1. Introduction
Silicon-based devices are nearing their physical limits, and there are increasing problems in conventional computing due to the von Neumann bottleneck [1–3]. To overcome these limitations, many new methods have been proposed based on the spin [4, 5], ionic transport [6, 7] and phase transition [8–10]. Among these, computing via memristive devices provides an attractive solution to low-power and high speed. In addition to the small size (<2 nm) [11], the memristor consumes relatively low energy and switch fast (<1 ns) [12]. The information is stored in a form of resistance in the memristor, and current is a product of input voltage and memristive conductance. Consequently, the in-memory computing is achieved by multiplication following Oham's law [13–15]. Due to its volatility and device character, the memristor attracts great interests in microelectronics [16, 17], informatics [18–20] and neuromorphic computing [21–23]. In addition to the material application, the memristor is theoretically used in chaos models and neuron models. The memristor-based models exhibit abundant dynamical behaviors, including extreme multiutility and neural firing patterns [24–30]. Ramamoorthy et al [25] observed the result of partial amplitude control and offset boosting in new asymmetric/symmetric memristive system. Liang et al [26] constructed a novel third-order neuron circuit by embedding a active memristor and a negative capacitor, and the abundant firing patterns were obtained.
The memristor was proposed theoretically by Chua in 1971 [31], and HP Labs produced the first memristor device based on the sandwich structure (Pt − TiO2 − Pt) [32]. The following memristor fabrications made great progress based on the different metal oxides, and the mechanisms are divided into several types, including electrical metallization, valence change memory effect, thermochemical memory effect, electrostatic effect [33–35] and phase change memory effect [36, 37]. However, there are remaining challenges for memristor manufacture, including parameter variability and cycling endurance [38, 39], and non-ideal performance is mainly avoided by the result of empirical trial and error [40, 41]. Hence, it is necessary to gain theoretical insights. Theoretically, an electric device is called a memristor if it meets the three fingerprints [42, 43]. The pinched hysteresis curves appear in the current–voltage plane, and the area of hysteresis decreases with increasing voltage frequency [43]. However, it is still limited to explain the principle of the fingerprints. In this paper, we disclose the essence of a memristor, and propose the condition for phase transition between the positive resistance, zero resistance and negative resistance.
2. Basic definitions of memristor
The memristor is divided into two categories, including the charge-controlled memristor φ = φ(q) and the flux-controlled memristor q = q(φ). For the charge-controlled memristor φ = φ(q), its derivative M(q) is 1 ) is rewritten as
$\begin{eqnarray}M(q)=\frac{{\rm{d}}\varphi (q)}{{\rm{d}}q}.\end{eqnarray}$
Obviously, M(q) has the same dimension as resistance, and thus it is called mem-resistance. Similarly, the derivative of the flux-controlled memristor q = q(φ) is defined as $\begin{eqnarray}W(\varphi )=\frac{{\rm{d}}q(\varphi )}{{\rm{d}}\varphi }.\end{eqnarray}$
Similarly, the dimension of W(φ) is the same as conductance, and it is called mem-conductance. To further confirm the dimension of M(q), equation ( $\begin{eqnarray}M(q)=\frac{{\rm{d}}\varphi (q)}{{\rm{d}}q}=\frac{{\rm{d}}\varphi (q)/{\rm{d}}t}{{\rm{d}}q/{\rm{d}}t}=\frac{v}{i}.\end{eqnarray}$
Similarly, W(φ) is rewritten as $\begin{eqnarray}W(\varphi )=\frac{{\rm{d}}q(\varphi )}{{\rm{d}}\varphi }=\frac{{\rm{d}}q(\varphi )/{\rm{d}}t}{{\rm{d}}\varphi /{\rm{d}}t}=\frac{i}{v}.\end{eqnarray}$
Therefore, the voltage–current relation of the memristor is $\begin{eqnarray}\left\{\begin{array}{l}v=M(q)i\\ \frac{{\rm{d}}q}{{\rm{d}}t}=i\end{array},\right.\left\{\begin{array}{l}i=W(\varphi )v\\ \frac{{\rm{d}}\varphi }{{\rm{d}}t}=v\end{array},\right.\end{eqnarray}$
where the v−i relation is for charge-controlled M(q), and the i−v relation is for the flux-controlled W(φ).Three fingerprints of the memristor [42]. If a device meets,
| (1) The pinched hysteresis loop appears in the voltage–current plane with a bipolar periodic signal, | |
| (2) The hysteresis area decreases with increasing of the driven signal frequency, | |
| (3) The hysteresis loop gets into a single-valued function when the frequency approaches to infinity. |
Then, it is a memristor. where Ron is the low resistance and Roff is the high resistance of the semiconductor layer. In addition, x is the state variable and m is the dimension parameter. The driven current i is set as $i=A\sin (\omega t)$, and the hysteresis curve is presented in figure 2. Apparently, the curves are symmetrical and pinched at the origin point, and the area shrinks with increasing frequency ω. Eventually, the memristive function is a single-valued function with a larger frequency.
