Nonlinear Properties of Ferroelectric Ceramics

T. N. Verbitskaya

Ferroelectrics are distinguished by a number of remarkable properties: high and ultrahigh values of dielectric permittivity, a substantial dependence of it on temperature and field strength, and the piezoelectric effect.

The distinctive features of ferroelectrics ensure their broad and varied application in technology. High values of $\varepsilon$ have led to the development and manufacture of low-voltage and high-voltage ferroelectric-ceramic small-sized capacitors.

The piezoelectric properties of ferroelectrics ensure their broad use in various acoustic equipment. Ceramic piezoelectrics are used as emitters and receivers of ultrasonic oscillations, vibration transducers, piezometers, and for other purposes.

Recently, great interest has been aroused by the nonlinear properties of ferroelectrics, manifested in the fact that, in the ferroelectric region, the polarization of the dielectric (and with it the dielectric permittivity) are nonlinear functions of the field strength. As the strength of the alternating field is increased, the polarization of the ferroelectric increases and reaches saturation; the dielectric permittivity increases, reaches a maximum, and then decreases.

The action of an electric field on ferroelectrics gives rise to processes analogous to those occurring under the action of a magnetic field on ferromagnets: the phenomenon of saturation and hysteresis appears. The curve of the dependence of the polarization of ferroelectrics on the electric-field strength has the same character as the curve of the dependence of magnetic induction on magnetic-field strength. This curve cannot be expressed even approximately by a linear equation. The physical nature of the nonlinear properties of ferroelectrics and magnetic materials is different; nevertheless, both magnetic materials and ferroelectrics can be used to replace electron tubes in electrical and radio-engineering circuits.

The degree of nonlinearity of different ferroelectrics is not the same. For some materials the nonlinearity appears noticeably in the region of weak fields; for others, in the region of sufficiently high field strengths.

The degree of nonlinearity of ferroelectrics is determined by the value of the alternating-field strength $E_{\text{max}}$ at which a change in the slope of the curve $P = f(E_{\sim})$ is observed, and the dielectric permittivity reaches its maximum value $\varepsilon_{\text{max}}$, and by the nonlinearity coefficient $K$, showing how many times the value of the dielectric permittivity increases from the initial value $\varepsilon_{\text{init}}$ to the maximum:

$$ K = \frac{\varepsilon_{\text{max}}}{\varepsilon_{\text{init}}} $$

For some ferroelectrics the value of the field strength $E_{\text{max}}$ is small and does not exceed values of several hundred volts per centimeter. For others, considerably higher fields are required in order for the dielectric permittivity to reach its maximum value.

For some materials, \(\varepsilon\) reaches a maximum in the region of weak fields, but the change in dielectric permittivity from the initial to the maximum value is small. The nonlinearity of such ferroelectrics is slight. Only the combination of a large value of the nonlinearity coefficient \(K\) and a small value of the field \(E_{\max}\) characterizes a ferroelectric as a material with high nonlinear properties.

In addition to the above, the nonlinearity of ferroelectrics is also determined by the slope of the reversible characteristic of the dielectric permittivity \(E = f(E_=)\). By reversible is meant, as is known, the dielectric permittivity measured in a weak alternating field with a simultaneously applied constant bias voltage. As the bias voltage is increased, all ferroelectrics show a decrease in \(\varepsilon\). The slope of the reversible characteristic for materials with high nonlinearity is considerable; the change in reversible \(\varepsilon\) (with an appropriate choice of alternating and constant voltage) may amount to several times. For materials with weakly expressed nonlinear properties, the reversible \(\varepsilon\) changes little.

Of all ferroelectrics known at present, the highest nonlinear properties in the region of weak fields are exhibited by Rochelle salt \([1—2]\). The dielectric permittivity of Rochelle salt changes by more than a factor of 20 when the field strength is varied from a few to hundreds of volts per centimeter; \(\varepsilon_{\max}\), at a field strength of about \(100\ \text{V/cm}\), reaches a value close to 20000. However, the ferroelectric properties of Rochelle salt, and with them its high nonlinearity, are preserved within a comparatively narrow temperature interval: from \(-15\) to \(+22.5^\circ\text{C}\), between the upper and lower Curie points. Rochelle salt has low mechanical strength and a low decomposition temperature, equal to \(+55^\circ\text{C}\). The nonlinear properties of Rochelle salt depend very sharply even on insignificant mechanical influences. Crystals of Rochelle salt are brittle and readily soluble in water. All this sharply limits the use of Rochelle salt as nonlinear elements, despite its high nonlinear properties.

Ceramic ferroelectrics are of great importance for technology. They are characterized by high mechanical and electrical strength. Their ferroelectric properties appear over a wide temperature interval. The compositions of ceramic ferroelectrics are varied, and along with this their properties are extremely diverse. For example, ferroelectrics have now been obtained whose Curie point lies at any temperature within the interval from \(+500\) to \(-200^\circ\text{C}\) \([3, 4]\). Many ceramic materials have been developed whose ferroelectric properties are retained in the working temperature interval from \(-60\) to \(+60^\circ\text{C}\). Ceramic ferroelectrics are distinguished both by a sharp dependence of \(\varepsilon\) on temperature, with the presence of a sharp maximum, and by a smoothed dependence. Their nonlinearity, at one and the same field strength, manifests itself differently.

