A review on different theoretical models of electrocaloric effect for refrigeration

Abstract

The performance parameters for characterizing the electrocaloric effect are isothermal entropy change and the adiabatic temperature change, respectively. This paper reviews the electrocaloric effect of ferroelectric materials based on different theoretical models. First, it provides four different calculation scales (the first-principle-based effective Hamiltonian, the Landau-Devonshire thermodynamic theory, phase-field simulation, and finite element analysis) to explain the basic theory of calculating the electrocaloric effect. Then, it comprehensively reviews the recent progress of these methods in regulating the electrocaloric effect and the generation mechanism of the electrocaloric effect. Finally, it summarizes and anticipates the exploration of more novel electrocaloric materials based on the framework constructed by the different computational methods.

Graphical abstract

Keywords

electrocaloric effect / effective Hamiltonian / phase-field modeling / different theoretical models

1 Introduction

Ferroelectric is a specific material that exhibits a spontaneous asymmetric crystal structure with permanent electric dipole moments [1]. Compared with the state when no electric field is applied, the polarization of ferroelectric materials changes under the action of the external electric field, thereby causing entropy change and temperature change, and the thermal effect caused by the change between the ordered and disordered states of the dielectric material during the application and removal of the electric field is called the electrocaloric effect (ECE). Due to its lightweight, high Carnot cycle efficiency, and environmental friendliness, electrocaloric materials (EMs) are superior to the traditional vapor compression refrigeration technology. They are expected to be widely used in integrated circuit thermal management [2,3].

Since the change of entropy and temperature of EM with the change of electric field, the focus of designing device cooling using ECE cycles is that EM can be depolarized under adiabatic conditions. In this way, EM will undergo two thermodynamic processes under the excitation and release of the electric field, i.e., the isothermal and adiabatic processes which consist of four steps (two isothermal entropy steps and two adiabatic temperature steps, as shown in Fig.1) that form an ideal refrigeration cycle [4,5]. The sub-image (i) is the initial state of Fig.1(a). The EM is under ambient conditions, no electric field is applied, and the system is disordered. When an electric field is subsequently used, the EM exhibits polarization and becomes an ordered state. At this time, the entropy in (ii) decreases. Because this process can be considered as an adiabatic process, the temperature of the system will increase. Then, the heat in (iii) is transferred to the environment through the heat sink, and the temperature of the system will decrease while the EM still maintains the polarization state under the electric field. After (iv), when the load device is connected to the system, the electric field is immediately removed, and the EM changes from an ordered state to a disordered state to reach (i), which will absorb the heat on the load, thereby achieving the goal of cooling. Repeating the whole process, the temperature of the load will continue to reduce [2]. This is the basic principle of the electrocaloric refrigerator. Thus, the change of isothermal entropy change (∆S) and adiabatic temperature change (∆T) are key parameters to measure the ECE of the EM for thermal cycling [6].

The ECE concept is generally thought to have been proposed by Granicher [7] or Hegenbarth [8] and was known for Kobeko and Kurtschatov’s experimental measurements of Rochelle salt in 1930 [9] but failed to publish their experimental data. Only much later, in 1963 [10], two American scientists repeated their experiments and measured ∆T to be a few mK under ambient conditions. Due to the limitation of the properties of polar materials such as ferroelectrics, the ∆T obtained by the study of the ECE is all less than 1 K in the next few decades [2]. This is mainly due to lower breakdown electric field of the bulk materials and the limited selection of materials [11]. The turning point came when two papers were published in the early 2000s, reporting giant electrocaloric responses in inorganic lead zirconate titanate thin films with a ∆T of 12 K [12] and organic materials in polyvinylidene fluoride (PVDF)-based polymers over 12 K [13], respectively, reactivating the research enthusiasm in ECE. Subsequently, numerous publications have been published on experimental EM, covering ferroelectric single crystals [14], ceramics [15], thin films [16], polymers [6], and ferroelectric inorganic-organic composites [17,18].

EM has not only made a series of advances in experimentation, theoretical calculation and modeling, but as a new research paradigm also plays a crucial role in the study of the large ECE of EM. Therefore, it is necessary to explore the physical mechanism of EM by using various computational scales and simulation methods to improve ECE performance and design ECE devices, including thermodynamic analysis [19–21], Monte Carlo atomistic simulations [22, 23], the first-principle-based effective Hamiltonian method [24–27], phenomenological approach [28–30], and finite element analysis [31−33]. Although the research and progress have been reviewed on EM in several articles, yet, most of the works have discussed it from an experimental point of view [2,10,34–40], and a review of EM-based on different theoretical models is still lacking [41–43].

This review first presents the research history of EM. It focuses on recent developments and applications from the first-principle-based effective Hamiltonian method to thermodynamic analysis to the phase-field method and finite element analysis in EM. Starting from the scale of materials, it introduces in turn the basic concept of the effective Hamiltonian method, the thermodynamic theory, phase-field simulation and finite element analysis. Then, it gives the basic equation and derivation for calculating ΔT using the indirect method. It analyzes and discusses the application of different methods in more detail, based on the methods for calculating ECE introduced in the previous section. Next, it discusses the application of the first principles effective Hamiltonian method by calculating two classical ferroelectric materials. After that, it gives a detailed summary of the ECE behavior of antiferroelectrics, and calculates the ECE under the electrical and mechanical boundary conditions by using the direct method. In thermodynamic and phase-field method applications, it enumerates the factors that influence and control ECE, such as the domain engineering, strain and stress effect, composition, the free energy barrier of different EM, and ECE under ultrafast electric fields. Numerical modeling-based finite element analysis can simulate the ECE of EM at a larger scale, and it reviews both fluid-based and solid-based EM coolers. Finally, based on the introduction of the calculation method of the ECE in the preceding sections, it summarizes these four methods. It is expected to develop and construct a set of different theoretical models on ECE calculation, which will provide a theoretical support for further exploration of high-performance EM. Fig.2 is the flow framework of this paper.

2 Theoretical methods

According to the literature, this section summarizes four methods for calculating the ECE at different scales: the effective Hamiltonian microscopic perspective theory based on the first-principle, the larger scale theory derived from Landau-Devonshire thermodynamics, phase-field simulation at mesoscopic scales, and finite element analysis in macroscopic aspects.

2.1 Effective Hamiltonian method

The significant advantage of first-principles calculations is that they can obtain a high accuracy and an atomic resolution of materials without any experimental data [44,45]. However, the downside is that their computations are expensive in time. So far, the first-principles calculations have mostly been able to simulate structures of hundreds of atoms, much smaller than the scales used in experiments. Another disadvantage is that first-principles calculations of materials properties are mostly limited to 0 K due to the adiabatic approximation. Temperature and electric field are two characteristic variables of ECE. Therefore, ECE requires ferroelectrics to exhibit different values of polarization magnitude at different temperatures and electric field strengths. Moreover, ECE can be calculated directly based on the first-principles simulation methods. Many theoretical models based on the first-principles have been developed to extend the accuracy and effective property parameters of first-principles calculations to finite temperature and larger atomic systems in ferroelectrics. There are three essential models, of which the effective Hamiltonian model [46,47] is the most widely used, while the other two shell model [48–50] and the bond-valence model [51, 52] are relatively less applied. Benefiting from the effective Hamiltonian model that extraordinarily successfully reproduced the ferroelectric phase transitions and phase diagrams, it has been widely used in many perovskites, such as BaTiO₃ (BTO), PbZrO₃ (PZO), BaSrTiO₃ (BST), PbZrTiO₃ (PZT), and BiFeO₃ (BFO) [53–61]. The cross-validation between simulations and experiments proves that this model approach is effective and promising [62,63]. Therefore, this paper will first introduce the effective Hamiltonian method based on the first-principle and explain the construction of the energy Hamiltonian operator for EM.

