Electrostrictive effect in ferroelectrics: An alternative approach to improve piezoelectricity

Electrostriction is a basic electromechanical phenomenon in all insulators or dielectrics. It describes the electric field/polarization-induced strain (⁠$Sij$⁠) that is proportional to the square of electric field (⁠$Ei$⁠)/polarization (⁠$Pi$⁠), expressed in the following equations:

where $Qijkl$ and $Mijkl$ are electrostrictive coefficients. Electrostriction is a four-rank tensor property; thus it can be observed in all crystal symmetries.¹ 

The strain induced by the electrostrictive effect is generally small when compared with that induced by piezoelectricity. Consequently, there has been limited attention focusing on the electrostrictive effect. In the 1980s, a systematic study on electrostriction was carried out on relaxor ferroelectrics with perovskite structures, such as Pb(Mg_1/3Nb_2/3)O₃ (PMN), Pb(Zn_1/3Nb_2/3)O₃ (PZN), and 0.9Pb(Mg_1/3Nb_2/3)O₃-0.1PbTiO₃ (PMN-0.1PT) single crystals/ceramics, in which a high electrostrictive strain was observed because of the high dielectric response of the relaxors.^2–7 Relaxors offer several advantages over ferroelectric materials, including low hysteresis in the strain-field response (excellent displacement accuracy), no remnant strain, reduced aging effects, and they do not require poling.³ Compared with classical ferroelectric ceramics, these materials are believed to have potential for use in actuator applications such as inchworms, micro-angle adjusting devices, and bistable optical devices, where reproducible and non-hysteretic deformation responses are required.³ In addition, in the 1990s, investigations on electrostriction were performed on polymers,^8–15 including ferroelectric polymers, dielectric elastomers, and polymer composites. Ultra-high electrostrictive strains were observed in these polymeric materials (>4% for polyvinylidene fluoride [PVDF] and >40% for silicone), giving them potential for use in actuator applications.¹⁵ 

On the other hand, ferroelectrics are the mainstay materials for piezoelectric transducer and actuator applications and have been reported to possess much higher piezoelectric responses when compared with non-ferroelectric materials.¹⁶ Typical ferroelectric single crystals and ceramics, including barium titanate (BaTiO₃, BT) (reported in the 1940s),¹⁷ lead titanate zirconate (Pb(Zr_1-xTi_x)O₃, PZT) polycrystalline ceramics (reported in the 1950s),¹⁷ and relaxor-PT based single crystals (reported in the 1980s to 1990s),^18–23 exhibited piezoelectric responses that are two to three orders higher than Quartz crystals (first piezoelectric crystal ∼2 pC/N). Similar to inorganic ferroelectrics, ferroelectric polymers such as PVDF also possess a higher piezoelectric response than their nonferrous counterpart.^24,25 Thus, it is desirable to understand the origins of high piezoelectric activity in ferroelectrics. For ferroelectrics whose paraelectric phase is centrosymmetric, the piezoelectric coefficient $dmij$ can be expressed as a derivative of strain to electric field

where $εij$ is the dielectric permittivity, and i, j, k, l, and m = 1, 2, or 3. Equation (1) is used as the expression of strain (⁠$Sij$⁠). Equation (2) indicates that the piezoelectric coefficients of these ferroelectrics originate from the electrostrictive effect, spontaneous polarization, and the dielectric response. Figure 1 shows that the piezoelectric coefficients can be recognized as the slope of the electrostrictive strain versus the electric field. Thus, we can conclude that the high piezoelectric response in ferroelectrics is established on the electrostrictive effect and is associated with a large “bias polarization” (spontaneous polarization). The electrostrictive effect plays a key role in the electromechanical behavior in ferroelectrics, investigations on which will benefit the exploration of high-performance piezoelectrics. In this paper, the investigations on the electrostrictive effect are surveyed, focusing on ferroelectric-related materials. First, the measurement methods of the electrostrictive effect are given and compared. Second, the origin of electrostriction is discussed for ionic crystals. Then, the relationships between the electrostriction and the crystal structure, dielectric response, elastic property and thermal expansion are surveyed. The electrostrictive properties of the ferroelectric materials are investigated with respect to temperature, composition, ferroelectric phase, and orientation. Finally, the contribution of electrostriction to the piezoelectric activity is discussed in detail for perovskite ferroelectrics, and we outline a possible approach for improving the piezoelectricity through the electrostrictive effect.

To determine the electrostrictive coefficients, five approaches have been proposed in the literature. These are based on the strain-polarization/electric field curves, dielectric permittivity-stress curves, the piezoelectric effect for the ferroelectrics, the lattice parameters, and the dielectric permittivity under a DC-biased electric field.

According to Eq. (1), the electrostrictive coefficients $Q$ and $M$ can be determined by measuring the strain-polarization (S-P) and strain-electric field (S-E) curves, respectively. The S-P and S-E curves are accepted as a method to determine the electrostrictive coefficients for materials with a high dielectric response, where the strains can be accurately measured.

Figures 2 and 3 show the measured results for a BaBi₂Nb₂O₉ ceramic and an [001] oriented PMN single crystal, respectively.^26,27 A linear relationship between the polarization and electric field was observed in Fig. 2(a), indicating that the BaBi₂Nb₂O₉ ceramic is a linear dielectric material. The strain followed a quadratic function with respect to both the electric-field and polarization, as shown in Figs. 2(b) and 2(c). On the contrary, PMN crystals are nonlinear dielectrics. Figure 3(a) shows the polarization saturation at high fields that deviate from a linear relationship. For nonlinear dielectrics, the electrostrictive strain is not a quadratic function of the electric field but follows a quadratic relation with respect to the polarization (Figs. 3(b) and 3(c)). Thus, the $Q$ coefficient is preferred over the M coefficient, for the investigation of the electrostrictive effect. For this reason, the following discussions on ferroelectrics focus on the $Q$ coefficients.

