A Charge-controlled memristance

If a memristor is charge-controlled, its state is controlled by the charge through the cell. Figure 11.3 shows a memristor biased by a current source I_in, and I_in can be any waveform. Integrating Eq. 11.9 yields the instantaneous w(t):

$\mathit{w}\left(\mathit{t}\right)={\mathit{w}}_0+{\mathit{\mu }}_{\mathit{v}}·\raisebox{1ex}{${\mathit{R}}_{\mathit{on}}$}\!\left/ \!\raisebox{-1ex}{$\mathit{D}$}\right.\mathit{·q}$ [11.16]

[11.16]

where w₀ is the initial state for state variable w. The memristor state will move to a higher position, w > w₀, if there is a positive charge injection. Likewise, negative charge injection will lower the memristor state; however, its state has a physical constraint: the state is bounded in between zero and total length D, namely 0 ≤ w ≤ D. Due to the physical constraint, the internal memristor state corresponds to the following effective Q range:

${\mathit{Q}}_{\mathit{LOW}}\underset{¯}{<}\mathit{q}\underset{¯}{<}{\mathit{Q}}_{\mathit{UP}}$ [11.17]

[11.17]

where Q_UP the upper limit of effective charge injection, and Q_LOW is the lower limit of effective charge injection. The expressions for them are listed as

$\left\{\begin{array}{c}\hfill {\mathit{Q}}_{\mathit{UP}}=\left(\mathit{D}-{\mathit{w}}_0\right)\mathit{D}/\left({\mathit{\mu }}_{\mathit{v}}{\mathit{R}}_{\mathit{on}}\right)\hfill \\ \hfill {\mathit{Q}}_{\mathit{LOW}}=-{\mathit{w}}_0\mathit{D}/\left({\mathit{\mu }}_{\mathit{v}}{\mathit{R}}_{\mathit{on}}\right)\hfill \end{array}\right.$ [11.18]

[11.18]

The charge-controlled memristor state (length of doped region) can be described as

$\frac{\mathit{w}}{\mathit{D}}=\left\{\begin{array}{ll}1\hfill & \hfill \mathit{q}>{\mathit{Q}}_{\mathit{UP}}\hfill \\ {\mathit{w}}_0/\mathit{D}+{\mathit{\mu }}_{\mathit{v}}·\raisebox{1ex}{${\mathit{R}}_{\mathit{on}}$}\!\left/ \!\raisebox{-1ex}{${\mathit{D}}^2$}\right.\cdot \mathit{q},\hfill & \hfill {\mathit{Q}}_{\mathit{LOW}}<\mathit{q}<{\mathit{Q}}_{\mathit{UP}}\hfill \\ 0\hfill & \hfill \mathit{q}<{\mathit{Q}}_{\mathit{LOW}}\hfill \end{array}\right.$ [11.19]

[11.19]

where w₀ is the initial state, D is the memristor length, μ_v is the average ion mobility and q is injected charges.

When actual charge injection goes beyond the upper limit of effective charge injection (Q_UP), the state will go no further after it reaches w = D. Likewise, the lowest state is at zero, even if charge injection goes lower than the lower limit of effective charge injection (Q_LOW). In the linear drift model, the memristor state will not go beyond total length D or go below zero. The state will be trapped at the boundary if injected charge is not within the effective Q range.

Equation 11.19 determines the charge-controlled memristance as

$\mathit{R}\left(\mathit{w}\right)=\mathit{M}\left(\mathit{q}\right)$ [11.20]

[11.20]

As Eq. 11.20 indicates, the resistance becomes charge dependent. Equivalently

$\mathit{M}\left(\mathit{q}\right)={\mathit{R}}_{\mathit{off}}-\left({\mathit{R}}_{\mathit{off}}-{\mathit{R}}_{\mathit{on}}\right)·\left(\frac{{\mathit{w}}_0}{\mathit{D}}+\frac{{\mathit{\mu }}_{\mathit{v}}{\mathit{R}}_{\mathit{on}}}{{\mathit{D}}^2}·\mathit{q}\right)$ [11.21]

[11.21]

where q is valid in the effective Q range. As a special case where w₀ = 0 and R_on is small enough such that (R_off − R_on) ≈ R_off, charge-controlled memristance can be simplified to

$\mathit{M}\left(\mathit{q}\right)={\mathit{R}}_{\mathit{off}}·\left(1-\frac{{\mathit{\mu }}_{\mathit{v}}{\mathit{R}}_{\mathit{on}}}{{\mathit{D}}^2}\mathit{·q}\left(\mathit{t}\right)\right)$ [11.22]

[11.22]

Since the charge is integral of the current with respect to time, the state change is only functional of the integrated charge regardless of the current waveform.

