3.3 Analysis
3.3.2 Optimising the fit parameters
For the subsequent part of this section following terminology is used (which is, to be more compact, different to the syntax in any other parts of the book): A pointPi in the parameter spacePfor a distinct conditioni is an ordered tuple with specific values for each external parameter. In this work, only bias voltage and frequency are varied.
However, there are other external parameters (e. g. temperature) which are held constant during the measurement and still used in the calculation of theoretical values for the fit.
Furthermore, there are parameters (e. g. signal oscillation amplitude) which are constant throughout the experiment but unused in the calculation of model values for the fit. In principle, there could be even more parameters (e. g. atmospheric pressure) which are simply not recorded during the measurement, since they are expected to be irrelevant or negligible for the measurement. For convenience, if a point in parameter space is written out, only its elements which are varied during the experiment are shown. Furthermore,
as an approximation, the external parameters themselves are assumed exact.
As a result, the measured data point for conditioni, given byPi, is expressed by Zi(meas)=
R(meas)i Xi(meas)
.
A (what is called) simulated data pointZi(sim), also consisting of resistance R(sim)i and reactanceXi(sim), is calculated utilising the given model for the whole system, which is basically two equations, one for resistance and reactance respectively, derived from the specific arrangement of components in the circuit and, of course, using the identical tuple of parametersPi as for the corresponding measurement condition. Except for the total serial resistance, which is an idealised lumped component, other components in the circuit are process-specific physical models dependent on external parameters. Their values for each specific conditioni are calculated and used in the equations for resistance
R(sim)and reactanceX(sim)of the whole system.
During an iteration of the fitting routine all parameters have defined values. Especially also the fit parametersF which are varied between the iterative steps and further fixed values, e. g. natural constants and literature values, denoted by C. Hence, for each iteration step simulated values for each condition are calculated and compared to the measured values. The comparison uses following squared residual function (as already mentioned, the weighting is according to Boukamp [12]) for each conditioni
δires2=
R(meas)i −R(sim)(Pi,Vi,F,C)2
+
Xi(meas)−X(sim)(Pi,Vi,F,C)2
σi(R)2
+
σi(X)2 ,
withσi(R)andσi(X)being the measurement error of the resistance and reactance of the data point for theith condition, respectively. Principally, not all functions use the same parameters from the tuplesPi,Vi,F,C, but for convenience (otherwise there would be even more indices) in these equations all functions get the full parameter tuple and are expected to use only those elements of it which they need. The tuple of all squared residuals ∆res2T
= (δ1res2, ...,δNres2) for allN conditions is minimised by the fitting routine by variation of the fit parameters in the tupleF.
Specifically for the simulated values, additionally to the fit parametersF and external parametersPi, the tuple of bias voltage dropsViover each serial universal Voigt-circuit element is needed. As indicated, the bias voltage dropsViare dependent on the condition i. This is the case, not only since the voltage is inside the tuple of external parameters, but also because the voltage drops are influenced, for example, by different temperatures. The voltage-dependent components in a serial piece of the system are, of course, not dependent on the total bias voltage applied to the entire sample, but instead only the local bias voltage drop over their specific piece, which cannot be straightforwardly extracted from the measurement. This bias voltage drop over each specific serial partkof totallyM serial parts is calculated for each conditioni through minimisation of the length (Euclidean
norm) of the tuple∆(Vi )by varying the voltage-drop tuple UiT from all its dc conduction processes together, at conditioni.2That isVi=min
|Ui|
is the total current dependent on the totally externally applied static bias voltageUbias,i. The above described calculations are triggered again and again by the fitting routine with changed fit parametersF up to the point that the length of the residual vector∆res2 is minimal.3 For simplicity, conditions with an externally applied bias voltage of0 Vare ignored in the fit.
As explained in section 2.4, an essential idea of the analysis introduced in this work is the combination of identical fit constants between different models, e. g. a shared permittivity for the conductive and resistive model or a joint concentration of traps for two different transport mechanisms. In the simple case of joint parameters, this is realised by using the same fit parameterfj∈ F in both models. Sometimes also derived quantities may be assumed identical: for example, the number of charges in the conduction band might be a shared quantity between resistive and capacitive properties of a depletion layer.
Then, also for performance reasons, the same function to calculate the respective value is only executed once (per condition) and its result used in both (or potentially even more joined) models.
2This minimisation is performed using therootmethod in theoptimizelibrary of theSciPy pack-age (version 0.17.0) which is implemented using a modification of the Powell hybrid method just as in MINPACK[77].
3This minimisation is performed using theleastsqboundmethod in theleastsqboundlibrary (version 1.1) which is a wrapper, extending theleastsqoptimiser in theoptimizelibrary of theSciPy package (which is, in version 0.17.0, implemented usingMINPACK’slmdifandlmderalgorithms) for possibility of boundaries as implemented in theMINUITpackage [77].
The presented novel approach of analysing immittance spectra utilising physical models dependent on external parameters is, in this section, exemplary applied on the metal/ta-C/p-Si/metal system. The unique features of the approach, especially the possibility to extract the capacitive and resistive properties of the individual serial pieces as coherent entities, enabled the verification of a particularly interesting hypothesis: The solely re-maining quantitative deviation of the Frenkel-Poole model (and its extensions) can be eliminated by a correct calculation of the internal field and that even without fundament-ally different assumptions.
