Comparison with the analysis of immittance data using specific

4.2 Evaluation of different models for the thin film

5.1.2 Comparison with the analysis of immittance data using specific

5.1.2.1 Introduction

For systems that have one piece with a single dominating transport process for each of the up to two types of charge carriers with opposing sign and arbitrary valence, specific process-independent solutions of Poisson-Nernst-Planck models, that are compared with the presented approach in this subsection, allow extracting relevant physicochemical parameters, for the corresponding piece, from immittance data.² It is not unusual to account for other pieces of the system with idealised lumped components, i. e. the selected specific process-independent solutions of Poisson-Nernst-Planck model is one component in a circuit consisting otherwise only of idealised components [115].

Process-specific versus process-independent In this context, process independence in-dicates (confer [119]) that it is neither distinguished nor necessary to know what the exact types of propagation of the charge carriers are, e. g. whether the charges move by small ions migrating through interstitial sites of a crystalline solid, electrons thermally excited into the conduction band, charges hopping through localised states or mobile ions in an electrolyte. The Poisson-Nernst-Planck models discussed in this work extract parameters including mobilities, charge carrier concentrations and their valence. Identifying the origins of, for example, the mobility is neither necessary to know for the Poisson-Nernst-Planck models, shown in this work, nor extracted by them. Subsequent analysis of the parameters can, however, be utilised to narrow down the processes, e. g. the mobility of ions migrating in a solid is expected to be much lower than electrons in a conduction band.

In the novel approach presented in this work, process-specific physical models de-pendent on external parameters are used to globally fit immittance spectra for multiple conditions. Each model carries an underlying physical concept of, to remain with the example of mobility, how the carriers propagate. The assumption of the specific process is actually what determines the distinct dependence on external parameters. The differences between the models described above are summarised as being process-independent or process-specific. In Table 5.1 examples for process-specific models important in CMOS technology are given.

2For an introduction in the use of Poisson-Nernst-Planck models in the field of immittance spectro-scopy, including a review, confer reference [119].

Bulk-limited conduction processes

Trap-filled limited / Space-charge limited conduction:

J_TFL/SPL(E,T)∝µ(E,T)^E_d5²

Table 5.1:Exemplary list after [23] (not intended to be exhaustive) ofprocess-speciﬁcphysical models dependent on external parameters for current transport processes in solids and at their interfaces, which are commonly found in dielectric films in CMOS technology. Each model is connected with an underlying concept and, consequently, certain assumptions. Furthermore, these models are non-linearly dependent on the applied field, underlining how important it was to allow for non-linear field-dependence in the conversion from Maxwell’s equation of total current density to an electrical equivalent circuit. The used symbols have following meaning:µ electron drift mobility,N_Cdensity of states in the condition band,eΦ_Ttrap energy level,ǫ^(dyn)_r dynamic relative permittivity (at frequencies of visible light),amean hopping distance,nelectron concentration in the conduction band,νfrequency of thermal vibration of electrons at trap sites, dthickness of dielectric thin film,A^∗effective Richardson constant,eΦ_BSchottky barrier height, hPlanck constant,m_T^∗tunnelling effective mass in the dielectric

The specific Poisson-Nernst-Planck models discussed in this work This work follows the usual conventions in the context of immittance spectroscopy regarding the term

‘Poisson-Nernst-Planck model’ which restricts it to specific solutions derived to analyse immittance spectra using lumped components, like the specific solutions derived in [116, 117, 120]. In general, Poisson-Nernst-Planck models may encompass any model, especially including those using finite-element methods, that describes the (spatially resolved) ion or charge flux considering the gradient of concentration, the influence of the electric field and, possibly, chemical reactions. Meaning that general Poisson-Nernst-Planck models are extremely versatile and limited only by the assumptions of an effective-medium and by disregarding the spatial extent of the charge carriers (which may, however, be specifically important for electrochemical experiments with ions in an electrolyte where agglomeration of molecules at the electrodes has a significant influence on the measured immittance)³.