The technology for manufacturing ceramic ferroelectrics is sufficiently simple; nonlinear elements made from them can be produced in the most varied shapes and sizes. They are reliable in operation, are distinguished by stability of properties, resistance to the action of moisture, the atmosphere, and various kinds of chemical reagents. They permit overheating and overload, withstand a long service life, and are resistant to the action of vibration and shocks. Ceramic elements have small weight and dimensions and can serve for creating compact apparatus; they are distinguished by variety of design and low cost.

Among ceramic ferroelectrics, barium titanate, the materials T-1700¹), T-7500, and radicond VK-1 have acquired the greatest importance in engineering. The technology of their mass production has been mastered; their characteristics are of the greatest interest.

Nonlinear properties of ferroelectrics in the region of normal temperatures

Barium titanate. The first of the ceramic materials discovered in the Soviet Union that was distinguished by ferroelectric properties was, as is known, barium titanate. According to the data of B. M. Vul and I. M. Goldman [5], the dielectric permittivity of barium titanate depends on the strength of the alternating field. As the latter is increased, \(\varepsilon\) rises smoothly and reaches a maximum value at a sufficiently high field strength, about \(8\text{—}10\ \mathrm{kV/cm}\). From our measurements it follows that the dielectric permittivity of barium titanate reaches a maximum at a field strength of about \(6\ \mathrm{kV/cm}\). Our specimens had a Curie point at \(120^\circ\mathrm{C}\) and were not identical to the specimens of M. B. Vul and I. M. Goldman.

Comparing the dependence \(\varepsilon=f(E_{\sim})\) in the region of weak fields for Rochelle salt and barium titanate, it is easy to see that the change of dielectric permittivity with field for Rochelle salt is manifested much more sharply than for barium titanate (Fig. 1).

The field \(E_{\max}\) for barium titanate is many times greater than \(E_{\max}\) for Rochelle salt. The dielectric permittivity of barium titanate in the range of field strengths from \(10\ \mathrm{V/mm}\) to \(E_{\max}\) changes by no more than a factor of 4–5. The magnitude of the field strength \(E_{\max}\) for barium titanate is sufficiently high that it makes difficult the use of this dielectric in circuits operating with the voltage switched on for a long time.

The reversible dielectric permittivity of barium titanate in the region of a weak alternating field (several hundred volts per centimeter) changes little when the constant biasing field is increased from zero to \(3 \div 4\ \mathrm{kV/cm}\) (Figs. 1a and b). Since the slope of the characteristic \(\varepsilon=f(E_{\sim})\) for barium titanate is small, increasing the alternating voltage to \(1\text{—}2\ \mathrm{kV/cm}\) does not lead to a noticeable increase in the steepness of the reversible dependence (Fig. 1b). A further increase of the constant voltage from \(4\ \mathrm{kV/cm}\) and above likewise does not lead to a noticeable change in the reversible \(\varepsilon\). Only when the magnitude of the alternating-field strength is chosen close to \(E_{\max}\) does the slope of the reversible characteristic become appreciable; the decrease of \(\varepsilon\) with increasing constant field reaches a factor of 4–5. However, the value of \(E_{\max}\) for barium titanate is \(6 \div 8\ \mathrm{kV/cm}\), which greatly hampers the use of nonlinear elements made of this material.

A substantial advantage of barium titanate over Rochelle salt is the preservation of nonlinear properties over a wide temperature interval from \(+120^\circ\mathrm{C}\) down to very low negative temperatures, its high mechanical properties and the associated resistance to the action of vibrations, shaking, and mechanical shocks, and, finally, ease of assembly. Vologdin proposed using first Rochelle salt and then barium titanate for frequency multiplication [6]. Demichev and Fogel showed the possibility of constructing a generator of audio and ultrasonic frequencies using barium-titanate elements; they realized a static frequency converter with a power of up to several kilowatts (with water cooling) [7].

¹) In nonlinear properties the material T-1700 practically does not differ from barium titanate; therefore we shall not dwell on it specially. In technological properties it is in many respects superior to barium titanate and is widely used for manufacturing small-size capacitors and piezoelements.

Scanavi and Neiman developed a voltage stabilizer with barium titanate capacitors [8].

Various applications of ferroelectric ceramics in electrical apparatus were considered by D. M. Kazarnovskii [9, 10]; he also carried out an analytical calculation of circuits with nonlinear capacitors [9, 10]. In all cases of using barium titanate as a nonlinear element

Fig. 1. Dependence of the dielectric permittivity of barium-titanate ferroelectrics, material T-7500, and varikond VK-1 on a—alternating and b—direct voltage.

