Phase Structures, Electromechanical Responses, and Electrocaloric Effects in K_0.5Na_0.5NbO₃ Epitaxial Film Controlled by Non-Isometric Misfit Strain

Abstract

Environmentally friendly lead-free K_1-xNa_xNbO₃ (KNN) ceramics possess electromechanical properties comparable to lead-based ferroelectric materials but cannot meet the needs of device miniaturization, and the corresponding thin films lack theoretical and experimental studies. To this end, we developed the nonlinear phenomenological theory for ferroelectric materials to study the effects of non-equiaxed misfit strain on the phase structure, electromechanical properties, and electrical response of K_0.5Na_0.5NbO₃ epitaxial films. We constructed in-plane misfit strain ($u_1-u_2$) phase diagrams. The results show that K_0.5Na_0.5NbO₃ epitaxial film under non-equiaxed in-plane strain can exhibit abundant phase structures, including orthorhombic $a_{1}c$, $a_{2}c$, and $a_1{a}_2$ phases, tetragonal $a_1$, $a_2$, and $c$ phases, and monoclinic $r_{12}$ phases. Moreover, in the vicinity of $a_{2}c-r_{12}$, $a_{1}c-c$, and $a_1{a}_2-a_2$ phase boundaries, K_0.5Na_0.5NbO₃ epitaxial films exhibit excellent dielectric constant ${\epsilon }_{11}$, while at $a_{2}c-r_{12}$ and $a_{1}c-c$ phase boundaries, a significant piezoelectric coefficient $d_{15}$ is observed. It was also found that high permittivity ${\epsilon }_{33}$ and piezoelectric coefficients $d_{33}$ exist near the $a_{2}c-a_2$, $a_1{a}_2-r_{12}$, and $a_{1}c-a_1$ phase boundaries due to the existence of polymorphic phase boundary (PPB) in the KNN system, which makes it easy to polarize near the phase boundaries, and the polarizability changes suddenly, leading to electromechanical enhancement. In addition, the results show that the K_0.5Na_0.5NbO₃ thin films possess a large electrocaloric response at the phase boundary at the $a_1{a}_2-r_{12}$ and $a_{1}c-a_1$ phase boundaries. The maximum adiabatic temperature change $\mathsf{\Delta }T$ is about 3.62 K when the electric field change is 30 MV/m at room temperature, which is significantly enhanced compared with equiaxed strain. This study provides theoretical guidance for obtaining K_1−xNa_xNbO₃ epitaxial thin films with excellent properties.

1. Introduction

Ferroelectric materials have electromechanical coupling properties due to the existence of spontaneous polarization [1,2] and have been widely used in electronic components such as capacitors, memories, actuators, et al. [3,4]. Traditional refrigeration cannot meet the local refrigeration of small devices. In contrast, solid-state refrigeration technology based on the electrocaloric effect, with the advantages of high-efficiency, environmental friendliness, lightweight, and easy miniaturization, is one of the ideal technologies to replace traditional compressor refrigeration [5,6]. Pb(Zr_xTi_1−x)O₃ (PZT) ceramics are among the most widely studied ferroelectric materials [7,8], and possess excellent performance due to their morphotropic phase boundary (MPB). The phase structure changes suddenly near the MPB, leading to enhanced electromechanical response [9,10,11]. In 2006, Mischenko et al. discovered the giant electrocaloric temperature change of 12 K in PbZr_0.95Ti_0.05O₃ ferroelectric thin films [12]. However, lead-based ceramics are harmful to human health and the environment [13,14]. Therefore, to meet the needs of sustainable human development, the research and development of lead-free piezoelectric materials have become the general trend.

In 2004, Saito et al. first prepared K_1−xNa_xNbO₃ (KNN) textured ceramics with excellent piezoelectric properties ($d_{33}=416$ pC/N) near the O-T phase boundary [15]. They then reported that modified and highly textured KNN-based lead-free piezoelectric ceramics possess high piezoelectric coefficients reaching 500~700 pC/N, equivalent to those in PZT [16,17]. Koruza et al. measured the electrocaloric effect of KNN-0.15ST and obtained the electrocaloric temperature change ($\mathsf{\Delta }T$) of 1.9 K under an electric field of 159 kV/cm and at a temperature of 340 K [18]. KNN-based ceramics have become promising lead-free ferroelectric materials and are the best substitutes for PZT materials, which have aroused extensive interest from researchers.

Over the past years, researchers have studied the composition and the resulting performance of bulk KNN materials and found a significant correlation between the phase boundary and the electromechanical performance [19,20,21,22]. In 2014, Wang et al. designed a new phase boundary with coexisting tetragonal and rhombohedral phases and found that the KNN-based ceramics near the phase boundary have excellent piezoelectric properties [23]. It is generally believed that the enhancement mechanism of the tetragonal and rhombohedral phase boundary of the KNN-based material is the polymorphic phase boundary (PPB). To recall, MPB depends on composition, whereas PPB is temperature-dependent. According to the PPB theory, the enhancement of the piezoelectric properties of KNN is caused by the orthorhombic-tetragonal (O-T) phase moving from about 200 °C to room temperature [24,25].

In recent years, with the rapid development of high-level integrated circuits, KNN-based bulk ceramics have been unable to meet the needs of device miniaturization [3,26]. However, there are few studies for high-performance KNN-based thin films. The theoretical exploration of thin films, such as the influence of misfit strain on its phase structure, electromechanical properties, and electrocaloric performance, are rarely reported, making the preparation lack theoretical guidance.

