K) [240]. To bridge the gap between this microscopic efficiency and the massive temperature spans required for macroscopic cooling, future material and device designs must adopt a spatial gradient approach. By chemically tuning the Curie temperatures of the active materials—such as through targeted elemental doping—and assembling them into a "layered regenerator" architecture, systems can seamlessly stitch together multiple narrow, efficient phase-transition zones. This gradient topology ensures that as the thermal load is pumped across the regenerator, the local active material is always operating at its peak thermodynamic efficiency, thereby effectively broadening the macroscopic working temperature range without sacrificing intrinsic energy performance [150,236]. Translating this macro-heterogeneous concept into practical thick-film architectures, recent studies have demonstrated the direct tape-casting of Zr- and Sn-doped BT mixtures [241]. Rather than employing alternating multilayer stacks, this direct compositional integration at the micrometer scale enforces a multiphase coexistence while enhancing the electric-field endurance of the thick film. By physically merging distinct relaxor states, this structural strategy mitigates intrinsic temperature sensitivity, converting isolated phase-transition peaks into a continuous, stable thermodynamic operational plateau that spans from 10 K to 60 K.
3.5 Materials conditions for self-actuated EC devices
As EC device topologies evolve, the reliance on external mechanical drivers introduces severe energy consumption and parasitic thermal mass. Consequently, research has shifted toward internalizing the actuation mechanism, where the core dielectric autonomously generates macroscopic bending or volume changes to physically switch thermal contacts. To achieve this self-oscillating "actuation-cooling" integration, the material must exploit a coupled volume-entropy response: synchronizing the configurational polarization entropy with intrinsic electromechanical displacement governed by defect and structural engineering.
3.5.1 Defect-engineered flexural strain for thermal switching
To execute self-driven thermal contact without external motors, materials must overcome rigid triaxial clamping and convert uniform lattice expansion into directional flexural bending. In oxygen-vacancy-rich systems such as PZT, reducing the ceramic thickness below 0.3 mm releases bulk physical clamping. Under an applied electric field, oxygen vacancies migrate to create a dynamic concentration gradient that asymmetrically pins domains on opposing surfaces. This biaxial-triaxial stress crossover yields a massive longitudinal strain of 1.3% at 70 kV/cm (Fig. 26A) [242].
This bending mechanism is geometrically amplified through explicit defect dipole engineering. In KNN ceramics, the introduction of acceptor dopants forms () defect dipoles that heavily populate the surface layers. During alternating field cycles, ferroelectric domains switch while these asymmetric defect dipoles remain orientationally locked. This establishes a severe stress difference across the layers, forcing the material into a unidirectional electrobending mode with a peak apparent strain of 3.2% (Fig. 26B) [243]. Alternatively, manipulating the short-range hopping of dipoles at ultralow frequencies induces a chemopiezoelectric effect. This translates local bond-length variations into an extreme asymmetric volume expansion up to 1.9%, yielding an effective d33* exceeding 6300 pm/V (Fig. 26C) [244]. In a self-oscillating device, this defect-driven giant bending acts as the internal mechanical driver that bridges the working body and thermal terminals.
3.5.2 Actuation-barrier flattening and mechanical-loss regulation
Efficient self-actuation requires triggering these massive deformations at low driving fields while strictly suppressing mechanical hysteresis. At the atomic scale, introducing heterovalent ions with divergent off-center displacements (e.g., Bi/Zr co-doping in KNN) physically disrupts long-range ordering, inducing an order-disorder phase transition. This structural heterogeneity produces a flattened Landau energy profile, minimizing polarization anisotropy and reducing the tetragonality ratio c/a to 1.009 [245]. This drastically lowers the activation energy for both polarization rotation and lattice distortion, allowing the crystal to seamlessly synchronize dipole entropy (cooling) with lattice strain (actuation).
However, massive volume changes inevitably generate domain wall friction and parasitic Joule heating, which negate net EC cooling. Resolving this contradiction requires a spatially decoupled gradient-acceptor-doping topology. By constructing a soft-core/hard-shell architecture, the defect-free central core maintains an active multiphase coexistence necessary for massive strain generation. Simultaneously, polar defect dipoles at the outer edges couple with spontaneous polarization to create a strong internal bias field (Ei ~3.8 kV/cm) (Fig. 26D) [246]. This field heavily pins non-180-degree domain wall sliding strictly at the boundaries, elevating the mechanical quality factor (Qm) to 340. This localized hardening effectively cuts off intrinsic frictional energy loss without clamping the elastocaloric volume expansion of the core.
3.5.3 Coupled actuation and cooling in integrated materials
For self-oscillating solid-state refrigeration, the strict boundary between EC and elastocaloric responses must be dissolved. By intentionally introducing asymmetric defect gradients, the inherent lattice strain of the dielectric is transformed into a macroscopic actuator. This allows the cooling medium to physically deflect and contact thermal terminals under the exact same alternating electric field that drives the entropy change.
This "system subtraction" eliminates external actuators and redundant interfacial materials, reducing solid-to-solid contact resistance issues. Furthermore, the management of mechanical loss dictates the practical viability of these self-driven heat pumps. High-strain materials are functionally obsolete for refrigeration if their internal Rayleigh dissipation outweighs the EC temperature drop. Future device engineering must adopt spatial gradient designs—such as soft-core/hard-shell microstructures—that confine mechanical softening to the device interior while heavily pinning the boundaries as lossless anchors. By converting structural imperfections (oxygen vacancy gradients, off-center displacements) into active thermodynamic drivers (flexural strain, lowered barriers), the material functions simultaneously as the refrigerant and the mechanical oscillator, enabling integrated, low-thermal-inertia solid-state cooling.
3.6 EC measurement and error calibration
Methodologically, the characterization of the EC effect is categorized into indirect thermodynamic derivations and direct calorimetric measurements [247].
3.6.1 Indirect measurement and thermodynamic assumptions
The indirect evaluation is anchored in classical thermodynamics. Under constant stress and electric field, or under constant stress and temperature, the intrinsic entropy and macroscopic polarization are mathematically derived from the first-order partial derivatives of the Gibbs free energy with respect to temperature and electric field, respectively:
Based on the Maxwell relation, taking the partial derivatives with respect to Ek and T at constant stress gives the pyroelectric coefficient . The EC effect can be regarded as the inverse process of the pyroelectric effect:
Derived from the fundamental differential of the Gibbs free energy (G) under constant stress (σ), the Maxwell relation establishes a direct coupling between the pyroelectric coefficient and ΔS. Accordingly, ΔS and the adiabatic temperature change (ΔT) are governed by the integral expressions:
Here, CE denotes the volumetric heat capacity under a specific electric field (E). Practically, this procedure extracts polarization (P) values across a temperature (T) series, fitting the P-T curves to execute numerical integration of the derivative ∂P/∂T. When utilizing the specific heat capacity per unit mass, the physical density parameter ρ must be incorporated into the formulation.
Despite experimental convenience, the indirect formulation dictates a rigid assumption of an ideal thermodynamic equilibrium state. In complex material systems, these indirectly derived expressions frequently trigger significant deviations, generating false extremums that contradict direct empirical data [248–251]. This systematic physical discrepancy is governed by four interacting mechanisms:
(1) Ergodicity breakdown and hysteresis loss
The indirect method inherently assumes ergodicity. Lisenkov and Ponomareva [252] supported via molecular dynamics that indirect evaluation is strictly valid only for nonhysteretic, reversible processes. However, for nonergodic relaxors—such as BNT-based ceramics—the ferroelectric-like hysteretic polarization cycle is accompanied by massive irreversible energy dissipation [250,253]. The Maxwell relation interprets this hysteresis loss as intrinsic configurational entropy fluctuation, thereby systematically overestimating the true EC yield [254].
(2) Kinetic overestimation of polarization
The polarization evolution is a dynamic process dictated by measurement frequency and field duration. Extended field exposure drives strong inter-domain coupling, irreversibly forcing polar microdomains to merge into macrodomains. This measurement-induced kinetic evolution overestimates the thermodynamic equilibrium polarization [255]. Furthermore, in multidomain systems, a portion of the input electrical energy is inevitably dissipated into non-caloric pathways (e.g., domain-wall friction heat), circumventing the ideal conversion between polarization and entropy [256].
(3) Mechanical clamping and size effects
The thermodynamic derivation requires isobaric boundary conditions (constant σ). In substrate-supported thin films, residual interfacial stress continuously evolves with T and E, mechanically clamping the ∂P/∂T response [257]. The breakdown of this mechanical assumption directly manifests in thickness-dependent studies. Bai et al. and Gao et al. [258,259] verified that varying the sample thickness alters the polarization-temperature response laws. Without introducing explicit stress-calibration tensors, the indirect calculation intrinsically outputs systemic deviations.
