NEW FERROELECTRICS AND ANTIFERROELECTRICS

G. A. Smolenskii

I. INTRODUCTION

The problem of ferroelectricity is one of the important problems of solid-state physics. The study of ferroelectric phenomena provides direct information about the nature of crystals. At the same time, the study of ferroelectric phenomena is also of independent interest, in view of the steadily increasing use of ferroelectric materials in a number of branches of technology.

The principal distinguishing feature of ferroelectrics is the presence of a phase transition of the first or second kind from a disordered state to an ordered polarized state.

As a result of the appearance of spontaneous polarization, ferroelectrics possess more or less pronounced anomalies of electrical and other properties in comparison with ordinary ionic crystals. In them, at temperatures below the phase-transition point, spontaneous deformations arise and the symmetry of the lattice is lowered.

The occurrence of spontaneous polarization is explicable within the framework of classical electrostatic interaction, since the internal field is determined by the interaction of dipoles caused by the displacement of ions. Of course, an essential role is played not only by long-range forces, i.e. dipole forces, but also by short-range forces. Spontaneous polarization arises only in the case when the elastic component of the restoring force that appears upon displacement of the ions is smaller than the dipole forces.

In contrast to ferroelectrics, in which ions of one kind at temperatures below the phase-transition temperature are displaced within a domain in one direction, in antiferroelectrics ions of one kind are displaced antiparallel to one another and the spontaneous polarization in such crystals is equal to zero. Ferroelectrics and antiferroelectrics are, as it were, the electrical analogues of ferromagnets and antiferromagnets, respectively. Depending on external conditions (temperature, pressure, electric field), in some crystals ions of one kind may be displaced either parallel or antiparallel to one another. Such crystals should expediently be classified not as ferroelectrics or antiferroelectrics, but rather it should be indicated that, under the given conditions, they are in a ferroelectric or antiferroelectric state.

All known ferroelectrics and antiferroelectrics may be divided into two main groups: 1) ferroelectrics and antiferroelectrics containing hydrogen (Rochelle salt, potassium phosphates and arsenates, and others), and 2) ferroelectrics and antiferroelectrics not containing hydrogen (barium titanate, lead titanate, potassium niobate, and others). The structure and properties of the ferroelectrics of these groups differ considerably from one another.

For all ferroelectrics and antiferroelectrics of the second group, an octahedral environment by oxygen ions of the smaller cation is characteristic, and therefore in what follows we shall call them ferroelectrics and antiferroelectrics of the oxygen-octahedral type.

The foundations of the physics of ferroelectric phenomena were developed by Kurchatov in the late thirties and early forties, in the study by him and his collaborators of Rochelle salt and its isomorphous crystals.^1 At that time Kurchatov put forward the idea that the occurrence of spontaneous polarization is quite explicable within the framework of the classical electrostatic interaction. The ferroelectrically active ion, i.e., the ion responsible for the occurrence of spontaneous polarization in Rochelle salt, potassium arsenates and phosphates, and other crystals containing hydrogen, is apparently the proton. However, determining the position of the proton in the complex structure of these crystals is difficult. In addition, Rochelle salt is characterized by brittleness and decomposes at comparatively low temperatures. In potassium arsenates and phosphates, spontaneous polarization arises at low temperatures. The circumstances listed hindered the successful development of studies of ferroelectrics and limited the field of their practical application.

The discovery by Wul and Goldman^2,3 of a new ferroelectric—barium titanate, crystallizing in a simple lattice of the perovskite type, having a sufficiently high Curie point (120° C), and distinguished by good mechanical strength and thermal stability—played an important role in the development of ideas about ferroelectric phenomena and served as the beginning of many works on the investigation of ferroelectrics of the oxygen-octahedral type and on the creation of a new group of materials.

After the discovery of the ferroelectric properties of barium titanate, for some time it was assumed that it was the only ferroelectric among crystals of the perovskite type, since the distance between titanium and oxygen ions in the barium titanate lattice is greater than the sum of their radii. This feature of barium titanate should lead to greater ionic polarizability upon displacement of titanium ions.

However, subsequently Smolenskii, Matthias, and others showed that barium titanate is one of many ferroelectrics of the oxygen-octahedral type. At present a considerable number of new ferroelectrics and antiferroelectrics are known which crystallize in structures of the perovskite, ilmenite, rare-earth sesquioxide, and pyrochlore types.

Let us consider the principal properties of the new ferroelectrics and antiferroelectrics.

II. NEW FERROELECTRICS AND ANTIFERROELECTRICS OF THE OXYGEN-OCTAHEDRAL TYPE

1) Strontium titanate

Strontium titanate crystallizes in the ideal cubic structure of the perovskite type (Fig. 1). Smolenskii^4 showed that polycrystalline samples of strontium titanate made from technical grades of titanium dioxide possess ferroelectric properties at very low temperatures (20–30° K). At the same time Hulm^5 published a communication in which the opposite is asserted.

Recently, Grenicher^6 reported that strontium titanate is not a ferroelectric down to 2° K, but in comparatively small external fields at temperatures below 50° K a hysteresis loop characteristic of ferroelectrics appears in it. Thus, according to Grenicher, in SrTiO$_3$ the ferroelectric state is induced by an external electric field. These investigations were

were carried out on very pure single crystals of strontium titanate. Taking into account Grenicher’s results, one can explain the discrepancies obtained in earlier works. Evidently, Hülm studied strontium titanate in weaker fields, while Smolenskii studied it in stronger fields.

2) Lead titanate

Lead titanate, as is known, crystallizes in a structure of the perovskite type. At room temperature the lattice of lead titanate is tetra-

Fig. 1. Arrangement of ions in the compounds CaTiO₃, SrTiO₃, BaTiO₃ with a perovskite-type structure.

gonal, with the ratio of the lengths of the unit-cell edges

\[ c/a = 4.141/3.891 = 1.0635^{7}. \]

The phase transition in lead titanate at \(500^\circ\mathrm{C}\) was discovered by Smolenskii\(^4\), and also independently by Shirane, Hoshino, and Suzuki\(^8\). The assumption that lead titanate is a ferroelectric with a high phase-transition point was first put forward by Jonker and Santen\(^9\). These authors found that the phase-transition temperature of the solid solutions \((\mathrm{Ba},\ \mathrm{Pb})\mathrm{TiO}_3\) and \((\mathrm{Sr},\mathrm{Pb})\mathrm{TiO}_3\) increases as the content of lead titanate in them is increased. On this basis they concluded that lead titanate possesses ferroelectric properties. However, lead titanate and solid solutions with a high content of lead titanate were not investigated by them.

Fig. 2. Dependence of the initial dielectric permittivity of a polycrystalline specimen of lead titanate on temperature.\(^4\)

The dependences of the dielectric permittivity and of the coefficient of linear expansion of polycrystalline lead titanate, according to Smolenskii’s data\(^4\), are shown in Figs. 2 and 3.

At the phase-transition point, on the curve representing the dependence of the coefficient of linear expansion on temperature, there is a minimum. This minimum is explained by the fact that, upon heating, the volume and linear dimensions

segnetoelectric change both through ordinary thermal expansion and through changes in the strains caused by a decrease in the spontaneous polarization. The volume of the body, owing to the decrease in spontaneous polarization upon heating, may decrease (positive volume electrostriction),

Fig. 3. Dependence of the relative change in length¹ and of the coefficient of linear expansion² of a polycrystalline lead titanate specimen on temperature⁴.

as in the present case, but it may also increase (negative volume electrostriction).

It should be noted that in barium titanate the volume spontaneous electrostriction is also greater than zero. However, the electrostriction in lead titanate is much greater than in barium titanate. This is also directly confirmed by neutronographic studies on determining ion displacements in the elementary cell of lead titanate¹⁰ (Table I).

Table I

* Oxygen ions located on lines parallel to the axis of spontaneous polarization are denoted \(O_1\).

X-ray structural studies⁸, ¹⁰ showed that above \(500^\circ\mathrm{C}\) the crystal has a cubic structure, and below it a tetragonal one. In lead titanate, just as in barium titanate, there occurs an increase of one axis \(c\), which evidently becomes the polar axis, and a decrease of the other two; the cell volume thereby increases. The heat of the phase transition of lead titanate is considerably greater than that of barium titanate and is equal to \(1150\ \text{cal}/\text{mol}\)¹¹.

