Magnetic flux periodicities and finite momentum pairing in unconventional superconductors [Elektronische Ressource] / vorgelegt von Florian Loder

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MagneticFluxPeriodicitiesandFiniteMomentumPairinginUnconventionalSuperconductorsDissertationzurErlangungdesDoktorgradesdermathematisch-naturwissenscha!lichenFakultat¨ derUniversitat¨ AugsburgvorgelegtvonFlorianLoder,November"##$Tagdermun¨ dlichenPruf¨ ung: "".%"."##$Erstgutachter: !iloKopp,Universitat¨ AugsburgZweitgutachter: DieterVollhardt,Universitat¨ AugsburgDrittgutachter: RaymondFres´ ard,LaboratoireCrismat-ENSICAEN&,Bld. duMarec´ halJuinF-%'#(#CaenFrankreichTitlepageillustration: Color-codedmomentum-spacemapofthedensityofstatesattheFermienergyinthepairdensitywavestate.ContentsIntroduction 60.1 AShortHistoryofFluxPeriodicitiesinConventionalSuperconductors 80.2 UnconventionalFluxPeriodicitiesinUnconventionalSuperconductors 13I hc!ePeriodicityinLoopsofNodalSuperconductors 181 hc!e orhc!!e-!ePhenomenon 211.1 NormalStateofaOneDimensionalRing . . . . . . . . . . . . . . . . . . 211.2 SuperconductingState . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Crossoverfromhc!e tohc!!eFluxPeriodicityins-WaveLoops 322.1 One-DimensionalRing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 MultichannelRing: Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . 353 FluxPeriodicitiesofNodalSuperconductors 443.1 SuperconductivityinaFlux-!readedCylinder . . . . . . . . . . . . . . 453.2 AnalyticSolutionandQualitativeDiscussion . . . . . . . . . . . . . . . . 493.3 NumericalSolutionford-WavePairingatT =# . . . . . . . . . . .
Publié le : jeudi 1 janvier 2009
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Source : D-NB.INFO/1008664901/34
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MagneticFluxPeriodicitiesandFinite
MomentumPairingin
UnconventionalSuperconductors
DissertationzurErlangungdesDoktorgradesder
mathematisch-naturwissenscha!lichenFakultat¨ der
Universitat¨ Augsburg
vorgelegtvonFlorianLoder,November"##$Tagdermun¨ dlichenPruf¨ ung: "".%"."##$
Erstgutachter: !iloKopp,Universitat¨ Augsburg
Zweitgutachter: DieterVollhardt,Universitat¨ Augsburg
Drittgutachter: RaymondFres´ ard,
LaboratoireCrismat-ENSICAEN
&,Bld. duMarec´ halJuin
F-%'#(#Caen
Frankreich
Titlepageillustration: Color-codedmomentum-spacemapofthedensityofstates
attheFermienergyinthepairdensitywavestate.Contents
Introduction 6
0.1 AShortHistoryofFluxPeriodicitiesinConventionalSuperconductors 8
0.2 UnconventionalFluxPeriodicitiesinUnconventionalSuperconductors 13
I hc!ePeriodicityinLoopsofNodalSuperconductors 18
1 hc!e orhc!!e-!ePhenomenon 21
1.1 NormalStateofaOneDimensionalRing . . . . . . . . . . . . . . . . . . 21
1.2 SuperconductingState . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Crossoverfromhc!e tohc!!eFluxPeriodicityins-WaveLoops 32
2.1 One-DimensionalRing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 MultichannelRing: Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 FluxPeriodicitiesofNodalSuperconductors 44
3.1 SuperconductivityinaFlux-!readedCylinder . . . . . . . . . . . . . . 45
3.2 AnalyticSolutionandQualitativeDiscussion . . . . . . . . . . . . . . . . 49
3.3 NumericalSolutionford-WavePairingatT =# . . . . . . . . . . . . . . 56
3.4 PeriodicityCrossoverforsmall) . . . . . . . . . . . . . . . . . . . . . . . 60
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
II RealSpace:!eBogoliubov-deGennesEquations 64
