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ChallengesinmaterialsanddevicesforResistive-Switching-basedNeuromorphicComputing

JavierdelValle1,JuanGabrielRamírez2,MarceloJ.Rozenberg3andIvanK.Schuller1

1DepartmentofPhysicsandCenterforAdvancedNanoscience,UniversityofCalifornia-

SanDiego,LaJolla,California92093,USA2DepartmentofPhysics,UniversidaddelosAndes,Bogotá111711,Colombia

3LaboratoiredePhysiquedesSolides,CNRS-UMR8502,UniversitéParis-Sud,Orsay91405,France

Abstract

Thistutorialdescribeschallengesandpossibleavenuesfortheimplementationofthecomponentsofasolid-statesystem,whichemulatesabiologicalbrain.Thetutorialisdevotedmostlytoacharge-based(i.e.electriccontrolled)implementationusingtransitionmetaloxidesmaterials,whichexhibituniquepropertiesthatemulatekeyfunctionalitiesneededforthisapplication.IntheIntroduction,wecomparethemaindifferencesbetweenaconventionalcomputationalmachine,basedontheTuring-vonNeumannparadigm,toaNeuromorphicmachine,whichtriestoemulateimportantfunctionalitiesofabiologicalbrain.Wealsodescribethemainelectricalpropertiesofbiologicalsystems,whichwouldbeusefultoimplementinacharge-basedsystem.InChapterII,wedescribethemaincomponentsofapossiblesolid-stateimplementation.InChapterIII,wedescribeavarietyofResistiveSwitchingphenomena,whichmayserveasthefunctionalbasisfortheimplementationofkeydevicesforNeuromorphiccomputing.InChapterIVwedescribewhytransitionmetaloxides,arepromisingmaterialsforfutureNeuromorphicmachines.TheoreticalmodelsdescribingdifferentresistiveswitchingmechanismsarediscussedinChapterVwhileexistingimplementationsaredescribedinChapterVI.ChapterVIIpresentsapplicationstopracticalproblems.WelistinChapterVIIIimportantbasicresearchchallengesandopenissues.Wediscussissuesrelatedtospecificimplementations,novelmaterials,devicesandphenomena.Thedevelopmentofreliable,faulttolerant,energyefficientdevices,theirscalingandintegrationintoaNeuromorphiccomputermaybringusclosertothedevelopmentofamachinethatrivalsthebrain.

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I.INTRODUCTIONWearelivingintheinformationage.Wehavepowerfulcomputersthatareinterconnectedtoevenmorepowerfuldatabasesthatprovideseeminglyendlessinformationandenableactionsatadistance.Thistechnologicalrevolutionisbasedontwopillarsdevelopedduringthepastcentury:hardwarecomputersandsoftwarecodes.Thefirstoneissupportedbyimpressivetechnologicalprogressinelectronics,includingtransistors,microprocessors,memories,LEDs,etc.Thesecondoneissupportedbyequallyimpressiveadvancesincomputerscience,includingprogramminglanguages,communicationprotocolsandcryptography,justtonameafew.Thesetwoavenuesoftechnologicalprogressmergedintotheinformationsuperhighwaythatweuseandcontinuetodevelopeveryday.“Moore’slaw”,thedoublingoftransistorsindensecircuitseverytwoyears,hasfueledtheexplosionintheuseandmanipulationofdataineverydaylife.However,itiscommonlyagreedthatinthenext15-25yearsa“Moore’scrisis”willdevelop,inwhichthecontinuousimprovementincomputationalpowerandthedecreaseincostwillend.1Thevastmajorityoftoday’scommercialcomputersfollowthevonNeumannarchitecturemodel.ThisdigitalelectroniccomputerwasdescribedbyJohnvonNeumannin1945basedonworkfromEckertandMauchly.2IndependentlyTuringwasdevelopinglogicalandpracticalideasonhowtoimplementaUniversalComputingmachine.3Thisso-calledTuring-vonNeumann(TvN)paradigmfeaturesseveralunitsandtheirinterconnectionsasshowninthefigure1.Itischaracterizedasastore-programmachine,asitkeepsboththedataandtheprograminstructionsinacommonstoragespace.ThemaincomponentsareaCentralProcessingUnit(CPU),aMemoryUnitconnectedtotheoutsideworldbyinputandoutputdevices.Theycanbeeasilyrecognizedinthefamiliardesktopmachinestoday.TheCPUisfurthercomposedofacontrolunit,anarithmeticandlogicunitandregisters.TheCPUfetchesdataandprograminginstructionsfromthememoryunit.Theseinstructionsanddataarefurtheracteduponbythecontrolunit,andtheoperationsarethenexecutedbythearithmeticandlogicunit.Inputandoutputdevicesaretheinterfaceswiththeoperatorortheexternalworld.Somepioneerideasregardingartificialintelligence(AI)gobacktoseminalthoughtsofJohnvonNeumann4whofirstpointedoutthatbrainscognitivepowerdoesnotemergefromaccuratedigitalcalculationsbutratherfromacollectiveformofcomputationinvolvinglargenumberofslow,impreciseandunreliableanalogcomponents.5,6Ourbrainsareamazinglymoreefficientatrecognizingpatternsthanpowerfuldigitalcomputers.Todistinguishtheimageofacatfromatigerisahardtaskforacomputer,butnotforachild.AIismakingstridesthanksto“neuromorphic”inspiredconceptsdevelopeddecadesago,suchasneuralnetworks7andconvolutionfilters,usingthetremendouscalculationalpowerofcurrentconventionalcomputers.8TheseAIsystemsareachievingremarkablefeats,frombeatingthebestchessandgoplayerstorecognizefacesinacrowd.9However,thehardwarerequirementstorunartificialneuralnetworksareextremelyhigh,bothforcalculationspeedandpowerconsumption,whichrepresentsasignificantimpedimentforconventionalcomputertechnology.Whilethebrainofachessplayerusesoftheorderof20watts,a

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supercomputerrequiresamilliontimesmoretofacethesamechallenge.Therefore,adifferentapproachtocomputing,probablyinspiredinneuromorphicconcepts,willundoubtedlyleadtomoreenergyefficientmachinesandanimprovementintheglobalpowerconsumption.10,11 Themassiveamountofinformationatourdisposalposesnewchallenges.Howtomakesenseofit?Howtorecognizepatterns?Howtomakedecisionsfromvastbutoftenconflicting,incompleteorimprecisedata?Thesechallengesopenanopportunityfortheemergenceofadisruptivetechnology.BreakingawayfromtheclassicalTuring-vonNeumannmachineparadigmtoimplementanewtypeofbio-inspired(“neuromorphic”)electronicdevicesorcognitivehardware6mayallowfortheimplementationofartificialneuralnetworks.Thesenewmachinesmaybebasedonnewtypesofdevices,suchasartificialsynapses,axons,dendritesandneurons,toenabletheconstructionof“neuromorphic”circuitswithAIcapabilities.Inaneuromorphiccomputer,contrarytoaTvNmachine,thereisnotaclear-cutseparationbetweentheunitexecutingtheoperations(calculationsorlogical)andthememory.Aneuromorphicsystemischaracterizedbysimpleunitswithahighdegreeofinterconnectivity(figure1),imitatingthebrain.TheunitsperformsimpleoperationsascomparedtothearithmeticandlogicunitofaTvNmachine.Theseconstituentunits,whichemulate“neurons”and“synapses”,areinterconnectedinaso-calledNeuralNetwork.AnotherkeydifferencebetweenTvNandneuromorphicmachinesisthattheformersoperateineminentlyserialmode,whilethelatterareintrinsicallyparallel.Thisfeaturehasmanyconsequences.PerhapsthemainoneisthattheoperationofTvNmachinesrequireahighdegreeofprecision.Sincetheiroperationisserial,theinitialdataundergoesalongsequenceofsuccessivemanipulations,whicharesusceptibletoerrorbuild-up.Hence,theyoftenrequirecalculationswith16significantfigures(“doubleprecision”)althoughtypicallytheprecisionneededfortheendresultisonlyonepartinathousand.Neuromorphicsystemsoperatinginparallelhaveunitsthatintegratetheinputofaratherlargenumberofinterconnectedunits.Thestatisticalnatureinherentintheiroperationmakesthesesystemslessvulnerabletoaccumulationoferrors.Inaddition,theoperationspeedofaserialsystemisgivenbythespeedofeachbasicoperationtimesthenumberofoperations.Thefirstisquitefastinmodernmachines(~10-9s)buttheformercanbequitelargeincomplexcalculations.Incontrast,parallelsystemsmayhaverelativelyslowoperatingunitsbutalargenumberofthemandsignificantlyshortercalculations,whichmayallowforfastercomputations.

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Figure 1. Comparison between von-Neumann and neural network computingarchitectures.Thepurposeofaneuromorphichardware12istoemulateabiologicalsystembymimickingthemostrelevantelectricfunctionsofbiologicalneuronaldevices.Thepurposeisnottoreproduceexactlyabiologicalsystembuttoextractessentialfeatures,whichmayallowforthedevelopmentofafunctioningmachine.Thus,wewilloutlinebelowthemostimportantpropertiesofthemaincomponentsofabiological-neuralsystemNeuronsThecentralelementsofbiologicalneuralcircuitsaretheneurons,13activeelementswhosefunctionistoproduceelectricsignals(typicallyspikes)asaresponsetoanexcitation.Thisresponseisnon-linear,whichgeneratesaspikeonlyiftheexcitationvoltageexceedsathreshold.Theresponseisalsotimedependent:theneuronismorelikelytofireifexcitationsignalsarrivemorefrequently.Thisbehavioriscommonlyreferredtoas“leaky,integrateandfire”(LIF).Themembraneofneuronactsasaleakycapacitorwhosevoltagebuildsupasincomingpulsesarrive,butslowlydischarges(leaks)overtime.Ifthethresholdoftheexcitatoryvoltageisexceeded,theneuronactivatesandfires(figure2).14Inbiologicalneuronstheintegrationprocessisbelievedtooccurwithinaregionofthecellbodycalledthe“axonhillock”.AxonsanddendritesNeuronsspanoutconnections,likecables,calleddendritesandaxons(figure2),whichprovideinter-neuronalconnectionsandcarryelectricimpulsesfromneurontoneuron.Dendritesarethereceivingconnectionsoftheneuron,gatheringalltheinputsignals,whileaxonscarryawaytheoutputgeneratedbytheneuron.Eachneuronmayhavefromonetohundredsofdendrites,andtheymayfurtherbranchintospinesthatcanreachtensofthousandsofconnections.Ontheotherhand,neuronssolelyhaveasingleormaybefewaxons.Thisresultsonahighly-interconnectednetworkofneurons,throughwhichtheelectricpulsesgeneratedbyoneneuronaresenttomanyothers.Itisimportanttopointoutthedifferencesbetweenaxonsanddendrites.Usuallytheyareconsideredasmeretransmissionline

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equivalents,however,(axonsinparticular)canalsoimplementinformationprocessingtasks.12,15Moreover,sincesynchronizationbetweenmultipleneuronsplayaroleinbrainactivity,long-rangeconnectivitythroughaxonsmayallowthecoordinatedbehaviorofremoteneurons.

