MagneticInductionofdx2y2+idxyOrderinHigh-TcSuperconductors

R.B.Laughlin

DepartmentofPhysics,StanfordUniversity,Stanford,CA94305

(Dated:April1,1997)

IproposethatthephasetransitioninBi2Sr2CaCu2O8recentlyobservedbybyKrishanaetal

[Science277,83(1997)]isthedevelopmentofasmalldxysuperconductingorderparameterphased

by=2withrespecttotheprincipaldx2y2onetoproduceaminimumenergygap∆.Theviolationofbothparityandtime-reversalsymmetryallowsthedevelopmentofamagneticmoment,thekeytoexplainingtheexperiment.TheoriginofthismomentisaquantizedboundarycurrentofIB=2e/hatzerotemperature.

PACSnumbers:74.25.B6,74.25.DN,74.25.Fy

InarecentpaperKrishanaetal

1

havereportedaphase

transitioninBi2Sr2CaCu2O8inducedbyamagneticfieldandcharacterizedbyakinkinthethermalconductivityasafunctionoffieldstrength,followedbyaflatplateau.Thehigh-fieldstateisalsosuperconducting.Theyar-guedfromtheexistenceofthisplateauthatheattrans-portbyquasiparticleswaszerointhenewstateandthatthisprobablyindicatedthedevelopmentofanenergygap.Thetransitionhasthepeculiarityofbeingeasilyinducedbysmallfields.Krishanaetalreporttheempir-icalrelationTc/p

B,althoughoverthelimitfieldrange

of0:6T<B<5T,andalsothatthetransitionsharpens

asTcisreduced.

Iproposethatthenewhigh-fieldstateistheparityand

time-reversalsymmetryviolatingdx2y2+idxysupercon-ductingstateproposedlongagobyme

2

,whichhasmany

propertiesincommonwithquantumhallstates,includ-ingparticularlychiraledgemodesandexactlyquantizedboundarycurrents.Theessentialpointofmyargumentisthatthestatemusthaveamagneticmomentinor-dertoaccountfortheexperiment,andthisispossibleonlyifitviolatesbothparityandtime-reversalsymme-try.Thedevelopmentofs+idorder,forexample,or

high-momentumCooperpairing

3

arebothruledoutfor

thisreason,asisarestructuringofthevortexlattice.

Myhypothesisleads,throughreasoningdescribedbe-

low,tothemodelfree-energyfunctional

F

L2=

1

6



3

(¯hv)2

1



eB

hc

tanh

2

(



2

)

4



(kBT)

3

(¯hv)2





(∆)

2

2

ln[1+e]+Z1



ln[1+ex]xdx

;(1)

whereistheinducedenergygapandv=

p

v2v2isthe

root-mean-squarevelocityofthed-wavenode.Therearethreekeystepsleadingtothisfunctional:

1.Theadoptionofconventionalquasiparticlesatfour

nodesasthelow-energyexitationspectrumoftheparentdx2y2state.

2.Thederivationofarelationbetweentheminimum

energytoinjectaquasiparticleinthebulkinteriorandaquantum-mechanicalboundarycurrent.

05

10152025

0123456

Tc(K)

B(Tesla)

3

3

3

3

3

3

3

3

3

3

FIG.1:Comparisonofmeasuredtransitiontemperaturever-susmagneticfield(diamonds)withEq.(5).

3.Aguessastothetemperaturedependenceofthis

boundarycurrentbasedonlegitimatebutmodel-dependentcalculations.

Thelastofthese,whichIshalldefendbelow,ispurephenomenology,sothisisatheoryofenergyscalesandnotatheoryofthetransition.Thevalueofthenodevelocityisfixedbyexperiment,inparticularphotoe-missionbandwidth

4

andthetemperaturedependenceof

thepenetrationdepthinYBCO

5,6

.FollowingLeeand

Wen

7

Ishallusethevaluesv1=1:1810

7

cm/secand

v1/v2=6:8,orhv=0:30eVA.Theuncertaintyinthisnumberisabout10%.Atzerotemperaturethefreeen-ergyisminimizedby

0=hv

r

2

eB

hc

;(2)

andhasthevalue

F0

L2=

1

3



30

(¯hv)2:(3)

AfinitetemperatureIfindaweaklyfirst-ordertransitiontoastatewith=0at

2

kBTc=0:520:(4)

ThisisplottedagainsttheexperimentinFig.1.Itwillbeseentoaccountforboththefunctionalformofthetransitiontemperatureanditsabsolutemagnitudewithnoadjustableparameters.

