# Compact Model for Bipolar and Multilevel Resistive Switching in Metal-Oxide Memristors

^{*}

## Abstract

**:**

_{x}films (30 nm) and bilayer TiO

_{2}/Al

_{2}O

_{3}structures (60 nm/5 nm) is demonstrated.

## 1. Introduction

^{2}[10]) with low power consumption (switching energy less than 10 fJ [11]) and possibility integration into cross-bar arrays.

_{x}Al

_{2}O

_{3}thin layers as the objects of modeling is due to the possibility of electric-field analog tuning of the nonvolatile resistance state in the range of seven orders of magnitude in combination with a bipolar resistive switching relatively to a given resistance state. In these structures, a 5 nm thick Al

_{2}O

_{3}layer oxide plays a role of an active (or switching) layer, while 30–60 nm 02.thick titanium oxide layer acts as a reservoir of oxygen vacancies. Moreover, the similar devices demonstrate high switching speed, low power consumption, wide dynamic range, high resistance to cyclic degradation, high scalability, and compatibility with CMOS technologies [25,26,27,28].

_{x}and on the basis of a bilayer structure TiO

_{2}/Al

_{2}O

_{3}, the current-voltage characteristics of which were used in this work to verify the proposed model; Section 5 discusses simulation results and their comparison with measured data; Section 6 provides general conclusions based on the results of the study.

## 2. Equivalent Circuit and Model Equations

_{SCL}is the current limited by the space charge; R

_{0}is the memristor resistance, determined by the equilibrium concentration of electrons in the dielectric film; C

_{B}is the equivalent capacitance characterizing the property of the resistive memory of the structure during bipolar switching; V

_{B}is the potential difference across the equivalent capacitance C

_{B}; I

_{B}is the voltage-controlled current source that characterizes the bipolar switching processes caused by electron transport and by processes of filling and releasing the energy levels of traps by electrons; R

_{DB}is the equivalent resistance characterizing the process of possible degradation of the memristor low-resistance state during bipolar switching; C

_{M}is the equivalent capacitance characterizing the property of the resistive memory of the structure during multilevel conductivity tuning; V

_{M}is the potential difference across the equivalent capacitance C

_{M}; I

_{M}is a voltage-controlled current source characterizing the processes of multilevel conductivity tuning caused by the transport of oxygen vacancies in the memristive structure; R

_{DM}is the equivalent resistance characterizing the relaxation process of the intermediate conductivity state of the memristor during multilevel switching.

_{SCL}, limited by the space charge. The memristor resistance, determined by the equilibrium electron concentration in the dielectric film, is given by expression (6) [15].

_{SCL}current in accordance with expression (2), depending on the state of the memristor, is determined either by the current in the high-resistance state I

_{H}, or by the current in the low-resistance state ${I}_{H}\frac{{R}_{OFF}}{{R}_{ON}}$, the smooth transitions between which in the process of bipolar switching are provided by smoothing functions ${F}_{H},{F}_{L}$, specified by expressions (4) and (5).

_{B}controlled by the current source I

_{B}and the equivalent resistance R

_{DB}, which characterizes the process of degradation of the low-resistance state of the memristor during bipolar switching. The change in the memristor conductivity during bipolar switching is described by differential equation (7), in which the recharge current ${I}_{B}$ of the equivalent capacitance ${C}_{B}$ is given by expressions (8) and (9). The direction of the current ${I}_{B}$ in accordance with expression (8) is determined by the sign of the voltage $sign\left(V\right)$ applied to the memristor contacts. In this case, the Heaviside function $\theta \left(\left|V\right|-\left({V}_{TFLP}+{V}_{FITP}\right)\theta \left(V\right)+\left({V}_{TFLD}+{V}_{FITD}\right)\theta \left(-V\right)\right)$ zeroes the current ${I}_{B}$ if the applied voltage $V$ is in the range

_{M}, voltage-controlled current source I

_{M}, and the equivalent resistance R

_{DM}, which characterizes the process of degradation of the memristor state during multilevel switching. The change in the memristor conductivity in the process of multilevel tuning is described by differential equation (10), in which the recharge current I

_{M}of the equivalent capacitance C

_{M}is given by expression (11). The direction and value of the current I

_{M}are determined by the voltage V applied to the memristor contacts and the equivalent fitting resistance R

_{FITM}. In this case, the Heaviside function $\theta \left(\left|V\right|-{V}_{MTH}\right)$ zeroes the current I

_{M}if the applied voltage V is in the range

## 3. Statistical Variation of Model Parameters

## 4. Materials and Methods

_{x}(bipolar switching) as well as on the basis of a bilayer film TiO

_{2}/Al

_{2}O

_{3}(multilevel conductivity tuning with bipolar switching), with structures shown schematically in Figure 2.