Zhong et al. pioneered the application of the effective Hamiltonian method to ferroelectrics materials [46,47], which only considered two degrees of freedom that are significantly affected by the ferroelectric properties, i.e., the local mode and the lattice distortion mode per unit cell. This model expands on the types of soft modes (local dipole and local strain) of cubic perovskites during the ferroelectric phase transition [46,56]. Later, to better present the rotation-related problems of oxygen octahedral in perovskite, a degree-of-freedom energy term on antiferrodistortive (AFD) was added to the original model [57]. Therefore, an effective Hamiltonian operator with three degrees of freedom is constructed to calculate the ECE properties, which are divided into the following two parts:

(1)E^{\mathrm{t}\mathrm{o}\mathrm{t}}= E^{\mathrm{F}\mathrm{E}}\left(\left\{\mathit{u}\right\},\left\{\text{η}\right\}\right)+E^{\mathrm{A}\mathrm{F}\mathrm{D}}\left(\left\{\mathit{u}\right\},\left\{\text{η}\right\},\left\{\text{ω}\right\}\right),

where u, η, and ω are the three independent degrees of freedom introduced by the effective Hamiltonian method, which are the local dipole, local strain, and AFD mode. The E^FE energy term contains the local dipole, the local strain, and its coupling portion, while the E^AFD term is concerned with the AFD mode energy and its interaction with local dipoles and strains.

Like the effective Hamiltonian proposed by Zhong et al. [46], the first term in Eq. (1) contains five terms, and takes the form as follows,

(2)\begin{array}{rl}{E}^{\mathrm{F}\mathrm{E}}\left(\left\{\mathit{u}\right\},\left\{\text{η}\right\}\right)=& E^{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}}\left(\left\{\mathit{u}\right\}\right)+E^{\mathrm{d}\mathrm{p}\mathrm{l}}\left(\left\{\mathit{u}\right\}\right)+E^{\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{t}}\left(\left\{\mathit{u}\right\}\right)\\ & +E^{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}}\left(\left\{\text{η}\right\}\right)+E^{\mathrm{i}\mathrm{n}\mathrm{t}}\left(\left\{\mathit{u}\right\},\left\{\text{η}\right\}\right).\end{array}

The contributions of these five energies come from the self-energy of the local mode, the interaction energy between long-range dipoles, the short-range coupling energy between adjacent soft modes, the local elastic energy, and the interaction between the local soft modes and local strains. The five constituent energy terms are described in detail in Eq. (3).

E^{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}}={\sum }_{i}E \left({\mathit{u}}_{\mathit{i}}\right)={\sum }_i\left[{\kappa }_2{\mathit{u}}_{\mathit{i}}^2+\alpha {\mathit{u}}_{\mathit{i}}^4+\gamma \left({\mathit{u}}_{\mathit{i}\mathit{x}}^2{\mathit{u}}_{\mathit{i}\mathit{y}}^2+{\mathit{u}}_{\mathit{i}\mathit{y}}^2{\mathit{u}}_{\mathit{i}\mathit{z}}^2+{\mathit{u}}_{\mathit{i}\mathit{z}}^2{\mathit{u}}_{\mathit{i}\mathit{x}}^2\right)\right],

E^{\mathrm{d}\mathrm{p}\mathrm{l}}\left(\left\{\mathit{u}\right\}\right)=\frac{Z^{\ast 2}}{{\epsilon }_{\mathrm{\infty }}}{\sum }_{i\ne j}\frac{{\mathit{u}}_{\mathit{i}}\cdot {\mathit{u}}_{\mathit{j}}-3\left({\hat{R}}_{ij}\cdot {\mathit{u}}_{\mathit{i}}\right)\left({\hat{R}}_{ij}\cdot {\mathit{u}}_{\mathit{j}}\right)}{R_{ij}^3},

E^{\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{t}}\left(\left\{\mathit{u}\right\}\right)=\frac{1}{2}{\sum }_{i\ne j}{\sum }_{\alpha \beta }{J}_{ij,\alpha \beta }{\mathit{u}}_{\mathit{i}\text{α}}{\mathit{u}}_{\mathit{j}\text{β}},

\begin{array}{rl}{E}^{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}}\left(\left\{\text{η}\right\}\right)=& \frac{N}{2}{B}_{11}\left({\text{η}}_1^2+{\text{η}}_2^2+{\text{η}}_3^2\right)+{NB}_{12}\left({\text{η}}_1{\text{η}}_2+{\text{η}}_2{\text{η}}_3+{\text{η}}_1{\text{η}}_3\right)\\ & +\frac{N}{2}{B}_{44}\left({\text{η}}_4^2+{\text{η}}_5^2+{\text{η}}_6^2\right),\end{array}

(3)E^{\mathrm{i}\mathrm{n}\mathrm{t}}\left(\left\{\mathit{u}\right\},\left\{\text{η}\right\}\right)= \frac{1}{2}{\sum }_i{\sum }_{l\alpha \beta }{B}_{l\alpha \beta }{\eta }_{i,l}{\mathit{u}}_{\mathit{i},\text{α}}{\mathit{u}}_{\mathit{i},\text{β}}.

The energy term E^self describes the double-well potential generated by the phase transition of ferroelectrics. Parameters κ₂, α, and γ are determined by the first-principles calculations. Z*, ε_∞, and {\hat{R}}_{ij} are the Born effective charge, the relative optical permittivity, and the distance between two local modes, which are also derived from the first-principles calculations. The J_ij,αβ describes the interaction parameters with two neighboring local modes, including a total of seven types. Their physical meanings are represented by the sketch diagram shown in Fig.3. N is the number of unit cells simulated in the term E^elas. The interaction between the soft modes is represented by the sum of the products of the coupling term B_lαβ, determined from the first-principles calculations. The Greek letters α and β represent different Cartesian components of the soft modes. The symbols i and j indicate that the dipoles between two-unit cells can traverse the entire simulation system.

The second term in Eq. (1) is the energy associated with the AFD modes [57,64,65], which can be written as

(4)\begin{array}{rl}{E}^{\mathrm{A}\mathrm{F}\mathrm{D}}& \left(\left\{{\mathit{u}}_{\mathit{i}}\right\},\left\{{\text{η}}_{\mathit{l}}\right\},\left\{{\text{ω}}_{\mathit{i}}\right\}\right)={\sum }_i[{\kappa }_{\mathrm{A}}{\text{ω}}_{\mathit{i}}^2+{\alpha }_{\mathrm{A}}{\text{ω}}_{\mathit{i}}^4+{\gamma }_{\mathrm{A}}({\text{ω}}_{\mathit{i}\mathit{x}}^2{\text{ω}}_{\mathit{i}\mathit{x}}^2\\ & +{\text{ω}}_{\mathit{i}\mathit{x}}^2{\text{ω}}_{\mathit{i}\mathit{z}}^2+{\text{ω}}_{\mathit{i}\mathit{y}}^2{\text{ω}}_{\mathit{i}\mathit{z}}^2)]+{\sum }_{ij}{\sum }_{\alpha \beta }{K}_{ij\alpha \beta }{\text{ω}}_{\mathit{i}\text{α}}{\text{ω}}_{\mathit{i}\text{β}}\\ & +{\sum }_i{\sum }_{\alpha }{K}^{\prime }{\text{ω}}_{\mathit{i},\text{α}}^3\left({\text{ω}}_{\mathit{i}+\text{α},\text{α}}+{\text{ω}}_{\mathit{i}-\text{α},\text{α}}\right)\\ & +{\sum }_i{\sum }_{\alpha \beta }{C}_{l\alpha \beta }{\text{η}}_{\mathit{l}}\left(i\right){\text{ω}}_{\mathit{i}\text{α}}{\text{ω}}_{\mathit{i}\text{β}}\\ & +{\sum }_{ij}{\sum }_{\alpha \beta \gamma \delta }{D}_{ij,\alpha \beta }{\mathit{u}}_{\mathit{j},\text{α}}{\text{ω}}_{\mathit{i},\text{α}}{\text{ω}}_{\mathit{j},\text{β}}\\ & +{\sum }_{ij}{\sum }_{\alpha \beta \gamma \delta }{E}_{\alpha \beta \gamma \delta }{\mathit{u}}_{\mathit{i}\text{α}}{\mathit{u}}_{\mathit{j}\text{β}}{\text{ω}}_{\mathit{j}\text{δ}},\end{array}

where {ω_i} represents the vector of octahedral motion magnitude and direction of BO₆ in unit cell i. The symbols i and j denote the summation of the unit cell on the B site occupancy and the six nearest unit cells in all the B occupants, respectively. The symbol {\eta }_l(i) is expressed in Voigt form as the lth component of the strain on the ith unit cell. The three Greek letters α, β, and γ, indicate the component size of the unit cell in the direction of x, y, and z, respectively. δ is the kronecker symbol. The symmetry requirement can simplify the system so that the parameters κ_A, α_A, and γ_A, as well as the matrices K_ijαβ, C_lαβ, and D_ij,αβ, can be determined by the local density approximation in the first-principles simulations, where the subscript A represents the AFD mode [44-66]. The fifth and sixth terms of Eq. (4) include the change in energy due to the degree of freedom of AFD motions and its coupling with the local modes [65]. Since the ECE characterizes the polarization response of the ferroelectrics under the electric field, the influence of the electric field on ECE cannot be ignored [43]. Therefore, the electric field term should be added to Eq. (2), as shown in Eqs. (5) and (6).

