A new penetration impedance technique

A new penetration impedance technique

Electrochtm#co Pergamon Acla, Vol. 41, Nos. 7/E. pp. 1011-1015, 1996 CouyMht 0 1996 Elwvicr Science Ltd. Printed in &eat Britain. All rights reserve...

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Acla, Vol. 41, Nos. 7/E. pp. 1011-1015, 1996 CouyMht 0 1996 Elwvicr Science Ltd. Printed in &eat Britain. All rights reserved 0013-4686/!36 315.00 + 0.00



Max-Planck-lnstitut fiir Festkorperforschung, HeisenbergstraDe 1, 70569 Stuttgart, Germany t National Institute of Chemistry, Hajdrihova 19, 61115 Ljubljana, Slovenia (Received 8 May 1995)

Abstract-To study the electrical properties of solid ionic conductors with a spatial resolution we developed a novel variant of the impedance technique based on the two-dimensionality of the cell-electrodes arrangement and resulting frequency dependent penetration depth of the excitation signal, ie the electrical signals spread deeper into the ionic conductor the lower the frequency. The technique was quantitatively verified by detecting the position of an additional Ag strip added to the ionic conductor. Using the advantage that at high frequencies only the region close to the working electrode is probed, the technique was used to study parent metal/ionic conductor contacts. In this context resistance and capacitance of the Ag/AgCI interface were measured and analysed in the light of ionic space charge effects. The difference of the described technique, when compared with the conventional impedance analysis, consists in the fact that the measured decay rate of electrical relaxation now includes also the information about the position in space at which the relaxation occurs. Key words: interfacial impedance, space charges, transport, measurement techniques.

INTRODUCHON Impedance spectroscopy is a technique which is able to distinguish between different processes due to different relaxation times. In the case of perfectly homogeneous materials the relaxation proceeds at every region within the material coherently, ie spatially in phase. Spatial variations of a resistivity[l] and/or of a dielectric constant of the material, however, introduce incoherencies, or if we are in the frequency domain, phase shifts, among electrical signals with respect to position coordinates. In this case the relaxation may be considered as a local process which depends on the local parameters, as long as the scale of consideration is larger than the Debye length. Although the relaxation occurs locally we usually measure only the cumulative signal, which is the sum of local values, summed up from one electrode to the other. Consequently, the information about local impedance, and therefore about the position dependent electrical properties of the material, is nearly completely lost. What remains, at least in a one-dimensional case, is an effective thickness of the region, where a particular relaxation occurs. This thickness may be calculated from the capacitance associated with the corresponding process, such as the thickness of the double layer or the overall thickness of the insulating grain boundaries in a polycrystalline material. On the other * Presented at the 3rd International Symposium on EIS, 7-12 May 1995, Nieuw-poort Belgium. t On leave from the National Institute of Chemistry, Ljubljana, Slovenia. 5 Author to whom correspondence should be addressed.

hand, the position of such regions is completely unknown. This deficiency of classical impedance spectroscopy renders the investigations of modern heterogeneous ionic conductors and electro-ceramics difficult. In these materials the electrical properties may change drastically with position, starting from the interior of the regions where, besides point defects, the crystal structure is perfect, and then approaching, spatially, the proper higher dimensional defects (dislocations, domain walls, grain boundaries, etc.)[Z]. Even in the case of electrode processes in the solid state it is advantageous to have techniques with spatial resolution. Contrary to liquid electrolytes, in which the use of a Luggin capillary is a standard procedure, one meets serious difficulties trying to apply such a reference electrode, which reduces the bulk contribution to the total impedance response. In addition, it is not easy to distinguish the electrode response from the responses of other boundaries located within the sample. In this paper we report on a novel impedance technique, which allows for the spatial resolution although the electrodes are fixed. The technique is based on the frequency dependent penetration depth of the excitation signal. Such frequency dependence is easy to achieve using the electrode configuration shown in Fig. 1. The working electrode (Ag) is in contact with the material under investigation (in our case AgCl) while the counter electrode (Si) is made completely blocking by using a thin dielectric layer (SiO,) which separates the counter electrode from the ionic conductor[3]. The important geometrical feature is the two-dimensionality of current and potential lines which leads to a relationship between the frequency