The three fingerprints of the HP memristor.
The HP memristor model [32] is
$\begin{eqnarray}\left\{\begin{array}{l}v=\left[{R}_{{\rm{on}}}x+{R}_{{\rm{off}}}(1-x)\right]i\\ \frac{{\rm{d}}x}{{\rm{d}}\tau }=mi\end{array},\right.\end{eqnarray}$
Figure 2. The hysteresis curves of the HP memristor with different frequency ω. |
3. Theorems for three fingerprints and phase transition
In this section, we disclose the symmetry principle ‘why the memristive hysteresis curves are symmetry at the origin point'. In addition, we explain the mechanism for the decreasing area of the hysteresis loop with the increasing of the frequency. Eventually, we propose the mechanism of phase transition between the positive resistance, zero resistance and negative resistance.
3.1. Mechanism for symmetry
For all memristors, the symmetry of the hysteresis loop is only determined by the driven input.
Taking the charge-controlled memristor in equation (
$\begin{eqnarray}\frac{{\rm{d}}M(q)}{{\rm{d}}t}=\frac{{\rm{d}}M(q)}{{\rm{d}}q}\frac{{\rm{d}}q}{{\rm{d}}t}=\frac{{\rm{d}}M(q)}{{\rm{d}}q}i.\end{eqnarray}$
Otherwise, if the derivate of M(q) does not exist, the function territory is divided into several differentiable parts between the non-differentiable points. In each small differentiable region, the derivate is the same as equation ( $\begin{eqnarray}M\left(q\left({t}_{2}\right)\right)-M\left(q\left({t}_{1}\right)\right)={\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(a)}{\int }_{{t}_{1}}^{{t}_{2}}i{\rm{d}}t,\end{eqnarray}$
where $a\in \left({t}_{1},{t}_{2}\right)$, and non-differentiable points are removed at t1, t2 for non-differentiable M(q). Equation (The v − i relation of the charge-controlled memristor is rewritten as
$\begin{eqnarray}v\left({t}_{2}\right)=\left[M\left(q\left({t}_{1}\right)\right)+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(a)}{\int }_{{t}_{1}}^{{t}_{2}}i{\rm{d}}t\right]i,\end{eqnarray}$
where M(q(t1)) is the resistance at q(t1), and the ${\rm{d}}M(q)\,/{\left.{\rm{d}}q\right|}_{q=q(a)}$ is the derivate value at q(a). Therefore, M(q(t1)) and ${\rm{d}}M(q)/{\left.{\rm{d}}q\right|}_{q=q(a)}$ are both specific values and make no contributions to the symmetry of the hysteresis loop, and the symmetry is only determined by the driven source i and its integration. □
Figure 3. The differential form of the memristive function. |
If mem-resistance is not determined by q[dM(q)/dq = 0], then equation (9 ) is rewritten as $v=M\left(q\left({t}_{1}\right)\right)i\to Ri$, and the memristor function M degenerates to a linear resistor R. The same conclusion is obtained in the flux-controlled memristor.
For all memristors, the hysteresis loops in the i − v (or v − i) plane are odd symmetry at the origin point with the sine driven signal.
Taking the charge-controlled memristor as an example, current i(t) is a sine function of t, and i(t) = 0 at t = 0. Therefore, i(t) is an odd function of t, and the integration of the odd function i(t) is an even function. Combining equation (
$\begin{eqnarray}\left\{\begin{array}{l}v\left({t}_{2}\right)=F(q,i,t)i,\\ F(q,i,t)=M\left(q\left({t}_{1}\right)\right)+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(a)}{\int }_{{t}_{1}}^{{t}_{2}}i{\rm{d}}t\end{array}.\right.\end{eqnarray}$
Obviously, F(q, i, t) is an even function of i(t), and thus v(t) is an odd function of i. Similarly, the i(t) is an odd function of v(t) for the flux-controlled memristor. □The simplest memristor model.
