## 2 Memristor: Models, Material, and Structure Design
### Memristor Modeling
The studies reported in [6,8] demonstrated the non-linear behavior of implemented memristors. These studies showed that the resistance of memristors does not change linearly with applied voltage or current. Therefore, memristor models are mathematical representations used to describe the behavior and characteristics of memristors. These models help in understanding how memristors function and how they can be utilized in various applications. They consider factors such as voltage, current, resistance, and capacitance to accurately simulate the behavior of memristors in different scenarios. By using these models, researchers and engineers can predict and analyze the performance of memristors in complex circuits or systems. Additionally, these models can also aid in the design and optimization of memristor-based devices, enabling the development of more efficient and reliable technologies. The model proposed in [6] considers the non-linear relationship between the dopant drift and the resulting resistance change in memristors. Understanding the I-V characteristics can significantly enhance the design and optimization of memristor-based circuits and systems. The non-linearity of the drift velocity in [19, 20, 21, 22] was simulated using the window function \(f(w)\), whose expression is given in Equation (3). This window function introduces a non-linear relationship between the applied voltage or current and the resulting resistance change in memristors, which aligns with the experimental findings. By incorporating this non-linearity into circuit design, engineers can better predict and control the behavior of memristor-based systems, leading to improved performance and efficiency. \[\frac{dw}{dt}=a\,f(w)\,V(t)^{m} \tag{3}\]  
where \(f(w)\) is the window function, '\(m\)' is an odd integer and '\(a\)' is a constant. In [8], the authors develop a more accurate physical model of the memristor, considering non-linear dopant drift and the voltage polarity's impact on the device behavior. They also propose a modified HP model to capture dynamic memristor behavior under various operating conditions accurately. A resistor is coupled in series with a Simmons tunnel barrier instead of connecting two resistors in series [20]. This modification allows for a more realistic representation of the memristor's behavior, particularly its resistance-switching  
Figure 2: TiO\({}_{2}\) memristor model [18]. dynamics. This model's switching behavior is non-linear and asymmetrical, and its state equation is as follows:  
\[\frac{dx}{dt}=\left\{ \begin{array}{l}C_{off}sinh\Big{(}\frac{i}{t_{off}}\Big{)} exp\Big{[}-exp\Big{(}\frac{x-a_{off}}{w_{c}}-\frac{|i|}{b}-\frac{x}{w_{c}} \Big{)}\Big{]}\\ C_{on}sinh\Big{(}\frac{i}{t_{on}}\Big{)}exp\Big{[}-exp\Big{(}\frac{x-a_{on}}{w_ {c}}-\frac{|i|}{b}-\frac{x}{w_{c}}\Big{)}\Big{]}\end{array} \right\} \tag{4}\]  
The state variable x represents the Simmons tunnel barrier width, and the fitting parameters are \(C_{off}\), \(C_{on}\), \(a_{off}\), \(a_{on}\), \(w_{c}\), and \(b\). The memristor's current thresholds are \(i_{off}\) and \(i_{on}\). The Simmons tunnel barrier model is more complicated, as shown in Equation (4), and as a consequence, it is computationally inefficient. This model introduces additional parameters and equations to describe the tunneling phenomenon accurately. However, its complexity requires more computational resources and time to solve compared to simpler models. In [21], the authors develop the ThrEshold Adaptive Memristor (TEAM) model to represent the same physical concept as the Simmons tunnel barrier model [19] in a more general and simpler manner. The TEAM model simplifies the description of the tunneling phenomenon by using a threshold adaptive approach, which reduces the computational complexity. This allows for more the efficient simulation and analysis of memristor behavior. A key advantage of the TEAM model is its ability to predict the memristor's electrical characteristics accurately.