Barium Titanate—A New Ferroelectric

A. V. Rzhanov

I. Introduction

The first of the known ferroelectrics—Rochelle salt itself—was, over the course of several years (in 1930–1934), studied in detail in the works of I. V. Kurchatov and his collaborators1 at the Physico-Technical Institute of the Academy of Sciences of the USSR. After this, in 1938, reports appeared[^2] on the discovery of ferroelectric properties in potassium phosphates and arsenates: \( \mathrm{KH_2PO_4} \), \( \mathrm{KD_2PO_4} \), and \( \mathrm{KH_2AsO_4} \). However, in these substances ferroelectric properties occur only at very low temperatures (below \(-151^\circ\mathrm{C}\) for \( \mathrm{KH_2PO_4} \) and below \(-122^\circ\mathrm{C}\) for \( \mathrm{KH_2AsO_4} \)), which makes their practical application impossible. The first practically important ferroelectric—barium titanate—was discovered in 1944 at the P. N. Lebedev Physical Institute of the Academy of Sciences of the USSR by B. M. Vul and I. M. Goldman[^3][^4] during an investigation of the dielectric properties of the titanates of metals of the second group of the periodic system of elements of Mendeleev.

In these works[^5] it was shown that the dielectric permittivity of the titanates of the alkaline-earth metals of the first subgroup, crystallizing in a lattice of the perovskite type, has values large in comparison with other types of lattices and increases with increasing radius of the alkaline-earth ion.

The values of the dielectric permittivity of various titanates are given in Table I.

B. M. Vul at once connected the sharp increase in the dielectric permittivity of barium titanate, as compared with other titanates of the perovskite type, with the circumstance that barium titanate is the only perovskite in which the distance between the titanium and oxygen ions is greater than the sum of their radii according to Goldschmidt (Table II).

The characteristic dependence of the dielectric permittivity on temperature and the presence of dielectric hysteresis allowed B. M. Vul to assign barium titanate to a new type of ferroelectrics. These investigations

...studies, which will be considered in detail below, were carried out on polycrystalline specimens of barium titanate obtained by the usual ceramic technology. From powdered BaCO$_3$ and TiO$_2$, taken in the stoichiometric ratio corresponding to barium metatitanate,

Table I

Subgroup	Alkaline-earth metal	Type of crystal lattice	Dielectric constant
I	Beryllium	—	70
I	Calcium	Perovskite	115
I	Strontium	Perovskite	155
I	Barium	Perovskite	$>1000$
II	Magnesium	Ilmenite	17
II	Zinc	—	30
II	Cadmium	Ilmenite	62

after thorough mixing, disks were pressed, which were then fired in a platinum or silite furnace at a temperature ranging, depending on the composition and amount of impurities, from 1370 to 1450° C. It was found in this connection$^{6}$ that impurities play

Table II

Titanate	Edge size of the elementary cube	Distance between Ti and O ions	Sum of the radii of Ti and O ions
Calcium titanate . . .	3.80	1.90	1.96
Strontium titanate . . .	3.89	1.95	1.96
Barium titanate . . .	3.97	1.99	1.96

a very substantial role. In particular, when specimens were prepared from chemically pure starting materials, a modification of barium titanate was discovered that did not possess ferroelectric properties.

An X-ray study carried out by A. N. Lyamina gave, for this modification, a structure close to rhombohedral, with axes $a=b=c=4.04$ Å and an angle close to 90°. This modification of BaTiO$_3$ had a dielectric constant of about 50 and all properties normal and close to those of other ceramics of this type. When small impurities were added to the initial pure products (2% Al$_2$O$_3$ or 1%

SrCO₃ by weight) a ferroelectric material was obtained, and X-ray analysis, carried out by V. P. Butuzov, gave at room temperature a tetragonal structure with axes \(a=b=3.98\,\text{Å}\) and \(c=4.02\,\text{Å}\).

Barium titanate obtained from technical materials was always ferroelectric and, at room temperature, possessed a tetragonal structure, which is evidently connected with the presence in it of a sufficient quantity of natural impurities.

Thus, the role of impurities, as follows from these works, evidently consists in the fact that their presence determines the crystallization of barium titanate into a lattice of the perovskite type, with the transition to which its ferroelectric properties are also connected. The small amount of the necessary impurities indicates the catalytic character of their action, since they are clearly insufficient to distort the lattice by the uniform introduction of impurities throughout the whole volume.

Soon after the publication of the works of B. M. Vul and his collaborators, other works devoted to the investigation of barium titanate also began to appear.

E. Megaw⁷, by the method of X-ray analysis, demonstrated a change in the crystal structure of BaTiO₃ at the transition point (the Curie point). She showed that at temperatures above \(120^\circ\text{C}\) barium titanate has an ideal cubic lattice of the perovskite type. At a temperature of \(20^\circ\text{C}\) the BaTiO₃ lattice is tetragonal, with an axial ratio

\[ \frac{c}{a}=1.0101 \]

(it is often still called pseudocubic in the foreign literature). The transition itself, occurring at \(120^\circ\text{C}\), consists in a homogeneous expansion along one of the axes, which also becomes the polar \(c\)-axis, and a contraction along the other two axes. The lengths of the axes change continuously with temperature, although nonlinearly, since the changes become more rapid near the Curie point. However, although the lengths of the axes change continuously, the coefficient of linear expansion has a discontinuity at the transition point and opposite signs for the directions along the polar axis and perpendicular to it. In Fig. 1 are given the curves of the change in the lengths of the axes of the unit cell as a function of temperature, according to the data of E. Megaw.

Thus, these works of E. Megaw were direct experimental proof of the presence in barium titanate of a second-order phase transition, on the basis of the conception of which V. L. Ginzburg constructed a theory of ferroelectric phenomena (see the theoretical review).

E. Megaw pointed out the coexistence, discovered by her, of cubic and tetragonal phases in a proportion depending on temperature, over a temperature interval of several degrees near the transition point. This coexistence was explained by her as the result of the presence of local stresses, which may accelerate or retard transitions, depending on their direction, in separate regions.

polycrystalline formation, owing to the insignificant difference in the energies of the general states in this temperature interval.

Interest in this new ferroelectric stimulated searches for the possibility of obtaining it in the form of a single crystal.

According to the somewhat fragmentary and incomplete data available in the literature, barium titanate crystals can be grown from a solution of \(\mathrm{BaCO_3}\) and \(\mathrm{TiO_2}\) in molten \(\mathrm{BaCl_2}\). For better crystal growth it is recommended to create a noticeable excess of \(\mathrm{BaCO_3}\). Thus, for example, according to the most complete recipe\(^8\), the author used, per one mole of \(\mathrm{BaCl_2}\), about \(0.53\) mole of \(\mathrm{BaCO_3}\) and \(0.26\) mole of \(\mathrm{TiO_2}\) (or, for \(50\) g of \(\mathrm{BaCl_2}\), \(25\) g of \(\mathrm{BaCO_3}\) and \(5\) g of \(\mathrm{TiO_2}\)). In this case, rather stringent requirements are imposed on the cooling regime. In this technique the melt was cooled from \(1200^\circ\mathrm{C}\) to approximately \(800^\circ\mathrm{C}\) over several hours (about 8). Some difficulties arise in selecting crucibles. Crucibles of pure platinum and of graphite were used, the entire process being carried out in a nitrogen atmosphere. In the case of platinum crucibles, a certain amount of platinum dissolved in the melt and entered the crystal. The crystal thereby acquired a color ranging from light yellow to reddish brown, the intensity of which could be reduced by annealing at \(200^\circ\mathrm{C}\) for several hours.