Compared with the bulk structures, due to the constraints of the boundary conditions imposed in thin films, the film lattice does not match the substrate and the film, resulting in in-plane misfit strain. It is well known that misfit strain can affect the phase structure of ferroelectric thin films [27], which can further affect the electromechanical properties and electrical response of thin films under critical thickness [28,29,30,31,32]. Thus, it can be seen that misfit strain can effectively regulate the physical properties of ferroelectric thin films [33]. Bai et al. [34] studied the effect of misfit strain on the phase structure and electromechanical properties of KNN thin films grown on cubic substrates under different compositions. Zhou et al. [35] studied the phase transition of KNN films under an external electric field through thermodynamic theoretical calculations, but they focused on the influence of extrinsic properties on the film. However, for typical KNN films grown on non-cubic substrates subjected to an in-plane non-equiaxed misfit strain, the correlations between the intrinsic phase structure, electromechanical, and electrocaloric properties are lacking.

Therefore, we used strain engineering to control the KNN thin film so that it can appear at the O-T phase boundary at room temperature, optimize the coexistence of two phases, and improve its performance significantly. Using nonlinear thermodynamic theory, we constructed the in-plane misfit strain phase diagram and studied the effects of non-equiaxed in-plane biaxial misfit strain on the phase structure, intrinsic electromechanical properties, and electrocaloric response of K_0.5Na_0.5NbO₃ epitaxial films at room temperature. This study provides theoretical guidance for optimizing the performance and experimental preparation of K_1−xNa_xNbO₃ thin films.

2. Theoretical Method

2.1. Thermodynamic Potential and Electromechanical Properties of Epitaxial Thin Films

Following the Landau-Devonshire theory applied to ferroelectric bulk at room temperature [36], conventional orthogonal coordinate systems with axes $x_1$ along [100], $x_2$ along [010], and $x_3$ along [001] are selected as references. The free energy density of ferroelectric bulk grown along (001) orientation can be described by a polynomial in polarization $p_i\left(i=1,2,3\right)$ and stress ${\sigma }_i\left(i=1,2,\mathrm{...6}\right)$, which is expressed in Voigt notation as [37]:

$$\begin{array}{l}G=a_1\left(p_1^2+p_2^2+p_3^2\right)+a_{11}\left(p_1^4+p_2^4+p_3^4\right)+a_{12}\left(p_1^2{p}_2^2+p_1^2{p}_3^2+p_2^2{p}_3^2\right)+\\ a_{111}\left(p_1^6+p_2^6+p_3^6\right)+{\alpha }_{123}{\left(p_1{p}_2{p}_3\right)}^2+{\alpha }_{111}(p_1^6+p_2^6+p_3^6)+\\ {\alpha }_{1111}(p_1^8+p_2^8+p_3^8)+{\alpha }_{1112}[p_3^2\hspace{0.33em}(p_1^6+p_2^6)+p_2^2\hspace{0.33em}(p_1^6+p_3^6)+p_1^2\hspace{0.33em}(p_3^6+p_2^6)]+\\ {\alpha }_{112}[p_3^2(p_1^4+p_2^4)+p_2^2(p_1^4+p_3^4)+p_1^2(p_3^4+p_2^4)]+\\ {\alpha }_{1122}(p_1^4{p}_2^4+p_1^4{p}_3^4+p_3^4{p}_2^4)+{\alpha }_{1123}[p_1^4{p}_2^2{p}_3^2+p_1^2{p}_2^4{p}_3^2+p_1^2{p}_2^2{p}_3^4]-\\ \frac{1}{2}{S}_{11}\left({\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2\right)-S_{12}\left({\sigma }_1{\sigma }_2+{\sigma }_1{\sigma }_3+{\sigma }_2{\sigma }_3\right)-\frac{1}{2}{S}_{44}\left({\sigma }_4^2+{\sigma }_5^2+{\sigma }_6^2\right)-\\ Q_{11}\left({\sigma }_1{p}_1^2+{\sigma }_2{p}_2^2+{\sigma }_3{p}_3^2\right)-Q_{44}\left({\sigma }_4{p}_2{p}_3+{\sigma }_5{p}_1{p}_3+{\sigma }_6{p}_1{p}_2\right)-\\ Q_{12}\left[{\sigma }_1\left(p_2^2+p_3^2\right)+{\sigma }_2\left(p_1^2+p_3^2\right)+{\sigma }_3\left(p_2^2+p_1^2\right)\right]-p_1{E}_1-p_2{E}_2-p_3{E}_3\end{array}$$

(1)

where $E_I\left(I=1,2,3\right)$ are the components of external electric fields; $a_1,a_{ij}$, and $a_{ijk}$ are the dielectric coefficients; $S_{ij}$ are the elastic compliance coefficients; and $Q_{ij}$ are the electrostrictive coefficients. Moreover, the first dielectric coefficient $a_1$ is influenced by temperature via:

$$a_1=\frac{T-T_C}{2{\epsilon }_{0}C},$$

(2)

where C is the Curie constant, ${\epsilon }_0$ is the vacuum dielectric constant, and $T_C$ is the Curie temperature of the involved material.

For thin films that are treated under the configuration of plane stress, assume the top surface is traction-free, then ${\sigma }_3={\sigma }_4={\sigma }_5=0$. Assume that the K_1−xNa_xNbO₃ epitaxial film grown on the anisotropic substrate is subjected to non-equal in-plane misfit axial strain [38], that is $u_1\ne u_2$, with zero shear strain component ($u_6=0$).