Figure2:Schematicelectricallyrelevantstructurea)andvoltagespiking,b)ofaneuronshowingtheleaky-integrateandfireafterathresholdpotentialisreached.

SynapsesAnotherkeyelementofbiologicalneuronalnetworksisthesynapse,theconnectionpointbetweenaxonsanddendrites(figure2).Thesesynapticconnectionsregulatethefractionoftheoutgoingsignaltravellingthroughtheaxonwhichistransferredtothedendrite.Inotherwords,theycontrolhowstronglyinterconnectedtwoneuronsare.Thisinterconnectionstrengthisknownas“synapticweight”andishowthememoryofthenetworkisstored.Theweightsalsodefinethenetwork’stopology,astheycontrolthepropagationofsignalsacrossthesynapses.Theseweightsarenotfixedbutcanchangeovertimedependingontheirprevioushistory.Thisproperty,knownas“synapticplasticity”,providesthelearningcapabilityofthenetwork.TheinitialstudyofthisfeatureinbiologicalsystemsisassociatedtoHebb,whosynthetizeditinthephrase“neuronsthatfiretogether,wiretogether.”16Therequirementforsynapticweightstomodifytheirvaluesaccordingtothecorrelationsintheelectricspikeactivityoftheirconnectedneuronsiscalled“spike-timing-dependentplasticity”(STDP).14,17Duetothismechanism,thesynapticweightofagivensynapseisreinforcedwhentheconnectedpre-neuronbecomesactivejustbeforethepost-neurondoes.Andconversely,theweightisdecreasedifthepost-neuronbecomesactivebeforethepre-neuron.Thefirsteffectiscalled“long-termpotentiation”(LTP)andthesecondoneis“long-termdepression”(LTD).14Theabovediscussionprovidesastartingpointfortheminimalfunctionalitiesexpectedfromelectronicneuromorphicdevices.Itseemsclearthatabio-inspiredsystemmaybecomposedoffourbasicartificialdevicesemulating:neurons,synapses,axonsanddendrites,wiredwithahighdegreeofinter-connectivity.AveryimportantissueisrelatedtotheenergyconsumptionofaTvNmachineascomparedtoabiologicalneuralsystem.TodevelopaTvNmachinethatisabletoperformsimilartasksasthehumanbrain,thenumberofdevicesmustbeincreasedbyseveralordersofmagnitudewithrespecttocurrentordinarycomputers.Whileitisnot

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obvioushowtocomparetheenergyconsumptionofsuchaTvNmachinewithabiologicalbrain,itiscommonlyacceptedthattheenergyconsumptionoftheformerisconsiderablelargerbymanyordersofmagnitude.Fromtheenergeticpointofviewtheincreasednumberofdevicesposestwomainchallenges:i)thelocalenergyconsumptionproducesdeleteriousthermalinteractionsbetweenthedevices,andii)theglobalenergyconsumptionbecomesprohibitive.Manyestimatesindicatethatafewofthesehigh-performanceTvNmachineswouldrequireaseriousfractionoftheworldenergyproduction.Thus,todevelopevermorepowerfulmachines,itisimperativetobreakawayfromtheTvNparadigm.ArtificialIntelligencemachinesthatattempttoemulatethecomputationalbehaviorofneuralsystemsareknownasNeuromorphicComputers.Severalimplementationshavebeenadvancedandappliedtothesolutionsofdifferentcomputationalproblems.Themostobviousoneistouseconventionalsiliconbasedtechnology(CMOS)toimplementselectedneuralfunctionalities.18Unfortunately,inordertoobtaineventhemostrudimentaryonesmanyclassicalCMOSdevicesareneeded,12whichgreatlyincreasesenergyconsumption.However,thesekindsofsystems,eveninareducedimplementation,allowprovingconceptualideasofneuromorphicmachines.Alternatively,completelynewredesignsofcomputationalmachineshavebeendevelopedinavarietyofconfigurations.Spintorqueoscillator-basedmachineshavebeendevelopedrecently,whichrelyonthetemporalevolutionofthosedevicesandhavebeenapplied,forinstance,tospeechrecognition.19Energyefficienthybridsuperconducting-magneticdevicesarebeingdevelopedtosimulatesynapses.20Andsemiconductor-superconductingdevices,usingfewphotonlight-emittingdiodesconnectedwithsuperconductingwires,havebeenproposedtoemulatespikingneurons.21Inthiscontext,aparticularlypromisingapproach,thatweshalldescribebelow,attemptstoemulateall-importantneuro-biologicalfunctionalitiesinasingleplatform,whichreliesontheelectricalresponseofcertaincarefullychosencompoundsbelongingtothefamilyofquantummaterials.22Inthisreviewarticle,weshalldiscussanddescribehowneuromorphicfunctionalitiesmayberealizedusingResistiveSwitching(RS),aphysicalphenomenoninwhichtheresistanceofadevicedependsontheappliedvoltageor/anditsprevioushistory.23–25Althoughthisphenomenonhasalreadybeenobservedinmanymaterials,26,27inthisreviewwewillfocusonthepromisingtransitionmetaloxides(TMO).28–34TMOsareattractiveastheyfeaturethetwomaintypesofresistiveswitching:volatileandnon-volatile.Involatileswitching,theresistanceofthematerialchangeswhenahighenoughvoltageorcurrentisapplied,returningtoitsoriginalvalueafterwards.Innon-volatileswitching,theresistanceofthematerialdoesnotgobacktoitsoriginalvalue,butitispermanentlymodified.Thesetwotypesofresistiveswitchingallowimplementingthebasichardwareelementsofneuromorphicsystemsthatweshalldenote:neuristors(whichemulatetheleaky,integrateandfirebehaviorofspikingneurons)andsynaptors(whichemulatethememorybehaviorofsynapses).35

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II.DEVICESANDFUNCTIONALITIESTheterminologyusedinthefieldofNeuromorphicComputingissomewhatconfusing.BecauseofthiswewilldefinethetermsfortheartificialingredientsofachargebasedNeuromorphicMachineandthecontextinwhichwewillusethemasfollows:Neuristorsareactiveelementsofthenetworkwhichemulatethecomputationfunctionalityofneurons.Theymustfeatureanon-linearactivationfunction,withanallornothingresponseintheformofanelectricspike.Volatileresistiveswitchingenablesthisfeaturethankstoitsthresholdbehavior:thematerialremainsnon-conductiveuntilathresholdvoltage(moreprecisely,anelectricfield)issurpassedandprovokesthecollapseoftheresistance.Oncethevoltageisremovedthedevicereturnstoitsinitial(non-conductive)restingstate,perhapswithacontrollabletimeconstantthatmaybeassociatedwiththerefractiveperiodofneurons.AkeyfunctionalitytheymustmimicistheLeaky,IntegrateandFire(LIF)behavior,whichimpliesthattheneuristormustkeepsomesortofshorttermmemorythatallowsittosumallinputs.35,36Suchmemorymightcome,forinstance,fromtherelaxationtimeoftheinsulator-metalphasetransition,asshownrecently.37Theneuristormusthaveoneorseveralinputandoutputtransmissionlines,correspondingtothedendritesandaxons,respectively.Theseaxonsanddendritescouldbeinprincipleimplementedbysimplemetallicinterconnects,whichseemstobetheapproachtakeninmostrealimplementations.However,weshouldnotdiscardthepossibilitytoincreasetheircomplexitybyaddingadditionalaxonicordendriticfunctionalitiestothem(suchassignalgainneededtokeeptheamplitudeofthepulseconstantalongtheaxon).Synaptorsplaytheroleofsynapses,controllingtheconnectivitybetweentheneuristors.Non-volatileRSisanidealeffecttoimplementsynapticfunctionalities:theconductivityofthesynaptorplaystheroleofthesynapticweightwhileitsnon-volatilitymimicssynapticmemoryandplasticity.Athin(<100nm)TMOlayersandwichedbetweentwometallicelectrodesistheeasiestwaytorealizeasynaptoralthoughwithrecentlithographytechniquesplanarsynaptorshavealsobeenimplemented.38Architectureisachallengingissuethatreferstothewayinwhichneuristorsandsynaptorsarewired.Currentfabricationtechniqueslimithardwarefabricationto2D,makingitcomplicatedtoachievethehuge3Dconnectivityofbiologicalsystems.Apartialsolutiontothisistheimplementationofcross-bararraygeometries,suchasshowninfigure3.Inthisgeometry,neuristorsaregroupedin“layers”,arowandacolumn,forinputandoutputintoasynaptormatrix.Thisarchitectureoflayersisquitesimilartothatofsoftwareimplementedneuralnetworksand,tosomeextent,torealbiologicalsystems.Eachneuristorhastwoterminalscorrespondingtothedendriteandtheaxon.Thegeometryisarrangedsoevery“axon”ofthepre-neuronlayerintersectsevery“dendrite”ofthepost-neuronlayer.Eachintersectionconsistsofasynaptor:AMetal/TMO/Metalnon-volatileRSelementplayingtheroleofasynapticconnection.Multilayer,“deep”,networkscouldbefabricatedbyconnectinginseriesseveralofthesearrays.

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Figure3:Cross-bararrayarchitecturetoimplementhardware-basedneuralnetworks.It consists of a synaptor-matrix device in between a pre-neuristor layer and a post-neuristorlayer.Thevariousneededneuromorphicfunctionalitiesmaybeimplementedbyacombinationofresistiveswitchingdevicesandmetallicconnections.39Resistiveswitchingallowsforimplementationofbasicneuronalandsynapticfunctionalitieswhereastheconnectivityisprovidedbythemetallicwiresandpads.MoredetaileddescriptionsofsuccessfulneuromorphicdevicesimplementedwithresistiveswitchingwillbegiveninsectionVI.III.PHENOMENAIII.1BasicsofResistiveSwitching(RS)NeuromorphicfunctionalitiescanberealizedinpracticebyusingandcontrollingtheResistiveSwitchingphenomenon.TheRSeffectconsistsofasuddenchangeoftheresistanceofasystemundertheapplicationofelectricstress,suchasapplicationsoftimedependentvoltage,currentand/ortemperature.Themicroscopicoriginandmechanismoftheresistancechangemaybeverydifferentand,accordingly,qualitativelydifferentfordifferenttypematerials.Asafirstclassification,therearetwotypesofRSswitching:volatileandnon-volatile.Intheformercasethephenomenonisasuddencollapseoftheresistanceundertheactionofanelectricstimulus,whichspontaneouslyreturnstotheinitialstate.40Thisrelaxationtakesplacegradually,acertaintimeaftertheelectricactiononthesystemisterminated.Ontheotherhand,thenon-volatileRSisasuddenchangeinresistancethatissemi-permanent.Itdoesnotreturntotheinitialstateaftertheelectricpulseisterminated,however,thechangemaybereversedbyapplicationofanewpulse.Thesenon-volatileresistiveswitchingbehaviorsmaybefurtherclassifiedintwosub-categories:unipolar(alsocallednon-polar)andbi-polar.31Theformeraresystemswheretheresistivechangecanbereversedbytheapplicationofasecondelectricpulseofeitherthesameoroppositepolarity.Incontrast,thebi-polarsystemsrequirethatthesecondpulsebeoftheoppositepolaritytothefirstone.Ithasbecomecustomarytocallthepulsethatproducetheresistancechangefromhightolowvalueasthe“set”process,andtheonethatreturnsittohighresistance,the“reset”one.Thecurrentstateofthesystemmaybereadwitha“read”pulse,whichisoftenasmallbiascurrentorvoltagethatshouldnotaffectthegivenresistivestate.

