A Voltage‐Driven Transport Model to Identify Ion Migration as the Rate‐Limiting Step in Memristive Switching

The physics behind the switching kinetics of memristors is gradually becoming clearer. The periods required for the onset of electromigration within memristors and the activation or deactivation of the low‐resistance state—referred to as the incubation and switching times—exhibit non‐linearity with applied voltage. This behavior prevails depending on the rate‐limiting step comprising nucleation and filament growth, electron transfer at the electrode/electrolyte interface, and ion migration through the electrolyte. Herein, a model is introduced for ion migration as the rate‐limiting step. This model analyses the incubation time and analytically correlates it with the electric field, diffusion coefficient, and temperature, facilitating the determination of threshold voltage and diffusivity from high to low resistance states for ion migration as the rate‐limiting step. By exploring parallel plate cells with Yttria‐Stabilized Zirconia (YSZ) of nanometer thickness, the application of the model is illustrated and the fundamental equations are applied to outstanding memristive cells in the literature. The applicability of this model to cells of various charge carriers is proposed, ranging from vacancies and electron transport to oxygen ions and metal cations, denoting its potential importance.


Introduction
Since their discovery, memristors have emerged as crucial devices in artificial neuromorphic systems, owing to their unique DOI: 10.1002/aelm.202300608ability to function both as memory and resistors. [1]The basic structure comprises two parallel metal electrodes separated by a solid-state nanometer-thick electrolyte, with its operation involving the application of voltage to switch between low and high-resistance states. [2]his process occurs due to the movement of charged species, such as ions or vacancies within the memristor, that alter the material's conductivity and, therefore, its resistance.
In the early years of the recent decade, investigating the insights of switching mechanisms remained at the forefront of research, [3] primarily due to the profound impact of kinetics on the memristor's operating window. [4]The analysis complexity arises from the fact that the flux of charged species triggers internal, interface, and migration reactions, which are challenging to differentiate and incorporate into a comprehensive model of voltage-driven transport. [5,6]he scientific community has categorized the mechanisms based on the species and phenomena involved during the operation of the devices.Memristors that rely on the conduction of oxygen anions (O 2 − ), metal cations (M + ), and oxygen vacancies (O vac ) are commonly referred to as redox-based resistive switching random access memory (ReRAM). [7]n the other hand, if the operation involves the reversible electrochemical reduction and oxidation of metal ions, typically from a metal electrode, it is known as an electrochemical cell (ECM). [8]nother category is valence change memory (VCM), where the switching process is based on the change in oxidation states (valence states) of specific cations within the solid-state material. [9]hermochemical (TCM) mechanisms [10] represent another type, wherein the memory cell relies on the flow of heat or thermal conductivity to achieve the switching process.Additionally, phase change memories (PCM) [11] form an additional classification where the cell operates based on the reversible phase change of material between its amorphous and crystalline states.
Experimentally, researchers employ both direct and indirect methods to characterize the transport mechanism in memristors.Direct characterization involves in situ observation of the cell during its operating conditions.For instance, Yang and coworkers investigated the activation of the memristive switching mechanism in an Ag/SiO 2 /Pt ECM cell using transmission electron microscopy (TEM). [12]By applying voltages ranging from 8 to 10 V, the low-resistance state (LRS) was activated, and conversely, applying -10 V deactivated the LRS, resulting in the high-resistance state (HRS).In the micrographs, they directly observed the forming of a conductive path (filament) consisting of Ag cations during the LRS state and the gradual fading of this filament when transitioning to the HRS state.
Another direct observation was made by Park and collaborators [13] in a Pt/SiO 2 /Ta 2 O 5 /TaO 2-x /Pt VCM cell.They found that during the activation of the LRS state, there is a predominant flow of oxygen vacancies in the direction of the current flux and a formation of a metallic cluster of Ta.To conduct their study, they employed advanced techniques such as Annular Dark-Field Imaging by Scanning Transmission Electron Microscope (HAADF-STEM) and Electron Energy Loss Spectroscopy (EELS).
By utilizing direct observations, researchers can closely track the reactions during the migration of charge carriers within the memristor cells.However, it is worth noting that these cuttingedge characterizations can be pretty costly, limiting their accessibility to only a few research groups with the necessary resources.
Another equally essential approach to elucidate the memristive mechanism is modeling the electrical response.It is worth noting that the memristor, one of the fundamental elements in electrical circuits along with resistor, capacitor, and inductor, can be effectively modeled with its basic circuit equations. [2]However, the unique characteristics of the electrolyte and interfaces at the nanoscale have required the development of more comprehensive models that encompass the underlying physics and chemistry of the mechanism. [14]n recent years, significant progress has been made in understanding the threshold potentials by modeling switching kinetics.In this sense, the work by Menzel et al. [14] presents a comprehensive investigation of the models that allow prediction [15] and distinguish [16] the contributions of three rate-limiting processes in different voltage ranges: (I) Nucleation and growth of filaments (analyzed through nucleation time and electron tunneling equations), (II) electron transfer during the reduction/oxidation of ions at interfaces (using Butler-Volmer equations), and (III) ion migration through the electrolyte toward interfaces (applying the Mott-Gurney law for ion hopping).Those models facilitate the determination of activation energies, transfer coefficients, and potentials associated with each rate-limiting step at various voltage levels.
Among the fundamental mechanisms of memristors, threshold kinetics emerges as a critical aspect since it identifies the minimum potential for the transition from the HRS state to the LRS state. [7]Physically, before the threshold potential, a critical process takes place involving the accumulation or redistribution of charged species (O 2 − , M + , or O vac ) to form the conductive filament.Before establishing this conductive filament, these charged species, including electrons and holes, play a critical role in the ongoing degradation of the electrolyte or metal oxide as it is subjected to continuous electrical stress.Given the significance of the threshold potential in characterizing memristors, it becomes crucial to establish a well-controlled characterization setup.For example, in current-voltage curves, inadequately con-trolled voltage ramps (V/s) may lead to an inability to appreciate the switching threshold accurately.
An alternative is the implementation of current-time curves under constant voltage.In the previously mentioned research work by Yang et al., [12] they implemented current-time curves to observe the formation of the conductive filament and observed a unique phenomenon known as the "incubation time" during the transition from the HRS state to the LRS state.This incubation time refers to the duration required before the transition process begins.Observation involves monitoring for an abrupt increase in current after the incubation time has elapsed.Interestingly, they also found that the duration of the incubation time is inversely dependent on both the applied voltage and the operating temperature.
In the study conducted by Tsuruoka et al., [17] it describes the incubation time in current-time curves for ECM Cu/Ta 2 O 5(15 nm) /Pt and Pt/Ta 2 O 5(15 nm) /Pt cells.The incubation time is characterized by a rapid and almost exponential decay in current over a short time.After incubation time, the LRS state is activated, and the current experiences a sudden and sharp increase until it reaches a stable value, indicating the equilibrium in the LRS state.
Building upon the existing knowledge, this study aims to introduce a novel model for switching kinetics, focusing on ion migration as the rate-limiting process.What sets this model apart is its exploration of the incubation time, rather than the switching time, as a critical parameter.The theoretical work for this model was laid by R. Kirchheim, [18] who applied it to voltage-driven transport phenomena like flash sintering [19] and Al cation migration, [20] and in a recent study, the authors of this work demonstrated the model experimentally for oxygen vacancy electromigration. [21]The model accurately determines the ratelimiting diffusion coefficient and threshold voltage, establishing a relationship between the incubation time, temperature, and applied electric field.
To demonstrate the model's general applicability, we employed a specific cell based on Yttria-Stabilized Zirconia electrolytes (YSZ).[24] YSZ naturally contains a high concentration of oxygen vacancies, which can be precisely controlled by varying the fraction of yttria into the Zirconia lattice.The number of oxygen vacancies is approximately half that of Y atoms, ensuring a stable and controllable vacancy concentration.This unique property allows YSZ electrolytes to be customized with desired yttria concentrations for memristors.Also, the compatibility of YSZ to a CMOS (complementary metal-oxide semiconductor) process enables its introduction into several processing stages of an integrated circuit, thus paving the way for enhancing the functionality of advanced integrated logic and memory technologies.
The cell configuration was Ru/YSZ (100 nm) /Au , operated at temperatures ranging from 100°C to 170°C.It is explored the implications of the model on the characteristic pinched hysteresis loop observed in current-voltage curves and compared the application of the model to other memristive cells reported in the literature, showcasing its versatility and ability to provide valuable insights into various memristor devices.

