Impact of this work on immittance-spectra analysis in other fields 7

the introduction. The referenced examples still use EECs and in consequence have to deal with all resulting drawbacks, especially circuit ambiguity. Since different arrangements of circuit components are equally likely, depending on the chosen structure, the extracted relevant parameters (e. g. the concentration of a toxin) do not necessarily represent the actual value. In the end, in every application the resistances and capacitances should be converted into meaningful parameters.

The application of the presented approach is possible directly for at least every solid part that is described in an EEC. If, for example, the electrodes are coated with a material exhibiting Frenkel-Poole conduction, its description should, hence, include the model accordingly. In order to do that, it is important to know which component(s) of a circuit represent which piece of the system. At least for resistors or capacitors, that should be converted to the relevant parameters, a one-to-one assignment between components and underlying processes is necessary.

Unlike in the case of a solid state system investigated here, in these very different biologic fields, models for conduction or polarisation processes are typically not available. The presented approach might, hence, not directly be applied. However, the basic concept of the novel approach, the introduction of process-specific physical models dependent on external parameters may still be beneficial. Though the use of different conditions is expected to be more restricted than for inanimate, inorganic systems, it might still be possible to trigger some reactions within the system. If for example, the generation of ions at the cell membrane is dependent on some stimulus or some concentration of a species that may be introduced, this dependence might be directly included in the describing circuit. The above mentioned biological scenarios are possibly preferentially described by spatially-extended simulations. Also in those, the explained concept may be included by using condition-dependent models for specific relevant properties, e. g. the number of ions at a membrane may still be calculated dependent on some concentration of a certain protein instead of simply fitting its value for each protein concentration separately.

Although, again, indirectly, the concept of this work may be successfully applied to inorganic electrochemical impedance spectroscopy. Possibly the best lumped-component descriptions of an electrochemical system, like a battery, are given by the specific solutions of Poisson-Nernst-Planck models. Although these models can lead to misleading

paramet-ers, if some processes or ionic species where omitted. As described in more detail in section 5.1.2, the basic concept of the novel approach, the introduction of process-specific physical models dependent on external parameters, can be combined with lumped Poisson-Nernst-Planck models. In this case the process-specific physical models are not included as circuit components, but to replace the fit values independent of external parameters, like the dis-sociation of the local species. As those parameters become external-parameter-dependent functions, global fitting of immittance spectra for different conditions, including auto-matic local weighting of parameters due to their specific regions of importance, becomes possible and the probability accidental description with effective parameters is reduced.

1.3 The exemplary experimental application

As already indicated above, this work focuses primarily on the development of a novel method to analyse experimentally obtained immittance spectra by using process-specific physical models dependent on external parameters. Ultimately, this allows the extraction of process-specific parameters which are inherently linked to a physical meaning (e. g. an acceptor concentration or a barrier height, instead of a resistance or a capacitance) and can, hence, be better compared between experiments. Furthermore, the introduction of models with dependence on external parameters eliminates circuit ambiguity which, as can be seen in Figure 1.1, also improves the comparability between different experi-ments. Moreover, the regions of importance for the respective processes are automatically weight by the parameter-dependence of the models. Additionally, external-parameter dependence enhances the association of circuit components with distinct pieces of the system under investigation and reduces the absorption of deviations, e. g. by an incom-plete model on one piece, into the fit parameters of another piece. Finally, if the same underlying physical properties are present in different models for the same piece they may be shared and, consequently, fitted jointly, even between capacitive and resistive properties.

The application of this novel approach on samples with ta-C films on p-type silicon sub-strates, with different doping concentration, is only an example. However, as explained below, a carefully chosen one.

The analysis of depletion layers in silicon, usually in form of capacitance-voltage meas-urements, is probably one of the most common applications of immittance spectroscopy in semiconductor physics. In comparison to many other systems investigated by immit-tance spectroscopy, which often rely on empiric models, depletion layers are relatively well-investigated and -understood phenomena which have theory-based models [138][176, pp. 245-297]. The latter is not only true for the capacitance of a depletion layer, but also for its resistance-voltage characteristic, i. e. its static current-voltage relation [176, pp. , 245-286]. As already explained, the novel approach in this work is based on introdu-cing physical models dependent on external parameters. In the case of the depletion layer, both resistive and capacitive properties are dependent on the external parameter voltage and both models are based on microscopic assumptions (which is what is meant in this work by physical model dependent on external parameters). Voltage-dependent immittance spectra, as those obtained in this work, not only contain the capacitance-voltage information, but also the full resistance-capacitance-voltage characteristic. Furthermore, the higher number of frequencies in the approach introduced in this work as compared to conventional capacitance-voltage analysis (often only one high- and one low-frequency measurement are performed [138, pp. 321-333, 388-389]) allows distinguishing and poten-tially identifying different serial parts in the system. This allows identifying potential parasitic contributions and, as also utilised in this work for the thin film, the introduction of separate descriptions for each serial piece of the complete system. For the depletion layer in silicon, well-known literature values exist for most model parameters (see section 2.4.1 for the sources of the literature constants for the depletion-layer model used in this work). The remaining parameters, which are determined by the fit, affect both, resistive

and capacitive properties. This enables evaluating yet another consequence of the novel approach: the simultaneous fit of parameters shared between the resistive and capacitive model. As a result of all above points, the presence of a silicon depletion layer in the experimental example system is a unique opportunity to compare the capabilities of the novel approach presented in this work with well-established conventional methods of analysing depletion layers. Interestingly, conventional capacitance-voltage analysis usually does not utilise the resistive information contained in the immittance measure-ment [138, pp.pp. 321-333, 388-389]. Although works often present resistive as well as capacitive analysis of the same system, the results are usually obtained in different set-ups (e. g. confer [18] and [49]). Consequently, the separately obtained measurement data is usually also independently analysed and only at the end both results are compared.

