View source for Memristor

{{Short description|Nonlinear two-terminal fundamental circuit element}}
{{Distinguish|Memistor|Memtransistor}}
{{Use dmy dates|date=May 2022|cs1-dates=y}}
{{Infobox electronic component
 |component         = Memristor
 |photo             = Memristor (50665029093) (cropped).jpg
 |photo_caption     = Memristor developed by the [[University of Illinois at Urbana-Champaign]] and [[National Energy Technology Laboratory]]
 |inventor          = [[Leon Chua]]
 | working_principle = Relating electric charge and magnetic flux linkage
 |symbol            = [[File:Memristor-Symbol.svg]]
 |terminal_number = 2
 |is_Linear = No
 |invention_Year = 1971
}}

A '''memristor''' ({{IPAc-en|ˈ|m|ɛ|m|r|ᵻ|s|t|ər}}; a [[portmanteau]] of ''memory resistor'') is a non-linear [[terminal (electronics)|two-terminal]] [[electronic component|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">{{cite journal |last1=Chua |first1=L. |title=Memristor-The missing circuit element |journal=IEEE Transactions on Circuit Theory |date=1971 |volume=18 |issue=5 |pages=507–519 |doi=10.1109/TCT.1971.1083337 |citeseerx=10.1.1.189.3614 }}
</ref>

Chua and Kang later generalized the concept to '''memristive systems'''.<ref name="chua76">{{Citation|last1=Chua|first1=L. O.|first2=S. M.|last2=Kang|title=Memristive devices and systems|journal=[[Proceedings of the IEEE]]|date=1 January 1976|volume=64|issue=2|pages=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==
[[File:Two-terminal non-linear circuit elements.svg|thumb|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 {{math|Φ{{sub|m}}(''t'')}} and the amount of electric charge that has flowed, {{math|''q''(''t'')}}:<ref name="chua71" />
<math display="block">f(\mathrm \Phi_\mathrm m(t),q(t))=0</math>

The ''magnetic [[flux linkage]]'', {{math|Φ{{sub|m}}}}, is generalized from the circuit characteristic of an inductor. It {{em|does not}} represent a magnetic field here. Its physical meaning is discussed below. The symbol {{math|Φ{{sub|m}}}} may be regarded as the integral of voltage over time.<ref>{{Citation|first=H. |last=Knoepfel|title=Pulsed high magnetic fields|location=New York|publisher=[[North-Holland Publishing Company|North-Holland]]|date=1970|at=p. 37, Eq. (2.80)}}</ref>

In the relationship between {{math|Φ{{sub|m}}}} and {{mvar|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 display="block">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 display="block">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 {{math|''M''(''q'')}}, which characterizes the device, and write it as a differential equation.

{| class="wikitable"
|+ Differential equation relationships between resistance, capacitance, inductance, and memristance
|-
! scope="col" | Device
! scope="col" | Symbol
! scope="col" | Characteristic property
! scope="col" | Units
! scope="col" | Unit ratio (V, A, C, Wb)
! scope="col" | Differential equation
|-
! scope="row" | [[Resistor]]
| {{mvar|R}}  || [[Electrical resistance|Resistance]] || [[ohm]] ({{math|Ω}}) || [[volt]]s per [[ampere]] ({{math|{{thinsp|V|/|A}}}}) || {{math|1=''R'' = d''V'' / d''I''}}
|-
! scope="row" | [[Capacitor]]
| {{mvar|C}} || [[Capacitance]] || [[farad]] ({{math|F}}) || [[coulomb]]s per volt ({{math|{{thinsp|C|/|V}}}}) || {{math|1=''C'' = d''q'' / d''V''}}
|-
! scope="row" | [[Inductor]]
| {{mvar|L}}  || [[Inductance]] || [[Henry (unit)|henry]] ({{math|H}}) || [[Weber (unit)|webers]] per ampere ({{math|{{thinsp|Wb|/|A}}}}) || {{math|1=''L'' = dΦ{{sub|m}} / d''I''}}
|-
! scope="row" | Memristor
| {{mvar|M}} || Memristance || ohm ({{math|Ω}}) || webers per coulomb ({{math|{{thinsp|Wb|/|C}}}}) || {{math|1=''M'' = dΦ{{sub|m}} / d''q''}}
|}

The above table covers all meaningful ratios of differentials of {{mvar|I}}, {{mvar|q}}, {{math|Φ{{sub|m}}}}, and {{mvar|V}}. No device can relate {{math|d''I''}} to {{math|d''q''}}, or {{math|dΦ{{sub|m}}}} to {{math|d''V''}}, because {{mvar|I}} is the time derivative of {{mvar|q}} and {{math|Φ{{sub|m}}}} is the integral of {{mvar|V}} with respect to time.

It can be inferred from this that memristance is charge-dependent [[electrical resistance|resistance]]. If {{math|''M''(''q''(''t''))}} is a constant, then we obtain [[Ohm's law]], {{math|1=''R''(''t'') = ''V''(''t'')/''I''(''t'')}}. If {{math|''M''(''q''(''t''))}} is nontrivial, however, the equation is not equivalent because {{math|''q''(''t'')}} and {{math|''M''(''q''(''t''))}} can vary with time. Solving for voltage as a function of time produces

<math display="block">V(t) =\ M(q(t)) I(t)\,.</math>

This equation reveals that memristance defines a linear relationship between current and voltage, as long as {{mvar|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 {{mvar|q}} does not cause [[small signal model|much]] change in {{mvar|M}}.

Furthermore, the memristor is static if no current is applied. If {{math|1=''I''(''t'') = 0}}, we find {{math|1=''V''(''t'') = 0}} and {{math|''M''(''t'')}} is constant. This is the essence of the memory effect.

Analogously, we can define a {{math|''W''(''ϕ''(''t''))}} as memductance:<ref name="chua71"/>

<math display="block">i(t)=W(\phi(t))v(t)\,.</math>

The [[power consumption]] characteristic recalls that of a resistor, {{math|''I''{{sup|2}}''R''}}:

<math display="block">P(t) =\ I(t)V(t) =\ I^2(t) M(q(t))\,.</math>

As long as {{math|''M''(''q''(''t''))}} varies little, such as under alternating current, the memristor will appear as a constant resistor. If {{math|''M''(''q''(''t''))}} increases rapidly, however, current and power consumption will quickly stop.

{{math|''M''(''q'')}} is physically restricted to be positive for all values of {{mvar|q}} (assuming the device is passive and does not become [[superconductive]] at some {{mvar|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 [[Semiconductor device modeling|device modeling]].<ref name="muthuswamyBanerjee2019">{{cite book|last1=Muthuswamy|first1=Bharathwaj|last2=Banerjee|first2=Santo|title=Introduction to Nonlinear Circuits and Networks|date=2019|publisher=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">{{cite report| author=Paul L. Penfield Jr. |contribution=1. Frequency-Power Formulas for Josephson Junctions |title=V. Microwave and Millimeter Wave Techniques |date=1974 |id=QPR No. 113 |pages=31–32|url=https://dspace.mit.edu/bitstream/handle/1721.1/56460/RLE_QPR_113_V.pdf }}</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 display="block">\begin{align}
             i_M(v) &= \epsilon\cos(\phi_0)v \\
&=W(\phi_0)v
\end{align}</math>
where {{mvar|ϵ}} is a constant based on the physical superconducting materials, {{mvar|v}} is the voltage across the junction and {{mvar|i{{sub|M}}}} 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">{{Citation|last=Langenberg|first=D. N.|title=Physical Interpretation of the <math>\cos\phi</math> term and implications for detectors|journal=Revue de Physique Appliquée|date=1974|volume=9|pages=35–40|doi=10.1051/rphysap:019740090103500|url=https://hal.archives-ouvertes.fr/jpa-00243770/file/ajp-rphysap_1974_9_1_35_0.pdf }}</ref><ref name="pedersen1972">{{Citation|last=Pedersen|first=N.F.|display-authors=et. al.|title=Magnetic field dependence and Q of the Josephson plasma resonance|journal=Physical Review B|date=1972|volume=11|issue=6|pages=4151–4159|doi=10.1103/PhysRevB.6.4151|bibcode=1972PhRvB...6.4151P|url=https://backend.orbit.dtu.dk/ws/files/3534398/NF.pdf}}</ref><ref name="pedersen1974">{{cite book | last1=Pedersen | first1=N. F. | last2=Finnegan | first2=T. F. | last3=Langenberg | first3=D. N. | title=Low Temperature Physics-LT 13 | chapter=Evidence for the Existence of the Josephson Quasiparticle-Pair Interference Current | publisher=Springer US | publication-place=Boston, MA | date=1974 | doi=10.1007/978-1-4684-2688-5_52 | pages=268–271| isbn=978-1-4684-2690-8 }}</ref><ref name="thompson1973">{{Citation|last=Thompson|first=E.D.|title=Power flow for Josephson Elements|journal=IEEE Trans. Electron Devices|date=1973|volume=20|issue=8|pages=680–683|doi=10.1109/T-ED.1973.17728|bibcode=1973ITED...20..680T}}</ref> A more comprehensive approach to extracting this phase-dependent conductance appeared with Peotta and Di Ventra's seminal paper in 2014.<ref name="peottaDiVentra2014">{{Citation|last1=Peotta|first1=A.|last2=Di Ventra|first2=M.|title=Superconducting Memristors|journal=Physical Review Applied|date=2014|volume=2|issue=3|pages=034011-1-034011-10|doi=10.1103/PhysRevApplied.2.034011|arxiv=1311.2975|bibcode=2014PhRvP...2c4011P|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 {{slink||Theory}}.

