ON THE POLARIZATION OF AN ATOMIC-IONIC DIELECTRIC IN AN EXTERNAL FIELD

R. L. MÜLLER

1. The properties of a dielectric depend on its physical state—crystalline, amorphous, or liquid. Still more essential are its chemical composition and its chemical structure—ionic, atomic-valence, or molecular. The degree of chemical homogeneity of the dielectric is also important.

Without dwelling here on a historical analysis of the development of our domestic and foreign research, it may be noted that the widest currency among investigators has been gained by the study of dielectrics from the standpoint of their structural, spatial-geometrical construction and that, to a certain extent for this reason, the modern theory of dielectrics is characterized by an insufficiently deep and concrete statistical analysis of the observed facts. Relatively little attention has been devoted to chemical phenomena in dielectrics.

It is to be hoped that the subsequent enrichment of the physico-statistical theory by concrete structural-chemical concepts will make it possible to obtain a more rigorous and complete picture of the processes in dielectrics and will contribute to a better understanding of the nature of the phenomena occurring in them.

If we speak of refractory glass-forming dielectrics, an important milestone in the development of the physical theory was the well-known work of Zachariasen [1], who proposed that ions in oxygen glasses be regarded as mutually bound by the same forces as in the crystalline state; in this connection, glasses were considered to possess a spatially disordered three-dimensional network of bonds. Warren’s X-ray investigations [2] supplemented the picture of the statistically averaged geometrically disordered structure of glasses and revealed the transition of the coordination of oxygen atoms around boron atoms from trigonal to tetrahedral upon the introduction of basic oxides into boric anhydride. A number of scientists made attempts to establish a more detailed structure of glasses [3]. In particular, the existence of weak and strong interactions in glasses led Fajans and Barber [4] to assume the presence in them of several structures in equilibrium (more and less strong). The change of a number of properties of glasses with temperature and pressure is ascribed by these authors to the accompanying quantitative change in the ratios of the different structures in the glass. Warren and Stevels developed the network theory of glasses as applied to a mixture of two types of oxides—glass-forming oxides \((\mathrm{SiO_2}, \mathrm{B_2O_3}\) and a few others, which give a network structure) and non-glass-forming oxides \((\mathrm{Na_2O}, \mathrm{CaO}\), etc.; basic oxides, which modify the glass-forming network) [5], [6].

It was noted that metallic cations are fixed near those oxygen ions which are not bridges \((—\mathrm{Si}—\mathrm{O}—\mathrm{Si}—)\) linking the network. The remaining oxygen ions are bridges in the network and mainly determine the degree of polymerization of a typically coordinated glass [7]. On the basis of such concepts were proposed

statistical theories of the physical properties of glasses, in particular their electrical conductivity [6], [8]. These and similar statistical calculations bear the stamp of mathematical abstraction, owing to the failure to take into account the real physicochemical structure of glass. The numerical values of the statistical coefficients in the resulting formulas, as a rule, are not calculated theoretically and are not compared with the corresponding coefficients of the observed empirical regularities. Thus, for example, the experimental-theoretical analysis of electrical conductivity is limited to consideration of the dependence of this quantity on temperature and field strength. Such a limited application of the theory has hitherto deprived the interpretation of the numerical energy quantities entering into the mathematical expressions of a serious basis [9].

1. Without touching upon the well-known discussion concerning the crystalline and network structure of glasses—structures which, in essence, are simultaneously present in the aggregate of glasses of various compositions—it is necessary to dwell on our works, which partly adjoin the foreign investigations noted here, but at the same time differ fundamentally in the formulation of the question of the chemical nature of glass-forming media.

The phenomenon of glass formation has long been associated with the increased viscosity of supercooled liquids. In elucidating the quantitative relations observed in these investigations, no answer was usually given to the question of the reasons for the presence, in certain substances, of such increased viscosity at solidification temperatures. Morey, for example, considered that, in the present state of our knowledge, the causes of the retardation of the processes of establishing equilibrium in glass-forming media cannot be revealed [10].

At one time serious attention was drawn to the existence of an illegitimate underestimation of the specific chemical features of dielectrics. Even Zachariasen, who with his conception of the network structure of glasses was the first to embrace, from a single point of view, the totality of glass-forming oxides and thereby came very close to the chemistry of the problem of glass formation, nevertheless left the properly chemical nature of this phenomenon essentially undisclosed. Other investigators to this day also do not touch upon the essence of the chemical nature of the structure of glassy dielectrics.

