# Structural and Parametric Identification of Knowm Memristors

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}-based resistive switching devices by HP Labs in 2008 [2]. Since then, any devices which exhibit specific properties, named “the fingerprints” [3], are usually referred to as memristors. Promising applications of memristive devices include non-volatile memory [4], logic circuits [5], sensing [6], cryptography [7], chaotic generators [8], and neuromorphic computing [9]. Development of the latter direction is performed from the standpoint of using memristors as synaptic connections in artificial neural networks [10], mimicking biological architectures in the nervous systems. The most recent progress in the study of memristors in bio-inspired circuits was made in [11,12].

_{2}memristor, arise by reaching nanoscales [2]. The creation of such devices requires high-level nanotechnology and advanced equipment. At the same time, the study of the nonlinear properties of memristors also requires expert knowledge in the field of dynamical systems, where researchers do not always have access to thin-film fabrication technology. Currently, this gap is filled with commercially available devices, among which are the memristors distributed by Knowm Inc.

_{2}memristors exponential dependence, but the memristivity function was not specified. In 2020, the first reports of chaotic circuits with Knowm memristors appeared. In [21,22], C.K. Volos et al. demonstrated chaotic modes of Shinriki’s circuit [23] modified by adding an SDC memristor. Despite studying the circuit equations, the complete memristive device model was not presented in these works. In [24], Minati et al. adapted Sprott’s jerk circuit [25] to exploit nonlinearities of SDC memristors for the appearance of chaotic attractors. In order to explain the observed dynamics, the authors applied the mean metastable switch (MMS) model of a memristor, recommended by the Knowm Inc. affiliated researchers [26,27].

^{−4}–10

^{−2}A. Meanwhile, the manufacturer strongly recommends limiting the current with a 50 kΩ series resistor at regular device operation under 1 V (maximum allowable voltage range of −5–3 V). Thus, the comprehensive investigation of the SDC memristors under low-current (less than 10

^{−5}A) operation is still in demand being a key to operational safety and energy efficiency of memristor-based systems. In this paper, we will explicitly show that the MMS memristor model cannot capture all of the significant switching properties of the Knowm devices in such operating conditions. Considering also the variability of memristive devices (see, e.g., [29]), it is of interest to create a method for constructing new memristor models.

_{2}devices in the low current operation regime. This work paid particular attention to the voltage snapback effect for which to describe a quantum mechanical model was proposed. In [31], the research was extended to Al

_{2}O

_{3}devices, and the conductance quantization description was refined as a quantum point-contact model. D. Niraula and V. Karpov in [32] proposed a comprehensive model, adopted for the low-current snapforward and snapback effects, as well as cycle-to-cycle switching variability, which was represented as particle dynamics in a finite number of double-well potentials. Unfortunately, operating with the description of processes in partial derivatives, these models are quite complex and not suitable for use in software circuit simulation environments. A much simpler, SPICE-suitable phenomenological model for the snapback effect in memristive devices was proposed by E. Miranda et al. [33]. The model parameters were selected for the Ta

_{2}O

_{5}-based structure. The disadvantages of this model follow from its discrete nature, the deviation from the concept of analog memristor, and the lack of accounting for cycle-to-cycle variability. Thus, we conclude that there is a strong need for compact models allowing adequate simulation of memristors in low-current switching regimes.

- The novel identification method is presented as a generalized process for a wide range of memristive elements.
- The proposed memristor model outperforms the existing ones in representing the switching threshold as a function of the state variables vector, making it possible to account for snapforward or snapback effects, frequency properties, and switching variability.
- The process and results of the parametric identification for the proposed memristor model are presented.

## 2. Materials and Methods

#### 2.1. Procedure for Identification of Memristive Elements

#### 2.2. Knowm Memristive Devices

_{2}Se

_{3}layer, where they facilitate the substitute ion of Ge on the Ge-Ge bond for Ag ions from the source layer. Areas where Ge-Ge dimers turn into Ag-Ge bonding sites form the self-directed channel. Since Ag has a tendency to agglomerate with other Ag atoms, the Ag-Ge sites constitute conductive clusters. By applying either a positive or negative potential across the device, one can vary concentrations of Ag within the clusters, establishing the mechanism for resistive switching.

