Application of Measurement Models to Impedance...

J. Electrochem. Soc., Vol. 142, No. 12, December 1995 �9 The Electrochemical Society, Inc. 4149

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A. Redondo, J. Power Sources, 22, 359 (1988). 5. M. S. Wilson and S. Gottesfeld, J. Appl. Electrochem.,

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(1993). 8. M. Uchida, Y. Aoyama, N. Eda, and A. Ohta, This Jour-

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Application of Measurement Models to Impedance Spectroscopy

II. Determination of the Stochastic Contribution to the Error Structure

Pankaj Agarwal, *'a Oscar D. Crisalle, and Mark E. Orazem** Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611, USA

Luis H. Garcia-Rubio Department of Chemical Engineering, University of South Florida, Tampa, Florida 33620, USA

ABSTRACT

Development of appropriate models for the interpretation of impedance spectra in terms of physical properties re- quires, in addition to insight into the chemistry and physics of the system, an understanding of the measurement error structure. The time-varying character of electrochemical systems has prevented experimental determination of the stochastic contribution to the error structure. A method is presented by which the stochastic contribution to the error structure can be determined, even for systems for which successive measurements are not replicate. Although impedance measurements are known to be heteroskedastic in frequency (i.e., have standard deviations that are functions of frequency) and time varying over the duration of the experiment, the analysis conducted in the impedance plane suggests that the standard deviations for the real and imaginary parts of the impedance have the same magnitude, even at frequencies at which the imaginary part of the impedance asymptotically approaches zero. On this basis, a general model for the error structure was developed which shows good agreement for a broad variety of experimental measurements.

This paper is part of a series intended to present the foundation for the application of measurement models to impedance spectroscopy. The basic premise behind this work is that determination of measurement characteristics is an essential aspect of the interpretation of impedance spectra in terms of physical parameters. The influence of the error structure on interpretation of impedance spectra is discussed elsewhere for various electrochemical and electronic systems. 1-7 In the previous paper of this series, it was shown that a measurement model based on Voigt circuit elements can provide a statistically significant fit to typical electrochemical impedance spectra. B Here a method is demonstrated in which the measurement model is used to

* Electrochemical Society Student Member. ** Electrochemical Society Active Member.

Present address: Department of Materials, Swiss Federal In- stitute of Technology (Lausanne), Lausanne, Switzerland.

identify the stochastic component of the frequency-dependent error structure of impedance data. A preliminary model for the stochastic component of the error is proposed. The third paper of this series addresses the use of the measurement model for identification of the bias component of the error structure. ~

Background The objective of impedance measurements is identifica-

tion of physical processes or parameter values appropriate for a given system. Interpretation of impedance spectra re- quires, in addition to an adequate deterministic model, a thorough understanding of the error structure for the measurement. The concept that the error structure plays an important role in the interpretation of experimental data is well established in the scientific literature. I~ While its ap-

4150 J. Electrochem. Soc., Vol. 142, No. 12, December 1995 �9 The Electrochemical Society, Inc.

plication to the interpretation of optical spectra is under development, knowledge of the error structure has been shown to allow enhanced interpretation of light scattering measurements in terms of particle size distribution or even particle classification, n'~2

Identification of the error structure for most radiation- based spectroscopic measurements such as light scattering can be accomplished by calculating the standard deviation of replicate measurements. The error analysis approach has been successful for light spectroscopy measurements because these systems lend themselves to replication and, therefore, to the independent identification of the different errors that contribute to the total variance of the measurements. In contrast, the stochastic contribution to the error structure of electrochemical impedance specfroscopy measurements generally cannot be obtained from the standard deviation of repeated measurements because even a mild nonstationary behavior introduces a nonnegligible time- varying bias contribution to the error.

The discussion of error structures in impedance spectroscopy, therefore, has been limited to a pr ior i predictions of measurement noise based on instrument noise ~3 and to standard assumptions concerning the error structure such as constant or proportional errors. ~4-~6 Zoltowski reports an experimental assessment of the stochastic noise at selected frequencies which showed that the standard deviation of the real and imaginary parts of the impedance (and admit- tance) are correlated. ~7 This result was used to defend the use of modulus weighting for regression of models to experimental data. ~749 Attempts to weight regressions by standard deviations determined from repeated impedance scans (e.g., Ref. 20) have not been successful because the standard deviation obtained from repeated impedance measurements includes both bias and stochastic contributions to the error structure. Our objective here is to present a procedure for assessing the stochastic contribution to the error structure of impedance measurements and to develop a model for the stochastic noise of impedance measurements.

Knowledge of the error structure is essential in assessing experimental technique and can play a role in determining the influence of instrumental strategies. For example, use of matched filters for the input of the frequency response analyzer has been proposed to be an appropriate technique for reducing the noise in impedance spectra. 2~ While the reduction of the noise in input signal can be readily seen by the display of an oscilloscope, a method for assessing the noise in the ult imate measurement is essential in deciding whether the filters contribute to reduction of noise in the impedance spectra.

Knowledge of the error structure is also needed for analysis of experimental data. For example, proper weighting during nonlinear regression of a model to impedance data is necessary to get unbiased parameter estimates. ~~ A weighted least squares strategy that includes the errors in the real and imaginary parts of the impedance is given by minimization of

(Zr, k - Zr, k) 2 (Zj,k -- Zj,k) 2 J = E ~ + ~ -' 2 ' [11

k O'r,k O'j,k

where Zr, k and Zi,k represent the real and imaginary part of the data, respectively, the caret signifies the corresponding model value, and cry, k and %,~ are the real and imaginary part of the standard deviation at each frequency. The use of the variance to weight data ensures that data points with "low noise" content are emphasized and the data points with "high noise" content are de-emphasized. Use of the experimentally determined variance for weighting has been shown to increase the amount and quality of the information that can be obtained from impedance measurements3

Classification of Measurement Errors The residual errors (e~J that arise when a model is re-

gressed to experimental data can be described as being composed of deterministic (systematic) and stochastic

(randomly distributed) contributions, Esyst and es~ooh, respectively. Thus, at any given frequency

Z - Z = ere~ = esy~t + e~toch [2]

where the caret signifies the model value for the complex impedance Z. The presence of stochastic errors estooh in any experimental data is inevitable. In this work, the systematic errors that result from model inadequacies are distinguished from the experimental errors that are propagated through the model and that could arise from a changing base line or from instrumental artifacts. Systematic errors are therefore defined to consist of contributions from the lack of fit of the model to the data (elo~) and a bias (%~as) in the experiment, i.e.