Meanwhile, as early as 15 years ago we noted, as a feature specific to glass-forming oxides, the presence of multivalent light atoms of boron, aluminum, silicon, and a few other elements with their short valence bonds, which predetermine increased quanta of activation energy at small valence displacements. The introduction into such oxides of basic oxides reacting with them is accompanied by partial destruction of the network structure, a decrease in the system in the number of valence bonds, and an increase in electrostatic dipole interactions [11]. The increased quasi-elastic forces in the covalently bonded atoms of glass-forming oxides determine not only the specific features of the rigid stability of the nonequilibrium amorphous structure of glasses, but also the whole totality of their physicochemical properties, such as viscosity, heat capacity, and critical temperature [12]. Conservative valence bonds explain the increase of Young’s modulus and the irreversible increase in density in glassy $B_2O_3$ and $SiO_2$ under pressure [13].

Thus, instead of a purely mechanical, abstract consideration of the network structure of refractory inorganic aluminoborosilicates both in the amorphous and in the crystalline states, an analysis is proposed of the phenomena from the point of view of the atomic-valence and ionic bonds existing in such systems between elementary particles. At the same time one cannot fail to note the fact that the published views of Warren

and Stevels only partly coincide with the approach to the phenomenon that we propose. There are substantial differences in the conclusions at which we arrive.

According to the statistical-mechanical ideas of Zachariasen, Warren, and Stevels, in the network structures under consideration ionic bonds are continuously broken and re-formed. Meanwhile, we have noted that valence branched bonds (in Si—O the energy of a single bond [15] is close to 100 kcal/mole) practically cannot be broken under ordinary experimental conditions, but only switch at sufficiently high temperatures. At temperatures below the critical ones, processes requiring excitation of paired-electron valence bonds, characterized by high quasielastic coefficients [12], cease. The assumption made by Zachariasen, Warren, and Stevels that a network linkage exists among ions that have no directional bonds may generally be regarded as illegitimate. On the contrary, the covalent interpretation of the linkage of atoms in glass-forming oxides makes Zachariasen’s network structure quite well substantiated.

Further, it seems to us that Stevels underestimates the role of intrinsically ionogenic constituent parts, denying the influence of such ions as Na\(^+\) and Ca\(^{++}\) on the vibrations of the ionized valence network of glass [7]. Without taking valence-chemical factors into account, it is impossible to explain the strengthening of the network when basic oxides are added to boric anhydride; nor is it possible to understand the sharp increase in electrical conductivity observed in this case and, at the same time, the equally steep increase in the chemical durability of boric glass [14]. Ionic-atomic-valence physicochemical conceptions also make it possible somewhat to deepen the understanding of certain physical phenomena observed in the field of polarization and electrical conductivity of dielectrics.

1. As is known, A. P. Aleksandrov, P. P. Kobeko, and E. V. Kuvshinskii [16] established the presence in borosilicates of considerable dielectric losses that are not explained by electrical conductivity. These losses are analogous to those caused by the rotation of polar molecules. Meanwhile, rotation of molecules in solid borosilicates is excluded. G. I. Skanavi proposed the hypothesis of thermal ionic polarization, which treats the statistical kinetics of polarization as the phenomenon of the hopping of cations from one potential well to a neighboring one over a low potential barrier, under conditions in which both potential wells are bounded externally by high potential barriers [17].

For all the physicomathematical justification of this hypothesis, one must note the absence of a clear picture of the mechanism of the process. It is not clear whether the cations undergo “significant displacements within such weak packings,” while at the same time not taking part in through conduction, or whether these displacements are comparable with the sizes of the particles themselves. Nor is the nature of the low and high potential barriers, which play such an essential role in the hypothesis under consideration, clear.

Meanwhile, in the light of the ionic-atomic-valence concept proposed by us, these questions receive answers, and the hypothesis of thermal ionic polarization receives its structural-chemical justification.

1. Owing to the low value of the dielectric constant in B\(_2\)O\(_3\) and SiO\(_2\), the degree of dissociation of polar groups B—O\(^{4/2}\)M\(^+\) or SiO\(_{3/2}\)O\(^{-}\)M\(^+\) is negligibly small and, practically, all cations enter into the composition of polar groups. Thus elastic ionic polarization is caused by small displacements of the equilibrium position of thermally vibrating cations relative to the negatively ionized atoms of the valence-atomic network that bind them. Such displacements of cations are nonactivated and are not accompanied by their leaving the potential wells they occupy.

In addition to such instantaneous polarization displacements within one and the same potential well of a given polar structural unit, activated displacements of cations are also possible, with transition from one potential well to another within one and the same structural element. In the borate structural element \(\mathrm{B}-\mathrm{O}^{4/2}\mathrm{M}^{+}\) there are four such potential wells, and in the silicate structural element \(\mathrm{SiO}^{3/2}\mathrm{O}^{-}\mathrm{M}^{+}\)—three.