J. JAMNIKet al. &,w/d,


yod Equivalent

Fig. 1. Schematic representation of a part of the sample with the corresponding equivalent circuit. and the spatial coordinate of the part of the film which is probed at that frequency (see the next section). The technique was verified by detection of the position coordinate of an artificial inhomogeneity, ie an additional Ag strip, built into the AgCl film. In the last section the application of the technique to the Ag/AgCl boundary is briefly touched upon[3,4]. THEORETICAL


In the case of the electrode configuration shown in Fig. 1, the electric current and potential lines within the ionic conductor, ie AgCl film, are frequency dependent. At high frequencies w, the impedance of the dielectric layer (SiO,) is relatively small, and consequently the current lines are concentrated in the vicinity of the working electrode. As w decreases, the current lines penetrate deeper into the ionic conductor and therefore larger and larger portions of the film are probed. In materials in which the dominant contribution to the current is electric field driven rather than diffusive, and at frequencies smaller than the reciprocal of the dielectric relaxation time of the ionic conductor, we may proceed in a manner similar to that given in Ref. [S]. In this case, the governing equation for the current I is Ohm’s law

E =r(x)I(x)


where @ denotes the electric potential difference between the film, at distance x for the working electrode, and the counter electrode. The symbol r denotes the resistance of the film with the conductivity Q, the thickness h and the width w, defined per unit length as r = (w/m)- ‘. The divergence of I(x), arising from the capacitance coupling of the conductor with the underlaying counter electrode, reads

aI -&=

a4 -Cjp

where c denotes the capacitance of the dielectric layer of thickness d and dielectric constant* .s&(c 5

.Q being the permittivity of free space). Combining both equations, and assuming that r # r(x) we obtain a time dependent diffusion equation for the electric potential CD with the solution is a complex wave @ 0~ eiorikx, where k = & number. The reciprocal value of the real part [Re(k)]-‘, describes the decay length and the reciprocal value of the imaginary part [Im(k)]- ‘, the wavelength of the excitation signal. For frequencies small enough that [Re(k)]-’ <
2 = iwcz2(x) - r(x),


a Ricatti equation, which, in general, is not solvable analytically. Therefore, in order to demonstrate the sensibility of Z(w) = z(x = 0, o) to the local behaviour of the sample conductivity r(x), we numerically calculated the frequency responses for two resistivity profiles with a reflective (Fig. 2(a), curve 1) and transmissive (Fig. 2(a), curve 2) type of electrical inhomogeneity. The corresponding frequency responses are depicted in Fig. 2(b), where the phase angle of the impedance is plotted vs frequency dependent decay length. In general, the decay length depends not only on the frequency, but due to r(x), also on the position. In order to avoid this position dependence we consider electrical inhomogeneities as perturbations and define the decay length according to a position independent background value (Fig. 2(a)), which in our case is [Re(k)]-’ = ,_/?(~cr,,~)-~‘~. This procedure can be shown to be justified in terms of the linearization of equation (3). For decay lengths which are small enough to ensure that the excitation signal does not reach the electrical inhomogeneities within the sample, the phase angle equals 45”, as in a perfectly homogeneous case. The appearance of the inhomogeneities is announced, so-to-speak, by the small peaks a and b (see Fig. 2(b)), while the more pronounced peaks A and B occur at the large decay lengths at which the whole electrically inhomogeneous region is probed. The reflective region results in an increased phase angle, and the inverse is true for the transmissive * .s$ is an effective dielectric constant, which takes into account also the space charge layers at the following interfaces: film/insulator and insulator/counter electrode. Consequently, it depends, in general, on bias and frequency. t To avoid singularities due to zero potential or current, which arise from particular boundary conditions, we may also define a local admittance in a similar manner.

A new penetration impedance technique


Frequency/Hz 10’ 103


I g 2 46 -

z 9 44. 8


bansmissive inhomogeneity







Distance z from the working electrode

% s42‘-d


2 F a 3% z e 36. 0






Decay length [Re(k)]-‘/pm

z b)

% 4


Decay length ‘TRe(t)F

Fig. 2. (a) Two arbitrary resistivity profiles in the case of (I) reflective and (2) transmissive inclusions. (b) Corresponding phase angles of the numerically calculated impedance presented as a function of decay length. Resistivity profile I results in phase-angle curve I, while resistivity profile 2 results in phase-angle curve II.

