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Point Contact Studies of Iron Based Superconductors

Permanent Link: http://ufdc.ufl.edu/UFE0045403/00001

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Title: Point Contact Studies of Iron Based Superconductors
Physical Description: 1 online resource (80 p.)
Language: english
Creator: Timmerwilke, John M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

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Subjects / Keywords: iron-based -- point-contact -- superconductivity
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This work presents spectroscopic measurements on Cobalt doped Barium Iron Arsenide at different dopings.  The technique used is Point-Contact Spectroscopy, PCS.  This technique is well known for being able to characterize the superconducting gap in detail.  The results are compared with an extended BTK model allowing for the extraction of the size, symmetry, and structure of the multiple gaps seen.  A strong change in the structures of the gaps is observed at different dopings.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by John M Timmerwilke.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Biswas, Amlan.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045403:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045403/00001

Material Information

Title: Point Contact Studies of Iron Based Superconductors
Physical Description: 1 online resource (80 p.)
Language: english
Creator: Timmerwilke, John M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: iron-based -- point-contact -- superconductivity
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This work presents spectroscopic measurements on Cobalt doped Barium Iron Arsenide at different dopings.  The technique used is Point-Contact Spectroscopy, PCS.  This technique is well known for being able to characterize the superconducting gap in detail.  The results are compared with an extended BTK model allowing for the extraction of the size, symmetry, and structure of the multiple gaps seen.  A strong change in the structures of the gaps is observed at different dopings.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by John M Timmerwilke.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Biswas, Amlan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045403:00001


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POINTCONTACTSTUDIESOFIRONBASEDSUPERCONDUCTORSByJOHNM.TIMMERWILKEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013JohnM.Timmerwilke 2

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IdedicatethistomyparentswithoutwhomIcouldneverhadreachedthispoint 3

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ACKNOWLEDGMENTS Iwanttothankrstmyadvisorandmentor,AmlanBiswas.Icannotcountthenumberoftimesproblemshavearisenonlytobesolvedwithaquickpieceofadvicefromhim.IalsoappreciateallthetimehevisitedtheLab,evenifasweoftenjoked,nothinggetsdonewhenheisphysicallypresent.ThestudentsandothermembersoftheBiswaslabdeservesomethanksaswell.Ithinkitisapointofimportancethatevenwhenworkingondifferentprojects,discussionbetweenlabmatesovertheprojectssooftenleadstobetterunderstandingofyourown.ToDr.JeenHyoungJeen,DanGrant,andInHaeKwakIrememberourusefulconversationsandIhopemyresponseswereusefulaswell.Ialsowanttothankthevariousundergraduateswhohavejoinedthelab,eitherviaaREUorasstudents.ChelseaMorien,KristenVoigt,TrevorSmith,RileyHowsden,IconMazzaccari,RayaJaved,AllesandraGallastegui,EverettGrimley,MariaViitaniemi,andGalinDragievIwasimpressedwiththediligenceyouappliedtoyourclassesandmakingyourselfusefulinthelab.IknowthatwhenIwasanundergraduateIdidnotapplymyselfassuch.IwanttothankDr.GregStewart,Dr.JunSungKim,GordonTamandBrendanFaethwhoproducedoursamplesused.Thisworkcouldnothavebeendonewithout,andtheyconsistentlysupplied.IwanttothankDr.PeterHirschfeldandYanWang,conversationswiththemprovedenlighteningandprovidedatheoreticalunderstandingtomotivatethiswork.IwanttothankDr.YoonseokLee,forhisadviceandwillingnesstoletusborrowhisbalance.Alsohishelpintheidenticationof`Features'.FinallyIwanttothanktheotherphysicsgraduatestudentformakingmytimeatUFsurprisinglyenjoyabledespitethedifculttaskofobtainingaPhd.IwouldputaspecialemphasisontheFrisbeeandRockclimbinggroupwithoutwhichIthinkIwouldhavelostmysanityintheprocess. 4

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IamcertainthatinthisIhaveforgottensomeoneofimportance,towhichImeantnodisrespect.Thenumberofpeoplewhohaveprovidedguidance,advice,andsupportislargerthanIcouldsimplylistorproperlyremember.IamremindedthatdespiteaPhDbelongingtoasingleperson,itisonlytheresultofmanypeopleworkingtogetherthatsuchathingispossible. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1SUPERCONDUCTIVITYINMATERIALS ..................... 11 1.1BasicSuperconductivity ............................ 11 1.2Fe-BasedSuperconductors .......................... 13 1.2.1Families ................................. 13 1.2.2ChemicalStructure ........................... 14 1.2.3PhaseDiagram ............................. 14 1.3SymmetryandStructureoftheGap/s .................... 15 1.3.1Thermalconductivity .......................... 17 1.3.2Penetrationdepth ............................ 18 1.3.3SpecicHeat .............................. 18 1.3.4NMR ................................... 19 1.3.5NeutronScattering ........................... 19 1.3.6ARPES ................................. 20 1.3.7STM ................................... 20 1.3.8PCS ................................... 21 1.3.9Summary ................................ 21 2POINTCONTACTSPECTROSCOPY ....................... 23 2.1PointContactTheory .............................. 23 2.1.1TransferHamiltonian .......................... 23 2.1.2BTKTheory ............................... 25 2.1.2.1IsotropicS-wave ....................... 30 2.1.2.2D-wave ............................ 31 2.1.2.3AnisotropicS-wave ..................... 31 2.1.2.4MultipleGaps ........................ 34 2.2Point-ContactApparatus ............................ 37 2.2.1C-axisorab-plane ........................... 39 2.2.2DataAcquisitionmethods ....................... 40 2.2.3TemperatureControl .......................... 41 2.2.4NormalizationProcedure ........................ 42 3POINTCONTACTSPECTROSCOPYOFOPTIMALLYDOPEDBaFe2)]TJ /F5 7.97 Tf 6.59 0 Td[(XCoXAs2 44 3.1ExperimentalMethods ............................. 44 3.1.1SamplePreparation .......................... 44 6

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3.1.2PointContactJunctionFabrication .................. 44 3.2Results ..................................... 45 3.2.1Summaryofoptimaldoping ...................... 51 4POINTCONTACTSPECTROSCOPYOFUNDERDOPEDBaFe2)]TJ /F5 7.97 Tf 6.58 0 Td[(XCoXAs2 .. 53 4.1ExperimentalProcedure ............................ 53 4.2Results ..................................... 54 4.3FutureWork ................................... 60 4.4SummaryofResultsforUnderdopedSamples ............... 61 5POINTCONTACTSPECTROSCOPYOFOVERDOPEDBa(Fe1)]TJ /F5 7.97 Tf 6.59 0 Td[(XCoX)2As2 62 5.1ExperimentalProcedure ............................ 62 5.2Results ..................................... 62 5.2.1Summary ................................ 67 6CONCLUSIONS ................................... 68 APPENDIX:NUMERICALINTEGRATIONCODE .................... 70 REFERENCES ....................................... 73 BIOGRAPHICALSKETCH ................................ 80 7

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LISTOFFIGURES Figure page 1-1Ba-122Structure ................................... 15 1-2FeSCsphasediagram ................................ 16 1-3sstructurewithanisotropy ............................. 17 2-1JouleHeatingConductance ............................. 25 2-2DerivativeoftheconductancewithVoltage .................... 26 2-3alphadiagram .................................... 30 2-4s-wavegapconductances .............................. 32 2-5)]TJ /F1 11.955 Tf 10.09 0 Td[(basedbroadening ................................. 33 2-6d-waveconductancecurves ............................. 34 2-7Anisotropicgapconductancecurves ........................ 35 2-8Conductancefortwogaps:ModeledconductancefortwogapswithvarioussymmetriesandZvalues .............................. 36 2-9Multiplegapartifacts ................................. 37 2-10PointContactSchematic ............................... 38 2-11ab-planeprobeschematicandpicture ....................... 40 2-12DataAcquisitionSchematic ............................. 41 3-1R-TplotofoptimallydopedBa(Fe1)]TJ /F5 7.97 Tf 6.59 0 Td[(xCox)2As2 .................. 45 3-2RawandnormalizeddI/dV-Vdata ......................... 46 3-3PCSdataatdifferenttemperatures ......................... 48 3-4TemperaturedependenceofnormalizedPCSdata ................ 49 3-5RawPCSdatafordifferentvaluesofZ ....................... 50 3-6RawPCSdatafordifferentjunctionsshowingreproducibility ........... 51 3-7NormalizeddI/dV-VcurvesfordifferentZvalues ................. 52 4-1R-TforunderdopedBaFe2)]TJ /F5 7.97 Tf 6.59 0 Td[(xCoxAs2crystal .................... 54 4-2RawPCSdataforunderdopedcrystal ....................... 55 4-3RawPCSdataforunderdopedcrystalsshowreproducibility ........... 56 8

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4-4PCSdataforsmallZ ................................. 57 4-5PCSdataforlargeZ ................................. 58 4-6NormalizationofPCSdatafromaunderdopedsample .............. 59 4-7NormalizedPCSdataandtsforaunderdopedsample ............. 60 5-1R-Tfortheoverdopedsample ............................ 63 5-2RawPCSdataforoverdopedcrystal ........................ 64 5-3NormalizedPCSdataandcorrespondingtsforaoverdopedsample ..... 65 5-4ComparisonoftwoLargeZjunctions ........................ 66 5-5NormalizedLargeZcurves ............................. 67 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyPOINTCONTACTSTUDIESOFIRONBASEDSUPERCONDUCTORSByJohnM.TimmerwilkeMay2013Chair:AmlanBiswasMajor:PhysicsThisworkpresentsspectroscopicmeasurementsonthematerial,Ba(Fe1)]TJ /F5 7.97 Tf 6.59 0 Td[(xCox)2As2atdifferentvaluesofx(dopings).Thesamplesweresuperconducting(hadzeroelectricalresistance)forallthedopingvaluesstudied.ThetechniqueusedisPoint-ContactSpectroscopy(PCS).Thistechniqueiswellknownforbeingabletocharacterizethesuperconductinggapindetail.Theresultsarecomparedwithamodelwhichenablestheextractionofthesize,symmetry,andstructureofthemultiplegapsseen.Astrongchangeinthestructuresofthegapsisobservedatdifferentdopings. 10

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CHAPTER1SUPERCONDUCTIVITYINMATERIALS 1.1BasicSuperconductivitySuperconductivitywasdiscoveredbyKamerlinghOnnesin1911whencoolingMercurydowntoliquidheliumtemperatures[ 1 ].Itwasafterwardsobservedinmanydifferentmaterials,thoughalwaysatalowtemperature(23.2KinNb3Gewasthehighesttemperaturefounduntil1986whencupratesuperconductorswerediscovered).Superconductivityisrecognizedbytwomajorcharacteristics,aninniteconductivityandtheexpulsionofmagneticeldsfromthesuperconductor(theMeissnereffect[ 2 ]).AmicroscopicunderstandingofsuperconductivitywasgivenbyBardeen-Cooper-Schrieffer(BCStheory)in1957[ 3 ].IntheBCStheory,thesuperconductivityinamaterialresultsfromthepairingoftwoelectronsinaboundstatecalledaCooperpair[ 4 ].Cooperpairsarecoherentandnonlocalized,whichgreatlylimitsthenumberofscatteringeventswhichcanoccur.Becauseofthissuppressionofscatteringtheelectronsinacooperpaircantravelunimpededthroughthematerial,resultinginsuperconductivity.IntheBCSmodelthispairingisdrivenbythecouplingoftheelectronstophononsassociatedwiththecrystallattice.Thesimpliedexplanationisthatastherstelectronpassesnearapointonthelattice,thechargeoftheelectrondistortsthepositivechargesofthelattice.Thisdistortionpersistsaftertheelectronhastraveledaway,sincethelatticedoesnotinstantlyrelax,andthepositivechargeofthedistortionattractsanotherelectrontothedistortion.Suchanattractioncouplestheelectrons,andasshownin[ 4 ]ifthereisapositiveattraction(negativeinteractionenergy)ofanymagnitude,formationofCooperpairsisguaranteedasitisthelowestenergygroundstate.Thisisnottheonlymethodwhichcanresultinpairing.ItiswellknownthatthecupratesuperconductorscannotbeexplainedusingtheBCSphononcoupling[ 5 ].UnlikeBCSsuperconductors,amicroscopictheoryofcupratesuperconductivityaswellas 11