4 !eBogoliubov-deGennesEquations 66
5 FluxPeriodicityinSuperconductingSquareFrames 69
6 Self-ConsistentTreatmentoftheMagneticField 76
6.1 Vector-PotentialinanAnnulus . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 LatticeFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7 FluxPeriodicityofJosephsonJunctions 81
*Contents
7.1 Current-PhaseRelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 Field!readedJunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8 Conclusions 93
III FiniteMomentumPairing 95
9 Introduction 96
10 Formalism 99
10.1 Gor’kovEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.2 ExtendedBogoliubovTransformation . . . . . . . . . . . . . . . . . . . . 102
11!ePairDensityWaveSolution 105
11.1 SolutionsoftheSelf-ConsistencyEquation . . . . . . . . . . . . . . . . . 105
11.2 !eSuperconductingCharge-StripeState . . . . . . . . . . . . . . . . . . 108
11.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
12 UnusualFluxPeriodicitiesinSuperconductingLoops 116
13 SummaryandOutlook 119
Appendix 124
A Current 125
A.1 ContinuityEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.2 CurrentinMomentumSpace . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.3 FreeEnergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B DopplerShi"andNodesoftheHankelFunctionAnsatz 129
C !eBogoliubov-deGennesEquations 132
C.1 MagneticFieldinDiscreteLattices . . . . . . . . . . . . . . . . . . . . . . 132
C.2 Mean-FieldHamiltonianinRealSpace . . . . . . . . . . . . . . . . . . . 134
C.3 Bogoliubov-deGennesEquations . . . . . . . . . . . . . . . . . . . . . . 135
C.4 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
C.5 InternalEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
D FormalismoftheExtendedBCS!eory 139
'Contents
D.1 Nearest-NeighborPairingInteraction . . . . . . . . . . . . . . . . . . . . 139
D.2 Gor’kovEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
D.3 BogoliubovTransformationMethod . . . . . . . . . . . . . . . . . . . . . 149
D.4 RealSpace: ExtendedBogoliubovdeGennesEquations. . . . . . . . . . 154
E SolutionsoftheLaplaceEquationinaSquareFrame 156
Publications 161
Bibliography 162
!anksto... 171
CurriculumVitae 172
(Introduction
WhenI+rstheardaboutsuperconductivity,thetheoryofsuperconductingquantum
interference devices (SQUIDs) had long been established and well understood. At
thistime,alsothetheoreticalconceptsofunconventionalsuperconductivityhadbeen
analyzed for years and the consequences of di,erent pairing symmetries had been
wellknown.!efabricationofSQUIDsfromhigh-temperaturesuperconductorswith
unconventionalpairingsymmetrieswascertainlyimportantforapplications,butdid
notarousehugeinterestfromthetheoryside. Nevertheless,high-temperatureSQUIDs
proved to be more challenging than expected, when experiments with YBa Cu O" * -!x
(YBCO)SQUIDs,conductedmostlybyChristofSchneiderin"##*–"##&inAugsburg,
revealedmagnetic.uxoscillationswithanumberofperiodsdi,erentfromtheexpected
valuehc!"e.
Fluxoscillationsinmultiplyconnectedgeometriesaregenerallyhighlyinteresting,
because they are a purely quantum mechanical e,ect and reveal much of the nature
ofthequantumstateofthesystem. Notonlyaretheycharacteristicforthefermionic
orbosonicnatureofthechargecarriers,buttheycontainvaluableinformationabout
quantummechanicalmany-bodystatesofthesystem,whichresultsfromthecomplex
interactionsbetweenthechargecarriersandthecrystallattice,andbetweenthecarriers
themselves.