(5)E^{\mathrm{t}\mathrm{o}\mathrm{t}}= E^{\mathrm{F}\mathrm{E}}\left(\left\{\mathit{u}\right\},\left\{\text{η}\right\}\right)+E^{\mathrm{A}\mathrm{F}\mathrm{D}}\left(\left\{\mathit{u}\right\},\left\{\text{η}\right\},\left\{\text{ω}\right\}\right)+E^{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}\left(\left\{\mathit{u}\right\},E^{\prime }\right) ,

(6)E^{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}\left(\left\{\mathit{u}\right\},E^{\prime }\right) =-Z^{\ast }{\sum }_i{E}^{\prime }\cdot \mathit{u},

where E^elec is related to the external field E^Ꞌ.

The effective Hamiltonian was first demonstrated in the BTO and PbTiO₃ (PTO) of traditional ABO₃ perovskite structures. However, this method is not suitable for perovskite structures in solid solution. Subsequently, Walizer [54] improved Eq. (5) so that the method could be adapted to a solid solution model, and the total system energy at this point will consist of the following two terms:

(7)E^{\mathrm{t}\mathrm{o}\mathrm{t}}= E^{\mathrm{a}\mathrm{v}\mathrm{e}}({\mathit{u}}_{\mathit{i}},\text{η})+E^{\mathrm{l}\mathrm{o}\mathrm{c}}\left(\left\{{\mathit{u}}_{\mathit{i}}\right\},\text{η},\left\{{\sigma }_j\right\},\left\{{\eta }_{\mathrm{l}\mathrm{o}\mathrm{c}}\right\}\right),

where E^{\mathrm{a}\mathrm{v}\mathrm{e}} describes the energy for a hypothetical solid solution ABO₃ system generated by the virtual crystal approximation (VCA) method to replace the real solid solution ABO₃ [54,66,67], which has the same expression as Eq. (4). E^{\mathrm{l}\mathrm{o}\mathrm{c}} represents the error energy term between the true solid solution and the expression of the solid solution using the VCA method, where the variable {σ_j} characterizes the atomic type at the jth site, for instance, σ_j = +1 (or = −1) indicates the existence of Sr (or Ba) atom for (Ba_1−xSr_x)TiO₃ system [54]. The variable {η_loc} represents the local strain caused by the different radius of Ba and Sr atoms. Thus the latter term η_loc will be represented in the form below at the ith unit cell on B site,

{\eta }_{\mathrm{l}\mathrm{o}\mathrm{c},1}\left(i\right)={\eta }_{\mathrm{l}\mathrm{o}\mathrm{c},2}\left(i\right)={\eta }_{\mathrm{l}\mathrm{o}\mathrm{c},3}\left(i\right)=\frac{\mathrm{\Delta }a}{8a}{\sum }_j{\sigma }_j,

(8){\eta }_{\mathrm{l}\mathrm{o}\mathrm{c},4}\left(i\right)={\eta }_{\mathrm{l}\mathrm{o}\mathrm{c},5}\left(i\right)={\eta }_{\mathrm{l}\mathrm{o}\mathrm{c},6}\left(i\right)=0,

where the difference in lattice parameter between the BTO and the virtual VCA-(Ba_0.5Sr_0.5)TiO₃ at 0 K is expressed by Δa. Thus, the second term to the right of Eq. (7) above, E^loc, can be written as

(9)\begin{array}{rl}{E}^{\mathrm{l}\mathrm{o}\mathrm{c}}\left(\left\{{\mathit{u}}_{\mathit{i}}\right\},\text{η},\left\{{\sigma }_j\right\},\left\{{\eta }_{\mathrm{l}\mathrm{o}\mathrm{c}}\right\}\right)=& {\sum }_{ij}\left[Q_{j,i}{\sigma }_j{\mathit{e}}_{\mathit{j}\mathit{i}}\cdot {\mathit{u}}_{\mathit{j}}+R_{j,i}{\sigma }_j{\mathit{f}}_{\mathit{j}\mathit{i}}\cdot {\mathit{v}}_{\mathit{i}}\right]\\ & +\frac{1}{2}{\sum }_i{\sum }_{l,\alpha ,\beta }{B}_{l\alpha \beta }{\eta }_{\mathrm{l}\mathrm{o}\mathrm{c},l}\left(i\right){\mathit{u}}_{\mathit{i},\text{α}}{\mathit{u}}_{\mathit{i},\text{β}},\end{array}

where i and j represent the ith unit cell and the mixed sublattice j site, respectively. Q_j,i is a quantitative parameter for local mode disturbances after materials alloying, while the variable R_j,i represents the parameter of the material affected by the inhomogeneous strain. The vectors e_ji and f_ji are both unit vectors, representing respectively the soft mode direction from point j at A site to the center at B site and the direction from point j to the initial displacement v_i, the latter term indicates an inhomogeneous strain. All of the above parameters can be calculated by the first-principles.

Using the above methods, an effective Hamiltonian model of the system can be constructed for a given ferroelectrics, by calculating the temperature dependence of electric polarization at a constant field, and then the system can be extended to a finite temperature range to calculate the properties of ECE and other aspects. The Monte Carlo (MC) method is based on the theory of thermodynamics and statistical mechanics, calculating the free energy of the system by numerical method sampling and statistical probability. This shows that the calculation results only apply to the system in equilibrium and cannot provide a dynamic explanation. However, the molecular dynamics (MD) method uses the numerical integral summation of Newton’s second law to calculate the thermodynamic parameters of the particle systems, while the degrees of freedom introduced in the effective Hamiltonian model will change with the evolution of time [68]. All roads lead to Rome. Therefore, the calculated results are independent of using different calculation methods for using the same Hamiltonian model and parameters. For example, Mani et al. [69] used the MD simulation to calculate the phase transition temperature for PbTiO₃ near 625 K [70], while the MC method gave a similar result of 630 K. When calculating the finite temperature properties of ferroelectric materials using the MC or MD method, both approaches can realize that the polarization of ferroelectrics changes with temperature, which forms the feasibility basis for achieving indirect calculation of ECE.

2.2 Landau-Devonshire theory

Landau [71] proposed the concept of order parameters based on thermodynamics [72] to provide a unified description of continuous phase transition in 1938, which is characterized by the change in the degree of order of material and the accompanying change in the symmetry of material. Usually, the phase below the critical temperature has low symmetry, high order, and nonzero order parameters, while the phase above the critical point has higher symmetry, lower order, and zero order parameters. In Landau theory, it is through the mean-field approximation that the average of the order parameters is converted to the smallest variational of the free energy of Landau, which breaks the ergodicity of the state space, leads to the symmetry breaking of the statistical system, and obtains the nonzero order, that is, the phase transition. Subsequently, based on the Landau’s phase theory, Devonshire [71] and Ginzburg included the effects of strain and polarization inhomogeneity on the phase transition of the structure, respectively, and constructed the Landau-Devonshire function and Ginzburg-Landau function successively.

