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This thesis is concerned with the efficient computation of Jacobian matrices of nonlinear vector maps using automatic differentiation (AD). Specifically, we propose the use of two directed edge separator methods, the weighted minimum separator and natural order separator methods, to exploit the structure of the computational graph of the nonlinear system.This allows for the efficient determination of the Jacobian matrix using AD software. We will illustrate the promise of this approach with computational experiments.

The task of the thesis was studding of tantalum capacitors with solid electrolytes properties. Ta – Ta2O5 – MnO2 capacitor by its construction represents MIS structure, where tantalum anode has metal conductivity and MnO2 cathode is semiconductor. Isolation layer consists of tantalum pentoxide Ta2O5 with relative permitivity r = 27. Dielectric thickness is typically in range from 30 to 150nm. The capacitor charge is not only stored and accumulated on electrodes but also in localised states (oxide vacancies) in isolation layer. The capacitor connected in normal mode represents MIS structure polarized in reveres direction when the applied voltage higher potential barrier between semiconductor - MnO2 cathode and isolation of Ta2O5. The transport of charge carriers via isolation layer is determined by Poole-Frenkel mechanisms and tunnelling. Poole-Frenkel mechanism of charge transport is dominant in low intensity of electric field. Tunnelling determines current at higher electric field intensity. During low intensity of electric field ohmic component is also presented which is determined by volume of resistance of impurities in isolation layer due to donor states of oxygen vacancies. Based on the modelling of measured VA characteristics is possible to estimate determine dielectric thickness of Ta2O5 and determine share of Poole-Frenkelov and tunnel current and charge transportation. The thesis is described charge transport and charge concentration on tantalum capacitor in low frequency area and analysis of capacitor behaviour at frequency band. The first impulse for the thesis was an effort to create equivalent circuit diagram of tantalum capacitor in respect of its physical and electrical behaviour. There is an opportunity to study and determine electric charge transport and its accumulation based on the equivalent circuit diagram structure. There is also a chance to define and trace potential barriers and charge distribution in the capacitor structure based on an measurement and carried out experiments. This methodology and analysis consists of electrical characteristic determination to create physical model of the capacitor describing it function, properties and behaviour.