Inthecaseofsuperconductivity,theobservationof.uxoscillationswithaperiodof
hc!"e wasamajorproofoftheexistenceofanattractivepairinginteractionbetween
electronsinthesuperconductingstate,andthusofthevalidityoftheBardeen-Cooper-
Schrie,er (BCS) theory.!e absence of oscillations with other periodicities in con-
ventionalsuperconductorsfurtherprovedthatinteractionsbetweenthepairsmustbe
negligibleinconventionalsuperconductors,whichisreasonableforaweaklyinteracting
system. Consequently,theobservationofdi,erent.uxperiodicities,asobservedinthe
YBCOSQUIDs,isasignforcorrelationsbetweenCooperpairs.
!e search for a theoretical description of such correlations, and for the origin of
theunconventional.uxoscillations,wasactivewhenIcametoAugsburgformyPhD
thesisin"##(andjoinedthefascinatingresearchinthis+eld. Becauseofthecomplexity
inthedescriptionofaSQUID,thefocusofmyattentionmovedtotheanalysisofthe
.ux periodicity of simple superconducting loops without Josephson junctions and it
soonturnedoutthatthestandardtheoryofsuchsystemsisnotadequateforloopswith
unconventionalpairingsymmetries. Workingouttheproperextensionofthetheory
&becamethereforethemaintopicofthisthesis. Itexplainswhythe.uxperiodicityof
loopsofnodalsuperconductorsistrulyhc!e andnothc!"e,asitisinsu/cientlylarge
loopsofconventionalsuperconductors,andtherebyprovidesananswertoapartofthe
periodicityproblemoftheSQUIDs.
Amathematicaltool,whichturnedouttobeespeciallysuitableforthedescription
of superconducting loops, is the Green’s function formalism and, in particular, the
Gor’kovequationsdescribingthesuperconductingstate.Inthecourseoftheinvestigation
of .ux periodicities, I realized that the Gor’kov equations in their general form can
have solutions which are clearly distinct from those of the conventional BCS theory.
!ese solutions are more of the Fulde-Ferrell-Larkin-Ovchnnikov type, but exist in
zeromagnetic+eld. Later,Irealizedthatsuchsolutionsarecurrentlywidelyanalyzed
inconnectionwithchargeorderedsuperconductingstates,referredtoas“pairdensity
wave”states,whichmightberealizedinsomehigh-temperaturesuperconductors.!is
connectionwasquiteexciting,andinthefollowingweestablishedasolidmicroscopic
model for the “pair density wave” state on the basis of the Gor’kov equations, which
becamethesecondlargetopicofmythesisandisanon-goingproject.
!isthesisisorganizedinthreeparts. PartIandpartIIareconcernedwiththe.ux
periodicityofsuperconductingloops. Ageneralintroductionwithahistoricoverview
ofthedevelopmentofthis+eldfollowsinthenexttwosectionsofthischapter. PartIII
containsthemodelforthe“pairdensitywave”statewithaseparateintroductionatthe
beginning. Finally,attheendofpartIII,Igiveasummaryofthecompletethesiswith
anoutlooktofutureprojectsonthetopicsofthisthesis. InpartsI–III,calculations
aregivenasdetailedasnecessarytofollowthediscussion. Forsomemostchapters,the
methodsusedandthecalculationsperformedaredescribedmoredetailintheappendix.
At the end of the appendix a collection of the most aesthetic wave functions of free
particlesinasquareframeisassembled,whichhavebeencalculatedduringmythesis.