Theassumptionofconventionalquasiparticlesat

d-wavenodesleadstotherepulsive

3

andfree-

quasiparticleentropytermsinEq.(1).Themodelhereisnotcritical,sinceonlythenodematters,soletususetheBCSHamiltonian

H=

X

ks

"kcy

kscks+

X

kk0

Vkk0cy

k"cy

k#ck0#ck0":(5)

Asusualweconsidervariationalgroundstatesoftheform

j>=

Y

k



uk+vkcy

k"cy

k#



j0>

jukj

2

+jvkj

2

=1;(6)

andminimizetheexpectedenergy

<|Hj>

=2

X

k

"kjvkj

2

+

X

kk0

Vkk0(ukv

k)(u

k0vk0)(7)

toobtain

uk=

s

12



1+

"k

p

"2

k+j2

kj



vk=

s

12



1

"k

p

"2

k+j2

kj



k

jkj

;(8)

where

k=

X

k0

Vkk0(u

k0vk0);(9)

or

k=

1

2

X

k0

Vkk0

k0

p

"2

k0+jk0j2

:(10)

EquivalentlywemaytakeEqs.(8)todefinej>intermsofkandminimizetheexpectedenergy

<|Hj>=

X

k

"k



1

"k

p

"2

k+jkj2



+

14

X

kk0

Vkk0





k

p

"2

k+jkj2

][

k0

p

"2

k0+jk0j2



;(11)

toobtainEq.(10).Regardlessofwhethertheextremalconditionismettheexpectedenergyofthequasiparticle

jk">=(u

kcy

k"+v

kck#)j>(12)

is

<k"|Hjk">=<|Hj>+

q

"2

k+jkj2:(13)

Theprototypicaldx2y2+idxystateis

"k=2t



cos(kxb)+cos(kyb)



(14)

k=x2y2



cos(kxb)cos(kyb)



+ixysin(kxb)sin(kyb):(15)

ThevelocityinEq.(1)isrelatedtothemodelparametersby

hv1=

p

8tbhv2=

p

2∆x2y2bv=p

v1v2:(16)

Assumingnowthattheextremalconditionrequiresxytobezero,sothatthenativegroundstatehasonlydx2y2order,andthenforcingtheminimumquasiparticleenergytobe∆,onefindsthattheenergyisminimizedwhen



2xy=





2

(q=hv)

2

;q=hv0;q>=hv



;(17)

whereqisthedistancetothenodeinsymmetrizedunits,andequals

<|Hj>=

X

k

("

2k+jkj

2

)



1

p

"2

k+jkj2



=

2



hvL

2

Z=hv

0



1

q



hv





q

3

dq=

L

2

6



3

(¯hv)2:(18)

Thequasiparticlecontributiontothefinite-temperaturefreeenergyunderthesecircumstancesis

Fquasi

L2=

4



kBT

3



Z1

0

ln[1+exp(

q

(¯hq)2+2

xy)]qdq:(19)

Letusnextconsiderthezero-temperaturemagnetic

moment,whichisduetoacirculatingboundarycurrentof

IB=2

e

h

0:(20)

Thisworksoutto0.13Aforagapof1.64meVinducedbyafield1Tesla.BoundarycurrentsofthismagnitudeareknowntoresultfromthedevelopmentofaT-violatingorderparameterofthissize

8

,sotheissueisnottheexis-

tenceofthesecurrentsortheirdisappearancewhenthesecondorderparametervanishesbutrathertheirspecificfunctionaldependenceonandsenseofcirculation.TandPmustbothbeviolatedfortheboundarycurrentstogenerateamoment.Thes+idstate,forexample,willnotworkbecauseitsreflectionsymmetryaboutthex-axisforcesthecurrentsatthe+yand-yedgestoflowinthesamedirection,whereasflowinoppositedirectionsisrequiredtogenerateamoment.