Describing the Electrical Setup and Response Characteristics
To illustrate the model's features for investigating the switching mechanism, we employed a constant voltage signal over time and conducted superimposed voltage sweeps using the initial constant voltage as a reference while monitoring the current (Figure 1a).After applying the constant voltage, the current showed a decay during the first few seconds due to the internal bias of the YSZ trapping electron carriers in a purely capacitive process (Figure 1b).This process involved charge accumulation mediated by O vac and O 2 − at adjacent electrodes, putting the cell in a HRS state.Then, the current reached a minimum value, indicating the attainment of the maximum electronic conductivity in YSZ, and remained constant until it increased sharply due to the onset of ionic conductivity (migration of O vac ).We will refer to the period until the onset of migration as the "incubation time" ( incubation ).A representative example of graphically determining the  incubation is provided in Figure 1b.During the abrupt increase in current, the growth rate remains constant.This section of the current curve allows for applying linear regression toward its initial minimum.The resulting X-axis value at the intercept gives us the value of  incubation .
As mentioned, the sharp increase in current marks the beginning of carrier migration from one electrode to the other through a purely conductive filament.As migration proceeds, the current reaches a maximum value, indicating the equilibrium of the migration reaction.This point corresponds to the "steady state," representing the LRS state.The time taken to transition from the HRS state to the LRS state at a specific constant voltage is defined as the "switching time." Once the steady state is achieved, we applied voltage sweep cycles superimposed on the initial constant voltage.At this stage, the characteristic pinched loop in the current curve, commonly studied in the switching mechanism, is observed (Figure 1c).It is important to note that the switching hysteresis is not observed when the voltage sweep cycles are applied at times shorter than the  incubation .

