HAL Id: jpa-00230059
https://hal.archives-ouvertes.fr/jpa-00230059
Submitted on 1 Jan 1990
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
BOUNDARIES IN ELECTRONIC AND SUPERCONDUCTING CERAMICS
D. Clarke
To cite this version:
D. Clarke. BOUNDARIES IN ELECTRONIC AND SUPERCONDUCTING CERAMICS. Journal de
Physique Colloques, 1990, 51 (C1), pp.C1-935-C1-943. �10.1051/jphyscol:19901147�. �jpa-00230059�
COLLOQUE DE PHYSIQUE
Colloque Cl, supplbment au n o l , Tome 51, janvier 1990
BOUNDARIES IN ELECTRONIC AND SUPERCONDUCTING CERAMICS
D.R. CLARKE
IBM Research Division, T.J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A.
Abstract The electrical properties of individual grain boundaries in a number of commercially important electronic ceramics and in superconducting yttrium barium cuprate are reviewed. In addition, a tech- nique of wide applicability for the preparation of bicrystal thin films by the templating of a grain boundary from a bicrystal substrate is introduced. Finally some of the notions pertinent t o the de- scription of grain boundaries in terms of crystal chemistry are discussed.
l
-
IntroductionA large number of electronic ceramics derive their unusual electrical characteristics from the properties of their grain boundaries. The general behavior of these boundaries has primarily been obtained by measuring the properties of a bulk, polycrystalline material and comparing them t o single crystals, thereby averaging over all the grain boundaries. Despite this limitation valuable information has been assembled from standard electrical measurements such as capacitance and current-voltage relations, and in ionic conductors by, for instance, complex impedance measurements. The latter technique has been especially revealing in separating the aver- age impedance behavior of grain boundaries from the grains themselves. However, in establishing the elec- trical properties of grain boundaries and trying to relate them to some of their other characteristics, techniques are required to measure individual and isolated boundaries. This is especially important in seeking to under- stand the behavior of electrical ceramics since the additives used to make the material often serve to promote densification (by the formation of a liquid phase a t the sintering temperature) and t o provide the necessary electrical doping. In some materials the additives fulfil1 both requirements, and in others it is far from clear which additive or combination of additives actually acts as the electrical dopant.
In this contribution the properties of grain boundaries derived from measurements of individual grain bound- aries, either made using bi-crystals or with probing methods across individual boundaries, are reviewed. From such measurements a clearer idea of the role of grain boundaries emerges, and in some cases, it is possible to distinguish between the behavior of the dopants added to enhance the electrical activity and those t o promote densification. Distinguishing the role of individual dopants is important if the design of improved materials is to be contemplated so that it will be possible t o deliberately select additives for the specific purposes of doping and/or densification. However, discerning the action of dopants is also important for an understanding of the electronic and atomic bonding, ie the crystal chemistry, which causes the observed electrical behavior. In il- lustrating these features, examples are drawn from work on the most commercially important electronic ce- ramics, with the exception of capacitors, namely the zinc oxide varistors, PTC (positive temperature coefficient) devices, MnZn ferrites, and the cuprate ceramic superconductors.
2
-
Measurement of Individual Grain Boundary PropertiesA number of methods for measuring individual boundaries have been employed but none has yet found widespread application. The.simplest, and most straightforward, is to place an electrical probe on either side of a boundary on a polished cross-section and make a two-probe I-V measurement. Such a method was used
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19901147
C 1-936 COLLOQUE DE PHYSIQUE
boundaries, and later to demonstrate that aging occurred with repeated pulsing. (A more recent elabc is to perform potentiometry using a scanning tunneling microscope /2/). Apart from the obvious lim that it is difficult to probe boundaries in fine grain (-1 pm) sized materials, the technique provides nc tional information about the boundary itself. Also, there is always the problem as to the extent th boundary is electrically isolated from other grains that might short or shunt the measurement current.
