Contactless method for studying the electrical properties of alkali halide crystals over a wide temperature-frequency range
Abstract Generated abstract
This paper evaluates a contactless torque method for measuring the electrical properties of spherical alkali halide crystals across very low frequencies and elevated temperatures. Building on Ogawa’s theory, the authors derive the relations between torque in a rotating electric field, conductivity, dielectric losses, relaxation time, and activation energy, and describe an automated apparatus operating from 10^-4 to 10^2 Hz and 20 to 400 degrees C. Experiments on spherical KCl specimens show relaxation peaks whose maxima shift with frequency, yielding an activation energy of about 0.92 eV. Conductivity values calculated from contactless torque measurements agree closely in magnitude and slope with direct current measurements, supporting the method’s applicability to ionic dielectrics over combined temperature and frequency ranges.
Full Text
CONTACTLESS METHOD FOR STUDYING THE ELECTRICAL PROPERTIES OF ALKALI-HALIDE CRYSTALS OVER A WIDE TEMPERATURE–FREQUENCY RANGE
A. P. NESTERENKO, Yu. M. ANNENKOV
(Presented by the scientific seminar of the electrophysical laboratory of the Research Institute of Radiation Physics, TPI)
Introduction
This method was first proposed by Hertz [1] in 1881. Lertes [2] applied it to the study of the electrical conductivity of certain organic substances.
In 1960, Tomoya Ogawa [3] developed the theory of the method as applied to dielectric losses, electrical conductivity (“$\sigma$”), and the dielectric constant ($\varepsilon$) for semiconductors and dielectrics.
As certain studies [3—7] have shown, the theory is in fairly good agreement with experiments carried out for a wide variety of substances.
Perhaps the only method for studying the electrical properties of semiconductors and dielectrics at infralow frequencies is the contactless method. It makes it possible to study polarization processes with both long and short relaxation times; to separate the imaginary and real parts of the dielectric permittivity; it is free from contact effects on the measured parameters; and, in operation, is simple, reliable, and easily automated.
Experiments performed earlier [3—7] were carried out over a wide frequency range at room temperature.
The present article is devoted to a practical verification of the application of this method for measuring the electrical properties of spherical alkali-halide crystals over a wide frequency ($10^{-4}$—$10^{2}$) Hz and temperature (20—400° C) range.
Theory
The principle of the contactless method [3] is that, in a rotating electric field, a specimen is suspended on a thin filament and is rotated about the suspension axis by a twisting torque proportional to the electrical conductivity (“$\sigma$”) or to the imaginary part of the complex dielectric constant ($\varepsilon''$), while the force exerted on the specimen by a linearly polarized field is proportional to the real part of the dielectric permittivity ($\varepsilon'$).
The expressions for the complex dielectric permittivity ($\varepsilon^{*}$), on the basis of Maxwell’s equations, are written in the form
\[ \varepsilon^{*}=\varepsilon' - i\varepsilon'' \tag{1a} \]
or
\[ \varepsilon^*=\varepsilon' - i\sigma/\omega . \tag{1б} \]
If the angular frequency \(\omega\) of the applied electric field is small, then the twisting torque \(\overline{T}\) in a specimen placed in a homogeneous field of strength \(\overline{E}_0\) is represented by the expression
\[ \overline{T}=V_s(\overline{P}_s^* \times \overline{E}_0), \tag{2} \]
where \(\overline{P}_s^*\) and \(V_s\) are the polarization vector and the volume of the specimen, respectively.
As is known, the polarization vector \(\overline{P}_s^*\) is given by the formula
\[ \overline{P}_s^*= \frac{(\varepsilon_s^*-\varepsilon_0^*)}{1+L(\varepsilon_s^*-\varepsilon_0^*)}\,\overline{E}_0, \tag{3} \]
where \(L\) and \(\varepsilon_s^*\) are the depolarization factor and the complex dielectric constant of the substance under study, respectively; \(\varepsilon_0^*\) is the complex dielectric constant of the medium in which the measurement is carried out.
