Variability in Resistive Memories

Alexander A. Dubkov; Alexey N. Mikhaylov; Bernardo Spagnolo; David Maldonado; Emili Salvador; Enrique Miranda; Francesca Campabadal; Francisco J. Alonso; Francisco Jim\'enez-Molinos; Juan B. Rold\'an

REVIEW 3 major objections 4 minor 255 references

Variability in Resistive Memories

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This review argues that cycle-to-cycle variability—the run-to-run scatter in switching voltages and resistances—is the fundamental obstacle to resistive memory development, and that a compact stochastic model with thermodynamically…

desk verdict A genuinely useful review of RRAM cycle-to-cycle variability with a strong modeling taxonomy; the original B/kBT extraction is shakier than the prose admits, but the synthesis deserves a serious referee. read the letter →

arxiv 2411.12369 v1 pith:NXGLRAS5 submitted 2024-11-19 physics.app-ph

Juan B. Roldán , Enrique Miranda , David Maldonado , Alexey N. Mikhaylov , Nikolay V. Agudov , Alexander A. Dubkov , Maria N. Koryazhkina , Mireia B. González

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Marco A. Villena Samuel Poblador Mercedes Saludes-Tapia Rodrigo Picos Francisco Jiménez-Molinos Stavros G. Stavrinides Emili Salvador Francisco J. Alonso Francesca Campabadal Bernardo Spagnolo Mario Lanza Leon O. Chua

This is my paper · ORCID

classification physics.app-ph

keywords cycle-to-cyclevariabilityresistiveswitchingRRAMcompactstochasticmodelLangevinequationfluctuation-dissipationrelationsconductivefilamenttime-seriesanalysis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that cycle-to-cycle variability is not a peripheral reliability problem but the central obstacle to resistive memory (RRAM) development, because the same stochastic ion and vacancy dynamics that enable resistive switching also produce run-to-run scatter in switching voltages, currents, and resistance states. It reviews experimental evidence across HfO2, ZrO2(Y), and 2D-material devices, then compares physical, stochastic, and behavioral modeling approaches, foregrounding a compact stochastic model. In that model, the conductive-filament length evolves as overdamped Brownian motion in a tilted periodic potential, with a white thermal noise source whose intensity is fixed by the Einstein relation and fluctuation–dissipation theorems. The authors claim the model reproduces switching-time statistics, the larger variability of the high-resistance state, and noise-induced phenomena such as stochastic resonance and transient bimodality, and conclude that EDA tools must treat RRAM parameters as correlated stochastic processes rather than fixed distributions.

What carries the argument

The central object is the compact stochastic model, a coarse-grained Langevin description of the conductive filament length $y(t)$ of a filamentary RRAM. Starting from a tilted periodic potential $U(y,V) = \Phi(y) - B V y$ and an overdamped Brownian-motion equation with white Gaussian thermal noise of intensity $2 k_B T \mu$, the authors coarse-grain over the lattice period to obtain effective drift and diffusion coefficients $v_{\mathrm{eff}}(V) = (2l/\tau_{\mathrm{kr}}) \sinh(BV/k_B T)$ and $D_{\mathrm{eff}}(V) = (2l/\tau_{\mathrm{kr}}) \cosh(BV/k_B T)$. The Fokker–Planck equation for the probability density has a Boltzmann stationary solution, and the mean first-passage time to switch states reduces to Kramers' time $\tau = \tau_0 \exp[(E_a - B|V_0|)/k_B T]$. The ratio $B/k_B T$ is estimated from the slope of $\lg \Theta$ versus $V_0$, and the same model yields predictions for stochastic resonance and transient bimodality.

What would settle it

Measure the mean switching time $\Theta(V_0)$ on a ZrO2(Y) device over a range wider than the 1.0–1.2 V window used here (say 0.8–1.5 V). If the plot of $\lg \Theta$ versus $V_0$ bends downward as described by $\lg \Theta = a_1 + (E_a/k_B(T_0 + A V_0^2)) \lg e - (B V_0/k_B(T_0 + A V_0^2)) \lg e$ with $A \gtrsim 100$, the constant-temperature premise fails and the extracted $B/k_B T = 30.5 \; \mathrm{V}^{-1}$ would be contaminated by Joule heating.

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Extended reading notes

Core claim

This review's central claim is that cycle-to-cycle variability is inherently rooted in the resistive-switching mechanism and must therefore be represented by stochastic, not deterministic, compact models. On the modeling side, the paper's distinctive contribution is a lumped compact stochastic model in which the state variable is the length of the conductive filament and the dynamics are governed by a Langevin equation with an effective drift $v_{\mathrm{eff}}(V) = (2l/\tau_{\mathrm{kr}}) \sinh(BV/k_B T)$ and an effective diffusion constant $D_{\mathrm{eff}}(V) = (2l/\tau_{\mathrm{kr}}) \cosh(BV/k_B T)$. Because the noise intensity is tied to dissipation through the Einstein relation and fluctuation–dissipation theorems, the model is thermodynamically consistent, and its stationary solution is the Boltzmann distribution. The paper shows that the model's parameters can be estimated from the slope of the measured mean switching time versus voltage, yielding, for example, $B_{\mathrm{RES}}/k_B T = 30.5 \; \mathrm{V}^{-1}$ for Au/Ta/ZrO2(Y)/Pt devices, and that the same framework reproduces stochastic resonance and transient bimodality observed in experiments.

Load-bearing premise

The model's key fitting step assumes the device stays at a constant temperature while the switching voltage is varied from 1.0 to 1.2 V; if the device heats up as the voltage rises, the extracted parameter is wrong and the model's quantitative predictions no longer follow.

Editorial extensions

If this is right

RRAM compact models in EDA tools must include stochastic noise sources whose intensity is tied to dissipation via fluctuation–dissipation relations, and the model parameters must be treated as correlated in time, not as independent random draws.
The model explains quantitatively why the high-resistance state varies more than the low-resistance state: the effective potential has a minimum at the LRS boundary but not at the HRS during reset, so the HRS distribution remains broad and nonstationary.
Extracted parameters such as $B/k_B T$ give circuit designers a direct route to switching-time statistics and to predicting noise-induced effects (stochastic resonance, transient bimodality) that could be used or suppressed in neuromorphic and security circuits.
The reviewed hierarchy of models (physical, stochastic, behavioral) provides a practical guide: deep physical models for device physics, compact stochastic models for circuit simulation, and time-series models for system-level forecasting of switching voltages.
Time-series ARIMA models reproduce the cycle-to-cycle 'memory' of the conductive filament, allowing prediction of set and reset voltages from previous cycles.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If variability is truly inherent to the switching mechanism, then attempts to eliminate it entirely may also suppress the switching itself; the practical design target is to confine and control variability, not to abolish it.
The constant-temperature assumption could be tested by measuring the $\lg \Theta$ versus $V_0$ slope at several well-controlled ambient temperatures; if the slope scales with $1/T$ as predicted, the extraction is consistent, and if not, Joule heating is likely contaminating the fit.
The same Langevin machinery could be extended to device-to-device variability by adding a quenched spatial noise term into the potential profile, which the paper sets up but does not develop; that would give a unified treatment of C2C and D2D variability.
The model's prediction of stochastic resonance in RRAMs suggests that a deliberately applied noise signal could be used as a control knob for switching reliability—a design idea the paper mentions but leaves as an implicit consequence.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: the reader's conditional verdict is about right. The core message—C2C variability is intrinsic to RRAM switching and has to be represented in compact models for EDA—is well supported by the literature they survey. The taxonomy of physical, stochastic, and behavioral models, plus the comparison in Table 6, is the real contribution. Someone entering the field or choosing a modeling strategy will save real time here.

What it does well: the mesoscopic/Landauer section is standard but correct, and the Langevin-to-Fokker-Planck derivation is internally consistent. The paper is also honest in the right places—it explicitly says the Stanford model's noise term is a fitting function without deep physical justification, and it concedes that set/reset voltage variations are not fully reproduced. The citations are broad and appropriate; self-citation is not a problem because much of the modeling work is theirs and they engage competing groups.

Soft spots, in proportion. The new ZrO2(Y) measurement is one device, 114 repetitions per voltage, no device-to-device error bars—fine as a demonstration, not a technology-level claim. The DMM section uses hand-picked sigma values; the text basically admits it is illustrative. The more substantive issue is the Section 4.3 extraction of B/kBT from the slope of lgTheta(V0). The slope reading requires T independent of V0, and the paper uses the absence of visible curvature to rule out Joule heating. That inference is too quick. With their own Ea/kBT0 around 40 and B/kBT0 = 30.5, modest heating (A = 50–100 K/V^2) produces an almost-straight lgTheta(V0) whose fitted slope is biased by 10–20%. So the reported BRES/kBT = 30.5 ± 0.6 is not robust at that precision, and the subsequent quantitative descriptions of stochastic resonance and transient bimodality inherit that fragility. The paper hedges with "qualitatively, or even quantitatively," so the overclaim is mild, but the parameter is load-bearing.

Who benefits: graduate students and circuit/model developers who need a map of variability modeling options, plus experimentalists who want better statistical presentation—Section 2.3 is a nice corrective to "representative curve" habits. It is a review, not a breakthrough, and the central argument holds up. I would send it to peer review: it deserves a serious referee, and the fix is tightening the quantitative claims with independent device statistics, held-out validation, or explicit uncertainty propagation rather than rejecting the synthesis.

Referee Report

3 major / 4 minor

Summary. This review manuscript surveys cycle-to-cycle (C2C) variability in resistive switching memories from experimental, physical, stochastic, and behavioral modeling perspectives. It presents original experimental data on 3000-cycle and 10^5-cycle HfO2-based devices, a Stanford-model variability extension, a quantum-point-contact (Landauer) treatment of HRS conduction, a compact stochastic model built from a Langevin/Fokker-Planck description with fluctuation–dissipation-consistent thermal noise, a charge-flux behavioral model with Monte Carlo parameter variability, the dynamic memdiode model with uncorrelated C2C parameter variability, and time-series (ARIMA) models of Vset and Vreset. The central thesis is that C2C variability is inherent to the switching mechanism and must therefore be incorporated into compact models for EDA tools. Original quantitative claims include the extraction of BRES/kBT = 30.5 ± 0.6 V^-1 from ZrO2(Y) data (Section 4.3) and the assertion that the resulting model quantitatively describes stochastic resonance, resonant activation, and transient bimodality.