A discharge tube can be modelled as a memristive device, with resistance being a function of the number of conduction electrons {{math|''n''{{sub|e}}}}.<ref name="chua76"/>

<math display="block">\begin{align}
v_\mathrm{M} &= R(n_\mathrm{e})i_\mathrm{M} \\
\frac{\mathrm{d}n_\mathrm{e}}{\mathrm{d}t} &= \beta n+\alpha R(n_\mathrm{e})i^2_\mathrm{M}
\end{align}</math>

{{math|''v''{{sub|M}}}} is the voltage across the discharge tube, {{math|''i''{{sub|M}}}} is the current flowing through it, and{{math|''n''{{sub|e}}}} is the number of conduction electrons. A simple memristance function is {{math|1=''R''(''n''{{sub|e}}) = ''F''/''n''{{sub|e}}}}. The parameters {{mvar|α}}, {{mvar|β}}, and {{mvar|F}} depend on the dimensions of the tube and the gas fillings. An [[#Experimental tests|experimental]] identification of memristive behaviour is the "pinched hysteresis loop" in the {{mvar|v-i}} plane.{{efn|For an experiment that shows such a characteristic for a common discharge tube, see {{cite AV media |people= Bharathwaj Muthuswamy |date= 2013-10-03 |title= A physical memristor Lissajous figure |via= YouTube |url= https://www.youtube.com/watch?v=hDfJoXrCSxk }} The video also illustrates how to understand deviations in the pinched hysteresis characteristics of physical memristors.}}<ref name="muthuswamy2014">{{cite book |doi=10.1109/ISCAS.2014.6865179 |chapter=Memristor modelling |title=2014 IEEE International Symposium on Circuits and Systems (ISCAS) |pages=490–493 |date=2014 |last1=Muthuswamy |first1=B. |last2=Jevtic |first2=J. |last3=Iu |first3=H. H. C. |last4=Subramaniam |first4=C. K. |last5=Ganesan |first5=K. |last6=Sankaranarayanan |first6=V. |last7=Sethupathi |first7=K. |last8=Kim |first8=H. |last9=Shah |first9=M. Pd. |last10=Chua |first10=L. O. |isbn=978-1-4799-3432-4 |s2cid=13061426 }}</ref><ref name="sah2015">{{Citation|last=Sah|first=M.|display-authors=et. al.|title=A Generic Model of Memristors with Parasitic Components|journal=IEEE TCAS I: Regular Papers|date=2015|volume=62|issue=3|pages=891–898}}</ref>

Thermistors can be modeled as memristive devices:<ref name="sah2015"/>

<math display="block">\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{\mathrm{d}T}{\mathrm{d}t}&=\frac{1}{C}\left[-\delta\cdot(T-T_0)+R(T)i^2\right]
\end{align}
</math>

{{mvar|β}} is a material constant, {{mvar|T}} is the absolute body temperature of the thermistor, {{math|''T''{{sub|0}}}} is the ambient temperature (both temperatures in Kelvin), {{math|''R''{{sub|0}}(''T''{{sub|0}})}} denotes the cold temperature resistance at {{math|1=''T'' = ''T''{{sub|0}}}}, {{mvar|C}} is the heat capacitance and {{mvar|δ}} is the dissipation constant for the thermistor.

A fundamental phenomenon that has hardly been studied is memristive behaviour in [[p-n junctions]].<ref name="chuaTseng74">{{Citation|last1=Chua|first1=L. O.|last2=Tseng|first2=C.|title=A memristive circuit model for p-n junction diodes|journal=[[International Journal of Circuit Theory and Applications]]|date=1974|volume=2|issue=4|pages=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 ''{{dfn|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">{{cite journal|last1=Strukov|first1=Dmitri B.|last2=Snider|first2=Gregory S.|last3=Stewart|first3=Duncan R.|last4=Williams|first4=R. Stanley|title=The missing memristor found|journal=[[Nature (journal)|Nature]]|date=2008|volume=453|issue=7191|pages=80–83|doi=10.1038/nature06932|pmid=18451858|url=http://www.ece.ucsb.edu/~strukov/papers/2008/Nature2008.pdf|bibcode=2008Natur.453...80S|s2cid=4367148}}</ref><ref>{{Citation|title=Memristor FAQ |url=http://www.hpl.hp.com/news/2008/apr-jun/memristor_faq.html |publisher=[[Hewlett-Packard]] |access-date=2010-09-03 }}</ref>

The HP Labs result was published in the scientific journal ''[[Nature (journal)|Nature]]''.<ref name="Williams08"/><ref name="Williams - Spectrum 2008-12">{{cite journal |last1=Williams |first1=R. S. |title=How We Found The Missing Memristor |journal=IEEE Spectrum |date=2008 |volume=45 |issue=12 |pages=28–35 |doi=10.1109/MSPEC.2008.4687366 |s2cid=27319894 |url=http://personal.delen.polito.it/mario.biey/SupportiDidattici/rnl/memristor/spectrum08.pdf |access-date=2018-03-26  |archive-date=2018-03-26  |archive-url=https://web.archive.org/web/20180326202533/http://personal.delen.polito.it/mario.biey/SupportiDidattici/rnl/memristor/spectrum08.pdf |url-status=dead }}</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">{{cite journal |last1=Chua |first1=Leon |title=Resistance switching memories are memristors |journal=Applied Physics A |date=28 January 2011 |volume=102 |issue=4 |pages=765–783 |doi=10.1007/s00339-011-6264-9 |bibcode=2011ApPhA.102..765C |doi-access=free }}</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">{{Citation |last=Clarke |first=P. |date=23 May 2012 |title=Memristor is 200 years old, say academics |url=http://www.eetimes.com/electronics-news/4373652/Academics-Memristor-is-200-years-old |work=[[EE Times]] |access-date=2012-05-25 }}</ref> However, there are doubts as to whether a memristor can actually exist in physical reality.<ref name="Meuffels_2012"/><ref name="DiVentra_2013"/><ref name="Sundqvist_2017">{{cite journal |last1=Sundqvist |first1=Kyle M. |last2=Ferry |first2=David K. |last3=Kish |first3=Laszlo B. |title=Memristor Equations: Incomplete Physics and Undefined Passivity/Activity |journal=Fluctuation and Noise Letters |date=21 November 2017 |volume=16 |issue=4 |pages=1771001–519 |doi=10.1142/S0219477517710018 |arxiv=1703.09064 |bibcode=2017FNL....1671001S |s2cid=1408810 }}</ref><ref>{{Cite journal|last=Abraham|first=Isaac|date=2018-07-20|title=The case for rejecting the memristor as a fundamental circuit element|journal=Scientific Reports|volume=8 |issue=1 |pages=10972|doi=10.1038/s41598-018-29394-7|pmc=6054652|pmid=30030498|bibcode=2018NatSR...810972A}}</ref> Additionally, some experimental evidence contradicts Chua's generalization since a non-passive [[nanobatteries|nanobattery]] effect is observable in resistance switching memory.<ref name="memristor_nanobattery">{{Citation|last=Valov |first=I. |date=2013 |title=Nanobatteries in redox-based resistive switches require extension of memristor theory |journal=[[Nature Communications]]|doi=10.1038/ncomms2784|arxiv=1303.2589 |bibcode=2013NatCo...4.1771V |volume=4 |issue=4 |pmid=23612312|display-authors=etal |pages=1771 |pmc=3644102}}</ref> A simple test has been proposed by Pershin and Di Ventra<ref name="Pershin_2018">{{cite journal|last1=Pershin|first1=Y. V.|last2=Di Ventra|first2=M.|date=2019|title=A simple test for ideal memristors|journal=Journal of Physics D: Applied Physics|volume=52|issue=1|pages=01LT01|arxiv=1806.07360|doi=10.1088/1361-6463/aae680|bibcode=2019JPhD...52aLT01P|s2cid=53506924}}</ref> to analyze whether such an ideal or generic memristor does actually exist or is a purely mathematical concept. Up to now,{{When|date=September 2024}} there seems to be no experimental resistance switching device ([[ReRAM]]) which can pass the test.<ref name="Pershin_2018"/><ref name="Kim_2019">{{cite journal|last1=Kim|first1=J.|last2=Pershin|first2=Y. V.|last3=Yin|first3=M.|last4=Datta|first4=T.|last5=Di Ventra|first5=M.|date=2019|title=An experimental proof that resistance-switching memories are not memristors| journal=Advanced Electronic Materials|doi=10.1002/aelm.202000010|doi-access=free |arxiv=1909.07238|s2cid=202577242}}</ref>

These devices are intended for applications in [[nanoelectronic]] memory devices, computer logic, and [[neuromorphic]]/neuromemristive computer architectures.<ref>{{Citation |last=Marks |first=P. |date=30 April 2008 |title=Engineers find 'missing link' of electronics |url=https://www.newscientist.com/article/dn13812 |work=[[New Scientist]] |access-date=2008-04-30 }}</ref><ref>{{Cite journal|last1=Zidan|first1=Mohammed A.|last2=Strachan|first2=John Paul|last3=Lu|first3=Wei D.|date=2018-01-08|title=The future of electronics based on memristive systems|journal=Nature Electronics|volume=1|issue=1|pages=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">{{Citation |date=1 November 2013|title=HP 100TB Memristor drives by 2018 – if you're lucky, admits tech titan|url=https://www.theregister.co.uk/2013/11/01/hp_memristor_2018/}}</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">{{Citation|url=http://www.hrl.com/hrlDocs/pressreleases/2012/prsRls_120323.html|title=Artificial synapses could lead to advanced computer memory and machines that mimic biological brains |publisher=[[HRL Laboratories]] |date=23 March 2012 |access-date=30 March 2012}}</ref>

[[Image:Memristor.jpg|thumb|right|225px|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 {{val|50|u=nm}}, or 150 atoms, wide.<ref>{{Citation|last=Bush |first=S. |date=2 May 2008 |title=HP nano device implements memristor |url=http://www.electronicsweekly.com/Articles/2008/05/02/43658/hp-nano-device-implements-memristor.htm |work=[[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, [[Leon Chua|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 [[Magnetoresistive RAM|MRAM]], [[phase-change memory]] and [[ReRAM]] are memristor technologies.<ref name="Mellor2011">{{Citation |last=Mellor |first=C. |date=10 October 2011|title=HP and Hynix to produce the memristor goods by 2013|url=https://www.theregister.co.uk/2011/10/10/memristor_in_18_months/ |work=The Register |access-date=2012-03-07}}</ref> Some researchers argued that biological structures such as blood<ref name="Courtland2011">{{cite web |last=Courtland |first=R. |date=1 April 2011 |title=Memristors...Made of Blood?|url=https://spectrum.ieee.org/blood-memristor |work=[[IEEE Spectrum]] |access-date=2012-03-07}}</ref> and skin<ref name="G K Johnsen et al">{{cite journal|last1=Johnsen|first1=G. K.|title=Memristive model of electro-osmosis in skin|journal=Physical Review E|date=24 March 2011|volume=83|issue=3|pages=031916|doi=10.1103/PhysRevE.83.031916|pmid=21517534|bibcode=2011PhRvE..83c1916J|s2cid=46437206}}</ref><ref name="McAlpine2011">{{Citation |last=McAlpine |first=K. |date=2 March 2011 |title=Sweat ducts make skin a memristor|url=https://www.newscientist.com/article/mg20928024.500-sweat-ducts-make-skin-a-memristor.html |journal=[[New Scientist]] |volume=209 |issue=2802 |pages=16 |access-date=2012-03-07|bibcode=2011NewSc.209...16M |doi=10.1016/S0262-4079(11)60481-8 }}</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">{{Citation |last=Clarke |first=P. |date=16 January 2012 |title=Memristor brouhaha bubbles under |url=http://www.eetimes.com/electronics-news/4234678/Memristor-brouhaha-bubbles-under |work=[[EETimes]] |access-date=2012-03-02 }}</ref> and that a broader definition of memristor is a scientifically unjustifiable [[land grabbing|land grab]] that favored HP's memristor patents.<ref name="PaulMarks2012">{{Citation |last=Marks |first=P. |date=23 February 2012 |title=Online spat over who joins memristor club|url=https://www.newscientist.com/article/mg21328535.200-online-spat-over-who-joins-memristor-club.html |work=[[New Scientist]] |access-date=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">
{{citation
 |last1=Meuffels |first1=P.
 |last2=Schroeder |first2=H.
 |date=2011
 |journal=[[Applied Physics A]]
 |title=Comment on "Exponential ionic drift: fast switching and low volatility of thin-film memristors" by D.&nbsp;B. Strukov and R.&nbsp;S. Williams in Appl. Phys. A (2009) 94: 515–519
 |doi=10.1007/s00339-011-6578-7
 |volume=105
 |issue=1
 |pages=65–67
 |bibcode=2011ApPhA.105...65M
|s2cid=95168959
 }}</ref> In 2012, Meuffels and Soni discussed some fundamental issues and problems in the realization of memristors.<ref name="Meuffels_2012">{{cite arXiv|last1=Meuffels|first1=P.|last2=Soni|first2=R.|date=2012|title=Fundamental Issues and Problems in the Realization of Memristors|eprint=1207.7319|class=cond-mat.mes-hall}}</ref> They indicated inadequacies in the electrochemical modeling presented in the ''[[Nature (journal)|Nature]]'' article "The missing memristor found"<ref name="Williams08"/> because the impact of [[concentration polarization]] effects on the behavior of metal−[[TiO2|TiO<sub>2−''x''</sub>]]−metal structures under voltage or current stress was not considered.<ref name="memristor_nanobattery"/>