In comparison to conventional current-voltage analysis, the knowledge of the immit-tance spectrum allows distinguishing distinct serial pieces of the system by their corres-ponding capacitive bypass. On the other hand, electrical-equivalent circuits consisting solely of idealised lumped components, which are usually used in immittance spectro-scopy, are ambiguous. Hence, a component of the circuit cannot be uniquely assigned to a piece in the system. The use of external-parameter-dependent immittance spectra allows a one-to-one association of different pieces in the system with separate components in the circuit while containing the full current-voltage information. In this section, the applied bias voltage is the varied external parameter. Consequently, voltage-dependent models for the resistances and (if necessary) capacitances are used to directly extract the physically relevant parameters.
The pure assignment of different serial pieces of a sample is possible even without the use of physical models. In fact, since the utilised model for a specific piece might not recognise all features, e. g. of the resistance-voltage behaviour, of the corresponding real-world element, the respective resistance and capacitance itself could be extracted undistorted by any assumptions, negligences or deficiencies of the applied model if their values would be redetermined for any condition (to account for a possible deviation from the typical frequency dependence of a constant phase element, even for different frequencies). For latter strategy of analysis, the correct arrangement of idealised-lumped components iscrucial, because without parameter dependence the circuit is (as mentioned above) unfortunately ambiguous. Also when models are not included, to arrive at the same one-to-one assignment between idealised lumped components and pieces in the system, the arrangement derived in chapter 2 must be used, i. e. each serial component is described by a universal Voigt-circuit element (that is like a conventional Voigt-circuit element, except for the capacitive part which might be complex instead of purely ideal and may contain several polarisation processes).
Voltage-dependent immittance spectra do not only allow the separation of serial pieces
in the system due to their corresponding capacitive bypasses, but they also include the full resistance-voltage behaviour (i. e. also the complete information of the current-voltage curve) of each piece. Hence, the presented approach allows the combined analysis of the resistive and capacitive properties, of each serial piece, as a coherent entity. Consequently, if, for example, a resistive model depends on a capacitive property of the same piece , e. g. its permittivity, it can be extracted directly from the corresponding capacitive part.
Similarly, specific features of a piece might influence both, the capacitive and resistive properties, of the corresponding piece. In such a case (which is quite frequent) where the same property might be determined separately from analysing either capacitive or resistive behaviour, the presented approach offers a third opportunity: the joint determination of the feature from both parts. For example in the analysis of experimental data in this work, the acceptor concentration that determines the capacitive and resistive properties of the depletion layer in the low-doped p-type silicon substrate was jointly fitted as a single parameter that could describe both parts well. The joint value for both parts of the thin-film permittivity is, as later explained, of paramount importance for the evaluation of the corrected Frenkel-Poole model, given in this work, and consequently discussed in its own section. The concept of combining parameters (and respectively correlating properties) determinative for the behaviour of both parts, capacitive and resistive, can be used to correlate or cross-check any value affecting both parts, e. g. the thickness of a film. The extraction of correct values for these parameters is, however, influenced by the assumed models. Thus, a comparison of the extracted parameters can also be used to validate the corresponding utilised model itself or its compatibility with the associated (resistive or capacitive) counterpart.
Both major benefits of the presented approach, the separation of the system into its serial pieces and the extraction of coherent capacitive and resistive behaviour, make clear that voltage-dependent immittance spectroscopy offers a valuable complement to conventional current-voltage analysis without the necessity to create specific samples.
With respect to conventional immittance analysis, the introduction of process-specific physical models dependent on external parameters leads to unambiguous circuits that allow the direct extraction of the statistically best guess for fit parameters of the model.
Furthermore, the respective models weight their region of dominance in the external-parameter-dependent immittance data. As a consequence, data from transition regions can become an equally well source of information as data from regions that are dominated by a single process.
The investigated samples show, according to their doping concentration to varying extent, a significant contribution of a depletion layer in silicon. In the first part of this chapter, the model for the depletion layer in silicon is explained. Certain simplifications in the model lead to identifiable deviations between the measured and fitted immittance spectra. The origin of these deviations could be associated with the presence of traps at the interface and fixed charges in the thin film. Conventional capacitance-voltage analysis is shown to highlight its differences to the proposed model. Additionally, the conven-tional analysis is found leading to similar parameter for (almost) identical assumptions.
Furthermore, for specific extreme cases, the magnitude of the defects at the interface or fixed charges in the thin film are estimated. Additionally, a depth profile of acceptor
the substrate has an acceptor concentration within the specification of the manufacturer, though a high number of defects at the interface or in the insulator lead to a strong dispersion that seems to require even higher frequencies (than the maximal frequency of1 MHzused in this work) to observe an undistorted capacitance-voltage curve which could be used to extract more accurate doping concentrations.
As already mentioned in the beginning, the benefit of a coherent entity of capacitive and resistive information is especially useful to validate an assumption about the per-mittivity in the resistive part. As more thoroughly discussed in section 5.6 (although the Frenkel-Poole model uses quite basic assumptions that might not be representing the real situation in the material very well) it was found, within this work, that the single remaining quantitative discrepancy in the Frenkel-Poole model, between the theoretically assumed and experimentally obtained barrier-lowering coefficientβ, can be explained by introducing a correct calculation of the superimposed internal field. This calculation requires the static permittivity of the material. Due to the above explained fact, that capacitive and resistive properties are jointly measured, the proposed method of analysis is uniquely suited to determine whether our correction is sound. For that, the fit quality and resulting fit parameter values are evaluated for three different models for the thin film in section 4.2.
4.1 Differences between the samples and experimental findings associated with the ta-C/p-Si interface effects
4.1.1 The different substrates, their associated properties and their