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Volatileresistiveswitchingisgenerallyobservedinsystemswhereametal-insulatortransitionisobserved(suchasMottinsulators).41Theresistancecollapseoccurswhenpartofthematerialchangesfromtheinsulatingtothemetallicphaseasaresultofanappliedvoltage.Ontheotherhand,non-volatileswitchingarisesfromtheredistributionofintrinsicdefectsorions,mainlyoxygenvacancies,whichbecomemobileinthepresenceofstrongelectricfields.Theirlocaldensitydistributionchangesandthusmodifiesthetotal(i.e.two-terminal)resistanceofthematerial.Inthissection,wewilldescribethemainmicroscopicmechanismsgoverningthedifferenttypesofswitching,whichareshownschematicallyinfigure4.

Figure 4: Schematic representation of the three main types of resistive switching:unipolar, bipolar and volatile (threshold) switching. Top panels show the current vsvoltage characteristics of each resistive switching type. The grey regions denotemetallicelectrodesandthecentralregiontheRSmaterial.III.2Originofnon-volatileresistiveswitching(NVRS)Thetwoqualitativelydifferenttypes(unipolarandbipolar)ofnon-volatileRSsystems,arisefromdifferenttypesofmicroscopicstructuralchanges.Theformeremergesfromfilamentarystructuresthatresultsfromasoftdielectricbreakdownorinitialelectro-formingprocess.31Incontrast,thelatterisobservedinmaterialsthatareconductorsandformhighlyresistive(Schottky)interfaceswiththeelectrodes.

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Theunipolarsystemstypicallyarebinaryoxides(withsomeexceptions)thatareinitiallyhighlyinsulating.Ontheotherhand,thebipolarsystemsoftenarecomplexoxidesandbadconductors,anddonotrequirethesoftdielectricbreakdown.42,43Amongthesesystemsthecupratesandparticularlythemanganiteshavereceivedsignificantattention.43,44Thisdistinctionisneverthelessnotstrictandhighlyoxygendeficientbinaryoxidesarealsooftenbipolar.Forinstance,TiO2-xhasbeenshowntobelongtothetwocategories,unipolarandbipolar,dependingonfabricationdetailsandtheoxygencontent.45–47III.2.1UnipolarRSUnipolar(sometimescallednon-polar)RSistypicallyobservedinsimplebinaryTMOssuchasTiO2,NiO,CuO,HfO2,Ta2O5,etc.26,32,33,48,49Thesematerialsintheirstoichiometricformareverygoodinsulators.Typicaldevicesarecapacitor-like,wherethedielectricisathinfilmoftheTMO.RSintheseinsulatorsisproducedbyelectroforming,aprocesswhichproducesasoftelectricbreakdownonthedielectric.Typically,MV/cmelectricfieldsareneeded,easilyachievablewithafewvoltsacross~100nmthinfilms.Topreventsampledamage,onemayusealoadresistorinseries,orimposeacurrentcompliance.Electroformingproducesamassivemigrationofions(suchasoxygenvacancies50),thatleavebehindfilamentarystructuresofpureconductivetransitionmetal.Thesefilamentsextendfromoneelectrodetotheother,havethicknessoftheorderofnanometers38andconductancequantizationhasbeenreported.51Thefilamentsmayalsoresultfromastablemetallicphaseofdifferentstoichiometry.OneexampleisfilamentsofmetallicTi4O7inamatrixofinsulatingTiO2,bothmembersoftheMagnelliseriesoftitaniumoxides.52Oncethinfilamentsareformed,theRSfollowsfromthesuccessiveinterruptionandreconnectionbyelectricfieldpulses.Thesetwoprocessesare,however,physicallyverydifferent.Jouleheatinginducesinterruptionofthefilament,i.e.resistivechangefromthelow-tohigh-resistancestate(reset).Whenacurrentflowthroughthefilament,thehighestheatingpointislocatedatitsthinnestpart.TypicalcurrentsinthemArange,dramaticallyincreasethelocaltemperature(~900K)renderingoxygenverymobilewhichproduceslocalre-oxidation.38Theconductivepathisbrokenjustatthelocalspotwhereithadaconstriction.Thereconnectionprocessisalsoeasytounderstand.Jouleheatingisnotimportantinthiscase,becausethecurrentisordersofmagnitudelowerinthehigh-resistancestate.However,thelocalelectricfieldacrossthetwotipsofthedisconnectedfilamentisverylarge.Thisstrongelectricfieldproducesanewdielectricbreakdownoftheoxidizedspotthathadbrokenthefilament.Oxygenvacanciesmigratetothespot,theoxideisreducedandthefilamentreconnects.Thisproducesthechangefromthehigh-tolow-resistancestate(set).Thus,theresetisdrivenbycurrent(Jouleheating),whilethesetisdrivenbyvoltage(electricbreakdown).Thetypicaltimescalesofthesetwophenomenaaredifferent,withtheformerbeingslower.Nevertheless,dependingonthegeometryofthe

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system,nanosecondswitchingmaybeachieved.Clearly,thephysicalprocessinvolvedintheswitchingareindependentoftheappliedpolarity,henceitsunipolarcharacter.III.2.2BipolarRSBipolarresistiveswitchingistypicallyobservedincomplexoxides,suchasPa0.7Ca0.3MnO3(PCMO),YBa2Cu3O7-x(YBCO) ,SrTiO3(STO),Zn-dopedamorphousSiOx(SZO),etc.,26,31,41,53–55butalsoinoff-stoichiometrybinaryoxides,suchasTiO2-x,HfO2-x,Ta2O5-x,etc.33,46,48Theselattersystemsmayalsoberealizedbydepositingacappinglayerofthetransitionmetalontopofitsoxide.AqualitativedifferencewithunipolarRSisthatsomebipolarsystemsmaynotneedanyelectroforming56,althoughstillrequireafewinitialelectriccyclesuntiltheydisplayreproducibleswitching.Unfortunatelythisadvantageisnotfoundinpristinehighlyinsulatingmaterials,whichrequireasoftelectricbreakdownbeforebecomingoperational.Thisisaclearpracticalinconvenientandovercomingitremainsanopenchallenge.Thebipolarbinaryoxidesaretypicallysmalldevices,madeofafewnmthickthin-film.Incontrast,complexoxidescanbesynthesizedinthin-filmformusingmodernsynthesistechniquesormaybestudiedinbulkform.AswewilldiscussbelowintheselattersystemstheRSactuallyoccursattheirSchottkyinterfaceswiththemetallicelectrodes.17,42,44,57ThephysicalphenomenabehindtheRSareperhapslessclearthanintheunipolarcase.Thesystemsaremorecomplexandatheoreticaldescriptionisabiggerchallenge.Nevertheless,theunquestionablemaincommonfeatureofallsystemsisthattheRSinvolvesthedriftofionicspeciesunderanelectricfield.26Unlikethepreviouscase,bipolarityrequiresreversingtheresistivestatewhichimpliesbackandforthmigrationofionswithinthematerial.However,tworelevantquestionsstillstand:1)Howcanionicmigrationchangesignificantlytheresistance?2)Howcantheswitchingspeedbesofast?53,26Theanswerstothesetwoquestionsarisefromgeneralconstraintsimposedonthephysicalmechanism.Therelevantionicdriftiscausedbytheelectricfieldacrossthedielectricmaterialproducedbytheexternalvoltageappliedtotheelectrodes.Thefieldisproportionaltothelocalvoltagedropsacrossthesystem.Ohm’slaw,impliesthatthefieldsisproportionaltothelocalresistivityρ(x),wherexdenotesaspatialcoordinateacrossthedielectricbetweentheelectrodes.Thus,thehighestfieldsappearatthehighestresistivityregions.Sincethebi-polardielectricsarepoorconductorstheyareexpectedtoformSchottkyinterfaceswithordinarymetalelectrodes.17,42,57ThisSchottkyinterfaceischaracterizedbyanelectrostaticbarrierthatarisesduetobandbendingtoequilibratethechemicalpotentialsandtheelectrostaticforces.58Theseinterfacesbecomedepletedofcarriersandthusthelocalresistivityofthedielectricisdramaticallyincreased,althoughwithoutastructuralchange.Thus,theRSeffectinthesetwo-terminalcapacitor-likedevicesisdominatedthehighlyresistive(Schottky)interfacesinanotherwiseconductivematerial.31Thisobservationisthekeytotheanswersofthetwoquestionsposedabove.Aswejustdescribed,upontheapplicationofanexternalvoltagethehighestelectricfieldsdevelopatthehighresistanceinterfaces.Asaconsequence,themostsignificantionicmigrationprocesstakesplacethere.Moreover,theionicdriftinduceslocal

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structuralchanges,leadingtosignificantmodulationinthe(two-terminal)resistanceofthedevices,andhence,theRS59(figure5).ThishelpsunderstandingthefastswitchingspeedofRSasthewidthoftheSchottkybarriersistypicallyafewnanometersandtheionicdriftspeedmaybeashighas~m/s.60Infact,thosehighspeedsmaybeeasiertoreachunderstrongelectricfieldsalonggrainboundariesandbythermaleffects.53Thus,atdriftvelocityofm/sittakesonlynanosecondstomigrateacrossaSchottkybarrier,whichaccountsforthefast-observedRScommutationspeeds.

Figure5.Toppanels:Kelvinprobemeasurements (Adaptedfromreference59)of thelocal resistance for a) an un-switched device, b) when the right electrode has beenswitchedandc)when the leftelectrodehasbeen switched.ThebrownareaswithintheblackinterfacesindicatearegionwheretheSchottkybarrierhasbeenlowereddueto resistive switching. Bottom panels: Schematic representation of bipolar resistiveswitching.

