Nonlinear Analysis: The Intermodulated Differential Immittance

Article pubs.acs.org/ac

Nonlinear Analysis: The Intermodulated Differential Immittance Spectroscopy Alberto Battistel* and Fabio La Mantia* Analytische Chemie − Zentrum für Elektrochemie, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780 Bochum, Germany S Supporting Information *

ABSTRACT: Intermodulation is used for the analysis of the nonlinear behavior of electrochemical and electronic systems. As a matter of fact, different than the passive elements, electrochemical systems have a highly nonlinear character, which can be used to obtain information on the reaction mechanism and structure of the double layer. The setup for measuring and analyzing the intermodulated sidebands is discussed in detail, using a commercial Schottky diode as the ideal system. A general intermodulated differential immitance spectroscopy technique was consequently defined as the analysis of the variation of the immittance elements as a function of the stimulus frequency, and its transfer function was called differential immittance spectrum. Through a simple model, it was possible to precisely calculate the flat band voltage and the doping level of the Schottky diode from a single differential immittance spectrum. The differential immitance spectra of a dummy cell containing passive elements demonstrated the resolution limits of the technique. at f p − fs and f p + fs. The output of the probe and the sidebands in the time domain (t-domain) represent the envelope of the amplitude modulation (AM) of the stimulus on the probe signal (i.e., the modulation of the amplitude of the probe output according to the frequency of the stimulus). This is the so-called phenomenon of the intermodulation, and it contains information coming from the stimulus, the probe, and the characteristics of the system. An important feature of the sidebands is that the intensity of these signals is in general higher than the intensity of second harmonics.1 In 1962, Neeb first applied the concept of the intermodulation to an electrochemical system,2 by using intermodulation polarography to study the nonlinear behavior of the polarization resistance of an electrochemical system. In the 70s, Rangarajan created a theoretical framework and a mathematical formalism in which to pose the second-order impedance spectroscopy.3 Rao et al. proposed to exploit the higher-order harmonics and the intermodulation signals to recover the kinetic parameters in corrosion science.4 They provided the mathematical treatment to obtain the corrosion current and the Tafel coefficients. A more detailed treatment was proposed by Devay and Meszaros.5 They suggested that the measurement of the intermodulation components is more advantageous than that of the harmonics components because it is not affected by instrumental distortions. Bosch et al. applied the intermodulation technique and the theoretical framework proposed by

I

f a linear system is perturbed by a single sinusoidal input, the output will be composed by a single sine wave with the same frequency of the input, scaled in modulus, and shifted in phase. In the frequency domain (F-domain) this can be visualized by both the input and output Fourier transform containing only one peak at the frequency of the input signal (Figure 1, panels a and b). If the system is nonlinear, as in electrochemical systems, the output Fourier transform shows additional peaks, corresponding to the higher-order harmonics (Figure 1c). The higher-order harmonics are intrinsically connected with the nonlinearity of the system; the intensity and the order depend on the nonlinearity, which can be approximated by higher order polynomials of the Taylor series. Eventually, the intensity of the higher order harmonics will be buried in the noise level. To avoid higher-order harmonics, a small input signal is used, as is done in electrochemical impedance spectroscopy (EIS). A spectrum is then obtained by spanning a large range of frequencies of the input signal. The intermodulation consists in perturbing the system with an input composed by two sine waves of different frequencies (Figure 1d), which will be called for simplicity stimulus (lower frequency, fs) and probe (higher frequency, f p). Under such conditions, the output of a linear system will be composed by two sine waves with the same frequencies, fs and f p, scaled in modulus and shifted in phase (Figure 1e), as for a single sinusoidal perturbation. However, for the nonlinear systems, the situation is considerably different: in addition to the higherorder harmonics of the stimulus and probe, symmetric peaks around the probe frequency are observed (see Figure 1f), which are called sidebands. The sidebands are generated by the second order term of the system’s Taylor series and are located © 2013 American Chemical Society

Received: March 27, 2013 Accepted: June 10, 2013 Published: June 10, 2013 6799

dx.doi.org/10.1021/ac400907q | Anal. Chem. 2013, 85, 6799−6805

Analytical Chemistry

Article

Figure 1. Schematic of the Fourier transform for a single (first row) and double input measurement (second row) on a linear (second column) and nonlinear system (third column). Δ represents the fundamental harmonic, □ the higher-order harmonics, and ○ the intermodulation sidebands.

