First-Principles Electronic Structure and Transport Calculations in Materials with Defects and Impurities

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First-Principles Electronic Structure and Transport Calculations in Materials with Defects and Impurities
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english
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Srivastava, Manoj K
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University of Florida
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Physics
Committee Chair:
Cheng, Hai Ping
Committee Co-Chair:
Hirschfeld, Peter J
Committee Members:
Monkhorst, Hendrik J
Rinzler, Andrew G
Phillpot, Simon R

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defect -- dft -- first-principles -- grain-boundary -- graphene -- low-symmetry -- nonorthogonal -- resistance -- scattering -- transport
Physics -- Dissertations, Academic -- UF
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Physics thesis, Ph.D.
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Abstract:
We present electronic structure and electron transport studies of materials with defects and impurities using density-functional theory. We develop a plane wave transport method based on density-functional theory for low symmetry nonorthogonal lattices. This is achieved by generalizing Choi and Ihm's algorithm for high symmetry lattices which requires the transport direction along a lattice vector that must be perpendicular to the basal plane formed by two other lattice vectors. This restriction is overcome in our method, allowing solutions to problems in which the transport direction is not along any lattice vectors. We apply our generalized transport method to calculate interface resistivity of grain boundaries in copper. Other than surface defects, we also study point defects such as single atom vacancy and impurities. Using electronic structure methods, we investigate adsorption of gold and iron clusters on perfect and defected graphene with a single vacancy. We focus on the size dependence of the electronic properties such as binding energy, charge transfer, magnetization, and density of states. Perfect graphene is found to be doped for Au clusters with an odd number of atoms and undoped with an even number of atoms. An odd-even oscillation in the magnetic moments is observed in Au-perfect as well as defected graphene system. While Fen clusters remain to be magnetic for all n, the spin of a single Fe atom on a defect site is very small due to a covalent bonding to C atoms.
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by Manoj K Srivastava.
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Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Cheng, Hai Ping.
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Co-adviser: Hirschfeld, Peter J.
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FIRST-PRINCIPLESELECTRONICSTRUCTUREANDTRANSPORTCALCULATIONSINMATERIALSWITHDEFECTSANDIMPURITIESByMANOJK.SRIVASTAVAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012ManojK.Srivastava 2

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Tomyparents 3

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ACKNOWLEDGMENTS IwouldliketoacknowledgeProf.Hai-PingChengwhoseguidancemadethisworkpossible.IamgratefultoDr.Xiao-GuangZhangfromOakRidgeNationalLaboratorywhoserolewaslikeaco-mentorinmyPh.D..Prof.ChengandDr.Zhangnotonlytaughtmephysicsbutalsomostvaluablelifelessons.IwouldalsoliketothankprofessorsattheDepartmentofPhysicswhoseclassesIenjoyed,speciallyProf.DmitriiMaslovandProf.JimFry.IamindebtedtoYanWang,post-doctoralresearcher,whowasalwaystheretoanswermyquestions.IwouldliketogivespecialthankstoEmmaPhillipswhosawupsanddownsofmyPhDandalwayssupportedmethroughdifculttimes.ManythankstomyfriendsVikas,DK,Atul,Shiboo,KP,Rahul,Bholu,Debu,Pooja,Rajib,andPriyawhohelpedmeineverywaypossible. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTIONANDBACKGROUND ...................... 10 1.1Materials .................................... 10 1.2DefectsandImpurities ............................. 12 1.2.1PointDefects .............................. 12 1.2.2DislocationsandGrainBoundaries .................. 13 2THEORETICALMETHODS ............................. 16 2.1Density-FunctionalTheory ........................... 17 2.2QuantumTransport ............................... 21 2.3SemiclassicalBoltzmannTransport ...................... 23 2.3.1EquationinCurrentPerpendiculartoPlane(CPP)Geometry ... 24 2.3.2SolutionofBoltzmann0sEquation ................... 26 3PLANEWAVETRANSPORTMETHODFORLOW-SYMMETRYLATTICES .. 29 3.1GeneralizedMethodforPlaneWaveTransport ............... 30 3.2ComplexBandStructurewithNonorthogonalUnitCell ........... 33 3.2.1CalculationofWavefunction ...................... 33 3.2.2BoundaryConditions .......................... 38 3.3ScatteringProblem ............................... 40 3.4CalculationofVelocityofBloch'sState .................... 41 3.5ChoosingtheImaginaryPartofEnergy ................... 41 3.6Summary .................................... 43 4ELECTRONTRANSPORTINMATERIALSWITHDEFECTS .......... 45 4.1TestCasesfortheGeneralizedMethod ................... 45 4.2GrainBoundaries ................................ 46 4.2.1Geometries ............................... 46 4.2.2ModelandCalculationDetails .................... 46 4.2.3Results ................................. 48 4.3Summary .................................... 49 5

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5STUDYOFADSORPTIONOFSMALLMETALCLUSTERSONGRAPHENE 54 5.1Method ..................................... 57 5.2ClustersonPerfectGraphene ......................... 58 5.2.1GroundStateGeometryandEnergeticsofGoldClusters ...... 58 5.2.2GroundStateGeometryandEnergeticsofIronClusters ...... 63 5.2.3ElectronicStructureandMagnetizationofGoldClusters ...... 64 5.2.4ElectronicStructureandMagnetizationofIronClusters ....... 66 5.3ClustersonDefectedGraphene ........................ 68 5.3.1GroundStateGeometryandEnergeticsofGoldClusters ...... 69 5.3.2GroundStateGeometryandEnergeticsofIronClusters ...... 70 5.3.3ElectronicStructureandMagnetizationofGoldClusters ...... 71 5.3.4ElectronicStructureandMagnetizationofIronClusters ....... 73 5.4Summary .................................... 74 6CONCLUSION .................................... 90 REFERENCES ....................................... 92 BIOGRAPHICALSKETCH ................................ 99 6