Insights into Ion Migration as Rate-Limiting in YSZ Electrolyte
In recent years, voltage-driven transport in solid electrolytes has gained significant relevance, especially with the emergence of memristors and phenomena like flash sintering [25] , which share essential transport characteristics.Flash sintering is a novel technology used for low-temperature densification of ceramic materials, accomplished by applying an intense electric field to a metal/ceramic/metal cell at a specific temperature.During this process, the electromigration of O 2 − , M + , or O vac occurs, leading to an experimental observation of an abrupt increase in current that eventually stabilizes at a stationary value, similar to what is observed in memristors.However, it is worth noting that flash sintering is an irreversible process, whereas the transition from the HRS to the LRS state in memristors is reversible.Despite this difference, if ion migration through the electrolyte is the ratelimiting step in both phenomena, memristive and flash sintering, they can be described using the theoretical model proposed by R. Kirchheim, where the ion motion is considered to be driven by joule heating and intense electric fields [18,19] .The validity of this model has been demonstrated through theoretical and experimental studies on the electromigration of aluminum cations in metals [20] and oxygen vacancies in YSZ [21] .The model includes an equation that correlates the observed  incubation in the current curves and the applied electric field, as follows: In this equation, D i is the diffusion coefficient, R is the ideal gas constant, T is the temperature, l is the thickness of the electrolyte, F is the Faraday constant, and U is the applied voltage.
To illustrate it Figure 2a illustrates the 3D current curve of the Ru/YSZ (100 nm) /Au cell at 120 °C, displaying different voltages ranging from 3 to 5.25 V.The  incubation , decreases as the applied voltage increases.Concurrently, the current in the LRS state also increases.
One notable feature of the model is the capability to calculate the threshold voltage necessary for the onset electromigration.In Figure 2a, when applying a voltage of 3 V, the current curve remains purely capacitive for up to 300 s.Graphically, it is challenging to determine if the transition from the HRS state to the LRS state will occur over a longer time.By rearranging Equation 1 in the following manner: where is found that the applied voltage, U applied , is proportional to the square root of the reciprocal of the incubation time, 1 √  .In Figure 2b, it is shown the plot of U applied versus 1 √  using the current data from Figure 2a.Extrapolating a straight line to the Y-axis intercept can obtain the minimum voltage required to transition from the HRS state to the LRS state (onset of migration).The determined value is approximately 2.4 ± 0.19 V. Based on the fitting, we can also determine that the  incubation when applying 3 V is ≈1210 s.
Figure 3a-d presents a schematic of the electromigration process within the Ru/YSZ (100 nm) /Au cell.Upon applying the voltage during the HRS state, an accumulation of v ö occurs at the Ru negative electrode, along with e − at Au, the positive electrode.Additionally, O 2 − migrates from the Ru/YSZ interface and accumulates at the Au/YSZ interface, resembling a redox reaction phenomenon (previous research [21,26] has provided evidence of the redox reactions at Au and Ru metal electrodes by physicochemical characterization).As the incubation time elapses, v ö starts to react at the Ru negative electrode as described by the following equation: The metallic Zr atom receives an extra electron due to the migration of O 2 − , leading to the formation of a Ru-Zr alloy.Once v ̇o are formed, they are transported to the positive electrode.At the Au electrode, v ̇o release their electrons: and this leads to a flux of v ö in the opposite direction, maintaining charge neutrality.The equilibrium of this reaction is observed in the steady state, and as mentioned earlier, the diffusion coefficient D i calculated by Equation 1 represents the rate-limiting step.This migration reaction leads to the formation of three distinct internal regions within the YSZ: one where v ̇o are generated, resulting in predominant n-type conduction; another where v ö are formed, leading to dominant p-type conduction and an intermediate transport i-region between these two regions.The resulting internal structure is analogous to p-i-n junctions.
An alternative way to verify the ionic conductivity is by reversing the direction of the electrical circuit and assessing the voltage drop (discharge) at constant current (Figure 3e).The discharging voltage curves were obtained for each applied voltage in Figure 2a at 2.2 μA cm −2 .The discharging voltage indicates the relaxation of the p-i-n regions, [27] and the plateau shape indicates the transport of ionic carriers, specifically O 2 − in this case.