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otherformsofunconventionalsuperconductivityhasnotbeenfoundandformsoneofthemajorchallengesofmoderncondensedmatterphysics.Tostudyothermechanismsforsuperconductivityandpairingweneedtoidentifymeasurablequantitieswhichwillbecomparablebetweenthedifferentmechanismsaswellasexhibitdenitivefeaturesassociatedwiththedifferentmechanisms.Itisexpected,thoughnotrequired,thatallformsofsuperconductivityresultfromtheformationofCooperpairs.WhenCooperpairsformtheyresultinagapinthedensityofstatesaroundtheFermisurfaceequaltotheenergyrequiredtobreakthepairing.Thisenergygapcanbeobservedinspectroscopicmeasurements.Thegapcanhaveamomentumdependenceassociatedwithit.ThroughBCStheory,theenergygapcanbetiedtothecriticaltemperatureandmostothersalientqualities,andprovidesanimportantenergyscaleforpairbreakingeventsandwhensuperconductivityissuppressed.Finally,ifagivencouplingmechanismandcrystalstructureisassumed,acorrespondinggapstructureandsymmetrycanbecalculated.Theenergygapisthereforeaprimequantitytomeasureandcharacterizeinthestudyofallsuperconductors.Themomentumdependenceforthegapofasuperconductorcanhavetwocauses,thesymmetryofthegapdenedbythecouplingmechanismandthestructureofthegapresultingfromelectronicstructureofthecrystal.ForBCStypecoupling,s-wavegapsymmetriesarerequired.Thesesymmetriesaremostlyisotropic,wherethegapremainsunchangedunderarotationofthecrystalaxisby90.Thed-wavegapsymmetry,commonlyseenincupratesandotherstronglycoupledsuperconductors,isgenerallyanisotropicandrequiresthata90rotationofthecrystalaxisisoutofphasebyradians.Therecanalsobep-wave,f-wave,andg-wavesymmetriesbutthes-waveandd-wavearethemostcommonincurrentlyknownsuperconductors[ 6 ].BeyondthissymmetrytherecanbemultiplesuperconductinggapswhentherearemultipleFermisheets,leadingtoanincreaseinthecriticaltemperatureofthesuperconductor,suchasinMgB2[ 7 ].Tunnelingmeasurementsareadirect 12

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spectroscopictechniquewhichcanbeusedtomeasuretheenergygapdirectly.Informationaboutthesizeandmomentumdependenceofthegapcanbeextractedfromtunnelingmeasurements[ 8 ].Wefocusedonpoint-contacttunnelingjunctionsthedetailsofwhicharediscussedlaterinchapter2. 1.2Fe-BasedSuperconductorsThehightemperatureiron-basedsuperconductors(FeSCs)werediscoveredin2008byKamiharaetal.[ 9 ]inFluorinedopedLaFeAsO,withaTcof26K.Sincethisdiscoverythenumberofcompoundsbelongingtothisgrouphasincreaseddramatically,withmultiplefamiliesdiscovered.Signicantworkhasbeenputintostudyingandexplainingthebehaviorseeninthesecompounds,butmeasurementshaveoftenresultedinconictingorambiguousresultswhencompared.ThesepreviousresultswillbediscussedlaterafteranintroductiontothebasicdetailsoftheFeSCs.FullreviewsoftheFeSCscanbefoundbyvariousgroups[ 10 13 ] 1.2.1FamiliesCurrentlythereare6recognizedfamiliesofcompoundsintheFeSCsbasedonchemicalstructure;the1111s,the122s,the111s,the11s,the21311s,andthe122*s.The1111familyarethosebasedontheLaFeAsOparentcompoundandthoughLacanbereplacedwithanotherrareearth,andFewithadifferenttransitionmetal,dopingisprimarilyontheoxygensite.The122familyhasaparentstructureMFe2As2,whereMisanalkalioralkalineearthmetal.Dopingmechanismsinthe122familycanbequiteelaborateasdopingateachsitehasshownsuperconductingbehaviorgivingrisetoseveralsetsofphasediagrams.The111familyislessextensivewiththeprimecandidateLiFeAs.The11familyisknownastheironchalcogenides,asunliketheotherfamiliesitdoesnotcontainanelementfromthepnictogengroupanexampleofaparentcompoundbeingFeSe.The21311familyisnewerandhasamorecomplicatedstructure,withfewcompoundshavingbeendiscoveredsofar.The122*familyisverysimilartothe122family,exceptitcontainsmissingatomsforexampleK0.8Fe1.6Se2and 13

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replacesthepnictogenatomwithachalcogenide.Thersttwofamiliesrequiredopingtoshowsuperconductivity,whilethelastfourdonot.ThereisanexceptioninKFe2As2,whichdoesshowsuperconductivitybuthascharacteristicsverydifferentthanthedoped122compounds.The122familycanbeproducedashighqualitysinglecrystals,makingitaprimecandidateforuseinexperiments.Italsocanbedopedinmanydifferentwaysviz.,holedopingontherareearthsite,electrondopingontheFesite,andisovalentdopingonthepnictogensite.ThislargesetofparameterswhichcanbechangedandstillinducesuperconductivitymakeitaprimematerialforstudytounderstandtheFeSCs.Ourworkconsistedprimarilyofmeasurementsonthe122family,specicallytheelectrondopedBa(Fe1)]TJ /F5 7.97 Tf 6.59 0 Td[(xCox)2As2andsofurtherdiscussionwillfocusonthismaterial. 1.2.2ChemicalStructureThebasicstructureofalltheFeSCsconsistoflayersofironandeitherapnictideorchalcogenide.ThisstructurecarriessomesimilaritytotheatCuOlayersseenincuprates,thoughthepnictidesaregenerallymore3Dthanthecuprates.TheFeAslayersareseparatedinthe1111,122,111,and122*familiesbyspacerlayers.ForBaFe2As2theFeAslayersformacagearoundtheBaatom,asshowninFigure1-1.Thematerialsoftengothroughatetragonal(Fmmm)toorthorhombic(I4/mmm)transitionastemperatureislowered,butthistransitionissuppressedwithincreasingdoping. 1.2.3PhaseDiagramTheundopedphasediagramoftheiron-basedSCsshowsantiferromagnetismandwithincreasingdopingtheAFMweakensandsuperconductivityarises.Athigherdopingsthesuperconductivityissuppressedonceagain,resultinginasuperconductingdome[ 14 ].Hence,thephasediagramcontainsasimilaritytothecuprates.Theantiferromagneticphaseincertainfamilies(primarilythe122family)showssignsofcoexistenceandcompetitionwiththesuperconductingphaseinunderdopedsamples[ 15 ].Thestructuraltransitioncoincideswiththemagnetictransitionintheparentcompound,butthetwotransitionsoftenseparateasdoped(thoughthisseparationisnot 14

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Figure1-1. TheBaFe2As2structure:LightgreenisArsenic,blueisIron,andyellowisBarium.Notethe2dlayersofFe-AswhichsurroundtheBaatoms trueforalltheFeSCs).ForCobaltdopedBaFe2As2thesuperconductingdomerangesfromdopinglevels2.5%to18%,withapeakTcof26Kat8%dopant[ 14 ].Between2.5%andabout6%dopingtherearesignsofacoexistenceofthesuperconductivityandtheAFMphase[ 15 ].Figure1-2showsthephasediagramforBa(Fe1)]TJ /F5 7.97 Tf 6.59 0 Td[(xCox)2As2whichshowsthediscussedfeatures. 1.3SymmetryandStructureoftheGap/sThereremainssomequestionaboutwhattheexactsymmetryisfortheFeSCs.Theredoesseemtobeclearevidencethatthesymmetryiss-wavewithmultiplegaps.Atleastanelectronandaholepocketareexpectedtobefound,leadingtotwoprimarymodels.TherstisthesmodelresultingfromspinuctuationsproposedbyMazinetal.[ 16 ].Thesecondisorbitaluctuationdrivens++model[ 17 ].Inbothofthesemodelstheanisotropyinthegapstructurecanvaryandeachmodelcanbecontinuously 15

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Figure1-2. PhasediagramofBa(Fe1)]TJ /F5 7.97 Tf 6.59 0 Td[(xCox)2As2:theredlinerepresentsthestructuraltransition,thebluelineistheAFMtransitionandthegreenlinethesuperconductingtransition.ForlowdopingsthereisaregionofoverlapbetweenthesuperconductingandAFMregions.ThemaximumTcof25Koccursnearx=0.07. transformedtotheother(gapstructureisthedirectk-dependenceofthegapsize).Ifthestructureshowslargevariationwithknodesinthes-wavecasecanbeinducedorremoved.Thisbehaviorisunlikethed-wavegapwhichrequiresnodesduetothesymmetry.Henceitcanbeextremelydifculttoconrmeitherthesors++symmetry,thoughextensivecharacterizationwouldshowthedifferences[ 18 ].Densityfunctionaltheory(DFT)calculationspredictstrongchangesintheFermisurfacewithdopingfortheFeSCs.Whichcancauseincreasinganisotropyofthegapstructureandcanevencausetheformationofaccidentalnodes(nodeswhicharenotrequiredbythesymmetryofthegap)[ 19 ].ThisbehaviormeansthatnodescanformorbesuppressedwithdopingastheFermisurfacechanges.Schematicallywecanshowthevariationfromanisotropicstonodalsmodelwithasignchange,asshownin 16

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Figure1-3. sstructurewithanisotropy:Threeschematicsofthemomentumdependenceofthegapwithssymmetry.Signchangesareshownbydifferentcolors(redandblue)Thefarleftshowstheisotropicsstructurewithaholeandelectronpocketwithaphasechange.Themiddleimageshowsthecasewhennodeshavejustformedontheelectronpocket.TherightimageshowsthedevelopmentofasignchangeonasingleFermisheetwithfurtheranisotropy. Figure1-3.Notethattheholepocketatthe)]TJ /F1 11.955 Tf 10.1 0 Td[(pointismostlyisotropicbuttheelectronpocketbecomesmoreanisotropicastheFermisurfaceisvaried.AnexampleofthisbehaviorispredictedinBa(Fe1)]TJ /F5 7.97 Tf 6.59 0 Td[(xCox)2As2whereMaitietal.[ 20 ]predictanisotropicgapatoptimaldopings,butallowforaccidentalnodestoformatdopingsawayfromoptimal.Experimentallythequestionofwhethertherearenodesorhighlevelsofanisotropyinthegapscanallbeaddressedtosomeextentbypoint-contactspectroscopy,andwewillfocusonthismatteringreaterdetailasfollowsinthefollowingsectionsandchapters.Firstwewilldiscussvariousothermeasurementsperformedbydifferentresearchgroupswhichrelatetothemeasurementofthegapsymmetryandsize. 1.3.1ThermalconductivityTanataretal.[ 21 ]andReidetal.[ 22 ]showedfromheattransportmeasurementsthatalineartermremainsinthethermalconductivityatdopingsawayfromtheoptimaldoping.InBa-122,optimallydopedsamplesshowamuchweakerlinearterm.Thistermimpliestheexistenceofanisotropyinthegapstructurewithdoping.FortheoverdopedcasethistermwouldbeexplainedbychangesintheelectronFermisheet.FortheunderdopedcasetheexplanationislessclearbecauseofthecoexistenceoftheSC 17

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phaseandtheAFMphase.Thethermalconductivityprovidesgoodevidenceofthechangeingapstructurewithdoping.AlthoughTanataretal.doesnotndevidencefornodesINunderdopedcompounds,Reidetal.haveobservednodalbehavior. 1.3.2PenetrationdepthPenetrationdepth()measurementscanshowthepresenceofnodesinthegapstructure,bylookingatthetemperaturedependenceof.Ifthesystemisfullygappedthenexponentialdependencewithtemperatureisexpectedwhile,ifapowerlawtisseenitisasignthatthegaphasnodes.Thebehavioratlowtemperaturescanalsotellthedifferencebetweenasingleandmultiplegapsbylookingforevidenceoftwodifferentsuperuiddensities.MagneticforcemicroscopywasusedtomeasurethepenetrationdepthandthussuperuiddensitybyLuanetal.forunderdopedBaFe1.9C00.1As2[ 23 ].Thedatatswelltoatwogapmodelwithfullygappedbands.SimilarresultswerefoundbyWilliamsetal.[ 24 ]usingmuonspinresonance,wherethebesttresultedfromtwofullgapsforBaFe1.852Co0.148As2.Gordonetal.havealsotakensuchmeasurementsandobservedaT2dependencewhichwouldnormallyimplyanodalgap,butcouldbeexplainedbymultipleisotropicgapsoraveryspecicratioofinter-bandtointra-bandscattering.[ 25 ].Latermeasurementsbythesamegrouphaveshowntheexponentincreasingtowardsthree[ 26 27 ].Thismakestheevidencefornodeslesslikelytobeobserved,buttheycouldbeliftedifsignicantscatteringisoccurring.ThemeasurementbyLuanetal.seemstoagreewithTanataretal.whichdoesnotndnodesintheunderdopedsample. 1.3.3SpecicHeatGofryketal.measuredthespecicheatofunder,optimal,andoverdopedBaFe2)]TJ /F5 7.97 Tf 6.58 0 Td[(xCoxAs2(x=0.09,0.16,0.206,0.21)samplesandtheynotedthatastrongresidualspecicheatcoefcientremainsatlowtemperatures[ 28 ].ThiseffecthadalsobeenseenbyGangetal.[ 29 ].Thistermissmallestfortheoptimallydopedsamplesandgrowslargerawayfromoptimaldoping.Ingeneralitisrecognizedthataresidual 18

