Point contacts in solid state ionics: finite element calculations and local conductivity measurements

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SOLID STATE

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ELSEVTER

Solid State Ionics 86-88

(1996)

1351-1356

Point contacts in solid state ionics: finite element calculations local conductivity measurements

and

J. Fleig”, J. Maier Max-Planck-Institut

fiir FestkBrperjorschung,

Heisenbergstr.

1, 70569 Stuttgart, Germany

Abstract Two aspects of point contacts in solid state ionics are discussed: firstly, the consequences of unintentionally existing ‘poor’ contacts at interfaces are considered with regard to the frequency dependent potential distribution and the impedance response. A quantitative description is given using finite element calculations to determine the exact potential distribution. Secondly, the use of point contacts as tools to resolve the local environment in solids is demonstrated in two examples, detection of a diffusion profile and the characterisation of a highly conductive surface layer in AgCI. Keywords:

Point contact;

Impedance;

Diffusion;

Surface conductivity

1. Introduction On one hand, microcontacts are often inherent structural elements at phase boundaries in composite materials, between grains and at electrodes. This results in geometrically imperfect (so-called ‘poor’) contacts and therefore in a distinct modification of the impedance, compared to samples with geometrically perfect interfaces (solid electrolyte and electrode in intimate contact). There are numerous experiments revealing an additional semicircle in the complex impedance plane due to poor contacts (arrays of point contacts) [l-lo]. Several models have been proposed to describe this behaviour in terms of two current paths, one of them being blocked (in the d.c.-case) by a gap capacitance [25,9,10]. However, a quantitative description of the

*Corresponding

author.

0167.2738/96/$15.00 Copyright PII SO167-2738(96)00316-5

01996

the

influence of the point contacts on the frequencydependent current lines, as well as on the impedance by solving the basic Laplace equation, is still missing. In this contribution the potential lines are calculated in two dimensions and the resulting impedance is analysed in terms of equivalent circuits. On the other hand, microcontacts are not only ‘unwelcome’ elements in materials science, but also useful means to characterize local effects. They are being used more and more to investigate electrode processes in solid state ionics [ 11,121. However, there are only very few examples using such point contacts to measure local bulk conductivities. A necessary requirement is the ability to measure very high impedances. In this contribution it is shown, exemplarily, for AgCl that local conductivity measurements in the pm-range are possible and that the technique is applicable to problems such as the measurement of a highly conductive surface layer, as

Elsevier Science B.V. All rights reserved

1352

J. Fleig, J. M&r

well as to determine in AgCl.

the diffusion coefficient

2. The impedance

of ‘poor’ contacts

I Solid State lonics

of Cd”

The main feature of geometrically imperfect contacts with respect to the impedance is the frequency dependence of the current lines; in the d.c.-case the current is constricted to a few locally established contacts, while for high frequencies the gaps or pores at the interfaces are permeable via dielectric displacement. The resulting frequency dependence of the potential lines gives rise to modified impedance spectra. For the simplest case of a single microcontact at an electrode/electrolyte interface, we calculated the exact potential distribution in two dimensions by using the finite element method to solve the Laplace equation for the electrical potential P 2

2

$++()

(1)

8Y

and considering the appropriate boundary conditions [13,14]. To avoid an overlapping of ‘true’ electrode

86-88

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effects and effects purely caused by geometry, an ideally reversible interface is assumed at the microcontact (no additional impedance at the points of contact such as transfer resistances). Fig. 1 shows the equipotential lines (absolute value) for different frequencies. (Since the phase of the potential changes within the sample a complex representation of the potential is used.) For the d.c.-case, the main potential drop occurs beneath the established contact area, leading to the well-known current constriction. However, the high frequency potential distribution is nearly identical to that of a perfect contact. It is worth noting that this requires not only some partly connected point contacts but ‘roofed’ point contacts. The complete impedance spectrum, calculated from these potential distributions [ 13,141, is shown in Fig. 2. Surprisingly, the relatively complicated frequency dependent potential distribution leads to a ‘simple’ impedance spectra, which can be ideally fitted by two par(R,C)-elements in series (Fig. 2, top). Comparing the parameters of this and other possible equivalent circuits, it becomes evident that only the series circuit in Fig. 2 reveals a simple meaning of circuit elements: R, and C, are identical to resistance and capacitance of a perfectly contacted sample,

perfect electrode

ri

/

lowfrequency

medium frequency

1a.c

contacted area:

Fig. 1, Equipotential lines of the absolute value of the potential for a sample with one poor contact calculated by the finite element method (line 1: potential 0, line 21: potential 1). The situation is shown for three typical frequencies (0.3, 120, 100 000 in arbitrary units).

