1.4 Structure of this book
2.1.5 Universal immittance
ǫ′(ω,V
d ,P)−iǫ′′(ω,V d ,P)
(e. g. confer [78, pp. 65-66] for the linear case). This work focuses on the analysis of immit-tance data with EECs. From this perspective, representing the real-valued contributions to the impedance as resistors has several advantages:
As a first consequence, a resistor, in an EEC, in that case always represents a component that does not introduce an out-of-phase component with respect to the applied field and, consequently, a capacitor always represents a purely out-of-phase component with respect to the applied field.
As explained in sections 2.1.2.1 and 5.4.1, the frequency range relevant for immit-tance spectroscopy is a subset of the low-frequency regimeLof any conduction process. Consequently, the delayed response of conduction processes and, hence, its imaginary part can be neglected. As a result, conduction processes, in an EEC, are always represented by process-specific, parameter-dependent resistors that have purely real phasors. Representing the loss of a polarisation process, that is a con-tribution in phase with the applied field and hence a part with real phasors, as a resistor as well means that all resistors in an EEC have always real valued phasors and are in phase with the applied field.
In the eyes of the author, resistors always representing contributions in phase with the applied field and capacitors representing always an out-of-phase contribution seems more intuitive, than using the identical symbol of a capacitor for an in-phase component and a component in quadrature to the applied field.
The second consequence is that all resistors are always components of loss, while all capacitors (and inductors) are always components that store their reactive power in electric (or respectively magnetic) fields. In the end, the symbols of the components in an EEC always have the above mentioned properties and one does not have to look at the designation of the capacitor to find out weather it is a lossy component or a regular capacitance-like component.
2.1.5 Universal immittance
The quantity in square brackets in equation (2.14) is (as mentioned in section 2.1.4) onlyalmost the admittance because it is, just as all other immittances, only defined for linear systems, i. e. where the response is a linear function of the stimulus.8 The used
8Conventional immittance, as used in the context of immittance spectroscopy, is based on linear systems theory [111, 110] and its linearisation is the fundamental property of impedance spectroscopy [99]. However,
formulation is more general and explicitly includes non-linear response to the stimulus.
In situations where this distinction is important, ‘universal’ is added to distinguish the regularly used (linear) immittance from the universal immittance allowing non-linear response that is derived in this work. Hence, the quantity in the brackets in equation (2.14) is the universal admittance, i. e. the inverse of the universal impedance
Ztot(ω,V,P) =¦
Figure2.5:The non-linear current-voltage characteristic of a diode in the limit for infinitesimal frequencies, i. e. purely resistive equilibrium current response. In both cases, the periodic stimulus is a sinusoidal voltage signal oscillating around a bias voltage of zero, illustrated as vertical sinus with hidden time axis. The resulting current response is given in frequency domain and in time domain right of the current-voltage characteristics with hidden time axis in the inset. In (a) the small-signal approximation is used. Consequently, the response is the same as if the oscillation took place on the shown tangent instead of the true, non-linear current-voltage characteristic.
In (b), on the other hand, large-signal analysis is performed. Hence, the oscillation follows the curvature of the non-linear response.
The response of a linear system contains only the fundamental frequencies of the stimulus as well as a constant offset (zero-frequency coefficient) if the oscillation is biased (see Figure 2.5a where the unbiased situation is illustrated). In consequence, a linear system does not add multiples of the fundamental frequency, the so-called higher harmonics (see Figure 2.5b and section 2.1.1.2 for the definition of harmonics). As a result, a sinusoidal stimulus with fundamental frequencyω0leads to a response with exclusively frequency ω0. This makes calculus with linear systems mathematically simpler since their response
there are and were always efforts, though different from the realisation presented in this work, to include higher harmonic responses. Possibly due to the easier handling of conventional immittance, poly-harmonic analysis is still rarely applied. Consequently, any immittance is always referring to the linear form and if higher harmonics are meant it is specifically mentioned.
can be described by a single phasor, a complex quantity that holds the information of the amplitude of the response (representing the magnitude of attenuation or gain) and the phase shift as compared to the stimulus.
