Transduction modality near instability in domain engineered relaxor ferroelectric single crystals

First principles computations by Fu and Cohen [5] indicated that the large strain values observed in FE rhombohedral PZNT crystals under a [001] electric field can be attributed to polarization rotation from the rhombohedral [111] to the tetragonal [001] polar axis. Amin and Cross [6] calculated the strain associated with F_R − F_O, F_R − F_T, and F_T − F_O transitions using Devonshire theory. The calculated strain associated with the F_R − F_T transition was 1%, in a good agreement with the experimental value of ∼1.1%. The presence of FE monoclinic (F_M) and FE orthorhombic (F_O) states between the F_R and F_T phases in the PMNT and PZNT systems have been confirmed by diffraction, dielectric, and optical measurements [7–11].

Vanderbilt and Cohen [12] extended the Devonshire theory of perovskite FEs to account for the observed monoclinic F_m phase. Using spherical coordinates instead of the customary Cartesian coordinates, the energy function (Gibbs free energy) was represented by an analytic function defined on the surface of a sphere and truncated at the 8th power of the polarization vector (order parameter) normalized to unity. The parameter space was thus a unit sphere spanned by the polar and azimuth angles α and β respectively, where 0 < α < π and 0 < β < 2π. The FE phases were obtained by numerically searching for constrained minima of the energy function as a function of orientation. The constraints were taken such that the order parameter must lie along a symmetry axis, i.e. [001, 111], or [011] for the FE tetragonal, rhombohedral, or orthorhombic phases respectively, or in a symmetry plane (mirror) for monoclinic FE phases. Three monoclinic phases are admissible M_A, M_B, and M_C as illustrated in figure 1.

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A two-dimensional representation of the FE phase diagram in terms of the linearly independent parameters α and ® was obtained by mapping the unit sphere onto a two-dimensional plane. This model can only describe inter FE transitions, and not transitions to the high temperatures paraelectric (PE) state. In this model, temperature, composition, elastic, and electric variables are not explicit, but it was the first to provide first principles calculations of the phase behavior observed in these crystals.

Significant progress has occurred over the last two and half decades in growth techniques, crystal quality and homogeneity, boule size, and commercial availability. The main focus has been in: (1) binary relaxor-FE with the general formula: (1 − x)Pb(B_1/3Nb_2/3)O₃-(x)PbTiO₃ , where B can either be Mg or Zn, with x∼ 0.06–0.08 for Zn compositions, and x∼ 0.28–0.32 Mg compositions, (2) ternary compositions in the system lead indium niobate-lead magnesium niobate-lead titanate (PINMT) with the general formula xPb (In_1/2Nb_1/2)O₃ -(1 − x − y)Pb (Mg_1/3Nb_2/3) O₃-(y) PbTiO₃ , with y∼ 0.24–0.30 and pseudo-binary lead magnesium niobate (1 − x)Pb(Mg_1/3Nb_2/3)O₃ -lead zirconate titanate (x)Pb(Zr,Ti)O₃ (PMNZT). The single crystal growth technique depends on the relaxor-FE system. Modified and continuous feed Bridgeman (CFB) are used to grow binary PMNT and ternary PINMT single crystals [13, 14], the flux method is used for PZNT [15], and the solid state conversion growth method for PMNZT [16]. The relaxor-FE single crystal compositions of interest in acoustic transduction are those with high electromechanical coupling coefficients. For instance, the electromechanical coupling factor ${\kappa _{33}}\,$ for length extensional vibration modes is ${\kappa _{33}} = \sqrt {{{{d_{33}^2Y_{33}^{\text{E}}}}/{{{\varepsilon _0}}}}{K^{\text{T}}}}$, where, d₃₃ is the piezoelectric coefficient, $Y_{33}^{\text{E}}$ the short circuit Young's modulus, and ${K^{\text{T}}}$ the free (unclamped) dielectric constant, and ${\varepsilon _0}$ the free space permittivity (8.854 × 10⁻¹² F m⁻¹).

Compositions of interest in acoustic transduction devices (both active and passive) are located near a FE rhombohedral F_R–FE tetragonal F_T degeneracy, the so called morphotropic phase boundary (MPB) where the electromechanical properties are maxima. The relaxor-FE single crystals reported in this review are the result of multiyear collaborative research projects with the crystal growers under ONR and DARPA contracts [13–16].

The MPB is found in several perovskite FE systems with the general formula A(B^I _1−xB^II _x)O₃ . Besides, PZNT, PMNT, and PINMT, the lead zirconate titanate (PZT) solid solution system is perhaps the best-known and most extensively investigated. It comprises most of the technologically important piezoelectric compositions used in the electronics industry today. These compositions have a Zr, Ti ratio ∼1 near the MPB. In the composition-temperature phase diagram, a MPB may be considered as the temperature trajectory of the free energy degeneracy of two FE phases with distinct symmetry subgroups of the higher temperature PE symmetry group. In the present case, these are a FE tetragonal phase F_T (4mm symmetry) with the polarization vector along the four-fold symmetry axis, and a FE rhombohedral phase F_R (3m symmetry), with the polarization vector along the three-fold symmetry axis. Both 4mm and 3m are polar subgroups of the centrosymmetric, higher temperature PE phase m3m. A FE orthorhombic phase F_O (2mm symmetry) with the polarization vector along the two-fold symmetry axis is also allowed.

A computed phase diagram for perovskite FEs with the general formula A(B^I _1−xB^II _x)O₃ , with 1 ⩾ x⩾ 0, a tetragonal (4mm), and a rhombohedral (3m) FE as end members is depicted in figure 2(a). The computations were carried using a modified 6th order Devonshire free energy density function (elastic Gibbs) under free elastic and electric boundaries, and a linear T_C versus composition across the composition field [17]. Numerical analysis of the full three-dimensional elastic Gibbs free energy density, suggested that a rhombohedral–tetragonal degeneracy (morphotropy) can be generated over a wide temperature range without any interleaving orthorhombic region for the PZT system, figure 2(b). However, the orthorhombic FE phase is always close to but metastable with respect to the rhombohedral and tetragonal phases [17]. A superposition of the experimental and phenomenological phase diagrams is illustrated in figure 2(c). Heitmann and Rosetti [18, 19] have used the concept of vanishing anisotropy upon approaching the MPB in conjunction with a 6th order Devonshire series to mimic the MPB. Recently, Lv et al [20] developed a 10th order Landau–Devonshire energy function with a maximum polarization constraint to describe the dielectric, piezoelectric, and FE properties of PINMT relaxor FEs.

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The Pb(Zn_1/3Nb_2/3)O₃-PbTiO₃ and Pb(Mgn_1/3Nb_2/3)O₃-PbTiO₃ phase diagrams near the MPB are illustrated in figures 3(a) and (b) respectively. Similar to the PZT system, the MPB separates tetragonal (4mm) and rhombohedral (3m) FE regions. The makeup of the MPB in the binary (1 − x)Pb(B_1/3Nb_2/3) O₃-(x)PbTiO₃ system has been the subject of several structural investigations using high-brilliance high-resolution synchrotron radiation. It has been found that morphotropic (1 − x)Pb(Mg_1/3Nb_2/3) O₃-(x)PbTiO₃ compositions with 0.31 ⩽ x⩽ 0.37 are complex mixed states (figure 3(b)) of rhombohedral, tetragonal, monoclinic and orthorhombic phases [21]. As pointed out by the authors, diffraction patterns of relaxor-FE systems such as (1–x)Pb(B_1/3Nb_2/3)O₃-(x)PbTiO₃ are extremely difficult to analyze, especially for compositions close to a MPB. Two major factors contribute to this complexity: (1) compositional inhomogeneity due to the concentration gradient typical of crystal growth from melt of incongruently melting compositions, and (2) the broad diffraction maxima characteristic of disordered relaxor-FE systems. This complicates peak deconvolution of overlapped or partially overlapped reflections in the reciprocal (Fourier) space. Overlapping reflections in Fourier space are common in pseudo symmetric structures, such as F_T, F_R, F_O, and F_M of the present systems.