Toexplainthephysicalchangethattakesplaceattheinterfacetwopossiblephenomenahavebeeninvoked.Electrostaticmodulationofthebarrierheightbyinandoutflowofionsintheinterfacialtraps.31,61Anotherpossibility,consistentwiththerobustuniversalityofthephenomenon,isthemigrationofoxygenvacancieswhichproducethepositivelychargeddefectsinallTMOs.Forinstance,inducingalargenumberofvacanciesmayintroducecarriersandturnaninsulatingbinaryoxideintoabipolarRSsystem.46,47Or,inmorecomplexandconductiveoxides,oxygenvacanciesaffecttheelectrictransport,sincetheyperturbtheO-M-ObondsformingtheconductionbandofTMOs.42Theelectrodechoicehasproventobeofcriticalimportance.Metal/InsulatorinterfacesgenerallyformSchottkybarriers,whileheavilydopedinsulatorsorsemiconductorstendtoformOhmiccontacts62.Inertelectrodes,suchasPt,PdorAu,donotchangethestoichiometryoftheoxide,creatingaSchottkybarrierinterfacesthatcanbetunedwithresistiveswitching.Ontheotherhand,reactivemetalssuchasTiorTainduceoxygenvacanciesintheunderlyingoxide,dopingthefilmandcollapsingthebarrier.ThisrendersOhmicinterfacesthatdonoteasilychangetheirresistancewhenavoltageisapplied63.WeshouldpointoutthatRSduetomotionoftheTMionshasalsobeenreportedinverythinfilmsofthebinarysystemsTaOx,HfOx

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andTiOx(seealsosectionIV.2.1).64,65III.3Originofvolatileresistiveswitching(VRS)Thevolatileresistiveswitchingisadifferentphysicalphenomenonfromthenon-volatileone.Itisobservedinmaterialsthatfeatureametal-insulatortransition(MIT)asthetemperatureislowered.41Theyevolvefromahightemperaturemetallicphasetoalowtemperatureinsulatingphase,withseveralordersofmagnituderesistivitychange.figure6ashowstheresistanceasafunctionofthetemperatureforaVO2thinfilm,withtheMITclearlyvisiblearound340K.

Figure 6: a) Metal insulator transition of a VO2 thin film. b) Structural transitionrevealedbyXRD.Adaptedfrom66.Differentmechanismshavebeenadvancedtoexplaintheoriginofthistransition.ThemostwidelyacceptedisbasedonthemechanismproposedbyN.Mottinthe40’s.Hearguedthatelectron-electronrepulsionduetoCoulombinteractionwouldfavorchargelocalizationoftheelectronsinapartiallyfilledband,renderinginsulatinganapriorimetallicmaterial.Thisinsulatingphaseiscommonlyknownas“Mottinsulator”.Repulsionenergybetweenelectronsbecomeslargeenoughtomakedoubleoccupationenergeticallycostly,limitingtheirhoppingbetweenneighboringsites.Astemperatureisincreased,thestabilityoftheMottinsulatorelectronicstatemay,however,becomeunfavorableascomparedtotheexpectedmetallicstate.Theensuingthermallydrivenmetal-insulatortransitioniscalledaMotttransition.Hence,keytothisphenomenonistheenergydifferencebetweentwoelectroniccompetingstates,whichcanbeaverysmallenergyscale.67,68SomematerialsareconsideredpureMottinsulatorssuchasGaTa4Se8,69andCr-dopedV2O3,70,71sincetheir1stordermetal-insulatortransitionoccursintheabsenceofanysymmetrybreaking(eitherstructuralormagnetic).However,quiteoftentheMITisaccompaniedbyaconcomitantstructuralphasetransition,andoccasionallybylowtemperaturemagneticordering.ExamplesofthesematerialsareVO2(structuraltransitionshowninfigure6b),V2O3,NbO2orthenickelatefamily(RENiO3).41SincethereisasymmetrychangethepreciseMITmechanismmaybemoreelusive.Thissymmetrychangeimpliestwopossibleadditionalmechanisms:thePeierlsandtheSlatertransitions.InaPeierlstransition,thelatticestructurechangeopensagapattheFermilevel,creatingandinsulatingphase.InaSlatertransition,themagneticorderingchangestheinteractionpotentialseenbytheelectrons,modifyingtheeffectivesymmetryofthelatticeandopeningthegapattheFermilevel.Inanumberofoxides(NbO2,VO2andV2O341)thedominantmechanismfortheMITisstillcontroversialandbeyondthescopeofthistutorial.

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IndependentlyofthephysicalmechanismbehindtheMIT,thereisasurprisinguniversalityintheelectricaltransportobservedinthesecorrelatedoxides.Themostnotableoneisthepossibilitytoinducethemetallicphasebyapplyingavoltageoracurrentinthematerial.Figure7showstheI-VcharacteristicsoffourdifferentMottinsulators.Thereisaremarkablesimilaritybetweenallofthem:theresistancesuddenlydropswhenathresholdvoltageisapplied,returningtoitsoriginalvalueoncethevoltageisremoved,yieldingareversible,volatileswitching.ThisresistiveswitchingcorrespondstopartofthesampleundergoingaMITintothemetallicstate,whichproducesasharpresistancedrop.Thereasonforthiselectricallytriggeredtransitiontotakeplaceisstillcontroversial.72ThesimplestexplanationisthatcurrentgeneratedJouleheatinglocallyraisesthetemperatureabovetheMITtransition,inducingtheresistiveswitching.ThereissomeexperimentalevidencethatthismightplayaroleintheI-VcharacteristicofVO2andV2O3smalldevices73–75dependingofthegapsizeratio76.However,suchandexplanationdoesnotholdforthecaseofCr-dopedV2O3thatdoesnothaveatemperature-drivenMIT.Thus,anotherexplanationisthatstrongelectricfieldsmightdestabilizetheinsulatingphase,makingthemetallicstateenergeticallymorefavorable.Thismayberevealedindifferentsystemsdependingonthegapsizeofthedevice.76,77BothmechanismsareverydifficulttodifferentiatefromtheI-Vpropertiesofthematerial,asbothwouldcreateathreshold-like,volatileresistiveswitching.72,76

Fig 7. I-V characteristics of three differentMott insulators: a) V2-XCrXO3 (X=0.3)78, b)V2O374c)NbO279andd)VO2

80AnothermechanismofelectricallytriggeredMITisbyfilamentarypercolationproducedbyelectro-thermaldomains(ETDs)74,similarlytonon-volatileresistiveswitching.Thisisexpectedindependentlyofthetriggeringmechanism:Jouleheating

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orelectricfield.Self-heatinginsuchadevicecanbestrongenoughtocauseanelectro-thermalbreakdown,i.e.,acurrentandtemperatureredistribution.Inanyofthosecases,thefirstpartofthesamplethatbecomesmetallicwilldragmorecurrentthroughitandlocallycreateamoreintenseelectricfield.Thiswill,forbothmechanisms,promotelongitudinalgrowthandeventuallyleadtotheformationoffilaments,asseeninfigure8forVO2andV2O3.

Figure8.ExperimentalobservationofametallicfilamentinthecurrentinducedMITina) V2O3, obtainedwith low-T SEM imagewidth corresponds to 10µm, from74 andb)VO2,obtainedwithnear-fieldscanningmicrowavemicroscopy,from81.Thecharacteristictimesofthevolatileswitching,potentiallycouldbeseveralordersfasterthanthenon-volatileswitching.Indeed,pump-probeexperimentshaveshownthatthetransitionfrominsulatortometalcantakeplaceinthepsrange.However,theimplementationofthesematerialsinrealdeviceswillimposeaconstraintinhowfastthetransitioncanbeinduced.TheparasiticcapacitanceandthelargeresistanceofthesematerialsintheirinsulatingstateleadstoRCconstantsoftheorderofthens.82Moreover,dependingonthedevicesize,itwilltakeacertainamountoftimeforheattobuildupacrosstheelectrodes.Thiseffectcanbeobservedinfigure9thatshowsresultsofavoltagepulseappliedacrossa100nmMetal/VO2/Metalstructure.Thetransitiondoesnottakeplaceimmediately,butittakesaround10nsforthemetallicphasetobeinduced.

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Figure9.Current (leftaxis,blacksquareddata)asa functionof timewhenavoltagepulseisapplied(rightaxis,dashedredline)toaVO2nanodevice(Inset).80IV.MATERIALSIV.1WhyTransitionMetalOxides?Resistiveswitchinghasbeenobservedinabroadrangeofsystems,rangingfromphasechangematerialstosolidsolutions.Wefocusthisreviewononeofthemostpromisingmaterialssystems,thetransitionmetaloxides.Thisgroupofoxideshasshownthehighestversatilityofallsystemswherethisphenomenonisobserved,astheymaycombinealmostuniversalresistiveswitchingwithotherunusualtransportpropertiesarisingfromelectron-electroncorrelations.TMOsdisplayahostofremarkablephenomenaandassociatedfunctionalitiesthatinclude:superconductivity,colossalmagnetoresistance,thermoelectricity,ferromagnetism,ferroelectricity,multi-ferroicity,metal-insulatortransitions,etc.Onemaywonderwhythisbewilderingvarietyofphenomena?Whatisspecialaboutthesecompounds?Afewoutstandingbasicfeaturesarise.Chemically,thetransitionmetalsoftenpresentmultiplevalencestates.Theseallowdifferentcoordinationwithoxygenwhichmayleadtoavarietyoflatticestructures.Inaddition,italsoallowsthemtochangetheelectronicoccupationintheactiveshell(orband).ThesemostrelevantaspectsleadtotheunusualelectrictransportpropertieskeytotheunderstandingoftheRSphenomena.OneofthemoststrikingaspectsofRSisitsubiquity.Infigure10weshowaperiodictablewherethetransitionmetalswhoseoxidesdisplaynon-volatileresistiveswitchingareindicated.

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Figure 10. Periodic table showing the elements whose corresponding binary oxidefeaturesnon-volatileresistiveswitching.33ThisubiquitypointstoarathergeneralcommonoriginfortheRSphenomenaandnotsomespecificmaterial-dependenteffect.Thecommonphysicaloriginofnon-volatileresistiveswitchingreliesonthefactthationscandriftintheseoxidesand,duetothemulti-valencestatesofthetransitionmetal,theresultingstructuresaremetastable.Oneimportantfeatureofthisphenomenonisthatitreliesoncreatingdefectsandinducingtheirmigration.Thus,anothercommonfeatureofallthesesystemsisthattheRScommutationbetweenahighandalowresistivestatedisplaysacertaindegreeofvariability.Therefore,non-volatileRSisnotinducedbyabonafidethermodynamicphasetransitionbutratherduetodifferentmetastablestructuralstates.WhilenotbelongingtothefamilyofTMOs,itisalsoworthtomentiontheimportantfamilyofsiliconoxides,whichhavedisplayedsimilarNVRSphenomenaasdescribedhereand.Fromapracticalpointofview,thesematerialshavetheadvantageofexcellentCMOScompatibility.83–86RegardingthevolatileRSeffect,transitionmetalsalsoplayaprominentroleandwemayconsiderthephenomenonuniversal,butinadifferentsense.VolatileswitchingisgenerallyobservedinMottsystems,whichexhibitametaltoinsulatortransition.Conventionalbandtheorypredictsthatahalf-filledbandshouldbeametal.However,theMottinsulatorstatemayariseduetothelocalizationofelectronsduetostrongon-siteCoulombrepulsion.Qualitatively,inamonovalentsystemwithnarroworbitals,theenergeticcostofdoublyoccupyingasitemaybesohighthatthesystemchoosestominimizeitsenergybykeepingoneelectronlocalizedateachatomicsite.Thepossibilityofeitherdelocalizingorlocalizinganoutershellelectronseemstobeaqualityoftransitionmetals.Theideaisthatd-orbitalsareneithertoobig,assandp’sthatyieldbands,nortoosmall,asf’sthatyieldlocalmagneticmoments.Insteadd-electrons“hesitate”betweenthesetwopossibilitiesandmayadopteitheronedependingonthedetailedsurroundingenvironment.