In lead titanate the birefringence \(\Delta n\) changes anomalously with temperature (Fig. 4). It is known that in barium titanate it increases with decreas-

In Figs. 2 and 3 are shown the temperature dependences of the dielectric constant and the coefficient of linear expansion, obtained during heating of the specimens at a rate of one degree per minute. From these figures it is seen that low-temperature phase transitions in lead titanate are not observed at temperatures down to \(-195^\circ\mathrm{C}\). However, if the temperature of the specimen is changed very slowly, then at \(-100^\circ\mathrm{C}\) a phase transition is observed\(^{12}\).

In the investigation of polycrystalline specimens of lead titanate it was not possible to obtain hysteresis loops. This is explained by the fact that at high temperatures the conductivity of the specimens is large, while at low temperatures, far from the transition temperature, the domains no longer orient themselves in permissible fields.

Fig. 4. Dependence of the double refraction of lead titanate on temperature\(^{10}\).

Single crystals of lead titanate were grown and investigated by Belyaev and Khodakov\(^{13}\). Fesenko succeeded in detecting the domain network of single crystals of lead titanate\(^{14}\). In addition, Fesenko determined the refractive index of lead titanate \((n = 2.65)\).

Blokhin, investigating the influence of temperature on the X-ray \(K\)-absorption spectrum\(^{15}\), came to the conclusion that lead titanate is a typical electronic semiconductor.

3) Cadmium titanate

In contrast to many ferroelectrics with a perovskite-type structure, cadmium titanate has a noncubic structure in the paraelectric region. At room temperature the structure of this titanate is orthorhombic. This feature of \(\mathrm{CdTiO_3}\) is explained by the small dimensions of the cadmium ion, which leads to a monoclinic displacement of the axes in the nonferroelectric region. According to Naray-Szabó, calcium titanate has an analogous distorted structure\(^{16}\).

The ferroelectric properties of cadmium titanate were discovered by Smolenskii\(^{4}\). On the curve \(\varepsilon = f(t)\) at \(50\text{--}60^\circ\mathrm{K}\) a maximum is observed. The Curie–Weiss constants of cadmium titanate at \(T > \theta\) proved to be considerably smaller than those of barium titanate and do not exceed \(40{,}000\text{--}55{,}000^\circ\mathrm{K}\). At temperatures below \(50\text{--}60^\circ\mathrm{K}\) hysteresis loops are observed. This ferroelectric has as yet been little studied.

4) Lead zirconate

A phase transition at 230°C in lead zirconate was discovered by Smolenskii^4 and Roberts^17 independently of one another.

Lead zirconate crystallizes in a perovskite-type structure, as was established by Gofman^18, Naray-Szabo^16 and Megaw^19.

Ueda and Shirane^20, as well as Sawaguchi^21, determined by X-ray methods the changes in the cell parameters with temperature and confirmed the presence of a phase transition at 230°C (Fig. 5). For \(T > \theta\) the elementary cell has cubic symmetry of the perovskite type; for \(T < \theta\) a rhombic superstructure is formed, composed of subcells of tetragonal symmetry. The parameters and volume of the cell change discontinuously at the transition point, and the cell volume upon transition to the paraelectric state increases by \(0.29 \, \text{Å}^3\), whereas in \(\mathrm{BaTiO_3}\) the cell volume decreases at the transition.

Fig. 5. Change in the subcell parameters of lead zirconate with temperature^21.

Fig. 6. Projection onto the plane \((0\,0\,1)\) of models of superstructures of lead zirconate: rhombic—solid heavy lines, pseudotetragonal—dashed heavy lines^2.

Sawaguchi, Maniwa and Hoshino^22 synthesized small single crystals of lead zirconate. Having studied the optical properties and carried out X-ray investigations of a single crystal, the authors showed that lead zirconate crystallizes at room temperature in a rhombic superstructure.

This structure has the following parameters: \(a = a_0 \sqrt{2} = 5.87\); \(b = a_0 2\sqrt{2} = 11.74\) and \(c = 2c_0 = 8.20 \, \text{Å}\), and contains eight “molecules” of \(\mathrm{PbZrO_3}\). The space group of the cell obtained in this way may be \(D_{2h}^{9}\) or \(C_{2v}^{8}\).

In Fig. 6, a model of the superstructure is shown as a projection onto the plane \((0\,0\,1)\), in accordance with the indicated space groups. In the case of the space group \(D_{2h}^{9}\), the authors assume that the lead ions are displaced along the rhombic \(a\) axis by \(0.2 \, \text{Å}\) antiparallel to one another, as shown in Fig. 6. In the second variant—the space group \(C_{2v}^{8}\)—the authors believe that, in addition, the lead ions are slightly displaced parallel to one another along the \(c\) axis. Thus, the appearance of superstructure lines on the X-ray patterns of \(\mathrm{PbZrO_3}\) is associated by the authors with antiparallel displacement of the lead ions. A small piezoelectric effect, discovered by Roberts^23 in prelimi-

tively polarized polycrystalline sample of PbZrO\(_3\), shows that the noncentrosymmetric space group \(C_{2v}^{8}\) is more probable than the group \(D_{2h}^{9}\). However, as Venevtsev\({}^{24}\) correctly notes, the true space group is still unknown.

The superstructure of PbZrO\(_3\) can also be described approximately with the aid of a pseudotetragonal packet structure with parameters \(4a_0 \times 4a_0 \times 2c_0\) or \(4a_0 \times 4a_0 \times c_0\)\({}^{22}\). A projection of a model of such a cell onto the plane \((001)\) is shown in Fig. 6.

The temperature dependences of the dielectric permittivity and of the coefficient of linear expansion of polycrystalline lead zirconate are shown in Figs. 7 and 8. Above the transition point the Curie–Weiss law is obeyed: \(\varepsilon = 1.15 \cdot 10^{5}/(T - 228)\). The phase transition in lead zirconate is accompanied by negative volume spontaneous electrostriction.

A jump-like change in the cell volume, substantial electrostriction, a sufficiently large transition heat \((440\ \text{cal}/\text{mol})\)\({}^{25}\), a sharp change in dielectric permittivity and its temperature hysteresis \((4^\circ\text{C})\) at the transition point, as well as the linear dependence of polarization on field strength above the transition point, show that at \(230^\circ\text{C}\) a first-order phase transition occurs in PbZrO\(_3\).

Fig. 7. Dependence of the initial dielectric permittivity of polycrystalline lead zirconate on temperature\({}^{4}\).

Lead zirconate has a number of peculiarities in comparison with barium titanate: 1) there is no hysteresis loop, 2) a superstructure is present, 3) the crystal volume decreases on cooling at the transition point (negative volume electrostriction), 4) the transition point is shifted into the region of lower temperatures when a constant electric field is applied, 5) the reversible dielectric permittivity increases in the region of the phase transition with increasing bias-field strength, 6) at high field strengths hysteresis loops anomalous for ferroelectrics are observed, and 7) the discharge current in the region of the phase transition upon heating a sample to which a constant voltage had previously been applied is absent.

Shirane, Sawaguchi, and Takagi\({}^{26}\), pointing to these peculiarities, put forward the supposition that lead zirconate with a minimal impurity content is an antiferroelectric. In addition, these authors believe that in the solid solutions \((\text{Pb},\text{Ba})\)ZrO\(_3\) and Pb\((\text{Zr},\text{Ti})\)O\(_3\), on lowering the temperature, a transition first occurs from the paraelectric state to the ferroelectric state, and then to the antiferroelectric state; while in the solid solutions \((\text{Pb},\text{Sr})\)ZrO\(_3\), the transition is from the paraelectric state directly to the antiferroelectric state.

Until now, in considering ferroelectrics and, in particular, barium titanate, we have assumed that the forces of long-range interaction are responsible for the occurrence of a phase transition from a disordered to an ordered polarized state. However, from the example of crystals in which orientational melting takes place, it is seen that in these cases phase transitions

are due to short-range forces. It is quite obvious that in some ferroelectrics as well the role of short-range forces may be great, and this will introduce considerable specificity into the behavior of such a ferroelectric in an electric field. It is not impossible that this is also true for lead zirconate. For a final solution of the question of the nature of the phase

Fig. 8. Dependence of the relative change in length \(\Delta l/l\) and of the coefficient of linear expansion \(\alpha\) of a polycrystalline sample of lead zirconate on temperature.\(^{4}\)

transition in \(\mathrm{PbZrO_3}\), it is necessary to carry out careful neutronographic and spectroscopic studies of single crystals of lead zirconate, and also to make use, in resolving this question, of the nuclear-resonance method.

