## 2 Memristor: Models, Material, and Structure Design  
### Memristor Modeling  
Memristor models are mathematical representations used to describe the behavior and characteristics of memristors. These models help in understanding how memristors store and retain information, as well as how they can be utilized in various applications. Some commonly used memristor models include the ideal memristor model, which assumes a linear relationship between charge and flux, and the nonlinear ion drift model, which considers the movement of ions within the memristor material. These models provide valuable insights into the electrical properties of memristors and enable researchers to design more efficient and reliable devices. Additionally, memristor models include the window function model, which considers the non-linear behavior of memristors near their switching thresholds; the adaptive memristor model, which incorporates the ability of memristors to adapt and change their resistance based on external stimuli; the diffusive memristor model, which considers the diffusion of charged species in the memristor material; the filamentary memristor model, which focuses on the formation and dissolution of conductive filaments; the spintronic memristor model, which incorporates the spin degree of freedom in addition to charge and flux; and the synaptic memristor model, which simulates the behavior of biological synapses. These models have significantly advanced our understanding of memristor devices and their potential applications. By studying different memristor models, researchers can gain insights into the underlying mechanisms of these devices and explore their potential for use in future technologies. The HP Labs-defined memristor physical model, which consists of a thin layer of TiO\({}_{2}\) sandwiched between two metal electrodes, is depicted in Figure 2[18]. This device exhibits resistance that can be increased or decreased depending on the polarity and magnitude of the applied voltage. In this experiment, one layer of TiO\({}_{2}\) was doped with oxygen vacancies, while the other was left undoped. The presence of oxygen vacancies introduces defects, which can alter the conductivity of the memristor. This yields an insulator in the undoped region and a conductor in the doped region, creating a distinct boundary between the two layers. This boundary enables the formation of a conductive filament, which can be controlled by applying different voltages across the electrodes, allowing for precise control over memristor resistance programming. The quantity and direction of the electric charges passing through a memristor determine the width of the doped zone w(t) and the strength of the conductive filament. The width can range from zero to the memristor's length (\(D\)). This control over the width of the doped zone enables the memristor to exhibit different resistance states, which allows for precise control over the flow of electric current through the device. By adjusting the width of the doped zone, the memristor can be switched between HRS and LRS, effectively controlling the flow of electrons. As a result, the memristor can be used as a programmable resistor in electronic circuits, offering potential applications in analog and digital fields. The total resistance of the memristor (\(R_{MEM}\)) varies when the virtual barrier between the doped and undoped zones shifts [2]. This change in resistance is due to the movement of charged particles within the doped zone, which alters the conductivity of the memristor. The expression for \(R_{MEM}\) is given as follows:  
\[R_{MEM}(x)=R_{on}(x)+R_{off}(1-x) \tag{1}\]  
where \(x=\mathrm{w}/D\in\)[0, 1], \(R_{on}\) is the memristor resistance when \(\mathrm{w}=D\), and \(R_{off}\) is the memristor resistance when the value \(\mathrm{w}\) is zero. The drift velocity is the speed of boundary movement between two nodes, expressed using the following state Equation [2]:  
\[\frac{dx}{dt}=k\,i(t)\ \ \ for\ k=\mu_{v}\frac{R_{on}}{D^{2}} \tag{2}\]  
where \(\mu_{v}\) is the dopant mobility.