I BaTiO$_3$; II T-7500; III VK-1.

it is necessary to use appreciable voltages in electrical and radio circuits. Practice poses the problem of finding new ceramic ferroelectrics whose nonlinear properties would manifest themselves to a considerably sharper degree in the region of weak fields. This is especially important when nonlinear elements are used in radio-engineering circuits.

Material T-7500. The name of the ferroelectric ceramic material T-7500, like that of a number of other ceramic materials, corresponds to the value of the dielectric permittivity. For material T-7500 the dielectric permittivity is 7500 at the Curie point (35–40°C). Below this temperature, material T-7500, down to low negative temperatures, retains

ferroelectric properties. In the region of room temperatures the dielectric permittivity reaches values of 4000–5000. At temperatures 20–30°C above the Curie point, the ferroelectric properties of material T-7500 are not manifested. The temperature dependence of the dielectric permittivity of material T-7500 and of other ferroelectrics is shown in Fig. 2.

Material T-7500 is used for the manufacture of various small-size ceramic capacitors, which are employed in cases where constancy of capacitance is not required—for example, when only the lower limit of the capacitance value of the capacitor is specified. From material T-7500 are made low-voltage capacitors in the form of thin-walled tubes and disks of small thickness, and high-voltage barrel capacitors.

Fig. 2. Temperature dependence of $\varepsilon$ of ferroelectrics in the region of weak fields.

1. Rochelle salt. 2. Barium titanate.
2. T—7500. 4. Varikond VK-1.

The nonlinear properties of material T-7500 at room temperature are manifested more sharply than in barium titanate; the dielectric permittivity reaches its maximum value at $E_{\max}=4\div5\ \text{kV/cm}$. However, the ratio of the greatest value $\varepsilon_{\max}$ to the initial value, measured at $E=10\ \text{V/mm}$ for material T-7500 under normal conditions, does not exceed 2–2.5. This is apparently connected with the fact that the Curie point of material T-7500 is close to room temperature and, along with this, the initial value of the dielectric permittivity, even in the region of weak fields, is rather high, about 3000–5000. The limiting values of $\varepsilon$, both owing to an increase of temperature up to the Curie point and owing to an increase of the field strength to $E_{\max}$, do not exceed 7000–8000 (these high values of dielectric permittivity are ensured only when pure raw materials are used).

The aging phenomenon, sharply manifested in T-7500 ferroelectric ceramics [11, 12], leads to instability of the dielectric-permittivity values of this material with time. The nonlinear properties of material T-7500 also do not remain constant with time. For an aged specimen the value of $E_{\max}$ is considerably greater than for an unaged one. The dielectric permittivity at the maximum in the first case is lower than in the second. The curves of the dependence of the reversible dielectric permittivity on the constant biasing field, $\varepsilon=f(E_=)$, for unaged and aged specimens are different—Fig. 1 в and 1 г. With an increase of the constant

of the biasing field in the first case the dielectric permittivity decreases noticeably, beginning with very small fields. In the second case a maximum appears in the dependence of $\varepsilon$ on the constant field.

The comparatively low nonlinearity of material T-7500 and the noticeable change of dielectric permittivity with time—aging—limit the use of this material for making nonlinear elements. At the same time, there arises the need to develop new ferroelectric-ceramic materials distinguished by stable, high nonlinear properties.

Varicond VK-1. The nonlinear properties of ferroelectrics are determined by the composition and structure of the crystal lattice of the material. As a result of our studies of the influence of composition on the electrical properties of ferroelectrics, a material with sharply expressed nonlinear properties was obtained [13]. This material was named varicond VK-1. Owing to its high nonlinear properties, varicond VK-1 is intended for the manufacture of nonlinear elements whose capacitance does not remain constant when the field strength changes.

Fig. 3. Dependence of the dielectric permittivity of ferroelectrics on the alternating-field strength.

1. Rochelle salt. 2. $\mathrm{BaTiO_3}$ according to the data of M. B. Vul. 3. T-7500. 4. Varicond VK-1.

In Fig. 3 the dependence of dielectric permittivity on the alternating-field strength is presented for varicond VK-1 and several other ferroelectrics.

The dielectric permittivity of varicond VK-1 changes sharply in the region of weak alternating fields and, at a field strength of about $700 \div 1000\ \mathrm{V/cm}$, reaches a maximum value. In its form the curve $\varepsilon = f(E_{\sim})$ of varicond VK-1 is very similar to the curve for Rochelle salt and differs noticeably from the analogous dependences for material T-7500 and barium titanate.

The hysteresis loops of the dependence of the charge on the plates of capacitors made of various ferroelectric-ceramic materials on the field strength also confirm the high nonlinear properties of varicond VK-1. If for barium titanate and material T-7500 only a barely perceptible deviation from linearity in the dependence of charge on field strength is outlined at $E = 1500\ \mathrm{V/cm}$, then for varicond VK-1 a saturation effect is already revealed rather clearly.