5) \(\mathrm{PbHfO_3}\)

Shirane and Pepinsky found that in polycrystalline \(\mathrm{PbHfO_3}\) with a perovskite-type structure two phase transitions are observed—at \(163^\circ\mathrm{C}\) and \(215^\circ\mathrm{C}\).\(^{27}\)

The dependence of the dielectric constant of a polycrystalline sample on temperature is shown in Fig. 9. The polarization over the entire temperature interval at \(T<\theta\) is practically independent of the field strength. At room temperature the unit cell is pseudotetragonal. The authors could not determine precisely the symmetry of the lattice (tetragonal with ratio \(c/a<1\), or orthorhombic). In the interval \(163\text{–}215^\circ\mathrm{C}\), \(\mathrm{PbHfO_3}\) has a tetragonal lattice with ratio \(c/a=0.997\), and above \(215^\circ\mathrm{C}\) it is cubic. On heating, the volume of the unit cell increases at the points of the phase transitions, i.e., both phase transitions are characterized by negative volume spontaneous electrostriction. In x-ray photographs of \(\mathrm{PbHfO_3}\) obtained at temperatures below \(215^\circ\mathrm{C}\), superstructure lines are observed. The authors suppose that \(\mathrm{PbHfO_3}\) is antiferroelectric at temperatures below \(215^\circ\mathrm{C}\).

b) Solid solutions \((\mathrm{Ba}, \mathrm{Pb})\mathrm{SnO}_3\)

Lead stannate, unlike lead zirconate and titanate, does not exist as a chemical compound\(^{24, 28}\). However, at a certain concentration (not less than 30%) of barium stannate it is possible to obtain solid solutions \((\mathrm{Ba}, \mathrm{Pb})\mathrm{SnO}_3\) with a perovskite-type structure. The unit-cell parameters of these solid solutions, according to Kalinina, are given in Table II.

Table II

The ferroelectric properties of these solid solutions were discovered by Smolensky and Agranovskaya\(^{29, 30}\). The dependence of the dielectric permittivity of the solid solutions \((\mathrm{Ba}, \mathrm{Pb})\mathrm{SnO}_3\) on temperature is shown in Fig. 10.

Fig. 9. Dependence of the initial dielectric permittivity of a polycrystalline \(\mathrm{PbHfO}_3\) sample on temperature\(^{37}\).

The temperature of the phase transitions, more precisely the temperature of the maximum of the dielectric permittivity, decreases as the concentration of barium stannate, which is not a ferroelectric, increases.

In the temperature region corresponding to the largest values of the dielectric permittivity, a change in the coefficient of linear expansion is observed, which indicates the presence of a phase transition. It is true, however, that, as in a number of other solid solutions, this transition is “smeared out.”

The spontaneous polarization of the solid solution \((\mathrm{Ba}_{0.3},\ \mathrm{Pb}_{0.7})\mathrm{SnO}_3\) at a temperature of \(-120^\circ\mathrm{C}\) reaches \(13 \cdot 10^{-6}\ \mathrm{coul}/\mathrm{cm}^2\).

Smolenskii and Myl'nikova have recently found that phase transitions are also observed in solid solutions \((\mathrm{Sr},\ \mathrm{Pb})\mathrm{SnO}_3\).

7) Tungsten trioxide

The ferroelectric properties of \(\mathrm{WO}_3\) were discovered by Matthias in 1949[^31]. Since then a considerable number of works have been published devoted to the investigation of the properties and structure of tungsten trioxide[^32–^41].

Fig. 10. Dependence of the initial dielectric permittivity of polycrystalline specimens of solid solutions \((\mathrm{Ba},\mathrm{Pb})\mathrm{SnO}_3\) on temperature. The numbers indicate the barium stannate content in molecular percent[^30].

Fig. 11. Dependence of the linear-expansion coefficient of a polycrystalline specimen of tungsten trioxide on temperature[^36].

However, up to the present time sufficiently verified data have not been obtained on the phase transitions and the positions of the ions in the \(\mathrm{WO}_3\) lattice at different temperatures. This is partly explained by the considerable conductivity of \(\mathrm{WO}_3\), which makes it difficult to investigate some properties of tungsten trioxide. Tungsten trioxide has been studied both on single crystals and on polycrystalline specimens.

The structure of \(\mathrm{WO}_3\) is a somewhat deformed \(\mathrm{ReO}_3\) structure. This structure may be regarded as a perovskite structure in which the calcium ion has been removed. At room temperature the unit cell of \(\mathrm{WO}_3\) is monoclinic and contains four “molecules.” Some investigators believe that at room temperatures \(\mathrm{WO}_3\) possesses ferroelectric properties[^31], while others assert that at these temperatures tungsten trioxide is an antiferroelectric[^41].

At \(740^\circ\)C there is a phase transition as a result of which the lattice changes from monoclinic to tetragonal\(^{32}\). In the region of the phase transition, during heating, a sharp decrease in the volume of the unit cell is observed—a positive volume spontaneous electrostriction (Fig. 11); the latent heat of transition reaches \(450\ \text{cal/mol}\) (Fig. 12). In works\(^{35}\) it is indicated that the domain structure disappears upon heating the crystal above \(710^\circ\)C. In paper\(^{33}\), on the basis of X-ray studies, the assumption is made that in the tetragonal region tungsten trioxide is an antiferroelectric.

Fig. 12. Dependence of the true heat capacity of polycrystalline tungsten trioxide on temperature\(^{36}\).

Fig. 13. Dependence of the dielectric permittivity and the tangent of the dielectric loss angle in weak fields of a polycrystalline sample of lead metaniobate on temperature\(^{42}\).

According to Mattias and Wood, at \(-50^\circ\)C in \(\mathrm{WO_3}\) there is also observed a phase transition from the monoclinic to a more symmetric phase, which is ferroelectric\(^{37}\). The dielectric permittivity of the single crystal at the temperature of liquid air reaches \(300^{31}\).

8) Lead metaniobate and metantalate

In 1953, Goodman showed that lead metaniobate \((\mathrm{PbNb_2O_6})\) is a ferroelectric with a phase-transition point of \(570^\circ\)C\(^{42}\).

On the basis of X-ray studies of small single crystals, the author asserts that the unit cell is orthorhombic, contains 40 “molecules,” and has the following dimensions: \(a = 25\), \(b = 25\), and \(c = 7\ \text{Å}\). In all probability, the niobium ion is octahedrally surrounded by oxygen ions, but the arrangement of the oxygen octahedra differs from their arrangement in perovskite.

The dependences of the dielectric permittivity and the relative change in length of a polycrystalline sample of lead metaniobate on temperature are shown in Figs. 13 and 14. For \(T > \theta\), the dielectric permittivity depends on temperature according to the Curie—Weiss law: \(\varepsilon = \dfrac{2.95 \cdot 10^5}{T - 530}\). Lead metaniobate has

positive spontaneous electrostriction. There are no low-temperature phase transitions in Pb(NbO₃)₂ down to −195°C.

The author observed hysteresis loops, but it was not possible to attain saturation of the polarization. At room temperature and \(E=60\ \text{kV/cm}\), the residual polarization is equal to \(0.6\cdot 10^{-6}\ \text{C/cm}^2\), and the coercive force is \(17\ \text{kV/cm}\).

Lead metaniobate is distinguished by properties that are of interest from the technical point of view. It can be used for the manufacture of piezoelements,

Table III

operating at elevated temperatures. Table III gives several properties of polycrystalline Pb(NbO₃)₂ in comparison with the properties of polycrystalline barium titanate according to Goodman’s data. The PbNb₂O₃ specimens were polarized by a constant field \(E=20\ \text{kV/cm}\) at 200–250°C for 0.5 hour.

After the discovery of ferroelectric properties in lead metaniobate, Smolenskii and Agranovskaya showed that spontaneous polarization also arises in lead metatantalate (PbTa₂O₆)\(^{29}\). The dependence of the dielectric permittivity and of the tangent of the dielectric-loss angle in weak fields is shown in Fig. 15. The transition from the nonpolar state to the polar one occurs at \(260^\circ\text{C}\). The spontaneous polarization of lead metatantalate at room temperature is equal to \(2.5\cdot 10^{-6}\ \text{C/cm}^2\). The structure of this compound has not yet been determined.

Fig. 14. Dependence of the relative change in length of a polycrystalline specimen of lead metaniobate on temperature\(^{42}\).