%s~ = elo~ + eblas [3]

In principle, improvement of a stationary model reduces the error associated with a lack of fit, but nonstationary behavior (%s) and instrumental artifacts ( e J can still contribute to the bias errors, i.e.

eb~ = en~ + ei~ [4]

The nonstationary contribution to the bias usually is observed most easily at frequencies that require the longest time for measurement. Instrumental artifacts may be seen at high frequency resulting from equipment limitations. Many, if not most, electrochemical systems show at least a mild nonstationary behavior due to changes in electrode properties during the course of an experiment. In contrast, solid-state systems, as a first approximation, may be assumed to be stationary. Impedance data can be corrupted by instrumental artifacts for both electrochemical and solid-state systems.

The emphasis in regression of models to impedance data is on reducing the error associated with a lack of fit, but it is evident that the residual errors must contain contributions associated with phenomena that are independent of the adequacy of the model, i.e.

Z - 2 = e~o~ + (ens + e i~ ) + e~o~h [5]

The object here is to quantify the stochastic errors in impedance measurements. The quantification of bias errors associated with experimental issues such as instrumental artifacts or nonstationary phenomena will be ad- dressed in a subsequent paper. 9

Identification of Error Structure The method for assessing the stochastic part of the error

structure is based on using a measurement model as a filter for nonreplicacy of impedance data. The measurement model is composed of a superposition of line shapes which can be chosen arbitrarily. A model composed of Voigt elements in series with a solution resistance has been shown to be a useful and general measurement model (see Fig. 1 in Ref. 8). While the line shape parameters may not be associated unequivocally with a set of deterministic or theoreti- cal parameters for a given system, the measurement-model approach has been shown to represent adequately the impedance spectra obtained for a large variety of electrochemical systems.8 The characteristic time constants for the line shapes and their dispersion are low frequency as compared to the noise. The line shape models therefore can be said to represent the low-frequency stationary components of the impedance spectra (in a Fourier sense). Re- gard]ess of their interpretation, the measurement model representation can be used to filter and thus identify the nonstationary (drift) and high frequency (noise) components contained in the same impedance spectrum.

It is not obvious that such an approach should work. It is well known, for example, that the impedance spectrum associated with an electrochemical reaction limited by the rate of diffusion through a stagnant layer (the Warburg impedance) can be approximated by an infinite number of resistance-capacitance (RC) circuits in series (the Voigt model). In theory, then, a measurement model based on the Voigt circuit should require an infinite number of parameters to describe adequately the impedance response of any

J. Electrochem. Sac., Vol. 142, No. 12, December 1995 �9

electrochemical system influenced by mass transfer. In practice, stochastic errors (or noise) in the measurement limit the number of Voigt parameters that can be obtained from experimental data. An infinite number of Voigt parameters cannot be obtained even from synthetic data because round-off errors limit the information content of the calculation. ~ The residual errors associated with fitting a Voigt model to experimental impedance data which are influenced by mass transfer, with appropriate weighting, can be made to be of the order of the stochastic noise in the measurement. 9 A Voigt circuit therefore can yield an appropriate measurement model for electrochemical impedance spectra.

It is evident that the measurement model composed of Voigt elements may not be the most parsimonious or effi- cient model for a given spectrum. However, it does have the advantage that the maximum number of elements and, therefore, parameters to be estimated, is limited by the noise frequency. In effect, in using the measurement model, one takes advantage of the noise present in any measurement, which limits the number of parameters that can be resolved, and the large number of measured frequencies as compared to the number of resolvable parameters.

Stochastic Contributions to the Error Structure The approach taken to assess the error structure of

impedance data is to identify the measurement characteristics experimentally rather than to assess the noise level from the published specifications of the component instrumentation. Pseudo-replicate impedance scans are obtained, recognizing that even stable electrochemical systems are usually nonstationary such that measurements cannot be repeated exactly. Two cases must be distinguished.

1. For stationary systems, the standard deviation (at a given frequency) of the residual errors resulting from a single fit of a measurement model to successive measurements can provide an estimate for the frequency-dependent standard deviation of the stochastic contribution to the error structure.

2. For nonstationary systems, the standard deviation of the stochastic contribution to the error structure is estimated from the standard deviation of the residual errors resulting from a fit of a measurement model to each individual successive measurement. Because a separate fit is used for each repeated spectrum, the measurement model can be said to act as a filter for the contribution of the lack of replicacy of successive measurements.

Stationary systems.--The procedure for calculation of the standard deviation of the stochastic error estoc h in sta- t ionary systems is illustrated by its application to impedance data for n-GaAs single-crystal wafer with a Ti Schottky contact and an Au/Ge/Ni ohmic contac t : Experi- mental data for five replicate experiments conducted at 320 and at 400 K are shown in Fig. 1. For stationary systems, e~ is equal to zero. Therefore Eq. 5 reduces to

Z - ~ = e ~ = e,o~ + em~ + es~oc~ [6]

where Z is calculated by regression of a measurement model to the combined set of replicate data. Under the as- sumption that the stochastic errors e~oc~ = %oc~,r + jesto~,~ follow a normal distribution ~

e~too~: ~ N(E,, ~r) [7a]

esto~hj ~ N(% ~j) [7b]

where the mean stochastic-error components er and ej are each equal to zero. The standard deviations ~r and ~ are not known a priori. For replicate experiments, the errors due to instrumental artifact can be assumed to be constant from one experiment to another. As one model was regressed to all data sets, the lack of fit is also constant. The frequency- dependent standard deviation ~, therefore, can be estimated by the standard deviation of the departure of the residual error ere s from the mean value at each frequency

The Electrochemical Society, Inc. 4151

9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 i l I O I I O O O O O O I I I O I I I O I I I O O I O | ( I I O | I f I !

