BaTiO₃-based piezoelectrics: Fundamentals, current status, and perspectives

Lead zirconate-titanate [Pb(Zr,Ti)O₃ (PZT)] is the most widely used piezoceramic material for electromechanical device applications. The toxicity and environmental impacts of lead have been known since the ancient times,¹ and many lead-free piezoelectrics were investigated over the past 50 years.² However, it was not until 2002 that directives regulating the use of Pb and other toxic elements in commercial products were introduced in the European Union.^3,4 These regulations have stimulated renewed interest in lead-free piezoelectrics. In 2004, Saito et al.⁵ reported alkaline-niobate-based lead-free piezoelectrics with properties comparable with those found in PZT. This report triggered an exponential increase in the number of both basic and applied research investigations related to lead-free piezoelectrics. The expanding research community soon revisited the pioneering work performed by Takenaka et al.⁶ in 1991 which had reported on alkaline-bismuth-titanate-based piezoelectrics. Thereafter, research efforts focused on strategies to improve the piezoelectric properties of these materials. These strategies have included searching for systems with morphotropic or polymorphic phase boundaries, microstructure optimization, and texturing. (Section IV provides details on different strategies to improve piezoelectric properties.)

BaTiO₃ (BT) was the first polycrystalline ceramic material ever discovered that exhibited ferroelectricity. During the 1950s, it was considered a serious candidate for piezoelectric transducer applications⁷ and the basic phenomenological theory for the monodomain, monocrystalline state was developed (see Sec. III for details).^8,9 However, PZT,² which possesses better piezoelectric properties and a higher Curie temperature $TC$⁠, was discovered soon after. This development diminished the interest in BT for piezoelectric applications.¹⁰ It was not until 2009 that Liu and Ren¹¹ reconsidered the potential of BT-based materials for piezoelectric applications. The outstanding piezoelectric properties found in Ca- and Zr-modified BT resulted in an even greater volume of work related to lead-free piezoelectrics (see Sec. V A for details). The discovery of other BT-based piezoelectrics with outstanding piezoelectric properties followed, albeit restricted to a limited temperature range (see Sec. V for details).

As of 2017, after 16 years of continuous worldwide research effort, the development of lead-free piezoelectrics is entering into a mature stage.¹² Nonetheless, several challenges remain. To date, no material encompasses the set of functional properties of PZT which are required for the broad range of piezoelectric applications in which it is implemented. Alkaline-niobate-based compositions possess piezoelectric properties that are generally lower than PZT, although the temperature stability of the piezoelectric properties is the best among lead-free materials.^13,14 Furthermore, the processing of materials within this family is still a major concern, due to a narrow sintering window in terms of temperature and dwell time, the high volatilization of alkalies, and early activation of surface diffusion that reduces the driving force for densification, among other factors.¹⁵ Alkaline-bismuth-titanate-based materials, on the other hand, feature only moderate small signal piezoelectric properties with acceptable temperature stability. Thus, they have been thoroughly investigated for applications working in the large signal off-resonance regime due to their giant strain. Early works demonstrate, however, that the giant strain typically occurs at too high electric field for commercial applications.^16,17 In contrast, BT-based materials have attracted attention due to their large small signal piezoelectric properties, although the stability of these properties is quite limited in temperature. To date, chemically modified and/or microstructurally engineered BT-based materials have resulted in a maximum $TC$ ∼ 160 °C.^18–21 

Consequently, a lead-free piezoelectric capable of replacing PZT in all commercial applications is still far from reality. However, the continuous research and development efforts have led to the commercialization of devices based on lead-free piezoelectrics and several prototypes have been tested.^12,22 Innovations based on lead-free piezoelectrics could position technology startups strategically or could assist manufacturers in diversifying their product portfolios.

From a scientific point of view, BT-based materials are a very interesting research platform since their phase diagrams involve phase transitions at relatively low temperature, making them attractive for probing mechanisms of enhanced piezoelectricity. These mechanisms include tricritical behavior, property divergence near interferroelectric phase boundaries, property enhancements due to polarization rotation and/or extension, among others. In addition, BT-based materials are quite versatile for fundamental studies since their functional properties are sensitive both to microstructure and chemical modifications. Despite their inherent advantages, the low temperature stability of piezoelectric properties in BT-based materials limits their technological implementation. Section VI discusses specific examples of the current technological relevance of BT-based piezoelectrics. Prototype piezodevices based on the family of BT materials include loud-speakers,²³ biocompatible nanogenerators,^24,25 self-powered fluid velocity sensors,²⁶ and intravascular ultrasonic transducers,²⁷ to name the most prominent examples.

Here, we review the current understanding of piezoelectricity in BT-based materials. We first briefly present the history and technological potential of BT-based materials. This is followed by an introduction to the fundamental concepts related to ferroelectricity and piezoelectricity. The most widely used approaches to engineer and optimize piezoelectric properties are then introduced. Chemical modifications are highlighted because binary and ternary solid solutions with different types of phase transitions constitute the most commonly used approach to tune the piezoelectric properties of BT-based materials. Subsequently, we assess the current state of understanding of the (Ba,Ca)(Zr,Ti)O₃, (Ba,Ca)(Sn,Ti)O₃, and (Ba,Ca)(Hf,Ti)O₃ systems. Detailed information on the synthesis, microstructure, phase diagrams, electromechanical and mechanical properties, reliability, and doping strategies is compiled. The technological relevance of BT-based piezoelectrics is assessed in terms of market potential. Furthermore, novel fields of application, as well as insights into toxicity and biocompatibility of this family of materials, are presented. Finally, potential lines of future research are suggested.

This section briefly outlines some of the key historical events leading up to the present day understanding of BT materials, with a focus on those discoveries connected with its use in piezoelectric applications. The current technological potential of BT-based materials for piezoelectric applications is also summarized. Readers interested in a more comprehensive background are referred to several excellent books and review articles describing the history of BT, as well as the broader fields ferroelectricity and piezoelectricity.^2,10,28–36

Three key discoveries gave birth to the fields of ferroelectricity and piezoelectricity in polycrystalline ceramic materials. The first was the finding of an extremely high relative permittivity in BT, which was more than ten times higher than that of other oxide compounds, such as TiO₂. The second was the recognition that the high relative permittivity was connected with the underlying phenomenon of ferroelectricity. The third was the discovery of the poling process for polycrystalline materials, which could produce a high remanent polarization, and continues to be of paramount importance for the application of ferroelectric ceramics as piezoelectric sensors, actuators, and transducers.² 

The early history of BT is somewhat clouded by the fact that its discovery in 1941 took place during the second world war.³⁷ Due to issues with the supply of Mica, the most widely used dielectric at that time, BT soon became an important strategic material for various military technologies.³⁶ This scenario led to the independent discovery of BT in the United States,^38,39 United Kingdom,³⁷ Russia,⁴⁰ and Japan⁴¹ in the search for new dielectric materials that involved the modification of TiO₂ with BaO. It is widely reported that the unusually high dielectric properties of BT were associated with ferroelectricity by von Hippel et al.⁴² in the United States, and by Wul and Goldman⁴³ in Russia, although according to Fousek,³¹ the first claim that BT was ferroelectric was made by Vul and Vereschtschagin.⁴⁴ BT constituted the first demonstration of ferroelectricity in a simple ternary oxide compound. This was not thought possible at that time. Ferroelectricity was believed to exist only in potassium dihydrogen phosphate and related hydrogen bonded crystals displaying ferroelectric phase transitions of the order disorder type. Barium titanate differed from these materials in that the crystal structure is considerably simpler, and the phase transition at the Curie point is one of the displace type. After these concepts were understood and several new ferroelectrics discovered, a classification based on their type of phase transitions became possible.⁴⁵ 

Upon termination of the second world war, the phase transitions and physical properties of BT became topics of intensive investigation.^42,46,47 Some of the most important scientific advances came from crystallographic investigations. Megaw,⁴⁸ Kay and Voudsen⁴⁹ in the United Kingdom, as well as Miyake and Ueda⁵⁰ in Japan, provided the first detailed understanding of the crystal structure and changes in symmetry on cooling through the phase transitions. Blattner et al.⁵¹ produced the first BT single crystals in Switzerland in 1947 via ternary melts. This method was improved by Matthias in the United States (formerly in Switzerland), after moving to the Laboratory for Insulation Research at M. I. T., which was led by von Hippel. During the 1940s, von Hippel and co-workers advanced the field of ferroelectricity by providing a thorough understanding of the ferroelectric switching, domain structure, and crystal structure evolution of BT.⁴⁷ 

Another major breakthrough that occurred during this early period was the discovery of the poling process. At the time, it was not yet recognized that the polarization direction in a randomly oriented ferroelectric ceramic could, in fact, be reoriented. Hence, the understanding of the poling process was a major breakthrough. This also enabled the implementation of polycrystalline ceramics into piezoelectric applications, as they were mechanically more robust and less expensive than single crystals. This process was first reported by Gray⁵² with the purpose to produce transducers. Following this discovery, several authors pushed forward the understanding of the poling process, electrostriction, and piezoelectricity in BT.^53–56 Forsbergh also investigated the evolution of domains under mechanical strain and electric fields.⁵⁷ Between 1947 and 1950, the first commercial products based on BT became available, which included phonograph pickups, transducers for ultrasonic generation, accelerometers, by-pass capacitors, and filter capacitors.^2,7,58,59

During the same period, theoretical approaches were put forward in an effort to explain the experimental observations.⁶⁰ Between 1949 and 1951, Devonshire developed the first phenomenological theory to reconcile relationships between the structural, dielectric, electromechanical, and thermal properties of BT (and generally of ferroelectrics).^8,9 The Devonshire theory was a particular form of the statistical thermodynamic theories of phase transitions put forward by Ginzburg, Landau, and Lifshitz.^61,62 Later on, in 1954, Remeika produced larger and higher quality single crystals.⁶³ This allowed further detailed characterization to be performed and correlated with the existing theoretical descriptions.⁶⁴ For instance, the seminal work of Merz⁶⁵ on nucleation and growth of ferroelectric domains was performed with Remeika's single crystals. Miller and Weinreich⁶⁶ extended these concepts. Together these early studies formed the basis for our current understanding of polarization dynamics.