(4) Thermal hysteresis near phase transitions
EC peaks typically anchor near structural phase boundaries, which inherently exhibit thermal lag during first-order transitions [260]. This microstructural asymmetry forces the dynamic lattice responses to diverge depending on the thermal history. Consequently, indirect EC values extracted from heating branches physically contradict those derived from cooling branches.
Given that indirect evaluation is susceptible to these interacting constraints, direct calorimetric measurements are preferable—especially when evaluating systems transitioning from nonergodic to ergodic states—to guarantee the physical validity of the underlying thermodynamic evolution.
3.6.2 Direct measurement and spatiotemporal resolution limits
Direct calorimetric measurement circumvents the ergodicity assumptions of indirect calculations by physically capturing field-driven thermal fluctuations. Based on the heat exchange mechanisms and spatial resolution, direct characterization techniques are systematically categorized into five primary paradigms:
(1) Differential scanning calorimetry
Modified differential scanning calorimeter (DSC) integrates high-voltage and ground terminals to record heat flow signals triggered by square-wave electric fields under isothermal conditions (Fig. 27A) [145]. The EC response is subsequently derived through the system's intrinsic thermal compensation curves, a protocol widely adopted for ceramics, single crystals, and MLCCs [116,120,121,148,154,171,172,174,218,233,261–263].
Engineering advantages: The system dictates extreme sensitivity to minute heat flow signals and provides excellent electrical insulation boundaries [264].
Physical limitations: First, the robust thermal isolation environment severely decelerates the heat dissipation/absorption processes, forcing prolonged measurement cycles. Second, extracting the absolute temperature change (ΔT) requires independent calibration of the specific heat capacity (Cp) under specific fields and temperatures. Third, dielectric breakdown poses a catastrophic risk to the instrumentation, and the geometry of the specimen is strictly constrained by the dimensions of the standard crucible.
(2) Contact thermometry
Quasi-adiabatic calorimeters utilizing thermocouples (TC) or thermistors (TM) offer the most direct contact-based approach to extract absolute temperature metrics (Fig. 27B) [210,248,249,264–267]. For thin-film configurations, substrate-integrated micro-thermistors enable in-situ profiling (Fig. 27C) [268].
Engineering advantages: This architecture provides ultra-fast temporal response and direct continuous data acquisition without complex intermediate thermodynamic conversions.
Physical limitations: The intrinsic thermal mass of the physical sensors inevitably absorbs a fraction of the EC heat. If the non-active region (e.g., unelectroded area) is substantial or the adiabatic environment fails, parasitic thermal conduction severely attenuates the recorded peak temperature. Consequently, this method is dependent on rigorous post-measurement heat transfer corrections [264].
(3) Heat-flux calorimetry
Tailored for thin films (e.g., PVDF copolymers) and miniaturized ceramics, this technique utilizes a specialized multilayer contact architecture [115,269]. The top layer integrates a reference heating resistor (functioning simultaneously as the driving electrode and calibration heat source), while the bottom layer comprises a heat flux sensor and a heat sink. Dynamically, a known Joule heat pulse is first injected to calibrate the signal area; subsequently, high-field application triggers the EC effect, and the sensor captures the transient EC heat flux for direct proportional conversion (Fig. 27D).
Engineering advantages: The built-in reference heater enables strict in-situ calibration, effectively circumventing systemic errors induced by interfacial thermal contact resistance and case-specific thermal scenarios.
Physical limitations: As a contact-dependent paradigm, the temporal resolution remains dictated by the thermal diffusion length across the multilayer interfaces. Mismatched heat sink baseline temperatures or degraded interfacial coupling inevitably distorts the high-frequency transient heat flux signals.
(4) Infrared (IR) thermography
Infrared detectors capture thermal radiation from the specimen surface to continuously map temperature fluctuations. This imaging technique is widely deployed in assessing PVDF copolymers and multilayer architectures (Fig. 27E) [270–277].
Engineering advantages: The non-contact architecture entirely eliminates parasitic thermal mass and mechanical clamping from physical sensors. It operates with high scanning speeds, enabling real-time visualization of dynamic thermal field distributions.
Physical limitations: In scenarios where the sample exhibits rapid environmental heat dissipation, the finite temporal resolution and sensitivity limits of the IR probe may fail to capture the transient thermal peak, resulting in underestimations of the intrinsic EC effect.
(5) Scanning probe calorimetry
To extract localized EC responses in thin films and miniaturized devices, Atomic Force Microscopy (AFM) probes are engineered into micro-thermal sensors (Fig. 27F) [278,279]. This topology operates either through electron-probe thermal radiation scanning or cantilever-integrated Wheatstone bridge physical contact.
Engineering advantages: It radically compresses the spatial resolution limits of thermal characterization. Contact probes integrated with modified Wheatstone bridges achieve an extraordinary temperature resolution approaching 8 mK [278], providing critical localized thermodynamic data for the inverse design of micro-cooling devices.
Physical limitations: The ultra-narrow scanning area renders radiation mapping susceptible to surface roughness artifacts. More critically, high-field operations introduce severe risks of charge injection and dielectric breakdown, wherein destructive leakage currents can cascade through the conductive AFM tip and compromise the entire analytical system.
3.6.3 Calibration strategies for direct temperature measurements
Direct calorimetric measurements inevitably suffer from parasitic thermal mass and environmental heat conduction, ensuring that the recorded signals are perpetually lower than the intrinsic EC response. To reconstruct the true underlying thermodynamic data, system engineering dictates two core calibration protocols: in-situ physical referencing and finite-element inverse calculation.
(1) Reference-heater calibration
The in-situ calibration methodology establishes a dynamic thermodynamic baseline by integrating a built-in reference heater (often concurrently serving as the excitation electrode) on the sample. Operationally, a small voltage V is applied to the top electrode resistor, injecting a known Joule heat . The sensor (e.g., heat flux sensor, TC, or IR probe) captures this heat, yielding a time-integrated electrical signal. The ratio of the known Q to this signal area defines the calibration coefficient. Subsequently, a high voltage triggers the EC effect, and the recorded attenuated signal is scaled proportionally by this coefficient. For films thicker than the thermal diffusion length, dual calibrations from both top and bottom electrodes must be averaged.
(2) Error breakpoints and evaluation
Although mathematically sound, the physical execution of this heat transfer protocol frequently triggers severe overestimation or underestimation of the intrinsic signal. This systematic deviation stems from the breakdown of four boundary conditions:
Size effects and active volume mismatch: As dictated by Ding et al. [280], polarization responses are dimension-dependent. The assumed joule-heating volume during calibration frequently deviates from the actual polarized active volume under high voltage, skewing the calculated volumetric entropy density.
Thermal asymmetry of surface vs. bulk heat sources: The reference resistor operates as a 2D surface heat source, whereas the EC ΔT is a 3D volumetric heat source. In environments dictating high heat exchange rates, the thermal dissipation pathways and transfer rates for surface Joule heat and internal EC heat diverge, amplifying baseline distortions.
Spatial gradients of material properties: The mass density and specific heat of the surface resistor inherently mismatch the underlying ferroelectric matrix. Furthermore, the physical distance to the external thermal sink (e.g., air, silicone oil) differs, and heterogeneous interfaces (such as adhesive and electrode layers) obstruct linear heat conduction.
Error cascading in MLCCs: When architectures are scaled into MLCCs, the aforementioned size effects, thermal asymmetries, and property mismatches are cumulatively multiplied across hundreds of parallel layers. To circumvent this cascaded parasitic effect, engineering protocols mandate a secondary calibration step: benchmarking the MLCC thermal signals against a pristine bulk ceramic of identical composition under equivalent driving fields.
(3) Finite-element inverse calibration
To bypass the intrinsic physical asymmetry of surface reference heaters, finite element analysis provides a deterministic inverse-calibration pathway (Fig. 28) [208,281]. The strategy models the exact transient thermal conduction process, capturing how the volumetric EC heat generation decays before reaching the physical sensor. By matching the simulated signal attenuation at the sensor coordinates with the empirically recorded signal, the model inversely calculates the true intrinsic adiabatic temperature change. The precision of this operation is strictly governed by rigid engineering prerequisites: it dictates the execution of ultra-fine mesh generation, a 1:1 geometric reconstruction of the testing topology (including parasitic adhesive layers), and the input of temperature-dependent real material parameters (thermal conductivity, heat capacity, and interfacial thermal resistance).