In combination with finite-element methods complex inhomogeneous media with arbitrary geometries may be modelled. However, though some properties are known, e. g. the distribution of grain sizes and their respective properties, the actual form of an underlying inhomogeneous material is usually unknown. Hence, such an approach is limited to simulate random structures with similar properties, e. g. generating a structure with identical grain-size distribution and corresponding phase and orientation properties, which only emulates the actual situation in a ‘real’ sample. Furthermore, the actual size of the sample might be too large while the important structures are too small to simulate a full-sized system in the necessary detail, i. e. a macroscopic piece of a nano-crystalline material. Instead, a smaller volume of only few detailed structures might be simulated and the result respectively scaled.

It should be possible to include any process that can be embedded within an effective-medium approach in such models, also those with non-linear dependence on the field like the Frenkel-Poole conduction process. It could be introduced as carrier generation rate of an element dependent on its local internal field (generation and recombination of charges was introduced in [120], but without utilising local properties as the internal field). Similarly, electrode limited processes, like Schottky emission, may be included in Poisson-Nernst-Planck models by respective boundary conditions or field-dependent rates of injection from the electrodes (electrode-discharge effects have been introduced in [117], but without dependence of the local field at the interface).

The general idea of this work, i. e. introducing process-specific physical models de-pendent on external parameters, can, as indicated in the above examples, be combined with finite-element models. It may even be the next logical step and possibly very fruitful, especially when dealing with inhomogeneous systems. However, such intriguing future advancements of this concept of analysing immittance data lie outside of the scope of this

3At interfaces between electrical and ionic conductors so-called electrical double layers form which have a very significant capacitive contribution and importance in many electrochemical applications. The structural and, consequently, the derived electrical properties of electrochemical double layers require a realistic modulation of the agglomeration of ions at the interface which should, as it turns out, recognise the finite size of the ions. For information about double layers and an overview of different descriptions see [173, pp. 27-34]

work, which primarily focuses on the first step of utilising external-parameter-dependent physical models, namely, analysis based on lumped components. As already mentioned, there are specific solutions of spatially extended Poisson-Nernst-Planck models that take the form of lumped components. Exactly those models are advertised to be used in the analysis of immittance spectra [119]. Consequently, the comparison of the novel approach presented in this work with Poisson-Nernst-Planck models is restricted to those that take the form of lumped components.

Historical development towards Poisson-Nernst-Planck models utilisable in the analysis of immittance spectra This is only a very short summary of the historical development of Poisson-Nernst-Planck models. It is focused exclusively on key points in the development of models specifically designed for the application in immittance spectroscopy. The first Nernst-Planck solutions that took time dependence into account and calculated polarisation response were developed in 1933 by Jaffé [74, 75]. These works did not take carrier generation or recombination into account, but instead assumed constant numbers for both. In the first work [74], the discharge at the electrodes was forbidden which makes the presence of a dc current impossible. In the second publication [75], another type of ion species (‘Ionen zweiter Art’) was introduced that could discharge at or traverse through the electrodes. In 1952 Chang together with the author of the first works refined the previous model by introducing more realistic boundary conditions that would be suited for the description of electrolytic solutions [22]. Therefore, the discharge at the electrodes is assumed to be a rate process. In the second publication of this series Jaffé and Rider [76] introduced different mobilities for positive and negatively charged ions since this let to a more satisfactory agreement with presented experimental results. The works of Jaffé and its coworkers were further refined in 1953 by J. Macdonald [116] especially by assuming partial dissociation of positive and negative ions from a neutral species and taking recombination of carriers into account. Furthermore, for the first time the Poisson equation was properly included. In 1978 the description of the situation in electrolytes and semiconductors was again improved [120]. Charge carriers were now assumed to have arbitrary valence and, further, to recombination to as well as generation from neutral centres, charges can be emitted or absorbed from immobile donor or acceptor like centres.

As illustrated, Poisson-Nernst-Planck models for the analysis of immittance data have an already more than eighty year old tradition and became incrementally refined, e. g.

to include dc bias [45] or to account for anomalous diffusion processes [122]. However, they have not yet found widespread application for the analysis of immittance data [119].

