ionic conduction in space charge regions

Pergamon Prog. Solid St. Chem., Vol. 23, pp. 171-263, 1995

Copyright @ 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0079-6786195 529.00

0079-6786(95)00004-2

IONIC CONDUCTION IN SPACE CHARGE REGIONS

Joachim Maier Max-Planck-Institut fur Festkiirperforschung, HeisenbergstraBe 1,

70569 Stuttgart, Germany

1. INTRODUCTION

1.1. General Intentions

Although the significance of boundary layers for the general behaviour of electronic systems was recognized very early, in the case of ionic conduction the main emphasis has been placed on the bulk properties. This was very fundamentally connected with the search for “super-ionic conductors”.

With regard to those properties of ionic conductors that quite obviously involve interfaces, as in the fields of catalysis, photography, sensors and electrode kinetics, intensive efforts are being made at the moment to obtain information concerning boundary zones.

In spite of a few earlier publications, the full significance of boundary layers with respect to ionic conductivity was first brought out by the experiments of Liang [l] who made a systematic study of the electrical properties of the two-phase system LiI-A1203 and found that the ionic conduction was anomalously high in comparison with that of the pure phases. Since then a whole range of similar effects have been reported in the literature. Especially the group of J. B. Wagner et al. played a major role in this respect [2]. Such solid electrolyte systems are known as “composite electrolytes” or “heterogeneous electrolytes” * [3-51 and have been the subject of intense discussion at specialist conferences.

The purpose of this article is to summarize the experimental and theoretical approaches in this field with emphasis on the author’s work on defect chemistry in space charges in particular ([3,4,6- 19]) and to discuss it within the context of overall developments. In order to keep this article within limits, for more details the reader is referred to the original publications (for details see [3,4,6-191).

Particular emphasis is placed on presenting the concept of space charge, which provides a natural bridge between bulk properties and those of the neighbouring phase. It hence possesses general importance and is able to explain many aspects of conductivity in heterogeneous systems. In other words it is a major purpose to elucidate the general significance of a defect chemistry in space charge regions and its relevance especially for transport properties. It is not the aim to give an overview over conducting effects in heterogeneous systems. Thus, aspects which concern interfacial migration due to a high mobility within the core layer will be only marginally touched upon [20].

Three interfaces that are of general importance in this context will be treated:

(a) the ionic conductor/insulator interface (MX/A) [6] (b) the interface between two different ionic conductors (MXfMX’) [7,16] (c) grain boundaries (MX/MX) [9] (d) the ionic conductor/gas interface [17]

The treatment is set out in the following manner: Firstly the principles and quantitative considerations of defect chemistry at interfaces are presented. The second step consists of the calculation

* The term “heterogeneous electrolyte” [3,4] is more comprehensive and also includes, e.g. those pure polycrystalline materials the grain boundaries of which contribute significantly to the overall ionic conductivity.

171

172 J. Maier

of the (parallel and/or perpendicular) conductivity of a boundary layer and a discussion of relevant experiments. Then - in order to be able to describe practical systems - there is a discussion of the conductivity effects in dispersed systems on the basis of a simple distribution topology and comparison with the particular experimental data.

This applies not only to the majority charge carriers (ions) but also to the minority ones (electrons) [4,11] and allows us to establish a general defect chemistry of space charge layers. Finally these considerations - both experimental and theoretical - are also applied to nanosystems [IO, 121 in which these boundary layer effects occur to a greater degree.

In addition to the conductivity aspects, implications for heterogeneous catalysis [18], phase boundary reactions and phase transformation will be outlined. Before this, however, we will return to the experimental findings with respect to the two-phase effect and its impact on ionic conductivity.

1.2. The SigniJicance of “Heterogeneous Doping”

There are, in principle, two ways of optimizing ionic conductivity: a search can be made, on the one hand, for new compounds and structures, and, on the other hand, modifications can be made to given materials. The classical method to achieve the latter, involves homogeneous doping: Here suitable materials with aliovalent ions are usually dissolved in the matrix in order to influence the concentration of charge carriers. In an analogous manner we will refer to influencing the conductivity by “physical” addition of coexisting second phases as “heterogeneous doping” [3a]. While the effect of homogeneous doping is attributable to the fulfillment of local electrical neutrality, for heterogeneous doping it is, as will be demonstrated in detail, the deviation from local electrical neutrality that is of great importance. The similarities in principle and the differences between the two methods are discussed in detail below.

After Liang’s experiment similar enhancement effects were discovered in a whole range of moderate ionic conductors, principally Li, Cu and Ag halides [ 1,3,4,8,2 l-481 (recently some alkali and alkali earth metal halides too [24,49-571). Besides Al203 other oxides such as SiOz, CeOz, ZrOz and BaTi have been found to be also effective. These “heterogeneous electrolytes” are usually prepared by fusing the ion-conducting matrix material. Typical volume compositions (VA) com- prise lo-40 v/o second phase. Typical mean particle diameters (2fA) are less than 1 p, typical increases in conductivity are one to two orders of magnitude.

Figure 1 refers to the classical experiment by Liang [l]. However, the conductivity here is plotted against the volume fraction of Al203 in anticipation of our treatment [3]. It can be seen that the conductivity increases linearly to a maximum value, that amounts to about 50 times the original conductivity (pure LiI); the insulating effect of the second phase becomes apparent at higher concentrations of Al203 and the conductivity falls drastically.

Figure 2 shows the effect of heterogeneous doping with y -Al203 on a range of ionic conductors with appreciable cationic mobility.

As can be seen from Fig. 3 Shahi and Wagner Jr. [59] also found considerable conductivity effects in two-phase mixtures of two coexisting ionic conductors, namely within the miscibility gap of the system AgBr-AgI. A more detailed study of the system AgCl-AgI is described in Chapter 3 [lq.

A range of different cases can be expected in such two-phase mixtures, even when X-ray methods cannot detect any global reactions of the two phases (cf. Fig. 1,2). The most important ones are:

(1) The simplest case is that the underlying structure is maintained up to that atomic layer which forms the layer of contact. As a consequence all the materials parameters can be assumed to behave in a step function way. This case will be considered in the following.

(2) Close to the interface the materials parameters may change more smoothly due to a structural adjustment or to gradient energy effects. This includes also elastic effects.

(3) Impurities may be injected which are mobile themselves (e.g. protons) and /or change the concentration of mobile defects.

(4) Higher dimensional defects such as dislocations may be formed to compensate the interfacial misfit (partial equilibrium) or simply due to non-equilibrium conditions.

(5) Thin layers of a third phase may be formed, the restricted width of which may be due to thermodynamic reasons (interfacial thermodynamics, see chapter 2.5.2) or due to kinetic

Ionic Conduction in Space Charge Regions 173

data for LiI:At203

10 20 30 LO 50 60

v* 100 -

Fig. 1. The Li+-conductivity of LiI [I] plotted against the volume fraction of Al203 of insulator phase [6]. The dotted lines represent the characteristicsexpected according to random distribution

[58] and are discussed in Chapter 2.3.3.

1.8 2.6 1.8 2.6 2.2 3.4 2.4 3.2 2.0 28

103T-'/K-l

Fig. 2. The effect of heterogeneous doping of various ionic conductors exhibiting cationic mobility by y-Al3O3, (mean particle size = 0.06 pm). It can be seen that the slope in the region of increasing conductivity can scarcely be distinguished from that in the extrinsic region of homogeneous samples (contamination with high valency cations) 1131. The data of AgLA1203

are taken from the literature [26].

174 J. Maier

-5

-6

I PW,, I y AgBrs$ ’

I ,

AgI Q25 0.50 0.75 AgBr

mole fraction _ of AgBr

Fig. 3. Specific conductivities in the system AgBr-AgI at room temperature according to Shahi and Wagner Jr. [59].

I immobile impur

mobile impurities I

SC\ i I

structural (elas)ic/plastic) effects (core, space charge)

SC\ I I

inter - I PM= scl 1

!

Fig. 4. Possible effects at the interface between an ionic conductor and a second phase considered as “inert” [13].

reasons.

All the heterogeneities described above can have a double influence in that they

(a) provide a “new” kinetic pathway themseives and/or (b) influence the conductivity - basically by affecting the point defect concentration - in the

adjacent boundary zones.

In the simplest approximation (see I) which will be chiefly considered here, the region of adjust-


I

EL/eV -

Fig. 5. The solid line (bisector of the angle) shows that the effective activation enthalpy (Et) of the (sukiently strong) heterogeneously doped ionic conductors AgCI, AgI, AgBr, TICI, LiI approximately equals the migration enthalpy of the cation vacancies (h+; n ). Also included in the figure are, for the purpose of comparison, other characteristic data: migration enthalpy of the counter defect (h-; q ), formation enthalpy (AH”; plus) and intrinsic activation enthalpy of

conductivity (AH”/2 + t+h+ + t-h_; t: transfer number).

ment is the space charge layer (scl), i.e. the region of redistribution of ionic and electronic point defects.

While a more or less independent migration enthalpy should occur in case a), it is to be expected in case b) that, to a first approximation, the migration part of the activation enthalpy is essentially the bulk value (this assumes that the elastic effects are not too large).

The space charge concept as the explanation of the “alumina-effect” is strongly supported by the fact that i) these effects are boundary layer phenomena; in particular, impurity effects, i.e. homogeneous doping have been ruled out (see Chapter 2, in particular 2.5.3) while ii) notwithstanding the effective activation enthalpy of the enhanced conductivity comes very close to bulk migration enthalpies (of the metal vacancies in Fig. 5) in the cited examples. This is shown in Fig. 5 together with other activation data. The relevance of other explanations will be discussed later, in particular those by Phipps [41] and Dudney [60] (Chapter 2.5.3). Especially the connection between point 1) and the space charge concept will be discussed there.

Apart from these experimental effects the space charge concept is of general nature and represents the natural extension of volume-defect thermodynamics based on the principle of local electrical neutrality.

Carl Wagner was the first to use the space charge concept to explain conductivity effects in semiconducting two-phase mixtures [61]. For ionic conductor systems T. Jow and J.B. Wagner Jr. [32] have discussed a semiquantitative space charge model. The indeterminacy of important parameters (effective space charge concentration) in their model makes it impossible to calculate absolute values, or the temperature dependence of the effect. A further weakness lies in the fact that no explicit account is taken of the coupled interfacial phenomena which are - as will be described

176 J. Maier

below - those that make possible appreciable deviation from volume defect chemistry. (As far as the dependence on geometrical parameters is concerned, the reader is referred to Chapter 2.3.3.) Most theoretical publications [58,62-651 have addressed the calculation of the superposition of given conductivity pathways in an heterogeneous system.

In the following it is shown how to derive quantitatively profiles of the defect concentrations, the partial conductivity and the effective (measured) specific conductivity for various boundary problems by starting from local thermodynamic and electrostatic relationships. The relationships derived reflect the dependence on geometric parameters, on temperature and on interfacial activity. The conductivity of two-phase mixtures is calculated on the basis of a greatly simplified, but relevant distribution topology (in contrast to e.g. random distribution). A series of accompanying experiments confirm that in particular the theoretical results obtained with AgCl and AgBr, where the necessary data are available, are in quantitative agreement with the experiment. The dependence on the activity of the second phase, the dependence on concentration, particle size and temperature, the impedance spectra and, with additional assumptions, the absolute values can be given. Also the dependence on the component activity follows from the model. Far-ranging conclusions are possible concerning the defect-inducing interfacial mechanism. Predictions can be made concerning changes in the mechanism of conductivity (TlCl), advantageous strategies and limits with respect to further optimization, as well as concerning effects on heterogeneous catalysis, boundary reactions, or sensing activities. It is possible to extend and verify experimentally the concepts to related effects in polycrystalline materials, to ionic conductor/gas interfaces and nanosystems in general. In addition, by consideration of the minority carriers, the establishing of a general defect chemistry of boundary layers and the extension to mixed conductors is possible. Also the characterization of space charge regions of a single interface by impedance spectroscopy is described.

The fundamental concepts will now be discussed on the basis of the ionic conductor / insulator interface as an example and assuming Frenkel defects in the ion conducting matrix “M+X-“. Other interfaces and other types of defects will then be considered. The changes in defect chemistry of the phase will be regarded as “perturbations”, i.e. changes in mobility and in thermodynamic standard potentials in MX will be neglected.

2. THE IONIC CONDUCTOR/INSULATOR INTERFACE (HETEROGENEOUS DOPING)

2.1. Defect Chemktry

In a Frenkel disordered solid (M+X-) disorder means that some of the cations have left their regular positions (regular cation: MM) * have occupied interstitial positions (vacant interstitial positions Vi, occupied interstitial positions: M;)* and have left behind vacancies (V,‘)*:

Mu+Vj=Mi+Vh (Rl)

In Eq. (Rl) Vi denotes a vacant interstitial site. In contrast to the regular particles, Mi and V&, are typical mobile ionic centres. When the number of electronic defects and of impurity ions is negligible the bulk concentrations of ionic defects must be equal t and fixed by the mass action constant (&) on account of the necessity for local electrical neutrality:

AFGO is the standard free enthalpy of the reaction (Rl)*, Ni and N, refer to the mole fraction of the building elements (Mi’ - Vi) and (Vh - MM).

* The Kroger-Vink notation is used [66]: In the symbol D;I, D denotes the species (M stands for cationic species, V for vacancy) occupying the position p (M: cation position; i: interstitial position). The symbol r denotes the relative charge (i.e. the absolute charge of D on p minus the absolute charge of the regular species on p). t Note that the results are not restricted to that approximation.


This does not apply any longer to boundary layers. Let us first consider a “free surface” * As Frenkel [67] has already emphasized and as was quantified very early in the electrochemistry of liquid electrolytes, there are deviations from local electroneutrality and, hence, from bulk stoichiometry in the regions close to the surface. The best way of regarding this in our case is to split the Frenkel reaction (RI) [67,68] into the separate reactions for formation of the defects (R2a,b) with the associated quantities AvG* and AiG* [3]

MM+VS +M,+Vf, (R2a)

M,+Vi=Vs+M: I Wb)

(This procedure used in Ref. [68,69] is not really sufficient in that the AvG* and AiG* values are not materials parameters (e.g. standard values) since they include a change in the electrical potential that depends on the concentrations. This is considered more accurately below.)

Here a regular cation is first transported to the “free”5 surface (surface site denoted by “s”) while a vacancy is formed, and only then to the interstitial site leading to the formation of the interstitial defect. It can be seen qualitatively that stabilization of the cation at the interface displaces the upper reaction to the right and the lower reaction to the left. This increases the concentration of vacancies and lowers that of the interstitial defects.

Figure 6a shows that the defect concentrations are identical in intrinsic bulk (X - CO), but as the free surface (hatched) is approached the profile diverges. However this effect - as can be estimated from the literature data on defect formation for say AgCl and AgBr - is not nearly large enough to explain the conductivity effects actually observed.

However, if a second phase is present, that - and this is the fundamental concept in this context [3,6] - can influence the chemical potential of the cation directly at the interface, then this effect can be very large (see Fig. 6b). The second phase (A) can exert a stabilizing effect (I) whereby cations are increasingly drawn from the space charge zones to the proper interfacial layer and vacancies are produced in the space charge region, or a destabilizing effect, whereby in the case of a material containing Frenkel defects the cations are increasingly driven into interstitial positions (II). In the case of a contacted surface reaction equations (R2a) and (R2b) are replaced by [3,6]:

MM+VA Q= Mk+Vb (R3a)

Mk+Vi sVA+M~ (R3b)

(A denotes an interfacial position here.) This means that the thermodynamic balance has to take account of a surface-surface interaction (transport of the cation from the free to the contacted surface):

M,+VA=MA+V~ (R4)

It suffices to consider (R3a) since the reaction equation (R3b) is redundant on account of the validity of the Frenkel reaction (Rl).

In the case of local equilibrium it is sufficient to transport one defect type virtually from the bulk to the interface and keep in mind the validity of the coupling equilibria (here (R 1)). The different steps - as illustrated in Fig. 7 - can be described by an electrochemical (E) reaction of the form

A(x) + B(x’) the equilibria of which can be determined by pa (x’ ) = DA (x’) (,k electrochemical potential, x and x’ denote two different loci). For the special cases A I B and x = x’ we arrived at the special cases of pure transport (T) and and chemical reaction (C), respectively [ 141. fij can be broken down into a chemical and an electrical term:

P(X) = Pj(X) + ZjF+(X) (2)

* It is usual in the literature to consider a “free” surface. Such a term has no meaning in the strict thermodynamic sense. It is necessary to consider a defined neighbouring phase, e.g. a gas phase in order to define the sample thermodynamically. Here it is assumed as a “model” that the gaseous neighbouring phase has, as will later be quantified. a comparatively slight influence on the interfacial ionic defect concentrations.

178 J. Maier

1 \ --- interface (MX/A)

b)

A-M; - MM A-M; -V,

AeM; - MM A-M; -V,

AeM; - MM A-M; --+V,

AaM; - M M A-M; - VI

A-phase s MX-phase A-phase s MX-phase

Fig. 6. a) The divergence of the defect profile at a free (. . .) or suitably contacted (- - -) surface. [3] b) A stabilizing (I) (destabilizing (II)) interaction leads to enrichment of M+ vacancy (or

interstitial) defects.

(pj: chemical potential, 4: electrical potential). As is already implicit in Eq. (1), here we will consider dilute defect concentrations, even though this limit is exceeded here and there. In this case it is possible to relate the chemical potential linearly to the mole fraction of defects ($ defects/g lattice molecules).

pj = @ + RT In Nj (3)

(pj : chemical standard potential). The concentration term stems from the configurational entropy, while the standard term contains all the concentration-independent energy and entropy parameters such as partial enthalpies of formation, partial vibrational entropy of the defects etc.

The equilibrium condition for a pure defect chemical reaction r is (x: reaction progress)

XjVjrpj E 6x@) = 0 )

and that for a pure transport of a defect type j from x to x + Sx.

(4)

S,fij = 0 = 6, (zjF+(x) + RT In Nj) (5)

In the following $ is independent of x as long as the structure does not change, the variable x represents the distance from the .interface; x = 0 designates that atomic position of MX that immediately adjoins the (idealized) interface.

The profile of the electrochemical functions of interest near a (ideal as defined above) interface is illustrated in Fig. 8a (see also Fig. 6, Appendix 1). Heed is also paid to the constancy of the electrochemical potentials of the charged components and the chemical potential of the neutral component. It is naturally also possible to describe the boundary layer situation for ions (Fig.


(gas space charge layer

0 E

x2 0 W3' Mf 0 T

-

-a @

Mf

T e+ el

0 e'

Fig. 7. For the solution of the basic defect-chemical problem at the interface the indicated steps have to be considered. In principle each equilibrium can be handled by considering a generalized electrochemical reaction (E) A(x) + B(x’) where the “species” (and thus /.I”) and/or the space coordinate (and thus 4) may change. The terms C, T refer to the special cases of pure chemical

reaction (x = x’) and pure transport (A = B), respectively.

8b) by parameters (E, E,, Ei) that are defined in an analogous manner to the Fermi energy (electrochemical potentials of the electrons) and the band edge (standard electrochemical potentials) in the band model for electrons. (The principal differences from the electronic picture of classical semiconductors (strict localization, single level scheme, migration threshold always present) are indicated in Fig. 8c (cross section parallel to interface)). The chemical representation in Fig. 8a is more useful for our general considerations that can handle ions and electrons simultaneously.

The solution of our problem, i.e. the calculation of the concentration profile and its reference to standard values, requires the following steps [3e,4], see also Appendix 1, Fig. 80, 8 1):

(1) Calculation of the bulk concentrations Nj,:

(2) (4

This is a typical bulk defect problem and uses the above relationships with the assumption of local electrical neutrality (ultimately a symmetry property since (1 &p/ax ]X_m = 0). This is the state-of-the-art and Eq. (1) is immediately arrived at for our example. Thus in the following we assume that we know Njm as a function of temperature (T), component activity (a), and doping content (C). (In the Brouwer approximation Njm = ~ja”jC”‘/II,KY ( T) .) Calculation of the local concentration increase <j = Nj / Nj,:

Eq. (5) yields:

The expression <j” is evidently independent of the type of defect (i,v;n (for e’), p (for h.).

This is illustrated in Fig. 9 for 1 Zj ( = 1. For positive values of (# - Cpoo) the concentrations of all negative defects are raised by the factor exp [-F(c#I - +,)/(RT)], while those of the positive defects are reduced by the same factor.

@I On account of the i) curl-free nature of the electrical field (E = -v4) in the absence of a time dependent magnetic field, and ii) the equality of the charge density (p) and the gradient of the dielectric displacement (VW&; EEO: absolute dielectric permeability), 4 is related to p via the Poisson equation and, hence, since to the defect concentration (for a one-dimensional

180 J. Maier

a) A- Phase

0 2h x- c)

A,Gi*

Fig. 8. a) The variation of the electrical, chemical and electrochemical potentials of components, ions, electrons and defects perpendicular to an “ideal” interface. (The component activity is fixed by an effective electrode. The contact phase is nucleophilic in effect.) [13] b) The band model

analogue perpendicular to the interface, c) parallel to the interface. [13]

problem) according to

(7)

Combination with Eq. (6) leads, since p(x) = CjzjFcj(x), to the differential equation (with Yj I -In cj)

- - _&4z,c,,e-s, d2'4j _ dx2 0

that is known in the literature as the Poisson-Boltzmann differential equation. In the following we will limit ourselves to the usual case of two major charge carriers [6], namely monovalent cation vacancies and interstitial defects. There are no problems - and this is important for the discussion of the extrinsic regime in ion conductors - in application to other charge carriers so long as the absolute charge numbers are the same. The complementary case and more general questions of stoichiometry, associated with appreciable concentrations of electrons must be discussed separately. In term of the concentration enhancement 5 we obtain

d2 ln ?Y?,” dE2 = f (5&v - Sv.i) = i (5i,v - C&t)

where 5 = x/h is introduced as local variable. h is the Debye length and is given by

h I EEORT 'I2 [ 1 2F2c, - (10)


---___ e’ -_-___.____*___-_

h’

#’ ,.I- _______-____-________-

,’ I ##’ / I I

1 2

5-

Fig. 9. The defect concentration profile at right angles to the boundary surface in a composition MX containing Frenkel defects. (Contact phase is nucleophilic.)[4]

If we consider a planar interface and assume that the extent of the ion conductor material is large with respect to h, then it is possible to integrate Eq. (9) to yield:

Tv,i = [

1 + 9”,ieXp-E ’ 1 - 9,,ieXp -5 1 = zsi;;’ (11)

The interfacial interaction is now clearly incorporated in the g-parameter introduced in [6]. This g-parameter is of great phenomenological utility [6] even if there may be no adequate model for further analysis. 9i is connected to <io via:

(12)

9j is zero if no boundary effects occur (point of zero charge, GO = 1), varies between 0 and +1 for an enrichment effect (with respect to j) and between 0 and -1 for a depletion effect (with respect to j). Hence we will designate this 9 parameter as the “degree of influence”. Its variation with respect to log TO is made evident in Fig. 10.

The expressions derived so far allow us to calculate the conductivity effect, whereby N,,io plays the role of a parameter. Although it is not decisive at this point we will now look at how the parameter iVj0 and hence Nj (x, T) can be related to standard materials constants (step 3) [3e,4]. As mentioned above it was sufficient to consider the reaction equation (R3a). Since we have reduced our problem - taking account of the electrical work required and assuming constant standard potentials in MX - to that of the question of the interface, we can, for a further analysis, imagine Vh and MM (in MX) localized in Eq. (R3a) immediately at the interface (x = 0) (“bulk structure” is still assumed here) while we assume the (quasi)particles Vi and MA in the proper contact layer with new standard potentials. In the simplest case this means at a distance of x = X.4 = s from the first MX layer (x = 0 ) (see Fig. 7). From the general equilibrium conditions it follows that:

PMA + PVO = -F(d’A - 40) (13)

with the abbreviation PMA foW(MA - VA) and x = XA; /bO for p(V& - MM) and x = 0; 40 for +(x=0); +A for +(x = xA). It further follows that

aMAN” = &KOA (14)

182 J. Maier

1

0.5

t Q) O

-0.5 -’

K -4 0 4 -2 2

kl5t) -

Fig. 10. The change of the degree of influence with respect to the concentration of defects in the first layer beneath the surface. [l3]

(where Kv = exp [ - @LA + p;)/(RT)] ;KOA = exp [-F(@A - &)/(RT)], a = activity).

If we assume that the electrical term In KOA is small or constant with respect to concentration - which is actually never strictly fulfilled [14] -, then two results can be derived immediately, these are associated in the literature with the names Kliewer and Kohler [68] and Pijppel and Blakely [69] and relate the defect concentrations with the enthalpy of defect formation for “free” surfaces. The fact that this is accomplished here so simply (see the original publication [68]) and that the limits of this model are indicated at the same time is a clear demonstration of the efficiency of defect chemical thermodynamics * . As a first approximation we can regard the activity term as being constant for the limiting case of a high cation density in the interaction layer (interfacial layer) t (lbc,o/c,ol << ~GCMA/CMA~), and Kliewer’s result follows already in an improved form

NV0 = exp [-(A& + const)/(RT)]

Analogously with (R3b) or simply with Eq. (1) it follows that

(15)

Nio = exp [-(AiG - const)/RT] (16)

The inadequacy of the Kliewer model in the general case is evident from the derivation and it can be seen that even the crudest approximation must take into account the entropy of mixing the surface cations. (In the limiting case low concentrations: UMA - [occupied surface sitesy[available surface sites]).

Piippel and Blakely showed that the fraction [occupied surface sites] / [unoccupied surface sites] is a better approximation for UMs at higher concentrations * . However, in our contect this result is helpful only in a semiquantitative sense because interactions that have been neglected a priori occur at the higher concentrations where deviations make themselves felt. (It is worth noting that also in the discussion of segregation effects the influence of the electric potential changes between x = 0 and XA have been ignored.)

