The complex resistivity $\hat{\rho}(\omega)$ of the vortex lattice in an untwinned crystal of 93-K $\rm YBa_2Cu_3O_7$ has been measured at frequencies $\omega/2\pi$ from 100 kHz to 20 MHz in a 2-Tesla field $\bf H\parallel c$, using a 4-probe RF transmissi

whichthevortexlatticeisweaklydisorderedbypinningtoalowdensityofpins[16,17].Amean- eldsolutionproposedbyOngandWu(OW)[18]isusedto tthemeasuredresistivityspectra.Bymeasuringthedampingviscosityη(T)independently,wereducethenumberofadjustableparameterstojust2numbersateachT.WeshowthattheOWsolutioncanachieveclose tstoρ1(ω)extendingover2decadesinω.The tsenableusto ndbothκandc66(towithinamultiplicativeconstant)ateachT.Thelatterexhibitsaremarkable4-decadecollapsetozerowithin1KofTm.

InSec.IIasummaryoftheexperimentalapproachisgiven.SectionIIIdiscussesskin-depthe ectsandthe‘volume’inductance.InSec.IV,wereportmeasure-mentsversusTandHatconstantω.Thecentralre-sults–thespectraofρ1andρ2/ω–arereportedinSec.V.SectionVIexplainsthe tsofthespectratotheweaklydisorderedlatticemodel,andSec.VIIdiscussesthetemperaturedependenceoftheshearmodulusnearTm.AppendixAdescribesthe4-probetechnique,whileAppendixBsummarizestheimpedancecalculationforasamplewithellipticalcross-section.AppendixCsumma-rizesthelinearresponseoftheweaklydisorderedlatticetoanaccurrent.

II.EXPERIMENTALDETAILS

AsinWOGT[10],aweak,alternatingcurrentJejωtisappliedwithintheCuO2plane(J a)inthepresenceofa eldH c.Theaveragevelocityresponseofthevorticesisdetectedasacomplexresistivityρ (ω)=v (ω)B/J0.BecausecontactresistancesonYBCOcrystalsaretyp-icallyafractionofanOhm,theyaddaT-dependentbackgroundthatdominatesthesamplesignal.Inanex-perimentinwhichthe eldHissweptatconstantT(asinWOGT),thislargebackgroundmaybesubtracted,byreferencetothezero- elddataateachT.However,whenTissweptatconstantH,thecontactbackgroundleadstosubstantialerrors.TheproblemisexacerbatedbytheTdependenceofthedielectric llinginthelongcoaxcables.Weminimizedthisbyusingspecialcableswithlow-densityPolytetra uoroethylene(te on)dielec-tric.Thehigherpropagationvelocity(0.85cversus0.66c)isalsoanadvantage.

The4-probedesignwasdevelopedtoremovetheback-groundaltogether.Thesample(impedanceZs=Rs+jωLs)shuntstheinnerandouterconductorsofboththeincidentandtransmissioncoaxialcable(seeAppendixA).IfZs Z0(thelineimpedance),theincidentsignalisstronglyre ected,andthetransmittedsignalishighlysensitivetosmallchangesinZs.Byphase-detectingthelatterwemaydetermineZs.Byinsertingbroad-bandtransformersinboththeincidentandtransmissionca-bles,wecanachievetrue4-probemeasurementatfre-quenciesupto50MHz(AppendixAdescribesthecali-brationprocedure).

Themeasurementsweremadeonadetwinnedcrystal

2

ofYBCO(ofdimensiona=1.11mm,b=0.41mm,andc=55µm),inwhichthedcresistivitydisplaysasharptransition(widthof0.2K)atTc=93.3K.TheRFresis-tivitywasmeasuredwiththeRFcurrentdensityJ a,andH c.Althoughallmeasurementswereperformedinascreenedroom,radiobroadcastsignalsarepickedupasweakresonancelinesinthespectra.Parasiticlineresonances,detectedasweakresonances,arethehardesttoeliminate.However,theycanbeminimizedbyempir-icallyadjustingthegroundingstrapcon guration.

Withthe4-probemethod,wemayvaryindependentlyeachofthevariablesH(0-8Tesla),ω(100kHzto20MHz),andT.Tocon rmthatweareinthelinear-responseregime,wecheckedthatcurvesofρ (ω)mea-suredwithincidentpowerat-15.5dBmandat4.5dBmarecloselysimilar(exceptforahighernoisecontentinthe-15.5dBmcurveforρ2).Wereportallmea-surementsintermsofthesample’scomplexresistivityρ (ω)=Zs(ω)(bc/a′)≡ρ1(ω)+jρ2(ω),wherea′=0.68mmistheseparationofthevoltagecontacts.Inplaceofρ2(ω),itispreferabletodiscussthe(speci c)induc-tancede nedbyLs≡ρ2/ω(notethatLs=Ls(bc/a′)).Dividingρ2byωisolatesthedivergentbehaviorofLsasω→0,therebyhighlightingthedramaticcollapseofLsatthelatticemeltingtemperature.

III.

SKIN-DEPTHANDVOLUME

INDUCTANCE

Inzeromagnetic eld,theimpedanceZs(ω)ofasuper-conductor(intheformofacylinder)maybecalculatedfrom2the2DHelmholtz2equation( 2+κ 2)A=0,whereκ =λ 2+2jδ (here,λistheLondonpenetrationlengthandδtheskin-depthdeterminedbythequasipar-ticleconductivity).

First,weconsiderthenormal-stateimpedance(T>Tc)bylettingλ→∞,andskin-depthδn=

replacingδbythenormal-state nedtotheskin-depth,andthee ectiveRs(Aω,)Jincreasesiscon-(Aisthecross-sectionarea).Foracircularcross-section,Zs(ω)isgivenby[20]

Zωµs(ω)=

0 δnJ0( x)2j

2j/δn,Jm( x)istheBesselfunctionoforderm,andr0( )istheradius(length)ofthesample.Thecaseofellipticalcross-sectionistreatedinAppendixB.

Equation1showsthat,atlargeω,Rs(ω)→ρ /(2πr0δn),asexpected.Moreimportantforourin-ductancediscussion,theRF eldenergyathighωiscon- nedtotheskin-depth.Therefore,theinductanceLs(ω)decreasesmonotonicallytozeroasω→∞.Asthisin-ductanceexpressestheRFenergystoredinthevolume