Physics-based compact modeling of electro-thermal memristors: Negative differential resistance, local activity, and non-local dynamical bifurcations
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Leon Chua's Local Activity theory quantitatively relates the compact model of an isolated nonlinear circuit element, such as a memristor, to its potential for desired dynamical behaviors when externally coupled to passive elements in a circuit. However, the theory's use has often been limited to potentially unphysical toy models and analyses of small-signal linear circuits containing pseudo-elements (resistors, capacitors, and inductors), which provide little insight into required physical, material, and device properties. Furthermore, the Local Activity concept relies on a local analysis and must be complemented by examining dynamical behavior far away from the steady-states of a circuit. In this work, we review and study a class of generic and extended one-dimensional electro-thermal memristors (i.e., temperature is the sole state variable), re-framing the analysis in terms of physically motivated definitions and visualizations to derive intuitive compact models and simulate their dynamical behavior in terms of experimentally measurable properties, such as electrical and thermal conductance and capacitance and their derivatives with respect to voltage and temperature. Within this unified framework, we connect steady-state phenomena, such as negative differential resistance, and dynamical behaviors, such as instability, oscillations, and bifurcations, through a set of dimensionless nonlinearity parameters. In particular, we reveal that the reactance associated with electro-thermal memristors is the result of a phase shift between oscillating current and voltage induced by the dynamical delay and coupling between the electrical and thermal variables. We thus, demonstrate both the utility and limitations of local analyses to understand non-local dynamical behavior. Critically for future experimentation, the analyses show that external coupling of a memristor to impedances within modern sourcing and measurement instruments can dominate the response of the total circuit, making it impossible to characterize the response of an uncoupled circuit element for which a compact model is desired. However, these effects can be minimized by proper understanding of the Local Activity theory to design and utilize purpose-built instruments.