To determine the $Q$ coefficients in ferroelectrics, domain reorientation (or domain wall motion) and electric field-induced phase transitions should be avoided. Figure 4 shows the phenomenon of domain reorientation in tetragonal ferroelectric crystals for the $Q33$ measurements along the [001] direction. For this condition, the contribution of 180° ferroelectric domain switching contributed to the strain and polarization is $2d33E$ and $2Ps$⁠, respectively, as shown in Fig. 4(a). Generally, $2d33E$ was much smaller than $4PS2Q33$⁠; thus the presence of the 180° ferroelectric domain switching mitigates the measured $Q$ value. On the contrary, the 90° ferroelastic domain switching made the measured $Q33$ larger than the real value of the lattice deformation. This was caused by polarization variations along the perpendicular direction, which was not considered in the measurement. Overall, domain switching impacted the electrostrictive coefficient measurements. Ferroelectric phase transitions also affected the measured results, because the direction of the spontaneous polarization varied during the ferroelectric phase transitions. To eliminate the impact of the domain wall motion and phase transition, the S-P curves should be measured at high temperatures (higher than the Curie temperature, $TC$⁠) and at high electric fields, because no ferroelectric domains exist in the paraelectric phase above the $TC$⁠, meanwhile, the domains could be clamped by high electric fields, increasing the difficulty in the domain switching.

Figure 5 shows the electrostriction results for [001] oriented PMN, PMN-0.28PT, and PZN crystals at temperatures above the temperature of maximum permittivity (⁠$Tm$⁠).²⁶ The S-P curves exhibited hysteretic behavior, which can be separated into hysteretic and reversible regions. The hysteretic region in the S-P curves can be attributed to either the micro-domain reorientation in relaxor materials (micro-domains exist at temperatures above $Tm$⁠) or the electric-field-induced micro-macro domain transition.^28,29 It should be noted here that the hysteretic region was also related to the measurement condition. For example, PMN-0.1PT thin film does not show obvious hysteretic S-P behavior at the temperatures above $Tm$⁠.³⁰ The difference between the results of Fig. 5 and Ref. 30 can be attributed to the substrate clamping effect of thin film and different measurement conditions (frequency and amplitude of electric field). The strain and polarization induced by the intrinsic lattice contribution at a high electric field accounts for the reversible S-P curve. By fitting the experimental data using $S33=Q33P32$⁠, the electrostrictive coefficient $Q33$ can be determined with respect to the amplitude of the electric field, as given in Fig. 6. In the electric field range of 10–80 kV/cm, the measured $Q33$ values for [001] oriented PMN-0.28PT, PZN, and PMN crystals increased with the electric field increasing, being in the ranges of 0.037–0.053 m⁴/C², 0.043–0.060 m⁴/C², and 0.018–0.027 m⁴/C², respectively. This was because the micro-domain reorientation (180°) reduced the measured $Q$ value at low electric fields, while the measured $Q$ approached the intrinsic value (lattice) at high electric fields.

Determination of electrostrictive coefficients from the relationship between dielectric permittivity and applied stress was based on the converse effect of electrostriction. The direct effect of electrostriction was given in Eq. (1) from which the electrostrictive coefficient $Qijkl$ can be expressed as follows:

Based on the elastic Gibbs free energy [Eq. (4)] and changing the order of differentiation in the free energy equation, Eq. (3) was transformed to Eq. (5) (Refs. 31 and 32)

where $G1$ is the elastic Gibbs free energy, $Xij$ is the stress, $S$ is the entropy, $T$ is the temperature, and $χkl$ is the inverse dielectric permittivity. Using the same method, the coefficient $Mijkl$ can be expressed as

where $G$ is the Gibbs free energy and $εkl$ is the dielectric permittivity. Equations (5) and (7) describe the converse effect of electrostriction. According to Eq. (5), the $Q$ coefficients can be written as follows:

In addition, for crystals with cubic symmetry, the hydrostatic electrostrictive coefficient $Qh$ can be written as³³ 

where $p$ is the applied hydrostatic pressure. Equations (8)–(10) were successfully employed to determine the $Q$ coefficients for materials with low electrostrictive strains. In these materials, it was difficult to accurately measure the S-P curves. Based on this approach, the $Qh$ coefficient for KMnO₃ was determined, as shown in Fig. 7.^11,33

It is important to note that Eqs. (4)–(10) are only applicable for linear systems. In ferroelectric materials, the inverse dielectric permittivity and stress may not follow a linear relationship because of nonlinear phenomena such as the depolarization, domain wall motion, and phase transitions.³² Thus, the effects of the nonlinear phenomena should be minimized under the applied stress/hydrostatic pressure to accurately measure the $Q$ coefficients in ferroelectrics. Figure 8 presents the stress-dependent dielectric properties for PMN-xPT crystals.^34,35 In single domain crystals, the stress perpendicular to the poling direction was found to stabilize the polarization state; at this time, the depolarization or domain wall motion was difficult to be induced. For this condition, the measured $Q33$coefficients were almost the same values as those measured from the S-P curves, as shown in Figs. 8(a) and 8(b). On the contrary, strong nonlinear phenomena existed when the compressive stress was along the poling direction, where a drastic increase in $χ33$ was caused by depolarization of the crystal, as shown in Fig. 8(c). For the case of hydrostatic pressure, $χkl$ and the pressure followed a linear relationship at low pressures, and the obtained $Qh$ was close to that derived from the piezoelectricity of the crystal. At high level of hydrostatic pressure, however, ferroelectric phase transition may be induced, leading to nonlinear characteristics; the measured $Qh$ deviates from the real value; therefore, great attention needs to be paid to this method for determination of electrostrictive coefficients of ferroelectrics.