As explained above, the novel approach presented in this work combinedly fits shared parameters and uses the voltage-dependent immittance spectra which contain both, the complete resistive and capacitive, information. Since a single measurement with the same set-up is utilised, to extract both kinds of data, resistive and capacitive information may be expected to be more likely consistent as compared to a case with two distinct set-ups.

The well-established conventional methods of analysis for depletion layers in silicon and their relatively well understood properties make them a suited candidate for a comparison between the conventional capacitance-voltage analysis and the approach presented in this work. This comparison, shown in section 4.1.2.1, suggests that the extracted parameters between both methods of analysis are in agreement.

In this work, not solely the depletion layer in silicon alone is investigated, but a com-plete metal/insulator/semiconductor (MIS) structure, where the thin-film material ta-C acts as fairly leaky insulator material.⁵ The analysis of a complete MIS structure is es-pecially valuable, firstly, since this is a very common and often analysed system in the semiconductor industry and, secondly, since the novel approach presented in this work can bring its full potential to bear for systems consisting of multiple serial pieces. While for the depletion layer in silicon both, capacitive and resistive, properties are depend-ent on voltage, the bulk-capacitance of the amorphous thin film material, with solely one atom species consequently connected by non-polar bonds, is not expected to be dependent on it. The resistance, on the other hand, is known to exhibit Frenkel-Poole conduction (e. g. confer to the voltage and often also temperature-dependent current-voltage analysis of [38], [168, 91]⁶, [90], [53], [132] and in particular Ronninget al.[156], Hofsäss [68] and Brötzmannet al.[15] who used the identical set-up for the synthesis of the ta-C film) which is again a process dependent on voltage. For this specific model a quantitative discrepancy between the experimentally obtained and theoretically pre-dicted barrier-lowering coefficient exists. According to the Frenkel-Poole theory (see [46]

5This material system was designated MASS for metal/amorphous semiconductor/semiconductor by Brötzmannet al.[14, 15].

6In these two latter works, the conduction process is interpreted as Schottky-emission process solely due to the deviation in the barrier-lowering coefficient. The authors were obviously unaware of the fact that this deviation from the theoretically predicted value is almost always observed for the Frenkel-Poole model, specifically also in systems where the current-voltage characteristic was proven to be bulk-limited, e. g. [129].

and the estimation in section 4.2), the dynamic permittivity (at optical frequencies) is the only value in the barrier-lowering coefficient which is not a natural constant. The usually observed too low barrier-lowering coefficient may be ascribed to a permittivity greater than the corresponding literature value. This was a typical quantitative argument against the correctness of the, in some eyes, too simple model which neither takes more modern physical concepts into account nor uses a realistic potential landscape (see full discussion in section 5.6). In this work, a correction of the Frenkel-Poole model or rather the commonly used calculation of the applied field in the Frenkel-Poole model is pro-posed. Instead of the external electrical field, the internal field is used. This correction is completely within the concept of classical electrodynamics and, hence, also within the scope of the Frenkel-Poole model, rather than an introduction of different physical con-cepts like the introduction of quantum-mechanical processes. Since the novel approach of analysis, presented in this work, measures and fits resistive and capacitive properties simultaneously and fits shared parameters like the permittivity jointly, the test of the correction was a uniquely suited task for the novel approach.

Tetrahedral amorphous carbon is a specifically suited material for both, studying the Frenkel-Poole model and finding a unified microscopic theory for ac as well as dc proper-ties. The latter mainly because it is a typical disordered system with the consequential well-known constant phase response. The former, since it is a relatively simple material, especially the in the mass-selective ion-beam deposition (MSIBD) grown variant which has a characteristically low number of foreign contaminants [101, pp. 211-213] and a high sp³content [154]. The amorphous structure of the thin ta-C film, which is assumed to be homogeneous, renders the material isotropic which specifically reduces the complexity of the long-range binding potential between the exiting electron and its trap, because it allows the assumption of a spherically symmetric potential (for distances sufficiently large than the lattice constant, see the specific discussion in section 5.6.4). Especially the MSIBD-grown ta-C which is particularly pure, even mono-isotopic, and has a rather high sp³content in comparison with other methods of synthesis, can be expected to consist almost entirely of carbon atoms. Consequently, as opposed to many other dielectrics exhibiting Frenkel-Poole conduction, polar bonds should play no role in the conduction process. Furthermore, again as distinguished from other representatives, typical bulk dielectric relaxations or resonances may be neglected below optical frequencies. Frenkel-Poole conduction may only occur in materials with sufficiently low mobility which is usually connected to some element of disorder [82]. Ta-C is a typical representative of disordered solids, including its property to exhibit a constant-phase behaviour with a constant phase around0.8πat low frequencies. The high number of defect states in a disordered material also has the advantage that space-charge layers at the interface, inside itself, in contact with other materials are highly unusual [80]. Consequently, using metals, Ohmic contacts are rarely a problem on disordered solids. In this experiment, Au/Cr and Al/Ti contacts, evaporated onto ta-C, resulted in indistinguishable voltage-dependent immittance spectra.

Although not the primary focus of this work, the used example system ta-C/p-Si is uniquely suited to illustrate the benefits of this novel method of analysis. Furthermore, the ta-C thin films in this work, synthesised by mass-selective ion-beam deposition, could

be a candidate to resolve what is said to be one of the major challenges of solid state physics [118], or even, the ‘most important unsolved problem in physics today’ [147]: a unified microscopic theory for the ac and dc properties of disordered dielectrics.