5.1.2.2 Advantages of Poisson-Nernst-Planck models over conventional EECs

Instead of resistances and capacitances, Poisson-Nernst-Planck models extract physico-chemical properties on the basis of underlying assumptions. Typical parameters defining a Poisson-Nernst-Planck model which can, consequently, be determined in fits (some parameters are coupled, i. e. not all of them can be jointly fitted, the corresponding counterpart should be known) are [115]: the concentration of dissociable entities, valence number ratio between the two species, their corresponding mobilities, generation and

recombination rates, reaction rates at the electrodes.⁴ If the extracted values are really connected to the underlying mechanisms (challenges achieving this are discussed below) they are much more meaningful than resistances and capacitances. Furthermore, there is no intermediate step of fitting EECs that may lead to loss of information.

5.1.2.3 The dangers of extracting effective values

In the last paragraph, it was established that the extracted parameters itself can be meaning-ful physicochemical parameters. This is, however, only the case if the extracted parameters actually correspond to some ‘real’ feature of the underlying process. For this, two con-ditions have to be fulfilled: First, the Poisson-Nernst-Planck model must be capable to describe the corresponding part of the system at all. Secondly, exclusively the immittance of this part must be associated with the fit parameters of the Poisson-Nernst-Planck model. Otherwise, effective parameters might be extracted which might not even give any indication that they have nothing to do with the real properties of the component.

The exclusive association is challenging, because currently used lumped-component Poisson-Nernst-Planck models do not include external-parameter dependence. Just as in the case of conventional EECs, it can, because no restrictions through specific depend-encies are imposed, not be ensured that the lumped Poisson-Nernst-Planck component really describes exclusively the intended piece of the system. If by accident, contributions of multiple components are interpreted by the Poisson-Nernst-Planck model, it might simply ‘absorb’ the distortion of the false contributions. It is entirely possible that these extractedeffectiveparameters might not be conspicuous at all. As a result, reasonable looking parameters might be extracted which might actually not correspond at all to any

‘true’ underlying property. Even if it could be ensured that only one part of the system is exclusively analysed with the Poisson-Nernst-Planck model, this part would have to fulfil certain requirements to extract relevant physicochemical parameters and not only effective ones. The part should at most have two types of charge carriers with opposing charge (they can be generated or recombined if necessary) whereas the charge injection may be limited by the electrodes (defined by specific rates). The boundary conditions have to be fulfilled. Without external-parameter dependence there might be no indication whether these requirements are not met or not.

An example that could exist in that form: assume a material where the transport is in parallel by two very different processes, a Frenkel-Poole conduction process and variable range hopping, both with electrons. Without external-parameter dependence, it is not possible to decouple two parallel Poisson-Nernst-Planck models. Additionally, without further knowledge only one transport process might be assumed. Fitting with a Poisson-Nernst-Planck model and assuming one mobile species, namely the electrons, effective parameters would be gained for the mobility, concentration, valence, etc. When varying the temperature, these effective parameters would change, however the involved underlying processes, their actual parameter and the fact itself that two processes are

4Here, the more common experimental situation is assumed, where the dimensions, e. g. the distance between the electrodes, are known and above parameters unknown. In principle it could also be the other way round.

involved would not be indicated. One could assume two mobile species (since they are not generated from each other with two concentrations and without generation and recombination), though they are actually not of opposing signs. With some knowledge, e. g. of the different mobilities or concentrations, the other parameters of the two sets may then possibly be associated with the different processes. Using the analytic solutions of Poisson-Nernst-Planck models for immittance spectroscopy, it would however not work for more species. So if an electrolytic solution with two solved species, e. g. NaCl and CaSO₄, is analysed, the parameters are inevitably effective values.

The result of the Poisson-Nernst-Planck models is compared with the novel approach presented in this work in the previous example. Of course, external parameters have to be varied, e. g. temperature and bias. One might assume a Frenkel-Poole conduction process. After fitting the experimental data and analysing the residuals one might find that this alone could not describe the experimental findings. Hence, variable-range hopping is added and one finds that the residuals do not require another process. With a bit more effort the fit does now agree with the experimental data, just as in the case of the Poisson-Nernst-Planck model before. However, the result contains more information as the models are more restrictive. Since the parameters of the Frenkel-Poole model were fitted for multiple temperatures and voltages, the activation energy and defect centre concentration may be extracted. Latter can then be compared with the density of centres determined form the mean hopping distance to determine whether or not the same centres that emit charges into the conductive band could principally also be those dominating the hopping contribution.