For ferroelectric single crystals, the electrostrictive coefficients can be calculated from the piezoelectric coefficients, dielectric permittivities, and spontaneous polarization, based on Eq. (2). For perovskite crystals with $3m$⁠, $mm2$⁠, and $4mm$ symmetries, detailed information on their calculations is given in the Appendix.

With this method, high temperature and high field (electric field or stress) measurements were not required. The full set of electrostrictive coefficients can be easily determined because the full set of piezoelectric coefficients can be derived using conventional approaches, such as resonance, ultrasonic, and the Berlincourt methods.³⁶ For ferroelectric ceramics, reliable information could not be derived using this method because the extrinsic contributions of the piezoelectric and dielectric properties from the domain wall motion (which is a nonlinear phenomenon) are quite high, being 10%−40%. Of particular importance is that the domain walls can be eliminated by poling along the direction of the spontaneous polarization in ferroelectric crystals. Thus, this method was preferred for ferroelectric crystals rather than ceramics.

The spontaneous strain in ferroelectrics can be inferred from the lattice parameters. The $Q$ coefficients can be obtained as the ratio between the spontaneous strain and the square of the spontaneous polarization. Berlincourt and Jaffe provided reliable $Q$ coefficients for BT crystal with this method³⁷ 

where $Q33$ and $Q31$ are the electrostrictive coefficients for the BT crystals (in the standard coordinate system with 4mm symmetry), $S3$ and $S1$ are spontaneous strains, and $P3$ is the spontaneous polarization.

Based on this notation, the $Qijkl$ coefficients for PZT and doped PMN single crystals were determined, where the lattice parameters were measured by powder diffraction (either X-ray or neutron diffraction).^38–41 The nonlinear phenomena were neglected in this approach as no electric field or stress was applied for this method.

Based on the theoretical Debye model, Pattnaik and Toulouse developed an expression for the dielectric permittivity involving the electrostrictive coefficient M with a cubic symmetry, as given in Eqs. (13) and (14) (Ref. 42)

where $ε′(ω)$ and $ε″(ω)$ are the real and imaginary dielectric permittivities, respectively, $εs$ and $ε∞$ are the static and infinite frequency limits of the permittivities, respectively, $ω$ is the angular frequency, $τ$ is the relaxation time, $s11E$ is the elastic constant, $E0$ is the DC bias field, $Yr$ and $Yi$ are proportional to the mechanical strain. Detailed information on the coefficients $Yr$ and $Yi$ are given in Ref. 42. The electrostrictive coefficient $M13$ can be extracted using Eqs. (13) and (14).^42,43 The issue with this method was whether or not a simple Debye model is available for the studied dielectrics.

In this section, the experimental methods for determining the electrostrictive coefficients were surveyed. The methods used to determine the electrostrictive coefficients are summarized in Table I, where the characteristics and issues with each method are presented. It should be noted that the electrostrictive coefficients for ferroelectrics can also be determined using a first-principle approach, where the calculated electrostrictive coefficients for LiNbO₃ and BaTiO₃ crystals were reported to be in good agreement with the measured results.^44,45

Electrostriction is present in all crystal symmetries because it is a fourth rank polar tensor. Electrostriction is a measure of the polarization induced by ions shifting away from their natural equilibrium positions, giving rise to variations in the lattice parameters (strain). In centrosymmetric crystals, the induced shifts of equivalent ions almost cancel each other out, while the difference in the shifts because of potential anharmonicity generates strain.³ The typical potential of an ion-pair is shown in Fig. 9, where the energy required to compress the ion-pair (⁠$UC$⁠) is higher than the energy required to extend the ion pair (⁠$UE$⁠).^46,47 Based on the potential characteristics, the relative displacement between positive and negative ions (polarization) is prone to occur via extension of the lattice instead of compression, as shown in Fig. 10. As a consequence, the longitudinal electrostrictive coefficient $Q33$ is generally positive for ionic crystals.⁴⁸ 

A quantitative analysis on the above phenomenon, proposed by Uchino et al.,⁴⁹ is presented in this section. The potential energy of the ion pair is the simplest rigid ion model with +q and −q ions and can be written as

where $r$ is the distance between a cation and its nearest-neighbor anion and $r0$ is the equilibrium position. $U(r)$ and $U(r0)$ are the potentials for the nearest neighbors separated by distances of $r$ and $r0$⁠, respectively. The first and second terms in the energy potential are the harmonic and anharmonic components, respectively, where the high order terms were ignored. The coefficients $f$ and $g$ are associated with the crystal structure and the characteristics of the ions (charge and radius). Under an electric field, the average induced strain $⟨x1⟩$ and polarization $P1$ can be calculated using Boltzmann distribution

where $q$ is the charge of the ions. Consequently, the electrostrictive coefficient $Q11$ is given by

It can be seen from Eq. (18) that a high $g/f$ corresponds to a high $Q11$⁠, where the $g/f$ ratio represents the comparison between the anharmonic and harmonic energy components. The coefficients $f$ and $g$ are both positive; thus, the longitudinal coefficient $Q11$ is also a positive coefficient.