Internal state w always comes back to the initial place, if the integral of current is zero over a certain time period. Figure 11.4 gives a brief demonstration taken from Ho et al. (2011). In Fig. 11.4(a), the current source I_in is composed of a positive pulse (t₀ to t₁) followed by a negative pulse (t₁ to t₂) with equal amplitude and width. In Fig. 11.4(b), starting from initial state w₀ at t₀, the state rises from t₀ to t₁ due to the positive pulse, causing the state rests at w_1. Based on Eq. 11.16, the value for w₁ is

${\mathit{w}}_1/\mathit{D}={\mathit{w}}_0/\mathit{D}+{\mathit{\mu }}_{\mathit{v}}·\raisebox{1ex}{${\mathit{R}}_{\mathit{on}}$}\!\left/ \!\raisebox{-1ex}{${\mathit{D}}^2$}\right.·\left({\mathit{I}}_{\mathit{A}}·\Delta \mathit{t}\right)$ [11.23]

[11.23]

where (I_A · Δt) is the charge injection by the positive pulse. From t₁ to t₂, the negative pulse follows, which moves the state from w₁ to w₂, and w₂ can be expressed in terms of w₁ as

${\mathit{w}}_2/\mathit{D}={\mathit{w}}_1/\mathit{D}+{\mathit{\mu }}_{\mathit{v}}·\raisebox{1ex}{${\mathit{R}}_{\mathit{on}}$}\!\left/ \!\raisebox{-1ex}{${\mathit{D}}^2$}\right.·\left(-{\mathit{I}}_{\mathit{A}}·\Delta \mathit{t}\right)$ [11.24]

[11.24]

where (−I_A · Δt) is the charge injected by the negative pulse. The state w₂ can be rewritten in terms of w₀ by substituting Eq. 11.23 into Eq. 11.24, which gives w₂ = w₀. This shows that the final state w₂ is the same as initial state w₀. The type of input waveforms in Fig. 11.4(a) are referred to as zero net-charge injection inputs, because the integral of the current over the corresponding time period is zero. Zero net-charge injection waveforms can be any waveforms, as long as the integral over a period is zero, and it can bring the state back to the original level regardless of the initial state. However, the state comes back only when the charge exerted on to memristor is within the effective q range. Otherwise, the state will not return to the initial state level.

B Voltage-controlled memristance

If a memristor is flux-controlled, the state of memristor is controlled by the flux across the cell. Figure 11.5 shows a memristor biased by a voltage source V_in, and V_in, can be any waveform. Denote β the off/on ratio (R_off = R_on β)_. Equation 11.9 can be rewritten as

$\frac{\mathit{dw}}{\mathit{dt}}=\frac{{\mathit{\mu }}_{\mathit{v}}}{\mathit{D\beta }-\mathit{w·}\left(\mathit{\beta }-1\right)}·{\mathit{V}}_{\mathit{in}}$ [11.25]

[11.25]

After certain manipulations using differential calculus, it becomes

${\mathit{w}}^2-\frac{2\mathit{·D\beta }}{\mathit{\beta }-1}\mathit{w}+\frac{2}{\left(\mathit{\beta }-1\right)}·\left({\mathit{\mu }}_{\mathit{v}}\mathit{·\phi }+\mathit{C}\right)=0$ [11.26]

[11.26]

where

$\mathit{C}=\mathit{D\beta ·}{\mathit{w}}_0-\left(\mathit{\beta }-1\right)/2·{\mathit{w}}_0^2$ [11.27]

[11.27]

and ϕ is the flux which is the integration of the voltage across the memristor. The solution to Eq. 11.26 would be

