Immittance1spectroscopy is a well-established and versatile method with a broad range of applications in a wide variety of different fields and subjects, ranging from science to industrial solutions [3]. Specific examples of its application are given in section 1.2.1.
Although immittance spectroscopy is widely applied, commonly used strategies of analysing its measured data do not exhaust its full potential: On the one hand, the rarely used [119], but at least in some situations physically correct, process-independent Poisson-Nernst-Planck models2might describeanycombination of processes by falling back onto effective parameterswhich, without any indication, do not necessary represent any actual underlying physical property of the system under investigation. On the other hand, the most common form of analysis is by electrical equivalent circuits (EECs). These are mere analogues which ‘are almost always assigned without regard to the physics of the system’
[110]. Due to this fact, their problem with circuit ambiguity in general (compare Figure 1.1) and the type of parameters which are obtained, EECs almost never represent the underlying physics and their parameters are likely irrelevant and incomparable between different experiments.
While analogues ‘may produce plots that are impressive in their fit to the experimental data, they do little to advance the science’ [110]. By introducing the dependence on external parameters in Maxwell’s extension of Ampère’s law, an EEC for a homogen-eous piece of material is derived that allows replacing idealised lumped components, like resistors and capacitors, by process-specific physical models dependent on external parameters. Utilising these models for measurements over a range of external parameters in combination with global fits, that do extract relevant physical parameters (e. g. ac-ceptor concentrations or bulk permittivities) describing all varied external conditions simultaneously instead of resistances and capacitancesper condition, allows identifying as well as understanding the underlying physical processes whilst extracting physically relevant parameters comparable between different experiments and more easily verifiable as reasonable in comparison to resistances and capacitances. Voltage-dependent immit-tance spectra, as utilised in this work, include the full information of a conventional
1Immittance, at first a coinage by Bode [11] to generalise the equivalent representations of impedance and admittance, was later further extended into a universal designation for any equivalent representation of impedance, explicitly including those not influenced by dimensions: e. g. admittance, complex permit-tivity, complex conductivity or modulus [113]. Since this work exclusively deals withelectricalimmittance spectroscopy, the prefix ‘electrical’ is generally omitted, for any type of immittance.
2This designation has its usual meaning in the context of immittance spectroscopy as given in reference [119] and explained in section 5.1.
NiTi DLC electrolyte
(a)Circuit for DLC on NiTi alloy in electrolyte by Chu [109]
NiTi DLC electrolyte
(b)Circuit for DLC on NiTi alloy in electrolyte by Suiet al.[174]
stainless steel DLC electrolyte
(c)Circuit for DLC on stainless steel in electrolyte by Maguireet al.[123]
stainless steel DLC electrolyte
(d)Circuit for DLC on stainless steel in electrolyte by Liuet al.[103]
Figure1.1:The challenges of circuit ambiguity using the example of diamond-like carbon: Di-verse variants to represent the DLC in an EEC, used to fit measured immittance spectra, are used. Subfigures (a)-(d) show four exemplary representatives, where the components that should according to the corresponding authors be associated with diamond-like carbon (DLC) coat-ing are coloured in green. In circuit (d) the parallel resistor is partially attributed to the DLC.
Since in conventional EECs the fit parameters are resistances and capacitances (inductances rarely play a role within the typical frequency range of immittance spectroscopy), comparison of the results of different works is rarely possible. Further, some arrangements of components may ignore the underlying physics and, as a result, the different components are not exclusively rep-resenting the suggested pieces but are rather a combination of different pieces of the system.
As a result, (accidental) usage of a circuit not representing the underlying physics may lead to misinterpretations.
current-voltage measurement. Additionally, they include the capacitive properties of the system. Consequently, even without introducing the presented novel approach of analysis, different serial pieces of the system3can principally be separated by their distinct capacitive bypass. As a result, the current-voltage behaviour of each separate piece can be extractedindividually. Bulk and electrode processes or the influence of high-resistive contacts are inherently separable and can be analysed individually without involving unconfirmed resistance models representing parasitic contributions around the actual piece of interest in the experiment. The introduction of process-specific physical models with dependence on external parameters is, of course, not limited to the resistive part.
Successfully fitting a serial piece simultaneously including external-parameter depend-ent models for both, resistive and capacitive part, can be seen as strong indication that the respective piece was correctly identified and, hence, its major underlying processes adequately understood. Furthermore, this is a convenient approach to study the compat-ibility of the involved dielectric and electric models at simultaneously obtained, hence self-consistent, data for the immittance of a serial piece. The approach becomes especially fruitful if both models share mutual parameters that may, consequently, be combined to single constants in the global fit. Although dielectric and electric behaviour is measured jointly, the processes responsible for each contribution can be quite different. Therefore, values of shared parameters obtained in such a way, that is furthermore not influenced by serial parasitic resistances and can due to the self-weighting effect of the models by their dependence on external parameters also include data from transition regions, may be con-sidered more reliable as compared to those extracted solely on the basis of conventional current-voltage analysis. As a result, in the analysis of heterostructures with a thin film of tetrahedral-amorphous carbon (ta-C) on different crystalline p-type silicon substrates performed in this work, as many mutual parameters as possible were fitted jointly. Next to a joined acceptor concentration that could describe both voltage-dependent models for the resistance as well as for the capacitance of a depletion layer forming in p-type silicon, fit parameters could also be combined for the bulk properties of the thin film.