Similar to the electrostrictive effect, Maxwell stress contributes to the strain, which is also proportional to the square of the electric field/polarization. As an electric field is applied to the electrodes on the dielectrics, the electrostatic force resulting from the free charges squeezes or stretches the materials. This electrostatic force is called the Maxwell stress and occurs in all insulators with an electric field applied to the deposited electrodes.⁵⁰ The longitudinal strain for dielectrics induced by a Maxwell stress is given by

where $SM$ is the longitudinal strain induced by the Maxwell stress, $s$ is the elastic compliance of the material, $ε$ is the dielectric permittivity. The difference between the strains induced by electrostriction and the Maxwell stress is that the former is caused by the potential anharmonicity of the ionic crystal, while the latter originates from free charges on the electrodes. However, in S-E loop measurements, these two contributions are generally mixed with each other. When determining the electrostrictive coefficient Q, if the strain induced by Maxwell stress is not separated from the total strain, the contribution of the Maxwell stress to the apparent electrostrictive coefficients (⁠$MM$ or $QM$⁠) can be written as follows:

It can be seen from Eqs. (20) and (21) that the longitudinal electrostrictive coefficient induced by the Maxwell stress is negative. The contribution of the Maxwell stress is significant in materials with either a large elastic compliance (such as soft polymers) or a low dielectric constant (such as CaF₂) [as expressed in Eq. (21)],¹¹ while it can be ignored in ferroelectric crystals and ceramics.

The values of the electrostrictive coefficients for various dielectrics span many orders of magnitude. The longitudinal coefficients $Q33$ and $Q11$ for soft elastomers are on the order of 10⁴ m⁴/C², while those for perovskite ferroelectrics are about 10⁻¹ to 10⁻² m⁴/C², as listed in Table II.^{11,37,51–56} The electrostrictive $Q$ coefficients have been analyzed with respect to the microstructures and macroscopic properties of the materials.

In this section, we focus on the hydrostatic electrostrictive coefficients $Qh$⁠, because the available $Qh$ data for various dielectrics were more comprehensive in comparison with other coefficients (such as $Q11$⁠, $Q12$⁠).

The microscopic characteristics, including the ionic radius, ionic charge, and crystal structure, are believed to be the dominant factors for electrostriction. As listed in Table II, the crystals with similar microscopic characteristics exhibited similar $Q$ values. In the simplest rigid model, the effect of the microscopic characteristics can be symbolized by the coefficients $r0$⁠, $f$⁠, $g$⁠, and $q$ in Eq. (18). For crystals with the formulas ABO₃ and ABF₃, the coefficients $r0$⁠, $f$⁠, and $g$ were assumed to be of the same order because they have the same perovskite structure. Thus, the difference in the $Q$ values between the ABO₃ and ABF₃ crystals was dominated by the difference in the valences $q$ of the ions. The valence ratio between F⁻ and O²⁻ was 1:2. Thus, the electrostrictive coefficient for ABF₃ was expected to be about four times larger than that for ABO₃, which is in good agreement with the measured results listed in Table II. Sundar and Newnham presented detailed expressions for the coefficients $f$ and $g$ according to the Born-Mayer potential with which the electrostrictive coefficient Q was accurately calculated for a LiF crystal giving the value of 0.53 m⁴/C², which is consistent with the experimental result.⁵⁷ 

The effects of the microscopic characteristics on the electrostrictive coefficient Q were further demonstrated with the degree order of the B-site cations in perovskite materials. Table III shows that the magnitude of the electrostrictive coefficient strongly depends on the level of order-disorder in the B-site cation arrangement. It was concluded that the electrostrictive coefficient $Q$ increased with the order level of the B-site cations, from disordered to partially ordered, and then to ordered structures. A schematic crystallographic model, proposed by Uchino, can be used to explain this phenomenon (Fig. 11).³ In this model, a large “rattling” space was expected for the smaller B-site cations (B2) in the disordered structures, because the larger B-site cations (B1) prop open the lattice framework. Meanwhile, less “rattling” space was expected in the ordered arrangement where the neighboring oxygen atoms were densely packed around the smaller B ions (B2). When an electric field was applied to the disordered perovskites, the B2 cations with a large rattling space could easily shift without distorting the octahedron framework. Therefore, a larger polarization and lower strains were expected for unit magnitude of the electric field, resulting in smaller electrostrictive coefficients and larger dielectric permittivities. On the contrary, in the ordered perovskites with a very small rattling space, the B2 ions could not move without distorting the octahedron, leading to a smaller dielectric permittivity and larger electrostrictive coefficients.

To obtain accurate electrostrictive coefficients from the microscopic characteristics, more complicated models have been proposed.^58–62 Though microscopic characteristics are the dominant factor for the $Q$ values, the relationship between the electrostriction and macroscopic properties, such as the dielectric properties, elastic properties, and thermal expansion, were also established for evaluation of the electrostriction.