When the alternating-field strength is increased from the initial value, equal to $10\ \mathrm{V/mm}$, to $E_{\max}$, the dielectric permittivity of varicond VK-1 increases by more than 6 times and reaches a maximum value of 15000–20000.

In the serial production of nonlinear elements from varicond, when the ceramic batch is manufactured in large quantity, the initial and maximum values of the dielectric permittivity decrease; however, the change of $\varepsilon$ from the initial value to the maximum remains sufficiently high

\[ \left(\frac{\varepsilon_{\max}}{\varepsilon_{\text{init}}}\ \text{not less than } 4 \div 5\right). \]

The reversal characteristic of a ferroelectric, \(\varepsilon = f(E_=)\), is extremely important in choosing a material for making nonlinear elements of a dielectric amplifier, for tuning the frequency of a circuit, and for other applications. In the case of ferroelectrics whose \(\varepsilon\) depends only weakly on the magnitude of the alternating voltage—for example, barium titanate—the effect of the magnitude of the alternating voltage on the slope of the reversal characteristics is insignificant. For the varikond VK-1, the form of the dependence of the dielectric permittivity on the constant biasing field is determined to a very large extent by the magnitude of the alternating voltage. The greatest steepness of the reversal characteristic is observed when the alternating voltage corresponds to \(E_{\text{max}}\), which corresponds to the maximum value of the dielectric permittivity \(\varepsilon_{\text{max}}\) on the curve \(\varepsilon = f(E_{\sim})\). At lower, as well as at higher, fields, the steepness of the reversal characteristic decreases. Effective regulation of the slope of the reversal dependence by the action of the alternating voltage became possible only after the creation of ferroelectrics with a sharply expressed dependence of the dielectric permittivity on the alternating voltage.

In Figs. 1-d and 1-v, graphs are presented of the dependences of the dielectric permittivity on the intensity of the alternating and constant biasing fields for the varikond VK-1. A comparison of these dependences quite clearly emphasizes the high nonlinearity of the characteristics of the varikond VK-1. For barium titanate, in the region of alternating fields up to \(1\ \text{kV/cm}\), when the constant biasing field is increased from zero to \(5 \div 6\ \text{kV/cm}\), the reversal dielectric permittivity decreases insignificantly, by \(3 \div 15\%\). For freshly prepared specimens of the T-7500 material under the same conditions, the reversal dielectric permittivity may decrease threefold; however, the slope of the reversal characteristics does not remain constant with time, and in the process of aging the character of the change of the reversal \(\varepsilon\) changes: the curve \(\varepsilon = f(E_=)\) passes through a blurred maximum (Fig. 1-g).

The varikond VK-1 exhibits a considerably stronger change of the reversal dielectric permittivity in the region of weak alternating fields. At an alternating-field intensity corresponding to \(E_{\text{max}}\), the reversal dielectric permittivity of the varikond decreases by a factor of \(4 \div 5\). In this case the magnitude of the constant biasing field changes from zero to \(2 \div 3\ \text{kV/cm}\). A further increase of the constant field does not cause noticeable additional changes in the dielectric permittivity. Increasing the alternating field above the value \(E_{\text{max}}\), or decreasing it below this value, leads to a reduction in the slope of the reversal characteristic.

If nonlinear elements are made in the form of thin-walled disks or tubes with a thickness of \(0.1 \div 0.5\ \text{mm}\), then the alternating-field voltage ensuring a sufficiently high steepness of the reversal characteristic will amount to tens and even units of volts.

Nonlinear properties of VK-1 varikonds at different temperatures

The use of nonlinear elements in a number of devices is possible if their nonlinear properties are retained over a wide temperature interval.

It is known that the dielectric permittivity of ferroelectrics changes sharply when the temperature changes, especially near the Curie point. These changes are especially significant when a weak alternating field is applied to the ferroelectric (Fig. 3). In the region of a strong alternating field

a different picture is found in the temperature dependence of the dielectric constant (Figs. 4, 5, and 6).

By a strong field, for each ferroelectric we shall mean values equal to and greater than \(E_{\max}\).

For Rochelle salt, instead of the usual two maxima appearing in weak fields near the Curie points (Fig. 4—1), in the temperature dependence of \(\varepsilon\) in a strong field, at a field strength \(E=50\ \mathrm{V/mm}\), one broadened maximum is observed (Fig. 4—2). For Rochelle salt \(E_{\max}\) is close to \(10\ \mathrm{V/mm}\). This broadened maximum covers the entire temperature interval from \(-15\) to \(+22.5^\circ\mathrm{C}\), corresponding to the ferroelectric region. The value of the dielectric constant within the indicated temperature interval remains high, about 3500. The data for Rochelle salt are taken from [2].

Fig. 4. Temperature dependence of the dielectric constant of Rochelle salt (1 and 2) and barium titanate (3 and 4) at different field strengths.