Thedx2y2+idxystatediffersfundamentallyfrom

s+idconventionals-wavestatesinnotbeingcontinu-ouslydeformabletoafermiseaonasamplewithedges,althoughitcanbesodeformedonatorus.ThisisthepropertyunderlyingWiegmann’sconceptofa“topolog-icalsuperconductor”

9

.Onatoruswemaywrite

j>=

Y

k

exp



iy

kkkk



j0>;(21)

where

k=



ck"

cy

k#



1=



0110



2=



0ii0



3=



1001



;(22)

asusual,andthuscontinuouslydeformthefermiseaj0>intoanyBCSsuperconductingstatej>welikeby

appropriatechoiceofthefunctionk.Onasamplewith

edges,however,thismakesnosensebecausekisnota

goodquantumnumber.Thebestwecandoissubstitutethetime-reversedorbitalpairsexp(ikx)sin(npiy/L),

whereListhesamplewidthandnisaninteger,fortheplanewavesexp(ikr)intheaboveexpression,inwhichcasewefindtheorderparameterktobeevenunderparityinthey-direction,apropertyfundamen-tallyincompatiblewithd+idpairing.Thusweconfront

aproblemsimilartotheoneencounteredinthevortexlattice-thesolutionofwhichwastheinventionoftheBogoliubov-DeGennesequations-namelythatexcitationofCooperpairsintotime-reversedorbitalpairsmakesno

senseinamagneticfield.Inthiscase,ofcourse,thevi-olationofparityandtime-reversalinvariancecomesnotfromanexternalmagneticfieldbutfromthevacuumit-self.

Thedx2y2+idxystateis,however,continuouslyde-

formableintoadoubly-occupiedLandaulevel.Thisisdemonstratedwiththefollowingsimplelatticeexample.Let

HHF=2t

X

k

{[

cos(kxb)+cos(kyb)



y

k3k

+



cos(kxb)cos(kyb)



y

k1k

+2msin(kx)sin(ky)Ψy

k2k



(23)

betheHartree-FockHamiltonianforadx2y2+idxysuperconductoronasquarelattice,wheremisacon-

stantcharacterizingthesizeoftheenergygap.ThentheHamiltonianUy()H

HFU(),where

U()=

Y

j

Uj()=

Y

j

exp





2



12(1)

`j

+(1)

`j+mj



(cy

j"cy

j#cj#cj")



;(24)

with`jandmjdenotingthex-andy-coordinatesofthej

th

latticesite,interpolatesbetweenHHFat=0and

alatticeLandaulevelHamiltonianat==4,allthewhilehavingthesameeigenvaluespectrumduetotheunitarityofU()

10,11

.Morespecifically,since

cy

j"cy

j#cj#cj"=iψy

j2 j;(25)

thesitetransformationsrepeatwiththepatternshowninFig.2,where

U1(



4

)=1U2(



4

)=

1iτ2p

2

U3(



4

)=

1+iτ2

p

2

U4(



4

)=iτ2;(26)

sowehaveforthetransformedbondHamiltonianfromsite2tosite1

Uy

1(

3+1

p

2

)U2=(

3+1

p

2

)(

1iτ2p

2

)=3;(27)

andsoforthfortheotherbonds.Thetransformationonthediagonalbondsgiveiτ3,asappropriateformagneticbandsonalattice.

4

3131

3+13+1

p

23

p

23

p

23

p

23

ssss

ssss

24

13

24

13

-



UUy

FIG.2:Illustrationofunitarytransformationbetweenthedx2y2+idxysuperconductingstateandafilledLandaulevelonalattice.

Letusnowimaginewrappingaribbonofdx2y2+idxy

superconductorintoaloopandadiabaticallyinsertingmagneticfluxhc/ethroughthisloopfollowingtheproce-dureusedinaquantumhallthoughtexperiment

12

.This

insertioncommuteswiththerotationofthesupercon-ductorintothequantumhallstatebyvirtueofthegapandthereforehasthesameeffectinthetwocases,i.e.to“pump”one"andone#quasiparticlefromoneedge

totheother.Theedgecurrentsineithercasemaybeidentifiedbyseparatingthisspectralflowinto

1.Transferofastatefromthechemicalpotentialat

theleftedgetothelowestavailableenergyinthebulkinterior.

2.Themirrorimageofthisattherightedge.