With decreasing frequency the wavelength associated with the excitation signal increases leading to a poorer space resolution, which is demonstrated by the peak broadening (see Fig. 2(b)). The positions of peaks in the phase-angle plots are not exactly the same as the positions of peaks in the r(x) profiles. Therefore the phase-angle plots are a helpful but only approximate tool for determining the positions of electrical inhomogeneities. The impedance measurements can be considered as a mapping of a position dependent resistivity r(x) to a frequency dependent impedance Z(w), with the formal description given by the Ricatti equation (3). Up to now we have not found the inverse transformation, ie the mapping Z(w) + r(x), which would be needed in order to determine r(x) from impedance measurements exactly. For a first approximation the phase-angle plots are useful. If we know, however, in advance, the approximate shape of inhomogeneities, we can model the response with several free parameters, eg the distance of the inhomogeneity from the working electrode, and then determine the parameters by fitting the model to the measurements. The sample shown in Fig. 1 actually involves inhomogeneities of such a type. The Ag/AgCl interface acts as a reflective type of inhomogeneity, whereas the additional Ag strip (see inset in Fig. 3)--with a negligible resistance when compared to AgCl-is an example of a transmissive inhomogeneity. Detailed modelling was published in Ref. [3]. The main features are indicated in the equivalent circuit in Fig. 1. The Ag/AgCl boundary is region.

Fig. 3. Phase angle of the impedance as a function of the decay length of the excitation signal. Curve 2 is obtained by fitting the measured values (circles) to the model described in the text, which takes into account the interfacial Ag/AgCl impedance and the presence of the additional Ag strip (see inset). Curve 1 displays the simulaied impedance response with the same values of parameters as curve 2, but without the presence of additional Ag strip. The inset shows schematically the structure of the corresponding sample.

described by an R,C, term, R, being the interfacial resistance and C, being the interfacial capacitance. The geometrical capacitance of the working electrode due to the underlying Si is taken into account by C,. For the purpose of modelling the impedance contribution of the additional Ag strip, two parallel R,-C, elements are combined with the geometrical capacitance of the strip C,,, (in our case C,,, = CJ. This combination is included in the transmission line at the position of the strip (see inset in Fig. 3). The description is completedC3, 43 by taking into account that the AgCl film in the actual sample extends on the right-hand side as well as on the lefthand side of the working electrode (Fig. 3). SAMPLE




As shown schematically in Fig. 1 we used as a counter electrode a silicon wafer with a thermally grown, 1OOnm thick, SiO, layer. On the top of the wafers, the working electrodes, 2 mm long and 20pm wide, were evaporated using photolithographic masks. Finally, AgCl films, about OSpm thick, were deposited on the sample by thermal evaporation and then annealed. For technological details see Ref. [3]. During the measurements the samples were placed in a shielding box under a continuous flow of argon. The impedance was measured by a HP 4284A LCR meter in the frequency range from 20Hz to 1 MHz; the amplitude of the excitation signal was 1OOmV. DISCUSSION



Measured impedances include contributions caused by the geometrical capacitance of the


1014 20

15 si G - 10 E 7 e measurement x model


I, 5




Re(Z)/MO Fig. 4. Measured impedance and the corresponding fit pre-

sented in a complex plane. The depressed semi-circle at the coordinate origin refers to the AgJAgCl boundary, whilst the inclined line corresponds to the bulk response of the film. For comparison a spectrum obtained by classical impedance spectroscopy of a sufficiently thin sample is shown schematically in the insert.

working electrode, CB, which, however, does not involve any information about the ionic conductor but masks its contribution at high frequencies. Owing to measurements before the evaporation of the AgCl layer (in that case Z(w) = (iwC,)-‘) we found that in the frequency regime of the measurements and without external voltage bias, the capacitance C, is frequency independent. Therefore, we fitted the model to the real part of the measured admittance, which is unaffected from the frequency independent C,. After that, for the sake of presentation clarity, the measured impedance was always freed from C, contribution algebraically[7]. (a) Scanning principle Figure 3 shows the phase angle of the measured impedance with respect to the decay length of the excitation signal (for comparison see Fig. 2). In the case of a completely homogeneous film one would expect a phase angle of 45”. The measured values strongly deviate from such an ideal value already at very short penetration depths of the signal indicating the influence of the Ag/AgCl boundary at which the charge transfer reaction occurs. With increasing penetration depth the phase angle also increases (Fig. 3, A + B). Then, at a certain decay length the phase angIe drops and reaches the minimum at 26pm, which may already be considered as a first approximation for the position of the additional silver strip, viz. 20pm f3%, as known from the sample preparation. The proper distance, viz. a value of z 23 pm, is obtained from the fitting procedure, as discussed in the last paragraph of the Theoretical Section. Since we knew the values of C, and Cad,, (and therefore also c) from the previous measurements, the free parameters in the fitting procedure were only R, and C, together with a dispersion parameter,+ the disl The interracial contribution was dispersive (see Fig. 4), so we used for the fitting the Cole-Cole equation, which is equivalent to a resistor and a constant phase element in parallel.