#### 2.3. Experimental Setup

#### 2.4. Modeling Criteria

- 1
- Correspondence of the model’s I-V curve and the switching dynamics in the time domain to the experimental data of real devices. Therefore, in Figure 3a,b, the SET transition process of the device from HRS to LRS demonstrates a sharp current increase with a shift of the voltage switching boundary (snapback) at the initial stage <1>, in case (c) the SET transition looks smooth throughout the entire section. The reverse RESET process, which switches the device to HRS, can either be instantaneous (a), snapforward effect, or significantly slower (b) and (c). The symmetry of the I-V curve relative to the diagonal of the II and IV quarters is often violated. This can also be visualized in the time domain when AC voltage is applied. In addition, there is a visible curvature of the <1> section due to the metal-semiconductor/insulator barrier between the electrodes and the inner layers of the devices.
- 2
- Nonlinearity of the switching function. The origin of this nonlinearity in memristive devices based on redox reactions is explained by the nonlinear movement of ion vacancies or defects, accelerated by Joule heating. This property is characterized in that the resistance switching time of the SET and RESET processes decreases by orders of magnitude if the applied voltage pulse increases only several times. Thus, this criterion tests the model for a nonlinear dependence of the switching time on the input voltage.
- 3
- Suitability for modeling the complementary serial connection of two elements. One of the distinguishing features of such a connection is the presence of a common LRS when AC voltage is applied. This criterion serves as a check for the consistency of the memristive device model.

- 4
- Ability to set several states of resistance. The criterion is to identify more than two states of resistance of the memristive device between the LRS and HRS, providing multi-bit data storage.
- 5
- Dependence of SET (or RESET) switching from the current state of the resistance. According to this criterion, the voltage required to set the device to a lower resistance state should depend on the high resistance value in the current cycle and vice versa. Thus, the switching kinetics should be power-dependent.
- 6
- Reliable simulation of the memory fading effect. This dynamical phenomenon is well known in the theory of nonlinear systems. With suitable periodic exposure, the previous “history” of the memristive device is gradually erased.

- 7
- Compact representation of a continuous mathematical model, which determines the suitability of using its discrete version in the processes of large-system simulation, including neural networks, as well as digital hardware emulators of memristive circuits.

#### 2.5. Candidate Memristor Models

#### 2.5.1. Mean Metastable Switch Memristor Model

_{OFF}, and the probability that a switch will change from HRS to LRS is defined as P

_{ON}. A metastable switch memristor can be represented as a set of N switching elements with dynamical evolution over discrete time intervals. The mean metastable switch memristor (MMS, [27]) model implements the limiting case when $N\to \infty $.

_{ON}is the threshold voltage for switching to the low resistance state, V

_{OFF}is the threshold voltage for switching to the high resistance state, $\beta =q/kT={V}_{T}^{-1}$ is the temperature parameter, q is the elementary charge, k is the Boltzmann constant, T is the absolute temperature, $\alpha =dt/\tau $ is the time parameter, and τ is the time constant of the memristor.

#### 2.5.2. Generalized Mean Metastable Switch Memristor Model

_{M}(V, t) (MMS model) and the Schottky diode I

_{S}(V):

_{S}(V) current is required to represent the Schottky barrier over the metal-semiconductor junction, and in turn, can be decomposed into forward and reverse bias components as follows:

_{f}

_{,r}and β

_{f}

_{,r}are positive parameters that specify the exponential behavior of the forward and reverse current flowing through the Schottky barrier.

## 3. Results

#### 3.1. Criterial Analysis of Candidate Models

_{ON}and V

_{OFF}. Like the similar mem-diode model [37] (whose equations are much more complicated to analyze), the MMS model uses the sigmoid switching function of the internal state variable X. Figure 4 shows the surface corresponding to the function of the right-hand side dX/dt(X, V). The β parameter in Equation (4) is responsible for the width of the sigmoid, 1/τ determines the maximum rise height along the sigmoid, the V

_{ON}and V

_{OFF}parameters set the coordinate of the center point of the sigmoid along the voltage axis V.