Significance. The review fills a real gap: it draws together experimental phenomenology, physical transport models, stochastic thermodynamics, and behavioral/statistical compact models into a single framework, and Table 6 provides a useful practical orientation for circuit designers. The derivation of the compact stochastic model from a Langevin equation whose noise intensity satisfies the fluctuation–dissipation relation is a genuine strength, as is the exact analytical mean-first-passage-time expression (Eq. 73). If the quantitative parameter extraction were secured, the paper would provide a practically usable recipe for generating correlated, physics-consistent variability in RRAM compact models. However, the original quantitative support is currently anchored to a single parameter-extraction procedure whose key assumption is not quantitatively justified, and the model-validation claims lack an independent holdout test.

major comments (3)

[Section 4.3, Eqs. (76) and (80)] The extraction of BRES/kBT = 30.5 ± 0.6 V^-1 from the slope of lg θ(V0) assumes T is independent of V0 over the measured 1.0–1.2 V window. The manuscript argues that because the measured dependence is indistinguishable from a straight line, the temperature increase is insignificant. That inference is not quantitative. Using the paper's own values (Ea/kBT0 = 40.3, B/kBT0 = 30.5 V^-1, T0 = 300 K), Eq. (80) with A = 50 K/V^2 already yields an almost-linear lg θ(V0) whose best-fit slope corresponds to B/kBT ≈ 27 V^-1, about 11% below the constant-T value; with A = 100 K/V^2 the apparent slope is about 24 V^-1, roughly 20% below. The absence of visible curvature therefore does not establish T(V0) = const, and the extracted value is not robust at the stated precision. Because this parameter is subsequently used to support quantitative descriptions of stochastic resonance and transient bimodality (final paragraph of Section 4.3), the quantitative component of the central claim is not yet secured. Please provide an independent bound on A, refit with T(V0), or explicitly reframe the stochastic-resonance and transient-bimodality statements as qualitative.
[Section 4.3, final paragraph] The claim that the compact stochastic model 'quantitatively describe[s]' stochastic resonance, resonant activation, and transient bimodality is not supported by an independent test. The parameters BSET/kBT and BRES/kBT are estimated from the same lg θ(V0) dataset that is used to demonstrate the model's capability, and no holdout validation is reported—for example, no prediction of a different observable, a different voltage range, or a different temperature. The agreement is therefore a consistency check rather than a quantitative confirmation. The authors should either supply a genuine predictive test or soften the wording to indicate model consistency with the observed phenomena.
[Section 4.3, ZrO2(Y) measurements] The new parameter extraction is performed on a single device: the text states 'All the measurements were carried out on the same memristive device,' with 114 repetitions per voltage amplitude. Consequently, the reported uncertainty 30.5 ± 0.6 V^-1 reflects only the statistical uncertainty of the averaged mean first-passage time for one device, not device-to-device variability. As written, Table 4 and the surrounding discussion present this as though it were a technology-level parameter. Please state explicitly that the value is single-device and, if possible, augment it with multi-device statistics or clearly restrict the claim.

minor comments (4)

[Abstract] The word 'adequation' is nonstandard; 'adequacy' or 'suitability' would be clearer.
[Section 4.3, text after Eq. (77)] The manuscript should state whether the uncertainty in BRES/kBT = 30.5 ± 0.6 V^-1 comes from the linear regression only or also from the repeatability across the 114 waveforms; the current description is ambiguous.
[Table 4] The row for Au/Ta/ZrO2(Y)/Pt lists BRES/kBT = 30.5 without an uncertainty, while the text reports ±0.6 V^-1; please make the table consistent with the text.
[Section 5.2, Table 5] The model script is useful but the distinction between 'normal' and 'lognormal' Gaussian additions is shown only by font color; please add an explicit text marker or column so the distinction survives monochrome printing.

Circularity Check

1 steps flagged · score 2.0 of 10

Only a minor in-sample 'prediction' in the TSSA demonstration; the central stochastic model is calibrated from data and backed by external experimental confirmations, so no significant circularity.

fitted input called prediction [Section 5.3 (Time-Series Modeling), Equations (102)-(103) and Figure 37]
"After some algebra to isolate the current set and reset voltages the final models are obtained, see Equation (102) for the set voltage Vsett = Vsett−1 − 0.8634εt−1 and (103) for the reset voltage Vresett = Vresett−1 − 0.5811εt−1 + 0.0924εt−2. These models can be used to predict the values of the series obtained in the laboratory for the devices under study, as shown in Figure 37."

The ARIMA coefficients in (102)-(103) were estimated from the same measured Vset/Vreset series whose ACF/PACF are analyzed in Figures 34-36. Figure 37 then plots the measured and 'predicted' values for that same series. Because the ε terms are the fitted residuals of the identical dataset, the displayed 'predicted' curves are the in-sample fit reconstituted, not an out-of-sample forecast. This is a fitted-input-called-prediction step, but it is a pedagogical demonstration in a review section and does not support the paper's central claims.

full rationale

The paper is primarily a review, and most of its modeling demonstrations are standard model calibrations rather than circular predictions. The central compact stochastic model is constructed from a Langevin equation with a fluctuation-dissipation-consistent thermal noise source (Section 4.2.2), and the key parameter BRES/kBT is extracted by a slope fit to measured mean-first-passage-time data (Equations 76-78, Figure 25). The subsequent statements that the model can 'qualitatively, or even quantitatively, describe' stochastic resonance, resonant activation, and transient bimodality refer to prior experimental studies (refs. 63, 190, 221), which are external evidence rather than a re-use of the same fitted quantity within this paper. I found one concrete reduction by construction: the ARIMA time-series models in Section 5.3 are fitted to the full experimental series and then presented as 'predicted' values on the same series; this is statistically forced, but it is a minor illustrative part of the review and not load-bearing. The skeptic's concern about the constant-temperature assumption in Equation (76) versus the possible T(V0) = T0 + AV0^2 dependence in Equation (80) is a legitimate parameter-identification robustness issue — even moderate Joule heating could bias B/kBT by 10-20% while leaving the plot nearly linear — but it is a fitting validity concern, not circularity. No load-bearing self-citation chain was found: the compact model equations are re-derived Kramers-type results, and the cited confirmations are experimental and externally falsifiable. Overall circularity is minimal.

Assumptions & free parameters 9 free parameters · 7 assumptions · 0 invented entities

No new physical entities are postulated. The models use established constructs from prior literature, such as the conductive filament, oxygen vacancies, the quantum point contact barrier, and the memristor state variable. The paper's original elements are fitted model parameters and statistical descriptions, not new forces, particles, or conserved quantities.

free parameters (9)

SM C2C variability parameters (delta0g, Tcrit, Tsmth) = delta0g = 0.5 nm, Tcrit = 450 K, Tsmth = 400 K
Tuned to reproduce the experimental cycle-to-cycle spread in Figure 14; the paper calls the gap variation a fitting function without significant physical explanation.
Stanford model I-V fitting parameters (I0, g0, V0, nu0, gamma0, alpha, beta) = Values in Tables 1 and 2, varying by set/reset
Extracted by fitting experimental I-V curves for HfO2 and VCM devices; different values for set and reset processes.
Series resistance Rseries = 22.3 ohm for TiN/Ti/HfO2/W stack
Extracted using the normalized voltage method; the Rseries value affects I-V curve shape and the resulting C2C variability.
Quantum point contact barrier height distribution (phi0i, sigma_i^2) = Not specified numerically; normal distribution assumed
Gaussian spread of the confinement barrier height is introduced to generate HRS variability, following refs [158,162].
Compact stochastic model B/kBT (BSET and BRES) = BRES/kBT = 30.5 +/- 0.6 V^-1 for ZrO2(Y); literature values in Table 4
Extracted from the slope of lg Theta(V0) and used to set the field-tilt potential and to simulate stochastic phenomena.
Joule heating coefficient A in T = T0 + A V0^2 = Illustrative values A = 100 and A = 400
Ad hoc values used to show how temperature rise bends the lg Theta(V0) curve; no direct temperature measurement is provided.
Charge-flux behavioral model parameters (G0, n) and Monte Carlo error terms = G1 = 14.31 S, G2 = -7.03 S, sigma = 152e-6 S, mu = -22.4e-6 S; n transition sigma = 0.07, mu = 0.10
Fitted to conductance versus pulse number data for ten cycles; used for the two-parameter Monte Carlo variability model.
DMM variability sigma values (sigioff, sigion, sigisb, sigvr) = 0.25, 0.1, 5e-6, 0.02
Hand-selected to produce the desired cycle-to-cycle spread in simulated I-V loops; no calibration against experimental statistics is shown.
ARIMA coefficients for Vset and Vreset = Vset: -0.8634; Vreset: -0.5811 and 0.0924
Fitted to measured Vset and Vreset series; they capture autocorrelation but no parameter uncertainties or holdout validation are reported.

assumptions (7)

standard math Landauer-Buttiker formalism with inverse parabolic barrier transmission coefficient
Used in Section 3.2, Equations (4) to (6), to model HRS conduction through a quantum point contact; standard mesoscopic transport.
standard math Fluctuation-dissipation relation D = kBT mu and equivalence between Langevin and Fokker-Planck equations
Section 4.2.2 derives the multiplicative noise intensity from the Gibbs equilibrium distribution; standard statistical physics.
domain assumption RRAM switching arises from field- and temperature-assisted migration of oxygen vacancies or metal ions forming and rupturing a conductive filament
Foundation of Section 2.1 and of the Stanford, stochastic, and memdiode models; well established in the cited literature.
domain assumption Cycle-to-cycle variability can be represented by white Gaussian noise on a lumped state variable (gap g or filament length y) with normal or lognormal parameter dispersion
Used in Sections 3.1, 4.2.3, and 5.2; adopted for compactness. The paper acknowledges that more complex noise such as 1/f noise and Levy flights exists.
domain assumption Active-region temperature T is constant over V0 = 1.0 to 1.2 V in the ZrO2(Y) MFPT experiment
Section 4.3, based on the straight-line lg Theta(V0) fit; this is load-bearing for the B/kBT extraction and is questioned by the temperature-dependence discussion around Equation (80).
ad hoc to paper Effective coarse-grained potential Ueff(y,V) = veff(V) y with veff and Deff given by Equations (66) and (67)
Derived from Kramer's escape over a periodic potential, but the linear-in-y effective potential and the neglect of spatial noise are simplifying assumptions for a compact model.
standard math ARIMA stationarity and ACF/PACF model selection assumptions
Section 5.3 applies standard time-series methodology, including first differencing to achieve stationarity and threshold bounds of +/-0.0619 for autocorrelation.

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Pith. "Pith review of Variability in Resistive Memories." pith.science (2026). https://pith.science/paper/NXGLRAS5

@misc{pith2026241112369,
  author       = {Pith},
  title        = {Pith review of: Variability in Resistive Memories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NXGLRAS5}},
  note         = {Machine review of arXiv:2411.12369}
}

read the original abstract

Resistive memories are outstanding electron devices that have displayed a large potential in a plethora of applications such as nonvolatile data storage, neuromorphic computing, hardware cryptography, etc. Their fabrication control and performance have been notably improved in the last few years to cope with the requirements of massive industrial production. However, the most important hurdle to progress in their development is the so-called cycle-to-cycle variability, which is inherently rooted in the resistive switching mechanism behind the operational principle of these devices. In order to achieve the whole picture, variability must be assessed from different viewpoints going from the experimental characterization to the adequation of modeling and simulation techniques. Herein, special emphasis is put on the modeling part because the accurate representation of the phenomenon is critical for circuit designers. In this respect, a number of approaches are used to the date: stochastic, behavioral, mesoscopic..., each of them covering particular aspects of the electron and ion transport mechanisms occurring within the switching material. These subjects are dealt with in this review, with the aim of presenting the most recent advancements in the treatment of variability in resistive memories.

Figures

Figures reproduced from arXiv: 2411.12369 by the authors.