$\mathit{w}\left(\mathit{t}\right)=\frac{\mathit{D\beta }}{\left(\mathit{\beta }-1\right)}·\left[1-\sqrt{{\left(1-\frac{\mathit{\beta }-1}{\mathit{D\beta }}{\mathit{w}}_0\right)}^2-\frac{2{\mathit{\mu }}_{\mathit{v}}\left(\mathit{\beta }-1\right)}{{\left(\mathit{D\beta }\right)}^2}\mathit{·\phi }\left(\mathit{t}\right)}\right]$ [11.28]

[11.28]

where w(t = 0) = w₀ is the initial condition. Note that Eq. 11.28 clearly indicates that w(t) is a function of the flux applied; it is indirectly dependent on the voltage across the memristor. The input voltage waveform with the same net flux injection leads to the same memristor state change. As a result, the flux-controlled memristor state can be described as

$\frac{\mathit{w}}{\mathit{D}}=\left\{\begin{array}{lll}1\hfill & \hfill & \mathit{\phi }\ge {\mathrm{\Phi }}_{\mathit{UP}}\hfill \\ \frac{\mathit{\beta }}{\mathit{\beta }-1}\left[1-\sqrt{{\left(\frac{\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}\right)}^2-\frac{\mathit{\phi }}{{\mathrm{\Phi }}_{\mathit{D}}}}\right],\hfill & {\mathrm{\Phi }}_{\mathit{LOW}}\hfill & <\mathit{\phi }<{\mathrm{\Phi }}_{\mathit{UP}}\hfill \\ 0\hfill & \hfill & \mathit{\phi }\le {\mathrm{\Phi }}_{\mathit{LOW}}\hfill \end{array}\right.$ [11.29]

[11.29]

where

$\left\{\begin{array}{c}\hfill {\mathrm{\Phi }}_{\mathit{LOW}}=-\frac{{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{R}}_{\mathit{off}}^2}\left({\mathit{R}}_{\mathit{off}}^2-\mathit{R}{\left({\mathit{w}}_0\right)}^2\right)\hfill \\ \hfill {\mathrm{\Phi }}_{\mathit{UP}}=\frac{{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{R}}_{\mathit{off}}^2}\left(\mathit{R}{\left({\mathit{w}}_0\right)}^2-{\mathit{R}}_{\mathit{on}}^2\right)\hfill \end{array}\right.$ [11.30]

[11.30]

and

${\mathrm{\Phi }}_{\mathit{D}}=\frac{{\left(\mathit{\beta D}\right)}^2}{2{\mathit{\mu }}_{\mathit{v}}\left(\mathit{\beta }-1\right)}.$ [11.31]

[11.31]

Due to the finite length D of the thin film, the internal memristor state is constrained as 0 ≤ w(t)/D ≤ 1, which corresponds to the following effective φ range:

${\mathrm{\Phi }}_{\mathit{LOW}}\le \mathit{\phi }\left(\mathit{t}\right)\le {\mathrm{\Phi }}_{\mathit{UP}}$ [11.32]

[11.32]

Memristor state, w, would not be more than D when the applied flux across the memristor is over the effective upper bound (Φ_UP), and it would not be lower than zero when the applied flux is smaller than the effective lower bound (Φ _LOW). In other words, if the applied flux is larger than the upper limit of the effective range, φ would be the upper limit of effective injection. Likewise, φ would be the lower limit of effective injection, if the applied flux is lower than the effective range.

As Eq. 11.29 indicates, the memristance works as a flux driven resistance, thus it implies that

$\mathit{R}\left(\mathit{w}\right)=\mathit{M}\left(\mathit{\phi }\right)$ [11.33]

[11.33]

and the flux-controlled memristance can be written as

$\mathit{M}\left(\mathit{\phi }\right)={\mathit{R}}_{\mathit{off}}\sqrt{{\left(\frac{\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}\right)}^2-\frac{\mathit{\phi }}{{\mathrm{\Phi }}_{\mathit{D}}}}$ [11.34]

[11.34]

The flux-control memristance, as indicated in Eq 11.34, is bounded between R_on and R_off.