Incubation Time in Outstanding Literature on Resistive Switching
In resistive switching, recent observations have highlighted the significance of analyzing current-time curves for discerning the threshold kinetics.In this section, some of them will be analyzed.Messerschmitt et al. [28] conducted a study on a VCM Pt/SrTiO 3-(600 nm) /Pt cell, where they explored the activating of the LRS state in current-time curves by applying constant voltages ranging from 1 V to 4 V.They obtained current curves similar to those shown in Figure 2a, and the incubation times from their study are presented in Figure 4a.Another work by Tsuruoka et al. [17] focused on the relationship between the threshold voltage and the incubation time in an ECM Cu/Ta 2 O 5(15 nm) /Pt. Figure 4a also includes incubation times for this cell, acquired by applying voltages between 2.5 and 5 V. Additionally, the incubation times from Buh et al., [29] where they conducted a study on a Pt/NiO (40 nm) /Pt cell, are also displayed in Figure 4a.Finally, the incubation times reported by Yang et al. [12] in an ECM Ag/SiO 2(200 nm) /Pt cell at high applied voltages of 10 and 26 V are also included.
To determine the threshold voltage for each cell, we employed the U field versus 1 √  plot introduced in Equation 2. Since the cells have different electrolyte thicknesses, we represented the applied voltage as the electric field.The results are illustrated in Figure 4b, where the linearity of the graph validates the model's applicability to these cells.Extrapolating the graph toward the intercept with the Y-axis at 0 enables us to find the threshold voltage.For the Cu/Ta 2 O 5(15 nm) /Pt cell, [17] the threshold voltage was determined to be 1.90 V, with an incubation time of ≈5750 s when applying 2 V.For the Pt/NiO (40 nm) /Pt cell, [29] the threshold voltage was found to be 2.98 V, with a maximum incubation time of ≈1600 s when applying 3 V.The Pt/SrTiO 3-(600 nm) /Pt cell [28] displayed a calculated threshold voltage of 0.78 V, while the authors observed a threshold voltage of 1.2 V for activating the LRS state.The slight discrepancy may be attributed to the diminishing changes in measured current near the threshold voltage, making distinguishing more challenging.For the Yang cell, [12]  Ag/SiO 2(200 nm) /Pt, the threshold voltage was determined to be 16.37 V.
To comprehend the model's applicability to various voltagedriven species migration systems, it is essential to understand its fundamental premise.Under the influence of an electric field, species undergo electromigration from one electrode to another, resulting in species concentration changes over time.Whether surpassing or falling below specific thresholds, these concen-tration shifts and trigger reactions propel the transport process.For the specific case of YSZ and the migration of oxygen vacancies and/or oxygen ions, previous research has extensively documented the boundaries and solutions of the model. [18,19,21]The model enables the determination of critical parameters like the unwanted electronic current maxima, the charge associated with vacancies (single or double ionized), and oxygen ions.It also offers insights into the thickness of the migration region.Solutions related to the transport of Al cations can be found in reference. [20]n the other hand, potential challenges for this model arise when considering ultra-thin electrolytes, so that electronic conduction is now predominant, and the role of ion migration, along with the diffusion properties of the ionic species of interest, could lead to under or overestimations of the incubation time.

Exploring Rate-Limiting Ion Migration in Fast Memristive Cells
This section delves into fast-switching cells where incubation times are in the ms-μs range.In these fast-switching cells, the incubation time closely approximates the switching time, enabling a direct analysis of threshold kinetics via Equation 1.
Recent studies in modeling memristors have led to a focused analysis of the relationship between switching time and applied voltage. [15,33,34]Researchers have identified distinct regions on the switching time versus voltage curve, each associated with intrinsic rate-limiting processes.In scenarios where the curve exhibits a pronounced slope, the rate-limiting step pertains to the nucleation and growth of the conduction filament.This phenomenon is investigated using nucleation time and electron tunneling equations.In cases of medium slopes, the electron transfer during the redox reactions of ions at interfaces governs the rate-limiting process.Such instances are analyzed by the application of Butler-Volmer equations.Soft slopes signify ion migration as the rate-limiting process, and its behavior aligns with the principles of the Mott-Gurney law for ion hopping.These approaches facilitate the determination of activation energies, transfer coefficients, and overpotentials that influence the migration reaction.
Figure 5a provides switching time versus voltage curves, featuring data extracted from various research papers that underscore ion migration as the rate-limiting step within a specific range of operating voltages.This compilation includes ECM cells based on the migration of Ag cations [15,[33][34][35][36] and a VCM cell involving oxygen vacancy migration. [37] Figure 5b-e, we present U field versus 1 √  plots derived from the data in Figure 5a.Given the substantial variation in Y-axis values, the data was distributed across four distinct graphs.Once again, the linearity observed in these U field versus 1 √  plots reinforce the model's validity within this specific range of operating voltages.Our analysis culminates in extracting the diffusion coefficient and threshold voltage of this behavior, as summarized in Table 1.Among the various cells, SiO 2 is a prominent electrolyte choice for facilitating Ag cation migration.Despite utilizing closely comparable thicknesses, disparities in threshold potentials and diffusion coefficients highlight the critical role between counter-electrodes and SiO 2 stoichiometry.
In the context of designing high-speed memristor devices, identifying the rate-limiting step of the migration reaction is essential for several reasons.Understanding what specific   [38] for the diffusion of Ag + cations in SiO 2 , is also included.c) Comparative of effective conductivity in the LRS state at equilibrium.
process limits the migration reaction allows the optimization of memristor design and materials.For example, if ion migration through the electrolyte is identified as the limiting factor using this model, researchers can focus on developing materials with improved ionic conductivity while also studying the systematic scaling down of their physical thickness.This could involve optimizing electrolyte composition, crystal structure, or defect density to enhance ion mobility and reduce resistance during migration.Alternatively, and more generally, identifying the limiting step in the pace of the migratory reaction allows us to focus our efforts.If electron transfer at interfaces or nucleation and growth are identified as limiting factors, optimization of electrode materials or engineering of interfaces can be pursued to minimize resistance, promote faster filament formation, and accelerate switching.Therefore, by considering ion-migration for even thinner YSZ materials, our model could deliver not only diffusion coefficients and threshold voltages for switching but also provide helpful parameters related to the operating conditions of memristive devices like power consumption, switching velocities and additional reliability parameters that are pretty important for scaled-down devices in advanced technology nodes.

