ARTICLE INFO
 Wei Wu, Ning Deng
 Memristor interpretations based on constitutive relations
 Journal of Semiconductors, 2017, 38(10): 104005
 http://dx.doi.org/10.1088/16744926/38/10/104005

Article history
 Received: 24 February 2017
 Revised manuscript received: 10 April 2017
The memristor was proposed by Chua as the fourth elementary circuit element in 1971 for the first time^{[1]}. He put forward a quadrangle with four fundamental circuit attributes: voltage v, current i, charge q and magnetic flux φ. The relationship between q and φ can interpret a memristor. Interests have been attracted since Strukov et al. implemented a TiO_{x}based memristor in 2008^{[2]} due to its emerging applications in the nonvolatile memories^{[3, 4]}, the chaotic circuits^{[5]} and the neuromorphic circuits^{[6, 7]}. Research is mainly concentrated on its modeling^{[8, 9]} and experimental realization^{[10, 11]}, but the fundamental interpretations on the memristor is limited. According to Chua^{[12]}, the memristor shows three fingerprints: (1) Pinched hysteresis loop; (2) Hysteresis lobe area decreases as frequency increases; (3) Pinched hysteresis loop shrinks to a singlevalued function at infinite frequency.
However, there are some queries on the memristors. For example, where is the magnetic flux φ in Strukov’s memristor? There is no obvious magnetic effect in the whole system. Another one is whether the memristor is truly the fourth elementary circuit element in the same manner with resistor, capacitor and inductor. For this query, Chua put forward a fourelement torus in 2003 to expand the elementary circuit elements into infinity^{[13]}. Wang proposed a triangular periodic table of elementary circuit elements, which is the core part of Chua’s fourelement torus^{[14]}. The triangular periodic table consists of three element series: resistive elements (with the unit of Ohm), capacitive elements (with the unit of Farad) and inductive elements (with the unit of Henry), as shown in Fig. 1.
Each element in the table can be interpreted by two circuit attributes:
Circuit elements with memory such as memristors are catching more and more attention due to their memory property and application in synaptic electronics^{[15]}. In the real world, the voltage v and the current i can be measured at any moment, which means they are transient physical quantities, while the charge q and the magnetic flux φ are related to the state of the element which store the information in the past. The resistive memristor can store information in the past because it is interpreted by the relationship between q and φ. The memory property comes from the property of the physical quantities. Elements related to state quantities have memory property while elements related to transient quantities have no memory. Assuming that their differential dv/dt and di/dt can be measured directly at any moment and dv/dt and di/dt are transient physical quantities. Then v and i are state quantities and can store information in the past. In that case, nonlinear resistors such as diodes can be used as memory element, as shown in Fig. 2.
The singlevalue function of v and i can generate a pinched hysteresis loop of dv/dt and di/dt under a sinusoidal excitation, which means the differential resistance of a nonlinear resistor can change with the working point v or i, just like that the resistance of a memristor can change with the working point q or φ. However, this case is not real in our world where the transient physical quantities are still v and i and q and φ are state quantities. That is why memristors have the memory property.
Memristors and diodes are taken as an analogy although they are different. For diodes, v or i determines the working point of the element and smallsignal impedance (differential resistance) determines the calculation result of the circuit. For memristors, q or φ determines the working point and its resistance can be seen as smallsignal impedance. That is why the hysteresis lobe area of a memristor decreases as frequency increases and shrinks to a singlevalued function at infinite frequency, as shown in Fig. 3.
When a sinusoidal current excitation of i = sinωt is applied, the working point of the memristor is the charge accumulated in the past q_{0}. The charge is
In the memristor theory, Chua used the relationship between q and φ to define it, but there is no obvious magnetic effect in Strukov’s memristor, only with a pinched hysteresis loop presented to prove that it is a theoretical memristor. That is where some researchers have questions. The charge q can be related to physical quantity i in circuit theory by its definition i = dq/dt, but there is no direct definition between v and φ. Faraday’s law of electromagnetic induction can be used to interpret the source of voltage v as v = dφ/dt although v may also come from chemical potential of a battery and other sources in real circuit. The memristor is imagined as a black box shown in Fig. 4.