Figure 12 shows log(Q_h) as a function of log(s/ε) for various materials, including ceramics and polymers, where a linear relationship was observed, demonstrating that the electrostrictive coefficient $Qh$ was proportional to $s/ε$⁠. Using a simple atomistic model, Newnham et al.^1,11 proposed a relationship between dielectric/elastic and the electrostrictive effects. As an electric field is applied, the cations and anions in a crystal structure are displaced in opposite directions by $Δr$⁠. This displacement is responsible for the electric polarization, dielectric permittivity, and the electrostrictive strain. To the first approximation,^1,11 the electrostrictive coefficient is written as

where $ε$ is the dielectric permittivity, x is the strain, and P is the polarization. For a certain electric field, a larger $Δr$ corresponds to a larger $ε$⁠.

The electrostrictive coefficients were also closely associated with the elastic compliance $s$⁠. Through the converse effect, electrostriction is proportional to the change in the inverse dielectric permittivity $χ$ with stress $X$⁠, which is expressed as

where $Q$ is the electrostrictive coefficient, $Δχ$ is the variation of $χ$ on the stress $X$⁠, and $x$ is the strain on the stress $X$⁠. To the first approximation, $Δχ$ is proportional to the relative variations in the lattice parameters.

Combining Eqs. (22) and (23), the electrostrictive coefficient $Q$ is thought to be proportional to $s/ε$⁠. For materials with a similar elasticity, the dielectric permittivity can be used to evaluate the level of the electrostrictive effect. Figure 13 illustrates $Qh$ as a function of the dielectric constant for perovskite materials, where log(Q_h) follows a linear relationship against log(ε), as the elastic constants are of the same order for these materials.

For ferroelectric-related materials, the Curie-Weiss constant $C$ can be used to evaluate the level of the electrostrictive coefficients, because the dielectric permittivity is sensitive to temperature and phase transition points. According to thermodynamic analysis, the product of the electrostrictive coefficient $Qh$ and the Curie-Weiss constant $C$ was constant for all perovskite ferroelectrics.³ Table III gives the experimental data for $QhC$⁠, including ferroelectrics, relaxors, and antiferroelectrics, where $QhC$ was (3.1 ± 0.5) × 10³ m⁴K/C² for perovskite ferroelectric materials.^{2,7,37,51,63–68}

The electrostriction and thermal expansion originate from similar mechanisms (i.e., potential anharmonicity), as such the thermal expansion and electrostrictive coefficients are thought to be interrelated.³ Based on the similar manner in deducing the electrostrictive coefficients (Sec. III), the thermal expansion can be written as⁴⁷ 

where $kB$ is Boltzmann's constant and $r0$ is the distance between a cation and its nearest-neighbor anion at equilibrium. $f$ and $g$ are the harmonic and anharmonic components, respectively. Similar to electrostriction, the thermal expansion coefficient was zero when the anharmonic effect was ignored (⁠$g=0$⁠).

The thermal expansion coefficient $α$ against the hydrostatic electrostrictive coefficient $Qh$ for various dielectrics with cubic symmetry is plotted in Fig. 14, based on the reported results.³ From this figure, the empirical relation can be written as $α=4.2×10−5Qh0.5$⁠, which can be used to evaluate the electrostrictive effect from the thermal expansion coefficient.

In this section, the relationships between electrostriction and the macroscopic characteristics were reviewed for dielectric materials. Although the relationships could be used to evaluate the electrostrictive effect in dielectrics, they did not give accurate results. Some important factors were not considered, such as the crystal anisotropy, boundary conditions, and phase transitions. For ferroelectric materials, the effects of the domain walls, domain configurations,^69–73 and the poling conditions⁷⁴ on the electrostriction should also be considered.

Extensive investigations on the electromechanical properties of ferroelectric single crystals, ceramics, and thin films have been a focus because their electric field-induced strains and piezoelectricity are much higher in comparison with other non-ferroelectric counterparts. In this section, the electrostrictive effects of perovskite ferroelectrics are discussed.

The reported electrostrictive coefficients for various perovskite ferroelectrics are listed in Table IV,^3,75–84 including the data for single crystals and ceramics. The properties of single crystals are very important for materials exploration, as they allow for easy interpretation of the polycrystalline ceramic properties. On the other hand, some crystals are difficult to be grown; their ceramic counterparts can be used to evaluate the single crystal properties. The difference in the electrostrictive effect between single crystals and polycrystalline ceramics is discussed here.

For perovskite ferroelectric materials, the longitudinal electrostrictive coefficients $Q33$ of the polycrystalline ceramics and thin films are generally much smaller than those of ⟨100⟩ oriented single crystals. This phenomenon can be attributed to the following two factors. First, the strong anisotropy of electrostriction presents in perovskite crystals, which will be discussed in Sec. V C. Second, it is difficult to eliminate the contribution of the domain wall motion in polycrystalline ceramics during the electrostriction measurements. As a result, the measured data for the polycrystalline materials include contributions from the domain wall motion and grain orientation. In addition, for ferroelectric thin films, the values of the electrostrictive coefficients were expected to be further affected by substrate clamping, which generally decreased the Q coefficients.⁸⁵ It should be noted that when compared with the electrostrictive coefficient $Q33$⁠, the hydrostatic coefficient $Qh$ was independent of the orientation for perovskite ferroelectric materials.⁸⁶ Meanwhile, $Qh$ was generally determined by measuring the inverse of the dielectric permittivity under hydrostatic pressure, where the influence of the domain wall was quite small when compared with the high field measurements. Thus, the hydrostatic electrostrictive coefficient $Qh$ of the polycrystalline ceramics was found to be the same to that of the single crystals.

In the following sections, the effects of the ferroelectric phase transition and crystal orientation on electrostrictive coefficients $Q$ are evaluated. The electrostrictive coefficients $Q$ are discussed for perovskite ferroelectrics with respect to electromechanical properties, such as the electric field-induced strain and piezoelectric coefficients, in order to find alternative ways to enhance the piezoelectric activity.

