半导体学报  2017, Vol. 38 Issue (10): 104005 PDF

#### ARTICLE INFO

Wei Wu, Ning Deng
Memristor interpretations based on constitutive relations
Journal of Semiconductors, 2017, 38(10): 104005
http://dx.doi.org/10.1088/1674-4926/38/10/104005

### Article history

Received: 24 February 2017
Revised manuscript received: 10 April 2017
Memristor interpretations based on constitutive relations
Wei Wu, Ning Deng
Institute of Microelectronics, Tsinghua University, Beijing 100084, China
Abstract: The attractive memristor is interpreted based on its constitutive relation. The memory property of the memristor is explained, along with the explanation on its three fingerprints: (1) Pinched hysteresis loop; (2) Hysteresis lobe area decreases as frequency increases; (3) Pinched hysteresis loop shrinks to a single-valued function at infinite frequency. Where the magnetic flux is in Strukov’s memristor is also introduced. Resistive elements including the memristor are taken as an example to argue that the constitutive relation determines the electrical property of a circuit element and diagram method is used to distinguish different elements in the resistive element series.
Key words: memristor     magnetic flux     constitutive relation     pinched hysteresis loop
1. Introduction

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 TiOx-based memristor in 2008[2] due to its emerging applications in the non-volatile 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 single-valued 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 four-element 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 four-element 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.

 Figure 1 A triangular periodic table of elementary circuit elements, with resistive elements series, capacitive elements series and inductive elements series.

Each element in the table can be interpreted by two circuit attributes: ${v^{\left( \alpha \right)}}\left( t \right)$ and ${i^{\left( \beta \right)}}\left( t \right)$ , where α and β can be any integer. When α/β is a positive integer, it represents α/βth-order differential w.r.t time t while when α/β is a negative integer, it represents α/βth-order integral w.r.t time t. So when α = β, ${v^{\left( \alpha \right)}}$ and ${i^{\left( \beta \right)}}$ can interpret a resistive element, such as resistor (vi) and memristor (v(–1)i(–1)). When α = β + 1, ${v^{\left( \alpha \right)}}$ and ${i^{\left( \beta \right)}}$ can interpret a capacitive element, such as capacitor $\left( {v - {i^{\left( { - 1} \right)}}} \right)$ and memcapacitor $\left( {{v^{\left( { - 1} \right)}} - {i^{\left( { - 2} \right)}}} \right)$ . When α = β – 1, ${v^{\left( \alpha \right)}}$ and ${i^{\left( \beta \right)}}$ can interpret an inductive element, such as inductor $\left( {{v^{\left( { - 1} \right)}} - i} \right)$ and meminductor $\left( {{v^{\left( { - 2} \right)}}- {i^{\left( { - 1} \right)}}} \right)$ . Memristor is in the resistive element series and in this paper, the resistive element series is taken as an example to propose some fundamental theories about memristor and other elements in this series.

2. Interpretations on memristor fingerprints

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, non-linear resistors such as diodes can be used as memory element, as shown in Fig. 2.

 Figure 2 Diagram of a non-linear resistor. A pinched hysteresis loop can also be generated in dv/dt–di/dt diagram.

The single-value 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 non-linear 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 small-signal 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 small-signal impedance. That is why the hysteresis lobe area of a memristor decreases as frequency increases and shrinks to a single-valued function at infinite frequency, as shown in Fig. 3.

 Figure 3 The change of resistance under a frequency-dependent sinusoidal current excitation. Red curve indicates the working section of the memristor. The resistance becomes constant when the frequency is infinite because the red curve shrinks to a dot.

When a sinusoidal current excitation of i = sinωt is applied, the working point of the memristor is the charge accumulated in the past q0. The charge is $q = {q_0} - \left( {1/\omega } \right){\rm cos}\omega t$ . ω is frequency and qq0 when frequency increases. The resistance varies in a quite small interval $({q_0} - 1/\omega,\,\, {q_0} + 1/\omega )$ , and becomes constant when ω is infinite. So the hysteresis lobe area on iv plane decreases as frequency increases and shrinks to a single-valued function at infinite frequency.

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 electro-magnetic 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.

 Figure 4 (Color online) An imaginary black box and closed loop with current i and voltage v. The charge q can be seen as the charge flows into the black box and the magnetic flux φ can be seen as the flux in the open space to generate the voltage source.

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 electro-magnetic 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 electro-magnetic 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 relation

Memristors 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 iv curve from qφ curve under different excitations, and vice-versa, as shown in Fig. 5.

 Figure 5 (Color online) Generating the pinched hysteresis loop in terms of constitutive relation under different excitations. A sinusoidal current excitation and a square wave current excitation are taken as examples.

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 iv curve is the common one in most papers while under a square wave current excitation, the iv 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 vi, so it differs from the memristor on the respect of electrical property. It is the same for all circuit elements in Chua’s four-element torus that constitutive relation can tell them apart.

The constitutive relation of an element can be divided into odd symmetric relation and non-odd symmetric relation. Odd symmetric relation means that the constitutive relation of the element is odd symmetric function while non-odd symmetric relation means that the constitutive relation is non-odd symmetric function. Take resistive elements series as an example, as shown in Fig. 6. The excitation is harmonic wave in the following examples.

 Figure 6 Odd symmetric relation and non-odd symmetric relation of a resistive element. (a) In odd symmetric relation case, the constitutive relation is not fixed. (b) In non-odd symmetric relation case, the constitutive relation is fixed.