**Figure 1.** b shows the typical current–voltage (I–V) characteristics during the forming process and the first complete RS cycle for a HfO2-based RRAM. In many cases, after device fabrication (in the pristine state) prior to the RS operation, the devices need to be formed in order to generate an oxygen vacancy (Vo) rich filamentary path. The forming process is performed by applying a positive voltage ramp until a sudden and abr… view at source ↗

**Figure 2.** for Ion (g), Ioff (h), Ireset (j), and Iset (k). The associated Pearson correlation coefficients (ρ) are given in the insets [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** a. Notice that the maximum and minimum voltages applied to the device in pulse operation mode are programmed to be the same as for the single sweep cycle in Figure 2a, being in the studied devices þ1.1 and 1.4 V, respectively. However, it is worth mentioning that the time, and therefore, the energy consumption during sweep and pulse modes differs. Figure 3b shows the evolution of Ion and Ioff at VREAD ¼0.1 V for … view at source ↗

Figures from the paper (34 more)

**Figure 4.** Figure 4: Evolution of the HRS and LRS currents at 0.1 V versus the number of cycles measured for a long resistive switching series of voltage ramps that lead to consecutive set and reset processes in filamentary RRAMs. Several cycle-to-cycle key variability issues are evidence…

**Figure 6.** Figure 6: shows.[96–98] The integration of 2D materials with all electronic components could extend their functionality and help to overcome fundamental device challenges such as the gate leakage current of the transistor, temperature dissipation, and transparency.[99–104] In t…

**Figure 5.** Figure 5: Schematic of a) 1T1R, b) 1S1R, and c) 1T2R memory cell. All these types of memory cells are placed in an array where WL is the word-line, BL is the bit-line, SL is the sense-line, and EL is the enable-line [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: Schematic of the main defect types that can be found in any exfoliated 2D material. These defects are (top): 1 thickness fluctuations, 2 defective bonding, 3 rests of polymer particles from the transfer scaffold, 4 wrinkles, 5 suspended 2D materials, 6 twin boundaries,…

**Figure 8.** Figure 8: b, where the darker regions identify the most probable electrical response of the device. According to this result, it is straightforward to conclude that the highlighted black solid line in Figure 8a is a better representative I–V curve for this device because it over…

**Figure 9.** Figure 9: Example of a correctly fitted Weibull distribution showing the parameters of the statistical distribution. Reproduced with permission.[250] Copyright 2021, Wiley-VCH. www.advancedsciencenews.com www.advintellsyst.com Adv. Intell. Syst. 2023, 5, 2200338 2200338 (10 of 4…

**Figure 10.** Figure 10: Example of different recommended methodologies to estimate and quantify the C2C variability; a) histogram, b) cycle-to-cycle progression, c,d) boxplot. a,b) Reproduced with permission.[251] Copyright 2022, American Chemical Society. c,d) Reproduced with permission.[12…

**Figure 11.** Figure 11: Example of a boxplot used to demonstrate the C2C and D2D variability of the device under study. Reproduced with permission.[120] Copyright 2020, Springer Nature. www.advancedsciencenews.com www.advintellsyst.com Adv. Intell. Syst. 2023, 5, 2200338 2200338 (11 of 46) ©…

**Figure 12.** Figure 12: Current versus voltage for experimental data (symbols) and simulated data (lines) for the electrochemical metallization cells described in the text. These simulations were obtained by employing the Stanford model using the parameters presented in [PITH_FULL_IMAGE:fig…

**Figure 13.** Figure 13: Current versus voltage set and reset cycles for a) experimental, b) simulated data for the investigated devices with unipolar RS. The SM was employed including the variability module using Equation (1) with the following parameters V0 ¼ 0.75 V, I0 ¼ 3 mA, g0 ¼ 0.167 n…

**Figure 14.** Figure 14: a) RLRS measured at 0.1 V versus cycle number for the experimental and simulated (inset) curves in the whole RS series, b) cumulative distribution functions for the values plotted in (a) [PITH_FULL_IMAGE:figures/full_fig_p012_14.png]

**Figure 15.** Figure 15: a (15b). As the δ0 g parameter rises, the current variations increase too. This fact arises since δ0 g stands for the gap deviation, which affects the final g value, that is responsible for the total current flowing across the device. In Figure 15c, the role of Tcrit …

**Figure 16.** Figure 16: a) Current versus voltage for experimental (symbols) and simulated data (lines) to reproduce the typical curve shape of valence change memory cells. Each color line represents a simulation obtained for a different series resistance; the selected value to reproduce the…

**Figure 17.** Figure 17: a) Schematic of the confinement potential barrier model for HRS. The dashed red solid line corresponds to a narrow (φN) constriction, while the blue solid line corresponds to a wide (φW) constriction. The blue boxes at both sides of the barrier are the cathode and ano…

**Figure 18.** Figure 18: a) Model curves for HRS and LRS using expression 6. As the confinement barrier height increases, the current that flows through the device decreases. I ¼ G0V corresponds to the complete absence of a tunneling barrier. b) Distribution of the I–V curves (N ¼ 100) for a …

**Figure 19.** Figure 19: a) Evaluation of the barrier height fluctuation using expression 9 for the current variation represented in Figure 18b. b) Evaluation of the barrier height fluctuation using expression 10. The difference between the two histograms arises because the data sets are not …

**Figure 20.** Figure 20: Theoretical approaches to modeling memristive devices. www.advancedsciencenews.com www.advintellsyst.com Adv. Intell. Syst. 2023, 5, 2200338 2200338 (18 of 46) © 2022 The Authors. Advanced Intelligent Systems published by Wiley-VCH GmbH 26404567, 2023, 6, Downloaded f…

**Figure 21.** Figure 21: Schematic representation of a memristive device with a conductive region with RON ROFF. The conductive region length can be selected as a lumped state variable y(t). www.advancedsciencenews.com www.advintellsyst.com Adv. Intell. Syst. 2023, 5, 2200338 2200338 (21 o…

**Figure 22.** Figure 22: The electric circuit with memristor and capacitor. www.advancedsciencenews.com www.advintellsyst.com Adv. Intell. Syst. 2023, 5, 2200338 2200338 (23 of 46) © 2022 The Authors. Advanced Intelligent Systems published by Wiley-VCH GmbH 26404567, 2023, 6, Downloaded from …

**Figure 23.** Figure 23: Linear part of the dependence of the logarithm of the average switching time lgΘ of a memristive device (in the transition from LRS to HRS) on the switching voltage V0 at a constant temperature (kBT ¼ const). www.advancedsciencenews.com www.advintellsyst.com Adv. Inte…

**Figure 24.** Figure 24: Examples of an I(t) waveform obtained under the influence of a constant bias voltage of þ1.0 V (red circles) and þ1.2 V (blue squares) for different time scales: a) 0–0.1 s, b) 0–9 ms, and c) 0–9 μs [PITH_FULL_IMAGE:figures/full_fig_p028_24.png]

**Figure 25.** Figure 25: Dependence of the mean first passage time θ on the value of the constant voltage V0 (at 300 K). www.advancedsciencenews.com www.advintellsyst.com Adv. Intell. Syst. 2023, 5, 2200338 2200338 (29 of 46) © 2022 The Authors. Advanced Intelligent Systems published by Wiley…

**Figure 26.** Figure 26: The dependence lg(Θ(V0)) at temperature changing according to the law described in Equation (80), and A ¼ 100 (red circles) and A ¼ 400 (blue squares). The dashed line denotes the Arrhenius dependence (Equation (76)) at a constant temperature T ¼ T0. Parameters Ea=kB…

**Figure 27.** Figure 27: It should be noted that in this device both potentiation and depression operations are considered to be in the LRS.[228] The model, as described by Equation (85), needs in principle only two parameters (i.e., G0 and n) to describe the device behavior during the depre…

**Figure 28.** Figure 28: Experimental values of n and G0 for each potentiation and depression set of cycles. Notice that n > 0 corresponds to potentiation, while n < 0 corresponds to the depression part. www.advancedsciencenews.com www.advintellsyst.com Adv. Intell. Syst. 2023, 5, 2200338 220…

**Figure 29.** Figure 29: Parameter n at transition t versus n at a transition of the same type (t-2). www.advancedsciencenews.com www.advintellsyst.com Adv. Intell. Syst. 2023, 5, 2200338 2200338 (32 of 46) © 2022 The Authors. Advanced Intelligent Systems published by Wiley-VCH GmbH 26404567,…

**Figure 30.** Figure 30: Simulated I–V curve obtained with the script and parameter configuration in [PITH_FULL_IMAGE:figures/full_fig_p033_30.png]

**Figure 31.** Figure 31: Impact in the I–V characteristics of including variability in model parameters one at a time: a) Ioff, b) Ion, c) Isb, and d) vr [PITH_FULL_IMAGE:figures/full_fig_p034_31.png]

**Figure 32.** Figure 32: I–V curves from 100 cycles simulations including variability in the 4 specified parameters as presented in [PITH_FULL_IMAGE:figures/full_fig_p034_32.png]

**Figure 33.** Figure 33: Comparison between the initial and final HRS current densities for different voltage and cycle situations: a) 1st cycle and Vmax ¼ 0.9 V, b) 1st cycle and Vmax ¼ 1.5 V, c) 2nd cycle and Vmax¼ 0.9 V and d) 2nd cycle and Vmax ¼ 1.5 V. www.advancedsciencenews.com www.adv…

**Figure 34.** Figure 34: a) Experimental I–V curves for several consecutive RS cycles showing the RS parameters Vset, Vreset, Iset, and Ireset measured under ramped voltage stress. b) Set voltage, c) set current, d) absolute value of the reset voltage, e) reset current versus cycle number alo…

**Figure 35.** Figure 35: ACF versus cycle lag for a) Vset and b) Vreset series plotted in [PITH_FULL_IMAGE:figures/full_fig_p037_35.png]

**Figure 36.** Figure 36: ACF and PACF versus cycle lag for the differentiated series. a) ACF and b) PACF for Vsett Vsett1 , c) ACF and d) PACF for Vresett Vresett1 . The ACF and PACF maximum and minimum threshold bounds are 0.0619 shown with blue dashed lines. www.advancedsciencenews.c…

**Figure 37.** Figure 37: a) Vset versus cycle number for the RS series under consideration. Measured values are shown by black lines and the modeled ones in blue, b) Vset series detail for the cycle interval [600-900]. c) Vreset versus cycle number for the RS series under consideration. Measu…

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Works this paper leans on

255 extracted references · 77 canonical work pages

[1]

Lanza, A

M. Lanza, A. Sebastian, W. D. Lu, M. Le Gallo, M.-F. Chang, D. Akinwande, F. M. Puglisi, H. N. Alshareef, M. Liu, J. B. Roldan, Science 2022, 376, 6597

2022
[2]

L. O. Chua, S. M. Kang, Proc. IEEE 1976, 64, 209

1976
[3]

Spiga, A

S. Spiga, A. Sebastian, D. Querlioz, B. Rajendran, Memristive Devices for Brain-Inspired Computing, Elsevier, Amsterdam, Netherlands 2020

2020
[4]

F. Pan, S. Gao, C. Chen, C. Song, F. Zeng, Mater. Sci. Eng. 2014, 83,1

2014
[5]

Variability Behavioral Models 5.1. Variability under the General Memristor Model Formalism In this section, we present an approach tackling with the variabil- ity of the parameters employed in a memristor model, the rela- tion between these parameters and the consequences on the model implementation. Speci ﬁcally, we consider a memristor representation de...