Devonshire first applied Landau’s phenomenological theory to BTO phase transitions [71]. The Landau’s phenomenological theory was subsequently used to investigate various ferroelectric properties [73–76]. The Gibbs free energy of a ferroelectric is expressed by Taylor expansion in terms of a power series of polarization P,

(10)G=G_0+\frac{1}{2}{a}_0\left(T-T_0\right)P^2+\frac{1}{4} b{P}^4+\frac{1}{6} c{P}^6,

where T, T₀, and E are the operating temperatures, the Curie-Weiss temperature of the system, and the externally applied electric field, respectively. The parameters a₀, b, and c are the coefficients in the Landau’s phase transition theory, and c is usually taken as positive or zero.

According to the thermodynamic theory, ferroelectric phase transitions can be divided into first- and second-order phase transitions. The specific heat capacity of the first-order ferroelectric phase transition will change abruptly at the phase transition point, accompanied by latent heat generation, and the spontaneous polarization intensity change is sharp. During the second-order phase transition process, only the specific heat capacity changes abruptly with temperature; there is no latent heat. The spontaneous polarization intensity varies continuously with temperature. It is obviously observed form Fig.4 and Eq. (10) that G is a function of polarization and temperature, where the symbol of the letter b is crucial. When b is negative, it corresponds to the first-order phase transition, and when b is positive, it corresponds to the second-order phase transition. The Landau-Devonshire coefficients are a fit to certain experimental data.

Since ΔS and ΔT are two important parameters characterizing ECE, to investigate ECE behavior in ferroelectrics, the calculation formulas of ΔS respectively are derived from Fig.5 and Eq. (10).

(11)\mathrm{\Delta }S= -{\left(\frac{\mathrm{\partial }G}{\mathrm{\partial }T}\right)}_P \mathrm{o}\mathrm{r}\mathrm{\Delta }S=-\frac{1}{2}{a}_0{P}^2,

and also ΔT can be obtained,

(12)\mathrm{\Delta }T= \frac{T}{2C_E}{a}_0{P}^2,

where ΔT = −T∙ΔS/C_E, C_E is the heat capacity.

In addition, ECE can also be calculated based on the Maxwell relations as follows,

(13){\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }E}\right)}_T={\left(\frac{\mathrm{\partial }P}{\mathrm{\partial }T}\right)}_E{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }E}\right)}_S=-\frac{T}{\rho C_E}{\left(\frac{\mathrm{\partial }P}{\mathrm{\partial }T}\right)}_E,

where ρ is the density of EM.

The ΔS and ΔT expressions of the EM can be obtained from Eq. (13),

(14)\mathrm{\Delta }S= {\int }_{E_1}^{E_2}{\left(\frac{\mathrm{\partial }P}{\mathrm{\partial }T}\right)}_E\mathrm{d}E, \mathrm{\Delta } T=-{\int }_{E_1}^{E_2}{\frac{T}{\rho C_E}\left(\frac{\mathrm{\partial }P}{\mathrm{\partial }T}\right)}_E\mathrm{d}E.

The relationship derived above to solve ECE characteristics is a well-known indirect method for measuring ECE, which has provided an essential theoretical basis for experimental and computational measurements [12,76–84]. It can be seen from the expressions characterizing ECE that EM requires a large polarization change rate and a considerable electric field breakdown resistance to obtain large ΔT and ΔS values.

2.3 Phase-field model

2.3.1 Introduction to EM phase-field method and construction of phase-field equation

The phase-field method is a differential equation developed based on thermodynamic theory to describe the evolution of mesoscopic scale over time, which has unique advantages in studying the phase transition of materials and simulating the evolution of material microstructures and is widely used to simulate random microstructures and complex microstructures in materials [85–87]. The Ginzburg-Landau equation that governs the evolution of domain configurations is derived from a simple, physically provable hypothesis to obtain the instantaneous microscopic morphology of the study system in time and space. Generally, the polarization intensity is taken as the order parameter for ferroelectric materials, and the Ginzburg-Landau equation with time is solved according to the principle of energy minimization, to obtain ferroelectric domain structure evolution with time [88].

(15)\frac{\mathrm{\partial }{\mathit{P}}_{\mathit{i}}\left(\mathit{r},t\right)}{\mathrm{\partial }t}=-L\frac{\mathrm{\partial }F}{\mathrm{\partial }{\mathit{P}}_{\mathit{i}}\left(\mathit{r},t\right)},i=1,2,3,

where {\mathit{P}}_{\mathit{i}}\left(\mathit{r},t\right) represents the magnitude of polarization of the ith component of polarization at time t and space r. L is the kinetic coefficient related to the domain wall motion, and F is the total free energy of the system, which is mainly composed of the following four parts, where f_Land, f_elas, f_elec, and f_grad are the corresponding densities of the energy terms of each part.

(16)F=F_{\mathrm{L}\mathrm{a}\mathrm{n}\mathrm{d}}+ F_{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}}+F_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}+F_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}}=\left(f_{\mathrm{L}\mathrm{a}\mathrm{n}\mathrm{d}}+f_{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}}+f_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}+f_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}}\right)\mathrm{d} V,

f_{\mathrm{L}\mathrm{a}\mathrm{n}\mathrm{d}}={\alpha }_i{\mathit{P}}_{\mathit{i}}^2+{\alpha }_{ij} {\mathit{P}}_{\mathit{i}}^2{\mathit{P}}_{\mathit{j}}^2+{\alpha }_{ijk}{\mathit{P}}_{\mathit{i}}^2{\mathit{P}}_{\mathit{j}}^2{\mathit{P}}_{\mathit{k}}^2,

(17)f_{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}}=\frac{1}{2}{c}_{ijkl}{e}_{ij} e_{kl} =\frac{1}{2}{c}_{ijkl}\left({\epsilon }_{ij}-{\epsilon }_{ij}^0\right)\left({\epsilon }_{kl}-{\epsilon }_{kl}^0\right),

(18)f_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}=-{\mathit{P}}_{\mathit{i}}\left(\mathit{r}\right)E_i\left(\mathit{r}\right) -\frac{1}{2} {\epsilon }_0 {\kappa }_{ij}^b \left(\mathit{r}\right)E_i\left(\mathit{r}\right)E_j\left(\mathit{r}\right),

(19)f_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}}=\frac{1}{2}{g}_{ijkl}{\mathit{P}}_{\mathit{i},\mathit{j}}{\mathit{P}}_{\mathit{k},\mathit{l}} ,

where α_i, α_ij, and α_ijk are the Landau coefficients corresponding to the expansion of polarization P to the sixth order. c_ijkl, e_ij, ε_ij, and {\epsilon }_{ij}^0 represent the elastic stiffness tensor, elastic strain, total elastic strain, and eigenstrain, respectively [89]. Tensor {\kappa }_{ij}^b is related to the background dielectric constant, and g_ijkl is the gradient coefficient of the polarization distribution in space [86].

2.3.2 Deviation of free energy

Section 2.2 shows that calculating and measuring the ECE of EM indirectly using Maxwell relations derived based on thermodynamics is easier and faster than a direct measurement. However, the ECE values calculated using Eq. (14) are only applicable to systems undergoing continuous phase transitions and are invalidated for first-order phase transition and relaxor ferroelectrics. Furthermore, in some cases, the Maxwell relation is used to calculate the ECE of EM. The polarization increases with the increase of temperature when the electric field strength is high. Therefore, the ECE calculated is negative, which is contrary to the actual situation. The resulting ECE is negative only for numerical processing rather than the properties of the material [90,91].

Another indirect method is proposed to solve the partial derivative of Gibbs free energy with respect to temperature. The advantage of this method is that it can make the curve smoother without jumping, which is more reasonable.