The simplest memristor is ideal, but it advances the understanding of the essence of the memristor. According to equation (9 ), the voltage–current relation of charge-controlled memristor is rewritten to 12 ) is so-called the simplest memristor model, and the dimension meets [k][q] = [M(q)]. The hysteresis curves of memristor equation (12 ) are shown in figure 4. Obviously, the simplest model exhibits pinched hysteresis loops with the external stimulus, and they are symmetrical at the origin point. Additionally, the area shrinks with increasing frequency ω, and there is a proportionate expansion of the curves with the increasing of amplitude (figure 4(b)).
$\begin{eqnarray}\left\{\begin{array}{l}v(t)=\left[M(q(0))+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(a)}{\int }_{0}^{t}i{\rm{d}}t\right]i\\ \displaystyle \frac{{\rm{d}}q}{{\rm{d}}t}=i\end{array}.\right.\end{eqnarray}$
For the simplest case, M(q(0)) = 0, and the differential item dM(q)/dq is set as a constant k. The simplest model is defined as $\begin{eqnarray}\left\{\begin{array}{l}v={M}_{1}(q)i=(kq)i\\ \frac{{\rm{d}}q}{{\rm{d}}t}=i\end{array}.\right.\end{eqnarray}$
For k = 1, equation (
Figure 4. The hysteresis curves of the simplest memristor model. (a) The hysteresis curves with different frequency ω of the current i. (b) The hysteresis curves with different amplitude A of the current i. The parameter is set as k = 1. The dynamics principle of the simplest memristor is a product of i and its integration q. |
Obviously, the dynamics is only determined by a product result of current i(t) and its integration (charge q(t)) for memristor equation (12 ) with k = 1. The resistance (or conductance) is a function determined by the charge q(or magnetic flux φ)), and it associates with the conducting mechanism of the memristor.
The non-differentiable memristive function.
Theorem 1 is still correct for the non-differentiable memristive function. The non-differentiable memristive function is obtained from equation (12 ), and it is given as, 13 ) is a memristor with the non-differentiable memristive function.
$\begin{eqnarray}{M}_{2}(q)=\left\{\begin{array}{ll}kq(t), & q\gt \max [q(t)]/2\\ -kq(t), & q\lt \max [q(t)]/2\end{array},\right.\end{eqnarray}$
where $\max []$ denotes the maximum value of q(t). Figure 5 shows the differences between the differentiable memristive function M1(q) and the non-differentiable memristive function M2(q). Apparently, there is a mutation for M2(q) when the q value exceeds half of the maximum value. It also exhibits a pinched hysteresis curve, and the loop area decreases with increasing frequency. Furthermore, the loop gets a single line with the further increasing of the frequency, and the enlarged area appears proportionately with the increasing of the amplitude, as shown in figure 5(b)(c). These results confirm the model (
Figure 5. The hysteresis curves of the non-differentiable memristive function. (a) The evolution of differentiable memristive function M1(q) and non-differentiable M2(q). (b) The hysteresis curves with different frequency of the current i. (c) The hysteresis curves with different amplitude of the current i. |
3.2. Mechanism for shrinking area
The shrinking area of the hysteresis curves results from the shrinking area enclosed by the drive signal and the time coordinate axis.
