- Intermodulated differential immittance spectroscopy (IDIS) is a nonlinear analysis technique that uses two input frequencies (a probe and stimulus signal) to perturb an electrochemical system. - For a nonlinear system, the output contains not only the probe and stimulus frequencies but also sidebands located at the sum and difference of the frequencies, due to intermodulation. - The technique defines a transfer function called the differential immittance spectrum, which can be calculated from the sideband amplitudes. This provides information about the system's nonlinearity. - Testing on a Schottky diode showed that the differential immittance spectrum could accurately determine the diode's flat band voltage and doping level from a single measurement.

1. 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, electro-
chemical 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.
If 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, fp). Under such
conditions, the output of a linear system will be composed by
two sine waves with the same frequencies, fs and fp, 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 higher-
order 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
at fp − fs and fp + 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 intermodu-
lation to an electrochemical system,2
by using intermodulation
polarography to study the nonlinear behavior of the polar-
ization 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 intermodu-
lation technique and the theoretical framework proposed by
Received: March 27, 2013
Accepted: June 10, 2013
Published: June 10, 2013
Article
pubs.acs.org/ac
© 2013 American Chemical Society 6799 dx.doi.org/10.1021/ac400907q | Anal. Chem. 2013, 85, 6799−6805

2. 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 co-
workers,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.
■ THEORETICAL BACKGROUND
In the intermodulated differential immittance spectroscopy
(IDIS), the system is perturbed by an input that contains two
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 fp, the stimulus response at frequency fs, and the two
sidebands generated by the intermodulation at frequencies fp −
fs and fp + 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
Ω =
Ω
Ω
Y( )
I( )
U( ) (1)
ω
ω
ω
=
Y( )
I( )
U( ) (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
ω
ω
ω
Ω = Ω
G
d ( , )
G ( )
U( ) (3)
ω
ω
ω
Ω = Ω
B
d ( , )
B ( )
U( ) (4)
ω ω ω
Ω = Ω + Ω
Y G B
d ( , ) d ( , ) jd ( , ) (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
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.
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3. 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.
■ 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 fp (fp
> 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
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.
■ 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
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.
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4. and the dopant concentration. The Mott−Schottky analysis is
based on the following equation (for n-type semiconductors):
ε ε
=
| |
− −
| |
−
⎛
⎝
⎜
⎞
⎠
⎟
C N
U U
k T
1 2
e e
SC
2
r 0 D
E fb
B
(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 Spectrosco-
py (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 out-
of-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
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
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
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.
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.
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5. 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
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:
ε ε
= −Ω
| |
− −
| |
−
⎛
⎝
⎜
⎞
⎠
⎟
B
N
U U
k T
d
1
2
e
2 e
r 0 D
E fb
B
3/2
(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.
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.
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
Ufb (V) ND (1017
cm−3
)
oscilloscope setup −0.535 ± 0.01 3.92 ± 0.08
lock-in setup −0.527 ± 0.006 3.88 ± 0.01
M-S analysis −0.536 ± 0.005 3.61 ± 0.01
tabulated data −0.53 not reported
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6. 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
ω
Δ =
A
tC (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 lock-
in 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 fp
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
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
.
■ 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
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.
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7. potential in the system (oscilloscope setup), or measure directly
the conductance and susceptance of the system by demodulat-
ing 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 < 0.1fp) 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
in surface-confined species, the corrosion behavior of metals,
and in other fields.
■ ASSOCIATED CONTENT
*
S Supporting Information
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.
■ AUTHOR INFORMATION
Corresponding Author
*A.B.: e-mail, alberto.battistel@rub.de. F.L.M.: e-mail, fabio.
lamantia@rub.de.
Notes
The authors declare no competing financial interest.
■ ACKNOWLEDGMENTS
The financial support by the Federal Ministry of Education and
Research (BMBF) in the framework of the project “Energies-
peicher” (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).
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