3.6.4 Cross-method validation and error boundaries
To establish an absolute data baseline for EC characterization, system engineering mandates cross-methodological evaluations on a singular target. The comprehensive joint characterization of PMN-PT MLCC (grain size ~1.2 µm) systematically exposed the quantitative deviation boundaries of four direct calorimetric paradigms [264]:
Driven under 80 kV/cm (80 °C), both DSC and TC measurements output a consistent peak ΔT of approximately 1.6 K. The exceptional temporal sensitivity of the TC method (response time 500–1000 ms) is governed by the minimal thermal mass of the sensor and unhindered physical thermal contact.
In contrast, the TM and quasi-adiabatic calorimeter (AC) recorded signal collapse. The TM registered only 1.4 K, an intrinsic attenuation triggered by the high thermal contact resistance of the probe's external glass insulation layer. Concurrently, the signal loss in AC was dictated by the alumina adhesive layer used to anchor the TC, which structurally obstructed the physical heat transfer pathway (delaying response to 1000 ms).
To counteract non-active parasitic volumes (e.g., insulating layers, electrical contacts), physical calibration is preferable. The AC method incorporated a temperature-dependent dynamic correction factor (1.232–1.285) formulated upon the ratio of the total subsystem heat capacity (MLCC + silver adhesive + copper wire) to the active dielectric capacity. Ultimately, relying on its insulation and high enthalpy sensitivity, only the DSC paradigm safely achieved the absolute benchmark of 2.67 K under extreme fields of 160 kV/cm.
This deviation governed by sensor temporal sensitivity is concurrently verified in inorganic high-entropy systems. Systemic evaluations by Du et al. [40] on giant EC materials revealed that under constant voltage slopes, IR thermography and in-situ heat flux sensors—possessing microsecond-level response times and extreme sensitivity—capture the transient EC peak preceding macroscopic contact TC. Consequently, both their raw signals and post-calibration absolute temperatures register marginally higher.
Furthermore, the rate at which the voltage is applied influences the measured signal by altering the time scale of heat release and heat loss. Consequently, parameters such as field ramp rate, rise/fall time, pulse width, and cycle period must be carefully evaluated relative to the detector response time and the sample thermal time constant when cross-comparing direct measurements.
(1) Physical divergence between direct and indirect evaluations. Macroscopically, the underlying phase-transition hysteresis governs the consistency boundary between indirect derivation and direct measurement. Nair et al. [270] confirmed that in canonical relaxor ferroelectrics characterized by low energy barriers and robust reversibility, the dissipation-free polarization switching dictates a precise convergence between Maxwell's calculations and direct calorimetric data. Conversely, in nonergodic systems or those exhibiting strong first-order phase transitions, irreversible structural evolution and massive domain-wall friction inevitably force indirect calculations to output false performance overestimations and systemic physical divergence [253].
(2) Engineering breakpoint: calibration boundaries and objective evaluation. In summary, the parasitic thermal mass of sensors and environmental heat dissipation constitute the objective physical basis for signal attenuation in direct measurements. However, the improper application of calibration strategies can introduce significant systematic deviations during inverse design. Deviating from the true dynamic heat transfer model of the material—such as relying solely on static volume ratios for compensation or neglecting interfacial thermal resistance gradients—can easily lead to an overestimation of the calibration factors, thereby outputting EC values that deviate from a reasonable physical range. Therefore, establishing rigorous 1:1 geometrically reconstructed finite element inversions, or employing multi-method and multiphysics cross-validation, serves as a crucial system engineering protocol for accurately evaluating the intrinsic EC ceiling and establishing reliable data benchmarks.
4 Theoretical and Data-driven Tools for EC Design
4.1 Phase-field modeling of domain dynamics and entropy change
In studies of the EC effect in ferroelectrics, phase-field simulation is used not only to track mesoscale domain evolution, but also to convert polarization distributions into macroscopic entropy change through rigorous thermodynamic models. The reliability of theoretical prediction therefore depends directly on the scientific validity and applicability range of the governing equations. This section traces the evolution of EC calculation from macroscopic phenomenology to microscopic statistical configurational models and discusses the theoretical value of PCHE, reversible-polarization correction, and weighted multiphase algorithms in complex systems.
4.1.1 Thermodynamic models for EC entropy calculation
(1) Gibbs free-energy approximation
In the early stage of EC theory, calculations were based mainly on the Landau-Ginzburg-Devonshire framework, which treats the material as a macroscopic continuous medium and obtains entropy change from the temperature derivative of the Gibbs free-energy density.
Under isothermal conditions, the entropy, the entropy change and the temperature change associated with the evolution of polarization P is typically written as [282,283],
where β is the temperature-dependent Landau expansion coefficient, inversely related to the Curie constant Θ, denotes the specific heat capacity per unit mass at constant pressure, Eappl represents the macroscopic applied electric field, denotes the vacuum permittivity, and acts as the temperature-dependent dielectric stiffness coefficient governing the quadratic polarization term in the Landau expansion, related to , which represents the intrinsic, temperature-independent Landau coefficient (inversely proportional to the Curie constant). ρ indicates the physical mass density of the material.
For simple first- or second-order ferroelectric transitions, this phenomenological expression can agree well with the Maxwell relation [283]. Its limitation is that it relies on a mean-field approximation, so that entropy change is attributed only to a change in the magnitude of the polarization vector. The intrinsic contribution of PCE is ignored. In fact, the EC response strongly relies on phase transitions; for instance, Weyland et al. [284] linked the EC effect to the latent heat of phase transitions through thermal measurements and mapped the phase diagram of the material system based on threshold electric fields. In high-entropy or multiphase systems, such as those near an R-O-T morphotropic phase boundary, distinct phase symmetries may correspond to very different microscopic disorder even when the total macroscopic polarization P is numerically similar. The phenomenological derivative therefore underestimates EC entropy change in multiphase-boundary systems because it cannot capture configurational entropy jumps arising from phase-state fluctuations.
(2) Configurational entropy and orientational degeneracy
To break the limitations of continuous mean-field approximations, statistical physics introduced discrete polar configurational entropy (polar entropy or dipole entropy) from statistical physics [40,49,140,285,286]. The core idea was to replace the entropy as a continuous function of polarization magnitude with entropy as a discrete function of the number of microscopic states. As outlined in Section 3.1, under Boltzmann statistics (as formulated by Du et al. [40]), the configurational entropy is , where is the orientational degeneracy of the system. This relation has been used to evaluate the upper EC limit of high-entropy materials with reasonably good agreement with experiment. Similarly, Chen et al. [131] employed a phase-configuration-driven summation formulation to calculate the total phase configurational entropy: .
Pirc et al. [140] proposed a related expression based on phase-configurational multiplicity, while further accounting for the microscopic dipolar-order contribution to polarization. Derived from thermodynamic and statistical-physics arguments, this model also provided a reasonably reliable prediction of the microscopic upper bound of the EC effect.
The derivation of the microscopic EC response originates from statistical mechanics. For a thermodynamic framework comprising a specific number of interacting dipolar entities, where each unit possesses Ω distinct equilibrium orientational states, the inherent dipolar entropy is strictly governed by the Gibbs-Shannon entropy of mixing:
Here, denotes the Boltzmann constant, represents the effective mean volume occupied by an individual dipolar unit. This characteristic volume can be phenomenologically evaluated using the Curie constant and the saturation polarization:
By substituting this effective volumetric parameter into the statistical mixing formulation, the intrinsic configurational entropy and the corresponding saturation adiabatic temperature change of the microscopic system are derived as:
For ferroelectric phases of specific symmetry, the number of equivalent polarization directions determines the intrinsic entropy ceiling: R phase: along equivalent <111> directions, Ω = 8; Orthorhombic phase (O): along equivalent <110> directions, Ω = 12; Tetragonal phase (T): along equivalent <001> directions, Ω = 6; Monoclinic phase (MA, MB, MC): along equivalent R-T/R-O/O-T directions, Ω = 24. The field not only constrains dipole orientation during polarization, but after field removal the dipoles can also relax toward several equivalent free-energy minima, which increases disorder. From this statistical perspective, the giant entropy change near multiphase boundaries arises mainly from the abrupt increase in system degrees of freedom when the material moves from a low-degeneracy to a high-degeneracy state. Although this formulation introduces phase-configurational effects explicitly, it still requires correction when the reversible polarization change between field-on and field-off states is limited by hysteresis or nonergodicity.
(3) Polarization correction in hysteretic systems
In high-entropy EC materials and relaxor ferroelectrics, one central physical question is whether the polarization that remains after field removal contributes to the useful cooling output. Theory resolves this issue by introducing a reversible-polarization correction appropriate for hysteretic systems [270,287]. Nair et al. [270] employed a Maxwell-relation-based expression for calculating the EC temperature change, with the fitting restricted specifically to the temperature-dependent polarization response along the field-decreasing path from Emax to E0. In the PST system investigated, the relatively relaxed state achieved after prolonged annealing likely facilitated the reasonably good agreement between the calculated results and the experimental.
