Quantum fluctuations and glassy behavior of electrons near metal-insulator transitions

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

Quantum uctuationsandglassybehaviorofelectronsnear

metal-insulatortransitions

arXiv:cond-mat/0403594v1 [cond-mat.str-el] 23 Mar 2004V.Dobrosavljevi´cDepartmentofPhysicsandNationalHighMagneticFieldLaboratory,FloridaStateUniversity,Tallahassee,FL32306,USAABSTRACTGlassybehaviorisagenericfeatureofelectronsclosetodisorder-drivenmetal-insulatortransitions.Deepintheinsulatingphase,electronsaretightlyboundtoimpurities,andthusclassicalmodelsforelectronglasseshavelongbeenused.Asthemetallicphaseisapproached,quantum uctuationsbecomemoreimportant,astheycontroltheelectronicmobility.Inthispaperwereviewrecentworkthatusedextendeddynamicalmean- eldapproachestodiscussthein uenceofsuchquantum uctuationsontheglassybehaviorofelectrons,andexaminehowthestabilityoftheglassyphaseisa ectedbytheAndersonandtheMottmechanismsoflocalization.Keywords:Electronglass,quantum uctuations,localization1.GLASSYBEHAVIORASAPRECURSORTOTHEMETAL-INSULATORTRANSITIONUnderstandingthemetal-insulatortransition(MIT)posesoneofthemostbasicquestionsofcondensedmatterphysics.IthasbeenbeenatopicofmuchcontroversyanddebatestartingfromearlyideasofMott,1andAnderson,2buttheproblemremainsfarfrombeingresolved.Quitegenerally,whenasystemisneitheragoodmetalnoragoodinsulator,boththelocalizedandtheitinerantaspectsoftheproblemareimportant.Inthisintermediateregime,severalcompetingprocessescanbesimultaneouslypresent.Asaresult,thesystemcannot

Figure1.Threebasicroutestolocalization

“decide”whethertobeametaloraninsulatoruntilaverylowtemperatureT isreached,belowwhichamoreconventionaldescriptionapplies.Thissituationistypicalofsystemsclosetoaquantumcriticalpoint,3whichdescribesazerotemperaturesecondorderphasetransitionbetweentwodistinctstatesofmatter.Understandingthenatureoflowenergyexcitationsintheintermediateregimebetweenametalandaninsulatorisofcrucialimportancefortheprogressinmaterialscience.

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

Theprimaryreasonfortheoreticaldi cultiesisrelatedtothefactthatboththeMottandtheAndersontransition ndthemselvesinregimeswheretraditional,perturbativeapproaches4cannotbestraightforwardlyapplied.Tomaketheproblemevenmoredi cult,simpleestimates1aresu cienttoappreciatethatinmanysituationsthee ectsofinteractionsanddisorderareofcomparablemagnitudeandthusbothshouldbesimul-taneouslyconsidered.Sofar,veryfewapproacheshaveattemptedtosimultaneouslyincorporatethesetwobasicroutestolocalization.

Anotheraspectofdisorderedinteractingelectronsposesafundamentalproblem.Verygenerally,Coulombrepulsionfavorsauniformelectronicdensity,whiledisorderfavorslocaldensity uctuations.Whenthesetwoe ectsarecomparableinmagnitude,onecanexpectmanydi erentlowenergyelectroniccon gurations,i.e.theemergenceofmanymetastablestates.Similarlyasinother“frustrated”systemswithdisorder,suchasspinglasses,theseprocessescanbeexpectedtoleadtoglassybehavioroftheelectrons,andtheassociatedanomalouslyslowrelaxationaldynamics.Indeed,boththeoretical5,6andexperimental7–11workhasfoundevidenceofsuchbehaviordeepontheinsulatingsideofthetransition.However,atpresentverylittleisknownastothepreciseroleofsuchprocessesinthecriticalregion.Nevertheless,itisplausiblethattheglassyfreezingoftheelectronsmustbeimportant,sincetheassociatedslowrelaxationclearlywillreducethemobilityoftheelectrons.Fromthispointofview,theglassyfreezingofelectronsmaybeconsidered,inadditiontotheAndersonandtheMottmechanism,asathirdfundamentalprocessassociatedwithelectronlocalization.Interestinunderstandingtheglassyaspectsofelectrondynamicshasexperiencedagenuinerenaissanceinthelastfewyears,primarilyduetoexperimentaladvances.Emergenceofmanymetastablestates,slowrelaxationandincoherenttransporthavebeenobservedinanumberofstronglycorrelatedelectronicsystems.TheseincludedtransitionmetaloxidessuchashighTcmaterials,manganites,andruthenates.Similarfeatureshaverecentlybeenreportedintwo-dimensionalelectrongases,andeventhreedimensionaldopedsemiconductorssuchasSi:P.