The depolarization factor \(L\) is determined by the shape of the specimen. In particular, for spherical, isotropic, nonpolar dielectrics \(L_x=L_y=L_z=\dfrac{1}{3\varepsilon_0}\), where \(L_x\), \(L_y\), \(L_z\) are the projections of the depolarization-factor vector on the coordinate axes \(X, Y, Z\), respectively.
For other specimen shapes, and also for anisotropic substances, \(L\) is determined experimentally. Since the dielectric losses of the surrounding medium are small, \(\varepsilon_0^* \simeq \varepsilon_0\).
From expressions (1a, 1б, and 3) it follows that
\[ \overline{P}_s^*= \frac{\varepsilon_s-\varepsilon_0-i\varepsilon''}{1+L(\varepsilon_s-\varepsilon_0)-iL\varepsilon''}\,E_0 \tag{4a} \]
or
\[ \overline{P}_s^*= \frac{\varepsilon_s-\varepsilon_0-i\sigma/\omega}{1+L(\varepsilon_s-\varepsilon_0)-iL\sigma/\omega}\,E_0 . \tag{4б} \]
The relations presented indicate that equation (4б) is used when the electrical conductivity \((\sigma^*)\) predominates over the imaginary part of the dielectric permittivity \((\varepsilon'')\), and conversely.
Let us consider the case corresponding to the condition \(\sigma>\varepsilon''\). We shall write the real and imaginary parts of the complex polarization \(\overline{P}_s^*\) explicitly. Then expression (4б) takes the following form:
\[ \overline{P}_s^*= \frac{(\varepsilon_s-\varepsilon_0)+L[(\varepsilon_s-\varepsilon_0)^2+(\sigma/\omega)^2]} {[1+L(\varepsilon_s-\varepsilon_0)]^2+(L\sigma/\omega)^2}\,E_0 -i\frac{\sigma/\omega}{[1+L(\varepsilon_s-\varepsilon_0)]^2+(L\sigma/\omega)^2}\,E_0 . \tag{5} \]
Since the external field \(E_0\), rotating about the \(Z\) axis, varies according to a sinusoidal law, it is represented by the components
\[ E_{0x},\ E_{0y},\ E_{0z} \]
\[ E_{0x}=E_x\exp(i\omega t), \]
\[ E_{0y}=E_y\exp i(\omega t-\alpha), \]
\[ E_{0z}=0. \]
Then the imaginary and real parts of the complex polarization will have the components \(P'_{sx};\ P''_{sx};\ P'_{sy};\ P''_{sy}\)
\[ P''_{\varepsilon x}=E_x\frac{\sigma_x/\omega}{[1+L_x(\varepsilon_x-\varepsilon_0)]^2+(L_x\sigma_x/\omega)^2}; \]
\[ P'_{s x}=E_x\frac{(\varepsilon_x-\varepsilon_0)+L_x[(\varepsilon_x-\varepsilon_0)^2+(\sigma_x/\omega)^2]}{[1+L_x(\varepsilon_x-\varepsilon_0)]^2+(L_x\sigma_x/\omega)^2}; \]
the \(Y\)-components are represented by analogous expressions.
