DIELECTRIC LOSS IN INSULATING SOLIDS CAUSED BY IMPURITIES AND COLOUR CENTRES BY J. VOLGER Philips Research Laboratories, N.V. Philips' Gloeilampenfabrieken, Eindhoven, Netherlands Received 28th January, 1957 Experiments at low temperatures on solids with various lattice defects have revealed the existence of dielectric relaxation phenomena due to these defects. The relaxation times are governed by activation energies far smaller than those normally found with diffusion or migration of ions. Typical measurements are given and discussed qualita- tively in relation to models of some lattice imperfections including colour centres. The study of dielectric properties of solids may, in some cases, give information concerning the imperfections of the lattice. The purpose of this paper is to report on a few investigations of dipolar relaxation due to imperfections of various types.It seems logical to distinguish between effects due to irregularities in the atomic arrangements only and those connected with " extra " electrons (or holes) in the solid. Though one cannot be very rigorous in this distinction, it has been used as a frame for this paper. One type of losses, however, will not be discussed, viz., the migration losses, i.e. both conduction and dipolar relaxation losses due to movements of ions over interatomic distances. These phenomena are related to diffusion. Migration losses are found, e.g., in alkali halides and glass, if the temperature is sufficiently high. The activation energy Q needed for the ionic jumps is of the order of magnitude of 0.5-1 eV.At low temperatures the ionic mobility is suppressed and migration losses can be excluded. Other dielectric loss mechanisms, however, may still exist. This is indicated by the experiments which will be discussed in the following. DEFORMATION LOSSES IN CRYSTALS Low-temperature losses have been studied in single crystals of clear quartz, mainly at frequencies of the order of 104 c/s. At room temperatune quartz exhibits a small loss factor, about 10-4, probably due to some mobile impurities such as Na+ ions. At lower temperatures the latter are frozen in and tan 6 decreases. Below 100" K, however, the tan 6 against T curves show sharp peaks depending upon the concentration and the nature of the impurities in the samples. Fig. 1 gives a few results obtained with crystals of different origins (cf.also ref. (1)). These low temperature losses can be characterized by specific relaxation times or, if we rely on the well-known formula (1) by specific sets of values TO and Q. These can be derived from an analysis of a number of tan 6 against T curves, each taken at constant frequency. Though this method is inferior to an analysis of tan 6 against o curves taken at constant temperatures, it quickly gives a valuable survey of the phenomena. The classical 7 = 70 ~ X P (QJkT), relaxation formulae in the approximation for small A€ € =€a + ~ 1 + w W Ae,k,, are applicable : (2) A€ or Em 1 +ow' tan 6 = - - (3)64 DIELECTRIC LOSS is the dielectric constant of the matrix in which dipoles with concentration N and moment p are embedded, giving rise to an increase A6 of the dielectric constant at zero frequency. The orientational polarizability depends on the internal field, affecting the factor f in but if N is small fequals 1 and AE < em.Then (em + 2)24nNp2 27kT ' E~ - EW = AE = FIG. l.-Sumey of tan 6 against T curves for a few monocrystals of clear quartz; measuring frequency 32 kc/s. ' A, specimen natural quartz from Brazil ; B, specimen synthetic quartz from Bell Telephone Lab. ; C, specimen synthetic quartz from General Electric Co., England. By comparison of results obtained with various crystals and using spectro- chemical analysis one can try to identify the different peaks. In fig. 1 has been indicated which impurity is thought to be responsible for the effects considered.To the temperature scale along the abscissa in fig. 1 an energy scale has been added, indicating the activation energies determining the relaxation times. This is permissible because the temperature T,,, where a maximum of the tan8 against Tcurve is found, is proportional to Q according to TO is found to be 10-13 sec for all processes found in clear quartz. p is of the order of 1 D. A further discussion of these relaxation processes, governed by such small values of Q (of the order of 0.1 eV or smaller) requires some model of the atomic situation at the lattice defect or built-in impurity. As has been remarked already,J . VOLGER 65 the ionic mobility in the usual meaning of the word is suppressed at the low temperatures applied. Our hypothesis is, however, that certain ions in the dis- turbed lattice maintain (e.g.by variation of valency angles) a possibility for dis- placements, though far more restricted and not exceeding 1 A, say, within the shell of their surroundings. Closely situated positions of relatively lowest energy might still occur. If the poten- tial barrier between these is not too high, dielectric loss may be expected at low temperature, provided the tran- sitions are