A mathematical framework for the analysis and modelling of memristor nanodevices

This work presents a set of mathematical tools for the analysis and modelling of memristor devices. The mathematical framework takes advantage of the compliance of the memristor's output dynamics with the family of Bernoulli differential equations which can always be linearised under an appropriate transformation. Based on this property, a set of conditionally solvable general solutions are defined for obtaining analytically the output for all possible types of ideal memristors. To demonstrate its usefulness, the framework is applied on HP's memristor model for obtaining analytical expressions describing its output for a set of different input signals. It is shown that the output expressions can lead to the identification of a parameter which represents the collective effect of all the model's parameters on the nonlinearity of the memristor's response. The corresponding conclusions are presented for series and parallel networks of memristors as well. The analytic output expressions enable also the study of several device properties of memristors. In particular, the hysteresis of the current-voltage response and the harmonic distortion introduced by the device are investigated and both interlinked with the nonlinearity of the system. Moreover, the reciprocity principle, a property form classical circuit theory, is shown to hold for ideal memristors under specific conditions. Based on the insights gained through the analysis of the ideal element, this work takes a step further into the modelling of memristive devices in an effort to improve some of the macroscopic models currently used. In particular, a method is proposed for extracting the window function directly from experimentally acquired input-output measurements. The method is based on a simple mathematical transformation which relates window to sigmoidal functions and a set of assumptions which allow the mapping of the sigmoidal to current-voltage measurements. The equivalence between the two representations is demonstrated through a new generalised window function and several existing sigmoidals and windows. The proposed method is applied on three sets of experimental measurements which demonstrate the usefulness of the window modelling approach and the newly proposed window function. Based on this method the extracted windows are tailored to the device under investigation. The analysis also reveals a set of non-idealities which lead to the introduction of a new model for memristive devices whose response cannot be captured by the window-based approach.Open Acces