Taking the charge-controlled memristor in equation (
$\begin{eqnarray}S=\left|{S}_{1}-{S}_{2}\right|=\left|{\int }_{\tfrac{T}{4}}^{\tfrac{T}{2}}v{\rm{d}}i-{\int }_{0}^{\tfrac{T}{4}}v{\rm{d}}i\right|.\end{eqnarray}$
Combing the equation ( $\begin{eqnarray}\left\{\begin{array}{l}{S}_{1}={\int }_{0}^{i\left(\tfrac{T}{4}\right)}\left[M(q(0))+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(b)}{\int }_{0}^{\tfrac{T}{4}}i{\rm{d}}t\right]i{\rm{d}}i\\ {S}_{2}={\int }_{i\left(\tfrac{T}{4}\right)}^{i\left(\tfrac{T}{2}\right)}\left[M\left(q\left(\displaystyle \frac{T}{4}\right)\right)+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(c)}{\int }_{\tfrac{T}{4}}^{\tfrac{T}{2}}i{\rm{d}}t\right]i{\rm{d}}i\end{array},\right.\end{eqnarray}$
where, $b\in \left(0,T/4\right)$, and $c\in \left(T/4,T/2\right)$. Then, if we set $\begin{eqnarray}\left\{\begin{array}{l}{S}_{i1}={\int }_{0}^{\tfrac{T}{4}}i{\rm{d}}t\\ {S}_{i2}={\int }_{\tfrac{T}{4}}^{\tfrac{T}{2}}i{\rm{d}}t\end{array},\right.\end{eqnarray}$
then, equation ( $\begin{eqnarray}\left\{\begin{array}{l}{S}_{1}={\int }_{0}^{i\left(\tfrac{T}{4}\right)}\left[M(q(0))+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(b)}{S}_{i1}\right]i{\rm{d}}i\\ {S}_{2}={\int }_{i\left(\tfrac{T}{4}\right)}^{i\left(\tfrac{T}{2}\right)}\left[M\left(q\left(\displaystyle \frac{T}{4}\right)\right)+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(c)}{S}_{i2}\right]i{\rm{d}}i\end{array},\right.\end{eqnarray}$
where Si1, Si2 are only determined by the amplitude and frequency of the current i. Thus, equation ( $\begin{eqnarray}\left\{\begin{array}{l}{S}_{1}=\left[M(q(0))+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(b)}{S}_{i1}\right]{\int }_{0}^{i\left(\tfrac{T}{4}\right)}i{\rm{d}}i\\ {S}_{2}=\left[M\left(q\left(\displaystyle \frac{T}{4}\right)\right)+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(c)}{S}_{i2}\right]{\int }_{i\left(\tfrac{T}{4}\right)}^{i\left(\tfrac{T}{2}\right)}i{\rm{d}}i\end{array}.\right.\end{eqnarray}$
Then, they are written as $\begin{eqnarray}\left\{\begin{array}{l}{S}_{1}=\frac{1}{2}\left[M(q(0))+{\left.\frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(b)}{S}_{i1}\right]\left[i{\left(\frac{T}{4}\right)}^{2}-i{(0)}^{2}\right]\\ {S}_{2}=\frac{1}{2}\left[M\left(q\left(\frac{T}{4}\right)\right)+{\left.\frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(c)}{S}_{i2}\right]\left[i{\left(\frac{T}{2}\right)}^{2}-i{\left(\frac{T}{4}\right)}^{2}\right]\end{array},\right.\end{eqnarray}$
and equation ( $\begin{eqnarray}\begin{array}{rcl}S & = & \displaystyle \frac{1}{2}\left\{M(q(0))\left[i{\left(\displaystyle \frac{T}{4}\right)}^{2}-i{(0)}^{2}\right]\right.-M\left(q\left(\displaystyle \frac{T}{4}\right)\right)\\ & & \times \,\left.\left[i{\left(\displaystyle \frac{T}{4}\right)}^{2}-i{\left(\displaystyle \frac{T}{2}\right)}^{2}\right]\right\}\\ & & +\displaystyle \frac{1}{2}\left\{{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(b)}{S}_{i1}\left[i{\left(\displaystyle \frac{T}{4}\right)}^{2}-i{(0)}^{2}\right]\right.\left.-{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(c)}{S}_{i2}\left[i{\left(\displaystyle \frac{T}{4}\right)}^{2}-i{\left(\displaystyle \frac{T}{2}\right)}^{2}\right]\right\}.\end{array}\end{eqnarray}$
The value i(0), i(T/4) and i(T/2) are the same value for different frequencies with the same amplitude, and the first item {*} is the same for different frequencies in equation (
Figure 6. The hysteresis curves and the current in half period T/2. (a) The hysteresis curves of the HP memristor. (b) The decreased area enclosed by the current and the coordinate axis with the increasing of the frequency. |
3.3. Mechanism for phase transition
The energy function E is determined by the current and the voltage of the memristor, and it is given as
$\begin{eqnarray}E={\int }_{{t}_{1}}^{{t}_{2}}v(t)i(t){\rm{d}}t,\end{eqnarray}$
where t1, t2 is finite time. v(t) is the voltage of the memristor, and the i(t) is the current of the memristor.Local activity condition [43]. For the memristors with local negative resistance or conductance, their energy meet
$\begin{eqnarray}E\lt 0\,(\mathrm{locally}).\end{eqnarray}$
If a mem-resistance function meet
$\begin{eqnarray}M(0)+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(a)}{\int }_{{t}_{2}}^{{t}_{1}}i{\rm{d}}t\lt 0,\quad a\in \left({t}_{1},{t}_{2}\right).\end{eqnarray}$
Then, it is a local-active memristor. According to equation (
$\begin{eqnarray}M(q)\lt 0.\end{eqnarray}$
Therefore, the local-active memristor meet equation (If the current of a driven signal meets
$\begin{eqnarray}M(0)+{\left.\displaystyle \frac{{\rm{d}}M(q)}{{\rm{d}}q}\right|}_{q=q(a)}{\int }_{{t}_{2}}^{{t}_{1}}i{\rm{d}}t=0,\quad a\in \left({t}_{1},{t}_{2}\right).\end{eqnarray}$
Then, the zero resistance is obtained in (t1, t2). Obviously, the resistance is 0 if the driven signal meets equation (The local activity of HP memristor.