2023
[6]

Discussion We have shown in Section 2 that variability is an essential issue in resistive memories operation and, as discussed along this manuscript, this issue can be tackled in multiple and intricate manners. From physical to behavioral models, combining sto- chastic approaches as well as purely mathematical tools, all con- verge to point out the import...

2023
[7]

Priority-2030

Conclusions Resistive memories have a simple structure and are compatible with CMOS technology; therefore, they can be integrated in the conventional processes of the current electronics industry. However, their huge potential in different applications could be limited by their inherent cycle-to-cycle variability. The variabil- ity issue is essential, and...

2023
[8]

J. S. Lee, S. Lee, T. W. Noh, Appl. Phys. Rev. 2015, 2, 031303

2015

Show all 255 references

[9]

Lanza, H.-S

M. Lanza, H.-S. P. Wong, E. Pop, D. Ielmini, D. Strukov, B. C. Regan, L. Larcher, M. A. Villena, J. J. Yang, L. Goux, A. Belmonte, Y. Yang, F. M. Puglisi, J. Kang, B. Magyari-Köpe, E. Yalon, A. Kenyon, M. Buckwell, A. Mehonic, A. Shluger, H. Li, T.-H. Hou, B. Hudec, D. Akinwan...

2019
[10]

Ielmini, R

D. Ielmini, R. Waser, Resistive Switching: From Fundamentals of Nanoionic Redox Processes to Memristive Device Applications , Wiley-VCH, Hoboken, NJ 2015

2015
[11]

Salinga, B

M. Salinga, B. Kersting, I. Ronneberger, V. P. Jonnalagadda, X. T. Vu, M. Le Gallo, I. Giannopoulos, O. Cojocaru-Mirédin, R. Mazzarello, A. Sebastian, Nat. Mater. 2018, 17, 681

2018
[12]

A. V. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii, R. S. Beach, A. Ong, X. Tang, A. Driskill-Smith, W. H. Butler, P. B. Visscher, D. Lottis, E. Chen, V. Nikitin, M. Krounbi, J. Phys. D. App. Phys 2013, 46, 139601

2013
[13]

Trentzsch, S

M. Trentzsch, S. Flachowsky, R. Richter, J. Paul, B. Reimer, D. Utess, S. Jansen, H. Mulaosmanovic, S. Müller, S. Slesazeck, J. Ocker, M. Noack, J. Müller, P. Polakowski, J. Schreiter, S. Beyer, T. Mikolajick, B. Rice, 2016 IEEE Inter. Electron Devices Meeting (IEDM), IEEE, Pi...

2016
[14]

von Witzleben, K

M. von Witzleben, K. Fleck, C. Funck, B. Baumkötter, M. Zuric, A. Idt, T. Breuer, R. Waser, U. Böttger, S. Menzel, Adv. Electron. Mater. 2017, 3, 1700294

2017
[15]

Funck, S

C. Funck, S. Menzel, ACS Appl. Electron. Mater. 2021, 3, 3674

2021
[16]

Lanza, R

M. Lanza, R. Waser, D. Ielmini, J. J. Yang, L. Goux, J. Su ˜ne, A. J. Kenyon, A. Mehonic, S. Spiga, V. Rana, S. Wiefels, S. Menzel, I. Valov, M. A. Villena, E. Miranda, X. Jing, F. Campabadal, M. Gonzalez, F. Aguirre, F. Palumbo, K. Zhu, J. B. Roldan, F. M. Puglisi, L. Larcher...

2021
[17]

Chou, Z.-J

C.-C. Chou, Z.-J. Lin, C.-A. Lai, C.-I. Su, P.-L. Tseng, W.-C. Chen, W.-C. Tsai, W.-T. Chu, T.-C. Ong, H. Chuang, Y.-D. Chih, T.-Y. J. Chang, in 2020 IEEE Symp. on VLSI Circuits , IEEE, Piscataway, NJ 2020, pp. 1 –2

2020
[18]

P. Jain, U. Arslan, M. Sekhar, B. C. Lin, L. Wei, T. Sahu, J. Alzate- vinasco, A. Vangapaty, M. Meterelliyoz, N. Strutt, A. B. Chen, P. Hentges, P. A. Quintero, C. Connor, O. Golonzka, K. Fischer, F. Hamzaoglu, in 2019 IEEE Inter. Solid- State Circuits Conf. - (ISSCC), IEEE, P...

2019
[19]

Carboni, D

R. Carboni, D. Ielmini, Adv. Electron. Mater. 2019, 5, 1900198

2019
[20]

Wei, et al., in 2016 IEEE Inter

Z. Wei, et al., in 2016 IEEE Inter. Electron Devices Meeting (IEDM) , IEEE, San Francisco, CA 2016, pp. 4.8.1 – 4.8.4, https://doi.org/10. 1109/iedm.2016.7838349

2016
[21]

Chen, IEEE Electron Device Lett

A. Chen, IEEE Electron Device Lett. 2015, 36, 138

2015
[22]

Lanza, C

M. Lanza, C. Wen, X. Li, T. Zanotti, F. M. Puglisi, Y. Shi, F. Saiz, A. Antidormi, S. Roche, W. Zheng, X. Liang, J. Hu, S. Duhm, K. Zhu, F. Hui, J. B. Roldan, B. Garrido, T. Wu, V. Chen, E. Pop, Adv. Mater. 2021, 33, 2100185. www.advancedsciencenews.com www.advintellsyst.com A...

2021 doi
[23]

S. Yu, Y. Wu, R. Jeyasingh, D. Kuzum, H.-S. Wong, IEEE Trans. Electron Devices 2011, 58, 2729

2011
[24]

Ambrogio, P

S. Ambrogio, P. Narayanan, H. Tsai, R. M. Shelby, I. Boybat, C. di Nolfo, S. Sidler, M. Giordano, M. Bodini, N. C. P. Farinha, B. Killeen, C. Cheng, Y. Jaoudi, G. W. Burr, Nature 2018, 558, 60

2018
[25]

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, D. S. Modha, Science 2014, 345, 668

2014
[26]

Alibart, E

F. Alibart, E. Zamanidoost, D. B. Strukov, Nat. Commun. 2013, 4, 2072

2013
[27]

Prezioso, F

M. Prezioso, F. Merrikh-Bayat, B. D. Hoskins, G. C. Adam, K. K. Likharev, D. B. Strukov, Nature 2015, 521, 61

2015
[28]

M. A. Zidan, J. P. Strachan, W. D. Lu, Nat. Electron. 2018, 1, 22

2018
[29]

Romero-Zaliz, E

R. Romero-Zaliz, E. Perez, F. Jiménez-Molinos, C. Wenger, J. B. Roldán, Electronics 2021, 10, 346

2021
[30]

Dalgaty, N

T. Dalgaty, N. Castellani, C. Turck, C. K.-E. Harabi, D. Querlioz, E. Vianello, Nat. Electron. 2021, 4, 151

2021
[31]

F. Hui, P. Liu, S. A. Hodge, T. Carey, C. Wen, F. Torrisi, D. Thanuja, L. Galhena, F. Tomarchio, Y. Lin, E. Moreno, J. B. Roldan, E. Koren, A. C. Ferrari, M. Lanza, Small 2021, 17, 2101100

2021
[32]

Sebastian, M

A. Sebastian, M. Le Gallo, R. Khaddam-Aljameh, E. Eleftheriou, Nat. Nanotechnol. 2020, 15, 529

2020
[33]

S. Yu, H. Jiang, S. Huang, X. Peng, A. Lu, IEEE Circuits Syst. Mag. 2021, 21, 31

2021
[34]

Dhar, Nat

P. Dhar, Nat. Mach. Intell. 2020, 2, 423

2020
[35]

Pérez, D

E. Pérez, D. Maldonado, C. Acal, J. E. Ruiz-Castro, F. J. Alonso, A. M. Aguilera, F. Jiménez-Molinos, Ch Wenger, J. B. Roldán, Microelectron. Eng. 2019, 214, 104

2019
[36]

Poblador, M

S. Poblador, M. B. González, F. Campabadal, Microelectron. Eng. 2018, 187–188, 148

2018
[37]

A. N. Mikhaylov, D. V. Guseinov, A. I. Belov, D. S. Korolev, V. A. Shishmakova, M. N. Koryazhkina, D. O. Filatov, O. N. Gorshkov, D. Maldonado, F. J. Alonso, J. B. Roldán, A. V. Krichigin, N. V. Agudov, A. A. Dubkov, A. Carollo, B. Spagnolo, Chaos, Solitons Fractals 2021, 144, 110723

2021
[38]

Dirkmann, J

S. Dirkmann, J. Kaiser, C. Wenger, T. Mussenbrock, ACS Appl. Mater. Interfaces 2018, 10, 14857

2018
[39]

J. B. Roldán, G. González-Cordero, R. Picos, E. Miranda, F. Palumbo, F. Jiménez-Molinos, E. Moreno, D. Maldonado, S. B. Baldomá, M. Moner Al Chawa, C. de Benito, S. G. Stavrinides, J. Su ˜né, L. O. Chua, Nanomaterials 2021, 11, 1261

2021
[40]

Aldana, P

S. Aldana, P. García-Fernández, R. Romero-Zaliz, M. B. González, F. Jiménez-Molinos, F. Gómez-Campos, F. Campabadal, J. B. Roldán, J. Phys. D: Appl. Phys. 2020, 53, 225106

2020
[41]

Padovani, L

A. Padovani, L. Larcher, O. Pirrotta, L. Vandelli, G. Bersuker, IEEE Trans. Electron Devices 2015, 62, 1998

2015
[42]

Waser, R

R. Waser, R. Dittmann, G. Staikov, K. Szot, Adv. Mater. 2009, 21, 2632

2009
[43]

Banerjee, A

W. Banerjee, A. Kashir, S. Kamba, Small 2022, 18, 2107575

2022
[44]

Matveyev, R

Y. Matveyev, R. Kirtaev, A. Fetisova, S. Zakharchenko, D. Negrov, A. Zenkevich, Nanoscale Res. Lett. 2016, 11, 147

2016
[45]

M. J. Lee, C. B. Lee, D. Lee, S. R. Lee, M. Chang, J. H. Hur, Y. B. Kim, C. J. Kim, D. H. Seo, S. Seo, U. I. Chung, I. K. Yoo, K. Kim, Nat. Mater. 2011, 10, 625

2011
[46]

Hwang, Y

H.-G. Hwang, Y. Pyo, J.-U. Woo, I.-S. Kim, S.-W. Kim, D.-S. Kim, B. Kim, J. Jeong, S. Nahm, J. Alloys Compd. 2022, 902, 163764

2022
[47]

C. Chen, C. Song, J. Yang, F. Zeng, F. Pan, Appl. Phys. Lett. 2012, 100, 253509

2012
[48]

F. Wu, P. Cao, Z. Peng, S. Ke, G. Cheng, G. Cao, B. Jiang, C. Ye, IEEE Trans. Electron Devices 2022, 69, 375

2022
[49]

Wang, D.-Y

S.-Y. Wang, D.-Y. Lee, T.-Y. Huang, J.-W. Wu, T.-Y. Tseng, Nanotechnology 2010, 21, 495201

2010
[50]