sine waves: the probe signal at angular frequency Ω and the stimulus signal at angular frequency ω. Following Figure 1f and removing the higher-order harmonics of the probe and stimulus, but keeping the sidebands, one can describe the output as composed by four sine waves: the probe response at frequency f p, the stimulus response at frequency fs, and the two sidebands generated by the intermodulation at frequencies f p − fs and f p + fs. If the input is the potential and the output is the current, the admittance at the angular frequencies Ω and ω can be calculated at once from

Devay, Meszaros, and Rao on a real corrosion system.1,6 They implemented the treatment, adding some considerations about the error arising from the uncompensated resistance and the double layer capacitance. So far the study of nonlinearity was restricted to the faradic reaction of the electrochemical system. Antaño-Lopez and co-workers were the first to apply the intermodulation technique to the study of the double layer.7 They implemented an original setup to perform the experiment and proposed a new name: the modulation of interface capacitance transfer function (MICTF) technique. The key point of this setup was to employ a lock-in amplifier to demodulate the intermodulation sidebands. Their idea was to prove that to consider the double layer capacitance as independent of the potential was an oversimplification. They applied their model to the study of a system with ion transfer and to the study of the interface between TiO2 and SnO2 in a dye-sensitized nanocrystalline solar cell (DSSC).8 Starting from the treatment of Antaño-Lopez and coworkers,7 this work is focused on the development of a technique for the analysis of the nonlinear behavior of electrochemical systems, using a diode as a relatively ideal nonlinear system; the diode is stable, reproducible, and the intermodulation can be predicted. In fact, the diode is equivalent to a conductance in parallel with a capacitance, which is dependent on the polarization potential. Additionally, a diode represents a fairly challenging benchmark for electrochemical instrumentation, due to its high impedance. Based on the intermodulation sidebands, we defined a general transfer function called differential immittance spectrum. Results from a dummy cell containing only passive elements (linear system) were used to validate the results and to show the resolution limit of our instrumentation.

Y(Ω) =

I(Ω) U(Ω)

(1)

Y(ω) =

I(ω) U(ω)

(2)

where I and U represent the Fourier transforms of the current and potential at the angular frequencies Ω or ω. We want to stress that the conductance, G, is equal to Re(Y), and the susceptance, B, is equal to Im(Y). From eqs 1 and 2, the differential conductance, dG, differential susceptance, dB, and differential admittance, dY, can be defined in a general sense as dG(ω , Ω) =

GΩ(ω) U(ω)

(3)

dB(ω , Ω) =

BΩ(ω) U(ω)

(4)

dY (ω , Ω) = dG(ω , Ω) + jdB(ω , Ω)

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(5)

where GΩ, BΩ, and YΩ are the Fourier transforms of the probe conductance, susceptance, and admittance at the stimulus angular frequency ω, respectively, and j is the imaginary unit. It has to be stressed that Re(dY) is equal to Re(dG) − Im(dB), and Im(dY) is equal to Im(dG) + Re(dB). The conductance

THEORETICAL BACKGROUND In the intermodulated differential immittance spectroscopy (IDIS), the system is perturbed by an input that contains two 6800



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and susceptance can be measured in the time domain by the lock-in amplifier, as done by Antaño-Lopez and co-workers.7 However, their Fourier transform at ω can be calculated directly from the sidebands rising from the intermodulation (see appendix A of the Supporting Information). The general term intermodulated differential immittance spectroscopy was used to describe both differential admittance and differential impedance. In appendix B of the Supporting Information, the treatment for the intermodulated differential impedance spectroscopy is reported. The differential conductance and the differential susceptance both respond to the Kramer− Kronig relations. The equivalent circuit of a diode in reverse bias is represented by a capacitance in parallel to a conductance. The capacitance takes into account the accumulation of charge in the space charge region, and its dependence on the polarization potential is given by the Mott−Schottky equation.9 The parallel conductance represents the leakage current due to thermo-emission of electrons and movement of the holes in the valence band; the conductance is weakly dependent on the polarization potential. The stimulus affects the potential across the diode, and as consequence, the value of the capacitance of the diode, which is the origin of the amplitude modulation of the probe current at the stimulus frequency. The Fourier transform of the current is very similar to that depicted by Figure 1f. Following eqs 3, 4, and 5 and the equivalent circuit of the Schottky diode, dG should be negative and imaginary, and dB should be negative and real (see appendix C of the Supporting Information). Both terms are correlated to the variation of the capacitance with the potential; however, the value of dG is proportional to ω, while the value of dB is proportional to Ω. From the Mott−Schottky analysis, it is possible to derive the flat band voltage and, knowing the relative permittivity εr, the dopant concentration of the semiconductor in the diode. The same information is obtained by the intermodulated differential admittance spectroscopy. We want to stress that Mott−Schottky analysis is done acquiring several EIS at different potential values and plotting the reciprocal of the square of the calculated capacitance against the potential, while the differential admittance spectrum is obtained at a single potential.