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Acomprehensivetablewithvariousmetaloxidesinwhichdifferenttypesofresistiveswitchinginobservedispresentedbelow(table1).Thistableisintendedasillustrationandguidanceforthereaderratherthananexhaustivelist.

Oxide ElectrodeTop/Bottom

Polarity ESet(kV/cm)

EReset(kV/cm)

Reference

MgO Pt/Pt Bipolar ∼1000 ∼400 87 Pt/Pt Unipolar ∼1000 ∼400 87

Al2O3 Ti/Ti Bipolar ∼250 ∼250 88SiOX Poly-Si/Poly-Si Unipolar ∼1000 ∼1000 83TiO2 Al/Ru Bipolar ∼300 ∼150 45 Al/Ru Unipolar ∼300 ∼150 45 Ni/Cu Bipolar ∼100 ∼80 89 Ni/Cu Unipolar ∼100 ∼80 89 Pt/Pt Unipolar ∼200 ∼400 52

VO2 Ti/Ti Irreversible ∼800 - 90 TiN/TiN Volatile ∼50(300K) - 82

V2O3 Ti/Ti Irreversible ∼1200 - 90 Ti/Ti Volatile ∼300(10K) - 75

CrO Pt/TiN Bipolar ∼460 ∼400 91MnO Ti/Pt Bipolar ∼90 ∼100 92Fe3O4 Pt/Pt Bipolar ∼400 ∼500 93CoO Pt/Pt Unipolar ∼130 - 94NiO Pt/Pt Unipolar ∼120 - 94NiO Pt/Pt Bipolar ∼200 ∼100 95Cu2O Pt/TiN Bipolar ∼80 ∼80 96

Pt/Pt Bipolar ∼50 ∼30 96 Pt/STO Bipolar ∼100 ∼100 96

ZnO TiN/Pt Bipolar ∼300 ∼400 97GaOX Pt/Pt Bipolar ∼200 ∼100 98

Pt/Pt Unipolar ∼200 ∼100 98GeOX Cu/W Bipolar ∼400 ∼200 99ZrOx Pt/p+-Si Unipolar ∼400 ∼300 100NbO2 TiN/TiN Voltatile ∼1000(300K) 101Nb2O5 Pt/p+-Si Unipolar ∼1000 ∼500 102MoOx Pt-Ir/Pt Bipolar ∼150 ∼100 103

Pt-Ir/Pt Unipolar ∼150 ∼50 103HfO2 Ti/Pt Bipolar ∼250 ∼250 104 Ti/Pt Unipolar ∼500 ∼250 104 Pt/TiN Bipolar ∼3000 ∼400 105 Pt/TiN Unipolar ∼3000 ∼200 105

TaOx Ti/Pt Bipolar ∼400 ∼120 104 Ti/Pt Unipolar ∼400 ∼150 104

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Ta/Pt Bipolar ∼1000 ∼200 106 Ta/Pt Bipolar ∼300 ∼300 107

ZnOx ZnS/ZnOx Bipolar ∼22 ∼22 108WOx TiN/W Unipolar ∼300 ∼600 109CeOX Al/Pt Bipolar ∼100 ∼500 110Gd2O3 Pt/Pt Unipolar ∼200 ∼100 111Yb2O3 Pt/TiN Bipolar ∼500 ∼500 112Lu2O3 Pt/Pt Unipolar ∼300 ∼150 113STO Au/Au Bipolar ∼0.1 ∼0.1 53PCMO - Volatile ∼10(20K) - 61PLCMO Ag/Ag Bipolar - - 56BFO Cu/Pt Bipolar ∼150 ∼150 114YBCO Au/Au Bipolar - - 115SZO Au/SRO Bipolar ∼15 ∼15 54

Table1.Examplesofoxidesinwhichresistiveswitchingisobserved.Thetypeofswitchingandsetandresetfieldsareindicated.

IV.2OthersystemsdisplayingresistiveswitchingAlthoughthisreviewismainlyfocusedinTMO,othermaterialsystemsshowresistiveswitching.Herewewillgiveabriefintroductiontotwoofsuchsystems:conductivebridgememoriesandphase-changematerials.IV.2.1CationDriftsystemsConductiveBridgememory(CBRAM)alsoknownaselectrochemicalmetallizationmemory(ECM),or“cation”systemsareproducedbythedriftofpositivemetallicanions(asopposedtooxygenvacancycationsinthecaseofTMO)23,32,116–118.Thesedevicesaregenerallycomposedbytwodifferentelectrodes:anelectrochemicallyactive(generallyCuorAg)andaninert(suchasPt,Au,CrorW)electrode.Thesetwoelectrodesareseparatedbyaninsulatorsuchasasolidelectrolytewithgoodionicconductivity(AgorCudopedsulfidesorchalcogenides:Ag2S119,120,GeSX121,GeSeX121,122orpuredielectrics(a-Si123,124,ZrO2125,HfOX126,SiO2123,127orAl2O3.

128)Apositivevoltageappliedtotheactiveelectrodeoxidizesthemetal(AgorCu),producingpositivemetallicionsthatdriftthroughtheinsulatorpushedbytheelectricfield(Figure11,pointB).Whentheanionsreachtheinertelectrode,theyarereduced,formingametallicdepositwhichgrowsfromtheinertelectrodetowardstheactiveelectrode,asshowninpointCofFigure11.Whenitreachestheoppositeelectrode,asuddenresistancedroptakesplace(PointD).Reversingthepolarityreversestheeffect:AgorCuwillbeoxidizedattheinertelectrodeinterfaceandpushedintheoppositedirection,breakingtheconductivepathbetweenelectrodes(PointE).ThisprocesscreatesabipolarNVRS.23,32,116–118

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Figure 11. Schematic representation of filament formation and bipolar resistiveswitchinginCBRAM.Adaptedfromreference117.

TheRSpropertiesofthesesystemsarestronglydependentonthemorphologicalpropertiesoftheinsulatingfilm.23,117,123Goodionicconductorsandgranularfilmsfacilitateionmotion,leadingtolongfilamentary/dendriticstructuresandlowformingvoltages.123,125Onthecontrary,denselypackedfilms(suchasSi)limitcationsupplyandinducetheformationofmetallicnanoclustersinsidetheinsulatingmatrix.125,126Theseclustersdeformorsplitwhenanexternalvoltageisapplied,modifyingtheresistanceofthedevice.123,126Volatileresistiveswitchinghasbeenrecentlyobservedinthistypeofsystems.Wangetal.126fabricatedcationdevicesconsistingonAgnanoparticlesembeddedindifferentoxidematrixesandsandwichedbetweentwoinertelectrodes.Thesedevicesspontaneouslyrecovertheirhighresistanceoncetheappliedvoltageisremoved,typicallyinamstimescale(Figure12a).UsingTEM,itwasarguedthatinterfacialtensionmightcausethiseffect:thenanoparticlesmergeintolargerclustersoncetheelectricfieldisremoved,breakingtheconductivepath(Figure12b).ThisshowsthatCBRAMdynamicsisacomplexphenomenonandseveralmechanisms(electrochemical,thermalandmechanical)mightbeatplay.126

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Figure12.Volatility in aCBRAMdevice. a) Current as a functionof time: thedevicereturnstoitsinsulatingresistancevalueafterafewms.b)HRTEMobservationoftherelaxation process: smaller Ag nanoparticles coalesce into larger ones when theexternalvoltageisremovedatt=5.0s.Adaptedfromreference126.IV.2.2Phase-ChangeMaterials(PCM)ThebasicfeaturethatenablesRSinthistypeofmaterialsisthattheycanexistintwovery different states: low resistance crystalline phase, or high resistance amorphousphase. Switching between both is possible by heating the sample and carefullycontrolling the cooling rate.129–133 This can be done either optically131,134–137 orelectrically,129–133,138–142 making phase change materials versatile for applications inphotonicsaswellasinelectronics.Theamorphousphasecanbeinducedbyapplyingashort, high voltage pulse to the sample: intense enough to drive it over itsmeltingpointandsharpenoughtoallowthermalquenchingintotheamorphousstate(RESETpulseinFigure13).Thecrystallinephasecanberecoveredbyapplyingalonger,lowervoltagepulsethatallowsrecrystallizationbyannealing(SETpulseinFigure13).

Figure13.a)TypicalstructureofPCMdevices.b)SchematicrepresentationoftheSET,RESETandreadpulsesusedinPCM.Adaptedfromreference132.Materials used for this type of memory should in general meet two requirements:large crystalline/amorphous resistivity ratios and fast re-crystallization times. 129–133Semiconducting alloys in pseudo-binary line between GeTe and Sb2Te3 (such asGe2Sb2Te5) show crystallization times in the 10-8 s range and up to 5 orders ofmagnituderesistivitychange.133,135Theirdiscoveryin1987135allowedthedevelopmentofcommercialtechnologiesbasedonPCM,suchasrewritableopticalstorage(CD-RW).131,133Threeothersystemswithfastcrystallizationtimeswerelaterdiscovered:Sb2Tedopedwith Ag, In or Ge136; GeSb137 and GeTeX alloys.138,139 The typical RS currents,voltagesandcharacteristic timescalesarecomparable to thoseobserved inTMO140,makingPCMadirectcompetitorforneuromorphiccomputingimplementations.141,142

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V.MODELSSofar,wehavepresentedthedifferentwaysthatresistiveswitchingmayappearandthematerialswherethephenomenaareobserved.Inthissection,wewillbrieflyreviewsomeofthetheoreticalmodelsthathavebeenproposedtoexplaintheexperiments.Table2showsthedifferenttypesofresistiveswitching,togetherwiththeiroriginandthemostnotablemodelsthatexplaintheirphenomenology(Pleasenotethatconductivebridgememoriesandphasechangematerialsarenotincludedinthetable).

