Wignercrystalinsnakednanochannels:OutlookO.V.Zhirov,D.L.ShepelyanskyBu - PDF document

Wignercrystalinsnakednanochannels:OutlookO.V.Zhirov,D.L.ShepelyanskyBudkerInstituteofNuclearPhysics,630090Novosibirsk,RussiaLaboratoiredePhysiqueTheoriqueduCNRS(IRSAMC),UniversitedeToulouse,UPS,F-31062Toulouse,FrancearticleinfoAvailableonline28January2012WignercrystalOrganicconductorsFrenkel…KontorovamodelAubrytransitionInvariantKAMcurves ContentslistsavailableatSciVerseScienceDirectjournalhomepage:www.elsevier.cPhysicaB0921-4526/$-seefrontmatter2012ElsevierB.V.Allrightsreserved. Correspondingauthor.E-mailaddress:dima@irsamc.ups-tlse.fr(D.L.Shepelyansky). PhysicaB407(2012)1909…1911 propertiesappearinspinglasses[11]whichhaverandomdis-orderedon-siteenergiesandinteractions.IncontrasttothatthesepropertiesoftheAubryphaseappearintheabsenceofanydisorderbeingofpurelydynamicaloriginofcantoriinapurelyperiodicpotential.ThestudiesofpropertiesofthequantumFrenkel…KontorovamodelhasbeenstartedinRef.Ref.andfurthersigni“cantlyadvancedinRef.Ref..Itwasshownshownthatquantum”uctua-tionsleadtomeltingofthepinnedphaseatsuf“cientlylargevaluesofdimensionlessPlanckconstant.Thistransitionisazero0quantumphasetransition.Atsmall1an0thephononmodeisfrozenbutquantumtunnelinggivestransitionsbetweenquasi-degenerateequilibriumclassicalcon-“gurationswhichcanbeviewedasinstantons.Atsmall1thedensityofinstantonsissmallandtheirinteractionsareweak.increasestheinstantondensitygrowsandaboveacertain1thequantummeltingofpinnedphasetakesplaceatzerotemperatureleadingtozerogap,appearanceofquantumphononmodeandquantumchainslidingsliding.TheresultsobtainedfortheWignercrystalinaperiodicpotentialpotentialinclassicalandquantumregimescon“rmthisqualitativepicture.At“xedamplitudeofperiodicpotentialtheclassicalWignercrystalispinnedatsmallchargedensitiesdensities.Indeed,at0wehaveaproblemofoneelectronwithzerokineticenergyandobviously,anelectronispinnedbyaperiodicpotential.2.SlidingWignersnakeThesituationisdifferentinthecaseofsnakednanochannel:noninteractingelectrons,correspondingtothelimit1,movefreelyinsidethewiggledchannelandpinningoftheWignercrystalappearsonlyaboveacertaincriticalchargedensityAnexampleofslidingandpinnedregimesisshowninFig.2.Thedataclearlyshowthattheslidingphaseat0.6hasasmoothhullfunctionandsounddispersionlawforsmalloscillationsofthecrystal.Incontrast,inthepinnedregimeat1.2thehullfunctionhasaformoffractaldevilsstaircaseandgappedspectrumofsmalloscillations.MoredetailedresultsondependenceofgaponchargeanddeformationaredescribedinRef.Ref..InRef.Ref.itisalsoshownthatformoderatedeformations1thechargepositionsinastaticcon“gurationaredescribedbyasymplecticdynamicalmap v¼vþ2a2ð1cos sin2 f¼fþ vþa2sin