Memristor

Not to be confused with Memistor or Memtransistor.

Memristor

Memristor developed by the University of Illinois at Urbana-Champaign and National Energy Technology Laboratory
Invented	Leon Chua (1971)
Electronic symbol

A memristor (/ˈmɛmrɪstər/; a portmanteau of memory resistor) is a non-linear two-terminal electrical component relating electric charge and magnetic flux linkage. It was described and named in 1971 by Leon Chua, completing a theoretical quartet of fundamental electrical components which also comprises the resistor, capacitor and inductor.<ref name="chua71">Chua, L. (1971). "Memristor-The missing circuit element". IEEE Transactions on Circuit Theory. 18 (5): 507–519. CiteSeerX 10.1.1.189.3614. doi:10.1109/TCT.1971.1083337. </ref>

Chua and Kang later generalized the concept to memristive systems.<ref name="chua76">Chua, L. O.; Kang, S. M. (1976-01-01), "Memristive devices and systems", Proceedings of the IEEE, 64 (2): 209–223, doi:10.1109/PROC.1976.10092, S2CID 6008332</ref> Such a system comprises a circuit, of multiple conventional components, which mimics key properties of the ideal memristor component and is also commonly referred to as a memristor. Several such memristor system technologies have been developed, notably ReRAM.

The identification of memristive properties in electronic devices has attracted controversy. Experimentally, the ideal memristor has yet to be demonstrated.<ref name="Pershin_2018"/><ref name="Kim_2019"/>

As a fundamental electrical component

Conceptual symmetries of resistor, capacitor, inductor, and memristor

Chua in his 1971 paper identified a theoretical symmetry between the non-linear resistor (voltage vs. current), non-linear capacitor (voltage vs. charge), and non-linear inductor (magnetic flux linkage vs. current). From this symmetry he inferred the characteristics of a fourth fundamental non-linear circuit element, linking magnetic flux and charge, which he called the memristor. In contrast to a linear (or non-linear) resistor, the memristor has a dynamic relationship between current and voltage, including a memory of past voltages or currents. Other scientists had proposed dynamic memory resistors such as the memistor of Bernard Widrow, but Chua introduced a mathematical generality.

Derivation and characteristics

The memristor was originally defined in terms of a non-linear functional relationship between magnetic flux linkage Φ_m(t) and the amount of electric charge that has flowed, q(t):<ref name="chua71" />

<math>f(\mathrm \Phi_\mathrm m(t),q(t))=0</math>

The magnetic flux linkage, Φ_m, is generalized from the circuit characteristic of an inductor. It does not represent a magnetic field here. Its physical meaning is discussed below. The symbol Φ_m may be regarded as the integral of voltage over time.<ref>Knoepfel, H. (1970), Pulsed high magnetic fields, New York: North-Holland, p. 37, Eq. (2.80)</ref>

In the relationship between Φ_m and q, the derivative of one with respect to the other depends on the value of one or the other, and so each memristor is characterized by its memristance function describing the charge-dependent rate of change of flux with charge.

<math>M(q)=\frac{\mathrm d\Phi_{\rm m}}{\mathrm dq} </math>

Substituting the flux as the time integral of the voltage, and charge as the time integral of current, the more convenient forms are;

<math>M(q(t)) = \cfrac{\mathrm d\Phi_{\rm}/\mathrm dt}{\mathrm dq/\mathrm dt}=\frac{V(t)}{I(t)}</math>

To relate the memristor to the resistor, capacitor, and inductor, it is helpful to isolate the term M(q), which characterizes the device, and write it as a differential equation.

Device	Characteristic property (units)	Differential equation
Resistor (R)	Resistance (V / A, or ohm, Ω)	R = dV / dI
Capacitor (C)	Capacitance (C / V, or farad)	C = dq / dV
Inductor (L)	Inductance (Wb / A, or henry)	L = dΦm / dI
Memristor (M)	Memristance (Wb / C, or ohm)	M = dΦm / dq

The above table covers all meaningful ratios of differentials of I, q, Φ_m, and V. No device can relate dI to dq, or dΦ_m to dV, because I is the derivative of q and Φ_m is the integral of V.

It can be inferred from this that memristance is charge-dependent resistance. If M(q(t)) is a constant, then we obtain Ohm's law R(t) = V(t)/I(t). If M(q(t)) is nontrivial, however, the equation is not equivalent because q(t) and M(q(t)) can vary with time. Solving for voltage as a function of time produces

<math>V(t) =\ M(q(t)) I(t)</math>

This equation reveals that memristance defines a linear relationship between current and voltage, as long as M does not vary with charge. Nonzero current implies time varying charge. Alternating current, however, may reveal the linear dependence in circuit operation by inducing a measurable voltage without net charge movement—as long as the maximum change in q does not cause much change in M.

Furthermore, the memristor is static if no current is applied. If I(t) = 0, we find V(t) = 0 and M(t) is constant. This is the essence of the memory effect.

Analogously, we can define a <math>W(\phi(t))</math> as menductance.<ref name="chua71"/>

<math>i(t)=W(\phi(t))v(t)</math>

The power consumption characteristic recalls that of a resistor, I²R.

<math>P(t) =\ I(t)V(t) =\ I^2(t) M(q(t))</math>

As long as M(q(t)) varies little, such as under alternating current, the memristor will appear as a constant resistor. If M(q(t)) increases rapidly, however, current and power consumption will quickly stop.

M(q) is physically restricted to be positive for all values of q (assuming the device is passive and does not become superconductive at some q). A negative value would mean that it would perpetually supply energy when operated with alternating current.

Modelling and validation

In order to understand the nature of memristor function, some knowledge of fundamental circuit theoretic concepts is useful, starting with the concept of device modeling.<ref name="muthuswamyBanerjee2019">Muthuswamy, Bharathwaj; Banerjee, Santo (2019). Introduction to Nonlinear Circuits and Networks. Springer International. ISBN 978-3-319-67325-7. </ref>

Engineers and scientists seldom analyze a physical system in its original form. Instead, they construct a model which approximates the behaviour of the system. By analyzing the behaviour of the model, they hope to predict the behaviour of the actual system. The primary reason for constructing models is that physical systems are usually too complex to be amenable to a practical analysis.

In the 20th century, work was done on devices where researchers did not recognize the memristive characteristics. This has raised the suggestion that such devices should be recognised as memristors.<ref name="muthuswamyBanerjee2019"/> Pershin and Di Ventra<ref name="Pershin_2018"/> have proposed a test that can help to resolve some of the long-standing controversies about whether an ideal memristor does actually exist or is a purely mathematical concept.

The rest of this article primarily addresses memristors as related to ReRAM devices, since the majority of work since 2008 has been concentrated in this area.

Superconducting memristor component

Dr. Paul Penfield, in a 1974 MIT technical report<ref name="penfield1974">Paul L. Penfield Jr. (1974). "1. Frequency-Power Formulas for Josephson Junctions". V. Microwave and Millimeter Wave Techniques (PDF) (Report). pp. 31–32. QPR No. 113.</ref> mentions the memristor in connection with Josephson junctions. This was an early use of the word "memristor" in the context of a circuit device.

One of the terms in the current through a Josephson junction is of the form:

<math>\begin{align}

            i_M(v) &= \epsilon\cos(\phi_0)v \\

&=W(\phi_0)v \end{align}</math> where <math>\epsilon</math> is a constant based on the physical superconducting materials, <math>v</math> is the voltage across the junction and <math>i_M</math> is the current through the junction.

Through the late 20th century, research regarding this phase-dependent conductance in Josephson junctions was carried out.<ref name="langenberg74">Langenberg, D. N. (1974), "Physical Interpretation of the <math>\cos\phi</math> term and implications for detectors" (PDF), Revue de Physique Appliquée, 9: 35–40, doi:10.1051/rphysap:019740090103500</ref><ref name="pedersen1972">Pedersen, N.F.; et al. (1972), "Magnetic field dependence and Q of the Josephson plasma resonance" (PDF), Physical Review B, 11 (6): 4151–4159, Bibcode:1972PhRvB...6.4151P, doi:10.1103/PhysRevB.6.4151</ref><ref name="pedersen1974">Pedersen, N. F.; Finnegan, T. F.; Langenberg, D. N. (1974). "Evidence for the Existence of the Josephson Quasiparticle-Pair Interference Current". Low Temperature Physics-LT 13. Boston, MA: Springer US. pp. 268–271. doi:10.1007/978-1-4684-2688-5_52. ISBN 978-1-4684-2690-8.</ref><ref name="thompson1973">Thompson, E.D. (1973), "Power flow for Josephson Elements", IEEE Trans. Electron Devices, 20 (8): 680–683, Bibcode:1973ITED...20..680T, doi:10.1109/T-ED.1973.17728</ref> A more comprehensive approach to extracting this phase-dependent conductance appeared with Peotta and DiVentra's seminal paper in 2014.<ref name="peottaDiVentra2014">Peotta, A.; Di Ventra, M. (2014), "Superconducting Memristors", Physical Review Applied, 2 (3): 034011-1–034011-10, arXiv:1311.2975, Bibcode:2014PhRvP...2c4011P, doi:10.1103/PhysRevApplied.2.034011, S2CID 119020953</ref>

Memristor circuits

Due to the practical difficulty of studying the ideal memristor, we will discuss other electrical devices which can be modelled using memristors. For a mathematical description of a memristive device (systems), see Theory.

A discharge tube can be modelled as a memristive device, with resistance being a function of the number of conduction electrons <math>n_e</math>.<ref name="chua76"/>

<math>\begin{align}

v_M&=R(n_e)i_M\\ \frac{dn_e}{dt}&=\beta n+\alpha R(n_e)i^2_M \end{align}</math>

<math>v_M</math> is the voltage across the discharge tube, <math>i_M</math> is the current flowing through it and <math>n_e</math> is the number of conduction electrons. A simple memristance function is <math>R(n_e)=\frac{F}{n_e}</math>. <math>\alpha,\beta</math> and <math>F</math> are parameters depending on the dimensions of the tube and the gas fillings. An experimental identification of memristive behaviour is the "pinched hysteresis loop" in the <math>v-i</math> plane. For an experiment that shows such a characteristic for a common discharge tube, see "A physical memristor Lissajous figure" (YouTube). The video also illustrates how to understand deviations in the pinched hysteresis characteristics of physical memristors.<ref name="muthuswamy2014">Muthuswamy, B.; Jevtic, J.; Iu, H. H. C.; Subramaniam, C. K.; Ganesan, K.; Sankaranarayanan, V.; Sethupathi, K.; Kim, H.; Shah, M. Pd.; Chua, L. O. (2014). "Memristor modelling". 2014 IEEE International Symposium on Circuits and Systems (ISCAS). pp. 490–493. doi:10.1109/ISCAS.2014.6865179. ISBN 978-1-4799-3432-4. S2CID 13061426.</ref><ref name="sah2015">Sah, M.; et al. (2015), "A Generic Model of Memristors with Parasitic Components", IEEE TCAS I: Regular Papers, 62 (3): 891–898</ref>

Thermistors can be modelled as memristive devices.<ref name="sah2015"/>

<math>\begin{align}

v&=R_0(T_0)\exp\left[\beta\left(\frac{1}{T}-\frac{1}{T_0}\right)\right]i \\ &\equiv R(T)i \\ \frac{dT}{dt}&=\frac{1}{C}\left[-\delta\cdot(T-T_0)+R(T)i^2\right] \end{align} </math>

<math>\beta</math> is a material constant, <math>T</math> is the absolute body temperature of the thermistor, <math>T_0</math> is the ambient temperature (both temperatures in Kelvin), <math>R_0(T_0)</math> denotes the cold temperature resistance at <math>T=T_0</math>, <math>C</math> is the heat capacitance and <math>\delta</math> is the dissipation constant for the thermistor.