2.EXTENDEDDMFTAPPROACHESFORDISORDEREDELECTRONS

Anumberofexperimentalandtheoreticalinvestigationshavesuggestedthattheconventionalpictureofdisor-deredinteractingelectronsmaybeincomplete.Mostremarkably,thecharacteristic“critical”behaviorseeninmanyexperimentscoversasurprisinglybroadrangeoftemperaturesanddensities.Thisismorelikelytore ectanunderlying“mean- eld”behaviorofdisorderedinteractingelectronsthantheasymptoticcriticalbehaviordescribedbyane ectivelong-wavelengththeory.Thusasimplemean- elddescriptionisneededtoprovidethe

Figure2.Indynamicalmean- eldtheory,theenvironmentofagivensiteisrepresentedbyane ectivemedium,rep-

resentedbyits“cavityspectralfunction” i(ω).Inadisorderedsystem, i(ω)fordi erentsitescanbeverydi erent,re ectingAndersonlocalizatione ects.

equivalentofaVanderWaalsequationofstate,fordisorderedinteractingelectrons.Suchatheoryhaslongbeenelusive,primarilyduetoalackofasimpleorder-parameterformulationforthisproblem.Veryrecently,analternativeapproachtotheproblemofdisorderedinteractingelectronshasbeenformulated,basedondynamicalmean- eldtheory(DMFT)methods.12Thisformulationislargelycomplementarytothescalingapproach,andhasalreadyresultinginseveralstrikingpredictions.

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

TheDMFTapproachfocusesonasinglelatticesite,butreplaces12itsenvironmentbyaself-consistentlyde-termined“e ectivemedium”,asshowninFig.2.Foritinerantelectrons,theenvironmentcannotberepresentedbyastaticexternal eld,butinsteadmustcontaintheinformationaboutthedynamicsofanelectronmovinginoroutofthegivensite.Suchadescriptioncanbemadeprecisebyformallyintegratingout12allthedegreesoffreedomonotherlatticesites.Inpresenceofelectron-electroninteractions,theresultinglocale ectiveactionhasanarbitrarilycomplicatedform.WithinDMFT,thesituationsimpli es,andalltheinformationabouttheenvironmentiscontainedinthelocalsingleparticlespectralfunction i(ω).Thecalculationthenreducestosolvinganappropriatequantumimpurityproblemsupplementedbyanadditionalself-consistencyconditionthatdeterminesthis“cavityfunction” i(ω).

ThepreciseformoftheDMFTequationsdependsontheparticularmodelofinteractingelectronsand/ortheformofdisorder,butmostapplications12tothisdatehavefocusedonHubbardandAndersonlatticemodels.TheapproachhasbeenverysuccessfulinexaminingthevicinityoftheMotttransitionincleansystemsinwhichithasmetspectacularsuccessesinelucidatingvariouspropertiesofseveraltransitionmetaloxides,13heavyfermionsystems,andKondoinsulators.14

Whenappropriatelygeneralizedtodisorderedsystems,13thesemethodsareabletoincorporateallthethreebasicmechanismsofelectronlocalization.Inparticular,theDMFTapproachisabletopresentaconsistentpicturefortheglassybehaviorofelectrons,anddiscussitsemergenceinthevicinityofmetal-insulatortransi-tions.Inthispaperwereviewrecentresultsobtainedinthisframework,anddiscusstheirrelevancetoseveralexperimentalsystems.

3.SIMPLEMODELOFANELECTRONGLASS

Theinterplayoftheelectron-electroninteractionsanddisorderisparticularlyevidentdeepontheinsulatingsideofthemetal-insulatortransition(MIT).Here,bothexperimental15andtheoreticalstudies16havedemonstratedthattheycanleadtotheformationofasoft“Coulombgap”,aphenomenonthatisbelievedtoberelatedtotheglassybehavior7–11,17oftheelectrons.Suchglassyfreezinghaslongbeensuspected18tobeofimportance,butveryrecentwork19,20hassuggestedthatitmayevendominatetheMITbehaviorincertainlowcarrierdensitysystems.TheclassicworkofEfrosandShklovskii16hasclari edsomebasicaspectsofthisbehavior,butanumberofkeyquestionshaveremainunanswered.

Asasimplestexample21displayingglassybehaviorofelectrons,wefocusonasimplelatticemodelofspinlesselectronswithnearestneighborrepulsionVinpresenceofrandomsiteenergiesεiandinter-sitehoppingt,asgivenbytheHamiltonian

H= ( t+εiδij)c

icj+V c

icicjcj.(1)

<ij><ij>

Thismodelcanbesolved21inaproperlyde nedlimitoflargecoordinationnumberz,12whereanextendeddynamicalmean- eld(DMF)formulationbecomesexact.Weconcentrateonthesituationwherethedisorder(ormoregenerallyfrustration)islargeenoughtosuppressanyuniformordering.Wethenrescaleboththehopping√z.Aswewillseeshortly,therequiredelementsandtheinteractionamplitudesastij→tij/

uctuationsthensurviveeveninthez→∞limit,allowingfortheexistenceoftheglassyphase.Withinthismodel:

TheuniversalformoftheCoulombgap16provestobeadirectconsequenceofglassyfreezing.

Theglassphaseisidenti edthroughtheemergenceofanextensivenumberofmetastablestates,whichinourformulationismanifestedasareplicasymmetrybreakinginstability.22

Asaconsequenceofthisergodicitybreaking,22thezero- eldcooledcompressibilityisfoundtovanishatT=0,suggestingtheabsenceofscreening16indisorderedinsulators.

Thequantum uctuationscanmeltthisglassevenatT=0,buttherelevantenergyscaleissetbytheelectronicmobility,andisthereforeanontrivialfunctionofdisorder.