8 .~888~8oo i% .... I o O o~f~gOno �9 o ~zu~ 16 : o o0~ �9 �9 400K

.1~ o

0 6 ~ �9 12 8

5 l O ~ " ; " ~

6 �9 �9 6 c o 2 8 A �9 I �9 4 "~ ,a

=" |~ ~s ",, ,�9 ". = E 1 .�9 2 E ~" o ~�9 .

1 10 100 1000 10000 100000

Frequency, Hz

Fig. 1. Real and imaginary parts of the impedance of single-crystal n-GaAs/Ti Schottky diode held at 320 K (open symbols) and 400 K (closed symbols). Five replicate measurements were made at each temperature. The circles represent the real part of the impedance and the triangles represent the imaginary part of the impedance.

N ,0. r^2 : ~ (Eres, ~ __ ires,r)2 [Sa]

k=l

~ = ~ [8b] N 1

k = l

where ~r 2 and ~ are the calculated variances for the real and imaginary components of the residual errors, respectively, N is the number of data points at a given frequency, and

e~es = mean (elof + e~) [9]

The above procedure yields the same results as obtained by calculating directly the standard deviation of the real and imaginary components of the data at each frequency. The discussion presented here is intended to emphasize the assumptions made and to illustrate the difference between the treatment used for stationary systems and that used for nonstationary systems.

The results of the calculation of the stochastic contribution to the error structure for the measurements presented in Fig. 1 are shown in Fig. 2 for data taken at 320 and 400 K. The standard deviation of the impedance measurement is a

1 0 0 0 0 0 0 ! . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . .

~~176 ~ ^ , oO~o~O A~e

I00000 ~ A ~~176 o o 320K

m ! a ., ~ �9 400K

10000 o i o 2~ 6 ~o .o I000 ma~ ..~ r A

r~ 100 A oa4 �9 @I �9

�9 - =- - _~ e=I &O�9 T T Io__ ~ �9 4

c 1 0 ~ = �9 �9 �9 �9 =o8~ , =o t I u~

1 % A

0 . 1 . . . . . . ' " . , , , , , . i . . . . . . d , , , , , , , , t , . . . . . .

1 10 100 1000 10000 100000

Frequency, Hz

Fig. 2. Unfiltered standard deviation of the data presented in Fig. 1 as a function of frec.luency. Circles represent the real part, and the triangles represent the imaginary part of the impedance.

4152 J. E l e c t r o c h e m . Soc. , Vol. 142, No. 12, December 1995 �9 The Electrochemical Society, Inc.

20 ' ' ' ' I ' ' ' ' I ' ' ' ' 1 ' ' '

15

7"-1o

5

0 5 10 15 20 Z~, kfl

Fig. 3. Six successive impedance measurements for a copper disk rotating at 1000 rpm in an alkaline 1 M chloride solution. The data were collected after 3 days of exposure.

strong function of frequency, and the real and imaginary parts are indistinguishable at 320 K and almost indist in- guishable at 400 K. The extent of the correlation of the real and imaginary standard deviations at 400 K is comparable to that reported by Zoltowski. 17 The correlation of real and imaginary noise and the observation of heteroskedasticity with respect to frequency are important results, and their significance is discussed in later sections.

Nonstat ionary s y s t ems . - -For most electrochemical systems, the nonstationary contribution to the bias error ens is not equal to zero. Since the system evolves with time, ens changes from one experiment to another as a set of replicate (or consecutive) experiments are performed. Hence, Eq. 8 leads to inaccurate estimates for r because the contribution of end to the standard deviation cannot be ignored.

One can identify three time scales over which an electrochemical system can change during the course of consecutive impedance experiments in which data are collected at several frequencies for each scan. The smallest time scale considered here is the time to collect a datum point at one frequency. The time required to collect one complete impedance scan constitutes the next time scale, and the third time scale is the total time elapsed from the start of the first experiment to the end of the last experiment.

If the system is evolving rapidly, changes can occur during the time in which one datum point is collected. Impedance spectroscopy may not be a feasible experimental technique for such systems. For systems showing a slower rate of change, the impedance at each frequency may be measurable, but significant change can occur between the start and end of a complete frequency scan. These types of nonstationarities result in the data being inconsistent with the Kramers-Kronig relations. The issues arising from these inconsistencies are discussed in the next paper in this series. 9

The third time scale considered here appears when s system under investigation is evolving slowly. In this case the change in the system during one complete scan is small and can be ignored, but nonnegligible differences can be seen between successive spectra. Such pseudo-stationary impedance scans are typically observed for even the most stationary electrochemical systems. The need for a method for analysis of such pseudo-stationary systems is illustrated below.

Six replicate (consecutive) impedance spectra obtained for a copper disk rotating at 1000 rpm in an alkaline (pH 11.5) 1 M chloride aqueous solution are shown in Fig. 3.3 The electrodes were held at the open-circuit condition. The electrical contact to the disk electrode (carbon brush and

0.2 '" '" '~ '" '" '~ ' " ' " 'q ' " ' " ' , '"'""1 ' " ""1

! o . o

I

t - 0 . 2 , ...... 1 . . . . . . .I ....... l ....... = i , . . ,J ...... J

1 0 - 1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5

F r e q u e n c y , H z

Fig. 4. Real residual errors }or the regression o} a single-measurement model to the data shown in Fig. 3.

stainless steel shaft) was cleaned with alcohol before and after each impedance scan without interrupting the rota- tion of the disk. The long (1% closure error) autointegration cycle of the frequency-response analyzer was used.