J. Fleig, J. Maier I Solid State lonics 86-88

0.000

0.010

0.005

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(1996) 1351-1356

the shape of a ‘sombrero’ (as shown in Ref. [14]) and being responsible for the second semicircle in the impedance plot. Similar calculations were also applied to a model of polycrystalline material where the grains are connected only by a few bridges. Fig. 3 shows again the main feature of poor contacts containing an insulating phase (pores, gaps): For low frequencies the conductivity of the sample is probed leading to current constriction at the bridges, while for high frequencies the dielectric constant determines the potential distribution. This results in a potential distribution at high frequencies which is nearly identical to that of a single crystal. If the calculated impedance spectrum is interpreted in terms of two par(R,C)-elements in series (Fig. 2), R, and C, are still the values of the corresponding single crystal as long as the volume fraction of the insulating phase is small. However, the fit of the low frequency semicircle is far from being perfect. In how far further information can be extracted from these nonidealities will be investigated in the future. The impedance dependence of R, however, still reflects the bulk activation enthalpy, provided the microstructure does not change.

0.015

Z(real) Fig. 2. Impedance spectrum in arbitrary units corresponding to the potential distribution given in Fig. 1, as calculated by the finite element method. The identity to the fit using the series equivalent circuit (top) is also given.

while

[ 13,141 R, is (as R, ) proportional to the bulk resistivity. The constant of proportionality, however, is a complicated shape factor. Therefore, the two resistances exhibit the same temperature dependence as often observed in experiments [710,13,15]. It is possible to understand this behaviour by dividing the sample into two frequency dependent regions, the one connected to the electrode exhibiting

electrode I

Y

elec rode II

Fig. 3. Potential distribution in a 2D-polyctystal consisting frequency case). The microstructure is given on top.

of 8 grains

which

are only partly

contacted

(1.h.s.: d.c.-case,

r.h.s.: high

1354

J. Fleig, J. Maier I Solid State Ionics 86-88

3. Point contacts to measure

local conductivities

In order to apply microcontacts for local conductivity measurements (as it is done in semiconductor physics to determine impurity profiles) on low conductive solid electrolytes two requirements are necessary: It should be possible (i) to establish reproducible, well-defined contacts of known shape and size [ 161 and (ii) to measure very high impedances reliably. Two kinds of point contacts are used here: A (silver coated) tungsten needle with tip radius 5 pm is pressed onto a AgCl single crystal (controlled by a microscope). Shape and size of the imprint were checked by AFM indicating the possibility to establish circular, nearly planar contacts (depth/diameter=O.15). Using a lithographic mask, silver points of 5 to 20 pm are evaporated onto the crystals. These microelectrodes were contacted by a tungsten needle of 1 ,um tip radius. The resistance between these point contacts and an extended silver counter electrode on the backside was measured [13,15]. In the case of samples characterized by a homogeneous conductivity g, this resistance (obtained from impedance measurements) is given by

R=p

(1996) 1351-1356

principle of measurement and the crucial part of the conductivity profile is given in Fig. 4. This profile was fitted using Fick’s law and a concentration dependent diffusion coefficient, resulting from the assumption that associates of silver-vacancies and Cd’+ are the diffusing particles [19,20]. The constancy of the profile up to about 200 ,um (not shown in Fig. 4), indicates the removal of a supersaturation by precipitation or segregation of a Cd-excess during cooling [21]. The diffusion coefficient of Cd*+ (counter diffusion of Cd*+ and Ag+) at 270°C was calculated to be 2.2X lo-” cm*/s (for concentration + 0) and was found to be in reasonable agreement with reference [22]. In this way, this relatively simple method to obtain diffusion coefficients could be applied to ionic conductors for the first time. In addition, thermodynamic data for the Cd*+-silver vacancy association reaction are obtained [13]. Another inhomogeneity of interest, the electrical characterisation of which is difficult by conventional means, is realised in samples exhibiting highly conductive surface layers. For a step-like conductivity profile perpendicular to the surface, it is