\(1\)—\(E_{\sim}=0.5\ \mathrm{V/mm}\);
\(2\)—\(E_{\sim}=50\ \mathrm{V/mm}\);

\(3\)—\(E_{\sim}=5\ \mathrm{V/mm}\);
\(4\)—\(E_{\sim}=600\ \mathrm{V/mm}\).

In the case of barium titanate, the temperature characteristic is also smoothed in strong fields. In a weak field (\(E=10\ \mathrm{V/mm}\)) the curve of the temperature dependence of the dielectric constant of barium titanate passes through three maxima at temperatures corresponding to the three phase transitions of the crystal lattice. The value of the dielectric constant in the region of weak fields and negative temperatures does not exceed several hundred, whereas at the point of the principal phase transition, at \(120^\circ\mathrm{C}\), it reaches several thousand. In strong fields (for barium titanate, at \(600\div800\ \mathrm{V/mm}\)) the values \(\varepsilon_{\max}\) at the temperatures of the second and third phase transitions of barium titanate become comparable with \(\varepsilon_{\max}\) at \(120^\circ\mathrm{C}\) (Fig. 4, curve 3). Over the entire investigated temperature interval from \(-180^\circ\mathrm{C}\) to \(+120^\circ\mathrm{C}\), at a field strength \(E=600\ \mathrm{V/mm}\), \(\varepsilon\) of barium titanate retains high values. The deviation of any value of \(\varepsilon\) in the temperature interval \(+80 \div -180^\circ\mathrm{C}\) from the mean, equal to 5000, does not exceed \(\pm 20\%\).

An especially sharp difference in the character of the temperature dependence of the dielectric constant in weak and strong fields is shown by varicond VK-1 (Fig. 5). In the region of weak fields, about \(5\ \mathrm{V/mm}\), with increasing temperature from \(-180^\circ\mathrm{C}\) to \(+80^\circ\mathrm{C}\), a continuous increase in the dielectric constant is observed (Fig. 5—1). The value of \(\varepsilon\) increases in this case by several tens of times. The Curie point of varicond VK-1 corresponds to \(80^\circ\mathrm{C}\); at this temperature \(\varepsilon\) reaches its maximum val-

decreases upon further heating. With an increase in the field strength, the dielectric constant retains high values over a broad temperature interval. The temperature dependence of $\varepsilon$ is smoothed out.

Fig. 5. Temperature dependence of the dielectric constant of VK-1 at various field strengths with a frequency of 50 Hz.

1. $E_{\sim}=5$ V/mm;  2. $E_{\sim}=50$ V/mm;
2. $E_{\sim}=160$ V/mm; 4. $E_{\sim}=260$ V/mm.

In the region of strong fields, around 200 V/mm, $\varepsilon$ of the varicond changes insignificantly in the temperature interval from $-60$ to $+60^\circ$C (Fig. 6, curve 2). The average value of the dielectric constant is close to 16000; any intermediate value differs by no more than $\pm 20\%$ from the average.

From consideration of the dependences presented in Figs. 5 and 6, it may be concluded that, over a broad temperature interval, varicond VK-1 retains ferroelectric properties. We established that, beginning at the Curie point and down to a temperature of $-195^\circ$C, the varicond is a ferroelectric with sharply pronounced nonlinear properties. Fig. 7a presents the dependence $\varepsilon=f(E_{\sim})$ for the varicond, measured at three temperatures: 60, 20, and $-60^\circ$C. In all the cases considered, the dielectric constant depends strongly on the field strength, and $E_{\max}$ does not exceed 250 V/mm.

The presence of high nonlinear properties of varicond VK-1 over a broad temperature interval makes it possible to use it as nonlinear elements in devices whose operating temperature varies over a wide—

within certain limits. When a definite field strength of the alternating field is chosen (for the varicond, about \(200\ \mathrm{V/mm}\)), reversible characteristics can be obtained over a wide temperature interval (from \(-60\) to \(+60^\circ\mathrm{C}\)) that differ comparatively little from one another (Fig. 7-b).

Fig. 6. Temperature dependence of the dielectric permittivity of varicond VK-1 at different field strengths.

1. \(E = 10\ \mathrm{V/mm}\).
2. \(E = 200\ \mathrm{V/mm}\).

In the case where the magnitude of the alternating field is less than \(E_{\max}\), for example, \(E = 40\ \mathrm{V/mm}\), the dielectric permittivity has different values at negative and positive temperatures; at the same time, the slope of the reversible characteristics of the dielectric permittivity, measured at \(+60\), \(+20\), and \(-60^\circ\mathrm{C}\), differs sharply (Fig. 7-b).

Fig. 7. Dependence of the dielectric permittivity of varicond VK-1 on field strength:

a) alternating field (\(\sim\));
b) constant biasing field at different temperatures (\(=\))

1. — \(t = 60^\circ\mathrm{C}\); 2. — \(t = +20^\circ\mathrm{C}\); 3. — \(t = -60^\circ\mathrm{C}\).