When hysteresis is significant, the reversibility of the entropy change is limited by the irreversible difference between the initial and final polarization states. To describe the adiabatic temperature change more realistically, the square-polarization term in the phenomenological expression is corrected from the saturation polarization Psat2 to an effective reversible polarization difference:
where Pmax is the polarization under the maximum applied field and Prem is the remanent polarization after field removal.
This correction defines the effective thermodynamic path of the EC effect clearly: only the portion of the polarization that disappears when the field is removed and returns to a statistically disordered state can contribute to a reversible EC temperature change. The correction therefore provides a crucial boundary condition for phase-field dynamics, prevents inflation of theoretical entropy change caused by neglect of polarization hysteresis, and sets a more realistic theoretical ceiling for high-performance ceramics operating under strong field. At a qualitative level, this framework has been extensively validated for predicting, or at least reliably assessing, EC trends in a broad range of material systems [208–210,216,234,235,266,288,289].
(4) Weighted multiphase configurational model
The limitations of the phenomenological derivative and the pure statistical expression are complementary. A more refined theory that combines them is the weighted multiphase-contribution model, which is currently among the most rigorous frameworks for evaluating high-entropy multiphase-coexisting systems [130,139].
This model abandons the assumption of a single homogeneous medium. Instead, it takes the phase fractions from phase-field simulation and combines phase-structure configurational entropy with reversible-polarization correction mathematically:
where is the volume fraction of phase k, such as T, R, or O phases; Ωk is the statistical degeneracy of that phase; and the term (Pmax2 – Prem2) restricts the contribution to reversible polarization only. In general, the sign of the adiabatic temperature change is opposite to that of the isothermal entropy change. The underlying reason is that the electric field alters the polarization state in a manner opposite to the isothermal entropy response. For a positive EC effect, the applied field stabilizes a more ordered polar configuration and thus lowers the total entropy. If the process is ideally adiabatic and the internal energy is conserved, the temperature must rise to maintain energy balance.
This formulation reveals two central thermodynamic mechanisms. First, the nonlinear amplification generated by multiphase weighting means that even a small fraction of R or O phase can enhance the total EC entropy strongly because these phases possess higher configurational degeneracy. Second, the total entropy contribution reflects a hierarchy between configurational entropy and reversible polarization response. Although the orthorhombic phase has the largest degeneracy, the R phase can dominate the total response because it often shows greater reversible polarization change under field. Giant EC response is therefore the product of deep coupling between high configurational degeneracy and high polarization sensitivity.
By linking real-space phase content directly to microscopic symmetry, this weighted framework resolves the predictive failure of traditional phenomenology in multiphase high-entropy systems. It not only matches experiment more closely, but also provides the mathematical basis for later phase-field studies of the gains produced by defects, textures, and gradient architectures.
Building upon the foundational framework established by Pirc et al. (Eq. 8), Liu et al. [130] incorporated phase-fraction weighting to evaluate random multiphase mixtures. The total phase-configurational entropy is mathematically expanded into the following summation:
Here, denotes the volumetric fraction of the ith phase, and represents the effective mean volume occupied by a single dipolar unit within that specific phase (with the absolute physical lower bound of defined by the individual molecular volume). Consequently, the macroscopic expressions for the EC properties can be reconstructed into the following weighted multiphase formulations:
Crucially, if the field-driven process induces a structural phase transition, the initial and final states will inherently diverge. Under such scenarios, the state-dependent phase fractions and orientational degeneracies (related to ) must be treated as asymmetric dynamic variables. To guarantee rigorous predictive precision, the mathematical derivation must explicitly integrate these evolving boundary conditions alongside the field- and temperature-dependent fluctuations of other relevant thermodynamic parameters.
4.1.2 Mesoscale domain evolution and heat-flow coupling
Phase-field dynamics serves a role far beyond simply solving thermodynamic equations numerically; it provides a virtual laboratory to dissect the precise mesoscale origin of EC enhancement. By solving time-dependent Ginzburg-Landau (TDGL) equations for polarization, strain, and electrostatic fields, phase-field simulation mathematically links tailored microstructures directly to their altered free-energy landscapes. This section categorizes the fundamental phase-field mechanisms across distinct material systems, revealing how local boundary conditions and spatial heterogeneity govern macroscopic entropy change.
(1) Phase-field dynamics in high-entropy ceramic electrocalorics
The central thermodynamic mechanism in high-entropy electrocalorics is the macroscopic flattening of the free-energy surface induced by extreme chemical complexity. To mathematically map this multi-element disorder in (Ba, Sr)(Hf, Sn, Zr, Ti)O3 (BSHSZT) ceramics, phase-field models introduce a spatially dependent Curie-Weiss temperature with a massive standard deviation to represent localized compositional fluctuations (Fig. 29A) [40]. Upon the introduction of compositional complexity, the flattened energy landscape governs the complete substantial of long-range stripe domains into dynamically active polar nanoclusters. The system evolves from macroscale, ordered R-, T-, and O- phase regions into nanoscale R/O, T/O, and C/T clusters, thereby increasing the polarization entropy associated with phase configurational disorder. Under an external electric field, the lowered activation barrier facilitates the cooperative merging of these degenerate, multi-phase nanoclusters. This process triggers a massive entropy release, yielding a theoretical adiabatic temperature change of 10 K and an isothermal entropy change of 15.0 J/(kg·K) at 100 kV/cm, a mechanism precisely predicted and validated by phase-field simulations.
A similar theoretical logic applies to alkali niobate high-entropy systems. In KNN-BT high-entropy systems, phase-field simulation utilizes a weighted configurational model that integrates phase volume fractions with reversible polarization response [139]. This formulation reveals a specific thermodynamic hierarchy: the superior polarization response of the R phase contributes more significantly to the total EC yield than the high configurational degeneracy of the O phase, while the O-phase configurational entropy remains higher than the polarization contribution of the T phase (Fig. 29B). Consequently, phase-field simulations suggest that an R-O coexisting configuration represents the optimal structural design for KNN systems, effectively coupling the high polarization sensitivity of the R-phase with the high configurational entropy of the O-phase to anchor a theoretical entropy ceiling beyond 70 J/(kg·K) (Fig. 18D).
(2) Phase-field dynamics in textured electrocalorics
Phase-field simulation is uniquely capable of isolating the anisotropic regulation of EC behavior dictated by crystallographic orientation. By defining the Landau polynomial in a local coordinate system and utilizing spatial rotation matrices with Euler angles, simulations map specific grain orientations to total free-energy constraints. Simulations comparing <001>-textured BST80 polycrystals against random orientations demonstrate that texture establishes a lower coercive field and forces a shift from first-order to second-order phase transition dynamics. Beyond increasing the absolute magnitude of ΔT, phase-field dynamics uniquely predict that texturing induces a significant shift of the peak temperature toward higher values (by up to 22 K), thereby physically broadening the operational window [290].
(3) Phase-field dynamics in PVDF copolymer electrocalorics
Extending from inorganic lattices to polymer chain conformations, phase-field models introduce space-dependent Landau coefficients to simulate geometric confinement and matrix disruption. In P(VDF-TrFE-CFE) terpolymers physically confined within a 2D polyamide (2DPA) framework, models utilize an interface distance function to weight the localized polarization response [291]. Phase-field dynamics reveals that the 2DPA template guides ordered conformational transformations by facilitating multi-site polar microstructures (Fig. 30A). This structure increases the conformational complexity and lowers the energy barrier for field-driven nonpolar-to-polar transitions (e.g., from TGTG' to TTTT). The simulation indicates that this reduced barrier enables a doubled cooling efficiency at fields as low as 400 kV/cm.
This interfacial confinement effect is geometrically amplified in nanovoid-engineered P(VDF-TrFE). Phase-field equations simulate 1 nm thick polarity zones surrounding distributed nanopores. These dense internal 2D polar interfaces severely weaken long-range dipole correlations and flatten the elastic Gibbs free energy, resulting in a giant configurational entropy [ΔS = 100 J/(kg·K) and ΔT = 20 K at 1000 kV/cm] driven by the synchronized collapse of degenerate short-range structures (Fig. 30B) [39]. Furthermore, chemical confinement via covalent double-bond defects in PVDF-based copolymers can be mapped using dual-set Landau coefficients. DFT and TDGL equations confirm that these defects lower the alpha-to-beta transition barrier by 10 kcal/mol, shattering 37 nm long-range domains into 23 nm entities (Fig. 30C) [41]. This targeted matrix disruption yields an extraordinary theoretical capacity of 80 J/(kg·K) at 1000 kV/cm.