X. Wei, H. Huang, C. Ye, W. Wei, H. Zhou, Y. Chen, R. Zhang, L. Zhang, Q. Xia, J. Alloys Compd. 2019, 775, P. 1301

2019
[51]

X. D. Huang, Y. Li, H. Y. Li, Y. F. Lu, K. H. Xue, X. S. Miao, Appl. Phys. Lett. 2020, 116, 173504

2020
[52]

Ismail, C

M. Ismail, C. Mahata, S. Kim, J. Alloys Compd. 2021, 892, 162141

2021
[53]

J.-H. Ryu, S. Kim, Materials 2020, 13, 4201

2020
[54]

Govoreanu, G

B. Govoreanu, G. S. Kar, Y.-Y. Chen, V. Paraschiv, S. Kubicek, M. Jurczak, in 2011 Inter. Electron Devices Meeting , IEDM, Washington, DC 2011, p. 31.6.1

2011
[55]

Perez, A

E. Perez, A. Grossi, C. Zambelli, P. Olivo, C. Wenger, IEEE J. Electron Devices Soc. 2017, 5, 64

2017
[56]

Y. Y. Chen, G. Ludovic, S. Clima, B. Govoreanu, R. Degreve, K. G. Sankar, A. Fantini, G. Groseneken, D. J. Wouters, J. Malgorzata, IEEE Trans. Electron Devices 2013, 60, 1114

2013
[57]

Poblador, M

S. Poblador, M. C. Acero, M. B. González, F. Campabadal, in IEEE Proc. 11th Spanish Conf. on Electron Devices (CDE 2017) , IEEE, Piscataway, NJ 2017, pp. 1 –4

2017
[58]

Kvatinsky, G

S. Kvatinsky, G. Satat, N. Wald, E. G. Friedman, A. Kolodny, U. C. Weiser, IEEE Trans. Very Large Scale Integr. VLSI Syst. 2014, 22, 2054

2014
[59]

Arumí, A

D. Arumí, A. Gomez-Pau, S. Manich, R. Rodríguez-Monta ˜nés, M. B. Gonzalez, F. Campabadal, IEEE Electron Device Lett. 2019, 40, 341

2019
[60]

Ielmini, Microelectron

D. Ielmini, Microelectron. Eng. 2018, 190, 44

2018
[61]

E. Covi, S. Brivio, A. Serb, T. Prodromakis, M. Fanciulli, S. Spiga, in Proc. IEEE Inter. Symp. on Circuits and Systems (ISCAS) , IEEE, Piscataway, NJ 2016, p. 393

2016
[62]

M. B. González, M. Maestro-Izquierdo, S. Poblador, M. Zabala, F. Campabadal, G. González-Cordero, S. Aldana, D. Maldonado, F. Jiménez-Molinos, J. B. Roldán, Mem-Elements for Neuromorphic Circuits with Arti ﬁcial Intelligence Applications , Elsevier, Amsterdam, Netherlands 2021

2021
[63]

Ambrogio, S

S. Ambrogio, S. Balatti, A. Cubeta, A. Calderoni, N. Ramaswamy, D. Ielmini, IEEE Trans. Electron Devices 2014, 61, 2912

2014
[64]

Abbaspour, S

E. Abbaspour, S. Menzel, C. Jungemann, J. Comput. Electron. 2020, 19, 1426

2020
[65]

Maldonado, S

D. Maldonado, S. Aldana, M. B. González, F. Jiménez-Molinos, F. Campabadal, J. B. Roldán, Microelectron. Eng. 2022, 265, 111876

2022
[66]

Mikhaylov, A

A. Mikhaylov, A. Belov, D. Korolev, I. Antonov, V. Kotomina, A. Kotina, E. Gryaznov, A. Sharapov, M. Koryazhkina, R. Kryukov, S. Zubkov, A. Sushkov, D. Pavlov, S. Tikhov, O. Morozov, D. Tetelbaum, Adv. Mater. Technol. 2020, 5, 1900607

2020
[67]

M. B. Gonzalez, J. M. Rafí, O. Beldarrain, M. Zabala, F. Campabadal, IEEE Trans. Device Mater. Reliab. 2014, 14, 769

2014
[68]

Y. Y. Chen, B. Govoreanu, L. Goux, R. Degraeve, A. Fantini, G. S. Kar, D. J. Wouters, G. Groeseneken, J. A. Kittl, M. Jurczak, L. Altimime, IEEE Trans. Electron Devices 2012, 59, 3243

2012
[69]

Huang, B

P. Huang, B. Chen, Y. Wang, F. Zhang, L. Shen, R. Liu, L. Zeng, G. Du, X. Zhang, B. Gao, J. Kang, X. Liu, X. Wang, B. Weng, Y. Tang, G.-Q. Lo, D.-L. Kwong, in IEEE IEDM Tech. Dig. , IEEE, Piscataway, NJ 2013, p. 597

2013
[70]

Balatti, S

S. Balatti, S. Ambrogio, Z. Wang, S. Sills, A. Calderoni, N. Ramaswamy, D. Ielmini, IEEE Trans. Electron Devices 2015, 62, 3365

2015
[71]

F. M. Puglisi, P. Pavan, L. Vandelli, A. Padovani, M. Bertocchi, L. Larcher, in 2015 IEEE Inter. Reliability Physics Symp. Proc. , IEEE, Piscataway, NJ 2015, p. 5B51

2015
[72]

M. B. Gonzalez, J. Martin-Martínez, M. Maestro, M. C. Acero, M. Nafría, F. Campabadal, IEEE Trans. Electron Devices 2016, 63, 3116

2016
[74]

E.; Ocola, L

Chakrabarti, B.; Chan, H.; Alam, K.; Koneru, A.; Gage, T. E.; Ocola, L. E.; Divan, R.; Rosenmann, D.; Khanna, A.; Grisafe, B.; T. Sanders, S. Datta, I. Arslan, S. K. R. S. Sankaranarayan, S. Guha, ACS Nano 2021, 15, 4155

2021
[75]

Y. Lu, K. Liu, J. Yang, T. Zhang, C. Cheng, B. Dang, L. Xu, J. Zhu, Q. Huang, R. Huang, Y. Yang, Adv. Electron. Mater. 2020, 6, 2000204

2020
[76]

Medeiros-Ribeiro, F

G. Medeiros-Ribeiro, F. Perner, R. Carter, H. Abdalla, M. D. Pickett, R. S. Williams, Nanotechnology 2011, 22, 095702

2011
[77]

H. Yeon, P. Lin, C. Choi, S. H. Tan, Y. Park, D. Lee, J. Lee, F. Xu, B. Gao, H. Wu, H. Qian, Y. Nie, S. Kim, J. Kim, Nat. Nanotechnol. 2020, 15, 574

2020
[78]

K. M. Kim, J. J. Yang, J. P. Strachan, E. M. Grafals, N. Ge, N. D. Melendez, Z. Li, R. S. Williams, Sci. Rep. 2016, 6, 20085

2016
[79]

M. A. Villena, J. B. Roldán, F. Jiménez-Molinos, E. Miranda, J. Su ˜né, M. Lanza, J. Comput. Electron. 2017, 16, 1095

2017
[80]

Z. Fang, H. Y. Yu, X. Li, N. Singh, G. Q. Lo, D. L. Kwong, IEEE Electron Device Lett. 2011, 32, 566

2011
[81]

Jiang, K

Y. Jiang, K. Zhang, K. Hu, Y. Zhang, A. Liang, Z. Song, S. Song, F. Wang, Mater. Sci. Semicond. Process. 2021, 136, 106131

2021
[82]

C. Pan, Y. Ji, N. Xiao, F. Hui, K. Tang, Y. Guo, X. Xie, F. M. Puglisi, L. Larcher, E. Miranda, L. Jiang, Adv. Funct. Mater. 2017, 27, 1604811

2017
[83]

W. Hu, Z. Yang, H. Ye, Adv. Eng. Mater. 2018, 20, 1700516

2018
[84]

Kormath Madam Raghupathy, T

R. Kormath Madam Raghupathy, T. D. Kühne, G. Henkelman, H. Mirhosseini, Adv. Theor. Simul. 2019, 2, 1900036

2019
[85]

C. Pan, E. Miranda, M. A. Villena, N. Xiao, X. Jing, X. Xie, T. Wu, F. Hui, Y. Shi, M. Lanza, 2D Mater. 2017, 4, 025099

2017
[86]

Joksas, A

D. Joksas, A. AlMutairi, O. Lee, M. Cubukcu, A. Lombardo, H. Kurebayashi, A. J. Kenyon, A. Mehonic, Adv. Intell. Syst. 2022, 4, 2200068

2022
[87]

K. S. Novoselov, O. A. Mishchenko, O. A. Carvalho, A. H. Castro Neto, Science 2016, 353, aac9439

2016
[88]

Sánchez Esqueda, H

I. Sánchez Esqueda, H. Zhao, H. Wang, J. Appl. Phys. 2018, 124, 152133

2018
[89]

C. Li, D. Belkin, Y. Li, P. Yan, M. Hu, N. Ge, H. Jiang, E. Montgomery, P. Lin, Z. Wang, W. Song, J. P. Strachan, M. Barnell, Q. Wu, R. S. Williams, J. J. Yang, Q. Xia, Nat. Commun. 2018, 9, 2385

2018
[90]

Y. Shi, L. Nguyen, S. Oh, X. Liu, F. Koushan, J. R. Jameson, D. Kuzum, Nat. Commun. 2018, 9, 5312

2018
[91]

Ambrogio, S

S. Ambrogio, S. Balatti, V. Milo, R. Carboni, Z.-Q. Wang, A. Calderoni, N. Ramaswamy, D. Ielmini, IEEE Trans. Electron Devices 2016, 63, 15081515

2016
[92]

C. H. Wang, C. McClellan, Y. Shi, X. Zheng, V. Chen, M. Lanza, E. Pop, H. S. Wong, in 2018 IEEE Inter. Electron Devices Meeting (IEDM), IEEE, Piscataway, NJ 2018

2018
[93]

Zhang, A

L. Zhang, A. Redol ﬁ, C. Adelmann, S. Clima, I. P. Radu, Y. Y. Chen, D. J. Wouters, G. Groeseneken, M. Jurczak, B. Govoreanu, IEEE Electron Device Lett. 2013, 35, 199

2013
[94]

Aluguri, T

R. Aluguri, T. Y. Tseng, IEEE J. Electron Devices Soc. 2016, 4, 294

2016
[95]

J. Woo, W. Lee, S. Park, S. Kim, D. Lee, G. Choi, E. Cha, J. Hyun Lee, W. Young Jung, C. Gyung Park, H. Hwang, in 2013 Symp. on VLSI Technology, IEEE, Piscataway, NJ 2013, pp. T168 –T169

2013
[96]

H. Y. Chen, S. Brivio, C. C. Chang, J. Frascaroli, T. H. Hou, B. Hudec, M. Liu, H. Lv, G. Molas, J. Sohn, S. Spiga, J. Electroceramics 2017, 39, 21

2017
[97]

J. Zhou, K. H. Kim, W. Lu, IEEE Trans. Electron Devices 2014, 61, 1369

2014
[98]

M. A. Lastras-Montano, K.-T. Cheng, Nat. Electron. 2018, 1, 466

2018
[99]