_{ON}= 5000 Ω, R

_{OFF}= 10

^{5}Ω, V

_{ON}= 0.2 V, V

_{OFF}= 0.1 V, τ = 0.0001, T = 298.5 K) I-V curve of a single memristor under conditions similar to the experiment in Figure 3.

_{ON}≠ V

_{OFF}.

_{ON}and V

_{OFF}.

_{ON}= 13,000 Ω, R

_{OFF}= 4.6·10

^{5}Ω, V

_{ON}= 0.17 V, V

_{OFF}= 0.1 V, τ = 6·10

^{−5}, T = 28.5 K, ϕ = 0.88, α

_{f}= α

_{r}= 10

^{−7}, and β

_{f}= β

_{r}= 8. In this case, for an expressed acceleration of switching processes, it was necessary to reduce the memristor time constant τ by order of magnitude and the temperature parameter T to a value with no physical meaning, which may be associated with the observation of quantum effects.

#### 3.2. The Modification of Memristor Model

_{ON}and V

_{OFF}on the internal state variable X, which is associated with the appearance of a section of negative differential resistance of the I-V curve in the first quarter when using a measuring circuit with a voltage divider. An array of experimental data of voltage depending on the variable X when switching SET is shown in Figure 8a, where one can see the snapbacks, i.e., abrupt current jumps with a slope inversely proportional to the load R

_{S}= 46.25 kΩ. A series-connected resistor R

_{S}allows stabilizing the LRS and the switching process of the SET device (the effects were studied in [30,39,40]), as well as revealing the fact that the V

_{ON}switching boundary has shifted from 0.22 V (V

_{ON,th}) to values less than 0.12 V in average (V

_{ON,tr}) close to V

_{OFF}~0.1 V (see Figure 8b).

_{1}is stable, as can be seen in Figure 3a,b. This observation necessitates modeling only the first jump, which can be performed using a continuous function, which in the conditions of this experiment (Figure 8a) takes the form:

_{ON}(X) function are shown in Figure 9. Depending on R

_{S}, the slope of the SET switching line changes correctly.

_{ON}and V

_{OFF}, as well as the deviation of the angles of the breaking lines α from the load value R

_{S}, increase.

_{ON}and V

_{OFF}boundaries adequately. The frequency f of the triangular voltage control signal is determined by the first derivative dV/dt, sinusoidal—the second derivative (from the equation d

^{2}V/dt

^{2}+ f

^{2}V = 0). In a measuring circuit based on a voltage divider, the memristor voltage is cut off in the SET process with the appearance of a negative differential voltage section, which complicates the static calculation of the frequency. In this case, one can use the dynamic estimation of the frequency F:

_{F}is the time coefficient of the state variable F, b

_{F}is the feedback coefficient, V is the voltage across the memristor.

^{2}V/dt

^{2}can often be derived analytically. However, in the general case, it requires numerical differentiation using finite difference methods.

_{ON}and V

_{OFF}from the state variable F at a

_{F}= 10 and b

_{F}= 0.9. As the signal frequency increases, one can see an increase in the angle α and the voltage thresholds V

_{ON}and V

_{OFF}.

_{ON}and V

_{OFF}. The Duffing oscillator equations are as follows:

_{Y}and a

_{Z}are time coefficients of the system, b

_{Z}is a feedback coefficient, and c

_{Z}is a signal coefficient.

#### 3.3. Parametric Identification

_{f,r}, β

_{f,r}, and ϕ from Equations (6) and (7) are determined based on the experimental data by reducing the value of the derivative calculated from the current I

_{M}of conductivity dG/dt to a minimum in the HRS section, which is much less noisy in comparison with the LRS (Figure 14b). The selection of optimal parameters leads to the rectification of the HRS and LRS sections on the I-V curve (Figure 14a) when displaying the current I

_{M}.

_{ON}and R

_{OFF}are determined using the Formula (5). The state variable X is calculated by normalizing the processed conductivity data at a frequency of 1 Hz, at which achieving the maximum and minimum memristor resistance is guaranteed. When calculating the state variable X from experimental data at high frequencies, it is necessary to use the values of the R

_{ON}and R

_{OFF}parameters determined at a frequency of 1 Hz.

_{ON,OFF}(X, F) of process A1.3.6.

