Memristor Device Modeling

Chapter
Part of the Analog Circuits and Signal Processing book series (ACSP)

Abstract

This chapter presents a physics-based mathematical model for anionic memristor devices. The model utilizes Poisson Boltzmann equation to account for temperature effect on device potential at equilibrium and comprehends material effect on device behaviors. A detailed MATLAB-based algorithm is developed to clarify and simplify the simulation environment. Moreover, the provided model is used to simulate and predict the effect of oxide thickness, material type, and operating temperatures on the electrical characteristics of the device. The value of this contribution is to provide a framework intended to simulate anionic memristor devices using correlated mathematical models. In addition, the model can be used to explore device materials and predict its performance.

Keywords

Memristor VCM Switching Vacancy Profile Temperature Oxide Thickness Parameter Potential Voltage Poisson Boltzmann

6.1 Introduction

As presented in Chap. 1, many memristive devices are fabricated and provided in many recent works, and it is challenging for researchers to carryout fair and accurate performance comparison among these devices. The reason is that the available memristors are fabricated using different device dimensions, oxygen vacancy concentrations, and oxide/electrode materials. Thus, it is important to have accurate physics-based mathematical model of memristive devices that can guide the fabrication process [1, 2].

Most of the recent works [1, 2, 3, 4, 5, 6] focus on representing memristors by general phenomenological models that reflect the electrical behavior of a memristive device regardless of its physical structure or material composition. Such models are sufficient for circuit-level simulations. On the other hand, physics-based models provide guidance and insights on optimizing the fabrication process. These models are limited to specific switching mechanism of the memristor and include some physical parameters that describe the active material characteristics in the device. To the best of our knowledge, there is no general physics-based mathematical model that can be used to simulate the behavior of all the existing memristive devices. The reason is that the switching mechanisms are different based on the memristor materials composition. As shown in Fig. 6.1, 1-D resistive switching in metal-oxide memristors is mainly explained through three different mechanisms: (i) fuse-antifuse, (ii) electrochemical metallization memory (ECM), and (iii) valence change memory (VCM) (detailed explanation is provided in Chap. 1).

This chapter presents a framework on modeling VCM-based memristive switching. Generally, VCM memristor provides quantum resistance which is beneficial for multilevel and neuromorphic computing applications (as presented in Chap. 4). One promising emerging application for bio-inspired neuromorphic devices is in image and pattern recognition [7]. The building blocks of this technology require memristor devices with variable levels of resistance and nonvolatility [8]. The variable resistance level in a memristor may be weighted or strengthened in response to several pulse stimuli in a direct comparison to synaptic plasticity change in the brain [9].

The physics-based modeling of anionic (VCM) memristive switching based on oxide materials has been traditionally made in the drift-diffusion approximation through the solution of a drift-diffusion time-domain bipolar model (Poisson, electron continuity equation, hole continuity equation (if exists)), accounting for the presence of mobile donors associated to vacancies. The mobile donor charge density appears in the Poisson equation, and an additional continuity equation for the mobile donor current density is added. Suitable boundary conditions are then used to close the system accounting for ohmic or Schottky contacts. This modeling framework is described in [10]. While [10] approximately treats mobile donors according to a band transport model and neglects nonlinear drift in high electric fields, [11] more correctly introduces a hopping model allowing for current flow due to the Poole–Frenkel effect and accounts for the nonlinear oxygen vacancies response to the strength of the local electric field. Such models are well known also in the field of transport model in amorphous and organic semiconductors. However, the clear algorithm and guidance for the researcher to understand, model, and simulate the behavior of anionic memristor devices are missing in the literature.

The work presented in [11] studied only the time evolution of the mobile vacancy distribution under an applied bias. On the other hand, the model presented this chapter utilizes Poisson–Boltzmann equation to account for temperature effect on device potential at equilibrium. The provided model is used to simulate and predict the effect of oxide thickness, material type, and operating temperatures on the electrical characteristics of the device.

