Minimizing energy dissipation during programming of resistive switching memory devices using their dynamical attractor states

TL;DR

The paper addresses energy dissipation during programming of resistive switching memristors by leveraging dynamical attractor states created via alternating-polarity pulse trains. It develops a theory based optimization using the voltage threshold adaptive memristor (VTEAM) model to design ad hoc pulse sequences that minimize Joule losses while driving the device to a desired attractor, and extends the framework to fast programming under time constraints. Key contributions include closed-form expressions for the energy metric $Q$, relations between attractor location $x_a$ and pulse parameters, and two design strategies for energy efficient and fast programming depending on the kinetic exponents $\alpha_{off}$ and $\alpha_{on}$. The approach supports low-power operation in crossbar and edge computing contexts and provides a pathway toward experimental validation with real memristive devices.

Abstract

Under certain conditions, applying a sequence of voltage pulses of alternating polarities across a resistive switching memory device induces a finite number of fixed-point attractors, known as dynamical attractors. Remarkably, dynamical attractors can be used to program analog values into the device state without supervision. Because different pulse sequences can produce the same trajectory solution for the state in the phase space, there is strong potential for optimization, particularly in regard to the energy cost of the programming phase, which this study addresses. Without loss of generality, the proposed theory-based energy minimization strategy is applied to the voltage threshold adaptive memristor model, known for its predictive capability and adaptability to fit a large number of resistance switching memory devices. The optimization design crafts ad-hoc pulse sequences, that minimize the energy required to program the device into a desired dynamical attractor state. The theoretical approach is also extended to cover situations, where a fast programming scheme should be adopted to serve time-critical electronics applications.

Figures (7)

• Figure 1: Illustration of the dynamical attractor concept for a first-order memristive device. (a) Pulse sequence, consisting, over each cycle, of a pair of pulses of alternating amplitudes $V_+$ and $V_-$ and corresponding widths $\tau_+$ and $\tau_-$, and let fall across the memristive device during the programming phase. $T$ is the pulse train period, $\tau_0$ the inter-pulse interval. (b) Schematic plot demonstrating the time-averaged flow of a time-averaged internal state variable, referred to as $\bar{x}$, for the device under the periodic pulse train excitation sketched in (a). The arrow pointing to the right (left) indicates a progressive increase (decrease) in the time-averaged state from cycle to cycle. The dot marks the position of an attractor, which the pulse train stimulus induces in the phase space of the memristive device. Adapted from pershin_2019_bif_anal_TaO_mod.
• Figure 2: Example of optimal control based upon the following parameter setting: $v_{off/on}=\pm 1$ V, $\alpha_{off}=3$, $\alpha_{on}=2$, $k_{off/on}=\pm 10^3$, $V_{\pm}^{max}=\pm 3$ V, $x_0=0.1$, $x_a=0.6$, $\epsilon=0.02$, $\tau_0=10$ µs. Here, plot (a) shows the control voltage pulse sequence, where $\hat{V}_{+}=V_+^{max}=3$V and $\hat{V}_{-}=V_-^{max}=-3$V according to Eqs. (\ref{['eq:amplitudes+']}) and (\ref{['eq:amplitudes-']}), respectively, while $\tau_{+}=6.3$ µs and $\tau_{-}=8.3$ µs as follows in turn from Eqs. (\ref{['eq:tp']}) and (\ref{['eq:tm']}). The pulse sequence admitted then an input period $T$ of $34.6$ µs, as computed via Eq. (\ref{['eq:24']}). Plots (b) and (c) depict the memristor state response to the pulse train in (a) over the initial phase and across the entire programming time, respectively. In (b) and (c), the smooth line represents the analytical solution according to Eq. (\ref{['eq:analytsol']}).
• Figure 3: Example of optimal control based upon the same parameter setting as in Fig. \ref{['fig:3']}, except for $x_a$, and $\epsilon$, here taken equal to $0.5$ and $0.1$, respectively. See the caption of Fig. \ref{['fig:3']} for the significance of the curves in plots (a), (b), and (c). For this simulation, according to Eqs. (\ref{['eq:amplitudes+']}) and (\ref{['eq:amplitudes-']}) $\hat{V}_{+}$ and $\hat{V}_{-}$ were respectively set to $V_+^{max}=3$V and $V_-^{max}=-3$V, as in Fig. \ref{['fig:3']}. As a result, using the formulas (\ref{['eq:tp']}) and (\ref{['eq:tm']}), $\tau_{+}$ and $\tau_{-}$ were in turn chosen equal to $25$ µs and $50$ µs. From equation (\ref{['eq:24']}), the pulse sequence period $T$ was thus found to amount to 95 µs.
• Figure 4: Example of optimal control based upon the same parameter setting as in Fig. \ref{['fig:3']}, except for $\alpha_{off/on}$, and $\epsilon$, here set equal to $1$ and $0.1$, respectively. Here the inequalities $V_{+}^{max}>2\,v_{off}/(2-\alpha_{off})$ and $V_{-}^{max}<2v_{on}/(2-\alpha_{on})$ hold true. Refer to the caption of Fig. \ref{['fig:3']} for information on the traces in plots (a), (b), and (c). For this simulation Eqs. (\ref{['eq:amplitudes+']}) and (\ref{['eq:amplitudes-']}) respectively set the optimal RESET and SET pulse amplitudes, i.e., in turn, $\hat{V}_{+}$ and $\hat{V}_{-}$, to $2\,v_{off}/(2-\alpha_{off})=2$V and $2\,v_{on}/(2-\alpha_{on})=-2$V, respectively. As a result, $\tau_{+}$ and $\tau_{-}$ were set to $250$ µs and $167$ µs, on the basis of Eqs. (\ref{['eq:tp']}) and (\ref{['eq:tm']}), respectively. Therefore, $T$ was found to be equal to $437$ µs according to equation (\ref{['eq:24']}).
• Figure 5: Minimizing Joule losses in the presence of constraints on pulse amplitudes and widths in the case where $\alpha_{\text{off}} < 2$ and $\alpha_{\text{on}} < 2$. The allowable variation ranges for $V_+$ and for $-V_-$ are $(v_{off},V_{+}^{max})$ and $(-v_{on},-V_{-}^{max})$, respectively. (a) A schematic illustration of the scenario where $V_+^{max}>2v_{off}/(2-\alpha_{off})$ and $|V_-^{max}|<2|v_{on}|/(2-\alpha_{on})$. (b) A schematic illustration of the scenario where $V_+^{max}>2\,v_{off}/(2-\alpha_{off})$ and $|V_-^{max}|>2|v_{on}|/(2-\alpha_{on})$. It is worth observing that, when both inequalities $\alpha_{\text{on}}<2$ and $\alpha_{\text{off}}<2$ apply, the contours of the function $Q=Q(V_+,V_-)$ are closed loci, which originates from the fact that in situations of this kind there always exists finite positive and negative pulse amplitudes, in turn $V_+$ and $V_-$, at which Joule losses are minimized under RESET and SET transitions, respectively, in the unconstrained case (refer to Fig. \ref{['fig:6']}(b)). On the other hand, in case $\alpha_{\text{on}}$ and/or $\alpha_{\text{off}}$ were larger than or equal to $2$, the contours of the function $Q=Q(V_+,V_-)$ would be open curves. In scenarios of this kind, if the first (second) inequality from the aforementioned pair held true, the most energetically-favourable SET (RESET) pulse amplitudes $V_-$ ($V_+$) would be negative (positive) infinity under unconstrained conditions (refer to Fig. \ref{['fig:6']}(a)). Refer to the text for additional information.