A fundamental phenomenon that has hardly been studied is memristive behaviour in pn-junctions.<ref name="chuaTseng74">Chua, L. O.; Tseng, C. (1974), "A memristive circuit model for p-n junction diodes", International Journal of Circuit Theory and Applications, 2 (4): 367–389, doi:10.1002/cta.4490020406</ref> The memristor plays a crucial role in mimicking the charge storage effect in the diode base, and is also responsible for the conductivity modulation phenomenon (that is so important during forward transients).

Criticisms

In 2008, a team at HP Labs found experimental evidence for the Chua's memristor based on an analysis of a thin film of titanium dioxide, thus connecting the operation of ReRAM devices to the memristor concept. According to HP Labs, the memristor would operate in the following way: the memristor's electrical resistance is not constant but depends on the current that had previously flowed through the device, i.e., its present resistance depends on how much electric charge has previously flowed through it and in what direction; the device remembers its history—the so-called non-volatility property.<ref name="chua11"/> When the electric power supply is turned off, the memristor remembers its most recent resistance until it is turned on again.<ref name="Williams08">Strukov, Dmitri B.; Snider, Gregory S.; Stewart, Duncan R.; Williams, R. Stanley (2008). "The missing memristor found" (PDF). Nature. 453 (7191): 80–83. Bibcode:2008Natur.453...80S. doi:10.1038/nature06932. PMID 18451858. S2CID 4367148.</ref><ref>Memristor FAQ, Hewlett-Packard, retrieved 2010-09-03</ref>

The HP Labs result was published in the scientific journal Nature.<ref name="Williams08"/><ref name="Williams - Spectrum 2008-12"> Williams, R. S. (2008). "How We Found The Missing Memristor" (PDF). IEEE Spectrum. 45 (12): 28–35. doi:10.1109/MSPEC.2008.4687366. S2CID 27319894.</ref> Following this claim, Leon Chua has argued that the memristor definition could be generalized to cover all forms of two-terminal non-volatile memory devices based on resistance switching effects.<ref name="chua11">Chua, Leon (2011-01-28). "Resistance switching memories are memristors". Applied Physics A. 102 (4): 765–783. Bibcode:2011ApPhA.102..765C. doi:10.1007/s00339-011-6264-9.</ref> Chua also argued that the memristor is the oldest known circuit element, with its effects predating the resistor, capacitor, and inductor.<ref name="Memristor200yearsold">Clarke, P. (2012-05-23), "Memristor is 200 years old, say academics", EE Times, retrieved 2012-05-25</ref> There are, however, some serious doubts as to whether a genuine memristor can actually exist in physical reality.<ref name="Meuffels_2012"/><ref name="DiVentra_2013"/><ref name="Sundqvist_2017">Sundqvist, Kyle M.; Ferry, David K.; Kish, Laszlo B. (2017-11-21). "Memristor Equations: Incomplete Physics and Undefined Passivity/Activity". Fluctuation and Noise Letters. 16 (4): 1771001–519. arXiv:1703.09064. Bibcode:2017FNL....1671001S. doi:10.1142/S0219477517710018. S2CID 1408810.</ref><ref>Abraham, Isaac (2018-07-20). "The case for rejecting the memristor as a fundamental circuit element". Scientific Reports. 8 (1): 10972. Bibcode:2018NatSR...810972A. doi:10.1038/s41598-018-29394-7. PMC 6054652. PMID 30030498.</ref> Additionally, some experimental evidence contradicts Chua's generalization since a non-passive nanobattery effect is observable in resistance switching memory.<ref name="memristor_nanobattery">Valov, I.; et al. (2013), "Nanobatteries in redox-based resistive switches require extension of memristor theory", Nature Communications, 4 (4): 1771, arXiv:1303.2589, Bibcode:2013NatCo...4.1771V, doi:10.1038/ncomms2784, PMC 3644102, PMID 23612312</ref> A simple test has been proposed by Pershin and Di Ventra<ref name="Pershin_2018">Pershin, Y. V.; Di Ventra, M. (2019). "A simple test for ideal memristors". Journal of Physics D: Applied Physics. 52 (1): 01LT01. arXiv:1806.07360. Bibcode:2019JPhD...52aLT01P. doi:10.1088/1361-6463/aae680. S2CID 53506924.</ref> to analyze whether such an ideal or generic memristor does actually exist or is a purely mathematical concept. Up to now, there seems to be no experimental resistance switching device (ReRAM) which can pass the test.<ref name="Pershin_2018"/><ref name="Kim_2019">Kim, J.; Pershin, Y. V.; Yin, M.; Datta, T.; Di Ventra, M. (2019). "An experimental proof that resistance-switching memories are not memristors". Advanced Electronic Materials. arXiv:1909.07238. doi:10.1002/aelm.202000010. S2CID 202577242.</ref>

These devices are intended for applications in nanoelectronic memory devices, computer logic, and neuromorphic/neuromemristive computer architectures.<ref>Marks, P. (2008-04-30), "Engineers find 'missing link' of electronics", New Scientist, retrieved 2008-04-30</ref><ref>Zidan, Mohammed A.; Strachan, John Paul; Lu, Wei D. (2018-01-08). "The future of electronics based on memristive systems". Nature Electronics. 1 (1): 22–29. doi:10.1038/s41928-017-0006-8. S2CID 187510377.</ref> In 2013, Hewlett-Packard CTO Martin Fink suggested that memristor memory may become commercially available as early as 2018.<ref name="memristor2018">HP 100TB Memristor drives by 2018 – if you're lucky, admits tech titan, 2013-11-01</ref> In March 2012, a team of researchers from HRL Laboratories and the University of Michigan announced the first functioning memristor array built on a CMOS chip.<ref name="HRLmemristor">Artificial synapses could lead to advanced computer memory and machines that mimic biological brains, HRL Laboratories, 2012-03-23, retrieved 2012-03-30</ref>

File:Memristor.jpg

An array of 17 purpose-built oxygen-depleted titanium dioxide memristors built at HP Labs, imaged by an atomic force microscope. The wires are about 50 nm, or 150 atoms, wide.<ref>Bush, S. (2008-05-02), "HP nano device implements memristor", Electronics Weekly</ref> Electric current through the memristors shifts the oxygen vacancies, causing a gradual and persistent change in electrical resistance.<ref name="Kanellos"/>

According to the original 1971 definition, the memristor is the fourth fundamental circuit element, forming a non-linear relationship between electric charge and magnetic flux linkage. In 2011, Chua argued for a broader definition that includes all two-terminal non-volatile memory devices based on resistance switching.<ref name="chua11" /> Williams argued that MRAM, phase-change memory and ReRAM are memristor technologies.<ref name="Mellor2011">Mellor, C. (2011-10-10), "HP and Hynix to produce the memristor goods by 2013", The Register, retrieved 2012-03-07</ref> Some researchers argued that biological structures such as blood<ref name="Courtland2011">Courtland, R. (2011-04-01). "Memristors...Made of Blood?". IEEE Spectrum. Retrieved 2012-03-07.</ref> and skin<ref name="G K Johnsen et al">Johnsen, G. K. (2011-03-24). "Memristive model of electro-osmosis in skin". Physical Review E. 83 (3): 031916. Bibcode:2011PhRvE..83c1916J. doi:10.1103/PhysRevE.83.031916. PMID 21517534. S2CID 46437206.</ref><ref name="McAlpine2011">McAlpine, K. (2011-03-02), "Sweat ducts make skin a memristor", New Scientist, 209 (2802): 16, Bibcode:2011NewSc.209...16M, doi:10.1016/S0262-4079(11)60481-8, retrieved 2012-03-07</ref> fit the definition. Others argued that the memory device under development by HP Labs and other forms of ReRAM are not memristors, but rather part of a broader class of variable-resistance systems,<ref name="Clarke2012">Clarke, P. (2012-01-16), "Memristor brouhaha bubbles under", EETimes, retrieved 2012-03-02</ref> and that a broader definition of memristor is a scientifically unjustifiable land grab that favored HP's memristor patents.<ref name="PaulMarks2012">Marks, P. (2012-02-23), "Online spat over who joins memristor club", New Scientist, retrieved 2012-03-19</ref>

In 2011, Meuffels and Schroeder noted that one of the early memristor papers included a mistaken assumption regarding ionic conduction.<ref name="Meuffels_2011"> Meuffels, P.; Schroeder, H. (2011), "Comment on "Exponential ionic drift: fast switching and low volatility of thin-film memristors" by D. B. Strukov and R. S. Williams in Appl. Phys. A (2009) 94: 515–519", Applied Physics A, 105 (1): 65–67, Bibcode:2011ApPhA.105...65M, doi:10.1007/s00339-011-6578-7, S2CID 95168959</ref> In 2012, Meuffels and Soni discussed some fundamental issues and problems in the realization of memristors.<ref name="Meuffels_2012">Meuffels, P.; Soni, R. (2012). "Fundamental Issues and Problems in the Realization of Memristors". arXiv:1207.7319 [cond-mat.mes-hall].</ref> They indicated inadequacies in the electrochemical modeling presented in the Nature article "The missing memristor found"<ref name="Williams08"/> because the impact of concentration polarization effects on the behavior of metal−TiO_2−x−metal structures under voltage or current stress was not considered. This critique was referred to by Valov et al.<ref name="memristor_nanobattery"/> in 2013.

In a kind of thought experiment, Meuffels and Soni<ref name="Meuffels_2012"/> furthermore revealed a severe inconsistency: If a current-controlled memristor with the so-called non-volatility property<ref name="chua11"/> exists in physical reality, its behavior would violate Landauer's principle, which places a limit on the minimum amount of energy required to change "information" states of a system. This critique was finally adopted by Di Ventra and Pershin<ref name="DiVentra_2013">Di Ventra, M.; Pershin, Y. V. (2013), "On the physical properties of memristive, memcapacitive and meminductive systems", Nanotechnology, 24 (25): 255201, arXiv:1302.7063, Bibcode:2013Nanot..24y5201D, CiteSeerX 10.1.1.745.8657, doi:10.1088/0957-4484/24/25/255201, PMID 23708238, S2CID 14892809</ref> in 2013.