electrically active. A proper name for such losses is perhaps defornzation losses, as they are as- sociated with slight local deformation rather than rearrangement of the lattice. It seems to us that in the experiments discussed we are dealing with such deformation losses, the more so as the dipole moments are small.Zt is useful to compare the di- electric investigations with recent measurements of the internal mecha- nical damping in a synthetic quartz crystal by Bommel (e.g. ref. (2)). Fig. 2 reproduces the published curve. The peak at about 55" K has been analysed as due to a relaxation pro- cess which is characterized by TO = 10-13 sec and Q = 0-056 eV and thus may be regarded as originating from the same lattice impurity which caused the dielectric effect characterized by these quantities, i.e. the peak at 38" K in fig. 1. lo+ 3 FIG. 2.-Internal damping in synthetic quartz crystal, measured by Bommel, e.g. re€. (2); measuring frequency 5 Mc/s. The effect at about 55" K is related to the peak at about 38" K in fig. 1. The effect at about 20" K is not considered in our paper.Theoretical work on the structural features of lattice defects (cf. also ref. (3)) and on the relaxation mechanisms, both dielectric and mechanical, in the disturbed lattice would be of a great value. DEFORMATION LOSSES IN GLASSES An experimental investigation of the low-temperature dielectric losses of various glasses, especially silica glasses, has recently been carried out by Stevels and the author.3 In these experiments ionic migration could again be excluded. At temperatures below 100" K the glasses showed losses of the deformation type, characterized by small activation energies, spread over a broad distribution function, in contrast to what had been found in quartz crystals. A typical tan 6 against T curve is shown in fig. 3. It has been proved that, generally, the deformation losses in glasses are related to the alterations of the SiO2 network, e.g.by the incorporation of extra oxygen or the substitution of silicon ions. Even fused silica shows deformation losses, probably caused by the presence of aluminium, which is an almost inevitable impurity. Boro-silicate glasses and glasses containing no silicon at all, such as phosphate glasses, also exhibit the low temperature loss under consideration. The same holds for silicates with structures like organic polymers, such as dimethylsiloxane (silicone) in the low-temperature glassy condition.4 Here the C66 DIELECTRIC LOSS deformation losses might be due to orientational relaxation of the side-chains of the molecules, which process may be governed by a small activation energy, as is so often used in restricted molecular rotation models.It should be noted that internal mechanical damping, analogous to dielectrical deformation losses, has been found in glasses, e.g. by Fine and co-workers 5 in fused silica and by Marx and Sivertsen 6 in commercial silicate glass. It is useful to try an analysis of the tan 6 against T curves in terms of distribution functions of activation energies. Let us return to fig. 3. According to a theory 6 0 0 5 01 015 0 2 0 2 5 0 3 035 0 4 045 0 5 0 5 5 Q (ev) FIG. 3.-Tan 6 against T of a glass specimen containing 70 mole % Si02 and 30 moIe % BaO ; measuring frequency 32 kc/s. The curve may also be considered as the distribution function f (Q) for the activation energies Q present. given by Gevers and du Pr&7 such a curve may be regarded as a representation of the distribution function f ( Q ) , as f ( Q ) is proportional to tan 6 with the T-axis interpreted as a Q-axis.The proof is based on the assumptions that f( Q) is a broad flat function and that p and TO do not depend on Q, such as to enable the integration to be carried out simply as follows : = nkTAf(Q)/2 with Q = - kT In 070. (7) In theory, A is inversely proportional to T. In order to adjust the Q-scale along the T-scale the value of WTO, i.e. of TO should be known. It appears that in glasses TO is again of the order of the inverse lattice frequency, perhaps somewhat greater than 10-13 sec which, however, is the value used, as it is hard to determine it exactly. Three remarks can be made : (i) As Imf(Q)dQ = N, the value Np2 can be derived from .oJ . VOLGER 67 For this glass eW = 6.1 and one finds from the curve after correction for the migra- tion losses at the high temperature side along the dashed extrapolated line, p2N = 8 x 1019, where p is expressed in D and N in cm-3. So if N = 1022 cm-3, which seems a reasonable figure for this glass, p would be of the order of 0.1 D only. (ii) The mean Q-value is 0.155 eV with a standard deviation of 0.090 eV, but the mode of the distribution function is 0.105 eV. (iii) Hence the distribution function is far from Gaussian. It is not logarithmic- normal either, but the 2/Q values are distributed almost normally. It is not yet clear whether much information can be derived from this feature. If, however, Q might be considered as being proportional to the square of a configura- tional co-ordinate x, characterizing the positions of the neighbouring