In semiconductor nanoflower-incorporated relaxor ferroelectric polymers, phase-field simulation was used to resolve the coupled distribution of electronic carriers and spatially inhomogeneous polarization near the ferroelectric-semiconductor heterointerface (Fig. 30D) [292]. The calculation shows that the hole concentration decreases markedly at the interface between the FeVO semiconductor and the polymer matrix, supporting the formation of a carrier depletion region. This depletion region produces an interfacial bias, which promotes local polar conformation of the polymer chains and increases the local polarization under zero field. Under an applied electric field, the same polar interface assists a field-induced polar transition that extends from the interface into the polymer matrix, resulting in enhanced overall polarization at 1000 kV/cm. Further simulation links this polarization enhancement to the depletion layer thickness, indicating that thicker depletion layers at the interface can further increase the polarization of nanocomposites with depletion-induced polar interfaces.
The same phase-field framework was then used to compare the field-induced entropy changes in nanosheet-incorporated relaxor ferroelectric polymer (RFP) and nanoflower-incorporated RFP. When the heterointerfaces are arranged in a randomly oriented, petal-like geometry, the polar structures with different orientations disturb the local polarization in the polymer matrix and form a more randomized polar structure than the orderly oriented nanosheets. This simulated polarization configuration corresponds to a higher microscopic degree of freedom and a high-entropy state under zero field. The calculated EC entropy change is therefore more pronounced in the terpolymer region near the nanoflower fillers than near the nanosheets. As a result, semiconductor nanoflower-incorporated RFP (SNF-RFP) exhibits greater simulated EC strength than SNS-RFP and the base terpolymer, consistent with the experimental comparison of EC strength.
(4) Phase-field dynamics for defect and strain engineering in electrocaloric
In bulk and thin-film ceramics, phase-field simulation explicitly decodes how mechanical boundary conditions and local defect fields modulate the thermodynamic phase transition. For bulk systems such as metal-free organic perovskites (MDABCO), simulations map mechanical boundary conditions by applying a 3D hydrostatic pressure (1 GPa), which physically shifts the free-energy minimum and lowers the Curie temperature to room temperature (Fig. 31) [283]. This prominent response is linked to the rapid change rates of the free energy barrier height with temperature. In epitaxial thin films, equiaxial misfit strain introduces T and O potential wells. Under an electric field along [111], the system undergoes a multi-domain to single-domain transition. Because the 71° domain switching exhibits the lowest energy barrier, the time-dependent evolution strictly follows a sequential flipping path (R4- to R3-(), then to R1+[111]), outputting a theoretical ΔS of −46.8 J/(kg·K) and ΔT of 20.5 K at 1000 kV/cm.
Simultaneously, the introduction of local B-site defects—such as (Li1/4Nb3/4)4+ in BNT-based ceramics—adds an independent defect electric field energy density to the total Landau free energy [208]. Phase-field models demonstrate that a moderate defect concentration exerts a strong pinning effect on domain walls. During polarization, these defects assist macrodomain alignment; upon field removal, they act as internal restoring forces that rapidly accelerate dipole dispersion. This targeted defect engineering forces the structure to relax from an ordered macrodomain state back to disordered polar nanoregions, generating a giant reversible entropy change with 1.49 K (Fig. 32A) over a wide temperature range.
(5) Phase-field dynamics in multilayer gradient electrocalorics
To break the limitation of a narrow operational window, uniform phase-transition points are deliberately eliminated. In Bi0.5Na0.5TiO3-SrTiO3 (BNT-ST) multilayer ceramics (Fig. 32B), phase-field models treat the high-temperature diffusion of Sr2+ as a spatially varying input parameter to define local Landau properties [235]. This parabolic compositional gradient establishes a continuous potential energy landscape. Dynamically, this gradient lowers the interfacial energy barriers and the activation energy for long-range domain formation. The system avoids a singular, whole-volume polarization flip; instead, the gradient dictates multistage successive phase transitions. This spatial asynchronous switching translates thermodynamically into a smoothed entropy peak, sustaining an EC strength of 0.41 K·mm/kV over a 52 K operative window. Gradient heterointegration broadens the operating window by distributing phase-transition temperatures across the active body. In phase-field modeling, this gradient can be represented as a spatially varying transition-temperature field, allowing local phase conversion to be linked to the total EC response.
(6) Phase-field dynamics in antiferroelectric electrocalorics
The phase-field logic of asynchronous transition is equally deterministic in AFE systems. In PZO-based ceramics, incorporating spatially dependent Landau coefficients and local random fields into TDGL equations allows phase-field models to capture a high density of polar boundaries, such as incommensurate antiphase boundaries. These boundaries serve as preferential, low-barrier nucleation sites. The simulated kinetics confirm that these polar boundaries physically interrupt the transient whole-volume polarization jump, converting the process into a regional nucleation and diffusion process (Fig. 33) [114,293]. This localized gradual diffusion prevents sharp entropy exhaustion, broadening the negative EC window to 75 K and delivering a peak ΔT of −13.05 K at 84 kV/cm.
4.1.3 Phase diagram reconstruction and thermodynamic potential parameterization
As established in the preceding sections, maximizing phase-configurational entropy dictates the strict integration of multiphase coexistence to increase macroscopic orientational degeneracy. However, relying solely on empirical trial-and-error to locate these exact phase boundaries across vast compositional spaces is systematically inefficient. To translate the theoretical demand for multiphase degeneracy into precise chemical doping strategies, phase-field modeling and first-principles thermodynamics serve as the indispensable inverse-design compass. The core of this methodology relies on the rigorous mathematical parameterization of the Landau-Ginzburg-Devonshire (LGD) thermodynamic potential. By converting chemical substitution into specific phenomenological coefficients, this framework explicitly maps the continuous evolution of the Gibbs free energy landscape. It quantitatively dictates the depth of potential energy wells, the topological flattening of activation barriers, and the precise intersections of distinct structural symmetries. Consequently, reconstructing temperature-composition phase diagrams provides the exact thermodynamic boundary conditions required to predict mesoscale domain evolution and optimize the theoretical EC entropy ceiling.
In the BST system (Fig. 34A) [147], the thermodynamic baseline for phase engineering is established by a high-order LGD potential expanded to the eighth-order of polarization. The structural evolution is mapped through the Gibbs free energy as a function of temperature calculated from the established thermodynamic potential, where the intersection of free energy curves for different symmetries identifies the precise first-order phase transition points. The theoretically reconstructed temperature-composition phase diagram unearths a systematic suppression of all transition temperatures as the Sr content increases, specifically manifested by the downward shift of the C-T, T-O, and O-R boundaries in temperature space [294]. The phase-field simulation captures a significant reduction in the spontaneous polarization modulus and a corresponding flattening of the energy barriers between polar states. Within this framework, the EC response is qualitatively maximized in the immediate vicinity of these structural instability boundaries, where the vanishing free energy difference between competing phases facilitates a massive field-induced entropy jump.
A parallel parameterization strategy drives the compositional inverse design of the BZT system (Fig. 34B) [155], where higher-order LGD coefficients are mathematically truncated to maintain thermodynamic stability under high Zr concentrations. The resulting temperature-composition phase diagram reveals a unique topological convergence of phase boundaries, characterized by an upward trend for the T-O and O-R transition temperatures which eventually intersect with the descending C-PE boundary [295–303]. This convergence forms a quintessential pinched phase transition or quadruple point near x = 0.15, where the energy landscape becomes degenerate with multiple structural symmetries coexisting in a narrow temperature-composition window [151]. Phase-field dynamics identify this quadruple node as the optimal configuration for triggering giant EC signals, as the synchronization of multiple phase transitions at this critical composition allows for a massive release of configurational entropy through access to a higher number of equivalent polarization states compared to single-phase regions. By strictly parameterizing the Gibbs free energy landscape, these theoretically reconstructed phase diagrams provide the exact chemical coordinates required to trigger high-entropy structural degeneracy and construct the essential physical baseline for simulating the subsequent breakdown of long-range ordered domains into active polar nanoregions.
Phase-field simulation has evolved from a phenomenological tool into a quantitative thermodynamic engine. Simulations across high-entropy oxides, textured solutions, confined polymers, and gradient multilayers collectively suggest: maximizing EC efficiency relies on the engineering of structural frustration and energy landscape flattening. Whether it is the multi-element chemical disorder in BSHSZT, the 2D confined interfaces in polymers, or the polar boundaries in PZO, the mesoscale objective remains identical—to shatter macroscopic long-range polar correlation into degenerate, short-range nanoscale entities. Phase-field dynamics suggests that these tailored microstructures lower the activation barrier for polarization reversal, allowing massive configurational entropy to be released under minimal driving fields. For the inverse design of future devices, phase-field theory dictates that optimization must focus on constructing continuous potential-energy landscapes and spatially decoupled structures, guiding the synthesis of low-thermal-inertia, high-efficiency EC refrigerants.