Figure 2. Schematic of the instrument setup. Generator 1 outputs the stimulus signal and generator 2 the probe signal. The potentiostat sends the potential output and the current output to the first two channels of the oscilloscope and current output to the input of the lock-in amplifier. The lock-in demodulates the current and sends the in-phase and out-of-phase components to the second two channels of the oscilloscope.

remaining two channels of the oscilloscope. The immittance and the differential immittance are obtained by the PC, using homemade Matlab-based software and calculating the Fourier transform of the signals recorded by the four channels of the oscilloscope. Investigated Systems. Two systems were investigated: a Schottky diode 80SQ040 (International Rectifier), as an ideal nonlinear system and a dummy cell composed by a 9.4 MΩ resistor in parallel with a 2 nF capacitor, as an ideal linear system. Cyclic voltammetry and electrochemical impedance spectroscopy (EIS) were performed on the Schottky diode using a Zahner Zennium (Zahner) potentiostat between 0 and 2 V. A scan rate of 10 mV s−1 was used for the cyclic voltammetry. The impedance spectra were measured between 100 kHz and 100 mHz, with 10 points per decade, using a 10 mV amplitude voltage sine wave, and an impedance spectrum was recorded each 100 mV. The diode was connected to the potentiostat using the IUPAC official setup, with the cathode attached to the working electrode and the anode to the reference and counter electrodes. For the measurement of the differential immittance, the probe frequency was kept constant at 1 kHz, and the stimulus frequency was scanned between 100 Hz and 100 mHz, at 10 points per decade. The amplitude of the probe and the stimulus were 20 and 40 mV, respectively. These are optimized values that provide good signal-to-noise ratios. Ten cycles of the stimulus signal were recorded with 20 points per period of the probe signal. A polarization voltage of 0.5 V was chosen for both systems.

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EXPERIMENTAL SECTION The Instrument. The instrument was composed by a potentiostat PG_310USB (HEKA Elektronik), a 2-channel lock-in amplifier HF2LI (Zurich Instruments), a 4-channel oscilloscope PicoScope 4424 (Pico Technology), two sine wave generators (included in the lock-in amplifier), and a personal computer equipped with Matlab. The main characteristic of the potentiostat is a wide bandwidth associated with a low noise level, so that only small distortions are introduced at high frequencies. Figure 2 shows a schematic of the instrument. Generator 1 provides the sine wave of the stimulus at frequency fs and generator 2 the sine wave of the probe at frequency f p (f p > fs); the latter was also used as a reference for the lock-in amplifier. The signals produced by the generators were summed and sent to the potentiostat, which was connected to the investigated system. The current and potential outputs of the potentiostat were sent to the first two channels of the oscilloscope; the current output was sent also to the lock-in amplifier, where it was demodulated according to the reference signal (the probe signal). The in-phase and out-of-phase components of the current were amplified and sent to the

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RESULTS AND DISCUSSION Cyclic Voltammetry and EIS on the Diode. Figure 3a shows the cyclic voltammogramm of the diode at a scan rate of 10 mV s−1, performed between 0 and 2 V. This voltage window corresponds to the inverse region of the diode; its flat band voltage is a located at −0.53 V. The leakage current is equal to ca. 400 nA at a 0.5 V polarization voltage. In the same voltage window an impedance spectrum was recorded each 100 mV. Calculating the parallel capacitance from the imaginary part of the admittance at 100 kHz for each potential, it was possible to use the Mott−Schottky analysis to obtain the flat band voltage 6801