9) Cadmium pyroniobate and strontium pyrotantalate

As a result of investigation of the properties of a series of niobates and tantalates of divalent metals, Wainer and Wentworth\(^{43}\) established that polycrystalline cadmium pyroniobate \((\mathrm{Cd}_2\mathrm{Nb}_2\mathrm{O}_7)\) is distinguished by a comparatively high dielectric permittivity (about 500 at room temperature) and by a large negative temperature coefficient of dielectric permittivity. Subsequent measurements by Cook and Jaffe\(^{44,45}\) showed that cadmium pyroniobate is a ferroelectric with a phase-transition point of \(-90^\circ\text{C}\). This

The compound crystallizes in a cubic structure \(E8_1\), of the pyrochlore type \((\mathrm{NaCaNb}_2\mathrm{O}_6\mathrm{F})\). The structural type is similar to that of distorted fluorite. However, instead of the composition \((A,B)_4X_8\), which would correspond in fluorite notation to \(AX_2\), the formula has the form \((A,B)_4X_7\), so that one of the eight anions in the fluorite-type structure is removed and the remaining oxygen ions are considerably displaced. There are eight “molecules” per unit cell.

Fig. 15. Dependence of the dielectric permittivity and the tangent of the dielectric-loss angle in weak fields of polycrystalline lead metatantalate on temperature \(^{29}\).

The framework of the structure consists of three-dimensional chains of octahedra joined at the corners. The seventh oxygen ion and the \(A\) ions occupy the free spaces between the octahedra. The configuration of the octahedra projected onto the \((010)\) plane is shown in Fig. 16. In contrast to perovskite, the \(O—B—O\) chains are arranged in a zigzag fashion along the \([111]\) direction. In the pyrochlore structure the \(A\) ions and the seventh anions may also be absent, just as the \(A\) ions may be absent in the perovskite structure, since they are not essential for the stability of the lattice, provided, of course, that electrical neutrality is maintained.

Detailed X-ray structural and optical study of cadmium pyroniobate and of a number of solid solutions based on it was carried out by John, Shirane, and Pepinsky \(^{46}\). The authors grew small single crystals \((0.03 \times 0.03 \times 0.10\ \mathrm{mm})\) from a melt of pure \(\mathrm{Cd}_2\mathrm{Nb}_2\mathrm{O}_7\). At room temperature the lattice constant of these crystals is the same as for polycrystalline samples: \(a = 10.372 \pm 0.001\ \text{\AA}\). Optical investigations reveal a phase transition from the cubic to a noncubic phase at \(-90^\circ\mathrm{C}\). Below the phase-transition temperature, distinct domains are not observed. X-ray diffraction patterns of single crystals at \(-150^\circ\mathrm{C}\) show a small distortion of the lattice, but it is not possible to determine the structure in the ferroelectric region. The authors believe that a displacement of cadmium ions relative to niobium ions should be observed.

Fig. 16. Configuration of octahedra in cadmium pyroniobate projected onto the \((101)\) plane \(^{46}\).

Single crystals grown from a melt of \(Cd_2Nb_2O_7\) with sodium fluoride added as a flux apparently have the following formula: \(Cd_{1.6}Na_{0.4}Nb_2O_{6.6}F_{0.4}\). The temperature of the phase transition of this crystal, in comparison with pure \(Cd_2Nb_2O_7\), decreases to \(-120^\circ C\). At \(-140^\circ C\) the lattice is tetragonal \((a = 10.378\ \text{Å}\) and \(c/a = 1.0011)\).

Fig. 17. Dependence of the initial dielectric permittivity of a polycrystalline cadmium pyroniobate sample on temperature at \(10^2\) Hz\({}^{47}\).

The dependence of the dielectric permittivity of a polycrystalline cadmium pyroniobate sample on temperature, according to Shirane and Pepinsky\({}^{47}\), is shown in Fig. 17. At \(-190^\circ C\) a second phase transition is observed. Similar results were obtained by Hulm\({}^{48}\). Above the Curie point the Curie—Weiss law is obeyed:

\[ \varepsilon = \frac{4.6 \cdot 10^4}{t - 150}. \]

At temperatures below \(-90^\circ C\) hysteresis loops are observed. At \(-173^\circ C\) and \(E = 25\ \text{kV}/\text{cm}\) the spontaneous polarization is \(1.8 \cdot 10^{-6}\ \text{C}/\text{cm}^2\). This value is two times smaller than that found by Cook and Jaffe.

The anomaly of the specific heat in the region of the phase transition at \(-90^\circ C\) was investigated by Danner and Pepinsky\({}^{49}\). From the curve \(C_p = f(t)\) the transition energy was determined:

\[ \Delta E = \int \Delta C_p\, dT = 18 \pm 2\ \text{cal}/\text{mole}, \]

which corresponds to an entropy change

\[ \Delta S = \int \frac{\Delta C_p}{T}\, dT = 0.09 \pm 0.01\ \text{cal}/\text{degree}\cdot\text{mole}. \]

Jonah, Shirane, and Pepinsky also studied several solid solutions with cadmium pyroniobate. In the system \(Cd_2Nb_2O_7 — Cd_2Ta_2O_7\) a continuous series of solid solutions of cubic structure is formed; in the system \(Cd_2Nb_2O_7 — Pb_2Nb_2O_7\) solid solutions are formed, apparently, with \(Pb_2Nb_2O_7\) content up to 90% (mol). \(Ca_2Nb_2O_7\) dissolves in \(Cd_2Nb_2O_7\) only up to 20% (mol).

Fig. 18. Dependence of the initial dielectric permittivity of a polycrystalline strontium protantalate sample on temperature\({}^{50}\).

Lead pyroniobate \((Pb_2Nb_2O_7)\) also crystallizes in the pyrochlore-type structure. However, at room temperature \(Pb_2Nb_2O_7\), in contrast to \(Cd_2Nb_2O_7\), has a rhombohedral structure: \(a = 10.570\ \text{Å}\), \(\alpha = 89^\circ 15'\)\({}^{46}\). In the temperature dependence of the dielectric permittivity of polycrystalline samples of this compound, a maximum is found at \(14—15^\circ K\)\({}^{47,48}\). However, below this temperature, at field strengths up to \(10—15\ \text{kV}/\text{cm}\), no hysteresis loops were observed. In the authors’ opinion, the question of a phase transition from the nonpolarized state to the polarized state for lead pyroniobate remains unresolved.

Recently Smolenskii, Isupov, and Agranovskaya\({}^{50}\) showed that polycrystalline strontium protantalate is a ferroelectric. The occurrence of spontaneous polarization in strontium protantalate is quite unexpected, since Hulm did not detect

of ferroelectric properties in Cd$_2$Ta$_2$O$_7$ and Pb$_2$Ta$_2$O$_7$ $^{48}$. By analogy with titanates, one would expect the appearance of ferroelectric properties primarily in the tantalates of lead and cadmium, and not in strontium tantalate.

The temperature dependence of $\varepsilon$ for a polycrystalline sample of Sr$_2$Ta$_2$O$_7$ is shown in Fig. 18. At $-180^\circ$C a second maximum of $\varepsilon$ is observed, evidently corresponding to a low-temperature phase transition. At temperatures below the first maximum hysteresis loops are observed. This ferroelectric is distinguished by a low value of the dielectric permittivity. The structure of strontium pyrotantalate is not known.

10) Tantalates and niobates of alkali metals

The ferroelectric properties of lithium tantalate and niobate were discovered by Matthias and Remeika $^{51}$. The authors believe that these compounds crystallize in a structure of the ilmenite type (rhombohedral lattice). This structure is best represented as a slightly distorted hexagonal close packing of oxygen ions. One third of the octahedral positions in lithium tantalate (niobate) are occupied by tantalum (niobium) ions, one third by lithium ions, and one third are vacant. In each unit cell there are two “molecules.” In the ilmenite structure the octahedra touch one another by edges and faces, and the symmetry is lower than in perovskite.

The transition point from the polarized state to the nonpolarized state in these compounds lies at temperatures above 450°.

In the temperature interval from $-100$ to $+450^\circ$C the dielectric permittivity, spontaneous polarization, and coercive field of a single crystal of lithium tantalate increase with increasing temperature. The spontaneous polarization of lithium niobate depends on temperature in an analogous manner. This kind of dependence led Matthias to the idea that in these ferroelectrics there is a lower phase-transition point $^{52}$.