Fig. 1. Change in the lengths of the axes and of the cubic root of the unit-cell volume as a function of temperature.

When appreciable amounts of platinum entered, non-ferroelectric modifications of \(\mathrm{BaTiO_3}\) of the hexagonal or monoclinic system were formed.

These modifications will be discussed in more detail below.

When a graphite crucible was used, the crystals acquired a bluish color, associated with the presence of reduced titanium. By heating to \(600\text{–}800^\circ\mathrm{C}\) in an oxygen atmosphere, the crystal could be decolorized.

Ferroelectric crystals of the tetragonal (pseudocubic) modification were obtained in the form of square plates or cubi-

... and it was found that, in the absence of special precautions, they consist of twins[^9]. Twinning depends to a large extent on impurities and stresses acting on the crystal, and occurs along the planes (101) and (011).

According to one of the recent works[^10], when attempts are made to grow a BaTiO$_3$ crystal from a melt in Na$_2$CO$_3$ and K$_2$CO$_3$ in a platinum crucible, crystals of hexagonal symmetry of composition Ba(Ti$_{0.75}$Pt$_{0.25}$)O$_3$ are obtained, with a density of $6.8 \pm 0.1\ \mathrm{g/cm^3}$. When a BaCl$_2$ melt is used in a platinum crucible, along with pseudocubic crystals, rhombic crystals of composition Ba$_4$Ti$_2$PtO$_{10}$ are obtained, with a density of $0.45 \pm 0.02\ \mathrm{g/cm^3}$. And finally, in corundum or graphite crucibles, pseudocubic BaTiO$_3$ crystals are obtained, with a density of $6.0\ \mathrm{g/cm^3}$. Both the hexagonal and the rhombic modifications are obtained by replacing part of the Ti$^{+4}$ ions with Pt$^{+4}$ ions, and neither has ferroelectric properties. The latter is attributed by the authors to the fact that, owing to distortion of the crystallographic cells, the Ti ion is in a state of stable fixation, in contrast to the case of the tetragonal (ferroelectric) modification, where its position at the center of the octahedron of oxygen ions is unstable and it is displaced in the direction of one of the oxygen ions.

II. DIELECTRIC PROPERTIES OF POLYCRYSTALLINE SPECIMENS

As already mentioned above, the anomalous properties of barium titanate were discovered and, for the most part, studied on polycrystalline specimens. Subsequent investigation of the properties of single crystals confirmed all the data obtained on polycrystalline specimens and yielded comparatively little that was fundamentally new. Moreover, experiments on single crystals, in view of the considerable contamination of the latter, often give results that are more difficult to interpret than those obtained on polycrystalline specimens, and require additional verification and refinement.

In view of all this, we shall follow the historical course and begin the description of the dielectric properties of barium titanate with the data obtained on polycrystalline specimens.

Polycrystalline specimens are ceramics in which there are individual microcrystals arranged chaotically relative to one another, and interlayers of a glassy phase; in good specimens obtained at a comparatively high sintering temperature (about 1450° C), there is relatively little of the latter—1–2% by weight.

At temperatures below the Curie point and in the absence of large external fields, the directions of the spontaneous electric moments of the individual microcrystals are distributed chaotically, which accounts for the absence...

of the total moment of the specimen. As we shall see later, under the influence of sufficiently large external fields the spontaneous moments of individual microcrystals can change their orientation, which changes the total polarization of the specimen.

After these preliminary remarks, let us turn to a description of the dielectric properties of polycrystalline specimens of barium titanate.

a) Dielectric permittivity as a function of temperature and field strength

The dielectric permittivity, measured in a small field (not exceeding 30 V/cm) as a function of temperature, is presented, according to the data of B. M. Vul5, in Fig. 2.

Fig. 2. Dependence of dielectric permittivity, measured in a small field, on temperature.

As is seen from the figure, the measurements were carried out from a temperature of \(+200^\circ\text{C}\) down to a temperature close to absolute zero and exceeding it by less than 2 degrees. At the temperature of liquid helium \((\sim 2^\circ\text{K})\), the dielectric permittivity of polycrystalline barium titanate is equal to 100; in solid hydrogen \((T = 14^\circ\text{K})\), \(\varepsilon = 114\); in solid nitrogen \((T = 60^\circ\text{K})\), \(\varepsilon = 165\); and in liquid oxygen \((T = 90.9^\circ\text{K})\), \(\varepsilon = 250\). Thus, for barium titanate the temperature coefficient of dielectric permittivity \(\frac{1}{\varepsilon}\frac{d\varepsilon}{dT}\) in the interval from \(2^\circ\text{K}\) to \(4.0^\circ\text{K}\) is about \(0.01\ \text{degree}^{-1}\).

At room temperature the dielectric permittivity of polycrystalline barium titanate prepared from technically pure materials is approximately 1400, and then changes relatively little up to \(+40^\circ\text{C}\). The maximum in this case lies at \(80^\circ\text{C}\), and the dielectric permittivity there is \(\varepsilon = 6600\).

The dielectric permittivity at the peak and at room temperature, as well as the position of the maximum, vary somewhat depending on the amount and composition of impurities. In particular, the position of the maximum can vary from \(80^\circ\text{C}\) to \(120^\circ\text{C}\). Since the dielectric permittivity of a polycrystalline specimen is determined by the statistical distribution of the directions of the polar axes of the microcrystals, and the latter may change somewhat during passage through the heating–cooling cycle, the value of the dielectric

permittivity near the peak fluctuates within the limits of \(\pm 5\%\) in different measurements of one and the same specimen.

B. M. Vul and L. F. Vereshchagin studied the dependence of the dielectric permittivity of barium titanate on pressure \(^{4}\). It was shown that \(\varepsilon\) increases with pressure, the relative increase decreasing as the pressure is increased. In the range from 300 to 2000 atm the relative change amounted to \(10^{-5}\ \mathrm{cm}^2/\mathrm{kg}\).

By means of a Schering bridge at a frequency of 50 cps, the dependence of the dielectric permittivity of barium titanate on the field strength at various temperatures was studied \(^{11}\). It was shown that at room temperature the dielectric permittivity of barium titanate

Fig. 3. Dielectric permittivity as a function of the field strength at different temperatures (measurements at a frequency of 50 cps).

Fig. 4. Dielectric permittivity as a function of temperature at different voltages on the specimen.

increases smoothly with the field strength up to a maximum at a field strength of several kilovolts per centimeter; then a tendency toward decrease is observed, which cannot be followed far, since breakdown occurs. The same type of dependence is also observed at the temperature of liquid oxygen (at \(-183^\circ\mathrm{C}\)), with the sole exception that here, in the accessible fields, no decrease in \(\varepsilon\) was detected. However, at high temperatures the dielectric permittivity shows no dependence on the field strength up to the fields attainable in the experiment (about \(6\ \mathrm{kV}/\mathrm{cm}\)).

