The State Change Per Cycle Map: a novel system-theoretic analysis tool for periodically-driven ReRAM cells

Abstract

Memristive devices are the subject of extensive studies nowadays. While the Dynamic Route Map is a powerful tool for analyzing the response of first-order memristors under DC stimuli, the development of an equivalent tool for investigating the response of these devices to AC stimuli is still an open question. Recently, Pershin and Slipko introduced a graphic method, which we name Time Average State Dynamic Route (TA-SDR), applicable to study first-order memristors subject to periodic rectangular pulse train-based stimuli. In this paper we introduce an alternative investigation tool, referred to as State Change Per Cycle Map (SCPCM), which is applicable in these very same scenarios. The novel analysis technique, inspired by the work of the French mathematician Henri Poincaré, reduces the investigation of a first-order non-autonomous continuous-time system to a simpler study of a first-order discrete-time map. A State Change Per Cycle Map defines precisely how the stimulus modulates each of the admissible device memory states over one input cycle. It is derivable either by means of numerical simulations, where a model for the ReRAM cell is available, or experimentally, in the case where the device memory state is accessible. While the predictive capability of a TA-SDR is limited to those case studies, where the AC periodic voltage signal applied across the device induces negligible changes in the respective memory state over each cycle, the conclusions drawn by analyzing a SCPCM have general validity, irrespective of the properties of the stimulus. The advantages of the novel analysis method for periodically driven ReRAM cells over the classical TA-SDR tool are highlighted through a number of case studies, some of which reveal the interesting capability of the ReRAM cell to display multiple oscillatory operating modes upon periodic stimulation via trains with a suitable number of SET and RESET pulses per period.

Theoretico-experimental analysis of bistability in the oscillatory response of a TaOx ReRAM to pulse train stimuli

“…Under these circumstances, the solution to the periodically forced ODE (Eq. 1) would exhibit bistability as long as the approximation, inherent to the time averaging method, would not corrupt the accuracy of the predictions, drawn via TA-SDR investigation, which may be verified by means of a SCPCM analysis (Ascoli et al, 2023b). Of course, ultimately, experimental tests shall have to be conducted to verify the emergence of local fading memory effects of this kind in the nano-device.…”

“…The period T of the stimulus is then τ + + τ − = 6.7667 • 10 6 s. The reset-set pulse width ratio r ≜ τ + /τ − amounts here to 1.5 • 10 -2 . (B) Δx k+1;k versus x k locus (Ascoli et al, 2023b), as extracted from the Poincaré map x k+1 P(x k ) of the periodically forced ODE (Eq. 1) in the excitation scenario under examination.…”

“…1) may evolve along one of two distinct locally stable oscillatory waveforms, upon forcing the device voltage v to follow an asymmetric periodic voltage stimulus v S , composed of a first-positive RESET 1 triangular pulse of amplitude vS,+ +0.31V, stretching across a time span τ + = 1 • 10 5 s, and of a second-negative SET triangular pulse of amplitude vS,− −0.2V, covering a time interval τ − = 6.6667 • 10 6 s, over each cycle of duration T = τ + + τ − = 6.7667 • 10 6 s [refer to plot (a)], as established by the circuit setup of Figure 2A. Plot (b) depicts the graph of the state change per cycle map (SCPCM) (Ascoli et al, 2023b), which is dictated by the vector field of the non-autonomous 2 first-order ODE (Eq. 1) under the specified excitation scenario.…”

“…The SCPCM (Ascoli et al, 2023b) provides the change Δx k+1;k ≜ x k+1 − x k , which the state x undergoes over the (k + 1) th input cycle for each k-value in the set {0, 1, 2, . .…”

“…5 Despite its high predictive accuracy, the SCPCM technique (Ascoli et al, 2023b) relies on numerical integrations, and, for this reason, is not employable to resolve this open question. It may be used, however, to verify the predictions from the time-averaging method, which holds validity only in periodic excitation scenarios, which induce small changes in the state per cycle Ascoli et al (2022).…”