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termisasignofpairbreakingeffects,eitherfromunconventionalsuperconductivityorfromdisorderanddefectsinthesamples.Thecaseofanitelowtemperatureresidualterminatwolevelglassysystemiswellknownanditresultsinaniteentropyatzerotemperature.ItispossiblethatsimilareffectscouldoccurintheFeSCssystems.TheviewtakenbyGofryketal.isthatthelineartermresultsfromnanoscaleelectronicinhomogeneities.Theyalsottedthefullspecicheattoatwogapmodel(havingpreviouslyshownthatasinglegapmodelisunabletosatisfytheoptimaldopedresults[ 30 ]),ndingthatthesmallergapsizedoesnotchangebymorethan10%betweendopings,whilethelargergapchangesby30%forunder-dopedand15%forover-dopedrelativetooptimaldopings.Kimetal.[ 31 ]measuredspecicheatoverseveraldopingsaswell(x=0.09,0.206,0.26,0.30)atlowtemperatureswitheldsuptoHc2.TheelddependenceofthespecicheatapproximatelyfollowsH0.5whichisasignofnodalsuperconductor.Specicheatisstronglyaffectedbydisorderinthesystem,especiallyinthecaseofnodalsuperconductivity[ 32 ]. 1.3.4NMRTheKnightshiftmeasuredforBa(Fe1)]TJ /F5 7.97 Tf 6.59 0 Td[(xCox)2As2decreasesforalldirections,whichisasignthatthesymmetryisspinsinglet[ 33 ].Forthisreasonourchoicesofsymmetryarelimitedtoeithers-waveord-wave. 1.3.5NeutronScatteringNeutronScatteringiscapableofmeasuringasignchangebetweentwodifferentFermisheetsusingaspinresonancewhichshouldbeexpectedbelow2inasinglebandsystem.Aspinresonancewouldbeagoodindicatorthatthecouplingiseitherd-waveors.InBaFe1.85Co0.15As2,measurementsoftheSCstateshowapeakaround10meV[ 34 ].Theexpectationfortheextendedspeakisthatsucharesonanceoccursonlynearthenodesandsotheeffectwouldbesmallcomparedtotheisotropicsmodel.Theobservationofsuchaclearpeakcanbeconsideredaclearindicationofthesmodel. 19

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1.3.6ARPESTwogapsofsizes3.1meVand7meVwereobservedinangleresolvedphotoemissionspectroscopy(ARPES)databyvanHeumenetal.[ 35 ].ThereisalsoworkbyTerashimaetal.[ 36 ]wherethefoundasinglegapof7mevinoptimallydopedsampleswhichisinducedbythenestingoftheFermisurfaces.Thegapsinthesemeasurementsshowlittleornosignofanisotropy.SinceARPESisadirectmeasurementofthegaps,thelackofanisotropyisproblematic.StilltherearereasonsthatARPESmightproveincorrectcomparedtobulkprobes.FirstARPESisprimarilyasurfaceprobe,sothatifthesurfacehasadifferentgapstructurefromthebulkthiswouldleadtoaninconsistencybetweenthetwodifferentmeasurements.InDFTcalculationsbyKemperetal.[ 19 ],anextraxybandisseenatthesurfacewhichleadstoamoreisotropicgapstructure.Intrabandscatteringnearthesurface(duetothesurfaceitself)couldalsocausethegaptobecomemoreisotropic.ThereisalsoapossibilitythattheresolutionofARPESmightnotbehighenoughtoresolvetwogaps,insteadleadingtoasinglebroadenedpeakwithanisotropiccharacter. 1.3.7STMVariousgroupshaveusedSTMtoprobethedensityofstatesofthesematerials,oftenseeingonlyasinglegap.Yinetal.[ 37 ]performedspectroscopicmeasurementsonBaFe1.8Co0.2As2.Theyonlyfoundevidenceofasinglegapofroughly6meV.Despitethereporteddoping,theTcwasmoreinlinewithanoptimallydopedratherthananoverdopedsample.Nishizakietal.[ 38 ]performedsimilarmeasurementsonBaFe1.86Co0.14As2ndingasinglegapof7.6meV.Masseeetal.[ 39 ]performedmeasurementsonthreedopingsBaFe1.92Co0.08As2,BaFe1.86Co0.14As2,andBaFe1.79Co0.21As2ndingsinglegapsof(4meV,5-7meV,6meV).IneachcaseaBCSlikegapcouldbeusedexcepttheratio2=KbTcislargecomparedtotheBCSvalue.ThereisalsoapaperbyZhangetal.[ 40 ]whichndsamuchlargergapstructureof30meVovernominaldopingsfromx=0-0.32.Teagueetal.[ 41 ]isoneofthefewgroupsto 20

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ndevidenceformutliplegapsusingSTMlookingatbothunderdoped(x=0.06)andoverdoped(x=0.12)samples.Theyfoundgapsof8meVand4meVfortheunderdopedand10and5meVfortheoverdopedsample.STMlikeARPESisasurfaceprobeandsoisverysensitivetosurfaceeffects.STMwouldnotjusthavemanyofthesameproblemsasARPESfromthesurface(suchassurfacereconstruction),buthassignicantissueswithsurfaceroughnessrequiringsmoothatomicallyatsurfacesformeasurement. 1.3.8PCSPointcontactmeasurementstakenbyTortelloetal.[ 42 ]showedclearevidencefortwogapofabout4.5meVand10meVintheab-plane.Thesegaps'shapesexcludead-wavesymmetry(BaFe1.80Co0.2As2)andtheyappearnodeless.TheTcreportedinthepaperis25K,whichisnotinlinewiththedopingreportedbutmoreinlinewithanoptimallydopedcrystal.PointcontactmeasurementshavealsobeentakenbySamuelyetal.[ 43 ]whoobserveasinglegapinBaFe1.80Co0.2As2of5meVandLuetal.[ 44 ]whoobtainedasharpconductancepeak,butnogapinBaFe1.80Co0.2As2.AgaintheTcforthesesamplesisabove20Kandmoreinlinewithanoptimallydopedsampleratherthanoverdoped.PCScanalsohaveissueswiththesurface,butislesspronetoerrorsduetoroughnessseeninSTMmeasurementsandismorelikelytoshowbulkeffectsasthetipcanpenetratethesurfacelayerslightly.PCSalsohastwosignicantadvantagesoverSTMmeasurements,viz.thejunctiontransparencycanbevariedanda-bplanemeasurementscanbetakenonthinsampleswithrelativeease. 1.3.9SummaryTheresultfrommultiplestudiesimplythatthesymmetryofBa(Fe1)]TJ /F5 7.97 Tf 6.59 0 Td[(xCox)2As2ismostlikelys-wave,either(s++ors).AtandnearoptimaldopingstherearemultiplenearlyisotropicgapsonmultipleFermisheets.Asonegoesawayfromoptimaldopingthegapsbecomemorenodal.ThereissomeconictingevidencefromSTMaboutthenumberofgapsandthesizeofthegapswhichvaryfrom10meVto3meV.Fromthe 21

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givendopingforthecrystalsexaminedviaSTMandPCSweshouldbeabletotellthedopingdependenceofthegapsymmetry,buttheTcseenfortheoverdopedsamplesareabove20K,whichimpliesfromthephasediagramthatthecrystalsusedweremuchclosertobeingoptimallydoped.Inadditionspecicheatmeasurementsbeingabulkprobeandcapableofgoingtomuchlowertemperaturesprovidesomeofthemostaccuratemeasurementsofsubgapstates,thoughitdoeshavedifcultyinidentifyingthesourceofsuchstates. 22

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CHAPTER2POINTCONTACTSPECTROSCOPY 2.1PointContactTheoryTunnelingSpectroscopyreliesonusingjunctionsbetweentwomaterialstoprobethedensityofstates(DOS)ofoneofthem.Probingthesuperconductordensityofstatescanbedonebycreatingajunctionbetweenanormalmetalandasuperconductingmaterial.Therearetwoprimarymodelsusedtoevaluatethemeasurementsacrosssuchjunctions. 2.1.1TransferHamiltonianTherstmethodistotreatthesystemexactlythesameastwonormalmetaljunctionswithalargebarrierseparatingthem.InthiscaseifavoltageVisappliedacrossthejunctionitshiftstheDOSofoneelectrodetobehigherthantheother.Ifthereareemptystatesoflowerenergyintheotherelectrodethenacurrentwillshowbetweenthetwoelectrodes.ThecurrentcanbecalculatedfromaTransferHamiltonian.UsingFermi'sGoldenRuleweexpressthecurrentasinEquation2-1.I(V)=Cjh ljHtj rij2Z1nl()nr(+eV)(fl())]TJ /F7 11.955 Tf 11.96 0 Td[(fr(+eV))d (2)Cisaconstant, listhewavefunctionintheleftelectrode, risintheright,Htisamatrixwhichcontrolsthetransferofanelectronfromtherighttoleftelectrodes.n()isthedensityofstatesoftheelectrodes,whilef()istheFermi-Diracdistribution.Thecurrentthendependsstronglyonthematrixterms,whichcontaintheunderlyingphysicsofthejunction.FormostmetalswhenvoltageandtemperaturearesmallthematrixtermsfromtheHamiltoniancanbeapproximatedasaconstant.InthislimititbecomesclearthatthecurrentisdirectlyproportionaltoaconvolutionofthedensityofstatesofthetwoelectrodeswiththeFermifunctions.Ifoneoftheelectrodeswerereplacedwithasuperconductor,thecurrentwouldshowclearevidenceofthesuperconductinggap.Moreimportantly,ifwendtheconductancedI/dVratherthanthecurrentwecansee 23

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thespectroscopicgapdirectly.NotethatinthistreatmentweassumethattheenergygainedbytheelectrontransferacrossthejunctionisexactlyeV.Thisassumptionisonlytrueforballisticjunctionswhereonlyelasticscatteringcanoccur.Thegeneralruleisthatforballistictransmissionthejunctionsizemustbesignicantlylessthanthemeanfreepathoftheelectron.Whichalsomeansthatthemomentaarenotscatteredinthejunctionitself.Ifthemeanfreepathiscomparabletothejunctionsize,measurementscanstillbetakenandthedatawillcontaininformationaboutthedensityofstatesbutthefeatureswillbesmearedovermomentumdirectionduetotheelasticscattering,andslightlyoverenergysinceonlyparallelmomentumneedstobeconservedattheinterface.Ifthejunctionissuchthatinelasticscatteringcanoccur(muchlargerthanthemeanfreepath)thennospectroscopicinformationcanbeobtainedfromthejunction.Theinelasticscatteringremovesanyrelationshipbetweenthecurrentmeasuredandthevoltageapplied.AgeneralruleisthatiftheresistanceofthejunctionisneartheSharvinresistancethenthecontactwillgivespectroscopicinformation(thoughitrequiresestimatingthesizeofthejunctionviaanothermethod).IfthejunctionistoolargethereisalsoastrongriskofJouleheatingoccurring.AdiscussionofthismattercanbefoundbyBaltzetal.[ 45 ].WhenJouleheatingoccursoneoftheclearsignsisaslopeofoppositesigntotheappliedvoltageawayfromanystructuresassociatedwithsuperconductivity,positivefornegativevoltagesandnegativeslopeforpositivevoltages.SeeFigure2-1foraclearconductancecurveshowingJouleheating.Thedownturnshownresultsfromtheheatingofthenormalmetalatthejunctionandtheincreaseinresistivitywithincreasingtemperature.Inpracticethetunnelingbehaviorisobservedinnormal-superconductorjunctions[ 46 ],howeverincertaincasesratherthanasuppressionofcurrentinsidetheenergyofthegapanenhancementisseeninstead.ThiswasrecognizedasresultingfromAndreevReection[ 47 ],wherethenormalcurrentinthemetalisconvertedtoasupercurrent.ToexplainthisbehaviorBlonder,Tinkham,andKlapwyjkdevisedatheory 24

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Figure2-1. JouleHeatingConductance:AconductancecurveinthepresenceofJouleheating.Whileitispossibletoseesomeofthefeaturesofthesuperconductinggaptheprimaryeffectoftheconductanceisthesharpdownturninconductancewithincreasingvoltage.Thisisaclearsignoftheheatingofthesamplewithincreasingvoltage. fortheinteractionsatthejunction(knownastheBTKtheory)[ 48 ].Itcanalsobeshownbythismethodthatthephononstructurecanbeprobedusingtunnelingspectroscopy.InthesecondderivativeofcurrentwithvoltageoneexpectstoseetheEliashbergfunction2F[ 49 ],whichcontainstheinformationaboutthephononsinthesuperconductor.AnexperimentalmeasurementofdI2/dV2isshowninFigure2-2. 2.1.2BTKTheoryIntheBTKmodeltheinterfaceistreatedasaniteDiracdeltapotentialbarrierU(x)=H(x)(ratherthanuseHasavariable,wewillusethemoregeneraldimensionlessconstantZdenedasH/~vf)whichweareapproachingformthenormalmetalside,andthereforewecanlimitourselvestothecaseofanapproachingelectron,thoughitisunderstoodthatasymmetricalsetofsimilarprocesseswouldoccurwerethecurrent 25

