Measurement Models for Electrochemical Impedance Spectroscopy ...
Measurement Models for Electrochemical Impedance Spectroscopy ... Measurement Models for Electrochemical Impedance Spectroscopy ...
4154 J. Electrochem. Sac., Vol. 142, No. 12, December 1995 9 The Electrochemical Society, Inc.0.2 [ ' ..... "I ...... "I ....... '1 .... '"'1 ........ I .......10 4' ''"'"1 ........ I ........ I ....... I ' '"""1 ' " .....10 2 .OO0.0IN--ItN--v9b0 o10 -2-0.2 ., ...... I ,, ..... ,I ........ I , ,,,,,.I , ,,,,,,,I , ,,,,,10 -1 10 0 101 10 2 10 `5 10 4 10 5Frequency, HzFig. ?. Imaginary residual errors for the separate regression ofmeasurement models to the dote shown in Fig. 3.The standard deviations are much smaller than seen inFig. 7, and the real and imaginary values are now equal.The method of regressing a measurement model to individualspectra serves as a filter for lack of replieaey, and, asmentioned above, the calculation of the standard deviationof the residual errors for the individual fits serves as a filterfor lack of fit.Model for the Error StructureIn this section a preliminary model for standard deviationof the error, ~, is proposed. Relatively little work hasBeen done in developing proper models for the error structureof impedance data. ~a ~6 Macdonald has proposed apower-law model for the frequency-dependent variance v isl) r = 0"~. ---- OL~ 4- ~I2r 12~0~ = ~ = ~ + ~[2~1 ~-0[10]where ar, ~r, and ~0 are parameters of the model for the errorstructure, Z~ and Z i are the real and imaginary part of theimpedance, respectively, v,, v i are the variance of the realand imaginary part of the impedance, respectively, and ~r- 0200 ' ' 'A%+ o o--200 i i i i i i-200 0 200e r, f~Fig. 10. The imaginary departures from the mean residual errorere, - ~,~, for the regressions shown in Fig. 9 and 10 as a function ofthe'corresponding real values.A-41 0 , ,,,,,,,I ........ i ........ I ........ I ........ I ......10 -1 10 0 101 10 2 10 5 10 4 0 5Frequency, HzFig. 11. Standard deviation of the residual errors presented inFig. 8 and 9 as a function of frequency. Circles represent the real andthe triangles represent the imaginary part of the standard deviation.The solid line represents the error structure for the n-GaAs sampleheld at 320 K. The dashed-dot line represents the contribution of theimaginary term to the error. The dashed-dot-dot line represents thecontribution of the real term, and the dashed line is the measuringresistor term.and ~j are the real and imaginary part of the standard deviation.Some symbols used in Eq. 10 have been changed toconform with the notation of this paper. Other authors havesupported use of modulus weighting on the basis of theobserved correlation between the standard deviations ofreal and imaginary components of impedance. :722For nonzero values of 6,, Eq. 10 does not conform to theexperimental evidence (e.g., Fig. 2 and 11) that the real partof the standard deviation is equal to the imaginary part. If,however, 6r is equal to zero, Eq. 10 yields a standard deviationthat is independent of frequency, a result that is also inconflict with Fig. 2 and 11.Theoretical development.--While it is evident that thestochastic contribution to the error structure is a functionof frequency, the most general formulation for the errorstructure can be written in terms of the measurement itself(as was done, for example, in Eq. 10). 15 Under the assumptionthat the fundamental impedance measurement in theinstrumentation is the magnitude IZl and the phase angle(b, the standard deviation ~ for the real and imaginary componentscan be expressed as8Zr8Zr(rr = ~ Ezi (Y6 + 0 Z 6 (rlzloz~(~J = Odp ~zl if* + O Z , (~z'[11]where a,z, and ~, are the standard deviation of the magnitudeand phase angle, respectively.The development of a preliminary model for the standarddeviation of measurements in the impedance plane wasbased on published instrument specifications. The error inthe phase angle was assumed to be a constant, and the errorin the magnitude was assumed to be proportional to thesignal with a term added to account for the poor signal-tonoiseratio experienced when there is mismatch betweenthe system impedance and the measuring resistor. Thus, theinitial postulate for the model development wasO'rz j+Lzq izl[12]