The late 1940s and early 1950s witnessed a large proliferation of new perovskite-structured ferroelectric materials, which was accompanied by a continuous improvement in dielectric and piezoelectric properties.²⁹ Some studies focused on the phase diagram and synthesis optimization of polycrystalline BT,^67,68 whereas others expanded the scope of research to encompass more complex BT-based solid solutions.^69–71 McQuarrie and Behnke⁶⁹ were the first to synthesize the quaternary system between BaTiO₃, BaZrO₃, CaTiO₃, and CaZrO₃ with the objective to investigate its structural and dielectric properties. During this same era and with similar purposes, the quaternary system between BaTiO₃, BaSnO₃, SrSnO₃, and CaSnO₃ was investigated by Coffeen.⁷⁰ Even with the development of these complex solid solutions, the poor temperature stability of the piezoelectric properties was not improved.

Between 1949 and 1954, a major breakthrough in the field of ferroelectrics and piezoelectrics occurred. In 1949, Waku and Hori discovered that PbTiO₃ and PbZrO₃ form a solid solution, PZT.⁷² The PZT system attracted the attention of Japanese scientists because of the variation in crystal structure with the Zr/Ti ratio and the apparent pseudo-cubic degeneration of the tetragonal and rhombohedral phases when approaching the composition Pb(Zr_0.55Ti_0.45)O₃.^73–76 It was not until 1954 though, that Jaffe et al.⁷⁷ recognized that PZT solid solutions featured outstanding electromechanical properties with a maximum response near the Pb(Zr_0.55Ti_0.45)O₃ composition. Moreover, this composition offered stable properties up to temperatures of 200 °C. This was attributed to the high Curie temperatures that characterize PZT solid solutions, which were found to lie between 220 °C and 490 °C, depending on the Zr/Ti ratio.⁷⁷ Soon after, Jaffe filed a patent that recognized the potential of PZT for transducers.⁷⁸ Jaffe also introduced the concept of a morphotropic phase boundary to the piezoelectric community⁷⁹ (discussed in Sec. IV B 2 a) and recognized its potential for a broad range of electromechanical devices.⁸⁰ This shifted the interest of the scientific community and industry towards this family of materials for piezoelectric applications. In the following decades, the interest in BT-based materials was mostly focused on their dielectric properties and their use in capacitor applications.

During the 1960s and 1970s, new solid state physics concepts arose to explain ferroelectricity in perovskite oxides, including lattice dynamical approaches and the soft mode theory.^31,33,81 During the 1970s through 1990s, there was a diversification of the ferroelectrics community working in PZT and BT-based materials, which was marked by the appearance of new fields associated with electro-optics and thermistors. In fact, BT-based materials have been indispensable in the understanding and technological applications of thermistors due to their positive temperature coefficient of resistance (PTCR).³³ 

During the 1980s and 1990s, the integration of ferroelectrics and miniaturization of devices began.³³ In part due to these technological drivers, the basic knowledge of piezoelectric materials continued to evolve. BT single crystals and ceramics proved themselves highly pertinent as model materials for research on strain mechanisms^82–85 as well as domain structure and microstructure engineering.^86–96 This trend, together with the governmental regulations targeting environmentally friendly piezoelectrics, was probably the impetus for research on (Ba,Ca)(Zr,Ti)O₃ for piezoelectric applications.^11,97 Clearly, the work by Liu and Ren¹¹ reignited the interest of scientists and some industries in this system (see Sec. VI for further details), which had been lost to a great extent since 1954.

It is important to note, however, that the technological utility of these materials as piezoelectrics is rather limited owing to their restricted temperature operational range. Thus, to date, this family of materials does not constitute an all-encompassing solution for lead-containing piezoelectrics. To highlight this fact, Table I provides well-established and impending piezoelectric applications, together with our view of the realistic potential of BT-based piezoelectrics for each technology. The reader interested in comparing the evolution of the technological applications based on piezoelectrics in the last 25 years may compare this table with Table 2 of Ref. 31. As it will become apparent throughout Secs. V and VI, however, there is still need for further research to make technology transfer of BT-based materials into industry a reality.

The piezoelectric effect represents a linear coupling between electrical and mechanical fields. When a piezoelectric material is subjected to external mechanical stress, electric charge is generated on the surface. The polarity of the generated surface charge depends on the direction of the applied stress, i.e., compressive or tensile. The conversion of mechanical to electrical energy is known as the direct piezoelectric effect. If, on the other hand, an external electrical field is applied to the piezoelectric material, the material dimensions are changed. Depending on the direction of the field, the material either contracts or expands. This property is known as the converse piezoelectric effect. Piezoelectricity is exhibited by crystals belonging to 20 of the 32 crystal classes.

Polar dielectrics, also referred to as pyroelectrics, belong to one of the ten crystal classes in which there exists a polar axis with a direction that is fixed by symmetry. The direction of the polar axis cannot change, for example, with temperature or pressure, unless the symmetry changes. A change in temperature, however, modifies the magnitude of the polarization. The relation between the changes in temperature and polarization is known as the pyroelectric effect.

Ferroelectrics are polar materials that possess at least two equilibrium orientations of spontaneous polarization in the absence of an electric field. Under the application of an electric field, the direction of the polarization vector can be reoriented. Hence, ferroelectric materials are both piezoelectric and pyroelectric and exhibit nonlinear phenomena associated with polarization reversal. The linear and nonlinear material responses of polar dielectrics and ferroelectrics are treated in Sec. III C. Most ferroelectric materials undergo a displacive structural phase transition from a high temperature prototype phase (often centrosymmetric) into one or more ferroelectric phases of lower symmetry. Barium titanate belongs to the family of perovskite materials with general formula ABO₃. In the schematic in Fig. 1, Ba²⁺ is located on the A-site at the corners of the cubic unit cell, while Ti⁴⁺ occupies the B-site in the cell center. The O^2– anions are located at the face centers of the unit cell and constitute BO₆ octahedra.^98,99 From a crystallographic point of view, the appearance of spontaneous polarization in barium titanate is related to a shift of the Ti⁴⁺ and O^2– ions relative to the Ba²⁺ ion at the origin. The resulting electric dipole moment is known as the spontaneous polarization, P_s. The appearance of spontaneous polarization is accompanied by changes in the dimensions of the unit cell. The strain relative to the cubic prototype due to these dimensional changes is termed the spontaneous strain, x_s.

The cubic phase of barium titanate belongs to the centrosymmetric crystal class $m3¯m$⁠. As such, it is both paraelectric and non-piezoelectric. The order parameter for the transition from the cubic prototype phase into the lower symmetry ferroelectric phases is the spontaneous polarization and is related to the amplitude of a transverse optic soft mode that condenses at the Curie temperature, $TC$⁠. The Curie principle asserts that the “proper” ferroelectric phase(s) that may appear in the crystal are those that are subgroups of the prototype phase. Thus, they must contain all symmetry elements common to the symmetry group of the prototype and the symmetry group of the order parameter. The spontaneous polarization is a polar vector quantity and its symmetry group is $∞m$⁠. Shuvalov¹⁰⁰ demonstrated that the cubic $m3¯m$ prototype phase of the perovskite crystals allows seven proper ferroelectric phases, one of tetragonal (⁠$4mm$⁠), one of rhombohedral (⁠$3m$⁠), one of orthorhombic (⁠$mm2$⁠), three of monoclinic (⁠$m$⁠), and one of triclinic (⁠$1$⁠) symmetry.

When cooled at ambient pressure, barium titanate undergoes a sequence of first-order phase transitions: cubic (⁠$Pm3¯m) →∼131 °C$ tetragonal (⁠$P4mm$⁠) $→∼0 °C$ orthorhombic (⁠$Amm2$⁠) $→∼¯90°C$ rhombohedral (⁠$R3m$⁠).⁹⁹ These transitions correspond to the appearance of spontaneous polarization directed parallel to the edge (tetragonal) of the pseudocubic unit cell of the perovskite structure, and its subsequent re-orientation along a face diagonal (orthorhombic) and body diagonal (rhombohedral). The transitions between phases are accompanied by strong dielectric softening, as indicated by three distinct maxima in the relative permittivity (Fig. 1). These phase transitions are also accompanied by distinct anomalies in thermal, mechanical, and piezoelectric properties, which are exploited in device applications.

