Probabilistic Memristive Networks: Application of a Master Equation to Networks of Binary ReRAM cells

Abstract

The possibility of using non-deterministic circuit components has been gaining significant attention in recent years. The modeling and simulation of their circuits require novel approaches, as now the state of a circuit at an arbitrary moment in time cannot be precisely predicted. Generally, these circuits should be described in terms of probabilities, the circuit variables should be calculated on average, and correlation functions should be used to explore interrelations among the variables. In this paper we use, for the first time, a master equation to analyze the networks composed of probabilistic binary memristors. Analytical solutions of the master equation for the case of identical memristors connected in-series and in-parallel are found. Our analytical results are supplemented by results of numerical simulations that extend our findings beyond the case of identical memristors. The approach proposed in this paper facilitates the development of probabilistic/stochastic electronic circuits and advance their real-world applications.

Microwave Components Realized by Additive Manufacturing Techniques

“…If one neglects low rate and/or forbidder transitions, we obtain the reduced transition scheme, which simplifies the solution of the master equation (3) (see Ref. [24] for some examples).…”

“…The dynamics of networks with discrete-state memristors can be imagined as a sequence of transitions between network states. Recently, we have introduced a master equation approach for the occupation probabilities of the network states [24] that can be used to describe circuits that include binary and multi-state memristors, resistors, voltage and current sources, and possibly some other components 1. In this previous work [24], the solution of the master equation was found analytically for networks of N in-series/in-parallel connected binary memristors driven by a constant voltage source.…”

“…It has been demonstrated in Ref. [24] that the master equation solution allows to calculate many quantities of interest including various mean switching times, mean current, resistance, etc.…”

“…There are two major advantages of the master equation compared to stochastic/Monte Carlo simulations: i) in principle, the master equation can be solved analytically (see Ref. [24] for examples), and ii) using the master equation, many network characteristics can be found in a single calculation without the need for averaging. In the case of symmetries in the circuit, the additional benefit of the master equation is its compactness.…”

Probabilistic Resistive Switching Device modeling based on Markov Jump processes

“…On the other hand, another category of inherently stochastic models is the kinetic Monte Carlo (kMC) based modeling approach [16], [17], which describes at microscopic level the field-enabled stochastic dispositions of ions within the switching layer. However, kMC is not a compact modeling approach, while there are only a few compact models developed on stochastic approaches of RS limited to the two-state (binary) switching of the device [18].…”