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

Weshouldstressthatalthoughthismodelallowstoexaminetheinterplayofglassyorderingandquantum uctuationsduetoitinerantelectrons,itistoosimpletodescribethee ectsofAndersonlocalization.Thesee ectsrequireextensionstolatticeswith nitecoordination,andandwillbediscussedinthenextsection.

Forsimplicity,wefocusonaBethelatticeathalf lling,andexaminethez→∞limit.Thisstrategyautomaticallyintroducesthecorrectorderparameters,andafterstandardmanipulations23theproblemreducestoaself-consistentlyde nedsinglesiteproblem,asde nedbyanthee ectiveactionoftheform

aβSeff(i)=

+1o βoa′2′a′dτdτ′[c i(τ)(δ(τ τ) τ+εi+tG(τ,τ))ci(τ)

2V2

a=bβo βob′dτdτ′δnai(τ)qabδni(τ).(2)

Here,wehaveusedfunctionalintegrationoverreplicatedGrassmann elds23cai(τ)thatrepresentelectronsonsiteiandreplicaindexa,andtherandomsiteenergiesεiaredistributedaccordingtoagivenprobabilitydistribution aaP(εi).Theoperatorsδnai(τ)=(ci(τ)ci(τ) 1/2)representthedensity uctuationsfromhalf lling.TheorderparametersG(τ τ′),χ(τ τ′)andqabsatisfythefollowingsetofself-consistencyconditions

G(τ τ)=

′′aa′χ(τ τ)=dεiP(εi)<δn

i(τ)δni(τ)>eff,

ab′qab=dεiP(εi)<δn

i(τ)δni(τ)>eff. aa′dεiP(εi)<c i(τ)ci(τ)>eff,(3)(4)(5)

3.1.Orderparameters

Intheseequations,theaveragesaretakenwithrespecttothee ectiveactionofEq.(2).Physically,the“hybridizationfunction”t2G(τ τ′)representsthesingle-particleelectronicspectrumoftheenvironment,asseenbyanelectrononsitei.Inparticular,itsimaginarypartatzerofrequencycanbeinterpreted24astheinverselifetimeofthelocalelectron,andassuchremains niteaslongasthesystemismetallic.Werecall23thatforV=0theseequationsreducetothefamiliarCPAdescriptionofdisorderedelectrons,whichisexactforz=∞.Thesecondquantityχ(τ τ′)representsan(interaction-induced)mode-couplingtermthatre ectstheretardedresponseofthedensity uctuationsoftheenvironment.Notethatverysimilarobjectsappearinthewell-knownmode-couplingtheoriesoftheglasstransitionindenseliquids.25Finallythequantityqab(a=b)isnothingbutthefamiliarEdwards-AndersonorderparameterqEA.Itsnonzerovalueindicatesthatthetimeaveragedelectronicdensityisspatiallynon-uniform.

3.2.EquivalentIn niteRangemodel

Fromatechnicalpointofview,aRSBanalysisistypicallycarriedoutbyfocusingonafreeenergyexpressedasafunctionaloftheorderparameters.InourBethelatticeapproach,onedirectlyobtainstheself-consistencyconditionsformappropriaterecursionrelations,23withoutinvokingafreeenergyfunctional.However,wehavefounditusefultomapourz=∞modeltoanotherin niterangemodel,whichhasexactlythesamesetoforderparametersandself-consistencyconditions,butforwhichanappropriatefreeenergyfunctionalcaneasilybedetermined.TherelevantmodelisstillgivenEq.(1),butthistimewithrandomhoppingelementstijandrandomnearest-neighborinteractionVij,havingzeromeanandvariancet2,andV2,respectively.Forthismodel,standardmanipulations23resultinthefollowingfreeenergyfunctional

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

F[G,χ,qab]=

12ln dεiP(εi) aaDc

iDciexp{ Seff(i)}, a=b2(βV)2qab(6)

withSeff(i)givenbyEq.(2).Theself-consistencyconditions,Eqs.(4-6)thenfollowfrom

0=δF/δG(τ,τ′);0=δF/δχ(τ,τ′);0=δF/δqab.(7)

WestressthatEqs.(3-5)havebeenderivedforthemodelwithuniformhoppingelementstijandinteractionamplitudesVij,inthez→∞limit,butthesameequationsholdforanin niterangemodelwheretheseparametersarerandomvariables.

3.3.Theglasstransition

Inourelectronicmodel,therandomsiteenergiesεiplayaroleofstaticrandom elds.Asaresult,inpresenceofdisorder,theEdwards-AndersonparameterqEAremainsnonzeroforanytemperature,andthuscannotserveasanorderparameter.Toidentifytheglasstransition,wesearchforareplicasymmetrybreaking(RSB)instability,followingstandardmethods.26,27Wede neδqab=qab q,andexpandthefreeenergyfunctionalofEq.(6)aroundtheRSsolution.Theresultingquadraticform(Hessianmatrix)hasthematrixelementsgivenby

2F

1

εi

4 +∞ ∞ dx x2/21etanh2π

W2+V2q,

(12)

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

sincethefrozen-indensity uctuationsintroduceanaddedcomponenttotherandompotentialseenbytheelectron.Asexpected,q=0foranytemperaturewhenW=0.Iftheinteractionstrengthisappreciableascomparedtodisorder,wethusexpecttheresistivitytodisplayanappreciableincreaseatlowtemperatures.Weemphasizethatthismechanismisdi erentfromAndersonlocalization,whichisgoingtobediscussedinthenextsection,butwhichalsogivesrisetoaresistivityincreaseatlowtemperatures.