-Introduction
#.$ AShortHistoryofFluxPeriodicitiesinConventional
Superconductors
!equantummechanicalwavefunctionofparticlesmovinginamultiplyconnected
geometryhastobeuniqueatallpositions.!isboundaryconditionleadstoadiscrete
energyspectrum,becausethephasedi,erenceofthewavefunctionaccumulatedona
closedpathhastobe"!kforsomeintegerk. Foracirculargeometry,thisphasewinding"number k represents the angular momentum hkof the particle. In the presence of a
r " "magnetic +eld B(r), an additional phase e!hc dr !A(r ) adds to the phase of the!r#
wavefunction,whereA(r)isthevectorpotentialgeneratingB(r), e theelectroncharge,
c the velocity of light, and h is Planck’s constant.!is leads to a phase di,erence of
"!(k"e0!hc)onaclosedpathC,where0= dr!A(r)isthemagnetic.uxinclosed!C
bythepathC. Becausephysicalquantitiesinthissystemareobtainedbysummingover
all possible k in a thermal average, they are periodic in 0 with a basic period of 0 ,#
where
0 = hc!e (#.%)#
isthe.uxquantuminthenormalstate. Inparticular,apersistentcurrent J(0)induced
bythemagnetic.uxiszerowhenever0!0 isaninteger.#
!ee,ectdescribedaboveispresentinanysystemwithsu/cientlystrongphaseco-
herence,andbestknownintheperiodicmodulationsoftheresistanceofamicroscopic
metalloop,predicted+rstbyEhrenbergandSidayin%$'1[%]andin%$($byAharonov
andBohm["]. Alreadytenyearsearlier,Londonpredictedthemanifestationofasimilar
e,ectinsuperconductingloops,wherethephasecoherenceisnaturallymacroscopic[*]:
themagnetic.uxthreadingtheloopisquantizedinmultiplesof0 ,becausetheinterior#
ofasuperconductorhastobecurrentfree. AlthoughLondondidnotknowtheexistence
of0 !".uxquantainsuperconductors,healreadyspeculatedthat,becausethesuper-#
currentmightbecarriedbypairsofelectronswithcharge"e,thesuperconducting.ux
quantum,andwithitthe.uxperiodicityofthesupercurrent,israther0 !".!ispoint#
ofviewbecamegenerallyacceptedwiththepublicationofthe‘!eoryofSuperconduc-
tivity’byBardeen, CooperandSchrie,er(BCS)in%$(-[']. Directmeasurementsof
magnetic.uxquanta0 !"trappedinsuperconductingringsfollowedin%$&%byDoll#
andNab¨ auer[(]andbyDeaverandFairbank[&],corroboratedlaterbythedetectionof
.uxlinesof0 !"inthevortexphaseoftypeIIsuperconductors[-,1].#
For thin superconducting loops with walls thinner than the penetration depth ",
+nite currents are.owing throughout the whole superconductor.!e magnetic.ux"is consequently not quantized, but only the quantity 0 = 0+2!c dr!J(r) called!!
.uxoid,introducedbyLondonforthispurposebycountingthe.uxinduced.!e.ux
0isthetotal.uxthreadingtheloop,whichcontainsthecurrentinduced.uxalready.
1!." AShortHistoryofFluxPeriodicitiesinConventionalSuperconductors
(a) (b)
Figure0.1:Schemeofthepairingofangular-momentumeigenstatesinaone-dimensionalmetalloop
for(a)! = 0and(b)! = hc/2e, asusedbySchriefferin[9]toillustratetheoriginofthe
! /2periodicityinsuperconductors.Pairedarestateswithequaleigenenergies! ,which0 k
leadstopairswithacenter-of-massangularmomentumq = 0in(a)andq = 1in(b).
2isaphenomenologicalconstantparametrizingthestrengthofthecurrentresponse
of the superconductor to a magnetic +eld and related to the penetration depth " as
" "2 = '!" !c through the London equation [*].!in loops react periodically to the
continuousvariable0,andthis.uxperiodicityisthemaintopicofthiswork.