As the transition between the cubic and tetragonal structure at $TC$ involves the appearance of spontaneous polarization, it is a paraelectric–ferroelectric transition. In barium titanate, this transition is of first-order. Above the Curie temperature, the dielectric susceptibility (⁠$η$⁠) follows the Curie-Weiss law:

where C is the Curie constant, $θ$ is the Curie-Weiss temperature, $ε′$ is the real part of dielectric permittivity, and $εo$ is the permittivity of free-space.⁹⁸ In contrast, the lower temperature transitions involve a re-orientation of the polarization direction and are often referred to as inter-ferroelectric transitions. From group theoretical analyses of the relations between phases at the tetragonal ↔ orthorhombic and orthorhombic ↔ rhombohedral transitions, these transitions are also of first-order.

The starting point for understanding the constitutive behavior of polar dielectrics is a form of the general field response theory constructed in the approximation of linear thermodynamics. It is assumed that the equilibrium state of the crystal is described by a set of three intensive thermodynamic variables comprising temperature (⁠$T$⁠), stress tensor (⁠$X$⁠, note that bold letters indicate tensors), and electric field vector (⁠$E$⁠), and their proper conjugate extensive variables of entropy (⁠$S$⁠), strain tensor (⁠$x$⁠), and dielectric displacement vector (⁠$D$⁠). According to the first law of thermodynamics the reversible change of the internal energy, $dU$⁠, is given by

where, from the second law of thermodynamics, the heat absorbed by the crystal is $∂Q=TdS$⁠. The work, $∂W$⁠, done on the crystal consists of both mechanical and dielectric contributions

and so the differential form of the internal energy becomes

wherein the range of letter indices is i,j =1, 2, and 3. Since three independent variables can be chosen in eight different ways from among the pairs of conjugate variables, there are seven thermodynamic functions in addition to Eq. (4), each of which can serve as an equation of state. In most cases, however, it is convenient to express Eq. (4) in a form where the independent variables are those under experimental control. Using the auxiliary function of enthalpy, $H=U+W$⁠, the differential form of the free energy, $G=H−TS,$ is obtained in terms of the independent variables of temperature, stress, and electric field^98,101

As Eq. (5) is an exact differential, it follows that the entropy, strain, and dielectric displacement are given by

where the subscripts indicate those variables held constant. The second-order partial derivatives of Eq. (5) with respect to temperature, stress, and electric field give the principal material property coefficients of heat capacity (⁠$CX,E$⁠), elastic compliance (⁠$sijklE,T$⁠), and dielectric permittivity (⁠$εijX,T$⁠), respectively. It is clear from Eq. (6) that these are

The second-order cross-partial derivatives of Eq. (5) describe coupled effects in the crystal. Because the order of differentiation can be reversed, the same numerical values of property coefficients are obtained for direct and converse effects. For example, the electrocaloric effect and the converse effect of pyroelectricity are described by the same property coefficient, $piX$⁠,

Similarly, the piezocaloric effect and the converse effect of thermal expansion are described by the same property coefficient, $αijE$⁠,

while direct and converse piezoelectric effects are described by $dijkT$

Naturally, under appropriate conditions all of the effects in Eqs. (7)–(12) can contribute to the entropy, strain, or dielectric displacement. Integrating these relations over small changes in temperature, stress, or electric field from an initial isothermal, stress-free, and zero-field state leads to a set of linear relations

Equations (13)–(15) describe the linear calorimetric, mechanical, and dielectric equations of state of the material. The relations are generally valid for linear polar dielectrics. They also hold for nonlinear dielectrics, such as ferroelectric materials as treated in Sec. III C, provided that the changes in independent variables are sufficiently small. However, some of the most important characteristics of ferroelectrics, such as strong electrostrictive coupling between spontaneous polarization and mechanical strain, and hysteresis in the polarization–electric field response, are inherently nonlinear. To account for these nonlinear phenomena, higher-order terms must be included in the equation of state. Higher-order terms also provide a description of possible divergences in material response, as observed in the proximity of paraelectric–ferroelectric and inter-ferroelectric transitions.

The phenomenological theory of barium titanate was developed in the classic work of Devonshire.^8,9 It provided the first unified description of the observed changes in crystal form and of the variations of the thermal, dielectric, mechanical, and piezoelectric properties on cooling through the sequence of phase transitions. The phenomenological theory assumes that the observed divergences in material responses originate in the nonlinear behavior of the dielectric response. It is further assumed that there exists a spontaneous strain due to electrostriction and that the stress-induced strain can be described in the linear approximation of Hooke's law. Coupling of the dielectric nonlinearities to mechanical and thermal properties automatically gives rise to associated anomalies in other responses. Taking into account the multicomponent nature of the order parameter, i.e., the polarization vector $P$⁠, allows the full matrix of tensor components describing the intrinsic properties of the monodomain, monocrystalline state to be determined in each of the ferroelectric phases.

It is usually convenient to express the free energy by choosing temperature, dielectric displacement, and stress as the independent variables. The dielectric displacement includes the free-space polarization, $Di=ε0Ei+Pi$⁠, but for most ferroelectric materials the free-space polarization can be neglected in comparison to material polarization, and, hence, $Di≈Pi$⁠. With these considerations in mind, the non-equilibrium free energy can be written as the sum of three parts

wherein $G0T$ is the reference energy of the cubic prototype phase. The first term on the right hand side of Eq. (16), $GL(T, P)$⁠, is a 2‐4-6 Landau polynomial describing the phase transitions among paraelectric and ferroelectric phases. The second term, $GC(P, X)$⁠, is the lowest-order (linear-quadratic) electrostrictive coupling energy. The third term, $GE(X)$⁠, is the elastic energy. The non-zero terms in the Landau polynomial, $GL(T, P)$⁠, are determined by the symmetry of the cubic (⁠$m3¯m$⁠) prototype phase, and the polynomial adopts the following form:

In accordance with the Curie-Weiss law [Eq. (1)], the coefficient of the quadratic term, $χ0=1/ε0C$⁠, is the dielectric stiffness and the coefficients of the quartic and sextic terms, $ξi(T)$ and $ζi(T)$⁠, are higher-order dielectric stiffnesses. The higher-order dielectric stiffnesses are, in principle, also analytical functions of temperature, and in accordance with the Landau theory¹⁰² are given by equations of the form

As $T→θ$⁠, the Landau polynomial becomes asymptotically accurate, and it is evident that near the Curie temperature the higher-order terms can be treated as constants, $ξi≅ξiθ$ and $ζi≅ζiθ$⁠. A general observation for ferroelectric perovskite crystals is that the range of validity of the Landau theory is wider than expected given the assumptions inherent to its formulation, such that the theoretical predictions can often be extrapolated with reasonable accuracy to temperatures well below the Curie temperature.¹⁰³ 

The electrostrictive strain associated with the distortion of the cubic prototype phase is assumed to be of the same order of smallness as the square of the spontaneous polarization

where $Qijkl$ are the electrostrictive coefficients of the cubic phase in polarization notation. The electrostriction tensor is symmetrical with respect to any permutations $i↔j$⁠, $k↔l$⁠, and $(ij)↔(kl)$ and there are three non-zero components in cubic crystals (in Voigt notation: $Q11$⁠, $Q12,$ and $Q44$⁠). We note here that the existence of a converse electrostriction remains a controversial subject with authors disagreeing on its existence.^104,105 The discussion of this phenomenon is outside the scope of this review.

The coupling energy, $GC(P, X)$⁠, is the product of an applied stress and the electrostrictive strain, and using Voigt notation, it follows from Eq. (19) that this is

In the approximation of Hooke's law, the stress-induced mechanical strain is

where $sijklP$ are the elastic compliances of the cubic phase (in Voigt notation $s11P$⁠, $s12P$⁠, and $s44P$⁠) at constant polarization. The elastic energy, $GE(X)$⁠, is the product of applied stress and induced mechanical strain, and again using Voigt notation, it follows from Eq. (21) that this is

Substituting Eqs. (17), (18), (20), and (22) into Eq. (16) gives the elastic Gibbs free energy, $ΔG1$⁠, which describes the nonlinear coupling between the thermal, dielectric, and mechanical responses of the crystal.

For barium titanate, the Landau polynomial [Eq. (17)] has four solutions of interest, which depend on the non-zero components of the polarization vector, $P={P1, P2, P3}$⁠. There is one trivial solution corresponding to the paraelectric cubic phase

and three others corresponding to differing directions of polarization of modulus, $P=P$⁠,

or the equivalent solutions obtained by interchanging $P1$⁠, $P2$⁠, and $P3$⁠. Inserting Eqs. (24)–(26) into the elastic free energy gives the non-equilibrium free energies of the tetragonal, orthorhombic, and rhombohedral phases, respectively. After applying these solutions to the elastic free energy [Eq. (16)], quantities of interest in each of the ferroelectric phases are computed by taking first and second partial derivatives with respect to temperature, polarization, and/or stress.