_{2}substrate with a 10 nm thick titanium adhesive layer. A titanium oxide layer TiO

_{x}(30 nm) was grown on the Pt-BE surface in an atomic layer deposition equipment TFS 200 (Beneq) at a temperature of T = 150 °C using titanium isopropoxide (Ti[OCH(CH

_{3})

_{2}]

_{4}) as a precursor and vapor water as an oxidizing agent. The values of the thickness of the grown layers were monitored in the course of scanning electron microscopy over a cross section of the structure formed by a focused ion beam on an FEI, Helios NanoLab system. The surface of the TiO

_{x}layer was examined using an atomic force microscope Dimension 3100, Veeco. Postdeposition annealing was performed in air for 30 s at a temperature of T = 150 °C. Top platinum electrodes (Pt-TE 50 nm) were deposited by magnetron sputtering at a temperature of T = 150 °C. The diameter of the upper electrodes was 350 μm.

_{2}substrate with a titanium adhesive layer (10 nm). A bilayer TiO

_{2}/Al

_{2}O

_{3}structure (60 nm/5 nm) was grown on the Pt-BE surface in an atomic layer deposition system TFS 200 (Beneq) at a temperature of T = 200 °C using trimethylaluminum Al(CH

_{3})

_{3}, tetrakis (dimethylamino) titanium C

_{8}H

_{24}N

_{4}Ti and water vapor. The thickness of TiO

_{2}layer is increased to 60 nm in comparison with the structure shown in Figure 2a because in a bilayer TiO

_{2}/Al

_{2}O

_{3}structure, the TiO

_{2}layer is used as an oxygen vacancies reservoir for the active Al

_{2}O

_{3}layer. Top platinum electrodes (Pt-TE 150 nm) were deposited by electron beam evaporation. The diameter of the upper electrodes was 100 μm. The process of fabricating the structure is described in more detail in [27].

## 5. Results and Discussion

_{P}= 0.1.

_{2}O

_{3}film: capture and release of electrons by traps (bipolar switching) and transport of oxygen vacancies (multilevel conductivity tuning). In this case, the TiO

_{2}film can be considered as an unlimited source of oxygen vacancies for Al

_{2}O

_{3}and, thus, the thickness d

_{R}of the TiO

_{2}film given in Table 2 is not considered as a parameter of the model (1)–(12).

_{2}/Al

_{2}O

_{3}structure, calculated for bipolar voltage pulses across the memristive element with a duration of 30 ms with amplitudes of 3 V, 5 V, and 7 V, exceeding the minimum threshold V

_{MTH}of multilevel conductivity tuning. The simulation results demonstrate an increase (at V > 0) and, accordingly, a decrease (at V < 0) in the memristor conductivity level with time during the action of positive polarity pulse (in the time interval 0–30 ms) and during the action of negative polarity pulse (in the time interval 30–60 ms), as well as an increase in the rate of change in conductivity with an increase in the amplitude of the applied voltage pulses.

_{M}across the capacitance C

_{M}of the equivalent circuit (Figure 1) corresponding to the memristor conductance level during multilevel tuning is shown.

_{2}/Al

_{2}O

_{3}structure (Figure 2b) [27], and Figure 8c shows the corresponding results of modeling this structure with parameters, given above in Table 2. Comparison of I–V characteristics in Figure 8c,d indicates sufficient agreement between the simulation results and experimental data.

## 6. Conclusions

_{x}films (30 nm) and bilayer TiO

_{2}/Al

_{2}O

_{3}structures (60 nm/5 nm) is demonstrated.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**I–V characteristics of a memristive element in linear (

**a**), logarithmic (

**b**) and semilogarithmic (

**c**) scales.

**Figure 4.**I–V characteristics of a memristive element, taking into account the statistical spread of parameters at D

_{P}= 0.1 in linear (

**a**), logarithmic (

**b**), and semilogarithmic scales (

**c, d**): simulation results (

**c**) and measurement results (

**d**) for memristive structure shown in Figure 2a (measurements were carried out for 35 samples of the memristor structure).

**Figure 6.**Transients of cyclic bipolar switching of memristive element taking into account the variability of parameters at D

_{P}= 0.1.

**Figure 8.**I–V characteristics of a memristive element with multilevel conductivity tuning without taking into account (

**a**) and taking into account (

**b**) the statistical variation of parameters, comparison of the simulation results (

**c**) with the measurement data (

**d**) [27] for the memristive structure shown in Figure 2b (measurements were carried out for one sample of the memristor structure).