Within this context, Meuffels and Soni<ref name="Meuffels_2012"/> pointed to a fundamental thermodynamic principle: Non-volatile information storage requires the existence of free-energy barriers that separate the distinct internal memory states of a system from each other; otherwise, one would be faced with an "indifferent" situation, and the system would arbitrarily fluctuate from one memory state to another just under the influence of thermal fluctuations. When unprotected against thermal fluctuations, the internal memory states exhibit some diffusive dynamics, which causes state degradation.<ref name="DiVentra_2013"/> The free-energy barriers must therefore be high enough to ensure a low bit-error probability of bit operation.<ref name="Kish_2014"> Kish, Laszlo B.; Granqvist, Claes G.; Khatri, Sunil P.; Wen, He (2014). "Demons: Maxwell's demon, Szilard's engine and Landauer's erasure–dissipation". International Journal of Modern Physics: Conference Series. 33: 1460364. arXiv:1412.2166. Bibcode:2014IJMPS..3360364K. doi:10.1142/s2010194514603640. S2CID 44851287. </ref> Consequently, there is always a lower limit of energy requirement – depending on the required bit-error probability – for intentionally changing a bit value in any memory device.<ref name="Kish_2014"/><ref name="Kish_2015">Kish, L. B.; Khatri, S. P.; Granqvist, C. G.; Smulko, J. M. (2015). "Critical remarks on Landauer's principle of erasure-dissipation: Including notes on Maxwell demons and Szilard engines". 2015 International Conference on Noise and Fluctuations (ICNF). pp. 1–4. doi:10.1109/ICNF.2015.7288632. ISBN 978-1-4673-8335-6. </ref>

In the general concept of memristive system the defining equations are (see Theory):

<math>\begin{align}

             y(t) &= g(\mathbf{x},u,t) u(t), \\
 \dot{\mathbf{x}} &= f(\mathbf{x},u,t),

\end{align}</math> where u(t) is an input signal, and y(t) is an output signal. The vector x represents a set of n state variables describing the different internal memory states of the device. ẋ is the time-dependent rate of change of the state vector x with time.

When one wants to go beyond mere curve fitting and aims at a real physical modeling of non-volatile memory elements, e.g., resistive random-access memory devices, one has to keep an eye on the aforementioned physical correlations. To check the adequacy of the proposed model and its resulting state equations, the input signal u(t) can be superposed with a stochastic term ξ(t), which takes into account the existence of inevitable thermal fluctuations. The dynamic state equation in its general form then finally reads:

<math>\dot{\mathbf{x}} = f(\mathbf{x}, u(t) + \xi(t), t),</math>

where ξ(t) is, e.g., white Gaussian current or voltage noise. On base of an analytical or numerical analysis of the time-dependent response of the system towards noise, a decision on the physical validity of the modeling approach can be made, e.g., would the system be able to retain its memory states in power-off mode?

Such an analysis was performed by Di Ventra and Pershin<ref name="DiVentra_2013"/> with regard to the genuine current-controlled memristor. As the proposed dynamic state equation provides no physical mechanism enabling such a memristor to cope with inevitable thermal fluctuations, a current-controlled memristor would erratically change its state in course of time just under the influence of current noise.<ref name="DiVentra_2013"/><ref name="Slipko_2013">Slipko, V. A.; Pershin, Y. V.; Di Ventra, M. (2013), "Changing the state of a memristive system with white noise", Physical Review E, 87 (1): 042103, arXiv:1209.4103, Bibcode:2013PhRvE..87a2103L, doi:10.1103/PhysRevE.87.012103, PMID 23410279, S2CID 2237458</ref> Di Ventra and Pershin<ref name="DiVentra_2013"/> thus concluded that memristors whose resistance (memory) states depend solely on the current or voltage history would be unable to protect their memory states against unavoidable Johnson–Nyquist noise and permanently suffer from information loss, a so-called "stochastic catastrophe". A current-controlled memristor can thus not exist as a solid-state device in physical reality.

The above-mentioned thermodynamic principle furthermore implies that the operation of two-terminal non-volatile memory devices (e.g. "resistance-switching" memory devices (ReRAM)) cannot be associated with the memristor concept, i.e., such devices cannot by itself remember their current or voltage history. Transitions between distinct internal memory or resistance states are of probabilistic nature. The probability for a transition from state {i} to state {j} depends on the height of the free-energy barrier between both states. The transition probability can thus be influenced by suitably driving the memory device, i.e., by "lowering" the free-energy barrier for the transition {i} → {j} by means of, for example, an externally applied bias.

A "resistance switching" event can simply be enforced by setting the external bias to a value above a certain threshold value. This is the trivial case, i.e., the free-energy barrier for the transition {i} → {j} is reduced to zero. In case one applies biases below the threshold value, there is still a finite probability that the device will switch in course of time (triggered by a random thermal fluctuation), but – as one is dealing with probabilistic processes – it is impossible to predict when the switching event will occur. That is the basic reason for the stochastic nature of all observed resistance-switching (ReRAM) processes. If the free-energy barriers are not high enough, the memory device can even switch without having to do anything.

When a two-terminal non-volatile memory device is found to be in a distinct resistance state {j}, there exists therefore no physical one-to-one relationship between its present state and its foregoing voltage history. The switching behavior of individual non-volatile memory devices thus cannot be described within the mathematical framework proposed for memristor/memristive systems.

An extra thermodynamic curiosity arises from the definition that memristors/memristive devices should energetically act like resistors. The instantaneous electrical power entering such a device is completely dissipated as Joule heat to the surrounding, so no extra energy remains in the system after it has been brought from one resistance state x_i to another one x_j. Thus, the internal energy of the memristor device in state x_i, U(V, T, x_i), would be the same as in state x_j, U(V, T, x_j), even though these different states would give rise to different device's resistances, which itself must be caused by physical alterations of the device's material.

Other researchers noted that memristor models based on the assumption of linear ionic drift do not account for asymmetry between set time (high-to-low resistance switching) and reset time (low-to-high resistance switching) and do not provide ionic mobility values consistent with experimental data. Non-linear ionic-drift models have been proposed to compensate for this deficiency.<ref name="Mitre_2012">Hashem, N.; Das, S. (2012), "Switching-time analysis of binary-oxide memristors via a non-linear model" (PDF), Applied Physics Letters, 100 (26): 262106, Bibcode:2012ApPhL.100z2106H, doi:10.1063/1.4726421, retrieved 2012-08-09</ref>

A 2014 article from researchers of ReRAM concluded that Strukov's (HP's) initial/basic memristor modeling equations do not reflect the actual device physics well, whereas subsequent (physics-based) models such as Pickett's model or Menzel's ECM model (Menzel is a co-author of that article) have adequate predictability, but are computationally prohibitive. As of 2014, the search continues for a model that balances these issues; the article identifies Chang's and Yakopcic's models as potentially good compromises.<ref name="E. Linn et al">Linn, E.; Siemon, A.; Waser, R.; Menzel, S. (2014-03-23). "Applicability of Well-Established Memristive Models for Simulations of Resistive Switching Devices". IEEE Transactions on Circuits and Systems I: Regular Papers. 61 (8): 2402–2410. arXiv:1403.5801. Bibcode:2014arXiv1403.5801L. doi:10.1109/TCSI.2014.2332261. S2CID 18673562.</ref>

Martin Reynolds, an electrical engineering analyst with research outfit Gartner, commented that while HP was being sloppy in calling their device a memristor, critics were being pedantic in saying that it was not a memristor.<ref name="Martin Reynolds">Garling, C. (2012-07-25), "Wonks question HP's claim to computer-memory missing link", Wired.com, retrieved 2012-09-23</ref>

Experimental tests

Chua suggested experimental tests to determine if a device may properly be categorized as a memristor:<ref name="chua76"/>

• The Lissajous curve in the voltage–current plane is a pinched hysteresis loop when driven by any bipolar periodic voltage or current without respect to initial conditions.
• The area of each lobe of the pinched hysteresis loop shrinks as the frequency of the forcing signal increases.
• As the frequency tends to infinity, the hysteresis loop degenerates to a straight line through the origin, whose slope depends on the amplitude and shape of the forcing signal.

According to Chua<ref name="MemristorExperimentalTests">Chua, L. (2012-06-13), Memristors: Past, Present and future (PDF), archived from the original (PDF) on 2014-03-08, retrieved 2013-01-12</ref><ref name="MemristorExperimentalTests2">Adhikari, S. P.; Sah, M. P.; Hyongsuk, K.; Chua, L. O. (2013), "Three Fingerprints of Memristor", IEEE Transactions on Circuits and Systems I, 60 (11): 3008–3021, doi:10.1109/TCSI.2013.2256171, S2CID 12665998</ref> all resistive switching memories including ReRAM, MRAM and phase-change memory meet these criteria and are memristors. However, the lack of data for the Lissajous curves over a range of initial conditions or over a range of frequencies complicates assessments of this claim.

Experimental evidence shows that redox-based resistance memory (ReRAM) includes a nanobattery effect that is contrary to Chua's memristor model. This indicates that the memristor theory needs to be extended or corrected to enable accurate ReRAM modeling.<ref name="memristor_nanobattery"/>

Theory

In 2008, researchers from HP Labs introduced a model for a memristance function based on thin films of titanium dioxide.<ref name="Williams08" /> For R_ON ≪ R_OFF the memristance function was determined to be

<math>M(q(t)) = R_\mathrm{OFF} \cdot \left(1-\frac{\mu_{v}R_\mathrm{ON}}{D^2} q(t)\right)</math>

where R_OFF represents the high resistance state, R_ON represents the low resistance state, μ_v represents the mobility of dopants in the thin film, and D represents the film thickness. The HP Labs group noted that "window functions" were necessary to compensate for differences between experimental measurements and their memristor model due to non-linear ionic drift and boundary effects.

Operation as a switch

For some memristors, applied current or voltage causes substantial change in resistance. Such devices may be characterized as switches by investigating the time and energy that must be spent to achieve a desired change in resistance. This assumes that the applied voltage remains constant. Solving for energy dissipation during a single switching event reveals that for a memristor to switch from R_on to R_off in time T_on to T_off, the charge must change by ΔQ = Q_on−Q_off.

<math>E_{\mathrm{switch}}

=\ V^2\int_{T_\mathrm{off}}^{T_\mathrm{on}} \frac{\mathrm dt}{M(q(t))} =\ V^2\int_{Q_\mathrm{off}}^{Q_\mathrm{on}}\frac{\mathrm dq}{I(q)M(q)} =\ V^2\int_{Q_\mathrm{off}}^{Q_\mathrm{on}}\frac{\mathrm dq}{V(q)} =\ V\Delta Q </math>

Substituting V = I(q)M(q), and then ∫dq/V = ∆Q/V for constant VTo produces the final expression. This power characteristic differs fundamentally from that of a metal oxide semiconductor transistor, which is capacitor-based. Unlike the transistor, the final state of the memristor in terms of charge does not depend on bias voltage.

The type of memristor described by Williams ceases to be ideal after switching over its entire resistance range, creating hysteresis, also called the "hard-switching regime".<ref name="Williams08"/> Another kind of switch would have a cyclic M(q) so that each off-on event would be followed by an on-off event under constant bias. Such a device would act as a memristor under all conditions, but would be less practical.

Memristive systems

In the more general concept of an n-th order memristive system the defining equations are

<math>\begin{align}

             y(t) &= g(\textbf{x},u,t)u(t), \\
 \dot{\textbf{x}} &= f(\textbf{x},u,t)

\end{align}</math>

where u(t) is an input signal, y(t) is an output signal, the vector x represents a set of n state variables describing the device, and g and f are continuous functions. For a current-controlled memristive system the signal u(t) represents the current signal i(t) and the signal y(t) represents the voltage signal v(t). For a voltage-controlled memristive system the signal u(t) represents the voltage signal v(t) and the signal y(t) represents the current signal i(t).

