DIELECTRIC CONSTANT AND MAGNETIC PERMEABILITY OF VARIOUS FERRITES AT MICROWAVE FREQUENCIES

In recent years, semiconducting and dielectric magnetic materials, especially ferrites—semiconductors with chemical composition \(m\mathrm{MeO}\cdot n\mathrm{Fe}_2\mathrm{O}_3\), where \(m\) and \(n\) are small integers and the symbol Me denotes a divalent metal (Ni, Co, Mn, etc.)—have acquired great practical importance.

A number of papers at the 2nd All-Union Conference on Magnetism in December 1951[^1] were devoted to the problem of the magnetic and electrical properties of ferrites. In the conclusion to his report[^1] at the conference, Ya. I. Dorfman emphasized that “the problem of ferromagnetic and antiferromagnetic semiconductors and dielectrics constitutes a special nodal problem of modern solid-state physics, and by no means a narrowly magnetic problem.” It therefore seems appropriate to present some new experimental data on the electrical and magnetic properties of ferrites in the most interesting range—at centimeter wavelengths.

In the work[^2], the dielectric constant \(\varepsilon\) and magnetic permeability \(\mu\) of magnesium, copper, cobalt, nickel, and manganese ferrites were determined at a wavelength of \(6.6\ \mathrm{cm}\). A specimen in the form of a thin disk \(1.5\ \mathrm{mm}\) thick and \(20\ \mathrm{mm}\) in diameter was introduced into a rectangular cavity resonator (wave \(TE_{104}\)) alternately either into the antinode of the electric field \(E\), or into the antinode of the magnetic field \(H\). The decrease in the resonator quality factor \(Q\) caused directly by these changes and the shift of the resonant frequency \(\Delta\omega_0\) were measured.

The effective dielectric constant and magnetic permeability may be written as the sum of their real and imaginary parts: \(\varepsilon=\varepsilon_1-j\varepsilon_2\), \(\mu=\mu_1-j\mu_2\), where \(\varepsilon_2\) and \(\mu_2\) characterize the losses. The loss tangent is equal to: \(\operatorname{tg}\delta_\varepsilon=\varepsilon_2/\varepsilon_1\) and, correspondingly, \(\operatorname{tg}\delta_\mu=\mu_2/\mu_1\).

As shown in[^3], the shift of the resonant frequency and the quality factor of the circuit when the specimen under study is introduced into it are related to \(\varepsilon_1\), \(\varepsilon_2\), \(\mu_1\), and \(\mu_2\).

by the following relations:

\[ -2\frac{\Delta\omega_a}{\omega}=(\varepsilon_1-1)\int_{\Delta V} E_a^2\,dv;\qquad \frac{1}{Q_\varepsilon}=\frac{\sigma}{\omega}\int_{\Delta V} E_a^2\,dv+\varepsilon_2\int_{\Delta V} E_a^2\,dv, \tag{1} \]

\[ -2\frac{\Delta\omega_a}{\omega}=(\mu_1-1)\int_{\Delta V} H_a^2\,dv;\qquad \frac{1}{Q_\mu}=\mu_2\int_{\Delta V} H_a^2\,dv, \tag{2} \]

where \(E_a\) and \(H_a\) are the field strengths in the resonator in the absence of the sample. Formula (1) applies to a sample placed in the antinode of the electric field, formula (2) to a sample placed in the antinode of the magnetic field. Relations (1) and (2) make it possible, from the measured values of \(Q\) and \(\Delta\omega_a\), to determine the required quantities—\(\varepsilon_1, \varepsilon_2, \mu_1, \mu_2\).

The results of the experiment are shown in Table I.

Table I

Dielectric constant and magnetic permeability of ferrites at a wavelength of 6.6 cm

The sign * marks ferrites subjected to “annealing,” i.e., slow cooling; the sign ** marks those subjected to “quenching”—rapid cooling from a temperature of 1200°C. Obviously, some of the ferrites (one of them—the copper ferrite—was cooled not at 1200°C, but at 1000°C).

It should be noted that this method is one of the most widespread methods for determining the electrical and magnetic properties of a substance in the centimeter range.

From Table I one may conclude that the dielectric constant of ferrites in the centimeter-wave range is of the order of 10, while the magnetic permeability is about 1.5. The loss tangent varies, depending on the chemical composition of the ferrite, within rather wide limits. The smallest losses are characteristic of \(\mathrm{CoO}\cdot\mathrm{Fe}_2\mathrm{O}_3\); the corresponding decrease in \(Q\) was so insignificant that it could not even be measured. To check this, measurements were repeated with samples of the same diameter, 20 mm, but of a different thickness—1 mm. The results of these measurements coincided with the data of Table I.

The data of work\(^2\) agree with the general character of the dependence of the dielectric constant of ferrites on frequency, investigated in work\(^4\); however, for lower frequencies. In\(^4\), by means of an alternating current, the capacitance of ferrite disks was measured and then the dielectric constant was calculated. Several typical results are given in Table II.

Table II

Dependence of the dielectric constant on frequency

The conductivity of sample 1 is \(1.8\cdot 10^{-6}\), of sample 2 is \(7\cdot 10^{-11}\), of sample 3 is \(10^{-3}\)

\[ \left(\text{in } \frac{1}{\mathrm{ohm}\cdot\mathrm{cm}}\right) \]

At low frequencies the measured value of the dielectric constant is very large, which is possibly connected with the inhomogeneity of the structure of ferrites (see also\(^5\)). With increasing frequency the dielectric constant falls very rapidly at first, and then, beginning with frequencies of the order of \(10^4\) c/s, already much more slowly, asymptotically approaching a value of the order of 10 at frequencies \(10^7\)—\(10^8\) c/s.

In\(^4\) there is also a reference to Gerschping’s work, in which at frequencies of \(4\cdot 10^9\) c/s the value of the dielectric constant of a number of ferrites was also obtained as being of the order of 10. (The measurements were carried out by a third commonly used method—with the aid of a coaxial line.)

It should be noted that \(\varepsilon\) and \(\mu\) of ferrites as polycrystalline materials depend not only on their chemical composition, but also on the dispersity of the system, the sizes and arrangement of the individual crystallites.

M. P.

References

1. Izvestiya AN SSSR, Ser. Phys. 14, No. 4 (1952); Uspekhi Fizicheskikh Nauk 46, 396 (1952).
2. T. Okamura, T. Fujimura, Muneyuki Date, Phys. Rev. 85, 1041 (1952).
3. J. C. Slater, Rev. Mod. Phys. 18, 441 (1946).
4. Möllgen, Zeits. angew. Physik 4, No. 6 (1952).
5. F. W. Brockman, P. Dowling, W. Steneck, Phys. Rev. 77, 85 (1950).