If the alternating field is chosen sufficiently large, for example, \(E = 120\ \mathrm{V/mm}\) or \(260\ \mathrm{V/mm}\), the values of the dielectric permittivity at temperatures \(+60\), \(+20\), and \(-60^\circ\) differ only slightly from one another; the slope of the reversible dependences \(\varepsilon = f(E)\) for VK-1 for the three indicated temperatures is also approximately the same (Fig. 7-b).

Thus, at an alternating-field strength of 120 V/mm and higher, nonlinear elements made of varikond VK-1 can yield reversible dependences of \(\varepsilon\) that change little over a wide temperature range.

The high nonlinearity of the varikond over a wide temperature range, combined with high mechanical strength, will ensure broad application of this material in technology.

Features of the Crystal Structure of Varikond VK-1

Barium titanate and varikond VK-1 are fairly close to one another in a number of physical characteristics. Both materials are prepared by the ceramic-technology method and are polycrystals. The crystal sizes are about 6–15 \(\mu\)m. Their density is about 6 g/cm\(^3\), and their mechanical strength is not less than 1000 kg/cm\(^2\). Barium titanate and varikond VK-1 are ferroelectrics over a wide temperature range; the Curie point of the former is 120°C, and of the latter 80°C. The dependences of \(\varepsilon\) on temperature and field strength for both ferroelectrics are analogous. At the same time, the degree of their nonlinearity is different. The nonlinear properties of ferroelectrics can be related to their crystal structure.

Earlier X-ray studies established that polycrystalline barium titanate, possessing ferroelectric properties, has a perovskite-type structure.

When the temperature changes, the crystal lattice of barium titanate undergoes changes. Starting at temperatures above the Curie point, the barium titanate lattice passes three times from one modification to another: from cubic to tetragonal, then to orthorhombic, and finally to rhombohedral. These transitions occur at the same temperature values at which the dielectric permittivity exhibits maxima. The first maximum occurs at the Curie-point temperature, 120°C, and is clearly manifested in the region of weak fields. The second and third maxima of \(\varepsilon\), at temperatures around zero and \(-80^\circ\)C, are clearly observed at elevated field strength.

Above the Curie point, barium titanate has an ideal cubic lattice. In the temperature interval from 120°C to zero, the lattice is tetragonal; at 20°C the ratio of the axes is

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

In composition, varikond VK-1 differs little from barium titanate (it consists mainly of barium titanate and a small amount of additives). The introduction of additives leads to a shift of the Curie point of the varikond to 80°C and to a significant change in the temperature dependence of the dielectric permittivity. For the varikond, much more sharply than in barium titanate, the second and third maxima appear in the region of a comparatively weak field in the curve \(\varepsilon = f(E_{\sim})\). If for barium titanate the transition points are separated from one another by almost 100°, then for the varikond the maxima are brought closer together and appear at temperatures of about \(+80\), \(+30\), and \(-30^\circ\)C. X-ray structural studies of varikond VK-1 and barium titanate were carried out over a wide temperature interval from 20 to 200°C.\(^1\) A certain difference is observed between the structures of barium titanate and the varikond (Fig. 8). The cubic lattice of barium titanate is retained up to the Curie point, 120°C, and that of the varikond up to 80°C. At temperatures above 120°C, the cubic cell of the varikond is somewhat larger than the cell of barium titanate; the cell parameters are as follows:

\(^1\) The X-ray investigations were carried out at the Research Institute by V. G. Prokhvatilov and E. I. Gindin. Debyegrams were taken in a chamber 114.6 mm in diameter by the asymmetric method, using copper radiation from an X-ray tube at a voltage of 30 kV and a current of 10 mA.

Below the Curie point the barium titanate lattice becomes tetragonal, with a gradually changing axial ratio. As can be seen from Fig. 8, our data for barium titanate agree well with Megaw’s data.

Fig. 8. Change in cell dimensions with temperature for BaTiO₃ and varicond VK-1.

1—1 and 2—2 according to Megaw’s data,

3—3 and 4—4 according to the data of V. G. Prokhvatilov and E. I. Gindin.

I — cubic phase.

II — tetragonal phase.

Samples of varicond VK-1 below the Curie point have a tetragonal structure of the perovskite type. The X-ray diffraction patterns were taken at temperatures of \(+20\), \(+37\), and \(+50^\circ\mathrm{C}\). The parameters of the crystal lattice at temperatures below the Curie point could be determined only on samples whose firing had been carried out at a temperature \(20 \div 30^\circ\) below the sintering temperature. For densely sintered samples, in X-ray diffraction patterns taken below the Curie point, the lines at large angles were strongly blurred and the parameters were not determined.

A specially carried out series of firings of the material with prolonged holding at the final firing temperature (from 2 to 24 hours) and an increase in the exposure time when taking the X-ray diffraction pattern did not lead to an increase in the sharpness of the lines on the X-ray diffraction patterns. Therefore it was necessary to abandon the supposition that the varicond samples had not been sufficiently sintered and that their crystal lattice was insufficiently formed.