6.2 Anionic Memristor Model

In this work, we consider a thin film closed system that consists of oxygen vacancies and free electrons. The nonlinear vacancies hopping model [11] coupled with the continuity equation [12] presented in 6.1 and 6.2, respectively, can be used to determine the switching time of such memristor device.
\begin{aligned} J_{V} (x) & = 2q_{v} f\,a\,\exp \left( {\frac{{ - U_{a} }}{{k_{B} T}}} \right)\sinh \left( {\frac{{q_{v} E_{v} \,a}}{{2k_{B} T}}} \right)N_{V} (x) \\ & \quad - q_{v} f\,a^{2} \,\exp \left( {\frac{{ - U_{a} }}{{k_{B} T}}} \right)\cosh \left( {\frac{{q_{v} E_{v} \,a}}{{2k_{B} T}}} \right)\frac{{\partial N_{V} }}{\partial x}. \\ \end{aligned}
(6.1)
$$\frac{{\partial N_{V} }}{\partial t} = - \frac{1}{{q_{v} }}\nabla \cdot J_{V} .$$
(6.2)
Where qv is the charge of oxygen vacancies, f is escape attempt frequency, Ua is the activation energy, and a is the hopping parameter. NV is the distribution of oxygen vacancies in 1-D, and t is drifting time. Jvis vacancy current density, kBis Boltzmann constant, and T is the temperature. The quantity Ev is the local electric field for an oxygen vacancy, and it is related to the external field E by Ev = (1 + XF) E where X is the bulk static permittivity and F is the Lorentz factor [11]. The escape attempt frequency describes how frequently an oxygen vacancy migrates from one position to another. Of course, higher escape attempt frequency decreases the switching time of memristor for a certain applied voltage. The activation energy defines the minimum amount of energy needed for a vacancy to move to the adjacent oxygen [13]. Thus, a lower activation energy enhances memristor switching time. Figure 6.2 shows a schematic of oxygen vacancy migration to model the hopping parameter. For a given activation energy specified for oxide material, if the oxygen vacancy NV can jump a distance X2, this gives shorter switching time than having a smaller hopping distance X1.
The model presented by 6.1 and 6.2 captures the high electric field in memristive devices. In addition, it includes the effect of temperature on oxygen vacancy profile in memristor. However, this model is not enough to calculate the resistance of the device for a given oxygen vacancy profile. To achieve this, 6.1 and 6.2 are coupled with Poisson equation [10] given in 6.3 and the drift current equation presented by 6.4 to calculate the current passing through the device.
$$\;\nabla^{2} P(x) = - \frac{q}{{\varepsilon_{r} \varepsilon_{o} }}\left( {N_{V} (x) - n(x)} \right)$$
(6.3)
$$J_{n} (x) = - q \cdot n(x) \cdot \mu_{n} \cdot \nabla P(x)$$
(6.4)
where P(x) is the potential at position x, ϵr is relative permittivity of the oxide, and ϵo is free space permittivity. n(x) is the electron density at position x, µn is the mobility of electrons, and q is the electron charge.
In semiconductor devices, calculating the concentrations of the charge carriers in the device is considered a very important initial step for creating a model of the chargers or the current. This can be achieved by studying the equilibrium condition of the device. The reason is that the equilibrium state is well known to be the reference condition of semiconductor devices. In addition, such devices can be considered to be operating in a quasi-equilibrium regime [14]. In this work, memristor mathematical model is enhanced by using Poisson–Boltzmann equation given in 6.5 to simulate the electrostatic potential at equilibrium.
$$\nabla^{2} P(x) = - \frac{q}{{\varepsilon_{r} \varepsilon_{o} }}\left( {N_{V} (x) - n\,\exp \left( {\frac{qP(x)}{{k_{B} T}}} \right)} \right)$$
(6.5)
This coupled equation has been traditionally used to model charge interactions in semiconductors [15]. As shown in Figs. 6.3 and 6.4, Boltzmann equation allows the model to calculate the local electrons density in static state depending on oxygen vacancy profile as well as device width and temperature. Thus, the model accounts for temperature effect on free electrons distribution in potential calculations. The analytical solution of Poisson–Boltzmann equation assumes that a 1-D device with mobile electrons and positive charged ions will reach thermal equilibrium through U-shape [16]. As the simulated TiO2–xmemristor has negative majority charge carriers, the electrons distribution profile will follow potential distribution shape.

In the proposed model, the ionic current continuity is persevered as we consider a closed system for the oxygen vacancies where the total number of vacancies is conserved in the oxide layer. This is achieved by using the continuity equation of oxygen vacancies 6.2. As for the electrons, the system is considered open as the interfaces are assumed to be purely ohmic to allow electrons transport. Moreover, the proposed model takes into account that the vacancy dynamics is slow versus electron dynamics, so that the model is solved iteratively starting from the mobile vacancies continuity equation in time domain and then calculating the electrons distribution based on the achieved device state. Suitable boundary conditions are used to account for ohmic contacts. The steps of the used algorithm can be summarized as follows.

1. 1.

An initial oxygen vacancies profile is assumed.

2. 2.

Equations 6.1 and 6.2 are solved for certain voltage and time duration to calculate the new vacancies profile.