_{ON,OFF}functions from the generator state variables Y and Z is determined based on the statistical characteristics of the experimental sample (data at a frequency of 1 Hz are shown in Figure 16a and Figure 17a). If one wants to achieve greater model accuracy, it is necessary to transform the distributions of the pseudo-random values Y and Z of the chaotic generator into an experiment-specific distribution. The solution of this problem by the example of the Zaslavsky web map is given in [47].

_{S}(see Figure 10).

## 4. Discussion

_{ON}function in the form of (8) under conditions of variable switching, one can see the deviations of the LRS values of variable X at frequencies of 10 and 100 Hz. In addition, the deviation of the experimental data points along the X-axis from the constant value of the model’s trajectory at negative voltage before the RESET switching is noticeable, which can also be seen in the third quarter of the I-V curve in Figure 17b. This observation can be explained by an error in describing the relationship between the ionic and electronic currents of the device by Equations (6) and (7).

_{2}device separate from Ag-doped TiO

_{2-x}-based memcapacitive device. Accounting for the snapback effect in our memristor model allows us to combine the features of negative differential resistance and resistive-switching in a single device, which can improve the scalability and reduce the power consumption of an artificial neuron. Consideration of variability between switching cycles as a chaotic process may also allow us to achieve greater similarity of simulations with the irregular firing of biological neurons. Note that in the case of artificial neural network development, during the memristor identification process one is required to use an extended set of criteria [49], which also includes the learning properties of memristive elements.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Stack of W-dopant Knowm memristor. materials (

**left**) and a graphical representation of the switching mechanism (

**right**). Reprinted from Ref [28].

**Figure 3.**Experimental I-V curves of devices with the property of resistive switching, examples (

**a**–

**c**) were obtained after forming the devices in different conditions.

**Figure 5.**I-V curve of the MMS model: (

**a**) a single element; (

**b**) a complementary series connection of two elements.

**Figure 8.**Experimental data: (

**a**) dependence of the threshold voltage on the internal state variable X during SET; (

**b**) I-V curve of one cycle with a length of 1 s.

**Figure 10.**Experimental I-V curve of the device at different frequencies of the sinusoidal control voltage.

**Figure 11.**Modification of the GMMS model: (

**a**) the values of the state variable F and (

**b**) the I-V curve of the memristive element at different frequencies of the sinusoidal control voltage.

**Figure 12.**Modification of the GMMS model: (

**a**) phase portrait of a chaotic generator and (

**b**) I-V characteristic of a memristive element at a frequency of 10 Hz.

**Figure 14.**(

**a**) Experimental I-V curves of the device at a frequency of 1 Hz when selecting the parameters of the Schottky barrier and (

**b**) processing the conductivity data.

**Figure 16.**Averaging of experimental I-V curves: (

**a**) average cycle and data of a sample of cycles at a frequency of 1 Hz; (

**b**) average cycles at different frequencies.

**Figure 17.**Comparison of the I-V curve of the experimental data and the device model at a frequency of 1 Hz: (

**a**) sample and (

**b**) a single cycle.

**Figure 18.**Visualization of the model (11) and experimental data of the device at a frequency of 1 Hz.

**Figure 19.**Visualization of the model (11) and experimental data of the device at a frequency of 10 Hz.

**Figure 20.**Visualization of the model (11) and experimental data of the device at a frequency of 100 Hz.

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**MDPI and ACS Style**

Ostrovskii, V.; Fedoseev, P.; Bobrova, Y.; Butusov, D.
Structural and Parametric Identification of Knowm Memristors. *Nanomaterials* **2022**, *12*, 63.
https://doi.org/10.3390/nano12010063

**AMA Style**

Ostrovskii V, Fedoseev P, Bobrova Y, Butusov D.
Structural and Parametric Identification of Knowm Memristors. *Nanomaterials*. 2022; 12(1):63.
https://doi.org/10.3390/nano12010063

**Chicago/Turabian Style**

Ostrovskii, Valerii, Petr Fedoseev, Yulia Bobrova, and Denis Butusov.
2022. "Structural and Parametric Identification of Knowm Memristors" *Nanomaterials* 12, no. 1: 63.
https://doi.org/10.3390/nano12010063