cos2whereareconjugatedaction-phasevariables,barmarkstheirvaluesafteriteration.Themapisimplicitbutsymplectic(seee.g..).ExamplesofPoincaresectionsofthismapattwovaluesofdeformationareshowninFig.3.Phasespaceregionwithscatteredpointscorrespondstochaoticdynamicswithpinnedphase,whilethesmoothinvariantKAMcurvescorrespondtotheslidingphase.InmanyaspectsthepropertiesoftheWignercrystalinsnakednanochannelsaresimilartothoseoftheFrenkel…KontorovmodelmodelandtheWignercrystalinaperiodicpotentialpotential:inthepinnedphasethereareexponentiallymanystaticcon“gurationsbeingexponentiallycloseinenergyandcorrespondingtothedynamicalglassphase.However,therearealsosomespeci“cfeatures:forrationalvaluesofdensities,whereaninteger,theWignersnakecanslidefreelysinceadisplacementdoesnotmodifytheCoulombenergyofelectroninteractions.InanalogywiththeresultspresentedinRef..,weexpectthatthequantumWignercrystalshowsazerotemperaturequantumphasetransitiongoingfromapinnedphaseat1toaslidingphaseat.However,adirectdemonstrationofthisfactrequiresfurthernumericalsimulationsusingquantumMonteCarlomethodsdescribedinRefs.fs.,12,13].3.DiscussionIntheabovesectionweconsideredtheWignercrystalinasnakednanochannelwithoutanyinternalpotential.Itisnaturaltoassumethatamorerealisticcaseofmolecularorganicconductors,asshownintheLittlesuggestioninFig.1,hasnotonlychanneldeformationbutalsoaperiodicpotentialinsidethechannel.Thusthecaseoforganicconductorscorrespondstoacaseofsnakedchannelwithaperiodicpotentialinsideit. Fig.1.(a)AschematicimageoftheLittlesuggestionforelectrontransportinorganicmolecules(after(after,5]).(b)AschematicimageofelectronWignercrystalwithchargese(points)slidinginasnakedsinusoidalnanochannel,dashedlinesshowforcedirectionsbetweennearbyelectrons. Fig.2.Hullfunction(a,b)andphononspectrum(c,d)forincommensurateelectrondensity233isshownat0.6(a,c)1.2(b,d).Heregivesthepositionsofelectronsat0.Heregivesthedimensionalamplitudeofsinusoidalchanneldescribedbyequation Fig.3.sectionforthedynamicalmap(1)at0.2(leftpanel),0.6(rightO.V.Zhirov,D.L.Shepelyansky/PhysicaB407(2012)1909…1911 ThecombinationofresultsofRefs.Refs.,3]showsthatforagivendeformationandamplitudeoftheperiodicpotentialwehavetheslidingphaseinacertainrangeofchargedensitiesWesupposethattheslidingKAMphasemaycorrespondtoeffectivesuperconductingbehaviorofelectrontransportinorganicconductors.Indeed,thepressurediagramoforganicconductorsshowninFig.4fromRef.Ref.showsthatsuper-conductivityexistsonlyisa“niterangeofpressure.WeassumethatpressuregivesvariationofeffectivechargedensityinsidethemolecularchannelsintheLittlesuggestioninFig.1.ThisleadsustotheKAMconceptofsuperconductivityofelectronswithoutattractiveforces:theWignercrystalofelectronsslidesfreelyinsideasnakedmolecularcrystalchannelifthechargedensityislocatedinsideofKAMphasede“nedbyEq.(2).Ofcourse,furtherstudiesarerequiredfordevelopmentofthisconcept.Infact.theslidingKAMphasecanbeviewedasasuper”uidphaseofelectrons.Indeed,weseethatintheKAMphasethereisaspectrumofexcitationswitha“nitesoundvelocity.ThusaccordingtotheLandaucriterioncriteriontheslidingofelectronswithvelocitiesissuper”uid.Hence,thetransitionfromtheslidingKAMphasetothepinnedAubryphasecorrespondstothetransitionfromsuper”uidtoinsulator.Inthissuper”uidliquidthechargecarrieshavechangeeandnot2easitisthecaseforBCSpairs.Maybeeffectofinteractionsbetweenelectronsinparallelsnakedchannelsshouldbetakenintoaccounttohave2e-pairs.Wenotethatitisknownthatrepulsiveinteractionscancreatesuper”uidphaseindisordered1dsystems,e.g.intherepulsiveHubbardmodelwithdisorderdisorder.TheexistenceofdynamicalspinglassphasewithpinnedWignercrystalshowsthatthereshouldbeveryslowrelaxationprocessescorrespondingtoveryslowtransitionsbetweenquasi-degeneratestaticequilibriumcon“gurations.Infacttheexperi-mentswithorganicconductorsshowveryslowvariationsofconductivitywhichtakeplaceofascaleofdays.Suchexperi-mentalresultshavebeenreportedatECRYS-2011byMiyagawaMiyagawaandMonceauMonceau.Usuallyitisarguedthattheglassyphaseappearsduetoimpurities.Wethinkthattheoriginofthisphenomenonisnotrelatedtodisorderandimpurities,whichpresenceshouldberathersmallinorganiccrystalsusedinin.IncontrastthisglassyphaseappearsasaresultofdynamicalspinglassphasedescribedinRefs.Refs.,12,13,3]whichexistsinpurelyperiodicstructureswithoutanyimpuritiesanddisorder.Finally,wenotethatinRef.Ref.itwasproposedtostudythedynamicalspinglasswithcoldionsinopticallatticeswhichcanmodeltheproblemofWignercrystalinaperiodicpotential.Suchexperimentswithcoldionsarenowunderactivediscussionsdiscussionsandtheirexperimentalrealizationisonthewayway.AcknowledgmentsThisworkissupportedinpartbyANRPNANOproject[1]E.Wigner,Phys.Rev.46(1934)1002.[2]I.Garcia-Mata,O.V.Zhirov,D.L.Shepelyansky,Eur.Phys.J.D41(2007)325.[3]O.V.Zhirov,D.L.Shepelyansky,Eur.Phys.J.B82(2011)61.[4]W.A.Little,Phys.Rev.A134(1964)1416;W.A.Little,Sci.Am.212(1965)21.[5]D.Jerome,Historicalapproachtoorganicsuperconductivity,in:A.Lebed(Ed.),ThePhysicsofOrganicSuperconductorsandConductors,Springer-Verlag,Berlin,2008,p.3.[6]B.V.Chirikov,Phys.Rep.52(1979)263.[7]A.J.Lichtenberg,M.A.Lieberman,RegularandChaoticDynamics,Springer,Berlin,1992.[8]O.M.Braun,Yu.S.Kivshar,TheFrenkel…KontorovaModel:Concepts,Methods,Applications,Springer-Verlag,Berlin,2004.[9]S.Aubry,Phys.D7(1983)240.[10]O.V.Zhirov,G.Casati,D.L.Shepelyansky,Phys.Rev.E65(2002)026220.[11]M.Mezard,G.Parisi,M.A.Virasoro,SpinGlassTheoryandBeyond,WorldScienti“c,Singapore,1987.[12]F.Borgonovi,I.Guarneri,D.L.Shepelyansky,Phys.Rev.Lett.63(1989)2010.[13]O.V.Zhirov,G.Casati,D.L.Shepelyansky,Phys.Rev.E67(2003)056209.[14]S.A.Brazovskii,Ferroelectricityandchargeorderinginquasi-1Dorganicconductors,in:A.Lebed(Ed.),ThePhysicsofOrganicSuperconductorsandConductors,Springer-Verlag,Berlin,2008,p.313.[15]L.Landau,Phys.Rev.60(4)(1941)356.[16]R.T.Scalettar,G.G.Batrouni,G.T.Zimanyi,Phys.Rev.Lett.66(1991)3144.[17]K.Miyagawa,Y.Muto,M.Kodama,Y.Shimada,K.Kanoda,CNMRandresistivitymeasurementsofchargeglassstateinacharge-frustratedorganic,reportatECRYS-2011,Cargese(seethisPhys.Bissue).[18]P.Monceau,Ferroelectricityinorganiclow-dimensionalsystems,reportatECRYS-2011,Cargese(seethisPhys.Bissue).[19]M.Johanning,A.F.Varon,C.Wunderlich,J.Phys.B:At.Mol.Opt.Phys.42(2009)154009.[20]A.Benassi,A.Vanossi,E.Tosatti,Nat.Commun.2(2011)236.[21]T.Pruttivarasin,M.Ramm,I.Talukdar,A.Kreuter,H.Haffner,NewJ.Phys.13(2011)075012. Fig.4.SchematicphasediagramofthefamilytakenfromRef.Ref.(seedetailsthere).O.V.Zhirov,D.L.Shepelyansky/PhysicaB407(2012)1909…1911