The response of a non-linear, periodic system is poly-harmonic, i. e. besides the fun-damental frequency higher harmonics are present, leading to a mathematically more complicated situation since multiple phasors have to be considered. Furthermore, addi-tionally to the static offset due to a biased stimulus, the zero-frequency coefficient of the response can be influenced by the asymmetry around the operation point. Since these are, for amplifiers, unwanted, but inevitable, and well-known effects they are summarised under the termharmonic distortion[166]. Unlike in the development of an amplifier, in science, the observation of harmonic distortion in the investigation of a system is neither good nor bad, since it is just another property of the system under investigation and gives information about its non-linear nature and, as such, it should be exploited although it is mathematically more difficult.
There is a distinct difference between the constant offset for linear responses and non-linear responses. In the linear case, it is just a consequence of the applied bias voltage, resulting in a steady response that is superimposed to the oscillation (as the stimulus in Figure 2.5a is oscillating around zero bias, the zero-frequency component equals zero, as can be seen in the frequency domain representation of the linearly assumed response). In the non-linear case on the other hand, asymmetry around the operation point alters the offset, in comparison to the linear case, as can be seen in the frequency domain representation of the non-linear response in Figure 2.5b.
To avoid the more complicated mathematics and more challenging experimental setups, it is quite common to linearise a non-linear response function, e. g.ξ(E,P), at a cer-tainoperation pointEO, i. e.ξE(lin)
O (P) = ∂∂ ξE E
O
. If the amplitudeEˆof the stimulus is approaching zero, e. g.lim approxima-tion of the value of the response funcapproxima-tion at the operaapproxima-tion pointEObecomes exact (this implies that the static value for the response is determined from the non-linear function).
If the amplitudes in an experiment are sufficiently small that this linear approximation, also called thesmall-signal approximation, is acceptable, the amplitude can be ignored.
This approach is calledsmall-signal analysisand currently the de-facto standard in the analysis of immittance spectroscopy (e. g. see [110, 7]). The contrary approach, that takes the value of the amplitude into account, is calledlarge-signal analysis.
In Figure 2.5 the difference between large- and small-signal analysis are illustrated at the example of the non-linear characteristic of a purely resistive diode (meaning that the frequency is almost zero). Since a purely resistive characteristic is assumed, the coefficients of the response function are all real values. The frequency of the oscillation is assumed to be sufficiently small, that the current response is in its equilibrium. The stimulus is a sinusoidal voltage oscillating around a bias voltage of zero. The finite amplitude of the stimulus reaches the points in the current-voltage characteristic indicated by the coloured contour. In Figure 2.5a small-signal analysis is performed. Hence, instead of following the
correct, non-linear current-voltage characteristic as marked in Figure 2.5b by the coloured contour, the response is determined by the resistance at the point of the bias voltage only, not taking the amplitude of the signal into account. This corresponds to assuming a linear relation, thus, as illustrated in Figure 2.5a, a tangent in the current-voltage characteristic.
As a result, the oscillating voltage does not follow the actually non-linear current-voltage-curve, but instead oscillates on the indicated tangent. As can be seen in the illustrated response, especially in its frequency domain representation, the current response has exclusively the same frequency as the voltage stimulus. From the increasing gap between the actual current-voltage characteristic and the tangent, the deteriorating accuracy of the approximation for increasing oscillation amplitude can be seen directly. In Figure 2.5b large-signal analysis is performed. In this case, the oscillation follows the non-linear current-voltage characteristic instead of the tangent. The difference is immediately visible from the form in the given current response in time domain. The frequency-domain representation of the current response, only shown to the seventh harmonic, has in this case finite coefficients for all multiples of the fundamental frequency. The finite value at the frequency zero, represents the static offset in the current response. It can be seen in the time-domain representation at the fact, that the area of the oscillating current above and below the abscissa is not equal. Further, the non-sinusoidal form in the time-domain representation leads to the harmonics at twice the fundamental frequency and higher. In summary, using the admittedly rather large amplitude in our example while assuming small-signal-approximation, so that the measurement setup gives only the fundamental frequency response, one would only measure the fundamental coefficient of the response in the frequency-domain-representing inset in Figure 2.5b and interpret this as the actual response that would than be assumed as in Figure 2.5a. All other values in the inset would be neglected. Depending on the non-linearity of the characteristic of the system, the chosen amplitude and the purpose of the analysis, it should be clarified whether this approximation would be appropriate or not.