The phase transition point in perovskite ferroelectrics, around where the dielectric and piezoelectric responses generally show peak values, has been extensively studied over the last 70 years. In this section, the electrostrictive coefficients $Q$ of perovskite ferroelectrics are discussed with respect to the phase transition points (induced by temperature or composition).

For typical ferroelectrics, such as BaTiO₃, PbTiO₃, and LiNbO₃, the electrostrictive coefficients $Q$ are temperature-independent. This notation is generally accepted based on thermodynamic analysis.⁵¹ Meanwhile, the $Q$ coefficients calculated from first principals at 0 K were consistent with the results measured at room temperature,^44,45 again revealing that the effect of temperature on the $Q$ coefficients is minimal for classical ferroelectrics. From the experimental aspect, the temperature-independent electrostrictive coefficients $Q$ were also observed in PZT and Bi_0.5(Na_0.82K_0.18)_0.5TiO₃ ferroelectric systems, as shown in Fig. 15.^83,87

To understand the temperature dependence of the electrostrictive behavior in relaxor/relaxor-PT single crystals, related parameters, including the piezoelectric coefficient, dielectric constant, spontaneous polarization, and electrostrictive coefficients, as a function of the temperature were investigated for single-domain PMN-xPT crystals, as shown in Fig. 16. The large variations in the dielectric and piezoelectric coefficients are closely associated with the PPT, where the increase in the dielectric and piezoelectric properties for PMN-0.28PT and PMN-0.32PT crystals with increasing temperature was attributed to the crystals approaching a PPT (⁠$TR−T$ and $TO−T$ are 95 °C and 80 °C, respectively), while the decrease in the dielectric and piezoelectric responses for PMN-0.37PT occurred because the PPT of PMN-0.37PT was below room temperature (∼$−45$ °C).²¹ The spontaneous polarization decreased with increasing temperature, caused by the pyroelectric effect.⁸⁸ Of particular significance is that the electrostrictive coefficients of the PMN-xPT crystals were insensitive to temperature and PPT, regardless of the strong temperature dependence of the piezoelectric/dielectric properties. The temperature-independent electrostrictive coefficients $Q11$ and $Q12$ were also observed in PMN crystals, as shown in Fig. 17.

It should be noted that the reported data for the PMN, PMN-PT, and Bi_0.5(Na_0.82K_0.18)_0.5TiO₃ systems were below the $TC$⁠. At temperatures above $TC$ (or above the Burns temperature $TB$ for relaxor ferroelectrics; for PMN and PMN-PT crystals, $TB$ ∼ 300 °C²⁹), the electrostrictive coefficient $Q11$ was found to decrease linearly with increasing temperature for BaTiO₃ and tin modified lead zirconate (PZS) crystals (Fig. 18),^89,90 which can be explained using the phenomenological theory.⁹¹ 

The MPB is an important characteristic for many perovskite ferroelectric solid solutions as it accounts for the enhanced dielectric, elastic, and piezoelectric properties. In this section, the effect of MPB on electrostriction is surveyed for perovskite ferroelectrics.

As listed in Table IV, the $Q$ coefficients were found to have similar values for compositions around the MPBs in PMN-PT and Bi_0.5Na_0.5TiO₃-BaTiO₃-K_0.5Na_0.5NbO₃ systems. However, Fig. 19 outlines the $Q$ coefficients for the PZ-xPT crystals, exhibiting a slight increase around the MPB composition.⁵¹ This was thought to be associated with the domain wall motion or phase transition during the electrostrictive measurements, because the $Q$ values for the PZT crystals were inferred from the measurements on the polycrystalline ceramics.

The electrostrictive coefficients (⁠$Q11C−Q12C$⁠)⁹² of [001] poled PMN-xPT and PZN-xPT crystals are derived from the data in Refs. 2^,18^,52^,93–97 and plotted in Fig. 20. The electrostrictive coefficients of the relaxor-PT crystals generally increase with increasing PT content, with no abnormal phenomenon observed around the MPB. Enhancement of the electrostrictive coefficient (⁠$Q11C−Q12C$⁠) with increasing PT content was attributed to the enhancement in the ordering degree of the relaxor-PT crystals by increasing the PT content, as discussed in Sec. IV. Vikhnin et al. theoretically explained the increase in the electrostrictive coefficient $Q$ as a function of the PT content by considering the quasi-local Ti-O stretching mode in PMN-PT solid solutions.⁶¹ It should be noted that for PZN-xPT crystals with x = 0.08 and 0.12, the electrostriction was comparable to that of the PT crystals, while the electrostrictive coefficients (⁠$Q11C−Q12C$⁠) for the PMN-xPT crystals were much lower than those of PT crystals, even in the composition range of x = 0.26–0.42. This result revealed that the B-site ordering degree for the PZN-xPT crystals was higher than that for the PMN-xPT crystals at the MPB composition.