4.2 First-principles analysis of local polar and thermal physics
Beyond mesoscale phase-field dynamics and domain evolution, first-principles calculations and atomic-scale structural analysis reveal the most fundamental origin of EC and pyroelectric response. By resolving band structure, lattice energetics, atomic displacement vectors, and local ionic ordering, they connect macroscopic thermodynamic behavior directly to electronic and atomic evolution.
4.2.1 Local structural distortion and energy landscapes
In ferroelectric matrices, the local environment modified by chemical substitution directly regulates polarization strength, phase-transition behavior, and intrinsic EC intensity. Density functional theory is particularly effective in quantifying local structural distortion and lattice-energy differences caused by isovalent substitution (Fig. 35A) [304].
In BT-based systems, introduction of isovalent ions such as Sr2+ on the A site and Sn4+ on the B site produces only small differences in lattice energy. This weak ionic positioning preference suppresses strong local structural heterogeneity and preserves a relatively ordered lattice on the atomic scale. By analyzing atomic displacement vectors in detail, one can separate the displacement component Δz parallel to the spontaneous polarization Ps from the perpendicular component Δy. DFT calculations show that in selected isovalently substituted systems both distortions remain small. This atomic-level minimization of distortion explains why a sharp phase transition and a large polarization derivative with respect to temperature can be retained even when the Curie temperature is shifted toward room temperature, thereby supporting giant intrinsic EC response.
4.2.2 Band structure, polarization, and pyroelectric response
Changes in electronic band structure clarify the microscopic mechanism by which heat-induced charge release enhances polarization-related thermal response. By calculating the density of states and band gap Eg in different crystal phases, first-principles theory reveals phase-dependent electronic structures directly (Fig. 35B) [305].
Electronic-structure analysis shows that the band edges in KNN-based systems are dominated by orbitals from B-site ions such as Nb and Zr together with oxygen. The calculated band gap in the tetragonal phase, about 2.13 eV, is smaller than that in the orthorhombic phase, about 2.3 eV. A smaller gap reduces the energy barrier for thermally exciting valence electrons into the conduction band. As a result, the tetragonal phase releases charge more readily under heating and exhibits stronger macroscopic pyroelectric response. DFT also indicates that the tetragonal phase has larger polarization amplitude than the orthorhombic phase, so band-gap reduction and polarization enhancement cooperate to create the thermodynamic advantage of the tetragonal phase in thermal-response applications.
4.2.3 Site ordering and multiphysics phase-transition barriers
In antiferroelectrics, the charge balance and spatial arrangement of B-site ions determine the energetic ground state and the entropy-change potential deeply. Combined first-principles calculations and atomic-scale characterization provide direct evidence for ordered structures and for their response to stress fields (Fig. 35C) [197].
In systems such as PMW, heterovalent B-site ions self-organize into a spontaneous chessboard pattern. This ordering is the microscopic origin of the antiferroelectric character and provides a large phase-transition enthalpy change. Under MLCC conditions, strong residual stress couples to this ordering. The analysis indicates that stress can regulate the AFE-FE transition barrier and shift the Curie temperature from 36 °C down to 19 °C, below room temperature. This polarization jump supported by B-site order allows the giant phase-transition entropy to be released fully under ambient conditions and can produce superposed positive and negative EC responses. On the atomic scale, this confirms strong intrinsic coupling among lattice distortion, electrostriction, and EC behavior.
4.2.4 Defect dipoles and phase-dependent polarization orientation
The configuration of () defect dipoles in BT-based system (Fig. 35D) [207] is established via A-site co-doping within a 1 × 2 × 2 supercell along the [001]c direction. This donor-acceptor architecture triggers a self-compensation mechanism that suppresses oxygen vacancy () formation, as dictated by local charge neutrality requirements. The resulting carrier activation energy (Ea) of 1.65 eV governs the suppression of charge transport, providing the thermodynamic stability necessary for high-field operation. Localized structural distortion is manifested by the intensified displacement of Ti4+ to 0.41 Å (pristine BT: 0.21 Å) and A-site cations to 0.24 Å (pristine BT: 0.13 Å), accompanied by an oxygen octahedral rotation of 5.28°.
At the electronic level, the expanded Ti 3d-O 2p orbital overlap area of 17.00 states (pristine BT: 4.57 states) reduces the repulsion between B-site ions and the oxygen framework. This electronic reorganization underlies the widening of the bandgap (Eg) to 1.829 eV (pristine BT: 1.486 eV), which serves as the physical basis for the enhanced dielectric breakdown strength of 152 kV/cm. The extrinsic polarizability of the () complex, measured at 3.44 × 10−37 F·m2, exceeds that of the intrinsic (Ti4+–O2-) dipole (0.92 × 10−41 F·m2), facilitating a saturation polarization of 34.2 μC/cm2.
In contrast to conventional domain-wall pinning, these defect dipoles function as nucleation centers for polarization switching, effectively lowering the activation energy barrier for domain reversal and resulting in a reduced coercive field of 1.56 kV/cm. This kinetic facilitation allows the system to circumvent the energy dissipation typically associated with ferroelectric hysteresis. The synergy between high polarizability, facilitated switching kinetics, and elevated breakdown strength enables the exhaustive release of configurational entropy. Consequently, the engineered system achieves a ΔT of 2.76 K and a ΔS of 3.10 J/(kg·K) at 70 °C.
4.2.5 Phonon spectra and thermal-transport constraints
Even when EC response is strong, thermal conductivity κ remains a key bottleneck for heat exchange in solid-state refrigeration, especially in organic ferroelectric polymers such as PVDF and related copolymers. Although some device-level studies indicate that the influence of thermal conductivity can be limited under specific conditions [46], structural verification of the link between structure and heat conduction remains essential from the perspective of materials design.
Phonon-dispersion calculations provide an important DFT route for this purpose (Fig. 36) [224,306,307]. By computing phonon spectra, one can assess anharmonicity and predict phonon transport. In EC research, phonon calculations are used both to confirm dynamic structural stability and to evaluate how polarization reversal changes phonon-scattering rates. Simulating group-velocity changes along particular phase-transition paths, for example, can be used to predict the dynamic evolution of thermal conductivity under field. Such phonon-level validation provides a fundamental structural basis for addressing the contradiction between strong polarization response and weak thermal conductivity in high-entropy materials and polymers.
Taken together, first-principles calculations and atomic-scale analysis show that optimization of EC and pyroelectric behavior has progressed from macroscopic phase-boundary design to the control of local distortion, band-gap engineering, defect-dipole energetics, and phonon dynamics. By simulating the effects of stress, electric field, and temperature on microscopic structure, DFT provides high-precision criteria for predicting polarization response. Future work will increasingly rely on these atomic-scale rules to design high-performance EC materials on demand while balancing intrinsic response strength with coupled electro-mechanical-thermal behavior. It should be noted that DFT results are used here as atomistic descriptors, including local distortion, defect-dipole orientation, band-gap-related leakage tendency, phonon stability, and switching-barrier trends. Device-level quantities such as ΔTspan, SCP, heat flux, and COP require mesoscale/domain and device heat-flow models and should not be inferred directly from DFT calculations alone.
4.3 ML for EC materials discovery
Phase-field modeling and density functional theory already provide high-fidelity descriptions of polarization reversal, phonon transport, and local configurational evolution in EC materials. Once the design problem is extended to realistic systems involving complex multielement substitution, high-entropy compounds, and broad-temperature multiphase coexistence, however, trial-and-error experimentation and ab initio simulation encounter a severe curse of dimensionality. ML now offers a cross-scale, data-driven route that reduces the number of candidates requiring experiments or high-fidelity calculations by establishing nonlinear mappings directly between large compositional or structural spaces and macroscopic thermal-response metrics such as adiabatic temperature change and isothermal entropy change.
4.3.1 From black-box prediction to physics-informed modeling
In the early stage of machine-learning applications to lead-free EC ceramics, especially BT-based systems, the main focus was on end-to-end predictive models. The core task was to encode composition, elemental attributes, measurement temperature, and electric field into descriptors to fit EC properties for regression or phase-classification models.
(1) Data-driven descriptors
To move beyond direct dependence on compositional ratios alone, later models incorporated derived descriptors with clear physical and chemical meaning. In models predicting ΔT for BT-based ceramics, features such as the Mendeleev number, ideal bond length, Shannon ionic radius, Pauling electronegativity, and the tolerance factor calculated from ionic radius were introduced (Fig. 37A) [308]. These physically informed descriptors, together with macroscopic variables such as electric field E and measurement temperature T, were then supplied to algorithms such as ε-support vector regression (ε-SVR). The authors have shared web application hosting the trained indirect/direct ΔT regression models via an open-access web platform [308].