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and a stimulus frequency scanning from 100 Hz to 100 mHz, with 10 points per decade. Ten cycles of the stimulus signal and 20 points per period of the probe signal were recorded with two channels of the oscilloscope. The Fourier transform was performed on the potential at angular frequencies Ω and ω and on the current at angular frequencies Ω, ω, Ω − ω, and Ω + ω, using a Blackman−Harris window function (see appendix D of the Supporting Information). The result of the Fourier transform of the current in the whole range of frequencies is reported in Figure 4b for the stimulus frequency of 10 Hz. The

Figure 3. (a) Cyclic voltammetry of the diode between 0 and 2 V at 10 mV s−1. (b) Mott−Schottky plot of the capacitance of the diode measured at 100 kHz and linear regression.

and the dopant concentration. The Mott−Schottky analysis is based on the following equation (for n-type semiconductors): ⎛ k T⎞ 1 2 ⎜UE − Ufb − B ⎟ = −2 εrε0|e|ND ⎝ | e| ⎠ CSC

(6)

where CSC is the capacitance of the Schottky diode, εr the relative dielectric constant, ε0 the permittivity of vacuum, e the charge of the electron, ND the concentration of dopants, UE the polarization voltage, Ufb the flat band voltage, kB the Boltzmann constant, T the absolute temperature. In Figure 3b, the Mott− Schottky plot for the diode is reported. From the linear regression, a flat band potential of −0.536 ± 0.005 V and a dopant concentration of 3.61 × 1017 ± 1015 cm−3 were calculated (εr = 11.68). The same results were achieved using lower frequencies (down to 1 kHz) and restricting the potential range from 0.2 to 2 V. Intermodulated Differential Immittance Spectroscopy (IDIS) on the Schottky Diode. For measuring the differential immitance spectra, two configurations are possible: the current and potential output of the potentiostat are connected to a 2-channel oscilloscope and the differential immittance is calculated from the Fourier transform of the current at Ω − ω and Ω + ω (oscilloscope setup) or the current output is demodulated at angular frequency Ω by a lock-in amplifier and the in-phase and out-of-phase components are recorded by an oscilloscope (lock-in setup). In the latter case, a 4-channel oscilloscope is necessary: potential, current, in-phase component, and out-of-phase component have to be recorded as a function of time, and the differential immittance is calculated from the Fourier transform of the in-phase and outof-phase components at ω. In the next paragraph, we will discuss the oscilloscope setup and its limitations. The differential immittance spectra were measured at a polarization voltage of 0.5 V, using a probe frequency of 1 kHz

Figure 4. (a) Nyquist plot of the EIS of the diode performed with a commercial instrument and that recorded with the oscilloscope setup. (b) Fourier transform of the current of the diode with a stimulus frequency of 10 Hz and a probe frequency of 1 kHz.

sidebands are located at 990 and 1010 Hz, as expected for the intermodulation effect. The impedance spectrum at 0.5 V can be calculated from the Fourier transform of the potential and current at ω. This is reported in the Nyquist plot of Figure 4a, together with the impedance measured previously at 0.5 V with the Zahner Zennium. The two curves are very close, thus indicating the good quality of data. The resistance measured by the IDIS is smaller because of the higher amplitude of the stimulus oscillation. By fitting the impedance spectrum with a capacitance parallel to a resistance, the values of 1.76 nF and 7.74 MΩ are obtained. The good quality of the fitting also confirms that the data respond to the Kramer−Kronig relations. The measurement of IDIS can be affected by the control loop of the potentiostat and bandwidth of the current follower. First, the control loop of the potentiostat can introduce a delay and an attenuation of the applied sine wave potential at high frequencies (the probe frequency) with respect to the 6802



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generated one; this error is easily compensated because the real applied potential is measured. The bandwidth of the current follower also introduces a delay and an attenuation of the measured current with respect to the real current flowing through the system. The correction of this distortion requires the knowledge of the transfer function of the current followers. In general, the lower is the current range (the higher is the current amplification), the lower is the bandwidth of the current follower. In this work, rather low current ranges (10 μA) had to be used to enhance the signal of the sidebands. The transfer function of the current amplifier is obtained by means of a calibrated resistor and measuring its impedance in the frequency range from 100 kHz to 100 mHz. The normalized admittance represents the transfer function of the current follower. In Figure 5a, the differential conductance, dG, and the differential susceptance, dB, are reported in their real and

dB = −Ω

1 2

−3/2 εrε0|e|ND ⎛ k T⎞ ⎜UE − Ufb − B ⎟ ⎝ | e| ⎠ 2

(7)