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Figure2-2. DerivativeoftheconductancewithVoltage:DirectmeasurementofdI2/dV2canshowfeaturesoftheEliashbergfunction,whichisrelatedtothephononspectralfunction.Thestrongestfeaturesaremarkedwitharrows. beingcarriedbytheholes.Theelectron,treatedasaplanewaveatthejunctioncanthenundergotransmissionorreectionattheinterface.BecausethewavefunctionofthesuperconductormustbeasolutiontotheBogulibov-deGennesequationtheinterfaceinteractionscanamixtheelectronintobothelectron-likeandhole-likequasiparticles.Becauseofthismixing,ratherthansimpletransmissionandreectionoftheelectronatthesurface,twoextraprocessescanoccur.Wecanhavetransmissionoftheelectronasahole-likequasiparticleandreectionoftheelectronasahole(AndreevReection).Theprobabilityofeachprocessoccurringwithanelectroncannowbecalculated 26

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bythesamemethodusedtocalculatescatteringofaparticleataninterface.Suchprobabilitieswilldependonlyontheenergyoftheelectron,thevalueofZ,andthesizeofthesuperconductinggap/s.Nowwereturntocalculatethetunnelingconductancefromtheprobabilitiesofeachprocessoccurringasthetotalprobabilityoftransmissionmultipliedbyaconstant.Wecanwriteatotaltransmissionasoneminustheprobabilityofallreections.WehavetotakecarethatsinceAndreevreectionchangesthesignofthechargecarrier,itenhancestheconductanceinsteadofsuppressingit.Assuchwecanrepresentthisconductanceas1+A()-B()whereA()istheprobabilityofAndreevreectionoccurringandB()istheprobabilityofnormalreectionoccurring.ThetotalcurrentacrossthejunctioncanthenbeexpressedasinEquation2-2.I(V)=CZ1(1+A())]TJ /F7 11.955 Tf 11.95 0 Td[(B())(fl())]TJ /F7 11.955 Tf 11.95 0 Td[(fr(+eV))d=Z1()(fl())]TJ /F7 11.955 Tf 11.96 0 Td[(fr(+eV))d (2)ThecomparisoncanthenbemadetothetransferHamiltonianmethod.InthetransferHamiltonianmethodweassumealargebarrierattheinterface,whichintheBTKmodelwecanmimicthisbylookingatlargeZ.Inthiscasetheconductance(dI/dV)oftheBTKmodelshowstheproperspectroscopicgap[ 48 ].TheeaseofcalculationcomparedtothetransferHamiltonianmethodformoretransparentbarriersmeansthatforpointcontactstheBTKmodelhasbecomethepreferredmethod.Theinterfaceofthejunctionisassumedtobesmalland`clean',suchthattheinteractionsarepurelyelastic.Inanactualjunctionthisisnotgoingtobethecase,asalargenumberofeffectscancausethejunctiontohavesomeamountofinelasticbehavior(surfaceroughness,impurities,orathinlayerofdifferentmaterialbetweentheN/Selectrodes).Theseinelasticscatteringsresultinnotonlyloweringthesizeoffeaturesobserved,butsincetheenergyofatunnelingparticleisnolongerexactlyeV,abroadeningoffeaturesaswell.Dynesetal.[ 50 ]includedamethodtoaccountforbroadeningduetonitelifetimesofthequasiparticlesbyreplacingtheenergywithacomplexenergy 27

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(E=E+i\inourequations.Althoughformostsuperconductorsthislifetimeenergy)]TJ /F1 11.955 Tf 10.1 0 Td[(willbesmall,itdoesgivetheproperbehaviorforinelasticscatteringeventsattheinterface.Becauseofthissimilarityinhowthelifetimeandthescatteringeventseffecttheconductanceithasbecomestandardtoinclude)]TJ /F1 11.955 Tf 10.1 0 Td[(asageneralscatteringterm.Whilenotexplicitlyshown,itisassumedthat)]TJ /F1 11.955 Tf 10.09 0 Td[(isincludedinlaterequationsforthecurrent.SofarwithinthebasicBTKmodelweassumedthatthegapwasisotropicandthatourjunctioncouldbetreatedasonedimensional.FortheFeSCsandotherhightemperaturesuperconductorstheseassumptionshavetoberelaxedasthegapstructureismorecomplicatedandourjunctionsdonotjustprobeasinglemomentum.Beforediscussingthedifferentgapstructuresitishelpfultodiscussthevariationinmomentumwhichapointcontactmeasures,sincethiswillhaveadirecteffectonhowaccuratelywecanmeasurevariationsinthegapstructure.Point-contactjunctionsprobeaconeofmomentumratherthanprobeasinglek.Tocalculatetheshapeofthecone,wecanstartbylookingatasimplerjunctionofnormalmetal-insulator-normalmetal.Thetransparencyofthejunctionfora1-DjunctionisdenedintermsofZinEquation2-3.1 1+Z2 (2)Wecanuseasimilartreatmentforplanewavesconnedona2-Dplanewithadiracdeltabarrierinterfaceatx=0,assumingthatthecomponentoftheFermimomentumparalleltotheinterfaceofthetwometalsarethesamethetransparencyasitvarieswithangleisinEquation2-4.Cos[]2 Cos[]2+Z2 (2)whereisdenedastheanglefromtheperpendicularoftheinterface.Thecurrentacrossanormal-normalinterfacecanthenbecalculated.I(V)=CZ=2)]TJ /F13 7.97 Tf 6.59 0 Td[(=2Z1Cos[]2 Cos[]2+Z2(fl())]TJ /F7 11.955 Tf 11.95 0 Td[(fr(+eV))Cos[]dd (2) 28

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Asimilartreatmentcangivetheconeina3-Dsystem,butthisisunnecessarytoincludeformostgapstructures.Withtheconeofmomentumwhichwillbeprobedestablished(andtheweightofeachmomentum)wecanextendthistoallowforgapswhicharenotisotropic.Insuchgapswewilldenethemomentumstructureintermsof,givingusanequationforthecurrent:I(V)=Z=2)]TJ /F13 7.97 Tf 6.59 0 Td[(=2Z1Cos[]2 Cos[]2+Z2(,)(fl())]TJ /F7 11.955 Tf 11.95 0 Td[(fr(+eV))Cos[]dd (2)Calculatingtheangulardependenceof(E,)wasdoneforgeneralanisotropicsuperconductorsbyKashiwayaetal.[ 51 ],bysolvingtheBogliubov-deGennesequationsforA(,)andB(,).Sincethegapofananisotropicsuperconductorcanchangesignforcertainmomentaourequationsbecomemuchmorecomplicated.Notonlymustweproperlyallowforanisotropyandnodesinthesuperconductinggap,wecanmeasureAndreevboundstateswhenthereisasignchangeonthegap.Theboundstatesresultwhenthequasiparticlesinthesuperconductorarereectedfromtheinterfaceandseeachangeinthesignofthegap.ThisgenerateszeroenergystatesattheFermisurface,resultinginazerobiasconductancepeak(ZBCP)[ 52 ].Thiscanbetakenintoaccountbydeningeachgapastwo(+and)]TJ /F1 11.955 Tf 7.08 1.8 Td[(),rotatedby+2betweenthem,whereistheanglebetweenthesymmetryaxisandtheinterfacenormal.Forad-wavegap,isshownintheFigure2-3.ToseetheseeffectsrefertoFigure2-6whichshowsa3meVgapatdifferentZvaluesandtwodifferentalpha(=4and0).Tosimplifywritingtheconductanceinthisformwedeneafewquantities.()=Cos[]2 Cos[]2+Z2 (2)(,)=)]TJ /F11 11.955 Tf 11.96 10.39 Td[(p 2)]TJ /F6 11.955 Tf 11.95 0 Td[(()2 () (2)=Sign(+()))]TJ /F7 11.955 Tf 11.96 0 Td[(Sign()]TJ /F6 11.955 Tf 7.09 1.79 Td[(()) (2) 29

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Figure2-3. alphadiagram:isdenedastheanglebetweenthesymmetryaxisandtheinterfacenormal.Ifis0thecurrentisinjecteddirectlyintoasinglelobe,whileatequalto=4itisinjectedintothenode.Theredandbluecolorsindicateachangeinsighofradians. containsthenormalweightofmomentawhichareprobedbythejunction.isrelatedtothedensityofstatesofthequasiparticlesandCooperpairs.isthephasedifferencebetweenthetwogaps(+and)]TJ /F1 11.955 Tf 7.09 1.79 Td[().willeitherbe0or,ifitiszerothennoboundstatesarebeingmeasured,ifitisthenboundstatesarebeingmeasuredandaZBCPshouldbeobserved.FollowingfromKashiwayaetal.[ 51 ]thenalcurrentforageneralgapstructure()iswritten.I(V)=Z=2)]TJ /F13 7.97 Tf 6.59 0 Td[(=2Z1(fl())]TJ /F7 11.955 Tf 11.95 0 Td[(fr(+eV)()Cos[]1+()+(,)2+(())]TJ /F3 7.97 Tf 6.58 0 Td[(1)+(,))]TJ /F3 7.97 Tf 6.25 1.07 Td[((,)2 1+(())]TJ /F3 7.97 Tf 6.59 0 Td[(1)+(,))]TJ /F3 7.97 Tf 6.25 1.07 Td[((,)ei2dd (2)Theresultscannowbemodeledbynumericcalculationsfordifferentsymmetries.SeeAppendixAforMathematicacodewhichcalculatestheconductance. 2.1.2.1IsotropicS-waveTheisotropics-wavegapisthegapwhichwouldbepredictedforapureBCSsuperconductor.Itremovesanyangulardependenceinduetochanginggapsizewithmomentum,leavingonlythedirectenergydependence.Theresultsareverysimilar 30

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tothosefortheoriginalBTKmodel,onlyslightlydecreasedduetotheeffectoftheintegrationovervariousmomenta.Resultsforagapof3meVareshownintheFigure2-4.TheZincreases(0,0.5,1,5)intheFigure2-4,andallcurvesareshownat4.2K.ThechangefromtheAndreevreectiondominatedbehaviortotunnelingbehaviorcanbeseendirectlyinthemodel.NotethatinthefullyAndreevreectionstate(Z=0)theconductancedoublesinsidethegapcomparedtothemostlytunnelingstate(Z=5)wheretheconductanceissuppressedinsidethegaptonearlyzeroandtwolargecoherencepeaksformattheedgesofthegap.Weseetheeffectofanincreasing)]TJ /F1 11.955 Tf 10.1 0 Td[(ona3meVisotropics-wavegapinFigure2-5.As)]TJ /F1 11.955 Tf 10.1 0 Td[(getslargerweseethefeaturesdecreaseinmagnitude,andatverylarge)]TJ /F1 11.955 Tf 10.1 0 Td[(theyhavebecomesmearedaway. 2.1.2.2D-waveAd-wavegapstructurehasastrongangulardependence,withfournodesandfourlobes,withadjacentlobeshaveadifferentphase.Wecanexpressthisasagapstructure(E,)=Cos[2()]TJ /F9 11.955 Tf 11.97 0 Td[()].Forthisstructure,weexpecttoseetwofundamentaldifferencesfromtheisotropics-wavecase.Firstthepresenceofnodesmeansthatforanymeasurementwehavequasiparticlestateswhichcanbeexciteddownto0mV,sounliketheatsubgapconductancesweseeintheisotropiccase,alld-wavecaseswillhaveaV-shape.Theotherfeatureisthatforcertainvalues(anythingbesides=0willhavesomeeffect)andlargeZvalues,thephasedifferenceisgoingtobeandwewillseeazerobiasconductancepeak(ZBCP).ToshowtheseeffectsrefertotheFigure2-6whichshowsa3meVgapatdifferentZvaluesandtwodifferentalpha(=4and0). 2.1.2.3AnisotropicS-waveFormostFeSCsthereisgoodreasontobelievethatthesymmetryissomesortofs-wave,butmaycontainlargeanisotropy.Thisanisotropymaybeenoughtocausenodesandpossiblyasignchangeofthegapforcertainmomenta.Asamodelforthisanisotropyweintroduceagapstructure(,)==(1+r)(1+rCos[2()]TJ /F9 11.955 Tf 12.14 0 Td[()]),,whichhadpreviouslybeenusedtodescribeextendeds-wavesymmetries[ 53 ].Forrvalues 31

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Figure2-4. S-wavegapconductances:ConductancesatvariousZvalueswhichwouldbeseenwithans-waveisotropicgapof3meV.WithincreasingZvalue,thereisacharacteristicchangefromanincreasedconductancetoadipat0mV. lessthan1,thisstructureshowsfeaturesassociatedwiththemoreisotropics-wave,i.e.atendencytoformatsurfacesinsidethegapthoughofasizelessthanthegapmagnitude.Nearrequalto1theseatsurfacesbecomesmallerandatrequalto1formanode,andatthispointtheconductancelooksverysimilartod-wavewhenalphaisequalto0.Forrgreaterthan1areassignphasechangeexistsandaZBCPwilloccurforselectmomentaorinotherwordsasrgetsmuchlargerthan1thedI/dVwillbecome 32