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LISTOFTABLES Table page 4-1Calculatedinterfaceresistivity(resistance-areaproduct)ofvariouscoppergrainboundaries.Unitfortheinterfaceresistivityis1016m2. ......... 50 5-1Anchoratom'spositionabovegrapheneplane(ha),distancefromNNCatom(dac),bindingenergies(BE1-BE3),chargetransfer(q),andtotalmagnetization(mcell)ofAuclustersadsorbedonperfectgraphene. ............... 76 5-2haandBE3ofAuclustersadsorbedonperfectgraphenewithinclusionofvanderWaalsinteractionsusingdifferentmethods. .................. 76 5-3WithvdWinteractions(optB88),ha,dac,BE1-BE3,q,andmcellofAuclustersadsorbedonperfectgraphene. ........................... 76 5-4ha,dac,BE1-BE3,q,andmcellofFeclustersadsorbedonperfectgraphene. 77 5-5ha,dac,BE1-BE3,q,andmcellofAuclustersadsorbedondefectedgraphene. 77 5-6ha,dac,BE1-BE3,q,andmcellofFeclustersadsorbedondefectedgraphene. 77 7

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LISTOFFIGURES Figure page 3-1Unitcell.a)AnonorthogonalunitcellusedingeneralizedPWCOND.Latticevectora3isnotperpendiculartobasalplanevectorsa1anda2.Thezdirectionisthetransportdirectionperpendiculartothebasalplane.b)AnorthogonalunitcellinafcclatticeusedinoldPWCONDcalculation. ............ 44 4-1Complexbandstructureofbulkcopperin[001]direction. ............ 51 4-2Complexbandstructureofbulkcopperin[111]direction ............. 51 4-3Topandsideviewsofgeometriesofgrainboundaries. .............. 52 4-4Totaltransmissionprobabilityintwo-dimensionalBZofgrainboundaries. ... 53 5-1OptimizedgeometriesofAuclustersadsorbedonperfectgraphene. ...... 78 5-2OptimizedgeometriesofFeclustersadsorbedonperfectgraphene. ...... 79 5-3Chargedensitydifferencen(r)andmagnetizationdensityofAuclustersonperfectgraphene. ................................... 80 5-4Totaldensityofstatesofperfectgraphenewithgoldclustersandprojecteddensityofstatesontotheporbitalsofgraphene. ................. 81 5-5Chargedensitydifferencen(r)andmagnetizationdensityofFeclustersonperfectgraphene. ................................... 82 5-6Totaldensityofstatesofperfectgraphenewithironclustersandprojecteddensityofstatesontotheporbitalsofgraphene. ................. 83 5-7OptimizedgeometriesofAuclustersondefectedgraphene. ........... 84 5-8OptimizedgeometriesofFeclustersondefectedgraphene. ........... 85 5-9Chargedensitydifferencen(r)andmagnetizationdensityofAuclustersondefectedgraphene. .................................. 86 5-10Totaldensityofstatesofdefectedgraphenewithgoldclustersandprojecteddensityofstatesontotheporbitalsofdefectedgraphene. ............ 87 5-11Chargedensitydifferencen(r)andmagnetizationdensityofFeclustersondefectedgraphene. .................................. 88 5-12Totaldensityofstatesofdefectedgraphenewithironclustersandprojecteddensityofstatesontotheporbitalsofdefectedgraphene. ............ 89 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyFIRST-PRINCIPLESELECTRONICSTRUCTUREANDTRANSPORTCALCULATIONSINMATERIALSWITHDEFECTSANDIMPURITIESByManojK.SrivastavaMay2012Chair:Hai-PingChengCochair:PeterJ.HirshfeldMajor:Physics Wepresentelectronicstructureandelectrontransportstudiesofmaterialswithdefectsandimpuritiesusingdensity-functionaltheory.Wedevelopaplanewavetransportmethodbasedondensity-functionaltheoryforlowsymmetrynonorthogonallattices.ThisisachievedbygeneralizingChoiandIhm'salgorithmforhighsymmetrylatticeswhichrequiresthetransportdirectionalongalatticevectorthatmustbeperpendiculartothebasalplaneformedbytwootherlatticevectors.Thisrestrictionisovercomeinourmethod,allowingsolutionstoproblemsinwhichthetransportdirectionisnotalonganylatticevectors.Weapplyourgeneralizedtransportmethodtocalculateinterfaceresistivityofgrainboundariesincopper.Otherthansurfacedefects,wealsostudypointdefectssuchassingleatomvacancyandimpurities.Usingelectronicstructuremethods,weinvestigateadsorptionofgoldandironclustersonperfectanddefectedgraphenewithasinglevacancy.Wefocusonthesizedependenceoftheelectronicpropertiessuchasbindingenergy,chargetransfer,magnetization,anddensityofstates.PerfectgrapheneisfoundtobedopedforAuclusterswithanoddnumberofatomsandundopedwithanevennumberofatoms.Anodd-evenoscillationinthemagneticmomentsisobservedinAu-perfectaswellasdefectedgraphenesystem.WhileFenclustersremaintobemagneticforalln,thespinofasingleFeatomonadefectsiteisverysmallduetoacovalentbondingtoCatoms. 9