The results of these measurements are presented in Fig. 3. The indicated special feature of barium titanate is perhaps still more clearly seen from Fig. 4, which gives the results of measurements of the dielectric permittivity as a function of temperature at various voltages,

applied to the specimen (the thickness of the specimen was equal to 1.2 mm). It is seen from the figure that the dielectric constant depends on the field strength only below the Curie point (the temperature of the maximum of \(\varepsilon\)), while above it it is the same for all fields.

b) Dielectric constant as a function of frequency

Measurements by B. M. Vul\(^5\) of the dielectric constant of barium titanate as a function of wavelength, up to 2 m, and by P. Novosil'tsev and A. Khodakov\(^ {12}\) at wavelengths of 109, 58, and 16 m showed that at these wavelengths the dielectric constant practically does not depend on frequency. Measurements by D. I. Mash\(^ {13}\) at a wavelength of 20 cm showed that barium titanate still retains a large value of the dielectric constant at this wavelength. Small changes in it could not be observed in this case, since under the experimental conditions (a specimen in the form of a small sphere) the investigated specimens could not be measured at low frequencies.

Fig. 5. Real and imaginary parts of the dielectric constant of barium titanate as a function of temperature \((\operatorname{tg}\delta = \varepsilon''/\varepsilon')\): 1) \(\varepsilon'\) at a frequency of 1.5 MHz, 2) \(\varepsilon'\) at a frequency of 9450 MHz, 3) \(\varepsilon''\) at a frequency of 9450 MHz, 4) \(\varepsilon''\) at a frequency of 1.5 MHz.

Very recently a work\(^ {14}\) appeared on the investigation of the dielectric properties of barium titanate at a frequency of 9450 MHz (about 3 cm) and at a frequency of 24000 MHz. At these frequencies considerable dispersion already occurs. If at a frequency of 1.5 MHz the dielectric constant of the specimens investigated in that work was 1500 at room temperature, then at 9450 MHz it was already 300, and at 24000 MHz—126. Correspondingly, \(\operatorname{tg}\delta\) changed, depending on frequency, from 0.015 to 0.53 and 0.59. At a frequency of 9450 MHz, the temperature dependence of the dielectric constant and of the tangent of the dielectric-loss angle was obtained, as presented in Fig. 5 together with the result of measurements at 1.5 MHz for comparison.

c) Dielectric hysteresis and aftereffects

Dielectric hysteresis of barium titanate was first discovered on polycrystalline specimens by B. M. Vul\(^ {15}\). With the aid of a cathode oscilloscope, according to the well-known circuit shown in Fig. 6,

observed between the instantaneous value of the polarization and the applied voltage. The oscillograms obtained at various temperatures are shown in Fig. 7. They show that dielectric hysteresis appears in barium titanate only below the Curie point (80°C), i.e., precisely where a sharp dependence of the dielectric constant on the field strength is found. However, in small fields the dielectric constant of barium titanate even below the Curie point depends linearly on the field strength[^5]. Here a complete analogy is observed with the magnetic behavior of ferromagnetics. In exactly the same way, by analogy with the case of ferromagnetics, the so-called “reversible” dielectric constant was measured, i.e., the dielectric constant measured in a small alternating field as a function of a simultaneously applied large constant field.

Fig. 6. Circuit for observing dielectric hysteresis in ferroelectrics.

It was found[^5] that the reversible dielectric constant does not depend on the strength of the polarizing constant field at temperatures considerably exceeding the Curie point and at low temperatures (−183°); at room temperature, however, the reversible dielectric constant decreased with increasing polarizing constant field. A. Bretteville[^16] repeated the oscillographic investigation of the dielectric properties of barium titanate. In these studies the dielectric constant was determined from the slope of the hysteresis loops at the value of the field \(E = 0\), i.e., at the crossing of the ordinate axis by the loops, and the dielectric losses from the area of the hysteresis loop. In the work there were found, besides the sharp maximum of the dielectric constant at 120°C, two smaller ones, located at −70°C and +10°C. The simultaneously obtained curve of the tangent of the dielectric-loss angle is shifted relative to the dielectric-constant curve in such a way that the maxima of \(\varepsilon\) correspond to minima of \(\tg \delta\). The magnitudes of both small maxima of \(\varepsilon\) depend on the field strength, the maximum at −70°C appearing only at field strengths greater than 200 V/cm.

The author relates the presence of these maxima to breaks in the thermal-expansion curve at 120, 10, and −55°C, discovered in earlier works. According to Bretteville’s data, all the maxima tend to shift toward lower temperatures as the field strength increases, while the temperature of the maxi-

...is related to the field strength by an empirical formula of the form

\[ E = E_0 \cdot e^{-bT_{\max}} . \]

Here it is necessary to make the reservation that, although this investigation is of some interest, nevertheless the dielectric constant measured by the author’s method has only a conditional character. Indeed, it represents a differential dielectric constant, determined at a certain point of the loop (at \(E = 0\)). The fact that precisely this point has been chosen in no way brings this value of the dielectric constant closer to the dielectric constant in weak fields, since for this it would have been necessary to measure it by this method along the so-called virgin curve of the hysteresis loop (starting from zero).

Fig. 7. Oscillograms of hysteresis loops of barium titanate at temperatures \(-183^\circ\) C, \(26.8^\circ\) C, \(73.8^\circ\) C, and \(115.5^\circ\) C and a field strength \(E = 25\) kV/cm.

Roberts’ work \(^{17}\) is devoted to the study of the dielectric constant, measured at a high-frequency small voltage, as a function of the magnitude of a simultaneously applied constant biasing voltage, i.e., of what in B. M. Vul’s work was called the reversible dielectric constant.

In contrast to B. M. Vul’s work, where the reversible dielectric constant was investigated over a wide temperature interval, Roberts studied it only near the Curie point (on both sides of it), where, owing to the strongly pronounced nonlinearity, the reversible...

BARIUM TITANATE—A NEW FERROELECTRIC

the dielectric permittivity depends strongly on the magnitude of the biasing constant voltage. Postulating the presence of a nonlinearity, expressed for the non-ferroelectric region (above the Curie point) in the form of the simple formula

\[ E=\alpha D+\beta D^3, \tag{1} \]

where \(D\) is the dielectric displacement, and introducing the dielectric permittivity at any field strength of the constant biasing field

\[ \varepsilon'=\frac{dD}{dE} \]

and the initial dielectric permittivity in the absence of a biasing field \(\varepsilon_1=1/\alpha\), equation (1) is easily transformed into the following form:

\[ E/E_0=\frac{1}{4}\left(\frac{\varepsilon_1}{\varepsilon'}-1\right)^{1/2} \left(\frac{\varepsilon_1}{\varepsilon'}+2\right), \tag{2} \]

where

\[ E_0=4(\alpha/3)^{3/2}\beta^{-1/2}. \tag{3} \]

According to equation (2), \(E_0\) is the field strength of the constant biasing field at which the dielectric permittivity decreases to one half of its initial value \(\left(\varepsilon'=\frac{1}{2}\varepsilon_1\right)\). Solving equation (3) with respect to \(\beta\):

\[ \beta=\frac{16}{27}\frac{\alpha^3}{E_0^2} =\frac{16}{27}\varepsilon_1^{-3}E_0^{-2}, \]

one can obtain values of \(\beta\) as a function of temperature. Equation (1) was also extended by Roberts to the ferroelectric temperature interval, which, although not entirely rigorously—because of the superposition of a nonlinear process of reorientation of the regions—nevertheless gives correct values of the field required to reduce \(\varepsilon'\) to one half of its initial value.

From the experimental data obtained it followed that the coefficient \(\beta\), which determines the nonlinearity of the dependence of polarization on field, is constant over a wide temperature interval on both sides of the Curie point. The author of the present review arrived at the same result, though obtained by another method, in his investigation of dielectric hysteresis carried out for the purpose of obtaining the temperature dependence of the spontaneous polarization\(^ {18}\).