Spontaneous polarization in sodium and potassium tantalates and niobates with a perovskite-type structure was discovered by Matthias $^{53}$. Single crystals of KTaO$_3$ and NaTaO$_3$ were grown from melts of Ta$_2$O$_5$ with KOH and Ta$_2$O$_5$ with NaOH $^{54}$. At room temperature the lattice of potassium tantalate has cubic symmetry, and that of sodium tantalate orthorhombic symmetry. The phase transition in KTaO$_3$ from the paraelectric state to the ferroelectric state occurs at very low temperatures $^{55}$. The dielectric permittivity of single crystals of potassium tantalate reaches its maximum value at 13.2°K. Below this temperature hysteresis loops are observed. In the interval 50—85°K the dielectric permittivity changes according to the Curie—Weiss law:
\[ \varepsilon = (6—8)\,10^4/(T-\theta'). \]

In NaTaO$_3$ the transition from the orthorhombic structure to the cubic structure occurs at 475°C $^{56}$. However, according to Isupov, this transition is apparently not connected with a transition from the paraelectric state to an antiferroelectric or ferroelectric one, but is caused by a change in the positions of the octahedra. Analogous transitions are observed in solid solutions (Ca, Sr)TiO$_3$ at a certain content of calcium titanate. At high temperatures the axes of the octahedra are parallel to one another, and the crystal has a cubic structure. When the temperature is lowered, the octahedra rotate slightly about the titanium ions and the symmetry of the crystal decreases; a naray-szabó-type packing structure of monoclinic symmetry arises. In NaTaO$_3$ at 475°C there apparently occurs an ordinary polymorphic transformation, in which no electric moment arises in the unit cell upon lowering the temperature.

At present, the existence in NaTaO$_3$ of a phase transition below which the unit cell would acquire an electric moment cannot be considered proven.

Single crystals of \(\mathrm{KNbO_3}\) were grown from a melt of \(\mathrm{K_2CO_3}\) and \(\mathrm{Nb_2O_5}\), or \(\mathrm{KOH}\) and \(\mathrm{Nb_2O_5}\) \({}^{51,57,58}\). Study of the temperature dependence of \(\varepsilon\) and \(\operatorname{tg}\delta\) of \(\mathrm{KNbO_3}\) single crystals \({}^{51,57,58,59,60}\), as well as x-ray and optical investigations of these crystals, showed the presence of three phase transitions at 435, 225, and \(-10^\circ\mathrm{C}\). The structure of \(\mathrm{KNbO_3}\) during cooling changes in the same way as in barium titanate: from cubic to tetragonal, from tetragonal

Fig. 19. Change of the unit-cell parameters of \(\mathrm{KNbO_3}\) with temperature \({}^{60}\).

to orthorhombic, and from orthorhombic to rhombohedral (Fig. 19). According to Wood, the unit cell has the following parameters: \(a=4.024\ \text{\AA}\) at \(500^\circ\mathrm{C}\); \(a=4.00\ \text{\AA}\) and \(c=4.072\ \text{\AA}\) at \(260^\circ\mathrm{C}\); \(a=4.045\ \text{\AA}\), \(b=3.984\ \text{\AA}\), \(c=4.045\ \text{\AA}\), and \(\beta=90^\circ21'\) (in monoclinic axes), or \(a=5.702\ \text{\AA}\), \(b=5.739\ \text{\AA}\), and \(c=3.984\ \text{\AA}\) (an orthorhombic cell containing two “molecules”) at \(25^\circ\mathrm{C}\).

The heats of transition \(\Delta Q\) (cal/mol) and the change in entropy at phase transitions (cal/deg·mol) of polycrystalline potassium niobate and, for comparison, of barium titanate are given in Table IV \({}^{59}\).

Table IV

Triebwasser and Halpern \({}^{61,62}\) showed that for \(T>\theta\) the Curie—Weiss law is obeyed: \(\varepsilon=\dfrac{2.68\cdot10^5}{T-350}\). The spontaneous polarization near the transition is equal to \(26\cdot10^{-6}\ \text{coul}/\text{cm}^2\). Cotts and Knight \({}^{63}\) investigated the nuclear magnetic resonance and nuclear quadrupole resonance of \(\mathrm{Nb}^{93}\) in a \(\mathrm{KNbO_3}\) single crystal in a magnetic field of 5250 oersteds.

Table V gives the transition temperatures determined by X-ray, optical, and electrical methods of investigation, as well as by the nuclear-resonance method1.

Table V

The table gives the transition temperatures during heating and cooling of the crystal. Cotts and Knight believe that the phase transitions in \(\mathrm{KNbO_3}\) are first-order transitions, since in all phase transitions a sharp change is observed in the quadrupole-resonance frequency and, consequently, in the magnitude of the quadrupole interaction. In addition, this work established a large magnitude of the quadrupole interaction, which is explained by the influence of covalent bonds.

The authors note that the Mason and Matthias model with six equilibrium positions for the central ion is not consistent with the large quadrupole interaction in the tetragonal phase and the presence of separate magnetic-resonance lines in the cubic phase.

Single crystals of \(\mathrm{NaNbO_3}\) were grown from a melt of \(\mathrm{Na_2CO_3}\) and \(\mathrm{Nb_2O_5}\), with sodium fluoride used as the flux2. In single crystals of \(\mathrm{NaNbO_3}\), four phase transitions are observed optically at 360, 470, 518, and \(640^\circ\mathrm{C}\)3. Below \(360^\circ\mathrm{C}\) the crystal structure is orthorhombic; above \(640^\circ\mathrm{C}\) it is cubic. In the temperature interval \(360\text{–}640^\circ\mathrm{C}\) the structure is pseudotetragonal. Not all phase transitions can be detected by the X-ray method. Superstructure lines are present on the X-ray diffraction patterns. The structure of \(\mathrm{NaNbO_3}\) was also studied in considerable detail by Vousden, who showed that niobium ions at room temperature are displaced in opposite directions along the \(a\) axis by \(0.11\,\text{Å}\)4.

At the points of the phase transitions, Cross and Nicholson observed changes in \(\varepsilon\) of single crystals: at \(360^\circ\mathrm{C}\) the dielectric constant changes discontinuously, while at 518 and \(640^\circ\mathrm{C}\) it changes only very slightly. At first, hysteresis loops could not be detected in \(\mathrm{NaNbO_3}\), and therefore sodium niobate was classified as an antiferroelectric. Cross and Nicholson showed that, depending on the direction and magnitude of the field, in a certain temperature interval both ordinary hysteresis loops and double loops characteristic of antiferroelectrics may be observed. Thus, sodium niobate possesses ferroelectric and antiferroelectric properties. The very small distortions of the structure and the slight changes at 518 and \(640^\circ\mathrm{C}\) do not exclude the possibility of assuming that above \(360^\circ\mathrm{C}\) sodium niobate becomes paraelectric.

The transitions observed in \(\mathrm{NaNbO_3}\) at 518 and \(640^\circ\mathrm{C}\) are apparently analogous to the transitions in \((\mathrm{Ca},\mathrm{Sr})\mathrm{TiO_3}\) and are associated with the small size of the sodium ion. This point of view was recently expressed by Krainik.

Table VI gives the temperatures of the phase transitions, the structure, and the cell parameters of new ferroelectrics and antiferroelectrics.

Table VI

III. On the Question of the Origin of Spontaneous Polarization in Crystals

Ginzburg^66 formulated, in general form, a criterion for the occurrence of ferroelectricity, consisting in the fact that the polarization energy arising upon displacement of the ions must exceed, in absolute value, the elastic energy.

In work^67 it was shown that a crystal becomes a ferroelectric only in the case when the force of electrostatic interaction arising upon displacement of the ions of one translational lattice from the equilibrium position is greater than the elastic component of the restoring forces, i.e., if

\[ e^{*}As > cs, \tag{1} \]

where \(e^{*}\) is the effective charge of the displaced ion, \(A\) is the internal-field constant, depending on the crystal structure, lattice parameters, charges and polarizabilities of the ions forming the lattice, \(s\) is the mean displacement of the ions, and \(c\) is the elastic-coupling coefficient. In this treatment the authors assumed that the motion of anharmonically vibrating ions in the absence of polarization occurs statistically independently of one another. In reality there exists a greater correlation in the motion of ions in crystals. Kozlovskii^68 considered another limiting case, when the ferroelectrically active ions form a rigid lattice. He showed that spontaneous polarization in a crystal may arise even if there is no condition for spontaneous displacement of an individual ion. This is determined by the fact that an ion, by being displaced and polarizing the crystal, creates favorable conditions for the displacement of neighboring ions. However, for a qualitative discussion one may use the inequality obtained in work^67. This condition for the occurrence of ferroelectricity is the most “severe,” since it was obtained from model representations in which the correlation in the motion of ions in the crystal is not taken into account.