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 ''{{dfn|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">{{Citation|last1=Di Ventra|first1=M. |last2=Pershin |first2=Y. V.|title=On the physical properties of memristive, memcapacitive and meminductive systems|journal=[[Nanotechnology (journal)|Nanotechnology]]|date=2013|volume=24|issue=25|pages=255201|doi=10.1088/0957-4484/24/25/255201|pmid=23708238|arxiv=1302.7063 |bibcode=2013Nanot..24y5201D|citeseerx=10.1.1.745.8657 |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 [[Thermodynamic free energy|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 rate|bit-error probability]] of bit operation.<ref name="Kish_2014">
{{cite journal|last1=Kish|first1=Laszlo B.|last2=Granqvist|first2=Claes G.|last3=Khatri|first3=Sunil P.|last4=Wen|first4=He|title=Demons: Maxwell's demon, Szilard's engine and Landauer's erasure–dissipation|journal=International Journal of Modern Physics: Conference Series|date=2014|volume=33|pages=1460364|doi=10.1142/s2010194514603640|arxiv=1412.2166|bibcode=2014IJMPS..3360364K|s2cid=44851287 }}
</ref> Consequently, there is always a lower limit of energy requirement – depending on the required [[Bit error rate|bit-error probability]] – for intentionally changing a bit value in any memory device.<ref name="Kish_2014"/><ref name="Kish_2015">{{cite book |doi=10.1109/ICNF.2015.7288632 |chapter=Critical remarks on Landauer's principle of erasure-dissipation: Including notes on Maxwell demons and Szilard engines |title=2015 International Conference on Noise and Fluctuations (ICNF) |pages=1–4 |date=2015 |last1=Kish |first1=L. B. |last2=Khatri |first2=S. P. |last3=Granqvist |first3=C. G. |last4=Smulko |first4=J. M. |isbn=978-1-4673-8335-6 }}
</ref>

In the general concept of memristive system the defining equations are (see {{slink||Theory}}):
<math display="block">\begin{align}
              y(t) &= g(\mathbf{x},u,t) u(t), \\
  \dot{\mathbf{x}} &= f(\mathbf{x},u,t),
\end{align}</math>
where {{math|''u''(''t'')}} is an input signal, and {{math|''y''(''t'')}} is an output signal. The vector <math>\mathbf{x}</math> represents a set of {{mvar|n}} state variables describing the different internal memory states of the device. <math>\dot{\mathbf{x}}</math> is the time-dependent rate of change of the state vector <math>\mathbf{x}</math> 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 {{math|''u''(''t'')}} can be superposed with a stochastic term {{math|''ξ''(''t'')}}, which takes into account the existence of inevitable [[thermal fluctuations]]. The dynamic state equation in its general form then finally reads:
<math display="block">\dot{\mathbf{x}} = f(\mathbf{x}, u(t) + \xi(t), t),</math>
where {{math|''ξ''(''t'')}} is, e.g., white [[Gaussian noise|Gaussian]] [[Noise (electronics)|current or voltage noise]]. On the basis 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., whether the system would 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">{{Citation|last1=Slipko|first1=V. A. |last2=Pershin |first2=Y. V.|last3=Di Ventra|first3=M.|title=Changing the state of a memristive system with white noise|journal=[[Physical Review E]]|date=2013|volume=87|issue=1 |pages=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 [[Probability|probabilistic]] nature. The probability for a transition from state {{math|{'''i'''} }} to state {{math|{'''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 {{math|{'''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 {{math|{'''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{{snd}}as one is dealing with probabilistic processes{{snd}}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 {{math|{'''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 heating|Joule heat]] to the surrounding, so no extra energy remains in the system after it has been brought from one resistance state {{math|'''x'''{{sub|''i''}}}} to another one {{math|'''x'''{{sub|''j''}}}}. Thus, the [[internal energy]] of the memristor device in state {{math|'''x'''{{sub|''i''}}}}, {{math|''U''(''V'', ''T'', '''x'''{{sub|''i''}})}}, would be the same as in state {{math|'''x'''{{sub|''j''}}}}, {{math|''U''(''V'', ''T'', '''x'''{{sub|''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 mobility|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">{{citation
 |last1=Hashem
 |first1=N.
 |last2=Das
 |first2=S.
 |date=2012
 |journal=[[Applied Physics Letters]]
 |title=Switching-time analysis of binary-oxide memristors via a non-linear model
 |url=http://www.mitre.org/sites/default/files/pdf/11_3987.pdf
 |archive-url=https://web.archive.org/web/20150924045230/http://www.mitre.org/sites/default/files/pdf/11_3987.pdf
 |url-status=dead
 |archive-date=24 September 2015
 |access-date=2012-08-09
 |doi=10.1063/1.4726421
 |volume=100
 |issue=26
 |pages=262106
 |bibcode=2012ApPhL.100z2106H
 }}</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">{{cite journal|last1=Linn |first1=E. |last2=Siemon |first2=A. |last3=Waser |first3=R. |last4=Menzel |first4=S. |date=23 March 2014 |title=Applicability of Well-Established Memristive Models for Simulations of Resistive Switching Devices  |arxiv=1403.5801|doi=10.1109/TCSI.2014.2332261|journal=IEEE Transactions on Circuits and Systems I: Regular Papers|volume=61|issue=8|pages=2402–2410|bibcode=2014arXiv1403.5801L |s2cid=18673562 |author3-link=Rainer Waser }}</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">{{Citation |last=Garling |first=C. |date=25 July 2012 |title=Wonks question HP's claim to computer-memory missing link|url=https://www.wired.com/wiredenterprise/2012/07/memristors/ |work=Wired.com |access-date=2012-09-23 }}</ref>

===Experimental tests===
[[Leon Chua|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">{{Citation |last=Chua |first=L. |date=13 June 2012 |title=Memristors: Past, Present and future |url=http://sti.epfl.ch/files/content/sites/sti/files/shared/sel/pdf/Abstract_Prof_Chua.pdf |access-date=2013-01-12 |archive-url=https://web.archive.org/web/20140308025032/http://sti.epfl.ch/files/content/sites/sti/files/shared/sel/pdf/Abstract_Prof_Chua.pdf |archive-date=8 March 2014 |url-status=dead }}</ref><ref name="MemristorExperimentalTests2">{{Citation
 |last1=Adhikari |first1=S. P.
 |last2=Sah |first2=M. P.
 |last3=Hyongsuk  |first3=K.
 |last4=Chua |first4=L. O.
 |date=2013
 |title=Three Fingerprints of Memristor
 |journal=[[IEEE Transactions on Circuits and Systems I]]
 |volume=60 |issue=11 |pages=3008–3021
 |doi=10.1109/TCSI.2013.2256171
|s2cid=12665998
 }}</ref> all resistive switching memories including [[ReRAM]], [[Magnetoresistive RAM|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 [[nanobatteries|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 {{math|''R''{{sub|on}} ≪ ''R''{{sub|off}}}} the memristance function was determined to be
<math display="block">M(q(t)) = R_\mathrm{off} \cdot \left(1-\frac{\mu_{v}R_\mathrm{on}}{D^2} q(t)\right)</math>
where {{math|''R''{{sub|off}}}} represents the high resistance state, {{math|''R''{{sub|on}}}} represents the low resistance state, {{mvar|μ{{sub|v}}}} represents the mobility of dopants in the thin film, and {{mvar|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 {{math|''R''{{sub|on}}}} to {{math|''R''{{sub|off}}}} in time {{math|''T''{{sub|on}}}} to {{math|''T''{{sub|off}}}}, the charge must change by {{math|1=Δ''Q'' = ''Q''{{sub|on}} − ''Q''{{sub|off}}}}.

<math display="block">\begin{align}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\end{align}</math>

Substituting {{math|1=''V'' = ''I''(''q'')''M''(''q'')}}, and then {{math|1={{intmath|int}}d''q''/''V'' = ∆''Q''/''V''}} for constant {{mvar|V}} to 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 {{math|''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 {{mvar|n}}-th order memristive system the defining equations are

<math display="block">\begin{align}
              y(t) &= g(\textbf{x},u,t)u(t), \\
  \dot{\textbf{x}} &= f(\textbf{x},u,t)
\end{align}</math>

where {{math|''u''(''t'')}} is an input signal, {{math|''y''(''t'')}} is an output signal, the vector {{math|'''x'''}} represents a set of {{mvar|n}} state variables describing the device, and {{mvar|g}} and {{mvar|f}} are [[continuous functions]]. For a current-controlled memristive system the signal {{math|''u''(''t'')}} represents the current signal {{math|''i''(''t'')}} and the signal {{math|''y''(''t'')}} represents the voltage signal {{math|''v''(''t'')}}. For a voltage-controlled memristive system the signal {{math|''u''(''t'')}} represents the voltage signal {{math|''v''(''t'')}} and the signal {{math|''y''(''t'')}} represents the current signal {{math|''i''(''t'')}}.

The {{em|pure}} memristor is a particular case of these equations, namely when {{mvar|x}} depends only on charge ({{math|1='''x''' = ''q''}}) and since the charge is related to the current via the time derivative {{math|1=d''q''/d''t'' = ''i''(''t'')}}. Thus for {{em|pure}} memristors {{mvar|f}} (i.e. the rate of change of the state) must be equal or proportional to the current {{math|''i''(''t'')}}.