Fig. 1.

In the case of a borate structural unit, what has been said may be illustrated schematically by Fig. 1. The cation \(\mathrm{M}^{+}\), bound in the polar structural element \(\mathrm{B}-\mathrm{O}^{4/2}\mathrm{M}^{+}\), is displaced by transition “\(\tau\)” from the first to the second position, while preserving its electrostatic bond with the central negatively charged boron atom. In such a transition the cation crosses the pass between oxygen atoms and moves somewhat farther away from the central boron ion. This distance will, however, be small in comparison with the removal of the cation upon dissociation. Accordingly, the activation energy \(E_p\) of such a displacement will be much smaller than the dissociation energy \(\Delta \Phi\). Using data on the sizes of ions and atoms, one can estimate approximately the value of the activation energy \(E_p = W_p - W_{ac}\) for sodium and potassium ions, where \(W_{ac}\) and \(W_p\) are the energies of the cations in the potential well and in the activated position, respectively (Table 1).

Table 1

The structural-element model considered is a certain first approximation to the actual structure of the isoelectronic surfaces of oxygen and boron atoms.

However, subsequent refinement of the structural model will not change the very fact that cations have four potential wells around a negatively ionized boron atom.

Analogous displacements of cations are observed around ionized oxygen atoms in silicates. In the mechanism considered for orientational displacement of cations, the low potential barrier of thermally activated polarization corresponds to the activation energy of displacement from one potential well to another within the given structural element. The bordering high potential barriers correspond to the energy of dissociation of the cation with its removal beyond the limits of the given structural ele-

ment. It also follows from what has been set forth that, in terms of the resultant polarization effect, the activated displacement of cations within structural polar units is identical to the activated rotation of polar molecules in liquids.

The activated polarization considered presupposes the presence in the dielectric of valence-atomic structural units and therefore is characteristic of iono-atomovalent dielectrics, and should not be observed in purely ionic crystalline bodies.

It is also natural that enhanced polarization and high dielectric losses exist precisely in borosilicates with readily displaced monovalent cations of alkali metals, and that such phenomena are absent in borosilicates with non-displacing divalent cations (simultaneous bonding of the latter with two ionized atoms). In the latter case the dielectric losses and permittivities are low.

Activated ionic polarization is manifested most strongly at low temperatures and at low concentrations of alkali oxides. With increasing temperature the absorption polarization current is masked by the through conduction current, owing to the larger temperature coefficient of the latter \((\Delta \Phi > E_p)\). Similarly, with increasing volume concentration of alkaline polar groups, the exponentially increasing conduction current masks the polarization current, which in this case grows linearly with the concentration of polar groups (Fig. 2). Therefore, elevated dielectric losses are observed at low temperatures in borosilicates poor in alkali oxides.

Fig. 2

1. Along with ionic polarization, modern physics also considers polarization caused by the displacement of valence electron shells relative to the nuclei. Electronic polarization in boron oxides essentially determines the dielectric permittivity. At the same time, its higher value in \(\mathrm{SiO_2}\) (3.8) compared with \(\mathrm{B_2O_3}\) (3.2) cannot be explained by denser packing of \(\mathrm{SiO_2}\) [18], for, as is seen from Table 2, the boron and oxygen atoms are packed more densely in \(\mathrm{B_2O_3}\) than the silicon and oxygen atoms in \(\mathrm{SiO_2}\).

Table 2

The difference in dielectric constants is most likely explained by the larger radius of the outer electron shell of silicon with tetrahedri-

with boron with trigonal coordination of oxygen atoms [19].

1. In a number of iono-atomic-valent dielectrics, within limited temperature ranges, anomalously increased polarizations and dielectric losses are observed, similar to those found in molecular ferroelectrics. Such a phenomenon has a complex character and must depend on subtle structural-chemical features in the constitution of the substance. As is known, the dielectrics under consideration include rutile, titanates, and certain other substances of definite composition and structure [17]. G. I. Skanavi proposed the idea that, in addition to the polarization of electronic and ionic displacement, titanates with the perovskite structure must also possess a specifically enhanced polarization caused by the “polarity of individual groups of the crystal lattice” [20].

From this point of view it is of interest to consider the iono-atomic-valent relation in the structural element of perovskite.