The torque \(\overline{T}_z\) is expressed by the equation
\[ \begin{aligned} \overline{T}_z={}&\frac{1}{2}V_sE_xE_y \left\{ \frac{\sigma_x/\omega}{[1+L_x(\varepsilon_x-\varepsilon_0)]^2+(L_x\sigma_x/\omega)^2} +\right.\\ &\left. +\frac{\sigma_y/\omega}{[1+L_y(\varepsilon_y-\varepsilon_0)]^2+(L_y\sigma_y/\omega)^2} \right\}\sin\alpha+\\ &+\frac{1}{2}V_sE_xE_y \left\{ \frac{(\varepsilon_x-\varepsilon_0)+L_x[(\varepsilon_x-\varepsilon_0)^2+(\sigma_x/\omega)^2]} {[1+L_x(\varepsilon_x-\varepsilon_0)]^2+(L_x\sigma_x/\omega)^2} -\right.\\ &\left. -\frac{(\varepsilon_y-\varepsilon_0)+L_y[(\varepsilon_y-\varepsilon_0)^2+(\sigma_y/\omega)^2]} {[1+L_y(\varepsilon_y-\varepsilon_0)]^2+(L_y\sigma_y/\omega)^2} \right\}\cos\alpha . \end{aligned} \tag{6} \]
For an isotropic spherical specimen, \(\varepsilon_x=\varepsilon_y\), \(\sigma_x=\sigma_y\), \(L_x=L_y\), \(E_x=E_y\), and the torque of the specimen produced by an alternating external field is represented by the equation
\[ \overline{T}^{R}_{z}=VE^2 \frac{\sigma/\omega}{[1+L(\varepsilon_s-\varepsilon_0)]^2+(L\sigma/\omega)^2}. \tag{7} \]
The torque produced by a linearly polarized field is represented by the real part of equation (6). Hence
\[ \overline{T}^{L}_{z}=0. \tag{8} \]
Differentiating equation (7) with respect to frequency (the maximum condition), we obtain the expression for \(\omega_{\max}\)
\[ \omega_{\max}=\frac{\sigma}{[1+L(\varepsilon_s-\varepsilon_0)]^2}. \tag{9} \]
From formulas (7) and (9) it follows that
\[ \overline{T}^{R}_{\max}=VE^2 \frac{1}{L[1+L^2(\varepsilon_s-\varepsilon_0)]^2}. \tag{10} \]
Formula (10) shows that \(\overline{T}^{R}_{\max}\) is not a function of “\(\sigma\).” At field frequencies \(\omega/\omega_{\max}\gg1\), the expression \(L^2(\varepsilon_s-\varepsilon_0)^2+(L\sigma/\omega)^2\ll1\); then formula (7) assumes the simple form
\[ \overline{T}^{R}_{z}=VE^2\sigma/\omega. \tag{11} \]
From equation (11) it is easy to determine “\(\sigma\)” knowing \(\overline{T}^{R}_{z}\), which is calculated or measured directly in the course of the experiment.
If, in the value of the torque \(\overline{T}^{R}_{\max}\) in (7), the frequency is written in the form \(f=\omega/\omega_{\max}\), then it takes the following expression:
\[ \overline{T}^{R}_{z}=V_s\overline{E}_{0}^{\,2} \left\{ \frac{1}{L[1+L(\varepsilon_s-\varepsilon_0)]} \right\} \frac{f}{1+f^2}. \tag{12} \]
Since \(\omega/\omega_{\max}=\omega\tau\) (\(\tau\) is the relaxation time), equation (12) can be written as follows
\[ \overline{T}_{z}^{R}=V_s E_0^2 \left\{ \frac{1}{L\left[1+L(\varepsilon_s-\varepsilon_0)\right]} \right\} \frac{\omega\tau}{1+(\omega\tau)^2}. \tag{13} \]
It follows from equation (13) that the torque \(\overline{T}_{z}^{R}\) is proportional to the imaginary part of the complex dielectric constant, and dispersion is observed at the frequency \(\omega_{\max}\). As Gross and Groner [5] showed, this dispersion may prove to be “false” in the case where “\(\sigma\)” predominates over the imaginary part of the dielectric permittivity \((\varepsilon'')\).
The criterion for the validity of the measured parameters is an experiment carried out by the contactless method and comparable with data obtained by other methods, in particular bridge methods. In the case where the experiments for some reason cannot be compared, the activation energy \((E_{\mathrm{act}})\) of the relaxation process, determined from the shift of the maximum in the frequency–temperature range, may serve as an estimate of validity,
\[ E_{\mathrm{act}}= \frac{\ln \omega_1-\ln \omega_2}{\dfrac{1}{T_2}-\dfrac{1}{T_1}}\,K, \tag{14} \]
where \(T_1\) and \(T_2\) are the temperatures \((K^\circ)\) at the peak maximum at the corresponding frequencies \(\omega_1\) and \(\omega_2\);
\(K\) is Boltzmann’s constant.
Experimental setup
In principle, the setup developed by us differs little from that previously proposed by Gross and Groner [4]. Its main distinction is that it makes it possible to carry out measurements not only in the frequency range, but also in the temperature range \((20\text{--}400^\circ\mathrm{C})\), and to make an automatic recording of the deflection of the light beam, directly associated with the rotation of the specimen in a rotating electric field.