The Gibbs free energy of ferroelectric thin film can then be obtained by using Legendre transformation [39], that is $\tilde{G}=G+u_1{\sigma }_1+u_2{\sigma }_2$, with $u_i=-\partial G/\partial {\sigma }_i$. For instance, the thermodynamic potential $\tilde{G}$ for K_1−xNa_xNbO₃ epitaxial films can be expressed by [38,39]:

$$\begin{array}{l}\tilde{G}={\alpha }_1^{*}{p}_1^2+{\alpha }_2^{*}{p}_2^2+{\alpha }_{11}^{*}\left(p_1^4+p_2^4\right)+{\alpha }_{12}^{*}{\left(p_1{p}_2\right)}^2+{\alpha }_{13}^{*}{P}_3^2\left(p_1^2+p_2^2\right)\\ +{\alpha }_3^{*}{p}_3^2+{\alpha }_{12}^{\ast }{\left(p_1{p}_2\right)}^2+{\alpha }_{13}^{\ast }{p}_3^2\left(p_1^2+p_2^2\right)+{\alpha }_3^{\ast }{p}_3^2+{\alpha }_{33}^{\ast }{p}_3^4\\ +{\alpha }_{123}{\left(p_1{p}_2{p}_3\right)}^2+{\alpha }_{111}(p_1^6+p_2^6+p_3^6)+{\alpha }_{1111}(p_1^8+p_2^8+p_3^8)\\ +{\alpha }_{1112}[p_3^2\hspace{0.33em}(p_1^6+p_2^6)+p_2^2\hspace{0.33em}(p_1^6+p_3^6)+p_1^2\hspace{0.33em}(p_3^6+p_2^6)]\\ +{\alpha }_{112}[p_3^2(p_1^4+p_2^4)+p_2^2(p_1^4+p_3^4)+p_1^2(p_3^4+p_2^4)]\\ +{\alpha }_{1122}(p_1^4{p}_2^4+p_1^4{p}_3^4+p_3^4{p}_2^4)+{\alpha }_{1123}[p_1^4{p}_2^2{p}_3^2+p_1^2{p}_2^4{p}_3^2+p_1^2{p}_2^2{p}_3^4]\\ +\frac{(u_1^2+u_2^2)S_{11}-2S_{12}{u}_1{u}_2}{2(S_{11}^2-S_{12}^2)}-E_3{p}_3-E_2{p}_2-E_1{p}_1\end{array},$$

(3)

where

$${\alpha }_1^{\ast }={\alpha }_1-\frac{u_1\left(Q_{11}{S}_{11}-Q_{12}{S}_{12}\right)+u_2\left(Q_{12}{S}_{11}-Q_{11}{S}_{12}\right)}{S_{11}^2-S_{12}^2},$$

(4)

$${\alpha }_2^{\ast }={\alpha }_1-\frac{u_2\left(Q_{11}{S}_{11}-Q_{12}{S}_{12}\right)+u_1\left(Q_{12}{S}_{11}-Q_{11}{S}_{12}\right)}{S_{11}^2-S_{12}^2},$$

(5)

$${\alpha }_3^{\ast }={\alpha }_1-\frac{Q_{12}\left(u_1+u_2\right)}{S_{11}+S_{12}},$$

(6)

$${\alpha }_{11}^{\ast }={\alpha }_{11}+\frac{S_{11}\left(Q_{11}^2+Q_{12}^2\right)-2Q_{11}{Q}_{12}{S}_{12}}{2\left(S_{11}^2-S_{12}^2\right)},$$

(7)

$${\alpha }_{33}^{\ast }={\alpha }_{11}+\frac{Q_{12}^2}{S_{11}+S_{12}},$$

(8)

$${\alpha }_{12}^{\ast }={\alpha }_{12}-\frac{S_{12}\left(Q_{11}^2+Q_{12}^2\right)-2Q_{11}{Q}_{12}{S}_{11}}{2\left(S_{11}^2-S_{12}^2\right)}+\frac{Q_{44}^2}{2S_{44}},$$

(9)

$${\alpha }_{13}^{\ast }={\alpha }_{12}+\frac{Q_{12}\left(Q_{11}+Q_{12}\right)}{S_{11}+S_{12}}.$$

(10)

Here, ${\alpha }_i^{*}$ and ${\alpha }_{ij}^{*}$ refer to the normalized dielectric constants. The material-specific coefficients (parameters) are listed in Table 1.

Based on the principle of minimum energy, the polarization components of the thin films at equilibrium (stable phase) can be computed as [40]:

$$\frac{\partial \tilde{G}}{\partial p_1}=0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\frac{\partial \tilde{G}}{\partial p_2}=0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\frac{\partial \tilde{G}}{\partial p_3}=0.$$

(11)