A significant, but technically more difficult, development was introduced by Laval and co-workers They used probes to measure the electrical properties of grain boundaries in TEM foils inside the ell microscope. In this way they were able to relate the voltage drop across any particular boundary crystallographic orientation. Studying MnZn ferrite materials /3/ they found the interesting result th voltage drop depended o n boundary misorientation; low Z boundaries had a voltage drop of <5mV, h boundaries had drops of >IOmV and boundaries covered in glass exhibited intermediate drops. A vari this method was introduced in the work of Olsson and Dunlop / S / investigating a commercial varist ramic. Prior t o thinning a disk of the material for TEM examination they deposited a metal through a m as to produce a set of multiple electrodes, each with a slightly different shape. Thinning from the opposit to preserve the electrode structure, they were able t o measure the I-V curves of individual boundaries were subsequently characterized in the TEM.
An alternative approach, which promises to be of wider applicability and offers the possibility of exar boundaries of a controlled crystallographic character, has recently been developed in our laboratory method relies o n making a bicrystal substrate of, for instance, an insulating material but having a pre mined crystallographic misorientation and then growing the desired material epitaxially on the sub (Figure l ) . The substrate then acts as a template for the depositing film with the grain boundary being cated as it grows. It is then possible to dope the grain boundary by diffusing the dopant species down the boundary or alternatively, a t least in principle, from the grain boundary in the bicrystal substrate. S1 approach has found its first application in the measurement of the transport critical currents across boundaries in superconducting yttrium barium cuprate. The technique is now also being applied to the ination of boundaries in other electronic ceramics. A variant on this technique is to form a bicrystal : by promoting grain growth in a sintered ceramic, for instance using controlled orientation "seeds", and the grains have coarsened sufficiently, cut out the large grain regions containing the grain boundary.
\
GRAINBOUNDARY
Fig.1 - Schematic diagram of a thin film bicrystal grown epitaxially o n a bicrystal substrate to templa boundary of characterized crystallographic misorientation into the bicrystal film.
3
-
ZnO Based VaristorsVaristors have become commodity electronic components in the years since their discovery by Matsuoki He found that zinc oxide powders sintered with additions such as bismuth oxide exhibit unusually largr linear I-V behavior, often expressed as I = P
.
Two features of the characteristic behavior are of par) interest. the origin of the varistor slope, alpha, and the shift of the leakage region to higher currents onliquid phase is wetting, the original suggestions were that the non-linear electrical behavior was due to the presence of a bismuth-rich intergranular phase a t the zinc oxide grain boundaries. ~ l t h o u ~ h a proportion of the grain boundaries are indeed found t o contain such a n intergranular phase, high resolution transmission electron microscopy demonstrated that most did not. Instead they were found to be enriched with bismuth.
Although commercial varistor compositions often contain a large number of elements, they all tend to contain bismuth o r some other rare-earth such as praseodymium. Considerable clarification as to the role of bismuth oxide came from some early studies by Morris and Cahn /7/ who made simple binary mixtures of zinc oxide and bismuth oxide. They found the resulting sintered material exhibited varistor behavior, although with a smaller exponent than that of commercial material, even in compositions for which the bismuth was below the solubility limit and with no intergranular phase. This work clearly implicated bismuth segregation to the grain boundaries a s being responsible for the varistor characteristics, a conclusion substantiated by the analytical electron microscopy studies of this author /8,9/.
The conclusion that the varistor behavior is due to the presence of a charge distribution at the grain boundary resulting from a segregation of bismuth ions to the boundary was further reinforced by the experiments of Lou /10/ who showed that sputtered layers of ZnO separated by Bi also showed a varistor characteristic. How- ever, recent experiments by Olsson and Dunlop / 5 / , in which the I-V curves for individual grain boundaries were measured and then examined in the TEM, have demonstrated that the phenomenon is somewhat more complicated. They found that boundaries with and without an intergranular phase exhibited varistor behavior, although the onset voltage was lower for the boundaries with glass and the slope was lower -33 than that of the segregated boundary -66 / l l / . The corresponding voltage drops a t the two types of boundary were 3.2 and 3.6 eV . This work also clearly demonstrates that the boundary region as a whole determines the electrical behavior as distinct from that at the boundary plane.