Figure 1 shows a vertical section of the measuring cell. The main parameters of the system are: quartz fiber \(l=1\ \mathrm{m}\); \(\varnothing=7\ \mu\mathrm{m}\); mirror \((0.5\times0.5)\ \mathrm{cm}\), weight \(0.2\ \mathrm{g}\), distance to the scale of the recorder measurement \(1\ \mathrm{m}\).
Figure 2 gives a schematic of the automatic tracking system for reading the angle of twisting of the specimen. The angle of deflection of the light “spot” and the temperature are recorded on the chart tape of an EPP-09-3M potentiometer. The heating rate—linear—can be regulated over wide limits from \(0.5^\circ\) to \(20^\circ\mathrm{C}\) per minute.
Using an EPV-2 potentiometer and a relay system, we can stabilize the temperature over the entire temperature interval with a maximum error of \((\pm 4^\circ\mathrm{C})\).
The electrodes are rotated by an MI-11F direct-current motor through a system of reducers and pulleys in the frequency range from \(10^{-4}\) to \(10^2\ \mathrm{Hz}\). The frequency is fixed with the aid of a tachogenerator and a pointer instrument (milliammeter), with a maximum error at frequencies from 2 to \(10^2\ \mathrm{Hz}\) of \((\pm 2\text{--}3\%)\). At low frequencies from 1 to \(10^{-4}\ \mathrm{Hz}\), the frequency is fixed with the aid of a stopwatch.
A constant voltage is applied to a pair of collinear vertical electrodes from a VS-2500 high-voltage source and is monitored by the source voltmeter. The electrodes are made of copper plates, each with an area of \(4\ \mathrm{cm}^2\) \((2\times2)\); the distance between them is \(2.7\ \mathrm{cm}\).
To minimize the influence of mechanical disturbances on the measurement system, the quartz filament is fastened to a vernier head of a supporting steel tube, which is mounted on a massive steel plate,
Fig. 1. Vertical section of the apparatus of the torque-measurement system. 1 — vernier head, 2 — supporting metal tube, 3 — quartz filament Ø 7 µm, 4 — chamber with partitions, 5 — optical system, 6 — adjustable mounting plate, 7 — massive metal shelf, 8 — bracket, 9 — suspended furnace chamber, 10 — furnace coil, 11 — thermocouple, 12 — contact rings, 13 — removable fluoroplastic disk, 14 — Teflon rod with contact brushes, 15 — metal support, 16 — rotating axis, 17 — pulley, 18 — supporting metal adjusting rods, 19 — ball bearing, 20 — specimen, 21 — electrode, 22 — furnace thermal insulation, 23 — screws for fastening the furnace chamber, 24 — thermal insulation, 25 — leveling screws, 26 — mirror
adjustable by three leveling screws in the horizontal and vertical planes. The metal shelf supporting the mounting plate is fastened on brackets (in the form of angle irons) built into a massive brick wall.
With a sensitive torque system, it is essential to isolate the specimen from the air flow produced by the rotating electrodes and by thermal convection. For this purpose, a choice was made
the spherical specimen is placed in a thin-walled quartz tube, by means of which, moreover, all-around heating of the specimen is carried out (see Fig. 1). In the second quartz tube there is placed a specimen with a thermocouple for monitoring and regulating the temperature. The second thermocouple is mounted directly in the temperature field of the furnace for more accurate regulation. Such a design of the apparatus is convenient in operation, makes it possible to change specimens easily and quickly, and does not require frequent adjustment of the suspension system.
Fig. 2. Diagram of the tracking system for recording the angle of twist of the specimen. \(D_1\), \(D_2\)—FD-3 photodiodes; UE-109—electronic amplifier; RD-109—reversible motor; \(T_1\)—toggle switch; \(R_1 \div R_5\)—resistances (rated values are selected when setting up the circuit); \(B_1\)—direct-current source (battery, \(U = 3.7\ \mathrm{V}\)).