In the case of an exact calculation it is also necessary to take account of the condition of global electrical neutrality (or continuity of the dielectric displacement, see Poisson equation), which provides a relationship between c,,a and the concentration c~s that appears as a new variable (cf. [7]). Activity coefficients must naturally be introduced if the concentrations are appreciable. Their calculation demands a detailed model of the core layer. In order to refer the concentration parameters to materials parameters it is necessary to calculate the value of KOA in Eq. (13). This is

* The more fundamental reason that we arrived at these results in such a straightforward way, lies in the possibility to proceed from the ionic level to the defect’s level simply by 6p(M+) = dp(Mi’) = -Sj@,,) [70]. t It must also be assumed that there is a high density of counterions. + This corresponds to simple statistics with exhaustible number of states (cf. Fermi-Dirac).


possible if it can be assumed that there is a charge-free zone characterized by a moderate dielectric constant between x = 0 and x = XA (see Chapter 3). The quantitative method to trace back Nis to true standard free enthalpy is set out in [3e,4] and in more detail in a forthcoming paper 1711 (see also Ref. [72]).

For the following treatment it suffices to start from the general form of Eq. (11). However, it is necessary to discuss the problem of the minority charge carrier [ 1 I J before we turn to the calculation of the conductivity contribution of the interface.

As discussed above, the concentration profile of electronic charge carriers in ionic conductors is also determined by the space charge potential 4 - $J~. The space charge potential itself depends on the charge density and hence, as a rule, on the (mobile!) majority charge carriers. Again, a more detailed picture must include core defect chemistry [71]. Thus the electron concentrations in ionic conductors are determined by the interaction of the ions with the interface (which may be referred to as “fellow traveller” effect). It means that in such cases the electronic carrier concentration is acid-base controlled rather than redox controlled (direct interaction). This yields in our example (n stands for e’; p for h’)

5” = 5;’ = 5” = (1 f 3 .exp-U2/(l - 3,exp-5)2 (17)

Here, as for Eq. (I l), the defect chemical equilibrium conditions have not been used explicitly, but are implicitly fulfilled [4]. In particular the constancy of the chemical potential of the neutral M component is assured * . Equation (17) is primarily of significance for semiconductors such as SnO2 and ZnO and particularly for their function as oxygen sensors t .

At this point it is worth establishing that the above relationships not only allow, in principle, the calculation of defect concentrations (together with local and overall specific conductivities) as a function of x and T, but also as a function of the component activities (a~). It is, therefore, possible to construct Brouwer diagrams of space charge layers. Recasting Eq. (11) as

(18)

we recognize that it is first necessary to calculate the activity dependence of the interfacial concentration (or 9). The bulk values (cf. also A) are calculated as usual by application of the laws of mass action to the defect reaction and the interaction reaction with a neighbouring phase which fixes the value of aM (e.g. parent metal electrode).

Figure 11 shows a Brouwer diagram calculated for volume (thick lines) and boundary layers (thin lines, x=const.). For the sake of simplicity the value of C,O (in contrast to C”O or cPo) is assumed to be constant here (in the sense of the Kliewer approximation). The fact that the Brouwer approximation (simplification of the electroneutrality equation by taking account of various majority charge carrier pairs in various activity ranges) coincides with the approximations with respect to the concentration necessary for the validity of Eq. (1 l), is a great advantage for the calculations required. In particular it can be seen from Fig. 11 that in regions of very high and very low activity values (when such regions are indeed thermodynamically realizable) the boundary layer defect concentrations approach the volume values at a constant value of x (note the reduction of h as a result of increase in the volume concentration).

As in the volume it is also possible to represent the “non-stoichiometry”, i.e. the detailed chemical composition, as a function of @.$, T and in this case also of x. Formulating locally we have:

[MI+,&]~(‘)+. Differences between E and 6 result from the electronic charge carrier concentrations. It is not possible to represent such charged non-stoichiometric compositions in conventional M-X diagrams, instead the charged particles have to be included as constituents (e.g. for E = 6 :

* PM c P(M+) + ~(e-1 = /.Ji + ~n = /.I: + & + RT In NtNn = $ + j& + RT In Ni,Nnm = Mtm + pnm = ~IM~

t The effects resulting from the profile of ionic defects are evtdently often neglected in semiconductor physics. Thus the unexplained finding that SnO2 functions well as an 02-sensor (influence on [e’] in the boundary zones) if F--impurities

are present (Fb) is probably a result of the increase in the electronic defect concentration (e’) in favour of the ionic (Vo“) 1731, and thus of an increase of the redox sensitivity.

184 J. Maier

-=:boundary layer QCV~)=O

. . ::

.‘. *:. I , I.., I I fi Log (activity of Xl -

Fig. 11. Brouwer diagram of the phase MX for bulk and space charge zone. It is assumed that c* e const. > ~~00. [4]

M+ - X--diagram; in general: M+ - X- - e’ - h’-tetrahedra [4]). The stoichiometric coefficients are given by:

&=Ni,(Si2_11)/Si+Elx,/Si=E(Kl,...;aM;4j) (19)

6=(Ni,+Npm)(SiZ-l)/5i=6(Kl,...;UM;9i) (20)

The bulk values (Ni,, A$,, Ed) E and 6 are determined by the temperature (K), the doping level and the nature of the neighbouring phases (UM, 9).

At the end of Chapter 2.2 after the discussion of conductivity contributions there is a more detailed investigation of the question of how far the example of a specific defect structure chosen here (Frenkel defect, monovalent defects) is capable of being generalized.

Finally mention should be made of the interesting application to semiconducting systems in which the surface potential and the volume defect chemistry are determined by the partial pressure of Oz. Here too it is possible to discuss Brouwer diagrams for boundary layers: As an example depletion layers at SnOz surfaces due to absorbed oxygen (i.e. c = c(Po,;x)) are considered (see Appendix 2, cf. also the next chapter, [3e,4]).

2.2. Conductivity Contribution of a Single Interface

If we replace defect mole fractions (Ni,“) by molar volume concentrations (ci,“) (in dilute solutions there is proportionality), then the conductivity profile W,,,(X) is obtained by multiplication by the molar charge and the mobility of the charge carrier ui,v. For small fields the latter quantity can be regarded to a first approximation as independent of the field and will be assumed to be independent of position in the following discussion (see, however, Chapter 2.5). Recall that as the /P-values, u is believed to follow roughly a step function behavior (step between x = 0 and x = XA).

First let us consider Fig. 12. This represents schematically a thought experiment carried out on an AlzO$MX bicrystal; as indicated the geometry is infinite in one direction in MX. An interaction layer (e.g. enrichment of M+) is accompanied by a space charge zone (e.g. enrichment of M+- vacancies). Integration of the conductivity profile from x = 0 into the bulk (xmax = d s+ A) gives the conductivity contribution of the defect under consideration (v,i) parallel to the interface. This yields [6,9].


Fig. 12. Thought experiment on a MX/A bicrystal. The geometry is exaggerated. B, the thickness of the interaction layer, is of the order of a monolayer. [6]

d

y”. I

V,I I ,-J .(x) dx = y”. + Ay”. V,I “.lrn V,I

0

with the bulk contribution Yv,i, = dFu,,icv,i, and the space charge contribution

AY/!i = lAov,i(X) dX = (2h)FU,,i [2cv,i_&] .

(21)

(22)

The result first given in Ref. [3a,b,6] is surprisingly simple and can be formally represented by an effective thickness of the space charge zone (2h) with an effective concentration (see term in brackets).

The total parallel conductance is given by summation over v,i and taking into consideration cell dimensions perpendicular to the x direction that were not considered in the definition of Yl (multiplication by electrode height/electrode distance). For the total space charge contribution to Yll we obtain

AY” = C,,i(I!h)U,,i (2Ccos) = (2h)(2Cm)9” (& - jf$) (23)

With respect to the contribution to the measured (effective) specific conductivity

AU! = AY”/d = (2h/d) (0”;~ + 6iL) , (24)

it should be remarked that - as can later be generalized - AC&!, can be obtained by multiplication of the mean space charge conductivity, &_, by the volume fraction of the interface (here: 43~ = 2h/d). Additional contributions (such as that of surface conductivity) can be introduced in an analogous manner (cf. Chapter 2.3). The values @rv,i determined via the space charge zone are obtained by comparison with Eq. (23).

For the sake of completeness (cf., however, Chapter 3,4) we will also include the result that would be obtained by measurement perpendicular to the interface, when the field dependence of mobility can be neglected (This is strictly speaking only guaranteed for small electrical fields). As shown in Ref. [9] integration over a(x)-’ and neglecting the depleted counter-defect (e.g. i) surprisingly * yield an additive breakdown of the form [9]

Z’=Z;+AZ’, (25)

* ZA stands for the total “background” value which is partly in parallel with the interfacial contribution (!)

186 J. Maier

where Z is the resistance multiplied by the electrode dimensions. The space charge contribution is naturally negative for a highly conducting interface (4, > 0) and it comes to

AZI _ 2h 2 4, ” -

_--- Fu, C”cg 1 + 9, * (26)

Again the additive form makes it simple to indicate the space charge effect by normal resistance elements and the simple form of Eq. (26) also allows an analysis into the (same) effective thickness (2A) and an effective charge carrier concentration (here a different one naturally).

Also the effective specific value (A&) can be written in the form (Q)L&).

For strong depletion effects (9 - - 1 and (1 + 4) - 2 (Q/c~)“*) Eq. (26) simplifies to

(27)

In Chapter (2.6) explicit use will be made of Eq. (27) [19,74]. A more detailed treatment of perpendicular effects is given in Ref. [75].

A priori the assignment of the factor 2 in Eq. (22) and Eq. (26) to an effective length and an effective concentration, respectively is arbitrary, but appropriate for the following reasons:

(a) Let us consider ZI and assume a highly conducting boundary layer (9 = I), so then we can interprete the result Z* = (d - 2A) / (Fuc, ) in such a manner that the thickness (2A) falls out of the total balance.

(b) It follows from Eq. (11) that the concentration profile for effects that are not too small (9 - 1) falls at x = 2h to a good approximation to a small value of 1.7 cm, independent of CO (as opposed to x = h). Therefore, 2h can for our purposes be identified as a reasonable thickness of the space charge layer [6].

Let us return to our own problem and consider Eq. (23) for large effects and the example of an enrichment of the vacancies (9 - 1). The result is

AY” = AY! = Fu,(2h)(c,c,o)“* a z&‘* , (28)

i.e. the effective concentration remains simply the geometric mean of bulk and interfacial concentration. (The contribution AY! is then -F(2h)uic,).

It is necessary to take account of the influence of the counter-defects in the case of an inversion effect, and for large effects (9 - 1) it is necessary to replace uV&$% by (u,w - uicm) * .

If the phase boundary effect is basically a depletion effect then Eq. (23) becomes (i designates the defect in question):

AY” = -(2h)FUi (Cm - &?$Z)

In an analogous manner to the majority charge carriers, when 9, - 1 Eq. (17) yields

AY” = &,,pF(2h)u,, [ *29,/ ( 1 T 9,) ~,,,a]

for the minority charge carriers [I I].

(29)

Wa)

Equation (30b) is simplified for large effects in the case of n-conducting space charge zones, becoming:

AY& = F(2h)u,,/G - F(2A)u,c,,, (30b)

Even when the depletion effects are negligible (second term = 0) Y$,, is still, in contrast to AYbn, a sensitive function of the bulk parameters and thus of possible impurities (cf. Chapter 2.3.10) since h a (c,,)‘l*.

At this point we will return to the effect of different simplifications with respect to defect structure some of which are made for the purpose of illustration and some on principle.

* Analogously i should be replaced by the majority defect in the case of an extrinsic sample.

(1)

(2)


The assumption of unit charge for defect 1 and 2 was not of any importance. In the case 1.~1 ) = IQ{ q z e 1 it is merely necessary to replace F by IzlF. The relationships only become complicated when the defects are no longer symmetrically charged. In that case (1.~1 I + 1221), the relationships which refer to both majority charge carriers must change (Eq. (23)). But the relationships for large effects, in which the depleted counter-defect (e.g. 2) is neglected (Eq. 28) remain valid. The reason lies in the fact that Eq. (27) can also be derived directly if the contribution of the counter-defect (2) to the charge density is neglected from the start (see Fig. 80, Appendix 1). However the Debye length (cf. also 5) then has a value individually defined for the majority charge carrier (1) *

[ I

112 hl =

EE~RT

2z;Fk,, (31)

Equation (9) can more generally be replaced by

d21n 51.2 dS:s,

= + (Cl.2 - 52.1) = ; (6.2 - r;T:I) = tG.2

(3)

Both equations agree with each other in the approximations for large effects. The example of an intrinsic Frenkel defect compound was an arbitrary choice. All relationships also apply to any defect pairs (e.g. Schottky defects, ionic and electronic defects, excess and defect electrons). Even the setting up of a complete defect equilibrium is only of importance for determining the bulk (c,) and interfacial concentrations (co) [4].

(4)

These conclusions are very important in this context, since if the temperature is not too high one of the majority charge carriers is generally an impurity defect. Thus at the limit of large enhancement effects the relationships are also applicable to the extrinsic region, without regard to the charge (for instance divalent contamination of M+X-) of the substituting particles and (with a certain loss of accuracy) without regard to the mobility (cf. Item 2). As far as the calculation of conductivity is concerned it is a fortunate situation that the impurity level does not need to be known since c, in Eq. (28) disappears on account of Eq. (31) (cf., however, Chapter 2.3.10). It is nevertheless important that the major defect 1 be sufficiently mobile and that immobilized profiles do not dominate. If, however, the immobile impurity defect determines the space charge density (depletion effect), the relationships change. In cases where there are several charge carriers at comparable concentrations (cf. the analogous bulk problem) it is no longer possible to analyse the basic differential equation

d2Yj ZjF2 - = --~jqcjbJe-% dx2 RTEEO

(5)

in the above way. Generally however it is possible, depending on the situation, to neglect as charge carriers all except two (oppositely charged) species (cf. Brouwer approximation). Limitations and idealizations affecting the basic validity of the above relationships will be discussed in Chapter 2.5.

Appendix 1 and in particular Fig. 80 are referred to for an overview of the quantitative relationships.

2.3. The Conductivity of Two-phase Mixtures (MX:insulator)

(32)

(33)

2.3.1, General relationships It is naturally a much more complicated problem to calculate the conductivity of a two-phase

MX:insulator mixture, since it is necessary to take the distribution topology into account. We are faced with a complex superposition of various transport paths (bulk, interface). If it were desirable or even possible to solve this problem exactly, it would be necessary to know the exact position of all pathways in space and the solution would be much too specific to be of general interest. The

* On the other hand a common Debye length defined via the “defect strength” can be written for small defects [7].

188 J. Maier

Fig. 13. The highly conducting boundary layer of an isolated A particle embedded in MX only does not make a perceptible contribution (a) to the overall conductivity. The contribution is large if continuous paths are produced, which occurs even at low A-concentrations when the

particles coherently arrange along the MX-grains (b) [3].

solution would only be applicable to the particular experiment whose details it would scarcely be possible to reproduce. As far as this problem is concerned it is necessary to simplify everything as far as possible in order to be able to make general statements without deviating too much from experimental reality (see Fig. 82, Appendix 1). Thus it is certainly realistic to test the predictions for their sensitivity to the model.

In the literature two sophisticated methods are generally applied to solve such transport problems in heterogeneous media [76]:

(a)

(‘4

The percolation theory, that yields threshold values at which the behaviour of the system changes qualitatively (e.g. the appearance of continuous paths) and that investigates micro- scopically the behaviour for other distributions particularly in the neigbourhood of these threshold levels. The effective medium theory, in which the particles are embedded in a continuous “effective medium” whose parameters have to be chosen in a self-consistent way.

A range of investigations have been reported in the literature, in which these theories have been applied to our problem with the assumption of random distribution of both phases in order to make the problems amenable to treatment [58,62-641. Even though it will be demonstrated below that such random distribution is not realized in practical experiments (in contrast to MX:MX’ in Chapter 3), these results nevertheless serve to check the flexibility of the relationships given below.

As can be seen from Fig. 13a a highly conducting boundary layer that can form round an isolated particle scarcely affects the conductivity (blockage of transport by poorly conducting matrix material). However, if continuous paths are formed (i.e. situation above the percolation threshold) then their contribution is large (Fig. 13b).

The model chosen for the following treatment is a symmetrical, three-dimensional network (simple cubic symmetry) of percolating paths (see also Fig. 14). As can be seen immediately all components lying perpendicularly in equipotential surfaces cancel out on average, while the remainder can be regarded as being combined to a single volume as far as their electrical effect is concerned since they are connected in parallel. Neglecting finite “transfer resistances” (see Chapter 2.3.2) it follows that

om = WWPaoa (34)

where a, is the specific mean conductivity of the possible transport regions (o( = L d space charge zone (whereby the bulk portion is subtracted as background), o( = MX 4 bulk (that extends up to the very interface at x = 0 as background), LY = A ? second phase, o( = w 2 interaction layer (possibly also third phase)), /Ia is a factor that measures the fraction of the volume that


Fig. 14. Top: optical micrograph of a hot-pressed AgCl : Al203 sample (rA = O.l5~(m) in the blocking region (width of image: 2 mm); below: scanning electron micrograph (AgCI : A1203 obtained by normal melting process, rA = 0.15~) in nonblocking (left) and blocking regions

(right) (image width: I5 m and 150 m). [13,3]

contributes to this quasi-parallel circuit (in ideal case: 0.2 . . . 0.7) and - as is demonstrated by model calculations - is (for a favorable distribution see below) only slightly dependent on the volume fraction fba_ A detailed discussion can be found in [6]. Taking into account blocking effects as a result of perpendicular portions of the network (lying in the equipotential surfaces)

and including capacitative effects, a relationship is found for the complex conductivity 0 $ [9]

that will be particularly useful for the discussion of grain boundary phenomena The corresponding equivalent circuit diagram is also reproduced in that chapter.

(see Chapter 4).

The above model is not just mathematically simple, since to a first approximation it is possible to neglect @/QL (see below) and an analytical treatment becomes possible, it is also a good representation of reality even at surprisingly low volume fractions as can be seen from optical and electron micrographs (Fig. 14). The situation is also compatible with EDX and Auger maps. The reason for the formation of continuous paths has its origin in preparative-“kinetic” phenomena but above all in the nature of the interactions at the interface (cf. Chapter 2.3.7) and in the smallness of the A1203 grains (typically: O.Ol- 0.5 pm) with respect to the grain size of the ionic conductor (l-100 pm). It is easily estimated that under certain circumstances very small proportions (- Iv/o) of A1203 are adequate to completely coat all ionic conductor grains. Then, the distribution is like a brick-layer model where the grain-boundaries are filled with coherent A1203-particles. Eq. (35) is a quantitative description of the complex conductivity in a brick-layer model which also includes space charge effects (first given in Ref. [9]). The situation also resembles a wetting of grains by a fluid. At the first glance it may appear that Eq. 34 may only be valid at a certain Q)A- value or a very restricted PA-range. This, however, is not the case. Rather, in each preparation process via the melting method the final morphology is determined anew and a change in the A-content can produce a change in the morphology such that a further increase of the volume fraction results in a decrease of the AgCl grain size. Thus more or less coherent monolayers are

(35)

190 J. Maier

formed * . Only at very large q-values the monolayer situation must be given up, agglomeration of Al203 and thus blocking effects occur (see Fig. 14 bottom, rhs). Some minor blocking effects already occur at moderate o+values. Those effects can be separated by impedance spectroscopy (see next chapter) and can be attributed to the increasing density of the particle arrangement. An ordered distribution (interfacial interactions!) and consideration of the actual size relationships reduce the first percolation threshold (formation of continuous paths) drastically t (in comparison with a typical IO%1 for random distribution and particles of comparable size). It is also possible to apply Eq. (34) to a pore model, that comes close to reality under certain circumstances at high Al203 levels and which assumes that the ionic conductor fills the pores of the Al203 matrix [39]. On account of the drastic differences in the a,-values, Eq. (34) is of utility even in the case of very low Al203 concentrations (the region of isolated particles) and especially in the region of very high concentrations (region of blocked paths) (&??/L@L +: 0, see below). In this sense such a picture describes a most favorable situation (quasi-parallel switching). A higher degree of isolated particles owing to specific preparational or geometrical conditions, or owing to a low surface- surface interaction leads to lower enhancement effects and to situations that are only accessible by more sophisticated descriptions [S&62,64] (see Chapter 2.3.3).

As will be described individually the above relationship also makes it possible to explain the impedance spectra, and the concentration, particle size and temperature dependencies in a satisfactory manner. Because it describes the maximum realistic effect Eq. (34) is naturally fundamental if the aim is to optimize the conductivity in regard to this two-phase effect. Other distributions not described by Eq. (34) are unfavourable with respect to conductivity. When we use this simple model for calculations in the following discussion we can even apply it with adequate approximation for a result derived from linear geometry, Eq. (28).

Neglecting the conductivity of the insulator phase we obtain [3,6]

(36)

The first term describes the conductivity attributable to the bulk, and the last term, that we will neglect in what follows, represents the contribution from newly produced migration mechanisms (actually surface conductivity, conductivity in the interaction layer, in any boundary phase etc.) (qW < PA). The middle term (&,I_) describes the important contribution from the space charge zone.

Here flA is the ratio of the surface to the volume of the isolating phase. As a result of a degree of uncertainty concerning the structure and density of y-Al203 in particular it is advantageous for practical purposes to express ~AL!A in terms of the (experimentally accessible) specific mass- related surface (G?;), the mass of the oxide and the volume of the sample. flA = 3/rA when the particles are spherical in shape. In all cases BL accounts for the surface that does not take any part in the conductivity effect. If a sufficiently large interfacial effect is assumed (~“0 > coo), then 2c,9,/ (I- 9,) becomes approximately (C,OC, ) ‘I2 and 2c, 9,/ ( 1 + 9,) approximately cbJ. The purely space charge term simplifies to

U,L = BL~AQ~A(~EEoRT)“‘(U,~- 24JC.x) - (37)

As can be seen the last term accounts for depletion effects with respect to the conductivity, but these are only important when the mobility of the depleted species is very much greater than that of the enriched species (Ui/uv -X cto/c$). If this is not the case we obtain with & = 1

0, = (1 - ~A)u, + BI_GAPA(EEoRT)“~u~~%~ (38)


Fig. 15. Impedance spectroscopy of Ag/AgCl : &03/ Ag at 25°C (cell constant: 0.306/cm).

Analysis of various system functions [4] (2 = complex impedance, c= I/ 8; k= complex

capacitance; $= 1 / b allows derivation of conductivity and capacity values for the intercept. [4]

2.3.2. Impedance spectra (frequency dependence)

Typical curves of the complex impedance $ and the reciprocal complex capacitance $ [4] 0 0

are reproduced in Fig. 15 for a two-phase AgCl : Al203 mixture. The analysis reveals that the high frequency branch is determined by the high-conductivity paths short-circuiting the bulk, whereas to a rough approximation the associated capacity is that of the bulk. This result is in complete agreement with the use of the quasi-parallel switching or brick-layer model (Eq. (34)) [4,9] since considering capacitive elements in the general Eq. (35) leads to:

S,= a, + jw.sm = 0,~ + jwE, (with j 3 a) (39)

Eq. (39) is also approximately valid for a random distribution. This (i.e. the assumption of percolating paths) is supported by the Monte Carlo simulations of Blender and Dieterich [77] on random AC networks, where appreciable deviations from the bulk permittivity of the ionic conductor only occurred in the neighbourhood of the percolation thresholds (cf. also Chapter 2.3.3).

The corresponding equivalent circuit diagram is to be found in Chapter 4 (Fig. 48). The low frequency effects reveal some (minor) blocking effects even in the regime of quasi-parallel-switching. This may be attributed to the increasing density of the AlzOs-arrangement representing a perpendicular hindrance for the (parallel) interfacial transport by narrowing the current paths and giving rise to an additional impedance characterized by a higher capacitance. This situation is analogous to a bad-contact-situation [16,78,79]. In the transition to the blocking regime the second semicircle

* In hot-press composites the grains are much larger and blocking effects occur earlier (see Fig. 14, top). Note that for the purpose of illustration in Fig. 14 bottom r.h.s. a large grain situation is displayed. t Two-phase systems with very low percolation thresholds have however been described in the literature (see R2).

192 J. Maier

‘A -

Fig. 16. Volume fraction dependence of the effective specific conductivity of AgBr: y-A1203 (rA = 0.03m) [8a].

increases and the relaxation times of the two branches approach each other. In the proper blocking regime mostly one depressed large semicircle remains. This transition is predominantly influenced by the density of the A1203 arrangement and the formation of multiple layers, aggregates and so on. It is worth noting that, in the regime of quasi-parallel-switching, the a-values used for the evaluation correspond to the high frequency semicircle; nevertheless also the effective d.c. conductivity which also includes the blocking effects is usually drastically enhanced compared to the AlzOs-free sample. An attempt to extract further information from the high frequency semicircle on the defect concentrations is given in Ref. [80].

2.3.3. Dependence on volume fraction Equation (36) immediately explains the proportionality of a, to the Al203 volume fraction

for LiI : Al203 (displayed in Fig. 1). Figure 16 reveals that similar relationships hold for AgBr [8a]. Similar characteristics have been found for CuCl, AgCl and TIC1 [3,4,32,33,81]. Naturally the approximate proportionality to volume fraction does not apply over the whole 43 range under all circumstances. At very low concentrations it must be primarily isolated particles that are present, that express themselves in lower conductivities (formally lower, but VA-dependent &values). Here it is relevant to apply the percolation or effective medium description. Blocking effects appear at very high concentrations, where insulator particles block the conductivity paths, the insulating effect of Al203 makes itself felt and jl falls rapidly when Eq. (34) is formally applied. The retention of Eq. (34) is not merely formal in nature but also useful, particularly for the case of blocking effects, when discussing T-dependence, since the effective &factors are almost independent of the temperature on account of the extremely small oA/o,,,L ratio. It must be mentioned again that the rough validity of the simple proportionality is maintained by a decrease of the ionic conductor’s grain size when the volume fraction is increased.

Here, it is also worth saying a word to the curved relationship given by Jow and Wagner [32] a,,, CC p/ ( 1 - cp) . The curvature due to (1 - q) is an artefact of the calculation which is performed with the apriori neglect of the AlzO3-volume, i.e. 47 K 1 or a,,, x 43. If we repeat the calculation by taking account of the AlzOs-volume, we obtain a,,, 0~ q/(1 - Q, + v,) = 43. The morphological model published recently [82] by Uvarov et al. is in general identical to the approach used in this work [3]. The authors also gave some interesting explicit examples for “non-ideal” distributions.