The blurring of the lines on the X-ray diffraction patterns at room temperature and in the interval from \(20^\circ\mathrm{C}\) to the Curie point can, in our opinion, be explained as follows. The lattice of varicond VK-1 near room temperature has an unstable crystalline structure. Slight transitions from one modification to another are possible in it. A comparatively small external action is sufficient to change the lattice parameters and its structure. Probably, just as in barium titanate, the varicond lattice undergoes three crystallographic changes when the temperature is varied. For barium titanate these changes occur at the same temperatures at which the dielectric constant passes through maxima. For varicond VK-1, \(\varepsilon\) passes through maxima at temperatures of about \(+80\), \(+30\), and \(-30^\circ\mathrm{C}\). It may be considered that at these same temperatures the varicond cell also undergoes changes. In this case the difference in energy of the two modification states for the varicond is much…

smaller than for barium titanate. In the temperature interval from \(+20^\circ\) to \(+80^\circ\mathrm{C}\), the dielectric permittivity of varikond passes twice through a peak, and, apparently, the lattice structure changes twice. Energetically these two lattices differ little from one another. In the roentgenograms in the indicated temperature interval one might have expected the presence of lines corresponding to two structures. If one assumes that the structures of these lattices are sufficiently close, then the appearance of broadened lines in the roentgenograms becomes understandable.

It may further be supposed that, if in a substance, upon a change in temperature, the transition of the structure from one modification to another occurs rather easily, then the action of an external electric field may also cause a change in the dimensions of the cell or its rearrangement. For varikond VK—1, changes in the lattice under the action of temperature occur more readily than for barium titanate. It may be considered that the action of an electric field will cause greater changes in the varikond lattice than in that of barium titanate. It was indicated above that the form of the curve of the temperature dependence of \(\varepsilon\) changes to a considerably greater degree for varikond than for barium titanate for one and the same change in the electric-field strength. The abrupt change in the dielectric permittivity of varikond with field strength can also be associated with a change in the structure of the substance. For a varikond specimen, a lower field must be applied than for barium titanate in order to cause rearrangement of the cells and, together with this, saturation of the polarization or a maximum of the dielectric permittivity. To confirm the suggestion stated above concerning the influence of the electric field on the structure of the substance, it would be of interest, in addition to studying the electrical characteristics, to obtain roentgenograms of ferroelectrics with and without application of an electric field. We were unable to carry out such experiments because of the absence of equipment.

Study of the coefficient of thermal expansion by the dilatometric method makes it possible to judge structural changes of the lattice of the substance [14]. It is known that at a temperature below the Curie point in ferroelectrics there arises not only spontaneous polarization, but also a spontaneous electrostrictive deformation associated with it. Longitudinal electrostriction in the direction of the spontaneous polarization or of the field is accompanied by transverse electrostriction of opposite sign; usually at the transition points there occurs a change in the volume of the body and, together with this, volume electrostriction is observed. On heating a ferroelectric, its dimensions change owing to ordinary thermal expansion, and upon transition from the ferroelectric region to the nonferroelectric region, or through the points of lattice transformations, also owing to changes in electrostrictive deformations caused by a change in the spontaneous polarization.

The dilatometric method is very sensitive and makes it possible to obtain data on small changes in the structure of a substance even in those cases when these changes cannot be detected by X-ray methods.

Dilatometric investigations were carried out by us on specimens of barium titanate and varikond in the region of positive temperatures from \(+20\) to \(+150^\circ\mathrm{C}\) \(^{1)}\). The data on the relative elongation \(\dfrac{\Delta l}{l}\) and the coefficient of linear expansion \(\alpha\) of barium titanate agree completely with those obtained earlier by V. I. Odelevskii [15] (the technique of the dilatometric measurements is described in [15]). In addition to ordinary dilatometric measurements,

\(^{1)}\) Dilatometric measurements of barium titanate and varikond were carried out in our laboratory by I. I. Kapichin, a diploma student of the Kiev Polytechnic Institute.

we studied the influence of an electric field, both alternating and constant, on the magnitude of the elongation and on the coefficients of linear expansion.

For barium titanate, in the temperature interval studied, one sharply expressed minimum was found at the Curie point in the curve of the coefficient of linear expansion versus temperature. For varicond, two minima were found: at the Curie-point temperature and near \(30^\circ\mathrm{C}\). These two minima are less deep than for barium titanate (Fig. 9).

Fig. 9. a) Dependence of the dielectric permittivity;
b) elongation and coefficients of linear expansion of varicond VK-1 and barium titanate on temperature.

\[ \frac{\Delta l}{l} \]

\[ \alpha \]

\[ \mathrm{BaTiO_2} \]

1 — × — × — × — 1

2 × — — × — — × — — 2

\[ \mathrm{VK\text{-}1} \]

\(\Delta l/l\) 3 ● — ● — ● — ● 3

\(\alpha\) 4 ● — — — ● — — — ● — — 4

The presence of a minimum in the curve of the coefficient of linear expansion versus temperature indicates the positive sign of the volume electrostriction of the ferroelectric.