The same author also discovered very peculiar aftereffect phenomena\(^ {18}\), characteristic of polycrystalline specimens of barium titanate. If a small field (\(5\ \mathrm{kV/cm}\)) is applied to a specimen for the first time, the hysteresis loop has a spindle-shaped form, which in the analogous case of ferromagnets is called a Rayleigh figure. With a further increase in voltage the figures approach more and more closely a typical hysteresis loop. If, however, the same small field of \(5\ \mathrm{kV/cm}\) is again applied to the specimen, but some time after the application of a large field (\(25\ \mathrm{kV/cm}\)), then the figure obtained will have the form of a hysteresis loop. In this case such characteristics sharply increase as

loops, such as residual polarization, coercive force, and hysteresis losses. The mutual arrangement of the loops at different field strengths also changes in the case of a preceding application of a high voltage: if in the first case the ends of all the loops form their own separate curve, then in the second case they all lie on the largest of the loops (Fig. 8). This phenomenon has an extremely long relaxation time, if that term is applicable in this case, since in order to obtain the original picture it is necessary to hold the specimen for several days after applying the high voltage. It should be noted that the discussion concerned the application of an alternating voltage, and depolarization by slowly lowering the alternating voltage to zero does not eliminate this phenomenon. Its cause evidently lies in the fact that, for rotation of the polar axis through \(90^\circ\), as is known from experiments with single crystals (see below), a rather considerable field is needed, and the disorientation after this proceeds very slowly.

Labels in the figure: “Primary”; “Repeated.”

Fig. 8. Successive photographs of hysteresis loops (after-effect) at a temperature of \(24^\circ\text{C}\), field strength \(E = 25\ \text{kV/cm}\), and a time interval between exposures of 10 minutes.

Hence it is clear that, under the action of a large alternating field, the majority of polar axes are oriented in the plane of application of the field, and this anisotropy remains even after it is removed. Therefore, upon subsequent application of a small field, polarization proceeds more easily, since the rotations through \(90^\circ\) have already been carried out.

The piezoeffect of polarized polycrystalline barium titanate specimens, discovered by the author, belongs to this same class of phenomena; it will be discussed in greater detail below.

г) Spontaneous polarization

In the above-mentioned work of A. Rzhanov\({}^{18}\), the temperature dependence of spontaneous polarization was studied. In this work, for the first time, the nonlinearity of the induced polarization near the Curie point was taken into account.

In all previous investigations of the spontaneous polarization of various ferroelectrics it was assumed that the induced polarization depends linearly on the field, and that the nonlinearity of the total polarization is determined only by the nonlinearity of the process of reorientation of domains. However, from consideration of the equation relating the polarization and the field outside the region of spontaneous polarization:

\[ E = 2\alpha P + 2\beta P^3, \]

it is easy to see that near the Curie point the nonlinearity of the principal polarization should appear in comparatively small fields. Indeed, according to the theory of V. L. Ginzburg (see the theoretical review), the coefficient \(\alpha > 0\) for \(T > \Theta\), \(\alpha < 0\) for \(T < \Theta\), and \(\alpha_\Theta = 0\). Thus, near the Curie point \((T=\Theta)\), where \(\alpha\) is small, the cubic term becomes comparable with the linear term even in relatively small fields.

It must be noted that this phenomenon, which in principle occurs in all ferroelectrics, is especially strongly expressed in barium titanate in the sense that here the nonlinearity of the induced polarization extends over a considerably wider temperature interval. This is connected with the circumstance that the region of sharp growth of \(\varepsilon\) with temperature (and, consequently, of small values of the coefficient \(\alpha\), inversely proportional to \(\varepsilon\)) in barium titanate extends over \(50\text{--}60^\circ\)C, whereas in Rochelle salt it extends over \(5\text{--}7^\circ\)C, and in \(\mathrm{KH_2PO_4}\) over \(10\text{--}15^\circ\)C.

For this reason, neglecting the nonlinearity of the induced polarization, both for Rochelle salt and for \(\mathrm{KH_2PO_4}\), did not introduce very substantial errors and, in any case, did not lead to finite values of the spontaneous polarization above the Curie point, as was obtained in the work of Hulm\({}^{19}\) with a single crystal of \(\mathrm{BaTiO_3}\).

In the author’s work\({}^{18}\) it was shown that both above and below the Curie point, at a considerable distance from it, the induced polarization, up to the fields attainable in the experiment, remains a linear function of the field, while the principal dielectric permittivity remains constant. Thus, far from the Curie point, below it, the spontaneous polarization can be obtained by simple subtraction of the linear induced polarization

\[ P'=\frac{\varepsilon-1}{4\pi}E \]

from the total.

The nonlinearity of the induced polarization, however, occurs only near the Curie point, approximately in the same temperature interval in which the principal dielectric permittivity has a sharp maximum.

Since this temperature region extends in the case of barium titanate over \(50\text{--}60^\circ\), obtaining part of the temperature course of the spontaneous—

of polarization near the Curie point becomes very difficult.

This difficulty was circumvented by the fact that, with the aid of the formulas of the theory of the phase transition in ferroelectrics, it proved possible to relate the temperatures on both sides of the Curie point at which the nonlinear induced polarization is the same. As is known, at temperatures above the Curie point spontaneous polarization is absent, and the nonlinear induced polarization is measured directly; the temperature below the Curie point which corresponds to the same value of the nonlinear induced polarization can be found by calculation. Therefore, the determination of the spontaneous polarization is reduced, also in the indicated case, to the simple subtraction of the induced polarization from the total polarization. This method of calculation is completely rigorous only for a single crystal, or, more precisely, for one domain of a single crystal.

Fig. 9. Temperature dependence of the spontaneous polarization of polycrystalline samples of barium titanate.

In the work under discussion, the object of investigation was polycrystalline samples of BaTiO\(_3\). However, in a strong field such samples can be polarized to a state of saturation, when the increase of the total polarization occurs only at the expense of the increase of the induced polarization. In this case the vector of the spontaneous polarization of each microcrystal must be directed along the cube axis nearest to the direction of the external field.

The greatest angle between the field and the nearest cube axis is approximately \(55^\circ\); it corresponds to the case when the field is directed along the diagonal of the cube. If the microcrystals are distributed chaotically and polarized along the axis nearest to the direction of the field, then the maximum value of the total spontaneous polarization \(P_{\max} \simeq 0.8 P_0\), where \(P_0\) is the spontaneous polarization of a single crystal.

Thus, polycrystalline samples subjected to the action of a strong field are, in dielectric respect, close to a single crystal and, consequently, the calculation considered above can be applied to them. The temperature variation of the spontaneous polarization obtained

thus for specimens with a Curie point at \(123^\circ\mathrm{C}\), is shown in Fig. 9.

Figure 10 shows the dependence of the square of the spontaneous polarization on temperature. It is evident from the figure that, in agreement with theory, the square of the spontaneous polarization depends linearly on temperature over a rather wide region near the Curie point.

d) Anomaly of the Specific Heat

As is known, the anomaly of the specific heat of ferroelectrics near the Curie point was first observed for Rochelle salt by P. P. Kobeko \(^{20}\). The assumption that there is a jump in the heat capacity was made on the basis of the notion that the destruction of the spontaneous polarization at the Curie point must be associated with the absorption by the lattice of additional heat. These considerations were fully confirmed both in the case of Rochelle salt and in the case of ferroelectrics of the type \(\mathrm{KH_2PO_4}\), \(\mathrm{KH_2AsO_4}\). The jump of the specific heat in the case of barium titanate \(^{5}\) was measured, at the suggestion of B. M. Vul, by V. A. Sokolov at the Institute of General and Inorganic Chemistry of the Academy of Sciences of the USSR.