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CHAPTER1INTRODUCTIONANDBACKGROUND 1.1Materials Mostmaterialsfoundinnatureorman-madecanbebroadlyclassiedintothreecategories-metals,insulators,andsemiconductors.Materialswhicharegoodconductorsofelectricityareknownasmetals.Mostoftheelementsintheperiodictable,forexample,sodium,potassium,copper,silver,gold,iron,cobalt,nickelbelongtothemetalcategory.Badconductorsofelectricityarecalledelectricinsulators,andexamplesofthoseincludebutnotlimitedtoair,glass,ceramics,andwood.Semiconductorsarethematerialswhichhaveconductivitiesbetweenmetalsandinsulatorsandarecharacterizedbysmallbandgaps,forexample,silicon,germanium,galliumarsenide,etc.. Metalshavebeenatthecenterofhumancivilizationbecauseoftheirusefulphysicalproperties.Metalsaremalleableandductile,thusprovidemechanicaladvantageoverotherclassofmaterials.Duetotheirhighelectricalconductivity,metalssuchascopperandaluminumareusedinelectricalcircuits.Inpresent-daytechnology,copperisthepopularchoiceofinterconnectinintegratedcircuits.Insulatorssuchasceramicberandmicroporussilicaareusedinheatinsulationathightemperatures,whileglassandaluminaareusedinhighvoltageelectricinsulation.Semiconductorsareusedtomakediodesandtransistors,andthusarethefoundationofpresent-dayelectronics. Inthelastthreedecades,thediscoveryofcarbonbasedmaterialshasdrawnwideinterestinthescienticcommunity,boththeoreticallyandexperimentally,duetotheirinterestingphysicalpropertiesaswellastechnologicalapplications.Carbonatomsareabletoformvariouskindsofhybridizationsuchassp,sp2,andsp3,whichgivesrisetodifferentinterestingallotropes.C60alsoknownasbuckminsterfullereneorbuckyball,azerodimensionalallotropeofcarbon,wasdiscoveredbyRichardSmalleyandhiscolleaguesatRiceUniversitynearlythreedecadesago[ 1 ].Thecarbonnanotube 10

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(CNT),anotherinterestingallotrope,discoverdbyIjimain1991[ 2 ],hascylindricalshapeandcanbeconsideredtobecreatedbyrollingoneatomthickgraphene.CNTsareclassiedbychiralvectorChwhichisgivenby(n,m),wherenandmareintegersdenotingnumberofunitcellvectorsinthetwodirectionsofhexagonalgraphenelattice.NanotubeswhichhaveachiralvectorChas(n,0)areknownaszigzagwhiletheoneswith(n,n)arecalledarmchair.Allothernanotubesareknownaschiralnanotubes.FullerenesandCNTshavenumerouspotentialapplicationsintheeldsoftransistors,solarcells,gassensors,hydrogenstorage,ultracapacitors,andbiopharmaceuticalsetc.[ 3 4 5 6 7 8 9 10 ]. Asinglelayerofcarbonatomsarrangedintwodimensionalhoneycomblatticeisknownasgraphene.Althoughrsttheoreticalstudyofgraphenegoesbackseveraldecades[ 11 ],itwasdiscoveredexperimentallyin2004byNovesolov[ 12 ].Graphenehaspeculiarproperties;itisazerobandgapsemiconductor,andhaslineardispersionaroundtheFermienergygivingrisetozeroeffectivemassofelectronsandholes.Duetomasslesschargecarriers,grapheneisanexcellentsystemtotesttheoriesofQuantumElectrodynamics.QuantumHalleffectwasobservedingraphenebyWilliamsetal.[ 13 ].AnotherinterestingpropertyofgrapheneistohaveminimumconductivityinthelimitofvanishingchargecarrierdensitywhichwasexperimentallydiscoveredbyChenetal.[ 14 ].Otherthantheusefulelectronicfeatures,inexperimentsperformedbyLeeetal.,graphenewasfoundtohaveexcellentmechanicalpropertiessuchashighbreakingstrengthandYoung0smodulus[ 15 ]. Graphenehaspotentialtobeaviablecandidateinthepostsiliconageelectronics.MOSFETusinggraphenewasachievedbyLemmeetal.[ 16 ].Liaoetal.reportedfabricationoftransistorswithfrequencyperformanceofmorethan300GHz[ 17 ].Theuseofgrapheneinambipolarelectronicsandthusindevicessuchasfrequencymultipliers[ 18 19 ],highfrequencymixers[ 20 ],anddigitalmodulators[ 21 22 ]hasalsobeendemonstrated.Changeofthecarrierdensityingraphenebythe 11

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adsorptionofmoleculesandatomsmakesgrapheneexcellentchoiceforgassensors.ThiswasshownforNH3,CO,H2O,andNO2moleculesbySchedinetal.[ 23 ].CompositematerialssuchasconductiveplasticswerecreatedbyStankovichetal.usinggraphene[ 24 ].Anotherpotentialapplicationofgrapheneisintheeldofhydrogenstorage[ 25 ]. 1.2DefectsandImpurities Nothinginthisworldisperfect,butthisimperfectioniswhatmakestheworldinteresting.Ageneraldenitionofdefectisthepresenceofaregionwithdifferentmicroscopicarrangementofconstituentionsascomparedtotheperfectcrystal.Acrystalcanhavefourkindsofdefects-pointdefects,linedefects,planardefectsandbulkdefects. 1.2.1PointDefects Anabsenceofatomsi.e.vacancies,andpresenceofextraatomsi.e.interstitialsconstitutepointdefectsinthecrystal.ApointdefectwithpositiveandnegativeionvacanciesisclassiedasSchottkytype,whileadefectwithvacanciesandinterstitialsisknownasFrenkeltype.Impuritiesarepresenceofforeignatomsinthecrystaleitherasasubstitutionalorinterstitial.Presenceofpointdefectsinmaterialsgiverisetointerestingphysicalproperties.Oneexampleofthisistheelectricalconductivityofioniccrystals.Ioniccrystals,suchasNaClorKCl,havehighresistivity.Astheyhavelargebandgaps,conductivityinsuchcrystalscannotbeduetoelectrons.Ithasbeenfoundthattheconductivityinthesecrystalsisthroughmotionofionswhichisincreasedbythepresenceofvacancieswhichfacilitatemovementofions[ 26 ].Thecolorofioniccrystalsisalsoexplainedthroughthepresenceofvacancies.Whenanegativechargedionisremovedfromthecrystal,electronslocalizearoundthevacancytomaintainchargeneutralityofthecrystal.Anelectronboundwithapositivelychargedcenterhasanenergyspectrum,andopticalabsorptionspectraproducedbyvariousexcitationsbetweentheenergylevelsofsuchelectrongivesrisetothecolorofthecrystal[ 26 ]. 12

