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2.6.5 Dielectric properties of materials

The absolute complex permittivity of a material is represented by the symbol , where = ′ − j″. This is related to the dimensionless relative complex permittivity _r, where _r = ′_r − j″_r, by the expression = _0r, ₀ being the permittivity of free space, a fixed constant given approximately by ₀ = 8.85 x 10⁻¹² F m⁻¹. In general, depends on temperature and, to a lesser extent, pressure. It is also frequency dependent, although ′ and ″ cannot vary independently with frequency, since their frequency variations are connected through the Kramers–Krönig relationship: a drop in ′ with increasing frequency is necessarily associated with a peak in ″. Except for exceedingly high applied fields, is independent of the magnitude of the applied electric field for all dielectric materials used in practice, excluding ferroelectrics.

A capacitor filled with a dielectric material has a real capacitance ′_r times greater than would have a capacitor with the same electrodes in vacuum. The dielectric-filled capacitor would also have a power dissipation W per unit volume at each point when, resulting from an applied voltage, a sinusoidal electric field of frequency f and root-mean-square value E exists at that point. This power dissipation is given by W = 2πfE²″. Thus ″ is a measure of the energy dissipation per period, and for this reason it is known as the loss-factor.

The complex permittivity is often represented in the Argand plane with ′ as abscissa and ″ as ordinate, giving a curve with frequency as parameter. The join of any point on this curve to the origin therefore represents the complex conjugate * of the complex permittivity where * = ′ + j″. Unfortunately, the use of the symbol * to represent complex permittivity is widespread and has become established in the literature, and care is needed if confusion over signs is to be avoided. The join to the origin makes an angle δ with the abscissa, such that tan δ = ″/ ′. Thus W may be rewritten as W = 2πfE²′ tan δ. Hence δ is known as the loss angle, and tan δ is known as the loss tangent.

The application of a sinusoidal voltage of root-mean-square value V to the dielectric-filled capacitor results in a current flow in the external circuit which leads the voltage by a phase angle or power-factor angle φ, where φ is the complement of δ. Thus, the power dissipation in the capacitor, given by IV cos φ, may also be expressed as IV sin δ. Since in most cases in engineering practice δ is small, sin δ tan δ and the power dissipation is given to a good approximation by IV tan δ. It should be noted that no such approximation is involved in the expression for W in the previous paragraph.

When the wavelength of electromagnetic radiation is in the optical region, the velocity v of propagation through a loss-free transmitting medium of refractive index n is given by v = c/n, where c is the velocity in free space. The velocity is also given by v = c/(μ_r′_r)^1/2 where μ_r, is the relative permeability. Thus for loss-free non-magnetic materials, for which μ_r = 1, _r′ = n². However, in general losses do occur, and the material is characterized by a complex refractive index given by = n − jk, where k is the absorption coefficient. Then _r = ², or _r′ – j_r″, = (n – jk)², from which it follows that _r′ = n² – k² and _r″ = 2nk. Nevertheless, when the loss is small, so that k << n, then _r′ n². The use of these relationships allows values of _r at high frequencies to be derived from optical measurements. As the frequency is reduced, specially designed interferometers (infra-red), free radiation methods (sub-millimetric wavelengths), wave-guides, coaxial lines and resonant cavities (centimetric wavelengths), and Q meters and bridges (radio frequencies to d.c.) have all been used. Time-domain spectroscopy, involving an analysis of the response of the medium to a step-function field, is capable in principle, and has had some success in practice, in giving a rapid measurement of over a very wide frequency spectrum.

The relative permittivity is directly related to the electronic, atomic and orientational polarization of the material. The first two of these are induced by the applied field, and are caused by displacement of the electrons within the atom, and atoms within the molecule, respectively. The third only exists in polar materials, i.e. those with molecules having a permanent dipole moment. Electronic and atomic polarization are temperature independent, but orientational polarization, depending on the extent to which the applied field can order the permanent dipoles against the disordering effect of the thermal energy of their environment, varies inversely with absolute temperature. All of these polarization mechanisms can only operate up to a limiting frequency, after which a further frequency increase will result in their disappearance. Because of the spring-like nature of the forces involved, this is accompanied by an absorption of the resonance type for electronic and atomic polarization, but for orientational polarization the disappearance, accompanied by a broader peak in the loss factor, is more gradual, because the mechanism involved is of the relaxation type, and may involve a broad distribution of relaxation times. Indeed, the decline in ′ may be so gradual that ″ may appear almost constant, and be correspondingly small, over a wide frequency range. This applies particularly to some polymers commonly used in engineering practice, many of which are polar. Those which are non-polar, usually with _r′< 2.5, show nearly constant values of ′ and ″ over the entire electrical frequency spectrum.