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Significant advances in compositional homogeneity have been achieved by JFE corporation [14] recent development of CFB techniques for growing relaxor-FE PINMT single crystals. This has resolved, or at least minimized the uncertainty in determination of phase symmetry from diffraction data.

Historically, similar arguments were presented in the early 70s regarding the MPB composition of PZT (figure 2(c)). Isupov [23] reported the possible existence of an extended region of mixed tetragonal and rhombohedral FE states for xPbZrO₃-(1 − x)PbTiO₃, Gur-Ari and Benguigui [24] suggested a coexistence region of the tetragonal and rhombohedral states for 0.49 ⩽ x⩽ 0.62. On the other hand, Kagekawa et al [25] noted that morphotropic xPbZrO₃-(1 − x)PbTiO₃ compositions prepared by a combination of wet-dry technique exhibited no coexistence of the tetragonal and rhombohedral states. Additionally, for compositions prepared by a dry technique, the observed (200) x-ray diffraction line broadening was attributed to compositional fluctuations rather than lattice strain.

Above the Curie temperature T_C, the prototypic high temperature, PE structure P_C from which the FE states are derived is cubic with space group Pm3m and a lattice constant a ∼ 0.4 nm (figure 4). The A-site can be occupied by a divalent cation such as Pb, Ba, Sr, Zr and the B-site by a tetravalent cation such as Zn, Nb, Mg, Nb, or Ti. There are two equivalent representations of the perovskite structure: (1) corner shared BO₆ octahedra, with the A²⁺ placed at ½, ½, ½ along the body diagonal as in figure 4(a), and (2) the A²⁺ is placed at the corners of the cubic unit cell (figure 4(b)). Below T_C, the B⁴⁺ cation is shifted along ±<100>_c, or ±<111>_c symmetry axes relative to the O²⁻ ₆ octahedra framework, developing a FE tetragonal P4mm phase with polarization vector along the four-fold symmetry axis, or a FE rhombohedral R3m phase with polarization vector along the three-fold axis. Other allowed FE phases are orthorhombic (Bmm2) with polarization vector along the two-fold symmetry axis, and monoclinic (m) with polarization vector in the mirror plane. Structural details for isomorphic barium titanate can be found in the classic textbook Ferroelectricity by Jona and Shirane [26].

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According to Shuvalov [27], knowing the spontaneous polarization symmetry group (∞ mm) and varying the direction of the spontaneous polarization with respect to the symmetry elements of G_p (the PE state symmetry group), one can find all possible FE groups G_F1, G_F2, resulting from G_P by applying Curie symmetry principle. The Curie symmetry principle may be stated as follows: The symmetry elements of the FE state group G_F consist of all the symmetry elements common to G_P and ∞ mm i.e., G_F = G_P ∩∞ mm such that G_F ⊂ G_P.

The number of orientation (domain) states associated with a phase transformation from a prototypic PE cubic P_C state, with a point group symmetry G_P (m3m) to a FE rhombohedral F_R point group (3m) or FE tetragonal F_T point group (4mm) G_F, is given by the index N of G_F in G_P. According to Aizu [28], there are six fully FE partially ferroelastic domain states present for the tetragonal (4mm) symmetry, eight for the rhombohedral (3m), twelve for the orthorhombic (2mm), and twenty-four for the monoclinic (m). Starting with a FE rhombohedral F_R single crystal, the engineered domains can be formed by applying a poling field (∼5–10 kV cm⁻¹) along the [001]_C, which will assemble the four F_R macro domain pattern with polarization vectors of each domain along one of the possible four [111]_C, [$\overline 1 11$]_C, [$1\overline 1 1$]_C, [$\overline 1 \overline 1 1$]_C, directions. The resulting macro-symmetry is 4mm and is often referred to as 4R due to the presence of only four of the rhombohedral variants. Similarly, a poling field along [011]_C will generate a macro-domain arrangement of <111> polarization vectors to form mm2 macro-symmetry that is often referred to as 2R. Domain engineered single crystal modes that were investigated for potential applications in sound projectors, and other transduction devices are discussed next. Unless otherwise stated, a right-handed Cartesian coordinate system is used as a reference frame. In figure 5, the crystallographic axes are embedded in the reference frame, and the applied uniaxial compression, drive field, and the induced strain directions are shown for the following domain engineered modes: (a) macro-symmetry 4mm, 33- length extensional (4R 33 mode), (b) macro-symmetry 2mm, 32- transverse length extensional (2R 32 mode), (c) macro-symmetry 2mm, 31-transverse length extensional (2R 31 mode), and (d) macro-symmetry 2mm, 33-mode length extensional (2R 33 mode).

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It was suggested [29] that the twin structure of domain engineered crystals with macro-symmetry 2mm, afforded by two domain states [111]_C and [$\overline 1 11$]_C are of the ferroelastoelectric type. A crystal is said to be ferroelastoelectric if it contains two or more domain states that can be switched by the simultaneous application of mechanical stress and electric field. Quartz is perhaps the best known ferroelastoelectric material. It belongs to the ferroic species 622F32, where 622 is the symmetry group of the high temperature phase, and 32 is that of the lower temperature. The letter F indicates ferroic behavior. Primary effects such as ferroelectricity, ferroelasticity, and ferromagnetism are forbidden in this symmetry species. Secondary ferroic effects such as ferroelastoelectricity are allowed and have been experimentally demonstrated [30]. The Dauphine or electric twin states are related by 180° rotation about the optic axis (three-fold symmetry axis). Hence, the positive a-axis in one twin component becomes the negative a-axis in the other twin component. Consequently, the piezoelectric coefficients d₁₁₁ and d₁₂₂ (=d₁₁₁) reverse signs in Dauphine twins. The driving potential for ferroelastoelectric state shifts can be synthesized in terms of the difference in one or more piezoelectric components Δd_ijk and the applied electric E_i and mechanical σ_jk fields. Cao and Amin [31] developed a theoretical model that enumerated the symmetry allowed ferroelastloelectric modes in crystals with macro-symmetry mm2. The driving potential inclusive of all the allowed modes for [111]_C and [$\overline 1 11$]_C state shifts under the combined application of mechanical stress and electric field was also developed. We list below the driving potential ΔG for one of the allowed modes, i.e. for a colinearly applied uniaxial compression σ₂, and electric field E₂ along the X₂ axis for 2mm domain engineered crystal (figure 5(b)). Experimental verification of this mode is presently underway,

$$\begin{align*}\Delta G = \Delta {d_{22}}{E_2}{\sigma _2}\end{align*}$$

where,

$$\begin{align*}\Delta {d_{22}} = \frac{{2\sqrt 3 }}{9}\left( {2{d_{15}} + 2{d_{31}} + {d_{33}} - 2\sqrt 2 {d_{22}}} \right)\end{align*}$$

and d_ij are the piezoelectric tensor coefficients for the 3m symmetry group [32].

The effect of an electric field applied along the X₃ direction (the domain engineering four-fold symmetry axis) of PZNT (4.5%PT) single crystals with macro-symmetry 4mm (figure 5(a)), was investigated by Park and Shrout [3, 4] and illustrated in figure 6(a). The mechanical compression–decompression response along the X₃ direction of PZNT (6%PT) single crystals is shown for several temperatures in figure 6(b) [22].