The first partial derivatives with respect to each of these variables give the electric field components, entropy, and mechanical strain components, respectively. Taking conditions of constant temperature and stress, these are

Putting the electric field components equal to zero

yields the equilibrium values, $P0,i$ determining the equilibrium spontaneous polarization $Ps,α=P0$ in each of the ferroelectric phases. The second partial derivatives give the material property coefficients. Taking these derivatives again with respect to temperature, polarization, and stress provides the excess heat capacity (⁠$ΔCP$⁠), the dielectric stiffnesses (⁠$χij$⁠), and the elastic compliances (⁠$sijP$⁠)

The dielectric stiffnesses are converted from absolute to relative values by multiplying by the permittivity of free-space, $ε0$⁠. The relative dielectric susceptibility coefficients (⁠$ηij$⁠) are given by

where $Aij$ and $Δ$ are the cofactor and determinant of $χij$ matrix, respectively. As with the linear thermodynamic theory, second-order cross-partial derivatives give other properties of interest. In particular, the piezoelectric charge coefficients (⁠$dij$⁠) are given by

wherein $−bij=∂2ΔG/∂Pi∂Xj$⁠. Using Maxwell relations, it can also be shown⁶ that the elastic compliances at constant electric field are

Equation (16) together with Eqs. (24)–(32) are sufficient to describe the equilibrium free energies in each of the ferroelectric phases and the corresponding matrices of thermal, mechanical, dielectric, and piezoelectric property coefficients.⁹ 

As a parameterized phenomenological theory, suitable experimental data must be available to determine the Curie and Curie-Weiss temperatures (⁠$TC$⁠, $θ$⁠), dielectric stiffness (⁠$χ0$⁠), higher-order stiffnesses (⁠$ξi$⁠, $ζi$⁠), the electrostrictive coefficients (⁠$Qij$⁠), and the elastic compliances (⁠$sijP$⁠). If a suitable set of coefficients is available from experiments, the intrinsic properties of the monodomain state can be self-consistently determined, as depicted in Fig. 2.

The phenomenological theory as summarized above is a useful tool for experimentalists because it provides a relatively simple framework within which the various physical properties of ferroelectric materials can be correlated. For example, the extrema in dielectric and piezoelectric coefficients near phase transitions, as illustrated in Fig. 2, have been described in terms of polarization rotation and extension mechanisms. These two terms emphasize the relative importance of the divergence in transverse or longitudinal dielectric susceptibility, respectively, on approaching either a paraelectric to ferroelectric or an inter-ferroelectric phase transition.^106,107 In addition, physical properties of a particular crystal as measured subject to one set of experimental conditions can be predicted under a different set of conditions, the latter being difficult to access in the laboratory. Alternatively, the physical properties of a given crystal as measured by different experimental methods can be compared with the theoretical predictions and so checked against one another for consistency. In those instances, where discrepancies between theory and experiment are observed, it is often possible to trace the differences in the quality of the specimens, to errors in the measurements, or to the occurrence of other physical phenomena in the crystal that were not anticipated at the outset of the experiments. In the latter case, embellishments to the low-order phenomenological theory described above may be required. Following this approach, a self-consistent set of physical properties characterizing the monodomain, single crystalline state can be established. For high-quality crystals for which reliable information is available, polycrystalline-averaging, such as the Voigt-Reuss-Hill method, may be used to estimate the properties of corresponding randomly oriented ceramic material.¹⁰¹ 

In the absence of external stimuli, such as an electric field or mechanical stress, the spontaneous polarization will be oriented along a family of crystallographic directions with the lowest free energy. For example, in the tetragonal (⁠$P4mm$⁠) phase of BT, the spontaneous polarization vector may be parallel to any of the six equivalent ⟨100⟩ directions. The regions of the ferroelectric where the polarization is uniformly oriented are called ferroelectric domains and the boundary between two domains is termed a domain wall.

The formation of domains in ferroelectric crystals is a consequence of the reduction in the elastic and electrostatic energies.^101,108,109 The walls separating domains with parallel but opposite polarization directions are called 180° domain walls, whereas the walls separating domains with polar vectors oriented at a given angle to one another are described as non-180° domain walls. For example, in the tetragonal phase of BT, both 90° and 180° domains contribute to the reduction of the energy due to depolarizing electric fields, but only 90° domains contribute to the minimization of elastic energy. The permitted domains for orthorhombic (⁠$Amm2$⁠) symmetry are 60°, 90°, 120°, and 180°, while for the rhombohedral (⁠$R3m$⁠) symmetry, the allowed domains are 71°, 109°, and 180°.¹¹⁰ 

In the virgin state, a ferroelectric material exhibits a random polydomain configuration. This leads to a macroscopic polarization equal to zero. Figure 3(a) represents schematically the virgin state of a ferroelectric material. At low electric fields, the relation between strain and polarization is nearly linear

where the linear increase with electric field is due to the intrinsic material response described above. Upon application of a sufficiently large external stimulus, such as an electric field or mechanical stress, domain walls can be displaced leading to domain switching. Domain switching is an extrinsic process, in the sense that it occurs on a length scale larger than that of a unit cell. The polarization loop, which arises as a consequence of switching, is the fingerprint of a ferroelectric material and involves the nonlinear increase of polarization with electric field (Fig. 3).¹¹¹ As discussed above, the angles between the polarization vectors of neighboring domains are restricted by crystal symmetry. Only an electric field can reorient 180° domain walls and no strain is generally associated with this type of domain switching (although higher order strain contributions were actually proven¹¹²). The non-180° domain walls, on the other hand, can be reoriented either by an electric field or a mechanical stress. This type of reorientation is generally termed ferroelastic switching and significantly contributes to the macroscopic strain of ferroelectrics.

Domain walls can be displaced reversibly or irreversibly. Reversible domain wall displacements refer to domain wall motion around a local energy minimum, while irreversible domain wall displacement arises from jumps of domain walls between local energy minima separated from one another by a potential barrier.¹¹³ At very high electric fields, the domain walls oriented along the field direction reach a saturation state, thereby restoring a linear change in polarization with electric field. The saturation state is characterized by a substantial change in the domain configuration as compared with the virgin state [Fig. 3(b)]. Once the linear increase in polarization sets in, a saturation of polarization, $Psat$⁠, is reached. The spontaneous polarization $Ps$ of the material may be estimated by extrapolating the linear polarization region near the saturation polarization down to zero electric field along the saturation polarization tangent.

When the field is reduced, some domain walls will back-switch. Hence, at zero electric field, a non-zero macroscopic polarization, or remanent polarization $Pr$⁠, is observed [Fig. 3(c)]. To bring the polarization back to zero an electric field must be applied in the opposite direction. The field at which the polarization regains a value of zero is called the coercive field $Ec$⁠. The polarization state at the coercive field is again random and has no net polarization [Fig. 3(d)]. The observed hysteresis in the P–E curve is thus a consequence of reversible and irreversible displacement of domain walls (i.e., switching). Switching is accompanied by energy dissipation in the material and produces undesired thermal losses.^112,114 The mechanisms of polarization switching have been studied in many different ferroelectric materials.^65,115

The re-orientation of spontaneous polarization and domain switching are accompanied by a mechanical strain. As a result, if the electric field is cycled, a strain-electric field hysteresis loop is also obtained, which resembles the shape of a butterfly (Fig. 3). At any given electric field strength, at which the strain-field response deviates from linearity, an engineering large signal piezoelectric coefficient, also referred to as normalized strain, can be defined as

Most of the strain developed in ferroelectrics under an electric field and far away from phase transitions arises from (i) the intrinsic converse piezoelectric effect, (ii) intrinsic electrostriction, and (iii) the extrinsic reversible and irreversible displacement of non-180° domain walls. Due to the importance of intrinsic and extrinsic effects these contributions are described in more detail in Sec. III E.

Several investigations in barium titanate, and related materials such as PZT, indicate that domain wall movement, among other extrinsic contributions, can be the major factor determining the dielectric and piezoelectric properties.^116–118 The extrinsic contributions are highly dependent on the crystal symmetry, defects, external stimuli, boundary conditions, and temperature. They can be strongly reduced under measurement conditions involving high dc electric fields where domain switching is saturated, high-frequency ac electric fields where domain wall motion cannot follow the alternating field, and/or at cryogenic temperatures where domain wall motion is thermally frozen. These conditions are often used to experimentally discriminate between intrinsic and extrinsic contributions. A common approach to estimate the irreversible domain wall displacement in ferroelectrics is provided by the Rayleigh model.^82,84,119 The Rayleigh model can be applied in low field conditions, where the density and structure of the domain walls remain mostly unchanged as the field is cycled.^82,119 This model relies on the assumption that the nonlinear behavior is a consequence of irreversible domain wall displacements.^84,119 The piezoelectric response as a function of electric field amplitude can be described by

where $dint$ represents the initial piezoelectric response as consequence of reversible domain wall displacement and intrinsic piezoelectricity, $α$ is the Rayleigh coefficient, and $E0$ is the amplitude of the applied alternating field. Similarly, the piezoelectric terms in Eq. (35) can also be replaced by the corresponding relative permittivity terms. The field-dependent component $αE0$ represents the hysteretic contribution of irreversible domain wall displacements into a new equilibrium positions when the applied electric field is sufficiently large to overcome the potential barrier between two equilibrium positions. The origin of the hysteresis (energy loss) is the pinning and unpinning processes of the domain walls during field cycling while the domain wall is displaced from one equilibrium position to another.^83,119,120 The Rayleigh model has also certain limitations. It cannot describe, for instance, the domain wall displacement of hard PZT. In such cases, or at electric fields higher than the Rayleigh regime, higher-order terms can be generally used to described the domain wall displacement.^83,119 Irreversible domain wall displacement strongly depends on the domain structure architecture and domain wall mobility. The most important variables controlling domain wall mobility are the crystal symmetry and concomitant spontaneous strain as well as pinning centers and microstructure.⁸⁴ Pinning centers alter domain wall mobility dramatically and are a result of defects such as vacancies, interstitials, among others.^121,122