atoms, it would mean that x is distributed normally around a mean value XO.DIELECTRIC LOSS IN INHOMOGENEOUS CONDUCTORS Solids may be rather good insulators for d.c. but may still contain insular regions of good conductivity. It has long been known (Maxwell, Wagner) that inhomogeneous materials show dielectric relaxation. This symposium on the molecular mechanism of rate process in solids is perhaps not the proper place for a discussion of these somewhat trivial effects. It might be useful, however, to give some consideration to this point, as it might elucidate certain conclusions of investigations with polycrystalline or ceramic substances.The disordered zones between adjacent crystallites are often barriers of rela- tively high resistivity. A simple linear barrier layer model with one thickness parameter proves to be satisfactory as a basis for a phenomenological theory.s9 9 Its results are that both overall conductivity and apparent dielectric constant vary with frequency according to simple Debye-like dispersion formulae, and that where G is the electrical conductivity, and where the subscripts s and denote values for d.c. and a.c. of infinite frequency respectively. €0 = 8.86 x 10-12F/m if Giorgi units are used, and If the intrinsic dielectric constant is the same for the barrier layer (1) and the bulk material (2), if the relative thickness dl of the layer is very small compared with unity, and if strong dispersion is found for (T, (9) and (10) can be written as with The ratio q/q is small and can be neglected, so (11) is almost identical with the usual relaxation loss formulae.An analysis of the dielectric loss angle in these inhomogeneous conductors gives in fact information on the conduction inside the well-conducting regions (grains). The curve for tan 6 is flattened and its maximum value is reduced if a distribution of T-values is present. In fig. 4 recent measurements of the loss factor of ceramic Fez03 are given. The sample had kindly been prepared by Mr. van Oosterhout of this laboratory. It contained 10-2 % Ti and had been fired in such a way that barrier layer resis- tance occurs. From the shift of peak temperature with measuring frequency it is derived that 0 2 1 ( T Z ~ exp (- Q/kT) with Q -= 0.20 eV.68 DIELECTRIC LOSS COLOUR CENTRE DIPOLE LOSSES Many a solid becomes coloured after irradiation by X-rays or fast electrons.The colour centres formed are spots of irregular charge distribution in the lattice where an electron or an electron hole is trapped. What dielectric properties do such centres exhibit? On the basis of a hydrogen-like electron orbit model the polarizability of the centre can be calculated, as has been done, e.g. for donor centres in germanium,lo but this is essentially " atomic " polarizability and orientational dipolar relaxation is not involved. However, irradiated quartz crystals (smoky quartz) investigated at low tem- peratures show marked dielectric losses of the relaxation type, which are neither manifest before irradiation nor after thermal bleaching of the specimens and which are most probably due to colour centres formed from impurities already present in the crystal before irradiation.1 Fig. 5 gives an example of the tan 6 against T curves found.A magnetic field of 3000 oersteds did not affect these losses. 4 0 0 D O E 200 100 0 0 FIG. 4.-Tan 6 against T and E against T of ceramic Fez03 containing some excess Fe; measuring frequencies 1 kc/s and 32 kc/s. We do not yet know with satisfactory precision how the static polarizability of the centres, nor how the relaxation time T varies with temperature. If we try to characterize r by a Boltzmann exponential the activation energy is of the order of 0-01 eV and the pre-exponential 70 is many orders of magnitude greater than 10-13 sec.From estimations of the concentration of centres it would follow that the dipole moment is far greater than in the case of the deformation losses dis- cussed in the preceding sections ; it might be about 10 D. An interesting feature is the partial disappearance of the loss peaks found in the clear crystals, after irradiation, as is clear from fig. 6 where the 38" K peak in the curve for the ir- radiated sample seems to be considerably reduced. Natural smoky quartz exhibits essentially the same effects. From the properties mentioned it is obvious that the nature of the colour centre dipole losses is quite different from that of the deformation losses. Our hypothesis is that we are dealing here with effects mainly due to the trapped electrons or holes themselves and that the dipolar relaxation indicates deviations from the simple hydrogen atom model for these.11 That such deviations occur is also known from paramagnetic resonance experiments.The trapped electron may in some cases, e.g. in F-centres in KCl, be considered as attached in turn to one of the cations which in a perfectly symmetrical way surround the anion vacancy.12 This leads to a model in which the electron or hole, at least at low temperature, is supposed to find energetically equivalent positions around the attracting centre proper, which positions could be stabilized by slight deformationsJ. VOLGER 69 of the lattice as a result of Coulomb forces, especially so when polar binding of the lattice occurs.The electrically active transitions between these equivalent positions can then be considered as a rate process (with small activation energy, apparently), though a tunnel effect may still play some role. FIG. 5.