===Pinched hysteresis===
[[File:Pinched crossing hysteresis.png|thumb|right|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">
{{citation
 |last1=Pershin |first1=Y. V.
 |last2=Di Ventra |first2=M.
 |date=2011
 |journal=[[Advances in Physics]]
 |title=Memory effects in complex materials and nanoscale systems
 |volume=60 |issue=2 |pages=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">
{{citation
 |last1=Biolek |first1=D.
 |last2=Biolek |first2=Z.
 |last3=Biolkova |first3=V.
 |date=2011
 |journal=[[Electronics Letters]]
 |title=Pinched hysteresis loops of ideal memristors, memcapacitors and meminductors must be 'self-crossing'
 |volume=47 |issue=25 |pages=1385–1387
 |doi=10.1049/el.2011.2913
|bibcode=2011ElL....47.1385B
 }}</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|Caravelli–Traversa–Di Ventra equation]], CTDV)<ref>{{cite journal |last=Caravelli  |display-authors=etal|arxiv=1608.08651 |title=The complex dynamics of memristive circuits: analytical results and universal slow relaxation |year=2017 |doi=10.1103/PhysRevE.95.022140 |pmid= 28297937 |volume=95 |issue= 2 |pages= 022140 |journal=Physical Review E|bibcode=2017PhRvE..95b2140C |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>{{cite journal |last=Caravelli  |display-authors=etal|arxiv=1608.08651 |title=Global minimization via classical tunnelling assisted by collective force field formation |year=2021 |doi=10.1126/sciadv.abh1542 |pmid= 28297937 |volume=7 |issue=52 |journal=Science Advances|page=022140 |bibcode=2021SciA....7.1542C |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">
{{cite arXiv
 |last=Mouttet |first=B.
 |date=2012
 |title=Memresistors and non-memristive zero-crossing hysteresis curves
 |eprint=1201.2626
 |class=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 function]]s. 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>

== Implementation of hysteretic current-voltage memristors ==
There exist implementations of memristors with a hysteretic current-voltage curve or with both hysteretic current-voltage curve and hysteretic flux-charge curve [arXiv:2403.20051]. Memristors with hysteretic current-voltage curve use a resistance dependent on the history of the current and voltage and bode well for the future of memory technology due to their simple structure, high energy efficiency, and high integration [DOI: 10.1002/aisy.202200053].

=== 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>
{{Citation
 |last=Fildes |first=J.
 |date=13 November 2007
 |title=Getting More from Moore's Law
 |url=http://news.bbc.co.uk/2/hi/technology/7080772.stm
 |publisher=[[BBC News]]
 |access-date=2008-04-30
}}</ref><ref>
{{Citation
 |last=Taylor |first=A. G.
 |date=2007
 |title=Nanotechnology in the Northwest
 |url=http://www.ieee-or.org/beeep/2007/sep/beeep_sep07.pdf
 |journal=Bulletin for Electrical and Electronic Engineers of Oregon
 |volume=51
 |issue=1 |page=1
}}</ref><ref>{{Citation
 |title=Stanley Williams
 |url=http://www.hpl.hp.com/people/stan_williams/
 |publisher=[[Hewlett-Packard|HP Labs]]
 |access-date=2011-03-20
 |archive-url=https://web.archive.org/web/20110719014357/http://www.hpl.hp.com/people/stan_williams/
 |archive-date=2011-07-19
 |url-status=dead
 }}</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 [[Nanoscopic scale|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|TiO<sub>2</sub>]] memristor, the resistance switching characteristics of titanium dioxide were originally described in the 1960s.<ref name="Argall1968">
{{Citation
 |last=Argall |first=F.
 |date=1968
 |title=Switching Phenomena in Titanium Oxide Thin Films
 |journal=[[Solid-State Electronics]]
 |volume=11 |issue=5 |pages=535–541
 |doi=10.1016/0038-1101(68)90092-0
 |bibcode=1968SSEle..11..535A
}}</ref>

The HP device is composed of a thin (50 [[nanometer|nm]]) [[titanium dioxide]] film between two 5&nbsp;nm thick [[electrode]]s, 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 carrier]]s, 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">
{{citation
 |last1=Terabe |first1=K.
 |last2=Hasegawa |first2=T.
 |last3=Liang |first3=C.
 |last4=Aono |first4=M.
 |date=2007
 |title=Control of local ion transport to create unique functional nanodevices based on ionic conductors
 |journal=[[Science and Technology of Advanced Materials]]
 |volume=8 |issue=6 |pages=536–542
 |doi=10.1016/j.stam.2007.08.002
 |bibcode=2007STAdM...8..536T
|doi-access=free
 }}</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''&nbsp;d''t'', 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">
{{citation
 |last1=Beck |first1=A.
 |date=2000
 |title=Reproducible switching effect in thin oxide films for memory applications
 |journal=[[Applied Physics Letters]]
 |volume=77 |issue=1
 |page=139
 |doi=10.1063/1.126902
 |bibcode=2000ApPhL..77..139B
|display-authors=etal}}</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" {{US Patent |7417271}}</ref>

In April 2010, HP labs announced that they had practical memristors working at 1 [[Nanosecond|ns]] (~1&nbsp;GHz) switching times and 3&nbsp;nm by 3&nbsp;nm sizes,<ref>
{{Citation
 |title=Finding the Missing Memristor - R. Stanley Williams
 | date=21 January 2010 |url=https://www.youtube.com/watch?v=bKGhvKyjgLY
}}</ref> which bodes well for the future of the technology.<ref>
{{Citation
 |last=Markoff |first=J.
 |date=7 April 2010
 |title=H.P. Sees a Revolution in Memory Chip
 |url=https://www.nytimes.com/2010/04/08/science/08chips.html?hpw
 |work=[[New York Times]]
}}</ref> At these densities it could easily rival the current sub-25&nbsp;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>{{cite journal |last1= Kavehei|first1=O. |last2= Iqbal|first2=A. |last3= Kim|first3=Y.S. |last4= Eshraghian|first4=K. |last5= Al-Sarawi|first5=S. F. | last6= Abbott|first6=D. |title= The fourth element: characteristics, modelling and electromagnetic theory of the memristor |journal= Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
|date=2010 |volume= 466 |issue= 2120 |pages= 2175–2202|doi=10.1098/rspa.2009.0553 |arxiv=1002.3210 |bibcode=2010RSPSA.466.2175K |s2cid=7625839 }}</ref>

However, hysteretic conductance in silicon was associated to memristive effects only in 2009.<ref>{{ citation | last1= Ben-Jamaa | first1= M. H. | last2= Carrara | first2=S. | last3= Georgiou|first3=J. | last4= Archontas|first4=N. | last5= De Micheli |first5=G. | title= Fabrication of memristors with poly-crystalline silicon nanowires | journal= Proceedings of 9th IEEE Conference on Nanotechnology | date=2009 | volume=1|issue=1|pages=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>{{cite journal |last1= Mehonic |first1=A. |last2= Cueff |first2=S. |last3= Wojdak |first3=M., … |last4= Kenyon |first4= A. J. |title= Resistive switching in silicon suboxide films |journal= Journal of Applied Physics |date=2012 |volume=111|issue=7 |pages=074507–074507–9 |doi= 10.1063/1.3701581|bibcode=2012JAP...111g4507M |url=https://hal.archives-ouvertes.fr/hal-01148232v2/file/Memristor%20manuscript%20_%20JAP_FINAL.pdf }}</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>
{{Citation
 |last1=Krieger |first1=J. H.
 |last2=Spitzer |first2=S. M.
 |date=2004
 |chapter=Non-traditional, Non-volatile Memory Based on Switching and Retention Phenomena in Polymeric Thin Films
 |title=Proceedings of the 2004 Non-Volatile Memory Technology Symposium
 |page=121
 |publisher=[[IEEE]]
 |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">
{{cite arXiv
 |last1=Erokhin |first1=V.
 |last2=Fontana |first2=M. P.
 |date=2008
 |title=Electrochemically controlled polymeric device: A memristor (and more) found two years ago
 |class=cond-mat.soft
 |eprint=0807.0333
}}</ref>

In 2010, Alibart, Gamrat, Vuillaume et al.<ref>{{cite journal | last1 = An | last2 = Alibart | first2 = F. | last3 = Pleutin | first3 = S. | last4 = Guerin | first4 = D. | last5 = Novembre | first5 = C. | last6 = Lenfant | first6 = S. | last7 = Lmimouni | first7 = K. | last8 = Gamrat | first8 = C. | last9 = Vuillaume | first9 = D. | date = 2010 | title =  An Organic Nanoparticle Transistor Behaving as a Biological Spiking Synapse| doi=10.1002/adfm.200901335 | journal = Advanced Functional Materials | volume = 20 | issue = 2| pages = 330–337 | arxiv = 0907.2540 | s2cid = 16335153 }}</ref> introduced a new hybrid organic/[[nanoparticle]] device (the [[NOMFET]] : Nanoparticle Organic Memory Field Effect Transistor), which behaves as a memristor<ref>{{cite journal | last1 = Alibart | first1 = F. | last2 = Pleutin | first2 = S. | last3 = Bichler | first3 = O. | last4 = Gamrat | first4 = C. | last5 = Serrano-Gotarredona | first5 = T. | last6 = Linares-Barranco | first6 = B. | last7 = Vuillaume | first7 = D. | date = 2012 | title =  A Memristive Nanoparticle/Organic Hybrid Synapstor for Neuroinspired Computing|doi=10.1002/adfm.201101935 | journal = Advanced Functional Materials | volume = 22 | issue = 3| pages = 609–616 | arxiv = 1112.3138 | 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>{{cite journal | last1 = Pavlov's | last2 = Transistors | first2 = Organic | last3 = Bichler | first3 = O. | last4 = Zhao | first4 = W. | last5 = Alibart | first5 = F. | last6 = Pleutin | first6 = S. | last7 = Lenfant | first7 = S. | last8 = Vuillaume | first8 = D. | last9 = Gamrat | first9 = C. | date = 2013 | title =  Pavlov's Dog Associative Learning Demonstrated on Synaptic-Like Organic Transistors| journal = Neural Computation | volume = 25 | issue = 2| pages = 549–566 | doi=10.1162/NECO_a_00377| pmid =  22970878| arxiv = 1302.3261 | bibcode = 2013arXiv1302.3261B | 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>
{{Citation
 |last1=Crupi |first1=M.
 |last2=Pradhan |first2=L.
 |last3=Tozer |first3=S.
 |title=Modelling Neural Plasticity with Memristors
 |journal=IEEE Canadian Review
 |url=http://www.ieee.ca/canrev/cr68/IEEECanadianReview_no68.pdf
 |volume=68 |pages=10–14
 |date=2012
}}</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>{{cite journal |last1=Erokhin |first1=V. |last2=Berzina |first2=T. |last3=Gorshkov |first3=K. |last4=Camorani |first4=P. |last5=Pucci |first5=A. |last6=Ricci |first6=L. |last7=Ruggeri |first7=G. |last8=Signala |first8=R. |last9=Schüz |first9=A. |title=Stochastic hybrid 3D matrix: learning and adaptation of electrical properties |journal=Journal of Materials Chemistry |date=2012 |volume=22 |issue=43 |page=22881 |doi=10.1039/C2JM35064E }}</ref>