Such purely data-driven models achieved notable statistical accuracy on specific datasets. For example, XGBoost models that combine Magpie elemental statistics with macroscopic variables such as Curie temperature Tc and dielectric constant ε can achieve a root-mean-square error near 0.38 K and R2 about 0.77 for ΔT prediction (Fig. 37B) [309]. Support-vector models trained on directly measured data have even reached R2 values around 0.93 [308]. Feature analyses in these models identify electric field, T-Tc, selected electronegativity/Mendeleev-related descriptors, and tolerance-factor terms as influential variables within the corresponding training spaces.
(2) Limits of black-box prediction
Despite their high fitting accuracy, purely data-driven models remain limited in physical generalization. The available EC datasets in these studies are limited in size: Ref. [308] curated 1836 indirect and 528 direct ΔT data points, while Ref. [309] assembled 97 ceramic materials and 4406 processed data points after excluding outliers. In addition, EC data are affected by substantial laboratory-to-laboratory variation because direct calorimetry and indirect inference from Maxwell relations are often mixed across different experimental setups.
Fundamentally, black-box models do not encode explicit thermodynamic evolution. Classification models cannot identify triple points in phase diagrams or predict how an external field shifts phase boundaries dynamically. They also neglect processing history and microstructural variables such as grain size, texture, and domain-wall density. Outside the training distribution, these models should be treated as screening tools with uncertain reliability rather than as extrapolative predictors.
(3) Physics-informed thermodynamic learning
To reduce the risk of poorly constrained extrapolation, several studies have combined ML surrogates with Landau-type thermodynamic models. The basic idea is not to let the model predict ΔT or ΔS directly. Instead, ML is used to predict low-level physical order parameters, after which the target thermal quantities are obtained from a thermodynamic model.
A representative architecture employs a serial combination of order-parameter surrogate and analytical coupling. Gaussian-process regression or support-vector methods are first used to build mappings from composition to Curie temperature Tc and spontaneous polarization P. The ML output is then inserted into a simplified Landau-Devonshire relation, for example (Fig. 38A) [158]. Through calibration on known samples, the physical applicability range of the model can be defined explicitly, and failure of purely black-box fitting in the extreme-field or nonlinear regime can be avoided.
A later strategy used support SVR with selected composition-derived features to predict Landau parameters, including T0, α0, α11, and α111, for the tetragonal proof-of-concept space (Fig. 38B) [159]. In that framework, composition-derived descriptors are transformed into the parameters governing the system energy landscape. After parameterization of the Gibbs free-energy expression, the coefficients were used to calculate polarization curves, dielectric spectra, dielectric tunability, pyroelectric response, and EC-related quantities; extension to piezoelectric coefficients would require additional phase-dependent parameters.
(4) Multiscale advantages and boundaries
By cutting the direct black-box mapping with the physics of Landau phase transitions, this white-box strategy offers a powerful route for high-dimensional screening. In the enormous A/B-site co-doping space of BT-based systems, it can identify candidate phases whose transition temperatures fall within the desired room-temperature operating window and can target the compositional crossover region in the phase diagram. Subsequent experiments confirm that compositions discovered in this way often exhibit weak relaxor features together with excellent and broad-temperature ΔT response under low field [158].
Even so, current PIML architectures still have limits. Most are built on the assumption of homogeneous single-domain media. Their Landau expansions are not fully adequate for the complex nonlinear polarization response of multiphase coexistence, and they neglect mesoscale contributions from grain boundaries, space charge, stress fields, and domain-wall motion [159]. A possible future direction is to test whether mesoscopic descriptors from phase-field simulations, including domain-wall density and multiphase weighting factors, can be coupled to Landau/ML workflows.
4.3.2 Active learning for multicomponent phase-diagram reconstruction
As lead-free EC ceramics move toward multielement substitution and high-entropy design, both ab initio methods and Calculation of Phase Diagram (CALPHAD) encounter prohibitive cost in systems with more than four components. Traditional grid search and gradient search become inefficient under the resulting combinatorial explosion. Ref. [160] used a closed-loop active-learning workflow combining elemental descriptors, classification/regression surrogate models, and maximum-variance experimental selection to refine multicomponent phase diagrams (Fig. 17A).
(1) Uncertainty-aware surrogate models
The central feature of active learning is that it does not depend on blindly expanding the dataset. Instead, it adopts an adaptive sampling mechanism based on uncertainty quantification. In the design of high-dimensional BT-based composition spaces containing co-doping on both the A and B sites, simple compositional ratios are first converted into intrinsic materials descriptors that represent radius, electronegativity, valence-electron number, and related structural and bonding information [160].
Comparison among surrogate models shows that universal kriging trained on measured phase-transition temperatures can capture real phase-boundary topology more faithfully than support-vector classifiers. Crucially, Gaussian-process and kriging models provide not only a prediction, but also a variance for that prediction. This variance functions as a measure of uncertainty and gives the optimization process directional guidance in a large unknown space.
(2) Phase-boundary localization by active learning
Once uncertainty is available, active learning selects the next experiment through utility functions. Under a maximum-variance criterion, the algorithm targets the region of the phase diagram with the highest predictive uncertainty and therefore the greatest information content. Expected improvement was used in Ref. [158] for selecting BT-based compositions with large predicted polarization; Ref. [160] used a maximum-variance criterion for phase-diagram refinement.
The strategy is especially effective in locating complex multiphase coexistence points. In one case, a complex pseudobinary solid solution was targeted by scanning a very large number of candidate combinations, and the algorithm identified a composition around the largest-variance point for the first synthesis cycle. The newly measured transition temperature was then fed back into the training set and the model was updated. For the BT-based pseudobinary phase diagram in Ref. [160], three new synthesis-and-characterization iterations were sufficient to establish the final validated phase diagram within the reported uncertainty.
In Ref. [158], expected-improvement selection identified BT-based EC candidates, including (Ba0.82Ca0.05Sr0.13)(Ti0.89Zr0.01Sn0.10)O3, after screening compositions with room-temperature-relevant Tc and large predicted polarization. For the third composition in Ref. [158] direct DSC measurements were compared with indirect estimates; the study reported ΔT of about 0.6 K at 20 kV/cm and a temperature span of about 30 K defined by 90% of the maximum ΔT. Taken together, Ref. [160] supports active-learning refinement of multicomponent phase diagrams, whereas Ref. [158] supports closed-loop selection of BT-based EC compositions; neither result alone establishes an experimentally validated “optimal EC phase boundary”.
4.3.3 Transfer learning for low-hysteresis polar topologies
EC, dielectric-energy-storage, and piezoelectric materials share a common condensed-matter foundation centered on lattice dynamics, local polarization nonlinearity, and phase-transition kinetics. Recent lead-free high-entropy relaxor studies for dielectric energy storage used random-forest models, descriptor reduction, and local-structure characterization to connect ion-derived features with near-linear polarization behavior and high energy-storage performance. These developments provide a valuable route for breaking the long-standing contradiction in EC materials between giant polarization entropy change and low hysteresis loss although their relevance to EC entropy change and ΔT has not been demonstrated.
(1) Cross-system physical fingerprints
In Refs. [310] and [311], small energy-storage datasets were treated with random-forest regression and feature reduction based on Pearson-correlation filtering and recursive feature elimination (Fig. 39A). The key innovation is the high-level quantification of complex polar topology. A/B-site-asymmetric atomic features, including valence-electron distance, Shannon ionic radius, and site-specific electronegativity, are extracted as the core inputs. These ion-derived descriptors encode selected aspects of A/B-site chemical disorder in the high-entropy relaxor design space. The weakly coupled, small polar clusters are achieved and display nearly linear polarization response at the macroscopic scale and therefore suppress the energy dissipation associated with polarization reversal (Fig. 39B) [311].
Transferred to EC design and treated as a hypothesis, the same enhanced recoverable-polarization logic offers a direct route toward giant EC response with low hysteresis. By maximizing recoverable polarization through high-entropy descriptors, one can construct shallow local free-energy barriers in EC lattices. Polar nanoregions with both large polarization amplitude and strong orientational disorder then reverse smoothly under electric field. This not only yields a large configurational entropy change and high ΔT, but also prevents rapid renucleation of long-range ferroelectric domains after field removal. Irreversible Joule-heating hysteresis can therefore be reduced toward its minimum.
(2) Generative design of multiscale networks
Random forests and active learning remain interpolation tools inside a known phase space. A possible next direction for EC materials design must instead be generative. Recent work in polymer dielectric energy storage has already shown the power of generative models for molecular inverse design (Fig. 40) [312].