Table2.Characteristicsofthedifferenttypesofresistiveswitching.VEnhancedstands

forVoltageEnhanced,ODriftstandsforOxygenDriftandRNetworkstandsforResistorNetwork.Notethatconductivebridgememoriesandphase-changematerials

arenotincludedinthetable.V.1Non-volatileresistiveswitching(NVRS)Belowwedescribebrieflysomeofthetheoreticalmodelsproposedtoexplainthenon-volatileRSeffectwhichprovidenumericalsupporttothequalitativedescriptionsdiscussedabove.Asdoneintheprevioussections,weshallmakethedistinctionbetweenunipolarandbipolarsystems.V.1.1UnipolarNVRSTheRandomCircuitBreaker(RCB)modelconsistsofaresistornetwork143(figure14)inwhicheachunittakesoneoftworesistancestates(high)rhor(low)rl,withrh>>rl.Themodelassumesthateachunitcanundergoatransition,followingtherulerh->rlifV>

Non-VolatileResistiveSwitching VolatileResistiveSwitching Unipolar BipolarOriginofSwitchingMechanism

Oxygenvacancygeneration

byelectroforming

Oxygenvacancydriftinduced

byanelectricfieldMetal-insulatorphasetransition

Materials Binaryoxides(mainly)

Complexoxides,off-stoichiometrybinary

oxides.Mottinsulators Mottinsulators

SetMechanism

Metalormetal-phasefilament

formationinducedbyelectricfield

Oxygenvacancy

driftinthebulk

Defectdriftin/out

interface

E-fieldinducedtransition

Heatinginducedtransition

ResetMechanism

Jouleheatinginducedreoxidation

offilamentaryweaklink

Oxygenvacancy

driftinthebulk

Defectdriftin/out

interface

Relaxationintoinsulatingphase

Relaxationintoinsulatingphase

Models RandomCircuitBreaker Memristor VEnhanced

ODriftRNetwork

E-FieldInducedRNetwork

HeatingInduced

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Vonandrl->rhifV>Voff.ThevalueVisthelocalvoltagedropateachelementandVonandVoffarethresholdvoltagevalues,withtheconditionthatVon>>Voff.

Figure 14. Random circuit breaker model, adapted from 143. Top panel shows thesimulated I-V characteristics of the model. Bottom panels show the distribution ofmetallicdomainswithintheoxidematrix.Theappliedvoltageisinitiallyrampedup,andeventuallyanavalancheoftransitionstakesplace(figure14).Atthispointthesystemhasbeen“electroformed”aslowresistancelinkshavepercolated.TheunipolarRSeffectisproducedastheappliedvoltageisrampedupagainfromzero.Thelocalvoltage-dropincreasesuntilalowresistiveelementreachesthelowerthresholdV>Voff(>>Von)andswitchesbacktotherhstate.Eventually,percolationislostatalast“critical”link(figure14panelsBtoC).Then,thetotalresistanceofthenetworkjumpstoahighresistanceRhigh.Forasufficientlylargedifferencebetweenrhandrlmostoftheappliedvoltagedropsacrossthefinallink.Thisthenswitchestothelowvaluerlwhentheexternalvoltageisincreasedagain.Thus,aftertheinitialelectroformingevent,thesuccessivehi-loresistiveswitchingisconcentratedontheon/offswitchingofacriticallink.TheRSeffectcanalsobemodeledbysolvingthedifferentialequationsthatdescribetheelectricandthermaltransportofaconductivefilamentsurroundedbyaninsulatingmatrix48.Themodelfurtherassumesthatthefilamentdissolvesbyoutdiffusionofitsconductiveelementsintotheinsulator.ThediffusionprocessisassumedtobeactivatedwhichleadstoadiffusionvelocitywithanArrheniusdependenceVD=VD0exp(-Ea/kBT),whereEaisanactivationenergy.Theelectrodesareassumedtobetheheatsinks.Asthetemperatureincreasesinthecenterofthefilamentthefilamentdissolves,whichfurtherincreasesthetemperaturebythelocalJouleheatinguntilthefilamentbreaks.V.1.2BipolarNVRS

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Anearlyqualitativemodelofbipolarsystems144anticipatedseveralfeatures,whichprovidedusefulguidance.Themodelassumedarelevantroleplayedbytheelectrode-materialinterfacesandtheionicmigrationwhichappearsinmanyofthelatermodels.Thememristormodel,46,145capturedagreatdealofattention(figure15).Itwasintroducedasthe“missingelement”,arisingfromsymmetryconsiderationsofelectriccircuittheory.Thissimplemodelwasformulatedtorationalizebi-polarRSexperimentsinTiO2-xdevices.

Figure 15. Top panel: Simulated I-V characteristic of the memristor model at twodifferentacfrequencies.Bottompanels:Schematicrepresentationofoxygenvacanciesdistribution for the ON and OFF states. w and w’ denote the size of the oxygendepletedlayer,whileDisthedistancebetweenelectrodes.Adaptedfrom46.

Itassumesthatthedielectricpartofthedevicehastworegions:alowconductivity,dopedTiO2-xoflengthw,andaninsulatingpureTiO2oflengthD-w,whereDisthedistancebetweentheelectrodes.Thus,themodelassumestwoseriesresistorsofhighandlowresistivitiesρoffandρon.ThisleadstoadeviceresistanceR=Roffw/D+Ron(1-w/D),whereRoffandRonaretheresistancesofthepristineandoxygendepletedfilms,respectively.Thecrucialmodelassumptionisthattheappliedvoltageacrossthedeviceproducesionic(oxygenvacancies)drift,whichleadstotheshiftofthelocationofthetwo-regioninterface.Thus,thedeviceisnolongerOhmicandtheI-Vcharacteristicsbecomesnon-linear.Thistypeofmemristivemodelappliestosmallnano-sizesystems,wheretheinsulatingpartissmallenoughandtheelectricfieldsthatcandeveloparelargeenoughtopromoteionicdriftofdefects.Thissimplemodel

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producesthegenericallyobservedpinchedhysteresisloopintheI-Vcharacteristics(figure15).Thistypeofmemristivemodelcanbeextendedtoincorporatethetemporalevolutionofwwiththeappliedvoltage.Thisleadstoafulldynamicalmodelwhich,inprinciple,maydescribethebehaviorofthememristoruponapplicationofarbitrarypulseandrampvoltageshapes.Thedrawbackofthismodelistheneedtomeasureandfitanextensivesetofdatawithad-hocmathematicalexpressionscontainingvariousfreeparameters.146Anotherbipolarmodelistheso-calledvoltageenhancedoxygenvacancydrift(VEOD)model42(figure16)formulatedtodescribeanunusualRSeffectinPt/PCMO/Ptdevices.147Thisbehaviordemonstratedthekey-roleplayedbythehighlyresistiveSchottkyinterfacesandtheircomplementarybehaviorinsymmetricdevices.

Figure 16. RS experimental data in manganite a) and cuprate b) devices driven bycurrentandvoltage,respectively.RSsimulationusingtheVEODmodelinacurrentc)andvoltage(d)controlledsimulation.Adaptedfrom42.Itshouldbementioned,thatmodelsofvoltage-enhanceddrift42,46naturallyleadtothepossibilityofshockwavepropagationoftheionicmotionduringresistivetransition.148,149V.2Volatileresistiveswitching(VRS)ThevolatileresistiveswitchingoriginatingfromadestabilizationofaMottinsulatorunderastrongelectricfieldpossessagreattheoreticalchallenge77,150withlimitedcontactwithexperiments.AnintriguingconsequencefromthetheoreticalstudiesisthattheelectricbreakdownofaMottinsulatorispredictedtooccuratMV/cmelectricfields,whileexperimentallyisobservedforfieldsofonlykV/cm.37,151Clearlymodelingthesesystemsrequiresconsideringrealisticsituationswhichincludeinevitableextrinsicandintrinsicdisorder.FurtherinsightmaybeobtainedfromphenomenologicalmodelstodescribethevolatileRSeffect.Themainproposed

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modelsareeitherbasedonresistornetworksoronsystemsofdifferentialequationsthatcouplethethermalbalancewiththeelectrictransport.V.2.1ResistornetworkTheresistornetworkmodelshowninfigure17wasintroducedtodescribetheuniversalvolatileRSobservedinVO2,152V2O3,74,153Cr-dopedV2O3,NiSe2-xSxandGaTa4Se8.37Inthismodel,eachresistorofthenetworkcanbeinoneoftworesistancestates:metalorinsulator.Theprobabilitytobeinthemetallicorinsulatingstatesisgivenbyafreeenergythatschematicallyrepresentsthefirstordernatureofthephasetransition.71Therearetwobasicvariationsofthismodel,dependingonthetriggeringmechanismoftheMIT:E-field37orJouleheating.74InthecaseofE-fielddriventransitions,anextratermisaddedtothefreeenergytotakeintoaccountthefieldinduceddestabilizationoftheinsulatingphase.71,77

Figure 17. a) Typical resistor network configuration used for simulations of resistiveswitching inMottmaterials. Each cell canbeeithermetallic or insulating (black andwhite).Bottompanelsshowtwosimulations inwhichfilamentformationisobservedfor: b) a Joule heating induced transition (adapted from74), and c) an E-field driventransition(adaptedfrom37).Themodelassumesthattheinsulatorismorestableatzerofieldthanthemetalandthattheyareseparatedbyanenergybarrier.TheresistivecellsofthenetworkmaychangetheirstatefollowinganArrheniustypethermalactivationwiththefurthercrucialassumptionthattheenergyoftheinsulatorisaffectedbythelocalelectricfieldactingontheresistivecell.ThismodelcapturesqualitativelytheexperimentalI-Vcharacteristicsandtherapiddecreasewithappliedfieldofthedelaytimeτdoftheresistancecollapse.VI.IMPLEMENTATIONS

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Resistiveswitchingintransitionmetaloxides,stronglycorrelated,andphase-changesystemsmayenablenovelneuromorphicelectronicfunctionalities.Asdiscussedabove,thenon-volatileRSenables“synaptic”functionalities,whilethevolatileRSenables“neuronal”functionalities.Thesetwofunctionalitiesaretheessentialcomponentsofanyneuralnetwork.ThisChapterdescribesomeoftherecentprogressintheimplementationofsuchphenomenaintorealdeviceswithcomputingcapabilities.VI.1SynapticfunctionalitiesThesimplestimplementationofanon-volatilesynapticfunctionistocontrolthesynapticweighti.e.theelectriccouplingbetweentwoelectronicneurons.Thisisachievedbyusingnon-volatileRStotunetheresistance(orconductance)ofthedevice.Asinbiologicalsystemsthatmodifythebrainsynapticconnectionswhenanewmemoryisacquired,thisismaybeaccomplishedinaneuromorphicsystembychangingtheresistanceoftheRSdevice.Inpractice,thiscanbeachievedusingelectricpulsesthatselectivelytargetdifferentsynapsesintheneuralcircuit.Besidesthesimplemodulationofasynapticintensity,othermoreelaboratesynapticfunctionalitiesarealsobeingcurrentlyproposedandinvestigated.STDPfunctionalityisattherootoflearningalgorithmsforspikingneuralnetworks.Figure18showsaconcreterealizationinadiffuse(SiOX:Ag)electrodeandadriftTiO2memristor-basedsynapse.126.STDPisobtainedbycarefuldesignoftheelectricpulsesemittedbytheneuristors(seefigure18a).Thisway,thevoltageacrossthememristorcontrolsthesynapticweightchange,bytherelativetimingbetweenpulses.

