Download Advanced Lab Course Impedance Spectroscopy M208 Content

Advanced Lab Course
Impedance Spectroscopy
M208
As of: 2015-04-01
Aim:
In situ characterization of electrochemical systems
Content
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INTRODUCTION
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BASIC CONCEPTS
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2.1 What is impedance
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2.2 FFT-Impedance Measurements
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2.3 Impedance of electrochemical systems
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EXPERIMENTS
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3.1 Measurements of a dummy cell
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3.2 Electrochemical oxidation of silicon
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HINTS FOR THE REPORT
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M208: Impedance Spectroscopy
1 Introduction
Impedance spectroscopy is a standard technique to characterize electrical systems. In this work a
special, more sophisticated kind of impedance spectroscopy will be used: The FFTimpedance
spectroscopy. This technique is useful in those cases where the speed is of importance, i.e. where the
system is non-stationary, though we will apply it to static systems here as well.
2 Basic Concepts
2.1 What is impedance
The impedance of an electrical system may be simply seen as its ac resistance. As in the dc case,
where the resistance R is defined as U/I, the impedance Z is also the ratio of the voltage U and the
current I, but in this case U and I are functions of time with frequency ω, for example:
U (t) = Vm sin(ωt)
and
I (t) = Im sin(ωt +ϕ).
In these equations Vm and Im are the amplitudes of the voltage and the current signals, respectively, ω
is the frequency, and ϕ is the phase shift between current and voltage. A phase shift of ϕ = 0 is only
found in purely ohmic systems. A quantity (F) changing with a sin (or cos) function can be treated as
a complex number, which can be described with the Euler equation:
F = cos(α) + i sin(α) = exp(iα).
Consequently the real (Re) and the imaginary (Im) parts of F are:
Re(F) = cos(α) and
Im(F) = sin(α).
And the amplitude of F is:
In this way, the impedance can be expressed as:
Z is independent of ω only for purely ohmic resistors. On the other hand, the impedance of a
capacitor is given by:
and the one of an inductor is
For the capacitor the phase shift between current and voltage is π/2 (and thus constant). This shows
the capacitors ability to store energy which is the reason why the response (the current) follows the
signal (voltage) retarded in time. Z is a complex quantity and can be represented in the complex
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M208: Impedance Spectroscopy
plane by plotting the imaginary part over the real part with ω as a parameter. Such a plot is called a
Nyquist plot; an example for an ideal capacitor is shown in Fig. 1.
Fig. 1
A plot of this impedance over frequency is shown in Fig. 2. This kind of plot (a variable vs
frequency) is called Bode plot.
Fig. 2.
The real part of the capacitor’s impedance is 0. The imaginary part becomes more negative for lower
frequencies and would increase to -infinity for ω→ 0; for ω →∞ it is zero. If the circuit consists of
more elements than a simple capacitor, the phase ϕ, in general, becomes frequency dependent as
well.
Let us now consider a slightly more complicated circuit (Fig. 3):
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M208: Impedance Spectroscopy
This circuit can be expressed by the following equation:
We see that for the case ω→∞ Z → RSer and for ω→ 0 Z → RSer + RPar . These are the limiting points
on the real axis for the Nyquist plot of the impedance of the circuit of Fig. 3, as can be observed in
Fig. 4. This can be explained as follows: For very low frequencies the capacitor is practically an
isolator and no current flows through it; in this way the impedance is equal to the sum of RSer + RPar.
For very high frequencies it becomes a good conductor and the resistor RPar is short-circuited, thus
only RSer contributes to the total impedance.
Fig. 4
As discussed, it is possible to obtain the values of the resistors of the circuit simply by analyzing the
Nyquist plot. In the same way it is possible to obtain the value of the capacitance. The present
Nyquist plot has a circular shape. The formula for a circle is A2 + B2 = r2, then the expression above
can now be written as
Re and Im are real and imaginary part, respectively, and
part equals the radius of the semicircle. Then we obtain:
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RPar
is the radius. The maximum imaginary
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M208: Impedance Spectroscopy
and on doing some arithmetic on this term:
ωmax is the frequency belonging to the maximum value of the imaginary part.
But this easy way of data evaluation is not possible if only a part of the semi circle can be measured.
This may be due to frequency limitation or due to noise. If this is the case, a fitting routine has to be
applied. "Bad" frequencies (i.e. data points for frequencies strongly deviating from the semicircle)
should be deleted before performing the fit.
2.2 FFT-Impedance Measurements
Since the impedance is the ratio of the voltage and the current, it could be obtained in principle by
simply dividing these two quantities. But there are many reasons in practice why this may yield
wrong results. If, for example, the U-I-characteristic of the measured system is not linear, the
impedance changes with the applied voltage. This is the case of a diode (Fig.5), which is a good
representative of electrochemical systems.