===Layered memristor===
In 2014, Bessonov et al. reported a flexible memristive device comprising a [[Molybdenum oxide|MoO<sub>x</sub>]]/[[MoS2|MoS<sub>2</sub>]] heterostructure sandwiched between silver electrodes on a plastic foil.<ref name="flexible_memristor">
{{citation
 |last=Bessonov |first=A. A.
 |date=2014
 |title=Layered memristive and memcapacitive switches for printable electronics
 |journal=[[Nature Materials]]
 |doi=10.1038/nmat4135
|display-authors=etal|bibcode = 2015NatMa..14..199B
 |pmid=25384168
 |volume=14
 |issue=2
 |pages=199–204}}</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, [[Transparency and translucency|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>{{Cite journal|last1=Ge|first1=Ruijing|last2=Wu|first2=Xiaohan|last3=Kim|first3=Myungsoo|last4=Shi|first4=Jianping|last5=Sonde|first5=Sushant|last6=Tao|first6=Li|last7=Zhang|first7=Yanfeng|last8=Lee|first8=Jack C.|last9=Akinwande|first9=Deji|date=2017-12-19|title=Atomristor: Nonvolatile Resistance Switching in Atomic Sheets of Transition Metal Dichalcogenides|journal=Nano Letters|volume=18|issue=1|pages=434–441|doi=10.1021/acs.nanolett.7b04342|pmid=29236504|bibcode=2018NanoL..18..434G|doi-access=free}}</ref> in the [[Deji Akinwande|Akinwande]] group at the University of Texas, first reported a universal memristive effect in single-layer [[Transition metal dichalcogenide monolayers|TMD]] (MX<sub>2</sub>, 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 [[Boron nitride|hexagonal boron nitride]], which is the thinnest memory material of around 0.33&nbsp;nm.<ref>{{Cite journal |last1=Wu |first1=Xiaohan |last2=Ge |first2=Ruijing |last3=Chen |first3=Po-An |last4=Chou |first4=Harry |last5=Zhang |first5=Zhepeng |last6=Zhang |first6=Yanfeng |last7=Banerjee |first7=Sanjay |last8=Chiang |first8=Meng-Hsueh |last9=Lee |first9=Jack C. |last10=Akinwande |first10=Deji |date=April 2019 |title=Thinnest Nonvolatile Memory Based on Monolayer h-BN |journal=Advanced Materials |volume=31 |issue=15 |pages=1806790 |doi=10.1002/adma.201806790 |pmid=30773734 |bibcode=2019AdM....3106790W |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 [[Chemical vapor deposition|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 MoS<sub>2</sub> or h-BN atomristors, indicating a new application of memristors for [[5G]], [[6G (network)|6G]] and THz communication and connectivity systems.<ref>{{Cite journal|last1=Kim|first1=Myungsoo|last2=Ge|first2=Ruijing|last3=Wu|first3=Xiaohan|last4=Lan|first4=Xing|last5=Tice|first5=Jesse|last6=Lee|first6=Jack C.|last7=Akinwande|first7=Deji|date=2018|title=Zero-static power radio-frequency switches based on MoS<sub>2</sub> atomristors|journal=Nature Communications|volume=9|issue=1|pages=2524|doi=10.1038/s41467-018-04934-x|pmid=29955064|pmc=6023925|bibcode=2018NatCo...9.2524K}}</ref><ref>{{Cite journal |date=June 2022 |title=Towards zero-power 6G communication switches using atomic sheets |journal=Nature Electronics |volume=5 |issue=6 |pages=331–332 |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>{{Cite journal |last1=Hus |first1=Saban M. |last2=Ge |first2=Ruijing |last3=Chen |first3=Po-An |last4=Liang |first4=Liangbo |last5=Donnelly |first5=Gavin E. |last6=Ko |first6=Wonhee |last7=Huang |first7=Fumin |last8=Chiang |first8=Meng-Hsueh |last9=Li |first9=An-Ping |last10=Akinwande |first10=Deji |date=January 2021 |title=Observation of single-defect memristor in an MoS<sub>2</sub> atomic sheet |journal=Nature Nanotechnology |volume=16 |issue=1 |pages=58–62 |doi=10.1038/s41565-020-00789-w |pmid=33169008 |bibcode=2021NatNa..16...58H |s2cid=226285710 |url=https://pure.qub.ac.uk/en/publications/observation-of-singledefect-memristor-in-an-mos2-atomic-sheet(1dd31266-03d2-4ba0-b1b7-71b701f6a9d6).html }}</ref>

===Ferroelectric memristor===

The [[ferroelectric]] memristor<ref>
{{citation
 |last=Chanthbouala |first=A.
 |date=2012
 |title=A ferroelectric memristor
 |journal=[[Nature Materials]]
 |volume=11
 |pages=860–864
 |doi=10.1038/nmat3415
 |pmid=22983431
 |issue=10
 |arxiv=1206.3397
 |bibcode=2012NatMa..11..860C
|s2cid = 10372470|display-authors=etal}}</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: {{nowrap|R<sub>OFF</sub> ≫ R<sub>ON</sub>}} (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<sub>ON</sub> or R<sub>OFF</sub>, 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">{{cite journal|last1=Ageev|first1=O. A.|last2=Blinov|first2=Yu F.|last3=Il’in|first3=O. I.|last4=Kolomiitsev|first4=A. S.|last5=Konoplev|first5=B. G.|last6=Rubashkina|first6=M. V.|last7=Smirnov|first7=V. A.|last8=Fedotov|first8=A. A.|title=Memristor effect on bundles of vertically aligned carbon nanotubes tested by scanning tunnel microscopy|journal=Technical Physics|date=11 December 2013|volume=58|issue=12|pages=1831–1836|doi=10.1134/S1063784213120025|bibcode = 2013JTePh..58.1831A |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>{{cite journal|last1=Il'ina|first1=Marina V.|last2=Il'in|first2=Oleg I.|last3=Blinov|first3=Yuriy F.|last4=Smirnov|first4=Vladimir A.|last5=Kolomiytsev|first5=Alexey S.|last6=Fedotov|first6=Alexander A.|last7=Konoplev|first7=Boris G.|last8=Ageev|first8=Oleg A.|title=Memristive switching mechanism of vertically aligned carbon nanotubes|journal=Carbon|date=October 2017|volume=123|pages=514–524|doi=10.1016/j.carbon.2017.07.090|bibcode=2017Carbo.123..514I }}</ref> that CNT memristive switching is observed when a nanotube has a non-uniform elastic strain Δ''L''0. 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>{{Cite journal|last1=Park|first1=Youngjun|last2=Kim|first2=Min-Kyu|last3=Lee|first3=Jang-Sik|date=2020-07-16|title=Emerging memory devices for artificial synapses|journal=Journal of Materials Chemistry C|volume=8|issue=27|pages=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>{{Cite journal|last1=Raeis-Hosseini|first1=Niloufar|last2=Park|first2=Youngjun|last3=Lee|first3=Jang-Sik|date=2018|title=Flexible Artificial Synaptic Devices Based on Collagen from Fish Protein with Spike-Timing-Dependent Plasticity|journal=Advanced Functional Materials|volume=28|issue=31|pages=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>{{Cite journal|last1=Park|first1=Youngjun|last2=Lee|first2=Jang-Sik|date=2017-09-26|title=Artificial Synapses with Short- and Long-Term Memory for Spiking Neural Networks Based on Renewable Materials|journal=ACS Nano|volume=11|issue=9|pages=8962–8969|doi=10.1021/acsnano.7b03347|pmid=28837313}}</ref> furthermore a natural silk fibroin demonstrated memristive properties;<ref>{{Cite journal|last1=Hota|first1=Mrinal K.|last2=Bera|first2=Milan K.|last3=Kundu|first3=Banani|last4=Kundu|first4=Subhas C.|last5=Maiti|first5=Chinmay K.|date=2012|title=A Natural Silk Fibroin Protein-Based Transparent Bio-Memristor|journal=Advanced Functional Materials|volume=22|issue=21|pages=4493–4499|doi=10.1002/adfm.201200073|s2cid=137399893 }}</ref> spin-memristive systems based on biomolecules are also being studied.<ref>{{Cite journal|last1=Cardona-Serra|first1=Salvador|last2=Rosaleny|first2=Lorena E.|last3=Giménez-Santamarina|first3=Silvia|last4=Martínez-Gil|first4=Luis|last5=Gaita-Ariño|first5=Alejandro|date=2020-12-16|title=Towards peptide-based tunable multistate memristive materials|journal=Physical Chemistry Chemical Physics|volume=23|issue=3|pages=1802–1810|doi=10.1039/D0CP05236A|pmid=33434247|hdl=10550/79239|s2cid=231595640 |hdl-access=free}}</ref>