(20)S=-\frac{1}{\rho }{\left(\frac{\mathrm{\partial }\mathrm{\Delta }G}{\mathrm{\partial }T}\right)}_E= -\frac{1}{\rho } \left(\frac{\mathrm{d}{\alpha }_1\left(T\right)}{\mathrm{d}T}{P}^2\right),

\begin{array}{rl}\mathrm{\Delta }S=& S\left(T,E=E^{\mathrm{a}\mathrm{p}\mathrm{p}}\right)-S\left(T,E=0\right)\\ \approx & -\frac{1}{\rho }\frac{\mathrm{d}{\alpha }_1\left(T\right)}{\mathrm{d}T}\left({P^2|}_{E=E^{\mathrm{a}\mathrm{p}\mathrm{p}}}-{P^2|}_{E=0}\right),\end{array}

(21)\begin{array}{rl}\mathrm{\Delta }T=& \frac{1}{c_p\rho }\frac{\mathrm{d}{\alpha }_1\left(T\right)}{\mathrm{d}T}\left({P^2|}_{E=E^{\mathrm{a}\mathrm{p}\mathrm{p}}}-{P^2|}_{E=0}\right)\\ =& \frac{T}{2{\epsilon }_{0}C{c}_p\rho }\left({P^2|}_{E=E^{\mathrm{a}\mathrm{p}\mathrm{p}}}-{P^2|}_{E=0}\right).\end{array}

However, the disadvantage is that in this case, when ΔS and ΔT are at their maximum, the electric field is added again, and their values will not change. Therefore, the ΔS and ΔT derived by this method cannot accurately reflect the real condition of temperature changes with the electric field.

In general, both electric field and temperature affect the specific heat capacity of a material. However, when solving by these two indirect methods, the specific heat capacity is set to a constant, which is bound to form an error with the actual situation.

2.4 Finite element method

To make use of the ECE, EM shall undergo thermodynamic cycles. A reverse Brayton cycle is formed through two adiabatic and two isothermal steps, as depicted in Fig.1(b), a simple schematic diagram of the cooler. As long as the application and removal of the electric field are controlled, the heat on the heat source is continuously transferred to the heat sink through the working medium, so that the target device at the cold source is cooled to achieve the purpose of refrigeration.

Heat needs a medium to transfer for the whole thermodynamic cycle on a prototype scale, and fluid is generally considered. Because the fluid flow can be regarded as incompressible, it is reasonable and reliable to use the Navier-Stokes equation to describe the flow process. The most available solutions are the finite element and finite difference methods. EM undergoes polarization and depolarization under the action of an electric field, resulting in an endothermic and exothermic phenomenon, to achieve cyclic refrigeration. The temperature can be determined by solving the partial differential heat transfer equations. For the constructed prototype, the energy equation of the heat transfer of the medium is [31,32,34]

(22){\rho }_E{C}_E\frac{\mathrm{\partial }{T}_E}{\mathrm{\partial }t}={\kappa }_E{\mathrm{\nabla }}^2{T}_E- {\dot{Q}}_E,

(23){\dot{Q}}_E={\rho }_E{T}_E{\left(\frac{\mathrm{\partial }{S}_E}{\mathrm{\partial }E}\right)}_{T_E}\left(\frac{\mathrm{\partial }E}{\mathrm{\partial }t}\right),

where ρ_E, C_E, κ_E, and T_E are the density, specific heat, thermal conductivity, and temperature of the EM, respectively. Besides, t denotes the time and, {\dot{Q}}_E refers to the internal heat source term caused by the ECE of EM. S_E and E in the latter formula represent the entropy of EM and the applied electric field.

In the numerical simulation [92], the internal heat source term is derived from the electric field-related heat energy. The dependence of entropy changes on the electric field is obtained from the experimental data [93]. The coefficient of performance (COP) is one of the key factors to explain the efficiency of heat pumps and refrigerators. The COP value refers to the ratio of the refrigerating effect to the work input. It can be written from the definition,

(24)\mathrm{C}\mathrm{O}\mathrm{P}=\frac{Q_c}{W_{\mathrm{i}\mathrm{n}}} ,

where Q_c represents the refrigerating effect and W_in indicates the power input.

3 Application of theoretical methods

3.1 Application of effective Hamiltonian method

The construction of the first-principles-based effective Hamiltonian method is only used to simulate the structural phase transition and domain structure evolution of the ferroelectrics in the early stages. With the continuous improvement of successors, it has been proven feasible and successful in exploring ferroelectric material ECE.

3.1.1 ECE on traditional ferroelectrics with effective Hamiltonian method

As is known, BTO and PTO, as the most typical ferroelectric materials, have the ABO₃ configuration, which has been widely and deeply discussed. Therefore, this section will summarize the two materials as the research object.

BTO is a widely investigated inorganic ferroelectric material. Under ambient conditions, BTO is a cubic structure. As the temperature decreases, it will undergo three phase transitions, i.e., from cubic phase to tetragonal phase to orthorhombic phase to rhombohedral phase, and the corresponding phase transition temperatures are 398, 281, and 202 K [94]. Because it has many phase transition temperatures, it provides more possibilities for BTO use in ECE. BTO is easily synthesized. Therefore, its ECE has been extensively studied experimentally [15,95,96] based on thermodynamic models [97,98]. Beckman et al. used MD to calculate the tetragonal to the cubic phase transition of BTO in 2012 [24]. As shown in Fig.6(a), the calculated ∆T is a good representation of the experimental results [99,100], which indicates that the MD model method applies to the ECE study. However, the electric field strength used then was enough to break down the material. The results obtained could only be used for qualitative analysis, not for quantitative analysis.

Another classic and widely studied inorganic ferroelectric material is PTO. Compared with BTO, it has no multiphase transition temperatures and maintains a tetragonal structure at a lower temperature until it transforms into a cubic structure at 740 K. Lisenkov et al. used the effective Hamiltonian method to study the ECE in the PTO through the MC method and obtained a ∆T of 1.3 K in Fig.6(b), which is consistent with the experimental values. The temperature change near 630 K is consistent with the testing phase transition position, but the simulated phase transition temperature is lower than the observed value [102,103]. With the continuous popularity of ECE research, more and more people know the effective Hamiltonian method. Bai et al. [95] also used this method to study the ECE behavior of PTO. Their research results are close to the experiment, and the phase transition temperature occurs near 670 K. At room temperature, the polarization will be reversed, reproducing other research results [104,105]. This makes the effective Hamiltonian method indispensable in simulating ferroelectric ECE.

3.1.2 ECE in antiferroelectrics with effective Hamiltonian method

Different from the conventional ferroelectrics, there are also ordered electric dipoles in the antiferroelectric materials from the micro perspective. Still, the polarization between adjacent dipoles is in reverse arrangement. Therefore, there is no net polarization in the macro view. For the phase-field method simulation [106,107] of antiferroelectric materials, a new order parameter will be introduced to distinguish the polarization reversal. The advantage of the effective Hamiltonian method is that each unit cell will define a soft mode in direction without introducing a new degree of freedom to describe the concept of polarization reversal. Because of the reverse polarization arrangement in antiferroelectricity, there will always be a negative ECE phenomenon when conducting ECE research. Lisenkov’s research group [108–110] studied the ECE of antiferroelectrics PZO using the effective Hamiltonian method. It can be seen from Fig.7(a) that when it is lower than T_c, the polarization is positively correlated with the temperature. At this time, (\mathrm{\partial }\mathit{P}/\mathrm{\partial }T)_E will lead to a negative value of ECE. PZO is no different from conventional ferroelectrics when it is higher than T_c.

Although negative ECE has no application value at present, the negative ECE phenomenon in the phase transition of antiferroelectrics will lead to a significant change in (\mathrm{\partial }\mathit{P}/\mathrm{\partial }T)_E and large ECE. This innovation has successfully made antiferroelectrics candidate materials for ECE research [111]. At present, an antiferroelectric phase transition occurs in NaNbO₃ [65,112], (Pb, La) (Zr, Sn, Ti)O₃ [113], and PZO thin films [114]. Using the antiferroelectric phase transition nearby, ECE will experience a massive jump from negative to positive values with temperature changes, resulting in an enormous ECE value. Kingsland et al. [110] proposed the idea of using antiferroelectric to realize the refrigeration cycle, with the ΔT up to 7.1 K, showing an excellent refrigeration potential.