The probability of the existence of highly charged ions in solids under ordinary conditions is known to be small [21]. It is natural to expect that in the course of thermal vibrations short-lived associations of such ions arise in the crystal lattice, with the formation of polarization bonds close to valence bonds. The possibility is not excluded that such ions as $\mathrm{Ti}^{4+}$ and $\mathrm{O}^{2-}$, on the average statistically in time, form complex ions with reduced effective charges or, at least, with charges distributed among several atoms of the complex, for example $[\mathrm{TiO}]^{2+}$ and $[\mathrm{TiO}_3]^{2-}$. Such average statistical formulas agree with the experimental data on the heat capacities of rutile and titanates [22]. The transformation of iono-valence bonds must be accompanied by an exchange of partners in the complex ions with a frequency within the limits $1 \ll \nu \ll 10^{13}\ \mathrm{sec}^{-1}$. These limits are determined by the fact that the relaxation period must be small, since X-ray structural analysis does not reveal complex ions during the exposure time. At the same time, the existence of such complex ions must be sufficiently long compared with the period of vibration of atoms and ions.

The formation of temporary complex structures is possible only for a definite structure of the solid. According to X-ray structural analysis data, in rutile and perovskites there is an average statistical coordination of six oxygen ions around the titanium ion. This determines the possibility of statistical switching of polarization valence bonds, with the formation in perovskites of three simultaneously realized rigid valence bonds and the production of polar complex ions:

\[ n[\mathrm{TiO}_{6/3}]^{2-} \longrightarrow n\overrightarrow{[\mathrm{TiO}_3]}^{2-} \]

and the production in rutile of one and three simultaneously realized bonds, likewise with the appearance of complex ions with dipole moments.

\[ n[\mathrm{TiO}_{6/3}] \longrightarrow \frac{n}{2}\,\overrightarrow{[\mathrm{TiO}]}^{2+} + \frac{n}{2}\,\overrightarrow{[\mathrm{TiO}_3]}^{2-}. \]

In the absence of an external field, the dipoles in the course of thermal motion are oriented statistically uniformly in all directions. In the presence of an external field, however, the orientation of such dipoles acquires a certain preferred direction.

In agreement with the above limits of the frequency of transformation of iono-valence bonds are the data on the relaxation periods of polarization in such dielectrics, $\tau = 10^{-8} \div 10^{-9}\ \mathrm{sec}$ [17].

At sufficiently high temperatures, the excitation of all valence bonds and the disappearance of rigid bonds has a loosening effect on complex ions, as a result of which dielectric losses should decrease. Such structural losses should also decrease at sufficiently low temperatures owing to the loss of intensity of thermal vibrations and the consequent hindrance of the activated transformation of ionic-valence bonds.

A certain verification of the considerations expressed here could be provided by experiments involving X-ray analysis of the structures of perovskite and rutile in a sufficiently strong electric field, as well as experiments on quenching the polarization field of such structures by rapid cooling in a strong electric field.

Fig. 3.

In titanates, the symmetry of switching is determined by the identity of the states of the oxygen ions. This predetermines a weakening of the effect with increasing polarizing action of such cations in titanates as magnesium ions (ilmenite structure) and the alkaline-earth metal ions closest to it in size (perovskite structure). This is confirmed by the increase in dielectric losses and dielectric permittivity in going from magnesium titanate to barium titanate. The barium ion, as the least polarizing, gives rise to the most significant anomalous dielectric losses.

It should also be noted that there must be a distinction between the excitation of permanent valence bonds in borosilicates and the excitation of oscillating valence bonds in titanates. The activation energy in the latter will be appreciably lower and may decrease not only in going from the magnesium cation to barium, but also upon introduction of other still less polarizing cations.

Conclusions

1. The polarization of ionic-atomic-valence dielectrics is characterized by features absent in purely ionic crystalline bodies.

2. Along with the practically instantaneous polarization caused by elastic displacements of cations in potential wells, there also exists a time-dependent activated ionic polarization as a result of the transition of cations from one potential well to another in each structural element. Such polarization may appear in the form of increased dielectric losses at low temperatures in borosilicates poor in alkali oxides.

3. Along with practically instantaneous electronic displacements in atoms, there also exists an orientational effect in polar structural elements in the form of a transformation of valence-ionic bonds by means of an activated displacement of valence electron pairs. The orientational effect should probably be used to explain the anomalously high polarizabilities and dielectric losses in bodies with the perovskite and rutile structures.

1. The presence, in a number of substances with an atomic–ionic structure, of a definite temperature range in which anomalously high dielectric losses are observed may be due to the fact that at lower temperatures the intensity of thermal vibrations decreases and, therefore, activated transformations of valence–ionic bonds are hindered, whereas at higher temperatures the valence bonds cease to be rigid and, as a result of the onset of all-round disordered vibrational motion of the atoms, the possibility of the formation of oriented dipolar complex ions is excluded.

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Kemerovo,
Mining Institute