Next,weexaminetheinstabilitytoglassyordering.Intheclassical(t=0)limitEq.(9)reducesto

1=1√ xβWeff(q),(13)2

withWeff(q)givenbyEq.(12).TheresultingRSBinstabilitylineseparatesalowtemperatureglassyphasefromahightemperature“badmetal”phase.Atlargedisorder,theseexperssionssimplify,andwe nd

TG≈1V2

2π

ρ(ε,t=0)=1

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

0.600

0.400

Vρ( ε )

0.200

0.000

–3.00–2.00–1.000.00

ε/V1.002.003.00

Figure3.Singleparticledensityofstatesintheclassical(t=0)limitatT=0,asafunctionofdisorderstrength.ResultsareshownfromasimulationonN=200sitesystem,forW/V=0.5(thinline)andW/V=1.0(fullline).Notethatthelowenergyformofthegaptakesauniversalform,independentofthedisorderstrengthW.ThedashedlinefollowsEq.(16).

Theergodicitybreakingassociatedwiththeglassyfreezinghasimportantconsequencesforourmodel.Again,usingtheclosesimilarityofourclassicalin niterangemodeltostandardSGmodels,22itisnotdi culttoseethatthezero- eldcooled(ZFC)compressibilityvanishesatT=0,incontrasttothe eld-cooledone,whichremains nite.Essentially,ifthechemicalpotentialismodi edafterthesystemiscooledtoT=0,thesystemimmediatelyfallsoutofequilibriumanddisplayshystereticbehavior29withvanishingtypicalcompressibility.Ifthisbehaviorpersistsin nitedimensionsandformorerealisticCoulombinteractions,itcouldexplaintheabsenceofscreeningindisorderedinsulators.

4.2.Arbitrarylatticesand nitecoordination:mean- eldglassyphaseofthe

random- eldIsingmodel.

Simplesttheoriesofglassyfreezing22areobtainedbyexaminingmodelswithrandominter-siteinteractions.Inthecaseofdisorderedelectronicsystems,theinteractionsarenotrandom,butglassinessstillemergesduetofrustrationintroducedbythecompetitionoftheinteractionsanddisorder.AswehaveseenfortheBethelattice,21randominteractionsaregeneratedbyrenormalizatione ects,sothatstandardDMFTapproachescanstillbeused.However,onewouldliketodevelopsystematicapproachesforarbitrarylatticesandin nitecoordination.Theseissuesalreadyappearontheclassicallevel,whereourmodelreducestotherandom- eldIsingmodel(RFIM).30ToinvestigatetheglassybehavioroftheRFIM,wedeveloped31asystematicapproachthatcanincorporateshort-range uctuationcorrectionstothestandardBragg-Williamstheory,followingthemethodofPlefka32andGeorgesetal..33Thisworkhasshownthat:

Correctionstoeventhelowestnontrivialorderimmediatelyresultintheappearanceofaglassyphaseforsu cientlystrongrandomness.

Thislow-ordertreatmentissu cientinthejoinedlimitoflargecoordinationandstrongdisorder.

Thestructureoftheresultingglassyphaseischaracterizedbyuniversalhysteresisandavalanchebehavioremergingfromtheself-organizedcriticalityoftheorderedstate.

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

5.QUANTUMMELTINGOFTHEELECTRONGLASS

Next,weinvestigatehowtheglasstransitiontemperaturecanbedepressedbyquantum uctuationsintroducedbyinter-siteelectrontunneling.Asinotherquantumglassproblems,quantum uctuationsintroducedynamicsintheproblem,andtherelevantself-consistencyequationscannotbesolveinclosedformforgeneralvaluesoftheparameters.Inthefollowing,wewillseethatinthelimitoflargerandomness,anexactsolutionispossible.

5.1.Quantumphasediagram

Themainsourceofdi cultyingeneralquantumglassproblemsrelatestotheexistenceofaself-consistentlydetermined“memorykernel”χ(τ τ′)inthelocale ectiveaction.Bythesamereasoningasintheclasicalcase,onecanalsoignorethistermsincethisquantityisalsobounded.

Figure4.Phasediagramasafunctionofquantumhoppingt,temperatureTanddisorderstrengthW.GlasstransitiontemperatureTGdecreasesonlyslowly(as1/W)inthestrongdisorderlimit.Incontrast,thecriticalvalueofthehopingelementtGremains niteasW→∞

Theremainingactionisthatofnoninteractingelectronsinpresenceofastrongrandompotential.Theresultinglocalcompressibilitythentakestheform

χloc(ε)=β

2βω).(17)

Here,ρε(ω)isthelocaldensityofstates,whichintheconsideredlargezlimitisdeterminedbythesolutionoftheCPAequation1,(18)ρε(ω)= ω+iη ε t2G(ω)

InthelimitW/t>>1,itreducestoanarrowresonanceofwidth =πt2P(0)～t2/W

ρε(ω)≈1

(ω ε)2+ 2

.(19)

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

Theresultingexpressionforthequantumcriticallineinthelargedisorderlimittakestheform

√tG(T=0,W→∞)=V/

4z2 +∞

dx

∞ +∞dy11+(x y)2cosh 2(1

∞

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

with

D(ωn)=2 yqEA/V4

Thisbehaviorre ectstheemergenceofsoft”replicon”modes22describinginourcaserepresentlowenergychargerearrangementsinsidetheglassyphase.At nitetemperatures,electronsundergoinelasticscatteringfromsuchcollectiveexcitations,leadingtothetemperaturedependenceoftheresistivitythattakesthefollowingnon-Fermiliquidform

ρ(T)=ρ(o)+AT3/2.