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Figure2-5. BroadeningEffects:)]TJ /F1 11.955 Tf 10.1 0 Td[(atvariousvaluesleadstoabroadeningoftheconductancecurves.Ingeneralsmall)]TJ /F1 11.955 Tf 10.1 0 Td[(ispreferablesincelarge)]TJ /F1 11.955 Tf 10.09 0 Td[(makesidentifyingfeaturesdifcult. veryclosetothed-waveconductance.IntheFigure2-7suchtransformationwithrareshownforlargeZandequalto=4(sowecanobservetheformationofZBCP).Therequalto1.5curveshowssomethingwhichlooksverydifferentfromalargeZcurve,makingitclearthatonehastobeverycarefulwhileidentingthefeaturesseeninacurveandlookingforreproducibilityovermanysuchcurves. 33

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Figure2-6. D-waveConductances:ShownatvariousZvaluesistheconductanceinad-wavegapfor=0(solidlines)and=4(dottedlines).NotethepeakfeatureseenforsmallZvaluesinbothcases,whileazerobiasconductancepeakfromaboundstateisonlyseenforthelargeZvalueat==4. 2.1.2.4MultipleGapsFormaterialswithmultipleFermisheets,therecanbemultiplesuperconductinggapswhichform.ThedI/dVmeasuredwillbealinearcombinationofthedI/dVfromthetwogapseachwithaspecicweightasshowninEquation2-11,wherew1istheweightforgap1.dI=dVtotal=w1dI/dV1+(1-w1)dI/dV2 (2)Itcannotbeassumedthatthe)]TJ /F1 11.955 Tf 10.1 0 Td[(andZvaluesarethesameforthetwogaps,thusforanisotropictwogapmodeltherearesevenindependentvariables(Z1,Z2,1,2,)]TJ /F3 7.97 Tf 6.77 -1.79 Td[(1,)]TJ /F3 7.97 Tf 6.77 -1.79 Td[(2,w). 34

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Figure2-7. Anisotropicgapvariation:Theconductanceexpectedforananisotropicgapof3meVatZ=1atvariouslevelsofanisotropy,withstructure(,)==(1+r)(1+rCos[2()]TJ /F9 11.955 Tf 11.96 0 Td[()]).whenrequals0thegapisthesameasanisotropiccase,atr=1itshowsnodes,andforr1thereisasignchangeacrossthemomentumandZBCP. Figure2-8showsaseriesofconductancecurvesfordifferentZvaluesinthetwogapmodel.Notethatthelargergapoftenappearsasashoulderontheconductancecurve.Ifthegapshaveanangulardependencethenumberofvariablesincreasesbyfour(r1,r2,1,2).Figure2-8showsafewcasesfortwogapswithangulardependence.Oftenthealphascanbetakenasthesamesinceitcorrespondswiththedifferencebetweenthestructureangleandtheelectroninjectionangle.ThismeanswhenttingaBTKmodeltoatwogap(oranygreaternumberofgaps)dI/dVcurveitbecomesextremelydifculttobecertainofthetswithhighaccuracy.Incertaincasesathirdgapmaybeincluded, 35

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Figure2-8. Showstheconductanceexpectedforasystemoftwogapsof3meVand6meVatZ=0andZ=1forsymmetrieswhichareisotropic,containanode,andcontainasignip. butthevalidityofthethirdgapbecomesquestionable.Ifathirdgapisincludeditiscombinedlinearlywithaweightsimilartotheadditionofthesecondgap.Itshouldbenotedthoughthatitispossibletoseemanyfeatureswhichcouldbeinterpretedasextragaps,butthesegapswillnotbeconsistentbetweenvariousmeasurements.Figure2-9showssignsoffourgaps,thoughonlythetwosmallergapsarelikelytoberealfromcomparisontoothermeasurements.ItisknownthatmanyconductanceartifactscanoccurinPCSmeasurements.Therearesomemodelswhichallowforinterferencebetweenthetwogapsandremovethelinearadditionofconductances.Golubovetal. 36

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Figure2-9. Multiplegapartifacts:PCSispronetoartifactsandsotobecertainofameasurement,comparisonsandconsistencymustbefoundbetweenmultiplemeasurements [ 54 ]haveonesuchpredictionfortheisotropicsmodel,however,ourPCSresultsarenotconsistentwiththismodel. 2.2Point-ContactApparatusThemechanismusedtocreateapointcontactcanvaryandtwotypesarecommonlyused;thevariableZmethod,whereatipisloweredontothesurfaceofthesuperconductor,orastaticpoint-contact,whereastaticphysicalconnectionisestablishedusingaconductingepoxy,paste,orsolder.Thesehavebeencalledtheneedleandanvilmethod(variableZ)andsoftpointcontact(static)byothergroups[ 55 ].Aslongasthecontactformedissmallthenbothmethodswillgiveequivalentresults.Eachmethoddoeshaveadvantagesanddisadvantagesthough.ThevariableZ 37

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Figure2-10. PointContactSchematic:Abasicschematicofac-axisvariablepointcontact.Thetipcanberaisedandloweredfromthesample methodprovidestheabilitytochangethejunctionduringandbetweenmeasurements,primarilybycontrollingthepressurewithwhichthetipispushedintothesample.Bychangingtheappliedpressure,thejunctionbarrierstrengthcanbechanged.RoughlyspeakingincreasedpressurewillresultinalowerZinthejunction;lesspressurewillresultinahigherZ.ThevariableZmethodthusallowsasinglesampletobeprobedovermultipleZvalues,whichmakesitveryusefulforidentifyingifthegapisnodalorhasasignchange.AschematicofthisdesignisshowninFigure2-10.Forcertainsuperconductorsthesurfacemaynotexhibitfullpropertiesofsuperconductivity,forexampleNiobiumoxidebecomessuperconductingatamuchlowertemperatureof1.38KthantheTcofniobiummetal(9.2K).Avariablepoint-contactsetupcansometimesusethepressureofthetiptopuncturethesurfacelayerandmeasurethepropertiesbeneath.Thedisadvantagesofthevariablesetupprimarilylieinthedifcultyofobtainingstablejunctions.Placingthetiponsmallcrystalscanbeextremelydifcult,sincethetipmustrstbealignedproperlyandtipsalsohaveatendencytobucklewhenpressureisapplied.Evenifthetipisrestingontheproperlocation,temperatureandheliumuctuationscancausethetiptomoveonthesamplechangingthejunctionduringameasurement.AstaticPCSjunctionprovideshighstabilityinthe 38

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junction,removingmostworriesaboutthejunctionpropertieschangingwithchangingtemperatureortime.Whichallowsrepeatedmeasurementsofthesamejunctionmanytimeswhileanothervariableischanged(T,MagneticField),orforstatisticalcomparison.ThestaticPCSmethoddoesnotapplysignicantpressuretothesamplesincetheconnectioncannotbemechanicallychangedonceestablished.ThemajordisadvantagesofthestaticPCSmethodarethereverseofthevariablemethod.Sincethejunctioncannotbechangedatlowtemperatures,onlyasingleZvaluecanbemeasuredonasinglecrystal.Ifasurfacelayerneedstobebrokenthrough,anditisnotdonebeforeestablishmentofthejunctionthenitcannotbedoneusingthestaticPCSmethod.FinallythesizeofastaticPCSjunctionwilloftenbelargerthanavariablejunctionbecausetheepoxyorpasteusedtocreatethejunctionwillspreadbeforeithascured.ThislargerjunctionmeansthatsometimesthedI/dVmaynotshowspectroscopicbehaviorandevenwhenitdoes,gettinghighZmeasurementsinsuchajunctionisextremelydifcult.WehaveusedthevariableZmethodforallourmeasurements. 2.2.1C-axisorab-planeBTKmodelscanbesolvedformeasurementsforbothc-axisandab-planejunctions.Dependingonsymmetry,therecanbeadistinctdifferencebetweenthetwo,withab-planejunctionsshowingthemorecomplexfeaturesunlessthegaphasastrongkzdependence.Forexampleind-wavesymmetry,forc-axisjunctionstheZBCPisnotgenerallyobservableduetothereectedquasiparticlemomentumanglestayingconstant.Forstaticpointcontactsthedifferencebetweenthetwotechniquesisnegligibleaslongasthejunctioncanbemade.Forvariablepointcontacts,thegeometryofthecrystalscanrequiredifferentprobesfordifferentmeasurements.Forc-axismeasurementsweusedthestandardvariablepointcontact,whereasharpenedtipisloweredontothesurfaceofthecrystaluntilajunctionforms(seeFigure2-10).Forthincrystalsthismethoddoesnotworkforab-planepointcontactsasthetipslidesoffthesideofthecrystal.Tomeasureab-planepointcontactsweusedaprobe 39

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Figure2-11. ab-planeprobe:(Left)Abasicschematicofanab-planevariablepointcontact.ThewirecanberaisedandloweredfromthesampletocontrolpressureandZ.(Right)Therealizationofthisprobemechanism. andmethodpreviouslyintroducedbyQazilibashetal.[ 56 ],whereatungstenwireisloweredperpendiculartothesideofathincrystal(Figure2-11).Thismethodcanstillcreateballisticjunctionswhenthecrystalisthinandthewireheightcanbecontrolledaccurately,limitingthecontactareatoasmallrectangleonthecrystalsurface. 2.2.2DataAcquisitionmethodsItwouldbeareasonablemethodtotakedirectI-VmeasurementsofthejunctionandcalculatethedI/dVthroughnumericaldifferentiation.Thisrequiresalargenumberofmeasurementstominimizethenoiseinvolvedintakingthederivative.Insteadweprefertotakeadirectmeasurementofthedifferentialconductance.Toacquiresuchdata,wegenerateasignalconsistingofaslowlyvaryingtriangularwavewithasmaller 40

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Figure2-12. DataAcquisitionSchematic:SchematicoftheelectronicsusedtocapturethedI/dV-Vdata higherfrequencysinusoidalmodulation.Whenthesignalispassedthroughaknownresistorandthenthejunction,weareabletomeasuretheDCvoltage,theIacrosstheknownresistor,andtheVacrossthejunction.ThisallowsustocalculateI/VattheDCvoltage;thedifferentialconductance.TomeasuretheIweuseanSR830lock-inamplier,theVismeasuredusingaKeithley2000multimeter,andtheDCvoltageusingaKeithley2182Ananovoltmeter.AschematicofthecircuitisshowninFigure2-12.BycontrollingthesizeoftheDCsignalandtheknownresistanceweareabletomeasuredI/dVdirectlyoverarangeofvoltages,sinceweareconcernedprimarilywiththegapfeatureswegenerallylimitourselvestodoublethegapfeatures.Incaseswherethereappearstobeachangingbackgroundtothemeasurementnotassociatedwiththesuperconductivity,alargerrangemightbemeasuredsothatwecanlookforthelinearrangeoftheconductance. 2.2.3TemperatureControlTemperaturewasvariedfrom300Kto1.9Kusingheatingelementsandpumpedliquidhelium.Mostmeasurementsweretakenat4.2K.TheheatingelementwascontrolledusingaLakeshore332temperaturecontroller.TheheatingelementconsistedoftwistedandcoiledManganin36AWGwire. 41

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2.2.4NormalizationProcedureInmanypointcontact(andothertunnelingmeasurements)thereisaconductancebackgroundthatisnotassociatedwiththesuperconductivity,butresultsfrompropertiesofthejunctionandelectrodes.WhenitbecomeslargethisbackgroundcancauseerrorsinttingthedI/dVdataormakettingimpossible.Hence,anormalizationprocedureisrequired.Therearetwonormalizationproceduresweused,dependingonifwehaddataforaspecicjunctionattemperaturesabovethecriticaltemperature.IfwehadthedI/dVdatafortemperaturesabovethecriticaltemperaturethenwecandividethelowtemperaturedatabythehightemperaturedata.ThisreliesontheassumptionthatthepropertiesofthejunctionwhicharenotrelatedtosuperconductivitydonotchangedramaticallyovertherangeoftemperaturesuptothesuperconductivityTc.Thisisthepreferablemethod,asthebackgroundremovalinthismethodmeanswedonotneedtoworryaboutchangesinthebackgroundofthesameenergyscaleasthesuperconductinggap.Thedownsidetousingthismethodisthatifapseudo-gapexisteditwouldberemovedbythistechniqueandcouldnotbeidentied,unlessamuchhighertemperaturecurvewasusedascomparison.Oftenbecauseoftheinstabilityofindividualjunctions,gettingmeasurementsofajunctionatlowtemperatureandabovethecriticaltemperatureisnotalwaysfeasible.Incaseswherethejunctionshiftsandwecannotobtainahightemperaturecurve,ratherthanmakingthedataunusablewegenerateanarticialbackground.Formostconductancemeasurementsthereisapointawayfromthegapfeatureswheretheconductancebecomeslinear.Byttingtothislineartermandthenthermallysmearingthetwecanproduceasmoothbackground.Wethendividebythissmoothedt.Afterdivisionbythebackgroundgenerated(eitherthedirectlymeasuremenedorhighvoltagetting),wehaveaconductancewhichisnearlynormalized.Ifitisnot,itisnowoffsimplybyaconstant,whichisnotproblematicsincethederivationoftheBTKmodelallowsforthecurvetobemultipliedbyaconstantwithoutimpactingthephysics.Toresolvethisissuewedividebyaconstantbringingthe 42