More details on eq 7 can be found in appendix C of the Supporting Information. Equation 7 can be used together with eq 6 for calculating the flat band voltage and the dopant level, with results equal to −0.535 ± 0.01 V and 3.92 × 1017 ± 8 × 1015 cm−3, respectively. Table 1 presents a summary of the results obtained with the different techniques compared with the tabulated data. They all show to be in good agreement. Table 1. Flat Band Potential, Ufb, and Dopant Concentration, ND, of the Diode Calculated with the Differential Admittance Measured by the Oscilloscope Setup and the Lock-in Setup, with the Mott-Schottky (M-S) Analysis, and Reported in the Datasheet oscilloscope setup lock-in setup M-S analysis tabulated data

Ufb (V)

ND (1017 cm−3)

−0.535 ± 0.01 −0.527 ± 0.006 −0.536 ± 0.005 −0.53

3.92 ± 0.08 3.88 ± 0.01 3.61 ± 0.01 not reported

The advantage of using the oscilloscope setup is of course a reduction of the costs and connections required. However, care has to be taken that high resolution in the F-domain is obtained, which is achieved by long time recording; this is necessary to visualize precisely the sidebands and separate them from the sidelobes of the probe frequency, as explained in more detail in appendix D of the Supporting Information. For this reason, up to 10 cycles of the stimulus signal are acquired. Moreover, to avoid the high frequency noise, 20 points per period of the probe signal are recorded. The restriction in the ratio Ω/ω rises from the fact that the number of points recorded at each stimulus frequency is equal to some 200 Ω/ω, and that due to limitations in the calculation power of the PC, files larger than 10 million points are difficult to handle. If lower frequencies have to be reached, the lock-in amplifier in combination with four low-pass filters has to be used to demodulate the current with respect to the probe frequency. Moreover, as will be shown below, the lock-in amplifier has a slightly higher resolution and better signal-to-noise ratio, with respect to the oscilloscope and could be necessary for extremely low values of the differential immittance. Use of the Lock-in Amplifier for Measuring the IDIS. A lock-in amplifier can be used to measure directly the conductance and the susceptance of the system at the probe frequency. Under these conditions, additionally to the current and potential at fs, the measured conductance and susceptance can also be recorded by a 4-channels oscilloscope (lock-in setup). After the acquisition, the differential conductance and susceptance can be calculated from the Fourier transform of the conductance and susceptance at angular frequency, ω, respectively, as explained in the Theoretical Background by eqs 3−5. However, a lock-in amplifier may be a source of distortion in the measured differential immittance: first, the phase of the demodulator of the current has to be adjusted to avoid delays between the real current flowing through the system and the measured current; moreover, the lock-in amplifier tries to cut off the intermodulation, as it considers it as noise. The advantage of using the lock-in amplifier is that the conductance and the susceptance are directly measured and it allows exploring a larger range of stimulus frequencies because

Figure 5. (a) Bode plot of the real (empty symbols) and imaginary part (solid symbols) of the differential conductance (● and ○) and of the differential susceptance (△ and ▲). (b) Bode plot of the real (○) and imaginary part (●) of the differential admittance.

imaginary part, as a function of the stimulus frequency. As expected (see appendix C of the Supporting Information), the value of dG is imaginary, negative, and increases with increasing fs, while the value of dB is real, negative, and constant with fs. In Figure 5b, the differential admittance, dY, is reported as a function of the stimulus frequency. dY is composed only by the imaginary part, while the real part remains mostly near 0. Im(dY) increases at higher stimulus frequencies, in accordance with the value of Im(dG) becoming larger. The differential susceptance for the diode is given by: 6803