From the computed polarization components $(p_1,p_2,p_3)$, the relative dielectric constants of the ferroelectric thin films are obtained as [41,42]:

$${\epsilon }_{ij}=1+{\eta }_{ij}/{\epsilon }_0,$$

(12)

where

$$\eta ={\chi }^{-1}={\left(\begin{array}{ccc}\frac{{\partial }^2\tilde{G}}{\partial p_1\partial p_1}& \frac{{\partial }^2\tilde{G}}{\partial p_1\partial p_2}& \frac{{\partial }^2\tilde{G}}{\partial p_1\partial p_3}\\ \frac{{\partial }^2\tilde{G}}{\partial p_2\partial p_1}& \frac{{\partial }^2\tilde{G}}{\partial p_2\partial p_2}& \frac{{\partial }^2\tilde{G}}{\partial p_2\partial p_3}\\ \frac{{\partial }^2\tilde{G}}{\partial p_3\partial p_1}& \frac{{\partial }^2\tilde{G}}{\partial p_3\partial p_2}& \frac{{\partial }^2\tilde{G}}{\partial p_3\partial p_3}\end{array}\right)}^{-1}.$$

(13)

The piezoelectric coefficient $d_{in}$ for the (001) orientation is calculated by [43]:

$$d_{in}=\frac{\partial s_n}{\partial p_1}{\eta }_{i1}+\frac{\partial s_n}{\partial p_2}{\eta }_{i2}+\frac{\partial s_n}{\partial p_3}{\eta }_{i3}.$$

(14)

In this work, the piezoelectric coefficients $d_{15}$ and $d_{33}$ will be analyzed. The in-plane normal strain $s_3$ and shear strain $s_5$ are derived [42]:

$$s_3=\frac{2u_m{S}_{12}}{S_{11}+S_{12}}+\left[Q_{12}-\frac{S_{12}(Q_{11}+Q_{12})}{S_{11}+S_{12}}(p_1^2+p_2^2)+(Q_{11}-\frac{2S_{12}{Q}_{12}}{S_{11}+S_{12}})p_3^2\right],$$

(15)

$$s_5=Q_{44}{p}_1{p}_3.$$

(16)

Table 1. Coefficients used in the computation for K_0.5Na_0.5NbO₃ thin films, where T represents the temperature at °C [36,44,45].

Coeff	Values	Units
${\alpha }_1$	$4.29\times {10}^7\times \left[\mathrm{Coth}\left(140/\mathrm{T}+273\right)-\mathrm{Coth}\left(140/657\right)\right]$	$C^{-2}{m}^{2}N$
${\alpha }_{11}$	$-2.7302\times {10}^8$	$C^{-4}{m}^{6}N$
${\alpha }_{12}$	$1.0861\times {10}^9$	$C^{-4}{m}^{6}N$
${\alpha }_{111}$	$3.0448\times {10}^9$	$C^{-6}{m}^{10}N$
${\alpha }_{112}$	$-2.7270\times {10}^9$	$C^{-6}{m}^{10}N$
${\alpha }_{123}$	$1.5513\times {10}^{10}$	$C^{-6}{m}^{10}N$
${\alpha }_{1111}$	$2.4044\times {10}^{10}$	$C^{-8}{m}^{14}N$
${\alpha }_{1112}$	$3.7328\times {10}^9$	$C^{-8}{m}^{14}N$
${\alpha }_{1122}$	$3.3485\times {10}^{10}$	$C^{-8}{m}^{14}N$
${\alpha }_{1123}$	$-6.2017\times {10}^{10}$	$C^{-8}{m}^{14}N$
$Q_{11}$	$0.16$	$m^4/C^2$
$Q_{12}$	$-0.072$	$m^4/C^2$
$Q_{44}$	$0.084$	$m^4/C^2$
$S_{11}$	$5.57\times {10}^{-12}$	$m^2/N$
$S_{12}$	$-1.57\times {10}^{-12}$	$m^2/N$
$S_{44}$	$13.1\times {10}^{-12}$	$m^2/N$
$C_{latt}$	$1.485\times {10}^6$	$J/m^{3}K$

Table 1. Coefficients used in the computation for K_0.5Na_0.5NbO₃ thin films, where T represents the temperature at °C [36,44,45].

Coeff	Values	Units
${\alpha }_1$	$4.29\times {10}^7\times \left[\mathrm{Coth}\left(140/\mathrm{T}+273\right)-\mathrm{Coth}\left(140/657\right)\right]$	$C^{-2}{m}^{2}N$
${\alpha }_{11}$	$-2.7302\times {10}^8$	$C^{-4}{m}^{6}N$
${\alpha }_{12}$	$1.0861\times {10}^9$	$C^{-4}{m}^{6}N$
${\alpha }_{111}$	$3.0448\times {10}^9$	$C^{-6}{m}^{10}N$
${\alpha }_{112}$	$-2.7270\times {10}^9$	$C^{-6}{m}^{10}N$
${\alpha }_{123}$	$1.5513\times {10}^{10}$	$C^{-6}{m}^{10}N$
${\alpha }_{1111}$	$2.4044\times {10}^{10}$	$C^{-8}{m}^{14}N$
${\alpha }_{1112}$	$3.7328\times {10}^9$	$C^{-8}{m}^{14}N$
${\alpha }_{1122}$	$3.3485\times {10}^{10}$	$C^{-8}{m}^{14}N$
${\alpha }_{1123}$	$-6.2017\times {10}^{10}$	$C^{-8}{m}^{14}N$
$Q_{11}$	$0.16$	$m^4/C^2$
$Q_{12}$	$-0.072$	$m^4/C^2$
$Q_{44}$	$0.084$	$m^4/C^2$
$S_{11}$	$5.57\times {10}^{-12}$	$m^2/N$
$S_{12}$	$-1.57\times {10}^{-12}$	$m^2/N$
$S_{44}$	$13.1\times {10}^{-12}$	$m^2/N$
$C_{latt}$	$1.485\times {10}^6$	$J/m^{3}K$