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The PZNT (6%PT) crystals have a high electromechanical coupling k₃₃ ∼ 0.93, a piezoelectric coefficient d₃₃ ∼ 2300 pC N⁻¹, and short circuit Young's modulus ∼9 GPa as deduced from small signal resonance—antiresonance experiments. The relationships between the strain magnitudes associated with inter-FE transitions that are polar subgroups of the symmetry group m3m, namely, the tetragonal 4mm, the rhombohedral 3m, and the orthorhombic 2mm were developed by Amin and Cross [6]. Assuming polarization continuity in the pseudo-cubic tetragonal (T), the rhombohedral (R), and the orthorhombic (O) states, i.e. P² _T = 2 P² _O = 3P² _R, the magnitude of the X₃ strain components are:

$$\begin{align}{S_{{\text{R}} - {\text{T}}}} = {\text{ }}2\left( {{Q_{11}}-{\text{ }}{Q_{12}}} \right){\text{ }}{P^2}_{\text{R}}\end{align} \tag{ 2.1 }$$

$$\begin{align}{S_{{\text{R}} - {\text{O}}}}{ = _{}}\left( {{Q_{11}}-{\text{ }}{Q_{12}}} \right){\text{ }}{P^2}_{\text{R}}\end{align} \tag{ 2.2 }$$

$$\begin{align}{S_{{\text{O}} - {\text{T}}}}{ = _{}}2{Q_{12}}{P^2}_{\text{O}}-{\text{ }}{Q_{12}}{P^2}_{\text{T}} = 0\end{align} \tag{ 2.3 }$$

where Q_ij are the electrostriction coefficients and P_R, P_O, P_R the polarization components along pseudo-cubic axes. Substituting Q₁₁ = 0.054 m⁴ C⁻², Q₁₂ = −0.027 m⁴ C⁻², and P_R = 0.25 C m⁻² in the above expressions we get for S_R–T a value of 1% which is in close agreement to the experimental values of 1.1% in figure 6(a). A possible transition path under field may occur via an intermediate monoclinic state M_A between the initial F_R (macro-polarized) and the final F_T (mono-polarized) states shown in figure 1. For S_R–O the calculated value of 0.50%, is close to the observed 0.45% (figure 6(b)). The forced orthorhombic state on compression will not be macro-polarized as the four equivalent <111> polarization vectors will rotate in the (111) mirror planes and eventually fold out into 90° and 180° domain states The reversible F_O − F_R transition upon decompression could be due to compensation charges on charged domain walls which pull the polarization vectors back to the F_R state. A possible transition path under compression may occur via an intermediate monoclinic state M_B between the initial F_R and the final F_O states as shown in figure 1. Further evidence will be discussed in the section dealing with x-ray and synchrotron studies. There is no strain difference associated with the F_O − F_T transition. In other words, for an observable difference in the strain, the condition 2 P² _O > P² _T must be fulfilled. There is no valid reason to accept this condition. At higher temperatures, the spontaneous strain and the critical stress magnitudes are decreased (figure 6(b)), as a consequence of decreased polarization levels and increased elastic softening respectively. Applied electric fields along the domain engineering axis, i.e. the four-fold symmetry axis, will stabilize the four rhombohedral (4R) domain ensemble to higher compression levels [22] as shown in figure 7.

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The transverse length extensional 32-mode geometry (2R 32 mode), with domain engineered macro symmetry 2mm (figure 5(b)) provides greater design flexibility in terms of mechanical loading (pre-stress) and electrical interconnect compared to the 33-mode tonpilz sound projector's design (4R 33 mode). Figure 8(a) illustrates the isothermal (20 °C) uniaxial compression–decompression response along the X₂ direction of domain engineered PZNT (6% PT) single crystal. The crystal has a high electromechanical coupling k₃₂ ∼ 0.91, a piezoelectric coefficient d₃₂ ∼ 2100 pC N⁻¹, and short circuit elastic compliance s^E ₂₂ ∼ 148 pm² N⁻¹ as deduced from small signal resonance–antiresonance experiments. The response of both the unbiased and dc biased crystal is shown in figure 8(a). The dc bias is applied along the X₃ axis, which is same as the [011] domain engineering (poling) axis. The 50 °C response is shown in figure 8(b) [29].

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In figure 8(a), starting with the initial F_R state at zero field (black trace), the strain increases with compression nearly linearly up to ∼12 MPa, then it is followed by a large increase ∼.45% associated with the F_R − F_o transition. A similar qualitative response is followed under a dc bias of 0.4 MV m⁻¹ (blue trace), except the transition occurs at a lower stress level ∼6 MPa. This is because both the applied field along the [011], and the compressive stress along the [100] will induce a F_R − F_o transition caused by polarization vectors [111] and [$\overline 1 11$]_C rotating in the mirror plane towards the [011] direction to form a mono-polarized F_O state. The inset depicts a plot of the intermediate stress vs. field levels responsible for F_R − F_O transition. Figure 8(b) depicts the response at 50 °C. Following the work of McLaughlin et al on 32-mode measurements of PMNT (32%PT) under stress and field [33], it is possible to construct a three-dimensional representation of the crystal stability in the stress-field-temperature parameter space as shown in figure 9. States below a phase boundary surface (marked with dashed outline) are stable F_R, those above are stable F_O. States which lie on the surface are poised near a F_R − F_O transition.

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This representation is useful for establishing design rules and drive parameters for sound projectors operating within the high coupling linear piezoelectric regime. Additional surface below that shown in figure 9 may be constructed to represent the crystal response under decompression, and the hysteretic effect associated with the reversible F_O − F_R transition. This is discussed further in section 4. There are a number of ways to cause the polarization vectors along [111] and [$\overline 1 11$]_C of the 2R domain engineered crystal to rotate: (1) a compressive stress only along X₂ will induce a F_R (macro-polarized)–F_O (mono-polarized) transition at ∼12 MPa in this composition, upon decompression, the F_R state is restored, because the F_O mono-polarized state is unstable under free electric and elastic boundaries; (2) an electric field only (∼0.75 m⁻¹) along the X₃ (the two-fold symmetry axis,) or (3) a combination of stress and field, i.e. any point on the stability surface (figure 9). A possible transition path under compression may occur via an intermediate monoclinic state M_B between the initial F_R and the final F_O states. Experimental evidence is presented in section 3 dealing with x-ray and synchrotron scattering experiments under stress and field.

There are differences in stress and field effects on the stability of the 2R domain ensemble of the 2mm macro-symmetry (2R 32 mode), compared to the 4mm (4R 33-mode). In the former both compressive stress and electric field will destabilize the 2R variants, while in the latter, compressive stress along the [001] direction has a destabilizing effect on the four F_R variants and drives them toward a F_O variant, whereas, applied electric fields along the [001] direction will stabilize the 4R ensemble to higher compression levels and at high enough electric field will destabilize them toward a single F_T variant.