3. 3.

Equation 6.5 is solved at zero boundary conditions to calculate the zero biasing electrostatic potential and free electrons distribution, using the vacancies profile obtained in step 2.

4. 4.

The vacancies and electrons distributions calculated in steps 2 and 3 are fed in 6.3 and 6.4 to calculate the current. Here, it is required to apply small voltage across the device during short time to sense the state of the device without disturbing it. Thus, the device is assumed to be operating in a quasi-equilibrium regime.

In the following sections, the proposed algorithm is used to achieve the following:

• Simulate TiO2–x memristor with different oxide thicknesses to study the effect of device width on the obtained OFF/ON resistance ratio.

• Investigate the effect of temperature using 10-nm TiO2–x memristor.

• Simulate 10-nm memristor behavior for different oxide materials; TiO2–x, ZnO1–x, and Ta2O5–x.

6.3 Effect of Oxide Thickness

The physics-based mathematical model presented in the preceding section is used to simulate 1-D TiO2–xmemristor using different device widths (oxide thickness). In these simulations, the normalized gap, that presents the ratio of the travelled distance to the device width, is decided based on the values of the device resistance [17]. For example, for the 10-nm memristor (Fig. 6.5a), the device is considered initially in the ON state, as represented by the blue curve. As the width of the applied voltage pulse (i.e., the switching time) increases, the migration distance of the vacancies increases and a new profile is obtained (Fig. 6.5a, green-curve → black-curve). For this particular device width, the resistance keeps changing until the oxygen vacancies travel a normalized gap about 0.7. As shown in Fig. 6.5b, this normalized gap decreases to 0.5 when the device width is doubled. For device widths 30 and 40 nm, the normalized gap diminishes to 0.2 and 0.18 as shown in Fig. 6.5c and d, respectively.

6.3.1 Zero Biasing Potentials

Poisson–Boltzmann equation given in 6.5 is used to simulate the electrostatic potential in memristive device when no voltage is applied across its terminals. This is achieved by solving 6.5 iteratively using zero boundary conditions. Figure 6.6a shows the simulated potential profiles (blue-curve → black-curve) that are corresponding to the oxygen vacancies profiles presented in Fig. 6.5a, respectively, for 10-nm device width. It is clear that the device has voltage barrier about 0.2 V when the device is ON and this barrier increases up to 0.47 V when the device is OFF. As for 20-nm device width, Fig. 6.6b shows that the voltage barrier of the ON state increases to 0.35 V while it is almost the same (0.48 V) when the device is OFF. This gives initial indication that the ON resistance of the simulated 20-nm memristor is higher than the ON resistance of the 10-nm device, while the OFF resistances for both device widths have close values. Figure 6.6c and d show the zero biasing potential profiles for simulated 30 and 40-nm TiO2–x memristor, respectively. It can be observed that the voltage barrier that corresponds to the ON profile of the 30-nm memristor is high (0.44 V). This value increases to 0.45 V when the device widths is increased to 40 nm. It can be observed that the simulated device has two approximately symmetric barriers when the oxygen vacancies are distributed through the inverted parabola profile. These two barriers become clearer for larger device widths as the resistance of the device increases.

6.3.2 OFF and ON Resistances

In this section, we calculate the resistance of the memristor that corresponds to each of the oxygen vacancies profiles presented in Fig. 6.5. To achieve this, a small voltage is applied by which the oxygen vacancies do not migrate and consequently the device state does not change (0.5 V is used in this work). Equations 6.3 and 6.4 are used to calculate the ON and OFF current values of the simulated device widths. In addition, the resistance values for each device width are determined using the ohmic relation between the current and the applied voltage as shown in Table 6.1. It is clear that the highest ROFF/RON ratio is obtained for a device width of 10 nm (~1.5E4). The reason is that it is easy to change the oxide stoichiometry in shorter device widths. It can be observe d that for device widths of 30 and 40 nm, the lowest obtained resistance values are in Megaohms. The relation between the device resistance and the oxide thickness is mainly determined by the solution of Poisson–Boltzmann equation. For the simulated 10- and 40-nm oxide thicknesses, the devices are assumed to have initially approximately equal oxygen vacancies concentration (fabrication dependent parameter), thus having close OFF resistance values as we consider high resistance material. As for the ON resistance, it will be easier to modify the stoichiometry of the insulator layer in shorter device width and consequently lower ON resistance can be achieved prior to full activation where Joule heating will take place. This explains the close barriers values of the ON and OFF states for the 30 and 40 nm which correspond to close OFF and ON resistance values and consequently small ROFF/RON ratio. The mathematical model presented in this chapter is validated with some sort of experimental data. Here, we compare the obtained resistance ratio rather than the resistance value at each state. The reason is that the oxide ON and OFF resistance ranges depend on the doping concentration (oxygen vacancies concentration) which varies based on the used synthesis procedure. The work reported in [18] presents experimental results of a 14-nm thick TiO2–xmemristor. The OFF/ON resistance ratio presented in [18] is about 1E3 which fits between the calculated ratios provided in Table 6.1 for 10- and 20-nm TiO2–x memristor devices. In addition, Gale et al. [19] fabricated a 40-nm TiO2–x memristor. The OFF/ON resistance ratio obtained in [19] agrees with modeling results provided in Table 6.1 for 40-nm devices.
Table 6.1