3.1.3 Effective Hamiltonian direct calculation ECE method

Experimentally, since ECE is a function of temperature and electric field, it is usually only necessary to measure polarization as the curve relationship between temperature and electric field, then calculate ΔS and ΔT according to the Maxwell relation, which is an approximation and is not suitable for first-order phase transitions and relaxor ferroelectrics. A more accurate approach is to measure the ΔT of the ECE directly, which avoids some errors in the calculation of the data. Of course, direct method measurement [115] requires higher adiabatic conditions of the material, which is difficult to achieve in actual measurement [116]. There are challenges to direct measures experimentally, and adiabatic conditions are easy to accomplish in theoretical research. Half of the research results are based on effective Hamiltonian methods for directly calculating ECE. ECE is calculated based on the effective Hamiltonian amount, and there are generally two types: the MD and the MC methods. The MD method first sets the electric field to equilibrium the system at a constant temperature, then obtains the temperature change of the system. MC simulation requires introducing demon energy, which represents the thermal energy term of the system [23], and uses the micro-regular method to solve the total energy.

Both the direct and indirect methods can calculate the ECE of ferroelectric materials. For the differences between the two ways, Nishimatsu and Wang’s group studied the ECE behavior of BTO, which is visible in Fig.8(a), showing a significant difference in the results calculated by the two methods [115–118]. However, Marathe et al. [118] conducted both the direct and indirect methods and came up with the same result. In addition, the entropy change calculated by the direct method starts from zero, while the initial entropy concept exists in the starting state of the indirect method.

3.1.4 Effect of electric and mechanical boundary conditions on ECE

The advantage of ferroelectric films is that when the film thickness exceeds the transition thickness d_t, its breakdown strength changes from an intrinsic state independent of thickness to an external state dependent on thickness [119]. This is also the reason that most solid-state cooling devices have been constructed with ferroelectric thin films in recent years. When it comes to the application of thin films, the influence of the substrate on the films has to be considered, including the stress and strain caused by different lattice constants of the films and the substrate and the depolarization field from the surface charge.

After the external field is excited by the ferroelectric film, a depolarization field will be formed on its upper and lower surfaces, inhibiting the oscillation of spontaneous ferroelectric polarization [120]. The effect of the depolarization field on ECE is discussed, that is, the influence of electrical boundary conditions on ECE performance is considered. There are two kinds, short circuit boundary and open circuit boundary. When studying PZO thin films, it was found that under short circuit conditions, PZO would change from an antiferroelectric phase structure to a ferroelectric phase structure, while under open circuit conditions, it was still an antiferroelectric phase [121,122]. In addition, PZO could form vortex domains [123–126]. Research shows that electrical boundary conditions affect ECE because the depolarization field can reduce the peak temperature to improve the performance of ECE at room temperature. However, the larger the depolarization field is, the better it is. Excessive depolarization field, in turn, will inhibit the performance of ECE [127]. Therefore, it is necessary to balance the electrical boundary conditions to adjust the performance of ECE.

Because the thin film is grown on the substrate, the lattice constants of the substrate and the film structure are not coherent, which will lead to the stress/strain of the film structure, resulting in different mechanical boundary conditions. Other mechanical boundary conditions will affect the ECE of EM, which has been explored theoretically and experimentally [122,129,130]. However, few studies on different mechanical boundary conditions use the effective Hamiltonian method [131,128], of which the free-standing BST film is the most typical. It can be seen from Fig.9(a) that the polarization, polarization derivative, specific heat, and ECE coefficient with temperature under different mechanical conditions have been studied. The results show that both compressive and tensile strain will enhance ECE response, which is consistent with other theoretical calculation results [132]. Therefore, applied strain is an effective method to regulate ECE.

3.2 Application of thermodynamic theory and phase-field method to ECE

The regions with the same spontaneous polarization direction in ferroelectrics are called ferroelectric domains. The free energy in ferroelectrics is composed of several energy terms, which will act on ferroelectric domains, thus affecting the performance of ECE. From the above theoretical analysis, it can be seen that the magnitude of the polarization change rate of EM near phase transition will affect ECE and significantly change the ferroelectric domain structure. Domain engineering is to control the performance of adjusting EM by changing domain structure and its number, which has been proved to be an effective way to adjust ECE. The advantage of the phase-field method is that it can simulate the formation and evolution of domain structures in Tab.1. The transformation of domain structures will bring about changes in ECE. Therefore, this method occupies a solid position in the regulation of domain structures.

3.2.1 Domain engineering

It is difficult to break through the performance of ECE before the giant ECE phenomenon in ferroelectric thin films (TF) is found. Since then, researchers have had a great enthusiasm for experimentally and theoretically researching ferroelectric thin films in ECE [133]. The thermodynamic theoretical model study of ECE is based on the assumption of a single ferroelectric domain [97,132–135]. Although a thermodynamic model can calculate the ECE performance of a single domain, the actual situation will have defects, surface effects, crystal orientation, and stress/strain considerations, making the single domain state too ideal. The analysis of the phase-field method is classified in the mesoscopic scale category, which is larger than the thermodynamic model and is suitable for general multidomain research. In some ultrathin ferroelectric films, a single domain will appear, while in thicker films, multiple domain structures are dominant [28,135–140].

Therefore, it is mainstream and practical to focus on studying the ECE behavior in multidomain ferroelectrics. Different from the ECE generated by the conventional ferroelectric phase transition, the domain structure transition will bring additional ECE, making it possible to become a giant ECE [88,146,147]. The temperature-induced multidomain change to a single domain in ferroelectrics will produce either positive or negative ECE. It is worth noting that in a given electric field, the cracked PTO will transition from a multidomain state to a single-domain state with the temperature change. At the same time, the polarization will suddenly increase, resulting in a negative ECE. The research shows that the transition temperature of domain structure is lower than T_c, which extends the operating temperature range of ferroelectric materials [75,76,141,142].

In addition to a single domain and multidomain, there is another type of ferroelectric nanodomain structure: vortex domain. The change of domain will also produce an entropy change for ECE cooling. The phase-field simulation observed that ∆T reached 16.6 K (−6.34 K) in the BIT nanoparticles [148] (PTO nanoparticles [149]), producing a large positive/negative ECE, thanks to the change of the vortex domain structure in the curling electric field. The number and type of ferroelectric vortex domain will show different behavioral characteristics under the action of an external field. In circular electric field and equiaxial mismatch strain, PTO with a double vortex domain has a large ECE in a wide temperature range, which is attributed to the transition from a dual vortex domain to a single vortex domain or from a single vortex domain to another vortex domains [150]. In the phase-field simulation of PST film in Fig.10(e), it is found that there is a transition from the needle domain to the vortex domain, and with the increasing temperature, because there is a large gradient in PST film [143,144,151], the total energy at the root of needle domain is locally concentrated, driving the transition from needle domain to vortex domain, thus enhancing ECE.

The generation of multidomain will bring domain walls. Karthik & Martin [136] studied the contribution of different domains to ECE based on the Landau-Ginzburg-Devonshire thermodynamic model. Their research results show that the existence of domain walls will affect the response of ECE [145,152]. Theoretical simulation can only be qualitatively analyzed, which is weak for ferroelectric systems with complex boundary conditions and mixed domain structures. Li et al. [152] considered the polarization surface-induced inhibition of BTO, and the ∆T generated by its ECE increased with the increase of domain wall density. When domain wall contribution is not considered, ∆T will suddenly decrease.

When considering the contribution of domain walls to material properties, domain walls can be divided into neutral and charged domain walls (CDWs). For ordinary ferroelectric perovskite materials, in the process of energy minimization, if the carrier concentration is insufficient, the charged domain wall cannot exist [153]. Domain walls can be charged when there is enough charge to compensate, such as through a bending mechanism or external charge defects. In some studies, CDWs of head-to-head (HH) and tail-to-tail (TT) have been observed [154]. Huang’s research group studied the ECE of the PTO films with HH and TT90° by utilizing the time-dependent Ginzburg-Landau (TDGL) method [146]. The results show that the ∆T near the ferroelectric domain wall is larger than that far away from the ferroelectric domain wall. The ∆T of the CDW system is proportional to the number of CDW and inversely proportional to the density of compensation charges.