Itistemptingtoexplainthe0 !".uxperiodicityofsuperconductingloopsbythe#
charge-"e Cooperpairscarryingthesupercurrent,butpairingofelectronsaloneisnot
su/cient. Atheoreticaldescriptionofthetrueoriginofthehalf-integer.uxquantawas
foundindependentlyin%$&%byByersandYang[%#],byOnsager[%%],andbyBrenig[%"]
onthebasisoftheBCStheory.!eyrealizedthattherearetwodistincttypesofelectron
pairing,whichleadtotwoclassesofsuperconductingwavefunctionsthatarenotrelated
by a gauge transformation. An intuitive picture illustrating these two types can be
found in Schrie,er’s book on superconductivity [$], using the energy spectrum of a
onedimensionalmetalring:!e+rstclassofsuperconductingwavefunctions,which
Londonhadinmindinhisconsiderationsabout.uxquantization,isrelatedtopairingof" "electronswithangularmomentahkand"hk,whichhaveequalenergiesinametalloop
withoutmagnetic.ux,asschematicallyshownin+gure#.%(a).!eCooperpairsinthis
statehaveacenter-of-massangularmomentum(pairmomentum)q =#.!epairing
wave functions of the superconducting state for all .ux values 0, which are integer" "multiplesof0 andcorrespondtoevenpairmomentahq="h0!0 ,arerelatedtothe# #
wavefunctionfor0=#byagaugetransformation.!esecondclassofsuperconducting
wave functions occurs for a .ux value 0 !" (or odd multiples of this value), where#" "electronstateswithangularmomentumhkand"h(k+%)arepaired,whichhaveinthis
caseequalenergy[+gure#.%(b)],andleadtoapairmomentumq=%.!ecorresponding
pairingwavefunctionisagainrelatedbyagaugetransformationtothosefor.uxvalues
$
LL??LLIntroduction
(a) (b)
Figure0.2:(a) Energy E(!) and (b) supercurrent J(!) as a function of flux ! for a conventional
superconductingloopatT = 0.TheminimainE(!)correspondtosuperconductingstates
withdifferentpairmomentaq. Thescreeningcurrentsinthesuperconductordrivethe
systemtotheclosestminimumforeachfluxvalue(blackpoints),ifthewallsoftheloop
arethickerthan".
0 which are half-integer multiples of 0 and correspond to the odd pair momenta#" "hq="h0!0 .#
!etwotypesofpairingdescribedabovearegenerallydi,erent. Forthesystemto
be0 !"periodic,itisfurtherrequiredthatthetwopairingtypesaredegenerate,which#
meansthattheyhavethesamefreeenergy. ByersandYang[%#],Onsager[%%]aswellas
Brenig[%"]showedthatthisisthecaseinthethermodynamiclimit,whichisobtainedfor
acontinuousdensityofstates. Inthiscase,theenergyE(0)asafunctionof0consists
ofaseriesofparabolaewithminimaatintegermultiplesof0 (correspondingtoeven#
pairmomenta q)andhalfintegermultiplesof0 (correspondingtooddpairmomenta#
q)[+gure#."(a)]. Iftheloopisthickerthan " andthe.uxisquantized,thenthesystem
isintheminimumclosesttothevalueoftheexternal.ux(blackpoints). Inmicroscopic
systemsthedegeneracyoftheevenandoddqminimaisli!ed,buttheirpositionis+xed
bythepairinginteractionandgaugeinvariance.!e.uxperiodicityinthinloopsisthen
notnecessarily0 !",butthesuperconducting.uxquantumisalways0 !".(!ereare# #
mechanismsleadingtofractional.uxquanta,towhichwereferlater.)!ecirculating
supercurrent J(0)attemperatureT =#isproportionalto!E(")!!0andformsa0 !"#
periodicsaw-toothpatterninthethermodynamiclimit,whichisshownin+gure#."(b).
Fromthe.uxperiodicityofE(0),itisstraightforwardtoderivethesame.uxperiod-
icityforallotherphysicalquantities[%*]. ForT >#, E(0)hastobereplacedbythefree
energy F(0). Aclearandunambiguousobservationof.uxoscillationsispossiblein
the.uxdependenceofthecriticaltemperatureT ofsmallsuperconductingcylinders.c
Suchexperimentshavebeendone+rstbyLittleandParksin%$&"[%'–%&].!eyactually
measuredtheresistanceRofthecylinderata+xedtemperatureT withinthe+nitewidth
%#
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