J. Electrochem. Soc., Vol. 142, No. 12, December 1995 9 The Electrochemical Society, Inc. 4155where e, ~, and ~ are constants, and R= is the value of thecurrent-measuring resistor. Parameters ~, ~, and % in principle,depend on the specific instruments being used forimpedance measurements. The expressions for the errors inthe real and imaginary components becomeO'r = ~rlZjl + ~rlZrl + % ~ iZ r]IZI~r i = %lZil + ~jlZrl + "y~ ~ IZil[13]To reconcile Eq. 13 with the observation that the real andimaginary standard deviations are equal, the following revisederror structure was proposedIZl ~% = ~= = ~ = ~lZil + ~lZrl + ~y Rm [14]While the form of Eq. 14 was suggested by the assumptionsgiven in Eq. ii, a recasting of Eq. 14 in polar coordinates;i.e.r IZ~I ~31Zrl i)iZ12+ ~+'y~ (IZ~l+lZ~l)( _ ,zqO-lz= = et ~IZI + 131Z~- + 'Y Rm/ (IZrl + IZ~l)[15]shows that the initial assumptions about the errors inphase angle and the magnitude are incorrect and that theerrors 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 deviationof the phase angle and the modulus are presented inFig. 13. The standard deviation of the phase angle reachesa 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 phaseangle of -90 ~ . The standard deviation of the modulustracks the value for the modulus only approximately. Thestandard deviation of the modulus is as high as 400,000 ~atlow frequencies (4% of the modulus) and reaches a value of0.01% of the modulus at high frequencies.Equation 14 is preliminary, but it has the desirable featuresof frequency dependence embedded in the measuredvalues for the impedance and of implicit agreement withthe experimental observation that the noise in the imaginaryand real parts of the impedance is equal. The validityof Eq. 14 as a model for the stochastic noise was establishedby comparison to experimental data, as discussed in thesubsequent section.1.0E+07 -1001.0E+06~ i -90-80= ! % -40O-3o1.0E+04-20-70-60-101.0E+03 ......................... 01 10 100 1000 10000 100000Frequency, HzFig. ] 2. Impedance response in the Bode Formot of a single-crystaln-GaAs/~ Schoffky diode held at 320 K. Five replicate measuremen~were made.1000000 10: , .,,.,. , ,,,,,., ........ . , ,,,,,,,j . ,,,,,,j!%o __ o 00%0 oI00000 ~ ~oo o@;Oo ~a,~ a~ o~ 10000 ,'2 ,9 ~ Q; a aa A o% ~l:o m a a 1"~ 1000 a %0~. ~ A A A A CO0 ~ o0.~ ~..~ a~ 100 A a A~ co '0m'~ a =" a,, s 0% oI 0 ~ o o "N: '.'........:'..:.... ,-,A0 AO"~ a % o 01 |o.1 9 ":"'" 49 eee 9 /0,01 ............................ A ....... -~' ~1~'""1 0.011 10 100 1000 10000 100000Frequency, HzFig. ] 3. Unfiltered standord deviation of the data presented inFig. 12 as a function of frequency. Open circles represent the standarddeviation of the modulus in ohms, closed circles represent thestandard deviation of the modulus as a percentage of the measuredmodulus, and triangles represent the standard deviation of the phaseangle in degrees.Comparison to experiment.--The error structure is developedhere for impedance data collected for an n-GaAs/Ti Schottky diode. This solid-state system was chosen forthis analysis because the standard deviation of the measuredimpedance had a broad range of values and because,to a first approximation, the sequential measurementscould be assumed to be replicate. Previous work 1'7 showedthat, while use of modulus weighting, proportional weighting,or no weighting gave only one electronic state, regressionof these data using the error structure to weight theregression allowed determination of four electronic states.The number of states and their energy level were confirmedby independent measurementsYExperiment.--Values for ~, ~, and ~/were calculated forimpedance data collected on a Solartron 1286 potentiostatand a Solartron FRA1250 frequency-response analyzer(FRA). The data used for the estimation of the error structureparameters were collected for the n-GaAs Schottkydiode discussed in Fig. 1 at temperatures ranging from320 K to 420 K. i The frequency range used for the experimentwas 1 Hz to 65 kHz. Data were collected frequency byfrequency using the long channel integration feature of theFRA, which completed a measurement at each frequencyon reaching a 1% closure error. The wide range of temperatureand frequency ensured a wide range of impedance values.Five replicate experiments were conducted at eachtemperature. The results of the experiments at 320 and400 K are shown in Fig. i. As a first approximation it wasassumed that the system was stationary, and the stochasticcomponent of the error e=toc ~ and its standard deviation, crwere calculated for data sets for the GaAs sample at 320,340, 360, 380, and 400 K, as described in the section onStationary systems,Regression procedure.-- Equation 14 was regressed to thestandard deviation values to obtain the values of cr ~, and~. The data were detrended to ensure that the mean residualerror for the regression was equal to zero. 27 Use of noweightingor proportional weighting in the regression gavepoor results. Following Eq. i, the regression was weightedby the estimated variance. The standard deviation of thestandard deviation for impedance measurements was obtainedby considering the standard deviation of the realand imaginary parts of the impedance to be independentobservations of the standard deviation at a given frequency.