Figure18.a)SchematicofthepulsesappliedtothecombineddeviceforSTDPdemonstration.Thelonglow-voltagepulseineachspiketurnsthediffusivememristorON,andtheshorthigh-voltagepulseswitchesthedriftmemristor.Whenthepost-spikeprecedesthepre-spike,thedeviceisreset(depressed),andwhenthepre-spikeprecedesthepost-spike,thedeviceisset(potentiated).Thetiming(!t)betweenthetwospikesdeterminesthevoltagedropacrossthedriftmemristor.b)DemonstrationofSTDP.Synapticweightchangeofamemristorasafunctionofthetimedifferencebetween(preandpost)spikesofthesystem.Theinsetshowsthequalitativeagreementwithabiologicalsystem.Figureadaptedfrom126.

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VI.2NeuronalfunctionalitiesAswedescribedabove,theneuron-likefunctionalitiesenabledbyvolatileRScanberealizedusingMottinsulatormaterials.Hereweshallbrieflydescribesomeoftheserecentimplementations.VI.2.1GenerationofelectricalspikesOneofthegoalsofaneuristorimplementationistoemulatethegenerationofanactionpotential.Thiscouldbemimicked35usingthenegativedifferentialresistanceofNbO2(figure19a),asitundergoesaninsulatortometaltransitionuponheating.ThiswasaccomplishedbyusingtwoNbO2volatilememristorscoupledtotwocapacitors(figure19b)followingtheconceptualmodelofHodgkinandHuxley(HH).ThisbiologicalmodeldescribestheresponseofNaandKchannelsintheneuronmembrane.TheRSthresholdbehavioroftheRSintheNbO2memristor,togetherwiththeRCconstantofthecapacitorsmimicsmanykeyfeaturesofbiologicalneuronssuchasspiking,signalgainandtherefractoryperiod(figure19c).

Fig.19NeuristorimplementationbasedonNbO2memristors.a)SEMofatypicalmemristorb)circuitimplementationoftheneuristortosimulateatwo-channelmodel.ThedeviceusestwonominallyidenticalNbO2Mottmemristorsbasedc)inputandoutputoftheneuristorbased-device.Topcurveshowsthedccurrentinput.BottomcurvesshowsthedeviceresponsefordifferentC1/C2(seeb)configurationswhichemulatedifferentfrequencyneuronresponses.35Thebasicfeatureofthecircuitisan“all-or-nothing”behavior,namely,asmalloutputsignaluntiltheinputovercomesagiventhresholdvaluethatproducestheoutputvoltagetodevelopalargespike(seefigure19c).Inaddition,thisdevicemayalsoproduceregulartrainsofvoltagespikesuponapplicationofaconstantdccurrentinput(seefigure19c).Interestingly,thenumber,frequencyandshapeofthespikescanbetailoredthroughtuningoftheassociatedcapacitors.Thispropertymaybeappealingforimplementationofsystemswithdifferenttypesofneuralcodingschemes.

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VI.2.2Leaky-integrate-and-fireAnotherwell-knownpropertyofneuronsistheleaky-integrate-and-fire(LIF).Inthismodel,theneuronactsasanaslossyintegratoroftheelectricsignalthatarrivestoitsbody(viaitsdendrites).Atacertainintegrationthreshold,theneuronfiresaspikethattravelsdownitsaxontoactonother“downstream”connectedneurons.Quiteremarkably,thisfunctionalitycanalsobeimplementedwithMottRSdevices.Inthiscase,thevolatileRSbehaviorisusedasdescribedinsectionIII.2.A3DMottinsulatorwithsmallenergygapundergoesaresistivetransitionwithinashortdelaytimeτduponapplicationofanabove-thresholdvoltageV>Vth.ThetimedelaydependsonthestrengthoftheappliedV.Whentheinputvoltageisterminated,theresistanceofthesystemsspontaneouslyrecoversitsoriginalvaluewithinacharacteristictimeτr.ThekeyfeaturethatleadstotheLIFbehavioristheresponseoftheMottRSuponapplicationofatrainofvoltagepulses.78,154Asingleabove-thresholdpulseappliedforτon<τd,doesnotproducetheresistivecollapse.Asubsequentpulseappliedafteratimeτoff,leadstotwopossibilities.Ifτoff>τrthesystemrelaxesanychangeinducedbytheinitialpulse,sothesecondpulsewillalsofailtoinducearesistivecollapse.However,ifτoff<τrtheeffectinducedbythefirstpulsewillnotbetotallyerasedbythetimethesecondpulsearrives.Furtherpulsesproduceacumulative(leaky-integration)effect,eventuallyproducingaresistivecollapseandanensuingcurrentspike(fire).37Theevolutionoftheresistornetworkmodel(sectionV.2)uponaninputofatrainofpulsescanbedescribedbyasetofdifferentialequationsanalogoustothebiologicalLIFmodel.37Moreover,giventhestrengthandfrequencyofanincomingtrainofpulses(spikes),amathematicalformulahasbeenderivedwhichpredictsthenumberofinputspikesthatwillproducethefireevent(i.e.resistivecollapse).37Thisprovidesaremarkablefunctionalitysimilartobiologicalneurons;theresponseofthesystemisfasterwhentheincomingtrainofspikesiseithermorefrequentand/orofstrongerintensity(seefigure20)

Figure20.Experimentallyobserveddecreaseofthenumberofincomingspikestosetafireevent(resistancecollapse)uponincreasingpulsefrequencyorwidth,inagreementwiththeLIFmodel.Adaptedfromreference35.

VI.3CrossbararchitectureHighconnectivityamongneuronsandselectiveaccesscanbeachievedpartiallybytheregularcrossbararchitectures.Thememristorcrossbararraysofferadvantages155in

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machinelearningimplementationsduetotwokeyfeatures156.First,thepossibilitytoimplementmatrixoperationsbasedonthesynapticweightinatwo-terminaldevice.Second,theoptionofonlinelearningbyusingasequenceofpulses155Typicalmemristorcrossbararraysarchitecturesareshowninfigure21.Eachnodeisatwoterminalmemristivedevice.Inputsareconnectedtothenetworkrowsandoutputstothecolumns.Theelectricpulsesattheinputcontrolthememristors(synaptic)weights.Thesedesignsareusedinseveralimplementationsfordataclusteringtechniqueslikeprincipalcomponentanalysisinlarge-scaledatasets155andpatternrecognition.157Animportantdisadvantageofthecrossbarimplementationsisthe“sneak-path”problemduetoanexcessofcurrent(“leakage”)whenoneofthenodesisinthehigh-resistancestate.Whenreadingsuchastate,thecurrentcanflowthroughanunintendedpaththatincludeslowresistancenodesgivinganincorrectreadout.Suchpossibilityisdepictedinfigure21.Severalwaystoovercometheseproblemshavebeenproposedwithmarginalsuccess.Forexample,groundingthefloatingterminalstodivertthecurrenttotheground,158amultistagereadingprotocol,159anunfoldedarchitectureusingareadlaneforindividualmemristors(figure21),diodeortransistorgating160andusingthenon-linearitycurrent-voltagecharacteristicofmetal-oxidememristors161,162havenotfullysolvedtheproblem.

Figure21a)TypicalCrossbararraygeometryforvariousimplementationsforneuromorphiccomputing.b)schematicofthesneak-pathproblemadaptedfrom155.Greenpathisthroughahighresistancestateandredisthroughthreelowresistancestates.VII.APPLICATIONSInthefollowingweprovideasummaryofcurrentapplicationsofneuromorphicarchitectures.Theseapplicationsarebasedincrossbargeometriesthatcombinedifferenttypesoffunctionalmemristors(includingphasechangeandmetalcationdevices).Theapproachistoemulateallpossibleexistingalgorithmsinneuralnetworksintoarraysofmemristor-basecellsthatallowsforvectoroperations.163–167VII.1Patternrecognitionanddatamining

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Patternrecognitionisatthecoreofneuromorphicapplicationsusingmemristorcrossbarsarrays.Forinstance,3x3matrixcrossbarrealizationhavebeentestedusingTiO2-xmemristiveelements,anduptoa12x12matrixhasalsobeenproposed,161seefigure22b.The3x3implementationdemonstratedafullyoperationalneuralnetwork,basedonanintegrated,transistor-freecrossbarwithTiO2-xmemristorwherethevariabilitywasreducedbyusingaAl2O3/TiO2-xheterostructure.Inthisapproach,asingle-layerneuralnetworkwasimplementedwithteninputsandthreeoutputs,fullyconnectedwith10x3=30synapticweights.Thegoalwastodetectapatternimagecomposedofa3x3pixelmatrixwithblackandwhiteelementsreproducingtheletters“z”,“v”and“n”(figure22c).Noisydatasetsfortrainingwerefabricatedbyflippingapixelrandomlyandusedastrainingandtesting(seefigure22a).Highfidelityconvergenceofnetworkoutputsduringthetrainingwasachievedaftersixiterationsfromdifferentinitialstatesasshowninfigure22d.

Figure22.Implementationofpatternrecognitionusingamemristorcrossbararray.a)Initialdatasetusedfortrainingthenetwork.b)Deviceusedtoimplementthecrossbararchitecture.c)threeoutputcategoriesinwhichtheinputdatawascategorizedandd)Performanceoftheneuralnetworkasafunctionofthenumberoftrainingsets.Adaptedfrom161.Memristors-basedcrossbararrayswereusedtoperformmatrixoperationsinanarrayofnineinputsandtwooutputs.Theinputswerefedwithadatasetcorrespondingtoabreastcancerstudyandthegoalwastosearchforprincipalcomponentsinthedatasets.Thiswasimplementedusingamemristor-basedneuromorphicchipusingTaOx/Ta2O5:Siheterostructure,wheretheSidopingisusedtotailorfilamentformation.Itcontrolstheion-hoppingdistanceanddriftvelocity,thusallowingimprovedtuningoftheRSprocessattheatomiclevel.