Overall, the influences of the PPT and MPB on the $Q$ electrostrictive coefficients were minimal in comparison with the effects on the dielectric and piezoelectric properties in perovskite ferroelectrics. This demonstrates that the magnitude of the electrostrictive coefficient does not strongly depend on various phases (ferroelectric, antiferroelectric, or non-polar phases) and phase transitions. For perovskite ferroelectrics, the phase transition (ferroelectric, antiferroelectric, paraelectric) is accompanied by atomic displacements that are dominated by motion of the B cations in the oxygen octahedron network.³² From the respect of crystal structure, the difference between the paraelectric, antiferroelectric, and various ferroelectric phases is reflected in the displacement of ions, including the direction and magnitude of the displacement. However, it is important to note that the displacements are relatively small in perovskite structures, being <5% of the lattice parameter, revealing that the ion-pair potential energy and lattice parameters [the parameters in Eq. (18)] were not greatly affected by these small displacements. As a result, electrostriction in perovskite crystals is not sensitive to the ferroelectric phases and phase transitions.

The orientation dependence of the electrostrictive coefficients $Q33*$⁹⁸ for BT and PMN-0.28PT crystals were calculated by the axis transformation using the data in Table IV, as shown in Fig. 21. The maximum electrostrictive coefficient $Q33*$ was along the ⟨100⟩ direction, while the moderate and minimum values were along the ⟨110⟩ and ⟨111⟩ directions, respectively. Similar anisotropic behavior was observed in PMN-0.32PT, PMN-0.37PT, and other perovskite crystals, due to the fact that the anisotropic factors $(Q11C−Q12C)/Q44C$ is much larger than 1 for those crystals, as listed in Table IV (⁠$Q33*$ will be an isotropic coefficient if $(Q11C−Q12C)/Q44C=1$⁠). Similar anisotropic behavior was also reported in other ionic crystals possessing oxygen octahedral structures (LiNbO₃, LiTaO₃, KNbO₃, and Ba₂NaNb₅O₁₅ crystals), as listed in Table V.⁵⁴ As demonstrated in Ref. 53, the anisotropy of the electrostriction in perovskite crystals led us to believe that it is inherently associated with the oxygen octahedra.

Figure 22 shows the variation in the oxygen octahedral structure as polarization appeared along different crystallographic directions. When a B-site cation moves along the [001] direction, an oxygen anion O₁ will be “pushed” upward to release the energy that originated from compressing the distance of B-O₁, as shown in Fig. 22(b). Consequently, the lattice parameter along the [001] direction is elongated by d. The strain along the [001] direction can be expressed as d/a. Figure 22(c) shows the scenario where a B-site cation moves along the [110] direction. In this case, the oxygen O₃ and O₄ anions will move along the [110] or $[11¯0]/[1¯10]$directions to release the B-O₃ and B-O₄ pair potential because the direction of B-O bond (⟨100⟩) is not parallel to the [110] direction. It should be noted that the electrostrictive coefficient $Q33*$ along the [110] direction is reduced by the motion of the O₃/O₄ ions along the $[11¯0]/[1¯10]$ directions, because the $[11¯0]/[1¯10]$ directions are perpendicular to the [110] direction. Analogous to the [110] direction, a similar scenario occurs for the [111] direction, as shown in Fig. 22(d). As a B-site cation moves along the [111] direction, the oxygen ions can move along the directions perpendicular to [111] direction to release the B-O pair potential, which mitigates the $Q33*$ along the [111] direction. The difference between the [111] and [110] directions is that in the former, the oxygen ions are easier to move along the perpendicular direction to release the compression energy. This is because the angle between the [111] and B-O chain directions is 54°, which is larger than the angle between the [110] and B-O bond directions (45°). Based on this approach, electrostrictive anisotropy of perovskite crystals has been explained qualitatively, which is related to the oxygen octahedral structure. The maximum electrostrictive coefficient $Q33*$ was along the ⟨100⟩ direction because the B-O chains align along the ⟨100⟩ direction. Using the same approach, the electrostrictive anisotropy for specific crystals can be explained. For example, for fluoride perovskite crystals, the maximum longitudinal electrostrictive coefficient $Q33*$ is along the ⟨100⟩ directions, as the B-F chains align along this direction; while maximum $Q33*$ is along the ⟨111⟩ directions for the fluorite crystal CaF₂, as the Ca-F chains align along the ⟨111⟩ directions, as shown in Fig. 23.

In addition to the longitudinal electrostrictive coefficient $Q33*$⁠, the other electrostrictive coefficients for perovskite crystals also showed strong anisotropic behavior. Two examples are: (a) In the coordinate system shown in Fig. 24(a), the $Q32*$ coefficient was positive, indicating that the extensions occurred simultaneously parallel and perpendicular to the applied electric field⁸⁶ and (b) the maximum shear coefficient for the perovskite crystal is presented in the coordinate shown in Fig. 24(b), being equal to $Q1313*=12(Q11C−Q12C)$⁠.

In general, an electric field-induced strain can be written as

where the first and second terms represent the piezoelectric and electrostrictive strains, respectively, $P(E)$ is the electric field-induced polarization and $g$ is the piezoelectric voltage coefficient. The strains related to the high order of $P$ were ignored in Eq. (25). For piezoelectric crystals without ferroelectricity, the piezoelectric strain is usually much higher than the electrostrictive strain, such as in quartz crystals. On the contrary, based on the phenomenological theory, the strain in ferroelectrics with centrosymmetric parent phases is induced by the electrostriction only.³² These ferroelectrics include LiNbO₃, BaTiO₃, PbTiO₃, KNbO₃, and other related materials. Thus, Eq. (25) can be expressed as Eq. (26) for these materials

where $ε$ is the dielectric permittivity. In Eq. (26), the anisotropy and nonlinear properties for the dielectric response were not considered. Based on Eq. (26), the figure of merit used to evaluate the level of strain induced by a certain electric field is $Qε2$⁠. It was observed that a large $Q$ coefficient does not necessarily give rise to large electric field-induced strains, because a high $Q$ coefficient generally corresponds to a low dielectric permittivity, as discussed in Sec IV B 1 (⁠$Q∝1/ε$⁠). It was reported that the $Q$ coefficient in the relaxor ferroelectric PMN was about one fifth that of a typical ferroelectric BT, while the dielectric permittivity of PMN was 5–10 times higher than that in BT. Thus, PMN exhibited a superior electric field-induced strain when compared with BaTiO₃. In addition, perovskite materials generally showed a larger electric field-induced strain because the dielectric permittivity for perovskites is significantly higher than other ionic crystals.