The above works mainly focus on studying ECE in the multidomain state of single-crystal ferroelectrics. The large heat capacity and thermal conductivity of ferroelectric ceramics meet the material requirements of refrigeration applications. There is a size effect in ferroelectric materials with first-order phase transition. Theoretically, a larger ECE can be obtained at lower temperatures by controlling the grain size of ferroelectric ceramic materials [155]. Hou et al. [155] used the phase-field model to study the ECE behavior of polycrystalline ferroelectrics. Compared with single-crystal ferroelectrics, the energy density of polycrystalline ferroelectrics is determined by the orientation of each crystal in the polycrystals. The P–E hysteresis loops at different temperatures drawn by indirect methods show that the ΔT calculated for coarse grains is more significant than that for fine grains. Fig.11 shows the domain morphology characteristics corresponding to the ECE of polycrystalline ferroelectrics. The large ECE can be attributed to the significant change rate of the polarization value of polycrystalline ferroelectrics with temperature. Because there exists the depolarization effect at the grain boundary of the dielectric material, leading to the formation of different ferroelectric domains, as the grain size becomes smaller, the depolarization effect increases [156–159].

3.2.2 Strain and stress effect

As mentioned above, the growth of the thin film depends on the substrate, and the lattice constant between the thin film and the substrate rarely appears as a coherent phenomenon, so the strain will inevitably be formed in the growth of the ferroelectric thin film. Strain/stress will affect the polarization along the electric field direction and the phase transition temperature, thus affecting ECE [146]. Among the parameters affecting ECE performance, strain engineering, including stress and strain, is a crucial regulatory variable [122]. As described in the previous section, under the action of tensile and compressive strain, the ferroelectric multidomain will be transformed into a single ferroelectric domain. In contrast, under compressive stress, the polarization direction of the transformed ferroelectric single domain is perpendicular to the electric field direction. Furthermore, tensile strain induces positive ECE, while compressive strain induces negative ECE [122].

Inspired by the concept that domain structure transition can induce a giant ECE [146], Hou et al. studied the ECE of PTO ferroelectric thin films at different adaptive strains using a phase-field model [147]. It can be seen from Fig.12 that the multidomain structure of the PTO film will switch to the single domain state with the increase of temperature, the shape of the P–E loop will change, and the curve of P versus T will suddenly rise, leading to the occurrence of a negative ECE. In addition, the domain structure transition temperature and the corresponding negative ECE can be controlled by the magnitude of the mismatch strain [28]. In addition to the traditional ABO₃-type ferroelectrics, Bi₄Ti₃O₁₂ in the Aurivillius phase also predicted positive and negative ECE through stress regulation [141].

The depolarization effect and strain engineering in ferroelectric nanostructures are conventional means to induce phase transition and regulate ECE. However, ferroelectrics with various nanoscale structures are not easy to synthesize and manufacture, significantly limiting their commercial use in solid-state refrigeration. Therefore, it is widely explored to find some novel ferroelectric nanostructures in theory, especially ferroelectric/paraelectric superlattice structures. The mismatch strain generated by their combination can induce many domains, including flux-closure domains, anti-vortexs, vortices, skyrmions, and polar mermon [160–163]. Ji et al. [144] formed a ferroelectric superlattice by inserting a paraelectric STO layer between the PTO layers, exploring the positive and negative coexistence of ECE and obtained that the number of vortex domains can adjust the ECE of the superlattice.

3.2.3 Composition

Sections 3.2.1 and 3.2.2 describe and summarize that adjusting ECE through domain engineering and strain engineering will not change the inherent T_c and spontaneous polarization of ferroelectrics. In contrast, it is also essential to regulate the ECE of ferroelectrics by changing the intrinsic properties of materials through chemical doping and a solid solution of ferroelectrics.

Landau parameters in the phase-field simulation of ferroelectric materials are based on experimental fitting. It is difficult to measure parameters without in situ XRD or synchrotron radiation equipment. The thermodynamic potential of BTO and STO is obtained by linear interpolation in the BST system of solid solution [164,165]. Still, the phase transition temperature calculated by linear interpolation does not agree with the experimental results. Wang et al. [166] referred to the work of Yu et al. [167] on the high-order coupling between pressure and polarization and replaced Ba²⁺ with Sr²⁺ with a smaller ionic radius. The lattice of the BTO structure of the matrix would collapse and shrink, which is equivalent to pre-compressing the BTO of the matrix, thus solving the problem of the landau potential energy of BST. Fig.13 shows the phase-field simulated ECE of BST of different components, and the calculated ΔT echoes the experimental measurements.

Generally speaking, if a single material cannot meet the requirements, two different materials will be considered to be combined to give full play to the advantages of the two materials. The same is true for ferroelectric composites. The composite of ceramic ferroelectric polymer nanomaterials can enable the composite materials to have both the extraordinary polarization of ferroelectric ceramics and the advantages of the high breakdown field strength of polymers, resulting in a large ECE. Because ferroelectric composites are composed of two phases, the variables that can adjust ECE include the structure type of matrix phase and reinforcement phase, their proportion, contact mode of the two-phase interface, etc. In recent years, ferroelectric polymers have been the majority of materials related to a large ECE. Zhang et al. [170] prepared composite ferroelectrics by using two kinds of ferroelectric materials with T_c at room temperature, ferroelectric polymer P(VDF-TrFE-CFE) as the matrix and nanometer Ba_xSr_1−xO₃ (x = 0.67) as the reinforcing phase. Because the matrix in the composite ferroelectric material has a high electric field breakdown field strength and the reinforcing phase has a large spontaneous polarization, combining the advantages of the two, the strength of ECE is far greater than that of single-phase ferroelectric materials. Qian et al. [171] compounded P(VDF-TrFE-CFE) with nano ferroelectric Ba(Zr_0.21Ti_0.79)O₃ (BZT) to obtain relaxor polymer nanocomposites. The results show that the phase transition temperature of this EM is under ambient conditions and there is a broadening phenomenon near room temperature [172,173]. In situ analysis and simulation of the interfacial thermodynamics and dynamics of the composite by piezoelectric response microscope and phase-field simulation showed that the local high electric field was induced around the interface formed by BZT nanofibers and polymers, and the enhancement of ECE performance was mainly due to the contribution of interfacial polarization. In addition, Liu et al. [174] also obtained the crucial role of the interface coupling effect in enhancing ECE when studying BTO/P(VDF-TrFE-CFE) ferroelectric polymer composites by combining experiments and calculations.

How to balance the parameters that affect ECE performance (such as ΔT, electric field strength, temperature application range and the ability to withstand high electric field strength), through appropriate composition and structure design, is one of the directions that researchers should pay attention to when looking for EM with an excellent ECE performance. Moreover, in terms of theoretical exploration, the parameters in the thermodynamic model and phase-field method are fitted with the behavior of materials under natural application conditions. Therefore, the accuracy of theoretical model construction is determined by the high-precision measurement of materials in time and space in experiments.

3.2.4 Free energy barrier of different materials

With the intensive study of ECE, EM has almost expanded to the entire material field, including classical inorganic perovskite ceramics, organic perovskite, organic polymers, molecular ferroelectrics, two-dimensional ferroelectric materials, etc. To further study the generation mechanism of ECE, based on the Landau-Ginzburg phase transition theory [175], Gao et al. selected several classical ferroelectric materials, including inorganic perovskite BTO, PTO, BFO [176,177], organic perovskite [MDABCO](NH₄)I₃ (MDABCO) [91], organic ferroelectric P(VDF-TrFE), molecular ferroelectrics ImClO₄, and 2D ferroelectric CuInP₂S₆ (CIPS). From the potential barrier diagrams of various materials near the phase transition temperature in Fig.14, it can be concluded that the energy barrier between the ferroelectric phase and the paraelectric phase will change with temperature, and the greater the rate of this change, the more ECE can be induced. In other words, as the polarization change induced by small temperature change increase, the ECE will increase.

3.2.5 ECE under ultrafast electric field

In an ideal cycle, the temperature decrease during depolarization when the electric field is released should be equivalent to the temperature increase caused by polarization when the electric field is applied. However, in practice, irreversible processes such as dielectric loss, friction, and entropy increase caused by using and removing electric fields make the actual refrigeration efficiency less than expected. Therefore, it is imperative to investigate the improvement of the temperature response of EM to the electric field to achieve higher cooling effectiveness [178–180].