In Fig. 11 the dependence of the heat capacity and of the dielectric permittivity on temperature is shown.

Fig. 10. Temperature dependence of the square of the spontaneous polarization.

The maxima of both curves lie at the same temperature \((80^\circ\mathrm{C}\)—the Curie point for these specimens).

The additional heat was then found to be equal to

\[ Q=\int (c_p-c_{p\varphi})\,dT=0.039\ \text{cal/g}. \]

The investigation of the temperature course of the specific heat was also carried out by Hippel^21, but, owing to insufficient experimental care, he did not detect any jump in the heat capacity.

Fig. 11. Dependence of heat capacity and dielectric permittivity on temperature.

On the other hand, Horwood, Popper, and Rushman^22, in their study of the temperature dependence of the heat capacity of polycrystalline specimens of barium titanate, measured a jump in the heat capacity near the Curie point, the magnitude of which was very close to that obtained by V. A. Sokolov.

On the basis of the formula proposed by V. L. Ginzburg (see the theoretical review), from the jump in the specific heat and the temperature coefficient of the dielectric permittivity, B. M. Vul estimated the spontaneous polarization.^5 At \(66^\circ\text{C}\) the spontaneous polarization was found to be equal to \(6.6 \cdot 10^{-6}\ \text{coul}/\text{cm}^2\), which is in good agreement with the value determined by measuring hysteresis loops.

e) Isomorphous mixtures of barium titanate

M. Eremeev and B. Kurchatov^23 showed that mixtures of Rochelle salt with substances isomorphous with it are ferroelectrics when the percentage of the non-ferroelectric component is not too large.

Exactly similarly, in the case of barium titanate, the addition to it of isomorphous substances, as A. K. Isihneli showed, shifts the Curie point into the region of lower temperatures and diminishes the magnitude of the dielectric permittivity at the peak.

Isomorphous mixtures exhibit the phenomenon of hysteresis; moreover, at large amounts of the non-ferroelectric component, both the value of the polarization itself and the degree of nonlinearity turn out to be considerably smaller than for barium titanate alone.

Although the addition of isomorphous substances to barium titanate lowers its ferroelectric properties, this sometimes turns out to be practically expedient, since at the same time the dielectric loss tangent is lowered and the possibility appears of controlling the position of the Curie point. This may prove especially expedient at ultrahigh frequencies.

Thus, for example, at a frequency of 9450 MHz a material consisting of 56% BaTiO\(_3\) and 44% SrTiO\(_3\) at room temperature has \(\varepsilon = 760\) and \(\operatorname{tg}\delta = 0.02^{24}\), and may be of considerable value for microwave engineering.

III. DIELECTRIC PROPERTIES OF SINGLE CRYSTALS

Although the anomalous properties of barium titanate have been studied in great detail on polycrystalline specimens, the study of the properties of single crystals is undoubtedly of great interest. As will be seen from what follows, investigation of single crystals makes it possible directly to see the structure of the regions (domains), to investigate the anisotropy of the dielectric permittivity, and to supplement purely electrical methods of investigation by optical ones.

As has already been indicated, all the results described below were obtained on single crystals whose degree of purity is evidently not very high. Therefore most of these results cannot lay claim to complete reliability, and some of them—for example, the almost complete absence of a peak of the dielectric permittivity in the direction of the spontaneous polarization \(\varepsilon_{\parallel}\) near the Curie point, and the difference between the values of the dielectric permittivities \(\varepsilon_{\parallel}\) and \(\varepsilon_{\perp}\) above the Curie point, where the anisotropy disappears—give rise to very great doubt.

The study of the properties of single crystals of barium titanate was begun only very recently, and its results must be regarded as very preliminary, requiring considerable refinement and verification.

a) Division into regions (domains)

In the study of single crystals it was found\({}^{25}\) that they, like other ferroelectrics, are divided into regions (so-called domains) with different directions of the polar axis, coinciding with the vector of spontaneous polarization. The cause of division into regions is clear from simple energetic considerations, since in this case the crystal as a whole has no electric moment and, consequently, creates no field around itself. It was found that in the case of barium titanate crystals this division into regions can be seen directly from the lines of cohesion of these regions, which, because of the change in birefringence in the stressed boundary layer, are perceived in polarized light as dark stripes on the surface of the crystal. These stripes were observed to be of two kinds: running parallel to the faces of the crystal and situated at an angle of 45° to these faces.

In crystals with a cubic external faceting, both kinds of stripes could be observed on one face; in crystals in the form of a plate, however, the larger surface usually proved to be covered with stripes situated at 45° to the faces, and the small lateral surfaces with stripes parallel to the faces.

When the crystal is heated to a temperature above the Curie point, both types of bands disappear, and the crystal becomes completely transparent and isotropic. Conversely, when the crystal is cooled after passing through the Curie point, they appear again, and the general pattern of their distribution, although not reproduced exactly, becomes approximately the same as before.

When the temperature is lowered near the secondary small maximum of $\varepsilon$ at a temperature of about $5^\circ$C, it is not the ferroelectric state that changes, but the position of the polar axis: it rotates spontaneously by $90^\circ$.

At a temperature near $-70^\circ$C this effect, in the authors’ opinion, is partially reversed. Changes in the distribution pattern of the bands can also be achieved by applying an electric field. If the crystal is observed in a direction perpendicular to the applied field, then the regions covered by bands at an angle of $45^\circ$ grow at the expense of those not covered by them; and in a sufficiently strong field the bands disappear, and the crystal becomes transparent. Observation in polarized light then gives the pattern of a uniaxial crystal observed perpendicular to the optical axis.

Fig. 12. Schematic representation of the division of a crystal into domains and of the directions of the spontaneous moments of the domains.

Similarly, when observing through semitransparent electrodes parallel to the applied field, the unshaded regions grow at the expense of those covered by bands at an angle of $45^\circ$; colored bands appear parallel to the faces of the crystal, and in a considerable field the crystal darkens until a central uniaxial cross appears, showing that the optical axis is parallel to the direction of observation and, consequently, to the applied field.

Thus, in this work it is shown that the optical axis of the crystal (or region) coincides with the direction of spontaneous polarization and may be rotated by a sufficiently strong field either by $90^\circ$ or by $180^\circ$. The bands at an angle of $45^\circ$ to the faces of the crystal (plane 101) are twin planes of regions of the crystal with different directions of the spontaneous moment.