In 2012, [[Sandro Carrara]] and co-authors have proposed the first biomolecular memristor with aims to realize highly sensitive biosensors.<ref>{{cite journal |last1=Milano |first1=G. |last2=Porro |first2=S. |last3=Valov |first3=I. |last4=Ricciardi |first4=C. |title=Recent Developments and Perspectives for Memristive Devices Based on Metal Oxide Nanowires |journal=Advanced Electronic Materials |date=2019 |volume=5 |issue=9 |page=1800909 |doi=10.1002/aelm.201800909 |doi-access=free |s2cid=139445142}}</ref> Since then, several memristive [[sensors]] have been demonstrated.<ref>{{cite journal |last1=Carrara |first1=S. |title=The Birth of a New Field: Memristive Sensors. A Review |journal=IEEE Sensors Journal |date=2021 |volume=21 |issue=11 |pages=12370–12378 |doi=10.1109/JSEN.2020.3043305 |bibcode=2021ISenJ..2112370C |s2cid=234542676 |url=https://ieeexplore.ieee.org/document/9286539}}</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">
{{citation
 |last1=Wang |first1=X.
 |last2=Chen |first2=Y.
 |last3=Xi |first3=H.
 |last4=Dimitrov |first4=D.
 |date=2009
 |title=Spintronic Memristor through Spin Torque Induced Magnetization Motion
 |journal=[[IEEE Electron Device Letters]]
 |volume=30 |issue=3 |pages=294–297
 |doi=10.1109/LED.2008.2012270
 |bibcode=2009IEDL...30..294W
|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">{{cite web
 |last1        = Savage
 |first1       = N.
 |date         = 16 March 2009
 |title        = Spintronic Memristor
 |url          = http://www.spectrum.ieee.org/semiconductors/devices/spintronic-memristors
 |work         = IEEE Spectrum
 |access-date   = 2011-03-20
 |archive-url  = https://web.archive.org/web/20101224201054/https://spectrum.ieee.org/semiconductors/devices/spintronic-memristors
 |archive-date = 2010-12-24
 |url-status     = dead
}}</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">{{cite journal |last1=Chanthbouala |first1=A. |last2=Matsumoto |first2=R. |last3=Grollier |first3=J. |last4=Cros |first4=V. |last5=Anane |first5=A. |last6=Fert |first6=A. |last7=Khvalkovskiy |first7=A. V. |last8=Zvezdin |first8=K. A. |last9=Nishimura |first9=K. |last10=Nagamine |first10=Y. |last11=Maehara |first11=H. |last12=Tsunekawa |first12=K. |last13=Fukushima |first13=A. |last14=Yuasa |first14=S. |title=Vertical-current-induced domain-wall motion in MgO-based magnetic tunnel junctions with low current densities |journal=Nature Physics |date=10 April 2011 |volume=7 |issue=8 |pages=626–630 |doi=10.1038/nphys1968 |arxiv=1102.2106 |bibcode=2011NatPh...7..626C |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">
{{Cite journal
    | doi = 10.1063/1.2345592
    | volume = 89
    | issue = 10
    | pages = 103517
    | last1 = Bowen | first1 = M.
    | last2 = Maurice| first2 = J.-L.
    | last3 = Barthe´le´my| first3 = A.
    | last4 = Prod’homme| first4 = P.
    | last5 = Jacquet| first5 = E.
    | last6 = Contour| first6 = J.-P.
    | last7 = Imhoff| first7 = D.
    | last8 = Colliex| first8 = C.
    | title = Bias-crafted magnetic tunnel junctions with bistable spin-dependent states
    | journal = Applied Physics Letters
    | date = 2006
    |bibcode = 2006ApPhL..89j3517B | url = https://hal.archives-ouvertes.fr/hal-00205009
 }}</ref>([[Resistive random-access memory|resistive switching]]).
The MTJ consists in a SrTiO3 (STO) tunnel barrier that separates [[half-metal|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-metal]]lic LSMO electrode, within the [[tunnel magnetoresistance|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">
{{Cite journal
| doi = 10.1063/1.2938696
| volume = 92| issue = 21| pages = 212115
| last1 = Halley| first1 = D.
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| last4 = Najjari| first4 = N.
| last5 = Henry| first5 = Y.
| last6 = Ulhaq-Bouillet| first6 = C.
| last7 = Weber| first7 = W.
| last8 = Bertoni| first8 = G.
| last9 = Verbeeck| first9 = J.
| last10 = Van Tendeloo| first10 = G.
| title = Electrical switching in Fe/Cr/MgO/Fe magnetic tunnel junctions
| journal = Applied Physics Letters
| date = 2008
|bibcode = 2008ApPhL..92u2115H }}</ref>
and 2009.<ref name="krzysteczko2009apl">
{{citation
 |last1=Krzysteczko |first1=P.
 |last2=Günter |first2=R.
 |last3=Thomas |first3=A.
 |date=2009
 |journal=[[Applied Physics Letters]]
 |title=Memristive switching of MgO based magnetic tunnel junctions
 |volume=95
 |issue=11 |page=112508
 |doi=10.1063/1.3224193
 |bibcode=2009ApPhL..95k2508K
 |arxiv=0907.3684
|citeseerx=10.1.1.313.2571
 |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">
{{citation
| volume = 109| issue = 8| pages = 013712–013712–5
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| last5 = Majjad| first5 = Hicham
| last6 = Bowen| first6 = Martin
| last7 = DaCosta| first7 = Victor
| last8 = Arabski| first8 = Jacek
| last9 = Doudin| first9 = Bernard
| title = Random barrier double-well model for resistive switching in tunnel barriers
| journal = Journal of Applied Physics| access-date = 2014-12-15
| date = 2011| url = http://scitation.aip.org/content/aip/journal/jap/109/8/10.1063/1.3561497
|bibcode = 2011JAP...109a3712D |doi = 10.1063/1.3530610 }}</ref>
and its impact<ref name="schleicherNC2014">
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 | title = Localized states in advanced dielectrics from the vantage of spin- and symmetry-polarized tunnelling across MgO
 | journal = Nature Communications| date = 2014-08-04|bibcode = 2014NatCo...5.4547S | pmid=25088937 | page=4547| doi-access = free}}</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">
  {{citation
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 | doi-access = free}}</ref> or multiferroicity.<ref name="pantelAM2012">
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=====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 [[Magnetoresistive RAM|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>
{{citation
 |last        = Huai
 |first       = Y.
 |title       = Spin-Transfer Torque MRAM (STT-MRAM): Challenges and Prospects
 |url         = http://www.cospa.ntu.edu.tw/aappsbulletin/data/18-6/33spin.pdf
 |journal     = AAPPS Bulletin|date=December 2008|volume=18|issue=6 |pages=33–40
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20120323215048/http://www.cospa.ntu.edu.tw/aappsbulletin/data/18-6/33spin.pdf
 |archive-date = 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'''&nbsp;=&nbsp;(''x''<sub>1</sub>,''x''<sub>2</sub>), where ''x''<sub>1</sub> describes the magnetic state of the electrodes and ''x''<sub>2</sub> denotes the resistive state of the MgO barrier. In this case the change of ''x''<sub>1</sub> is current-controlled (spin torque is due to a high current density) whereas the change of ''x''<sub>2</sub> 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">
{{citation
 |last1=Krzysteczko |first1=P.
 |last2=Münchenberger |first2=J.
 |last3=Schäfers |first3=M.
 |last4=Reiss |first4=G.
 |last5=Thomas |first5=A.
 |date=2012
 |journal=[[Advanced Materials]]
 |title=The Memristive Magnetic Tunnel Junction as a Nanoscopic Synapse-Neuron System
 |volume=24
 |issue=6 |pages=762–766
 |doi=10.1002/adma.201103723|pmid=22223304
 |bibcode=2012APS..MAR.H5013T
 |s2cid=205242867
 }}</ref>

====Spin memristive system====
A fundamentally different mechanism for memristive behavior has been proposed by Pershin and [[Di Ventra]].<ref>{{cite web|url=http://physics.ucsd.edu/~diventra/|title=Massimiliano Di Ventra's Homepage|website=physics.ucsd.edu}}</ref><ref name="Pershin2008">
{{citation
 |last1=Pershin |first1=Y. V.
 |last2=Di Ventra |first2=M.
 |date=2008
 |title=Spin memristive systems: Spin memory effects in semiconductor spintronics
 |journal=[[Physical Review B]]
 |volume=78
 |issue=11 |page=113309
 |doi=10.1103/PhysRevB.78.113309
|bibcode=2008PhRvB..78k3309P |arxiv=0806.2151 |s2cid=10938532
 |url=https://scholarcommons.sc.edu/phys_facpub/24
 }}</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>
{{citation
 |last1=Pershin |first1=Y. V.
 |last2=Di Ventra |first2=M.
 |date=2008
 |title=Current-voltage characteristics of semiconductor/ferromagnet junctions in the spin-blockade regime
 |journal=[[Physical Review B]]
 |volume=77
 |issue=7 |page=073301
 |doi=10.1103/PhysRevB.77.073301
|bibcode=2008PhRvB..77g3301P |arxiv=0707.4475 |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>
{{Citation
 |last=Campbell |first=K.
 |date= January 2017
 |title=Self-directed channel memristor for high temperature operation
 |journal=Microelectronics Journal
 |volume=59
 |pages=10–14
 |doi=10.1016/j.mejo.2016.11.006
|arxiv=1608.05357
 |s2cid=27889124
 }}
</ref>
The SDC device is the first memristive device available commercially to researchers, students and electronics enthusiast worldwide.<ref>
{{Citation
 |title=Knowm Memristors
 |url=http://knowm.org/product/bs-af-w-memristors/
 |publisher=[[Knowm Inc]]
}}</ref>
The SDC device is operational immediately after fabrication. In the Ge<sub>2</sub>Se<sub>3</sub> active layer, Ge-Ge homopolar bonds are found and switching occurs. The three layers consisting of Ge<sub>2</sub>Se<sub>3</sub>/Ag/Ge<sub>2</sub>Se<sub>3</sub>, 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 {{convert|350|C|F}}, the device has significantly higher processing and operating temperatures at {{convert|250|C|F}} and at least {{convert|150|C|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 {{convert|150|C|F}}.

== Implementation of hysteretic flux-charge memristors ==
There exist implementations of memristors with both hysteretic current-voltage curve and hysteretic flux-charge curve [arXiv:2403.20051]. Memristors with both hysteretic current-voltage curve and hysteretic flux-charge curve use a memristance dependent on the history of the flux and charge. Those memristors can merge the functionality of the arithmetic logic unit and of the memory unit without data transfer [DOI: 10.1002/adfm.201303365]. 

=== Time-integrated Formingfree memristor ===
Time-integrated Formingfree (TiF) memristors reveal a hysteretic flux-charge curve with two distinguishable branches in the positive bias range and with two distinguishable branches in the negative bias range. And TiF memristors also reveal a hysteretic current-voltage curve with two distinguishable branches in the positive bias range and with two distinguishable branches in the negative bias range. The memristance state of a TiF memristor can be controlled by both the flux and the charge [DOI: 10.1063/1.4775718]. A TiF memristor was first demonstrated by [[Heidemarie Schmidt]] and her team in 2011 [DOI: 10.1063/1.3601113]. This TiF memristor is composed of a BiFeO<sub>3</sub> thin film between metallically conducting electrodes, one gold, the other platinum. The hysteretic flux-charge curve of the TiF memristor changes its slope continuously in one branch in the positive and in one branch in the negative bias range (write branches) and has a constant slope in one branch in the positive and in one branch in the negative bias range (read branches) [arXiv:2403.20051]. According to Leon O. Chua [Reference 1: ''10.1.1.189.3614''] the slope of the flux-charge curve corresponds to the memristance of a memristor or to its internal state variables. The TiF memristors can be considered as memristors with a constant memristance in the two read branches and with a reconfigurable memristance in the two write branches. The physical memristor model which describes the hysteretic current-voltage curves of the TiF memristor implements static and dynamic internal state variables in the two read branches and in the two write branches [arXiv:2402.10358].

The static and dynamic internal state variables of a non-linear memristors can be used to implement operations on non-linear memristors representing linear, non-linear, and even transcendental, e.g. exponential or logarithmic, input-output functions.

The transport characteristics of the TiF memristor in the small current – small voltage range are non-linear. This non-linearity well compares to the non-linear characteristics in the small current – small voltage range of the basic former and present building blocks in the arithmetic logic unit of von-Neumann computers, i.e. of vacuum tubes and of transistors. In contrast to vacuum tubes and transistors, the signal output of hysteretic flux-charge memristors, i.e. of TiF memristors, is not lost when the operation power is switched off before storing the signal output to the memory. Therefore, hysteretic flux-charge memristors are said to merge the functionality of the arithmetic logic unit and of the memory unit without data transfer [DOI: 10.1002/adfm.201303365]. The transport characteristics in the small current – small voltage range of hysteretic current-voltage memristors are linear. This explains why hysteretic current-voltage memristors are well established memory units and why they can not merge the functionality of the arithmetic logic unit and of the memory unit without data transfer [arXiv:2403.20051].

==Potential applications==
Memristors remain a laboratory curiosity, as yet made in insufficient numbers to gain any commercial applications.