The measurement of recoil curves and bias-field-dependent small signal properties are alternative approaches to estimating extrinsic properties.^123–125 A recoil curve is a subloop at a given electric field and polarization state. In this method, it is assumed that the extrinsic irreversible contributions can be estimated by the remaining polarization upon electric field removal. The extrinsic reversible contribution is calculated as the difference between the total polarization and remaining polarization. For the correct estimation of reversible and irreversible processes, the recoil curve should be obtained at sufficiently low electric field to assure that irreversible processes remain constant. Probing the extrinsic contributions via recoil curves near the coercive field is not reliable. This is because an electric field of a low magnitude such as the one used during the recoil loop is high enough to produce irreversible domain wall displacements near the coercive field.¹²⁴ 

Bias-field-dependent small signal properties are measured by superimposing a sinusoidal ac electric field in the kHz range to a base waveform in the Hz range. For a proper estimation, the amplitude of the modulating wave should be as low as possible so that domain walls are only displaced reversibly. The base waveform should be ideally as large as required to achieve the saturation of the material. This ensures that the intrinsic contributions are properly estimated upon saturation of switching. The measured property with zero bias-field is comprised of intrinsic and irreversible extrinsic contributions, under the assumption that reversible switching is considerably reduced at zero bias-field. This measurement has to be interpreted pointwise since the measured reversible domain wall motion will be a function of the domain configuration imposed by the base waveform. In other words, it provides information on the reversible domain wall movement at a given irreversible domain state defined by the base waveform. Bias-field dependent capacitance measurements provide information on both 180° and non-180° switching processes, whereas bias-field-dependent measurements of $d33$ provide information only on non-180° switching processes.^124,125 The latter considers that 180° domain wall displacements do not contribute to the strain of the material at low electric field.¹²⁶ 

Microstructural engineering is an effective strategy to tune functional properties. Grain size has a profound effect on the dielectric and piezoelectric properties of ferroelectrics.^127–134 Grain size can be tailored by modification of powder particle size and/or the sintering process. Another related approach to optimize properties is by engineering the domain configuration. Domain engineering is a method to produce a stable and desirable domain size in order to maximize the piezoelectric and dielectric properties.¹³⁵ It has been applied to single crystals^94,136 but has not been widely used to tailor properties in polycrystalline materials. Both approaches, however, share similarities since the grain size and domain size in polycrystalline materials are related.

An optimized grain size generally results in increased small signal and large signal electromechanical properties, increased relative permittivity, decreased Curie temperature, and decreased coercive field as compared with the material with non-optimized grain size.^127,128,132 The optimal grain size results in a maximized density of domain walls that are capable of switching with minimal back switching.^{117,130–132} Grains that are too large often reduce piezoelectric and dielectric properties as a consequence of excessive back switching.¹³¹ In contrast, grains that are too small generally lead to suppression of ferroelectricity due to reduced lattice distortion.^137–139 At very small grain sizes, domains may also become unstable, since the spontaneous polarization is compensated by charges at grain boundaries and/or by polarization gradients.¹³⁹ Thus, depending on the material, there is a threshold grain size at which ferroelectricity ceases.

Increasing grain size in BT yields an enlargement of domain size [Fig. 4(a)] and reduction of the piezoelectricity [Fig. 4(b)]. A strong increase in functional properties in BT occurs at grain sizes between 1 μm and 2.3 μm and domain sizes below 100 nm.^{90,117,129,132} In contrast, a critical grain size between 10 nm and 100 nm induces a paraelectric state in BT at room temperature.^137–139 Several studies of BT have determined that a fine domain structure through engineered synthesis routes can greatly enhance its piezoelectric and dielectric properties.^86,88–91 For BT at room temperature, $d33$ values between 350 pC/N and 500 pC/N and $d31$=−185 pC/N were achieved.^{86,89,90,129,140,141} Thus, the properties of engineered BT can be comparable with the most widely investigated solid solutions that will be treated later on.

Texturing is also an effective way to enhance piezoelectric and dielectric properties, as well as to improve their temperature stability.^5,143–146 In contrast to other approaches, it does not lower the temperature range at which the ferroelectric state remains stable.¹⁴⁷ Templated grain growth and reactive templated grain growth are the most commonly used texturing techniques.¹⁴⁸ In lead-free ferroelectrics possessing the perovskite structure, orientation degrees up to 90% were achieved in various systems. A high degree of grain orientation produces enhancements of $d33$ between 1.4 and 2.1 times with respect to the random polycrystalline material.¹⁴⁸ The maximum piezoelectric and dielectric properties achievable in textured polycrystalline materials are limited by the properties of the corresponding single crystal.¹⁴⁹ This is schematically represented in Fig. 5.

BT materials with [001], [110], and [111] texture feature improved electromechanical properties compared with random polycrystalline materials.^95,150,151 Vriami et al.¹⁵¹ used [001]-oriented BT plate-like particles synthesized via the molten salt method to improve piezoelectricity. With an optimum incorporation of 1 wt. % templates, they achieved a 97% textured material and a $d33$ = 274 pC/N. Note, however, that leakage current increased substantially with template addition.¹⁵¹ Kobayashi et al.¹⁵⁰ synthesized [111]-textured BT via electrophoretic deposition under a magnetic field of 12 T. They claimed 100% texture, which led to an enhancement of 30% in $d33*$ and reduced the strain hysteresis. Texturing is also applicable in materials with engineered grain and domain sizes, although it increases the processing complexity and costs.¹⁵² Wada et al.^92,95,96 indicated that [110]-textured BT, with an average grain size of 75 μm and average domain size of 800 nm, has a $d33$ of 788 pC/N, clearly highlighting the potential of this approach. Texturing maximizes the piezoelectric and dielectric properties since it increases the fraction of domains oriented along the electric field.^147,148

One of the recent strategies to improve the properties of alkaline-niobate- and alkaline-bismuth-titanate-based piezoelectrics involves the design of heterostructure composite^153–157 and core-shell materials.^17,158 The development of BT-based core-shell bulk materials has been widely used to improve dielectric properties for capacitor applications.^17,159 Nonetheless, to date, this microstructural engineering approach remains in its infancy to tailor piezoelectric properties.¹⁷ The core-shell microstructures in bulk materials are generally a consequence of partial homogenization as a result of limited diffusion^160–162 or formation of a liquid phase with an excess of solute that precipitates or crystallizes upon cooling.^163–165 

The main research in alkaline-bismuth-titanate-based materials is directed towards their application in off-resonance actuators working at high electric fields.^16,17 In this context, the large signal strain has been attributed to an electric field-induced phase transition.¹⁶ Composites were designed to aid this electric field-induced phase transition due to the different polarity of the chosen ferroelectric constituents.^153–155 Recently, alkaline-bismuth-titanate-based composites also were tested for temperature-stable small signal piezoelectric applications. In this case, a semiconductor was selected as seed material. It was proposed that the free charges of the semiconductor agglomerate at grain boundaries, thereby stabilizing the domains of the matrix. This results in considerably improved stability of the piezoelectric properties.^156,157

The core-shell approach was also successful in obtaining promising alkaline-niobate-¹⁵⁸ and alkaline-bismuth-titanate-based^17,166 piezoelectrics suitable for off-resonance actuators working at high electric fields. Tuning the core-shell microstructure, grain size, and density was used to increase the large signal strain output and reduce the electric field required to induce the high strain.¹⁶⁰ It was also proven that the mechanisms of core-shell materials can be quite complex¹⁶⁷ and that the electrical charges at the interface between core and shell have an important role in determining the piezoelectric properties.¹⁶⁸ To date, no core-shell materials were specifically developed for small signal piezoelectric applications. Neither have BT-based core-shell materials been engineered for piezoelectric properties. Therefore, the potential of core-shell piezoelectrics remains largely unexplored.

Owing to their simplicity, chemical modifications constitute the most common approach for tuning dielectric and electromechanical properties of ferroelectrics. In ferroelectrics, the term doping typically refers to ion substitutions made at concentrations lower than 3 to 5 at. %. Despite the low concentration, dopants can alter properties significantly. Chemical modification typically refers to substitutions greater than 5 at. % and extending well into the chemical mixing regime to form solid solutions.¹⁰ Modifiers are used to raise or lower the Curie temperature and/or to alter the nature and sequence of phase transitions from the high temperature prototype phase. Both dopants and modifiers are discussed in detail below.

In perovskites, doping generally involves the replacement of either the A- or B-site cations. Although the dopant concentration may be very low, they can considerably alter the functional properties by modifying the extrinsic contributions.^121,169 Isovalent or aliovalent dopants can be added prior to calcination to modify properties or also as a sintering aid to facilitate the densification process. The dopants used for a particular ferroelectric are selected based on properties to be controlled for a target application.

Aliovalent dopants can have a lower or higher oxidation number as compared to the lattice site, referred to as acceptor or donor doping, respectively. Acceptor doping is compensated by an increase in the concentration of oxygen vacancies and/or electron holes. This can lead to the formation of complex defect dipoles between dopant ions and oxygen vacancies that locally modify the elastic and electrostatic fields. These complex defects serve as pinning centers, which hinder domain wall motion and lead to ferroelectric hardening. Hardening is reflected in pinched ferroelectric hysteresis loops and development of an internal bias-field. It is also accompanied by a reduction of the dielectric and piezoelectric response as well as a decrease in the dissipation factor.

Acceptor-doped ferroelectrics with charge-compensating oxygen vacancies form the technological basis for hard piezoceramics.^170,171 It is well documented that domain configurations, switching processes and thus electromechanical properties are highly dependent on the defect dipoles and their alignment along the spontaneous polarization direction of ferroelectric phases.^172,173 This phenomenon has been termed a symmetry-conforming short-range ordered (SC-SRO) configuration of point defects.¹⁷⁴ The SC-SRO model indicates that, at equilibrium, the local symmetry surrounding point defects will conform to the symmetry as the host crystal. In the case of ferroelectrics, the local symmetry is that of the defect dipoles, which align along the spontaneous polarization direction.¹⁷⁴ Since the ferroelastic switching responsible for the strain does not involve diffusion of ions, symmetry conforming defect dipoles act as a restoring force for reversible domain switching and concomitant strain. This strain mechanism has been used to describe the recoverable strain of several doped single crystals^174,175 and ceramics.^176–178 It is worth noting though that this mechanism is based on a balance between kinetics (ion migration) and thermodynamics.¹⁷⁹ Therefore, the effectiveness of this mechanism to produce a restoring force after cycling, different temperatures, and frequencies should be assessed for each specific case. Some studies have indeed established certain stability of the reversible strain^180–182 but more work in this area is required.

Replacing the A- or B- site cations with a donor dopant is compensated by free electrons, a decrease in oxygen vacancies, and/or formation of cation vacancies. Donor doping leads to ferroelectric softening, resulting in a high remanent polarization, high dielectric loss, as well as an increase in dielectric and piezoelectric properties. These features are ascribed most commonly to an increased mobility of domain walls.^2,183

Doping can be used as an effective strategy to tune several functional properties. A good example for aliovalent doping can be found for semiconductors based on barium titanate, which exhibit a PTCR. In PTCR materials, the electrical conductivity abruptly changes from a conductive to a non-conductive state when the temperature is raised, and the change in conductivity can be used in sensors, self-regulating heaters, overcurrent protection, and time delay circuitry. In barium titanate, the PTCR behavior is achieved by introducing donor dopants, such as Y³⁺ or La³⁺. These A-site donor dopants, or B-site donor dopants such as Nb⁵⁺ and Ta⁵⁺, may also be used to suppress the magnitude of the relative permittivity at $TC$ for high temperature dielectrics, although they may need to be compensated with acceptor dopants to avoid semiconducting properties.^2,10

A summary of recent works that realized promising doped BT-based materials with $d33$ values above 200 pC/N is given in Table II. Table II include the Curie temperature (⁠$TC$⁠), relative permittivity (ε), piezoelectric strain coefficient (⁠$d33$⁠), planar electromechanical coupling coefficient (k_p), and mechanical quality factor (Q_m). For comparison purposes, we note here that pure BT (with no specific engineering approaches) typically features a $TC$∼130 °C, room temperature $εr′$∼1700, $d33$∼190 pC/N, and $kp$∼0.35.² Note that all dopants presented in Table II were added before calcination, i.e., not as sintering aids. Li⁺ is a classical dopant used to lower the sintering temperature and promote densification through liquid phase sintering. Kimura et al.¹⁸⁴ demonstrated that Li-doped BT nanopowders obtained by a solvothermal method can be sintered successfully at 1100 °C, which is between 200 °C and 250 °C lower than the sintering temperature generally employed for BT.² The samples featured a $d33$=260 pC/N and a $TC$=130 °C. Yang et al.¹⁸⁵ demonstrated the possibility to sinter LiF-doped BT at 1100 °C leading to an optimum $d33$ = 270 pC/N for 0.04 at. % LiF. Zhu et al.¹⁸⁶ revealed that 0.04 at. % LiF doped BT with an excess of 0.025 at. % Ba features a $d33$=340 pC/N, while keeping the same sintering temperature of 1100 °C. Nonetheless, in both studies, the Curie temperature was decreased to below 95 °C. If 0.02 at. %, Ca²⁺ excess is added instead of Ba²⁺ excess to 0.04 at. % LiF-doped BT, an even higher $d33$ = 361 pC/N is obtained. Concomitantly, a lower $TC$=70 °C is also obtained. Co-doping BT with 0.04 at. % Sn⁴⁺ and 0.002 at. % Y³⁺ leads to promising properties with a high $d33$=454 pC/N and a moderate $TC$=84.7 °C. Nonetheless, the sintering temperature of 1420 °C was relatively high.¹⁸⁷ Doping with A(Cu_1/3Nb_2/3)O₃ (A = Ca²⁺, Sr²⁺, Ba²⁺) revealed that this combination of cations lowers $TC$ considerably and does not increase $d33$ above 350 pC/N.¹⁸⁸ 

The perovskite crystal structure admits a wide range of isomorphous ion substitutions. In fact, there are numerous binary ferroelectric systems showing complete solubility across the sub-solidus phase diagram. Typical examples of A-site modifiers in barium titanate include Sr²⁺, Ca²⁺, Bi³⁺, and Cu²⁺.^10,190 Typical isovalent B-site modifications to improve barium titanate piezoelectric properties include Zr⁴⁺, Hf⁴⁺, and Sn⁴⁺.^2,191–198 More details about the effect of these modifiers on the pseudo-binary phase diagrams, synthesis, and piezoelectric properties are given below.

The technologically most relevant solid solution systems are those that have compounds of different symmetry located as opposing end members. In such systems, a change in composition leads to a change in Curie temperature and often to a change in the order of the paraelectric to ferroelectric transitions from first-order to second-order, or in some cases, to diffuse and/or relaxor ferroelectric behavior. The phase diagrams of these systems are usually constructed under the assumption that atomic diffusion providing equilibration of the system is thermally frozen such that single-phase regions of differing symmetry are separated by line boundaries. These boundaries may be crossed either by an isothermal change in composition or by a change in temperature at a fixed composition. It is typically the extrema in physical properties occurring near the phase boundary lines that are utilized in device applications. The physical phenomena underlying the disparate phase change characteristics displayed near transition lines in solid solution systems thus remain of intense interest.

Despite the fact that most research since 1950s has utilized the property softening associated with these transitions to enhance piezoelectric properties,⁷ we note that there does not seem to be a consensus as to whether the phase transitions occurring at the boundary lines should be more properly regarded as polymorphic or morphotropic.¹⁹⁹ For example, Goldschmidt²⁰⁰ proposed a distinction between systems that feature compositionally driven polymorphs from those in which the transition between polymorphs occurs as a consequence of changes in temperature or pressure. The transition line in materials with compositionally driven polymorphs was termed a morphotropic phase boundary (MPB), whereas the transition line occurring as a result of changes in other thermodynamic variables was termed a polymorphic phase boundary (PPB). However, in the ferroelectrics literature, the term morphotropic is usually reserved for systems that display nearly temperature-independent (e.g., vertical) boundary lines in the composition-temperature plane.⁷⁹ The archetypal example of such a system is PZT. Though widely used, this definition is somewhat arbitrary, since even in lead-based systems the MPB is not strictly vertical, and depending on the particular system, displays varying degrees of inclination and curvature. Nonetheless, we will adopt this convention here. We will refer to an MPB only to denote a nearly vertical phase boundary and use the term PPB to denote a phase boundary showing substantial inclination and/or curvature. We note the latter type of phase boundary is more typical of the BT-based solid solutions, as discussed below in more detail.