Interestingly,veryrecentexperiments37ontwodimensionalelectrongasesinsiliconhaverevealedpreciselysuchtemperaturedependenceoftheresistivity.Thisbehaviorhasbeenobservedinwhatappearstobeanintermediatemetallicglassphaseseparatingaconventional(Fermiliquid)metalathighcarrierdensity,fromaninsulatoratthelowestdensities.

Anotherinterestingfeatureofthepredictedquantumcriticalbehaviorrelatestodisorderdependenceofthecrossoverexponentφdescribinghowthegapscale ～δrφvanishesasafunctionofthedistanceδrfromthecriticalline.Calculations38showthatφ=2inpresenceofsiteenergydisorder,whichforourmodelplaysaroleofarandomsymmetrybreaking eld,andφ=1initsabsence.Thisindicatesthatsitedisorder,whichiscommonindisorderedelectronicsystems,producesaparticularlylargequantumcriticalregion,whichcouldbetheoriginoflargedephasingobservedinmanymaterialsnearthemetal-insulatortransition.|ωn|.

5.3.E ectsofAndersonlocalization

Aswehaveseen,thestabilityoftheglassyphaseiscruciallydeterminedbytheelectronicmobilityatT=0.Moreprecisely,wehaveshownthattherelevantenergyscalethatdeterminesthesizeofquantum uctuationsintroducedbytheelectronsisgivenbythelocal“resonancewidth” .Itisimportanttorecallthatpreciselythisquantitymaybeconsidered2asanorderparameterforAndersonlocalizationofnoninteractingelectrons.Veryrecentwork13,24demonstratedthatthetypicalvalueofthisquantityplaysthesameroleevenataMott-Andersontransition.Wethusexpect togenerallyvanishintheinsulatingstate.Asaresult,weexpectthestabilityoftheglassyphasetobestronglya ectedbyAndersonlocalizatione ects,aswewillexplicitlydemonstrateinthenextsection.

6.GLASSYBEHAVIORNEARTHEMOTT-ANDERSONTRANSITION

Onphysicalgrounds,oneexpectsthequantum uctuations39associatedwithmobileelectronstosuppressglassyordering,buttheirprecisee ectsremaintobeelucidated.Notethateventheamplitudeofsuchquantum uctuationsmustbeasingularfunctionofthedistancetotheMIT,sincetheyaredynamicallydeterminedbyprocessesthatcontroltheelectronicmobility.

Toclarifythesituation,thefollowingbasicquestionsneedtobeaddressed:(1)DoestheMITcoincidewiththeonsetofglassybehavior?(2)Howdodi erentphysicalprocessesthatcanlocalizeelectronsa ectthestabilityoftheglassphase?Inthefollowing,weprovidesimpleandphysicallytransparentanswerstobothquestions.We ndthat:(a)Glassybehaviorgenerallyemergesbeforetheelectronslocalize;(b)Andersonlocalization2enhancesthestabilityoftheglassyphase,whileMottlocalization1tendstosuppressit.

Inordertobeabletoexamineboththee ectsofAndersonandMottlocalization,weconcentrateonextendedHubbardmodelsgivenbytheHamiltonian

H= ijσ( tij+εiδij)c i,σcj,σ+U ini↑ni↓+ ijVijδniδnj.

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

Here,δni=ni ni representlocaldensity uctuations( ni isthesite-averagedelectrondensity),Uistheon-siteinteraction,andεiareGaussiandistributedrandomsiteenergiesofvarianceW2.Inordertoallowforglassyfreezingofelectronsinthechargesector,weintroduceweakinter-sitedensity-densityinteractionsVij,whichwealsoalsochoosetobeGaussiandistributedrandomvariablesofvarianceV2/z(zisthecoordinationnumber).Weemphasizethat,incontrasttopreviouswork,21weshallnowkeepthecoordinationnumberz nite,inordertoallowforthepossibilityofAndersonlocalization.Toinvestigatetheemergenceofglassyordering,weformallyaverageoverdisorderbyusingstandardreplicamethods,40andintroducecollectiveQ- eldstodecoupletheinter-siteV-term.40Amean- eldisthenobtainedbyevaluatingtheQ- eldsatthesaddle-pointlevel.Theresultingstabilitycriteriontakestheformsimilarasbefore

1 V2

j[χ2ij]dis=0.(23)

Here,thenon-localstaticcompressibilitiesarede ned(fora xedrealizationofdisorder)as