Onlythelowest-energybulkstatemattersbecausetheanti-crossingrulepreventsanyhigher-energystatesfromflowingtothechemicalpotential.TheedgecurrentisthengivenbyEq.(20),where0isthedifferenceinenergybetweenthislowest-energybulkexcitationandthechemicalpotential.Thisresultisexact.

Fluxadditionalsoinducesbulksupercurrent,formal

gaugetransformationsbeingnotsoinnocuousinasu-perconductor,butthisiseasilyremovedbycausingtheringcircumferenceLtodiverge,sincetheenergyinques-tionis

Ebulk=

h

2

2m

(

2

L

)

2

Z

ns(~r)d~r;(28)

wherensisthesuperfluiddensity,whichfallsoffas1/L.Equivalentlyonemaysaythatthereisisaphysicaldif-ferencebetweencurrentalreadypresentandcurrentin-ducedbytheinjectedflux.

Thefinalmatterforconsiderationisthereduction

ofthismomentbythermalexcitationofquasiparticles.Thisis,unfortunately,sensitivetodetailsandthusdiffi-culttocalculatewithsufficientaccuracytodescribethephasetransition.ItcanbeunderstoodsimplyintermsofthefluxHamiltonianobtainedbyrotatingEq.(23)by==4.Thisconsistsofupperandlowerquantum

hallbandswithoppositequantizations,thesebeingman-ifestedprimarilyinthestatesofenergynear=4tm.

EvaluatingtheHallconductanceofthismodelbytheKuboformulainthelimitofsmallmwefindthat

11

xy=

e

2

h

Z1

0

tanh





2

p

1+x



dx

(1+x)3=2:(29)

ThestrongquenchingeffectatkBToccursbecausefreequasiparticlescontributeaHallconductanceoppo-sitetothatofthegroundstate.Assumingnowthatvariesslowlyinspaceandequalszeroatthesampleedge,wemayintegrateinfromtheedgetoobtain

IB'2

e

h

kBT

Z1

0

ln



cosh(



2

p

1+x)



dx

(1+x)2:

(30)

TheversionofthisappropriatetoEq.(17)is

IB'4

e

h

kBTln



cosh(



2

)



:(31)

ThisisquiteclosetothefunctionalformappearinginEq.(1)intherangeofinterest,saturatestolinearityinatzerotemperature,becomesexponentiallyquenchedfortemperatureskBT>>∆,butgivesnophasetran-sition.ThephenomenologicalfunctionIchoseismerelyanapproximationtothisoneconstrainedtobeanalyticinandodd.TheproportionalityofTcto0,however,isexpectedongeneralgroundsbecausethereisnootherenergyscaleintheproblem.

ThecompleteabsenceofthermaltransportaboveTc

intheexperimentisnotexplainedbythermalactivationtoagapoforder0,asthisissimplytoosmalltofreezeoutallthequasiparticles.Thiscriticism,however,ap-pliesequallywelltoanytheoryoftheeffectonewouldcaretoconsider,foritisphysicallyunreasonableforagapmuchlargerthankBTctodevelopspontaneously.Ithereforebelievethatabsenceoftransportisaneffectofenhancedscatteringandtrappingofquasiparticlesinthenewstateandisadetailtobeworkedoutoncethesymmetryofthesecondorderparameterisestablished.Thereiscertainlythepotentialforviolentscatteringinthedx2y2+idxystategiventheinhomogeneityofthemagneticfieldduetothevortexlatticeandthepossi-bilitythatthetransitionisweaklyfirst-order,butitisamistaketousethisasacriterionfordecidingwhetherthesymmetryIhaveidentifiedistherightone.

IwishtoexpressspecialthankstoC.M.Varmaforalertingmetothelargemomentcarriedbythisclassofsuperconductor,andtoN.P.Ong,F.D.M.Haldane,J.Berlinsky,C.Kallin,andA.Balatskyforhelpfuldiscus-sionandcriticism.ThisworkwassupportedprimarilybytheNSFundergrantNo.DMR-9421888.AdditionalsupportwasprovidedbytheCenterforMaterialsRe-searchatStanfordUniversityandbyNASACollabora-tiveAgreementNCC2-794.

5

R.B.Laughlin:http://large.stanford.edu

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