Examination of the bismuth concentration at the grain boundaries has also provided some insight into the long term degradation and aging of varistors. Both Chiang et al. /12/ and Olsson and Dunlop /13/ have shown that after prolonged application of a d.c. voltage (to simulate aging), the bismuth concentration profile is shifted in the direction of the applied field and away from being peaked at the grain boundary dcfined crystallographically. At the same time, although the leakage current had increased the actual varistor charac- teristic was unchanged. Such a finding suggests that it is the potential barrier that determines the varistor be- havior rather than its location at the grain boundary. If this is so then the function of thc grain boundary is simply to provide a high diffusivity internal surface that facilitates the distribution of potential determining ionic species. What remains to be understood is how relatively minor additions over and above thc main dopant can further increase the value of the exponent alpha.
4
-
PTC CeramicsThe majority of PTC ceramics are based o n barium titanate but with small additions of aliovalent ions. Barium titanate itself is an insulator having a ferro-electric transition at 130°C but becomes semiconducting by the substitution of trivalent ions such as Y, Sm or La for barium. When appropriately prepared ceramic samples exhibit a very large PTC effect at the ferroelectric transition temperature, often with the resistance increasing by many orders of magnitude. The fact that the PTC effect is associated with thc grain boundaries was dra- matically demonstrated by Goodman /14/. He showed that a single crystal of B ~ , , , , S ~ . , , T i O , , which has a negative temperature coefficient, exhibited a large positive temperature coefficient when ground up and sintered to form a polycrystalline sample. The range of the PTC effect and the temperature of the onset are routinely manipulated in the manufacture of these materials by a combination of substitutional doping and by heat treatment on cooling from the temperature of densification.
The established explanation for the P T C effect is that, as originally proposed by Hewyang /15/, there is a double-sided electrical barrier at the grain boundary with a barrier height that depends on both the charge and inversely on the dielectric constant. H o d v e r , the data from some PTC compositions does not appear t o be in agreement with Heywang's barrier layer model. The reason for this is not well understood but may well be because these materials have a discrete intergranular phase at the grain boundaries. This is certainly the case in the materials examined by Nemoto and Oda, where a siliceous intergranular phase, approximately 1 pm thick, was present /16/, and also in the work of Ihrig in Sb-doped BaTiO,. Indeed, doped BaTiO, compos-
Cl-938 COLLOQUE DE PHYSIQUE
that a discrete intergranular phase may be expected. In such cases a more appropriate description m:
a space charge limited boundary layer as proposed by Nemoto.
5
-
Cuprate SuperconductorsIn marked contrast to the behavior of conventional superconductors, grain boundaries in the recently ered cuprate superconductors, at least at the present stage of their materials development, behave nc crease the transport critical current but rather as weak links. In conventional superconductors it is und that the grain boundaries act to pin fluxons and so the materials design strategy has been to decrease tl size and introduce a high density of interfaces. In the new superconductors the transport critical currc orders of magnitude smaller than that attainable in both single crystals and in epitaxial, single crysti The application of an external magnetic field further deceases the attainable Jc in bulk material.