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highandlowenergyconductance(awayfromthegapfeatures)toone.AtthispointwecandirectlycompareourdatatotheextendedBTKmodel. 43

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CHAPTER3POINTCONTACTSPECTROSCOPYOFOPTIMALLYDOPEDBAFE2)]TJ /F5 7.97 Tf 6.58 0 Td[(XCOXAS2 3.1ExperimentalMethods 3.1.1SamplePreparationOursamplesareoptimallydopedsinglecrystalsofBaFe2)]TJ /F5 7.97 Tf 6.59 0 Td[(xCoxAs2(x=0.15),withatransitiontemperature(Tc)of24Kandatransitionwidthof1Kasobservedinfourproberesistancemeasurements,showninFigure3-1.ThecrystalsweregrownusingtheuxmethodwithanIndiumux,similartothatusedin[ 57 ].Inthismethodthecrystalconstituentsareheatedanddissolvedinaux.Asthesolutioniscooledtheuxstaysliquidwhileaprecipitateofthedesiredcrystalforms.AratioofBa:(Fe,Co):As:Inof1:2:2:20washeatedfromroomtemperatureto1100Cat75C/hr,thencooledto600Cat5C/hr,beforenallybeingreturnedtoroomtemperatureat75C/hr.Electricalcontactsweremadeusingasinglepartsilverepoxy,whichrequiredheatingto200Cfortwohours. 3.1.2PointContactJunctionFabricationPCSmeasurementsrequirethatthejunctionbeballisticornearlyballistic,suchthattheinjectedelectronsdonotgothroughinelasticcollisionswithinthejunction.Astheareaofajunctionincreasesthelikelihoodofinelasticcollisionsoccurringincreasesresultinginlargevaluesof)]TJ /F1 11.955 Tf 10.1 0 Td[(andalossofspectroscopicinformation.However,ourmeasurementsshowedenergyresolvedspectroscopicdataandthuswecanconcludethat,similartothecaseofstaticPCSwherethecontactareaisalsolarge,ourjunctionisdominatedbyoneormanysmallermicro-junctionsatthenanometerscale.Thestabilityofourjunctionsatdifferenttemperatureswasstillanissue,aswedesiredjunctionswhichwerestableaboveTcdowntoourminimumtemperatureof4.2K.Micro-junctionsinaninsulatingmaterialareknowntoberesponsivetopulsesofhighvoltage,whichcaneithercreateordestroynewjunctions[ 42 ].Hence,wedecidedtoapplyavoltage 44

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Figure3-1. R-TplotofoptimallydopedBa(Fe1)]TJ /F5 7.97 Tf 6.58 0 Td[(xCox)2As2:Thefourproberesistanceversustemperatureoftheoptimallydopedsample.Theinsetshowsthephasediagrammarkedwiththedopingofthesample. ofabout100mVduringtheprocessofmovingthewireontothesample.Wefoundthisprocessincreasedthestabilityofthejunctionduringthecoolingprocessandthereafter. 3.2ResultsFigure3-2ashowsdI/dV-VdataforlowZmeasurementsonaBaFe2)]TJ /F5 7.97 Tf 6.59 0 Td[(xCoxAs2(x=0.15)crystal.ThehighZmeasurementdoesnotshowtheindividualsuperconductingfeatures,butratheraV-shapedconductancecurvewithabroadpeaklikefeaturebetweenfrom20meVtoabout40meV.ThesefeaturesaretiedtothesuperconductivitysincetheydisappearwhenthetemperatureisincreasedabovetheTcofthesample.However,inrepeatmeasurementswheresuchfeaturesareseen,theenergyatwhichthesefeaturesappearvariesindifferentjunctionsbyroughly20meV(rangingbetween20meVoutto60meV,thoughpeaksnear20meVweremorecommon).TheZwasthenreducedbypressingthetungstenwirefurtheronthesinglecrystal.ThedI/dV-V 45

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Figure3-2. RawandnormalizeddI/dV-Vdata:(a)dI=dV)]TJ /F7 11.955 Tf 11.95 0 Td[(VcurveforahighZmeasurement.(b)Overlayed(dI/dV-V)pointcontactspectraforalowZjunctionat4.2Kand21K.Thetwogapsarevisibleat6meVand10meV.The21Kconductancecurveshowsthebackgroundconductanceinthenormalstate. curvesforthelowZjunctionshavetobenormalizedtoenablecomparisontoboththeBTKmodelandbetweenmeasurementsatdifferenttemperatures.Thenormalizationmethodreliesontheassumptionthattheconductancegoestoaconstantvalueatenergiesofabout2andgreaterassuggestedbytheextendedBTKmodel.Wehavenormalizedourdataintwosteps.First,thedI/dV-VcurveisdividedbythevalueofdI/dVatV=30meV,whichismorethantwicethevoltageatwhichgapfeaturesarepresent,toobtainoverlayeddI/dVcurves.Figure3-2bshowsthedI/dV-VdataforalowZjunctionat4.2Kand21Kwhichhavebeenmultipliedbyaconstantsotheyareoverlayed,inwhichweseeevidenceforatwo-gappedsystemwithapproximategapvaluesof101meVand61meVasmarkedbytheverticallines.ThesegapscanbecomparedtopointcontactmeasurementstakenbyTortelloetal.foraslightlyover-dopedBa-122sample,wheretheyfoundgapsof4.4meVand9.9meVinthea)]TJ /F7 11.955 Tf 12.01 0 Td[(bplane[ 42 ].TocomparethedatatotheBTKmodelweshouldobtainaatdI/dV-VcurveattemperaturesaboveTc.Thereisaslightpeakfeatureseeninthe4.2Kdatanear0meV,whichisalsoseeninafewothercurvesandwillbediscussedlater.ItisclearinFigure3-2bthatat21Kthesuperconductingfeatureshavebeensuppressedbutthe 46

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conductanceisnotat.Hence,inthesecondstepofournormalizationprocedureweremovetheparabolicbackgroundobservedinthecurveat21KbydividingthedI/dV-VdataatallthetemperaturesbythedI/dV-Vcurveat21KtoobtainnormalizeddI/dV-Vcurves.Thisnormalizationprocessfacilitatesaccurateidenticationofthegapstructureandsize.Werstshowtherawconductancemeasurementsattemperaturesbetween5Kand21KinFigure3-3.Thesedatashowthatdespiteincreasingtemperaturesthejunctionconductanceat30mVstayednearlyconstantiswithin0.055%.Toomuchshiftintheconductancecouldbeasignofjouleheatingwhichwewishtoavoid.Therawdataalsomakesclearthatthesubsequentnormalizationprocedureisnotundulyaffectingtheresultsinthenormalizedcurves.ThenormalizeddI/dV-VcurvesareshowninFigure3-4.Thesecurveswerethenttoatwogaps-waveBTKmodel.Figure3-4alsoshowsthecorrespondingts(solidlines)fortemperaturesbetween5Kand17K.Theextractedgapsizesare6.6meVand10.3meVat5K,whichissimilartotheestimatesfromFigure3-3bi.e.beforenormalizationwiththe21Kdata.OurthasaZvalueof0.15forthesmallgapand0.45forthelargegap,and)]TJ /F1 11.955 Tf 10.1 0 Td[(valuesof4.4meVand4.4meV(thesame)]TJ /F1 11.955 Tf 10.1 0 Td[(oneachgap)respectivelyat5K.Increasingthetemperaturecausesthegapsizetodecreaselinearlyuntilgapfeaturescouldnolongerbeidentiedat19K.ThisbehaviorisdifferentfromthatobservedbyTortelloetal.[ 42 ].TheinsetofFigure3-4showsthemagnitudeofthetwogaps(obtainedfromthets)versustemperature.Ournormalizeddatashowlargeattopfeatureswhichareconsistentwithanisotropics-wavesymmetryinalowZmeasurement,whichhadalsobeenobservedbyTortelloetal[ 42 ].ToconclusivelyverifythisgapsymmetrywetestedhowthepointcontactspectravariedwithachangingZ.Anewpointcontactjunction(differentfromtheoneusedforFigures3-2through3-4)wasformedandweconrmthereproducibilityofthePCSdataatlowZ.Therawdatanormalizedbyaconstantat30mVforjunctionswithdifferentZareshowninFigure3-5,theredline(21Kconductancedataforthe 47

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Figure3-3. PCSdataatdifferenttemperatures:TherawPCSdataoftheoptimaldopedjunctionfrom5Kto21K junction)wasusedtonormalizethedata.Figure3-6showsmoreZdependentdataforadifferentjunctionwhichshowsimilarfeatures.Bothgraphsshowsimilarfeatures,whichshowsthatourlaterconclusionsabouttheZdependenceareconsistentovermanymeasurementsTheZvariationwasmeasuredonstablejunctionsstartingwithalowZthenslowlyretractingthetipcausinganincreaseinZ.ShowninFigure3-7arethenormalizeddatawithdifferentZ:Z=0.15,0.17,0.22forthesmallgapandZ=0.45,0.47,0.63forthelargegap.Fromthets,weconsistentlycalculatealargerZvalueforthelargergapandasmallerZvalueforthesmallergap.ThebehaviorunderincreasingZisonlyconsistentwithanisotropicfullygappedsystemsincethegapfeaturesshowabroadatbehaviorinsidethegap.Theextractedgapsinthismeasurementare6.9meVand10.3meVindependentofourZvalues.WedondthatasZincreaseswehaveanincreasing)]TJ /F1 11.955 Tf 6.77 0 Td[(.WealsonotethatwhilethelowZmeasurementscouldonlybettedbyatwogapsystem,thelargestZdatacanbettedwithasinglegapwithmagnitudebetweenthemagnitudeofthetwogapsandasignicantincreasein)]TJ /F1 11.955 Tf 6.77 0 Td[(.ForthisreasonwefounditpreferabletoextractthenumberofgapsfromlowZmeasurementsandthe 48

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Figure3-4. TemperaturedependenceofnormalizedPCSdata:dI/dV-Vdataatdifferenttemperaturesfrom5Kto17K.TheextendedBTKmodeltsareshownassolidlines.Theinsetshowsthevariationofthetwogapsasafunctionoftemperature. symmetryofthegapsfromhigherZmeasurements.ThesmallpeakwhichappearsatzerobiasinthelowZmeasurementsshouldnotbeinterpretedasaZBCPresultingfromasignchangeintheorderparametersincesuchapeakshouldincreaseinsizewithincreasingZ.Infact,thepeakdisappearsasweincreaseZandwemustlookforanotherexplanationforit.OnepossibleexplanationisaJosephsonpeakresultingfromtunnelingwithinthesample.TotestwhetherthepeakatzerobiasisduetotheJosephsoneffectweappliedamagneticeldof5Tonacurveshowingsucha 49

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Figure3-5. RawPCSdatafordifferentvaluesofZ:RawdatashowingtheZdependenceoftheoptimallydopedsample.Thesolidredcurveisthe21KbackgroundandthescatterdiagramsshowdI/dV-VfordifferentZvaluejunctions. conductancepeak.Theelddidnotremovethepeaknorsplitit,onlycausingittobroadenveryslightly.Itisalsopossiblethatitiscausedbymagneticimpurities,butaspointedoutbyShanet.al.onadifferentpnictide[ 58 ],thiswouldnotbecorrelatedwiththesuperconductingtransitionandsothepeakwouldnotbesuppresseduponreachingTc.Onefeasibleexplanationisthatthepeakisassociatedwithpressurebeingappliedbythetiponthesample.Thisreasonwasgivenasapossibleexplanationforasimilar 50

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Figure3-6. RawPCSdatafordifferentjunctionsshowingreproducibility:FurtherdatashowingtheZdependence,butwithnobackgroundavailabletoshownormalization.Thechangefromapeaktodipisstillclear. peakseeninthe1111compoundsbyYatesetal.[ 59 ].Wendthistobeareasonableexplanationforthepeakobservedinourmeasurements. 3.2.1SummaryofoptimaldopingOurmeasurementsshowthatBaFe2)]TJ /F5 7.97 Tf 6.58 0 Td[(xCoxAs2(x=0.15)isatwogapsuperconductor.TheshapesofthepointcontactspectraasafunctionofZshowthattheindividualgapsdonotcontainasignipasthecharacteristicZBCPwasnotseen.Theexistenceofnodesisalsounlikelysincethespectrashowfeaturesassociatedwithfullgaps.Wedo 51