It follows from inequality (1) that the occurrence of spontaneous polarization is favored by a large internal-field constant in the crystal, a large effective charge, and a small elastic-coupling coefficient of the displaced ion. The internal-field constant is the larger, the greater the electronic polarizability and the charges of the ions in the crystal, and also the greater its density. In addition, it depends substantially on the crystal structure. If one assumes that the central cation in the octahedra is displaced, then the elastic-coupling coefficient of this ion will be the smaller, the larger the size of the octahedron and the smaller the size of the displaced ion. The elastic-coupling coefficient also depends on the character of the bonds. These conclusions fully coincide with the considerations expressed earlier, made on the basis of a generalization of experimental data^69.

It is important to note that in the case when \(c < 0\), i.e., when the ions have several local minima of potential energy in the cell, then at a sufficiently low temperature spontaneous polarization must always arise, since \(e^{*}As - cs > 0\) for arbitrarily small \(e^{*}As\).

In work^67 it was assumed that the ferroelectrically active ion responsible for the occurrence of spontaneous polarization is the central ion. Comparatively recently Venevtsev and Zhdanov^70 suggested that the ferroelectrically active ion in compounds with a perovskite-type structure \(ABO_3\) may be both the \(B\) ions and the \(A\) ions.

At present, not knowing the character of the motion of ions in ferroelectrics, one cannot say with certainty that one or another ion is ferroelectrically active. Apparently, it is necessary to take into account the anharmonicity of the vibrations of all ions in the lattice, i.e., the ions of all translational lattices are, to some extent, ferroelectrically active*). True, the possib—

*) The case when all ions of the crystal lattice vibrate anharmonically was considered by Pasynkov.

the fact that, owing to the special nature of the given structure, the anharmonicity of the vibrations of the ions of one translational lattice is expressed to a greater degree than in the ions of the other translational lattices. Such an assumption was made for titanium ions in barium titanate. It is known that the sum of the radii of the titanium and oxygen ions is less than the distance from the center to the vertex of the octahedron in the unit cell of barium titanate, and therefore the titanium ion performs sufficiently pronounced anharmonic vibrations.

X-ray and neutron-diffraction investigations show that in ferroelectrics (\(\mathrm{BaTiO_3}\), \(\mathrm{PbTiO_3}\)) virtually all ions are displaced, only some more and others less. However, from the fact that the titanium ion in barium titanate and the lead ion in lead titanate are displaced by a larger amount than the other ions, one cannot conclude that they are precisely the ferroelectrically active ions in these ferroelectrics.

In considering the criterion of ferroelectricity, one must not forget that the emergence of spontaneous polarization may be due not only to the spontaneous inelastic displacement of ions, but also to the spontaneous displacement of the electronic shells of ions. As is known, in the electronic theory of Jaynes and Wigner \(^{71,72}\), the spontaneous displacement of the electronic shells of ions determines the emergence of spontaneous polarization in barium titanate.

However, if one assumes that spontaneously displaced ions are responsible for the emergence of ferroelectricity, then the ferroelectrically active ions must vibrate anharmonically; in the particular case they must “jump” within the limits of the unit cell from one potential well to another. Such peculiarity in the vibrations of ferroelectrically active ions may be explained by the partially homeopolar character of the bonding of ions in crystals, or by the “looseness” of the crystal structure when the bonds are substantially ionic.

The first point of view is expressed by a number of authors. However, different authors evaluate the role of partially covalent bonds in these crystals in different ways. In papers \(^{67,73}\) it is indicated that in this kind of semipolar compounds the elastic-bond coefficient is, in all probability, comparatively small, and the anharmonicity of the vibrations is sufficiently pronounced. Megaw \(^{72}\) believes that spontaneous polarization arises because of a sharp increase in the covalent character of the bonds at the Curie point. This point of view is criticized by Venevtsev and Zhdanov, who consider that the shortened Ti—O distances in barium titanate are explained not by a significant interaction of these particles, but by large internal fields acting on these ions. Megaw, in considering the question of the phase transition from the paraelectric state to the ferroelectric one, does not explain the behavior of the crystal in the paraelectric region (the increase of polarization with decreasing temperature).

The second point of view was expressed by Venevtsev and Zhdanov \(^{74}\). The primary condition for the ferroelectric activity of an ion in crystals with a perovskite-type structure, in the authors’ opinion, consists in the fact that such an ion should be “free” in the cell. They consider that when

\[ t=\frac{R_A+R_O}{\sqrt{2}(R_B+R_O)}>1 \]

(\(R_i\) are the radii of the corresponding ions) the ferroelectrically active cation is cation \(B\), while for \(t<1\) it is cation \(A\). For ferroelectrics, values of \(t\) both greater and less than unity are observed; for antiferroelectrics the value of \(t\) is only less than unity; for antiferroelectrics the value of \(t\) is always less than unity.

The conditions formulated by Venevtsev and Zhdanov are not fulfilled for some compounds. The authors suppose that in \(\mathrm{NaNbO_3}\), whose geometrical factor \(t\) is noticeably less than unity (\(t=0.86\)), the Na ions should be displaced antiparallel. Whereas from the X-ray data of Vousden \(^{65}\) it follows that the Nb ions are displaced antiparallel. According to the classification of ferroelectrics and antiferroelectrics proposed by Venevtsev and Zhdanov ...

bium, cadmium titanate should be an antiferroelectric. However, Smolenskii^4 and Odelevskii observed a hysteresis of dielectric polarization versus field strength in cadmium titanate.

Recently Krainik^75 considered the question of the relative stability of the ferroelectric and antiferroelectric phases in crystals with a perovskite-type structure, taking into account the electrostatic energy of the crystal lattice and using the works of Zaucer^76 and Takagi^77. From this qualitative consideration it follows that the role of the geometric factor must not be overestimated. He showed that an increase in the relative stability of the antiferroelectric state in comparison with the ferroelectric one, upon a decrease in the geometric factor, is a general regularity for a number of solid solutions with a high content of PbZrO\(_3\) \((t = 0.90)\). However, in CdTiO\(_3\), despite the smaller value of the geometric factor \((t = 0.87)\), not the antiferroelectric state but the ferroelectric one is realized. Krainik explains this by the fact that the electronic polarizability of the Cd\(^{2+}\) ion is smaller than that of the Pb\(^{2+}\) ion, and also by the fact that the homeopolar character of the Cd—O bond is apparently expressed to a lesser degree than in the Pb—O bond. Owing to this, the dipole moment associated with the displacement of the Cd ion (angular moment) is small in comparison with the central dipole moment. In this case, as follows from the works of Zaucer and Takagi, the absolute minimum of the electrostatic energy is observed for a parallel arrangement of the dipoles.

Moreover, as Krainik rightly notes, at sufficiently small values of the geometric factor \(t\) one cannot assume that the ion \(A\) is free in the cell. It is known that CdTiO\(_3\) crystallizes above the Curie temperature in an orthorhombic structure. As a result of the small size of the ion (Cd\(^{2+}\)), a distortion of the ideal cubic structure occurs owing to alternating tilting of the oxygen octahedra, and the Cd\(^{2+}\) ion may find itself in a “clamped” state.

It is quite obvious that, in order to clarify the question of the ferroelectric activity of one ion or another in the lattice, it is necessary to take into account all types of interaction between ions. In this connection it is appropriate to point out that Devonshire^78 and Syrkin^79 investigated the form of the potential relief for ions in barium titanate. Syrkin calculated the expansion coefficients of the potential energy of an ion in a crystal for a small displacement, proceeding from elementary laws of interaction, without any a priori assumptions about the form of the potential relief. It follows from the calculations that only the oxygen ions have several local minima of potential energy in the cell. The position of the titanium ion is stable at the center of the unit cell. This applies to an even greater degree to the Ba\(^{2+}\) ion. These calculations are valid for a purely ionic crystal. Determination of the form of the potential relief for ions in crystals with a partly homeopolar bond, to which barium titanate apparently belongs, presents great difficulties.

Thus, the question of whether all ions in the lattice, or the ions of one translational lattice, are ferroelectrically active remains unresolved. However, on the basis of a generalization of experimental data, one may formulate certain conditions that ferroelectrics of the oxygen-octahedral type must satisfy. In works^69,73 it was shown that all known ferroelectrics and antiferroelectrics crystallize in structures in which the cations, sufficiently small in size, located in the oxygen octahedron—that is, the central ions—have the electronic structure of a noble-gas atom after loss of \(s\)- and \(d\)-electrons; ions of this kind are formed from atoms with an unfilled penultimate shell, and it is assumed that the bond between the oxygen ions and the central ion is not purely ionic, but to one degree or another intermediate—

ly. Therefore, in reality some of these electrons are delocalized. In all probability, in semipolar compounds of this kind the coefficient of elastic coupling is small, and the anharmonicity of the vibrations of the ions is clearly expressed.