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high frequency signals do not need to be recorded. The setup in this paper consists of a 2-channel lock-in amplifier; before starting the experiment, the software automatically forces the lock-in to detect the phase shift between the generated and the applied probe signal (the delay introduced by the control loop of the potentiostat) and then adds it to the shift in phase of the current follower (measured as in paragraph 4.2) and sets it as the phase of the demodulator. In this way, the distortion between the measured and the real current flowing through the system is removed, apart from a proportional factor that can be calculated later from the transfer function of the potentiostat. A lock-in amplifier is designed to demodulate the input signal with respect to a reference signal, which can be external or internal. The demodulation consists in measuring the correlation between the input and reference signal (in a mathematical sense). The correlation can be then used to calculate the in-phase and out-of-phase components of the input signal with respect to the reference signal. After the demodulation, the in-phase and out-of-phase components pass through a low-pass filter and an amplifier. The series of the demodulation and low-pass filter is equivalent to a band-pass filter. Because of the low-pass filter, the lock-in amplifier introduces a distortion to the differential immittance through its own transfer function. The low-pass filter has a bandwidth that is given by the time constant of the lock-in amplifier and the order of the filter: a long time constant and a high-order filter attenuate all the signals away from the reference, including the intermodulation effect and a short time constant and a low order filter allow demodulating the sidebands properly at expenses of high noise level. This effect generates a limitation on the higher stimulus frequency that can be measured: in the proposed setup, with a fourth-order filter (24 db/octave), the upper limit in fs is equal to 0.1fp. If we name Δω the bandwidth of the lock-in in rad s−1, the relevant parameter that controls the transfer function of the lock-in amplifier with respect to the stimulus frequency is given by ω/Δω. The relation between the time constant, tC, and the bandwidth is

Δω =

A tC

Figure 6. Bode plots of the lock-in amplifier transfer function H, experimental data (○), and fourth-order inverse polynomial fit (line): (a) absolute value and (b) phase-shift of H.

oscilloscope setup, with the Mott−Schottky analysis and reported in the database. The Ideal Linear System: The Dummy Cell. A dummy cell composed by passive elements is the ideal linear system. A dummy cell consisting of a 9.4 MΩ resistor in parallel to a 2 nF capacitor was built and tested. The differential admittance was measured at 0.5 V potential using a probe signal of 1 kHz, having amplitude of 20 mV and a stimulus signal ranging from 100 Hz to 100 mHz, with 40 mV of amplitude. The measurement was performed with both oscilloscope setup and lock-in setup. In Figure 7a, the Fourier transform of the current signal in a large range of frequencies is reported for fs equal to 10 Hz. For comparison, the same result for the Schottky diode is reported. It can be immediately observed that, while for the Schottky diode the intermodulation sidebands are clearly visible, for the dummy cell, the intermodulation sidebands are completely buried under the noise level. The differential admittance of the dummy cell is a measure of the noise level of the device. In Figure 7b, the noise level of the IDIS for the oscilloscope setup and for the lock-in setup, measured though the differential admittance of the dummy cell, is shown. We want to stress that the noise level is very low and less than 1% of the measured value in the diode. Also, it can be observed that in the range between 1 and 10 Hz, the lock-in setup works better than the oscilloscope setup. Under these conditions, the limit of detectability of the differential admittance is equal to ca. 20 nS V−1.

(8)

where A is a constant that depends on the low-pass filter order. The transfer function of the lock-in, H(ω/Δω), was obtained from the ratio of the differential admittance measured from the lock-in setup and the one measured by the oscilloscope setup for the Schottky diode, recorded for different values of Δω and ω/Δω. In Figure 6, the measured transfer function of the lockin amplifier is reported in the Bode representation. The experimental data (empty dots) were fitted with a fourth-order inverse polynomial (line), which was thereafter used for the proper correction. For ω/Δω = 1, the attenuation is 3 db, and the phase-shift is ca. 90 degrees. For ω/Δω > 1, H attenuates and strongly delays the intermodulation signal; however, larger frequencies are measurable. For ω/Δω < 1, H tends to unity and the phase-shift tends to 0 degrees, but frequencies near f p are not accessible. A good compromise was obtained with ω/ Δω = 0.2. The homemade software automatically sets Δω = 5ω each time the stimulus frequency is changed. In this way, it is possible to measure dG and dB with higher precision than by using the oscilloscope setup. In Table 1, the analysis of the differential immitance of the diode measured with the lock-in setup, maintaining Δω = 5ω is reported and is in very good agreement with the previous results obtained with the

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CONCLUSIONS The intermodulation effect can be used to study the nonlinearity of electrochemical and electronic systems. To measure it, it is possible to proceed with a simplified approach, using a 2-channel oscilloscope to record the current and the 6804



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in surface-confined species, the corrosion behavior of metals, and in other fields.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information contains four appendixes: A, correlation between differential admittance and sidebands; B, correlation between differential impedance and differential admittance; C, differential admittance of the Schottky diode; and D, the window function and the sidelobes. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*A.B.: e-mail, [email protected]. F.L.M.: e-mail, fabio. [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support by the Federal Ministry of Education and Research (BMBF) in the framework of the project “Energiespeicher” (Grant FKZ 03EK3005) and the funding of the Centre for Electrochemical Sciences (CES) by the European Commission and the state North Rhine-Westphalia (NRW) in the framework of the HighTech.NRW program are gratefully acknowledged.