Sivan, Y

M. Sivan, Y. Li, H. Veluri, Y. Zhao, B. Tang, X. Wang, E. Zamburg, J. F. Leong, J. X. Niu, U. Chand, A. V. Y. Thean, Nat. Commun. 2019, 10, 5201

2019
[100]

C. H. Wang, C. McClellan, Y. Shi, X. Zheng, V. Chen, M. Lanza, E. Pop, H. S. P. Wong, in 2018 IEEE Inter. Electron Devices Meeting (IEDM), IEEE, Piscataway, NJ 2018, pp. 22 –25

2018
[101]

R. Yang, H. Li, K. K. Smithe, T. R. Kim, K. Okabe, E. Pop, J. A. Fan, H. S. P. Wong, in 2017 IEEE Inter. Electron Devices Meeting (IEDM) , IEEE, Piscataway, NJ 2017, pp. 19 – 25

2017
[102]

K. Zhu, C. Wen, A. A. Aljarb, F. Xue, X. Xu, V. Tung, X. Zhang, H. N. Alshareef, M. Lanza, Nat. Electron. 2021, 4, 775

2021
[103]

J. Ma, H. Liu, N. Yang, J. Zou, S. Lin, Y. Zhang, X. Zhang, J. Guo, H. Wang, Adv. Mater. 2022, 34, 2202371

2022
[104]

S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutiérrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston- Halperin, ACS Nano 2013, 7, 2898

2013
[105]

Lembke, A

D. Lembke, A. Kis, ACS Nano 2012, 6, 10070

2012
[106]

N. Li, Q. Wang, C. Shen, Z. Wei, H. Yu, J. Zhao, X. Lu, G. Wang, C. He, L. Xie, J. Zhu, Nat. Electron. 2020, 3, 711

2020
[107]

Akinwande, N

D. Akinwande, N. Petrone, J. Hone, Nat. Commun. 2014, 5, 5678

2014
[108]

H. Zhao, Z. Dong, H. Tian, D. DiMarzi, M. G. Han, L. Zhang, X. Yan, F. Liu, L. Shen, S. J. Han, S. Cronin, Adv. Mater. 2017, 29, 1703232

2017
[109]

M. A. Villena, F. Hui, X. Liang, Y. Shi, B. Yuan, X. Jing, K. Zhu, S. Chen, M. Lanza, Microelectron. Reliab. 2019, 102, 113410

2019
[110]

X. Jing, F. Puglisi, D. Akinwande, M. Lanza, 2D Mater. 2019, 6, 035021

2019
[111]

F. Hui, W. Fang, W. S. Leong, T. Kpulun, H. Wang, H. Y. Yang, M. A. Villena, G. Harris, J. Kong, M. Lanza, ACS Appl. Mater. Interfaces 2017, 9, 39895

2017
[112]

Cheng, K

P. Cheng, K. Sun, Y. H. Hu, Nano Lett. 2016, 16, 572

2016
[113]

G. H. Shin, C. K. Kim, G. S. Bang, J. Y. Kim, B. C. Jang, B. J. Koo, M. H. Woo, Y. K. Choi, S. Y. Choi, 2D Mater. 2016, 3, 034002

2016
[114]

X. Zhao, Z. Fan, H. Xu, Z. Wang, J. Xu, J. Ma, Y. Liu, J. Mater. Chem. C 2018, 6, 7195

2018
[115]

Choudhary, M

S. Choudhary, M. Soni, S. K. Sharma, Semicond. Sci. Technol. 2019, 34, 085009

2019
[116]

B. Tang, H. Veluri, Y. Li, Z. G. Yu, M. Waqar, J. F. Leong, M. Sivan, E. Zamburg, Y. W. Zhang, J. Wang, A. V. Thean, Nat. Commun. 2022, 13, 3037

2022
[117]

Kumar, D

M. Kumar, D. K. Ban, S. M. Kim, J. Kim, C. P. Wong, Adv. Electron. Mater. 2019, 5, 1900467

2019
[118]

X. Yan, Q. Zhao, A. P. Chen, J. Zhao, Z. Zhou, J. Wang, H. Wang, L. Zhang, X. Li, Z. Xiao, K. Wang, Small 2019, 15, 1901423

2019
[119]

W. Huh, S. Jang, J. Y. Lee, D. Lee, D. Lee, J. M. Lee, H. G. Park, J. C. Kim, H. Y. Jeong, G. Wang, C. H. Lee, Adv. Mater. 2018, 30, 1801447

2018
[120]

Y. Shen, W. Zheng, K. Zhu, Y. Xiao, C. Wen, Y. Liu, X. Jing, M. Lanza, Adv. Mater. 2021, 33, 2103656

2021
[121]

Y. Shi, C. Pan, V. Chen, N. Raghavan, K. L. Pey, F. M. Puglisi, E. Pop, H. S. Wong, M. Lanza, in 2017 IEEE Inter. Electron Devices Meeting (IEDM), IEEE, Piscataway, NJ 2017, pp. 5 – 4

2017
[122]

Y. Li, S. Long, Y. Liu, C. Hu, J. Teng, Q. Liu, H. Lv, J. Su ˜né, M. Lu, Nanoscale Res. Lett. 2015, 10, 420

2015
[123]

S. Chen, M. R. Mahmoodi, Y. Shi, C. Mahata, B. Yuan, X. Liang, C. Wen, F. Hui, D. Akinwande, D. B. Strukov, M. Lanza, Nat. Electron. 2020, 3, 638

2020
[124]

Y. Shi, X. Liang, B. Yuan, V. Chen, H. Li, F. Hui, Z. Yu, F. Yuan, E. Pop, H. S. P. Wong, M. Lanza, Nat. Electron. 2018, 1, 458

2018
[125]

B. Yuan, X. Liang, L. Zhong, Y. Shi, F. Palumbo, S. Chen, F. Hui, X. Jing, M. A. Villena, L. Jiang, M. Lanza, Adv. Electron. Mater. 2020, 6, 1900115

2020
[126]

R. Ge, X. Wu, M. Kim, J. Shi, S. Sonde, L. Tao, Y. Zhang, J. C. Lee, D. Akinwande, Nano Lett. 2018, 18, 434

2018
[127]

Krishnaprasad, D

A. Krishnaprasad, D. Dev, S. S. Han, Y. Shen, H. S. Chung, T. S. Bae, C. Yoo, Y. Jung, M. Lanza, T. Roy, ACS Nano 2022, 16, 2866

2022
[128]

R. Xu, H. Jang, M. H. Lee, D. Amanov, Y. Cho, H. Kim, S. Park, H. J. Shin, D. Ham, Nano Lett. 2019, 19, 2411

2019
[130]

Maldonado, S

D. Maldonado, S. Aldana, M. B. González, F. Jiménez-Molinos, M. J. Ibá ˜nez, D. Barrera, F. Campabadal, J. B. Roldán, Microelectron. Eng. 2022, 257, 111736

2022
[131]

X. Guan, S. Yu, H.-S. Philip Wong, IEEE Electron Device Lett. 2012, 33, 1405

2012
[132]

P. Chen, S. Yu, IEEE Trans. Electron Devices 2015, 62, 4022

2015
[133]

Maldonado, F

D. Maldonado, F. Aguirre, G. González-Cordero, A. M. Roldán, M. B. González, F. Jiménez-Molinos, F. Campabadal, E. Miranda, J. B. Roldán, J. Appl. Phys. 2021, 130, 054503

2021
[134]

https://nano.stanford.edu/downloads/stanford-rram-model (accessed: December 2022)

2022
[135]

Jiang, Y

Z. Jiang, Y. Wu, S. Yu, L. Yang, K. Song, Z. Karim, H.-S. P. Wong, IEEE Trans. Electron Devices 2016, 63, 1884

2016
[136]

González-Cordero, J

G. González-Cordero, J. B. Roldán, F. Jiménez-Molinos, J. Su ˜né, S. Long, M. Liu, Semicond. Sci. Technol. 2016, 31, 115013

2016
[137]

Huang, X

P. Huang, X. Y. Liu, B. Chen, H. T. Li, Y. J. Wang, Y. X. Deng, K. L. Wei, L. Zeng, B. Gao, G. Du, X. Zhang, J. F. Kang, IEEE Trans. Electron Devices 2013, 60, 4090

2013
[138]

Huang, D

P. Huang, D. Zhu, S. Chen, Z. Zhou, Z. Chen, B. Gao, L. Liu, X. Liu, J. Kang, IEEE Trans. Electron Devices 2017, 64, 614

2017
[139]

Jiménez-Molinos, M

F. Jiménez-Molinos, M. A. Villena, J. B. Roldán, A. M. Roldán, IEEE Trans. Electron Devices 2015, 62, 955

2015
[140]

Bocquet, D

M. Bocquet, D. Deleruyelle, H. Aziza, C. Muller, J.-M. Portal, T. Cabout, E. Jalaguier, IEEE Trans. Electron Devices 2014, 61, 674

2014
[141]

Romero-Zaliz, A

R. Romero-Zaliz, A. Cantudo, E. Pérez, F. Jiménez-Molinos, C. Wenger, J. B. Roldán, Electronics 2021, 10, 3141

2021
[142]

Pérez, A

E. Pérez, A. Javier Pérez-Ávila, R. Romero-Zaliz, M. Kalishettyhalli, E. Pérez-Bosch, J. B. Roldán, F. Jiménez-Molinos, C. Wenger, Electronics 2021, 10, 645

2021
[143]

Maldonado, A

D. Maldonado, A. I. Belov, M. N. Koryazhkina, F. Jiménez-Molinos, A. N. Mikhaylov, J. B. Roldán, Phys. Status Solidi A 2022, 2200520. https://doi.org/10.1002/pssa.202200520

2022 doi
[144]

Maldonado, C

D. Maldonado, C. Aguilera-Pedregosa, G. Vinuesa, H. García, S. Due ˜nas, H. Castán, S. Aldana, M. B. González, E. Moreno, F. Jiménez-Molinos, F. Campabadal, J. B. Roldán, Chaos, Solitons Fractals 2022, 160, 112247

2022
[145]

Maldonado, A

D. Maldonado, A. M. Roldán, M. B. González, F. Jiménez-Molinos, F. Campabadal, J. B. Roldán, Microelectron. Eng. 2019, 215, 110983

2019
[146]

Reuben, M

J. Reuben, M. Biglari, D. Fey, IEEE Trans. Nanotechnol. 2020, 19, 508

2020
[147]

J. B. Roldán, F. J. Alonso, A. M. Aguilera, D. Maldonado, M. Lanza, J. Appl. Phys. 2019, 125, 174504

2019
[148]

J. B. Roldán, D. Maldonado, F. J. Alonso, A. M. Roldán, F. Hui, Y. Shi, F. Jiménez-Molinos, A. M. Aguilera, M. Lanza, in 2021 IEEE Inter. Reliability Physics Symp. (IRPS) , IEEE, Piscataway, NJ 2021

2021
[149]

Imry, Directions in Condensed Matter Physics (Eds: G

Y. Imry, Directions in Condensed Matter Physics (Eds: G. Grinsteing, G. Mazenko), World Scienti ﬁc, Singapore 1986, pp. 101 –163

1986
[150]

Mesoscopic Physics and Electronics (Eds: T. Ando, Y. Arakawa, K. Furuya, S. Komiyama, H. Nakashima), Springer-Verlag, Berlin, Germany 1998