The element in Fig. 6(a) has an odd symmetric constitutive relation. Then we can find that every order diagram $\left( { \cdots\!, \, {\rm d}v/{\rm d}t \text{–} {\rm d}i/{\rm d}t, v \text{–} i, \mathop \smallint \nolimits^ v{\rm d}t \text{–} \mathop \smallint \nolimits^ i{\rm d}t, \cdots } \right)$ is always odd symmetric, which means that every order relation can each be the constitutive relation of this element. This kind of element can be called “unipolar element” and the relationship between attributes that are easiest to measure are adopted as the constitutive relation of this element. The relationship between v and i is adopted in this example and this element is a non-linear resistor. For unipolar capacitive elements, relationship between v and q can be used and for unipolar inductive elements, constitutive relation can be the relationship between φ and i.

By the same token, the elements with non-odd symmetric constitutive relation can be called “bipolar element” as shown in Fig. 6(b). The element with non-odd symmetric φq curve can generate a pinched hysteresis loop in lower order diagram $\left( {v \text{–} i \to {\rm d}v/{\rm d}t \text{–}{\rm d}i/{\rm d}t} \right)$ and a multi-value curve in higher order diagram (viφq, there remains $\mathop \smallint \nolimits^ \varphi {\rm d}t$ after one cycle of q). In this case, the constitutive relation of the element is fixed, and only the relationship between φ and q can be adopted as its constitutive relation.

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 vi curve by oscilloscope, and the shape of the curve is quite varied. If an odd symmetric vi 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 non-odd symmetric vi curve are listed, as shown in Fig. 7.

 Figure 7 Different kinds of non-odd symmetric v–i curve. (a) Single-value function. (b) A pinched hysteresis loop. (c) An open multi-value curve

The element in Fig. 7(a) is a non-linear resistor. The constitutive of the resistor is fixed. The element in Fig. 7(b) shows a pinched hysteresis loop and after generating the higher-order diagram (viφq), a single-value function of φq can be obtained and this element is a memristor with fixed constitutive relation (φ, q). The element in Fig. 7(c) has an open multi-value curve and after generating lower-order diagram (vi→dv/dt–di/dt), a single-value 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. Conclusion

Memristors should have equal status with other circuit elements in Chua’s four-element 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 frequency-dependent property can be achieved. It can also be considered as a resistor with a special vi curve, not the same as a traditional resistor. Specific electronic systems can be designed to implement a memristor, such as Strukov’s TiOx-based memristor. The resistance of the memristor is changed by ion drifting and it is a resistor with state-changeable property essentially.

References
 [1] Chua L O. Memristor—the missing circuit element. IEEE Trans Circuit Theory, 1971, 18(5): 507 doi:10.1109/TCT.1971.1083337 [2] Strukov D, Snider G, Stewart D. The missing memristor found. Nature, 2008, 453: 80 doi:10.1038/nature06932 [3] Ho P W C, Almurib H A F, Kumar T N. Memristive SRAM cell of seven transistors and one memristor. J Semicond, 2016, 37(10): 104002 doi:10.1088/1674-4926/37/10/104002 [4] You Z Q, Hu F, Huang L M. A long lifetime, low error rate RRAM design with self-repair module. J Semicond, 2016, 37(11): 115004 doi:10.1088/1674-4926/37/11/115004 [5] Kokate P P. Memristor-based chaotic circuits. IETE Techn Rev, 2009, 26(6): 417 doi:10.4103/0256-4602.57827 [6] Jo S H, Chang T, Ebong I. Nanoscale memristor device as synapse in neuromorphic dystems. Nano Lett, 2010, 10(4): 1297 doi:10.1021/nl904092h [7] Kim H, Sah M P, Yang C. Neural synaptic weighting with a pulse-based memristor circuit. IEEE Trans Circuits Syst I, 2012, 59-I(1): 148 [8] Shinde S S, Dongle T D. Modelling of nanostructured TiO2-based memristors . J Semicond, 2015, 36(3): 034001 doi:10.1088/1674-4926/36/3/034001 [9] Biolek Z, Biolek D, Biolkova V. SPICE model of memristor with nonlinear dopant drift. Radioengineering, 2009, 18(2): 210 [10] Kumar A, Baghini M S. Experimental study for selection of electrode material for ZnO-based memristors. Electron Lett, 2014, 50(21): 1547 doi:10.1049/el.2014.1491 [11] Ho P W C, Hatem F O, Almurib H A F. Comparison between Pt/TiO2/Pt and Pt/TaOx/TaOy/Pt based bipolar resistive switching devices . J Semicond, 2016, 37(6): 064001 doi:10.1088/1674-4926/37/6/064001 [12] Adhikari S P, Sah M P, Kim H. Three fingerprints of memristor. IEEE Trans Circuits Syst I, 2013, 60(11): 3008 doi:10.1109/TCSI.2013.2256171 [13] Chua L O. Nonlinear circuit foundations for nanodevices I: the four-element torus. Proc IEEE, 2003, 9(11): 1830 doi:10.1109/JPROC.2003.818319 [14] Wang F Z. A triangular periodic table of elementary circuit elements. IEEE Trans Circuits Syst I, 2013, 60(3): 616 doi:10.1109/TCSI.2012.2209734 [15] Kuzum D, Yu S, Wong H S. Synaptic electronics: materials, devices and applications. Nanotechnology, 2013, 24(38): 382001 doi:10.1088/0957-4484/24/38/382001