An important issue, especially since most available instruments used in immittance spectroscopy are still only designed for linear system response, is that one has to recognise the non-linearity of the response of the system. In an experimental setup capable of measuring non-linear response that is straight forward, since the presence of harmonics in the response can be seen directly. If the setup expects linear response, one may utilise the above given definition of the small-signal analysis: the immittance should be measured as a function of the signal amplitude, preferably under conditions where the non-linearity of the system under investigation is (expected to be) most pronounced. For decreasing amplitude the measured immittance should converge to the zero-oscillation-amplitude limit. Since the signal-to-noise ratio might become quite small at the lower end of oscil-lation amplitudes this characterisation of the amplitude dependence of the immittance might be performed with a sufficiently high number of repetitions.
Oscillation amplitudes smaller than the thermal voltageVT=kBqT should generally be small enough to neglect the oscillation amplitude and assume small-signal analysis to be valid [7, p. 5]. However, for cryogenic measurements such small amplitudes are almost impossible and, further, for measurements at room temperature hard to handle (usually
causing larger errors). With the above mentioned measurement of the immittance as a function of the oscillation amplitude, amplitudes that are sufficiently low to assume small-signal approximation but larger than the thermal voltage and, hence, in a usable voltage range can be determined. Otherwise, performing large-signal analysis is recommendable.
As already mentioned, in some cases, e. g. usually when working with high-power amp-lifiers [100], the non-linearity of the response has to be considered. As the understanding of the underlying physics is (as usually and understandably in that field) of lower priority, it is an accepted solution to add additional components (resistors, capacitors, etc.) in the EEC to model the harmonic response [159]. Instead of using such a descriptive approach, it can in some cases, especially when a physical model for the system under investigation is known, be advantageous to work with the universal immittance as defined in this work (or rather in the corresponding paper [3]). The direct insertion of the field or voltage and external-parameter dependent process-specific physical model then accounts for the higher harmonics.
The additional information from the non-linear part of the response might in general also be used, e. g. as additional verification that the correct model was selected. There are several ways to include the additional information from the non-linear character of the response in the analysis. The most direct one is possible if the set-up is capable of measuring the amplitude of the harmonics directly. Then the amplitudes of the harmonics, calculated from the assumed non-linear process-specific physical model, can be fitted to the measured amplitudes by adjustment of the system-specific parameters (of course, only those, not in literature). If unrealistic values are required for the system-specific parameters or the model cannot explain the observed amplitudes at all, the model should be modified or exchanged.9In many cases the experimental setup allows only the determination of the linear parts of the response, i. e. measurement of a regular (linear) immittance. In this case a non-linear model can still be beneficial, since part of the applied power is in the harmonics. The non-linear model can be utilised to calculate the correct fraction of the part of the power in the response with fundamental frequency. The fit of the linear immittances then considers this attenuation. The use of the knowledge of the underlying non-linear model presented last is, actually, just the correct application of small signal analysis. None the less, it is effective in ascertaining the correct underlying physics and rarely done. Usually, immittance spectroscopy is performed oscillating only around a single bias voltage, i. e. the operation point is not varied. Also variations of other external parameters are rare. By choosing a sufficiently small amplitude, the linear immittance at a certain bias voltage (or applied field) can be determined in good approximation. To utilise the known non-linearity of the model, multiple bias voltages are applied (static offsets of the stimulus are chosen). Then, a global fit (for all different bias voltages and other varied external parameters at once) of the calculated to the measured immittances gives the system-specific parameters. In that way all advantages of the traditional linear analysis, like the easier mathematics of linear system analysis as well as the possibility to
9This point is about the dismissal of clearly unfounded models. Of course, one may always have to accept deviations up to a certain degree, since any model is to some extent an idealisation of the real situation.
use conventional immittance measurement setups, are combined with the benefit that underlying physical processes are verifiable. If the model fits physically useful system-specific parameters can be extracted. The selection of an unsuited model cannot describe the development of the immittance with bias voltage and would lead to large differences between the measured and the corresponding calculate immittances at a high number of bias voltages. All three above explained approaches utilise the external parameter-dependent process-specific physical models instead of idealised lumped components.
Hence, the underlying physical processes can be identified and the physically useful system-specific parameters extracted.