The pure memristor is a particular case of these equations, namely when x depends only on charge (x = q) and since the charge is related to the current via the time derivative dq/dt = i(t). Thus for pure memristors f (i.e. the rate of change of the state) must be equal or proportional to the current i(t) .

Pinched hysteresis

Example of pinched hysteresis curve, V versus I

One of the resulting properties of memristors and memristive systems is the existence of a pinched hysteresis effect.<ref name="DiVentraPershin2011"> Pershin, Y. V.; Di Ventra, M. (2011), "Memory effects in complex materials and nanoscale systems", Advances in Physics, 60 (2): 145–227, arXiv:1011.3053, Bibcode:2011AdPhy..60..145P, doi:10.1080/00018732.2010.544961, S2CID 119098973</ref> For a current-controlled memristive system, the input u(t) is the current i(t), the output y(t) is the voltage v(t), and the slope of the curve represents the electrical resistance. The change in slope of the pinched hysteresis curves demonstrates switching between different resistance states which is a phenomenon central to ReRAM and other forms of two-terminal resistance memory. At high frequencies, memristive theory predicts the pinched hysteresis effect will degenerate, resulting in a straight line representative of a linear resistor. It has been proven that some types of non-crossing pinched hysteresis curves (denoted Type-II) cannot be described by memristors.<ref name="Biolek2011"> Biolek, D.; Biolek, Z.; Biolkova, V. (2011), "Pinched hysteresis loops of ideal memristors, memcapacitors and meminductors must be 'self-crossing'", Electronics Letters, 47 (25): 1385–1387, Bibcode:2011ElL....47.1385B, doi:10.1049/el.2011.2913</ref>

Memristive networks and mathematical models of circuit interactions

The concept of memristive networks was first introduced by Leon Chua in his 1965 paper "Memristive Devices and Systems." Chua proposed the use of memristive devices as a means of building artificial neural networks that could simulate the behavior of the human brain. In fact, memristive devices in circuits have complex interactions due to Kirchhoff's laws. A memristive network is a type of artificial neural network that is based on memristive devices, which are electronic components that exhibit the property of memristance. In a memristive network, the memristive devices are used to simulate the behavior of neurons and synapses in the human brain. The network consists of layers of memristive devices, each of which is connected to other layers through a set of weights. These weights are adjusted during the training process, allowing the network to learn and adapt to new input data. One advantage of memristive networks is that they can be implemented using relatively simple and inexpensive hardware, making them an attractive option for developing low-cost artificial intelligence systems. They also have the potential to be more energy efficient than traditional artificial neural networks, as they can store and process information using less power. However, the field of memristive networks is still in the early stages of development, and more research is needed to fully understand their capabilities and limitations. For the simplest model with only memristive devices with voltage generators in series, there is an exact and in closed form equation (Caravelli-Traversa-Di Ventra equation, CTD)<ref>Caravelli; et al. (2017). "The complex dynamics of memristive circuits: analytical results and universal slow relaxation". Physical Review E. 95 (2): 022140. arXiv:1608.08651. Bibcode:2017PhRvE..95b2140C. doi:10.1103/PhysRevE.95.022140. PMID 28297937. S2CID 6758362.</ref> which describes the evolution of the internal memory of the network for each device. For a simple memristor model (but not realistic) of a switch between two resistance values, given by the Williams-Strukov model <math>R(x)=R_{off} (1-x)+R_{on} x</math>, with <math>dx/dt=I/\beta-\alpha x</math>, there is a set of nonlinearly coupled differential equations that takes the form:

<math> \frac{d\vec{x}}{dt} = -\alpha \vec{x}+\frac{1}{\beta} (I-\chi \Omega X)^{-1} \Omega \vec S </math>

where <math>X</math> is the diagonal matrix with elements <math>x_i</math> on the diagonal, <math>\alpha,\beta,\chi</math> are based on the memristors physical parameters. The vector <math>\vec S </math> is the vector of voltage generators in series to the memristors. The circuit topology enters only in the projector operator <math>\Omega^2=\Omega</math>, defined in terms of the cycle matrix of the graph. The equation provides a concise mathematical description of the interactions due to Kirchhoff 's laws. Interestingly, the equation shares many properties in common with a Hopfield network, such as the existence of Lyapunov functions and classical tunnelling phenomena.<ref>Caravelli; et al. (2021). "Global minimization via classical tunnelling assisted by collective force field formation". Science Advances. 7 (52): 022140. arXiv:1608.08651. Bibcode:2021SciA....7.1542C. doi:10.1126/sciadv.abh1542. PMID 28297937. S2CID 231847346.</ref> In the context of memristive networks, the CTD equation may be used to predict the behavior of memristive devices under different operating conditions, or to design and optimize memristive circuits for specific applications.

Extended systems

Some researchers have raised the question of the scientific legitimacy of HP's memristor models in explaining the behavior of ReRAM.<ref name="Clarke2012" /><ref name="PaulMarks2012" /> and have suggested extended memristive models to remedy perceived deficiencies.<ref name="memristor_nanobattery"/>

One example<ref name="memresistor"> Mouttet, B. (2012). "Memresistors and non-memristive zero-crossing hysteresis curves". arXiv:1201.2626 [cond-mat.mes-hall].</ref> attempts to extend the memristive systems framework by including dynamic systems incorporating higher-order derivatives of the input signal u(t) as a series expansion

<math>\begin{align}

             y(t) &= g_0(\textbf{x}, u)u(t) + g_1(\textbf{x}, u){\operatorname{d}^2u\over\operatorname{d}t^2} +
                       g_2(\textbf{x}, u){\operatorname{d}^4u\over\operatorname{d}t^4} + \ldots +
                       g_m(\textbf{x}, u){\operatorname{d}^{2m}u\over\operatorname{d}t^{2m}}, \\
 \dot{\textbf{x}} &= f(\textbf{x}, u)

\end{align}</math>

where m is a positive integer, u(t) is an input signal, y(t) is an output signal, the vector x represents a set of n state variables describing the device, and the functions g and f are continuous functions. This equation produces the same zero-crossing hysteresis curves as memristive systems but with a different frequency response than that predicted by memristive systems.

Another example suggests including an offset value <math>a</math> to account for an observed nanobattery effect which violates the predicted zero-crossing pinched hysteresis effect.<ref name="memristor_nanobattery"/>

<math>\begin{align}

             y(t) &= g_0(\textbf{x},u)(u(t)-a), \\
 \dot{\textbf{x}} &= f(\textbf{x},u)

\end{align}</math>

Implementations

Titanium dioxide memristor

Interest in the memristor revived when an experimental solid-state version was reported by R. Stanley Williams of Hewlett Packard in 2007.<ref> Fildes, J. (2007-11-13), Getting More from Moore's Law, BBC News, retrieved 2008-04-30</ref><ref> Taylor, A. G. (2007), "Nanotechnology in the Northwest" (PDF), Bulletin for Electrical and Electronic Engineers of Oregon, 51 (1): 1</ref><ref>Stanley Williams, HP Labs, archived from the original on 2011-07-19, retrieved 2011-03-20</ref> The article was the first to demonstrate that a solid-state device could have the characteristics of a memristor based on the behavior of nanoscale thin films. The device neither uses magnetic flux as the theoretical memristor suggested, nor stores charge as a capacitor does, but instead achieves a resistance dependent on the history of current.

Although not cited in HP's initial reports on their TiO₂ memristor, the resistance switching characteristics of titanium dioxide were originally described in the 1960s.<ref name="Argall1968"> Argall, F. (1968), "Switching Phenomena in Titanium Oxide Thin Films", Solid-State Electronics, 11 (5): 535–541, Bibcode:1968SSEle..11..535A, doi:10.1016/0038-1101(68)90092-0</ref>

The HP device is composed of a thin (50 nm) titanium dioxide film between two 5 nm thick electrodes, one titanium, the other platinum. Initially, there are two layers to the titanium dioxide film, one of which has a slight depletion of oxygen atoms. The oxygen vacancies act as charge carriers, meaning that the depleted layer has a much lower resistance than the non-depleted layer. When an electric field is applied, the oxygen vacancies drift (see Fast ion conductor), changing the boundary between the high-resistance and low-resistance layers. Thus the resistance of the film as a whole is dependent on how much charge has been passed through it in a particular direction, which is reversible by changing the direction of current.<ref name="Williams08"/> Since the HP device displays fast ion conduction at nanoscale, it is considered a nanoionic device.<ref name="Terabe2007"> Terabe, K.; Hasegawa, T.; Liang, C.; Aono, M. (2007), "Control of local ion transport to create unique functional nanodevices based on ionic conductors", Science and Technology of Advanced Materials, 8 (6): 536–542, Bibcode:2007STAdM...8..536T, doi:10.1016/j.stam.2007.08.002</ref>

Memristance is displayed only when both the doped layer and depleted layer contribute to resistance. When enough charge has passed through the memristor that the ions can no longer move, the device enters hysteresis. It ceases to integrate q=∫I dt, but rather keeps q at an upper bound and M fixed, thus acting as a constant resistor until current is reversed.

Memory applications of thin-film oxides had been an area of active investigation for some time. IBM published an article in 2000 regarding structures similar to that described by Williams.<ref name="Beck2000"> Beck, A.; et al. (2000), "Reproducible switching effect in thin oxide films for memory applications", Applied Physics Letters, 77 (1): 139, Bibcode:2000ApPhL..77..139B, doi:10.1063/1.126902</ref> Samsung has a U.S. patent for oxide-vacancy based switches similar to that described by Williams.<ref>Stefanovich, Genrikh; Cho, Choong-rae; Yoo, In-kyeong; Lee, Eun-hong; Cho, Sung-il; Moon, Chang-wook (2006) "Electrode structure having at least two oxide layers and non-volatile memory device having the same" U.S. patent 7,417,271</ref>

In April 2010, HP labs announced that they had practical memristors working at 1 ns (~1 GHz) switching times and 3 nm by 3 nm sizes,<ref> Finding the Missing Memristor - R. Stanley Williams</ref> which bodes well for the future of the technology.<ref> Markoff, J. (2010-04-07), "H.P. Sees a Revolution in Memory Chip", New York Times</ref> At these densities it could easily rival the current sub-25 nm flash memory technology.