χij= ni/ εj,(24)

whereniisthelocalquantumexpectationvalueoftheelectrondensity,and[···]disrepresentstheaverageoverdisorder.Obviously, 2thestabilityoftheglassphaseisdeterminedbythebehaviorofthefour-ordercorrelation(2)functionχ=[χij]disinthevicinityofthemetal-insulatortransition.Weemphasizethatthisquantityisto

j

becalculatedinadisorderedHubbardmodelwith niterangehopping,i.e.inthevicinityoftheMott-Andersontransition.Thecriticalbehaviorofχ(2)isverydi culttocalculateingeneraql,butwewillseethatsimpleresultscanbeobtainedinthelimitsofweakandstrongdisorder,asfollows.

rgedisorder

Asthedisordergrows,thesystemapproachestheAndersontransitionatt=tc(W)～W.The rsthintofsingularbehaviorofχ(2)inanAndersoninsulatorisseenbyexaminingthedeeplyinsulating,i.e.atomiclimitW t,wheretoleadingorderwesett=0andobtainχij=δ(εi µ)δij,i.e.χ(2)=[δ2(εi µ)]dis=+∞diverges!Sinceweexpectallquantitiestobehaveinqualitativelythesamefashionthroughouttheinsulatingphase,weanticipateχ(2)todivergealreadyattheAndersontransition.Notethat,sincetheinstabilityoftheglassyphaseoccursalreadyatχ(2)=V 2,theglasstransitionmustprecedethelocalizationtransition.Thus,forany niteinter-siteinteractionV,wepredicttheemergenceofanintermediatemetallicglassphaseseparatingtheFermiliquidfromtheAndersoninsulator.Assumingthatnearthetransition

χ(2) A

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

4

3

2

1

1.002468

W/t01012t/VFermi Liquid

Metallic Glass

Insulating Glass

0.0024

W/V68χ(2)

dωImG(εi,ω,W)1.50.5Figure5.Phasediagramforthez=3Bethelattice,validinthelargedisorderlimit.Theinsetshowsχ(2)ingthismethod,wehavecalculatedχ(2)asafunctionofW/t(forthislatticeathalf- llingEF=√2 εi

G(εi,ω,W)==1 εi1 0(27) ∞

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

t/VW/VFigure6.PhasediagramfromTypicalMediumTheoryofAndersonlocalization,43givingα=1.TheintermediatemetallicglassyphaseshrinksasdisorderWgrows,parethistotheBethelatticecaseFig.5,whereα=∞.

6.2.Lowdisorder-Motttransition

InthelimitofweakdisorderW U,V,andinteractionsdrivethemetal-insulatortransition.Concentratingonthemodelathalf- lling,thesystemwillundergoaMotttransition1asthehoppingtissu cientlyreduced.SincefortheMotttransitiontMott(U)～U,nearthetransitionW t,andtoleadingorderwecanignorethelocalizatione ects.Inaddition,weassumethatV U,andtoleadingorderthecompressibilitieshavetobecalculatedwithrespecttotheactionSelofadisorderedHubbardmodel.Thesimplestformulationthatcandescribethee ectsofweakdisorderonsuchaMotttransitionisobtainedfromthedynamicalmean- eldtheory12(DMFT).Thisformulation,whichignoreslocalizatione ects,isobtainedbyrescalingthehoppingelements√t→t/

3πtc(1 otc(W)

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

Figure7.SchematicphasediagramforanextendedHubbardmodelwithdisorder,asafunctionofthehoppingelementtandthedisorderedstrengthW,bothexpressedinunitsoftheon-siteinteractionU.Thesizeofthemetallicglassphaseisdeterminedbythestrengthoftheinter-siteinteractionV.

thefullsolutionoftheMott-Andersonproblem.TherequiredcalculationscanandshouldbeperformedusingtheformulationofRef.13,24,butthatdi culttaskisachallengforthefuture.However,basedongeneralargumentspresentedabove,weexpectχ(2)tovanishasoneapproachestheMottinsulator(W<U),buttodivergeasoneapproachestheMott-Andersoninsulator(W>U).NearthetetracriticalpointM(seeFig.2),wemayexpectχ(2)～δW αδtβ,whereδW=W WMott(t)isthedistancetotheMotttransitionline,andδt=t tc(W)ingthisansatzandEq.(23),we ndtheglasstransitionlinetotaketheform

δt=tG(W) tc(W)～δWβ/α;W～WM.(31)

Wethusexpecttheintermediatemetallicglassphasetobesuppressedasthedisorderisreduced,andoneapproachestheMottinsulatingstate.Physically,glassybehaviorofelectronscorrespondstomanylow-lyingrearrangementsofthechargedensity;suchrearrangementsareenergeticallyunfavorableclosetothe(incom-pressible)Mottinsulator,sincetheon-siterepulsionUopposescharge uctuations.Interestingly,veryrecentexperimentsonlowdensityelectronsinsiliconMOSFETshaverevealedtheexistenceofexactlysuchanin-termediatemetallicglassphaseinlowmobility(highlydisordered)samples.37Incontrast,inhighmobility(lowdisorder)samples,44nointermediatemetallicglassphaseisseen,andglassybehavioremergesonlyasoneenterstheinsulator,consistentwithourtheory.SimilarconclusionshavealsobeenreportedinstudiesofhighlydisorderedInO2 lms,7–11wheretheglassyslowingdownoftheelectrondynamicsseemstobesuppressedasthedisorderisreducedandonecrossesoverfromanAnderson-liketoaMott-likeinsulator.Inaddition,theseexperiments37,44providestrikingevidenceofscale-invariantdynamicalcorrelationsinsidetheglassphase,consistentwiththehierarchicalpictureofglassydynamics,asgenerallyemergingfrommean- eldapproaches22suchastheoneusedinthiswork.