The 31-mode stability under the influence of compressive stress and applied electric field of domain engineered 2R ensemble with 2mm macro-symmetry (2R 31 mode) (figure 5(c)) was investigated for a morphotropic PZNT composition with 5.5% PT content [34]. The room temperature uniaxial compression–decompression response at 0.0 and 1.5 MV m⁻¹ is illustrated in figure 10(a). The stability of the F_R state up to 90 MPa compression under zero field is believed to be the highest observed compression level among any crystal investigated. Also shown in figure 10(a) is the stress level (∼30 MPa) required to induce a stable, high coupling F_R state over high compression levels, for a 1.5 Mv m⁻¹ dc biased F_O initial state. Therefore, both conditions for large drive, and high compression are satisfied. These are desired design features for sound projectors operating at high drive fields. The strain versus field along the [011] is depicted in figure 10(b). The strain associated with F_R − F_O transition is given by, S_R–O [011] = [0.25 (Q₁₁− Q₁₂) + Q₄₄)] P_R ² = 2010 micro strain, in a good agreement with observed 2200. The phase transition from F_R to F_O under field proceeds by rotation of the 2R polarization vectors in the mirror symmetry plane to a final stable (under field) mono-domain FE orthorhombic F_O state, having the polarization vector along the [011] symmetry axis.

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The measured room temperature compression–decompression response did not exhibit any instabilities up to 90 MPa compression. This is no longer the case at 60 °C, as shown in figure 11. At ∼55 MPa compression, two transitions are identifiable (inset for details): first a transition from F_R to F_M near 55 MPa, followed by a second transition from F_M to F_T near 65 MPa compression. The phase transition from F_R to F_T under compression proceeds by rotation of the 2R polarization vectors in the mirror symmetry plane to an intermediate F_MA state at ∼55 MPa, upon a further increase of the compression, the onset of the final F_T state occurs near 65 MPa, with polarization vectors along the ±[100] symmetry axis. The calculated strain associated with F_R − F_T transition is 2300 macrostrain, compared to an observed 1500. The lower observed strain was attributed to a mixed state or possibly a partial delamination of the strain gauge at 60 °C.

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The 33-mode stability of domain engineered single crystals with macro-symmetry 2mm (2R 33 mode) under compressive stress and field revealed some unexpected results [35]. The observed anomaly near 50 MPa compressive stress in figure 12(a) was attributed to a FE rhombohedral F_R–FE tetragonal F_T transition. On decompression, however, the F_T state remained stable and did not revert back to the initial F_R. The phase transition from F_R to F_T under compression proceeds by the rotation of the 2R polarization vectors in the mirror symmetry plane leading to a final stable FE tetragonal F_T state, having the polarization vectors along the ± [100] symmetry axis. The F_T state is comprised of 180° tail-to-tail charged domain walls that remain stable after decompression. Using the values Q₁₁, = 0.066 m⁴ C⁻², Q₁₂= −0.032 m⁴ C⁻², and Q₄₄ =0.015 m⁴ C⁻¹ for the electrostriction coefficients [17], and 0.24 C m⁻² for P_R, as determined from a room temperature hysteresis loop measurement, the calculated strain associated with F_R − F_T phase transition is: S_R–T [011] = −0.5(Q₁₁ − Q₁₂ − 2 Q₄₄) P_R ² = −1958 micro strain, in a reasonable agreement with the experimental observation of ∼−2200 micro strain in figure 12(a).

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The strain-field response under different pre-stress levels is depicted in figure 12(b). For pre-stress levels up to 41 MPa the response remained linear, with a piezoelectric coefficient d₃₃= 1121 pm V⁻¹ as calculated from the strain derivative with respect to field. At a pre-stress level of 62 PMa, which is sufficiently large to stabilize the F_T state with 180° tail-to-tail charged domain wall structure (net zero polarization,) the strain derivative with respect to field is zero up to a field level ∼0.5 MV m⁻¹ and the F_R state becomes stable for fields >0.5 MV m⁻¹. The 33-mode stability under compressive stress and field of domain engineered single crystals fabricated from a ternary PINMT composition, with macro-symmetry 2mm revealed similar behavior as shown in figures 13(a) and (b).

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The role of charged domain walls in driving the reversible inter FE F_R − F_O transition is evident in the 33- and 32-modes with 4mm (4R 33 mode) and 2mm (2R 32 mode) macro symmetries. However, this is not the case for the 33-mode geometry with 2mm macro symmetry (2R 33 mode) as demonstrated in crystals fabricated from two different ternary systems, i.e. PMNZT and PINMT. In both systems, a uniaxial compression causes the 2R domain state to rotate in the mirror symmetry plane, leading to a stable 180° tail-to-tail (2T) domain state, under free electric and elastic boundaries. This is further discussed in the conclusion.

Transduction modality near an inter-FE instability in domain engineered single crystals was developed by Finkel et al nearly a decade ago [36, 37]. The motivation was to harness the large strain and polarization changes associated with inter-FE transitions in domain engineered crystals using low drive field or low mechanical excitation. For practical purposes, the choice was to further investigate this modality in the 2mm domain engineered ternary PINMT single crystals configured in the 32-mode geometry (2R 32 mode). Figure 14 illustrates the uniaxial compression–decompression response at several dc bias fields of domain engineered PINMT single crystal configured in the 32-mode. The basic idea is to mechanically clamp the crystal just below the critical stress (10 MPa at 0 bias) that is required to induce the for F_R − F_O transition. At this bias stress level, small oscillations in stress will drive the crystal in and out of the transition. Alternatively, referring to the three-dimensional diagram (figure 9), bringing the F_R state to any point close to phase boundary surface, e.g. by a mechanical compression, makes it possible for small mechanical, electrical, or thermal oscillations to drive the crystal through the transition cycle F_R − F_O −F_R. The associated large output can, therefore, be harnessed by a proper circuit design.

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The drive field and generated strain for the 32-mode PINMT single crystal have been investigated as a function of loading rate. A 5000 micro-strain amplitude oscillation was generated under a drive field frequency of 1–2 Hz, and amplitude ∼0.1 MV m⁻¹ as shown in figure 15(a). The crystal withstood continuous drive sweeps, either harmonic or rectangular field ramp (figure 15(a) inset), switching for nearly 10⁵ cycles before it fractured. Compared to the results of Shrout and Park [3, 4] in figure 2(a), the present drive field is more than an order of magnitude lower. A compact flextensional low frequency, high source level sound projector using the transduction modality near inter-FE instability in PINMT is shown in figure 15(b).

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In another development, Dong et al demonstrated the outstanding energy harvesting capabilities of this transduction modality [38, 39]. This promising performance gain of has been demonstrated in other device concepts. This is the subject of the next sections.

The effort in crystal growth of high electromechanical coupling crystals has been focused on four main compositions, namely, binary or pseudo-binary morphotropic FE rhombohedral F_R such as PMNT, PZNT, PMNZT, and more recently, the CFB growth of ternary PINMT. The CFB method provides significant improvements in (1) compositional uniformity, both in-plane and axial, over large size boules, (2) stability of the electromechanical properties under compressive stress, large drive, and temperature, and it promises the potential of cost reduction.

The small signal electromechanical properties of 33- and 32-modes bars fabricated from a 28 mole % PT morphotropic composition are listed in table 1. A three-dimensional representation of 32-mode stability of 2mm domain engineered PINMT single crystals (2R 32 mode) in the stress-temperature field parameter space [40] is depicted in figure 16.

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The AF–FE and the FE–PE modalities have been demonstrated to significantly enhance some applications. These applications are briefly discussed as they have relevance to the interphase modalities of the relaxor single crystals. The AF–FE transduction in modified polycrystalline lead zirconate titanate stannate to generate large strain was discussed in [41, 42]. This modality does not rely on bringing the AF state close to instability using a bias stress prior to triggering the transition. Instead, the transition is triggered by applying a significantly large field to the AF state that induces the FE state in compositions just to the AF side of the phase boundary. Pan, Zhang, Bhalla, and Cross [42] have demonstrated in modified lead zirconate titanate stannate, strain levels up to 0.85% at fields ∼6 MV m⁻¹. This drive level is ∼30 times larger compared to the field required to generate the same strain level in domain engineered crystals poised near an inter FE transition.