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Pointdefects,suchasvacanciesandimpuritiesingraphene,havedrawnconsiderableinterestrecentlyandhavebeenextensivelystudied[ 27 28 ].Singleatomordoubleatomvacanciesongraphenechangetheelectronicstructureofgrapheneandgiverisetointerestingproperties.Graphenewithasinglevacancydefectisfoundtobemagneticwithmageneticmoment1B[ 29 ].Somevacancytypedefectsdesignedingraphenecanopenbandgap[ 30 ],andthereforeareusefulforgraphenebasedelectronics.Innumericalsimulations,vacancieshavebeenfoundtoincreaseelectronscattering,andthusdecreasetheconductivityofgraphene[ 31 32 ].Impuritiescanalsobeusedtomodifytheelectronicstructureofgraphene.PotassiumisusedtochemicallydopegrapheneandtheconductivityofgrapheneasafunctionofimpuritydensityisstudiedbyChenetal.[ 14 ].Density-functionalcalculationsofadatomsonthesurfaceofgrapheneisperformedbyChanetal.[ 33 ].AtheoreticalunderstandingofchargedimpurityscatteringingrapheneisprovidedbyAdametal.[ 34 ].UsingquantumMonteCarlomethodmagneticimpuritiesonthesurfaceofgraphenearestudiedbyHuetal.[ 35 ]. 1.2.2DislocationsandGrainBoundaries Otherthanpointdefects,arealcrystalcanalsohavelinedefectssuchasdislocationsoflatticeplanes.DislocationsarealwayscharacterizedbyanonzeroBurgersvector.Edgeandscrewdislocationsarethesimplestlinedefectspossibleinthecrystal,althoughmuchmorecomplicatedcombinationsofbothtypesofdislocationsarepresentintherealcrystals.Usingdislocations,onecanexplainwhythepuricationofcrystalthroughsomeprocesssuchasannealingyieldsrelativelylowerstrengthcomparedtothatofthedefectivecrystal.Infact,onemightexpecttheexactopposite,astheannealedcrystalshouldapproachtothelimitofperfectcrystalandthereforeshouldhavehighstrength.Itwasfoundthatdislocationscanmoveeasilythroughlatticeaslongastheydonotencounterpointdefects.Inanotsowellpreparedcrystal,therearemorepointdefectswhichpreventmovementofdislocations,thusprovidingthecrystalhigherstrength.Intheannealedcrystal,pointdefectsareremovedthusmore 13

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movementofdislocationsispossible,whichresultsinlowerstrengthofthecrystal.Insomerecentworks,moleculardynamicssimulationsofdislocationprocessinaluminumwereperformedbyYamakovetal.[ 36 ].DislocationsinlightemittingdiodesbasedonGaNwerestudiedbyLesteretal.[ 37 ].Ingraphene,carefullyengineereddislocationswerereportedtoactasmetallicwiresduetoself-doping[ 38 ]. Beyondonedimensionaldefects,higherdimensionaldefects,suchasplanardefects,arealsopossibleinthecrystal.Grainboundaries(GBs)areexamplesofplanardefectswhichareformedbythemisorientationbetweentwograinsofthecrystal.Dependingontherotationaxisandboundaryplane,GBsareclassiedintotwotypes-tiltGBandtwistGB.Inrealcrystals,generalGBswhicharecombinationoftiltandtwistGBsareoftenpresent[ 39 ].IntiltGB,axisofrotationliesinthesameplaneastheboundary,whileitisperpendiculartotheboundaryplanefortwistGB.GBsarealsoclassiedbasedonthemagnitudeoftherotationangle.Iftheangleofrotationislessthan150theyarecalledlowanglegrainboundaries(LAGB),whilehighanglegrainboundaries(HAGB)havearotationanglelargerthan150.TherearealsoGBscorrespondingtosomespecialorientationswhichhavesmallerinterfacialenergiesascomparedtoHAGBs.AllGBsaredescribedbycoincidentsitelattice(CSL)model[ 40 ].Uponrotationoflatticesaboutacommonaxissomelatticepointsofbothgrainscoincide,andCSLisformed.AnimportantquantitytodescribesuchGBsis,whichisdenedastheratioofthevolumeoftheCSLlatticetothatoftheprimitiveunitcellofthecrystal.UndertheframeworkofCSL,isthemostsymmetricGBwhichisformedbyarotationof600oftwograinsoffacecenteredcubiclatticeabout<111>direction.OtherexamplesofGBsinclude,,,,,,,,and. TheGBschangethephysicalpropertiesofthematerialbyaffectingitselectricalconductivity,thermalconductivityandmechanicalstrength.OneoftheapplicationsofGBsisinstrengtheningofmetalsastheystopthemotionofdislocationsinthe 14

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crystal.Butitcomesatthecostofincreasedelectronscattering,thusreducingtheelectricalconductivity.Asuitabletradeoffofthesetwoeffectscaninprincipleimprovethequalityofmetals.Forexample,inanexperimentbyLuetal.[ 90 ],theintroductionoftwinboundarieshasbeenfoundtoincreasethemechanicalstrengthofcopper,whilekeepingitselectricalconductivityclosetothatofthepurecopper.Innanowires,scatteringbyGBsisfoundtobesignicantandvariousstudiesofsizedependenceofresistivityhavebeencarriedout[ 41 42 43 ].Asthetechnologyismovingtowardsmakingsmallerwidthnanowires,it'snotchoice,butrathernecessityforustounderstandtheGBdefects.Inanotherexoticeldofhightemperaturesuperconductors,GBshavebeenfoundtobethereasonforthesuppressionofmaximumcriticalcurrentanddependenceofcriticalcurrentwithmisorientationanglehasbeenstudied[ 44 45 46 ]. Inthisthesis,wepresenttheresultsofourcomputationalstudyofgrainboundarydefectsincopper,andpointdefectssuchassinglevacancyandimpuritiesingraphene.Weemployelectronicstructureandelectrontransportmethodsbasedondensity-functionaltheoryinourcalculations.Wedevelopaplanewavetransportmethodforlowsymmetrylattices,andapplyittocalculateinterfaceresistanceofvariouscoincidentgrainboundaries.Electronicstructurecalculationsforthegoldandironclusteradsorptiononperfectgrapheneanddefectedgraphenewithsinglevacancyareperformed.Wediscussthesizedependenceoftheelectronicpropertiessuchasbindingenergy,chargetransfer,magnetization,anddensityofstates. 15

