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Ginzburg–LandautheoryFromWikipedia,thefreeencyclopediaInphysics,Ginzburg–Landautheory,namedafterVitalyLazarevichGinzburgandLevLandau,isamathematicalphysicaltheoryusedtodescribesuperconductivity.Initsinitialform,itwaspostulatedasaphenomenologicalmodelwhichcoulddescribetype-Isuperconductorswithoutexaminingtheirmicroscopicproperties.Later,aversionofGinzburg–LandautheorywasderivedfromtheBardeen-Cooper-SchrieffermicroscopictheorybyLevGor'kov,thusshowingthatitalsoappearsinsomelimitofmicroscopictheoryandgivingmicroscopicinterpretationofallitsparameters.Contents[hide]1Introduction2Simpleinterpretation3Coherencelengthandpenetrationdepth4FluctuationsintheGinzburg–Landaumodel5ClassificationofsuperconductorsbasedonGinzburg–Landautheory6Landau–Ginzburgtheoriesinstringtheory7Seealso8References8.1PapersIntroduction[edit]BasedonLandau'spreviously-establishedtheoryofsecond-orderphasetransitions,GinzburgandLandauarguedthatthefreeenergy,F,ofasuperconductornearthesuperconductingtransitioncanbeexpressedintermsofacomplexorderparameterfield,ψ,whichisnonzerobelowaphasetransitionintoasuperconductingstateandisrelatedtothedensityofthesuperconductingcomponent,althoughnodirectinterpretationofthisparameterwasgivenintheoriginalpaper.Assumingsmallnessof|ψ|andsmallnessofitsgradients,thefreeenergyhastheformofafieldtheory.whereFnisthefreeenergyinthenormalphase,αandβintheinitialargumentweretreatedasphenomenologicalparameters,misaneffectivemass,eisthechargeofanelectron,Aisthemagneticvectorpotential,andisthemagneticfield.Byminimizingthefreeenergywithrespecttovariationsintheorderparameterandthevectorpotential,onearrivesattheGinzburg–Landauequationswherejdenotesthedissipation-lesselectriccurrentdensityandRetherealpart.Thefirstequation—whichbearssomesimilaritiestothetime-independentSchrödingerequation,butisprincipallydifferentduetoanonlinearterm—determinestheorderparameter,ψ.Thesecondequationthenprovidesthesuperconductingcurrent.Simpleinterpretation[edit]Considerahomogeneoussuperconductorwherethereisnosuperconductingcurrentandtheequationforψsimplifiesto:Thisequationhasatrivialsolution:ψ=0.Thiscorrespondstothenormalstateofthesuperconductor,thatisfortemperaturesabovethesuperconductingtransitiontemperature,TTc.Belowthesuperconductingtransitiontemperature,theaboveequationisexpectedtohaveanon-trivialsolution(thatisψ≠0).Underthisassumptiontheequationabovecanberearrangedinto:Whentherighthandsideofthisequationispositive,thereisanonzerosolutionforψ(rememberthatthemagnitudeofacomplexnumbercanbepositiveorzero).Thiscanbeachievedbyassumingthefollowingtemperaturedependenceofα:α(T)=α0(T-Tc)withα0/β0:Abovethesuperconductingtransitiontemperature,TTc,theexpressionα(T)/βispositiveandtherighthandsideoftheequationaboveisnegative.Themagnitudeofacomplexnumbermustbeanon-negativenumber,soonlyψ=0solvestheGinzburg–Landauequation.Belowthesuperconductingtransitiontemperature,TTc,therighthandsideoftheequationaboveispositiveandthereisanon-trivialsolutionforψ.FurthermorethatisψapproacheszeroasTgetsclosertoTcfrombelow.Suchabehaviouristypicalforasecondorderphasetransition.InGinzburg–Landautheorytheelectronsthatcontributetosuperconductivitywereproposedtoformasuperfluid.[1]Inthisinterpretation,|ψ|2indicatesthefractionofelectronsthathavecondensedintoasuperfluid.[1]Coherencelengthandpenetrationdepth[edit]TheGinzburg–Landauequationspredictedtwonewcharacteristiclengthsinasuperconductorwhichwastermedcoherencelength,ξ.ForTTc(normalphase),itisgivenbywhileforTTc(superconductingphase),whereitismorerelevant,itisgivenbyItsetstheexponentiallawaccordingtowhichsmallperturbationsofdensityofsuperconductingelectronsrecovertheirequilibriumvalueψ0.Thusthistheorycharacterizedallsuperconductorsbytwolengthscales.Thesecondoneisthepenetrationdepth,λ.ItwaspreviouslyintroducedbytheLondonbrothersintheirLondontheory.ExpressedintermsoftheparametersofGinzburg-Landaumodelitiswhereψ0istheequilibriumvalueoftheorderparameterintheabsenceofanelectromagneticfield.Thepenetrationdepthsetstheexponentiallawaccordingtowhichanexternalmagneticfielddecaysinsidethesuperconductor.TheoriginalideaontheparameterkbelongstoLandau.Theratioκ=λ/ξispresentlyknownastheGinzburg–Landauparameter.IthasbeenproposedbyLandauthatTypeIsuperconductorsarethosewith0κ1/√2,andTypeIIsuperconductorsthosewithκ1/√2.TheexponentialdecayofthemagneticfieldisequivalentwiththeHiggsmechanisminhigh-energyphysics.FluctuationsintheGinzburg–Landaumodel[edit]Takingintoaccountfluctuations.ForTypeIIsuperconductors,thephasetransitionfromthenormalstateisofsecondorder,asdemonstratedbyDasguptaandHalperin.WhileforTypeIsuperconductorsitisoffirstorderasdemonstratedbyHalperin,LubenskyandMa.ClassificationofsuperconductorsbasedonGinzburg–Landautheory[edit]IntheoriginalpaperGinzburgandLandauobservedtheexistenceoftwotypesofsuperconductorsdependingontheenergyoftheinterfacebetweenthenormalandsuperconductingstates.TheMeissnerstatebreaksdownwhentheappliedmagneticfieldistoolarge.Superconductorscanbedividedintotwoclassesaccordingtohowthisbreakdownoccurs.InTypeIsuperco