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Figure3-7. NormalizeddI=dV)]TJ /F7 11.955 Tf 11.96 0 Td[(VcurvesfordifferentZvalues:ThecalculatedZvaluesare0.15,0.150.17,0.22(forthesmallergap)and0.45,0.45,0.47,0.63(forthelargergap)forthelowZ,midZ,andhighZcurves,respectively.Thegapsof6.9meVand10.3meVaremarkedbydashedlines.ThetwolowestZcurvesdifferonlybyachangein)]TJ /F1 11.955 Tf 6.78 0 Td[(.) notseeevidenceofnodes.However,wecannotruleoutthepresenceofasmallamountofanisotropy. 52

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CHAPTER4POINTCONTACTSPECTROSCOPYOFUNDERDOPEDBAFE2)]TJ /F5 7.97 Tf 6.59 0 Td[(XCOXAS2Givenasamplewithawellcharacterizedoptimaldopingonemightaskwhybeconcernedwithnon-optimaldopingswhenthesuperconductivityisnotasrobust.Firstwehavetorememberthatsuperconductivityoftenisincompetitionwithotherphaseswhichhaveadistincteffectonthecoupling(theAFMphaseandspinuctuationmodelsintheFeSCs).Secondthatthesuperconductingfeaturesareheavilyinuencedbytheelectronicstructureofthecrystal,andthisstructurewillbeinuencedbythedopingforbothelectronandholedopings.Inthischapterandthenextweconcernourselveswithdopingsawayfromoptimal.InthecaseofBaFe2)]TJ /F5 7.97 Tf 6.59 0 Td[(xCoxAs2,itispredictedbycurrentmodelsthatthegapstructureshouldmeasurablychangewithdoping,somethingwhichneedstobetestedexperimentally[ 12 ]. 4.1ExperimentalProcedureTheunderdopedsamplewasproducedusingasimilarmethodtotheoptimaldopedsample,usingaselfuxratherthanIn.Theresistanceversustemperatureshowsatransitionof16Kwitharangeof1K.Fromthisproperty,weestimatethetruecompositionofthesampletobeasBaFe1.9Co0.1As2.ForthiscrystaltheresistancebelowTcdoesnotnotgotozeropossiblyduetocontactresistanceeffects.Sincewewanttoperformresistancemeasurementsonthesamecrystalonwhichwesubsequentlycarryoutpointcontactmeasurements,weneedtokeepasignicantpartofthecrystaluntouchedbythefourprobemeasurementcontacts,whichoftenleadstocontactresistanceproblems.However,ourresistancemeasurementswereperformedmainlytochecktheTcofaspeciccrystal.FromtheratiosusedintheuxweexpectedthecrystaltobeBaFe1.9Co0.1As2,sotheTcfromresistanceseemsreasonable.Theab-planepointcontactspectroscopy(PCS)techniquedevelopedbyQazilbashetal.[ 56 ]wasusedtotakeconductancevs.voltagecurves.NormalizationofthecurveswasdonebydividingbyacurvemeasuredabovetheTcwhennosuperconductingfeatureswere 53

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Figure4-1. R-TforunderdopedBaFe2)]TJ /F5 7.97 Tf 6.58 0 Td[(xCoxAs2crystal:R-Tfortheunderdopedsample,BaFe1.9Co0.1As2.Theinsetshowsthelocatonofthesampleinthephasediagram. seen.Thecurvewasthenmultipliedbyaconstantvaluesothatathighvoltages(30mV)thenormalizedconductancegoesto1. 4.2ResultsInFigure4-2therawconductanceforthreejunctionsofdifferentZareshown.Therawconductanceshowsabasicsimilaritytotheconductancesfortheoptimallydopedcrystalsviz.,atlowZtherearesignsoftwogapsandatlargeZthecurveshowsadipatzerobiasanddoesnotshowazerobiasconductancepeak.Thereisasmallpeakat0mV,butunliketheoptimallydopedcompoundwhereasmallchangeintheZ 54

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Figure4-2. RawPCSdataforunderdopedcrystal:ThreerawdI/dV-Vcurvesfortheunderdopedsample. valuecausedthepeaktodisappear,fortheunderdopedsamplethispeakstayswhileAndreevreectiondominatestheconductance.Forthisreason,justfromtherawdatawecanconcludethattheunderdopedcompoundshowssignsofmoreanisotropicandpossiblynodalbehavior.Tocheckforconsistencyitisusefultolookovermultiplejunctionsshowingsimilarbehavior.Figures4-3through4-5showsetsofconductancecurveswhichshowsimilarfeaturesatdifferentconductancevaluesbutcouldnotbenormalizedbecausearelatedhightemperaturecurvewasnotavailable.Ineachcaseweseesimilarresultstotherawdatashownbefore,wherethereisasmallcentralpeak 55

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Figure4-3. RawPCSdataforunderdopedcrystalsshowreproducibility:Thebasicshapeoftheconductancecurvefortheunderdopedsampleishighlyreproducibleandconsistentovermanymeasurements.Eachcolorconsistsofavoltagesweepforanewjunctiononthesamesample. at0mVforlargeconductanceswhichturnstoadipatsmallconductances.Thereisoftenashoulderinthelargeconductancesaround10mVandforthesmallconductanceabroadincreasedconductanceissometimesseenoutsideof10mV,belowwhichtheconductancehasacharacteristicdipfeature.Thereisadipintheconductancebetween20meVand25meVattheedgeofthegapfeatureswhichdoesnotchangeasZisvaried.Thesedataweretakenat4.2Kandunlessotherwisespeciedfurthercurvesweretakenatthistemperatureaswell.Totthedataweneedtonormalize 56

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Figure4-4. PCSdataforsmallZ:AsmallZcurvewhichshowsthepeakduetoananisotropicgap.Thedoublingresultsfromthevoltagebeingsweptovertherange. itand,inthiscasewehavea16Kconductancecurvecorrespondingwiththeshownjunctions.ThenormalconductanceandoneofthesuperconductingcurvesareshowninFigure4-6.Thecurveshavebeenmultipliedbyaconstanttobringthemto1at30mV.Oncenormalizedthecurvesweretusingatwogapmodelwithgapstructure(,)==(1+r)(1+rCos[2()]TJ /F9 11.955 Tf 12.12 0 Td[()]).ThenormalizeddataandtteddataareshowninFigure4-7.ThenormalizedtsmakeclearthatthegapstructureofunderdopedBaFe1.9Co0.1As2isnotisotropicandinfact,containsastronganisotropy,sincethecurvesdonotatten,butratherresultinapeakfeatureatzerobias.Thehighest 57

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Figure4-5. PCSdataforlargeZ:AtlargeZtheunderdopedsampleshowsadipindI/dV-VwithnoZBCP,whichmakesitunlikelythatthereisasignchangeonthegaps.Thedoublingresultsfromthevoltagebeingsweptovertherange. conductancecurvehastwogapsof6.11.5and13.11.5meV.IthasaZof0.1forthesmallergapand0.33forthelarger,with)]TJ /F1 11.955 Tf 10.1 0 Td[(of8and13.Thelargergapslightlydominatescomparedtothesmallergapwithaweightof0.45forthesmallergap.Thervalueassociatedwiththeanisotropyofthegapsis1.0forthesmallergapand0.5forthelargergap.Thisanisotropyisverydifferentfromtheoptimalcasewererwaszeroforbothgaps.Anrvalueof1.0foragapmeansthattherearesubgapstatesdownto0mV.is=4forthecurve,butunlessrgoesover1thisvalueofcausesinsignicanteffectsontheconductancecurvesforlarge)]TJ /F1 11.955 Tf 10.1 0 Td[((thosegreaterthanhalfthegapsize). 58

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Figure4-6. NormalizationofPCSdatafromaunderdopedsample:Theredcurveshowsthe16KconductanceoverlayedwithoneoftheZdependentdI/dV-Vcurves. ThetofthemiddleconductancecurveisthesameastherstexceptthattheZforthelargergaphasincreasedto0.4.Thesmallestconductance,associatedwiththelargestZ,didtbesttoasingleanisotropicgap.Thegapsizewas6.1meVandZwas1.0.Therwas1.0andthe)]TJ /F1 11.955 Tf 10.1 0 Td[(was8.ForthisreasonwecanseethatatlargeZweonlyseeoneofthegaps,inthiscasethesmallergap.SincelargeZmeasurementsapproachthetunnelinglimitthisobservationmayexplainwhySTMisunabletoobservemultiplegaps.ThereisalsoaslightangularbackgroundtothelargeZcurvewhichwasnot 59

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Figure4-7. NormalizedPCSdataandtsforaunderdopedsample:ThemodiedtsareshownforthreecurvesofdifferentZvalues. removedbynormalizationandcouldberemovedbydividingbyaline,butwefeltthiswasunnecessarytoachieveameaningfult. 4.3FutureWorkForunderdopedFeSCsitisknownthattheresistivityhasastronginplaneanisotropyintheresistivitywhenstrainisapplied[ 60 ].ThereisalsoevidencethatwhenpressureisappliedwithatipinPCS,acentralpeakoccurs(asseeninouroptimallydopedsamplesandbyYatesetal.[ 59 ].Weproposethatthepressurechangebythetipmaybeconnectedtotheobservedinplaneanisotropyoftheresistivity.To 60

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characterizethisbehaviorweplantoperformPCSonanadaptedprobewhichcanapplystrainacrossthesamplewhileconcurrentlymeasuringthepointcontactspectra.Twoprimaryprobeshavebeendevelopedforthisexperiment:onewhichappliesstrainbyacantilevercontrolledwithastaticscrewandusesthescrewcontrolledPCSmechanismforvariableZPCS.TheotherusesstaticPCSandascrewtocompressthesamplebetweentwoplates.Onlypreliminarydatahasbeenobtainedusingtheseprobes. 4.4SummaryofResultsforUnderdopedSamplesTheresultsreportedinthischapterindicatethatwithdopingthereisaverystrongchangeintheanisotropyofthegapstructurecomparedtotheoptimallydopedcase.Thereislikelyaformationofnodesonthesmallergap.Thelargergapincreasesinsizeslightlygoingfrom10meVto13meV.Thesmallergapisroughlythesamesizeforbothoptimallyandunderdopedsamples. 61

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CHAPTER5POINTCONTACTSPECTROSCOPYOFOVERDOPEDBA(FE1)]TJ /F5 7.97 Tf 6.58 0 Td[(XCOX)2AS2 5.1ExperimentalProcedureOuroverdopedsamplewasproducedinasimilarmethodtotheoptimallydopedsampleusingtheuxmethod.RatherthanIndiumux,aselfuxofFeAswasusedsimilarto[ 61 ].Resistancemeasurementsshowabroadtransitionwhichstartsat15Kandendsat14K(seeFigure5-1).Thismeansourdopingisapproximatelyx=0.11forBa(Fe1)]TJ /F5 7.97 Tf 6.58 0 Td[(xCox)2As2.Fromtheratiosofelementsincludedintheuxitwasexpectedthatthedopingwouldbex=0.15.PCSontheoverdopedsampleswasperformedthesamewayastheoptimallydopedsamples,usingthemethodestablishedbyQazilbashetal.[ 56 ].Forthesesamples,formingjunctionsabovetheTcwhichwerestableprovedmoredifcultsowewereunabletoprovideatemperaturedependencenornormalizebytheregularprocedure.Insteadwewereforcedtonormalizealldatabyttingtoalinearconductancetermatlargevoltages,thenthermallysmearingthelineartermandusingthisasourbackground. 5.2ResultsInFigure5-2youcanseeasetofrawdatatakenforvariousZvalues.Notethatunliketheoptimaldopeddatatheconductanceat0mVhasaclearpeakfeature.Thisistherstclearsignthatthegapstructureishighlyanisotropicandcontainsnodesfortheoverdopedsample.Tobecertainweattemptedtstovariousgapstructures.Thehighestconductancemeasurementshowsthelargestnumberoffeatures.Intherawdataitattensaround25mVandlarger,thenincreasesinconductanceatroughlyaconstantslopetillaround12mV,beforeanupturnintheslope.At7mVitattensshortlybeforegoingtothepeakat0mV.Thisdatawiththeatteningat7mVandthepeakshowsclearevidencefortwogaps(thechangeinslopeat12mVcouldbeinterpretedasanothergap).ThenormalizeddataandthecorrespondingmodiedBTKtsare 62

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Figure5-1. R-Tfortheoverdopedsample:theinsetshowsthelocationofthesampleinthephasediagram. showininFigure5-3.Ourbesttoccurswhenamuchlargergapisincluded.ThebesttforthelowestZdataoccurswhenwetakethreegaps,alargeisotropic18meVgap,asmallerisotropic10meV,andasmallanisotropic3.5meVgapwithnodes.The18meVgaphasaweightroughly1/10thoftheweightforthetwosmallergaps.The3.5meVgapandthe10meVgapseemreasonablefromcomparisontotheoptimaldopedcase.InthiscasealltheZvaluesforeachgapare0.The18meVgapcouldpossiblybearesultofthelargeVfeaturesseenintheoptimallydopedcase.GapsofthissizearecorroboratedbythesecondsmallZconductanceshown.Thegapsforthebesttareverysimilar,theonlydifferencebeinganincreasedZvalueforthemiddlegapfrom0to 63