Experiment
The experiment was carried out on KCl specimens grown by the Kyropoulos method from “chemically pure” grade salts. The specimens had the shape of a sphere \(1.0\ \mathrm{cm}\) in diameter. The field strength on the plates of the rotating capacitor was constant \((E = 3.7 \cdot 10^4\ \mathrm{V/m})\). The investigations were carried out in air. The rate of heating of the specimens did not exceed \(1^\circ\mathrm{C}\) per minute.
Figure 3,a shows the dependence of the angular deflection of the specimen in a rotating electric field under isochronous heating from \(20^\circ\mathrm{C}\) to \(310^\circ\mathrm{C}\) at three fixed frequencies \((0.01;\ 0.1;\ 2)\ \mathrm{Hz}\). As the frequency of the external field increases, the maximum of the peak shifts into the high-temperature region; at the same time the half-width of the maximum increases considerably, while its amplitude remains constant. The character of the peak is relaxational. The activation energy of the process, calculated from formula (14), is \(0.92\ \mathrm{eV}\). The peaks are strictly symmetric.
Figure 3,b gives the temperature dependence of the conductivity \(\sigma\) in KCl (“chemically pure”). Curve 4 is calculated from formula (11). Its calculated parameters \((\omega\) and \(\Theta^\circ)\) were obtained from the data of Fig. 3,a. The dash-dotted vertical lines show to what values of the twist angles
of the sample and to the frequencies of the rotating electric field correspond the calculated conductivity values (curve 4, Fig. 3, b). The parameters \(\omega\) and \(\Theta^\circ\) taken above the horizontal cutoff line cannot be used in calculating “\(\sigma\),” since the applicability condition for formula (11) will not be satisfied.
The maximum limiting value of the angle \(\Theta^\circ\), taken on the relaxation curve, at which formula (11) can still be applied, must be obtained at a frequency differing from the frequency of maximum dispersion by a value greater [3].
\[ f_i = 1/\sqrt{3}. \]
In the case when the peak expresses not the frequency but the temperature dependence of \(\Theta^\circ\), the value of “\(\sigma\)” should be calculated from experimental data obtained at temperatures below \(T_i\):
\[ T_i \simeq (0.8—0.97)T_{\max}, \]
where \(T_{\max}\) is the temperature corresponding to the maximum of the curve peak (K°);
\(T_i\) is the temperature at which the dispersion curve begins to deviate from the exponential law.
The nonconstancy of the values \(T_i\) is determined by the increase in the area of the dispersion curve with increasing temperature due to the thermal function.
The torque \(\overline{T}_{z}^{R}\) created in the sample by the electric field is equal to the mechanical resistance torque \(T_{\text{mech}}\) arising when the thread is twisted. It is determined by the formula [8].
\[ T_{\text{mech}}=\frac{\Pi^2\cdot J\cdot \Theta}{T_0}, \tag{15} \]
where \(J\) is the total moment of inertia of the sample and the mirror;
\(T_0\) is the period of oscillation of the sample without the field.
Fig. 3: a) temperature–frequency dependence of the rotation angle of the sample \(\Theta^\circ\) in a rotating electric field \((E=2.3\cdot 10^3\ \text{V/m})\) for KCl “analytical grade”: 1 — at frequency \(0.01\ \text{Hz}\); 2 — \(0.1\ \text{Hz}\); 3 — \(2\ \text{Hz}\); b) temperature dependence of the conductivity “\(\sigma\)” in KCl “analytical grade”: 1 — curve calculated from the data of Fig. 3, a by formula 11; 2 — experimental curve obtained from measurements of “\(\sigma\)” on direct current.
In the present experiment \(T_{\text{mech}}\) or
\[ \overline{T}_{z}^{R}=\Theta^\circ\cdot 1.0\cdot 10^{-9}\ \text{N m} \]
\[ T_0=3.37\ \text{s}. \]
Substituting the value \(\overline{T}_{z}^{R}\) into formula (11), we can determine “\(\sigma\)”:
\[ \sigma=\frac{\overline{T}_{z}^{R}\cdot \omega}{V\cdot E^2} \]
or, in final form, for a KCl “analytical-grade” sample,
\[ \sigma=10^{-11}\cdot \Theta\cdot \omega_{\text{opt}}. \tag{16} \]
Curve 2 (Fig. 3, b) expresses the temperature dependence of the electrical conductivity measured with direct current. Curve 1 (Fig. 3, b) was obtained from data on measuring the twisting angle of the specimen by the contactless method. Curves 1 and 2 (Fig. 3, b) in fact coincide both in magnitude and in slope. The activation energy of the process in both cases is equal to 0.92 eV.