Table VII gives cations with the electronic structure of a noble-gas atom and the hydrogen ion, as well as their ionic radii (in angstroms) for coordination number six. The boxed entries are ions with a noble-gas electronic structure arising from atoms after the loss of \(s\)- and \(d\)-electrons. The central ions of ferroelectrics and antiferroelectrics known at the present time are given in boldface.

Table VII

An exception to the condition indicated above is constituted by certain crystals containing \( \mathrm{Pb}^{2+} \) ions. In these crystals, spontaneous polarization may also arise when oxygen octahedra are filled by ions that do not have the electronic structure of a noble-gas atom, for example, by \( \mathrm{Sn}^{4+} \) ions. This exception is explained in work \(^{73}\) by the influence of the strongly polarizable lead ions on the character of the bonds in such crystals. Venevtsev and Zhdanov \(^{74}\) explain this by the fact that in these crystals the ferroelectrically active ions are not the central ions, but the \( \mathrm{Pb}^{2+} \) ions.

The considerations set forth above are valid not only for ferroelectrics, but also for crystals with a high dielectric constant that are not ferroelectrics. Indeed, only those crystals have a dielectric constant greater than 40 and a negative temperature coefficient of dielectric constant in which, for sufficiently large octahedron dimensions, the central ion has the electronic structure of a noble gas after the loss of \(s\)- and \(d\)-electrons. On the basis of these consid-

NEW FERROELECTRICS AND ANTIFERROELECTRICS

...seems possible to seek materials with a high dielectric constant.

The definite structure of the electron shell of the central ion of ferroelectrics and antiferroelectrics determines, to a known extent, a definite character of the bonds in these oxygen compounds. However, there is no unanimity on the question of the nature of the bonds in ferroelectrics. Matthias^80 believes that the closed shell of the central ion determines ionic bonds in these crystals. Venevtsev and Zhdanov^74 suppose that ferroelectrics with a perovskite-type structure are compounds with a predominantly ionic character of bonding. Smolenskii, Megaw, and Buzden, as was already noted above, assert that ferroelectrics are ionic compounds with a certain share of covalent bonds. The latter point of view is confirmed by the works of Blokhin^81, who investigated the X-ray \(k\)-absorption spectrum of barium titanate and lead titanate. Cottse and Knight^63 arrived at an analogous conclusion, investigating nuclear magnetic resonance and nuclear quadrupole resonance of \(Nb^{93}\) in a single crystal of \(KNbO_3\).

Megaw established that \(d\)-electrons play a substantial role in the formation of homeopolar bonds. Therefore the temperatures of phase transitions of niobates of alkali metals are considerably higher than the transition temperatures of tantalates. Niobium and tantalum atoms in the ground state have the electronic configurations \(4d^4 5s\) and \(5d^3 6s^2\), respectively. Consequently, niobium has a greater possibility of forming bonds by \(d\)-electrons than tantalum.

In considering the question of the nature of the bonds in oxygen-octahedral ferroelectrics, it is useful to touch upon the question of the nature of the hydrogen bond, with whose presence the appearance of spontaneous polarization in some compounds containing hydrogen is associated. There are also two points of view on the nature of the hydrogen bond. Earlier the opinion was widespread that the hydrogen bond is determined by the simple electrostatic attraction of dipoles or residual charges of the interacting groups. In a number of works^82,83 it has been convincingly shown that the indicated forces, although they do play an essential role, are in principle insufficient for explaining many features of the hydrogen bond. From the quantum-mechanical treatment it follows that in the complex \(A—H \ldots B\) there occurs the formation of a weak donor–acceptor bond, caused by the unshared electrons of atom \(B\).

Thus, in all ferroelectrics, both containing hydrogen and not containing hydrogen, the bonds are not purely ionic.

The further accumulation of experimental facts and the development of the theory of ferroelectric phenomena should lead to a refinement and development of our ideas about the conditions for the occurrence of spontaneous polarization in crystals.

IV. NEW FERROELECTRICS AND ANTIFERROELECTRICS WITH HYDROGEN BONDING

Ferroelectrics containing hydrogen have been studied less intensively than ferroelectrics of the oxygen-octahedral type. Nevertheless, in recent years ferroelectric and antiferroelectric properties have been discovered in a number of crystals with hydrogen bonding.

1) \(NH_4H_2PO_4\) and \(ND_4D_2PO_4\)

Mason and Matthias^84, studying the dielectric, piezoelectric, and elastic properties of ammonium dihydrogen phosphate and of deuterium-substituted ammonium dihydrogen phosphate, expressed the supposition that both compounds are

antiferroelectrics. These crystals have a tetragonal structure at room temperature, and below the transition temperature an orthorhombic one. At the transition points, at \(-125^\circ\mathrm{C}\) for \(\mathrm{NH_4H_2PO_4}\) and at \(-31^\circ\mathrm{C}\) for \(\mathrm{ND_4D_2PO_4}\), a sharp change is observed in the properties of the crystals along the \(a\) axis (Fig. 20), and on cooling a large latent heat of transition is released,^85 but at \(T < \theta\) no hysteresis loops are found. The authors suppose that the antiferroelectric axis lies along one of the crystallographic \(a\) axes. These crystals are expediently used in electroacoustic transducers.

Fig. 20. Temperature dependence of the initial dielectric permittivity along the \(a\) and \(c\) axes of an \(\mathrm{ND_4D_2PO_4}\) single crystal.^84

2) \(\mathrm{CsH_2AsO_4}\)

The ferroelectric properties of all tetragonal dihydrogen phosphates and dihydrogen arsenates of K, Rb, and Cs were known earlier, with the exception of cesium dihydrogen arsenate. In 1953 Fraser and Pepinsky^86 found that cesium dihydrogen arsenate is also a ferroelectric with a transition point of \(-130^\circ\mathrm{C}\). At \(T < \theta\) typical hysteresis loops are observed.

Consideration of the properties of other ferroelectric dihydrogen arsenates and dihydrogen phosphates makes it possible to draw certain generalizations. The larger the alkali cation, the higher the transition point and the smaller the dielectric permittivity. Table VIII gives the temperatures of phase transitions in phosphates and arsenates.

Table VIII

3) \((\mathrm{NH_4})_2\mathrm{H_3JO_6}\)

In tetragonal disubstituted ammonium periodate, according to Baertschi,^87 in the temperature interval from \(-20\) to \(-30^\circ\) a phase transition occurs, associated with a pronounced dielectric anomaly. The latent heat of transition reaches \(350\ \mathrm{cal/mol}\). However, no hysteresis loops were found at \(T < \theta\), and therefore Busch, Kenchig, and Meier^88 expressed the supposition that this compound is an antiferroelectric. This consideration is confirmed by the fact that at \(T < \theta\) superstructure lines appear on X-ray photographs. The periods of identity in the direction perpendicular to the trigonal axis are doubled.

4) \(\mathrm{Ag_2H_3JO_6}\)

Grenicher, Meier, and Petter^89 found that disubstituted silver periodate is also an antiferroelectric.

The authors found that the hexagonal unit cell of this silver salt, containing one “molecule,” has the following dimensions at \(20^\circ\mathrm{C}\): \(a=5.932\) and \(c=12.685\ \text{\AA}\). At a temperature of \(-46^\circ\mathrm{C}\) a phase transition is observed, accompanied by “anomalies” of the specific heat and dielectric permittivity. However, no dielectric hysteresis or piezoelectric charge was detected. Superstructure lines are observed on the X-ray diffraction patterns, showing that the identity periods both in the \(c\) direction and in the \(a\) direction double when the crystal is cooled below the transition temperature.

5) \(\mathrm{LiNH_4C_4H_4O_6\cdot H_2O}\) and \(\mathrm{LiTlC_4H_4O_6\cdot H_2O}\)

Ferroelectric properties of lithium ammonium tartrate were discovered by Merz\({}^{90}\), and also by Matthias and Hulm\({}^{91}\). This salt has an orthorhombic structure at \(t\sim20^\circ\mathrm{C}\). The dielectric permittivity at

Fig. 21. Temperature dependence of the resonance frequency and piezomodulus \(d_{25}\) of a single crystal of \(\mathrm{LiNH_4C_4H_4O_6\cdot H_2O}\).\({}^{90}\)

room temperature is small along all three axes. When the temperature is lowered, \(\varepsilon_a\) and \(\varepsilon_c\) change very little, while \(\varepsilon_b\) passes through a sharp maximum at \(-167^\circ\mathrm{C}\). The spontaneous polarization at low temperatures is equal to \(0.21\cdot10^{-6}\ \text{coul}/\text{cm}^2\).