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

{{Memory types}}
Memristors can potentially be fashioned into [[non-volatile memory|non-volatile solid-state memory]], which could allow greater data density than hard drives with access times similar to [[Dynamic random-access memory|DRAM]], replacing both components.<ref name="Kanellos">
{{Citation
 |last=Kanellos |first=M.
 |date=30 April 2008
 |title=HP makes memory from a once theoretical circuit
 |url=http://news.cnet.com/8301-10784_3-9932054-7.html
 |work=[[CNET]] News
 |access-date=2008-04-30
}}</ref> HP prototyped a crossbar latch memory that can fit 100 [[gigabit]]s in a square centimeter,<ref name="EETimes">{{Citation |last=Johnson |first=R. C. |date=30 April 2008 |title='Missing link' memristor created |url=http://www.eetimes.com/electronics-news/4076910/-Missing-link-memristor-created-Rewrite-the-textbooks- |work=[[EE Times]] |access-date=2008-04-30 }}</ref> and proposed a scalable 3D design (consisting of up to 1000 layers or 1 [[petabit]] per cm<sup>3</sup>).<ref>
{{Citation|date=January 22, 2010
|url=https://www.youtube.com/watch?v=bKGhvKyjgLY&t=37m45s
 |title=Finding the Missing Memristor - R. Stanley Williams|work=Youtube}}</ref> In May 2008 HP reported that its device reaches currently about one-tenth the speed of DRAM.<ref>
{{Citation
 |last=Markoff |first=J.
 |date=1 May 2008
 |title=H.P. Reports Big Advance in Memory Chip Design
 |url=https://www.nytimes.com/2008/05/01/technology/01chip.html
 |work=[[New York Times]]
 |access-date=2008-05-01
}}</ref> The devices' resistance would be read with [[alternating current]] so that the stored value would not be affected.<ref>
{{Citation
 |last=Gutmann |first=E.
 |date=1 May 2008
 |title=Maintaining Moore's law with new memristor circuits
 |url=https://arstechnica.com/old/content/2008/05/maintaining-moores-law-with-new-memristor-circuits.ars
 |work=[[Ars Technica]]
 |access-date=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>{{Citation
 |last=Palmer |first=J.
 |date=18 May 2012
 |title=Memristors in silicon promising for dense, fast memory
 |url=https://www.bbc.co.uk/news/science-environment-18103772
 |newspaper=[[BBC News]]
 |access-date=2012-05-18
}}</ref>

Memristors have applications in [[Programmable logic device|programmable logic]]<ref>Snider, Gregory Stuart (2004) "Architecture and methods for computing with reconfigurable resistor crossbars" {{US patent|7203789}}</ref> [[signal processing]],<ref>Mouttet, Blaise Laurent (2006) "Programmable crossbar signal processor" {{US Patent|7302513}}</ref> [[super-resolution imaging]]<ref>{{cite journal |last1=Dong |first1=Zhekang |last2=Sing Lai |first2=Chun |last3=He |first3=Yufei |last4=Qi |first4=Donglian |last5=Duan |first5=Shukai |title=Hybrid dual-complementary metal–oxide–semiconductor/memristor synapse-based neural network with its applications in image super-resolution |journal=IET Circuits, Devices & Systems |date=1 November 2019 |volume=13 |issue=8 |pages=1241–1248 |doi=10.1049/iet-cds.2018.5062|doi-access=free }}</ref> [[physical neural network]]s,<ref>Snider, Greg (2003) "Molecular-junction-nanowire-crossbar-based neural network" {{US patent|7359888}}</ref> [[control theory|control systems]],<ref>Mouttet, Blaise Laurent  (2007) "Crossbar control circuit" {{US Patent|7609086}}</ref> [[reconfigurable computing]],<ref>Pino, Robinson E. (2010) "Reconfigurable electronic circuit" {{US Patent|7902857}}</ref> [[in-memory computing]],<ref>{{cite journal |last1=Ielmini |first1=D |last2=Wong |first2=H.-S. P. |title=In-memory computing with resistive switching devices |journal=Nature Electronics |date=2018 |volume=1 |issue=6 |pages=333–343 |doi=10.1038/s41928-018-0092-2 |hdl=11311/1056513 |s2cid=57248729 |hdl-access=free }}</ref> [[brain–computer interfaces]]<ref>Mouttet, Blaise Laurent (2009) "Memristor crossbar neural interface" {{US Patent|7902867}}</ref> and [[RFID]].<ref>Kang, Hee Bok (2009) "RFID device with memory unit having memristor characteristics" {{US Patent|8113437}}</ref> Memristive devices are potentially used for stateful logic implication, allowing a replacement for CMOS-based logic computation<ref>{{cite journal |last1=Luo |first1=Li |last2=Dong |first2=Zhekang |last3=Duan |first3=Shukai |last4=Lai |first4=Chun Sing |title=Memristor-based stateful logic gates for multi-functional logic circuit |journal=IET Circuits, Devices & Systems |date=20 April 2020 |volume=14 |issue=6 |pages=811–818 |doi=10.1049/iet-cds.2019.0422|doi-access=free }}</ref> Several early works have been reported in this direction.<ref>{{cite journal |last1=Lehtonen |first1=E. |last2=Poikonen |first2=J.H. |last3=Laiho |first3=M. |title=Two memristors suffice to compute all Boolean functions |journal=Electronics Letters |date=2010 |volume=46 |issue=3 |pages=230 |doi=10.1049/el.2010.3407 |bibcode=2010ElL....46..230L }}</ref><ref>
{{cite book
 |last1=Chattopadhyay |first1=A.
 |last2=Rakosi |first2=Z.
 |date=2011
 |doi=10.1109/VLSISoC.2011.6081665
 |isbn=978-1-4577-0170-2
 |pages=200
|chapter=Combinational logic synthesis for material implication
 |title=2011 IEEE/IFIP 19th International Conference on VLSI and System-on-Chip
 |s2cid=32278896
 }}</ref>

In 2009, a simple electronic circuit<ref>
{{citation
 |last1=Pershin |first1=Y. V.
 |last2=La Fontaine |first2=S.
 |last3=Di Ventra |first3=M.
 |date=2009
 |title=Memristive model of amoeba learning
 |journal=[[Physical Review E]]
 |volume=80 |issue=2 |page=021926
 |doi=10.1103/PhysRevE.80.021926
 |pmid=19792170
 |bibcode=2009PhRvE..80b1926P
 |arxiv=0810.4179
|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">
{{citation
 |last1=Saigusa |first1=T.
 |last2=Tero |first2=A.
 |last3=Nakagaki |first3=T.
 |last4=Kuramoto |first4=Y.
 |date=2008
 |title=Amoebae Anticipate Periodic Events
 |journal=[[Physical Review Letters]]
 |volume=100 |issue=1 |page=018101
 |doi=10.1103/PhysRevLett.100.018101
 |pmid=18232821 |bibcode=2008PhRvL.100a8101S
|hdl=2115/33004
 |s2cid=14710241
 |url=https://lib-repos.fun.ac.jp/dspace/bitstream/10445/4400/2/Final20071109.pdf
 |hdl-access=free
 }}</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, [[Massimiliano Versace|Versace]] and Chandler described the MoNETA (Modular Neural Exploring Traveling Agent) model.<ref>
{{cite web
 |last1=Versace |first1=M.
 |last2=Chandler |first2=B.
 |date=23 November 2010
 |title=MoNETA: A Mind Made from Memristors
 |url=https://spectrum.ieee.org/robotics/artificial-intelligence/moneta-a-mind-made-from-memristors/0
 |archive-url=https://web.archive.org/web/20101125205953/http://spectrum.ieee.org/robotics/artificial-intelligence/moneta-a-mind-made-from-memristors/0
 |url-status=dead
 |archive-date=25 November 2010
 |work=[[IEEE Spectrum]]
}}<br />{{cite journal
 |last1=Versace |first1=M.
 |last2=Chandler |first2=B.
 |date=2010
 |title=The brain of a new machine
 |journal=[[IEEE Spectrum]]
 |volume=47 |pages=30–37
 |doi=10.1109/MSPEC.2010.5644776
 |issue=12
|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>{{Citation
 |last=Snider  |first=G.
 |date=2011
 |title=From Synapses to Circuitry: Using Memristive Memory to Explore the Electronic Brain
 |journal=[[IEEE Computer]]
 |volume=44 |issue=2 |pages=21–28
 |doi=10.1109/MC.2011.48
|s2cid = 16307308|display-authors=etal}}</ref> Application of the memristor crossbar structure in the construction of an analog soft computing system was demonstrated by Merrikh-Bayat and Shouraki.<ref>
{{Citation
 |last1=Merrikh-Bayat |first1=F.
 |last2=Bagheri-Shouraki |first2=S.
 |last3=Rohani |first3=A.
 |date=2011
 |title=Memristor crossbar-based hardware implementation of IDS method
 |journal=[[IEEE Transactions on Fuzzy Systems]]
 |doi=10.1109/TFUZZ.2011.2160024
 |volume=19 |issue=6 |pages=1083–1096
|arxiv=1008.5133|s2cid=3163846
 }}</ref> In 2011, they showed<ref>
{{cite arXiv
 |last1=Merrikh-Bayat |first1=F.
 |last2=Bagheri-Shouraki |first2=S.
 |date=2011
 |title=Efficient neuro-fuzzy system and its Memristor Crossbar-based Hardware Implementation
 |eprint=1103.1156
 |class=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 theory|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>
{{cite journal
 |last=Chua |first=L.
 |date=2013
 |title=Memristor, Hodgkin-Huxley, and Edge of Chaos
 |journal=[[Nanotechnology (journal)|Nanotechnology]]
 |volume=24 |issue=38 |pages=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, [[Massimiliano Di Ventra|Di Ventra]], Pershin, and Chua extended<ref name="DiVentra2009">
{{citation
 |last1=Di Ventra |first1=M.
 |last2=Pershin |first2=Y. V.
 |last3=Chua |first3=L.
 |title=Circuit elements with memory: memristors, memcapacitors and meminductors
 |journal=[[Proceedings of the IEEE]]
 |volume=97 |issue=10 |pages=1717–1724
 |date=2009
 |arxiv=0901.3682
 |doi=10.1109/JPROC.2009.2021077
|bibcode=2009arXiv0901.3682D
 |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 calculus|fractional derivative]]s.<ref>
{{Citation
 |last1=Abdelhouahad |first1=M.-S.
 |last2=Lozi |first2=R.
 |last3=Chua |first3=L.
 |title=Memfractance: A Mathematical Paradigm for Circuit Elements with Memory
 |journal=[[International Journal of Bifurcation and Chaos]]
 |volume=24 |issue=9 |pages=1430023 (29 pages)
 |doi=10.1142/S0218127414300237
 |date=September 2014
|bibcode = 2014IJBC...2430023A |url=https://hal.archives-ouvertes.fr/hal-01322396/file/Lozi_Memfractance_IJBC_2014_personnal_file_for_HAL.pdf
 }}</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>
{{Citation
 |last1=Prodromakis |first1=T.
 |last2=Toumazou |first2=C.
 |last3=Chua |first3=L.
 |title=Two centuries of memristors
 |journal=[[Nature Materials]]
 |volume=11 |issue=6 |pages=478–481
 |doi=10.1038/nmat3338
 |pmid=22614504
 |date=June 2012
 |bibcode=2012NatMa..11..478P
}}</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 TiO<sub>2</sub> which was later claimed by researchers from Hewlett Packard to be evidence of a memristor.<ref name="Argall1968"/>{{citation needed|date=January 2020|reason=2008 claim by HP needs citing}}