Each spectrum shown in the figure was found by the method presented in Ref. 9 to be consistent with the Kramers-Kronig relations, but a careful analysis of the data using the measurement-model approach showed that the data were not replicate, i.e., the system changed meas- urably from one experiment to another2 To demonstrate that the impedance scans were not replicate, a measurement model with eight Voigt elements was regressed (using a complex nonlinear least squares regression algorithm 7'8 under modulus weighting) to the combined data set (i.e., a single measurement model was regressed to all six data sets simultaneously). The normalized residual errors obtained from the regression are shown in Fig. 4 and 5. The sinu- soidal character of the residual errors is caused by

_ 0 . 0

I

L

- 0 . 2 5 10

F requency , Hz

Fig. 5. Imaginary residual errors for the regression of a single- measurement model to the data shown in Fig. 3.


r - -

1 0 0 0

500

I

: o o

:. o o

- 1 0 0 0 - 5 0 0 0

%, Q

-500

- 1 0 0 0

| , , , , .

, , , I , , , ,

5 0 0 1 ) 0 0

Fig. 6. The imaginary departures from the mean residual error ere, i -- ere, for the regression shown in Fig. 4 and 5 as a function of the' corresponding real values.

errors associated with the lack of fit of the model. The time- varying character of the measurements is not readily apparent in Fig. 3, but examination of the residual errors shows that the system changed from one experiment to the other and that the residuals for the six experiments can be distinguished from one another.

The measurement-model analysis shown in Fig. 4 and 5 shows that the standard deviation obtained using Eq. 8 must contain, in addition to the desired stochastic contribution to the error structure, a significant contribution from a nonstationary bias component. The standard deviations that are corrupted by such bias errors show several distinctive features. The real and imaginary departures from the mean residual error are correlated, as shown in Fig. 6. The standard deviations calculated using Eq. 8 are shown in Fig. 7 as a function of frequency. Examination of real and imaginary values suggests that they are not equal. Both observations are counter to results obtained for sta-

1 0 4 . . . . . . . . . ' . . . . . . . . ' . . . . . . . . ' . . . . . . . . ' . . . . . . . ' ........ 10 2

- - 2 - - ; ' - -

b

I 0 - 4 _ 'I ....... I . . . . . . . . j . . . . . . . . i . . . . . . . . i . . . . . . 1 0 10 0 10 1 1 0 2 10 `5 10 4 10 5

Frequency, Hz

Fig. 7. Unfiltered standard deviation of the data shown in Fig. 3 as a function of frequency. Circles represent the real, and the triangles represent the imaginary part of the standard deviation. The solid line represents the error structure for the n-GaAs sample held at 320 K (see Eq. 14). The dashed line represents the contribution that is proportional to I Z.i I, the dashed-dot line represents the contribution that is proportional to I Zrl, and the dashed-dot-dot-dot line is the cantri- 2 bution proportional to IZI /R,. The discontinuity apparent in the solid line and the dash-dot-dot-dot line is caused by a change of the value of the current measuring resistor at 100 Hz.

~ L

N

N I

I L. N

0 . 2 . . . . ' ' I . . . . . . . '1 " ...... I ' " ' " 1 ' ....... I ' " ' " "

0 . 0 ~

-0.2 ........ I .... ,,,I ....... J , ,,,,,,,I , ,,,,,.I ......

1 0 - 1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5

F r e q u e n c y , Hz

Fig. 8. Real residual errors for the separate regression of measurement models to the data shown in Fig. 3.

tionary data, where the real part of the standard deviation was observed to be equal to the imaginary part.

For nonstationary systems, the stochastic component of the error cannot be calculated by taking the standard deviation of the residual errors directly. A procedure to filter out the nonstationary component of the error is outlined in the next section. The significance of the lines in Fig. 7 is discussed in later sections.

Identi f ication of the noise m o d e L - - F o r nonstationary data, a measurement model is regressed to each data set using the maximum number of parameters that can be resolved from the data. For this work, a complex nonlinear least squares program was used that was developed in- house? '8 As a model for the stochastic contribution to the error structure is not known a priori, modulus weighting is typically used for the regression. The regressed parameters for the measurement model for each data set are slightly different because the system changes from one experiment to the other. Hence, by regressing the measurement model to individual data sets separately, the effects of the change of the experimental conditions from one experiment to another are incorporated into the measurement model parameters. The standard deviations of the real and imaginary residual errors therefore can be obtained as a function of frequency and provide a good estimate for the standard deviation of the stochastic noise in the measurement.

The data set from Fig. 3 is used to illustrate the technique. The normalized real and imaginary residual errors for six regressions are shown in Fig. 8 and 9. The real and imaginary residual errors are randomly distributed about the mean value at a given frequency. The plot of the imaginary vs. real departures from the mean residual error, shown in Fig. 10, further suggests that the residual errors at a given frequency are not correlated. The cyclic behavior of the residual errors with frequency is caused by the lack of fit of the model. The technique of estimating the standard deviation of the stochastic contribution by calculating the standard deviation of the residual errors is in effect a filter for lack of fit. Using the measured error structure to weight the regression of data that are free of bias errors, the residual errors for the measurement model can be made to be of the same order as the standard deviation of the measurement. 9

The real and imaginary standard deviations of the residual errors are shown as a function of frequency in Fig. 11.

4154 J. Electrochem. Sac., Vol. 142, No. 12, December 1995 �9 The Electrochemical Society, Inc.

0 . 2 [ ' ..... "I . . . . . . "I . . . . . . . '1 . . . . '"'1 ........ I . . . . . . .