Four representative pseudo-binary phase diagrams for barium titanate-based solid solutions are illustrated in Fig. 6, which introduces a nomenclature to differentiate among the various topologies of phase diagrams in BT systems that are referred later in the text. Note that we will refer to many phase diagrams in this and subsequent sections as “pseudo-binary,” “pseudo-ternary,” and so on. The term “pseudo” is used to indicate that the end members on the phase diagram of interest are not the individual components of the system. As an example, the 50 mol. % isopleth in the three-component system BaO-SrO-TiO₂ may be referred to as pseudo-binary system comprising the oxide compounds BaTiO₃ and SrTiO₃ as end members. The solid solution between barium titanate and strontium titanate (BSrT) is represented by the phase diagram of Fig. 6(a).²⁰¹ Both the line of paraelectric–ferroelectric transitions and the lines of the inter-ferroelectric transitions shift to lower temperatures with increasing strontium concentration, and thus BSrT solid solution is described here as having a phase diagram of the “shifting” type. The second phase diagram in Fig. 6(b) is representative of the solid solution between barium titanate and barium zirconate (BZT).^193,194 Note, however, that this type of phase diagram is also displayed by solid solutions between barium titanate and barium hafnate (BHT),^194,198 as well as between barium titanate and barium stanate (BSnT).¹⁹⁶ The B-site cations Zr⁴⁺, Hf⁴⁺, and Sn⁴⁺ have similar effects as modifiers, albeit Sn⁴⁺ generally modifies the phase boundaries more strongly than Zr⁴⁺ and Hf⁴⁺.²⁰² With sufficiently high B-site dopant concentration, these systems exhibit relaxor features, which were attributed to local chemical heterogeneity in BSnT.²⁰³ The line of paraelectric–ferroelectric transitions decreases with zirconium (hafnium, tin) concentration, but the lines of the inter-ferroelectric transition increase, such that stability fields of the four phases (cubic-tetragonal-orthorhombic-rhombohedral) converge on approaching the line of Curie points. The BZT (BHT, BSnT) solid solution may, therefore, be described as having a phase diagram of the “pinching” type. The third phase diagram in Fig. 6(c) is representative of the solid solution between barium titanate and calcium titanate (BCT).¹⁹⁰ The line of paraelectric–ferroelectric transitions decreases slightly with calcium concentration, but the stability fields of the orthorhombic and rhombohedral phases vanish, at the expense of the tetragonal phase, at comparatively low substitution levels. The BCT solid solution may, therefore, be described as having a phase diagram of the “disappearing phase” type. A very similar phase diagram, except with an increase in Curie temperature, occurs in the solid solution between barium titanate and lead titanate.²⁰⁴ It is evident that other pseudo-binary systems can be formed by choosing appropriate isopleths within the solid solution fields of the pseudo-ternary or pseudo-quaternary systems comprising the four end-member compounds (barium titanate, barium zirconate, strontium titanate, and calcium titanate) just considered. For example, by choosing such an isopleth in the pseudo-ternary diagram formed by barium titanate, barium zirconate, and calcium titanate, the phase diagram depicted in Fig. 6(d) is obtained, which is a schematic representation of the BZT-BCT pseudo-binary phase diagram. Section V A 2 a discusses the details of this phase diagram.

Because of the similarity of the diagram in Fig. 6(d) with the so-called morphotropic systems, such as lead zirconate titanate (PZT), lead magnesium niobate-lead titanate (PMN-PT), or lead zinc niobate-lead titanate (PZN-PT), this diagram was first regarded as a morphotropic system.¹¹ However, this similarity is evident only in a portion of the diagram above room temperature. A careful comparison of the diagrams in Figs. 6(a), 6(b), and 6(d) reveals that the phase diagram of BZT-BCT is not topologically distinct from those of other binary solid solutions. In fact, the phase diagrams of BZT (BHT, BSnT) and BZT-BCT are both diagrams of the pinching type, differing only in the slopes of the interferroelectric transition lines. Further, as discussed above, since a morphotropic transition should occur only in systems for which the inter-ferroelectric transition lines are nearly vertical, all of the transitions between phases indicated on the diagrams in Fig. 6 should be regarded as polymorphic based on the nomenclature proposed by Goldschmidt²⁰⁰ and Jaffe.⁷ 

The different phase diagrams in Fig. 6 can be rationalized by considering how the various solutes (Sr²⁺, Ca²⁺, Zr⁴⁺, Hf⁴⁺, and Sn⁴⁺) interact with the host barium titanate. The first, and perhaps most important, of these interactions is the modification of the paraelectric–ferroelectric phase transition temperature. It has been argued that the variation of the transition temperature with solute concentration can be divided into four regimes.²⁰⁵ In the first regime, the solute concentration is very low, and the solute atoms are widely separated, such that the distortion fields around the solute atoms do not interact. In this regime, the transition temperature is essentially insensitive to the presence of the solute. In the second regime, at somewhat higher solute concentrations, the distortion fields still do not overlap, but can couple indirectly by interactions through the structure of the host crystal. In this regime, the transition temperature changes linearly with solute concentration. In the third regime, as the solute concentration is further increased, the distortion fields begin to overlap more significantly, producing conjugated fields and a more dramatic change in transition temperature. In the fourth regime, at even higher solute concentrations, the average distance between solute atoms becomes comparable with the interatomic separation distance in the host crystal, and a homogenous field develops. In this fourth regime, the transition temperature is again linearly proportional to concentration, but with a proportionality constant that is different from that in the second regime. The net result is that the variation of the paraelectric–ferroelectric transition temperature is linear with concentration in the chemical mixing regime, but weakens at low solute concentrations, resulting in a plateau effect.²⁰⁵ Such an effect is evident in the phase diagram of BZT depicted in Fig. 6(b). The concentration range of this effect, in turn, depends on the type of solute (isovalent or aliovalent) and the corresponding nature and strength of the defect field. It is typically in the range of a few mole percent but may be much less. Thus, such an effect may be present in the other diagrams provided in Fig. 6. However, it may not be visible on the scale of the figure or transition temperatures may not have been measured at sufficiently low solute concentrations.

The linear dependence of a transition temperature on solute concentration in the chemical mixing regime can be described by adding to the Landau polynomial of Eq. (17), the Gibbs free energy of the solute interaction

where $ϑi$ are the interaction coefficients. Eq. (17) then becomes

and considering only the lead terms, this reduces to

wherein $λ1=2ϑ1/χ0$⁠, and the $ξi0=ξi(θ)$ with $λ2=4ϑ2/ξ10$ and $λ3=2ϑ3/ξ20$⁠. Comparing Eqs. (17) and (38), it is seen that the Curie-Weiss temperature is a linear function of composition

where $θ(0)$ is the Curie-Weiss temperature in pure barium titanate (x = 0) and decreases with solute concentration for $λ1$> 0. Similarly, the quartic terms in Eq. (38) also take on a dependence on solute concentration

A point on the composition-temperature diagram where the transition between paraelectric and ferroelectric phases passes over from first-order to second-order is known as a tricritical point (TCP) and occurs when the sum of the quartic terms in Eq. (38) are equal to zero. Thus, following Eqs. (40) and (41), the Gibbs free energy of the solute interaction also provides for the possibility of such a point. For example, a TCP has been reported in the BSrT system,²⁰⁶ with the location indicated in Fig. 6(a).

If, on the other hand, the solute concentration is sufficiently high as to reduce the paraelectric–ferroelectric transition temperature to values approaching absolute zero, the simple predictions in the mean-field approximation of the Landau theory no longer apply. Near absolute zero, quantum mechanical effects play a significant role, and the dependence of the transition temperature on composition strengthens. This phenomenon is known as the quantum saturation (QS) effect.²⁰⁷ It has been shown that for this situation, the free energy in Eq. (38) is modified to

where $θS$ is the saturation temperature below which quantum effects become important. The steepening of the paraelectric to ferroelectric transition temperature versus composition curve due to quantum saturation effects is also observed in the BSrT system [Fig. 6(a)]. A quite similar situation occurs when barium titanate is placed under hydrostatic pressure.^208,209

With reference to the phase diagrams displayed in Fig. 6, attention has thus far been focused on phenomena that occur close to the lines of the paraelectric–ferroelectric transitions, namely, the expected variation of the paraelectric–ferroelectric transition temperature with solute concentration and the possible occurrence of a TCP. To describe the different topologies of these diagrams, it is now necessary to account for the inter-ferroelectric transitions, and hence to consider explicitly the solutions to the Landau polynomial given by Eqs. (24)–(26), which correspond to the tetragonal, orthorhombic, and rhombohedral phases, respectively. Writing the Landau polynomial of Eq. (17) in terms of the modulus of polarization, P, the non-equilibrium free energy for each of these phases is

wherein it is understood that, as discussed above, the Curie-Weiss temperature is a function of composition and that the dielectric and higher-order stiffness coefficients may also depend on composition and/or temperature. Inspection of Eqs. (43)–(45) reveals that the topology of the energy surface, and hence the transitions between ferroelectric phases, as well as the anisotropy in dielectric, piezoelectric, and elastic properties, are controlled by the differences between higher-order dielectric stiffness coefficients. After minimizing the free energies of Eqs. (43)–(45) with respect to $P$⁠, the free energies are equal if it turns out that $ξ1=ξ2$⁠, $ζ1=ζ2$⁠, and $ζ3=2ζ2$⁠. It is thus convenient to rewrite the free energy in terms of three parameters, $B2=ξ1−ξ2$⁠, $C2=ζ1−ζ2$⁠, and $C3=ζ3−2ζ2$⁠, which describe the anisotropy of the energy surface. As such, the composition and/or temperature dependences of these terms primarily control the phase diagram.