Figure 17a provides a schematic picture of a a,,, - Q)A characteristic over the whole range. The extent of the various regions depends on the experimental conditions, i.e. on the type, size and concentration of the phases involved and on the preparative details. A distinction can be made


1)

mtinous paths blocking effec

1 b)

I cl

2 ‘E

"0 m d)

0 Ll 0

9, -

t?A -

Fig. 17. The occurrence of various regions (isolated particles, continuous paths, blocking effects) in a u,,, - QI* characteristic and specific cases depending on the conditions (exp. examples from the top LiI : A1203, AgCl : Al2O3; LiBr : AlzO3, /3-AgI : “fly ash”; B-AgI : A1203, AgCl : SiO2

[1,6,35,45]) (see text).

between the following extreme cases: When continuous paths are formed very early (promoted by favourable interfacial effects and

rA values small in comparison with the grain size of the ionic conductor) it is to be expected that there will be an extension of the quasi-parallel circuit region (Fig. 17b), however here percolation thresholds can be masked by small homogeneous doping effects (see Chapter 2.5.3) that are of importance at low VA levels. A concave increase is likely (Fig. 17~) if isolated particles still con- tinue to occur in the high concentration range (large particles, low surface-surface interaction, unfavourable distribution such as random distribution, see below); a convex shape (Fig. 17d) ought to be obtained if a blocking effect is significant very early (small particles and unfavourable distribution). All these patterns are to be found in the literature (see figure legends). Such unfavorable morphologies can easily be realized by hot pressing (see Fig. 14 top).

Naturally a concave initial characteristic ought also to be realized if random distribution is assumed. This is confirmed by the Monte Carlo simulations of Bunde, Dieterich and Roman [62,63,76]. Figure 18 describes a typical situation in two dimensions [63]; as can be seen percolating paths are formed but the greater part of the highly conducting regions still remains isolated. Quantitative comparisons are not possible here, particularly since it has been assumed that the particle sizes are identical and that the boundary layer is homogeneously conducting. The qualitatively similar behaviour of o,,,(Q)A) also follows from the effective medium theory [58] (cf. Fig. 1) (see also Chapter 3).

The assumption of a completely regular arrangement of equi-distant second-phase particles yields a similar curve that is naturally displaced towards lower conductivity [65].

A further topological model, namely the pore model of Poulsen and Moller [39], appears to approach reality closely at high values of Q)A in particular, as Auger scanning investigations carried out by us have revealed.

The rather fractal character of the surface of y- (also r~-) Al203 on a larger scale can be approximately accounted for by the use of BET surface density calculations. Extremely narrow fjords, that are to be expected on account of a certain self similarity of the structure and which are not detected by BET measurements, are also likely to be short-circuited as a result of the nano-size effects to be expected (cf. Chapter 2.5.3). It is certain, however, that the surface determined by gas adsorption will not be precisely identical with, but rather larger than the contact surfaces accessible to the melt.

194 J. Maier

Fig. 18. The formation of continuous paths in the Monte Carlo model of Roman, Bunde and Dieterich.[63]

t -

lo- LiI +40m/o Al,& Poulsen. Msller (19851

7g : 25’C cbnst = 106‘ 300°C const =5x10*

b 6

i

\

20 60 100 140 Qi I m*g-’ -

Fig. 19. Literature results [39] with respect to the dependence of u,,, on the (mass) specific surface area of the insuiator phase in the system LiI-A1203, at a suitable scale for two different

temperatures. [13]

2.3.4. Dependence on particle size As we were able to demonstrate for AgCl and AgBr [3,8a] a,,, is approximately proportional to

RA (BET method). The values of effective radii using y-Al203 (Meller, rA = 0.03ym) are calculated from aA. The effective value is in very good agreement with the geometrical radius @A = 0.15pm) in the case of Al203 (Fluka) that, according to our results and according to the manufacturer, contains 96% a-Al203 but possesses - like pure y-Al203 - a large number of surface OH groups. A proportionality such as is required by Eq. (38) was also found here.


030 035 040

h(Vi)/eV -

Fig. 20. The effective activation energy of enhanced conductivity in (heavily) heterogeneously doped (cation) electrolytes vs. bulk migration enthalpy of cation vacancy [14].

In the case of LiI-Al203 Poulsen and Mraller [39] also found a proportionality between the effective conductivity and the specific surface area of the oxide (see Fig. 19). Comparable results have been described by J.B. Wagner et al. [32,46]. Of course this most simple relationship is not always maintained, for the reasons given above (see [24]).

2.3.5. Temperature dependence Whereas the above parameters basically only affect the formation of continuous paths, the tem-

perature dependence ought to tell us something about the mechanism on which the increased interfacial conductivity is based. The temperature dependence is mainly composed of the temperature dependence of the mobility and of the concentration of the dominant charge carrier types in the space charge zone at x = 0. For the quasi-parallel switching region and for sufficiently high second phase contents Eq. (38) yields [6,13]:

_&!%h z -R

al/T

As has been demonstrated experimentally and is understandable from the above discussion EJ_ the same value is also expected to be measured in the blocked region (resulting in a parallel shifting; #I < B(ideal); afl/aT = 0) [8a] (see below, Fig. 25). This was recently also confirmed by computer experiments for random distribution [64] (see also Ref. [79]).

The relative change in the charge carrier concentration on temperature change should be small when there are strong interactions. If a Kliewer-Khhler-type model is taken as an approximation then it follows that

EL = hj + $AjHi = hj + i (AjH{ + AH&) . (41)

The formation term is obtained from half the standard enthalpy that is necessary to create defect j by charge transfer from the bulk to the interface (AjHi (cf. Eq. (R3a) and Eq. (R3b)). This value is reduced by the (negative) interaction enthalpy (AH&) with respect to the (positive and) comparably small enthalpy of formation with reference to a free surface (AH&) [6], and thus negligible to a first approximation. The reader is referred to the next chapter for a more quantitative discussion. In

J. Maier

addition the above model requires supplementings by several additional correction terms as a result of the electrical potential and the limited number of interfacial sites. If the plausible assumption is made that A,Hi > 0 then this latter effect in particular should lead to an additional reduction

of -Rf$F (cf. “saturation effect” in Chapter 2.3.6). Hence, to a reasonable approximation we obtain * for a large effect

EL = hj (42)

It is important to note that Eq. (42) also holds for a frozen interfacial concentration (frozen “degree of influence”). The independence of the concentration term from temperature is also supported by Ref. [80].

The above finding now explains in a very simple manner the result presented at the start (see Fig. 5) and again plotted in Fig. 20, namely that the activation energy of conductivity of all the examples cited there approximately approaches (generally slightly higher) the enthalpy of migration of the cation vacancies. This not only supports the model of “heterogeneous doping”, it also reveals that cations are evidently stabilized (internally adsorbed) at the interface in all the cases cited in Fig. 20 and that the vacancies produced as a result of this reaction are responsible for the increased charge carrier transport. The slight deviation in Fig. 20 may be explained by the neglected formation contribution. Reference should be made to Chapter 2.3.7 with regard to experiments on semiconductors, melts and aqueous systems, that confirm this mechanism (cf. also the discussion of the influence of the interfacial activity of the second phase).

It should be added that according to Eq. (36) the remaining bulk contribution to the conductivity at high temperatures is slightly less than the conductivity of the pure phase and that this brings about the (slight) parallel shift in the (T- vs. T-curves that follow. In addition it can be seen from the form of Eq. (38) that all curves for different PA values should intersect at a common a-T-point (a* = Bc&o).

2.3.6. Experiments on AgCl and AgBr: quantitative description In order to obtain information concerning absolute values detailed experiments have been car-

ried out on AgCl and AgBr, for which all the necessary data are available [3,6,8,81]. Figure 21 shows results for AgCl : Al203 (13v/o, 0.15 pm , Fluka) as a function of temperature. The typical conductivity increase at low temperatures is evident. The bulk effect predominates at higher temperatures, the conductivity is merely slightly reduced with respect to that of the pure ionic conductor on account of the “lost volume” (second phase) (cf. (1 - qA)U, in Eq. (38)). If now the assumption is made that the interface at x = 0 is saturated with charge carriers (Ag+ vacancies) - maximum effects of this type are not unknown for electronic charge carriers [83] and have been discussed for the grain boundaries of ionic ceramics [84,85] -, this allows the calculation of the conductivity of the two-phase mixture without the use of an adjusting parameter. Here it should be noted that strictly speaking such an assumption goes beyond the limits of the model (Boltz- mann distribution!) (see also Chapter 2.5.2). As can be seen from Fig. 2la, in which the calculated maximum contribution is indicated, this starting point leads to a surprisingly good agreement with the experiment - considering the crudity of the model. If now the bulk contribution is taken into account according to Eq. (36) together with a (very weak) temperature dependence of ~“0, then it is possible to produce an accurate description of the conductivity curve over the whole temperature range (Fig. 21b). Figure 22 demonstrates that this applies to other concentrations in the nonblocking range and for various particle sizes. The curve drawn in for a homogeneously doped material reveals the principal differences and analogies discussed in Chapter 2.3.9.

The experimental results obtained with AgCl : Al203 (different volume fractions, different particle sizes) by another research group (Khandkar and Wagner [33]) can also be quantitatively described by this relationship [33,6] (Fig. 23). For a more detailed discussion of the effects in the polycrystalline halide the reader is referred to Chapter 4.

* The situation is somewhat more complicated in the case where the counter-defect is of importance (see TICI); however, for sufficiently large effects the exact relationship reduces to Eq. (42).


t -1

- E-i -* “$

2 -3 2

-4

-5

-6

-i

-t

I’

i-

(b)

I I I I I I

1.0 1.4 1.8 2.2 2.6 3.0 3.4 y/K-l -

Fig. 21. Quantitative analysis of an AgCl : Al203 sample (rA = 0.15~, $A = 13%) [6]. (a) Absolute calculation of the space charge contribution assuming a saturation effect (- : /?L = 0.5; - - - : BL = 1). (b) If the bulk contribution is included it is possible to fit the curve

perfectly by only assuming a slight T dependence of co.

The experiments on AgBr [8a] can be understood in an analogous manner. Here too the assumption of a saturation effect makes it possible to describe the absolute value of the total conductivity as a function of temperature, specific surface area and concentration of the second phase (Fig. 24).

The blocking region reveals itself particularly by a reduction in the conductivity in the high temperature regime. In agreement with the discussion above the a,,,(T) curves in Fig. 25 are subject to parallel displacement (Chapter 2.3.3, 2.3.5). Accordingly it is possible to take account of the number of blocked conductivity paths by reduced T-independent p-values. Such blocking effects also appear at low temperatures (see Fig. 26b), thus Fig. 26 also displays the blocking of the bulk pathways where a separation of series and parallel effects is not possible due to the high impedances involved.

The temperature dependence of C,O that was introduced as a fitting parameter - to obtain better agreement between theory and experiment - is very small in all cases. It amounts to ca. 0.06 eV in our experiments with AgCl:AlzOs; it is ca. 0.00-0.02 eV in the published measurements on AgCl:Al203 as it is for our measurements on AgBr:AlzOj. It would be stretching the model too far to interpret precisely the slight differences in the case of AgCl. A thermodynamic discussion on the basis of the Kliewer model is included below even though it is not possible to rule out that the very low activation energy of C,O is the result of a frozen-in interfacial interaction. Although this model is not strictly speaking adequate, as is demonstrated in Chapter 2.1 (particularly in the

198 J. Maier

t Ag Cl: Alz03

10 exp. results and -2 - theor. calculations

T E

-0 -4- ‘C

3 -

2 -6-

-8l 1 I I I\ I 1 2 3

lo3 T” / K” -

Fig. 22. The theoretical description (continuous line) also succeeds for other voiume fractions and particle sizes (“7”: 7v/o,O.O3 /MI; “10”: lOvlo,O.O3 /.w, “13”: 13 V/O, 0.15 p). For comparison: single crystal (- - -), polycrystal (. . .I, homogeneous (0.003 m/o CdClz doped AgCl (- - -). (Note

the “knee” in the conductivity curve.) [3fl

t

I E

$ 2 h

1 I

ol

-1 -

-2 -

-3 -

-4 -

-5 -

AgCl : Al203 Experiments, KHANDKAR and

\ WAGNER :

‘\

\ ‘1.

l VA= lvlo 0 to,= tov/o

b. ---pure AgCl

Absolute Calculation, this work

Fig. 23. Quantitative description of the experimental results from [33] according to [a].


IT E u

‘G .

exp. results and theor: calculations

-6 -

2 3

lo3 T-’ / K’ -

Fig. 24. Quantitative description (-) of the results for AgBr:AlzOj (saturation effect). The volume fractions (qA) are given by the numbers (e.g. 5 denotes 5v/o). (The mean grain radius is 0.15 m for “(IO)” otherwise it is 0.06 /.m~; “0” is pure polycrystalline AgBr.) The filled symbols refer to an organosilylated (lOv/o, 0.06 m) Al203 [8a]. The other (open or half-filled) symbols

correspond to the parameters of the respective solid line.

case of large effects), for the thermodynamic description of the interface, it will be used in the following in order to provide a feeling for the effects that occur and to provide a semiquantitative comparison with published work relating to the “free” surface.

In the case of AgCl we find a great deal of scatter in the literature for a “free” surface (R2a) [86,87,14] (see Chapter 2.3.6).

A,@ = (0.5. ..0.7) eV - (8... 1) kT. (43)

If we estimate a value for A,Hi from the difference between -W2)aln cvol al/T and h, and a value for A& from the absolute value (Chapter 2.35) then for the enthalpy and entropy of the interfacial interaction we obtain reasonable values of the order of [6]:

AH&, = - (0 4 0 6) eV . . . . . w AS& = - (5... 1) x 10e4 eV/K (45)

which are consistent with appreciable bonding. Less reliable data are available for AgBr, but a first estimate [8a] leads to comparable results (- (0.05. . . 1) eV and ( 10e4.. . 10m3) eV/K), which is not unexpected in view of the similarity of the crystallography and defect chemistry of the two phases. More reliable data should be available from thin film data and tracer profile experiments [lOb,89,126]. Evaluating these results leads to a significant discrepancy in the entropy values. However, the discussion about the nature of the majority carriers in the thin film experiments is not yet finished [71] (see Chapter 6). It has already been emphasized that the agreement in the absolute value may be somehow accidental in view of the approximations involved. One approximation not yet mentioned, is the continuity of the concentration function. Strictly speaking the distribution is discrete on an atomistic level. This leads to the fact that the actual maximum effect may be

200 J. Maier

I - -Ii- -23

exp: l 15vlo

0 15vlo _ blocking

1 0 19vlo effects

A catc. :

4 - e,=~~=l,~~:~=o.s

‘a B, BJ& = -- id=

8, Ed

1.5 2.0 2.5 3.0 3.5

lo3 T-‘/K-’ -

Fig. 25. The results in the blocked region (here: AgBr:Al24, hollow symbols) can be understood in terms of smaller constant /?-factors (theory: - - -). The nmnhers indicate volume fractions [Sal.

Fig. 26. At low temperatures the high impedance does not allow separation of parallel and serial effects [8 11.

AgCWo Al 2 02

1. (0) -00

2. (a) -10 2. ( l ) -20 4. (0) -20 5. ( 0) -40

2.0 3.0 4.0 5.0

10” T-‘/K-’ -


AgCI:y-A1203 (tp,,=O.O7; rp=0.03pm)

- 1 absolute calculation (0 = bid

-2 calculation, plpid=O.l

pure alumina:

. melted and pressed o grindedand pressed

SILYLATED alumina + grindeland pressed

1

2

'. +

-8 I I I I 1.5 2.0 2.5 3.0 3.5

10 3T-‘/ K-’ -

Fig. 27. The conductivity effect of Al203 in AgCl is drastically reduced by the use of deactivated (organosilylated) oxide, blocking effects occur simultaneously. [3]

greater than assumed above: Following Eq. (38) the mole number of adsorbed particles per area

is (2E@TV;‘F-*) *‘* in the case of a maximum effect. A rough estimate of the same quantitity

in the case of a monolayer is simply ( V$VA)-“~ which may be up to a factor of 10 larger. Thus the first expression is not strictly an upper limit (see Chapters 2.5.3 and 5).

2.3.7. The defect-inducing interfacialprocess. Attempts have been made to modify the interfacial activity of the second phase [3,8a] in order

to obtain further information concerning the interfacial effect and the defect-inducing step. For this purpose investigations have been carried out to elucidate: a) the different influences of various oxides, b) the differing effects of various modifications of a given oxide such as Al203, c) the effect of chemical modifications of the surface of a given oxide modification (Y-Al2O3).

In a first exercise two-phase mixtures containing o(- Al203 that had been heated to a very high temperature (> 1200°C) were investigated [8a]. In accordance with the carefully carried out preparation and the conditions employed it had to be assumed, in accordance with the literature, that the interfaces were free from the OH groups that normally coat the surface of most oxides. The absence of a marked two-phase effect (see also [36]) points to the importance of these surface groups [8a].

Zhao et al. [48] have reported that different modifications of A1203 exert different effects. In order to obtain more reliable information we modified the surface of Al203 by reaction with (CH3)3SiCl:

CH3

-OH+ (KH)3)3sc1 - -HC1 -0SiCH3

CH3 (W

As is well known the nucleophilic effect should be greatly reduced by shielding of the lone pairs of electrons of the OH group. A difficulty of principle should be pointed out here: Changing the surface activity and, hence, the interactions naturally alters the distribution topology too. This is also true for the above-mentioned experiments with ~~-Al203. However, the blocking effects occuring due to a non-favorable distribution are particularly evident above in the high temperature behavior. Thus, by parallel-switching the graph in Fig. 27 (see previous chapter) we can approximately deconvolute

202 J. Maier

Lil :y -Al203 (q+, =020; rA=0.03Wn)

- 1 alumina : pure - 2 alumina : pretreated

with CH3Li

\ I -. 1 I 1.5 20 2.5 31) 35

103T-l/K”-

Fig. 28. The conductivity effect of Al203 in LiI is drastically reduced by pretreatment with CHjLi.

The latter has a higher conductivity than pure AlzO3, but this is negligible in comparison with o(LiI:Al203). [3]

the two effects. This indicates that the use of Al203 pretreated in such a manner visibly reduces the local conductivity effect and, hence, the surface activity. A large reduction of the effect is also observed when AgBr is the ionically conducting matrix.

An analogous result is brought about in the system LiI:A1203, by the use of Al203 whose surface protons have been exchanged for Li+ by reaction with CH3Li:

-OH + CH3Li -CH4 -0Li (R6)

This pretreatment causes a reduction of the effect in both LiI and AgBr. The intrepretation, however, is not that clear since different partially counteracting effects are possible: They include an increase of cation activity of the surface, size effects, increase of basicity, structural changes (“Li20”, LiOH”) etc.

Experiments with SiOz (small effect) and MgO (large effect, if not reaction) * support the conclusion that the activity falls as the acidity of the HO-M group is reduced. In addition but not independent of this evidently, however, the OH groups are of fundamental importance (whether formulated as adsorbed Hz0 or as functional OH groups). The often observed effect of water on LiI ought probably to be mentioned too in this context [36,37].

While these experiments are affected by the inherent uncertainty concerning distribution topology (tendency to agglomeration), the results of our wetting experiments - at least with respect to an AgCl melt - provide clear evidence concerning interfacial activity (Figs. 29-31). While in the case of polycrystalline a-Al203 the surface interaction in contact with a silver chloride melt is small and, a finite wetting angle can be observed after solidification, the surfaces of y-Al203 and MgO are completely wet. The melt also readily penetrates the pores (see Fig. 29 left). In the case of polycrystalline SiOz (see Fig. 29 right) the melt has - in accordance with the acidity - little tendency to wet (Fig. 29 right). Similar results are obtained with chemically pretreated samples.

* The experiments with MgO suggest a local reaction to AgzO/Ag.


Fig. 29. Solidified AgCI melts on y-Al203 (left) and SiO2 pressings.

Fig. 30. Solidified AgCl melts on the basal plane of cu-AlzO3, wetting angle: = 45” (left). The hexagonal symmetry is revealed in the plan view (right).

Fig. 31. Solidified AgCl melt on the prism plane of uAl20~. Wetting angle: = 90” (left). The two-fold symmetry is revealed in the plan view (right).

The black coloration in Fig. 29 is a result of the illumination. (On account of the elevated electronic concentration the rate of reduction is probably raised in the case of a surface-active substrate.)

Experiments with defined crystallographic surfaces indicate an interaction of the cation with oxygen. If AgCl is fused on the basal plane surface of corundum, where the (partially covalent) oxygen atoms basically form the surface, then after solidification the angle of contact is ca. 4S”,

204 J. Maier

Table I. Energy and entropy values of the contact between AgCl (I) and ~A1203 surfaces [88]

(hkil) Iri,& s;s

(Jm-*) (10-4Jm-2K-1)

(0001) 3.21 f 0.5 13 2 2

(ioio) 3.75 + 0.6 21 f 3

(ll?!O) 4.58 -c 0.7 35 + 5

while it is much higher (ca. 90”) in the case of the prism plane surface (Figs. 30,3 1 left). (In the plan view the solidified drops reveal the hexagonal symmetry (Fig. 30 right) and the two-fold symmetry (Fig. 30 right) that mirror the crystallographic planes.) The same diversity of surface activity is also met in the thin-film experiments described in Chapter 6.

Recently [88], accurate in-situ measurements of the wetting angle and the interfacial tension between liquid AgCl and different crystallographic planes of a-Al203 have been performed as a function of temperature. The results show the strongest interaction for the (1000) plane (see Table I). Structure calculation performed by Tasker [89] reveal negatively charged regions that really invite the cation to be adsorbed here (Fig. 31). Interestingly, at very high temperature the melting behavior is determined by the interaction entropy (see Table I) that shows a ranking inverse to that of the enthalpy (as also obvious from Fig. 31). (The latter finding is less interesting in this context, but of significance for the sinter theory [28].) In agreement with the observations in the composites, immediate spreading is observed if humidity is not carefully removed.

Consequently we formulate our defect-inducing mechanism as follows (see Fig. 33).

(Nu...V)A+MM ti (Nu...M)k+VE, (R7)

Nu stands for a nucleophilic surface group such as an OH group that adsorbs the cation. Since we are involved with space charge phenomena the oxonium group -OHM+ formed in the case of an OH-group does not of course have to give up the proton. It is worthy of note that -0Hf groups have been discussed in a completely analogous manner in colloid and sensor chemistry (pH ISFETs) for contact with aqueous medium at high proton activities. In this context too there are very similar orders of oxide activity ((y-AlzO3) > (SiO2) > ((Y-AlzO3)) that express themselves in corresponding zero charge pH values (e.g. pH”(y-AlzOs) = 9; pH”(SiO2) = 3 [90,91]). It will certainly be worthwhile to vary the surface acidity by the formation of mixed crystals of Al203 and SiO2.

The interfacial effects, described briefly in Chapter 2.4, 2.6,4, and 5 provide a strong confirmation of the defect-inducing reaction formulated. In particular the effects of basic gaseous species (Chapter 5) are analogous to those of the OH-groups.

An intensive search of the literature has led us in the meanwhile to examples of surface interactions of oxides such as Al2O3, SiO2 and metakaolinite with melts or aqueous solutions of AgNOs, LiNOs and CuClz, which also support the above mechanism [92-95l.A discussion of the application of further techniques is given in Chapter 2.5.4.

It has still not been clarified to what extent the method of preparation produces OMt groups (implying the formation of HX) in addition to OHM+ groups. On the one hand, it is always possible to detect traces - or in the case of undried oxides significant amounts too - of gaseous HX (or X2 when X = I, instability of HI!), that are attributable to a simple acid base reaction; on the other hand, in the case of dried oxides there is no difference detectable from the thermal reaction of the pure ionic conductor in (moist) air. Acid is evolved to a limited degree in the case of the system AgNOs:AlzOs even at temperatures well above the melting point [93].


Fig. 32. Calculated surface structures of a-Al203 (0001) according to Tasker [89]. The expected adsorption site of an Ag+ ion is indicated (a) (0001) A1203 surface; (b) (IOiO) Al203 surface;

(c) (1 120) Al203 surface. [88]

2.3.8. Experiments with TlCl: influence of the counter-defect. We are now in a position to discuss the various equilibrium cases possible in terms of the defect

type and the degree of mobility between the metal ion vacancy and the counter-defect [3d]. If, as in the case of Schottky defect, LiI (u(V,“) > u(Vr’)) the conductivity is determined by the

metal vacancy, then the simplest possibility, namely a continuous increase in the total conductivity, is realized if the mechanism under consideration is valid (Fig. 34 top). In the case of Frenkel defect silver halides the counter-defect (u(Ag;) is more mobile (u(Agi’) > u(V,g’)). For this reason significant heterogeneous doping (with 4, > 0) must pass through a minimum value of conductivity, corresponding to an ionic P-N transition (here: V-I junction) analogous to the p-n transition of semiconductors (cf. Fig. 34 center). Indications for such ionic p-n transitions have been described for the system PbO/ZrOz in [96]. If there is homogeneous doping by cations of higher valence - as is to be actually expected in the lower temperature range - then the upper profile is relevant once again (#(Vi,) > u(N&)). The analogous case for Schottky defects is found in TlCl which is intrinsically a Cl- conductor. When the temperature is not too low the mobility differences may be so large (see Fig. 34 bottom), that primarily depletion boundary effects are to be expected (as is also the case for silver halides at higher temperatures). Such a case will be treated in Chapter 2.6. An interesting variant is encountered in the case of anti-Frenkel defects, such as are found in alkaline earth halides (cf. Chapter 2.3.12). Here, because of the validity of the corresponding mass action

206 J. Maier

a) b) Ag+ - Ag+q Ag+ -

AgCl Al203 A&l AI203

Fig. 33. Possible interactions of the mobile cation with nucleophilic centres. The experiments described in the text favour the right-hand mechanism. Figure 33b also indicates the nucleophilic

influence of “water”. [6]

process, the stabilization of the cation is identical with the destabilization of the anion (cf case II in Fig 6b). Since u(F;) > 0~‘) it is the upper profile in Fig. 34 that applies. However, as described in Chapter 2.3.12, F- stabilization is also possible and even probable due to the amphoteric nature of the surfaces.