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J. Electrochem. Soc., Vol. 142, No. 12, December 1995 9 The <strong>Electrochemical</strong> Society, Inc. 4155where e, ~, and ~ are constants, and R= is the value of thecurrent-measuring resistor. Parameters ~, ~, and % in principle,depend on the specific instruments being used <strong>for</strong>impedance measurements. The expressions <strong>for</strong> the errors inthe real and imaginary components becomeO'r = ~rlZjl + ~rlZrl + % ~ iZ r]IZI~r i = %lZil + ~jlZrl + "y~ ~ IZil[13]To reconcile Eq. 13 with the observation that the real andimaginary standard deviations are equal, the following revisederror structure was proposedIZl ~% = ~= = ~ = ~lZil + ~lZrl + ~y Rm [14]While the <strong>for</strong>m of Eq. 14 was suggested by the assumptionsgiven in Eq. ii, a recasting of Eq. 14 in polar coordinates;i.e.r IZ~I ~31Zrl i)iZ12+ ~+'y~ (IZ~l+lZ~l)( _ ,zqO-lz= = et ~IZI + 131Z~- + 'Y Rm/ (IZrl + IZ~l)[15]shows that the initial assumptions about the errors inphase angle and the magnitude are incorrect and that theerrors 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 <strong>for</strong>mat(phase angle and modulus) in Fig. 12. The standard deviationof the phase angle and the modulus are presented inFig. 13. The standard deviation of the phase angle reachesa 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 phaseangle of -90 ~ . The standard deviation of the modulustracks the value <strong>for</strong> the modulus only approximately. Thestandard deviation of the modulus is as high as 400,000 ~atlow frequencies (4% of the modulus) and reaches a value of0.01% of the modulus at high frequencies.Equation 14 is preliminary, but it has the desirable featuresof frequency dependence embedded in the measuredvalues <strong>for</strong> the impedance and of implicit agreement withthe experimental observation that the noise in the imaginaryand real parts of the impedance is equal. The validityof Eq. 14 as a model <strong>for</strong> the stochastic noise was establishedby comparison to experimental data, as discussed in thesubsequent section.1.0E+07 -1001.0E+06~ i -90-80= ! % -40O-3o1.0E+04-20-70-60-101.0E+03 ......................... 01 10 100 1000 10000 100000Frequency, HzFig. ] 2. <strong>Impedance</strong> response in the Bode Formot of a single-crystaln-GaAs/~ Schoffky diode held at 320 K. Five replicate measuremen~were made.1000000 10: , .,,.,. , ,,,,,., ........ . , ,,,,,,,j . ,,,,,,j!%o __ o 00%0 oI00000 ~ ~oo o@;Oo ~a,~ a~ o~ 10000 ,'2 ,9 ~ Q; a aa A o% ~l:o m a a 1"~ 1000 a %0~. ~ A A A A CO0 ~ o0.~ ~..~ a~ 100 A a A~ co '0m'~ a =" a,, s 0% oI 0 ~ o o "N: '.'........:'..:.... ,-,A0 AO"~ a % o 01 |o.1 9 ":"'" 49 eee 9 /0,01 ............................ A ....... -~' ~1~'""1 0.011 10 100 1000 10000 100000Frequency, HzFig. ] 3. Unfiltered standord deviation of the data presented inFig. 12 as a function of frequency. Open circles represent the standarddeviation of the modulus in ohms, closed circles represent thestandard deviation of the modulus as a percentage of the measuredmodulus, and triangles represent the standard deviation of the phaseangle in degrees.Comparison to experiment.--The error structure is developedhere <strong>for</strong> impedance data collected <strong>for</strong> an n-GaAs/Ti Schottky diode. This solid-state system was chosen <strong>for</strong>this analysis because the standard deviation of the measuredimpedance had a broad range of values and because,to a first approximation, the sequential measurementscould be assumed to be replicate. Previous work 1'7 showedthat, while use of modulus weighting, proportional weighting,or no weighting gave only one electronic state, regressionof these data using the error structure to weight theregression allowed determination of four electronic states.The number of states and their energy level were confirmedby independent measurementsYExperiment.--Values <strong>for</strong> ~, ~, and ~/were calculated <strong>for</strong>impedance data collected on a Solartron 1286 potentiostatand a Solartron FRA1250 frequency-response analyzer(FRA). The data used <strong>for</strong> the estimation of the error structureparameters were collected <strong>for</strong> the n-GaAs Schottkydiode discussed in Fig. 1 at temperatures ranging from320 K to 420 K. i The frequency range used <strong>for</strong> the experimentwas 1 Hz to 65 kHz. Data were collected frequency byfrequency using the long channel integration feature of theFRA, which completed a measurement at each frequencyon reaching a 1% closure error. The wide range of temperatureand frequency ensured a wide range of impedance values.Five replicate experiments were conducted at eachtemperature. The results of the experiments at 320 and400 K are shown in Fig. i. As a first approximation it wasassumed that the system was stationary, and the stochasticcomponent of the error e=toc ~ and its standard deviation, crwere calculated <strong>for</strong> data sets <strong>for</strong> the GaAs sample at 320,340, 360, 380, and 400 K, as described in the section onStationary systems,Regression procedure.-- Equation 14 was regressed to thestandard deviation values to obtain the values of cr ~, and~. The data were detrended to ensure that the mean residualerror <strong>for</strong> the regression was equal to zero. 27 Use of noweightingor proportional weighting in the regression gavepoor results. Following Eq. i, the regression was weightedby the estimated variance. The standard deviation of thestandard deviation <strong>for</strong> impedance measurements was obtainedby considering the standard deviation of the realand imaginary parts of the impedance to be independentobservations of the standard deviation at a given frequency.
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