Latter could be used to verify a correction (suggested in this work) in the calculation of the superimposed field from the externally applied potential difference within the physical basis of the Frenkel-Poole model (i. e. without the need of introducing any more recent physical concepts like quantum mechanics or more elaborate local environ-ments), that unexpectedly resolves an over 50 year old [80, 163] quantitative deviation of the Frenkel-Poole model (a semi-classical conduction model based on the process of introducing charges into the conductive band4by field-assisted thermal excitation from charged traps [46]). Through many enhancements of the Frenkel-Poole model over the years (contributions by different researchers are reviewed in section 5.6.1), especially a three-dimensional description of the local potential landscape first introduced by
Jon-3In this work, the word ‘piece’ designates a distinct unit of a potentially larger system, i. e. consisting of multiple pieces. The word ‘part’ is used to reference a subgroup of elements in a circuit, often the representation of a certain piece in the circuit.
4The term ‘conductive band’ is used troughout this work as a generalisation that shall indicate that it can be the conduction band in an n-type material, the valence band in a p-type material and, further for both cases, it can be the corresponding mobility edge if the material is disordered.
scher [80], a consistent description for small fields could be derived that resolved one of two major [144] quantitative differences between experimental data and the description of the model. The up to now [163] remaining deviation of an, in comparison to the predicted barrier-lowering coefficient, too shallow slope extracted from experimental data, could in this work be explained by using the, within the concept of classical elec-trodynamics required, internal field (i. e. the field inside a polarisable material) for the linear superimposed field in the local potential landscape around the trap, instead of the external field. That such a simple correction, which does not resolve any of the many remaining fundamental limits of the Frenkel-Poole model (discussed in section 5.6.2), especially concerning the description of the local potential landscape, could eliminate the one remaining quantitative deviation of the Frenkel-Poole model may at first seem surprising. However, a first attempt in understanding why the basic description might be sufficient is given in section 5.6.4. Verifying this hypothesis is a unique opportunity to test the capabilities of voltage-dependent immittance spectroscopy in a context where usually conventional current-voltage analysis is applied. Especially in this case, where the presence of permittivities (which are basicallythefundamental dielectric property) within the resistive part is in question, the unification of fit parameters mutually shared between resistive and capacitive models, which is an integral part of the proposed novel approach of analysing immittance spectra using process-specific physical models depend-ent on external parameters, can unfold its full potdepend-ential. Since it is found that a model, correctly calculating the internal field, with a connection of the permittivity can describe the system equally well as a model where both parameters can be chosen freely (in fact, in the free case, the fitting routine even chooses comparable values for the permittivity in resistive and capacitive part) this is a strong indication that the correction proposed in this work was indeed the missing feature of the Frenkel-Poole model that resolves the deviation between experimentally obtained and predicted barrier-lowering coefficients.
Since it is actually quite surprising that such a simple correctionwithin the conceptof the Frenkel-Poole model could actually be responsible, instead of introducing more elaborate physical concepts or refinements around the crude assumption of the local potential landscape, this work also re-examines the structure of defects in undoped covalently bound semiconductors that may potentially represent those trap centres in the Frenkel-Poole theory. The central potential of such traps is indeed highly oriented and neither spherically symmetric nor at all well described by a Coulomb potential. Furthermore, the central part of such potentials is indeed sensitively dependent on the distinct material. In fact, the Frenkel-Poole conduction is observed for asubstantial range of materials, with essentially only one similarity: a low mobility. Already this empirical finding suggests that a feature which is very different for all of these materials, namely the core potential of structural defects, should either not be a dominant property in the process or it should be an attribute of the model which is varied accordingly with the different specimens. The answer to the question why such a ‘simple’ model might describe the situation correctly is assumed to lie within the simplicity of the model itself. The Frenkel-Poole model is restricted to sufficiently high temperatures at which a significant amount of charge carriers can be thermally excited over relatively high barriers (e. g. in comparison to doping-levels in silicon). The relevant maximum of the barrier for the charges must be sufficiently
distant from the immobile oppositely charged trap centre that their respective interaction can be described by a Coulomb potential with the bulk permittivity of the surrounding medium. From the calculation of binding energies for deep level traps, which besides also gives the potential landscape of these traps, it is known that the binding energy, which is a parameter in the Frenkel-Poole model, is dominated by the core potential. However, any defect that is charged when empty (structural defects, e. g. dangling bonds, can fall into that category) has also an, usually assumed Coulombic, outer part of the potential.
This outer potential was found by Grimmeiss and Skarstam [55] to show a hydrogenic characteristic and dominate the properties of excited states for deep centres.
The Frenkel-Poole conduction is obviously dominant in a regime (of voltage and temperature) where the thermal excitation over the barrier is sufficiently prominent to significantly exceed the contributions of any other current-transport process. Exactly then, the dominant contribution is by those charge carriers that have enough energy to overcome the final, that is the outer, barrier by the Coulombic interaction. The measurement in that regime only perceives those charges which fulfil these requirements (i. e. the high enough thermal energy) and since the charges overcome the barrier thermally, a process for which only the highest point of the barrier is decisive, the knowledge of the core potential may be restricted to the binding energy, which is a parameter in the Frenkel-Poole model. In summary, the Frenkel-Poole model is observed in a region where its assumptions, though very crude, are representing exactly the remaining relevant properties. The discrepancy in the barrier-lowering coefficient can be explained by an omitted factor in its calculation. Interestingly, Poole already explains in a footnote of his publication [149, third footnote on p. 128] that he omitted this factor since he is only interested in a qualitative description and it seems to have been forgotten ever since.
1.2 A broader perspective - general impact of this work on the