1998
[151]

Van Houten, C

H. Van Houten, C. Beenakker, B. Van Wees, Semiconductors and Semimetals, Vol. 35, Academic Press, Cambridge, MA 1992, pp. 9–112

1992
[152]

Agrait, A

N. Agrait, A. Levy Yetati, J. van Ruitenbeek, Phys. Rep. 2003, 377, 81

2003
[153]

Ihn, Semiconductor Nanostructures, Quantum States and Electronic Transport, Oxford University Press, Oxford, UK 2010

T. Ihn, Semiconductor Nanostructures, Quantum States and Electronic Transport, Oxford University Press, Oxford, UK 2010

2010
[154]

Datta, Lessons from Nanoelectronics: A New Perspective on Transport, 2nd ed., World Scienti ﬁc, Singapore 2017

S. Datta, Lessons from Nanoelectronics: A New Perspective on Transport, 2nd ed., World Scienti ﬁc, Singapore 2017

2017
[155]

Landauer, J

R. Landauer, J. Phys.: Condens. Matter 1989, 1, 8099

1989
[156]

Krans, J

J. Krans, J. van Ruitenbeek, V. Fisun, I. Yanson, L. De Longh, Nature 1995, 375, 767

1995
[157]

J. V. Ruitenbeek, M. Morales Masis, E. Miranda, In Resistive Switching (Eds: D. Ielmini, R. Waser), Wiley 2016, https://doi. org/10.1002/9783527680870.ch7

2016 doi
[158]

Milano, M

G. Milano, M. Aono, L. Boarino, U. Celano, T. Hasgawa, M. Kozicki, S. Majumdar, M. Menghini, E. Miranda, C. Ricciardi, S. Tappertzhofen, K. Terabe, I. Valov, Adv. Mater. 2022, 34, 2201248

2022
[159]

Miranda, J

E. Miranda, J. Su ˜né, Microelectron. Reliab. 2004, 44,1

2004
[160]

Miranda, C

E. Miranda, C. Walczyk, C. Wenger, T. Schroeder, IEEE Electron Device Lett. 2010, 31, 609

2010
[161]

J. B. Roldán, E. Miranda, G. González-Cordero, P. García-Fernández, R. Romero-Zaliz, P. González-Rodelas, A. M. Aguilera, M. B. González, F. Jiménez-Molinos, J. Appl. Phys. 2018, 123, 014501

2018
[162]

Cartoixa, R

X. Cartoixa, R. Rurali, J. Su ˜ne, Phys. Rev. B 2012, 86, 165445

2012
[163]

Büttiker, Phys

M. Büttiker, Phys. Rev. B 1990, 41, 7906

1990
[164]

Hu, Phys

P. Hu, Phys. Rev. B 1987, 35, 4078

1987
[165]

Miranda, A

E. Miranda, A. Mehonic, W. Ng, A. J. Kenyon, IEEE Electron Device Lett. 2018, 40, 28

2018
[166]

Avellán, E

A. Avellán, E. Miranda, D. Schoeder, W. Krautschneider, J. Appl. Phys. 2005, 97, 014104

2005
[167]

Miranda, J

E. Miranda, J. Sune, IEEE Trans. Nanotechnol. 2020, 19, 837

2020
[168]

Aguirre, J

F. Aguirre, J. Su ˜né, E. Miranda, Micromachines 2022, 13, 330

2022
[169]

Salvador, M

E. Salvador, M. B. Gonzalez, F. Campabadal, J. Martin-Martinez, R. Rodriguez, E. Miranda, Solid-State Electron. 2021, 185, 108040

2021
[170]

R. Hull, P. Keblinski, D. Lewis, A. Maniatty, V. Meunier, A. A. Oberai, C. R. Picu, J. Samuel, M. S. Shephard, M. Tomozawa, D. Vashishth, S. Zhang, Appl. Phys. Rev. 2018, 5, 011302

2018
[171]

D. B. Strukov, G. S. Snider, D. R. Stewart, R. S. Williams, Nature 2008, 453, 80

2008
[172]

Chua, Appl

L. Chua, Appl. Phys. A 2018, 124, 563

2018
[173]

Y. N. Joglekar, S. J. Wolf, Eur. J. Phys. 2009, 30, 661

2009
[174]

Shang, W

Y. Shang, W. Fei, H. Yu, IEEE Trans. Circuits Syst. I Regul. Pap. 2012, 59, 1906

2012
[175]

E. Linn, A. Siemon, R. Waser, S. Menzel, IEEE Trans. Circuits Syst. I Regul. Pap. 2014, 61, 2402

2014
[176]

Ielmini, IEEE Trans

D. Ielmini, IEEE Trans. Electron Devices 2011, 58, 4309

2011
[177]

J. L. Borghetti, D. B. Strukov, M. D. Pickett, J. J. Yang, R. S. Williams, J. Appl. Phys. 2009, 106, 124504

2009
[178]

Pickett, D

D. Pickett, D. B. Strukov, J. L. Borghetti, J. J. Yang, G. S. Snider, D. R. Stewart, R. S. Williams, J. Appl. Phys. 2009, 106, 074508

2009
[179]

Kim, S.-J

S. Kim, S.-J. Kim, K. M. Kim, S. R. Lee, M. Chang, E. Cho, Y.-B. Kim, C. J. Kim, U.-I. Chung, I.-K. Yoo, Sci. Rep. 2013, 3, 1680

2013
[180]

Kim, S.-H

S. Kim, S.-H. Choi, W. Lu, ACS Nano 2014, 8, 2369

2014
[181]

Marchewka, B

A. Marchewka, B. Roesgen, K. Skaja, H. Du, C.-L. Jia, J. Mayer, V. Rana, R. Waser, S. Menzel, Adv. Electron. Mater. 2016, 2, 1500233

2016
[182]

V. G. Karpov, D. Niraula, I. V. Karpov, R. Kotlyar, Phys. Rev. Appl. 2017, 8, 024028

2017
[183]

A. N. Malakhov, N. V. Agudov, Chaos 1994, 4, 665

1994
[184]

Valenti, A

D. Valenti, A. Carollo, B. Spagnolo, Phys. Rev. A 2018, 97, 042109

2018
[185]

Carollo, B

A. Carollo, B. Spagnolo, D. Valenti, Sci. Rep. 2018, 8, 9852

2018
[186]

Carollo, B

A. Carollo, B. Spagnolo, D. Valenti, Entropy 2018, 20, 485

2018
[187]

Naous, M

R. Naous, M. Al-Shedivat, K. N. Salama, IEEE Trans. Nanotechnol. 2016, 15, 15

2016
[188]

Stotland, M

A. Stotland, M. Di Ventra, Phys. Rev. E 2012, 85, 011116

2012
[189]

Jiang, D

H. Jiang, D. Belkin, S. E. Savel ’ev, S. Lin, Z. Wang, Y. Li, S. Joshi, R. Midya, C. Li, M. Rao, M. Barnell, Q. Wu, J. J. Yang, Q. Xia, Nat. Commun. 2017, 8, 882

2017
[190]

Langevin, C

P. Langevin, C. R. Acad, Science 1908, 146, 530

1908
[191]

N. V. Agudov, A. V. Safonov, A. V. Krichigin, A. A. Kharcheva, A. A. Dubkov, D. Valenti, D. V. Guseinov, A. I. Belov, A. N. Mikhaylov, A. Carollo, B. Spagnolo, J. Stat. Mech: Theory Exp. 2020, 2020, 024003

2020
[192]

Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Springer, New York City, USA 1984

H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Springer, New York City, USA 1984

1984
[193]

N. V. Agudov, A. A. Dubkov, A. V. Safonov, A. V. Krichigin, A. A. Kharcheva, D. V. Guseinov, M. N. Koryazhkina, www.advancedsciencenews.com www.advintellsyst.com Adv. Intell. Syst. 2023, 5, 2200338 2200338 (44 of 46) © 2022 The Authors. Advanced Intelligent Systems published b...

2023 doi
[194]

N. V. Agudov, A. A. Dubkov, B. Spagnolo, Physica A 2003, 325, 144

2003
[195]

A. V. Yakimov, D. O. Filatov, O. N. Gorshkov, D. A. Antonov, D. A. Liskin, I. N. Antonov, A. V. Belyakov, A. V. Klyuev, A. Carollo, B. Spagnolo, Appl. Phys. Lett. 2019, 114, 253506

2019
[196]

Sánta, Z

B. Sánta, Z. Balogh, A. Gubicza, L. Pósa, D. Krisztián, G. Mihály, M. Csontos, A. Halbritter, Nanoscale 2019, 11, 4719

2019
[197]

Sánta, Z

B. Sánta, Z. Balogh, L. Pósa, D. Krisztián, T. N. Török, D. Molnár, C. Sinkó, R. Hauert, M. Csontos, A. Halbritter, ACS Appl. Mater. Interfaces 2021, 13, 7453

2021
[198]

R. F. Voss, J. Clarke, Phys. Rev. B 1976, 13, 556

1976
[199]

G. N. Bochkov, Y. E. Kuzovlev, Sov. Phys. Usp. 1983, 26, 829

1983
[200]

Larentis, F

S. Larentis, F. Nardi, S. Balatti, D. C. Gilmer, D. Ielmini, IEEE Trans. Electron Devices 2012, 59, 2468

2012
[201]

Ielmini, Semicond

D. Ielmini, Semicond. Sci. Technol. 2016, 31, 063002

2016
[202]

J. J. Yang, M. D. Pickett, X. Li, D. A. A. Ohlberg, D. R. Stewart, R. S. Williams, Nat. Nanotechnol. 2008, 3, 429

2008
[203]

D. V. Guseinov, A. N. Mikhaylov, Y. V. Pershin, IEEE Trans. Circuits Syst. II Express Briefs 2022, 69, 18021806

2022
[204]

S. A. Gerasimova, A. I. Belov, D. S. Korolev, D. V. Guseinov, A. V. Lebedeva, M. N. Koryazhkina, A. N. Mikhaylov, V. B. Kazantsev, A. N. Pisarchik, Sensors 2021, 21, 5587

2021
[205]

W. Yi, S. E. Savel ’ev, G. Medeiros-Ribeiro, F. Miao, M. X. Zhang, J. J. Yang, A. M. Bratkovsky, R. S. Williams, Nat. Commun. 2016, 7, 11142

2016
[206]

R. L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics , Springer, New York City, USA 1992

1992
[207]

H. B. Callen, T. A. Welton, Phys. Rev. 1951, 83, 34

1951
[208]

G. F. Efremov, J. Exp. Theor. Phys. 1969, 28, 1232

1969
[209]

M. S. Gupta, Phys. Rev. A 1978, 18, 2725

1978
[210]

G. N. Bochkov, Y. E. Kuzovlev, J. Exp. Theor. Phys. 1977, 45, 125

1977
[211]

Jarzynski, Phys

C. Jarzynski, Phys. Rev. Lett. 1997, 78, 2690

1997
[212]

Kubo, Rep

R. Kubo, Rep. Prog. Phys. 1966, 29, 255

1966
[213]

A. A. Dubkov, P. Hänggi, I. Goychuk, J. Stat. Mech: Theory Exp. 2009, 2009, P01034

2009
[214]