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CHAPTER2THEORETICALMETHODS ItwassuggestedcenturiesagobyIndianandGreekphilosophersthateverythingaroundusisconstitutedofatoms.Inthelate19thcentury,Rutherford,throughhisexperiment,demonstratedthatthestructureofanatomiscomposedofanucleusandelectrons.Bohrdescribedtheelectronsintheatomasclassicalparticlesrevolvingaroundthenucleusinquantizedorbits.Although,themodelproposedbyBohrwasincorrect,itpavedthewayforthediscoveryofquantummechanics.ThroughthepioneerworkofdeBroglie,Schrodinger,Heisenberg,andDirac,thebasicfoundationofquantummechanicswaslaid.Inquantummechanics,theSchrodingerequationissolvedtogetthewavefunction,andphysicalpropertiesofanysystemcanbecalculatedonceitswavefunctionisknown.ThesolutionofSchrodinger'sequationisreadilyobtainedforsingleelectronsystems,howeverwhenthenumberofelectronsincreases,solvingSchrodinger'sequationbecomesadauntingchallenge.Thus,eventhoughquantummechanicsprovidesthetheoreticalsetup,itsapplicationtothereallifeentitiessuchasatoms,moleculesandsolidsrequiresvarioustheoreticaltools. TherstcalculationforatomswithmanyelectronswasperformedbyD.R.Hartree[ 47 ].InHartree'smethod,Shrodinger'sequationdescribingthemotionofelectronunderthepotentialofnucleusandotherelectronsissolvedinself-consistentapproach.InHartree'swork,wavefunctionofthemany-particlesystemwasnotantisymmetrized,butFocktookcareofthatdeciencyanddevisedamethodwhichisknownasHartree-Fockmethod[ 48 ].CellularmethodbyWignerandSeitz[ 49 50 ],augmentedplane-wavemethod(APW)byJ.C.Slater[ 51 ],Green'sfunctionmethodofKorringa,KohnandRostoker(KKR)[ 52 53 ],andorthogonalizedplane-wavemethod(OPW)byHerring[ 54 ]areexamplesofafewmajortheoreticalmethodsdevelopedtostudythemanyelectronproblem. 16

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Furtheradvancedmethodssuchasdensity-functionaltheory,quantumMonteCarlomethods,andmany-bodyperturbationmethodshavealsobeendevelopedtotacklemany-electronproblem.Ourstudyisbasedonthedensityfunctionaltheoryinwhichtheproblemofinteractingsystemisreplacedwithanon-interactingproblemwhichhasthesamegroundstatedensityasthatoftheoriginalinteractingsystem.Density-functionaltheoryhasbeenphenomenalindealingwithlargenumberofmaterialssuchasmetals,insulators,polymers,andbiologicalsystems. 2.1Density-FunctionalTheory Hamiltonianofaphysicalsystemcomprisedofelectronsandnucleihastheform[ 55 ] H=~2 2meXir2iXi,IZIe2 jriRIj+1 2Xi6=je2 jrirjjXI~2 2MIr2I+1 2XI6=JZIZJ jRIRJj, (2) whereloweranduppercaseindicessuchasiandIcorrespondtoelectronsandnuclei,respectively.riandearepositionandchargeofithelectron,whileRIandZIrepresentthoseofIthnucleus.MassofelectronsandnucleiaregivenbymeandMI.Equation 2 incorporatesalltheinteractionsnamelyelectron-ion,electron-electron,andion-ioninteractionsinthesystem.Inordertounderstandphysicalbehaviorofasystem,oneneedstosolvethecelebratedSchrodinger'sequationwiththeaboveHamiltonian.Otherthanafewsimplesystems,suchasHatomorH+2molecule,obtainingsolutionofSchrodinger'sequationforreallifesystemisanimpossibletask,andapproximationsareneededtosolvetheHamiltonian.Asthenucleiareheavierthantheelectrons,theymoveslowerascomparedtotheelectronsandthuswecanneglecttheirkineticenergy.ThisisknownasBorn-Oppenheimerapproximation.Theion-ioninteractiontermisjustaconstantindescribingelectronsandcanbeneglectedaswell.Withtheapproximations,theHamiltonianbecomes H=1 2Xir2iXi,IZI jriRIj+1 2Xi6=j1 jrirjj. (2) 17

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InaboveequationwehaveusedHartreeatomicunits(~=me=e=1).Inamorecompactnotation,whereT,VextandVeerepresentkineticenergy,externalpotentialofnucleus,andelectron-electroninteractionenergy,respectively,aboveequationcanbewrittenas H=T+Vext+Vee (2) ThebasicfoundationofdensityfunctionaltheoryislaidupontheworkofHohenbergandKohn[ 56 ]whoshowedthatanypropertyofaninteractingparticlesystemcanbedeterminedbythegroundstatedensityn0(r). Hohenberg-Kohntheoremsarestatedas1.ForasystemofinteractingparticlesunderanexternalpotentialVext(r),thegroundstatedensityn0(r)uniquelydeterminestheexternalpotentialVext(r),uptoaconstant.Asthepotentialisknown,theHamiltonianofthesystemisfullydetermined,thereforethewavefunctionofthesystemcanbecalculated. 2.ForanexternalpotentialVext(r),wecandeneafunctionalfortheenergyE[n]intermsofthedensityn(r).Groundstateenergyofthesystemistheglobalminimumofthefunctional,andthegroundstatedensityn0(r)isthedensitythatminimizesthefunctional. AlthoughtheHohenberg-Kohntheoremshowedthatthereexistsadensityfunctionaltodeterminethepropertiesofthesystem,theydidnotgiveawaytoconstructthefunctional.Kohn-Sham[ 57 ]suggestedanapproachinwhichtheproblemofinteractingsystemisreplacedbyanauxiliaryindependentparticleproblemwiththeassumptionthatthegroundstatedensityofauxiliarysystemisidenticaltothatoftheoriginalinteractingsystem. InKohn-Sham'sapproach,densityofauxiliarysystemisgivenby n(r)=NXi=1ji(r)j2,(2) 18