Silicon dioxide memristor

It seems that memristance has been reported in nanoscale thin films of silicon dioxide as early as the 1960s .<ref>Kavehei, O.; Iqbal, A.; Kim, Y.S.; Eshraghian, K.; Al-Sarawi, S. F.; Abbott, D. (2010). "The fourth element: characteristics, modelling and electromagnetic theory of the memristor". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 466 (2120): 2175–2202. arXiv:1002.3210. Bibcode:2010RSPSA.466.2175K. doi:10.1098/rspa.2009.0553. S2CID 7625839.</ref>

However, hysteretic conductance in silicon was associated to memristive effects only in 2009. <ref>Ben-Jamaa, M. H.; Carrara, S.; Georgiou, J.; Archontas, N.; De Micheli, G. (2009), "Fabrication of memristors with poly-crystalline silicon nanowires", Proceedings of 9th IEEE Conference on Nanotechnology, 1 (1): 152–154</ref> More recently, beginning in 2012, Tony Kenyon, Adnan Mehonic and their group clearly demonstrated that the resistive switching in silicon oxide thin films is due to the formation of oxygen vacancy filaments in defect-engineered silicon dioxide, having probed directly the movement of oxygen under electrical bias, and imaged the resultant conductive filaments using conductive atomic force microscopy. <ref>Mehonic, A.; Cueff, S.; Wojdak, M. , …; Kenyon, A. J. (2012). "Resistive switching in silicon suboxide films" (PDF). Journal of Applied Physics. 111 (7): 074507–074507–9. Bibcode:2012JAP...111g4507M. doi:10.1063/1.3701581.{{cite journal}}: CS1 maint: multiple names: authors list (link)</ref>

Polymeric memristor

In 2004, Krieger and Spitzer described dynamic doping of polymer and inorganic dielectric-like materials that improved the switching characteristics and retention required to create functioning nonvolatile memory cells.<ref> Krieger, J. H.; Spitzer, S. M. (2004), "Non-traditional, Non-volatile Memory Based on Switching and Retention Phenomena in Polymeric Thin Films", Proceedings of the 2004 Non-Volatile Memory Technology Symposium, IEEE, p. 121, doi:10.1109/NVMT.2004.1380823, ISBN 978-0-7803-8726-3, S2CID 7189710</ref> They used a passive layer between electrode and active thin films, which enhanced the extraction of ions from the electrode. It is possible to use fast ion conductor as this passive layer, which allows a significant reduction of the ionic extraction field.

In July 2008, Erokhin and Fontana claimed to have developed a polymeric memristor before the more recently announced titanium dioxide memristor.<ref name="Erokhin2008"> Erokhin, V.; Fontana, M. P. (2008). "Electrochemically controlled polymeric device: A memristor (and more) found two years ago". arXiv:0807.0333 [cond-mat.soft].</ref>

In 2010, Alibart, Gamrat, Vuillaume et al.<ref>An; Alibart, F.; Pleutin, S.; Guerin, D.; Novembre, C.; Lenfant, S.; Lmimouni, K.; Gamrat, C.; Vuillaume, D. (2010). "An Organic Nanoparticle Transistor Behaving as a Biological Spiking Synapse". Advanced Functional Materials. 20 (2): 330–337. arXiv:0907.2540. doi:10.1002/adfm.200901335. S2CID 16335153.</ref> introduced a new hybrid organic/nanoparticle device (the NOMFET : Nanoparticle Organic Memory Field Effect Transistor), which behaves as a memristor<ref>Alibart, F.; Pleutin, S.; Bichler, O.; Gamrat, C.; Serrano-Gotarredona, T.; Linares-Barranco, B.; Vuillaume, D. (2012). "A Memristive Nanoparticle/Organic Hybrid Synapstor for Neuroinspired Computing". Advanced Functional Materials. 22 (3): 609–616. arXiv:1112.3138. doi:10.1002/adfm.201101935. hdl:10261/83537. S2CID 18687826.</ref> and which exhibits the main behavior of a biological spiking synapse. This device, also called a synapstor (synapse transistor), was used to demonstrate a neuro-inspired circuit (associative memory showing a pavlovian learning).<ref>Pavlov's; Transistors, Organic; Bichler, O.; Zhao, W.; Alibart, F.; Pleutin, S.; Lenfant, S.; Vuillaume, D.; Gamrat, C. (2013). "Pavlov's Dog Associative Learning Demonstrated on Synaptic-Like Organic Transistors". Neural Computation. 25 (2): 549–566. arXiv:1302.3261. Bibcode:2013arXiv1302.3261B. doi:10.1162/NECO_a_00377. PMID 22970878. S2CID 16972302.</ref>

In 2012, Crupi, Pradhan and Tozer described a proof of concept design to create neural synaptic memory circuits using organic ion-based memristors.<ref> Crupi, M.; Pradhan, L.; Tozer, S. (2012), "Modelling Neural Plasticity with Memristors" (PDF), IEEE Canadian Review, 68: 10–14</ref> The synapse circuit demonstrated long-term potentiation for learning as well as inactivity based forgetting. Using a grid of circuits, a pattern of light was stored and later recalled. This mimics the behavior of the V1 neurons in the primary visual cortex that act as spatiotemporal filters that process visual signals such as edges and moving lines.

In 2012, Erokhin and co-authors have demonstrated a stochastic three-dimensional matrix with capabilities for learning and adapting based on polymeric memristor.<ref>Erokhin, V.; Berzina, T.; Gorshkov, K.; Camorani, P.; Pucci, A.; Ricci, L.; Ruggeri, G.; Signala, R.; Schüz, A. (2012). "Stochastic hybrid 3D matrix: learning and adaptation of electrical properties". Journal of Materials Chemistry. 22 (43): 22881. doi:10.1039/C2JM35064E.</ref>

Layered memristor

In 2014, Bessonov et al. reported a flexible memristive device comprising a MoO_x/MoS₂ heterostructure sandwiched between silver electrodes on a plastic foil.<ref name="flexible_memristor"> Bessonov, A. A.; et al. (2014), "Layered memristive and memcapacitive switches for printable electronics", Nature Materials, 14 (2): 199–204, Bibcode:2015NatMa..14..199B, doi:10.1038/nmat4135, PMID 25384168</ref> The fabrication method is entirely based on printing and solution-processing technologies using two-dimensional layered transition metal dichalcogenides (TMDs). The memristors are mechanically flexible, optically transparent and produced at low cost. The memristive behaviour of switches was found to be accompanied by a prominent memcapacitive effect. High switching performance, demonstrated synaptic plasticity and sustainability to mechanical deformations promise to emulate the appealing characteristics of biological neural systems in novel computing technologies.

Atomristor

Atomristor is defined as the electrical devices showing memristive behavior in atomically thin nanomaterials or atomic sheets. In 2018, Ge and Wu et al.<ref>Ge, Ruijing; Wu, Xiaohan; Kim, Myungsoo; Shi, Jianping; Sonde, Sushant; Tao, Li; Zhang, Yanfeng; Lee, Jack C.; Akinwande, Deji (2017-12-19). "Atomristor: Nonvolatile Resistance Switching in Atomic Sheets of Transition Metal Dichalcogenides". Nano Letters. 18 (1): 434–441. Bibcode:2018NanoL..18..434G. doi:10.1021/acs.nanolett.7b04342. PMID 29236504.</ref> in the Akinwande group at the University of Texas, first reported a universal memristive effect in single-layer TMD (MX₂, M = Mo, W; and X = S, Se) atomic sheets based on vertical metal-insulator-metal (MIM) device structure. The work was later extended to monolayer hexagonal boron nitride, which is the thinnest memory material of around 0.33 nm.<ref>Wu, Xiaohan; Ge, Ruijing; Chen, Po-An; Chou, Harry; Zhang, Zhepeng; Zhang, Yanfeng; Banerjee, Sanjay; Chiang, Meng-Hsueh; Lee, Jack C.; Akinwande, Deji (April 2019). "Thinnest Nonvolatile Memory Based on Monolayer h-BN". Advanced Materials. 31 (15): 1806790. Bibcode:2019AdM....3106790W. doi:10.1002/adma.201806790. PMID 30773734. S2CID 73505661.</ref> These atomristors offer forming-free switching and both unipolar and bipolar operation. The switching behavior is found in single-crystalline and poly-crystalline films, with various conducting electrodes (gold, silver and graphene). Atomically thin TMD sheets are prepared via CVD/MOCVD, enabling low-cost fabrication. Afterwards, taking advantage of the low "on" resistance and large on/off ratio, a high-performance zero-power RF switch is proved based on MoS₂ or h-BN atomristors, indicating a new application of memristors for 5G, 6G and THz communication and connectivity systems.<ref>Kim, Myungsoo; Ge, Ruijing; Wu, Xiaohan; Lan, Xing; Tice, Jesse; Lee, Jack C.; Akinwande, Deji (2018). "Zero-static power radio-frequency switches based on MoS₂ atomristors". Nature Communications. 9 (1): 2524. Bibcode:2018NatCo...9.2524K. doi:10.1038/s41467-018-04934-x. PMC 6023925. PMID 29955064.</ref><ref>"Towards zero-power 6G communication switches using atomic sheets". Nature Electronics. 5 (6): 331–332. June 2022. doi:10.1038/s41928-022-00767-1. S2CID 249221166.</ref> In 2020, atomistic understanding of the conductive virtual point mechanism was elucidated in an article in nature nanotechnology.<ref>Hus, Saban M.; Ge, Ruijing; Chen, Po-An; Liang, Liangbo; Donnelly, Gavin E.; Ko, Wonhee; Huang, Fumin; Chiang, Meng-Hsueh; Li, An-Ping; Akinwande, Deji (January 2021). "Observation of single-defect memristor in an MoS₂ atomic sheet". Nature Nanotechnology. 16 (1): 58–62. Bibcode:2021NatNa..16...58H. doi:10.1038/s41565-020-00789-w. PMID 33169008. S2CID 226285710.</ref>

Ferroelectric memristor

The ferroelectric memristor<ref> Chanthbouala, A.; et al. (2012), "A ferroelectric memristor", Nature Materials, 11 (10): 860–864, arXiv:1206.3397, Bibcode:2012NatMa..11..860C, doi:10.1038/nmat3415, PMID 22983431, S2CID 10372470</ref> is based on a thin ferroelectric barrier sandwiched between two metallic electrodes. Switching the polarization of the ferroelectric material by applying a positive or negative voltage across the junction can lead to a two order of magnitude resistance variation: R_OFF ≫ R_ON (an effect called Tunnel Electro-Resistance). In general, the polarization does not switch abruptly. The reversal occurs gradually through the nucleation and growth of ferroelectric domains with opposite polarization. During this process, the resistance is neither R_ON or R_OFF, but in between. When the voltage is cycled, the ferroelectric domain configuration evolves, allowing a fine tuning of the resistance value. The ferroelectric memristor's main advantages are that ferroelectric domain dynamics can be tuned, offering a way to engineer the memristor response, and that the resistance variations are due to purely electronic phenomena, aiding device reliability, as no deep change to the material structure is involved.

Carbon nanotube memristor

In 2013, Ageev, Blinov et al.<ref name="Rubashkina2013">Ageev, O. A.; Blinov, Yu F.; Il’in, O. I.; Kolomiitsev, A. S.; Konoplev, B. G.; Rubashkina, M. V.; Smirnov, V. A.; Fedotov, A. A. (2013-12-11). "Memristor effect on bundles of vertically aligned carbon nanotubes tested by scanning tunnel microscopy". Technical Physics. 58 (12): 1831–1836. Bibcode:2013JTePh..58.1831A. doi:10.1134/S1063784213120025. S2CID 53003312.</ref> reported observing memristor effect in structure based on vertically aligned carbon nanotubes studying bundles of CNT by scanning tunneling microscope.