Considerable circumstantial evidence for the grain boundaries acting as weak links has existed for sor but the most direct evidence comes from the direct transport measurements across individual boundarie experiments carried out a t IBM. The approach taken was to grow bicrystal thin films, by epitaxy on a b substrate, and then pattern the films (by laser ablation and photolithography techniques) t o make the c pads and junctions so as to be able to form four-probe measurements, both across the grain boundarq the grains on either side. The majority of the experiments were conducted on [OOl] oriented grains of barium cuprate by epitaxial growth on [ l 1 l ] oriented strontium titanate substrates. In each case the conducting current flowed parallel t o the C u - 0 planes of the cuprate crystal structure and perpendicula plane of the grain boundary. The experiments have had three principal results: The demonstration th;
boundaries d o indeed act as weak links; That in an cxternal magnetic field they can behave as SQUIL the critical transport current is dependent o n the crystallographic misorientation across the grain b o ~ Although relatively small misorientations
(>
10') cause a drop of about 30 in the critical current dens actual values are still quite large and significantly larger than that attainable in bulk, polycrystalline Imaterials. For instance, at a misorientation of -20' the grain boundary critical current density is A/cm2. This compares with the best aligned microstructure, that produced by melt processing, o x103 A/cm2 over a length of
-
1 cm, and of-
103 A/cm2 for bulk ceramic samples. It is, as yet, f;clear what is actually limiting the attainable critical current densities of the latter bulk ceramics. A intergranular phase can probably be ruled out since the common impurities that lead to intergranular in other ceramics are readily soluble in the yttrium barium cuprate crystal structure. (The exception i material is off-composition in which case an intergranular phase consisting of the liquid phase can be fc More subtle is the possibility of incomplete formation of the superconducting phase o r non-stoichiorr the grain boundaries. Some evidence for the former comes from our recent work on ceramic proces yttrium barium cuprate /18/. In this work the decomposition rate of barium carbonate was identifiec fecting both the rate of densification and also, if its decomposition products could not escape, could le lowering of the superconducting transition temperature (as measured magnetically). A barium carbon:
species has been detected o n the (intergranular) fracture surfaces by XPS/UPS and, more recen electron energy loss spectroscopy from grain boundaries in thin TEM foils made from bulk ceramics.
Very little is known about the detailed atomic structure of grain boundaries in yttrium barium cupratc attention has been focused on the twins in the structure (presumably because there were a number 01 not yet confirmed, relating the twin density t o the superconducting transition temperature) and o n ascer whether a distinct intergranular phase was present (to account for the low critical currents in polycry:
materials). In addition, microscope studies are limited by the susceptibility of the material t o both e irradiation damage and environmental degradation. The latter is now established as being due t o a Ic action with carbon dioxide and moisture to form a carbonate-like phase a t the boundary. Neverthele important first steps have been taken; one is the demonstration that small crystallites, growing in a ( Ba rich melt, will rotate to adopt preferential, special rnisorientations /18/. The other is that low angl boundaries are observed to contain dislocations whose Burgers vectors account for the angular misoriel of the boundary /19/. F o r instance, low angle [l001 tilt boundaries in yttrium barium cuprate consis array of dislocations having a closure failure of 1.17 nm, the lattice parameter of the unit cell in the [ 0 rection. At misorientations of greater than -5' the dislocations appear t o have dissociated into three p each with =1/3 [001] (in accord with the expectation that this lowers the elastic energy density).
Burgers vector corresponds to the repeat distance of the underlying perovskite unit cell. At still large]
Fig. 2
-
High resolution transmission electron micrographs of a) 5" and b) 7.5" tilt boundaries in yttrium barium cuprate. In both the boundary is parallel to the [001] plane in one of the grains. In the 5" boundary each dislocation has dissociated into three partials separated by a distance p corresponding to c/3. In the 7.5' boundary the defects appear to have amorphous cores. (Photographs courtesy of D. Smith and M.Chisholm
Cl-940 COLLOQUE DE PHYSIQUE
cores have an amorphous appearance. Such an observation is unprecedented but may well have imp consequences for understanding current transport across high angle grain boundaries as well as for the g structural description of high angle grain boundaries.
6
-
Crystal Chemistry of Grain BoundariesAlthough incomplete, the preceding summaries of grain boundaries in.a number of important electric ramics is representative of the apparent lack of any direct relationship between electrical properties to boundary structure and the way in which electrical dopants are accommodated a t or near t o a grain b o u ~ Whilst work seeking a connection between these will continue, it is perhaps worthwhile to contemp somewhat different approach.