2.2. Electrocaloric Effect Temperature Change

The electrocaloric effect refers to the phenomenon of temperature change caused by external electric fields or entropy change caused by isothermal conditions of dielectric materials under adiabatic conditions [46,47]. Based on the principle of entropy conservation, the isothermal entropy change $\mathsf{\Delta }S$ and adiabatic temperature change $\mathsf{\Delta }T$ can be computed to characterize the electrocaloric performance of ferroelectric thin films. For instance, the system isothermal entropy change $S_{total}$ is the sum of dipole entropy $S_{dip}$ and lattice entropy $S_{latt}$, that is [28,48]:

$$S_{total}\left(E,T\right)=S_{dip}\left(E,T\right)+S_{latt}\left(T\right),$$

(17)

where the dipole entropy is attributed to the electric dipole in the film, and it is a function of the polarization $P_i$, which in turn is related to the applied electric field. In contrast, the lattice entropy depends on temperature and is independent of the applied electric field:

$$S_{dip}(E,T)=-{\left(\frac{\tilde{\partial }G(E,T)}{\partial T}\right)}_E,$$

(18)

$$d{S}_{latt}(T)=\frac{C_{latt}}{T}dT,$$

(19)

where $C_{latt}$ is the heat capacity per unit volume of the thin film. During the electrocaloric process, the total entropy of the system remains zero under an adiabatic electric field; $T_i$ and $T_f$ denote the initial temperature and the final temperature, respectively; $E_i$ and $E_f$ represent the initial electric field and the final electric field, respectively.

$$dS=d{S}_{dip}+d{S}_{latt}=0,$$

(20)

$$\mathsf{\Delta }{S}_{latt}=S_{latt}\left(T_f\right)-S_{latt}\left(T_i\right)=C_{latt}{\displaystyle {\int }_{T_i}^{T_f}\frac{1}{T}dT\approx C_{latt} \left(T_i\right)In\left(\frac{T_f}{T_i}\right)} ,$$

(21)

$$\mathsf{\Delta }{S}_{dip}=S_{dip}\left(E_f,T_f\right)-S_{dip}\left(E_i,T_i\right)=\frac{\partial \tilde{G}\left(E_i,T_i\right)}{\partial T}-\frac{\partial \tilde{G}\left(E_f,T_f\right)}{\partial T}.$$

(22)

The final temperature of the material can be calculated from the formula:

$$T_f=T_{i}exp\left\{-\frac{1}{C_{latt}}\left[\frac{\partial \tilde{G}\left(E_i,T_i\right)}{\partial T}-\frac{\partial \tilde{G}\left(E_i,T_i\right)}{\partial T}\right]\right\}.$$

(23)

Moreover, the expression of the final adiabatic temperature change can be written as [49,50,51]:

$$\mathsf{\Delta }T=T_f-T_i=T_{i}exp\left\{-\frac{1}{C_{latt}}\left[\frac{\partial \tilde{G}\left(E_i,T_i\right)}{\partial T}-\frac{\partial \tilde{G}\left(E_f,T_f\right)}{\partial T}\right]\right\}-T_i.$$

(24)

3. Results and Analysis

The nonlinear thermodynamic model is applied to study the phase structure, electromechanical properties, and electrocaloric response of K_0.5Na_0.5NbO₃ epitaxial film at room temperature under non-equiaxed in-plane misfit strain. The correlation coefficients used in the calculation are shown in Table 1. With these parameters and the nonlinear thermodynamic model, the phase structure and dielectric properties of K_0.5Na_0.5NbO₃ bulk material were accurately repeated [36], indicating that the calculated parameters are reliable. For instance, at 25 °C, when the equiaxed compressive strain gradually transforms to tensile strain, the phase structure of the K_0.5Na_0.5NbO₃ film changes with the routes $c\to r_{11}\to a_1{a}_1$, which is consistent with the phase structure corresponding to the phase diagram ($u_1-u_2$) along the diagonal line [35], which also validates the correctness of the calculation results.

First, we studied the effect of non-equiaxed in-plane misfit strain on the phase structure of K_0.5Na_0.5NbO₃ epitaxial film at room temperature (25 °C). We constructed the phase diagram over in-plane strains $u_1-u_2$, as shown in Figure 1a, and the corresponding changes in the polarization components are shown in Figure 1b–d. When subjected to equiaxed misfit strain, the phase structure corresponds to the diagonal line of the phase diagram in Figure 1a. It can be found that when the equiaxed compressive strain gradually transforms to tensile strain, the phase structure of K_0.5Na_0.5NbO₃ film changes sequentially as $c\to r_{11}\to a_1{a}_1$, which is consistent with the literature results, as shown in Figure 2a [35], which shows the correctness of the calculation results.

Interestingly, when $u_1$ and $u_2$ are not equal, which corresponds to in-plane non-equiaxed misfit strain, the symmetry of the phase structure of the K_0.5Na_0.5NbO₃ thin film is obviously broken at room temperature, leading to the emergence of a rich variety of phase structures. There are seven phase structures according to the applied misfit strains, and the polarization characteristics of each phase are featured in

Table 2. Phase structures and characteristic polarization components of K_0.5Na_0.5NbO₃ thin films at room temperature and under no electric field.