The frequency at which these mechanisms drop out is related to the inertia of the moving entities involved. Typically, electronic polarization persists until a frequency of about 10¹⁶ Hz, atomic polarization until about 10¹³ Hz, while the dispersion for orientational polarization may lie anywhere within a wide frequency range, say 10²–10¹⁰ Hz, depending on the material and its temperature. In addition to these polarization mechanisms, the existence of interfacial effects such as macroscopic discontinuities in the material, or blocking at the electrodes, causes the trapping of charge carriers, and such phenomena, as well as the inclusion in the dielectric of impurities giving rise to conducting regions, result in behaviour classified under the general heading of Maxwell–Wagner effects. These give rise to an effective polarization and associated loss, the frequency behaviour of which is similar to that of orientational polarization, with a dispersion region which may lie in the region of 1 Hz or lower.

When orientational polarization is operative, it is usually the dominant polarization mechanism present. The classical theory of this mechanism is due to Debye. For a single relaxation time τ, the variation of _r with angular frequency ω is given by the Debye equation, (_r − _∞)/(_s − _∞) = (1 − jωπ)/(l + ω²τ²), where _s and _∞ are the relative permittivities at frequencies much lower and much higher (but not high enough to involve any reduction in atomic or electronic polarizations) respectively than the anomalous dispersion region. Equating real and imaginary parts gives

          (′_r − _∞) / (_s − _∞) = 1/(1 + ω²τ²)   and   ″/(_s − _∞) = ωτ/(1 + ω²τ²)

If ″ is plotted against ′, the Cole–Cole plot results. This is a semicircle if the Debye equation is obeyed. Frequently experimental results yield a circular arc, rather than a semicircle, with its centre below the abscissa. Such behaviour can be expressed as a suitable distribution of relaxation times, though no satisfactory physical reason for doing so has yet been established. There is a variety of other shapes obtained in practice, such as the skewed arc in which the high frequency end of the arc approximates to a straight line. Anything other than a perfect semi-circle is now taken as evidence of co-operative effects within the dielectric.

The permittivity of many substances changes not only with frequency and temperature, but also with specimen age and history. Two specimens of nominally the same material may have significantly different permittivities because of different manufacturing processes, different amounts of oxidation, and different inclusions, some of which might have been deliberately introduced, e.g. anti-oxidants. For such reasons, tables of values should be used as an indication of the magnitudes to be expected, and not as a source of precise data which can be repeated by accurate measurements on particular test specimens, except in cases in which the physical and chemical state of both the reference material and the test specimen are very closely specified. The properties of ferroelectric materials depend on so many factors that it is inappropriate to include them in tables of data. Generally, they have permittivities of the order of a thousand, strongly dependent on applied voltage and temperature, and exhibit considerable power loss.

References

C. J. F. Bötcher (1973) Dielectrics and Static Fields, Vol. 1, 2nd edn, Elsevier Scientific Publishing Company, Amsterdam.
C. J. F. Bötcher and P. Bordewijk (1978) Dielectrics in Time Dependent Fields, Vol. 2, 2nd edn, Elsevier Scientific Publishing Company, Amsterdam.
V. V. Daniel (1967) Dielectric Relaxation, Academic Press, London.
H. Fröhlich (1958) Theory of Dielectrics, 2nd edn, Clarendon Press, Oxford.
Nora E. Hill, Worth E. Vaughan, A. H. Price, Mansel Davies (1969) Dielectric Properties and Molecular Behaviour, van Nostrand Reinhold Company Ltd., London.
A. R. von Hippel (1954) Dielectrics and Waves, Chapman & Hall, London.

Tables of relative permittivity and loss tangent

Temperature (t) is in °C, and frequency (f) in Hz. Temperature coefficient of _r′ is denoted by a = 10⁵ d_r′/_r′ dt and density in g cm⁻³ by d. For non-cubic crystals, the symbols , ||, indicate measurements with field respectively perpendicular to and parallel to the c-axis. Ranges of quantities are indicated by the numerical limits of the range, separated by a solidus. For commercial materials, the values should be regarded as examples only, since some vary greatly with composition and purity. This applies also to the loss angle of some pure materials, which may depend on traces of impurity. The ranges of _r′ and tan δ, however, are intended to indicate not these variations, but only the variation within the stated ranges of temperature and/or frequency. However, because data relating to different temperatures and frequencies often have to be taken from more than one source, even for what is nominally the same material, it is commonly impossible to be certain of the cause of the variations.

Solids