Later it was found<ref>Il'ina, Marina V.; Il'in, Oleg I.; Blinov, Yuriy F.; Smirnov, Vladimir A.; Kolomiytsev, Alexey S.; Fedotov, Alexander A.; Konoplev, Boris G.; Ageev, Oleg A. (October 2017). "Memristive switching mechanism of vertically aligned carbon nanotubes". Carbon. 123: 514–524. Bibcode:2017Carbo.123..514I. doi:10.1016/j.carbon.2017.07.090.</ref> that CNT memristive switching is observed when a nanotube has a non-uniform elastic strain ΔL0. It was shown that the memristive switching mechanism of strained СNT is based on the formation and subsequent redistribution of non-uniform elastic strain and piezoelectric field Edef in the nanotube under the influence of an external electric field E(x,t).

Biomolecular memristor

Biomaterials have been evaluated for use in artificial synapses and have shown potential for application in neuromorphic systems.<ref>Park, Youngjun; Kim, Min-Kyu; Lee, Jang-Sik (2020-07-16). "Emerging memory devices for artificial synapses". Journal of Materials Chemistry C. 8 (27): 9163–9183. doi:10.1039/D0TC01500H. S2CID 219912115.</ref> In particular, the feasibility of using a collagen‐based biomemristor as an artificial synaptic device has been investigated,<ref>Raeis-Hosseini, Niloufar; Park, Youngjun; Lee, Jang-Sik (2018). "Flexible Artificial Synaptic Devices Based on Collagen from Fish Protein with Spike-Timing-Dependent Plasticity". Advanced Functional Materials. 28 (31): 1800553. doi:10.1002/adfm.201800553. S2CID 104277945.</ref> whereas a synaptic device based on lignin demonstrated rising or lowering current with consecutive voltage sweeps depending on the sign of the voltage<ref>Park, Youngjun; Lee, Jang-Sik (2017-09-26). "Artificial Synapses with Short- and Long-Term Memory for Spiking Neural Networks Based on Renewable Materials". ACS Nano. 11 (9): 8962–8969. doi:10.1021/acsnano.7b03347. PMID 28837313.</ref> furthermore a natural silk fibroin demonstrated memristive properties;<ref>Hota, Mrinal K.; Bera, Milan K.; Kundu, Banani; Kundu, Subhas C.; Maiti, Chinmay K. (2012). "A Natural Silk Fibroin Protein-Based Transparent Bio-Memristor". Advanced Functional Materials. 22 (21): 4493–4499. doi:10.1002/adfm.201200073. S2CID 137399893.</ref> spin-memristive systems based on biomolecules are also being studied.<ref>Cardona-Serra, Salvador; Rosaleny, Lorena E.; Giménez-Santamarina, Silvia; Martínez-Gil, Luis; Gaita-Ariño, Alejandro (2020-12-16). "Towards peptide-based tunable multistate memristive materials". Physical Chemistry Chemical Physics. 23 (3): 1802–1810. doi:10.1039/D0CP05236A. hdl:10550/79239. PMID 33434247. S2CID 231595640.</ref>

In 2012, Sandro Carrara and co-authors have proposed the first biomolecular memristor with aims to realize highly sensitive biosensors.<ref>Milano, G.; Porro, S.; Valov, I.; Ricciardi, C. (2019). "Recent Developments and Perspectives for Memristive Devices Based on Metal Oxide Nanowires". Advanced Electronic Materials. 5 (9): 1800909. doi:10.1002/aelm.201800909. S2CID 139445142.</ref> Since then, several memristive sensors have been demonstrated.<ref>Carrara, S. (2021). "The Birth of a New Field: Memristive Sensors. A Review". IEEE Sensors Journal. 21 (11): 12370–12378. Bibcode:2021ISenJ..2112370C. doi:10.1109/JSEN.2020.3043305. S2CID 234542676.</ref>

Spin memristive systems

Spintronic memristor

Chen and Wang, researchers at disk-drive manufacturer Seagate Technology described three examples of possible magnetic memristors.<ref name="Xiaobin2009"> Wang, X.; Chen, Y.; Xi, H.; Dimitrov, D. (2009), "Spintronic Memristor through Spin Torque Induced Magnetization Motion", IEEE Electron Device Letters, 30 (3): 294–297, Bibcode:2009IEDL...30..294W, doi:10.1109/LED.2008.2012270, S2CID 39590957</ref> In one device resistance occurs when the spin of electrons in one section of the device points in a different direction from those in another section, creating a "domain wall", a boundary between the two sections. Electrons flowing into the device have a certain spin, which alters the device's magnetization state. Changing the magnetization, in turn, moves the domain wall and changes the resistance. The work's significance led to an interview by IEEE Spectrum.<ref name="Neil2009">Savage, N. (2009-03-16). "Spintronic Memristor". IEEE Spectrum. Archived from the original on 2010-12-24. Retrieved 2011-03-20.</ref> A first experimental proof of the spintronic memristor based on domain wall motion by spin currents in a magnetic tunnel junction was given in 2011.<ref name="Chanthbouala2011">Chanthbouala, A.; Matsumoto, R.; Grollier, J.; Cros, V.; Anane, A.; Fert, A.; Khvalkovskiy, A. V.; Zvezdin, K. A.; Nishimura, K.; Nagamine, Y.; Maehara, H.; Tsunekawa, K.; Fukushima, A.; Yuasa, S. (2011-04-10). "Vertical-current-induced domain-wall motion in MgO-based magnetic tunnel junctions with low current densities". Nature Physics. 7 (8): 626–630. arXiv:1102.2106. Bibcode:2011NatPh...7..626C. doi:10.1038/nphys1968. S2CID 119221544.</ref>

Memristance in a magnetic tunnel junction

The magnetic tunnel junction has been proposed to act as a memristor through several potentially complementary mechanisms, both extrinsic (redox reactions, charge trapping/detrapping and electromigration within the barrier) and intrinsic (spin-transfer torque).

Extrinsic mechanism

Based on research performed between 1999 and 2003, Bowen et al. published experiments in 2006 on a magnetic tunnel junction (MTJ) endowed with bi-stable spin-dependent states<ref name="BowenAPL2006"> Bowen, M.; Maurice, J.-L.; Barthe´le´my, A.; Prod’homme, P.; Jacquet, E.; Contour, J.-P.; Imhoff, D.; Colliex, C. (2006). "Bias-crafted magnetic tunnel junctions with bistable spin-dependent states". Applied Physics Letters. 89 (10): 103517. Bibcode:2006ApPhL..89j3517B. doi:10.1063/1.2345592.</ref>(resistive switching). The MTJ consists in a SrTiO3 (STO) tunnel barrier that separates half-metallic oxide LSMO and ferromagnetic metal CoCr electrodes. The MTJ's usual two device resistance states, characterized by a parallel or antiparallel alignment of electrode magnetization, are altered by applying an electric field. When the electric field is applied from the CoCr to the LSMO electrode, the tunnel magnetoresistance (TMR) ratio is positive. When the direction of electric field is reversed, the TMR is negative. In both cases, large amplitudes of TMR on the order of 30% are found. Since a fully spin-polarized current flows from the half-metallic LSMO electrode, within the Julliere model, this sign change suggests a sign change in the effective spin polarization of the STO/CoCr interface. The origin to this multistate effect lies with the observed migration of Cr into the barrier and its state of oxidation. The sign change of TMR can originate from modifications to the STO/CoCr interface density of states, as well as from changes to the tunneling landscape at the STO/CoCr interface induced by CrOx redox reactions.

Reports on MgO-based memristive switching within MgO-based MTJs appeared starting in 2008<ref name="HalleyAPL2008"> Halley, D.; Majjad, H.; Bowen, M.; Najjari, N.; Henry, Y.; Ulhaq-Bouillet, C.; Weber, W.; Bertoni, G.; Verbeeck, J.; Van Tendeloo, G. (2008). "Electrical switching in Fe/Cr/MgO/Fe magnetic tunnel junctions". Applied Physics Letters. 92 (21): 212115. Bibcode:2008ApPhL..92u2115H. doi:10.1063/1.2938696.</ref> and 2009.<ref name="krzysteczko2009apl"> Krzysteczko, P.; Günter, R.; Thomas, A. (2009), "Memristive switching of MgO based magnetic tunnel junctions", Applied Physics Letters, 95 (11): 112508, arXiv:0907.3684, Bibcode:2009ApPhL..95k2508K, CiteSeerX 10.1.1.313.2571, doi:10.1063/1.3224193, S2CID 15383692</ref> While the drift of oxygen vacancies within the insulating MgO layer has been proposed to describe the observed memristive effects,<ref name="krzysteczko2009apl" /> another explanation could be charge trapping/detrapping on the localized states of oxygen vacancies<ref name="BertinJAP2011"> Bertin, Eric; Halley, David; Henry, Yves; Najjari, Nabil; Majjad, Hicham; Bowen, Martin; DaCosta, Victor; Arabski, Jacek; Doudin, Bernard (2011), "Random barrier double-well model for resistive switching in tunnel barriers", Journal of Applied Physics, 109 (8): 013712–013712–5, Bibcode:2011JAP...109a3712D, doi:10.1063/1.3530610, retrieved 2014-12-15</ref> and its impact<ref name="schleicherNC2014">

Schleicher, F.; Halisdemir, U.; Lacour, D.; Gallart, M.; Boukari, S.; Schmerber, G.; Davesne, V.; Panissod, P.; Halley, D.; Majjad, H.; Henry, Y.; Leconte, B.; Boulard, A.; Spor, D.; Beyer, N.; Kieber, C.; Sternitzky, E.; Cregut, O.; Ziegler, M.; Montaigne, F.; Beaurepaire, E.; Gilliot, P.; Hehn, M.; Bowen, M. (2014-08-04), "Localized states in advanced dielectrics from the vantage of spin- and symmetry-polarized tunnelling across MgO", Nature Communications, 5: 4547, Bibcode:2014NatCo...5.4547S, doi:10.1038/ncomms5547, PMID 25088937</ref> on spintronics. This highlights the importance of understanding what role oxygen vacancies play in the memristive operation of devices that deploy complex oxides with an intrinsic property such as ferroelectricity<ref name="garciaS2010">
 Garcia, V.; Bibes, M.; Bocher, L.; Valencia, S.; Kronast, F.; Crassous, A.; Moya, X.; Enouz-Vedrenne, S.; Gloter, A.; Imhoff, D.; Deranlot, C.; Mathur, N. D.; Fusil, S.; Bouzehouane, K.; Barthelemy, A. (2010-02-26), "Ferroelectric Control of Spin Polarization", Science, 327 (5969): 1106–1110, Bibcode:2010Sci...327.1106G, doi:10.1126/science.1184028, PMID 20075211, S2CID 206524358</ref> or multiferroicity.<ref name="pantelAM2012">
 Pantel, D.; Goetze, S.; Hesse, D.; Alexe, M. (2012-02-26), "Reversible electrical switching of spin polarization in multiferroic tunnel junctions", Nature Materials, 11 (4): 289–293, Bibcode:2012NatMa..11..289P, doi:10.1038/nmat3254, PMID 22367005</ref>

Intrinsic mechanism

The magnetization state of a MTJ can be controlled by Spin-transfer torque, and can thus, through this intrinsic physical mechanism, exhibit memristive behavior. This spin torque is induced by current flowing through the junction, and leads to an efficient means of achieving a MRAM. However, the length of time the current flows through the junction determines the amount of current needed, i.e., charge is the key variable.<ref> Huai, Y. (December 2008), "Spin-Transfer Torque MRAM (STT-MRAM): Challenges and Prospects" (PDF), AAPPS Bulletin, 18 (6): 33–40, archived from the original (PDF) on 2012-03-23</ref>