For high-temperature polyimide composite dielectrics, Ref. [312] used a generative diffusion framework with global and a local equivariant graph neural network (EGNN) encoders to design organic fillers with target EHOMO and ELUMO. The local EGNN channel encodes adjacent polar chemical bonds, whereas the global channel captures long-range dipolar or van der Waals interactions that govern condensed-matter behavior. By concatenating atomic features with target energy-level parameters such as EHOMO and ELUMO and then applying Langevin denoising dynamics, the model can generate previously unknown organic molecular fillers that combine a very large band gap with strong electron affinity. The generated fillers introduce deep charge traps and reduce leakage current density in PI-based composites at high temperature.
This result suggests a transferable strategy for EC polymers, but its effectiveness must be tested under EC cycling conditions. Flexible PVDF-based copolymers are limited mainly by space-charge injection, leakage current, and Joule heating under high field, all of which directly offset the cooling released during depolarization. If the concept of artificial intelligent generated deep-barrier trap fillers is introduced into EC composites, artificial charge-pinning networks could be constructed in the amorphous polymer region. The sustainable field limit would increase, and Joule-heating loss in high-frequency cycling could be reduced.
Notably, current EC datasets are small and heterogeneous, with ΔT and ΔS reported under different fields, temperatures, waveforms, sample geometries, and direct or indirect protocols. Negative or non-optimal results remain sparse, which biases composition screening toward reported high-response systems. ML models for EC inverse design should therefore report feature definitions, uncertainty, validation outside the training family, and transferability across ceramic, polymer, and composite systems.
Future data-driven workflows should remain tied to explicit physical constraints rather than prompt-level targets alone. For EC polymers and composites, useful generative searches should combine dipolar-disorder descriptors, leakage-current barriers, thermal-conductivity constraints, and synthesis feasibility. This constraint-based route is more compatible with EC inverse design than unconstrained generation of nominally high-performance compositions.
5 Challenges and Perspectives
EC refrigeration should be evaluated as a coupled materials-device-system problem. Intrinsic ΔT and ΔS define the available thermodynamic response, but useful cooling depends on whether this response can be transported, accumulated, and electrically recovered within a finite cycle. Temperature span, SCP, and COP are therefore controlled by active mass fraction, thermal impedance, interfacial contact resistance, charge-recovery efficiency, mechanical compliance, and manufacturability. This coupling defines the central boundary for translating EC materials into deployable solid-state cooling modules.
This review identifies several design boundaries that now govern the transition from laboratory EC response to deployable solid-state refrigeration. Configurational disorder can enlarge reversible entropy change but also increases phonon scattering. High-entropy chemical substitution can raise polarization activity but must remain within thermodynamically stable solid-solution windows. Defect and interface engineering can reduce leakage and switching barriers, but they also introduce reliability and strain-coupling constraints. These tradeoffs define the perspectives discussed below.
5.1 Tradeoff between polar disorder and lattice heat transport
High configurational entropy effectively flattens the local free-energy landscape and maximizes polarization configurational entropy under low field by disrupting long-range ferroelectric coupling. Yet this design strategy also creates a severe condensed-matter contradiction: the lattice disorder required for easy polarization reversal inevitably enhances phonon scattering and therefore degrades intrinsic thermal conductivity.
In real refrigeration cycles, poor thermal conductivity not only delays the extraction of useful cooling, but also promotes heat accumulation under high-frequency alternating fields. The central problem for future EC materials is therefore how to achieve a precise physical compromise between polarization disorder and ordered phonon transport. Future studies should move beyond uniform solid-solution designs and examine hierarchical microstructures with local order but macroscopic disorder, or interfacial architectures that create anisotropic ballistic phonon channels. In this effort, phonon-dispersion analysis and anharmonic-scattering evaluation from first principles will provide indispensable bottom-level criteria.
5.2 Thermodynamic stability of chemical-configurational high-entropy systems
In CCHE systems based on complex multielement occupation of the A and B sites, pursuit of peak dielectric response must be accompanied by strict thermodynamic stability criteria. Strongly forced mixing of many aliovalent or isovalent species can readily approach or exceed the solid-solubility limit. Once that thermodynamic boundary is crossed, energetic instability can cause nanoscale secondary-phase precipitation or severe local compositional segregation.
This microscopic chemical heterogeneity contributes little to useful reversible polarization. Instead, it becomes a trapping center for space charge and a leakage-current pathway, thereby accelerating fatigue and electrical breakdown under strong-field cycling. Accurate phase diagrams for multicomponent systems, together with high-fidelity models that combine CALPHAD and mesoscale phase-field dynamics, are therefore necessary prerequisites for advancing lead-free high-entropy EC materials toward engineering use. Only within a framework of thermodynamic long-range structural stability does the EC entropy reservoir acquire practical device significance.
5.3 Generative models for multiscale EC optimization
The parameter space of high-entropy oxides, complex lead-free compounds, and organic-inorganic composites grows exponentially in both composition and structure, so conventional trial-and-error strategies and intuition-driven chemistry have reached an efficiency ceiling. As argued in Section 4.3, EC materials research should move from discriminative prediction toward generative inverse design.
Future high-throughput discovery will not rely only on high-accuracy surrogates that map composition to physical quantities such as Curie temperature and polarization. It will also require generative architectures, including diffusion models and equivariant graph neural networks, that can cross the boundary of known crystallographic databases. Under constraints such as maximizing fluctuating local polarization while minimizing phase-transition barrier, these models could create new candidate polar topologies, including dipolar-glass-like networks or tailored polymer-chain architectures. When combined with active-learning loops based on Bayesian uncertainty quantification and with automated synthesis, characterization, and modeling, such self-driving materials laboratories could redefine the logic of EC materials discovery and allow atomically precise design of broad-temperature, high-capacity refrigeration media.
5.4 Strain antagonism under electrical, thermal, and mechanical coupling
EC materials in real devices are not subjected to a single uniform electric field, but to a coupled mechanical–electrical–thermal environment. Polarization reversal is always accompanied by anisotropic macroscopic and local strain because of electrostriction and inverse piezoelectricity. In selected self-oscillating topologies, this electromechanical volume change can cooperate constructively with the entropy change. In more common stacked all-solid-state architectures and in constrained thin films, however, the coupling often becomes antagonistic on thermodynamic grounds.
When EC materials are clamped by rigid electrodes or substrates, the local elastic stress field generated by electrostriction changes the potential-energy boundary for polarization reversal directly. Residual stress can suppress the large configurational entropy associated with phase transitions and reduce useful cooling output. Repeated accumulation of such stress can also initiate and propagate microcracks, thereby determining the mechanical fatigue life of the device. A central theoretical challenge is therefore to move beyond the idealized stress-free assumption, quantify how local stress fields alter phase-transition barriers, polarization hysteresis, and thermodynamic cycling paths, and identify strain-engineering routes, such as epitaxial mismatch and interfacial lattice relaxation, that can reconfigure electro-mechanical coupling in a favorable way.
5.5 Miniaturized manufacturing and interfacial thermal resistance engineering
At the final level of system implementation, most EC prototypes remain limited to single-layer thick films, millimeter-scale bulks, or simple arrays. To enter targeted active cooling in microelectronic chips and ultracompact wearable temperature control, the refrigeration medium is likely to require further miniaturization and dense integration in MLCC-type architectures.
This geometric downscaling introduces physical and engineering difficulties. Submicrometer dielectric layers can generate ultrahigh electric fields under very low voltage and thereby reduce system power consumption, but at the same time they impose near-limit requirements on manufacturing. The field still lacks a mature commercial processing loop capable of combining high-density submicrometer tape casting with ultrathin co-firing of base-metal electrodes and ceramics while suppressing cross-interface elemental diffusion and interfacial oxygen-vacancy defects precisely. Even more critically, high-density stacking introduces a large number of electrode-ceramic heterogeneous interfaces. Phonon scattering at these interfaces can amplify the thermal boundary resistance. Resolving interlayer thermal resistance and achieving high-throughput directional heat transfer in ultra-miniaturized architectures therefore represents one of the most important deep-water challenges on the path from laboratory prototypes to manufacturable integrated all-solid-state refrigerators.
Overall, EC refrigeration is moving from isolated materials optimization toward coupled design across materials, devices, and operating scenarios. Progress will depend on criteria that connect polarization-barrier control, thermal-impedance management, charge recovery, manufacturing limits, and interfacial stability. Under this systems-level constraint, EC materials can be evaluated not only by intrinsic entropy change, but by their ability to deliver useful ΔTspan, SCP, and COP in deployable solid-state cooling architectures.
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