We can now return to thallium(I) chloride, which forms an interesting test for the theory on account of its intrinsically dominant anion conduction [8b]. No effect is observed at moderate temperatures (2 100°C) if TlCl is doped heterogeneously with Al203 [8b], (Fig. 35, curves 5 and 1).

This is absolutely consistent with the expected depletion layer (short-circuited by the bulk). However when measurements are carried out at lower temperatures (lower intrinsic concentrations, greater Debye lengths), then a dominant Tl+ conduction is evidently set up (curve 5) as the result of the formation of an adequate number of Tl+ vacancies, as is shown by the low temperature activation energy in Fig. 5 [8b]. The fact that there is no depression of the conductivity with respect to the Arrhenius branches (“knee”; Koch-Wagner-effect) in the higher temperature region, which must be the case for suitable homogeneous doping (and which can be simulated by doping with PbClz: Fig. 35, curve 2), demonstrates that an impurity effect is not decisive. On account of the more favourable parameters this is somewhat more significant than for AgCl (cf. CdClz doping in Fig. 22).

As is illustrated in Fig. 36 it is also possible to describe the conductivity of TlCl quantitatively over the whole temperature range for various grain sizes and concentrations in the second phase. A value fitted for cvo (4 x 10m5 moYcm3) is appreciable below the saturation level, so that it must be assumed that there is - in comparison to above - evidently a markedly weaker (possibly frozen-in) interfacial effect. However, &o and, hence, the space charge potential is of the same order of magnitude. It is not clear if this is accidental or if this term can play the role of a limiting factor. The conductivity profiles for various temperatures perpendicular to the interface calculated using the above cvs values are plotted in Fig. 37. It can be seen that the depletion zones become more relevant as the temperature is increased. The extent of the boundary layer is reduced at the same time. We can also recognize additional fundamental eiements in Fig. 35: 1) Comparison of curves 2,4 and 5 shows that the heterogeneous effect is much greater than any possible maximum homogeneous effect (with PbC12).

The reason for this is likely to be that the (homogeneously distributed) Pb,, defects associate strongly with the Tl vacancies and largely immobilize these, while in heterogeneous doping this does not take place on account of the separation in space [Sb]. Furthermore it can be seen (curve 3), that -as already observed in the case of AgCl and AgBr - the nominally pure polycrystalline TlCl also


total * . . . . . . . . . . . . . . . . . . . . . ;. . . . . . . . . . . . . . . . . . . . . . . . . .

%l *,____________~~_~_~~_________

/ I I/ I

k-- Fig. 34. Profiles of the partial and total specific conductivities perpendicular to the interface (assuming 91 = -92 > 0) depending on the relationship between the mobilities of the ionic defects 1 and 2. (With respect to conductivity the enrichment, inversion and depletion boundary

layers are involved.) Ionic P-N transitions occur in the middle case [3d].

exhibits a heterogeneous effect (see chapter 4). The fact that there is no reduction in conductivity at higher temperatures also reveals that this is not a homogeneous effect. This phenomenon will be discussed in more detail in Chapter 4. Finally curves 4 and 6 reveal the superimposition of homogeneous and heterogeneous effects, that are required by the above consideration: Curve 6 describes the conductivity of heterogeneous (Al2O3) and homogeneous (PbC12) doped material, curve 4 describes the conductivity of polycrystalline TlCl(PbCl2). As expected the homogeneous effect predominates at higher temperatures, the heterogeneous effect at low ones [ 131.

2.3.9. Homogeneous and heterogeneous doping: d@erences and similarities. Since in addition to the classical (homogeneous) doping the heterogeneous doping represents

the second basic mechanism to influence a given material with respect to the defect chemistry, the basic similarities and differences between homogeneous and heterogeneous doping [13] will be summarized here.

The creation of additional metal vacancies in M+X- by heterogeneous doping according to the reaction equation given in Chapter 2.3.7 (R7) is similar to homogeneous doping (Zn > 0) by cations of higher valence (N2+ in the place of M+) (or by anions of lower valence ):

NX2 + 2Mrvr +%MX+N,+Vh (Rg)

The increase in the number of metal vacancies then usually takes place homogeneously (within

208 J, Maier

1.4 3.0 4.6

lo3 T-l/ K-’ b

Fig. 35. The (effective) conductivity of a pure (curve 1 (single crystal), curve 3 (polycrystal)), homogeneously (curve 4 (polycrystal)) and heterogeneously (curve 5) doped thallium(I) chloride.

Curve 6 refers to material that is both homogeneously and heterogeneously doped [8b].

1: pure, monocryst.

2 : horn. doped, mcr.

(0.05 m/o PbCl 2)

3 : pure, polyctyst,

4 : horn. doped, per.

(005 m/o PbCl,)

5 : het. doped

-10 - 6 : horn. & het. doped

(0.05 m/o & 10 v/o1

2.2 3.0 3.6 4.6 lo3 T-’ / K-’ -

Fig. 36. Quantitative analysis of the results for various volume fractions and particle sizes with T1CI:A1203 [8bj.


T=350K

T=30OK *___s. ____________________._______________

T= 27OK _____-_--

____________.*~-----’

*_*_ __.__-__s

I I I I I

0 2 4 s 8

Fig. 37. Calculated profiles for a(TlCl : AlzOx) at various temperatures [8b]

the sample) and linearly with the doping content on account of the local electrical neutrality applying to the bulk. If it is assumed that interstitial metal ions are more mobile, then homogeneous doping will first reduce the conductivity as a result of a depletion of the M; defects; as the doping concentration (D) increases it passes through a minimum (a(M;) = a(V,‘)) at

&in = (Ui - U,) Cm (UiU~)-*‘2 ; (46)

finally the vacancy conduction predominates at higher levels (cf. Fig. 38a). The analogous behaviour is to be expected (at constant D) in the a-T curve (Fig. 39) on transition from high to low temperatures.

Quite analogously in the case of heterogeneous doping there is also a change at a critical coordinate from M; conduction to Vt conduction in the case of heterogeneous doping (Fig. 38b)

(47)

If attention is focussed on the (integral) a, value, it is possible to calculate the critical concentration of second phase at which the global change in mechanism occurs from Eq. (36). It can be seen from Fig. 39 that intrinsic conductivity dominates at higher temperatures for both doping mechanisms. (The factor BDJ(l - VA) remains in the case of heterogeneous doping (see above).) The difference in the intermediate temperature range is significant. In the case of the homogenous effect a depletion of M; defects is clearly evident in the form of a “knee” (i.e. Koch-Wagner-effect), while in the heterogeneous case when the boundary layers are not too extended the depletion layers are at least partly short-circuited by the bulk. The difference in association behaviour has already been discussed in the previous chapter. An extension to heterogeneously doped extrinsic (i.e. heterogeneously and homogeneously doped) materials is straightforward.

2.3.10. Electronic conductivity in heterogeneous electrolytes. Following the remarks in Chapter 2.3.1 we will now return once again to the problem of electronic

conduction in this system [I 11. If it is assumed (arbitrarily) that the cation is stabilized then, in the

210 J. Maier

1 -5 L min 0

Fig. 38. u for a) homogeneously (higher valence cation impurity) doped MX as a function of the doping concentration [13]; b) heterogeneously doped MX (9, > 0) as a function of site. [13]

I - --born. doped (zD ~0)

l/T -

Fig. 39. Differences and analogies in the T dependence of homogeneously and heterogeneously doped material (ZD > 0; 9” > 0; see Fig. 35). [13]

case of a pure space charge effect with respect to the electrons, it must be expected that n-conduction increases and p-conduction decreases because S(Vr,.r’) = 9(e’) = -9(h.) > 0 (see Chapter 2.1). Wagner-Hebb polarizations of LiI:AlzOs [34,37] and AgI:A1203 [46] indicate an increase in the p- conduction with respect to the pure material. The effect is very weak in CuCkAl2O3 [32] but the tendency is analogous. We have carried out long-term measurements on a polarization cell

+C]AgCl : A12031Ag-

that indicated that the p-conduction is also increased here. We observed appreciable effects on the minority charge carrier at 3Oo’C, a temperature at which a two-phase effect on the majority charge carrier can no longer be detected. Reasonable stationary characteristics are impossible to obtain at


AgCl : y-Al,O, (0.03 pm J

30-o 0 0 :4v/o 0 : IOv/o

Qc 300 ‘C

2 20- lzl 0

10 - 0 Kl

00 1 I 1

0.1 0.2 0.3

T/V -

Fig. 40. Results of the polarization experiments at 300°C where for two compounds distinct time dependencies occur (number symbols indicate the sequence in time) [l 11.

a C

. cr

&$t : y- Al,0,(4v/0,0.03 yrn) 3 _ 215C

It

2- /

: 1 - /

d 0’

w-c cc0 _--- I 1 1

0.4 0.6

Fig. 41. Stationary results of the polarization experiments on AgCl : 4v/o Al203 samples at 215°C. The waiting period per value was approximately 4 weeks; the depolarization leads to the

initial state. [I I].

that temperature (see Fig. 40). Evidently we were observing a continuous flux of impurities, being important for the minority conduction, but being completely negligible with respect to the ionic majority charge carriers.

This is not unexpected; after all, in the case of AgCl the concentration of the minority charge carrier lies more than 7 orders of magnitude lower than that of the ionic defects. The fact that the effects are greater for samples containing higher concentrations, makes it likely that the diffusion source is to be found in the Al203 particles.

However, it is possible to record stationary E-i-characteristics at lower temperatures (see 2 1 YC, Fig. 41), at which it is necessary to take account of space charge effects (see Fig. 22). The long waiting periods are due to the low value of the chemical diffusion coefficient. Figure 41 is characteristic for the current increase in the p-region. An increase in p-bulk-conduction over that of pure

212 J. Maier

AgCl[97] (and thus a decrease of n-bulk-conduction) as a result of impurities may be understood in terms of an incorporation of O-defects (on a very minor level!) according to

0 + Clc1 =O&,+h’+Cl (R9)

The existence of Ok, impurities is supported by our observations of homogeneous doping effects by AgzO (cf. AgzS effect [98,99]).

A closer consideration (particularly of the n-component) requires that we refine our discussion and take space charge effects into account.

For a positive space charge potential (Ag+ adsorption) Eq. (30) and Eq. (34) yield

um,eon = urn - urnjon = B (1 - VA) (6 + +o)

+BLVA~~AU~ ~EEORTC,O (cnaolcvoo) (48)

for the specific electronic conductivity of a nonpolarized two-phase sample. It can be seen that space charge contribution depends greatly on bulk values.

The next step is to extend the Wagner-Hebb analysis, that was derived for homogeneous materials, to the case of an inhomogeneous initial distribution (Fig. 42). The situation of a single interface in a thought experiment is shown in Fig. 42. Since it is possible to assume contact equilibrium (@/ax)&+ = (g/ax) Fe’ = (a/ax) PAg = 0) the problem can be treated (approximately) as a parallel problem (y = direction of measurement). It can be shown that the regional current densities can be superposed analogously to Eq. (34) to yield an effective current density i,,, and that an excess current density can also be separated. The latter contribution is obtained from the specific electronic excess conductivity Aaeon in the following way [l 11:

(49)

(LX, L, = dimensions of and distance of electrodes in x and y direction). The integration in the stationary state over the chemical potential (the dashes indicate electrode contact) usual in the Wagner-Hebb analysis is followed by another integration over the x coordinate. The above approximations now yield the usual evaluation formula for a two-phase sample

aim,eon /an = ~m,eonlLy VW

(q: polarization potential), although with the mean specific conductivity given in Eq. (48). Since, to a good approximation, C,O and c yoo in Eq. (48) are independent of the silver activity, the known relationship

1-exp-$,)+o$,(expg-l)] p

is also obtained - formally - but again with the effective non-constant a-values (that refer here to silver activity 1), which are defined according to Eq. 48 [l 11.

When account is taken of the relatively large voltage realized in the experiment, Eq. (50b) simplifies to

&.0&F rlF RT

= a;,, + f&P exp - . RT (51)

It is possible to determine a,& and a&, from the plot reproduced in Fig. 43. Comparison with

the literature values for pure AgCl (O’ KP) (T,,, give what is at first sight the remarkable result that

both effective values are increased by about a power of ten. While increasing the p-conductivity

urn,,, = (1 - PA)Uco, = uca, should reflect the homogeneous effect * , a,,, ought to be greatly affected by the space charge potential (cf. Fig. 9):

* Series effects are neglected.

Ionic Conduction in Space Charge Regions

Fig. 42. Thought experiment of a Wagner-Hebb polarization parallel to a y-Al~O$AgCI interface (initial state) and semiquantitative equivalent circuit diagram (see [lOO]) [I I].

213

lo-5exp(~) -

Fig. 43. The mean specific conductivities are determined from the slope and intercept (215°C) [I l]

cm.n z &, (1 - VA) + BLR&&=&;:~ . (52)

If the value obtained in the independent analysis of the majority charge carriers (Chapter 2.3.6) is substituted for cvo and the literature value for pure AgCl for cvm (at 215°C extrinsic effects do not play any role for the majority charge carrier), then it follows that a,,, = 35 x a&,,,. Hence the measured effective value is greater than the bulk value by a factor of about 35 and thus the

bulk value is smaller than the conductivity of pure silver chloride ( a&, = (l/4)&) as would be

expected for an impurity effect that increases the p-concentration. Thus, the result of a simultaneous increase in n- and p-conduction can be explained without problem by means of the space charge effect as a result of Ag+ adsorption. It is not necessary to assume structural changes (although these certainly take place to some degree). It is interesting to note that Valverde-Diez and Wagner jr. [loll recently found the same qualitative effect in the system AgI$-AlzOs, but did not interpret

214 J. Maier

0 t

AgCl Ia61 l/y-A$O,

-xl-

20 40 60

x/nm -

Fig. 44. Idealized approximate quantitative concentration profile for majority and minority charge carriers at an AgCl (4.~ = l)/pAl2O3 interface (215°C). [l l]

it further. There the n-conduction is appreciably higher than for AgCl, as the Al203 effect on the ions is aiso greater (cf. Eq. 17).

In view of the inaccuracies and approximations in the model, measurements and literature values (for example the neglect of valence changes [loo]) a better agreement (l/l0 instead of 114) is not to be expected in the above analysis. The calculated conductivity profiles for all relevant carriers (a,~ = 1, 215%) are illustrated in Fig. 44.

Since the above example involved ionic and electronic carriers, the general case of a mixed conductor can be tackled (see next paragraph). Important conclusions can also be drawn with respect to semiconductors, where often the ionic point defects are in majority. At least during preparation, if not during the measurements those defects may be mobile enough to create ionic space charges. In those cases the electric fields at the boundaries may even be determined by the neighbouring phase with the ions (acid-base interaction) rather than with the electrons (redox interactions). This is completely neglected in semiconductor physics. A phenomenon like this may explain why (slightly) F-doped SnOz is advantageous as an 02-sensor material over an acceptor doped SnOr. In the second case the space charge effect may be primarily acid-base controlled in contrast to a redox-control that is desired [73,102]. It is also clear that generally “heterogeneous doping”is able to tune the degree of the mixed conduction.

2.3.11. Complete defect chemistry of AgCl contacted with y-AIzO3. The above discussed calculations and experiments provided the first complete picture of the

defect chemistry of a (mixed) ionic conductor not only in the bulk but also at a given interface (here interface AgCYy - A&03). From the theoretical foundations given in chapter 2.3.1 the temperature and the activity dependence of the majority and the minority carriers can be obtained.

As regards the Cj - aAs or Cj - Pc12 relationships an independence of the surface concentration


Fig. 45. (a) The defect concentration for V& (large body), for e’ (small body) as a function of Ag-activity and distance from the interface to y- AgCl [3e]; (b) The defect concentration for V& (large body), for e’ (small body) as a function of temperature and distance from the interface to y- AgCl [3e]; (c) The (parallel) excess conductance of the interface of the interface AgCUy-Al203 as a function of temperature and silver acivity [3e].

has been assumed as far as the ionic carrier is assumed. This is reasonable since effects of Ag or Cl2 on the AgCl/A1203 interface are only expected to a very minor extent. Then, however, according to the equilibria to be considered the minority carriers depend on UAs or Pclz (For details see [3e]).

Figure 45a shows as an example the defect concentration of V& and e’ as a function of distance and Ag-activity, Fig. 45b the same quantities as function of distance and temperature and Fig. 4% gives, as a measure of the interface concentration, the parallel excess conductance AYll as a function of T and aAs.

It should be stressed again, that in addition to the solution of the classical defect-chemical problems, namely to give the defect-concentration as a function of the contolling parameters T, P, C, (C = doping level), normally

Cjm = LxjPNJCMJII,K,y”( T)

now also the situation in the space charge regions can be computed as [3e,104]

(53)

Cj = f( P, C, T; X) . (54)

2.3.12. The conductivity of composites involving other ionic conductors (LiI - LiCl - CuCl - AgI - NaCl - CaF2 - BaF2 - PbF2 -SKI2 etc.)

Our present knowledge of defect parameters unfortunately makes it impossible to make quantitative space charge estimates for calculation of all systems investigated so far in the literature. Thus, in the case of the compounds listed above, the relative mobility data are unknown or unsat- isfactory sometimes even contradictory. In the case of CuCI, for instance, not even the enthalpy of migration of Cu+ vacancies is known * . Jow and Wagner found considerable effects in CuCl on admixture of y-Al203 [32]. We, in our laboratory, have used BaTi as the second phase [4].

* If the above model is assumed to apply for CuCl : Al203 also, a value of about 0.4 eV (= EL, see Eq. (42)) is probable.

216 J. Maier

When the difference in the geometric parameters according to Eq. (36) is taken into acount the effect is comparable and independent of whether the (relatively insulating) BaTi used is n- or p-conducting. We also studied single interface effects by applying edge electrodes at the interface CuCYor-Al203 [4]. In the case of LiI and AgI the data on defect parameters is contradictory. For AgI (and under certain conditions also for LiI[ 151) it is not even certain exactly what modification is present in the two-phase mixture (AgI : fi or y or (/!I and y)). However, the above mechanism is supported in both cases by the activation energy of the dispersions as well as by the analogous behaviour of polycrystalline and thin film material. More recent measurements [25] reveal that the appreciable differences in the activation energies that are observed for heterogeneously doped and polycrystalline (extrinsic) AgI disappear if one waits long enough, whereby a (B/r) phase change occurs. There are also recent reports of the appearance of an amorphous AgI-phase in the composites [103] (This will be discussed in Chapter 2.5.2). The detailed impedance spectroscopic evaluation by Soltis et al. [80] on AgI composites do also not give a clearer picture about the penetration depth of the interfacial effects; similarly, results by Schmidt et al. [104] only indicate a strong interaction but do not allow a distinction between a pure interfacial interaction or a chemical reaction.

In the case of LiI - and on the basis of the mobility data of Jackson and Young [75] - Eq. (38) yields values that are about half an order of magnitude lower than those determined experimentally. The mobility values of Haven [105] are even lower, while mobilities from thin- film experiments are very much higher [lOa]. However, the latter data are probably debased by complicated effects [15]. Here, it is to be recalled (see Chapter 2.3.6) that the estimate based on Eq. (38) is not reflecting the maximum effect possible due to the discreteness of the problem. Thus, it is not allowed to rule out space charge effects in this way. In addition the LiI used apparently always contains water (see Chapter 2.5.3). Even extremely carefully treated material evolves methane on contact with CH3Li (see also chapter 2.5.4). Moreover, the very high critical second phase concentrations (at the transition to blocking regime) should involve extensive nano-size effects (very small effective particle distances) which again enhance the overall conductivities (see Chapter 6). With the latter assumption the experimental values can be verified by the calculations. It it also worth mentioning that a hexagonal modification of LiI can be observed if present as a thin film on SiOz under suitable conditions (s. Chapter 2.5.2, Chapter 6). The interpretation of contact resistances in LilSOClz battery systems [106] by depletion of Li+-vacancies in the LiCl layer formed is also worth mentioning in this context.

Recently some work has been published on heterogeneously doped alkali earth metal halides exhibiting anti-Frenkel disorder [51-531. The above mechanism would lead to the expectation of an increase in the interstitial conduction (e.g. Fi) in CaF2 due to internal adsorption of Ca*+. Khandkar et al. [51] interpreted their experimental results in this manner. Vaidehi et al. [52] argued for an increase in the anion vacancies as a result of adsorption of the halide ions. Although these investigations did not succeed in making an adequate separation of possible space charge effects from superimposed effects, which it was perhaps not possible to achieve in any case on account of the high preparation temperatures necessary, it seems nevertheless not implausible to assume anion adsorption. The ambivalent behaviour of oxide surfaces is well known and the mobility of the cations when preparing these compounds may be too low (no molten state), to be able to participate in competitive adsorption. Careful investigations are required here. This is especially valid in cases where no unambiguous structural data are available. Moreover, a detailed impedance spectroscopic analysis and especially the extraction of the relevant activation energies demands deconvolution of parallel and perpendicular effects as described in Chapter 2.3.2. Such measurements in our laboratory support the assumption of F--adsorption in PbFz-composites [57].

We have other work in progress [56] that suggests an increase of Vh conduction in CaF2 and PbF2 if the grains have been pretreated in SbFs. This would be completely in line with our chemical model. In the case of the super-Lewis acids such as SbFs or BF3 we definitley expect a stabilization of F- at the surface in form of SbF; surface anions and an increase of the F--vacancy concentration (see Chapter 4, 5). With SbFs there is, however, the complication of a compound formation in a certain temperature range. In all these cases also the oxygen and the water solubility has to be considered [65].

Meanwhile a huge amount of conductivity anomalies in heterogeneous systems have been re-


-0.11 I I I I I I 0 10 20 30 40 SO

time (t)/min -

Fig. 46. Relative pressure increase during the dehydrochlorination with different catalysts at

150°C (q: volume fraction; r: radius of alumina particles; N: number of pellets) [IS].

ported (involving Nasicon, NaCI, LiF, glasses, polymers etc.[53]) and partly been interpreted by space charge effects. In most cases the data set was not complete enough to allow for a reliable quantitative analysis.

Generally speaking the interfacial situation can be very complicated especially under extreme chemical conditions. Thus the proton conductivity reported in [107] in systems such as Li2S04:A1203 or RbNOJ:A1203 [108] under fuel cell conditions (Hz!) may be due to the formation of hydroxide (+H# or NH3) [108]. Besides the purely electrical effects, designed composites also offer the possibility of multifunctionality. In particular organic/inorganic composites may lead to a combination of favorable electrical and mechanical properties [ 1091. But also electrical effects of dispersions in ionically conducting polymers are expected due to the above reasons.

2.4. The Catalytic Activity of AgCllAI203 Composites

Beyond conductivity effects, the increased defect concentration is expected to influence sensitively the rate of chemical or electrochemical reactions. In Ref. [54] Simkovich and C. Wagner reported on an increase of the rate of elimination of HCl from tertiary-butyl chloride

(CH3)3CCI - I-ICI + (CH3)2C(CH2) (55)

when CdCl2 doped AgCl is present in the reaction chamber as a heterogeneous catalyst. The rate constant is roughly proportional to [Cd2+] and thus to the vacancy concentration. Besides, the possibility of an acidic effect of the Cd& centers, a basic effect of V& is probable, the latter by relieving the detachment of the protons; the first by relieving the detachment of Cl-. In Ref. [18] we reported on similar experiments using AgClly-A1203 composites. The relative pressure increase during reaction (55) can be nicely used to follow the reaction kinetics.

As Fig. 46 shows the effect is tremendous. In the presence of AgClly-Al203 the reaction at T = 423 K has reached equilibrium already after 20 min, whereas not even a significant reaction start is observed after such a waiting period if pure AgCl is used.

The effect of pure y-Al203 is not insignificant but much lower if related to the specific area and can be partly masked by subsequent reaction of all pellets with (CH3)jSiCl (for details see [18]).

The effective rate constant key can be determined from the time evolution of the overall pressure.

Figure 47 clearly shows the proportionality of key to the numbers of pellets reflecting the active area. The constancy of key (K # number of surface sites) suggests that adsorption is rate limiting.

218 J. Maier

1 0.3

a” \t ; 0.2

.

a” I 0.1

a \t

0

N=7 l l

l T

n

A

A

0 10 20 30 40 time (t)/min. -

Fig. 47. A suitable plot according to Ref. [18] yields k,a from the slope. The behavior is shown for different number of (identical) pellets (N) [18].

The density of such active surfaces can be calculated by the space charge considerations. Ac-

cording to [18] one obtains for k,tr

k&T oc RA~~A (EEoc~)~‘~ exp - A%* /RT) exp (L&* IR) (56)

Figure 48 shows the temperature dependence of kes. The Arrhenius behaviour allows one to calculate the effective activation enthalpy.

The parallel shifts are not caused by different activation entropies, but rather by the different geometric parameters. If we take acount of those according to Eq. (67) we end up with a unique value for AS* [ 181. If we evaluate mass action constants, a unique line is obtained independent of the geometric parameters as it should be, for a catalytic effect. More details can be found in [18].

Since interfacial reactions lead very sensitively to changes in defect densities, the use of composite materials for such processes seems to be very promising.

Recent investigation of the effect of heterogeneous doping on electrode reactions, namely on response time of a chlorine sensor cell (AgCl-formation cell)

Pt, Cl2 1 A&l] Ag . (57)

have been studied. Significant effects were found to be due to humidity possibly but not necessarily pointing in the discussed direction [17].

2.5. Approximations and Limitations of the Ideal Space Charge Model

2.5.1. Inherent approximations In spite of the success in describing the conductivity effects in the composites it is necessary to bear

in mind the approximations involved, not to speak of the uncertainty involved in the experimental data and the parameters brought in. The above treatment assumes that mobility, dielectric constant, molar volume and standard potential are independent of locality. This can be accepted to a first approximation so long as no appreciable structural influence is to be expected from electric fields, elastic or plastic effects (cf. also the assumption of transport parallel to the phase boundary). The same approximation is used successfully in the field of semiconductor physics, nevertheless from


2.20 2.40 2.60 2.60

103T-‘/K-l

Fig. 48. Arrhenius plots of the effective rate constant and of the equilibrium constant of the elimination under various conditions (qx volume fraction: N: number of pellets and r: 0.06~).

semiconductor physics one is familiar with local changes of the band gap at interfaces. A refinement to those cases (which is probably more important here than for semiconductor physics since the ionic constituents are affected) could be very complex and requires more precise experimental information. In the case of a simultaneous change in c(x) and u(x) for one and the same charge carrier it follows from

Au = IzI F @AC + SAu) (58)

(ii, 2 are the arithmetic means of the values before and after the change) that, for a given Au, the space concentration effect is the more important the lower the bulk concentration and the higher the mobility.