The analysis of differing types of phase diagrams becomes more transparent if Eqs. (43)–(45) are expressed in a simple general form, taking the polarization vector P to be nP, where n is unit vector in the direction of polarization and P =|P| is its absolute value^210,211

with

wherein the three invariants of the form $(n12+n22+n32)m=1$ and are therefore isotropic. It is thus evident from Eqs. (47)–(49) that the inter-ferroelectric transitions will be dictated by the three anisotropy parameters, $B2(T,x)$⁠, $C2(T,x)$⁠, and $C3(T,x)$⁠. After inserting Eqs. (47)–(49) into Eq. (46) and minimizing the free energy with respect to n and P, the equilibrium free energies of the tetragonal, orthorhombic, and rhombohedral phases are obtained. Assuming that the Landau polynomial is asymptotically accurate on approaching the line of Curie points, the polarization becomes small, and the sextic contributions can be neglected. The lead anisotropic term $B2(T,x)$ then plays a dominant role. Under these conditions, the paraelectric–ferroelectric transition will be of the type cubic to tetragonal if $B2(T,x)<0$ and of the type cubic to rhombohedral if $B2(T,x)>0$⁠; the truncated Landau polynomial cannot produce a stable minimum for the orthorhombic phase at small values of P.²¹² 

The phase diagrams represented by BSrT, BZT (BHT, BSnT), and BZT-BCT all display a convergence region. Taking $B2(T,x)$ to be a linear function of composition and temperature, as in the low-order approximation described above, then gives an equation for a line in the phase diagram along which the anisotropy of the energy surface is dramatically reduced

or simply,

As the temperature is lowered from the convergence point (x_CP, T_CP), the polarization increases, and so the sextic terms in Eq. (49) begin to play a more important role. It is therefore expected that the inter-ferroelectric transitions between tetragonal, orthorhombic, and rhombohedral phases occur in the region of the phase diagram around the line in Eq. (51). The particular topology of the phase diagram will then depend on the variation of the sextic terms, $C2(T,x)$ and $C3(T,x)$⁠, with composition and temperature. In fact, the topologies of the shifting, pinching, and polymorphic phase diagrams (Fig. 6) can all be adequately reproduced by simply changing the slope of the line, $λ/ξ′$ along which the lead contribution to the anisotropy of the energy surface is reduced.^213–215 This reduction leads to a rotational instability of the polarization, which has important consequences for the domain structure and electromechanical properties. For example, as can be appreciated from Fig. 2, near the tetragonal-orthorhombic phase boundary, the rotational instability is reflected in a divergence of the transverse dielectric susceptibility and a strong softening of the shear compliance. The implications of such instabilities for the conformal miniaturization of the domain structure to nanoscale dimensions were pointed out²¹⁶ and have been recently reviewed.²¹⁷ Gao et al.²¹⁸ confirmed experimentally the miniaturization of the domain structure near phase boundaries in BaSn_0.105Ti_0.895O₃. Convergent beam electron diffraction revealed that the nanometer-sized domains feature different local symmetries.

In addition, the simple low-order theory described above also accounts for the possibility of one or more TCPs. As noted above, a tricritical transition occurs when the entire quartic contribution in Eq. (48) to the free energy vanishes

The variation of the difference between Curie and Curie-Weiss temperatures can then be easily determined from the standard thermodynamic relation for first-order transitions

Since, to a first approximation, $θ(x)$ and all the Landau coefficients may be taken as linear functions of x, it is apparent that the value of $Δ(x)$ varies as x². Hence, the temperature difference $Δ(x)$⁠, which is a measure of the first-order character of the transition, becomes vanishingly small, asymptotically approaching zero as (x – x_cr) → 0.

The application of the simple analysis above is complicated in real systems by two main factors. First, the type of solute also plays an important role in determining whether the phase transitions are sharp or diffuse. The influence of differing types of solutes has been divided into three main classes: (i) solutes that cannot generate electric fields locally, but modify the transition temperature, (ii) solutes that generate random local electric fields that interact with the host crystal and (iii) solutes that are thermally frozen in non-equilibrium configurations and which persist below the transition temperature.²¹⁹ Although the isovalent ion substitutions (Sr²⁺, Ca²⁺, Zr⁴⁺, Hf⁴⁺, and Sn⁴⁺) may be expected to act as solutes of type (i), it is well-known that perovskite crystals contain rather high densities of defects in the form of uncompensated oxygen vacancies, thereby generating random fields normally attributed to solutes of type (ii). Further, depending on the thermal process history, compositional heterogeneities corresponding to solutes of type (iii) may not always be eliminated. A second complicating factor concerns two-phase coexistence. The inter-ferroelectric boundary lines drawn in Fig. 6 describe two phases in equilibrium at the same composition under the condition that atomic diffusion is thermally frozen. Under these conditions, there always exists a metastable two-phase coexistence region adjacent to the boundary line, as the transitions between the ferroelectric phases are all of first-order. However, if atomic diffusion providing equilibration of the system is not thermally frozen, then the free energy of mixing must be explicitly taken into account. For example, if the enthalpy of mixing is positive, a miscibility gap may exist, leading to rather more extensive two-phase fields along the boundary lines.²²⁰ Either of these two factors can produce strong smearing of the phase transitions, making the accurate determination of transition temperatures, phase convergence, and TCPs—i.e., compared with the corresponding homogeneous system—experimentally quite difficult. Many of the controversies in the literature concerning the phase diagrams in Fig. 6 are likely to have their origins in measurement uncertainties or errors in analysis arising from smearing of the phase transitions due to defects or phase coexistence.

The synthesis parameters and formation mechanisms of BT-based materials generally follow similar trends. BaTiO₃ powder is produced by the solid state reaction between BaCO₃ and TiO₂ in the temperature range between 800 °C and 1300 °C.^221,222 The reaction consists of several stages, whereby the Ba²⁺ and O^2– ions first diffuse into TiO₂ and the perovskite BaTiO₃ phase is formed. The product, however, hinders further diffusion of Ba²⁺, which results in the formation of the intermediate Ba₂TiO₄ phase. Finally, in the case of a stoichiometric starting mixture, this intermediate phase reacts with remaining TiO₂ to form the BaTiO₃ phase.

The formation of solid solution compositions can be more complex. As an example, let us consider the synthesis and formation mechanism of BZT. Bera et al.²²³ investigated the formation mechanism of BaTi_0.60Zr_0.40O₃ produced via the solid state route from BaCO₃ (D₅₀=2.09 μm), TiO₂ (D₅₀=0.35 μm), and ZrO₂ (D₅₀=10.27 μm) powders. From differential scanning calorimetry and thermal gravimetric analysis, they observe that BaCO₃ begins to decompose at 556 °C, although the most significant decomposition occurs at 979 °C. X-ray diffraction indicates that BT starts to form at 700 °C and BaZrO₃ (BZ) begins to form at 800 °C. In contrast to BT, formation of intermediary phases was not detected by the authors. Note that this is a similar temperature range, between 600 °C and 700 °C, reported for the formation of BaSnO₃.²²⁴ BZT forms due to interdiffusion between BT and BZ that begins at 1300 °C, but secondary phases can be still resolved even at 1600 °C.²²³ Cation diffusion has been recognized as the limiting step in the synthesis of BZT²²³ and in similar systems such as BaSnO₃.²²⁴ Thus, solid solutions of BT-based piezoelectrics have been generally calcined at temperatures between 1100 °C and 1300 °C to form a perovskite phase. Generally, sintering between 1400 °C and 1560 °C under atmospheric conditions has been performed.^{191–197,201} Under these conditions, secondary phases may remain.²²³ 

Kalyani et al.¹⁹¹ investigated the effect of the electric field magnitude on the poling behavior of BaZr_0.02Ti_0.98O₃. A maximized $d33$ value of 357 pC/N was obtained when the material was subjected to a poling field of 1.7 kV/mm. For electric fields lower than 1.7 kV/mm, BaZr_0.02Ti_0.98O₃ features coexistence of tetragonal and orthorhombic phases. For poling fields larger than 1.7 kV/mm, the electric field stabilized a single rhombohedral phase thereby diminishing the piezoelectric properties.

Yao et al.²⁰¹ considered the relationship between the phase diagram and the $d33$ activity in the BSnT, as displayed in a contour plot in Fig. 7. Near the convergence region at 11 mol. % Sn⁴⁺ and 40 °C, an extremely large $d33$=697 pC/N was found. So far, this is the largest $d33$ value reported for a binary BT-based system. The high temperature variations of $d33$ values with variation of composition and temperature in BSnT are clearly highlighted in this figure.

Figure 8 shows the variation of (a) $TC$ and (b) $d33$ as a function of Zr⁴⁺, Sn⁴⁺, and Hf⁴⁺ content at room temperature with data adapted from several works.^{191,193,195–197,201} With increasing modifier content, $TC$ is clearly reduced in all systems. It is observed that Zr⁴⁺ and Hf⁴⁺ affect $TC$ quite similarly, whereas Sn⁴⁺ decreases $TC$ more strongly. The $d33$ values are similarly modified with Zr⁴⁺ and Hf⁴⁺ modification below 4 at. %. For higher modifier content, Zr⁴⁺ induces much higher $d33$ values. Sn⁴⁺ generally leads to the highest $d33$ values with a maximum value of 425 pC/N at 4 at. % Sn⁴⁺.