As the name implies, the actuation duration of the ultrafast electric field is so fast, reaching nanosecond or even picosecond level. Compared with the constant electric field, the dynamic response mechanism of ferroelectrics in an ultrafast electric field is unknown. To understand the dynamics of materials in ultrafast electric fields, Akamatsu et al. developed a dynamical phase-field modeling method [181]. Different from the traditional time-dependent phase-field evolution equation, the polarization is further extended to the second derivative concerning time, which has the characteristic of intrinsic oscillation [182], which can be written as

(25)\mu \frac{{\delta }^2{\mathit{P}}_{\mathit{i}}\left(\mathit{r},t\right)}{\delta t^2}+\gamma \frac{\delta {\mathit{P}}_{\mathit{i}}\left(r,t\right)}{\delta t}+\frac{\delta F}{\delta {\mathit{P}}_{\mathit{i}}\left(\mathit{r},t\right)}=0,

where \mu and \gamma are kinetic coefficients related to domain wall motion in the above formula. At present, these values are only measured through experiments.

Unlike conventional constant electric fields, the electric field pulse positively correlates with the action time, as depicted in Fig.15(b). However, the ΔT induced by a constant electric field and electric field pulse share a precise physical mechanism: applying and removing electric field disorder/arranging dipoles in ferroelectrics and causing changes in the structure and temperature of materials. In Ref. [183], the double-well of the free energy curve was exploited from the ferroelectrics of the bulk BTO and solid solution BST to the paraelectric, employing the ultrafast electric field pulse (the frequency reaches the order of GHz) to realize the ECE based on the modified phase-field modeling. The results show that the ECE at the same strength on a constant electric field can be obtained by controlling the duration of the electric field pulse. In addition, with the increased electric field pulse duration, the domain flipping from a paraelectric state to a ferroelectric state is a gradual process rather than the instantaneous switching of domains in a constant electric field. Due to the brief period of the electric field, achieving an ultrafast cooling rate of 10⁸ K/s has proven to be more efficient. Furthermore, a plurality of electric field pulses stimulate ferroelectrics and realize cyclic ultrafast refrigeration. With the continuous improvement of GHz and even THz technologies, this research is expected to provide an essential theoretical support for ultrafast cooling.

3.3 Application of finite element analysis

When using finite element analysis to simulate the ECE, the scale of materials generally involved will be the prototype size in the experiment. The most common prototype of ECE uses a heat transfer fluid pump to deliver heat, most of which are based on the principle of active thermal regeneration. This section reviews numerical modeling on electrocaloric coolers, including both fluid and solid bases.

3.3.1 Fluid-based ECE models

Yao’s group [31] built a three-dimensional microscale active electrocaloric regenerator (AER) based on fluid heat transfer using COMSOL Multiphysics software in 2014. The structure diagram of AER is shown in Fig.16(a). Its working principle is that the diaphragm at the left and right ends of the AER is driven reversely in the electrostatic field, leading to the heat transfer movement of the fluid to achieve refrigeration. The cooling efficiency of AER is measured by setting the temperature of the cold side and hot side within the given external temperature range. Running the prototype at 15 K and 20 Hz, Guo et al. obtained a cooling power density of 3 W/cm² and a COP of 31% [31].

In 2021, Shi et al. [33] designed a rotating electrocaloric refrigeration model machine based on a fluid working medium by using the finite element software ANSYS, which proved that it had a large cooling capacity. Fig.16(b) shows the structural unit and model of the system, in which the EM is P(VDF-TrFE-CFE). The positive and negative heat pulses are used to conduct ECE simulation, and the entire model and environment is set as adiabatic conditions without heat exchange. When the equipment operates at a 10 K temperature span and 10 s cycle, a cooling power of 290 W is calculated. At this time, COP is 5.5. Subsequently, this team continued to use the rotary electrocaloric refrigeration device to study the influence of different heat transfer fluid media on refrigeration efficiency [184]. Their research results show that using water as the heat transfer medium, the cooling power and COP are the largest. Within the temperature range of 10 K, the cooling capacity and COP can reach 9.89 W/cm³ and 10.5, respectively.

Based on the above two limited analysis results on the ECE model, it can be concluded that, first, if the temperature span is kept constant, the cooling efficiency and COP will increase linearly with the size of the applied electric field. In other words, regardless of the breakdown field strength, the higher the electric field, the higher the temperature span and the cooling efficiency. Then, the cooling power will increase in a short electric field cycle.

In addition to constructing a 3D ECE model, a 2D model is also a research hot spot. Aprea et al. [185] studied the influence of different EM on AER performance, including PTO, P(VDF-TrFE-CFE)/BST polymer, PMN-PT thin films, PbLa(Zr, Sn, Ti)O₃ and Pb(Ba)ZrO₃ [186–188]. Among them, the COP of PTO is the highest when the elastocaloric effect occurs. Additionally, when the PTO shows the multicaloric effect, the refrigeration efficiency will be improved. Moreover, they also analyzed the influence of Al₂O₃-water nanofluids [188,189] as a heat transfer fluid on ECE.

3.3.2 Solid-based ECE models

Zhang’s research group also simulated the prototype of the rotating electrocaloric based on finite element analysis [190]. The EM is a P(VDF-TrFE-CFE) terpolymer as shown in Fig.17(a). The system temperature remains constant and is set to adiabatic conditions. This model comprises two electrocaloric disk units, up and down, rotating coaxially at the same speed but in opposite directions to ensure that heat is always transferred between electrocaloric. Fig.17(b) plots the relationship between ∆T_ECE and the temperature span, and it can be seen that for a temperature span of 20 K and a COP of 57%, the cooling power density is 37 W/cm³. Excitingly, in 2018, the authors experimentally selected BTO-based multilayer capacitor ceramics (MLCCs) as EM to reproduce the model constructed by finite element analysis [191]. At an electric field of 165 kV/cm, the maximum temperature span was measured to be 2 K.

Since a single EM cannot meet the requirements of the cooling strength of the ECE, the ceramic-based MLCC came into being. Smullin et al. [192] proposed incorporating the concept of MLCC into the heat pump model. The EM is designed as an ECE prototype in the form of an MLCC, and the current simulation can achieve the ability to know the operating temperature range and cooling efficiency, as well as the size of the MLCC, which can quickly predict the electrothermal conversion area of the prototype and the number of stacked samples. Subsequently, the research on ceramic-based MLCCs in ECE became a hot spot.

4 Conclusions and prospects

It has been over a hundred years since the discovery of ferroelectrics, among which the research results of the ECE of ferroelectric materials have been remarkable in the past two decades, constantly witnessing the breakthrough of a higher ECE of new materials, and the increasing improvement of theoretical calculation methods. In the global trend of pursuing dual carbon goals, ECE, as an environment-friendly solid-state refrigeration technology, will continue to play an essential role in the application of environmental and energy fields. The characteristic index of EM is the characterization of ΔT. Based on theoretical calculation, this paper comprehensively expounds the thermodynamic basis of EM, systematically introduces the calculation of ΔT from the four methods of the effective Hamiltonian method, thermodynamic simulation, phase-field method, and finite element analysis, and the regulation of ECE from three aspects: domain engineering, strain engineering, and composition strategy.

Although significant progress has been made in basic research of ECE, there is still a long way to go from laboratory to industrialization. With the improvement of cluster computing power and algorithms, computing science has become the third paradigm science following experimental and theoretical science. Compared with the trial-and-error method of traditional material design, computational materials science has a higher research cycle and efficiency in new materials, and theoretical calculation is a low-cost method which has become an essential tool for screening materials with excellent performance. The material parameters in the phase-field method are usually fitted from the experimental results, and the first principle calculation does not rely on any experimental data to obtain the material performance parameters. Therefore, the development of a theoretical model framework from the first-principles to the phase-field method and then to the finite element analysis not only provides a theoretical research basis for the future multiscale full-chain design of ferroelectric materials but also provides more accurate guidance for the experiment of new ferroelectric materials.

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