The FE–PE transduction modality was motivated by the need for inexpensive, compact, uncooled, handheld IR imaging systems in the medium wavelength range 7.5–14 m, corresponding to the thermal black body maximum near room temperature. Bandgap infrared (IR) detectors such as mercury cadmium telluride operate at low temperature around 77 K to obviate thermal noise, they require cooling systems, and they are expensive. This modality utilizes the pyroelectric effect near a FE–PE phase transition where it is maximum. The vertically integrated technology for system production utilizing this modality was pioneered in the 80s and 90s by GEC Marconi in the United Kingdom using polycrystalline FE lead scandium tantalate (PScT) and Texas Instruments (TI) in the United States using polycrystalline FE barium strontium titanate (BST).

We briefly describe the TI hybrid approach [43–45]. This approach relies on the large polarization change with temperature (pyroelectricity) of a poled and biased FE material (BST) poised near its FE–PE transition temperature T_C (Curie temperature). Figure 17(a) shows the polarization change with temperature and the pyroelectric response of FE barium strontium titanate (BST) wafer with a Curie temperature T_C near room temperature ∼23 °C. The wafer is biased at 2.5 V/mil (0.1 MV m⁻¹) to prevent depoling. The detector consists of a densely populated two-dimensional focal plane array fabricated from BST pixels (320 × 240) on a size scale 50 × 50 × 25 μm³ per pixel, directly bonded to the readout integrated circuit (ROIC) as shown in figure 17(b). Figure 17(c) shows the array implementation of pixel unit cells, where each pixel is represented by a capacitance connected to a CMOS inverter pre-amplifier with a large feedback resistor, a gain stage, and address switch.

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An antireflection Ge window covers the package and allows IR transmission in the range 7.5–14 m spectral region. An IR lens projects the IR image on the focal plane array. The array is mounted on a thermoelectric cooler for optimum operation near T_C, and a chopper provides thermal modulation at 30 Hz. The ROIC filters, amplifies, samples, and multiplexes the detector signal one row at a time, and delivers the output at a standard RS 170 rate. The imaging system (handheld, helmet, or dashboard mounted) operates at room temperature, thereby eliminating the need for cryogenic cooling. A useful measure of the system sensitivity is the noise equivalent temperature difference (NETD). This is the temperature difference at the scene ΔT_scene, required to generate an rms signal-to-noise (S/N) ratio = 1. An impressive NETD of less than 0.02 K was achieved.

Transduction modality near an instability in domain engineered single crystals provides a significant advantage for generating large strain at a much lower field when compared to the traditional linear-mode piezoelectric and the AF–FE transduction modalities. For applications in infrared point detectors and thermal imaging arrays, its benefit is not well established, simply because of the lack of experimental data. The large polarization change associated with the F_R − F_O transition, and the F_R − F_T transition, that can be triggered by small oscillations in mechanical stress, electric field, or temperature, provides the potential for their applications in IR imaging technology. Unlike BST imaging arrays, the requirement for dc bias is obviated.

In this section x-ray diffraction studies are reviewed. These include studies of domain engineered single crystals under the combined application of mechanical compression and electric field [46], and recent grazing incidence [47] and synchrotron investigation using high brilliance high resolution beam [48]. The diffraction experiment geometry and frames of references are depicted in figure 18. The diffraction circle for the ${{\theta } /{{2\theta }}}$, and the ${{\omega }/{{2\theta }}}$ scan-modes (reciprocal space mapping) is shown. The measurement reference frame (X₁, X₂, X₃) is depicted in figure 18 bottom along with the crystallographic axes of the domain engineered [011] poled, 32-mode single crystal with (macro-symmetry 2mm). The applied electric field E₃ and the uniaxial compression $\sigma 2$ directions are marked. The two FE rhombohedral F_R variants P_r [$\bar{1}$11] and P_r [111] under free elastic and electric boundaries are depicted on the upper left. Note, the original pseudo-cubic <100> axes are taken such that the [001] axis is along X₃, and the right-handed coordinate system is completed with X₁ and X₂ along the [100], and [010] respectively. The mono-polarized FE orthorhombic state F_O under E > E_C, or $\sigma > {\sigma _c}$ is shown on the upper right.

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${{\theta }/{{2\theta }}}$ x-ray diffraction scans were conducted using Cu K_a radiation for a [011] poled PINMT single crystal, with ∼29% PT content on the rhombohedral R3m side of the MPB. The diffraction pattern in figure 19(a) is taken with 19 MPa compression and 0 MV m⁻¹ field applied to the crystal (sufficient to induce the F_R − F_O transition). The red trace is that of the F_O state (040) and (400) reflections. The application of −0.18 MV m⁻¹ electric field was sufficient to pull back and stabilize the F_R state, as demonstrated by the rhombohedral (220) and (202) reflections (blue trace.) The integrated intensity of the rhombohedral (220) and the orthorhombic (040) peaks scanned at 19 MPa compression (the critical stress) as a function of ±0.20 MV m⁻¹ harmonic drive field are shown in figures 19(b) and (c). The results clearly demonstrate the switching between the F_R and the F_O states as the crystal being cycled by the electric field under compression. The electric field dependence of the rhombohedral lattice spacing d (220) and the orthorhombic d (040) is shown in figure 19(d), both states behave as linear piezoelectric.

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Reciprocal space maps figure 20(a) of the pseudo-cubic (0 2 $\bar{2}$) reflection at 0 MPa clearly show the crystal to be predominately rhombohedral, with two clear (220) and (2 0 $\bar{2}$) reflections. At 20 MPa the two rhombohedral variants of the domain engineered crystal collapse into one in the orthorhombic state, with the (040) and (040) reflection present. The mono-polarized orthorhombic state has its polarization vector along the orthorhombic a-axis, the X₃ direction as shown in the upper right corner of figure 19(a) The calculated lattice parameters of the FE rhombohedral state at 0 MPa are a = 0.404 16 (7) nm and α = 89.87 (3). And those of the FE orthorhombic state at 20 MPa are, a = 0.574 96 (11) nm, b = 0.571 804 (3) nm, and c = 0.401 76 (59) nm, the numbers between () represent the standard deviation in the least significant digit(s).

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The axial strain along X₂, and the transverse components along X₃ and X₁ for the F_o state (macro symmetry 2mm) are displayed versus the ±0.20 MV m⁻¹ harmonic drive field in figure 21(a) at 19 MPa compression. The crystal exhibited a slight fatigue after ∼40 million cycles (inset). The volume fraction of the F_O phase is depicted in figure 21(b), this is qualitatively similar to the x-ray intensities of figure 19(c). Figure 21(b) confirms that the entire crystal transforms cooperatively between F_O and F_R. Arguably, the volume fraction cannot be negative as shown in the region below the zero crossings in figure 21(b). This negative portion of volume fraction is attributed to the simple model used in the calculation with experimental data input. A more involved model would need to account for domain shape and propagation.