In recent years there has been considerable attention given t o the crystal structure of grain boundaries.
was the work describing the structure of grain boundaries in terms of a small number of distinct polyl units. This description was elaborated in terms of periodically repeating structural units /20/ and mc cently in terms of the quasi-crystalline packing of structural units /21/. In metals such packings probablj little consequence other than to fill space and t o provide accommodation of the overall crystallography grain boundary (the bicrystallography of Pond /22/). However, the existence of such structures ma!
have important consequences for the crystal chemistry of boundaries in ionic and covalent solids since in solids the structure will, in some senses, dictate the possible electronic bonding. The importance of c chemistry is recognized t o be very much a nascent concept but one that may help clarify the electrical a(
of dopants, the types of grain boundary that can be most effectively doped and the types of site that ds ions might occupy.
The starting point for a description of grain boundaries based o n their crystal chemistry is to assume th, polyhedral arrangement of atoms a t grain boundaries, discussed in terms of hard sphere models for met also appropriate t o the description of boundaries in ionic solids. This assumption appears reasonable c basis of strong repulsive nature of the Coulombic interaction between like ions and is reinforced by the puted grain boundary structures in NiO by Duffy and Tasker /23/. Eight possible polyhedral structural based on the assumption that these are the smallest polyhedra into which no atom can be fitted interst have been described by Ashby e t al. /24/. Pond et al. /25,26/ identify five such units (some of whic be compounded to a total of eight). In cubic metals the polyhedral holes are triangulated, fol Archimedian void coordination polyhedra. These voids may be described a s the tetrahedron, the octahe the trigonal prism, the capped trigonal prism, the square prism and the pentagonal bipyramid. In ada other, irregular, polyhedron are also observed t o exist in computer simulations of metals in greater propc when any but the raost symmetrical orientations are examined. These polyhedral units are now assumc a first assumption, t o be the coordination polyhedra for binding a dopant substitutionally or interstitial]
boundary in ionic solids. The question becomes whether the rules of crystal chemistry can help to pi where a dopant ion will reside and what its contribution to the boundary charge might be. The implic sumption behind much of the work in the literature is that o n account of their size, and so as t o minimiz elastic strain energy, dopant ions are located either interstitially in polyhedral holes formed a t the boundary o r substitutionally a t sites immediately adjacent to the boundary plane. Such qualitative, intui based, assumptions are, however, of limited value in elucidating dopant sites because they describe e strain but lack any bonding information. Therefore, it is proposed that additional insight may be gain1 applying the principles of crystal chemistry, as codified in Pauling's rules, to grain boundaries. Accordi this view, dopants (segregants) will only be found in grain boundary sites for which standard rules of c.
chemistry are obeyed.
The first rule states that for filling of space by coordinating polyhedra each cation must be surrounded coordinated polyhedron of anions (and vice versa), with the cation-anion distance in the polyhedra deterr;
by the radius sum and the coordination number of the cation by the radius ratio of each species. This appi treats the ions as rigid spheres, each of constant radii, and packs them according to their relative size polyhedral descriptions of grain boundaries conform t o this rule, in that they fill the grain boundary spacl to the extent that they have been couched in terms of packing of hard spheres, but the approach neglecl possibility that the ionic size depends o n the ion coordination.
The polyhedral description of boundaries is clearly appropriate in the simplest boundaries. For instance, in many oxides, such as strontium titanate, aluminum oxide and zinc oxide, where the crystal structure may be conveniently thought of as close- packed oxygen ions, with the cations in tetrahedral o r octahedral sites. There are clearly certain boundary orientations in which the oxygen sub-lattice will continue unaffected across the boundary plane with only the cation sites misoriented. An example of such a boundary is the basal twin boundary in alumina, and at an extreme, stacking faults in alumina which are found t o be faults in the cation sub-lattice but not in the oxygen sub-lattice. Consideration of boundary space filling polyhedra as sites for doping as a function of boundary misorientation is consistent with the findings of Pinet-Berger and Lava1 /3/, referred to earlier, that high Z boundaries in a MnZn ferrite exhibited a larger barrier than low Z boundaries.