IN-- 0.0 I

t �9 N-- v

-0.2 ., ...... I ,, ..... ,I ........ I , ,,,,,.I , ,,,,,,,I , ,,,,, 10 - 1 10 0 101 10 2 10 `5 10 4 10 5

Frequency, Hz

Fig. ?. Imaginary residual errors for the separate regression of measurement models to the dote shown in Fig. 3.

The s tandard deviat ions are much smal ler than seen in Fig. 7, and the real and imaginary values are now equal.

The me thod of regressing a measuremen t model to indi- v idua l spectra serves as a f i l ter for lack of replieaey, and, as ment ioned above, the ca lcula t ion of the s tandard devia t ion of the res idual errors for the ind iv idua l fits serves as a f i l ter for lack of fit.

M o d e l for the Error Structure In this section a p re l iminary model for s t andard devia-

t ion of the error, ~, is proposed. Rela t ive ly l i t t le work has Been done in developing proper models for the error s t ruc- ture of impedance data. ~a ~6 Macdona ld has proposed a power - l aw mode l for the f r equency-dependen t va r iance v is

l) r = 0"~. ---- OL~ 4- ~ I 2 r 12~0 [10]

~ = ~ = ~ + ~[2~1 ~-0

where ar, ~r, and ~0 are paramete rs of the mode l for the error s tructure, Z~ and Z i are the real and imaginary par t of the impedance, respectively, v,, v i are the var iance of the real and imaginary par t of the impedance, respectively, and ~r

2 0 0 ' ' '

%

A + o o

- 0

A

- - 2 0 0 i i i i i i

- 2 0 0 0 2 0 0

e r, f~

Fig. 10. The imaginary departures from the mean residual error ere, - ~,~, for the regressions shown in Fig. 9 and 10 as a function of the'corresponding real values.

10 4 ' ' ' " ' " 1 . . . . . . . . I . . . . . . . . I . . . . . . . I ' ' " " " 1 ' " . . . . .

10 2 .

O O

0 o

b

1 0 - 2

- 4 1 0 , , , , , , , , I . . . . . . . . i . . . . . . . . I . . . . . . . . I . . . . . . . . I . . . . . .

10 -1 10 0 101 10 2 10 5 10 4 0 5

Frequency, Hz

Fig. 11. Standard deviation of the residual errors presented in Fig. 8 and 9 as a function of frequency. Circles represent the real and the triangles represent the imaginary part of the standard deviation. The solid line represents the error structure for the n-GaAs sample held at 320 K. The dashed-dot line represents the contribution of the imaginary term to the error. The dashed-dot-dot line represents the contribution of the real term, and the dashed line is the measuring resistor term.

and ~j are the real and imaginary par t of the s tandard deviation. Some symbols used in Eq. 10 have been changed to conform wi th the no ta t ion of this paper. Other authors have suppor ted use of modulus weight ing on the basis of the observed corre la t ion be tween the s tandard deviat ions of real and imaginary components of impedance. :722

For nonzero values of 6,, Eq. 10 does not conform to the exper imenta l evidence (e.g., Fig. 2 and 11) tha t the real par t of the s tandard devia t ion is equal to the imaginary part. If, however, 6r is equal to zero, Eq. 10 yields a s tandard devia- t ion that is independen t of frequency, a resul t that is also in confl ic t wi th Fig. 2 and 11.

T h e o r e t i c a l d e v e l o p m e n t . - - W h i l e it is ev ident tha t the s tochast ic cont r ibut ion to the error s t ruc ture is a funct ion of frequency, the most genera l fo rmula t ion for the error s t ruc ture can be wr i t t en in terms of the measuremen t i tself (as was done, for example, in Eq. 10). 15 Unde r the assump- t ion tha t the fundamen ta l impedance measurement in the ins t rumenta t ion is the magni tude IZl and the phase angle (b, the s tandard devia t ion ~ for the real and imaginary components can be expressed as

8Zr 8Zr (rr = ~ Ezi (Y6 + 0 Z 6 (rlzl

oz~ (~J = Odp ~zl if* + O Z , (~z'

[11]

where a,z, and ~, are the standard deviation of the magnitude and phase angle, respectively.

The development of a preliminary model for the standard deviation of measurements in the impedance plane was based on published instrument specifications. The error in the phase angle was assumed to be a constant, and the error in the magnitude was assumed to be proportional to the signal with a term added to account for the poor signal-to- noise ratio experienced when there is mismatch between the system impedance and the measuring resistor. Thus, the initial postulate for the model development was

O ' r z j Lzq izl +

[12]


where e, ~, and ~ are constants, and R= is the value of the current-measuring resistor. Parameters ~, ~, and % in principle, depend on the specific instruments being used for impedance measurements. The expressions for the errors in the real and imaginary components become

O'r = ~rlZjl + ~rlZrl + % ~ iZ r]

IZI ~r i = %lZil + ~jlZrl + "y~ ~ IZil

[13]

To reconcile Eq. 13 with the observation that the real and imaginary standard deviations are equal, the following revised error structure was proposed

IZl ~ % = ~= = ~ = ~lZil + ~lZrl + ~y Rm [14]

While the form of Eq. 14 was suggested by the assumptions given in Eq. ii, a recasting of Eq. 14 in polar coordi- nates; i.e.

[15]

r IZ~I ~31Zrl i) iZ12+ ~ + ' y ~ (IZ~l+lZ~l)

( _ ,zq O-lz= = et ~IZI + 131Z~- + 'Y Rm/ (IZrl + IZ~l)

shows that the initial assumptions about the errors in phase angle and the magnitude are incorrect and that the errors in phase angle are not independent of frequency. These conclusions can be confirmed experimentally. The "stationary" data set of Fig. 1 is presented in Bode format (phase angle and modulus) in Fig. 12. The standard deviation of the phase angle and the modulus are presented in Fig. 13. The standard deviation of the phase angle reaches a maximum of 2 ~ at a frequency between i0 and 60 Hz (corresponding to a phase angle between -20 and -70 ~ respectively) and reaches a value as low as 0.02 ~ at a phase angle of -90 ~ . The standard deviation of the modulus tracks the value for the modulus only approximately. The standard deviation of the modulus is as high as 400,000 ~at low frequencies (4% of the modulus) and reaches a value of 0.01% of the modulus at high frequencies.