Temperature Dependence of the Incubation Time
Figure 6a depicts the current versus time curves of the Ru/YSZ (100 nm) /Au cell under a constant 3 V across tempera-tures ranging from 120 to 170 °C.By employing Equation 1, the diffusion coefficient was calculated, and its corresponding Arrhenius plot is illustrated in Figure 6b.It is important to note that in this scenario, we must account for U is equal to the U applied − U threshold , [21] which leads to a modified equation: Within this analysis, the activation energy Ea, is ≈1.10 eV, corresponding to the ionic conductivity in the YSZ. [39]The data collected under 4 V from our previous work [21] is also incorporated, and the alignment between diffusion coefficients across various voltages agrees with the model's predictions, wherein the motion of a planar front is the determinant of velocity.In instances where diffusion governs the ratelimiting step, the applied voltage does not affect the D i itself.Instead, the variation in current and  incubation pertains to the increase of mobile charge carriers.Extrapolation of these diffusion coefficients to elevated temperatures show congruence between our values and those obtained through Secondaryion mass spectrometry measurements [39] in YSZ with varied stoichiometry.
In the context of the Ag/SiO 2 /Pt cell by Yang et al., [12] the relationship between incubation time and temperature was explored by employing the analysis of Equation 6.In Figure 6a, the Another aspect within the curves presented in Figure 6a is the behavior of the steady-state current over long times (LRS state).It is observed that as the temperature increases, the current reaches higher values.These observations can be examined by the effective conductivity, with the expression: where i steady − state is derived from current curves at long times and L A represents the sample volume factor.Figure 6c shows the evolution of  eff.for the Ru/YSZ 100nm /Au and Pt/SrTiO 3-(600 nm) /Pt [28] cells under different operating voltages.
The activation energy for the Ru/YSZ 100nm /Au cell decreased from 1.4 eV (at 3 V) to 0.67 eV (at 4 V).This change in conductivity and activation energy can be attributed to the higher concentration of charge carriers within the conductive filament at higher voltages.
Interestingly, our findings are consistent with those obtained in the Pt/SrTiO 3-(600 nm) /Pt cell. [28]Their exploration of effective conductivity within the LRS state under varying applied voltages shows a similar decrease in activation energy at higher operat-ing voltages.In the context of this model, operating at higher voltages increased the concentration of charge carriers, facilitating their movement and leading to a decrease in activation energy.

The Impact of Constant Voltage Offset
The second phase of the experiment, as outlined in Section 2.1.1,involves the application of superimposed voltage sweeps during the stable LRS state.By visualizing Figure 7a, these superimposed voltage signals were conducted with the initial constant voltage ranging from 0 to 4 V at 140 °C.The subsequent voltage sweep was ± 5 V, operating at 30 Hz.
The response currents, as shown in Figure 7b, reveal a correlation with the initial constant voltage.When the initial constant voltage remains below 2 V, the current remains mainly flat, aligned with the anticipated behavior of a polarizable capacitor (HRS state).
In contrast, when the initial constant voltage surpasses the 2 V threshold, the pinched hysteresis loop characteristic of the switching mechanism emerges.The current curve observed during the transition from the HRS to the LRS state resembles that The initial constant voltage was set to 4 V. e) Relationship between series voltage and frequency calculated using Equation 8. f) Set-current of HRS and LRS states plotted against series resistance.g) Frequency-dependent variations in HRS and LRS resistances.Forward and reverse resistances measured above 100 Hz are also included.Operational Regimes and Circuit Model of Ru/YSZ (100 nm) /Au Cell.h) Conductive filament development (dotted line) with frequency-dependent i-region and Au oxidation.i) Desynchronization of conductive filament, i-region, and Au oxidation with increasing frequency.j) Absence of conductive filament, coupled with reduced reactive capacitance at high frequency, enhances charge trapping.k) Proposed electrical model for the Ru/YSZ (100 nm) /Au cell, illustrating parasitic resistances and components.
of a p-i-n diode, showing a soft transition near the threshold voltage.
Considering the p-i-n regions formed within the Ru/YSZ (100 nm) /Au cell, as discussed in Section 2.1.2,the observed behavior can be illustrated as follows: Figure 7c-f illustrates the stages and their phenomena for the Ru/YSZ (100 nm) /Au cell behavior based on internal p-i-n junction, which governs the current-voltage characteristics depicted in Figure 7b.Of particular significance is the presence of two serial resistances: Rs i , associated with the resistance of the i-region, and Rs ox , to the redox process at the electrodes.Rs i , Rs ox, as well as the ion migration (O 2 − , v Ȯ, and v Ö) and redox reactions all depend on the applied voltage and frequency.These phenomena exhibit limitations in tracking their corresponding frequency-induced variations at higher frequencies, a topic we will investigate later.

