There are three fundamental two-terminal passive circuit elements, i.e., resistors R, capacitors C and inductors L, which can be found and introduced from any basic circuit books. Chua first theoretically postulated the existence of the fourth passive electrical circuit element in 1971 [1], named as memristor, whose memristance (i.e., an acronym for memory resistance) M can be defined by the relation of magnetic flux φ and electrical charge q, namely, M = d φ / d q . Together with the definitions of three commonly used basic circuit elements R, C and L, and definitions of the electric current and Faraday’s law, memristance closes the loop among four variables v, i, φ and q with six equations. Under the linear situation of memristor, M is seen as a constant and memristance is the same as the resistance. In fact, the memristance depends on the amount of charge that passed through the memristor, resembling a resistor with time-varying resistance.

No great progress was developed in the research field of memristor until the group of Williams at Hewlett-Packard (HP) lab successfully fabricated the first nanoscale memristive device based on titanium dioxide thin film in 2008 [2]. Since then, many researchers or scholars have focused on the memristor research [3–7] and its numerous potential applications in nonvolatile random access memories [8, 9], neuromorphic systems [10, 11], chaotic circuits [12], reconfigurable logics [13, 14] and RF/microwave devices [15, 16].

In order to emulate these memristor applications, various memristor nonlinear drift models using SPICE have been presented before fabricated realization [17–24]. Window functions need to be used to tackle the boundary conditions of the memristors [17–22]. In ref. 17, a switching window function was proposed, where the current i is involved, but it has limited scalability. Two different window functions were presented to mimic nonlinear effects in refs. 18 and 20. However, as the boundary between the doped and undoped layers approaches either end of the memristive device, it cannot be excited by external stimulus. In refs. 20 and 22, the piecewise functions were utilized to construct the window functions, which were relatively complicated when applying in the memristive devices.

In this paper, a memristor model using a new window function, called cosine window function, is proposed along with its versatile characteristics. To illustrate the significance of the proposed window function, main characteristics of state-of-the-art ones in recent years are listed in Table 1 for comparisons. Then, the SPICE behaviors of the linear and nonlinear memristor model are investigated through PSpice simulation. Furthermore, the characteristics of the composite SPICE behaviors are both studied when two memristors are connected in series and in parallel. The relationships among flux, charge, voltage, current, and memristance are simulated and analyzed using PSpice.

The concept of memristive devices or systems was generalized by Chua and Kang using the following equations [25], v ( t ) = M ( x ( t ) ,   i ( t ) ) ⋅ i ( t ) (1) d x ( t ) d t = f ( x ( t ) , i ( t ) ) (2) where v is the voltage, i is the current, and M ( x ( t ) , i ( t ) ) is the instantaneous memristance that depends on the internal state variable x and current i of the memristor. The state variable x, ranging from 0 to 1, is the normalized width of the memristor doped region x ( t ) = w ( t ) / D , with D being the whole switching bilayer thickness of the memristor, as seen in Figure 1.

At time t, the width of the doped region w depends on the amount of charge that passed through the memristor. Thus, the time derivative of w is proportional to the current i, which can be expressed by d w ( t ) d t = μ v R O N D i ( t ) (3) where μ v is the average dopant mobility, R O N denotes the low resistance at high dopant concentration region. If we set k = μ v R O N / D 2 , Eq. 3 can be changed as follows, d x ( t ) d t = k ⋅ i ( t ) (4) The Eqs. 3, 4 is actually an expression of a linear drift model of memristor, but it does not take the boundary effects of the memristor into account. Nonlinear dopant drift phenomena exist at the boundaries of the physically realized nanoscale memristor, therefore, an appropriate window function f(x) need to be introduced in ref. 4, i.e., d x ( t ) d t = k ⋅ f ( x ) ⋅ i ( t ) (5) where f(x) ensures zero drift at the boundaries, i.e., f (0) = f (1) = 0.

On the other hand, the memristance can be expressed by M = R O N ⋅ x ( t ) + R O F F ( 1 − x ( t ) ) (6) where R O F F denotes the high resistance at low dopant concentration region. In this paper, a new cosine window function is proposed as follows, f ( x ) = 1 − ( cos ( π x ) ) 20 ⋅ p (7) where p is the control parameter. This new and simple cosine window function is demonstrated in Figure 2 for various values of p. As the parameter p rises, a unity window function f(x) will occur, and thus the model tends to a linear model. Here, the parameter p is set with a large range of variation, because the cosine window function is not changed significantly near the region of boundary.

Using this new cosine window function, the behaviors of the memristor can be observed in Figures 3–6 with versatile characteristics. Here, we set R O N = 100 Ω, R O F F = 16 kΩ, w ₀ = 5 nm, D = 10 nm, and μ v = 10^{–14 m2}/Vs, thus the initial state variable x ₀ is equal to 0.5 at t = 0. When the voltage biased at two sides of memristor is set as v ( t ) = sin ( 2 π t ) , i.e., the frequency of the AC voltage f = 1 Hz, the calculated dynamical properties of the normalized doped state variable x ( t ) with different values of p are illustrated in Figure 3A. For instance, when p = 16, the state variable periodically saturates between the normalized boundary “1” from 0.295 + N to 0.705 + N s, where N is a natural number starting from 0. On the other hand, according to (6), the curves of the corresponding memristances can be obtained as seen in Figure 3B. When the memristor biased voltage is set as v ( t ) = - sin ( 2 π t ) , the calculated dynamical properties of the normalized doped state variable x ( t ) with different values of p is shown in Figure 3C, and the corresponding memristance curves are plotted in Figure 3D.

Actually, the minimum state variable x ( t ) corresponding to maximum memristance is dependent on the frequency of the biased AC voltage. From Figure 3C, we can see that the state variable does not arrive the normalized boundary “0”. Through decreasing the frequency of the biased voltage, two edge states, i.e., saturation (w = D) and depletion (w = 0), can be reached when the memristor device is biased with positive voltage sinusoidal stimulus and reversed polarity stimulus, respectively. Figure 4 shows the influence of the frequency variation of the biased voltage on the state variable x and memristance M. As the frequency of the AC voltage f decreases from 1 Hz to 1/3 Hz, the minimum value of state variable x will approach 0 and the corresponding maximum memristance will be 16 kΩ, under the condition of p = 8.

Figure 5 illustrates a nonlinear calculated current-voltage (i-v) curve with p set as 8, where a periodic pinched hysteresis loop with hard switching emerges. This memristive model has zero-crossing property in a form of i-v Lissajous figure, which represents no current through the system when the voltage value is zero. Figure 6 shows the versatility of our model, where the i-v characteristic curves are plotted for eight different values of p. This model demonstrates high flexibility for emulating the behaviors of the practical HP memristors, which can be beneficial in characterizing memristive devices before fabrication.

The HP memristor model using cosine window function and its characteristics have been addressed in the above section. To further investigate the relationships among charge, flux, and memristance of composite memristors connected in different topologies, simulated SPICE behaviors of single and double memristor circuits will be presented in details in this section.

A new window function has been created and applied into the HP memristor model in this paper. To validate its advantages, versatile characteristics and SPICE behaviors of the memristor have been demonstrated. Moreover, we also investigate the composite SPICE behaviors when two memristors are connected in series and in parallel with same or opposite polarities, and make comparisons with those of the individual memristors. Their relationships among flux, charge, voltage, current, and memristance are analyzed using PSpice simulation.