Phase	Polarization
$c$	$p_1=p_2=0,p_3\ne 0$
$a_1$	$p_1\ne 0,p_2=p_3=0$
$a_2$	$p_2\ne 0,p_1=p_3=0$
$a_{1}c$	$p_1\ne p_3\ne 0,p_2=0$
$a_{2}c$	$p_2\ne p_3\ne 0,p_1=0$
$a_1{a}_1$	$p_1=p_2\ne 0,p_3=0$
$a_1{a}_2$	$p_1>p_2\ne 0,p_3=0 or p_2>p_1\ne 0,p_3=0$
$r_{11}$	$p_1=p_2\ne 0,p_3\ne 0$
$r_{12}$	$p_1\ne p_2\ne p_3\ne 0,p_1>p_2 or p_1\ne p_2\ne p_3\ne 0,p_2>p_1$

. Among these phases, the monoclinic phase corresponds to the center of the phase diagram, and other phases are distributed around it. The pattern of the overall phase diagram is symmetric about the line $u_1=u_2$. The tetragonal phase exists in the region subjected to larger compressive in-plane strain. It can be seen from Figure 1b that in the transition from compressive strain to tensile strain, the plane polarization component $p_3$ gradually decreases to zero, making the $c$ phase disappear.

At the same time, $u_1\ne u_2$ induces $p_1\ne p_2,$ leading to unequal energies of the two phases $a_{1}c$ and $a_{2}c$. The $a_{2}c$ phase is more likely to form in the region with $u_2>0$, while $a_{1}c$ phase is more likely to form in the region $u_1>0$. At the same time, the tetragonal $a_2$ phase is located in the region in the phase diagram with a large tensile normal strain $u_2$, while the $a_1$ phase exists in the region with a large tensile normal strain $u_1$. The $a_1{a}_2$ phase is located in the region with large tensile normal strains $u_1$ and $u_2$.

From the diagram showing polarization components, it can be seen that the non-equiaxed in-plane misfit strains $u_1$ and $u_2$ can affect the polarization of the film at room temperature. Figure 1b indicates that the in-plane polarization $p_1$ increases with the increase in the tensile strain $u_1$. Figure 1c indicates that the change in direction of polarization $p_2$ is parallel to the strain $u_2$ and becomes larger as it increases. In contrast, Figure 1d shows that the out-of-plane polarization $p_3$ exists stably in the region with compressive strain and increases with the increase in compressive strains $u_1$ and $u_2$.

Next, we observe the change of the polarization components with the misfit strains where the conditions with room temperature and no applied electric field are imposed. Figure 2b–d depict the relationships between polarization components and strains at $u_1=-1\%$, $u_1=0$, and $u_1=1.5\%$, respectively.

It is found that in Figure 2b, at $u_1=-1\%$, as the misfit strain $u_2$ increases from compressive strain to tensile strain (−2~2%), the film undergoes a sequence of phase transitions $a_{1}c\to c\to a_{2}c$, and the orthorhombic phase accounts for the majority of which the polarization components $p_1$ and $p_2$ experience a sudden change in the $a_{1}c-c$ phase. In contrast, the polarization component $p_3$ decreases continuously with the increase in strain. Figure 2c shows that the tetragonal phase degenerates while the monoclinic $r_{12}$ phase emerges, and the strain $u_2$ of $a_{1}c$ phase boundary extends to −0.55%. As the strain $u_1$ becomes tensile and as the misfit strain $u_2$ increases, the thin films undergo a sequence of phase transformations $a_{1}c\to a_1\to a_1{a}_2$, and the polarization component $p_1$ has no noticeable change. Still, the abrupt change point of the polarization component $p_3$ has dropped to the region with compressive strain. Figure 1 and Figure 2 illustrate that, compared with the equiaxed misfit strain, the non-equiaxed misfit strain causes a change in the symmetry of the phase structure, thus inducing a rich variety of phase structures. Moreover, multiple phases co-exist near the phase boundary, leading to easy polarization, enhancing the performance, and effectively controlling the electromechanical properties of K_0.5Na_0.5NbO₃ epitaxial film.

To explore the influence of non-equiaxed misfit strain on the electromechanical properties of K_0.5Na_0.5NbO₃ epitaxial film, we calculated the dielectric constant ${\epsilon }_{ij}$ and piezoelectric coefficient $d_{ij}$ of K_0.5Na_0.5NbO₃ epitaxial film under non-equiaxed misfit strain, as shown in Figure 3, Figure 4, Figure 5 and Figure 6.

Figure 3a–c show the stack distribution of the dielectric constants ${\epsilon }_{11},{\epsilon }_{22},{\epsilon }_{33}$ on the phase diagram when the non-equiaxed misfit strains ($u_1,u_2$) interact at room temperature. Figure 3a demonstrates that near the $c-a_{1}c,a_{2}c-r_{12},a_2-a_1{a}_2$ phase boundaries exhibit excellent transverse permittivity ${\epsilon }_{11}$, mainly due to the polarization component $p_1$ gradually decreasing to 0 as the strain $u_1$ decreases, as shown in Figure 1b. In Figure 3b, the scores of permittivity ${\epsilon }_{22}$ are mainly distributed near the multi-phase boundaries $c-a_{2}c,a_{1}c-r_{12},a_1-a_1{a}_2$, in the range of low compressive strain $u_2$. In the region of Figure 3c with a larger tensile strain $u_1$, application of $u_2$ induces large ${\epsilon }_{33}$, indicating that non-equiaxed misfit strain enhances the dielectric constant near the phase boundaries with the coexistence of multi-phase at room temperature, which is consistent with the discovery in the KNN-based thin film by Lou et al. [52] where a monoclinic phase emerges at room temperature with a noticeable performance enhancement near the phase boundary.