If novel migration processes at the phase boundary itself (which in principle are contained in the third term of Eq. (36)) are dominant in a, there ought, however, to be marked deviations from the bulk values of the migration enthalpies. It appears very unlikely, but cannot be completely excluded, that the prefactor alone would change. In the discussed cation conducting composites partial blocking of fast pathways by vacancy conducting regions would also be in line with the temperature dependence. In view of the effective activation enthalpy (migration enthalpy of vacancies even at high temperatures) however, these regions should be space charge regions again. Reproducibility, values of the capacitance and especially the experiments in the previous chapter speak against this complicated situtation. A refinement would be to seek a possible change in the standard thermodynamic values on the basis of structural relaxations or elastic effects. But it is not possible to take account of such effects at the present time because the accuracy of measurement does not allow detailed conclusions on these lines. Very recent measurements of the overall thermodynamics of the composite electrolytes have been performed with the electrochemical cell

Pb I PbClz I A&l: A1203 1 Ag .

The results proved, that deviations of the formation enthalpy and entropy of the composite from the values for pure AgCl are very minor and can be attributed to the surface interactions and/or thermoelastic strain due to different expansion coefficients [S 11.

220 J. Maier

The most important aspects of the problem of the underlying simple distribution topology (cf. also one-dimensional treatment) and the assumption of constant b-factors have already been discussed. This also applies to the assumption of a Boltzmann distribution for effects close to saturation (see also Chapter 2.5.2).

It has also to be borne in mind that the discreteness of the atomistic problems is not adequately appreciated in the continuous approach as described in Chapter 2.3.6.

The assumption of semi-infinite boundary conditions having one finite limit is a fundamental point. The Debye length is rather large particularly at low temperatures so that these conditions can sometimes be broken. As shown in Chapter 6 this may be associated with a further increase in conductivity (cf. “nano-size factor”). As shown on page 29, the assumption of a pure material is not critical for the evaluation.

2.5.2. Space charge induced boundary phases The effect of high concentrations occuring in space charge effects close to the interface deserves

a special consideration. Naively, one may assume that an increase of defect concentrations over the Boltzmann dilution limit would result always in a harmless correction to the calculated behavior. There may, however, exist conditions, where the situation changes qualitatively. In spite of the positive standard free enthalpy of defect formation, there is an equilibrium concentration of point defects in any ionic crystal due to the initially steep increase of configurational entropy with the number of defects, resulting in a minimum in the total free enthalpy (plotted as a funtion of the defect number) at very small defect numbers. Most defect-defect interactions and also structural changes that can also be expressed in terms of defect-defect interaction lead to an additional contribution in G falling with the defect numbers. In many cases this results in a second minimum at comparably high defect numbers. Under certain conditions, especially at high temperatures, this minimum may become the absolute minimum. Then a transition occurs from a low-disordered state to a highly disordered state [llO,lll]. Such transitions may be superionic transitions such as the transition from the /3-AgI phase to the oc-AgI phase, or the higher order transition in PbF2 where the cation-sublattices are considered to be molten. In materials where the completely molten state is lower in free energy, an overall melting takes place. The qualitative reasons for the transition is simple: At low defect concentration the formation of a new defect does not depend on the number of already existing defects. This changes at high concentration values. If the presence of already present defects facilitates the formation of new ones, a positive “feed-back” results and a disordered lattice becomes very favorable in free energy. Recently this behaviour could be quantitatively interpreted for different ionic conductors by a cube root law [1 111 just reflecting the mean defect-defect distance.

Such an instability is due to defect-defect interactions. Since high defect concentrations are expected under some cirumstances in space charge regions, a local phase transformation at the boundary may occur [1 111 (see also Ref. [112]) (even though the nature of the defect-defect interactions is different from those in the bulk). In the superionic phase the Debye length is small, space charge effects have probably vanished, also the defect profile in the adjoining bulk phase is not as steep as before due to the screening effect of the superionic phase with respect to the surface interaction. Such effects probably account for another class of boundary conductors, whose small activation enthalpies point to a liquid-like surface conduction, but may also explain the occurrence of amorphous phases in AgI : Al203.

In particular since charge and counter charge are separated in space charge regions such effects do not unavoidably happen. In AgCl and AgBr especially (also CuCI, TlCl, LiI) the activation enthalpies as well as X-ray-studies [6&l] point to a “normal” space charge behavior.

2.5.3. Discussion of alternative models Every competitive model has to explain the fact that for the compounds given in Figs. 5, 20 the

activation energy of the conductivity corresponds to the bulk value of the enthalpy of migration of the metal vacancies.

First let us search for hypothetical alternatives that could also fulfil this requirement. The most obvious alternative is an impurity effect as a result of higher valency cations. The classical argument


of Liang to exclude such a possibility lies in reference to the observed dependence on the volume fraction in the second phase. A few percent by volume would certainly be sufficient to maintain a saturation of homogeneous impurity. This argument, however, is not sufficient, there are several possibilities to explain such behaviour [3e]:

(a) Small quantities of A13+ could be liberated during preparation and then remain (more or less) homogeneously frozen in the material at the temperature of measurement.

(b) The number of high valency cations available could be determined by the surface of the second phase, Thus it could be imagined that foreign cations on the surface of the Al203 (or possibly segregated out) could be responsible for the effect or A13+ cations which could have been -to pick out one hypothesis - introduced into the material as a result of taking part in an acid- base reaction with the OH groups according to the following hypothetical overall reaction:

6 1 -OH + Al203 + 6AgAs - 6 1 -0Ag + 2HzO + 4Vis + 2Al&, . CR101

Solubilities in LiI as a result of effective foreign cations have been demonstrated to be very low [4]. Impurity effects such as those described above have been excluded for a number of reasons in the case of TlCl, in particular:

(a) the extent of this effect in comparison to the effect of PbC12, (b) the significant absence of the Koch-Wagner-effect (“knee”) in the log-l/T curve.

For these reasons very detailed experiments have been carried out in AgCl and AgBr to refute such a mechanism [3fl, The results are illustrated in Fig. 44 and Fig. 50:

(i) The conductivity effect is not dependent on the duration of preparation (Fig. 50). (ii) Even with AlC13 it is not possible to produce such a homogeneous effect in AgCl (Fig. 50).

(iii) If the A1203 particles are removed from the two-phase mixture (dissolution of AgCl in NH3, separating and washing the aluminium oxide phase with HCl, combining the two solutions in excess HCI, filtering off and drying the precipitated AgCl, preparing samples and measuring it), the conductivity effect disappears (Fig. 50). This demonstrates the evident importance of the physical presence of Al203 particles. (It is not possible to exclude certain inhomogeneous influence of A13+ ions in the space charge zone completely.)

If the case of an independent conductivity mechanism as in the case of effects discussed in the previous chapter the only remaining possibility of directly observing the enthalpy of migration of metal vacancies in the bulk [4] is that highly conducting paths, that determine the conductivity

(o,,, = o,;n,p&, are “interrupted” by only moderately conducting parts of specific conductivity CJ

(a,,, = &,athQ)patho,+, CC (T, since the path is in series to them). The major part of the pathway would have to be extremely conductive in such a case so that the few blocking regions determine the path resistance, on the one hand (i.e. weighted for the bulk fraction and the &factors), and, on the other hand, the effective conductivity of the complete pathway remains still high with respect to the pure substance.

Now the observed energy of activation is approximately the enthalpy of migration of the metal vacancies, but not that of the bulk (at moderate temperatures the enthalpy of formation is appreciable for the bulk material), so that it has to be assumed in such a case that, in contrast to the remaining material, these blocking ionic conductor regions are suitably doped or that there are specific thin-layer effects, such as were discussed in Chapter 6, operative in these regions. In the last case at least - if not under much more general conditions, as suggested by calculations carried out by Blender and Dieterich [77] - the assigned effective capacity would be expected to be much higher than that found experimentally. Severe non-reproducibility and new percolation thresholds ought also to be observed, neither of which are found experimentally. Neither does it seem possible to interpret consistently the effects about to be described in two-ionic conductor mixtures, in polycrystalline materials (e.g. NH3 effect) or thin-layer effects by other mechanism than a space charge mechanism. Note in this context especially the effects induced by gaseous species, the effect on heterogeneous catalysis or the behavior of the minority species. It is not completely possible, however, to exclude a complicated superimposition of various influences, particularly the

222 J. Maier

x: IOmin. melted 0: 30min. melted

-6 -

I I I I I I.8 2.6 3k

* 103T -‘/ K-’

Fig. 49. The conductivity effects in AgBr:AhO, (after sintering!) are not a function of time [3fl

xxx 4Cl, pure 000 4Cl + lmlo AXI3 --- A&l +IOv/oAl2O3 l l l AgCI, precipitated 0 D 0 AgCl : A1203 MUS A@

-5 - I

-7 - X

I I 6 , I 1

1.4 2.2 ) ,()3T_l,

21 3.

Fig. 50. The effect of homogeneous doping of AgCl with AU3 and the removal of Al203 from AgCI:A1203 [3fl.

fjord effects in fractal structures in all cases. At this point, the work of two authors deserves special attention viz. the work of Phipps et al.

and the work of Dudney. Phipps and Whitmore [41,42] investigated the boundary between LiI and SiO2 and found a fast pathway with a novel activation enthalpy. Even if we neglect the fact that the use of point electrodes may modify the effective activation enthalpy, this result does not contradict the overall scheme. At the contact to a SiOz-crystal no strong surface interactions and thus no strong space charge effects are expected. Thus proper surface pathways with independent migration-values should, if existing, show up. The situation becomes clearer in the discussion of


LiI thin films in Chapter 6.

223

Dudney performed systematic experiments on AgCl composites giving evidence for the importance of stress effects and dislocations [60]. External stresses give rise to an increase in the conductivity [5]. Such effects are well known to play a role in freshly prepared materials either polycrystalline samples or thin films [60,13]. This will be treated in detail in Chapter 5 and 6. A careful annealing is able to heal out most of these effects. Even modifiying the boundary conditions the main question in this context is not the existence of stable dislocations after annealing: It is possible that a stable configuration of Al203 layers in an AgCl matrix may involve stable dislocations, in the same way as small angle grain boundaries may be attributed to a disclocation arrangement. The main question is, if these dislocations provide a novel pathway with novel migration parameters, or if space charge effects play a major role. In our view an overwhelming amount of evidence as already given above and as will be given below confirm the latter point. It is also known in particular from dislocations in AgCl that the charges of space charge zones involved can have the same sign as discussed above [87]. In the silver halides even the non-stationary effects of sample preparation, e.g. pressing, are likely to be space charge effects as laid down in Chapter 5. A more direct indication of space charge effects by impedance spectroscopy of a single AgCl-oxide interface is discussed in Chapter 2.6.

2.54. Discussion of other methods of measurement The analysis of the surface at solid-solid interfaces is fundamentally more difficult than of free

surfaces [113]. The removal of discrete layers is always associated with in-situ changes in the boundary conditions. Although experiments at ceramic interfaces suggest concentration profiles and basically support the mechanisms proposed here these sources of error cannot be ignored [114]. In the case of the systems described here there are other problems in addition to the small width of the layer; these include the radiation sensitivity of the silver halides (STEM, SAM, XPS, LEED etc.), their sensitivity to moisture in the case of LiI and to oxygen in the case of CuCl. IR and microwave experiments suggest an interfacial interaction [95,115]. We have initiated IBIQ experiments, b-NMR and (pulsed field gradient), NMR, high resolution infrared measurements and ESR investigations from which we hope to obtain further information. The interpretation of the NMR spectra so far obtained is controversial [2 1,22,90,28,116]. Some authors are in agreement that the majority of Li ions in LiI:AlzOs and LiBr- H20:AlzO are in an environment that differs from that of the bulk, which could be the result of the formation of an extended interaction layer (possibly even a third phase that is only stable in the boundary layer) and/or of the formation of space charge zones in the thin-film region. It is not possible to eliminate the possibility of additional effects, especially in the case of these two very water-soluble substances, whose bulk defect chemistry itself is in dispute. This applies particularly when inadequately dried aluminium oxide is used, as in Ref. [116].

X-ray experiments have been carried out by almost every author who has published results on the systems under consideration here. The results confirm that no global chemical reaction has occurred but do not provide the basis for adequate conclusions concerning local interactions. Interestingly in AgI : Al203 (in contrast to AgCl : Al203 and AgBr: Al203 [81]) some amounts of amorphous phases have been detected [103].

As already reported recently the standard free enthalpy has been measured by e.m.f. techniques [8 11. Only small deviations are found which can, if meaningful at all, be contributed to the surface- surface interaction, and/or to thermoelastic strains.

Wagner jr.‘, group has carried out calorimetric determinations on AgI-Al203 two-phase mixtures (and on the two ionic conductor system AgI:AgBr) (see Chapter 2.3.3) [27,114]. The authors found perceptible excess enthalpies in comparison to “mechanical” mixing and interpreted the findings in terms of defect formation in the above sense. The values obtained are rather large, so elastic effects and acid-base interactions cannot be left out of consideration. Qualitatively the effect on the temperature of the /?/cx transformation by the addition of Al203 takes place in the same manner as by homogeneous doping with high valency cations and can be correlated with the space charge model [25,26].

Thermoelectric measurements on the system AgI:AlzOs have also been interpreted in terms of

224 J. Maier

Ag+ vacancies by Shahi and Wagner [46]. Evidence for space charge zones at the surfaces of silver halides has been obtained by L. Slifkin’s

research group [87,117,118]: These authors doped AgCl and AgCl homogeneously with MnCl2, whose rate of diffusion is great enough to reach equilibrium within a week, but low enough to bring about perceptible changes in the Mn profile during the experiment. Individual layers were removed by precise etching with great spatial resolution using suitable aqueous solutions. The Cl profile was monitored simultaneously. This gentle method (in contrast to SIMS) yielded a profile that was in agreement with the calculated space charge profile. As regards the quantitative evaluation s. Chapter 6.

2.6. Detection of a Single Ionic Space Charge Layer by Impedance Spectroscopy

Recently we succeeded in investigating ionic space charge effects at a single interface oxide/AgX for the first time [19]. As has been discussed above, space charge effects at a single interface are difficult to measure by conductance experiments. In a measurement perpendicular to the interface an enhancement layer cannot be seen even in a parallel experiment, the conductance effect is too small to be detected (if not very thin films are used as in chapter 6). On the other hand, depletion effects resulting in the occurence of a thin insulating layer, would not be measured in a parallel experiment but might be seen in a series experiment. In order to investigate the above discussed ionic space charge effects in such a way, two conditions have to be fulfilled:

(1) Second phase materials have to be found which act with respect to the cations as a base similarly as y-Al203 (i.e. sufficiently), but nevertheless exhibit a high electronic conductivity to be used as electrodes (in contrast to the insulating alumina).

(2) Measurements involving the silver halides have to be performed at elevated temperatures, where the interstitial conductivity significantly prevails over the vacancy contribution such that an accumulation of cations at the interface corresponds to strong depletion effects (see Fig. 31, bottom).

(It should also be noted that the effects of insulating layers are, if at all, not reliably seen in dispersed systems, since they are partly short-circuited by the better conducting bulk * , whereas the detection of highly conductive layers requires the high interfacial density of composites.) Recently such measurements succeeded for the first time. RuOz, Fe304, Nb~05_~, Lao.sSrs.sCo03 turned out to be applicable phases. They were, in polycrystalline or monocrystalline forms, contacted with AgCl or AgBr single crystals, the other side of which was contacted by sputtered Pt. Details are found in Ref. [ 191.

In addition to the bulk impedance three further impedance signals were detected with all the oxides, one can be unambiguously attributed to the insufficient contacts (point contacts [78,79]), a further one (at the lowest frequencies) to a Wagner-Hebb polarization. The remaining semi-circle indicates the depletion layer due to the following reasons:

(1) The signal does not appear if Ag or Pt is used as electrode material rather than the above oxides.

(2) The signal depends on interfacial conditions but not on the thickness of the solid electrolyte. (3) The signal is bias dependent. (4) The temperature dependence of the corresponding resistance corresponds to the temperature

dependence of the bulk value. This is in agreement with the extra resistance calculated in Eq.(28). Since the T-dependence of Ji;;; is small, the T-dependence of the space charge resistance = AZ1 is given by

AZ’ a (Ui * c~)-’ (5%

and thus roughly by half of the Frenkel enthalpy

(-Rain ui/al/T u O.leV,-Rain c,/al/T = AFH0/2;AFHo/2(AgCl)

* Blocking effects giving rise to a low frequency impedance branch, are visible but are difficult to evaluate due to the complex microstructure.

Ionk Conduction in Space Charge Regions 225

-1

-2

-3

-4

-5

-6

-7

eV

-8

I-12 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

lOOO/(T/[Kl) -

t

z U

2 WI

A

Fig. 51. Capacitance and resistance of the interfacial impedance response as a function of temperature (Ag I AgCl I Nb205) [I91

= 0.74eV, AFH/2(AgBr) = 0.6 - 0.7eV).

Figure 51 shows data for AgBr where a measured value of 0.75 eV fulfills exactly the expectations.

(5) Even more indicative, since very significant, is the temperature dependence of the corresponding capacitance. Whereas C-values are normally more or less independent of temperature, space charge capacitances are expected to increase strongly with temperature due to decreas- ing Debye length.

As well known from electrochemistry, the space charge capacitance is inversely proportional to h. Since h DC &* and since the AgX-bulk is intrinsic (i.e. cm CC exp-AFHo/2) the temperature dependence is predicted to be AFH’/~. Figure 51 confirms the prediction with EA = 0.36eV (AFH’(AgBr) = 1.2 - 1.4eV).

The absolute values of the space charge response are not very meaningful due to the insufficient contact. On the other hand, this complication is certainly fortunate in to make the space charge effect measurable [19,119]. At present we are investigating the response using different designs, special point electrodes as well as a new method to scan the distance from the interface by frequency variation [75,120]. In this “penetration impedance technique” the interfacial behavior can be nicely and sensitively detected. The AglAgCl contact studied is probably characterized by an ionic inversion layer as a consequence of Ag+ injection by the Ag-metal.

So far the bias dependence for the oxidelAg-contact could not be reasonably evaluated since a superposition of the space charge polarization with the Wagner-Hebb effect (stoichiometric bulk polarization) occurs. Momentarily, theoretical calculations are being performed to understand better the complete impedance response in a mixed conductor from a fundamental point of view [75]. Nevertheless, these measurements provide clear evidence for the expected ionic space charge effect. Also in our laboratory ionic field effects at the boundary of AgCl to insulating oxides are under study. However, it may well be that ionic field effects in these systems cannot be detected due to a comparably high core disorder [75].

226 J. Maier

3. INTERFACES BETWEEN TWO IONIC CONDUCTOF& (CONDUCT. ANOMALIES IN MISCIBILITY GAPS)

Enormous conductivity effects have also been detected in two-phase mixtures, i.e. misciblk. gaps involving ionic conductors: first in 8-AgI : AgBr [44,45] (see Figs. 3, 52c), but later on also in the system AgI : &Cl [161 (see below) and in other chemically more complex ones (see Fig. 3) [53]. In Frenkel disordered systems, we would expect a double effect if the above heterogeneous doping mechanism is taken as the explanation. The silver ion can pass not only from one ionic conductor to the interface, it can also pass into the boundary layer of the neighbouring ionic conductor. The situation is analogous to the contact potential formed at a contact between two electronic conductors (see also [121] and Fig. 52a). According to Fig. 6b one ionic conductor (MX’) represents, as it were, a cationophile, while conversely the other (MX) has a destabilizing effect on the cations. If Frenkel equilibria are assumed this transformation can be described adequately by transfer of a defect Vi,:

V&(AgX) + AgAs(AgX’) = Va,(AgXl + AgA,(AgXl . @11)

Equivalently the process can be described by the transfer of M; or by a homogeneous Frenkel equilibrium.

The latter formulation is most suitable from the point of view of the majority charge carriers

AgA, (AgX) -t Vi (AgX’) e Agi’ (AgX’) + Vi, (AgX) (Bl2)

In this way the vacancy concentration is increased in one ionic conductor (MX) and that of the interstitial particles in the other (MX’) (Fig. 52). When there is contact between two Schottky ionic conductors large effects (i.e. beyond the level of intrinsic concentrations) are not to be expected in the absence of the possibility of accumulating charge at the phase boundary (see Chapter 4) or of introducing ions interstitially at least in the boundary layer, so that the negative results obtained with the system LiBr : LiI are by no means surprising [122].

The thermodynamic calculation [7] closely follows that in Chapter 2.1. An even more complete phenomenological treatment is possible, as the bulk thermodynamics in both materials is defined. The variation of the relevant functions is reproduced in Fig. 53. Equation (11) relates the concentration of the defects in the bulk with that in the interface (cf. Fig. 52b), x = 0 and x’ = 0. The heterogeneous Frenkel equilibrium, referred to x = 0 and x’ = 0, now interrelates both concentration parameters cvo (= cv(x = 0)) and cio (= ci(x’ = 0)) by application of the equilibrium condition ,SVjfij = 0. For sufficiently dilute defects

E’ c;o (Go - Cm)* = c,o -=- E Go (a _ c’_)2 - 40 - (604

is obtained (dashed parameters refer to MX’). The condition of global electrical neutrality, which is equivalent to the continuity of the dielectric displacement (absence of charge accumulation at the interface) on account of the Poisson equation, makes it possible to calculate the remaining free concentration parameters and the potential jump across the interface. This gives (c”: standard potential) for semi-infinite boundary conditions:

‘5’ aaat G; ‘I2 E’ cio KTexp- RT =r7. 1

A 1 G; is the free standard enthalpy of the heterogeneous Frenkel reaction, that can be related toayhe standard enthalpy of transfer of V’ As and Ag,: and the bulk values. K takes account of the potential jump at the interface and can be calculated by assuming a charge carrier-free contact zone (characterizable by an effective dielectric constant) [7].

The contribution to the normalized parallel conductivity, AYII, such as comes into the bicrystalline experiment illustrated in Fig. 59, is calculated as:

AY” = (4R2T2&;K_$‘4 (uv + u;) (61)


---k&j- MX’ T MX’

b) Ln I

3 vi

I’ , ,’ ‘.., tvl;

.@ *... . . . .

l- 40 -=rr...

-*....... .....l,..-- . . . . _---

a... *.. Q.. Mt. *.* . ..*__*

/’ .@

’ %4 5.

1 F-4 f 00 1 f’-

.x. 5 '... e' . . . . 0 .%. %. - . . . . . . .

. . . . . .*.....

u ,I . . . N... k

s O/'

. . . . 3 I I I . . . .

20 LO 60 80 "..._. 0

loo. qAgBr, -

Fig. 52. a) Process expected during the establishment of a contact equilibrium between two Frenkel defective compounds (heterogeneous Frenkel equilibrium). b) Concentration profile set up as a consequence of a). c) Experimental results from [59] in the miscibility gap of the system

AgI:AgBr at 25°C suitably plotted. [7]

The electrochemical equilibrium “constant” 1”;F* characterizes the product of the pure chemical constant of the heterogeneous Frenkel reaction (standard chemical work) and the electrical potential factor (electrical work).

An estimate using realistic data for the free energy of transfer reveals that it is possible to explain conductivity effects of the order of magnitude under discussion [7]. In addition, we have to be aware of the possibility of charge storage in the core (see next chapter).

Since both phases possess similar conductivities the superposition of possible conductivity paths is much more difficult here than it was in Chapter 2. However, on account of the nature of both phases, the present case probably adequately fulfils the conditions of similar grain form and size and random distribution of both phases (see Chapter 2.3.3). Thus, here, a Monte Carlo calculation as laid down by Roman et al. [63] would seem profitable.

In fact when the data in the original publication (see Fig. 3) of Shahi and Wagner [59] are plotted in a suitably modified manner as in Fig. 52c, they exhibit the typical and calculated profile for random distribution (cf. Fig. 17). However, on account of the double function of both phases, the 1.h.s. region of Fig. 17c appears - crudely put - as a mirror image.

If realistic grain sizes of the order of 10m2 . . . IO-‘pm (or even less) are assumed (cf. preparation by cooling homogeneous mixtures in the region of the miscibility gap) and Eq. (34) is used to make

228 J. Maier

I 1 1 I I I

-5 h 00 x 2x y-

Fig. 53. The variation of relevant electrochemical functions perpendicular to the interface between two ionic conductors. [7]

+

Fig. 54. Bicrystalline experiment (MX/MX’) schematically (0: Vb, x: Mi; T: thickness of the interaction layer). [7]

a crude estimate of the maximum effect, then the experimental results are explicable, in particular if the nano-effects are also taken into account(Chapter 6).

Recently the conduction mechanism and the effects of microstructure on the enhancement have been thoroughly studied for the system AgCl-jl AgI by using impedance spectroscopy [16].

The conductivity effects as a function of the volume content are given in Fig. 55 for one temperature. The microstructure has been studied by etching experiments and Auger maps [123].


1

-‘- 1 0T:30°C I

I 1 l T=49*C I

t-- miscibility gap -1

I _e”l I I

I I I I I , I , 1,

0 IO 20 30 40 50 60 70 80 90 100

Fig. 55. Conductivity of AgCl/AgI-mixtures as a function of the volume fraction at lower temperatures (AgCIIB-AgI).

In agreement with that impedance spectroscopy shows two similar high frequency semicircles exhibiting conductivities higher than the bulk (completely in agreement with the equivalent circuit). The bulk is not observed, since short-circuited (Also not observed are perpendicular blocking effects.). The ratio of resistance and capacitances allow us to derive the fractions of homo- and heterojunctions (ri and 12).