G. A. Smolenskii et al. previously pointed out the connection between the dependence \(\varepsilon=f(E_{\sim})\) and the magnitude of electrostriction when studying the ferroelectric properties of certain solid solutions [16].

In our view, the two minima in the curve \(\alpha=f(t)\) correspond to two points of the phase transition of the crystal lattice of varicond VK-1.

The application of an electric field with an intensity of about \(200\ \mathrm{V/mm}\) has a noticeable influence on the form of the temperature characteristic \(\varepsilon\) of varicond and has an insignificant effect on the dependence \(\varepsilon\) of barium titanate. An analogous influence is exerted by the electric field on the temperature characteristic of the relative elongation and of the coefficient of thermal expansion of ferroelectrics. For barium titanate, at these field intensities no change is observed in the quantities

\[ \frac{\Delta l}{l} \]

and \(\alpha\), measured with the field applied and without the field. For varicond, the influence of a transverse electric field on the thermal-expansion curve has been established quite clearly. We have shown that a comparatively small transverse field, both alternating and constant, shifts the curve of the temperature dependence \(\alpha=f(t)\) of varicond VK-1 from the position of this curve obtained without applying a field (Figs. 10, 11, 12). A constant electric field leads to a shift

Fig. 10. Temperature dependence of the relative elongation \(\Delta l/l\) and of the coefficient of linear expansion \(\alpha\) for varicond VK-1 without a field and with a field applied.

\[ \begin{aligned} 3&\text{—}3\ \Delta l/l,\\ 4&\text{—}4\ \alpha \end{aligned} \left\}\ E=0;\qquad \begin{aligned} 5&\text{—}5\ \Delta l/l,\\ 6&\text{—}6\ \alpha \end{aligned} \left\}\ E=200\ \mathrm{V/mm}. \]

Fig. 11. Temperature dependence of the relative elongation \(\Delta l/l\) and of the coefficient of linear expansion \(\alpha\) for varicond VK-1 without a field and with an alternating voltage applied.

\[ \begin{aligned} 3&\text{—}3\ \Delta l/l,\\ 4&\text{—}4\ \alpha \end{aligned} \left\}\ E=0,\qquad \begin{aligned} 7&\text{—}7\ \Delta l/l,\\ 8&\text{—}8\ \alpha \end{aligned} \left\}\ E_{\sim}\ne 0. \]

to transverse compression of the specimen (Fig. 10), and alternating—to elongation (Fig. 11). These data may be regarded as preliminary; however, even now it may be assumed that the action of an electric field causes a change in the structure of the ferroelectric, and that this change is the stronger, the higher the nonlinear properties. It may be assumed that the more readily structural transformations of a substance from one modification to another take place, the higher the nonlinearity of ferroelectrics will be.

Fig. 12. Temperature dependence of \(\Delta l/l\) and \(\alpha\) for varicond VK-1 in a field and without application of a field.

\[ \begin{array}{ll} 3\!-\!3\ \Delta l/l,\quad 4\!-\!4\ \alpha & \} \ E=0;\\[2mm] 5\!-\!5\ \Delta l/l,\quad 6\!-\!6\ \alpha & \} \ E_{\sim}\ne 0;\\[2mm] 7\!-\!7\ \Delta l/l,\quad 8\!-\!8\ \alpha & \} \ E_{\sim}\ne 0. \end{array} \]

Conclusion

In connection with the high nonlinear properties of varicond VK-1, a real possibility has arisen of using this material to replace electron tubes in various radio devices. The preservation of high nonlinear properties over a wide temperature range further expands the possibilities for its application.

Nonlinear elements made of varicond are used in frequency multipliers, dielectric amplifiers of low- and high-frequency and direct-current signals, voltage stabilizers, for frequency modulation; moreover, in dielectric frequency modulators a considerable frequency deviation is achieved. The dependence of the dielectric permittivity on the constant biasing field makes it possible to regulate the frequency of an oscillatory circuit, as well as the magnitude of the alternating current in a circuit, by means of a controlling constant voltage. Nonlinear elements made of varicond can serve for tuning the input circuits of receivers, for providing variable selectivity in intermediate-frequency stages, and for trimming various filters.

Cases of application of nonlinear elements in automation and telemechanics deserve attention. The considerable steepness of the hysteresis curve of varicond will make it possible to use them for creating memory elements of computing devices. Ferroelectric ceramic materials with high nonlinear properties will find wide application in apparatus and devices—

devices that must operate under conditions of large accelerations and vibration and from which absolute reliability of operation is required.

We have noted only a few cases of the application of variconds. Further study of their properties, the production of materials with higher nonlinearity, as well as theoretical and experimental investigations of radio-engineering and electrical-engineering circuits with nonlinear elements made of varicond, will define new fields for their application.

The use of variconds in combination with transistors, magnetic elements, and other representatives of nonlinear elements will lead to the creation of reliable, inexpensive, fail-safe, and durable apparatus and devices.

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Errata