## Abstract

All 2-terminal non-volatile memory devices based on *resistance switching* are *memristors*, regardless of the device material and physical operating mechanisms. They all exhibit a distinctive “fingerprint” characterized by a *pinched hysteresis loop* confined to the first and the third quadrants of the *v*-*i* plane whose contour shape in general changes with both the amplitude and frequency of any periodic “sine-wave-like” input voltage source, or current source. In particular, the pinched hysteresis loop shrinks and tends to a straight line as frequency increases. Though numerous examples of voltage versus current pinched hysteresis loops have been published in many unrelated fields, such as biology, chemistry, physics, etc., and observed from many unrelated phenomena, such as gas discharge arcs, mercury lamps, power conversion devices, earthquake conductance variations, etc., we restrict our examples in this *tutorial* to solid state and/or nano devices where copious examples of published pinched hysteresis loops abound. In particular, we sampled arbitrarily, one example from each year between the years 2000 and 2010, to demonstrate that the memristor is a device that does not depend on any particular material, or physical mechanism. For example, we have shown that *spin-transfer magnetic tunnel junctions* are examples of memristors. We have also demonstrated that both *bipolar* and *unipolar* resistance switching devices are memristors. The goal of this *tutorial* is to introduce some fundamental circuit-theoretic concepts and properties of the memristor that are relevant to the analysis and design of *non-volatile* nano memories where binary bits are stored as resistances manifested by the memristor’s continuum of equilibrium states. Simple pedagogical examples will be used to illustrate, clarify, and demystify various misconceptions among the uninitiated.

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## Notes

- 1.
A function is

*piecewise-differentiable*if its derivative is uniquely defined everywhere except possibly at isolated points. - 2.
Just as

*memristor*is an acronym for*memory resistor*,*memristance*is an acronym for*memory resistance*. Similarly*memductance*is an acronym for*memory conductance*. - 3.
Also known as a

*Lissajous figure.* - 4.
Note the preceding memristor fingerprint property is stated for the case \(R(q) > 0\).

- 5.
To avoid clutter, we will often write Memristance

*M*(*q*) and Resistance*R*(*q*) interchangeably. Likewise, we will often write Conductance \(G(\varphi )\) for Memductance \(W(\varphi )\). Similarly, we use the terms memristance and resistance, as well as memductance and conductance, to mean the same thing. - 6.
To avoid clutter, we usually write only the term resistance, or conductance, with the understanding,

*mutatis mutandis,*that the same follows for the dual case. - 7.
We henceforth adopt the standard notation

*x*to denote a*state variable*in*mathematical system theory*, where*x*may be a vector \(\mathbf x = ( x_{1},x_{2},\ldots ,x_{n})\). This will be the case for many*non-ideal*memristors found in practice. - 8.
The terminology “chord resistance” had been widely used by neuro-biologists, including Hodgkin and Huxley [3], for similar geometrical interpretations.

- 9.
A state \(x=x_{0}\) is said to be an

*equilibrium point*of a dynamical circuit if \(\frac{dx(t)}{dt} =0\) at \(x = x_{0}\). It is said to be*locally asymptotically stable*if it always returns to its original position whenever subjected to small perturbations, such as a small current pulse. An equilibrium point is said to be*stable*if any drift from its original position due to any perturbation to the state variable*x*is confined to a neighborhood of radius of about the same size as that of the perturbation. In other words, it does not diverge to infinity, as would be the case for an unstable equilibrium point. Neither does it return to its original position, as would be the case if the equilibrium point is asymptotically stable [2]. - 10.
For a strictly-passive memristor, defined by \(R(q) > 0\), there is no mathematical difference between a charge-controlled memristor and a flux-controlled memristor except for the choice of the independent variable. However, for a

*locally-active*memristor, defined by \(R(q) <0\) at some point on the \(\varphi \)-*q*curve, the difference becomes important because the \(\varphi \)-*q*curve in this case is no longer a single-valued function, and therefore does not have an inverse function. - 11.
This memristor is not charged-controlled because its memristance is infinite at all points on the horizontal segment where the memductance \(G_{0}\) is equal to zero.

- 12.
The two memory states are chosen sufficiently far apart in practice to enhance robustness and reliability.

- 13.
Measurement instrument companies could exploit the high market potentials of automated pinched-hysteresis-loop measuring instrumentations, and their memristance extractions.

- 14.
Exactly the same theory of the memristor can be used to identify a memcapacitor (acronym for memory capacitor), and a meminductor (acronym for a memory inductor), from the table of axiomatically-defined circuit elements [2, 6], as depicted in Fig. 11, and presented at the 2008 Berkeley Symposium on Memristors and Memristive Systems (Part 1, towards the end of the opening lecture) on Nov. 21, 2008 (proceedings of this symposium were videotaped and available via YouTube), as well as further elucidated in [13, 14].

## References

Chua, L.O.: Memristor—the missing circuit element. IEEE Trans. Circuit Theory

**CT-18**, 507–519 (1971)Chua, L.: Nonlinear circuit theory. In: Moschytz G.S., Neirynck J. (eds.) Modern Network Theory – An Introduction: Guest Lectures of the 1978 European Conference on Circuit Theory and Design, p. 81. Georgi, St. Saphorin, Switzerland (1978)

Jack, J.J.B., Noble, D., Tsien, R.W.: Electric Current Flow in Excitable Cells. Oxford University Press, Oxford (1975)

Itoh, M., Chua, L.: Memristor oscillators. Int. J. Bifurc. Chaos

**18**, 3183–3206 (2008)Diao, Z., Pakala, M., Panchula, A., Ding, Y., Apalkov, D., Wang, L.-C., Chen, E., Huai, Y.: Spin-transfer switching in MgO-based magnetic tunnel junctions. J. Appl. Phys.

**99**, 086510 (2006)Chua, L.O.: Nonlinear foundations for nanodevices. I. The four-element torus. Proc. IEEE

**91**, 1830–1859 (2003)Duan, X., Huang, Y., Lieber, C.M.: Nonvolatile memory and programmable logic from molecule-gated nanowires. Nano Lett.

**2**, 487–490 (2002)Bruce, J.W., Giblin, P.J.: Functions and Singularities. Cambridge University Press, Cambridge (1999)

Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S.: The missing memristor found. Nature

**45380**, 80–83 (2008)Chua, L.O., Kang, S.M.: Memrisive devices and systems. Proc. IEEE

**64**, 209–223 (1976)Waser, R., Aono, M.: Nanoionics-based resistive switching memories. Nat. Mater.

**6**, 833–840 (2007)Chua, L.: Device modeling via nonlinear circuit elements. IEEE Trans. Circuits Syst.

**CAS-27**, 1014–1044 (1980)Chua, L.O.: Introduction to Memristors. IEEE Expert Now Education Course, IEEE (2010)

Di Ventra, M., Pershin, Y.V., Chua, L.O.: Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc. IEEE

**97**, 1717–1724 (2009)

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Chua, L. (2019). Resistance Switching Memories are Memristors. In: Chua, L., Sirakoulis, G., Adamatzky, A. (eds) Handbook of Memristor Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-76375-0_6

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