Equation 14 is preliminary, but it has the desirable features of frequency dependence embedded in the measured values for the impedance and of implicit agreement with the experimental observation that the noise in the imaginary and real parts of the impedance is equal. The validity of Eq. 14 as a model for the stochastic noise was established by comparison to experimental data, as discussed in the subsequent section.

1.0E+07 -100

~ i -90

-80 1.0E+06

-70

-60

= ! % -40 O

-3o 1.0E+04

-20

-10

1 .0E+03 ......................... 0 1 10 100 1000 10000 100000

Frequency, Hz

Fig. ] 2. Impedance response in the Bode Formot of a single-crystal n-GaAs/~ Schoffky diode held at 320 K. Five replicate measuremen~ were made.

1000000 10 : , .,,.,. , ,,,,,., ........ . , ,,,,,,,j . ,,,,,,j

!%o __ o 00%0 o I00000 ~ ~oo o

@; Oo ~a

, ~ a ~ o ~ 10000 ,'2 , �9 ~ Q; a a a A o % ~l: o m a a 1

"~ 1000 a %0 ~. ~ A A A A CO 0 ~ o

0 . ~ ~. .~ a~ 100 A a A~ co '0 m'~ a =

" a,, s 0% o I 0 ~ o o "N

"~ a % o 01 | : ' . ' . . . . . . . . : ' . . : . . . . ,-, A 0 AO

o.1 �9 " : " ' " 4 �9 eee �9 /

0 , 0 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . A ....... -~ ' ~1~'""1 0.01

1 10 100 1000 10000 100000

Frequency, Hz

Fig. ] 3. Unfiltered standord deviation of the data presented in Fig. 12 as a function of frequency. Open circles represent the standard deviation of the modulus in ohms, closed circles represent the standard deviation of the modulus as a percentage of the measured modulus, and triangles represent the standard deviation of the phase angle in degrees.

Comparison to experiment.--The error structure is developed here for impedance data collected for an n-GaAs/ Ti Schottky diode. This solid-state system was chosen for this analysis because the standard deviation of the measured impedance had a broad range of values and because, to a first approximation, the sequential measurements could be assumed to be replicate. Previous work 1'7 showed that, while use of modulus weighting, proportional weighting, or no weighting gave only one electronic state, regression of these data using the error structure to weight the regression allowed determination of four electronic states. The number of states and their energy level were confirmed by independent measurementsY

Experiment.--Values for ~, ~, and ~/were calculated for impedance data collected on a Solartron 1286 potentiostat and a Solartron FRA1250 frequency-response analyzer (FRA). The data used for the estimation of the error structure parameters were collected for the n-GaAs Schottky diode discussed in Fig. 1 at temperatures ranging from 320 K to 420 K. i The frequency range used for the experiment was 1 Hz to 65 kHz. Data were collected frequency by frequency using the long channel integration feature of the FRA, which completed a measurement at each frequency on reaching a 1% closure error. The wide range of temperature and frequency ensured a wide range of impedance values. Five replicate experiments were conducted at each temperature. The results of the experiments at 320 and 400 K are shown in Fig. i. As a first approximation it was assumed that the system was stationary, and the stochastic component of the error e=toc ~ and its standard deviation, cr were calculated for data sets for the GaAs sample at 320, 340, 360, 380, and 400 K, as described in the section on Stationary systems,

Regression procedure.-- Equation 14 was regressed to the standard deviation values to obtain the values of cr ~, and ~. The data were detrended to ensure that the mean residual error for the regression was equal to zero. 27 Use of no- weighting or proportional weighting in the regression gave poor results. Following Eq. i, the regression was weighted by the estimated variance. The standard deviation of the standard deviation for impedance measurements was obtained by considering the standard deviation of the real and imaginary parts of the impedance to be independent observations of the standard deviation at a given frequency.


Table I. Parameter estimates for the error-structure model (Eq. 14) obtained by regression to replicate data for a single-crystal n-GaAs/Ti Scholtky diode.

Parameter

Unfiltered Filtered No moving Three-point moving Five-point moving No moving Average Average Average Average

Three-point moving Five-point moving Average Average

cr 3.29-+ 0.13 • 10 -4 2.46_+ 0.48 • 10 -4 8.2-+ 5.6 • 10 3 8.12_+ 0.020 • 10 -4 9.45-+ 0.50 • 10 4 (_+4.0%) (-+20%) (-+69%) (_+0.23%) (-+5.3%)

1.20-+ 0.01 • 10 3 1.24_+ 0.03 x 10 -3 1.59-+ 0.07 • 10 -3 9.33-+ 0.011 • 10 4 1.49_+ 0.045 • 10 -3 (-+0.88%) (+-2.8%) (-+4.1%) (-+1.1%) (+-3.0%)

2.833 2.832 2.785 2.306 2.209 -+0.0021 • 10 -4 _+0.0062 _+ 10 ~ +-0.011 • 10 4 -+0.0017 _+ 10 -4 _+0.0090 • 10 4

(0.074%) (0.22%) (0.39%) (0.072%) (0.41%)

X3/~ 4.74 1.77 1.64 15.2 2.18

9.43 -+ 0.64 • I0 -4 (+_6.8%)

1.12 _+ 0.052 • 10 -3 (-+4.7%)

2.268 -+ 0.010 • 10 .4 (0.46%)

1.83

The square of the standard deviation of the standard deviations for the real and imaginary components of the impedance was used, therefore, to weight the regression for identification of the error structure. In addition, three- and five-point moving averages were used to increase the sample size for the standard deviation while retaining the general trends. The weighted X2 statistic (normalized by the degrees of freedom for the regression)

