Memristor Technology: Synthesis and Modeling for Sensing and Security Applications pp 93-104 | Cite as
Memristor Device Modeling
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 Boltzmann6.1 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].
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 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.
An initial oxygen vacancies profile is assumed.
- 2.
Equations 6.1 and 6.2 are solved for certain voltage and time duration to calculate the new vacancies profile.
- 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.
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 TiO_{2–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 TiO_{2–x} memristor.
Simulate 10-nm memristor behavior for different oxide materials; TiO_{2–x}, ZnO_{1–x}, and Ta_{2}O_{5–x}.
6.3 Effect of Oxide Thickness
6.3.1 Zero Biasing Potentials
6.3.2 OFF and ON Resistances
OFF and ON resistance of TiO_{2–x} memristor modeled with different device widths
Oxide thickness | ON resistance | OFF resistance (MΩ) | R_{OFF}/R_{ON} |
---|---|---|---|
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
OFF/ON resistance ratio for 10-nm TiO_{2–x} memristor modeled with different operating temperatures
Device temperature (°C) | R_{OFF}/R_{ON} |
---|---|
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 Ta_{2}O_{5–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
Simulation parameters for the different oxide materials
Oxide material | f (Hz) | U_{a} (eV) | a (nm) | ϵ_{r} |
---|---|---|---|---|
TiO_{2–x} | 1E + 13 | 1.2 | 0.15 | 12 |
ZnO_{1–x} | 1E + 13 | 1.285 | 0.325 | 9 |
Ta_{2}O_{5–x} | 1E + 13 | 2.5 | 0.4 | 8 |
Simulated electrical characteristics for different oxide materials
Oxide material | Switching voltage (V) | OFF/ON resistance ratio |
---|---|---|
TiO_{2–x} | 2.4 | 1.5 E4 |
ZnO_{1–x} | 1.15 | 2.3 E6 |
Ta_{2}O_{5–x} | 2 | 9 E5 |
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