Although considerable agglomeration occurs, the microstructure can be simplified as follows: One recognizes that interfacial pathways consist of homo- and heterojunctions. Conduction via pure percolating pathways consisting of heterojunctions only is improbable due to the only moderate conductivity ratios (- 10 : 1) in view of their low probability (Fig. 56).

To a crude approximation this means that

where 1 and 2 indicate the two hetero- and homoeffects. Fig. 57 shows that both ratios are roughly comparable and diverge in the center of the phase diagram; at nearly equal volume fractions (of course depending on given size and distribution) the f2 - 0, Ii - 1 and h/Ii - co.

At high temperature where AgI is always in the a-form no second phase effect is obvious. Here the conduction occurs via highly conducting Ix-AgI grains resulting in a typical percolation law (Fig. 58).

Figure 59 gives the activation enthalpies as a function of volume fraction of AgI for three temperatures. In curve a, EA is close to 0.3 eV the migration enthalpy of the vacancies. This does not reveal much on the mechanism since also the solid solutions, i.e. the pure phases, exhibit these values.

However, this value is also maintained in the gap at higher temperatures (curve b), where the pure phases exhibit higher values (intrinsic). This points to a vacancy mechanism in both homo- and heterojunctions. At temperatures where AgI exists only in the a-AgI phase the activation enthalpy is much lower in agreement with the above consideration.

Fig. 60 finally shows how a vacancy conduction can be realized in both homo- and heterojunctions. This is in line with a transfer of Ag+ from /3-AgI to AgCl. [124,16]

230 J. Maier

Fig. 56. Two-dimensional model for two phase mixtures, where the different grains are modelled as squares of different coulors. Light bars denote the boundary regions occuring due to homo-

contacts, dark ones denote those occuring due to hetero-contacts [16].

This two-phase effect also opens up a range of possibilities as well as posing many interesting questions, with respect to both the direction of future research projects (e.g. multilayers) both from an experimental (multilayers) and from a theoretical point of view (molecular dynamic simulation of contact and phase equilibrium). Meanwhile many other, especially fluoride, systems have been investigated [53]. Of special interest are conductivity variations due to morphological changes in these systems. On account of chemical complexity and thermodynamic uncertainty, however, a quantative analysis has not been possible.

4. GRAIN BOUNDARIES (CONDUCTIVITY ANOMALIES IN POLYCRYSTALLINE MATERIALS)

Grain boundaries (and the same is true for single dislocations) constitute another important heterogeneity in solid systems. As already mentioned several times (AgCl, AgBr, TlCl) similar phenomena occur in pure polycrystalline materials to those in ionic conductor/oxide mixtures. Boundary layer phenomena were also postulated by Khandkar and Wagner to explain the particle size dependence of the conductivity of polycrystalline AgCl samples [33]. Although this dependence


-40 50 60 70 80 SO lob

100 cp W&J

Fig. 57. Ratio of the resistances RI and R2, weighted by the factor 10 and of the prefactors of the CPA-element C; and C;, (which are capacitances for a - t), as a function of the volume

fraction of the solid solution AgI,V.V (AgClo.~Io.%) [16].

0.5 * , . . . . , , ,

o-

-0.5

-1

-1.5

-2

-2.5

-3 -3.5 !.,,/., 0 :

-4 -3 -2 -1 0 1 2

Fig. 58. Double logarithmic plot for two phase mixtures AgCllor-AgI at T = 182°C [16].

and the temperature dependence found can both still be explained in terms of dissolved impurities, the absence of a conductivity “knee” (Koch-Wagner effect) at high temperatures (cf. Figs. 22, 35) in the silver halides and particularly in the case of TlCl points to space charge effects. On account of the symmetry of the properties (e.g. chemical potential of the charge carrier, see Fig. 61) on both sides of the grain boundary and the observed activation enthalpy, it has to be assumed that an accumulation of Ag+ or Tl+ at the grain barrier is favoured on crystal-chemical grounds or because of the chemical effect of the segregated impurities at the grain boundary. The result is symmetrical space charge zones enriched with metal vacancies on both sides of the grain boundary (cf. Fig. 61).

Even though elevated grain boundary conductivity is normally attributed to core effects (“pipe

232 J. Maier

11J1J.y(Agbs)

0 20 40 60 60 100

1.I.I. I I

temperature ranges

-25 - 70

-0-100 - 130 0.6 : I

-t-150 - 190 I

a *+e*-*- \ Ct.*.

0.1 : )' c \ q -*

0 ..*'.r.'c.."'*'... 0 20 40 60 60 100

lOO.x(Agl)

Fig. 59. Activation energies in the three different temperature regions as a function of the mole fraction x and the volume fraction with respect to the miscibility gap Magi,.

diffusion”) [20], the phenomenon of space charges is not unknown in the context of grain boundaries, where a measurable mobility of ionic defects is expected to be of importance [114]. In the present case according to the previous discussion it is once again the activation energy that provides evidence of dominance of the second mechanism (according to Fig. 6b). As a driving force for the charge accumulation (ionic segregation) we consider, in addition to the simple osmotic pressure effect, a structural, i.e. chemical, stabilization of the ions in the proper interfacial layer. On the basis of very similar results in other ionic conductor systems it is possible to explain in this manner a whole range of phenomena, that were previously not understood [9,12].

The contributions to the conductivity parallel and (approximately) perpendicular to a grain boundary are described by Eq. (22) and Eq. (26). There are no difficulties in including separate core effects in the manner illustrated in Fig. 62. While it is immediately obvious that, in the case of parallel effects, the impedance can be described by parallel switching of a capacitor to the appropriate resistor (integration over a(x) +&IX(X)!), one would not intuitively expect that in the case of the impedance perpendicular to the interface, but has nevertheless been found to be a good approximation in many experiments (cf. [9]) (see equivalent circuit diagram Fig. 63).

Figure 64 illustrates the results of measurements on a bicrystal. Apart from the bulk semicircle at high frequency it is also possible to recognize a low frequency semicircle which takes into account the transfer impedance through the interface. The latter disappears by intermediate annealing, the former does not. The comparably low capacitance of the interfacial effect rules out space charge depletion layers but is fully consistent with “current narrowing effects” [9,78,79]. Such a bicrystal experiment cannot be used to demonstrate a possible increased parallel conductivity in the case of silver halides; the effect is too small for this.

However, the situation is different in the case of polycrystalline material. Here the portion of both parallel and perpendicular paths is considerable, and the superposition of all the effects is complex. If the topology given in Fig. 65 is taken as a first approximation, the bulk and boundary layers can - as described in detail in Chapter 2 - be described by Eq. (35) [9].

On account of the symmetry of translation of the brick-layer model the corresponding equivalent circuit diagram is simply represented by Fig. 63. (The grain boundary elements can be split into core and space charge elements as described in [9].) A s in the two-phase mixtures the capacitance of the parallel paths can be regarded as being very small.


b)

I AgX AU

E

AgX AM

Fig. 60. Energetic profiles at the phase boundary AgXIAgX’, when partial transfer of Ag+ from AgX to AgX’ is assumed (a: top, b: bottom). a) Chemical ~1, electrochemical ,ij and electrical potential 9. b) Ionic analogue to a semiconductor’s band scheme: Ei,Ev denote the one particle level of interstitial and vacancy (standard electrochemical potentials) respectively; E is the analogue to the Fermi-level (electrochemical potential). Energetic profile at the interface AgX/AgX (grain boundary). It is assumed, that accumulation of silver ions is possible at the contact of two grains, leading to an increased silver vacancy concentration in the space charge

layers. p, jj # c) and Ei, Evil E d) have the same meaning as in a), b).

After separation of the core (gi) and space charge contributions (SC), the impedance spectra can be represented, as an approximation [9], by

A 12 2 A” urn =u, +-Q)L CL

3 g-1 2 -1 m = CT, + -ippT~

C 3 1 A -I 2 0, = K urn + -pLu; ) 3

(63a)

(63b)

’ I *-I + -qhcq, 3 1 + fq$j $ii-’ for W << T&-l

234 J. Maier

a) MX

V

Log c .c 3 _ ..2 ./.

. . . _.....

. ..’ . ...‘.

‘. . . .._._... 0.’ i Log a

b)

space coordinate -

Fig. 61. Profiles of the defect concentrations, the total specific conductivity (top) and relevant electrochemical functions (bottom) perpendicular to an ionic conductor grain boundary in which

the accumulation of cations is favoured (9, > 0, ai > uv). [13]

Fig. 62. Measurement of conductivity perpendicular and parallel to a grain boundary by alter- nating the contacts in the desired direction. The actual grain boundary (core thickness: 2w) is

surrounded by a zone of space charge (effective thickness: 2A). [9]


Fig. 63. Simplified equivalent circuit diagram (see [9] for a more accurate one) for both the highly conducting paths which short-circuit a grain and approximately for the macroscopic sample.

-0 1 2 3 4 5 6

1cPx real part /R W

Fig. 64. Plots of the complex impedance (AgBr: bicrystal) (I grain boundary) for various sinter times (300°C). (Measurement temperature: 96”C, cell constant: 1.03km) [9]

Here it has been assumed that the relaxation times (T = EEO / a) for the bulk, space charge zone and core region increase in this order and are well separated from each other (In a better approximation the specific nature of the crossing points has to be considered.).

Figure 66 illustrates the impedance spectroscopic analysis (see first 2 plot) of freshly prepared AgCl at 25°C. Two semicircles can be seen as in the case of the bicrystal experiment. If the sample here is also annealed at 400°C and cooled, the remeasurement reveals - at the first glance surprisingly - that the right-hand semicircle has become, as expected, smaller (mainly grain growth); but the left-hand semicircle has increased [9]. This divergence in the behavior can only be explained by assuming that in the fresh sample highly conducting parallel layers (parallel to the bulk) contribute significantly to the high frequency semicircle and that these disappear with heat treatment (as the perpendicular blocking boundaries do). At the same time the capacity of the high frequency process remains unaffected (see diameter of B plot). This confirms the expected anisotropy of the grain boundary in a very impressive manner and in particular - on account of the activation energy - the space charge zone as a conduction path.

These reflections are supported by a quantitative analysis:

236 J. Maier

(1)

(2)

(3)

h

AZ

w ‘t u Ax

Fig. 65. Simplified topology of a polycrystal for calculation of superposition of the effects occurring. [9]

The maximum frequency of the high frequency semicircle actually yields the bulk dielectric constant. If such an E value is also assumed as an approximation for the grain boundary region (low frequency branch), then Eq. (63) yields a effective grain boundary thickness of 10 A, that suggests a blockade by the actual grain boundary core (possibly with participation of the V- I transition). Obviously bad contact effects as observed in the bicrystal experiment are less serious. The conductivity values of samples prepared in the same manner (~“0 = const) can be correlated approximately with the grain size according to Eq. (28, 35) [4] (see also Fig. 67), so that it is not necessary to introduce additional effects by way of explanation, even though it is probable that they occur [60,4,9,13] and allow further refinement of the data. The experimental curves can then be described quantitatively by c,s values that are about one order of magnitude below those for heterogeneously doped samples in agreement with the previous chapter.

As can be seen from Fig. 67 grain boundary effects can nearly reach the same order of magnitude as two-phase effects. The lower and less T-dependent values of C,O and Q)L mean, however, that the conductivity increases are usually smaller and react sensitively on annealing, whereas in the case of the two-phase mixtures the particles of Al203 fix the interface density, determine and enhance the interfacial potential. We found the strongest grain boundary effect (Fig. 67) with respect to c,s in a AgCl material prepared by decomposing the diamine complex (Ag(NH&Cl). Evidently some NH3 remains adsorbed at the grain boundaries in spite of the temperature of preparation, this - not unexpectedly - like the OH groups in Al203 acts as a nucleophile on account of its single lone pair of electrons and can stabilize Ag+ ions according to

(I’. . . INHs )gb + &A, = (Ag.. . INH3),b + V,+,g’ . (64)

Figure 67 shows that conductivity levels are higher than for a sample of comparable grain size, prepared by chlorinating Ag particles, and can, like Al203 samples, be described in terms of a saturation effect. The differences disappear within a heating/cooling cycle. Of the same significance for the interpretation are very recent results [56] suggesting an enhanced Vk conduction in polycrystalline CaF2 and PbF2 by activating the grains with SbFs which provides a strong attractor for F--ions. The impedance measurements also provide a nice configuration of the equivalent circuit: The first semicircle shows the vacancy migration energy (space charge) and the second the bulk activation energy (current narrowing).

In very fine-grained AgCl samples (0,x) at lower temperatures (see Fig. 67) we observe an additional flattening of the curve in the Arrhenius presentation, that is not attributable to electronic


real part / 10v5R, 10’°F-’

Fig. 66. Plots of complex impedance and of complex inverse capacitance for a polycrystalline AgCl sample (25°C) before and after sintering (400°C) (cell constant: 0.076 l/cm). [9,12]

-1 I 1 I I

I single crystal

2 exp and calculated for AgCl Al,03 (10 v/o.O6pml

3 exp and calculated fof AgCl Al203 1 lOv/o.O.O3~m

., 4 exp,AgCI+3Oppm CdCL2

Oh annealed ) grain size - 10 km

0

4 q (~.annealed ) 1

. from Ag (NH3)zCL g 5 -1O~ml30-40km)

-9- 0precipitated.g~ (05*02)pm

x cnwxpltated.gs (07’-02)wm I I I I I

1.8 2.6 34 4.2 103T-‘/ K-’ -

Fig. 67. Experimental results on polycrystalline AgCl on heating (hollow symbols) and cooling (solid symbols). [12,13]

conduction. The nano-size effect discussed in the Chapter 6 provides a possible explanation.

238 J. Maier

I ’ 1 I 0 20 40 -

time / min

Fig. 68. Conductivity effect of the (CN)z contact phase at room temperature.

5. THE INTERFACE IONIC CONDUCTOR/GAS (A NEW SENSOR PRINCIPLE FOR ACID-BASE ACTIVE GASES)

The above considerations with respect to the ammonia effect are fully confirmed by experiments of thin films exposed to NHs. In epitaxial films on various substrates [ 171 is a completely analogous increase in the conductivity of the AgCl phase in the low temperature region. As is to be expected from the electronic structure of the gas molecule and the chemistry of silver, the same effect is also observed with (02 (INC - CNI) (Fig. 68).

If for the sake of a higher effect, free-standing very porous thin films are used, the overall effect is a reduction of conductivity due to series effects. It is, however, not yet certain whether this effect is actually attributable to a V-I transition or to core processes. There is a conductivity reduction also in the intrinsic region at higher temperatures, at which interstitial conduction predominates. In any case this effect can be used as a new sensor principle to detect acid-base active gases [17,102]. Figure 69 shows the anology to the conventional boundary sensors for the detection of redox-active gases in a thermochemical and a semiconductor-like boundary picture.

Figure 70 illustrates the single-valued dependence of the NH3 pressure and the reversibility of the response. It was not our ambition to optimize the response time of the signal, however, a much faster response time (- 10s) can be achieved by using microelectrodes [79].

The quantitative relation between the conductance change and the ammonia pressure can be approximately described by a simple pathmodel: a series switching of heterogeneous pathways, each consisting of elements which are non-available for NH3 (resistivity p), of those which are available for NH3 but not occupied (p*) and of elements which are occupied by NH3 (p’). The path resistance then is given [17] by

R,,th = (&tip + (Nk - Nku) P* + (N - N/c) P> x/A (65)

where A is the cross section of the path and x the width of one site (free or occupied), N = $ total sites, Nk = # sites available for adsorption, Nk= = $4 sites occupied by NH3. If P is the equilibrium NH3 pressure, AR, the maximum resistance change (i.e. AR for P - 03, saturation), then the relationship AR(P) is given by

P(AR,/AR) = l/K + P (66)

Figure 70 (insert) shows that this relationship is fulfilled for K = adsorption constant 2: 17.


t I 02 e-

al Me0

239

space coordinate -

Fig. 69. Correspondence between the nucleophilic gas effect for AgCl and the sensor effect for SnO2 [14]. Bottom row: The interfacial effect of an adsorbed acid-base active gas on mobile cations, compared with the effect of redox-active gases (here oxidizing) on electronic carriers. The situation is represented in a thermochemical language (p, /J, 4) and in a semiconductor-like picture using energy and Fermi-levels corresponding to p” and fi functions. The r.h.s. column shows the changes in the conductivity (i, v denote interstitial and vacancy defects; p, n denote

electron hole and conduction electron).

This adsorption of silver ions by nucleophilic gas molecules is analogous to the electron capture process of chemisorbed redox active gases (e.g. 02) in oxide semiconductors. As mentioned in the previous section, we are investigating the influence of SbFs vapor and of BF3 (strong Lewis-acid) vapor on the conductivity of thin film of CaF2.

Useful relationships for boundary sensors as derived from the space charge concept outlined in this paper, are given in Appendix 3.

6. NANOSYSTEMS (MESOSCOPIC IONIC CONDUCTIVITY EFFECTS)

There are two reasons why space charge effects ought to occur more strongly if the (ionic conductor) particles, which until now have always been assumed to be large in comparison with the Debye length, are made successively smaller, or if the thickness of conductive films is successively reduced:

(1) Trivially, the portion of the boundary layer increases. (2) The behaviour changes qualitatively in the case of very small dimensions.

The latter point is of fundamental importance for the understanding of nanosystems (extremely thin films, nanocrystalline materials, high densities of dislocations, grain boundaries, high interfacial densities in heterogeneously doped materials (high Al203 concentrations)) [ 10,12,13].

This is illustrated for a symmetrical film of thickness L in Fig. 7 1. The defect profile in the centre of the sample approaches the bulk value as long as L approximately remains > 4h ; this no longer applies if L falls below - 4h, the sample is then charged throughout and considerably influenced by the boundary phase. This means mathematically that the boundary conditions are no longer semi-infinite but are finite in both directions. The influence of the right-hand boundary layer is experienced even in the immediate context of the left-hand boundary layer. The influence of these effects on the parallel conductivity will be considered below. The published literature contains a whole range of experimental evidence on this topic but no satisfactory quantitative approach. In what follows an ideal linear problem (Fig. 71) will be assumed for this purpose. (It is critical to

240 J. Maier

b t/Ihl -

2 4 6 a lo 12 14 e 120 22 24 26

N 2.0-

8

8 .

1.0 -

1.6-

plNll,l/bw - I I 1 1 I 1 I I 1 J

0 1 2 3 L 5 6 7 .t/thl -

Fig. 70. The resistance changes of porous free-standing AgCl-films as a response to different NH3 partial pressures as a function of time. The response of (CN)2 is also shown. The insert indicates that the stationary response excitation relationship can be understood by a Langmuir- adsorption (K: mass action constant; AR: stationary resistance change upon exposition to NH3;

the lower index m refers to the saturation pressure).

note that fulfilment of the approximations made, such as structural invariance, neglect of elastic effects on thermodynamic and transport properties, may be dubious at such small dimensions.)

For mathematical reasons a symmetrical film is assumed with the limits x = 0 and x = L (?& = i&) for the calculation. The relative concentration in the centre of the sample (x* = 4, z* = L/2h) would be g*. The first integration of the Poisson-Boltzmann differential equation can still be carried out analytically - assuming two majority charge carriers 0’ = 1,2) with the same absolute charge (z = zr = -22). We obtain [lOa,12]

dln<j _ F - (SigIlZj) [ (cj + <y’) - (<T + ri*-‘)I”’

where c; and <jo are implicitly related to each other via $* = L/2h according to

z* = 2e [f (C7’; 5) -f (<i*-‘, arcsindm)]

Elliptical integrals of the first order are designated f here [lo]. (In a very limited number of cases (sufficently large samples, see [lo]) it is also possible to approximate g* by superpos- ing the electrical potentials of two undistorted profiles [125], from which it follows that <T =

exp [89jsign(zj) exp(-LIZA)]). For small effects IYol = 1 &I - t& 1 /(RT) K 1 or 5 CK e) it is possible to integrate Eq. (67)

further after Taylor expansion with the result [10,12]


0 2 4 norm.space coordinate -

Fig. 71. Defect profile (9” > 0) of an “ideal” thin film for various thicknesses (L). The bulk value is not reached when L < 4A [lo].

(69)

The corresponding conductivity increase as a function of the film thickness is given by (whereby 21.2 = f-2)

Au/,(L) = F.zFc, (2 In 501.2) tanh & ( (UI - ~2) - A (~1 + ~2)) (70)

with the abbreviation

A = : [ 1 + (L/h) / sinh (L/h)]. (71)

The last term in Eq. (70) is important in the case where defects 1 and 2 have similar mobility. Aa,

is always positive under these conditions. In the normal case of different u values (e.g. UI z+ ~2) and for values of L that are not too large the equation simplifies to [lo]

Au:(L) = + zFulc, In co1,2 CC0

Ftanh&. (72)

Even though the equation for small effects is not very relevant to our aims, it does exhibit the intuitively expected variation of Aa,! with L (see also Fig. 73~).

Unfortunately it is not possible to produce a rigorous explicit solution for the case of large effects (510 > e) that is more important in practice. However, the following procedure is possible: The deviation from the bulk values of the concentration of defect 1 (cl) that will be regarded arbitrarily here as the enriched defect, will be approximated by the difference from concentration value of the counter defect (~2). The variation of the two concentrations and the quality of the approximation (for a very unfavourable situation) is shown in Fig. 72. For not too thin films (L 2 4h) the relative error is given by [lOa]

Error= (1 -91)/(1 +9r) . (73)

The reason that this approximation is serviceable for large effects is that coo and c2 can be neglected close to the interface while at large distances from the interface c2 approaches cm = czrn because of the condition of electrical neutrality. Since

242 J. Maier

I x=0 (74)

the calculation of AU,,, only requires knowledge of the first differential of 4 at x = 0 . This yields

Ll2 LIZ L/2 Ask a

I (cl -c,) dxa

I IPI dx a

I 0 0 0

[lo] the important relation

Aui = 2: [~RTE&~ (c,o - c;)]“’

whereby c: is accessible via Eq. (68).

(75)

Figure 73 reveals various potentialities of the Y-L characteristics. When there are negligible interfacial effects, the defect profile is a horizontal line. The conductivity (a Yll = Lam) is proportional to the film thickness (case I). The Y-L curve still remains a line if space charge effects become appreciable (i.e. large effects or smaller film thickness) but it is displaced in a parallel manner. The space charge contribution appears as a finite intercept (case II). This overall value is naturally identical with the value which is infinite in one direction according to Eq. (22), that is approximated for large effects (just as Eq. (75)) for c* -K c,) by Eq. (28) (Yll = La, +const.ul,/?$. If L I 4 then it is not only the bulk value that is affected by the thickness variation and in the ideal case the characteristics vary in the complicated manner shown in Fig. 73 bottom (case III).

(Y/i N La, + const.ul cl0 - c, (L)). 7 It is worth mentioning that the different cases may be observed by considering different contact

phases (at constant L) but also with a given contact phase when successively increasing the local resolution.

In the following text different literature results are discussed reflecting the above-mentioned cases. Figure 74 shows an experimental example of normal bulk behaviour corresponding to case I. A plot is illustrated of the conductivity of a LiI film on a SiO2 substrate investigated by us [15]. Surprisingly a hexagonal modification [15] is found for evaporated films (on account of the ratios of the radii the normal cubic modification lies at the edge of stability!), that gentle annealing (> 85’C) and pressure increase (2 10m3atm) change irreversibly to the usual rock-salt structure. As is to be expected from the discussion in Chapter 2.3.7, it is not possible to detect an appreciable surface effect at least at this resolution (the layer thickness increases by -35 nm/min). Stability, structure and conductivity considerations show that the bulk of the hexagonal phase is not responsible for u effect in LiI/oxide dispersions [15].

Several authors have observed a weak, constant surface effect (not comparable with the y- Al203 effects in the dispersions) in evaporated AgCl and AgBr films (usually mica or NaCl), that is dependent on substrate, orientation and temperature [126-1301 as shown in Fig. 75. This corresponds to case II. It has to be mentioned again how important careful annealing is to obtain

I

0 distance - distance -

Fig. 72. Profile (91 > 0) of the concentrations of the majority charge carrier, their difference and the excess value [lOa].


thickness 2

Fig. 73. Dependence of the normalized parallel conductivity (Y” = CT,!&) on the film thickness: a) (in contrast to b, c) no boundary effect detectable; b) film thickness large in comparison with the extent of the boundary layer; c) film thickness comparable with the Debye length [IO,1 31.

reliable stationary values. This is especially stressed by Ref. [ 13 1] and is analogous to the situation in freshly prepared ceramics and composites.

N. Starbov [129] in particular has evaluated a series of such results - by application of the Kliewer model - in order to obtain individual defect parameters. In his experiments, due to careful annealing the bulk conductivity of AgCl-films is very close to that observed in single crystals (in contrast to other authors), so that here equilibrium conditions may be assumed. However, incorrect relationships were used for evaluation due to a lacking quantitative theory.

Reconsideration [lob] by us according to the above space charge theory outlined here now yields a fair agreement with the data of Farlow et al. [87] for a “free” surface obtained by the profile analysis using the signal of 54Mn&-defects, as described in Chapter 2.5.4 (see Table 1 for (111) surface). Slight differences remain not least because of interactions with the substrate (see Eq. (41)) [4,14]. Large interactions as in the case of y-Al203 are not expected. It should be mentioned, however, that a very recent reevaluation of the tracer data resulted in somewhat different data [90]. Also, recent experiments on AgCl surfaces in our laboratory are not fully consistent with the assumption of an Agi-accumulation at AgCl-surfaces [132,133]. More careful experiments are needed to clarify this problem. The possible difference in mechanism compared to the AgCI/A1203 dispersions, however, is not a problem at all.