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wherei(r)aretheorbitalsofnon-interactingauxiliarysystemcontainingNparticles.Kineticenergyoftheauxiliarysystemcanbewrittenas Ts[n]=1 2NXi=1hijr2jii.(2) TheHartreeenergyofthesystemisgivenby EHartree[n]=1 2Zdrdr0n(r)n(r0) jrr0j.(2) IntheKohn-Shamapproach,theenergyfunctionalfortheinteractingsystemisconstructedbycombiningthekineticenergy,energyduetoexternalpotential,andHartreeenergyas EKS=Ts[n]+ZdrVext(r)n(r)+EHartree[n]+Exc[n](2) whereExc[n]istheexchange-correlationenergy,whichcontainsallmany-bodyeffectsofexchangeandcorrelation.Solutionofthenon-interactingauxiliarysystemforthegroundstatecanbeobtainedbyminimizingthetotalenergyfunctionalwithrespecttoKohn-Sham'sorbitalsas EKS i(r)=Ts i(r)+Eext n(r)+EHartree n(r)+Exc n(r)n(r) i(r)=0,(2) withtheconstraint hijji=ij.(2) Thisgives (HKSi)i(r)=0,(2) where HKS(r)=1 2r2+VKS(r),(2) 19

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VKS(r)=Vext(r)+EHartree n(r)+Exc n(r)Vext(r)+VHartree(r)+Vxc(r). (2) Equation 2 2 arethecelebratedKohn-Sham'sequations,whichcanbesolvedself-consistentlytogetthegroundstatedensityoftheinteractingsystem.Sofarinthetheory,otherthantheansatz,nofurtherapproximationsaremadeandtheseequationswouldyieldtheexactsolutionoftheinteractingsystemiftheexchange-correlationpotentialVxc(r)wereknown.But,wedonotknowVxc(r)andapproximationsareneededtomodelit. Oneoftheattemptstomodelexchangeandcorrelationfunctionalisthelocalspindensityapproximation(LSDA),whereasolidisconsideredasclosetothehomogeneouselectrongas,andthusVxc(r)ofthesolidistakentobesameasthatofthehomogeneouselectrongas.Inmathematicaltermstheapproximationcanbewrittenas[ 55 ] Exc[n,n]=Zd3rn(r)homxc(n(r),n(r)),(2) whereandrefertoupanddownspin,andhomxc(n(r),n(r))istheexchangecorrelationpotentialforthehomogenouselectrongas.Thistermcanbefurtherseparatedintoexchangeandcorrelationterms.Theexchangepotentialforthehomogenouselectrongasisgivenby x=3 4(6n )1=3.(2) ThecorrelationpartiscalculatedbyMonteCarlomethods. AlthoughLSDAfunctionalsworkremarkablywellwithhomogeneoussystemssuchassolids,forinhomogeneoussystemssuchasatomsandmoleculesfunctionalsbasedongeneralizedgradientapproximation(GGA)givebetterresults.InGGAfunctionals,exchangecorrelationisnotonlyfunctionalofthedensitybutalsoofthemagnitudeofgradientofthedensity.TheexchangecorrelationenergyinGGAapproximationcanbe 20

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writtenas[ 55 ] Exc[n,n]=Zd3rn(r)xc(n,n,jrnj,jrnj,..)Zd3rn(r)homx(n)Fxc(n,n,jrnj,jrnj,..), (2) whereFxcisadimensionlessnumberknownasexchangeenhancementfactor.ExpansionofFxcisgivenintermsofreduceddensitygradientssmwhichisdenedas sm=jrmnj (2kF)mn=jrmnj 2m(32)m=3(n)(1+m=3).(2) Usingsm,Fxcanbewrittenas Fx=1+10 81s21+146 2025s22+....(2) VariousexchangecorrelationfunctionalsbasedonnumerousformsofFxhavebeenproposed,mostnotablyofwhicharebyBecke(B88)[ 58 ],PerdewandWang(PW91)[ 59 ],andPerdew,Burke,andEnzerhoff(PBE)[ 60 ].Forsmallvaluesofs(0
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betweentwoelectrodes.Conductanceofthesampleiscalculatedintermsofthetransmissionprobabilitiesofthechannelsandgivenby G=2e2 hXkTk(2) whereTkisthetransmissionprobabilityofchannelk,andthefactoroftwoisduetothespin.Therearetwoprevalentmethodsintheeldtocalculatetransmissioncoefcients:Green'sfunctionformalismandscatteringformalism. Forasystemofsemi-inniteelectrodesconnectedtoadevice,theGreen'sfunctionisgivenby[ 64 ] G(E)=(ESH)1,(2) whereHandSareHamiltonianandoverlapmatrix,andEistheenergyparameter.ThetransmissioncoefcientcanbecalculatedbyusingCaroli'sformula[ 65 ] Tk=Tr[LG+CRGC],(2) whereGCistheGreen'sfunctionofthedeviceregionandisgivenby GC=(ESCCHCCLR)1(2) and, L=R=i[+L=RL=R].(2)L=Ristheselfenergyoftheleft/rightelectrode. Inthescatteringformalism,theopenboundaryconditionsoftheelectrodesarerepresentedinanincidentwaveandthetransportproblemissolvedastheproblemofthescatteredwavefunction.ABloch'sstatefromtheelectrodesundergoesreectionandtransmissionuponincidenceontheresistiveregion.Forastatekfromtheleft 22