To further understand the influence of non-equiaxed strains on the dielectric properties, we selected the region of the phase diagram with high dielectric properties to study the relationship between specific strain and polarization characteristics and electromechanical properties, as shown in Figure 4a–c. Figure 4a shows the relationship between the dielectric constant ${\epsilon }_{11},{\epsilon }_{22},{\epsilon }_{33}$ and the in-plane strain $u_2$, when $u_1=-1\%$. Recall that the relationship between the corresponding polarization and the phase structure has been described in Figure 2b. In the figure, it can be seen that with the increase in strain $u_2$, the dielectric constant ${\epsilon }_{33}$ increases slowly, and there is no apparent change near the phase boundary, mainly because the out-of-plane polarization component $p_3$ of the film remains relatively stable when the strain is applied. However, in the region of the phase diagram corresponding to the coexistence of the orthogonal and tetragonal phases, the lateral permittivity ${\epsilon }_{11}$ and ${\epsilon }_{22}$ exhibit peak values when the applied strains are $u_2=-1.44\%$ and $u_2=0.88\%$, respectively. A stable region with a permittivity ${\epsilon }_{11}$ of about 500 is also observed in the $a_{2}c$ phase.

For the case with $u_1=0$, when the applied electric field is 0, at 25 °C, the phase diagram corresponding to its polarization component is shown in Figure 2c. A sharp peak in the permittivity appears near the $a_{1}c-r_{12}$ phase boundary, which can be seen as the abrupt change in the polarization component from Figure 1c. In addition, at this phase transition point, the in-plane polarization $p_2$ gradually decreases to 0, and its slope changes discontinuously. At the same time, the dielectric constant ${\epsilon }_{33}$ is located at the point where the slope of the out-of-plane polarization $p_3$ decreases, and due to the slight decrease in the in-plane polarization $p_1$, the dielectric constant remains basically unchanged. It can maintain good stability at room temperature, and the peak value of ${\epsilon }_{33}$ throughout the process is greater than ${\epsilon }_{22}$.

We also calculated the relationship between the dielectric constant and strain, as shown in Figure 4c, when the tensile strain is $u_1=1.5\%$. Similar to Figure 4b, a phase enhancement effect is also observed. The peaks of ${\epsilon }_{22}$ and ${\epsilon }_{33}$ are located at the orthogonal and tetragonal phase boundaries. The strain difference between these peaks is small due to the sudden change in the corresponding polarization component, where the discontinuity of ${\epsilon }_{33}$ is shifted towards the direction of compressive strain.

Next, we studied the influences of non-equiaxed misfit strain on the piezoelectric coefficients $d_{15}$ and $d_{33}$ of the K_0.5Na_0.5NbO₃ epitaxial film. Figure 5a,b show the cloud distributions of the piezoelectric coefficients $d_{15}$ and $d_{33}$ when the applied electric field is 0 at room temperature. It can be found that the piezoelectric coefficients are all zero across the phase structures $a_2$,$a_1{a}_2$, and $a_1$. This is mainly attributed to the distribution of polarization components $p_3$ in Figure 1d, which is consistent with the piezoelectric coefficients only when the asymmetric structure is met.

The distribution of the extreme value of the piezoelectric coefficient $d_{15}$ is similar to that of the dielectric constant ${\epsilon }_{11}$, which exists at the phase boundaries of $c-a_{1}c$ and $a_{2}c-r_{12}$ phases. The distribution of the piezoelectric coefficient $d_{33}$ is similar to the dielectric constant ${\epsilon }_{33}$ and follows a similar trend, which undergoes enhancement when the polarization component $p_3$ experiences a sudden change. While it is most well known in the PZT film, which has tetragonal and rhombohedral phases coexisting at MPB, there is a certain difference in the mechanism of the two. MPB is due to the polarization reversal generated by the material itself, which causes the enhancement effect, and the polarization component will change discontinuously near the phase boundary. For the KNN film, it is regulated by external strain.

In Figure 6a, when $u_1=-1\%$, the relationship between the piezoelectric coefficient and the misfit strain $u_2$, when it is changed from compressive strain to tensile strain, the phase structure undergoes a transformation: $a_{1}c\to c\to a_{2}c$. It is found that the piezoelectric coefficient $d_{15}$ attains a peak value near the $a_{1}c-c$ phase boundary, and the strain $u_2=-1.44\%$ is mainly caused by the sharp change in the slope of in-plane polarization $p_1$. $d_{15}$ is rising continuously. Along the diagonal region of the phase diagram, the influence of the phase boundary on the electromechanical performance is minimal because the polarization components at the phase transition point are all continuously changing with no sudden change in the slope.

Figure 6b, c show the relationship curves of the piezoelectric coefficient and strain $u_2$ when the misfit strain $u_1$ is 0 and 1.5%, respectively. It can be seen the piezoelectric coefficient $d_{15}$ is close to 0 with no noticeable change in the polarization in the phase diagram where the polarization component $p_1$ is in a stable state. When the misfit strain $u_1$ is not present, the piezoelectric coefficient attains peak values near the $r_{12}-a_1{a}_2$ phase boundary, while away from the peak, it changes slowly in the orthorhombic $a_{1}c$ phase. When subjected to tensile strain $u_1$, the piezoelectric coefficient $d_{33}$ peaks at $a_{1}c-a_1$, which is similar to the trend of the dielectric constant ${\epsilon }_{33}$ because its value is the product of the polarization component and the dielectric constant.