There is a relation between charge-controlled and flux-controlled memristance, since flux and charge injection both can change memristor state. The memristance change by charge or by flux should be identical for the same memristor state change, thus

$\mathit{M}\left(\mathit{q}\right)=\mathit{M}\left(\mathit{\phi }\right)$ [11.35]

[11.35]

Therefore, the q–φ relationship can be expressed as

$\mathit{q}\left(\mathit{\phi }\right)=\frac{2{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{R}}_{\mathit{off}}}\left[\left(\frac{\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}\right)-\sqrt{{\left(\frac{\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}\right)}^2-\frac{\mathit{\phi }}{{\mathrm{\Phi }}_{\mathit{D}}}}\right]$ [11.36]

[11.36]

According to the definition of memductance from Eq. 11.2, the memductance is

$\mathit{W}\left(\mathit{\phi }\right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathit{R}}_{\mathit{off}}$}\right.·\sqrt{{\left(\frac{\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}\right)}^2-\frac{\mathit{\phi }}{{\mathrm{\Phi }}_{\mathit{D}}}}.$ [11.37]

[11.37]

Based on Eq. 11.5, the inverse of memductance gives memristance. Since memductance has flux as the control variable, the inverse of that gives flux-controlled memristance, which is the same as in Eq. 11.34.

Similar to the charge-controlled scenario, the memristor state change is also a function of the voltage integration, regardless of the waveform shape of the bias voltage. The change to the memristor states would be the same, as long as the flux injections (integration of their voltages) remain the same. In addition, a zero net-flux injection input, one whose integrated voltage over the time is zero, pushes the state of the memristor back to the initial level if the flux exerted on to the memristor is within the effective φ range, as in Eq. 11.32.

By utilizing Eqs 11.33 and 11.34, the applied flux needed across the memristor to move the state of memristor from an initial state w₀ to an arbitrary state w is

$\mathit{\phi }\left(\mathit{t}\right)=\frac{{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{R}}_{\mathit{off}}^2}\left[{\left(\mathit{R}\left({\mathit{w}}_0\right)\right)}^2-{\left(\mathit{R}\left(\mathit{w}\left(\mathit{t}\right)\right)\right)}^2\right].$ [11.38]

[11.38]

If the applied voltage is a square-wave pulse with amplitude V_A and width T_w, the flux across the memristor is described as

$\mathit{\phi }\left(\mathit{t}\right)={\displaystyle {\mathit{t}}_{\int }^0\mathit{v}}\left(\mathit{\tau }\right)\partial \mathit{\tau }=\left\{\begin{array}{cc}\hfill {\mathit{V}}_{\mathit{A}}\mathit{·t}\hfill & \hfill \mathit{if}\mathit{t}\le {\mathit{T}}_{\mathit{w}}\hfill \\ \hfill {\mathit{V}}_{\mathit{A}}·{\mathit{T}}_{\mathit{w}}\hfill & \hfill \mathit{if}\mathit{t}>{\mathit{T}}_{\mathit{w}}\hfill \end{array}\right.$ [11.39]

[11.39]

The time it takes to move the state of memristor from w₀ to w can be derived. The input voltage magnitude is V_A when time is between zero and T_w, and flux is accumulating in this time range. When time goes beyond T_w, the voltage magnitude is zero, so there is no more flux increment beyond time T_w. Hence, the total flux injection for a square-wave pulse is V_A · T_w,,which is the amplitude times the width. The total flux injection determines the change of memristor state. With a given pulse magnitude V_A, the required width needed to move memristor state from w₀ to w is

${\mathit{T}}_{\mathit{w}}=\frac{{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{V}}_{\mathit{A}}{\mathit{R}}_{\mathit{off}}^2}\left|{\left(\mathit{R}\left({\mathit{w}}_0\right)\right)}^2-{\left(\mathit{R}\left(\mathit{w}\right)\right)}^2\right|$ [11.40]

[11.40]

As a special case, the required time needed to move the memristor state from w = 0 to w = D is the same as to move from w = D to w = 0, and the expression is

${\mathit{T}}_{\mathit{w}}=\left|\frac{{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{V}}_{\mathit{A}}{\mathit{R}}_{\mathit{off}}^2}·\left({\mathit{R}}_{\mathit{off}}^2-{\mathit{R}}_{\mathit{on}}^2\right)\right|$ [11.41]

[11.41]

Furthermore, conisder a constant resistor R_x and a memristor connected in series and biased using a voltage source, as shown in Fig. 11.6. The voltage at node x is proved to be