The combination of intrinsic (spin-transfer torque) and extrinsic (resistive switching) mechanisms naturally leads to a second-order memristive system described by the state vector x = (x₁,x₂), where x₁ describes the magnetic state of the electrodes and x₂ denotes the resistive state of the MgO barrier. In this case the change of x₁ is current-controlled (spin torque is due to a high current density) whereas the change of x₂ is voltage-controlled (the drift of oxygen vacancies is due to high electric fields). The presence of both effects in a memristive magnetic tunnel junction led to the idea of a nanoscopic synapse-neuron system.<ref name="krzysteczko2012advmat"> Krzysteczko, P.; Münchenberger, J.; Schäfers, M.; Reiss, G.; Thomas, A. (2012), "The Memristive Magnetic Tunnel Junction as a Nanoscopic Synapse-Neuron System", Advanced Materials, 24 (6): 762–766, Bibcode:2012APS..MAR.H5013T, doi:10.1002/adma.201103723, PMID 22223304, S2CID 205242867</ref>

Spin memristive system

A fundamentally different mechanism for memristive behavior has been proposed by Pershin and Di Ventra.<ref>"Massimiliano Di Ventra's Homepage". physics.ucsd.edu.</ref><ref name="Pershin2008"> Pershin, Y. V.; Di Ventra, M. (2008), "Spin memristive systems: Spin memory effects in semiconductor spintronics", Physical Review B, 78 (11): 113309, arXiv:0806.2151, Bibcode:2008PhRvB..78k3309P, doi:10.1103/PhysRevB.78.113309, S2CID 10938532</ref> The authors show that certain types of semiconductor spintronic structures belong to a broad class of memristive systems as defined by Chua and Kang.<ref name="chua76"/> The mechanism of memristive behavior in such structures is based entirely on the electron spin degree of freedom which allows for a more convenient control than the ionic transport in nanostructures. When an external control parameter (such as voltage) is changed, the adjustment of electron spin polarization is delayed because of the diffusion and relaxation processes causing hysteresis. This result was anticipated in the study of spin extraction at semiconductor/ferromagnet interfaces,<ref> Pershin, Y. V.; Di Ventra, M. (2008), "Current-voltage characteristics of semiconductor/ferromagnet junctions in the spin-blockade regime", Physical Review B, 77 (7): 073301, arXiv:0707.4475, Bibcode:2008PhRvB..77g3301P, doi:10.1103/PhysRevB.77.073301, S2CID 119604218</ref> but was not described in terms of memristive behavior. On a short time scale, these structures behave almost as an ideal memristor.<ref name="chua71"/> This result broadens the possible range of applications of semiconductor spintronics and makes a step forward in future practical applications.

Self-directed channel memristor

In 2017, Kris Campbell formally introduced the self-directed channel (SDC) memristor.<ref> Campbell, K. (January 2017), "Self-directed channel memristor for high temperature operation", Microelectronics Journal, 59: 10–14, arXiv:1608.05357, doi:10.1016/j.mejo.2016.11.006, S2CID 27889124 </ref> The SDC device is the first memristive device available commercially to researchers, students and electronics enthusiast worldwide.<ref> Knowm Memristors, Knowm Inc</ref> The SDC device is operational immediately after fabrication. In the Ge₂Se₃ active layer, Ge-Ge homopolar bonds are found and switching occurs. The three layers consisting of Ge₂Se₃/Ag/Ge₂Se₃, directly below the top tungsten electrode, mix together during deposition and jointly form the silver-source layer. A layer of SnSe is between these two layers ensuring that the silver-source layer is not in direct contact with the active layer. Since silver does not migrate into the active layer at high temperatures, and the active layer maintains a high glass transition temperature of about 350 °C (662 °F), the device has significantly higher processing and operating temperatures at 250 °C (482 °F) and at least 150 °C (302 °F), respectively. These processing and operating temperatures are higher than most ion-conducting chalcogenide device types, including the S-based glasses (e.g. GeS) that need to be photodoped or thermally annealed. These factors allow the SDC device to operate over a wide range of temperatures, including long-term continuous operation at 150 °C (302 °F).

Potential applications

Memristors remain a laboratory curiosity, as yet made in insufficient numbers to gain any commercial applications. Despite this lack of mass availability, according to Allied Market Research the memristor market was worth $3.2 million in 2015 and was at the time projected to be worth $79.0 million by 2022.<ref>"Memristor Market Expected to Reach $79.0 Million by 2020, Globally - Allied Market Research". Archived from the original on 2017-02-26. Retrieved 2017-02-25.</ref> In fact, it was worth $190.0 million in 2022.<ref>"Groundbreaking Memristor Technology Set to Disrupt Electronics Industry, Fueling a Projected $2.6 Billion Market Growth by 2028". 2023-09-27.</ref>

A potential application of memristors is in analog memories for superconducting quantum computers.<ref name="peottaDiVentra2014"/>

Memristors can potentially be fashioned into non-volatile solid-state memory, which could allow greater data density than hard drives with access times similar to DRAM, replacing both components.<ref name="Kanellos"> Kanellos, M. (2008-04-30), "HP makes memory from a once theoretical circuit", CNET News, retrieved 2008-04-30</ref> HP prototyped a crossbar latch memory that can fit 100 gigabits in a square centimeter,<ref name="EETimes">Johnson, R. C. (2008-04-30), "'Missing link' memristor created", EE Times, retrieved 2008-04-30</ref> and proposed a scalable 3D design (consisting of up to 1000 layers or 1 petabit per cm³).<ref> "Finding the Missing Memristor - R. Stanley Williams", Youtube, 2010-01-22</ref> In May 2008 HP reported that its device reaches currently about one-tenth the speed of DRAM.<ref> Markoff, J. (2008-05-01), "H.P. Reports Big Advance in Memory Chip Design", New York Times, retrieved 2008-05-01</ref> The devices' resistance would be read with alternating current so that the stored value would not be affected.<ref> Gutmann, E. (2008-05-01), "Maintaining Moore's law with new memristor circuits", Ars Technica, retrieved 2008-05-01</ref> In May 2012, it was reported that the access time had been improved to 90 nanoseconds, which is nearly one hundred times faster than the contemporaneous Flash memory. At the same time, the energy consumption was just one percent of that consumed by Flash memory.<ref>Palmer, J. (2012-05-18), "Memristors in silicon promising for dense, fast memory", BBC News, retrieved 2012-05-18</ref>

Memristors have applications in programmable logic<ref>Snider, Gregory Stuart (2004) "Architecture and methods for computing with reconfigurable resistor crossbars" U.S. patent 7,203,789</ref> signal processing,<ref>Mouttet, Blaise Laurent (2006) "Programmable crossbar signal processor" U.S. patent 7,302,513</ref> super-resolution imaging<ref>Dong, Zhekang; Sing Lai, Chun; He, Yufei; Qi, Donglian; Duan, Shukai (2019-11-01). "Hybrid dual-complementary metal–oxide–semiconductor/memristor synapse-based neural network with its applications in image super-resolution". IET Circuits, Devices & Systems. 13 (8): 1241–1248. doi:10.1049/iet-cds.2018.5062.</ref> physical neural networks,<ref>Snider, Greg (2003) "Molecular-junction-nanowire-crossbar-based neural network" U.S. patent 7,359,888</ref> control systems,<ref>Mouttet, Blaise Laurent (2007) "Crossbar control circuit" U.S. patent 7,609,086</ref> reconfigurable computing,<ref>Pino, Robinson E. (2010) "Reconfigurable electronic circuit" U.S. patent 7,902,857</ref> in-memory computing,<ref>Ielmini, D; Wong, H.-S. P. (2018). "In-memory computing with resistive switching devices". Nature Electronics. 1 (6): 333–343. doi:10.1038/s41928-018-0092-2. hdl:11311/1056513. S2CID 57248729.</ref> brain–computer interfaces<ref>Mouttet, Blaise Laurent (2009) "Memristor crossbar neural interface" U.S. patent 7,902,867</ref> and RFID.<ref>Kang, Hee Bok (2009) "RFID device with memory unit having memristor characteristics" U.S. patent 8,113,437</ref> Memristive devices are potentially used for stateful logic implication, allowing a replacement for CMOS-based logic computation<ref>Luo, Li; Dong, Zhekang; Duan, Shukai; Lai, Chun Sing (2020-04-20). "Memristor-based stateful logic gates for multi-functional logic circuit". IET Circuits, Devices & Systems. 14 (6): 811–818. doi:10.1049/iet-cds.2019.0422.</ref> Several early works have been reported in this direction.<ref>Lehtonen, E.; Poikonen, J.H.; Laiho, M. (2010). "Two memristors suffice to compute all Boolean functions". Electronics Letters. 46 (3): 230. Bibcode:2010ElL....46..230L. doi:10.1049/el.2010.3407.</ref><ref> Chattopadhyay, A.; Rakosi, Z. (2011). "Combinational logic synthesis for material implication". 2011 IEEE/IFIP 19th International Conference on VLSI and System-on-Chip. p. 200. doi:10.1109/VLSISoC.2011.6081665. ISBN 978-1-4577-0170-2. S2CID 32278896.</ref>

In 2009, a simple electronic circuit<ref> Pershin, Y. V.; La Fontaine, S.; Di Ventra, M. (2009), "Memristive model of amoeba learning", Physical Review E, 80 (2): 021926, arXiv:0810.4179, Bibcode:2009PhRvE..80b1926P, doi:10.1103/PhysRevE.80.021926, PMID 19792170, S2CID 9820970</ref> consisting of an LC network and a memristor was used to model experiments on adaptive behavior of unicellular organisms.<ref name="amoebaexp"> Saigusa, T.; Tero, A.; Nakagaki, T.; Kuramoto, Y. (2008), "Amoebae Anticipate Periodic Events" (PDF), Physical Review Letters, 100 (1): 018101, Bibcode:2008PhRvL.100a8101S, doi:10.1103/PhysRevLett.100.018101, hdl:2115/33004, PMID 18232821, S2CID 14710241</ref> It was shown that subjected to a train of periodic pulses, the circuit learns and anticipates the next pulse similar to the behavior of slime molds Physarum polycephalum where the viscosity of channels in the cytoplasm responds to periodic environment changes.<ref name="amoebaexp"/> Applications of such circuits may include, e.g., pattern recognition. The DARPA SyNAPSE project funded HP Labs, in collaboration with the Boston University Neuromorphics Lab, has been developing neuromorphic architectures which may be based on memristive systems. In 2010, Versace and Chandler described the MoNETA (Modular Neural Exploring Traveling Agent) model.<ref> Versace, M.; Chandler, B. (2010-11-23). "MoNETA: A Mind Made from Memristors". IEEE Spectrum.
Versace, M.; Chandler, B. (2010). "The brain of a new machine". IEEE Spectrum. 47 (12): 30–37. doi:10.1109/MSPEC.2010.5644776. S2CID 45300119.</ref> MoNETA is the first large-scale neural network model to implement whole-brain circuits to power a virtual and robotic agent using memristive hardware.<ref>Snider, G.; et al. (2011), "From Synapses to Circuitry: Using Memristive Memory to Explore the Electronic Brain", IEEE Computer, 44 (2): 21–28, doi:10.1109/MC.2011.48, S2CID 16307308</ref> Application of the memristor crossbar structure in the construction of an analog soft computing system was demonstrated by Merrikh-Bayat and Shouraki.<ref> Merrikh-Bayat, F.; Bagheri-Shouraki, S.; Rohani, A. (2011), "Memristor crossbar-based hardware implementation of IDS method", IEEE Transactions on Fuzzy Systems, 19 (6): 1083–1096, arXiv:1008.5133, doi:10.1109/TFUZZ.2011.2160024, S2CID 3163846</ref> In 2011, they showed<ref> Merrikh-Bayat, F.; Bagheri-Shouraki, S. (2011). "Efficient neuro-fuzzy system and its Memristor Crossbar-based Hardware Implementation". arXiv:1103.1156 [cs.AI].</ref> how memristor crossbars can be combined with fuzzy logic to create an analog memristive neuro-fuzzy computing system with fuzzy input and output terminals. Learning is based on the creation of fuzzy relations inspired from Hebbian learning rule.