It should be noted that domain switching can also be used to enhance the electric field-induced strain in ferroelectric crystals. Interesting results were reported for [001] oriented BT crystals with Mn dopants or preloaded stress in which ∼1% strain was induced with an electric field.^99–101 With a decrease in the applied electric field, Mn dopants or preloaded stress made the domains switch back to the directions that were perpendicular to the applied electric field, as shown in Fig. 25. Theoretically, the maximum strain induced by domain switching in [001] oriented tetragonal crystals can be expressed as $S3=(Q11−Q12)Ps2$ (as depicted in Fig. 25), which is much larger than that induced by lattice deformation. Through domain switching, the level of maximum strain is only related to the electrostrictive coefficients and the spontaneous polarization and not to the dielectric permittivity. However, the strain induced by domain switching is generally accompanied by a large hysteresis, limiting its potential application in electromechanical devices.

Analogous to the electric field-induced strain, the intrinsic piezoelectricity of ferroelectrics with centrosymmetry parent phases comes from the electrostrictive effect. Compared with the extrinsic contribution (domain wall motion), the electrostrictive contribution showed low hysteresis and high stability with respect to the frequency and electric field.^69,70,102 In addition, the sign of the piezoelectric coefficient $d$ can also be determined by the sign of the electrostrictive coefficient $Q$⁠. For example, the longitudinal piezoelectric coefficient $d33$ has a positive value for ferroelectric crystals and ceramics, whereas $d33$ has a negative value for ferroelectric polymers, because the electrostrictive coefficient was positive for ionic crystals and negative for polymers. Furthermore, the anisotropy of the electrostriction plays an important role in the piezoelectric anisotropy in ferroelectric crystals. To illustrate this scenario, the detailed relationship between the piezoelectric and electrostrictive coefficients are given for relaxor-PT crystals, whose parent phase and symmetry were cubic and $m3¯m$⁠, respectively.

According to Eq. (2) and the transformation of the coordinate system, the piezoelectric coefficient and dielectric permittivity for [001] poled rhombohedral crystals can be written as

where the symbols R and C indicate the rhombohedral 3m and cubic $m3¯m$ coordinate systems in which the property coefficients were measured (see details in the Appendix). An important characteristic in the relaxor-PT crystals is that the transverse dielectric permittivity $ε11$ is much higher than the longitudinal dielectric permittivity $ε33$⁠.²⁰ Thus, Eqs. (27) and (28) can be simplified for relaxor-PT crystals as

where $ε33[001]$ is the longitudinal dielectric permittivity for [001] poled rhombohedral crystals, $ε11R$ is the transverse dielectric permittivity measured in the standard coordinate system for rhombohedral 3m symmetry (see details in the Appendix), and $PSR$ is the spontaneous polarization of the rhombohedral crystals. Based on the same approximation, the piezoelectric coefficient $d33$ of [001] poled orthorhombic relaxor-PT crystals and [111] poled tetragonal relaxor-PT crystals can be written as

where $Q33R$ and $Q31R$ are the electrostrictive coefficients measured in the standard coordinate system for a rhombohedral 3m symmetry.⁹² 

As expressed in Eqs. (30)–(32), the longitudinal piezoelectric response of domain engineered relaxor-PT crystals is proportional to the product of the longitudinal dielectric permittivity $ε33$ and the electrostrictive coefficient (⁠$Q33−Q31$⁠). In Eqs. (30)–(32), the coefficient $d33$ was not only attributed to the longitudinal $Q33$ but also to the transverse electrostriction $Q31$⁠, because the polarization direction changed as the applied electric field deviated from the polar direction for domain engineered crystals. Table VI gives the piezoelectric coefficients, dielectric permittivities, and electrostrictive coefficients for the domain-engineered PMN-xPT crystals.¹⁰³ The piezoelectric coefficients for [001] poled rhombohedral and orthorhombic crystals were found to be much higher than those of [111] poled orthorhombic and tetragonal counterparts, though the latter possessed higher dielectric constants. This phenomenon can be explained by the anisotropy of the electrostriction, where the highest electrostrictive coefficients $Q33*$ were observed along the ⟨100⟩ directions in perovskite crystals. Thus, the high longitudinal piezoelectric response in perovskite crystals is expected in the ⟨100⟩ poled domain-engineered rhombohedral or orthorhombic crystals, because both the dielectric permittivity and electrostrictive coefficient are enhanced. On the contrary, for ⟨111⟩ and ⟨110⟩ poled domain-engineered tetragonal crystals, the dielectric permittivity is enhanced, but the electrostrictive coefficient greatly decreased. As a result, the piezoelectric coefficient in ⟨111⟩ poled tetragonal crystals doesn't benefit significantly from the engineered domain configuration, since $d33*/d33$ is much smaller in comparison with their rhombohedral and orthorhombic counterparts, as listed in Table VI.