et al.

tance of the additional strip from the working electrode and the film resistivity r. We believe that a small systematic error may be expected mainly due to the two-dimensionality of the model (we neglected the z-axis, ie the bonding pads of the Ag strips, etc). The other source of error may be the assumption that the current within the ionic conductor flows only in the x-direction, and the current within the dielectric only in the y-direction. This assumption might be questionable at very short decay lengths. If the decay length increases further, the phase angle approaches again towards the ideal value of 45”. (6) Interfacial impedance When the impedance of the sample is presented in a complex plane, it resembles the response of a partially blocking electrode immersed in a supporting electrolyte (Randles equivalent circuit; see for example Ref. [8]). A supporting electrolyte is often used in liquid electrochemistry since it depresses the response of the bulk, and may lead to a dominancy of the interfacial response, which is of interest in such measurements, at least at high frequencies. The use of supporting electrolytes is hardly feasible in solids. However, using the two-dimensional electrode configuration (Fig. 1) instead, we obtain a similar effect (see Fig. 4). When compared to classical impedance spectra of a solid ionic conductor (see inset in Fig. 4) the order of appearance of bulk and interfacial response is reversed. In this manner we succeeded to measure the impedance of the Ag/AgCl boundary[3] which is nearly impossible using classical impedance spectroscopy. The reason is, that even in comparatively thin samples (a few tenths of millimetre), the bulk response dominates. (c) Limitations The spatial resolution of this technique is limited only to one dimension. In order to satisfy the assumption that the ionic current mainly flows in the x-direction, the technique is applicable to samples which are thinner than the shortest penetration depth.

CONCLUSIONS We presented a new technique which makes it possible to measure the impedance with a spatial resolution. The key to the technique is in a twodimensional electrochemical cell configuration. The tuning of the frequencies corresponds to the tuning of the penetration depth of the excitation signal and thus to a kind of scanning of the sample from the contact to the working electrode up to the bulk. The feasibility of the technique was verified by introducing an artificial electrical inhomogeneity into the ionic conductor. Finally, it was used for studying the Ag/AgCl boundary, the response of which is diflicult to measure by classical impedance spectroscopy due to usually prevailing bulk contributions. Acknowledgements-The help of the technology group of MPI-FKF Stuttgart headed by H.-U. Haber meier is grate-

A new penetration impedance technique fully acknowledged. This work was partly funded by the “Intemationales Biiro”, Forschungszentrum Jiilich GmbH, Jiilich within the German-Slovene scientific co-operation programme.

3. 4. 5. 6.

REFERENCES 7. 1. J. Jamnik, M. GabergEek and S. Pejovnik, Efectrochim. Acta 35,423 (1990). 2. J. Maier, in Superionic and Solid Electrolytes, Recent Trends (Edited by L. Laskar and S. Chandra), p. 137.



Academic Press, New York (1989); Progr. Solid State Chem., 23, 171 (1995). J. Jamnik, J. Maier and H.-U. Habermeier, Physica B 204, 57 (1995). J. Jamnik, Ph.D. thesis Ljubljana (1994). I. D. Raistrick, in Impedance Spectroscopy (Edited by J. R. Macdonald), p. 58. John Wiley, New York (1987). S. A. Gajar and M. W. Geis, J. electrochem. Sot. 139, 2833 (1992). J. Jamnik, J. Maier and S. Pejovnik, Solid State Ionics 80, 19 (1995). J. R. Macdonald and D. R. Franceschetti, in Impedance Spectroscopy (Edited by J. R. Macdonald), p. 103. John Wiley, New York (1987).