-Effect of irradiation upon low-temperature dielectric losses of monocrystal of natural quartz from Brazil. The peaks at 50°K and 60°K are probably due to the presence of alkali ions (lithium). t I I 1 I80 FIG. 6.-Effect of irradiation upon low temperature dielectric losses of monocrystal of synthetic quartz. The peak at 38" K is due to the presence of aluminium.70 DIELECTRIC LOSS In glasses, though colour centres of the same kind and in appreciable con- centration can be formed by irradiation, no trace of colour centre dipole losses has been found.However, the essential feature of the model above, i.e. the perfect symmetry, is absent in the glass. DONOR CENTRE DIPOLE LOSSES IN SEMI-CONDUCTORS In semi-conductors the free electrons or holes are at low temperatures trapped at the donor or acceptor centres and here again we may look for deviations from the spherical charge distribution. Recent dielectric investigations at low temperature FIG. 7.-Tan 6 against T curves and E against T curves at two measuring frequencies for Fe203 with some excess Fe (left-hand curves for 32 kc/s ; right-hand curves for 100 kc/s). The variations at about 30" K correspond to the dispersion phenomenon shown in fig. 8, and are due to electrons bound at the donor centres. FIG. 8.-Tan 6 and E at 20" K as functions of the frequency, for Fez03 with some excess Fe.The dashed curve is a single Debye curve with the same maximum value as the experimental one. on various semi-conductors indeed seem to indicate such deviations. As an ex- ample, results of measurements on Fez03 with excess Fe, will be discussed. A ceramic sample has been prepared by Mr. van Oosterhout and was, apart from its non-stoichiometric composition, very pure. The effects found in it were not shown by a sample of stoichiometric composition fired from the same raw material.J . VOLGER 71 Fig. 7 gives tan 6 against T curves and E against T curves for two different frequencies. It is seen that E varies in the dispersion regions in accordance with the dispersion formulae (2), though somewhat flattened.AE decreases with increasing temper- ature. (At about 100” K a new phenomenon sets in, but this is an inhomogeneity effect due to the increased conductivity, such as has been dicussed earlier.) From the curves it would appear that AE does not vary exactly proportionally to T-1, but rather like T-4. A provisional analysis of the r-values on the basis of (I) gives a Q-value of about 0.005 eV and TO rn 2 x 10-7 sec. This result may throw some light on the conduction mechanism in oxide semi-conductors like Fe2O3. It has often been suggested that a conduction electron in Fe2O3, once loosened from the donor centre “jumps ” from the potential well at one iron ion to that of the next ion, every time changing an Fe3f into a Fe2+ and vice versa. This, however, as is indicated by our dielectric measurements, seems also to be the case for the trapped electron in the vicinity of the centre, with this difference that the trapped electron is not free to move away into the lattice.Its transitions may be regarded as embryonic conduction. The mobility of the loosened conduction electron can be estimated on the basis of this picture. If the relaxation time for the transitions of the trapped electron may be used and if numerical factors of the order of magnitude 1 are disregarded, the mobility p is Fig. 8 gives tan 6 and E as functions of the frequency at 20” K. p =$/ekTr, (1 3) which is a strongly temperature-dependent quantity. p would be very small (about 10-5 cm2/V sec at room temperature) but this result is not unreasonable in comparison with the conclusions of electrical investigations by Morin.13 The extrapolation of our T-values to higher temperatures, however, is subject to uncertainty . Thanks are due to Mr. D. Hofman for his most valuable help withthe measurements. 1 Volger, Stevels and van Amerongen, Philips Res. Reports, 1955, 10, 260. 2 Bommel, Mason and Warner, Jr., Physic. Rev., 1955, 99, 1894 ; 1956, 102, 64. 3 Volger and Stevels, Pliilips Res. Reports, 1956, 11, 470. 4 Volger, Supplhtent au Bulletin de l’lnstitut Infernational du Froid, aniiexe, 1955, 2, 89. 5 Fine, van Duyne and Kenney, J. Appl. Physics, 1954, 25,402. 6 Marx and Sivertsen, J. Appl. Plzysics, 1953, 24, 81. 7 Gevers, Philips Res. Reports, 1946, 1, 197, 297, 361, 447. 8 KOOPS, Physic. Rev., 1951, 83, 121. 9 Volger, Physica, 1954, 20, 49. 10 d‘Altroy and Fan, Physic. Rev., 1956, 103, 1671. 11 Volger and Stevels, Philips Res. Reports, 1956, 11, 79. 12 Kahn, and Kittel, Physic. Rev., 1953, 89, 315. 1; Morin, Physic. Rev., 1954, 93, 1195. Hutchinson and NobIe, Physic. Rev., 1952, 87, 1125.