,

movable point electrode

1 u.2d

(2)

(d=diameter of the planar point). Since the main potential drop occurs in a small region around the point, it is possible to deduce local conductivities within the sample. The applicability of the method was proved by measuring five AgCl single crystals with different impurity levels using both conventional ‘macroscopic’ electrodes and point contacts. The conductivities are identical within an error range of 15%. To measure impedances as high as 500 CL!, a self-made impedance adapter was used [ 17,181. The method was applied to determine a conductivity profile in AgCl caused by an inhomogeneous distribution of Cd’+. This distribution is a frozen-in situation of an impurity diffusion experiment. The

extended cou rx er electrode

260

280

300

depth/pm

Fig. 4. Top: A schematic experimental set-up to measure a conductivity profile by using microelectrodes. Bottom: The diffusion coefficient of the dopant is calculable from the conductivity profile shown.

J. Fleig, J. Maier I Solid State Ionics 86-88

possible to give the point contact resistance analytically in dependence of the bulk conductivity cBB’ the surface conductivity a,, the layer thickness w and the point diameter d [23]. Varying the point diameter opens up the opportunity to calculate the ratio pa/us and w (provided point diameters are in the range of the layer thickness). Furthermore, the activation energy of the surface conductivity is deducible if bulk conductivity and diameter are given [13,16]. This is shown in the following example: A silver chloride single crystal surface etched with Na,S,O,solution was polished with a very fine grained emery paper. The surface roughness of the sample was in the order of 100 nm, as checked by AFM. This leads to a highly conductive surface layer. The resulting point resistance was measured for different point diameters and temperatures, and compared to the known resistance without this layer. Thus, the conductivity ratio as /a, could be calculated to be = 100 while the layer thickness w turned out to be =4 pm. These results were confirmed by other measurements [ 13,161. Fig. 5 shows the derived temperature dependence of the surface conductivity. It can be described by an activation energy of 0.42 eV, which is similar to those known from highly disordered thin films. This leads to the assumption that higher dimensional defects give rise to highly conductive surface paths,

(1996) 1351- 1356

1355

possibly via a space charge mechanism [3]. Annealing experiments are discussed elsewhere [16].

4. Conclusions A quantitative description of the impedance of geometrically imperfect electrode contacts exhibiting two semicircles in the impedance plot is given. The impedance spectra should be analysed in terms of two serial par(R,C)-elements to allow for a simple interpretation of circuit elements: The high frequency resistance and capacitance are identical to those expected for perfect contacts. This means that R, and C, can be used to evaluate materials parameters as if the electrodes were ideal. In the case of a single interface, the low frequency capacitance enables a rough estimation of the gap thickness. In all cases, R, and R, show identical temperature dependence. Therefore, the occurrence of a second semicircle exhibiting similar temperature dependence is a strong indication for the existence of a poor contact. In this way, earlier heuristic models are put on a firm basis. Point contacts are tools to obtain information about the local conductivity on a scale of ,um. This could be evidenced by two examples: Point contacts can be used to detect impurity profiles in solid electrolytes and thus, to calculate diffusion coefficients as shown for the counter diffusion coefficient of Cd*+ and Ag+. Furthermore, the point contact resistance is very sensitive to highly conductive surface layers and can be used to characterise layers which otherwise are difficult to investigate. By varying point diameter and temperature, a layer caused by mechanical treatment could be detected on AgCl: The conductivity increase is about 100 while the thickness is about 4 ,um.

Acknowledgments

-5

,““,““,““l”“l”“I”“I”‘~

2.7

2.8

2.9

3.0

3.1

3.2

3.3

3.4

1 QQQuiT Fig. 5. Temperature dependence of the conductivity of a highly conductive surface layer on AgCl, derived from point impedance measurements.

The authors gratefully acknowledge the help of M. Winter, Rechenzentrum Universitat Stuttgart in the implementation of the finite element program FLUXEXPERT and the advice of G. Notzel, Institut fiir Physikalische Chemie, Tiibingen for his support in constructing the ‘high impedance adapter’. The help of the technology group of the MPI Stuttgart headed

J. Fleig, J. Maier I Solid State Ionics 86-88

1356

by H.-U. Habermaier and many discussions with J. Jamnik and J. Werner (both MPI, Stuttgart) are also gratefully acknowledged.

[ll]

[12] [13] [14] [IS] [16]

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