Low-Power Control of Resistance Switching Transitions in First-Order Memristors

TL;DR

The paper addresses reducing Joule heating during resistance-switching in first-order memristors by formulating a Pontryagin-based optimal-control framework that converts the time-integrated energy cost into a state-domain problem via a variable transformation. It applies the framework to the VTEAM model and, in the Appendix, the dynamic-balance model, deriving unconstrained and constrained optimal voltage protocols that can be as simple as a single square pulse or as complex, state-dependent waveforms depending on device parameters like $\alpha_{off/on}$ and threshold dynamics. Key findings include closed-form optimal amplitudes $V^*_{off/on}$ for $\alpha<2$, a set of Case-based constrained solutions with maximal or state-dependent voltage profiles, and a shortest-programming-time bound $T^{min}_{off}$ that governs feasibility; invariants of motion appear for certain regimes, informing the structure of optimal strategies. The results suggest substantial energy savings for programming large memristor arrays, offering practical guidance for low-power ReRAM and in-memory computing applications and motivating experimental validation and extensions to broader memristive models.

Abstract

In many cases, the behavior of physical memristive devices can be relatively well captured by using a single internal state variable. This study investigates the low-power control of first-order memristive devices to derive the most energy-efficient protocols for programming their resistances. A unique yet general approach to optimizing the switching transitions in devices of this kind is introduced. For pedagogical purposes, without loss of generality, the proposed control paradigm is applied to a couple of differential algebraic equation sets for voltage-controlled devices, specifically Kvatinsky's Voltage ThrEshold Adaptive Memristor mathematical description and Miranda's and Sune's dynamic balance model. It is demonstrated that, depending upon intrinsic physical properties of the device, captured in the model formulas and parameter setting, and upon constraints on programming time and voltages, the optimal protocol for either of the two switching scenarios may require the application of a single square voltage pulse of height set to a certain level within the admissible range across a fraction or entire given programming time interval, or of some more involved voltage stimulus of unique polarity, including analogue continuous waveforms that can be approximated by trains of square voltage pulses of different heights, over the entire programming time interval. The practical implications of these research findings are significant, as the development of energy-efficient protocols to program memristive devices, resolving the so-called voltage-time dilemma in the device physics community, is a subject under intensive and extensive studies across the academic community and industry.

Figures (10)

• Figure 1: Schematic illustration of a scenario, where $f(x,V)$ becomes equal to $0$ within the programming phase, for an arbitrary non-volatile memristor model. The original time interval $[t_i,t_f]$, of duration equal to a pre-defined programming time $T=t_f-t_i$, is made up of a first time interval $[t_i,t_s]$, of width $T_c=t_s-t_i<T$, where $f(x,V)$ is non-zero, featuring a unique sign -- assumed here to be positive (negative) for $V>(<)0$ so as to induce a monotonic increase (decrease) in the state $x$ -- and the device undergoes a resistance switching transition, which may be of SET (RESET) or RESET (SET) nature, depending on the physical origin of its state $x$, and of a second time interval $(t_s,t_f]$, of width $T-T_c$, where $f(x,V)$ is identically null, and $x$ keeps unchanged, as is the case for non-volatile memristors. As for all the simulation results, shown in this paper, we assumed $t_i=0$, resulting in the identity $T = t_f$.
• Figure 2: Unconstrained control of the RESET (SET) resistance switching transition in a VTEAM memristor subjected to a square voltage pulse: (a) programming time $T_{off(on)}$ from Eq. (\ref{['eq:Toff:VTEAM:1']}), ((\ref{['eq:Ton:VTEAM:1']})) and (b) switching energy $Q_{off(on)}$ from Eq. (\ref{['eq:Qoff:VTEAM:1']}), ((\ref{['eq:Qon:VTEAM:1']})) against the positive dimensionless parameter $V_0/v_{off(on)}$ for some values, assigned to the exponent $\alpha_{off(on)}$.
• Figure 3: Comparison between the RESET energy costs incurred during the application of the constant voltage-based and optimal voltage-based switching strategies to a VTEAM memristor, for a couple of values, specifically $2$ and $3$, assigned to the parameter $\alpha_{off}$. (a) RESET switching energy $Q_{off}$ and (b) pulse amplitude $V_0$ against programming time $T$. The VTEAM model parameter setting, employed in these investigations, reads as follows: $w_i=0.1$, $w_f=0.9$, $w_{on}=0$, $w_{off}=1$, $v_{off}=1$ V, $k_{off}=10^5$ s$^{-1}$, $G_{min}=10^{-5}$ S, $G_{max}=10^{-3}$ S, $V_1=1$ V, $V_2=5$ V. $T^{min}_{off}$ is found to be equal to $1.84$ µs for $\alpha_{off}=2$ and to $0.460$ µs for $\alpha_{off}=3$.
• Figure 4: Comparison between the RESET energy costs due to the application of the constant voltage-based and optimal voltage-based control paradigms to a VTEAM memristor for $\alpha_{off}=1$. (a) RESET switching energy $Q_{off}$ and (b) pulse amplitude $V_0$ against programming time $T$. For the optimal switching control protocol, only the case $T>T_{off}^*$ is visualized in plot (b). To avoid clutter, in fact, the case $T_{off}^{min}<T<T_{off}^*$ is accounted for in plot (c), showing the dependence of the optimal control voltage $\hat{V}(w)$ from Eq. (\ref{['eq:VTEAM:Vhat:7']}) upon the device state $w$ for a number of values assigned to the programming time. The values for $T$, labeling the loci in (c), are indicated in (b) as short vertical lines crossing the horizontal axis. (d) Time course of the memristor state $w$ upon assigning a couple of values (one value) from the interval $T_{off}^{min}<T<T_{off}^{*}$ ($T>T_{off}^{*}$), specifically $6$ µs and $8$ µs ($50$ µs), to the programming time $T$, according to the optimal voltage-based switching control protocol. The time evolution of $w$ under the constant voltage-based switching control protocol is also shown for one case, i.e. when $T=50$ µs. Except for $\alpha_{off}$, set here to $1$, the very same parameter setting as reported in the caption of Fig. \ref{['fig:8']} was assumed for the numerical analysis this plot illustrates. Notably, here $V_{off}^{*}=2$ V, $V_2=5$ V, $T_{off}^{min}=5.493$ µs, and $T_{off}^{*}=21.972$ µs.
• Figure 5: Summary of the proposed switching control protocol for inducing the most energetically-favorable RESET transitions across a first-order memristive device modelled via the VTEAM mathematical description. It recommends a voltage $\hat{V(w)}$, to be applied across the device for any value the state $w$ assumes as it increases from some initial value $w_i$ to some final value $w_f$ across the respective existence domain $[w_{on},w_{off}]$, on the basis of the available programming time $T$, for each of two possible RESET control regimes, depending critically upon $\alpha_{off}$, $V_1=v_{off}$, and $V_2$. Each of the two panels (a) and (b) includes a qualitative sketch for the time course of the optimal control voltage waveform, including a graphical indication, based on the use of numbered circles, for any case from section \ref{['sec:3a3']}, it refers to, at the critical points, separating the regions, the $T$ axis is partitioned into, and at one arbitrary point within each of these regions. In any scenario, where $T_c<T$, $\hat{V}$ is enforced to $0$ for the $(T-T_c)$-long remainder of the programming phase. A similar control scheme for reducing Joule losses across the device, the VTEAM model is fitted to, as it undergoes SET switching transitions, could be easily conceived.

Theorems & Definitions (2)

• Remark 1
• Remark 2