Voltage Sweeps Frequency and Multilevel Switching
Figure 8 shows current curves corresponding to ±5 V sweeps at frequencies ranging from 5 Hz to 100 kHz for the Ru/YSZ (100 nm) /Au cell at 140 °C.At frequencies below 50 Hz, an apparent decrease of the applied voltage and current is observed as frequency increases, as illustrated in Figure 8a,b.
The apparent decrease in the applied voltage can be explained if we consider the presence of internal resistance R series , including that designated as Rs ox , which we associate with the progressive redox reactions at the Au electrode.The internal series voltage (I × R series ) can be extracted from the measurements using the relationship: where V applied is the applied voltage, V series is the voltage due to R series , and I is the set current.The calculated values of V series are illustrated in Figure 8e.It is noteworthy that V series tends to approach zero beyond 100 Hz, as higher frequencies render redox reactions unable to sustain Au oxidation because they are slower, avoiding the development of Rs ox and eliminating any parasitic internal voltage drop (Figure 8c,d).
In the case of the effect that frequency has on the current and resistance of the LRS and HRS states, the motion of oxygen vacancies, which is highly frequency dependent, may be the cause.In particular cases, the modulation of the maximum current in memristors is an essential aspect of multilevel switching. [40,41]Typically, the test equipment adjusts the compliance current I cc (maximum) by incorporating diverse resistors into the circuit's configuration.However, different investigations have demonstrated that self-limiting behavior can be introduced by integrating at least two materials with distinct resistances within the oxide electrolyte.Hardtdegen et al. [42] observed multilevel switching utilizing a Ti/TiO 2 /HfO 2 trilayer as the electrolyte, Kyung et al. [43] employed a Ta 2 O 5 /TaO x bilayer, while Fantini et al. [44] showcased the potential of metal/electrolyte junctions such as TiN/Hf/HfO 2 to induce this effect.Turning to our Ru/YSZ (100 nm) /Au cell, the series resistances and multilevel switching stem from the variations in conductivity within the internal p-i-n regions, as stated in Section 2.1.2.A comparative analysis of maximum SET/RESET currents against the calculated R series from Equation 8 is depicted in Figure 8f.The influence of series resistance is most pronounced during the RESET process.
As the frequency ranges from 5 to 50 Hz, we notice a decrease in the hysteresis area, moving toward an almost reversible behavior.With increasing frequency, a predictable reduction in the hysteresis width is anticipated. [45]This change can be attributed to the fact that the transported O vac struggle to keep up with the rapid changes in the applied signal's speed (frequency).Figure 8g shows the resistance of the HRS state and LRS state as a function of frequency, where it is observed that starting from 100 Hz, the hysteresis width between the resistance of the HRS and LRS states disappears.
Interestingly, within 200 Hz to 100 kHz (Figure 8c,d), the hysteresis area experiences an increase, resembling characteristics often associated with ferroelectric materials. [46]In ferroelectrics, this phenomenon primarily stems from internal polarization.For our Ru/YSZ (100 nm) /Au cell, the internal local polarization emerges from the internal p-i-n regions, which remain constant during the stable LRS state.Furthermore, the change in the hysteresis loop's direction correlates with a more significant contribution from electron trapping at the interfaces between YSZ and the metal.
In summary the Ru/YSZ (100 nm) /Au cell shows three distinct operational regimes, each of which exhibits memristive properties that are frequency-dependent: Memristive behavior in the frequency range of 5-50 Hz, the internal p-i-n junction behavior from 100-500 Hz, and capacitive behavior by significant electron trapping within 1-100 kHz.These operational regimes are visually represented in Figure 8h-k, alongside a proposed circuit model designed to identify the parasitic resistances in the Ru/YSZ (100 nm) /Au cell.It is evident that at higher frequencies, the conductive filament, i-region, and Au oxidation cease to be prominent, resulting in a capacitive behavior.Simultaneously, electron trapping becomes more pronounced due to the reduced impedance of the cell.