Fig. 5
From Fig. 5 tan α is the direct ratio between U and I and thus represents a resistance. However, the
relevant resistance for an ac measurement is dU/dI and it is clearly visible that the ratio of U1 to I1 for
the respective point on the characteristic yields another value than the slope on the characteristic
itself. That is why in this experiment small signals are used, i.e. a constant voltage is applied and an
ac-voltage signal with small amplitude is superimposed. The corresponding differential change in the
current is measured and the differential resistance or impedance is obtainable. To use small signals is
important to fulfill the condition of linearity, meaning that the ratio dU/dI represents the correct
resistance on any point of the characteristic (see above). In a simple measurement to obtain an
impedance spectrum, the impedance is measured repeatedly for different frequencies until values for
all the interesting frequencies for the investigated system have been tried. This is the general way of
doing impedance spectroscopy. An advanced modification of this method is the Fast Fourier
transform (FFT) impedance spectroscopy. The advantage of this is that the FFT gives us the
possibility to measure all the frequencies at once, separating them later by the FFT-routine. The "Fast
Fourier Transform" method is a mathematical algorithm for doing a Fourier transform i.e.
transforming a function from the time-domain into the frequency-domain. (f(t)⟶F(ω)) in a
particularly quick way. The time for the measurement is theoretically one period of the lowest
frequency signal (but practically two periods are applied to avoid measuring artifacts which may
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M208: Impedance Spectroscopy
occur due to transient effects). The mathematical expression for the transformation of the current I(t)
into the frequency domain for example is:
In the same way a corresponding equation holds for the voltage and yields FU(ω). This now enables
us to calculate the impedance Z(ω) from the ratio of the Fourier transforms of current and voltage as a
function of the frequency ω.
From the measurement of I(t), the impedance response to the various frequencies contained in the
signal can be separated.
2.3 Impedance of electrochemical systems
The characteristic of a junction between two materials is dependent on the band structure of both
materials. When describing a metal-semiconductor-contact, for example, the relative position of the
Fermi energies of metal and semiconductor determines whether the contact will be ohmic or
Schottky-type. In an electrolyte solution the expression “energy band”, of course, cannot be applied
in its conventional meaning. But there are energy levels in a solution which correspond to chemical
reactions in the electrolyte. Typically there is a level belonging to the oxidizing reaction, and one
belonging to the reducing reaction; these levels are not sharp but "smeared out" because of the
thermal energies in the system. The Silicon acetic-acid-contact (which will be used in the
experiment), for example, is Schottky-type and the “band diagram” may be depicted as shown below
in Fig.6.
Fig. 6
The bent region in the semiconductor is the so-called space charge region (SCR). In this example it
contains a negative charge. The SCR also has a capacity CSCR. On applying an anodic potential an
oxidizing electrochemical reaction starts and the band bending is reduced. As soon as an oxide layer
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M208: Impedance Spectroscopy
emerges on the Si-electrode, most of the voltage drops over this layer and thus the capacitance is
governed by this contribution.
3 Experiments
3.1 Measurements of a dummy cell
First the basics of FFT-impedance measurements shall be investigated by measurements on a
“dummy cell.” Worldwide there are not so many FFT impedance spectrometers in use (as compared
to several 1000 normal spectrometers), so be careful! The dummy cell could be composed of any
arrangement of electrical elements (resistors, capacitors or inductors). The task in this experiment is
to determine which are the components of the cell. Make the Nyquist and Bode plots of the system.
3.2 Electrochemical oxidation of silicon
The measurement described above may be performed by FFT-impedance spectroscopy but, as the
dummy cell is a static system, it could also be measured by subsequent application of the single
frequencies just as well. Normally (if time is not an issue) this is the better way in such case, because
several measurements can be averaged, improving the signal to noise ratio. Electrochemical oxidation
means that an electrochemical reaction causes an oxide to grow. Investigating the oxide formation in
situ during the reaction reveals the real potential of the FFT technique. For the experiment a p-Si
sample is used. The "native" oxide, which is always present in Si in air atmosphere, has to be
removed by a HF-etch before the oxidation. The oxidation process is quite fast and in order to gain a
denser mesh of measurement points it is advisable to use the so called “batch mode” (using a preprogrammable set of parameters for all measurements) that the TRW-program provides.
With the measured data do the following analyses:
1. Plot the different parameters with time (serial resistance, parallel resistance and capacitance).
Explain the behavior of the curves.
2. Plot the oxide layer thickness versus the time. Apply the standard capacitor formula:
𝐴
𝑑
2
For SiO2 a εr = 3.9 can be assumed. The sample area is 1 cm . Explain the shape of the curve.
For anodic oxidation a continuity equation can be set up
𝐶 = 𝜀𝑟 𝜀0
β is a growth parameter, α an etch rate and JOx is the oxidizing current density. ds/dt is the
change in oxide thickness. Take two subsequent values from the beginning and two from the
end of the experiment and calculate β. A literature value for β is
α can be considered as 0 in this case. Please explain the difference of the obtain values from
the one of the literature.
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M208: Impedance Spectroscopy
3. The oxide density is 2.1 g/cm3. Considering the thickness of the oxide layer, the time and the
used current it is possible to calculate the valence of Si in the oxide. In this case the valence of
Si can be interpreted as the number of charges that are needed to build one SiOx-molecule.
Calculate the valence of Si in the oxide after 1 min of oxidation. Discuss the result.
4 Hints for the report
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Please write down all the used parameters, like current, voltage, time, temperature, electrolyte,
etc., which will be given by the advisor.
Indicate the configuration of the measuring system.
Add the plots of the original data in the report.
Do not cite web sites. To get more information about impedance spectroscopy take a look in
books or scientific articles.
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