In general, the enhancement of electromechanical properties often occurs near the O-M, O-T, and T-M phase boundaries, which is consistent with the existence of morphotropic phase boundaries in the KNN system and is regulated by external strain. At the same time, when the in-plane polarization $p_1$ changes suddenly, the dielectric constants ${\epsilon }_{11}$ and piezoelectric coefficient $d_{15}$ attain peak values, and when the in-plane polarization $p_2$ changes, the dielectric constants ${\epsilon }_{22}$ follow the change. When the dielectric constants ${\epsilon }_{33}$ and piezoelectric coefficient $d_{33}$ are at peaks, the slope of the out-of-plane polarization $p_3$ often changes abruptly. When the value of $p_3$ is 0, the piezoelectric coefficients $d_{15}$ and $d_{33}$ are both zero. By adjusting the magnitude of the misfit strain, the position of the polymorphic phase boundary at room temperature can be controlled, thereby adjusting the electromechanical properties of the thin film system.

Finally, we studied the effect of non-equiaxed misfit strains on the adiabatic temperature change $\mathsf{\Delta }T$ in the electrocaloric response of K_0.5Na_0.5NbO₃ thin film. At room temperature, an electric field change $\mathsf{\Delta }E$ of 20 MV/m along the [001] direction is applied from an initial electric field of 1 MV/m. Figure 7a shows that at room temperature, the phase boundaries $a_1{a}_2-r_{12}$, $a_{2}c-a_2$, and $a_{1}c-a_1$ shift towards tensile strain compared to the case with no applied electric field (as indicated by the white dashed line versus phase boundary enhancement), which is due to the applied electric field along the direction of the out-of-plane polarization $p_3$, leading to enhancement. At the same time, large adiabatic temperature changes appear near the $a_1{a}_2-r_{12}$, $a_{2}c-a_2$, and $a_{1}c-a_1$ phase boundaries because the polarization $p_3$ near these phase boundaries is zero when there is no external electric field, when compared (cross-referenced) to Figure 1d; when an external electric field in the [001] direction is applied, $p_3$ increases, causing a large entropy change, resulting in a large adiabatic temperature change $\mathsf{\Delta }T$.

There is a difference in the extreme value of the adiabatic temperature change near the ferroelectric phase, indicating that the enhancement is not caused by the Curie temperature of the ferroelectric material. When the strain $u_2$ remains constant, the enhancement of phase boundaries in Figure 7a shows an upward trend with the increase in the in-plane strain $u_1$, possibly due to the continuous decrease in polarization $p_3$ under this condition. To further understand the electrocaloric effect under the applied electric field and the in-plane misfit, we calculated the characteristics of the adiabatic temperature change with the misfit strain under different applied electric fields when $u_1$ and $u_2$ were 2% and −0.45%, respectively, as shown in Figure 7b, c. The study found that with the increase of the applied electric field, $\mathsf{\Delta }T$ increases and remains unchanged upon reaching the peak value. In Figure 7b, as $u_1$ increases, the adiabatic temperature change shows an upward trend. When $u_1$ is about −0.7%, $\mathsf{\Delta }T$ experiences a small mutation, which is caused by the emergence of the polarization component $p_1$. In Figure 7c, the adiabatic temperature change can reach 3.62 K subjected to $\mathsf{\Delta }E=30\mathrm{MV}/\mathrm{m}$, which is further improved compared to the equiaxed misfit strain. The peak of $\mathsf{\Delta }T$ is clearly seen at the phase transition in Figure 7c, indicating that an appropriate non-equiaxed misfit strain and an applied electric field can enhance the electrocaloric effect at room temperature. This provides a specific guiding significance for the stable operation of the electrocaloric refrigeration device at room temperature.

4. Conclusions

In summary, this article uses the nonlinear Landau-Devonshire thermodynamic theory to study the effects of non-equiaxed in-plane misfit strains on the phase structures, electromechanical properties, and electrocaloric effect of K_0.5Na_0.5NbO₃ epitaxial thin films grown on anisotropic substrates at room temperature. It is found that at room temperature, misfit strain can induce orthorhombic phases ($a_1{a}_1,a_1{a}_2,a_{1}c,a_{2}c$), tetragonal phases ($c,a_1,a_2$), and monoclinic phases ($r_{11},r_{12}$). Due to the sudden change in the slope of the in-plane and out-of-plane polarization components, the electromechanical properties and electrocaloric effects are enhanced near the O-M, T-M, and O-T phase boundaries. The phase boundary generated by strain engineering is different from MPB in classical PZT. Among these phase structures, there are excellent dielectric constant ${\epsilon }_{11}$ and piezoelectric coefficient $d_{15}$ near the $a_{1}c-c$ phase boundary. At the same time, the applied electric field along the [001] direction can shift the $a_{1}c-a_1$ phase boundary in the direction of strain increase. When the electric field changes to 30 MV/m, the adiabatic temperature change $\mathsf{\Delta }T$ can reach about 3.6 K when the film is in the monoclinic phase $r_{11}$. This work provides theoretical guidance for experimental research on controlling lead-free K_1−xNa_xNbO₃ thin films by strain engineering.