**Table 1.**Model parameters for a memristor with the structure shown in Figure 2a.

Parameter | Symbol | Value | Unit |
---|---|---|---|

Dielectric film thickness | $d$ | 3 × 10^{−8} | m |

Area of memristive element | $S$ | 7.07 × 10^{−8} | m^{2} |

Cross-sectional area of the filament (mean value) | ${S}_{F}$ | 3 × 10^{−16} | m^{2} |

Equilibrium concentration of electrons in the film | ${n}_{0}$ | 1.3 × 10^{16} | m^{−3} |

Electron mobility | ${\mu}_{n}$ | 5 × 10^{−4} | m^{2}/(V*s) |

Relative dielectric constant of oxide film | $\epsilon $ | 160 | – |

Threshold voltage of SET process (mean value) | ${V}_{TFLP}$ | 1.9 | V |

Threshold voltage of RESET process (mean value) | ${V}_{TFLD}$ | −1.2 | V |

Minimum threshold for multilevel switching of memristor conductance | ${V}_{MTH}$ | 2.7 | V |

The ratio of the resistances of a memristor in high-resistance and low-resistance states (mean value) | $\frac{{R}_{OFF}}{{R}_{ON}}$ | 50 | – |

Fitting parameter | ${K}_{M}$ | 26 | – |

Fitting parameter | ${V}_{FITP}$ | −0.2 | V |

Fitting parameter | ${V}_{FITD}$ | −0.2 | V |

Fitting parameter | ${R}_{FITM}$ | 5 × 10^{8} | Ohm |

Fitting parameter | ${V}_{BF}$ | 1 | V |

Fitting parameter | ${V}_{MP}$ | 2.5 | V |

Fitting parameter | ${V}_{MD}$ | 35 | V |

Fitting parameter | ${I}_{FITB}$ | 4 × 10^{−9} | A |

Elementary charge | $q$ | 1.6 × 10^{−19} | C |

Boltzmann constant | ${k}_{B}$ | 1.38 × 10^{−23} | J/K |

Dielectric constant of vacuum | ${\epsilon}_{0}$ | 8.85 × 10^{−12} | F/m |

Absolute temperature | $T$ | 300 | K |

Relative change in the values of randomly varied parameters of the memristor model | ${D}_{P}$ | 0.1 | – |

**Table 2.**Model parameters for a memristor with the structure shown in Figure 2b.

Parameter | Symbol | Value | Unit |
---|---|---|---|

Thickness of Al_{2}O_{3} dielectric film | $d$ | 5 × 10^{−}^{9} | m |

Thickness of TiO_{2} dielectric film | ${d}_{R}$ | 6 × 10^{−8} | m |

Area of memristive element | $S$ | 1 × 10^{−8} | m^{2} |

Cross-sectional area of the filament (mean value) | ${S}_{F}$ | 3 × 10^{−16} | m^{2} |

Equilibrium concentration of electrons in the film | ${n}_{0}$ | 1.0 × 10^{1}^{1} | m^{−3} |

Electron mobility | ${\mu}_{n}$ | 5 × 10^{−4} | m^{2}/(V*s) |

Relative dielectric constant of Al_{2}O_{3} film | $\epsilon $ | 10 | – |

Threshold voltage of SET process (mean value) | ${V}_{TFLP}$ | 1.5 | V |

Threshold voltage of RESET process (mean value) | ${V}_{TFLD}$ | −1.5 | V |

Minimum threshold for multilevel switching of memristor conductance | ${V}_{MTH}$ | 2.7 | V |

The ratio of the resistances of a memristor in high-resistance and low-resistance states (mean value) | $\frac{{R}_{OFF}}{{R}_{ON}}$ | 20 | – |

Fitting parameter | ${K}_{M}$ | 6.7 × 10^{−7} | – |

Fitting parameter | ${V}_{FITP}$ | −0.2 | V |

Fitting parameter | ${V}_{FITD}$ | −0.2 | V |

Fitting parameter | ${R}_{FITM}$ | 1 × 10^{8} | Ohm |

Fitting parameter | ${V}_{BF}$ | 1 | V |

Fitting parameter | ${V}_{MP}$ | 1.3 | V |

Fitting parameter | ${V}_{MD}$ | 500 | V |

Fitting parameter | ${I}_{FITB}$ | 4 × 10^{−9} | A |

Absolute temperature | $T$ | 300 | K |

Relative change in the values of randomly varied parameters of the memristor model | ${D}_{P}$ | 0.1 | – |

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**MDPI and ACS Style**

Ryndin, E.; Andreeva, N.; Luchinin, V.
Compact Model for Bipolar and Multilevel Resistive Switching in Metal-Oxide Memristors. *Micromachines* **2022**, *13*, 98.
https://doi.org/10.3390/mi13010098

**AMA Style**

Ryndin E, Andreeva N, Luchinin V.
Compact Model for Bipolar and Multilevel Resistive Switching in Metal-Oxide Memristors. *Micromachines*. 2022; 13(1):98.
https://doi.org/10.3390/mi13010098

**Chicago/Turabian Style**

Ryndin, Eugeny, Natalia Andreeva, and Victor Luchinin.
2022. "Compact Model for Bipolar and Multilevel Resistive Switching in Metal-Oxide Memristors" *Micromachines* 13, no. 1: 98.
https://doi.org/10.3390/mi13010098