• 1 (Yexpt~i -- Ymodel,i) 2 i

w a s i m p r o v e d fo r r e g r e s s i o n s u s i n g a m o v i n g a v e r a g e va lue fo r t he va r i ance . T h e u s e of a m o v i n g a v e r a g e d id n o t a p - p e a r to i n f l u e n c e the fi t of t h e m o d e l to cases w h e r e the s t a n d a r d d e v i a t i o n fo r t he i m p e d a n c e w a s large , b u t t he f i t w a s q u a l i t a t i v e l y i m p r o v e d f o r cases w h e r e the s t a n d a r d d e v i a t i o n of t he i m p e d a n c e w a s smal l . T h e u t i l i ty of t he moving average should depend on the sampling rate. The five point moving average worked well for the ten points/ decade sampling rate used here. The increased standard deviation for the parameter estimates obtained using the moving average reflects the corresponding increased value for the average standard deviation used to weight the regression. By giving a better estimate for the standard deviation of the fitted quantity, the moving-ayerage approach yielded more reliable estimates for the confidence intervals of the error-structure model parameters.

Results of regression.--The parameter estimates obtained are given in Table I. The results of the regression for GaAs at 320 K are shown in Fig. 14, and the results for 400 K are presented in Fig. 15. The solid line represents the model for the error structure given by Eq. 14. The dashed lines represent the contribution of the different terms in Eq. 14. The jog in the line for the model corresponds to the frequency at which a change was made in the value of the current measuring resistor.

The filtering algorithm described in the section Identifi- cation of the noise model was applied to the above data set to give a better estimate for the stochastic errors. Equa- tion 14 was regressed to the filtered errors. The resulting parameters for the error structure model are shown in Table I. The filtered errors for GaAs at 320 and 400 K, with the new model, are shown in Fig. 16 and 17, respectively. The standard deviations for the filtered data (Fig. 14) were smaller than for the unfiltered case (Fig. 12), and the real and imaginary parts of the standard deviations are closer. The regression results in Fig. 16 and 17 show that the agreement of the model for the error structure (Eq. 14) to the experimental standard deviations is good. Similar agreement was found at all temperatures. The parameters from Table I, for filtered errors, were used to predict the errors for corrosion of copper data shown in Fig. 3. The results are presented in Fig. ii. The model shows a good agreement with the experimentally obtained standard deviations. The validity of the equation is supported since a three-parameter model provides a good agreement for solid-state systems as well as for electrochemical systems, for data collected under various experimental conditions, and for errors ranging in magnitude from 10 -3 to i0 ~ ~.

Discussion and Conclusion The measurement model provides much more than a pre-

liminary analysis of impedance data in terms of the number of resolvable time constants and asymptotic values, as suggested, for example, by Zoltowski. 23'24 As shown here, the measurement model can be used as a filter for lack of replicacy that allows accurate assessment of the standard deviation of impedance measurements. As is discussed by many authors, 1'15,2~ this information is critical for selection of weighting strategies for regression. This information also provides a quantitative basis for assessment of the quality of fits and can guide experimental design. In the subsequent paper of this series, 9 the measurement model is used to assess the bias component of the error structure.

The results presented here show that impedance measurements are heteroskedastic (in the sense that the standard deviations are funetior~s of frequency). In spite of the apparent complexity, a definite structure for the errors is resolved. In the impedance plane, the standard deviations for the real and imaginary parts of the impedance are equal, even at frequencies sufficiently high or low that the imaginary part of the impedance asymptotically approaches zero. This result is consistent with the results presented by Zoltowski. 17

Aside from the obvious impact on the parsimony of the model for the error structure (only three parameters are

b"

10 6

10 4

10 2

10 0

- 2 10

- 4 10

' ' " " 9 . . . . . . . . I . . . . . . . '1 ' '"'"'1 ' ' .....

-

"%

i % . , * , i l i i l [ i I. l i l l J d , * l i i l , , l , i l i l i l d , , , , i , I0 ~ I01 I02 I0 ~ 104 105

Frequency, Hz

Fig. 14. Unfiltered standard deviation of the 320 K data presented in Fig. I as a function of frequency. Circles represent the real and the triangles represent the imaginarypart of the standard deviation. The solid line represents the error structure with unfiltered parameters shown in Table I. The dashed-dot line represents the contribution of the imaginary term to the error. The dashed-dot-dot line represents the contribution of the real term, and the dashed line is the measuring-resistor term.


106

104

102,

100

10 - 2

- 4

' ' ' " ' " I ' ' ' ' " " I ' ' ; ' " " I ' ' ' " ' " I ' ' ' " I

s

i

t \ -"

f

1 0 = , , , , . , I , , , , , , , , i , ,,,,,ul , , , , , , , , I , i , . . . . 100 101 102 105 104 05

Frequency, Hz

Fig. 15. Unfiltered standard deviation of lhe 400 K data presented in Fig. 1 as a function of frequency. Symbols and lines as defined in Fig. 14.

' ' ' " ' " 1 ' ' ' " ' " t ' ' ' " ' " l ' ' ' " " ~ 106

104

102

100

10 - 2

10 - 4 1

1 , i i %.--

.4

f \ 14

!