If it is assumed that the number of larger polyhedra per unit area of boundary increases with crystallographic misorientation, and that dopants can only be inserted into those polyhedra that contain a sufficiently large interstice, then the charge per unit area, and hence potential of the boundary, will also increase with boundary misorientation.
The second rule, commonly referred t o as the electrostatic valence principle, states that the total strength of the valence bonds to an anion from the neighboring cations in a polyhedron will be such that the net ion charge is equal to zero, namely the polyhedron has no net electrical charge. The computations by Duffy and Tasker of the [001] tilt grain boundaries in NiO are consistent with this rule. The authors showed, for instance, that the energy of a (310) twin boundary could be lowered by removal of a discrete formula unit per unit cell in the boundary plane, an operation in which the net charge was unchanged. Furthermore, it not only lowered the boundary energy t o less than that for the dissociation into free surfaces but also prevented the overlap of ions having the same charge when ionic radii were used t o represent the ions. Application of the valence sum rule to computed boundary structures leads t o some interesting insights, as may be seen, again using the computations of Duffy and Tasker as a n example. The undoped boundaries have of course n o net charge.
However, if one assumes that the ions in NiO maintain a coordination of six, calculation of the valence sum for each polyhedra in the (310) twin reveals a zero net charge whereas those in the (320) boundary alternate between being + 2 / 3 and -2/3. Thus, if a cationic dopant were to be inserted into the (320) boundary the local charge balance would indicate that the site in the center of polyhedron B would be preferred. However, since anions are generally larges than cations, the size of the polyhedron A is larger than that of polyhedron B, and so they differ not only in local charge but also in size. Thus, although a n appropriately sized cation could conceivably fit in the interstice of polyhedron A, it would not satisfy the valence sum rule and so there are apparently conflicting requirements for doping of this particular boundary.
Fig. 3
-
Calculated ionic positions at a (320) [001] tilt boundary in NiO. Although the apparent size of the two repeating polyhedral holes is the same, the larger size of the oxygen ions suggests that hole A would be the favored one for doping if size alone was the doping criterion. However, application of the valence sum rule indicates that the holes A and B have formal charges of +2/3 and -2/3 respectively. Thus, a cationic dopant would favor hole B, contrary t o that predicted by size. (redrawn after Duffy and Tasker).Pauling's third rule states that, the sharing between two anion polyhedra of edges, and particularly faces, de- creases the stability. This rule is a consequence of the fact that such sharing tends t o bring cations closer to-
Cl-942 COLLOQUE DE PHYSIQUE
is demonstrated by the work of Carter and Shaw on twin boundaries in a spine1 /27/. The (1 11) t1 special boundary in the sense that the anion sublattice is continuous across the boundary whereas tht sublattice is not. Two possible geometrical structures for the twin can be contemplated, one in wk octahedra containing aluminum ions share faces and the other in which they share edges. In conform Pauling's third rule the only boundary observed by the authors was the one in which the edges are sha Therefore Pauling's three rules can be used to give an indication and insight of how crystal chemis provide a means to establish the location of dopants at a grain boundary. Clearly, detailed computati necessary to establish the predicted dopant sites and their associated charge, but considerations such 2
outlined here provide a starting point for those computations as well as for interpreting the results of spectroscopies applied to intergranular fracture surfaces. In the absence of detailed computations presume to introduce a number of possible corollaries to the rules of crystal chemistry for grain bou~
One is that dopant ions will prefer sites in the boundary where their coordination is similar to that the, in crystals of their own oxides. Secondly, dopant ions will substitute into lattice sites forming the boul they can also locally adopt the same coordination.