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

7.CONCLUSIONS

Recentyearshavewitnessedenormousrenewedinterestinthemetal-insulatortransition.Scoresofnewandfascinatingmaterialsarebeingfabricated,withpropertiesthatcouldnotbeanticipated.Acommonthemeinmanyofthesesystemsisthepresenceofboththestrongelectron-electroninteractionsanddisorder,asituationwhichproveddi culttoanalyzeusingconventionaltheoreticalmethods.Inthispaper,wehavedescribedanovelapproachtothisdi cultproblem,andshownthatitcancapturemostrelevantprocesses.Thisformulationcaneasilybeadaptedtomanyrealisticsituationsandwillopennewavenuesforthedevelopmentofmaterialsscienceresearch.

ACKNOWLEDGMENTS

TheauthorwouldliketoacknowledgehiscollaboratorsA.A.Pastor,M.H.Horbach,D.Tanaskovi´c,D.Dali-dovich,L.Arrachea,andM.J.Rozenberg,withwhomitwasapleasuretoexplorethephysicsofelectronglasses.IalsothankS.Bogdanovich,S.Chakravarty,J.Jaroszynski,D.Popovi´c,Z.Ovadyahu,J.Schmalian,andG.Zimanyiforusefuldiscussions.ThisworkwassupportedbytheNSFgrantDMR-9974311andDMR-0234215,andtheNationalHighMagneticFieldLaboratory.

REFERENCES

1.N.F.Mott,Metal-Insulatortransition,Taylor&Francis,London,1990.

2.P.W.Anderson,“Absenceofdi usionincertainrandomlattices,”Phys.Rev.109,pp.1492–1505,1958.

3.S.Sondhi,S.Girvin,J.Carini,andD.Shahar,“Continuousquantumphasetransitions,”Rev.Mod.Phys.69,p.315,1997.

4.P.A.LeeandT.V.Ramakrishnan,“Disorderedelectronicsystems,”Rev.Mod.Phys.57,p.287,1985.

5.J.H.Davies,P.A.Lee,andT.M.Rice,“Electronglass,”Phys.Rev.Lett.49,pp.758–761,1982.

6.M.PollakandA.Hunt,“Slowprocessesindisorderedsolids,”inHoppingTransportinSolids,M.PollakandB.I.Shklovskii,eds.,pp.175–206,Elsevier,Amsterdam,1991.

7.M.Ben-Chorin,D.Kowal,andZ.Ovadyahu,“Anomalous elde ectingatedAndersoninsulators,”Phys.Rev.B44,pp.3420–3423,1991.

8.M.Ben-Chorin,Z.Ovadyahu,andM.Pollak,“Nonequilibriumtransportandslowrelaxationinhoppingconductivity,”Phys.Rev.B48,pp.15025–15034,1993.

9.Z.OvadyahuandM.Pollak,“Disorderandmagnetic elddependenceofslowelectronicrelaxation,”Phys.Rev.Lett.79,pp.459–462,1997.

10.A.Vaknin,Z.Ovadyahu,andM.Pollak,“Evidenceforinteractionsinnonergodicelectronictransport,”

Phys.Rev.Lett.81,pp.669–672,1998.

11.A.Vaknin,Z.Ovadyahu,andM.Pollak,“Aginge ectsinanAndersoninsulator,”Phys.Rev.Lett.84,

pp.3402–3405,2000.

12.A.Georges,G.Kotliar,W.Krauth,andM.J.Rozenberg,“Dynamicalmean- eldtheoryofstronglycorre-

latedfermionsystemsandthelimitofin nitedimensions,”Rev.Mod.Phys.68,p.13,1996.

13.V.Dobrosavljevi´candG.Kotliar,“Dynamicalmean- eldstudiesofmetal-insulatortransitions,”Phil.Trans.

R.Soc.Lond.A356,p.1,1998.

14.M.J.Rozenberg,G.Kotliar,andH.Kajueter,“Transferofspectralweightinspectroscopiesofcorrelated

electronsystems,”Phys.Rev.B54,p.8542,1996.

15.J.G.MasseyandM.Lee,“Low-frequencynoiseprobeofinteractingchargedynamicsinvariable-range

hoppingboron-dopedsilicon,”Phys.Rev.Lett.77,p.3399,1996.

16.A.L.EfrosandB.I.Shklovskii,“Coulombgapandlowtemperatureconductivityofdisorderedsystems,”

J.Phys.C8,pp.L49–51,1975.

17.G.Martinez-Arizala,C.Christiansen,D.E.Grupp,N.Markovic,A.M.Mack,andA.M.Goldman,

“Coulomb-glass-likebehaviorofultrathin lmsofmetals,”Phys.Rev.B57,pp.R670–R672,1998.

18.D.BelitzandT.R.Kirkpatrick,“Anderson-Motttransitionasaquantum-glassproblem,”Phys.Rev.B52,

p.13922,1995.