I i l l i l l l l j I ' i l l l i i i t I i l l ' l tJ t I l l l i l J i I i l a i l

)0 101 10 Z 105 104 05 Frequency, Hz

Fig. 17. Filtered standard deviation of the 400 K data presented in Fig. 1 as a function of frequency. Symbols and lines as defined in Fig. 16.

needed in Eq. 14 as compared to six in Eq. 13), the equality of the real and imaginary standard deviations has implica- tions for the regression of models to impedance data. That the information content of the imaginary part of the impedance can be obscured by noise at the asymptotic tails influences the manner in which the Kramers-Kronig relations can be applied to assess the bias contribution to the measurement. ~ In addition, the equality of the real and imaginary standard deviations becomes a criterion for selection of appropriate weighting strategies. Among the commonly applied weighting strategies, for example, proportional weighting does not conform to this observation, but "no-weighting" and modulus weighting do conform and may be useful weightings for preliminary regressions. While the heteroskedastic nature of the measurements shown here suggests that the no-weighting strategy is inappropriate for experiments under potentiostatie modula- tion, this is not a general result since the standard deviation of some measurements [e.g., electrohydrodynamic imped-

106 ........ , ........ I ........ , ........ I ....... 104~ J %',%

102 "~. %,%,

10 ~

- 2 " - 10

10 4 ....... J ........ , ........ i ........ , . . . . 10 U 101 102 103 10 4

Frequency, Hz

Fig. 16. Filtered standard deviation of the 320 K data presented in Fig. 1 as a function of frequency. Symbols and lines as defined in Fig. 14 with exception that model parameters are those given as Filtered in Table I.

ance spectroscopy (EHD)] is, to a first approximation, independent of frequency. 6'7

Selection of inappropriate weighting strategies may have a severe impact on the amount of information that can be obtained by regression of models to impedance data since the weighting strategies discussed here can differ by many orders of magnitude. To illustrate the influence of weighting, normalized weightings, defined by

[ o'/IZl ~2 to,z, = \(r l}~vJ [16a]

for modulus weighting or by

~ o = [16b]

1.0E+01 100

0 +00 m 1,0E-0

I0 o ~ I . O E - 0 2 " ~" i ~ .oE-o3 ..

~1.0E-04 1 "

~ .'_'_iiii'i iii n - N 1.0E-06 ~.

E1.0E_07 0"1 i Z Z

1.0E-08 ~ t No Weighting

1.0E-09 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.01 t 10 100 1000 10000 100000

Frequency, Hz

Fig. 18. Comparison of modulus weighting and no-weighting strategies to the weighting by the error structure given in Eq. 14. The terms plotted are defined by Eq. 16. The solid lines represent the comparison of the modulus weighting strategy to weighting by the standard deviation of the experiment determined by the methods of this paper. The dashed lines represent the corresponding comparison for the no-weighting weighting strategy. The upper set of lines were obtained for the data collected at 320 K, and the lower set of lines were obtained for the data collected at 400 K.


for no weighting, are shown in Fig. 18 for the semiconduc- tor da ta of Fig. 1. Modulus and no-weighting options are thereby compared to the weighting by ~2 where a is given by Eq. 14. Weighting according to Eq. 14 has been shown to yield more detailed information about the Schottky diode than could be obtained by modulus- or no-weighting options, and the s tructural information so obtained was confirmed by independent measurements} Thus, Fig. 18 can be said to provide a comparison of the modulus- and no- weighting options to the opt imal weighting strategy for this system. The jog in the line for the mode] corresponds to the frequency at which a change was made in the value of the current measuring resistor. The data taken at 320 K show a fully resolved spectrum over the measured frequency range (see Fig. 1 and 12). In this case, the no- weighting option underweights the high-frequency response by nine orders of magnitude as compared to the low frequency data. The modulus weighting is better, but still underweights the high-frequency end by three orders of magnitude. The consequence is that information contained in the high-frequency data is not extracted by the regression procedure. The spectrum is less fully resolved at 400 K, and the no- and modulus-weighting options are closer to the optimal weighting strategy for this system. In this case, the modulus weighting overweights the high frequency end by a factor of only ten; whereas the no-weighting option under weights the high frequency end by the same factor.

The model presented as Eq. 14 provides good agreement to the experimentally measured stochastic noise for a broad range of experimental systems. The systems have included, in addition to those discussed here, studies of electrochemistry at metal hydride electrodesy corrosion of copper, 3'7 transport across biological membranes, 2~ and electronic transitions in large bandgap materials such as ZnO and ZnS. 26 The stochastic noise seems to he primarily a function of the instrumentation used, since use of parameter values in Eq. 14 that were obtained with the semiconducting system provided a good prediction of the error structure for the electrochemistry at a rotating copper disk. The carbon electrical contact to the rotating electrode and the reference electrode did not make a significant addi- tional contribution to the noise. With the exception of experiments in which gas evolution was seen on the electrodes, similar agreement was seen for other experimental systems mentioned above.

The model for the error structure is still under development. The role, for example, of the amplitude of the poten- tial perturbation has not been incorporated into the model. The lack of importance of the perturbation amplitude apparent in the work to date may be a testament to the effi- ciency of the autointegration feature of the FRA. The error structure has been developed here for potentiostatic modu- lation and may need to be refined for experiments conducted under galvanostatic control. Finally, all systems studied here have had a small ohmic contribution to the impedance. A large ohmic resistance may have a striking impact on the error structure of impedance measurements given the observed equality of the standard deviation for the real and imaginary parts of the impedance.

The model parameters presented in Table I are not neces- sarily those that would be obtained on all systems or in other laboratories. The parameters can depend on the experimental design, the type of instruments used, the manner in which the instruments are used, and, in some cases, the system under study. The contr ibution of this work is a systematic model development s trategy for impedance measurements that incorporates a quanti tat ive assessment of the stochastic contr ibution to the error structure.

Acknowledgment The work performed at the University of Florida (RA.

and M.E.O) was supported by the Office of Naval Research and by Gates Energy Products. The work performed at the University of South Florida (L.H.G.R.) was supported by the National Science Foundation under Grants No. RII- 8507956 and INT-8602578. This paper was written while one author (M.E.O) was on sabbatical leave at the UPR 15 du CNRS "Physique des Liquides et Electrochimie," in Paris, France. The use of their facilities to prepare portions of this work and the helpful suggestions of Dr. Claude Deslouis and Dr. Bernard Tribollet are greatly appreciated.

Manuscript submitted April i0, 1995; revised manuscript received Aug. 4, 1995.

University of Florida assisted in meeting the publication costs of this article.

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