Voltage Sweeps and Temperature Dependence
The effect of temperature on the current-voltage curves for the Ru/YSZ (100 nm) /Au cell is illustrated in Figure 9a,b.The experimental setup was a temperature range from 100 to 170 °C, with a constant voltage of 4 V.The superimposed voltage sweep remains at ± 5 V at a frequency of 30 Hz, as in the preceding section.
With ascending temperatures, both the current response and the hysteresis increased.The emergence of the pinched hysteresis loop aligns closely with the temperature range of 110°C to 120°C, corresponding to the lower threshold temperature for the onset of electromigration (Section 2.1.5).
The increase in the hysteresis window and the current response with temperature can be attributed to higher ion diffusivity, as stated in Equation 1.On the other hand, it is also observed that the increase in the operating temperature reduces the threshold voltage for the LRS state, as shown in Figure 9c.
At 110 °C, the required voltage for activating the LRS state (V set ) is ≈2.7 V; with the temperature increasing to 170 °C, this value decreases to approximately 2 V.In contrast, the voltage for deactivating the LRS state (V reset ) remains at 1.55 V.
As temperature increases, the probability of oxygen vacancy creation within the material, particularly YSZ in this context, increases.This phenomenon aligns with the solid-state theory and can be quantified using the equation [47] : Here, E v is the energy required for vacancy creation, k B is Boltzmann's constant, and T is the temperature.The decrease in V set results from the increased probability of creating oxygen vacancies with increasing temperature, facilitating their migration.The V reset constancy can be attributed to maintaining the initial HRS state through the constant application of 4 V. Figure 9d shows the conductance of the HRS and LRS states within an Arrhenius plot.The curves 120 to 170 °C were fitted, excluding 100 and 110 °C due to the incomplete formation of the pinched hysteresis loop.The activation energy for the HRS state was calculated at 68.5 kJ mol −1 or 0.71 eV, while for the LRS state, it amounts to 66.5 kJ mol −1 or 0.69 eV.

Cooling the Ru/YSZ (100 nm) /Au Cell while Sustaining the Switching Mechanism
In this experiment, we explore the cooling process of the Ru/YSZ (100 nm) /Au cell from 170 °C down to 100 °C while ensuring the continuous operation of the switching mechanism.The cell was subjected to the same conditions: a constant 4 V, a ±5 V superimposed sweep, and a 30 Hz frequency.Unlike the previous section, where the voltage signal was turned off during temperature transitions, the voltage signals remained active throughout the cooling process.The schematic representation and resulting current curves are depicted in Figure 9e,f.A notable observation arises as the sample is gradually cooled to 100°C: the SET voltage remains constant at 2 V across all temperatures (Figure 9g).This contrasts with the previous section, where the SET voltage exhibited a range of values from 2.7 V at 100 °C to 2 V at 170 °C.Also, the pinched hysteresis loop persists up to 100°C as the temperature is lowered without discontinuing the applied voltage signal.Additionally, the RESET voltage maintains its value at 1.55 V throughout.Constructing an Arrhenius plot based on the conductance data reveals a consistent curve that spans the entire temperature range (Figure 9h).The extracted activation energy for the HRS and LRS states is 60.2 kJ mole −1 or 0.65 eV.These slightly diminished values can be anticipated, as the initial energy input from the potential is not required at each temperature.The initial temperature of 170 °C might have induced additional electronic/ionic charges, effectively "locking" the SET/RESET voltages.

Conclusion
In this study, we investigated the fundamental aspects of memristive switching kinetics in metal/electrolyte/metal cells, where the rate-limiting process depends on the diffusion of metal atoms, oxygen vacancies, or oxygen ions through the electrolyte.
The exemplification of the model on the memristive switching behavior of the Ru/YSZ (100 nm) /Au cell elucidates different voltage-and frequency-dependent phenomena, encompassing electronic conduction, charge trapping, redox processes, valence changes, oxygen vacancy migration, ion conduction, and conductive filament formation/annihilation.The parameters mentioned above, such as the diffusion constant of the electrolyte and the threshold voltage that governs the behavior, were calculated by applying the model to different cells highlighted in the literature.Identifying the ion migration through the electrolyte as the rate-limiting step has essential implications for memristor design.Researchers can explore modifications to electrolyte composition, crystal structures, scaled-down physical thickness, or defect densities to improve ion mobility and facilitate faster switching.
The used model highlights the nature of the incubation time necessary for the onset of electromigration, which exhibits a linear correlation with the inverse of the applied voltage squared, regardless of the species transported, as explained in Equation 1.We emphasize the importance of identifying the switching reaction in current measurements over time to examine critical parameters such as threshold voltage and LRS state stability rather than relying solely on I-V curve sweeps.Lastly, the exploration of thermal activation of electromigration not only provides insights into charge carriers through activation energy calculations but also sheds light on potential transitions to alternate transport mechanisms, contributing to a more profound understanding of the overall behavior.

Experimental Section
Fabrication of the Device Structure: The thin-film device was fabricated on p-type (100) silicon wafers and followed a Metal/Insulator/Metal structure, with Ru as the lower electrode, YSZ as the electrolyte, and Au as the upper electrode.
To fabricate the Ru and YSZ, Atomic Layer Deposition (ALD) was employed in a Beneq TFS-200 system.First, a 40 nm Al 2 O 3 layer was deposited to enhance adhesion to the Ru layer and electrically isolate it from Si.The Al 2 O 3 deposition occurred at 250 °C, utilizing trimethylaluminum (TMA, Strem chemicals) as the precursor and water (H 2 O) as the oxidizing agent.
Following this, 100 nm of YSZ was grown at 250 °C by alternately depositing 4 cycles of ZrO 2 (growth rate of 0.8 Å per cycle) and 1 of Y 2 O 3 (growth rate of 1.1 Å per cycle).The precursors for YSZ were tetrakis(ethylmethylamino)zirconium (TEMAZr, Strem chemicals), and tris(methylcyclopentadienyl)yttrium (Y(MeCp) 3 , Strem chemicals) combined with H 2 O. Finally, the upper Au electrode was thermally evaporated in circular patterns of 760 μm diameter using a shadow mask on a JEOL JEE-400 PVD system, resulting in a 30 nm thick Au layer.
The comprehensive material characterization of the fabricated cells had been thoroughly presented in the previous works. [26,48,49]The analysis includes the structural, surface, vibrational, and optical properties.
Electrical Characterization: Electrical measurements were conducted in an air atmosphere from 100 to 170 °C.For the electrical circuit connection, gold-plated needle probes affixed to micromanipulators (Semiprobe) were employed for both the lower and upper electrodes.To apply voltage signals, a source meter unit (SMU, Keithley 2450) and a wave function generator (Rigol G1022Z) were utilized.The response current was then measured using the same SMU and an oscilloscope (Tektronix DPO2004B).All electrical measurements were performed inside a Faraday cage to mitigate the influence of external electrical interference (noise).