${\mathit{V}}_{\mathit{x}}={\mathit{V}}_{\mathit{in}}\frac{{\mathit{R}}_{\mathit{x}}}{{\mathit{R}}_{\mathit{x}}+\mathit{M}\left({\mathit{\phi }}_{\mathit{in}}-{\mathit{\phi }}_{\mathit{x}}\right)}$ [11.42]

[11.42]

To prove Eq. 11.42, the input-output relationship of the divider circuit will be derived by solving φ_x in terms of φ_in analytically. Note that φ_in is the input flux injection, φ_x is the flux accumulated at node x, and φ_in − φ_x is the flux across memristor, M. Based on Kirchhoff’s Current Law (KCL), the KCL equation at node x implies that all the net charges into node x would be zero. Hence, the charges (integral of current) that went through the memristor, q_x, should be the same as the charges that went through the resistor, so q_x = φ_x/R_x. Accordingly, the flux across the memristor is φ_in − φ_x, and replacing φ by φ_in − φ_x and q by q_x in Eq. 11.36 yields the charges that went through the memristor:

$\frac{2{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{R}}_{\mathit{off}}}\left[\left(\frac{\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}\right)-\sqrt{{\left(\frac{\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}\right)}^2-\frac{{\mathit{\phi }}_{\mathit{in}}-{\mathit{\phi }}_{\mathit{x}}}{{\mathrm{\Phi }}_{\mathit{D}}}}\right]=\frac{{\mathit{\phi }}_{\mathit{x}}}{{\mathit{R}}_{\mathit{x}}}$ [11.43]

[11.43]

in which φ_x has an unique solution of

${\mathit{\phi }}_{\mathit{x}}\left(\mathit{t}\right)=\frac{2{\mathrm{\Phi }}_{\mathit{D}}{\mathit{R}}_{\mathit{x}}}{{\mathit{R}}_{\mathit{off}}}·\left[\frac{{\mathit{R}}_{\mathit{x}}+\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}-\sqrt{{\left(\frac{{\mathit{R}}_{\mathit{x}}+\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}\right)}^2-\frac{{\mathit{\phi }}_{\mathit{in}}\left(\mathit{t}\right)}{{\mathrm{\Phi }}_{\mathit{D}}}}\right]$ [11.44]

[11.44]

Since voltage V_x is the total derivate of φ_x, V_x is derived as

${\mathit{V}}_{\mathit{x}}={\mathit{V}}_{\mathit{in}}·{\mathit{R}}_{\mathit{x}}/\left[{\mathit{R}}_{\mathit{off}}\sqrt{{\left(\frac{{\mathit{R}}_{\mathit{x}}+\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}\right)}^2-\frac{{\mathit{\phi }}_{\mathit{in}}}{{\mathrm{\Phi }}_{\mathit{D}}}}\right].$ [11.45]

[11.45]

To verify that Eq. 11.45 is indeed the same as Eq. 11.42, we evaluate memristance M(φ_in − φ_x) by using Eqs 11.44 and 11.34. The result yields

$\mathit{M}\left({\mathit{\phi }}_{\mathit{in}}-{\mathit{\phi }}_{\mathit{x}}\right)={\mathit{R}}_{\mathit{off}}·\left({\overline{\mathrm{\Phi }}}_{\mathit{x}}-{\mathit{R}}_{\mathit{x}}/{\mathit{R}}_{\mathit{off}}\right)$ [11.46]

[11.46]

where

${\overline{\mathrm{\Phi }}}_{\mathit{X}}=\sqrt{{\left(\frac{{\mathit{R}}_{\mathit{x}}+\mathit{R}\left({\mathit{w}}_0\right)}{{\mathit{R}}_{\mathit{off}}}\right)}^2-\frac{{\mathit{\phi }}_{\mathit{in}}}{{\mathrm{\Phi }}_{\mathit{D}}}}$ [11.47]

[11.47]

Finally, substituting Eq. 11.46 backing to Eq. 11.42 shows that it is exactly equal to Eq. 11.45. Therefore, the memristor series-connect resistor circuit in Fig. 11.6 indeed behaves as a voltage divider.