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The Gibbs free energy difference at free boundaries $\Delta$ G between the FE rhombohedral F_R and orthorhombic F_O states, can be written in terms of the energy difference components near the critical stress (19 MPa) and field (0.18 MV m⁻¹) as follows,

$$\begin{align*}\Delta G & = - \Delta {P_3}{E_3} + 1/2\Delta {\kappa _3}E_3^2 - \Delta {d_{32}}{E_3}{\sigma _2} - \Delta {\varepsilon _2}{\sigma _2}\nonumber\\ & \quad + 1/2\Delta \left( {1/Y} \right)\sigma _2^2\,\left( {{\text{J}}\,{{\text{m}}^{ - {\text{3}}}}} \right).\end{align*}$$

The first term in the right-hand side is due to the polarization difference between the orthorhombic and rhombohedral states. The second, third, fourth, and fifth terms are the dielectric permittivity, piezoelectric, transition strain, and elastic contributions respectively. The calculated ΔG of ∼0.07 × 10⁶ J m⁻³, is in the same order of magnitude as that derived from a modified Devonshire theory for morphotropic PZT compositions [17]. Such a small difference in energy accounts for the agility of the F_R − F_O inter-FE transition by an electric field, mechanical stress, or a combination of both. The transition from a macro-polarized F_R ensemble to a mono-polarized F_O state occurs via polarization rotation in a symmetry (mirror) plane of the two rhombohedral variants. Because the mono-polarized F_O state is not stable under free elastic or electric boundaries, it reverts back to the original macro-polarized F_R ensemble (lower energy state) upon the release of the applied field and/or the mechanical stress. The calculated latent heat ΔQ from the Clausius equation is $\Delta Q = T\Delta S = T\Delta \varepsilon \,{\left[ {\partial {\sigma _{\text{C}}}/\partial T} \right]_{\text{E}}} = T\Delta P{\left[ {\partial {E_{\text{C}}}/\partial T} \right]_\sigma }$ = 1.5 × 10⁵ J m⁻³, in a close agreement to theoretical values [18, 19].

Application of either mechanical stress along X₂, or electric field along X₃ will lead to a slight monoclinic distortion (M_B or M_C) of the rhombohedral symmetry, via polarization rotation in symmetry planes. Upon further increase of the loading, the final state will be orthorhombic. The monoclinic distortion is below the resolution limit of the present x-ray diffraction system. Further experiments using a high resolution-high brilliance Synchrotron source have overcome this obstacle and revealed some interesting results as explained below.

The diffraction pattern of pseudo symmetric structures such as FEs is characterized by partial overlap of diffraction maxima. This is due to the relatively small strain associated with the PE–FE phase transition. Relaxor FEs such as PMNPT pose an additional problem due to the broad nature of the Bragg peaks. The existence of a strained surface layer with different property and or symmetry from the bulk crystal, or ceramic, because of mechanical processing, such as griding, or polishing has long been known [49–51]. For instance, early x-ray diffraction studies of polycrystalline tetragonal barium titanate by Subbaro et al [49], demonstrated the drastic effect of polishing on the reorientation of 90° ferroelastic domains. Almost a complete switching of c- to a- domains occurred as deduced from the diffraction pattern. A subsequent diffraction study by Amin et al [50], on semiconducting FE barium titanate confirmed the same. Additionally, extensive plastic deformation of barium titanate because of grinding was inferred from x-ray line broadening and the FE–PE transition temperature [51]. Furthermore, symmetry and structural information from a diffraction experiment are limited by the penetration depth of x-rays, which is only a few tenth of a micron below the crystal surface. Therefore, it may be difficult in many cases to deconvolute the diffraction pattern to unequivocally deduce the correct symmetry of the crystal. There is a plethora of publications in the open literature that address these issues in relaxor-FE single crystals using x-ray and synchrotron diffraction. We summarize below the results from two recent diffraction experiments under the application of electric field and mechanical stress.

In situ electric field study of surface effects in domain engineered [011] poled rhombohedral PMNPT by grazing incidence x-ray diffraction was conducted by Finkel et al [46] In this study the diffraction data was collected as a function of depth (up to 0.2 µm below the surface), electric field, and temperature. The d-spacing of the probed hkl reflections were calculated from the measured peak position, assuming a pseudo cubic structure with lattice parameter 0.4040 nm. The results showed that at depth up to the maximum accessed (∼0.2 µm), the d-spacings were below and have not reached the bulk value. This is indicative of a strained surface layer, with a strain gradient across. For instance, based on d (010) measurements at 40 °C, the strain varied from ∼0.5% at a depth of ∼0.005 µm to ∼1% at ∼0.200 µm below the surface. The measured depth dependence of the d-spacing of selected reflections, and strain calculation as a function of field, identified those planes which have a higher sensitivity to field, hence providing an opportunity for strain optimization in devices such as energy harvesting and magnetoelectric (ME) composites. Recent synchrotron and optical investigation by Finkel et al [48] of domain engineered [011] poled rhombohedral PINMPT single crystal demonstrated that near a compressive stress of 30 MPa the two rhombohedral twins transformed to a twined monoclinic M_B structure based on symmetry arguments. At a higher compressive stress, the twins collapsed to a single phase dubbed M^* which could be described by a monoclinic symmetry. Finally, one should keep in mind, as discussed earlier, that structural information derived from x-ray diffraction is penetration depth limited to a few microns below the crystal surface. Penetration depth calculations for PZNT domain engineered single crystals showed that the x-ray intensity drops to only 10% of the incident intensity at a depth of 15 µm from the surface, giving rise to a penetration of 10–20 µm [52].

Piezoelectric materials can be used to convert mechanical work to electrical energy. Mechanical work is done when a force is applied to a surface of the crystal. As the crystal surface displaces, the force acting over a displacement does work. Forces on a surface are expressed as tractions (force per unit area), and the rate mechanical work is done is given by equation (4.1),

$$\begin{align}{\dot W_m} = \mathop {{{\mathop \int \nolimits}}}\limits_{} {t_i}{\dot u_i}{\text{d}}S\end{align} \tag{ 4.1 }$$

where ${\dot W_m}$ is the rate work is done by the mechanical force as the surface displaces (J s⁻¹). ${t_i}\,$ are the components of the traction vector (N m⁻²) and are equal to the dot product of the stress tensor just beneath the surface with the unit normal to the surface. ${t_i} = {\sigma _{ji}}{n_j}$, ${\dot u_i}$ are the corresponding displacement rate vector components (m s⁻¹), and $S$ is the surface area (m²). The work done in a loading and unloading cycle is found by integrating equation (4.1) over the period of the cycle. The work rate is written in terms of a volume integral by substitution of the stress and unit normal, and use of the divergence theorem. When the material behavior is rate independent, the rates can be replaced with increments; and when it is path independent (no hysteresis, a good example being linear elastic), the work done can be written in terms of the end points of the path. This gives equation (4.2) where the final term comes from the linear elastic case where ${\sigma _{ij}} = {c_{ijkl}}{\varepsilon _{kl}}$ and ${c_{ijkl}}$ are the components of the stiffness tensor,

$$\begin{align}\frac{W}{V} = \int\limits_1^2 {\sigma _{ij}}d{\varepsilon _{ij}} = \frac{1}{2}{\sigma _{ij}}{\varepsilon _{ij}}.\end{align} \tag{ 4.2 }$$

Consider applying a compressive load cycle to a linear elastic material. Positive work is done in the loading half of the cycle (work is done on the material by the load), and an equal amount of negative work is done during the unloading cycle (work is done by the material on the load). The work done in the loading half of the cycle is the area under the load–displacement curve and the work per unit volume is the area under the corresponding stress–strain curve. The amount of work done on the loading half of the cycle to a given maximum load depends on the stiffness. Less work is done on a stiffer material because the resulting displacement and strain components are smaller. Upon unloading, work is done by the material on the load. The work done on a linear elastic material in a closed loading–unloading cycle is zero.