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Figure 7. (a) Fourier transform of the current of the diode (line) and of the dummy cell (○) with a stimulus frequency of 10 Hz and a probe frequency of 1 kHz. (b) Bode plot of the absolute value of the differential admittance measured by the oscilloscope setup (○) and the lock-in setup (line).

REFERENCES

(1) Bosch, R. W.; Bogaerts, W. F. Corrosion 1996, 52 (3), 204−212. (2) Neeb, R. Naturwissenschaften 1962, 49 (19), 447−447. (3) (a) Rangarajan, S. K. J. Electroanal. Chem. Interfacial Electrochem. 1974, 56 (1), 1−25. (b) Rangarajan, S. K. J. Electroanal. Chem. Interfacial Electrochem. 1974, 56 (1), 27−53. (c) Rangarajan, S. K. J. Electroanal. Chem. Interfacial Electrochem. 1974, 55 (3), 337−361. (d) Rangarajan, S. K. J. Electroanal. Chem. Interfacial Electrochem. 1974, 55 (3), 363−374. (4) Rao, P. G.; Mishra, A. K. J. Electroanal. Chem. Interfacial Electrochem. 1977, 77 (1), 121−125. (5) Devay, J.; Meszaros, L. Acta Chim. Acad. Sci. Hung. 1980, 104, 311. (6) Bosch, R. W.; Hubrecht, J.; Bogaerts, W. F.; Syrett, B. C. Corrosion 2001, 57 (1), 60−70. (7) (a) Antaño-Lopez, R.; Keddam, M.; Takenouti, H. Electrochim. Acta 2001, 46 (24−25), 3611−3617. (b) Antaño-Lopez, R.; Keddam, M.; Takenouti, H. Corros. Eng., Sci. Technol. 2004, 39 (1), 59−64. (c) Larios-Durán, E. R.; Antaño-López, R.; Keddam, M.; Meas, Y.; Takenouti, H.; Vivier, V. Electrochim. Acta 2010, 55 (21), 6292−6298. (8) Cachet, H.; Keddam, M.; Takenouti, H.; Antaňo-Lopez, R.; Stergiopoulos, T.; Falaras, P. Electrochim. Acta 2004, 49 (16), 2541− 2549. (9) Dewald, J. F. J. Phys. Chem. Solids 1960, 14 (0), 155−161.

potential in the system (oscilloscope setup), or measure directly the conductance and susceptance of the system by demodulating the current with a lock-in amplifier and recording the signals with a 4-channel oscilloscope (lock-in setup). Both configurations have advantages and disadvantages. The oscilloscope setup is less complicated, but a limitation rises in the maximum value of Ω/ω that can be explored, mainly due to the limited calculation power of the personal computer. The lock-in setup has the advantage to make lower stimulus frequencies accessible; however, it has a limited higher stimulus frequency ( fs < 0.1f p) due to the correlation between the time constant of the lock-in amplifier and noise rejection. Moreover, the transfer function of the lock-in amplifier, H, has to be measured. We proposed to measure H by comparison of the differential immittances of a Schottky diode obtained by the lock-in setup and the oscilloscope setup in the range of stimulus frequencies that are available for both configurations. The differential immittance of a diode (an ideal nonlinear system) was measured. From the data, it was possible to obtain the flat band voltage and the dopant concentrations, which showed good agreement with the values tabulated and obtained with classical techniques (such as the Mott−Schottky analysis). Measuring the differential admittance on a dummy cell composed by passive elements, it was possible to quantify the detectability limit of the setups; the error on the values of dY obtained for the diode was estimated to be less than 1%. We can foresee the importance of such a technique to study the reaction mechanism of electro-catalytic reactions, the transport and trapping of carriers in semiconductors, the electron transfer 6805


Nonlinear Analysis: The Intermodulated Differential Immittance

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