OFF and ON resistance of TiO2–x memristor modeled with different device widths

Oxide thickness

ON resistance

OFF resistance (MΩ)

ROFF/RON

10-nm

0.13 kΩ

2

15E3

20-nm

23 kΩ

3

130

30-nm

0.4 MΩ

6

15

40-nm

2 MΩ

8

4

6.4 Temperature Effect

The proposed Poisson–Boltzmann equation is used to study how the temperature of the VCM device can affect its resistance. 10-nm TiO2–xmemristor is simulated at the following temperatures: 25, 65, 85, and 125 °C. These temperature values are of interest for electronics devices [20, 21]. As shown in Table 6.2, ROFF/RON ratio decreases by approximately one order of magnitude when the temperature of the device increases from 25 to 125 °C. The simulation results show that increasing the temperature causes the OFF resistance to decrease (from 2 MΩ to 47 KΩ) and consequently the obtained resistance ratio decreases as well. The iterative solution of Poisson–Boltzmann equation shows that the effect of the temperature on the device resistance is minimal when the oxygen vacancies are spread through the device width. Thus, the ON resistance of the memristor stays in the same range. The results presented
Table 6.2

OFF/ON resistance ratio for 10-nm TiO2–x memristor modeled with different operating temperatures

Device temperature (°C)

ROFF/RON

25

15E3

65

4E3

85

1E3

125

5E2

in Table 6.2 agree with the experimental data reported in [22]. These experimental results show how the temperature affects the obtained resistance ratio of Ta2O5–x memristor device. It is shown that increasing the temperature causes the OFF resistance to decrease and consequently the obtained resistance ratio decreases as well.

6.5 Effect of Oxide Material

The physics-based model presented in Sect. 6.2 includes some physical parameters that describe the active material characteristics in memristive devices. These parameters are escape attempt frequency f, activation energy Ua, hopping parameter a, and oxide relative permittivity ϵr. To study the effect of these parameters, the proposed model is used to compare the electrical behavior of three different oxide materials; Ta2O5–x, ZnO1–x, and TiO2–x. The simulation parameters of these materials are provided in Table 6.3 [22, 23, 24]. The devices are simulated with oxide thickness of 10 nm and assumed to be initially ON with the same oxygen vacancies profile. In order to have fair comparison between the used oxide materials, the same time duration (10-ns) is applied to the three simulated devices. The applied voltage is tuned for each device until the full OFF state is reached. As shown in Table 6.4, the ZnO1–xmemristor requires the least amount of voltage to switch OFF. The reason is that the hopping parameter for ZnO1–x is bigger than its value for TiO2–x, and its activation energy is lower than what is reported for Ta2O5–x. The calculated application voltages for TiO2–x, Ta2O5–x, and ZnO1–xmemristors agree with the experimental voltage ranges reported in [18, 22, 25], respectively. In addition, the ZnO1–x and Ta2O5–x provide higher OFF/ON resistance ratio as they have lower relative permittivity than TiO2–x [10, 26, 27]. The ON to OFF vacancies profiles and the corresponding zero biasing potential profiles for ZnO1–x and Ta2O5–x are presented in Fig. 6.7 [28].
Table 6.3

Simulation parameters for the different oxide materials

Oxide material

f (Hz)

Ua (eV)

a (nm)

ϵr

TiO2–x

1E + 13

1.2

0.15

12

ZnO1–x

1E + 13

1.285

0.325

9

Ta2O5–x

1E + 13

2.5

0.4

8

Table 6.4

Simulated electrical characteristics for different oxide materials

Oxide material

Switching voltage (V)

OFF/ON resistance ratio

TiO2–x

2.4

1.5 E4

ZnO1–x

1.15

2.3 E6

Ta2O5–x

2

9 E5

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