Moreover, the expression for the required input flux to move the memristor state from an initial to w for the divider circuit shown in Fig. 11.6 is

${\mathit{\phi }}_{\mathit{in}}\left(\mathit{w}\right)=\frac{{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{R}}_{\mathit{off}}^2}·\left|{\left(\mathit{R}\left({\mathit{w}}_0\right)+{\mathit{R}}_{\mathit{x}}\right)}^2-{\left(\mathit{R}\left(\mathit{w}\right)+{\mathit{R}}_{\mathit{x}}\right)}^2\right|$ [11.48]

[11.48]

Equation 11.48 is derived from Eq. 11.38. The φ in Eq. 11.38 is the flux across the memristor. The φ would be φ = φ_in − φ_x for the case of a memristor series connected to a resistor. Since φ_x is already derived in Eq. 11.44, substituting φ = φ_in − φx into Eq. 1.38 yields Eq. 11.48. For a square-wave voltage supply, the time to move the state of memristor from w₀ to w with a given amplitude V_A is

${\mathit{T}}_{\mathit{w}}=\frac{{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{V}}_{\mathit{A}}{\mathit{R}}_{\mathit{off}}^2}·\left|{\left(\mathit{R}\left({\mathit{w}}_0\right)+{\mathit{R}}_{\mathit{x}}\right)}^2-{\left(\mathit{R}\left(\mathit{w}\right)+{\mathit{R}}_{\mathit{x}}\right)}^2\right|$ [11.49]

[11.49]

As a special case, to move the state from w₀ = 0 to w = D requires the same T_w as that needed to move from state w = 0 to w = D, and the T_w expression is

${\mathit{T}}_{\mathit{w}}=\frac{{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{V}}_{\mathit{A}}{\mathit{R}}_{\mathit{off}}^2}·\left|{\left({\mathit{R}}_{\mathit{off}}+{\mathit{R}}_{\mathit{x}}\right)}^2-{\left({\mathit{R}}_{\mathit{on}}+{\mathit{R}}_{\mathit{x}}\right)}^2\right|$ [11.50]

[11.50]

Furthermore, a memristor has an effective flux restriction due to finite length D. Equation 11.32 shows the effective φ range for a single memristor case. Similarly, the case of a memristor resistor connected in series would also have an effective range. The effective range for φ_in across both memristor and resistor is

${\overline{\mathrm{\Phi }}}_{\mathit{LOW}}\le {\mathit{\phi }}_{\mathit{in}}\le {\overline{\mathrm{\Phi }}}_{\mathit{UP}}$ [11.51]

[11.51]

where φ_in is integral of V_in and

${\overline{\mathrm{\Phi }}}_{\mathit{LOW}}=-\frac{{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{R}}_{\mathit{off}}^2}\left({\left({\mathit{R}}_{\mathit{off}}+{\mathit{R}}_{\mathit{x}}\right)}^2-{\left(\mathit{R}\left({\mathit{w}}_0\right)+{\mathit{R}}_{\mathit{x}}\right)}^2\right)$ [11.52]

[11.52]

${\overline{\mathrm{\Phi }}}_{\mathit{UP}}=-\frac{{\mathrm{\Phi }}_{\mathit{D}}}{{\mathit{R}}_{\mathit{off}}^2}\left({\left(\mathit{R}\left({\mathit{w}}_0\right)+{\mathit{R}}_{\mathit{x}}\right)}^2-{\left({\mathit{R}}_{\mathit{on}}+{\mathit{R}}_{\mathit{x}}\right)}^2\right)$ [11.53]

[11.53]

Finally, a zero net-flux input voltage pattern will insure that the state of the memristor comes back to the initial position for the circuit shown in Fig. 11.6. This property, where the state comes back to initial position, holds true for both a single memristor cell and a memristor series connected with a resistor cases. This contributes important design guidance for ensuring read stability, as discussed in detail in later sections. To prove this, simply set φ_in equal to zero in Eq. 11.48 and solve possible solutions for R(w); two possible solutions exist: one is w = w₀, and the other is outside of the memristor physical range. Therefore, the state will be back to the initial level as w = w₀. However, this is true only when the effective flux range condition as in Eq. 11.51 is satisfied.

A brief summary on the above-mentioned memristor characteristics are listed into 14 properties in the Appendix at the end of this chapter.