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electrodewhichisincidentonthedeviceregionweget[ 83 ] =8>>><>>>:k+Xk0rkk0k0z0Xk0tkk0k0zl(2) Thetransmissionandreectioncoefcientsaredeterminedbyimposingthecontinuityofthewavefunctionaswellasitsderivativeattheboundarybetweenelectrodesandthedeviceregion.Usingtheboundaryconditions,transmissioncoefcientsaregivenby T=Tr[TyT]=Xk,k0jTk,k0j2(2) whereTk,k0=p Ik=Ik0tkk0,andIkistheprobablitycurrentcarriedbytheBloch'sstatek. Inourwork,wehaveusedscatteringformalismusingcomplexbandstructuretothequantumtransportproblem,whichisdiscussedingreatdetailinchapter3. 2.3SemiclassicalBoltzmannTransport Insemiclassicaltransport[ 66 67 68 ],electronsaretreatedasclassicalparticlesexceptthattheyobeyFermistatistics.Adistributionfunctionfs(r,k,t)isdenedforelectrons.Physicalinterpretationofthedistributionfunctionisthenumberofelectronswithpositionr,wavevectorkandspinsattimet.Ifweapplyaeld,electronswillbeoutofequilibrium,butwaitsufcientlylongenoughsothatthesystemhasreachedsteadystate,wehave df dt=0(2) Therearethreecontributionstothetotalchangeofthedistributionfunction:changecausedbymovementofelectronsi.e.drift,forceontheelectronsduetotheappliedeld,andscatteringofelectronsbythedefectsinthelattice.Combiningtheseeffects,wecanwrite df dtdrift+df dteld+df dtscattering=0(2) 23

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Wecancalculatethesetermsindividuallyas df(r,k,t) dtdrift=v(k).rrf(r,k,t), df(r,k,t) dteld=e ~rkf(r,k,t).E,df(r,k,t) dtscattering=Xk0Pkk0[f(r,k0,t)f(r,k,t)], wherevandEarethevelocityofelectronandtheappliedeld.Pkk0istheprobabilityoftheelectrontoscatterbetweenstateskandk0.Combiningallabovetermsweget v(k).rrf(r,k)+e ~rkf(r,k).E+Xk0Pkk0[f(r,k0)f(r,k)]=0(2) AboveequationistheBoltzmannequationthatneedstobesolvedinordertogetthedistributionfunction.Inthefollowing,wefurthersimplifyEquation 2 forourgeometryandwritedowntheboundaryconditionsneededtogettheuniquesolutionoftheequation. 2.3.1EquationinCurrentPerpendiculartoPlane(CPP)Geometry Weconsideralayeredsystemwhichishomogeneousinthexandydirectionsbutnonuniformintheperpendicularzdirectionwhichmaybeduetodifferentlayersofmaterials,interfacesetc..InthiscasethedistributionfunctionasobtainedfromEquation 2 dependsonz.Inordertogetthedistributionfunctionforthewholesystem,wesolveEquation 2 foreachlayeranduseboundaryconditionsoftomatchsolutionsfromthelayers[ 66 67 68 ].IfweapplyanelectriceldEinasystemwhichisinhomogeneousinthezdirection,currentisgivenbyJ(z)=(z)"(z) 24

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where(z)isthelocalconductivityand"(z)isthelocaleld.Wecandeneadeviationfunctionh(r,k)fromtheequilibriumasf(r,k,t)=f0(k)+(k)h(r,k), wheref0istheequilibriumdistributionfunctionandDeltafunction(k)ispresentbecausethestatesattheFermisurfaceareinvolvedinthetransportinmetalsatroomtemperature. WecanwriteEquation 2 intermsofthedeviationfunctionfortheCPPgeometryas vz(k)@h(z,k) @zXk0Pkk0[h(z,k0)(0k)h(z,k)(k)]=evz(k)@f0 @k@V @z,(2) wherebecauseofsmallE,higherordertermsareneglectedintheeldterm.Letusassumethatthescatteringisisotropic,thusPkk0=(kk0) n, wherenisthedensityofstatesattheFermienergyandisthescatteringtime.UsingaboveexpressionforPkk0,wecanderive Xk0Pkk0h(z,k)(k)=h(z,k)(k)=(2) Xk0Pkk0h(z,k0)(0k)=(z)(k)=(2) AboveequationscanbepluggedintoEquation 2 toyield vz(k)@h(z,k) @z+h(z,k)(z) =evz(k)@V @z(2) LetusdeneanisotropicdistributionfunctionashA(z,k)=h(z,k)(z)and=eV.Usingthesevariables,Boltzmann0equationinthecaseofcurrentperpendiculartothe 25

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planegeometrybecomes: vz(k)@hA(z,k) @z+hA(z,k)(z) =vz(k)@ @z(2) Asthedistributionfunctionisanisotropic,itmustsatisfy XkhA(z,k)(k)=Xk[h(z,k)(z)](k)=0(2) ThedistributionfunctioniscalculatedbyobtainingthesolutionofEquation 2 whichalsosatisesEquation 2 ,andisdiscussedinthenextsection. 2.3.2SolutionofBoltzmann0sEquation ThegeneralsolutionoftheBoltzmann0sequationforCPPgeometryforthelayeriandspinsisgivenby[ 66 67 68 ] hAs(z,k)=Js isevisz(k)isFis(k)e(z=visz(k)is)+Fis(k0)e(z=visz(k0)is) visz(k)visz(k0)visz(k) visz(k) visz(k)visz(k0)kk0,(2) andgeneralizedpotentialcanbeobtainedas (z)=isJsJs isez+Fis(k0)e(z=visz(k0)is) visz(k) visz(k)visz(k0)kk0,(2) where,isandFis(k)areconstantstobedeterminedusingboundaryconditions.InaboveexpressionsJsisthecurrentdensityandisisthebulkconductivitywhichisgivenby is=e2 VXk(visz(k))2is(k)(2) BoundaryconditionsfortheBoltzmann0sequationaredeterminedbyuxconservationateachinterface.Letusdenethedistributionfunctionsfortheelectronsgoingin+zandzdirectionsinlayeriofthematerialash+,ji(z,k?)andh,ji(z,k?),herejreferstothebandindexindicatingmorethanoneBloch'sstateatagivenk?.Therelationshipbetweendistributionfunctionsinthelayeriandi+1canbeestablishedbytheknowledgeofreection(T+i,T+i)andtransmissionprobabilities(T++i,Ti) 26