The spontaneous moments themselves (or the polar axes) of the regions are arranged parallel to the faces of the cube and perpendicular to one another in neighboring regions separated by a band running at an angle of $45^\circ$ to the faces of the cube. The general pattern of the distribution of the bands and of the directions of the spontaneous moments is shown in Fig. 12, borrowed from $^{25}$. Regions of one direction of spontaneous polarization need not necessarily be

BARIUM TITANATE—A NEW FERROELECTRIC

pass through the whole crystal, but may be confined to only a part of it. In this case the stripes at \(45^\circ\) close into spindle-shaped forms, shown in Fig. 12.

b) Dielectric constant

The dielectric constant of a single crystal proves to be substantially different when measured in directions perpendicular and parallel to the polar axis. In the former case it is considerably greater than in the latter. The temperature dependence of the dielectric constant is also different in the two cases\({}^{25}\), as is seen in Fig. 13. Here the polar axis coincides with the \(z\)-axis (001). As was already indicated, the absence of a temperature maximum at the Curie point for the dielectric constant \(\varepsilon_{\parallel}\) (along the \(z\)-axis), and the difference between the values of \(\varepsilon_{\parallel}\) and \(\varepsilon_{\perp}\) above the Curie point, where according to X-ray data there is no anisotropy, seem completely inexplicable and are evidently connected with the considerable contamination of the crystal by impurities.

The curves presented have only a qualitative character, since, in addition to contamination, the division into regions had not been entirely eliminated in the crystal, and only most regions of the crystal had their polar axes oriented along the \(z\)-axis (001). In one of the works\({}^{26}\), for a crystal consisting of a single region, it was found that

\[ \varepsilon_{\perp}=2\cdot 10^{5},\quad \text{and}\quad \varepsilon_{\parallel}=3\cdot 10^{2}. \]

\( \varepsilon \) — Along the \(x\)-axis (100)

\( \varepsilon \) — Along the \(y\)-axis (010)

\( \varepsilon \) — Along the \(z\)-axis (001)

Fig. 13. Temperature dependence of the dielectric constant of a single crystal of \(\mathrm{BaTiO_3}\) for three crystallographic directions.

The anisotropy of the dielectric permittivity was demonstrated very clearly by the following experiment[^25]. At room temperature a small alternating field was applied to the crystal in a direction perpendicular to the polar axis, and the dielectric permittivity \(\varepsilon_{\perp}\) was measured (Fig. 14). Then, in the same direction, an additional constant field was applied. As this constant field was gradually increased, an ever larger number of regions changed the orientation of their polar axes (and, consequently, the direction of the spontaneous moment) by \(90^\circ\) into a position in which the polar axis becomes parallel to the applied field. This is facilitated, as is evident from Fig. 13, by the gradual decrease of the dielectric permittivity

Fig. 14. Change in the dielectric permittivity, measured in a small alternating field, with increasing strength of the constant field applied in a direction coinciding with the direction of the alternating field.

to the value \(\varepsilon_{\parallel}\) of about 600 (at field \(E = 0\)), which corresponds to a complete rotation of all regions by \(90^\circ\). If one then begins to reduce the strength of the constant field, and subsequently changes its direction, only the sign of the polar axis changes (it rotates by \(180^\circ\)), and not its angle with the field. If, in doing so, the entire cycle of field-strength variation is traversed, then the magnitudes of the polarization and of its dependence on the field give a hysteresis loop, while the dielectric permittivity measured in the small alternating field superposed on the constant field gives the dielectric permittivity at different points of the loop.

Very recently, work has been carried out[^27] on the dielectric anisotropy of a single crystal of barium titanate consisting of one region and, apparently, considerably purer. According to these data, the difference between the dielectric permittivities \(\varepsilon_{\perp}\) and \(\varepsilon_{\parallel}\) is considerably smaller than had been obtained earlier; their ratio-

Barium Titanate—A New Ferroelectric

ratio is approximately \(20:1\). Thus, for example, at \(20^\circ\mathrm{C}\), \(\varepsilon_{\perp} \simeq 4000\), while \(\varepsilon_{\parallel} \simeq 200\). The temperature dependence for both dielectric constants has a sharp maximum at the Curie point, and above the Curie point both dielectric constants \(\varepsilon_{\perp}\) and \(\varepsilon_{\parallel}\) have identical values. These data, therefore, agree much better with theoretical ideas and with data obtained by X-ray analysis.

c) Dielectric hysteresis and spontaneous polarization

The hysteresis loops can be demonstrated more clearly with the aid of the arrangement described above, which is commonly used for such observations. The hysteresis loops of a barium titanate single crystal have been investigated by this method by many authors. In some works these investigations were only qualitative in character, with the aim of showing the presence of hysteresis loops and the change in their form as a function of temperature—in particular, their disappearance at temperatures above the Curie point. In one such work\(^{16}\) it was possible to record several hysteresis loops for a single crystal consisting of a small number of regions. On the inclined branches of the loops, steps corresponding to the reorientation of one large region are distinctly visible, or even the entire loop is reduced to a single abrupt jump (Fig. 15).

Fig. 15. Hysteresis loops of a single crystal with a small number of regions (domains).

In the work of Hulm\(^{19}\), dielectric hysteresis in an alternating field of frequency \(50\) cycles was studied in order to measure the spontaneous polarization of barium titanate and its dependence on temperature. The spontaneous polarization was obtained by extrapolating the saturation branches of the hysteresis loops—when it may be assumed that all regions are oriented along the field—to a field value equal to zero. According to Hulm’s data, the spontaneous polarization at room temperature is, for a single crystal, \(16 \cdot 10^{-6}\ \mathrm{coul}/\mathrm{cm}^{2}\). According to other data\(^{25}\), it is about \(12 \cdot 10^{-6}\ \mathrm{coul}/\mathrm{cm}^{2}\). The difference is evidently determined by the different degree of purity of the crystal. With increasing temperature, the spontaneous polarization, according to Hulm’s data, changes hardly at all up to \(110^\circ\mathrm{C}\), and then begins to decrease with temperature. The temperature course of the spontaneous polarization near the point

Curie could not be obtained by this method with sufficient reliability, since in this work the nonlinearity of the induced polarization near the Curie point was not taken into account (see above). In connection with this, hysteresis occurs over a certain temperature interval even above the Curie point, and the author has to invoke an experimentally wholly unsupported notion of false hysteresis at temperatures above \(100^\circ \mathrm{C}\), due to the nonlinearity of the conductivity, in order to avoid the absurd conclusion that spontaneous polarization exists also above the Curie point.

g) Anomaly of the heat capacity near the Curie point

Measurements of the specific heat of a single crystal have recently been made \(^{28}\). The additional heat, calculated from the jump in heat capacity, was found to be

\[ Q=\int c_p\,dT=0.2\ \mathrm{cal/g}=47\ \mathrm{cal/mol}, \]

i.e., approximately five times greater than for a polycrystal. The authors treat their results from the standpoint of the theory of the internal field \(F=E+fP\) and calculate the Lorentz internal-field factor \(f\) from the relation

\[ Q=\frac{1}{2}\,f\,P_0^2, \]

where \(P_0\) is the spontaneous polarization, which must be determined by an independent method. Using Hülm’s results for the spontaneous polarization, the authors find for the internal-field coefficient the value \(f=0.044\), which agrees well with the value of the same coefficient calculated from the Curie–Weiss law for the dielectric permittivity near the Curie point:

\[ f=\frac{\Theta}{C}=0.049, \]

where \(\Theta\) is the Curie temperature \((393^\circ \mathrm{C})\) and \(C\) is the Curie constant \((8000^\circ \mathrm{C})\).

d) Refractive index and electrical conductivity

Proceeding from the fact that the dielectric constant \(\varepsilon\), measured at a certain frequency, must lie between the square of the refractive index and the static dielectric constant:

\[ n^2 < \varepsilon < \varepsilon_\infty, \]

and also from the fact that both \(\varepsilon\) and \(\varepsilon_\infty\) have sharp peaks near the Curie point, one might have expected a similar temperature dependence for the refractive index as well. Indeed, it turned out\({}^{29}\) that the dependence of the refractive index on temperature has a maximum at the Curie point (Fig. 16). On the other hand, one may expect that the activation energy \(E\), which determines the temperature dependence of the electrical conductivity according to the formula

\[ \sigma = A(T)\cdot e^{-\frac{E}{2kT}}, \]

should change at the Curie point, where a change in the type of lattice occurs.