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Figure23.ResultsoftheprincipalcomponentanalysisimplementedinainTaOx/Ta2O5:Simemristorcrossbargeometryshowinganalmost-perfectclusteringofabreast-cancerdataset.155,168Thisworkshowsthatunsupervisedlearningrulescanbesuccessfullyimplementedinactualdevicesasdepictedinfigure23.Thememristorbasedresultsarealmostidenticaltostandardcovariancematrixapproaches.Thisopensnewroadsforfurtherimplementationsofneuromorphiccomputingchips.VII.2AnaloguecomputingandotherimplementationsMemristor-basedcrossbararchitectureshavebeenproposedandrecentlyusedtoperformsparsecodingandimageprocessing169,170.Sparsecodingrepresentationisalearningmethodthataimstofindasparserepresentationofaninputdatasetandthereforeconstructingadictionarythatallowsforidentificationofpatternsinnewdatasets.Thebasicelementsarenotorthogonal,meaningthattheymaycontainsimilarinformationandthereforeareoversampled.Thisisdevelopedtoemulatethefunctionalityofbiologicalsystemswhichprovidesthemeanstoanalyzecomplexdatasetsfrominputneurons.Someapplicationsofsparsecodingarecompressedsensingandsignalrecovery.171,172Animplementationofthis,usingWOX-basedanalogmemristorsisshowninfigure24(a).A32x32crossbararraywasusedtoimplementasparsecodingalgorithm169.Theoperationalprincipleconsistsoffindingthebasicrepresentationofanelementasschematicallyrepresentedinfigure24(b).Themaintaskisusinganimageinputtodecomposeitandrepresentitwithaminimalnumberof“dictionary”elements.Theexperimentalrealizationofsuchalgorithmisdepictedinfigure24(c-d).Infigure24(c)theinputimageisa5x5inputmatrixwherethecolorcodecorrespondtodifferentconductancestates(weighs)thatarerepresentedingreyscale.Thereare20dictionaryelements,eachcontaining25weighs.Thereconstructedimageisshowninfigure24(c)afterthestabilizationofthesparse-codingnetworkisreached(figure24(d)).Morecompleximageswerereconstructedprovingtheconceptofsparsecodingimplementationusingmemristor-basedcrossbararchitectures.

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Otherrecentimplementationsofcrossbararraysbasedonhafniumoxide170areasbigas128x64cells.Theimplementationsdescribedaboverelaysonfeatureextractionfromreal-spacedatasets,suchasimages.Anotherimportantapplicationistheprocessingoftemporalinformation.Inthiscase,thealgorithmsarebasedinaneuralnetwork-basedcomputingparadigmcalledreservoircomputing.ThesetypeofsystemshavebeenrecentlyimplementedwithWOxmemristorshavingshort-termmemorythatallowstoimplementthedynamicreservoir.Applicationsofsuchsystemshavebeendemonstratedforhand-writtendigitrecognition.173

Figure24.a)A32x32crossbararrayofWOX-basedanalogmemristors.Eachconductancestateiscontrolledbyionredistribution,b)Insparsecoding,adictionaryofoversampledelementsisconstructedwhichallowforidentificationofelementsinadataset.Asanexampletheimageofaclockisused.c)Originalandreconstructedimageusedtoprovethesparsecodingconcept.d)Membranepotential(voltageoutputofaneuron)of20neuronsaftereachiteration.Thedashedlineindicatedthethreshold.After10iterationstheneuronsreachequilibrium.Adaptedfrom169.VIII.CONCLUSIONSANDPERSPECTIVEWeconcludethistutorialdiscussingsomeopenquestionsandissuesrelatedtothephysicalimplementationofRSforneuromorphicsystems.Ofcourse,therearealsomanychallengesrelatedtosoftwareimplementation,whicharebeyondthescopeofthisarticle.Itisimportanttoemphasizethatenergy-efficientNeuromorphicComputationalsystemshavealreadybeenimplementedforsomelimitedapplications.Thus,thereisaclearproofofconcept,whichshowsthatthisapproachshouldbeabletoproduceaviablecomputationalmachine.However,thereareseveralbasicresearchissuesineachconstituentofsuchasystem,whichimplythatmuchresearchisneeded

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beforeageneralpurpose,energy-efficient,computationalmachinewhichrivalsthehumanbraincanbeimplemented.VIII.1ImmediateChallengesImmediatedevelopmentofresistivememoriesfacestwomajorproblems:variabilityandintegrationwithcurrenttechnologicalplatforms.CurrentRSdevicesshowagreatdegreeofvariability,bothbetweendifferentdevicesanddifferentcycles174–177.Thefundamentalreasonistheirsmallsize,inthenanometerscale,comparabletoimportantstructurallengthssuchasthegrainsizeorthedomainlengthinMottinsulators.NVRSishighlydependentondefectsastheyenhancevacancyformationandionmobility,facilitatingthefilamentformation.Similarly,imperfectionsinthefilmactasnucleationcenters,reducingthethresholdvoltageneededtotriggertheMIT.Slightlydifferentconfigurationsineachdevicewillresultinagreatvariabilitybetweenthem.Thecycle-to-cyclevariabilityiscausedbytheirreproducibilityoftheRSprocessinthesamedevice,i.e.thedevicewillnotswitchatthesamevoltages,orwillnothavethesameresistanceeverytimeitiscycled175–177.Thatisalsocausedbythesmallsizesinvolved:filamentstendtogrowinanextremelythinfashion,typicallyafewnanometersacross.SetandResetprocessescommonlytakeplaceinadestructivefashion:partofthefilamentmeltsanddisconnectsanditisveryunlikelythattheexactsameconfigurationwillberepeatedwhenthefilamentisreformed.AmongtheTMO,HfOXandTaOXhavebeendemonstratedtobeparticularlyreliable,showinglessvariabilitybetweencycles65,170,177,178.Itisuncleartowhatextentthisdevice-to-deviceandcycle-to-cyclevariabilitycanaffecttheperformanceofRSforNeuromorphicComputing.Somefeaturesofbiologicalneuralsystemsarefaulttolerantanddonotrequireextremeprecisiontooperateprecisely.Querliozetal.179showedthatdevicevariationscanbeovercomebyadjustingtheneuronfiringrate.Ontheotherhand,functionalitiessuchasspikefiringareveryrepeatablebetweendifferentcells.Whetherthisvariabilityposesagreatproblemtorealimplementationsisstillnotclear.ShorttermapplicationsofRSwillcertainlybeintegratedtogetherwithexisting,maturetechnologiessuchasCMOS.Nanofabricationproceduresmightnotbecompatiblewithexistingtechnologiesmightbeacomplicatedtask,assomematerialsmightnotbesuitedforthat.NVRScommonlyusestwoelectrodesofdifferentmetals:oneactiveandoneinert,generallyPtorPd.ThesematerialsareincompatiblewithCMOStechnology,sootherelectrodes,suchasTiNorRumustbeused180.ForVRS,thechallengeisevenbigger:materialssuchasNbO2,VO2orV2O3aregrownathightemperature(ashighas700C)inordertodisplayusefulMITproperties181.SuchhightemperaturesproducedeleteriousdopantredistributionintheCMOStransistors.NewfabricationroutesforMottinsulatorsclosetoroomtemperatureareneededtoovercomethisproblem.VIII.2Longtermchallenges

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Apartfromtheseimmediatechallenges,longtermimplementationofneuromorphicdeviceswillrequiretoaddressotherissues.AselectionofthesearelistedbelowGeneral:

1. ShouldcurrentNeuromorphicsoftwareparadigmsguidethedevelopmentofNeuromorphichardware?

2. Canageneralpurpose,realistic,energyefficientNeuromorphicmachinebedesigned?

3. WhichisthemostpromisingapproachfortheimplementationofNeuromorphicfunctionalities:charge,spin-torque,superconductingorhybridsystems?

4. Canthecharge-basedapproacheventuallyscaletoproduceacomputationalmachinewithsufficientelements,flexibilityandfunctionalitytorivalthehumanbrain?

Materials:

1. Candifferentneuromorphicfunctionsbeimplementedinthesameorsimilarcompatiblematerialssystems?

2. Arestronglycorrelatedsystemsthebestcandidatestosearchfornon-linearphenomenaandnovelpropertiesusefulfortheimplementationofneuromorphicfunctionalities?

3. Withinthecharge-basedsystemsinwhichmaterialsisitpossibletointegratethewidestrangeofrelevantfunctionalities?

4. Whichmaterialssystemsarethemostpromisingfortheimplementationofresistiveswitching,volatileandnon-volatile?

5. Whatistheroleofdisorderandhowfaulttolerantshouldasystembe?6. Whatistheroleofdisorderandheatdissipationasasystemisdownscaled?

Phenomenaanddevices:

1. Whichisthemostusefulswitchingmechanismforincorporationintoaneuromorphicsystem?

2. Canionicmotionbecontrolledpreciselyenoughtoproducestatisticallyreproducibleresultsaftermanycycles?

3. Whatistheroleofthermodynamicsatthesmallscale?4. Howfastcanthedynamicsofswitchingbeandisthisanimportantissue?5. Canpureelectricswitchingbeimplementedorathermalcomponentwill

alwaysbepresent?6. Canthermalswitchingbemorepracticalandeasiertocontrolthanelectric

switching?7. Candirectimagingofchangesinneuromorphicdevicesbeimplementedand

studiedin-operandoinrealtime?8. Isthereaminimumnumberofinterconnecteddevicesthatproduceemergent

functionalities,whichdonotarisefromthesimplesumofindividualcomponents?

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Architecture

1. Whatistheroleofarchitecture?2. Arethereanycleverartificialarchitectureswhichcanaddressand/orilluminate

importantissuesrelatedtointerconnectivity?3. Howshouldthedifferentcomponentsofaneuromorphicsystembe

interconnected?4. Can3Dstacksofcrossbararraysbeimplemented?5. Willthermaldissipationbeanissue?

Thistutorialdescribedthebasisandimportantissueswhicharisetowardstheimplementationofenergyefficient,general-purpose,neuromorphiccomputer.ThepapershowsevidencethatResistiveSwitchingisafeasiblemechanismandaseriouscompetitortowardstheimplementationofaneuromorphicmachine.TheexamplesdescribedaboveillustratesthatneuromorphicfunctionalitiesusingResistiveSwitchinghavealreadybeendevelopedalthoughforverylimitedapplications.Themainaimandultimategoalofscalingandincorporatingsufficientnumberofdevicesintoanenergyefficientneuromorphicmachinewhichrivalsthebrainremainsanopenproblem.ACKNOWLEDGEMENTSThismultidisciplinaryreviewpaperintegratesandincludesmajorcomponentsinthefieldsof;quantummaterials,bio-inspiredelectronicsandneuromorphiccomputation.Thedevelopmentofbio-inspiredhybrids(IKS)wassupportedbytheVannevarBushFacultyFellowshipprogramsponsoredbytheBasicResearchOfficeoftheAssistantSecretaryofDefenseforResearchandEngineeringandfundedbytheOfficeofNavalResearchthroughgrantN00014-15-1-2848.TheinternationalcollaborationbetweenUCSDandCNRS(IKS,MJR)andamajorefforttodevelopanenergy-efficientneuromorphiccomputerisfundedthroughanEnergyFrontierResearchCenterfundedbytheU.S.DepartmentofEnergy,OfficeofScience,BasicEnergySciencesunderAward#DE-SC0019273. MJRacknowledgesfruitfulcollaborationsanddiscussionswithC.Acha,L.Cario,B.Corraze,I.H.Inoue,E.Janod,P.Levy,M.J.Sanchez,P.StoliarandF.Tesler,andthesupportfromtheLIACNRS-UCSD.JGRacknowledgesupportfromFAPAprogramthroughFacultaddeCienciasandVicerrectoriadeInvestigacionesofUniversidaddelosAndes,BogotáColombiaandColcienciasNo.120471250659andNo.120424054303.JdVthanksFundaciónRamónArecesfortheirfunding.

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