At room temperature the piezomodulus is sufficiently large, \(d_{25}=20\cdot10^{-8}\) CGSE units, and at the transition point it increases to \((10\,000—20\,000)\,10^{-8}\) CGSE units (Fig. 21).

Lithium ammonium tartrate has certain special features compared with Rochelle salt. The spontaneous polarization in \(\mathrm{LiNH_4C_4H_4O_6\cdot H_2O}\) is directed along the \(b\) axis, whereas in Rochelle salt it is along the \(a\) axis. In this ferroelectric only one phase transition has been detected. Instead of four “molecules” of water as in Rochelle salt, lithium ammonium tartrate has one “molecule” of water.

The occurrence of spontaneous polarization at very low temperatures (\(\sim10^\circ\mathrm{K}\)) in \(\mathrm{LiTlC_4H_4O_6\cdot H_2O}\) was discovered by Matthias and Hulm\({}^{91}\). The spontaneous polarization in this crystal is directed along the \(a\) axis and at \(1.3^\circ\mathrm{K}\) is approximately \(0.14\cdot10^{-6}\ \text{coul}/\text{cm}^2\).

6) \((\mathrm{CN}_3\mathrm{H}_6)\mathrm{Al}(\mathrm{SO}_4)_2 \cdot 6\mathrm{H}_2\mathrm{O}\)

A new class of ferroelectrics with hydrogen bonds was recently discovered by Holden, Matthias, Merz, and Remeika[^92]. Guanidinium aluminum sulfate hexahydrate is a ferroelectric over a wide temperature range. This is the first ferroelectric with hydrogen bonding that has a transition point in the high-temperature region \((>200)\). The authors note that above \(100^\circ\) the crystal begins to lose water of crystallization. The crystal has a trigonal structure. The ferroelectric axes are the trigonal axes. The space group is \(C_{3v}(2) — P31m\), and the unit cell contains three “molecules.” This ferroelectric is readily grown from aqueous solutions by the usual method.

At room temperature the saturation polarization is approximately \(0.35 \cdot 10^{-6}\ \mathrm{coul}/\mathrm{cm}^2\), and the coercive field is \(\sim 1500\ \mathrm{V}/\mathrm{cm}\). With decreasing temperature the spontaneous polarization and the coercive field increase. The hysteresis loops are symmetrical and rectangular; therefore the crystal will evidently be of technical interest. The crystal is distinguished by a low dielectric permittivity \((\varepsilon_{\parallel}=15,\ \varepsilon_{\perp}=5)\).

The authors write that it is possible to obtain isomorphous crystals in which \(\mathrm{Al}^{3+}\) is replaced by \(\mathrm{Ga}^{3+}\) and \(\mathrm{Cr}^{3+}\), \((\mathrm{SO}_4)^{2-}\) by \((\mathrm{SeO}_4)^{2-}\), and \(\mathrm{H}_2\mathrm{O}\) by \(\mathrm{D}_2\mathrm{O}\)[^93].

7) \((\mathrm{CH}_3\mathrm{NH}_3)\mathrm{Al}(\mathrm{SO}_4)_2\,12\mathrm{H}_2\mathrm{O}\) and other alums

A peak in the curves \(\varepsilon=f(t)\) of ammonium ferric \((-180^\circ\mathrm{C})\) and ammonium aluminum alums \((-220^\circ\mathrm{C})\) was first reported by Granier[^94].

Pepinsky, Jona, and Shirane[^95], studying the optical and electrical properties, as well as the structures, of a number of alums, found that they are ferroelectrics or antiferroelectrics at low temperatures.

In particular, methylammonium aluminum sulfate dodecahydrate \((\mathrm{CH}_3\mathrm{NH}_3)\mathrm{Al}(\mathrm{SO}_4)_2 \cdot 12\mathrm{H}_2\mathrm{O}\) possesses ferroelectric properties. For \(T>\theta\), the Curie–Weiss law is obeyed,

\[ \varepsilon=\frac{C_w}{T-\theta}, \]

where \(C_w=1000^\circ\mathrm{K}\). The peak of the dielectric permittivity is accompanied by a sharp peak of the tangent of the dielectric loss angle. The coercive field \(E_c\) increases very rapidly with decreasing temperature: at \(-119^\circ\mathrm{C}\), \(E_c=5\ \mathrm{kV}/\mathrm{cm}\), and at \(-142^\circ\mathrm{C}\), \(E_c=15\ \mathrm{kV}/\mathrm{cm}\). The spontaneous polarization at \(-127^\circ\mathrm{C}\) reaches \(0.6 \cdot 10^{-6}\ \mathrm{coul}/\mathrm{cm}^2\).

The authors synthesized and investigated a large number of alums \(M_{\mathrm{I}}[M_{\mathrm{III}}(\mathrm{SO}_4)_2]\cdot 12\mathrm{H}_2\mathrm{O}\), where \(M_{\mathrm{I}}=\mathrm{K}^{1+},\ \mathrm{NH}_4^{1+}\), \(M_{\mathrm{III}}=\mathrm{Al}^{3+},\ \mathrm{Cr}^{3+},\ \mathrm{Fe}^{3+},\ \mathrm{Ce}^{3+}\), and \(\mathrm{In}^{3+}\). In addition, \(\mathrm{SO}_4\) was replaced by \(\mathrm{SeO}_4\), and \(\mathrm{H}_2\mathrm{O}\) by \(\mathrm{D}_2\mathrm{O}\).

8) \((\mathrm{NH}_4)_2 \cdot \mathrm{SO}_4\).

The ferroelectric properties of ammonium sulfate were recently discovered by Matthias and Remeika[^96], although the anomalous dependence of \(\varepsilon\) of this compound on temperature had been known for a comparatively long time[^97].

Ammonium sulfate has an orthorhombic layered lattice with the following unit-cell parameters: \(a=5.97;\ b=10.60;\ c=7.76\ \text{Å}\)[^98]. The ferroelectric axis is the \(a\) axis. Spontaneous polarization appears at \(-50^\circ\mathrm{C}\) and at \(-58^\circ\mathrm{C}\) reaches \(0.25 \cdot 10^{-6}\ \mathrm{coul}/\mathrm{cm}^2\). At this temperature the coercive field is \(2000\ \mathrm{V}/\mathrm{cm}\).

There is no water of crystallization in ammonium sulfate, and therefore in this compound a hydrogen bond of the type \(\mathrm{N—H—O}\) is realized.

9) \((\mathrm{NH}_4)_2\mathrm{Cd}_2(\mathrm{SO}_4)_3\).

Quite recently Jona and Pepinsky\(^{99}\) discovered ferroelectric properties in \((\mathrm{NH}_4)_2\mathrm{Cd}_2(\mathrm{SO}_4)_3\). This double salt crystallizes in a structure of the langbeinite type \([\mathrm{K}_2\mathrm{Mg}_2(\mathrm{SO}_4)_3]\); above the temperature of the phase transition \((\theta=-184^\circ\mathrm{C})\) it has a cubic lattice with parameter \(a=10.35\,\text{Å}\). The cells contain 4 “molecules.”

Unlike most ferroelectrics, \((\mathrm{NH}_4)_2\mathrm{Cd}_2(\mathrm{SO}_4)_3\) is characterized by a low dielectric permittivity at the transition point. In Fig. 22 the dependence of \(\varepsilon\) of a single crystal on temperature is shown, measured in weak fields along the direction \([111]\). Below the transition temperature hysteresis loops are observed. At \(-190^\circ\mathrm{C}\) the spontaneous polarization is \(0.3\cdot10^{-6}\ \mathrm{coul}/\mathrm{cm}^2\), and the coercive force is \(25\ \mathrm{kV}/\mathrm{cm}\).

Fig. 22. Temperature dependence of the dielectric permittivity of a single crystal of \((\mathrm{NH}_4)_2\mathrm{Cd}_2(\mathrm{SO}_4)_3\), measured along the direction \([111]\) in a weak field\(^{99}\).

Table IX gives the temperatures of the phase transitions of new ferroelectrics and antiferroelectrics with hydrogen bonding.

Table IX

Thus, in recent years a considerable number of new ferroelectrics have been discovered. Some antiferroelectrics have also been found and investigated.

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