In this context it should be mentioned that in completely analogous experiments on the electronically conducting ZnO films, whose conductivity has also been measured as a function of the partial pressure of oxygen, Gdpel and Lampe [134] were able to obtain bulk and interfacial conductivities as a function of temperature and partial pressure of oxygen (P). In the case of the bulk it was found that a 0~ P-'14, which corresponds to a simple defect model in the case of acceptor doped material, while a characteristic exponent of -0.15+ 0.03 was obtained for the interface. If the experimentally determined space charge potential and a low mobility for the extrinsic (acceptor) defects is assumed then it is simple from Eq. (38) (compensating influence between effective concentration and effective thickness) to derive an exponent of -l/8=-0.13 that lies very close to

244 J. Maier

evaporation timelmin -t

Fig. 74. Reciprocal resistance of evaporated hexagonal LiI films on SiO2 as a function of layer thickness (WC), rate of evaporation: -35 nm/min). [IS]

0 364V E 333K

200 400 600 600

{ill} AgCL .400K

364i

film thickness/ nm -

Fig. 75. Parallel conductivity of AgBr and AgCl films (111) as a function of the thickness, according to Baetzold [ 1271 and Baetzold and Hamilton [ 1261.

that observed [3e]. (see Table II) But back to our ionic conductors: Very recent experiments by Schreck et al. [135] suggest that

these authors have also been able to make measurements in the extreme region of very thin layers of LiI (evaporated onto sapphire), whose expected conductivity characteristics are reproduced in the lower curve in Fig. 73 (case III). The experimental results of these authors are reproduced in Fig. 76; the similarity with the expected curve is very evident.


Table II. Defect parameters for AgQ(I 11) surfaces*

245

AjH,“, Conductivity Profile analysis Aj Si [129] Eq. 38 [87]* AiH{ /meV 370 800 780 a&/k 4.7 IO 9.2 A.,Hi /meV 880 600 690 A,S;/k 2.0 0 0.6

# The evaluation relies on an Agj accumulation which is in agreement with the literature but which is still under debate (see text) [132,133]. * A recent reevaluation resulted particularly in different entropy values. [118].

Table III. Par dependence of the conductivity of ZnO

Charact. Exponent exp. [ 1341 theoret. [3e] Bulk -0.25 -l/4

Interface - 0.15 f 0.03 -118

14

iE 6 12- ,Q/’

“b IO-

:‘/ Y c a 0 2 % 6 4 0 gP 0 /

G’O

j$ 2-i __-- _-

__-- __--

__-- __-- oo-- ’ I 1

1000 2000 3( DO

THICKNESS / [ A I

Fig. 76. Theoretical fitting of the conductivity data for LiI layers on sapphire after [135] by assumption of ideal interfacial behaviour and the approximation of symmetrical boundary

conditions [lOa].

As revealed by the results of our numerical analysis in Fig. 76 the space charge model that has been developed provides a good description of the results [lo] when the parameters 7&s (v here means Li vacancy) and h are set at & = 20 and h = 62OA. It is implicitly assumed here that approximately symmetrical boundary conditions can be introduced on account of the low activity of the sapphire (see above).

In Fig. 77 the defect concentration in the centre of the sample is plotted against film thickness in order to illustrate the thin-film effects. The transition to the actual region of nano-size effects is visible at x E 4h.

The first of the values obtained above, corresponding to a interfacial potential of 75 mV, certainly corresponds to the expectations for an inert phase such as sapphire. The obtained value of 620 2 for the Debye length is only reasonable for an extremely pure sample and would be amply large enough to interpret the results in LiI:AlzOs dispersions consistently. The activation energy for thin film conduction is very similar to that in the dispersions, which in general indicates that for LiI the temperature sensitivity of the interfacial concentration is very low. The attributing of the results of the measurement by Phipps et al. [41,42,136] on LiI/SiOz, which revealed a much lower activation

246 J. Maier

L/A -

l 2 3 4 5 6 7 0 I I I I I I I

Zo=3=ln~0

A= 620A zc-- 3

moderate thin films i

* N

Fig. 77. The plot of the normalized defect concentration in the centre of the film reveals the transition to the region of very thin films (L g 4h) [lOa].

energy, to a qualitatively different interfacial pathway * is consistent with this. Nevertheless, if the Debye length obtained is used to calculate charge carrier concentrations and

then mobilities are calculated via the measured specific conductivities then the results obtained are comparable with those for silver halides but about two orders of magnitude greater than those determined by doping experiments t . Unfortunately it is not possible to decide conclusively which result is the more reliable. What is certain is that the large apparent h-values are not artefacts of the assumption of symmetrical boundary conditions involved or the approximations made in the derivation of Eq. (75). Probably island formation and other possible non-ideal conditions, such as the presence of grain boundaries, dislocations, or non-equilibrium bulk structures may be involved as in the case of the thin-layer experiments described [15].

On the other hand doubts have already been expressed concerning the doping experiments. It seems rather odd that the enthalpy gradation of the Schottky defect reaction for LiI deviates significantly from both the CsI-RbI-KI-NaI-LiI series and the LiF-LiCl-LiBr-LiI series [137] and from the correlation to the melting point [127].

An interesting application of nano-effects that has already been mentioned occurs in micropoly- crystalline systems where the grain size is comparable to or even less than the Debye length (see Fig. 78a).

It is possible to define a “nano-size factor” (g) [12] that describes the ratio between the true conductivity increase to the value obtainable using semi-finite boundary conditions. For large effects (symmetrical boundary conditions, linear problem) [12] this becomes:

g = [4h/L] [(co - c*) /crJ]1’2 (76)

and can be calculated numerically for given values of h IL and co/cm with the aid of Eq. (61). The numerical result will not be dealt with here, rather the following estimate will be made: If the effect is large and the film thickness is not very much smaller than the Debye length, then g = 4h /L. If L = 0.4h , e.g. the conductivity is increased by a further order of magnitude, under the above idealized conditions.

Such nano-effects are probably responsible for the increased a-values of nanocrystalline AgCl samples at lower temperatures (see Fig. 67; 0,x). At such temperatures a comparatively high h /L

* On account of the extremely low surface activity of the silicon dioxide used this is in agreement with the discussion in Chapter 2.5, 2.5.3. In addition it must be considered that these authors used point electrodes, but evaluated the experimental results in a conventional manner taking account of a simple cell constant. t See [lOa] for a more precise discussion.


I

247

a)

f

0 0,

b)

X-

Fig. 78. Schematic conductivity profile of a nanocrystalline sample of MX/MX’. . sequences (a) and a nanopolycrystalline sample (b). The dotted lines indicate the bulk values. [12,13]

value can result from the association of intrinsic and extrinsic charge carriers and/or as a result of a high disclocation density. Since at these temperatures the temperature dependence of h ought to be determined by the negative enthalpy of association, it is possible that the negative temperature coefficient of the specific conductivity at low temperature sometimes reported in the literature may be attributable to such effects.

The above influences ought to play a role in two-phase mixtures at high A1203 concentrations and be partially responsible for some quantitative discrepancies (see Chapter 2.3.12). Assuming a typical surface density of 100 m2 /g for y-Al203 and complete wetting in a two-phase mixture of LiI: 4Ow/o y-Al203 would result in a mean oxide separation of ca. 50 A!

Starting from these considerations we have developed a method of deliberately preparing a nano-sized two-phase system AgBr:y-Al203 by coprecipitation of Al(OH)3 and AgBr ([Ag(NH3)Br + Al(N03)3] solution + (NH4)Br solution). However, disadvantageous agglomeration has so far prevented us from preparing a sample that was superior to a conventional one with respect to conductivity.

Two-phase mixtures or heterostructures containing two coexisting Frenkel defective ionic conductors are probably more promising, since here the phase distribution may prevent the grains from growing. The expected defect chemical behaviour is indicated in Fig. 78b.

In this context a very recent impedance study of highly conducting nanocrystalline CaF2 (grain size - 10 nm) has to be mentioned, too [138].

7. OUTLOOK

As can be seen from the considerations discussed above, the “heterogeneous doping” mechanism yields strong effects when the concentration of charge carriers in the pure material is relatively low and the mobility relatively high. The second phase must possess a suitable surface activity

(1 J Ai > A .G large if it is an insulating phase, while in the case of two ionic conductors the standard chemical potentials of the relevant defects (and thus AiGL,) must be sufficiently different. Taking realistic values for molar volume, volume fraction and possible grain size as well as for the dielectric constant (non-ferroelectric compounds) a crude estimate reveals that the conductivity effect achievable in R-‘cm-i has the same order of magnitude as the mobility of the enriched defect in cm2V-ls-l.

lg (0id [a-‘cm-l]) - lg (u/ [cm*V-‘s-i]) + 1 (77)

(Here effects stemming from the discreteness of the profiles (see chapter 2.3.6) or from the smallness of the system (6) leading to additional enhancements have been neglected.) From this it follows that (in the absence of strong structural effects and nano-size effects) “superionic conductors” cannot be created as a result of the ideal heterogeneous doping effect, but that all conductivity properties and all properties that are influenced by defect concentrations can be changed and controlled. The above mechanism also allows non-stoichiometries to be created in the boundary regions, which it is not possible to extend into the bulk. Beyond that an extension of preparation into the nano- size range and also making use of possible structural effects, in particular those leading to new interphases, can result in further surprises.

This is of particular importance for sensor applications:

(a)

(b)

The effect of gases on the surface potential can provide information concerning their concentration as a result of space charge effects due to the chemical interaction. This principle con- stitutes the ionic conductor analogue of the well-known semiconductor surface sensor mode. While the latter refers to the detection of redox activity, here it is - in the general sense - acid-base activity that is decisive. The transfer resistance, which plays a decisive role in the response and selectivity of electrochemical sensors - particularly at lower temperatures - (and often has a limiting effect in battery-powered systems), is largely controlled by the defect concentration at the interface, so that the incorporation of suitable heterogeneity (e.g. multilayers, nanosystems) ought to be able to bring about a reduction of the transfer resistance.

Space charge layers are also of similar importance for other elementary processes, such as the sorption steps, that play an important role in heterogeneous catalysis in particular (cf. current experiments in the field of organic elimination reactions, Chapter 2.5.4).

The importance of (electronic) space charges for chemical reactions was recognized very early on, but in our opinion its full importance has never been properly appreciated. In the case of solid state reactions reacting particles perceive the boundary layer right up to complete consumption - especially in this final phase of the reaction (see Chapter 6). Similarly, space charge effects should play an eminent role in chemical transport along interfaces, creep and sintering phenomena.

Other observations indicate considerable two-phase effects in the efficacy of photoelectrodes [139].

Put crudely, taking account of heterogeneities and in particular of multiphase systems in ion conductivity research wins a “practical” degree of freedom for materials research. If we consider the great advances reached by the preparation of small particles and microstructuring that have been made in the field of thin-film technology in particular we must expect further interesting developments in this field in the future.

8. SUMMARY

The defect chemistry of boundary layers (i.e. the potentials, defect concentrations and non- stoichiometries as a function of position, temperature and component activity) for the case of “dilute solutions” and structural invariance has been investigated with the aid of defect chemical equilibria and the Poisson equation, while taking into account a chemical coupling to the contact phase. Particular emphasis has been given to ionic conduction - although the relationships derived are applicable to other defect pairs and provide a generalization for the case of the mixed conductor. On account of the three contact problems that are treated, (i) ionic conductor/insulator, (ii) ionic conductor l/ionic conductor 2, (iii) ionic conductor l/ionic conductor 1 (grain boundary), three distribution processes are discussed:

(i) unilateral transfer of defects from and to the insulator boundary, (ii) transfer of defects from on ionic conductor boundary layer to that of its neighbour,

(iii) bilateral transfer of defects to or from the grain boundary.


In all cases it is possible to define not only the profile of the majority charge carrier but also that of the minority charge carrier. The interfacial interactions are taken into account in an interaction parameter, denoted as “degree of influence”, that can be based on standard thermodynamic quantities in the framework of an interfacial model. This makes it possible to construct Brouwer diagrams for boundary layers and more generally a defect chemistry of boundary regions.

It was possible to determine the conductivity contribution of the space charge zones parallel and - less precisely - perpendicular to the interface in all cases; these results are used to interpret experiments involving defined contacts.

In order to be able to compare the results with the author’s experiments and with other experimental results from the literature, calculations were made of the effective specific conductivities of heterogeneous systems (two-phase mixtures, polycrystalline materials). The simple distribution topology assumed not only makes it possible to derive analytical expressions, it is also realistic on the basis of interfacial interaction and particle size distribution and fulfilled to a satisfactory degree (at least in our experiments), as revealed by microstructure investigations.

The relationship obtained makes it possible to explain in a quantitative manner the increase in conductivity of two-phase mixtures (particularly ionic conductor-Al203 dispersions) that have been the subject of so much discussion in the literature. The dependence on grain size, volume fraction, surface activity and temperature can be described. Furthermore, the frequency distribution of impedance measurements can also be understood.

If a saturation effect is assumed it is possible to describe the absolute values quantitatively in the case of our experiments with AgCl and AgBr for which all the necessary parameters are known.

In agreement with this model of “heterogeneous doping” we have found that the chloride ionic conductor TlCl is converted to a Tl+ conductor at lower temperatures by the addition of Al203 (a sufficient creation of thallium(I) vacancies). Here too it is possible to describe the results -with a lower interfacial concentration, but a similar space charge potential - for various particle sizes and volume fractions over the whole temperature range. Comparative experiments with PbC12 throw light (in agreement with the calculations) on the similarities and differences in principle between the (classical) homogeneous doping and heterogeneous doping (as described here). Experiments on both homogeneously and heterogeneously doped materials could be interpreted.

The various possible concentration and conductivity profiles (enrichment, depletion, inversion layers) and the occurrence of ionic P-N transitions (e.g. vacancy-interstitial transitions) are discussed as a function of the graduation of the mobilities of the defects involved and the type of disorder.

Due to the enhanced defect concentrations such composite electrolytes have been shown to be efficient heterogeneous catalysts for organic reactions. As for the conductivity the rate constant can be quantitatively expressed in terms of space charge effects.

Variation of the insulating compound, the use of various modifications and changing its surface activity by means of chemical pretreatment make it possible to draw conclusions concerning the defect-inducing interfacial mechanism. These and other experiments published in the literature can be explained consistently if it is assumed that the free pairs of electrons of OH-groups enable the latter to act as nucleophilic groups for the stabilizing interaction with cations. The analogy with the effect of OH-groups of solids in contact with aqueous solutions is very evident. Further support is provided by the observed dependence of the conductivity effect on the basicity of the M-OH unit.

Wetting experiments on various substrates (various oxides, various crystallographic planes) with molten AgCl are in complete agreement with the mechanism assumed.

The same also applies to our demonstration that similarly nucleophilic molecules such as NH3 and (CN)2 exert a significant conductivity effect.

This makes it possible to extend the arguments to the ionic conductor/gas interface and to construct novel chemical sensors that can measure the partial pressure of complex acid-base active gases.

Comparable experiments with respect to modifying the properties of fluoride conductors by appropriate solid, liquid or gaseous phases are described.

Alternative models for explanation and the application of further experimental methods are the subject of detailed discussion. In particular impurity effects are excluded in the case of AgCl and

250 J. Maier

AgBr on the basis of detailed experiments. The use of surface active electronically conducting oxides (e.g. RuOl, Nb20s) allowed us to study

single ionic space charge layers by impedance spectroscopy. Particularly conclusive are the temperature changes of the resistive and capacitive part of the interfacial response upon temperature. Microelectrode impedance methods and a “penetration impedance method” allow for a higher sensitivity in this respect.

The analysis of the Wagner-Hebb technique is extended to two-phase systems with space charge zones and makes it possible to determine the electronic charge carrier in AgCI:A1203. We succeeded in explaining the simultaneous increase in n- and p-conduction (also observed for &AgI [loll). A further extension enables us for the first time to give a quantitative picture of the complete defect chemistry of an ionic solid which includes boundary effects.

The increase in conductivity can also be calculated and related to standard values in the case of contact between two ionic conductors. An estimate enables us to interpret reasonably the increase in conductivity reported in the literature for the miscibility gap in the system /GAgI-AgBr. A combined impedance spectroscopic and microstructural study let us understand details of the conduction pathway (hetero-, homojunctions) in AgI-AgCI two phase mixtures.

Our experimental results on pure, polycrystalline ionic conductors (AgCl, AgBr and TlCl) support the conclusion that the processes related to two-phase effects occur at the grain boundaries, i.e. - highly conducting (parallel) contributions as a result of space charge zones in the environment of the grain boundaries in addition to the “normal” blocking influence of these higher dimensional defects (in the perpendicular direction). On the basis of the activation energies it is to be presumed that (on account of structural peculiarities or impurities that have separated out) in these compounds cations accumulate in the core and vacancies are enriched in the space charge zone. Here too it is possible to calculate the specific effective conductivity and the specific impedance of bi- and polycrystalline samples. The evaluation of the impedance measurements, provides significant confirmation of these presumptions and may clarify a number of hitherto mysterious phenomena reported in the literature. The adsorption of NH3 to the grain boundaries increases the effect in absolute agreement with model predictions.

In addition an investigation has been made in this work of the consequences of a successive reduction of sample size on defect chemistry and conductivity. Since for very small dimensions the defect profile no longer extends to the bulk value, account has to be taken of complicated finite boundary conditions (corresponding to a higher sensitivity to neighbouring phases). Here too it proved possible for the first time to provide closed expressions for the relevant conductivity contributions, allowing a posteriori explanation of thin film results. In this way, individual defect parameters could be precisely evaluated. Mesoscalic effects - described by a “nano-size factor” - result in additional conductivity effects beyond the space charge contributions discussed above. Also interfacial phase transitions preceding bulk effects have to be taken account of.

To make a long story short: This article should have made clear, that the establishment of a consistent defect chemistry of boundary regions is possible in a first approximation, that it can explain many anomalies, especially with respect to conductivity (but also with respect to chemical kinetics catalysis, sensor and sintering science), and that it can be used in further studies to design materials systems with special properties. Much refinement is possible and I apologize to many colleagues for not having been able to adequately appreciate all the detailed work in this “inhomogeneous field”.

[l] C.C. Liang, J Electrochem. Sot. 120 (1973) 1298. [2] J.B. Wagner, High Conductivity Conductors Solid Ionic Conductors, T. Takahashi, ed., World

Scientific, Singapore, 1989; A. Shukla, N. Vaidehi, K.T. Jacob, Proc. Indian Acad. Sci. 96 (1986) 533.

[3] a) J. Maier, Ber. Bunsenges. Phys. Chem. 88 (1984) 1057; b) phys. stat. sol. B 123 (1984) K89, K187; c) Mater, Sci. Monogr. 28A (1985) 419; d) Structure Relations in Fast Zon and Mixed


Conductors, F. W. Poulsen et al., eds. Riser Nat. Lab., 1986, Roskilde p. 153; e) Solid State Zonics 32Z33 (1989) 727; f) Solid State Zonics 18119 (1986) 1141.

[4] J. Maier, J Electrochem. Sot. 134 (1987) 1524; Solid State Zonics: Materials and Applications, S. Chandra, B.V.R. Chowdari, eds., World Scientific, Singapore, 1992, p. 111.

[5] N.J Dudney, Ann. Rev. Mat. Sci. 19 (1989) 103. [6] J. Maier, J: Phys. Chem. Solids 46 (1985) 309. [7] J. Maier, Ber. Bunsenges. Phys. Chem. 89 (1985) 355. [8] a) J. Maier, Mater Res. Bull. 20 (1985) 383; b) J. Maier, B. Reichert, Ber Bunsenges. Phys.

Chem. 90 (1986) 666. [9] J. Maier, Ber Bunsenges. Phys. Chem. 90 (1986) 26.

[IO] a) J. Maier, Solid State Zonics 23 (1987) 59; b)phys. stat. sol. (a) 112 (1989) 115. [1 1] J. Maier, Ber. Bunsenges. Phys. Chem. 93 (1989) 1468; Ber. Bunsenges. Phys. Chem. 93 (1989)

1474; Solid State Zonics 28-30 (1988) 1073. [12] J. Maier, S. Prill, B. Reichert, Solid State Zonics 28-30 (1988) 1465. [13] J. Maier, Mater Chem. Phys 17 (1987) 485; Superionic Solid and Solid Electrolytes, Recent

Trends, A.L. Laskar, S. Chandra, eds., Academic Press, New York, 1989, p. 137; in Science and Technology of Fast Zon Conductors, H.L. Tuller, Ed. Plenum Press, New York 1989, p. 345.

[14] J. Maier, Angew. Chem. Znt. Ed. Engl. 32(3) (1993) 3 13; Angew. Chem. Znt. Ed Engl., 32(4) (1993) 528; in Defects in Insulating Materials, 0. Kanert and J.-M. Spaeth, eds., World Scientific, Singapore, 1992, p. 2.

[ 151 J. Wassermann, T.P. Martin, J. Maier, Solid State Zonics 28-30 (1988) 15 14. [16] U. Lauer, J. Maier, Ber Bunsenges. Phys. Chem. 96, (1992) 111; Solid State Zonics 51 (1992)

209. [17] J. Maier, U. Lauer, Ber Bunsenges. Phys. Chem. 94 (1990) 973; U. Lauer, J. Maier, W. Gopel,

Sensors and Actuators B 2 (1990) 125; Solid State Zonics 40141 (1990) 463. [ 181 P. Murugaraj and J. Maier, Solid State Zonics 32133 (1989) 993; Solid State Zonics 40/41(1990)

1017. [19] U. Lauer and J. Maier, J Electrochem. Sot. 139(5) (1992) 1472. [20] A. Atkinson, R.I. Taylor, Phil. Msg. A43 979; Solid State Zonics 28-30 (1988) 1377. [21] T. Asai, S. Kawai, Solid State Zonics 20 (1986) 225. [22] T. Asai, C.-H. Hu, S. Kawai, Mater. Res. Bull. 22, (1987) 269. [23] L. Chen , Z. Zhao, G. Wang, Z. Li, Kexue Tonghao 26 (1981) 308. [24] M.R.-W. Chang, K. Shahi, J.B. Wagner, J: Electrochem. Sot. 131 (1984) 1213. [25] P Chowdhary, V.B. Tare, J.B. Wagner, J Electrochem. Sot. 132 (1985) 123. [26] P. Chowdhary, J.B. Wagner, Mater Lett. 3 (1985) 78. [27] F? Chowdhary, A. Khandkar, J.B. Wagner, The Electrochemical Society Meeting, October

1985, Las Vegas. [28] R. Dupree, J.R. Howells, A. Hooper, F.W. Poulsen, Solid State Zonics 3/4 (1981) 277. [29] P. Dubec, J.B. Wagner, Mater Lett. 2 (1984) 302. [30] A. Hooper, J: Power Sources 9 (1983) 161. [31] T.R. Jow, C.C. Liang, Solid State Zonics 9110 (1983) 695. [32] T. Jow, J.B. Wagner, J: Electrochem. Sot. 126(1979) 1963. 1331 A. Khandkar, J.B. Wagner, The Electrochem. Sot. Meeting, San Francisco, 1983; A. Khand-

kar, Thesis, 1985, Arizona State University. [34] CC. Liang, A.V. Joshi, N.E. Hamilton, 1 Applied Electrochem. 8 (1978) 445. [35] 0. Nakamura, J.B. Goodenough, Solid State Zonks 7 (1982) 125. [36] S. Pack, B. Owens, J.B. Wagner, J: Electrochem. Sot. 127 (1980) 2177. [37] F.W. Poulsen, N.H. Andersen, B. Kindl, J. Schoonman, Solid State Zonics 9/10 (1983) 131. [38] F.W. Poulsen, Solid State Zonics 2 (1981) 53. [39] F.W. Poulsen and P.J. Moller in: lot. cit. [3d], p. 159. [40] F.W. Poulsen in: lot. cit. [3d], p. 67. [41] J.B. Phipps, D.H. Whitmore, Solid State Zonics 9/10 (1983) 123. [42] J.B. Phipps, D.H. Whitmore, J Power Sources 9 (1983) 373.

252 J. Maier

[43] PM. Skarstadt, D.B. Merritt, B.B. Owens, Solid State IO&S 314 (1981) 277. [44] K. Shahi, J.B. Wagner, 1 Phys. Chem. Solidrs 43, (1982) 713. [45] K. Shahi, J.B. Wagner, J Solid State Chem. 42 (1982) 107. [46] K. Shahi, J.B. Wagner, d Electrochem. Sot. 128 (1981) 6. [47] J.B. Wagner, Muter. Res. Bull. 15 (1980) 1691. [48] Z. Zhao, C. Wang, L. Chen, Wtdi Xuebao 33 (1984) 1205; Z. Zhao, C. Wang, S. Dai, I.,.

Chen, Solid State Zonics 9/10 (1983) 1175. [49] S. Fujitsu, M. Miyayama, K. Koumoto, H. Janagida, T. Kanazawa, J Mater. sci. 20 (1985)

2103. [50] S. Fujitsu, K. Kuomoto, H. Janagida, Solid State Zonics 18/19 (1986) 1146. [Sl] A. Khandkar, VB. Tare, J.B. Wagner, Rev. Chim. Min. 23 (1986) 274. [52] N. Vaidehi, R. Akila, A.K. Shukla, K.T. Jacob, Muter. Res. Bull. 21 (1986) 909. [53] E. Hartmann, VV Peller, G.I. Rogalski, Solid State Zonics 28-30 (1988) 1098, V. Trnovcova,

C. Bttrta, PI? Fedorov, 1. Zibrov, Materials Science Forum 76 (199 1) 13, S. Adams, K. Har- iharan, J. Maier, Solid State Ionics, bf 75 (1991) 193, Y. He, Z. Chen, Z. Zhang, L. Wang and L. Chen in Materials for Solid State Batteries, B.V.R. Chowdhari and S. Radhakrishna, eds., World Scientific, Singapore, 1986, p. 333., J. R. Stevens and B.E. Mellander, Solid State Ionics 21 (1986) 203, S. Skaarup, K. West and B. Zachau Christiansen, Solid State Ionics 28-30 (1989) 975, J. Plocharski, W. Wieczorek, J. Przyluski and K. Such, 1 Appl. Phys. A 49 (1989) 55, W. Wieczorek, K. Such, H. Wycislik and J. Plocharski, Solid State Zonics 36 (1989) 255, J. Plocharski and W Wieczorek, Solid State Zonics 28-30 979; (1988) Y. Saito, T. Asai, K. Ado and 0. Nakamura, Mat. Res. Bull. 23 (1988) 1661, Y Saito, T. Asai, 0. Nakamura and Y. Yamamoto, Solid State Zonics 35 (1989) 241, Y. Saito, J. Mayne, K. Ado, Y. Yamamoto and 0. Nakamura, Solid State Zonics 40141 (1990) 72, J. Mayne, Y. Saito, H. Kageyama, K. Ado, T. Asai and 0. Nakamura, Solid State Zonks 40/41 (1990) 67, Y. Saito, K. Ado, T. Asai, H. Kageyama, 0. Nakamura and Y. Yamamoto, Solid State Ionics 53-56 (I 992) 728,O. Nakamura and Y. Saito, Solid State Zonks: Materials and Applications, S. Chandra, S. Singh and PC. Srivastava, eds., World Scientific, Singapore, 1992, p. 101; B. Wnetrzewski, J.L. Nowinski and W. Jakubowski, Solid State Ionics 36 (1989) 209, P Wes- lowski, W. Jakubowski and J.L. Nowinski, phys. stat. sol. (a) 115 (1989) 81, J.L. Nowinski, P Kurek and W. Jakubowski, Solid State Zonics 36 (1989) 213; P Kurek, J.L. Nowinski and W. Jakubowski, Solid State Zonics 36 (1989) 243; H. Husono and Y. Abe Solid State Zonics, 44 (1990) 293; B. Kumar, J.D. Schaffer, M. Nookola, L. G. Scanlou, d Power Sources 47 (1994) 63; S.N. Reddy, AS. Chary, K. Saibabu, T. Chiranjiui, A. Brune, J.B. Wagner, Solid State Zonics 25 (1987) 165, K. Singh, S.S. Bhoga Solid State Zonics 39 (1990) 205; S.S. Bhoga, K. Singh Solid State Zonics 40/41 (1990) 27, S. Chaklanobis, K. Shahi, R.K. Syal Solid State Zonics 44 (1990) 107, L. Chen, C. Gros, R. Castaynet, P. Hagenmuller Solid State Zonics 31 (1988) 209.