Figure 1 .
Figure 1.Schematic of the experiment for activating the memristive mechanism in Ru/YSZ (100 nm) /Au cell.a) Voltage signal applied over time.b) Response current during the applied constant voltage.The log-log plot is shown in the inset.c) Current characteristics by applying the voltage sweep cycles.The curves were taken at 140 °C with a constant voltage of 4 V.The superimposed voltage sweep was from −5 to 5 V.

Figure 2 .
Figure 2. a) 3D current curve as a function of time at different applied voltages to the cell Ru/YSZ (100 nm) /Au at 120 °C.b) Plot of the U applied versus the square root of the inverse of the  incubation .

Figure 3 .
Figure 3. a-d) Schematic of the internal reaction during the electromigration process in the switching mechanism of the Ru/YSZ (100 nm) /Au cell.e) Relaxation voltage at a constant current of 2.2 μA cm −2 after turning off the applied voltage at 120 °C.

Figure 4 .
Figure 4. a) A comparison of the incubation time for activating the switching mechanism in different memristive cells.b) U field versus 1 √  plot for the

Figure 5 .
Figure 5. a) Comparison of switching times from diverse, fast memristive cells.b-e) U field versus 1 √  plots for threshold voltage determination.

Figure 6 .
Figure 6.a) Current-time curves under a constant 3 V at temperatures between 120 and 170 °C.b) Comparative analysis of calculated diffusion coefficients.Data from McBrayer et al.[38] for the diffusion of Ag + cations in SiO 2 , is also included.c) Comparative of effective conductivity in the LRS state at equilibrium.

Figure 7 .
Figure 7. a) Voltage signals employed for investigating the switching mechanism in the Ru/YSZ (100 nm) /Au cell.The experimentation occurred at 140 °C, utilizing constant voltages between 0 and 4 V and a voltage sweep of ±5 V at 30 Hz. b) Distinctive current response during the experimental procedure.c-f) A phenomenological model for the memristive hysteresis loop for the Ru/YSZ (100 nm) /Au cell, attributed to the internal p-i-n junction within the YSZ.The behavior of its i-region and redox reactions is influenced by both voltage and frequency.Panels (c) and (d) replicate previous Figure 3a-d.The impact of the superimposed voltage sweep becomes evident in panels (e) and (f), wherein this additional voltage input directly modulates the i-region.

Figure 8 .
Figure 8. a-d) Variation in the current response for the Ru/YSZ (100 nm) /Au cell under superimposed ±5 V sweeps at 140 °C as a function of frequency.The initial constant voltage was set to 4 V. e) Relationship between series voltage and frequency calculated using Equation8.f) Set-current of HRS and LRS states plotted against series resistance.g) Frequency-dependent variations in HRS and LRS resistances.Forward and reverse resistances measured above 100 Hz are also included.Operational Regimes and Circuit Model of Ru/YSZ (100 nm) /Au Cell.h) Conductive filament development (dotted line) with frequency-dependent i-region and Au oxidation.i) Desynchronization of conductive filament, i-region, and Au oxidation with increasing frequency.j) Absence of conductive filament, coupled with reduced reactive capacitance at high frequency, enhances charge trapping.k) Proposed electrical model for the Ru/YSZ (100 nm) /Au cell, illustrating parasitic resistances and components.

Figure 9 .
Figure 9. Temperature effect on the current response to the voltage sweeps in the applied to the Ru/YSZ (100 nm) /Au cell.a) Applied signal, b) Currentvoltage curves between 100 -170 °C with constant 4 V and superimposed ±5 V sweeps at 30 Hz. c) Temperature dependence of SET and RESET voltages.d) Arrhenius plot of the HRS and LRS states.Second experiment.e) Applied signal, f) Current response of the Ru/YSZ (100 nm) /Au cell from 170 to 100 °C while maintaining an active voltage signal.Conditions include a constant 4 V, a superimposed ±5 V sweep at 30 Hz. g) Temperature-dependent Set and Reset Voltages.h) Arrhenius plot illustrating the HRS and LRS states' behavior across temperatures.

Table 1 .
The calculated diffusion coefficient and the threshold voltage from the cells in Figure5.