Electrical work is done on a material by moving surface charge from one location on the surface to another in the presence of an electric potential difference between those two points. The rate electrical work is done on the material, ${\dot W^e}$ (J s⁻¹), is given by equation (4.3)

$$\begin{align}{\dot W^e} = {\mathop \int \nolimits}\phi \dot \omega {\text{d}}S\end{align} \tag{ 4.3 }$$

where $\varphi$ is the electric potential (V) (recall one volt is one Newton-meter per Coulomb or Joule/Coulomb), and $\dot \omega$ is the rate surface charge density is changing (C m⁻²s⁻¹). The surface charge density is equal to the normal component of electric displacement, $\dot \omega = {\dot D_j}{n_j}$. With this substitution, using the divergence theorem, and assuming rate and path independence, gives equation (4.4),

$$\begin{align}\frac{W}{V} = \int\limits_1^2 {E_m}{\text{d}}{D_m} = \frac{1}{2}{E_m}{D_m}.\end{align} \tag{ 4.4 }$$

Equations (4.2) and (4.4) are used to identify ideal loading paths for mechanical energy harvesting using piezoelectric materials.

A linear piezoelectric material has a higher stiffness under open circuit conditions, denoted $c_{_{ijkl}}^D$, than under short circuit conditions because charge cannot flow during open circuit loading. This generates a back electric field that restricts the mechanical displacement via the piezoelectric effect. Under short circuit loading the stiffness is lower, denoted $c_{_{ijkl}}^E$, because charge flows and the displacement is not restricted by a back electric field. Figure 22 illustrates an ideal loading cycle for converting mechanical work to electrical work. In this cycle, the material is loaded under open circuit conditions. The material is loaded mechanically, O-A in figure 22(a). This induces an electric field in the material, O-A in figure 22(b). Next, electrical work is done as the charge is drained into a circuit (A-B in figure 22(b)). The material mechanically displaces at constant stress (A-B in figure 22(a)). Additional mechanical work is done during this displacement. The material is then mechanically unloaded under open circuit conditions (B-C), and electrical work is again done as the charge is drained (C-O) as depicted in figure 22(b). In this ideal case, the area within the two curves is identical. The mechanical loop is traversed clockwise, corresponding to work being done on the material. The electrical loop is traversed counter clockwise, corresponding to electrical work being done by the material on a circuit.

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In developing these idealized cycles it was assumed that there is no material hysteresis (loss) and that the material behavior is rate independent. In general, neither assumption is valid for FE materials. Upon changing the load, either electrically or mechanically, domain walls move. This can be accompanied by some diffusion of charge carriers. This results in material hysteresis affecting the loading cycle. The hysteresis shows up as heat being generated. Under sinusoidal loading it is often represented as a loss tangent ($\tan \delta$). Relaxor FE crystals are domain engineered to minimize the driving force for domain wall motion. This results in the material being very low loss.

When the material behavior is dissipative, heat is generated as either a stress or an electric field is cycled. Under cyclic loading, this heat will build up unless it is removed through convection, conduction, or radiation. The dissipative processes in oxide FEs are generally related to domain wall motion and a small amount of local electrical conduction. In the case of phase transformations this can be associated with losses in the process, or a difference in energy reversibly stored within the different crystal structures (enthalpy of transformation).

As discussed in section 1, part E, loading relaxor FE single crystals in the vicinity of a phase transformations can enhance the magnitude of the electro-mechanical energy conversion. This is discussed for both the [001] cut (4R 33 mode) and the [011] cut (2R 32 mode). Sufficiently high compressive loading of the [001] cut and poled crystal in the <001> direction drives it from the 4R symmetry to the 4O symmetry. This reduces the polarization from the initial remanent polarization of around 0.38 C m⁻² to near zero polarization, a very large polarization change in a single cycle. The effect is different in the [011] cut and poled crystal. A sufficiently high compressive loading of the [011] cut and poled crystal in the [100] direction drives it from 2R symmetry to 1O symmetry. This increases the polarization in the <011> direction.

Idealized open circuit and short circuit loading and unloading curves are shown for the [011] cut in figure 23 with the mechanical loading along the [100] direction and the polarization along the [011] direction. Under short circuit loading and unloading there is a phase transformation from 2R to 1O and back to 2R. This phase transformation produces a hysteresis loop in the loading–unloading curve that is shown as the white shaded area in figure 23(a) associated with loading path O-A'-B'-C'-E'-A'-O. In this crystal cut, compressive stress increases the polarization. Under open circuit loading, the back electric field that opposes the polarization change under open circuit conditions is positive for compressive loading and negative for compressive unloading.

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The ideal energy harvesting cycle with a phase transformation present involves loading the material under open circuit conditions (path O-A) allowing electrical work to be done by the linear piezoelectric effect (path A-B), allowing additional electrical work to be done by the phase transformation (path B-C) unloading under open circuit conditions (path C-E), and again allowing electrical work to be done by the linear piezoelectric effect and the reverse phase transformation (path E-F-O) as shown in figures 23(a) and (b). The shaded area in figure 23(a) is equal to the shaded area in figure 23(b) and represents the energy per unit volume converted from mechanical to electrical.

This R-O phase transformation is temperature dependent. As the material is cycled and heats up, the threshold for the forward and reverse phase transformations change. A brief discussion of heat added and heat generated by dissipative processes provides insight into self-heating and keeping the material or device cool.

The addition of heat from external sources or from internal heat generation results in an increase of temperature and entropy of the material. The rate of thermal energy added to the material,$\dot Q$ (J) is given by equation (4.5)

$$\begin{align}\dot Q = {\mathop \int \nolimits}\dot r{\text{d}}V - {\mathop \int \nolimits}{\dot q_i}{n_i}{\text{d}}S\end{align} \tag{ 4.5 }$$

where $\dot r$(J m⁻³) is the rate of thermal energy addition from external sources per unit volume, ${\dot q_i}$ (J m⁻²) are the vector components of the outward heat flux, and ${n_i}$ are the components of the unit normal to the surface. The negative sign is to account for thermal energy being added to the material.

The rate work is done on the material plus the rate heat is added are equal to the rate of increase of the kinetic and internal energy of the material.

The expression for energy balance is given by equation (4.6)

$$\begin{align}{\dot W^m} + {\dot W^e} + \dot Q = \dot K + \dot E\end{align} \tag{ 4.6 }$$

where $\dot K$ is the rate of increase of kinetic energy, and $\dot E$ is the rate of increase of internal energy. Energy can be stored within a material as strain energy, electrical energy, thermal energy, or configurational energy. In the relaxor FE single crystals, the configurational energy included domain wall energy that changes with domain wall density, and energy associated with different crystal structures when phase transformations take place.

Upon substitution, the energy balance is written as equation (4.7)

$$\begin{align}& {\mathop \int \nolimits}{t_i}{\dot u_i}{\text{d}}S + {\mathop \int \nolimits}\phi \dot \omega {\text{d}}S + {\mathop \int \nolimits}\dot r{\text{d}}V - {\mathop \int \nolimits}{\dot q_i}{n_i}{\text{d}}S\nonumber\\ & \quad = \frac{{\text{d}}}{{{\text{d}}t}}{\mathop \int \nolimits}\frac{1}{2}\rho {\dot u_j}{\dot u_j}{\text{d}}V + \frac{{\text{d}}}{{{\text{d}}t}}{\mathop \int \nolimits}\frac{1}{2}\rho e{\text{d}}V\end{align} \tag{ 4.7 }$$

where $\rho$ is the mass density and $e$ is the internal energy density. The traction vector is written in terms of the stress and the unit normal, and the surface charge density is written in terms of the electric displacement and the unit normal to give equation (4.8)

$$\begin{align}& {\mathop \int \nolimits}{\sigma _{ij}}{n_j}{\dot u_i}{\text{d}}S - {\mathop \int \nolimits}\phi {D_j}{n_j}{\text{d}}S + {\mathop \int \nolimits}\dot r{\text{d}}V - {\mathop \int \nolimits}{\dot q_i}{n_i}{\text{d}}S\nonumber\\ & \quad = \frac{{\text{d}}}{{{\text{d}}t}}{\mathop \int \nolimits}\frac{1}{2}\rho {\dot u_j}{\dot u_j}{\text{d}}V + \frac{{\text{d}}}{{{\text{d}}t}}{\mathop \int \nolimits}\frac{1}{2}\rho e{\text{d}}V.\end{align} \tag{ 4.8 }$$