The corresponding experimental curve appears to confirm these considerations (Fig. 17).

Fig. 17. Dependence of the logarithm of the electrical conductivity of a single crystal on reciprocal temperature.

According to the data of Eremeev and Kurchatov\({}^{23}\), the activation energy \(E\) below the Curie point is equal to \(1.75\ \mathrm{eV}\). Near the Curie point, however, the activation energy decreases to \(E = 1.15\ \mathrm{eV}\), and then increases again.

Analogous results were obtained at Dnepropetrovsk University by E. V. Sinyakov\({}^{30}\) on polycrystalline samples of \(\mathrm{BaTiO}_3\). However, since similar breaks in the curve of the logarithm of electrical conductivity as a function of reciprocal temperature were also obtained by him for samples of \(\mathrm{CaTiO}_3\), \(\mathrm{BeTiO}_3\), \(\mathrm{ZnTiO}_3\), and \(\mathrm{MgTiO}_3\), which have no phase transition and do not possess ferroelectric properties, the author associated the appearance of the break in the curve with the influence of impurities.

Apparently, in the case of a single crystal the amount of impurities is also rather significant, and their influence on the temperature dependence of the electrical conductivity requires additional study.

IV. PIEZOEFFECT OF BARIUM TITANATE

a) Piezoeffect of a single crystal

Despite the relatively large number of works devoted to the properties of \(\mathrm{BaTiO_3}\) single crystals, complete quantitative data on the piezoeffect of single crystals have not yet been published anywhere. The natural resonance frequency of a piezoelectrically excited plate made from a \(\mathrm{BaTiO_3}\) single crystal has two sharp minima at both lower maxima of the dielectric permittivity (\(+5\) and \(-70^\circ\mathrm{C}\))\(^{31}\), which, in the authors’ opinion, indicates the presence of a certain transition at these temperatures. On the other hand, on approaching the Curie point no decrease in the resonance frequency is observed, but at the Curie point itself it cannot be established at all, since, according to these studies, the crystal abruptly passes into a non-piezoelectric state\(^{25}\).

In attempts at a quantitative estimate of the piezoeffect of a single crystal, a serious obstacle is the division of the crystal into regions with different directions of spontaneous polarization, as a result of which the results cannot be unambiguous, but represent a superposition of the quadratic and linear piezoeffects\(^{25}\). According to the same work, the modulus of the linear piezoeffect in the direction of the polar axis is at least one order of magnitude larger than the anomalous modulus \(d_{14}\) of Rochelle salt.

b) Piezoeffect of polarized polycrystalline specimens

Polycrystalline specimens in the normal state have no piezoeffect, which is connected with the chaotic distribution of the polar axes of the individual crystallites in the specimen. The piezoeffect of polarized specimens was first discovered by the author of the present review in the investigation of dielectric hysteresis and the phenomena following it\(^{18}\).

In the first experiments the presence of the piezoeffect was established from maxima in the frequency dependence of losses and from the characteristic behavior of the capacitance near the natural resonance frequency of the specimens. Quantitative measurements of the piezoeffect in the dynamic regime were subsequently made directly from the frequency dependence of the current through the specimen.

In addition, measurements of the piezoeffect were also made in the static testing regime by measuring the charge arising on the specimen when it was compressed.

In these investigations it was shown\(^{32}\) that the piezomodulus of polarized specimens depends, both in magnitude and in stability with time, on the strength and duration of application of the polarizing field. The magnitude of the piezomodulus decreases after removal of the polarizing field rather rapidly in the first days, and then slowly.

BARIUM TITANATE—A NEW FERROELECTRIC

approaching a stable value amounting to 50 to 80% of the initial value, depending on the manufacturing technology and the composition of the specimens.

After this initial fall, the value of the piezomodulus remains unchanged, which was observed over a period exceeding one year. The value of the piezomodulus reaches saturation, and its dependence on time has only the character described after polarization by a field of strength 20–25 kV/cm for one hour.

c) Static tests of piezoelectric elements

The synthetic piezoelectric elements obtained by the method described above were also tested under static conditions. The static tests consisted in measuring the charge arising on the specimen electrodes during its mechanical compression. It was shown that the values of the piezomodulus of such specimens under compression in the direction of the preceding polarization and in the perpendicular direction differ from one another (in both cases the charge on the electrodes was measured in the direction of polarization). Namely, it was found that the piezomodulus in the direction of the preceding polarization (which may be called the modulus \(d_{11}\)) is equal, for specimens of one grade, to \(3.2 \cdot 10^{-6}\) electrostatic units, while the piezomodulus in the perpendicular direction (modulus \(d_{12}\)) is equal to \(1.4 \cdot 10^{-6}\) electrostatic units, i.e., more than two times smaller.

Fig. 18. Temperature behavior of the piezomodulus and dielectric permittivity, obtained by the static method.

An attempt was made to study the temperature behavior of the piezomodulus \(d_{11}\). The results obtained with one of the specimens are shown in Fig. 18 together with the curve of dielectric permittivity as a function of temperature.

c) Piezoelectric vibrations of bars

The piezoelectric properties in the dynamic regime can most simply be studied on long, thin, and narrow bars in which longitudinal compressional vibrations are excited. With a suitable choice of the dimensions and of the method of fastening the specimen, it is possible to avoid the excitation of other types of vibrations and their interaction with the one being studied, which could distort the results. To excite longitudinal vibrations the bars were clamped at the middle in a special knife-edge clamp.

As is known, the natural resonance frequency of such a bar is determined by the formula

\[ f_R=\frac{1}{2l}\sqrt{E/\rho}=\frac{1}{2l}v, \]

where \(l\) is the length of the bar, \(E\) is Young’s modulus, \(\rho\) is the density, and \(v\) is the velocity of propagation of mechanical vibrations. It follows from this that \(2f_Rl=v=\mathrm{const}\).

In our experiments it was found that the velocity of propagation of vibrations remains constant to within \(1\%\) when the length of the bars is varied from \(25\ \mathrm{mm}\) to \(5\ \mathrm{mm}\), and is equal to \(4.3\cdot10^5\ \mathrm{cm/sec}\). For a density \(\rho\) equal to \(6.2\ \mathrm{g/cm^3}\), Young’s modulus, according to the data obtained, is \(1.15\cdot10^{12}\ \mathrm{dyn/cm^2}\).

Since the measurements \(^{33}\) were carried out on specimens with electrodes, the value obtained corresponds to Young’s modulus in the absence of an electric field \(E_{\mathrm{e}}\).