===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 [[capacitor]]s and [[inductor]]s, whose properties depend on the state and history of the system.<ref name="DiVentra2009"/>

In July 2014, the MeMOSat/[[LabOSat]] group<ref name="labosat1">
{{citation |last=Barella |first=M. |date=2016 |chapter=LabOSat: Low cost measurement platform designed for hazardous environments |doi=10.1109/SASE-CASE.2016.7968107|title=2016 Seventh Argentine Conference on Embedded Systems (CASE) |pages=1–6 |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>{{Cite news
 |title= Probaron con éxito las memorias instaladas en el satélite argentino "Tita"
 |url= http://www.telam.com.ar/notas/201407/71782-satelite-argentino-tita-inti-comision-nacional-de-energia-atomica-memorias-prueba.html
 |work= [[Telam]] |date=21 July 2014
}}</ref> Since then, seven missions with different devices<ref name="labosat2">
{{citation |last=Barella |first=M. |date=2019 |title=Studying ReRAM devices at Low Earth Orbits using the LabOSat platform|journal=[[Radiation Physics and Chemistry]] |volume=154 |pages=85–90 |doi=10.1016/j.radphyschem.2018.07.005 |bibcode=2019RaPC..154...85B |doi-access=free }}
</ref> are performing experiments in low orbits, onboard [[Satellogic]]'s [[Ñu-Sat]] satellites.<ref>{{Cite web|url=http://www.unsam.edu.ar/escuelas/ciencia/labosat/|title=UNSAM - Universidad Nacional de San Martín|website=www.unsam.edu.ar}}</ref><ref>{{Cite news
 |title= Qué hace LabOSat, el laboratorio electrónico dentro de los nanosatélites Fresco y Batata
 |url= https://www.lanacion.com.ar/tecnologia/que-hace-labosat-el-laboratorio-electronico-dentro-de-los-nanosatelites-fresco-y-batata-nid1911090
 |work= [[Telam]] |date=22 June 2016
}}</ref> {{clarify|date=January 2020|reason=What is the relation of these devices to memristors?}}

On 7 July 2015, Knowm Inc announced Self Directed Channel (SDC) memristors commercially.<ref>
{{Cite news
 |title= Startup Beats HP, Hynix to Memristor Learning
 |url= http://www.eetimes.com/document.asp?doc_id=1327068
 |work= [[EE Times]] |date=5 July 2015
}}</ref>
These devices remain available in small numbers.

On 13 July 2018, MemSat (Memristor Satellite) was launched to fly a memristor evaluation payload.<ref>{{Cite news
 |title= MemSat
 |url= http://space.skyrocket.de/doc_sdat/memsat.htm
 |work= Gunter Space Page |date=22 May 2018
}}</ref>

{{anchor|Lithionics}}In 2021, [[Jennifer Rupp]] and Martin Bazant of [[MIT]] started a "Lithionics" research programme to investigate applications of [[lithium]] beyond their use in [[battery electrode]]s, including [[lithium oxide]]-based memristors in [[neuromorphic computing]].<ref>{{Cite web|url=https://eepower.com/news/mit-and-ericsson-collaborates-to-research-new-generation-of-energy-efficient-computing-networks/|title=MIT and Ericsson Collaborates to Research New Generation of Energy-Efficient Computing Networks - News|website=eepower.com}}</ref><ref>{{Cite web|url=https://www.allaboutcircuits.com/news/mit-and-ericsson-set-goals-zero-power-devices-new-field-lithionics/|title=MIT and Ericsson Set Goals for Zero-power Devices and a New Field—"Lithionics" - News|website=www.allaboutcircuits.com}}</ref>

In May 2023, TECHiFAB GmbH <nowiki>[https://techifab.com/]</nowiki> announced TiF memristors commercially. [arXiv: 2403.20051, arXiv: 2402.10358] These TiF memristors remain available in small and medium numbers.

In the September 2023 issue of [[Science (journal)|''Science Magazine'']], Chinese scientists Wenbin Zhang ''et al.'' described the development and testing of a memristor-based [[integrated circuit]].<ref>{{Cite journal |last=Zhang |first=Wenbin |date=14 September 2023 |title=Edge learning using a fully integrated neuro-inspired memristor chip |url=https://www.science.org/doi/10.1126/science.ade3483|journal=Science |volume=381 |issue=6663 |pages=1205–1211 |language=en |doi=10.1126/science.ade3483|pmid=37708281 |bibcode=2023Sci...381.1205Z |s2cid=261736380 }}</ref>

==See also==
{{Portal|Electronics}}
* [[3D XPoint]]
* [[Electrical element]]
* [[Hybrid Memory Cube]]
* [[List of emerging technologies]]
* [[Neuromorphic engineering]]
* [[Trancitor]]

==Footnotes==
{{notelist}}

==References==
{{reflist|30em}}

==Further reading==
* {{cite journal|editor1-last=Chen|editor1-first=Dongmin|editor2-last=Chua|editor2-first=Leon O.|editor3-last=Hwang|editor3-first=Cheol Seong|editor4-last=Wang|editor4-first=Shih-Yuan|editor5-last=Waser|editor5-first=Rainer|editor6-last=Williams|editor6-first=R. Stanley|editor7-last=Yang|editor7-first=Jianhua|title=Special Issue: Memristive and Resistive Devices and Systems|journal=[[Applied Physics A]]|date=March 2011|volume=102|issue=4}}
* {{cite journal|editor1-last=Mazumder|editor1-first=P.|editor2-last=Kang|editor2-first=S. M.|editor3-last=Waser|editor3-first=R.|title=Special Issue: MEMRISTORS: DEVICES, MODELS, AND APPLICATIONS|journal=[[Proceedings of the IEEE]]|date=June 2012|volume=100|issue=6|pages=1905{{endash}}2092|doi=10.1109/JPROC.2012.2197452}}
* {{cite book|editor1-last=Tetzlaff|editor1-first=Ronald|title=Memristors and Memristive Systems |date=2013|publisher=[[Springer Science & Business Media]]|isbn=978-1-4614-9068-5|url=https://books.google.com/books?id=OOW5BAAAQBAJ|doi=10.1007/978-1-4614-9068-5}}
* {{cite book|editor1-last=Adamatzky|editor1-first=Andrew|editor2-last=Chua|editor2-first=Leon|title=Memristor Networks|date=2013|publisher=[[Springer Science & Business Media]]|isbn=978-3-319-02630-5|url=https://books.google.com/books?id=Mxu4BAAAQBAJ|doi=10.1007/978-3-319-02630-5|s2cid=39739718}}
* {{cite journal |last1=Atkin |first1=Keith |title=An introduction to the memristor |journal=Physics Education |date=May 2013 |volume=48 |issue=3 |pages=317–321 |doi=10.1088/0031-9120/48/3/317 |bibcode=2013PhyEd..48..317A |s2cid=121268844 }}
* {{cite journal |last1=Gale |first1=Ella |title=TiO<sub>2</sub>-based memristors and ReRAM: materials, mechanisms and models (a review) |journal=Semiconductor Science and Technology |date=1 October 2014 |volume=29 |issue=10 |pages=104004 |doi=10.1088/0268-1242/29/10/104004 |arxiv=1611.04456 |bibcode=2014SeScT..29j4004G |s2cid=5686212 }}
* {{cite journal |last1=Traversa |first1=Fabio Lorenzo |last2=Di Ventra |first2=Massimiliano |title=Universal Memcomputing Machines |journal=IEEE Transactions on Neural Networks and Learning Systems |date=November 2015 |volume=26 |issue=11 |pages=2702–2715 |doi=10.1109/TNNLS.2015.2391182 |pmid=25667360 |arxiv=1405.0931 |citeseerx=10.1.1.747.5690 |s2cid=1406042 }}
* {{cite journal |last1=Caravelli |first1=Francesco |last2=Carbajal |first2=Juan Pablo |title=Memristors for the curious outsiders |journal=Technologies |date=January 2019 |volume=6 |issue=4 |pages=118 |doi=10.3390/technologies6040118|arxiv=1812.03389|s2cid=54464654 |doi-access=free }}
* {{cite journal |last1=Maan |first1=Akshay Kumar |last2=Jayadevi |first2=Deepthi Anirudhan |last3=James |first3=Alex Pappachen |title=A Survey of Memristive Threshold Logic Circuits |journal=IEEE Transactions on Neural Networks and Learning Systems |date=August 2017 |volume=28 |issue=8 |pages=1734–1746 |doi=10.1109/TNNLS.2016.2547842 |pmid=27164608 |arxiv=1604.07121 |s2cid=1798273 }}
* {{cite journal | vauthors=((Ghosh, M.)), ((Singh, A.)), ((Borah, S. S.)), ((Vista, J.)), ((Ranjan, A.)), ((Kumar, S.)) | journal=IEEE Transactions on Electron Devices | title=MOSFET-based memristor for high-frequency signal processing | volume=69 | issue=5 | pages=2248–2255 | date= 2022 | url=http://dx.doi.org/10.1109/ted.2022.3160940 | issn=0018-9383 | doi=10.1109/ted.2022.3160940| bibcode=2022ITED...69.2248G | s2cid=247889089 }}
* {{Citation | vauthors=((Singh, A.)), ((Borah, S. S.)), ((Ghosh, M.)) | year=2021 | title=Simple grounded meminductor emulator using transconductance amplifier | publisher=IEEE}}

==External links==
{{Commons category|Memristors}}
* {{youTube|id=n3XzuBt54ig |title= Finding the missing memristor}}
* [http://memlinks.eu Interactive database of memristor papers (2013)]
* {{cite news |title=Machine Dreams |first=Tom |last=Simonite |date= 21 April 2015 |access-date=5 December 2017 |url=http://www.technologyreview.com/featuredstory/536786/machine-dreams |publisher=Technology Review}}
* [https://www.sztucznainteligencja.org.pl/leon-chua-a-bulb-versus-google-go-player/ "Leon Chua: A bulb versus Google go player"] - (in Polish) an interview with Leon Chua, the creator of memristor
* [https://www.sztucznainteligencja.org.pl/en/leon-chua-a-bulb-versus-google-go-player/ "Leon Chua: A bulb versus Google go player"] - (in English) an interview with Leon Chua, the creator of memristor

{{emerging technologies|topics=yes|electronics=yes}}
{{Electronic components}}
{{Differentiable computing}}

{{Authority control}}

[[Category:Electrical components]]
[[Category:American inventions]]
[[Category:Electronic circuits in computer storage]] <!-- exact wording may be wrong -->
[[Category:Experimental electrical components]]