For HP memristor, its V − I relation is given as 27 ) is rewritten as
$\begin{eqnarray}\left\{\begin{array}{l}v=\left[{R}_{{\rm{on}}}q+{R}_{{\rm{off}}}(1-q)\right]i\\ \frac{{\rm{d}}q}{{\rm{d}}t}=i\end{array}.\right.\end{eqnarray}$
According to theorem 3, the local activity condition is $\begin{eqnarray}\left\{\begin{array}{l}{R}_{\mathrm{off}}+\left({R}_{\mathrm{on}}-{R}_{\mathrm{off}}\right){S}_{{t}_{0}}\lt 0\\ {S}_{{t}_{0}}={\int }_{0}^{{t}_{0}}i{\rm{d}}t,{t}_{0}\in (0,T]\end{array},\right.\end{eqnarray}$
where T is the period of the driven signal, and equation ( $\begin{eqnarray}{S}_{{t}_{0}}\gt \frac{{R}_{{\rm{off}}}}{{R}_{{\rm{off}}}-{R}_{{\rm{on}}}},\end{eqnarray}$
where Roff/(Roff − Ron) is called the phase transition threshold, and the zero resistance is obtained if St0 is equal to the threshold. When the integration St0 is less than the threshold, the HP memristor is a passivity memristor. Otherwise, it is a local-active memristor. Obviously, the phase transition between the local activity and passivity is determined by a combination of the driven signal and the material parameters (Roff, Ron). When the material parameters Roff, Ron are fixed, different driven inputs enable the transition in positive resistance, local zero resistance and local negative resistance. For further confirmation, we investigate the phase transition for the HP memristor, as shown in figure 7. For Roff = 10, Ron = 5, the threshold is 2. The memristor is passive when the integration of the driven input is smaller than the threshold value 2, as shown in figure 7(a1)(a2). With increasing St0, there is a local zero resistance area, as shown in figure 7(b1)(b2). Furthermore, the memristor gets a local activity memristor when St0 is larger than 2, and the negative resistance area appears, as shown in figure 7(c1)(c2). There is an obvious phase transition between the passivity and local activity (Phase transition point 1 and 2).
Figure 7. The resistance phase transition of the HP memristor, and the amplitude is adjusted to change the integration St0. (a) Positive resistance with A = 1. (b) Local zero resistance with A = 2. (c) Local negative resistance with A = 3. Points 1 and 2 are phase transition points (zero resistance). For Roff = 10, Ron = 5, the threshold is 2 (dash and dot line). |
4. Conclusion
In this paper, we disclose the mechanism of the three fingerprints for a memristor, and explain them based on the differential form of the memristor function. The symmetry is only determined by the driven source, and the simplest memristor model is presented to clarify the essence of the memristor. The resistance is determined by the integration of the driven signal. The shrinking area of the hysteresis curves results from the shrinking area enclosed by the drive signal and the coordinate axis with increasing frequency. Furthermore, the appropriate input results in phase transition of the memristor, and the resistance switches between the positive resistance, local zero resistance and local negative resistance. This switch phenomenon is confirmed in the HP memristor. These theorems advance the understanding of the memristor mechanism, and they show the potential application to function as a single neuron and synaptic connection.