W. T. Coffey, Y. P. Kalmykov, J. T. Waldron, World Scientiﬁc Series in Contemporary Chemical Physics , World Scienti ﬁc, Singapore 1996

1996
[215]

L. O. Chua, IEEE Trans. Circuit Theory 1971, 18, 507

1971
[216]

V. A. Slipko, Y. V. Pershin, M. Di Ventra, Phys. Rev. E 2013, 87, 042103

2013
[217]

Biolek, M

D. Biolek, M. Di Ventra, Y. V. Pershin, Radioengineering 2013, 22, 945

2013
[218]

S. H. Jo, K. H. Kim, W. Lu, Nano Lett. 2009, 9, 496

2009
[219]

D. O. Filatov, M. N. Koryazhkina, A. S. Novikov, V. A. Shishmakova, M. E. Shenina, I. N. Antonov, O. N. Gorshkov, N. V. Agudov, A. Carollo, D. Valenti, B. Spagnolo, Chaos, Solitons Fractals 2022, 156, 111810

2022
[220]

F. Pan, J. Jang, V. Subramanian, presented in 70th Device Research Conf., 2012 , IEEE, University Park, PA, pp. 217 – 218, June 2012

2012
[221]

O. N. Gorshkov, I. N. Antonov, A. I. Belov, A. P. Kasatkin, A. N. Mikhaylov, Tech. Phys. Lett. 2014, 40, 101

2014
[222]

A. V. Emelyanov, K. E. Nikiruy, V. A. Demin, V. V. Rylkov, A. I. Belov, D. S. Korolev, E. G. Gryaznov, D. A. Pavlov, O. N. Gorshkov, A. N. Mikhaylov, P. Dimitrakis, Microelectron. Eng. 2019, 215, 110988

2019
[223]

N. K. Upadhyay, W. Sun, P. Lin, S. Joshi, R. Midya, X. Zhang, Z. Wang, H. Jiang, J. H. Yoon, M. Rao, M. Chi, Q. Xia, J. J. Yang, Adv. Electron. Mater. 2020, 6, 1901411

2020
[224]

M. A. Ryabova, D. O. Filatov, M. N. Koriazhkina, D. A. Antonov, I. N. Antonov, O. N. Gorshkov, A. A. Kharcherva, A. A. Dubkov, J. Phys. Conf. Ser. 2021, 1851, 012003

2021
[225]

Corinto, P

F. Corinto, P. P. Civalleri, L. O. Chua, IEEE J. Emerging Sel. Top. Circuits Syst. 2015, 5, 123

2015
[226]

Picos, S

R. Picos, S. G. Stavrinides, M. M. Al Chawa, C. de Benito, S. Duenas, H. Castan, E. Hatzikraniotis, L. O. Chua, Electronics 2022, 11, 1672

2022
[227]

Maldonado, M

D. Maldonado, M. B. Gonzalez, F. Campabadal, F. Jimenez- Molinos, M. M. Al Chawa, S. G. Stavrinides, J. B. Roldan, R. Tetzlaff, R. Picos, L. O. Chua, Chaos, Solitons Fractals 2020, 139, 110288

2020
[228]

M. M. Al Chawa, R. Picos, R. Tetzlaff, in presented at 2020 IEEE Int. Symp. on Circuits and Systems (ISCAS) , IEEE, Piscataway, NJ 2020, pp. 1 –1

2020
[229]

M. M. Al Chawa, R. Picos, R. Tetzlaff, IEEE Trans. Circuits Syst. I Regul. Pap. 2021, 68, 3631

2021
[230]

M. M. Al Chawa, R. Tetzlaff, R. Picos, in 27th IEEE Inter. Conf. on Electronics, Circuits and Systems (ICECS) , IEEE, Piscataway, NJ 2020, pp. 1 –4

2020
[231]

Frascaroli, S

J. Frascaroli, S. Brivio, E. Covi, S. Spiga, Sci. Rep. 2018, 8, 7178

2018
[232]

Brivio, J

S. Brivio, J. Frascaroli, S. Spiga, Appl. Phys. Lett. 2015, 107, 023504

2015
[233]

Picos, J

R. Picos, J. B. Roldan, M. M. Al Chawa, F. Jimenez-Molinos, E. Garcia-Moreno, in 2016 Conf. on Design of Circuits and Integrated Systems (DCIS) , IEEE, Piscataway, NJ 2016, pp. 1 –6

2016
[234]

S. Yu, P. Y. Chen, IEEE Solid-State Circuits Mag. 2016, 8, 43

2016
[235]

Y. Dai, Y. Zhao, J. Wang, J. Xu, F. Yang, AIP Adv. 2015, 5, 017133

2015
[236]

Blasco, N

J. Blasco, N. Ghenzi, J. Su ˜né, P. Levy, E. Miranda, Microelectron. Reliab. 2015, 55,1

2015
[237]

D. J. Wouters, S. Menzel, J. A. J. Rupp, T. Hennen, R. Waser, Faraday Discuss. 2019, 213, 183

2019
[238]

Salvador, M

E. Salvador, M. B. Gonzalez, F. Campabadal, R. Rodríguez, E. Miranda, in 2021 IEEE 32nd Inter. Conf. on Microelectronics (MIEL), IEEE, Piscataway, NJ 2021, pp. 73 –76

2021
[239]

S. Long, X. Lian, T. Ye, C. Cagli, L. Perniola, E. Miranda, M. Liu, J. Su ˜né, IEEE Electron Device Lett. 2013, 34, 623

2013
[240]

W. C. Luo, J. C. Liu, H. T. Feng, Y. C. Lin, J. J. Huang, K. L. Lin, T. H. Hou, in Inter. Electron Device Meeting , New York City, USA 2012, p. 9.5.1

2012
[241]

Aldana, P

S. Aldana, P. García-Fernández, A. Rodríguez-Fernández, R. Romero-Zaliz, M. B. González, F. Jiménez-Molinos, F. Campabadal, F. Gómez-Campos, J. B. Roldán, J. Phys. D: Appl. Phys. 2017, 50, 335103

2017
[242]

E. Y. Wu, B. Li, J. H. Stathis, Appl. Phys. Lett. 2013, 103, 152907

2013
[243]

E. Y. Wu, E. J. Nowak, R. Vollertsen, L. K. Han, IEEE Trans. Electron Device 2000, 47, 2301

2000
[244]

C. Acal, J. E. Ruiz-Castro, A. M. Aguilera, F. Jiménez-Molinos, J. B. Roldán, J. Comput. Appl. Math. 2019, 345, 23

2019
[245]

M. Xie, S. L. Ho, J. Qual. Maint. Eng. 1999, 5, 50

1999
[246]

F. J. Alonso, D. Maldonado, A. M. Aguilera, J. B. Roldan, Chaos, Solitons Fractals 2021, 143, 110461

2021
[247]

Rodriguez, D

N. Rodriguez, D. Maldonado, F. J. Romero, F. J. Alonso, A. M. Aguilera, A. Godoy, F. Jimenez-Molinos, F. G. Ruiz, J. B. Roldan, Materials 2019, 12, 3734

2019
[248]

P. J. Brockwell, R. A. Davis, Introduction to Time Series and Forecasting, 2nd ed., Springer, New York City, USA 2002

2002
[249]

Bisgaard, M

S. Bisgaard, M. Kulahci, Time Series Analysis and Forecasting by Example, Wiley, Hoboken, NJ 2011

2011
[250]

D. C. Montgomery, C. L. Jennings, M. Kulahci, Introduction to Time Series Analysis and Forecasting , Wiley Series in Probability and Statistics, Wiley, Hoboken, NJ 2015

2015
[251]

Poblador, M

S. Poblador, M. Maestro-Izquierdo, M. Zabala, M. B. Gonzalez, F. Campabadal, Microelectron. Eng. 2020, 223, 111232

2020
[252]

J. B. Roldan, D. Maldonado, C. Aguilera-Pedregosa, F. J. Alonso, Y. Xiao, Y. Shen, W. Zheng, Y. Yuan, M. Lanza, IEEE Trans. Electron Devices , in press 2022, https://doi.org/10.1109/TED. 2022.3197677. www.advancedsciencenews.com www.advintellsyst.com Adv. Intell. Syst. 2023, 5...

2022
[253]

Lanza, F

M. Lanza, F. Palumbo, Y. Shi, F. Aguirre, S. Boyeras, B. Yuan, E. Yalon, E. Moreno, T. Wu, J. B. Roldan, Adv. Electron. Mater. 2021, 2100580

2021
[254]

Z. Ma, J. Ge, W. Chen, X. Cao, S. Diao, Z. Liu, S. Pan, ACS Appl. Mater. Interfaces 2022, 14, 21207

2022
[255]

W. Kim, S. Menzel, D. J. Wouters, Y. Guo, J. Robertson, B. Roesgen, R. Waser, V. Rana, Nanoscale 2016, 8, 17774

2016
[256]

Garcia-Moreno, R

E. Garcia-Moreno, R. Picos, M. M. Al-Chawa, in 2015 IEEE Inter. Autumn Meeting on Power, Electronics and Computing (ROPEC) , IEEE, Piscataway, NJ 2015, pp. 1 – 4

2015
[257]

Rodriguez-Fernandez, J

A. Rodriguez-Fernandez, J. Su ˜né, E. Miranda, M. B. Gonzalez, F. Campabadal, M. M. Al Chawa, R. Picos, in 32nd Conf. on Design of Circuits and Integrated Systems (DCIS) , IEEE, Piscataway, USA 2017, pp. 1 –4. Juan B. Roldán received his degree in physics and Ph.D. degree from...

2017

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Lanza, A

M. Lanza, A. Sebastian, W. D. Lu, M. Le Gallo, M.-F. Chang, D. Akinwande, F. M. Puglisi, H. N. Alshareef, M. Liu, J. B. Roldan, Science 2022, 376, 6597

2022

[2] [2]

L. O. Chua, S. M. Kang, Proc. IEEE 1976, 64, 209

1976

[3] [3]

Spiga, A

S. Spiga, A. Sebastian, D. Querlioz, B. Rajendran, Memristive Devices for Brain-Inspired Computing, Elsevier, Amsterdam, Netherlands 2020

2020

[4] [4]

F. Pan, S. Gao, C. Chen, C. Song, F. Zeng, Mater. Sci. Eng. 2014, 83,1

2014

[5] [5]

Variability Behavioral Models 5.1. Variability under the General Memristor Model Formalism In this section, we present an approach tackling with the variabil- ity of the parameters employed in a memristor model, the rela- tion between these parameters and the consequences on the model implementation. Speci ﬁcally, we consider a memristor representation de...

2023

[6] [6]

Discussion We have shown in Section 2 that variability is an essential issue in resistive memories operation and, as discussed along this manuscript, this issue can be tackled in multiple and intricate manners. From physical to behavioral models, combining sto- chastic approaches as well as purely mathematical tools, all con- verge to point out the import...

2023

[7] [7]

Priority-2030

Conclusions Resistive memories have a simple structure and are compatible with CMOS technology; therefore, they can be integrated in the conventional processes of the current electronics industry. However, their huge potential in different applications could be limited by their inherent cycle-to-cycle variability. The variabil- ity issue is essential, and...

2023

[8] [8]

J. S. Lee, S. Lee, T. W. Noh, Appl. Phys. Rev. 2015, 2, 031303

2015