In 2013 Leon Chua published a tutorial underlining the broad span of complex phenomena and applications that memristors span and how they can be used as non-volatile analog memories and can mimic classic habituation and learning phenomena.<ref> Chua, L. (2013). "Memristor, Hodgkin-Huxley, and Edge of Chaos". Nanotechnology. 24 (38): 383001. Bibcode:2013Nanot..24L3001C. doi:10.1088/0957-4484/24/38/383001. PMID 23999613. S2CID 34999101.</ref>

Derivative devices

Memistor and memtransistor

The memistor and memtransistor are transistor-based devices which include memristor function.

Memcapacitors and meminductors

In 2009, Di Ventra, Pershin, and Chua extended<ref name="DiVentra2009"> Di Ventra, M.; Pershin, Y. V.; Chua, L. (2009), "Circuit elements with memory: memristors, memcapacitors and meminductors", Proceedings of the IEEE, 97 (10): 1717–1724, arXiv:0901.3682, Bibcode:2009arXiv0901.3682D, doi:10.1109/JPROC.2009.2021077, S2CID 7136764</ref> the notion of memristive systems to capacitive and inductive elements in the form of memcapacitors and meminductors, whose properties depend on the state and history of the system, further extended in 2013 by Di Ventra and Pershin.<ref name="DiVentra_2013" />

Memfractance and memfractor, 2nd- and 3rd-order memristor, memcapacitor and meminductor

In September 2014, Mohamed-Salah Abdelouahab, Rene Lozi, and Leon Chua published a general theory of 1st-, 2nd-, 3rd-, and nth-order memristive elements using fractional derivatives.<ref> Abdelhouahad, M.-S.; Lozi, R.; Chua, L. (September 2014), "Memfractance: A Mathematical Paradigm for Circuit Elements with Memory" (PDF), International Journal of Bifurcation and Chaos, 24 (9): 1430023 (29 pages), Bibcode:2014IJBC...2430023A, doi:10.1142/S0218127414300237</ref>

History

Precursors

Sir Humphry Davy is said by some to have performed the first experiments which can be explained by memristor effects as long ago as 1808.<ref name="Memristor200yearsold" /><ref> Prodromakis, T.; Toumazou, C.; Chua, L. (June 2012), "Two centuries of memristors", Nature Materials, 11 (6): 478–481, Bibcode:2012NatMa..11..478P, doi:10.1038/nmat3338, PMID 22614504</ref> However the first device of a related nature to be constructed was the memistor (i.e. memory resistor), a term coined in 1960 by Bernard Widrow to describe a circuit element of an early artificial neural network called ADALINE. A few years later, in 1968, Argall published an article showing the resistance switching effects of TiO₂ which was later claimed by researchers from Hewlett Packard to be evidence of a memristor.<ref name="Argall1968"/>

Theoretical description

Leon Chua postulated his new two-terminal circuit element in 1971. It was characterized by a relationship between charge and flux linkage as a fourth fundamental circuit element.<ref name="chua71"/> Five years later he and his student Sung Mo Kang generalized the theory of memristors and memristive systems including a property of zero crossing in the Lissajous curve characterizing current vs. voltage behavior.<ref name="chua76"/>

Twenty-first century

On May 1, 2008, Strukov, Snider, Stewart, and Williams published an article in Nature identifying a link between the two-terminal resistance switching behavior found in nanoscale systems and memristors.<ref name="Williams08"/>

On 23 January 2009, Di Ventra, Pershin, and Chua extended the notion of memristive systems to capacitive and inductive elements, namely capacitors and inductors, whose properties depend on the state and history of the system.<ref name="DiVentra2009"/>

In July 2014, the MeMOSat/LabOSat group<ref name="labosat1"> Barella, M. (2016), "LabOSat: Low cost measurement platform designed for hazardous environments", 2016 Seventh Argentine Conference on Embedded Systems (CASE), pp. 1–6, doi:10.1109/SASE-CASE.2016.7968107, ISBN 978-987-46297-0-8, S2CID 10263318 </ref> (composed of researchers from Universidad Nacional de General San Martín (Argentina), INTI, CNEA, and CONICET) put memory devices into a Low Earth orbit.<ref>"Probaron con éxito las memorias instaladas en el satélite argentino "Tita"". Telam. 2014-07-21.</ref> Since then, seven missions with different devices<ref name="labosat2"> Barella, M. (2019), "Studying ReRAM devices at Low Earth Orbits using the LabOSat platform", Radiation Physics and Chemistry, 154: 85–90, Bibcode:2019RaPC..154...85B, doi:10.1016/j.radphyschem.2018.07.005 </ref> are performing experiments in low orbits, onboard Satellogic's Ñu-Sat satellites.<ref>"UNSAM - Universidad Nacional de San Martín". www.unsam.edu.ar.</ref><ref>"Qué hace LabOSat, el laboratorio electrónico dentro de los nanosatélites Fresco y Batata". Telam. 2016-06-22.</ref>

On 7 July 2015, Knowm Inc announced Self Directed Channel (SDC) memristors commercially.<ref> "Startup Beats HP, Hynix to Memristor Learning". EE Times. 2015-07-05.</ref> These devices remain available in small numbers.

On 13 July 2018, MemSat (Memristor Satellite) was launched to fly a memristor evaluation payload.<ref>"MemSat". Gunter Space Page. 2018-05-22.</ref>

In 2021, Jennifer Rupp and Martin Bazant of MIT started a "Lithionics" research programme to investigate applications of lithium beyond their use in battery electrodes, including lithium oxide-based memristors in neuromorphic computing.<ref>"MIT and Ericsson Collaborates to Research New Generation of Energy-Efficient Computing Networks - News". eepower.com.</ref><ref>"MIT and Ericsson Set Goals for Zero-power Devices and a New Field—"Lithionics" - News". www.allaboutcircuits.com.</ref>

In the September 2023 issue of Science Magazine, Chinese scientists Wenbin Zhang et al. described the development and testing of a memristor-based integrated circuit, designed to dramatically increase the speed and efficiency of Machine Learning and Artificial Intelligence tasks, optimized for Edge Computing applications.<ref>Zhang, Wenbin (2023-09-14). "Edge learning using a fully integrated neuro-inspired memristor chip". Science. 381 (6663): 1205–1211. Bibcode:2023Sci...381.1205Z. doi:10.1126/science.ade3483. PMID 37708281. S2CID 261736380.</ref>

See also

• 3D XPoint
• Electrical element
• Hybrid Memory Cube
• List of emerging technologies
• Neuromorphic engineering
• Trancitor

Further reading

• Chen, Dongmin; Chua, Leon O.; Hwang, Cheol Seong; Wang, Shih-Yuan; Waser, Rainer; Williams, R. Stanley; Yang, Jianhua, eds. (March 2011). "Special Issue: Memristive and Resistive Devices and Systems". Applied Physics A. 102 (4).
• Mazumder, P.; Kang, S. M.; Waser, R., eds. (June 2012). "Special Issue: MEMRISTORS: DEVICES, MODELS, AND APPLICATIONS". Proceedings of the IEEE. 100 (6): 1905–2092. doi:10.1109/JPROC.2012.2197452.
• Tetzlaff, Ronald, ed. (2013). Memristors and Memristive Systems. Springer Science & Business Media. doi:10.1007/978-1-4614-9068-5. ISBN 978-1-4614-9068-5.
• Adamatzky, Andrew; Chua, Leon, eds. (2013). Memristor Networks. Springer Science & Business Media. doi:10.1007/978-3-319-02630-5. ISBN 978-3-319-02630-5. S2CID 39739718.
• Atkin, Keith (May 2013). "An introduction to the memristor". Physics Education. 48 (3): 317–321. Bibcode:2013PhyEd..48..317A. doi:10.1088/0031-9120/48/3/317. S2CID 121268844.
• Gale, Ella (2014-10-01). "TiO₂-based memristors and ReRAM: materials, mechanisms and models (a review)". Semiconductor Science and Technology. 29 (10): 104004. arXiv:1611.04456. Bibcode:2014SeScT..29j4004G. doi:10.1088/0268-1242/29/10/104004. S2CID 5686212.
• Traversa, Fabio Lorenzo; Di Ventra, Massimiliano (November 2015). "Universal Memcomputing Machines". IEEE Transactions on Neural Networks and Learning Systems. 26 (11): 2702–2715. arXiv:1405.0931. CiteSeerX 10.1.1.747.5690. doi:10.1109/TNNLS.2015.2391182. PMID 25667360. S2CID 1406042.
• Caravelli, Francesco; Carbajal, Juan Pablo (January 2019). "Memristors for the curious outsiders". Technologies. 6 (4): 118. arXiv:1812.03389. doi:10.3390/technologies6040118. S2CID 54464654.
• Maan, Akshay Kumar; Jayadevi, Deepthi Anirudhan; James, Alex Pappachen (August 2017). "A Survey of Memristive Threshold Logic Circuits". IEEE Transactions on Neural Networks and Learning Systems. 28 (8): 1734–1746. arXiv:1604.07121. doi:10.1109/TNNLS.2016.2547842. PMID 27164608. S2CID 1798273.
• Ghosh, M., Singh, A., Borah, S. S., Vista, J., Ranjan, A., Kumar, S. (2022). "MOSFET-based memristor for high-frequency signal processing". IEEE Transactions on Electron Devices. 69 (5): 2248–2255. Bibcode:2022ITED...69.2248G. doi:10.1109/ted.2022.3160940. ISSN 0018-9383. S2CID 247889089.
• Singh, A., Borah, S. S., Ghosh, M. (2021), Simple grounded meminductor emulator using transconductance amplifier, IEEE

External links

• Finding the missing memristor on YouTube
• Interactive database of memristor papers (2013)
• Simonite, Tom (2015-04-21). "Machine Dreams". Technology Review. Retrieved 2017-12-05.
• "Leon Chua: A bulb versus Google go player" - (in Polish) an interview with Leon Chua, the creator of memristor
• "Leon Chua: A bulb versus Google go player" - (in English) an interview with Leon Chua, the creator of memristor