Up to this point it has been assumed that the crystal structure of the grain boundary determines its cl state. However, it is conceivable that a dopant introduced into the boundary will alter the local crysta ture. Evidence for such a change has been seen in iron [001] twist boundaries where the introduction
% gold alters the primary dislocation structure from a square < l 10> dislocation network to two dl square dislocation networks of <110> and < l o o > type / 2 8 / . Similarly, bismuth segregation to bou in copper causes extensive facetting. The most direct evidence for a relationship between chemical cc tion and structure is undoubtedly that of "chemical twinningf', a concept originally introduccd by Hyc /29/. They envisaged a process in which a large proportion of a dopant or impurity ion is accommod a twin plane since a different, often larger site, with a different coordination is present at the twin plai in the adjacent crystal structure. Possibly the simplest example is reflection twinning on (1 13) of t h ~ structure in which cation sites must be eliminated in order to avoid overlap of like sites with the rer cation occupying the center of a trigonal prism formed by the nearest anion neighbors. In the most e case, a periodic repetition of the twin plane leads to the formation of a new compound, chemically anc turally distinct from the initial compound.
Acknowledgements
It is a pleasure to acknowledge helpful discussions with my colleagues Drs. D. A. Smith and R. H. J. H2 References
/ l / Einzinger, R., Ber. Dt. Keram. Ges., 52 (1975) 244.
/2/ Kirkley, J. R., Washburn, S. and Brady, M. J., Phys. Rev. Lett., 60 (1988) 1546.
/ 3 / Pinet-Berger, M. H. and Laval, J. Y., in "Ceramic Microstructures 86" edited Pask and Evans, 657.
/4/ Berger, M. H. and Laval, J . Y., this volume.
/ S / Olsson, E. and Dunlop, G. L., J. Appl. Phys., In Press.
/6/ Matsuoka, M., Jap. J. Appl. Phys. 10 (1971) 736.
/7/ Morris, W. G. and Cahn, J. W. in "Grain Boundaries in Engineering ~ a t e r i a l s " , edited by J. W. P et al (Claitors, Baton Rouge, 1975).
/8/ Clarke, D. R., J. Appl. Phys., 49 (1978) 2407.
/9/ Clarke, D. R., J . Appl. Phys., 50 (1979) 6829.
/10/ Lou, L. F., J. Appl. Phys. 50 (1979) 555.
/ l l / Olsson, E. and Dunlop, G. L., Proc. 2nd Int'l Varistor Conf., Schenectady, 1988.
/12/ Chiang, Y.-M., Kingery, W. D. and Levinson, L. M., J. Appl. Phys. 53 (1982) 1765.
/13/ Olsson, E. and Dunlop, G. L., "High Tech. Ceramics", edited by P. Vincenzini (Elsevier, Amste~
1987) 1765.
/14/ Goodman, G . , J. Am. Ceram. Soc., 46 (1963) 48.
/ 15/ Heywang, W., J. Am. Ceram. Soc., 47 (1 964) 484.
/ 16/ Nemoto, H. and Oda, I., Adv. Ceram., 1 (198 1) 167.
/18/ Smith, D. A., Chisholm, M. F. and Clabes, J., Appl. Phys. Lett.
/19/ Chisholm, M. F. and Smith, D. A., Philos. Mag. A59 (1989) 181.
/20/ Sutton, J. P. and Vitek, V., Phil. Trans. R. Soc., A309 1983) 1.
/21/ Rivier, N. and Lawrence, A. J. A., Physica B, 150 (1988) 190.
/22/ Pond, R. C., and Vlachavas, D. S., Proc. Roy. Soc., A386 (1983) 95.
/23/ Duffy, D. M. and Tasker, P. W., Philos. Mag. 47 (1983) 817
/24/ Ashby, M. F., Spaepen, F. and Williams, S., Acta Metall., 26 (1978) 1647.
/25/ Pond, R. C., Vitek, V. and Smith, D. A., Acta Cryst., A35 (1979) 689.
/26/ Pond, R. C., Smith, D. A. and Vitek, V., Scripta Metall. 12 (1978) 699.
/27/ Carter, C. B. and Shaw, T. M., Philos. Mag. 55 (1987) l . /28/ Sickafus, K. E. and Sass, S. L., Scripta Met., 18 (1984) 165.
/29/ Hyde, B. G., Andersson, S., Bakker, M., Plug, C. M. and O'Keefe, M., Prog. Solid St. Chem. 12 (1974) 273