In summary, a good fit of a Poisson-Nernst-Planck model does not necessarily mean that the extracted parameters are really representing some underlying property. Any deviations could be absorbed by effective parameters. It is, hence, necessary to acquire some previous knowledge about the system under investigation. This could be the case for electrolytic solutions. Still, including external-parameter dependence can be utilised also in electrolytes. Similar to the field-dependent emission of a charge carrier from a trap, the dissociation of weak electrolytes has a field-dependence [144].

5.1.2.4 Describing parallel charge transport processes

Poisson-Nernst-Planck models can describe transport processes with one mobile charge by assigning its mobility, concentration of charge carriers, valence etc. As mentioned above, these parameters describe the transport of charges independent of the actual process itself. Meaning that, for example, subsequent tunnelling through localised states (that is hopping) would be assigned an appropriate mobility. Actual underlying process-specific parameters, e. g. the mean hopping distance, would not be extracted. Also two charge species with opposing sign can be well described. However, again without extracting the process-specific parameters. Except for an electrolytic solution with no more than one dissociated electrolyte species (it is one specific case the lumped Poisson-Nernst-Planck models were designed for) would be described well and the parameters would be appropriate for the underlying process. More then two charge transport processes or more than two types of ions in the solution cannot be described without fitting effective

values. Lumped Poisson-Nernst-Planck models cannot be arranged in parallel because their values would be coupled.

5.1.2.5 Decoupling multiple parallel transport processes and extracting process-specific parameters

There are two possibilities that enable decoupling of multiple different transport pro-cesses and extraction of process-specific underlying parameters: using the novel approach presented in this work or using a combination of its main idea, i. e. including process-specific physical models dependent on external parameters, with Poisson-Nernst-Planck models.

Using the novel approach presented in this work First, the transport processes depend-ent on external parameters is included as parallel resistors in the universal Voigt-circuit element that is supposed to describe the respective piece of the system (other parts are similarly modelled). Secondly, the parameters that cannot or should not be taken from literature or other sources are globally fitted. In the example given above, the concentra-tion of defects that participate in the hopping process (derived from the mean hopping distance), the density of traps from which electrons are thermally excited into the con-ductive band as well as the corresponding energy barrier can be obtained. A comparable concentration may indicate that the involved traps could be identical. If possible, inter-process correlations should be utilised, e. g. the permittivity from the parallel capacitance can be used as joined constant for the permittivity in the barrier-lowering coefficient.

Combining the main idea of the novel approach presented in this work and Poisson-Nernst-Planck models The second method of analysis to decouple the parallel transport pro-cesses in the given example is a combination of the general idea of the presented approach with the Poisson-Nernst-Planck models. Sometimes specific physical processes may be as-sumed to determine certain internal properties of the system. In the example given above, the internal property of the number of charge carriers in the conduction band were sup-posed to follow a specific process, the field-assisted thermal emission (i. e. Frenkel-Poole model). Also the process of injection at the electrodes and its dependence on external para-meters, like temperature and applied field, might be known. The central idea of this work, utilising process-specific physical models with their dependence on external parameters, can also be used to extend Poisson-Nernst-Planck models. Including these process-specific properties in conventional EECs allows decoupling of parallel processes or finding correct arrangements of components to obtain a one-to-one assignment of physical processes and circuit components, while also allowing global fitting over immittance spectra of various different conditions. Using the corresponding external-parameter-dependent physical processes to calculate the properties used in the Poisson-Nernst-Planck models, e. g. the temperature and field dependence of the concentration of charge carriers of one species, of its mobility or the injection rate at the electrodes, evokes similar advantages

Im Dokument A novel approach of immittance-spectra analysis and how it resolves a decade-old deviation of the Frenkel-Poole model (Seite 163-172)