The divergence theorem is applied to convert the surface to volume integrals, a comma in front of an index indicated the partial derivative with respect to that index, mechanical equilibrium, ${\sigma _{ji,j}} + {f_i} = \rho {\ddot u_i}$, and quasi-static charge equilibrium, ${D_{j,j}} = {\rho _v}$, are applied, the internal displacements are assumed to be homogeneous such that $\frac{{\text{d}}}{{{\text{d}}t}}\int {\frac{1}{2}\rho {{\dot u}_j}{{\dot u}_j}{\text{d}}V = \int {\rho {u_j}{{\ddot u}_j}{\text{d}}V} }$, and the resulting volume integral is reduced to equation (4.9)

$$\begin{align}& {\mathop \int \nolimits}\left( {{\sigma _{ij}}{{\dot \varepsilon }_{ij}} - {f_j}{{\dot u}_j}} \right){\text{d}}V + {\mathop \int \nolimits}\left( {{E_j}{{\dot D}_j} + \phi {{\dot \rho }_v}} \right){\text{d}}V + {\mathop \int \nolimits}\dot r{\text{d}}V\nonumber\\ & \quad = {\mathop \int \nolimits}{\dot q_{j,j}}{\text{d}}V = \frac{{\text{d}}}{{{\text{d}}t}}{\mathop \int \nolimits}\frac{1}{2}\rho e{\text{d}}V.\end{align} \tag{ 4.9 }$$

Relaxor FE single crystals have very low electrical conductivity, and in energy harvesting applications they are typically used well below their resonant frequency such that body charges and body forces can be neglected. In the absence of body forces and body charges this reduces to equation (4.10)

$$\begin{align}{\mathop \int \nolimits}\left( {{\sigma _{ij}}{{\dot \varepsilon }_{ij}}} \right){\text{d}}V + {\mathop \int \nolimits}\left( {{E_j}{{\dot D}_j}} \right)\,{\text{d}}V + {\mathop \int \nolimits}\dot r{\text{d}}V & = {\mathop \int \nolimits}{\dot q_{j,j}}{\text{d}}V\nonumber\\ & = \frac{{\text{d}}}{{{\text{d}}t}}{\mathop \int \nolimits}\frac{1}{2}\rho e{\text{d}}V.\end{align} \tag{ 4.10 }$$

Shrinking the volume to a small enough size that the fields within can be treated as uniform, yet large enough that there can still be domains within the volume as long as the volume is large enough that it is a good representation of the volume average behavior leads to equation (4.11)

$$\begin{align}\rho \dot e = {\sigma _{ij}}{\dot \varepsilon _{ij}} + {E_j}{\dot D_j} + \dot r - {\dot q_{j,j}}.\end{align} \tag{ 4.11 }$$

This expression is valid even when the strain and electric displacement increments contain irreversible (dissipative) terms. This includes strain and polarization changes associated with domain wall motion or phase boundary motion where the domain wall motion generates heat, and similarly strain and polarization changes associated with phase transformations where there may be elimination of domain walls and generation of new domain walls as the crystal transforms from a rhombohedral structure to an orthorhombic structure and back. This annihilation and regeneration of the domain structure in the rhombohedral phase may be a contributor to the hysteresis in the phase transformation from a rhombohedral engineered 2R domain structure to the electric field or stress induced orthorhombic 1O single domain single crystal structure. Evidence of the evolution of the domain structure during this phase transformation is shown in figure 24, from Liu et al [53].

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The second law of thermodynamics states that the increase of thermal energy within any volume of material is always greater than or equal to the rate heat is being added from external sources, equation (4.12)

$$\begin{align}T\dot S \unicode{x2A7E} \dot Q\end{align} \tag{ 4.12 }$$

where $\dot S = {\mathop \int \nolimits}\rho \dot s{\text{d}}V$, thus the rate of generation of thermal energy is either equal to or exceeds the rate of heat addition, equation (4.13),

$$\begin{align}{\mathop \int \nolimits}\rho \dot s{\text{d}}V \unicode{x2A7E} {\mathop \int \nolimits}\left( {\dot r - {{\dot q}_{j,j}}} \right)\,{\text{d}}V.\end{align} \tag{ 4.13 }$$

When some portion of the strain rate and electric displacement rate are dissipative (open hysteresis loops), some of the external mechanical and electrical work generate heat within the material and is not available for harvesting. Further insight is gained by separating the strain rate and electric displacement rate into a reversible part and an irreversible part, equations (4.14) and (4.15)

$$\begin{align}{\dot \varepsilon _{ij}} = \dot \varepsilon _{ij}^r + \dot \varepsilon _{ij}^{ir}\end{align} \tag{ 4.14 }$$

$$\begin{align}{\dot D_i} = \dot D_i^r + \dot D_i^{ir}.\end{align} \tag{ 4.15 }$$

Substituting these into the energy equation gives equation (4.16)

$$\begin{align}\rho \dot e = {\sigma _{ij}}\left( {\dot \varepsilon _{ij}^r + \dot \varepsilon _{ij}^{ir}} \right) + {E_j}\left( {\dot D_i^r + \dot D_i^{ir}} \right) + r - {\dot q_{j,j}}\end{align} \tag{ 4.16 }$$

where, upon substitution, the increase of internal energy is given by equation (4.17)

$$\begin{align}\dot \Phi = \rho \dot e = {\sigma _{ij}}\dot \varepsilon _{ij}^r + {E_j}\dot D_j^r + \rho T\dot s\end{align} \tag{ 4.17 }$$

where $\dot S$ is the rate of entropy generation per unit mass. The rate of increase of thermal energy is thus given by equation (4.18)

$$\begin{align}T\dot S = \rho T\dot s = {\sigma _{ij}}\dot \varepsilon _{ij}^{ir} + {E_j}\dot D_j^{ir} + \dot r - {\dot q_{j,j}}.\end{align} \tag{ 4.18 }$$

In an adiabatic process (no heat transfer to or from the surroundings), the last two terms of equation (4.16) (the external heat added) are zero. The first term is the stress operating on the irreversible portion of the strain rate, and the second term is the electric field operating on the irreversible portion of the electric displacement rate. These are equal to the rate of thermal energy generation by dissipative processes. Just as the extraction of electrical energy during a cycle gives a hysteresis loop in the mechanical loading cycle, the conversion of mechanical or electrical energy to thermal energy results in hysteresis in the mechanical and electrical cycles. Under short circuit loading where there is no electrical work done, the area within the stress–strain hysteresis loop that occurs in a loading–unloading cycle is equal to the heat generated. In the domain engineered crystals under either [001] (4R 33 mode) or [011] (2R 32 mode) compressive stress loading, there is minimal driving force for domain wall motion as long as the stress and electric field are small enough that there is no field induced phase transformation. This is the reason the crystals have very low loss. In driving the 32 mode crystals through the phase transformation with a [011] electric field or a [100] compressive stress, the domain walls associated with the rhombohedral system are eliminated during the forward F_R − F_O phase transformation and then regenerated during the reverse F_O − F_R transformation. This is likely part of the hysteresis that is associated with the phase transformation.