Discussion
Studying the temperature dependence of “$\sigma$” and $\operatorname{tg}\delta$ by the contactless method is of considerable interest from the standpoint that relaxation processes characterized by the relaxation time “$\tau$” are mainly a function of temperature.
Equations (12–13) show that the torque $T_z^R$ arising in a spherical specimen has a purely relaxation character, regardless of whether the imaginary part “$\varepsilon^*$” or the electrical conductivity “$\sigma$” is responsible for the relaxation process. Both $\varepsilon''$ and “$\sigma$,” when one of them predominates, have the same character of frequency dependence; that is, both “$\sigma$” and $\varepsilon''$ vary according to one and the same law:
\[ \frac{\omega\tau}{1+(\omega\tau)^2}, \]
where $\tau$ is the relaxation period of the process. According to [9],
\[ \tau=\tau_0 e^{-\frac{E}{KT}}, \tag{17} \]
where $E$ is the activation energy,
$\tau_0$ is the frequency factor.
Naturally, formula (12), in the form in which it is written, is not applicable for analyzing the torque of a specimen under continuous change of its temperature at a constant frequency. Let us express formula (12) through the temperature-frequency dependence in explicit form. To do this, we write the values $f=\omega/\omega_{\max}$ through the expression $f=\omega\tau$, where $\tau=1/\omega_{\max}$. From formulas (12) and (17) we have
\[ \overline{T_z^R} = VE_0^2 \left\{ \frac{1}{L[1+L(\varepsilon_s-\varepsilon_0)]} \right\} \times \]
\[ \times \frac{\omega\tau_0 e^{-\frac{E}{KT}}} {1+\left(\omega\tau_0 e^{-\frac{E}{KT}}\right)^2}. \tag{18} \]
There arises a need to check the correctness of formula (18). The results are presented in Fig. 4, which shows the dependence of $T_z^R$ on temperature at the frequency $\omega=2$ Hz. Curve 1 is experimental; curve 2 is calculated by formula (18). As can be seen, the calculated and experimental curves correlate well.
Fig. 4. Dependence of the torque $T_z^R$ on temperature at $f=\mathrm{const}$ (2 Hz): 1 — experimental curve; 2 — curve calculated by formula (18).
Thus, the theory developed by Tomoya Ogawa [3] is applicable for studying “$\tau$” and $\operatorname{tg}\delta\,\varepsilon''$ of ionic dielectrics of the ShGK type, both in the frequency and in the temperature interval.
As the authors [3–7] have shown, the contactless method can be used to study not only “\(\sigma\),” \(\operatorname{tg}\delta\), \(\varepsilon\), but also the anisotropy of substances, the Hall effect, and magnetic permeability.
By selecting specimens of the appropriate shape, the polarity of the external field and its frequency, it is possible purposefully to exclude the imaginary or real parts of the dielectric permittivity [3].
REFERENCES
- Hertz H., Wied Ann. Vol., 13, 1881, p. 226.
- Lertes P. Z. Physik, 4, 313 (1920); 6, 56 (1921).
- Ogawa T. J. Appl. Physik, 32, 1961.
- Gross L. E. and Groner C. F. IEEE. Transactions on Instrumentation and Measurement. Vol. IM-13, No. 4, 1964, p. 312—318.
- Gross L. E. and Groner. J. Appl. Phys., 40, 1969.
- Fittipaldi F. and Pauciulo L. J. Appl. Phys., 37, 1966.
- Carelli A., Fittipaldi F. and Pauciulo L. J. Phys. and Chemistry of Solids, 29, No. 2, p. 297—299, 1967.
- C. Kittel, R. Thomson. Physics of the Solid State. “Mir,” Moscow, 1969.
- A. A. Brandt. Investigation of Dielectrics at Superhigh Frequencies. Fizmatgiz, Moscow, 1963.