Glassy behavior is a generic feature of electrons close to disorder-driven metal-insulator transitions. Deep in the insulating phase, electrons are tightly bound to impurities, and thus classical models for electron glasses have long been used. As the meta

19.V.Dobrosavljevi´c,E.Abrahams,E.Miranda,andS.Chakravarty,“Scalingtheoryoftwo-dimensional

metal-insulatortransitions,”Phys.Rev.Lett.79,pp.455–458,1997.

20.S.Chakravarty,S.Kivelson,C.Nayak,andK.Voelker,“Wignerglass,spin-liquids,andthemetal-insulator

transition,”Phil.Mag.B79,p.859,1999.

21.A.A.PastorandV.Dobrosavljevi´c,“Meltingoftheelectronglass,”Phys.Rev.Lett.83,p.4642,1999.

22.M.Mezard,G.Parisi,andM.A.Virasoro,SpinGlasstheoryandbeyond,WorldScienti c,Singapore,1986.

23.V.Dobrosavljevi´candG.Kotliar,“Strongcorrelationsanddisorderind=∞andbeyond,”Phys.Rev.B

50,p.1430,1994.

24.V.Dobrosavljevi´candG.Kotliar,“Mean eldtheoryoftheMott-Andersontransition,”Phys.Rev.Lett.

78,p.3943,1997.

25.H.Z.Cummins,G.Li,W.M.Du,andJ.Hernandez,“Relaxationaldynamicsinsupercooledliquids:

experimentaltestsofthemodecouplingtheory,”PhysicaA204,p.169,1994.

26.J.deAlmeidaandD.J.Thouless,“Stabilityofthesherrington-kirkpatricksolutionofaspinglassmodel,”

J.Phys.A11,p.983,1978.

27.M.MezardandA.P.Young,“Replicasymmetrybreakingintherandom eldisingmodel,”Europhys.Lett.

18,pp.653–659,1992.

28.R.G.PalmerandC.M.Pond,“Internal elddistributioninmodelspinglasses,”J.Phys.F9,p.1451,

1979.

29.F.Pazmandi,G.Zarand,andG.T.Zimanyi,“Self-organizedcriticalityinthehysteresisoftheSherrington-

Kirkpatrickmodel,”Phys.Rev.Lett.83,pp.1034–1037,1999.

30.T.Nattermann,“Theoryoftherandom eldIsingmodel,”cond-mat/9705295,1997.

31.A.A.Pastor,V.Dobrosavreljevi´c,andM.L.Horbach,“Mean- eldglassyphaseoftherandom eldIsing

model,”Phys.Rev.B66,p.014413(14),2001.

32.T.Plefka,“Convergenceconditionofthetapequationforthein nite-rangedIsingspinglassmodel,”J.

Phys.A15,pp.1971–1978,1982.

33.A.Georges,M.Mezard,andJ.S.Yedida,“Low-temperaturephaseoftheIsingspinglassonahypercubic

lattice,”Phys.Rev.Lett.64,p.2937,1990.

34.N.Read,S.Sachdev,andJ.Ye,“LandautheoryofquantumspinglassesofrotorsandIsingspins,”Phys.

Rev.B52,p.384,1995.

35.J.MillerandD.A.Huse,“Zero-temperaturecriticalbehaviorofthein nite-rangequantumIsingspinglass,”

Phys.Rev.Lett.70,p.3147,1993.

36.D.DalidovichandV.Dobrosavljevi´c,“LandautheoryoftheFermi-liquidtoelectron-glasstransition,”Phys.

Rev.B66,p.081107(4),2002.

37.S.BogdanovichandD.Popovi´c,“Onsetofglassydynamicsinatwo-dimensionalelectronsysteminsilicon,”

Phys.Rev.Lett.88,p.236401(4),2002.

38.L.Arrachea,D.Dalidovich,V.Dobrosavljevi´c,andM.J.Rozenberg,“MeltingtransitionofanIsingglass

drivenbymagnetic eld,”Phys.Rev.B(inpress),2004.

39.A.A.PastorandV.Dobrosavljevi´c,“Meltingoftheelectronglass,”Phys.Rev.Lett.83,pp.4642–4645,

1999.

40.V.Dobrosavljevi´c,D.Tanaskovi´c,andA.A.Pastor,“Glassybehaviorofelectronsnearmetal-insulator

transitions,”Phys.Rev.Lett.90,p.016402(4),2003.

41.R.Abou-Chacra,P.W.Anderson,andD.Thouless,“Aselfconsistenttheoryoflocalization,”J.Phys.C6,

p.1734,1973.

42.A.D.Mirlin,“Statisticsofenergylevelsandeigenfunctionsindisorderedsystems,”Phys.Rep.326,p.259,

2000.

43.V.Dobrosavljevi´c,A.Pastor,andB.K.Nikoli´c,“TypicalmediumtheoryofAndersonlocalization:Alocal

orderparameterapproachtostrongdisordere ects,”Europhys.Lett.62,pp.76–82,2003.

44.J.Jaroszy´nski,D.Popovi´c,andT.M.Klapwijk,“Universalbehavioroftheresistancenoiseacrossthe

metal-insulatortransitioninsiliconinversionlayers,”Phys.Rev.Lett.89,p.276401(4),2002.

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