A closed loop provides external power supply for the black box. The current in the loop is i and the voltage v drops on the black box. Then the charge q in the memristor definition can be seen as the charge flows into the black box through current i. So we have i = dq/dt and it fits the memristor’s definition. For the magnetic flux φ, the magnetic field H can be assumed in the open space whose change leads to the voltage drop according to electromagnetic induction. Then the voltage dropping on the black box can be calculated as v = dφ/dt, which fits the memristor’s definition. The imaginary magnetic flux φ can be seen as the flux in memristor theory for the source of voltage. In real cases, the voltage supply can be obtained from electromagnetic induction or some other sources. So the flux φ in memristor theory may not correspond to the real physical quantity but a mathematical quantity as the integral of voltage. That is why no obvious magnetic effect in Strukov’s memristor can be found but it is still a real memristor.
3. Circuit element interpretations based on the constitutive relationMemristors differ from traditional resistors on the respect of constitutive relation. The constitutive relation between q and φ interprets a memristor and determines its electrical property. The graph method in Ref. [14] is used to generate i–v curve from q–φ curve under different excitations, and viceversa, as shown in Fig. 5.
It can be seen in Fig. 5 that given the constitutive relation between q and φ, the relationship between v and i is different under different excitations. Under a sinusoidal current excitation, the i–v curve is the common one in most papers while under a square wave current excitation, the i–v curve is very different. That means only the constitutive relation can describe the electrical property of an element accurately, without considering the external excitation. The constitutive relation of a traditional resistor is v–i, so it differs from the memristor on the respect of electrical property. It is the same for all circuit elements in Chua’s fourelement torus that constitutive relation can tell them apart.
The constitutive relation of an element can be divided into odd symmetric relation and nonodd symmetric relation. Odd symmetric relation means that the constitutive relation of the element is odd symmetric function while nonodd symmetric relation means that the constitutive relation is nonodd symmetric function. Take resistive elements series as an example, as shown in Fig. 6. The excitation is harmonic wave in the following examples.
The element in Fig. 6(a) has an odd symmetric constitutive relation. Then we can find that every order diagram
By the same token, the elements with nonodd symmetric constitutive relation can be called “bipolar element” as shown in Fig. 6(b). The element with nonodd symmetric φ–q curve can generate a pinched hysteresis loop in lower order diagram
The common resistor seen in our daily life is a unipolar element. So we define it by relation between two attributes easiest to measure: v and i. As the diode is a bipolar element, we can only use the relationship between v and i as its fixed constitutive relation.
For a resistive element, it is easy to measure the v–i curve by oscilloscope, and the shape of the curve is quite varied. If an odd symmetric v–i curve is obtained after measurement, then the element can be defined as a traditional resistor with constitutive relation (v, i). Here three simple kinds of nonodd symmetric v–i curve are listed, as shown in Fig. 7.
The element in Fig. 7(a) is a nonlinear resistor. The constitutive of the resistor is fixed. The element in Fig. 7(b) shows a pinched hysteresis loop and after generating the higherorder diagram (v–i→φ–q), a singlevalue function of φ–q can be obtained and this element is a memristor with fixed constitutive relation (φ, q). The element in Fig. 7(c) has an open multivalue curve and after generating lowerorder diagram (v–i→dv/dt–di/dt), a singlevalue function of dv/dt–di/dt can be obtained. Then the element is defined as a differential resistor with constitutive relation (dv/dt, di/dt).
4. ConclusionMemristors should have equal status with other circuit elements in Chua’s fourelement torus. However, memory property makes it much more important in novel applications such as synaptic electronics. In essence, the memristor is a resistive element with constitutive relation between φ and q. Then a pinched hysteresis loop and frequencydependent property can be achieved. It can also be considered as a resistor with a special v–i curve, not the same as a traditional resistor. Specific electronic systems can be designed to implement a memristor, such as Strukov’s TiO_{x}based memristor. The resistance of the memristor is changed by ion drifting and it is a resistor with statechangeable property essentially.
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