[54] G. Simkovich, C. Wagner, J Catalysis 1 (1962) 521. [55] TL. Wcn, R.A. Huggins, A. Rabenau, W. Weppner, Rev. Chim. Min. 20 (1983) 40. [56] Y. Saito, J. Maier, in preparation; Y. Saito. K. Hariharan, J. Maier, Proc. 4th Znt. Symp. on

Syst. with Fast Ionic Transport, Warsaw, 1994, in press. [57] K. Hariharan, J. Maier, J Electrochem. Sot., submitted. [58] A.M. Stoneham, E. Walde, J.A. Kilner, Muter Res. Bull. 14 (1979) 1661. [59] K. Shahi, J.B. Wagner, Appl. Phys. Lett. 37 (1980) 757. [60] N.J. Dudney, 1 Am. Ceram. Sot. 70 (1987) 65. [61] C. Wagner, 1 Phys. Chem. Solids 33 (1972) 1051. [62] A. Bunde, W. Dieterich, E. Roman, Solid State Zonics 18119 (1986) 147. [63] H.E. Roman, A. Bunde, W. Dieterich: lot. cit. [3d], p. 165. [64] H.E. Roman, A. Bunde, W. Dieterich, Phys. Rev. B34 (1986) 331. [65] J.C. Wang, N.J. Dudney, Solid State Zonics 18/19 (1986) 112. [66] H. Schmalzried, Solid State Reactions. Verlag Chemie, Weinheim, 198 1. [67] J. Frenkel, Kinetic Theory of Liquids. Oxford University Press, New York, 1946.


[68] K.L. Kliewer, J.S. K6hler, Phys. Rev. A 140 (1965) 1226; C. Newey, P. Pratt, A. Lidiard, Phil. Mug. 3 (1958) 75.

[69] R.B. Poeppel, J.M. Blakely, Surf: Sci. 15 (1969) 507. [70] J. Maier, in Science and Technology ofFast Zon Conductors. H.L. Tuller, M. Balkanski, Eds.

Plenum Press, New York, 1989, p. 299. [71] J. Jamnik, J. Maier and S. Pejovnik Solid State Zonics, in press. [72] J.R. Macdonald, D.R. Franceschetti and A.P. Lehnen, J Chem. Phys. 73 (1980) 5272; J.R.

Macdonald and A.P. Lehnen, Cryst. Latt. Def 9 (1982) 149. [73] J. Maier, W. G6pe1, J Solid State Chem. 72 (1988) 293; W. Gijpel, J. Maier, K. Schierbaum,

and H.-D. Wiemhafer, Solid State Zonks 32133 (1989) 440. [74] J. Maier, U. Lauer, Solid State Zonics 51 (1992) 209. [75] J. Jamnik, J. Maier and S. Pejovnik, Solid State Zonks 75 (1995) 51; Electrochimicu Acta, 14

(1993) 1975; J. Jamnik, J. Maier and S. Pejovnik, Proc. 1st Znt. Slovenian-German Seminar ofJoints Projects in Mat. Sci. and Technof., Ljubljana, October 94, to be published; J. Fleig and J. Maier, Proc. 1st Znt. Slovenian-German Seminar of Joints Projects in Mat. Sci. and Technol., Ljubljana, October 94, to be published.

[76] H. BGttger, V.V. Bryksin, Hopping Conduction in Solids. VCH, Weinheim, 1985. [77] R. Blender, W. Dieterich, Solid State Zonks, 28-30 (1988) 82. [78] M. Kleitz, H. Bernard, E. Fernandez, E. Schrouter, Adv. Cerum. Sci. Tech. Zircon&z 3 (1981)

310; H. Rickert, U. Schwaitzer, Solid State Zonks 9/10 (1983) 689; H. J. Queisser, J.H. Werner, Mat. Rex Sot. Symp. Proc. (1988) 53; H. J. M”oller, H. P. Strunk, J. Werner eds., Polycrys- tulline Semiconductors, Springer, Berlin, 1989.

[79] J. Fleig, Ph D-Thesis, Tiibingen-Stuttgart, 1995 (detailed calculation of such effects by finite elements); J. Fleig, J. Maier, in preparation.

[80] R.E. Soltis, E.M. Logothetis, A.D. Brailsford, J.B. Wagner, 1 Electrochem. Sot. 135 (1988) 2380; A.D. Brailsford, Solid State Zonics 21 (1986) 159; N.F. Uvarov, E.F. Hairetdinov, J.M. Reau, P. Hagenmuller, Solid State Comm. 79 (1991) 635.

[81] N.F. Uvarov and J. Maier, Solid State Zonics, 62 (1993) 251. [82] N.F. Uvarov, VP. Isopov, V. Sharma, A.K. Shukla Solid State Zonks 51 (1992) 41. [83] G.E. Pike, private communication. [84] Y.M. Chiang, A.F. Henriksen, W.D. Kingery, D. Finello, J Am. Cerum. Sot. 64 (1981) 385. [85] P. Sutton, private communication. [86] K.L. Kliewer, J Phys. Chem. Solids 27 (1966) 705. [87] G. Farlow, A.B. Blose, Sr.J. Feldott, B.D. Lounsberry, L. Slifkin, Radiation Efl 75 (1983)

1; Sr. J. Feldott, A.B. Blose, B.D. Lounsberry, L. Slifkin, J. Zmug. Sci. 29 (1985) 39; R.A. Hudson, G.C. Farlow, L.M. Slifkin, manuscript.

[88] U. Riedel, J. Maier, and R. Brook, J. Eur. Cerum. Sot. 9(3) (1992) 205; Solid State Zonks, M. Balkanski, T. Takahashi, H.L. Tuller, Eds. Elsevier Science Publishers, Amsterdam, 1992 p. 259.

[89] P.W. Tasker Advances in Ceramics 10 (1984) 176. [90] L. Bousse, N.F. de Rooij, P. Bergveld, Trans. Electron. Dev. 30 (1983) 1263. [91] A. van der Berg, P. Bergveld, D.N. Reinhondt, E.J.B. Sudhtilter, Sens. Actuut. 8 (1985) 129. [92] E.A. Daniels, S.M. Rao, 2. Phys. Chem. N. F. 137 (1983) 247. [93] S.M. Rao, Ph. D. Thesis, 1983, Poona University. [94] J.L. White in: Proc. 4th Nut. Conf Cluys and Clay Min., Znt. Ser. Monogr. Earth Sciences, p.

133. [95] C. Zipelli, J.C.J. Bart, C. Petrini, S. Galvagno, C. Cimino, 2. Anorg, ulZg. Chem. 502 (1983)

199. [96] J. Maier, G. Schwitzgebel, Muter. Res. Bull. 17 (1982) 1061; J. Maier, Solid State Ionics 32/33

(1989) 727. [97] J. Mizusaki, K. Fueki, Solid State Ionics 6 (1982) 55. 1981 0. Stasiw, J. Teltow, Ann. Phys. (Lpz.) I (1947) 261. 1991 D. Miiller,phys, stat. sol. 1312 (1965) 775.

254 J. Maier

[IOO] J. Maier, 2. Phys Chem. N.F. 140 (1984) 191; J Am. Ceram. Sot., 76(5) (1993) 1212; J Am. Gram. SOC 76(5) (1993) 1218; J Am. Ceram. Sot., 76(5) (1993) 1223; 1 Am. Ceram. SOC., 76(5) (1993) 1228.

[loll N. Valverde-Diet, J.B. Wagner, Jr., Solid State Ionics, 28-30 (1988) 1697. [102] J. Maier, Solid State lonics, 62 (1993) 105. [103] N.E Uvarov, M.C.R. Shastry and K.J. Rao Rev. SolidState Sci, 4 (1990) 61. [104] J.A. Schmidt, J.C. Bazan, L. Vito, Solid State fonics 27 (1988) 1. (1051 Y Haven, Rec. Tran. Chem. 69 (1950) 1471, 1505. [106] M. Mogensen, J Power Sources 20 (1987) 53; M. Gaber%ek, J. Jamnik, S. Pejovnik, 1

Electrochem. Sot 140 (1993) 308. [107] B. Zhu, B.E. Mellander, in: C. Singhal and H. Iwahara, eds. Solid Oxide Fuel Cells (The

Electrochemical Sot. Inc., Pennington, NJ, 1993) 156. [108] T Unruh, Ph D-Thesis, Saarbrticken, FRG (1995) [log] E. Gianellis, Solid State Ionics, in press. [llO] R.A. Hubermann, Phys. Rev. Lett. 32 (1974) 1000. [ 1111 N. Hainovsky, J, Maier, Solid State Zonks, in press; Phys. Rev., in press. [112] R. Lipowski, Phys. Rex Lett. 49 (1982) 1575; V.N. Bondarev and A.B. Kuklov, Solid State

Ionics 44 (1991) 145. [113] G. Ertl, J. Kiippers, Low Energy Electrons and Surface Chemistry. VCH, Weinheim, 1985. [114] W.D. Kingery, Muter. Sci. Monogr 28A (1985) 25. [115] J.-E. Gerner, Diploma Thesis, 1984, University of Konstanz. [116] J.H. Strange, S.M. Rageb, R.C.T. Slade, Phil. Msg. A 64 (1991) 1159. [117] R.A. Hudson, G.C. Farlow and L.M. Slifkin Phys. Rew. B 36 (1987) 4651. [ 1181 L. Slifkin, personal communication; S.K. Wonnell and L.M. Slifkin Phys. Rev. B 48 (1993)

78. [119] J. Fleig, J. Jamnik, J. Maier, in preparation. [120] J Jamnik, H.-U. Habermeier, J. Maier, Physica B 204 (1995) 57. [121] M. Kleitz, Solid State Ionics 3/4 (1981) 513. [122] S. Gupka, S. Patnaik, K. Shahi, Solid State Ionics 31 (1) (1988) 5. [123] U. Lauer, Ph D thesis, University of Tubingen, 1991. [I 241 F. Granzer, 1 Zmag. Sci. 33 (1989) 207. [125] J.T. Overbeek in: Colloid Science I, H. R. Kruyt, Ed. Elsevier, Amsterdam, 1952. [126] R.C. Baetzold, J.F. Hamilton, Surf: Sci. 33 (1972) 461. [127] R.C. Baetzold, 1 Phys. Chem. Solids 35 (1974) 89. [I281 H.A. Hoyen, J Appl. Phys. 47 (1976) 3784. [129] N. Starbov, J Inf: Rec. Mater 13 (1985) 307. [I 301 Y.T. Yan, H.A. Hoyen, Surj Sci. 36 (1973) 242. [131] S. Mtihlherr, K. Lauger, N. Nicoloso, E. Schreck, K. Dransfeld, Solid State Ionics 28-30

(1988) 1495. [ 1321 J. Jamnik, Ph D-Thesis, Ljubljana-Stuttgart, 1994. [133] J. Fleig and J. Maier, in preparation. [134] W. Gopel, U. Lampe, Phys. Rev. B 22 (1980) 6447. [ 1351 E. Schreck, K. Lauger, K. Dransfeld, Z. Phys. B62 (1986) 33 1. [136] J.B. Phippq D.L. Johnson, D.H. Whitmore, Solid State Ionics 5 (1981) 393. [137] J. Corish, P.W.M. Jacobs, Surf: Def Prop. Solids 2 (1973) 160. [138] W. Puin, I? Heitjans, Proc. Int. Conf: Nunocryst. Muter., Stuttgart, 1994, in press; W. Puin,

l? Heitjans, J. Maier, in preparation. [139] S. Chandra, private communication. [ 1401 W. Gopel, Progr Surf: Sci. 20 (1986) 1.


(9 (2)

Fig. 79. Profiles of thermodynamic functions, defect concentrations and conductivities for various contact problems. [ 131

APPENDIX

Thermodynamics, defect concentrations, conductivities

Example of a boundary layer Brouwer diagram of a semiconductor [14]. SnOz is an n- conductor under all P-T conditions that have been investigated (here P stands for partial pressure of oxygen, total pressure = 1 atm). The intrinsic ionic majority defect is the completely ionized oxygen vacancy - at least at higher temperatures [73]. Adsorbed oxygen brings

256 J. Maier

I ~OW&EFECT CCMCENTRA- I NO STRUCTURAL CHANGES cqbl: = 0)

p= l XP[-fq-qbd/(f?T)) 8 z; (~determiner concentration

~_T~_~#N~~T ~hmcomont for fi defect@)

UAGNETK: FIELDS @%x2)* * -tFjF/(CQ

-i

& f.2.c. $4 RTCC.

I I IO

THCK SAWLES (Lb D&y.-longthaAa<c

( Wj 3 +ti F(6-&)/(RTI ? THlN SAMPLES SEE

MINCXWY CHARGE CARRIERS: e.g. G,f S(e’)= S&= G(ti) = g+ bnrtruction of Brouwer-diagrams for space charge layers, calculation of the non-stokhiometry [f$,X~l

G I I ‘C

I 1 3

FURE PARALLEL-TRANSPDRT h f40 FIELD EFFECTS on 1

6&c,v I= 0 a 0

AY”‘-~tiXl-~dX =EjZ F[PAJ[2C~Bjfl_8j~U~ t!

=IjZ ujF[eff. thkknesd[eff. CO~C.)*~L

I CORRELATKIN BETWEEN Go AND STAfWARD POTENTIALS

I J

<AO)‘u 0,0_ -0 1 (depletion) I aMs * o,e_ - 1

I

~Y”=-~2A~rFu&, -G 1 AY”42h_)z_F(u_Jc,,c,-u.c,)

=(~cc,RT)~~u . UE--~ 1 l 0 =(2cc,RTI”b~~&.~~

I (AO’) = <Aa!) (enhancement)

PERPENDlCULAR CONTRlBUTlON IF u-Const. M;jb’-$o&)dx

ZFucJke) tinport8nt for MX/MX I MXIMX’

Fig. 80. Calculation of concentration and conductivity effects for a space charge zone (infinite in one direction, one-dimensional). [13]


EXAMPLE: FRENKEL DISORDER

MM+Vi m MT +Vi (AFGol

BULK: local electroneutrality 81 mass action law; if no electronic carrier important - c, =KF/c = const Cxl (Znst =*(*I&) if intrinsic (extrinsic))

SPACE CHARGE REGION:

V,+) + MM(x) @ &Ix) + MM(QD)

transport equilibrium: Axpj = 0 - Cj(X) =f[cj,,cjol

INTERFACE

‘tY

M,(O) + y (int) m i/h(O) + MI tint)

a) electrochemical equilibrium: A&I> = 0 (x : reaction progress)

b) continuity of Eta/ax)@ _ global electroneutrali c) model for potential jump (e.g. charge free zone)

( d) additional intercorrelation with neighbour-phase)

EXAMPLE: MX/MX’

c,(0)/c”= [x(E’k) ex p - A,,GLl(RT j]“: We 1 (cl (0)/c’) x accounts for potential jump; Aa& refers to heterogeneous Frenkel-reaction)

FREE SURFACE

a) a&) = const -Cjo = f[AjG*I X= const e model of Kliewer and Kohler :

Cj a exp [-Aj G*/(RT)] b) consideration of surface configurational entropy

and of limited number of surface sites ----) model of Pdppel and Blakely 28

c) more accurate see text

CONTACTED SURFACE IN A KLIEWER-TYPE MODEL

cjO a exp_lAj Go+ AsI GoI / (RT)

As] Go = ARGO (M’(s) * M’(int)} T-dependence:

-R WY/T)C,, = ASI Ho + Aj Ho (+correction terms)

Fig. 8 1. The problem of linking bulk and interfacial concentrations to standard parameters. [13]

258 J. Maier

OVERALL CONDUCTIVITY:

om= P (000, oscl, ocore, OA, . . . ) The percolation operator reacts sensitively on the detailed distribution

HIGH LAYER CONDUCTIVITY (scl,core) = oA , ooo

Om=Ea Pa (pa a, (quasi-parallel-switching) (a denotes region =,L(scl,core,A); p counts measured percolating pathways) moderate regime (cf. experiments): ( a/&pa) Pa = 0 = (a/aT) Pa

blocking regime (highqA-values): ( W&p,) Paf 0 =(a/aT) Qa

VERY LOW VA-VALUES, RANDOM DISTRIBUTION, FRACTALISED SURFACES, SIMILAR CONDUCTIVITIES:

detailed percolation theory, Monte-Carlo-simulations effective medium theory

INTERRUPTED HIGHLY CONDUCTIVE PATHWAYS

0, = 1~ pa kpa [ Ia* Ph* vp;X@ 0,;’ ] -’

CUBE-MODEL (cf. polycristalline materials)

om=(o,o++ PLl'~LoLl'OL')/(OL'+PLI~Lo,)

FREQUENCY DEPENDENCE

a,- sa = a,+ 47 w &a

QUASI-PARALLEL-SWITCHING (oA=o)

om = ko(l-~A)Ocu + PL%cl + PL%oreocore ($2~: surface-to-volume ratio of A-particles)

omL= PL%I <AO>SCI = PL~PAQAAY” = PLqAQAzj 2 F (ZA)(ZC,8j/(I_8j))Uj

= fiL~A~A(2EE,RT)“2h_ co-U+ ‘&&if 8_ - 1

= PL~ARA(2&&,,RT)1’2U_ Jc_, , if a (counter defects 1 = 0

T-DEPENDENCE (e--l,OmL-)OmL-1 EL:-R(a/alb) OmL = h_ + +(A-H”+AslH”) + corr. =h_

ELECTRONIC CONDUCTIVITY (e_ --) l,OmL-’ OmL-1 Om,e=poo(l-~A)oeao + PL~ARA Ue (2~~ RTc-,)“~ (ce,/c_J

Fig. 82. The problem of calculating the effective specific conductivity in relevant heterogeneous systems. [ 131

about a depletion of the space charge zones as a result of electron capture (e.g. 0;) [140]. The variation of the defect concentrations for the bulk (thick line) and boundary layer (thinner line: x/h = const; thinnest line: const = 0) is illustrated schematically in Fig. 84.

Since it is also possible to determine the electrical potential at a constant site within the space charge zone from the equilibrium conditions (Eq. (4)) (see also Figs. 79, 8), the law of mass action

P'12 [Vi] [e’]’ = &(T) . (78)


1 POISSON-BOLTZMANN-EQUATION 1

SYMMETRICAL FILM \ pco =Q) J Ip/(z_F)I =c_

TWO SYM. CHARGE CARRIERS . 1 (2+=-z_ = 2)

1

(d/dg)W =-2(cosh W -coshW”)“2 W” I W (sample center)

small effects I

large effects (c_nc+) I

(d/dg)W = -(W2 -W*2)“2

s;-*=lS w++w2

I 3 am,!’ = (2h/L)zFc, x(~ln!&) tgh(L/2h) x[(u,-U-I-A(u_+u,)] where A=(W,/4)[1+(L/h)/sinh (~/h)]

0~2’ = (4h/L)zFu_ cm x[(c_o+ ~_;‘I-([: +<:-‘)]“2 where L/(21)= 2 cx [T(g”_ jnz/2)-

T (c_“j Arcsin -I and ‘F(k,r) E ida(l-k2sin2a)-1’2

up ( independent of czunier defects)

Fig. 83. Conductivity effects in microsystems (finite in both directions, symmetrical, one- dimensional). [ 131

applies to all x 2 0 (insofar as it is possible to assume Boltzmann distribution). On account of the sign of the band bending it can be seen that the relative change in the concentration of oxygen vacancies in the boundary layers with the partial pressure of oxygen is negligible at even lower partial pressures (i.e. [V;;] 2: const in Eq. (78)), than is possible in the bulk (extension of the -l/4 region).

The “distance law” In [Vo”], - In [VO”], = 2 (In [e’l, - In [e’l,) (cf. Fig. 8, Eq. (4)) is automatically obeyed by Eq. (78). The effective contribution to the specific conductivity for such a depletion layer is described by Eq. (29) for large effects. It follows that

Aoll = const’pm-/2 - const”pml)/2 m (79)

260 J. Maier

(3)

t (-l/6) ---_ l .

-,---____----~--~~‘-‘----

% ‘. .-

2

**.. ‘. “*...*

5 \. ‘e2

E p--_----_-_-_-_--_--_-_-_-_

G *a.. l + + *.

5.. l ..> L*.

i:

‘%, l .*: Q/ (01 *..* ._

. .._ 0. .~~~~~~~11)~~ l *

5

‘..* 0. .* l * 5 0. . . 2[V;l =[A’1

5.. l . l *

t ‘..,* ‘0. f

E (-l/4)

8

‘*.. . ..* :*..>... . . ‘. 5 0. ‘. 0.. 0..

log (oxygen partial d;essure) -

Fig. 84. Log-log plot of defect concentrations against partial pressure of oxygen for SnO2 (Brouwer diagram) for bulk and boundary layer (arbitrary absolute values).

A: acceptor impurity. [14]

(moo and 1110 are the characteristic exponents for x=00 and x = 0 as in Fig. 84). If both coincide then exactly half of the exponent for the bulk would be expected for the boundary layer conductivity [102]. The reason for this lies in the interaction of effective thickness (h CC c;~‘~) and effective charge carrier concentration - as already explained in the text for the example of ZnO (Chapter 6) (see [14]). Conductivity-partial pressure characteristics in boundary layer sensors [102]: The space charge effect can be characterized by:

Aa = f [C, T, 9(c,,, c,)l (80)

The chemical effect is hidden in 9, the “degree of influence” which characterizes the gas- surface interaction and which itself is a function of the bulk concentration (c,) and the defect concentration immediately beneath the surface (cg). As derived in the text, in the case of an enhancement effect, the extra conductivity in a measurement parallel to the interface is found to be (pure material)

Ao” = &‘%12 (81)

where Nk,, is the characteristic exponent in the first layer adjacent to the proper surface. In the case of a depletion effect, the extra resistance (in a measurement perpendicular to the boundary) is approximately

&,l = /3p-Nd2, (824 if the equilibrium with the bulk is not established, and

&,A = Yp-(N&+Nk.,) (82b)

if overall equilibrium is achieved. If the measurement of the depletion effect is performed parallel to the interface the extra conductivity can be expressed as

Ao” = 6 _ rpN”/lz (83a)

if only boundary equilibrium is established, and

&r/l = <d’kd _ ,,pM2 PW


if also bulk equilibrium is established. Eq. 81 to Eq. 83 are only valid for a simple defect chemistry where only k determines the electrically dominating effect. Due to profile effects (anisotropy), the parameters (x, 8, y, S, E, <, n are not necessarily related in a simple manner to each other (see text).

Explanation of the most important symbols

C

E

G

H

I

fN-0 M

MX

L, Ly

N

s

V

Y

Z

a

c, [I g h

i

Y

u

X

.Y

z

A, @

Y

4

R

capacitance, doping level

activation energy; energy parameter in band model

free enthalpy

enthalpy

electric current

(electrochemical) equilibrium constant

metal ion in MX

model ionic conductor (M+X-)

electrode dimensions in x and y directions

mole fraction, doped cation

entropy

vacancies

normalized conductivity (only contains x dimension)

normalized resistance (only contains x dimension)

activity, area

molar concentration with respect to volume

micro size factor (see text)

enthalpy of migration

electrical current density

imaginary unit

particle radius

mobility

positional coordinate in MX (perpendicular to surface)

positional coordinate in MX (parallel to surface)

charge number

reaction operator (r designates reaction)

electrical potential

electrical energy parameter (= zF(@ - GW))

degree of influence (see text)

specific area (per unit volume)

262 J. Maier

B E

=0

5 rl

K

h

P(P) V

5 4 X

P

u

-r

0

A

F

L

M

S

SA

eon

ion

m

n

p SC

gb

gi

V

W

a

E

distribution factor (see text)

relative dielectric permeability

electrochemical potential per particle

absolute dielectric permeability

normalized concentration (=N/N,)

overpotential

electrical contribution to k = KK

Debye length

(electro)-chemical potential

stoichiometric coefficient

normalized positional coordinate (=x/A)

volume fraction

reaction progress

charge density

specific conductivity

relaxation time

angular frequency

The following suffixes denote:

insulating phase; interfacial site

Frenkel reaction

boundary layer

metal ion site; metal ion (as structural element)

surface; surface site

surface interaction reaction

electrons

interstitial ion (as structural element)

ions

charge carrier variable

effective value

conduction electron

electron hole

space charge zone

grain boundary (gi & SC)

grain boundary core (interior)

vacancy (as structural element)

interaction layer

regional variable

non-stoichiometry

0

03

132

A

b

*

II I


first layer of MX (x = 0)

bulk

(positive, negative) majority charge carriers

The following supefixes represent:

complex number

partial charge, non-stoichiometry

standard value

positive effective charge

negative effective charge

centre of sample

parallel to interface

perpendicular to interface

ionic conduction in space charge regions

Documents