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Figure5-2. RawPCSdataforoverdopedcrystal:RawZdependentdI/dV-VdatashowingtheclearnodalbehavioroftheoverdopedsamplesfordifferentZvalues 0.2andageneralincreasein)]TJ /F1 11.955 Tf 6.77 0 Td[(.Thelowconductancecurveisaninterestingproblem.First,itonlyshowsonefeatureclearly.Second,thecurvecanbetinmultipleways,eitherwithasmallZvaluewithnodesoralargeZvaluewithazerobiasconductancepeak(ZBCP).IfitisttoasinglesmallZgap,wendthebesttoccursforagapof6.5meVwithaZof0.IfitisttoaZBCPwithalargeZwendagapof10meVforananisotropicgapwithr=3andZequalto3.7.WecanthenidentifythelowconductancecurveasalargerZmeasurementsinceitisunlikelyforachangeinZtocauseachange 64

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Figure5-3. NormalizedPCSdataandcorrespondingtsforaoverdopedsample:Normalizeddata(symbols)andmodiedBTKts(redlines)forjunctionwiththreedifferentZvalues. inthesizeofthegapandtheothertwoconductancecurvesdonotshowsignsofa6.5meVgap.Whilethepreviousexampleshowsclearevidencefornodesandextremelylikelyevidenceforaphasechange,therecanbecasesweretheZBCPisnotseen.InFigures5-4and5-5weseethesmallZmeasurementwehadpreviouslyshownandanotherlowerconductancemeasurementwhichdoesnotshowthesignsofaZBCP.Itisinterestingthatthisonlyoccursforcasesweretheconductanceislow.Evenwhenitdoesoccur,theshapeat0mVisstillinlinewithnodalbehavior.Thisbehavioris 65

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Figure5-4. ComparisonoftwoLargeZjunctions:alargeZvalueneednotshowtheZBCPinalljunctions explainedifweassumethatthejunctionformeddidnotconsistofasinglepointcontactjunctionbutmanyjunctions.InsuchacaseeachjunctionwillhavedifferentandZvalues,andtheconductancewemeasureisaweightedaverageofthevariousjunctions.Asweretractthetipfromthesample,largeZjunctionsaregoingtobethersttobreak.AsthesejunctionsbreakthesmallZjunctionswillbecomedominant.IfforthesmallZjunctionisnear0,thenaswemoveawaytheZBCPwilldisappear. 66

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Figure5-5. NormalizedLargeZcurves:NormalizedLargeZcurvesforthedatashowninFigure5-4. 5.2.1SummaryOurmeasurementsoftheoverdopedcrystalsshowclearevidencethatthegapstructurecontainslargeanisotropy,causingnodestoformandlargeenoughforasignchangetooccurresultinginafourlobestructure.Therearetwogaps,ofroughly10meVand3.5meV.Thereisevidenceofafeatureat18meVforsmallZmeasurementsandthoughitisttedtoanisotropicgap,wedoubtitisanactualgapbutratherthelargescalefeaturesseenintheoptimalmeasurements. 67

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CHAPTER6CONCLUSIONSInthisthesisIhavereportedZdependentpointcontactspectroscopymeasurementsonBa(Fe1)]TJ /F7 11.955 Tf 13.24 0 Td[(xCox)2As2samplesofthreedifferentdopings;optimallydoped(x=0.075),underdoped(x=0.05)andoverdoped(x=0.11).PrimarilyweshowthatintheBa(Fe1)]TJ /F7 11.955 Tf 12.03 0 Td[(xCox)2As2compoundthereisastrongtransformationofthegapstructurewithdoping,fromananisotropicstructureforunderdopedsampleschangingtoanisotropicstructureatoptimaldopingsthentoahighlyanisotropicstructureathigherdopings.ThisbehaviorismadeclearbytheZdependenceandthecomparisonbetweenthehighZandlowZconductancevs.voltagecurves.Ourdatacanbecomparedtopredictionsfromtheory[ 12 19 20 ]whichshowthatsuchanincreaseinanisotropyisexpectedawayfromoptimaldoping.Theycanalsobecomparedtootherexperimentalworkwhichshowsimilartrendsintheanisotropy[ 21 22 31 ].Wealsoseethatthelargergapmagnitudestaysroughlyconstantwithdopingatabout10meV(increasingslightlyto13meVfortheunderdopedsamples)andthesmallergapvariesbetween7meVatoptimaldopingto3.5meVwhenoverdoped.Thisbehaviorisinterestingsincethegapsizeisrelatedtothestrengthofthemechanismwhichcausessuperconductivityandgapsizedoesnotshowasignicantvariationoverthedopingrange.Oftenacharacteristiccouplingconstant2 kbTcisdiscussedforsuperconductorsandhasasetvalueinBCSsuperconductivity.InallofourmeasurementsthiscouplingconstantvalueislargerthantheexpectedBCSvalue(3.52)andisgenerallyasignofstronglycoupledsuperconductivity(5.42forthesmallestgapintheoverdopedsamplesand18.8forthelargestintheunderdopedsamples).Suchastrongcouplingconstantisnotunheardof.Forexample,manycupratestendtowardshighervaluesof2 kbTcastheTcissuppressed.[ 62 63 ]ThelargeratiomightalsobeexplainedfordopingsawayfromtheoptimalamountbytheintroductionofscatteringsiteswhichlowerTcbutdonotchangethegapsizedramatically. 68

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Thechangesingapanisotropyweobservedcannotbeusedtoidentifythepairingmechanismdirectlyalthoughtheyareconsistentwithcalculationsforthesmechanismwithlargeanisotropy.Thisobservationdoesnotruleoutas++mechanismwithlargeanisotropy,asthetechniquecannotdifferentiaterelativephaseofthetwogapsbutonlymeasurestheanisotropyofasinglegap.Hence,variablebarrierstrength(Z)pointcontactspectroscopy(PCS)providesavaluablemethodofmeasuringimportantfeaturesinthegapstructureforsuperconductors,andinthecaseofunconventionalsuperconductorsitisquiteusefulforcharacterizationwhenothermethodsareinconsistentorprovedifcult.OurresultsmakeclearthatthegapstructureofBa(Fe1)]TJ /F7 11.955 Tf 12.65 0 Td[(xCox)2As2changesstronglywithdopingandprovidesimportantinsightintothebasisofsuperconductivityintheFeSCs. 69

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APPENDIX:NUMERICALINTEGRATIONCODE Z1=0.00;Z2=2.70;Z3=0;==4;1=3.5;2=10.0;3=18;r1=1;r2=3.0;w1=0.0;w2=1.0;)]TJ /F1 11.955 Tf 6.78 0 Td[(1=1.0;)]TJ /F1 11.955 Tf 6.78 0 Td[(2=7.3;)]TJ /F1 11.955 Tf 6.78 0 Td[(3=4.1;T=4.2;nenergy=40000;n=Table[0.001i,fi,0,nenergyg];ntheta=100;c=0.08617*T;=Table[)]TJ /F9 11.955 Tf 9.3 0 Td[(=2+i=ntheta,fi,0,nthetag];1a=1(1+r1Cos[2()]TJ /F9 11.955 Tf 11.96 0 Td[()])=(1+r1);1b=1(1+r1Cos[2(-)]TJ /F9 11.955 Tf 11.95 0 Td[()])=(1+r1);2a=2(1+r2Cos[2()]TJ /F9 11.955 Tf 11.95 0 Td[()])=(1+r2);2b=2(1+r2Cos[2(-)]TJ /F9 11.955 Tf 11.95 0 Td[()])=(1+r2);3a=3*Abs[Sign[)]TJ /F9 11.955 Tf 11.95 0 Td[(]];3b=3*Abs[Sign[-)]TJ /F9 11.955 Tf 11.95 0 Td[(]]; 70

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1a=Table[If[1a[[i]]==0,0,((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(1)-((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(1)^2)]TJ /F1 11.955 Tf 11.96 0 Td[(Abs[1a[[i]]]^2)^0.5)/Abs[1a[[i]]]],fi,1,ntheta+1g];1b=Table[If[1b[[i]]==0,0,((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(1)-((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(1)^2)]TJ /F1 11.955 Tf 11.96 0 Td[(Abs[1b[[i]]]^2)^0.5)/Abs[1b[[i]]]],fi,1,ntheta+1g];2a=Table[If[2a[[i]]==0,0,((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(2)-((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(2)^2)]TJ /F1 11.955 Tf 11.96 0 Td[(Abs[2a[[i]]]^2)^0.5)/Abs[2a[[i]]]],fi,1,ntheta+1g];2b=Table[If[2b[[i]]==0,0,((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(2)-((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(2)^2)]TJ /F1 11.955 Tf 11.96 0 Td[(Abs[2b[[i]]]^2)^0.5)/Abs[2b[[i]]]],fi,1,ntheta+1g];3a=Table[If[3a[[i]]==0,0,((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(3)-((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(3)^2)]TJ /F1 11.955 Tf 11.96 0 Td[(Abs[3a[[i]]]^2)^0.5)/Abs[3a[[i]]]],fi,1,ntheta+1g];3b=Table[If[3b[[i]]==0,0,((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(3)-((n+i)]TJ /F1 11.955 Tf 6.78 0 Td[(3)^2)]TJ /F1 11.955 Tf 11.96 0 Td[(Abs[3b[[i]]]^2)^0.5)/Abs[3b[[i]]]],fi,1,ntheta+1g];1=Table[Limit[Cos[q]^2=(Cos[q]^2+Z1^2),q![[i]]],fi,1,ntheta+1g];2=Table[Limit[Cos[q]^2=(Cos[q]^2+Z2^2),q![[i]]],fi,1,ntheta+1g];3=Table[Limit[Cos[q]^2=(Cos[q]^2+Z3^2),q![[i]]],fi,1,ntheta+1g]; 1=(Sign[1a]-Sign[1b])=2; 2=(Sign[2a]-Sign[2b])=2; 3=(Sign[3a]-Sign[3b])=2;con1=(1+1Abs[1a]^2+(1)]TJ /F6 11.955 Tf 9.3 0 Td[(1)Abs[1a1b]^2)=Abs[1+(1)]TJ /F6 11.955 Tf 9.3 0 Td[(1)1a1bExp[i 1]]^2;con2=(1+2Abs[2a]^2+(2)]TJ /F6 11.955 Tf 9.3 0 Td[(1)Abs[2a2b]^2)=Abs[1+(2)]TJ /F6 11.955 Tf 9.3 0 Td[(1)2a2bExp[i 2]]^2;con3=(1+3Abs[3a]^2+(3)]TJ /F6 11.955 Tf 9.3 0 Td[(1)Abs[3a3b]^2)=Abs[1+(3)]TJ /F6 11.955 Tf 9.3 0 Td[(1)3a3bExp[i 3]]^2;I11=Sum[1[[i]]*Cos[[[i]]]con1[[i]]=ntheta,fi,1,ntheta+1g];I12=Sum[1[[i]]*Cos[[[i]]]=ntheta,fi,1,ntheta+1g];I21=Sum[2[[i]]*Cos[[[i]]]con2[[i]]=ntheta,fi,1,ntheta+1g];I22=Sum[2[[i]]*Cos[[[i]]]=ntheta,fi,1,ntheta+1g];I31=Sum[3[[i]]*Cos[[[i]]]con3[[i]]=ntheta,fi,1,ntheta+1g];I32=Sum[3[[i]]*Cos[[[i]]]=ntheta,fi,1,ntheta+1g]; 71

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m1=I11/I12;m2=I21/I22;m3=I31/I32;x1=Table[fn[[i]],w1*m1[[i]]+(w2)*m2[[i]]+(1-w1-w2)m3[[i]]g,fi,1,nenergy+1g];x11=Join[Reverse[Table[-x1[[i]][[1]],fi,1,Length[x1]g]],Drop[Table[x1[[i]][[1]],fi,1,Length[x1]g]],1];x12=Join[Reverse[Table[x1[[i]][[2]],fi,1,Length[x1]g]],Drop[Table[x1[[i]][[2]],fi,1,Length[x1]g]],1];V=Table[-30+0.1i,fi,0,600g];T=Sume(x11[[i]])]TJ /F16 5.978 Tf 5.76 0 Td[(V)=c c(1+e(x11[[i]])]TJ /F16 5.978 Tf 5.76 0 Td[(V)=c)2x12[[i]]0.001,fi,1,Length[x11]g;ListPlot[Table[fV[[i]],T[[i]]g,fi,1,Length[V]g]] 72

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BIOGRAPHICALSKETCH JohnwasborninasmallruraltowninIllinois.Fromayoungageheshowedinterestinscienceperformingmanyexperiments.LaterheattendedtheUniversityofIllinoisatUrbana-ChampaignwherehereceivedaBScinPhysics.HethencametoUFinFallof2007werehehasworkedforAmlanBiswassinceSpringof2008.HereceivedhisPhDinphysicsinSpringof2013. 80