The magnitude of the piezoelectric modulus can be calculated if the difference between the resonance and antiresonance frequencies is known, \(\Delta f=f_A-f_R\), as well as the dielectric constant \(\varepsilon\):

\[ d^2=\frac{\pi}{64\rho l^2}\cdot\frac{\Delta f}{f_R^3}\,\varepsilon . \]

The resonance and antiresonance frequencies were determined from the frequency dependence of the current through the specimen. The curves for one of the specimens are given in Fig. 19. For specimens prepared by different technological methods and of different composition, the magnitude of the piezomodulus for longitudinal vibrations \(d_{12}\) lay between \(2.5\cdot10^{-6}\) and \(1.3\cdot10^{-6}\) electrostatic units; moreover, the scatter of values for different specimens of the same type did not exceed \(10\%\). Measurements in the dynamic regime of the thickness effect (modulus \(d_{11}\)) gave results coinciding with the results of static measurements, i.e. \(d_{11}\simeq 2d_{12}\).

In addition to the longitudinal and thickness vibrations of bars, the radial vibrations of disks were subjected to preliminary investigation. In this case the piezomodulus has an intermediate value between \(d_{11}\) and \(d_{12}\), while the product of the resonance frequency by the radius remains constant.

From the results obtained it is seen that the material under investigation can be used successfully in piezoelectric transducers, sensitively

ness of which is determined by the piezomodulus \(d\). Such transducers are the piezotelephone and the piezoloudspeaker, which operate in the short-circuit regime. The piezomodulus of barium titanate \(d_{12}\) (and, even more so, \(d_{11} \simeq 2d_{12}\)) is of the same order as

Fig. 19. Current curve through the specimen as a function of frequency near the specimen’s natural resonance frequency.

the anomalous modulus \(d_{14}\) of Rochelle salt and is approximately 100 times greater than the modulus of quartz.

The quality factor of the piezoresonators studied depends strongly on the method of clamping. In the knife-edge clamps mentioned above, the quality factor of bars in longitudinal vibrations was about 300.

During the course of these investigations a number of works appeared devoted to the study of the piezoelectric properties of polarized specimens of polycrystalline barium titanate.\(^{34}\) The results coincide in their main features with those described above, apart from small numerical discrepancies due to certain differences in the manufacturing technology and in the degree of purity of the specimens. The only substantially new phenomenon mentioned in these investigations is the discovery of shear vibrations of specimens upon application of a high-frequency field in a direction perpendicular to the direction of the preceding polarization.

V. PRACTICAL USE OF THE ANOMALOUS PROPERTIES OF BARIUM TITANATE

The high dielectric constant of barium titanate, together with the relative simplicity and low cost of manufacturing ceramic capacitors from it, provides grounds for using it in compact blocking capacitors of large capacitance and, in general, in all those cases where a large capacitance is important and there are no strict requirements on its temperature stability.

Considerably more interesting is the use of barium titanate as a nonlinear element. The beginning of such an application of barium titanate was laid by Prof. V. P. Vologdin2, who constructed a frequency multiplier with its aid. The principle of operation of this multiplier consists in the fact that, owing to the nonlinearity of the dependence of the capacitance of a barium titanate capacitor on voltage, when a sinusoidal voltage is applied to it the current will contain higher harmonics, which can be separated out.

Since the nonlinearity of the dependence of capacitance on voltage is preserved up to very high frequencies ($\sim 10^7$ cycles), such a method of frequency multiplication can be applied in radio engineering.

The dependence of the capacitance of a barium titanate capacitor at high frequency on a simultaneously applied constant bias voltage (“reversible dielectric constant,” see above) makes it possible to use these capacitors for variable-frequency generators (television) and for frequency modulation.

The large piezoelectric effect of polarized barium titanate specimens, with its good stability and independence of temperature over a wide temperature range, as well as their great mechanical strength, give grounds for supposing that such piezoelectric elements will find wide application in the very near future.

In addition to the positive qualities of synthetic piezoelectric elements indicated above, they also possess a whole series of advantages over natural piezocrystals. Among these must be included the fact that piezoelectric oscillations of various types and with various moduli are obtained here not by different oriented cutting of a piezoresonator from a piece of crystal, but much more simply—by different directions of application of the alternating voltage relative to the direction of polarization by a constant voltage. In this case it is easy to obtain piezoresonators of such types of oscillations as cannot at all be made from natural crystals (for example, disks, cylinders, spheres, and parts of a sphere with radial oscillations).

Piezoelectric elements made of barium titanate can be successfully used for making a piezotelephone and a piezoloudspeaker, a pressure indicator, a piezo-sensor, etc.

From the far from complete list of possible applications of barium titanate given above, it follows that it should find wide application.

in the most diverse branches of electronic engineering and radio engineering. It seems to us that our branch institutes should widely expand work on designing various specific types of instruments and equipment based on the use of the anomalous properties of barium titanate, first discovered and investigated in our country.

References

1. I. V. Kurchatov, Ferroelectrics, GTTI (1933).
2. G. Busch, Helv. Phys. Acta 11, 269 (1938).
3. B. M. Vul and I. M. Goldman, DAN 46, 154 (1945).
4. B. M. Vul and L. F. Vereshchagin, DAN 48, 662 (1945).
5. B. M. Vul, Elektrichestvo 3, 12 (1946).
6. B. M. Vul and I. M. Goldman, DAN 60, 41 (1948).
7. H. D. Megaw, Proc. Phys. Soc. 58, 133 (1946).
8. B. Matthias, Phys. Rev. 73, 808 (1948).
9. H. Kay and R. Rhodes, Nature 160, 126 (1947).
10. H. Blattner, H. Grünicher, W. König, W. Merz, Helv. Phys. Acta 21, 341 (1948).
11. B. M. Vul and I. M. Goldman, DAN 64, 179 (1945).
12. P. Novosiltsev and A. Khodakov, ZhTF 17, 651 (1947).
13. D. I. Mash, ZhETF 17, 537 (1947).
14. J. Powles, Nature 162, 614 (1948).
15. B. M. Vul and I. M. Goldman, DAN 57, 21 (1946).
16. A. Bretteville, J. of Amer. Ceramic Soc. 29, 303 (1946); Phys. Rev. 73, 807 (1948).
17. S. Roberts, Phys. Rev. 71, 891 (1947).
18. A. Rzhanov, ZhETF, in press.
19. J. Hulm, Nature 160, 127 (1947).
20. P. P. Kobeko, ZhETF 1, 228 (1931).
21. A. Hippel, R. Breckenridge, Tisza, Laszlo, Ind. and Eng. Chem. 82, No. 11 (1946).
22. M. Harwood, O. Popper and Rushman, Nature 160, 58 (1947).
23. M. Eremeev and B. Kurchatov, ZhETF 2, 329 (1932).
24. J. Powles, Nature 162, 655 (1948).
25. B. Matthias and A. Hippel, Phys. Rev. 73, 1378 (1948).
26. G. Danielson, B. Matthias, J. Richardson, Phys. Rev. 74, 986 (1948).
27. W. Merz, Phys. Rev. 75, 687 (1949).
28. H. Blattner, W. Merz, Helv. Phys. Acta 21, 210 (1948).
29. G. Busch, H. Flury, W. Merz, Helv. Phys. Acta 21, 212 (1948).
30. E. V. Sinyakov, Dissertation, Dnepropetrovsk State University, 1948.
31. H. Blattner, B. Matthias, W. Merz, P. Sherrer, Experientia 3, 148 (1947).
32. A. Rzhanov, ZhETF.
33. A. Kharkevich, Theory of Transducers, Gosenergoizdat, 1948.
34. W. Mason, Phys. Rev. 74, 1134 (1948).
35. V. P. Vologdin, Elektrichestvo 8, 28 (1946).

1. UFN, Vol. XXXVIII, No. 4. ↩

2. 35. ↩