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oftheelectronsincidentattheinterfacezi.T+i(k,k0)isthereectionprobabilityofanelectroninstatek0goingin+zdirectionuponreectionfromtheinterfacezitravelinginzdirectioninthestatek.T++i(k,k0)isthetransmissioncoefcientoftheelectrongoingin+zdirectioninstatek0togoto+zdirectioninthestatek.SimilarconventionsholdtrueforT+iandTi.Conservationofuxyields h+,ji+1(z+i,k?)=NRXj0,k0?T+i(jk?,j0k0?)h,j0i+1(z+i,k0?)+NLXj0,k0?T++i(jk?,j0k0?)h+,j0i(zi,k0?)(2) h,ji(zi,k?)=NLXj0,k0?T+i(jk?,j0k0?)h+,j0i(zi,k0?)+NRXj0,k0?Ti(jk?,j0k0?)h,j0i+1(z+i,k0?)(2) Intheaboveexpressions,NLandNRrepresentnumberofBlochstatesk0totheleftandrightoftheinterfacezi.Ifthelayersareperiodicindirectionperpendiculartotransportsothatduringtransmissionandreectionthemomentumofelectronsparalleltotheinterfaceisconserved,thenEquation 2 2 canbesimpliedas h+,ji+1(z+i,k?)=NRXj0T+i(j,j0)h,j0i+1(z+i,k?)+NLXj0T++i(j,j0)h+,j0i(zi,k?)(2) h,ji(zi,k?)=NLXj0T+i(j,j0)h+,j0i(zi,k?)+NRXj0Ti(j,j0)h,j0i+1(z+i,k?)(2) Theseequationscanbefurthersimpliedbyconsideringunitincidentuxin+zandzdirections.Then,wehave NRXjT++(j,j0)+NLXjT+(j,j0)=1(2) NLXjT(j,j0)+NRXjT+(j,j0)=1(2) Byconsideringunituxleavingtheinterface,weget NLXj0T++(j,j0)+NRXj0T+(j,j0)=1(2) 27

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NRXj0T(j,j0)+NLXj0T+(j,j0)=1(2) ThereisonecaseinwhichBoltzmann0sequationcanbesolvedsimply.Letusimaginethattherearenointerfaces,whichmeansthattheelectronicstructureofallthelayersaresame.Astherearenointerfaces,thecurrentwillbeconservedi.e.@J(z)=@z=0.So,weget hA(z,k)=vz(k)@(z) @z(z)(2) ThenEquation 2 impliesthat(z)@(z)=@zisindependentofz,andwehave (z)@(z) @z=JV en<(vz)2>(2) AsthevoltagedropacrossthesampleiseV=Rdz@(z)=@z,theresistanceofthesampleisgivenby RA=Rdz1 (z) e2n<(vz)2>=Zdz(z)(2) whereAistheareaofthesampleperpendiculartothetransportdirectionand(z)isthelocalresistivity. 28

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CHAPTER3PLANEWAVETRANSPORTMETHODFORLOW-SYMMETRYLATTICES ThestudyofquantumtransportofelectronsthrougharesistiveregionconnectedbetweenelectrodeshastypicallybeenbasedonthemesoscopictheoryofLandauerandButtiker[ 62 63 ].Forrst-principlescalculations,theprevalentapproachintheeldisbasedonaGreen'sfunctionformalismusinglocalizedbasissets[ 69 70 71 72 ],thetight-bindingHamiltonian[ 73 74 ],orotherrealspacebasedmethods[ 75 76 ].AnalternativeapproachtotheGreen'sfunctionmethodisbasedonscatteringtheory,wherebytheopenboundaryconditionsarerepresentedinanincidentwave,andthequantumtransportproblemissolvedastheproblemofthescatteredwavefunctionduetotheresistiveregion.Inthecaseofnoninteractingelectrons,thescattering-basedapproachisshowntobecompletelyequivalenttothenonequilibriumGreen'sfunctionmethod[ 77 ]. Someoftheearliestrst-principlesquantumtransportcalculationsusethescatteringapproach,andtheelectrodesareapproximatedusingajelliummodel[ 78 79 ].Amorerigorousrst-principlesmethod[ 80 81 ]basedonthescatteringtheoryandthepseudopotentialsisimplementedwithinthePWCONDpartoftheQuantumEspressopackage[ 82 ],andhasbeenappliedtostudyanumberofquantumtransportproblems[ 83 84 85 86 ].Thescatteringapproachtoquantumtransportalsohassuccessesinthestudyofspintronics,inwhichlayeredsystemsarestudiedusingthelayerKorringa-Kohn-Rostoker(layer-KKR)approach[ 77 87 ].Avariantofthescatteringapproachistoinvokeasourceandsinkrepresentedbyacomplexpotential[ 88 ],ormoredirectlytocomputeageneralizedcomplexbandstructure[ 89 ]. Thelayer-KKRandPWCONDcodesareverysimilarintheirquantumtransportapproach.Theyshareacommonfeaturethatbothusetwo-dimensionalplanewave PendingpublicationinPhysicalReviewBbytheAmericanPhysicalSociety. 29

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basis,whichisparticularlysuitedforlayeredmaterials,andisalsoanecessaryingredientforplane-wavebasedquantumtransportcode.Bothalsohaveseverelimitations.Thelayer-KKRmethodcanonlybeimplementedwithinthemufn-tinortheatomicsphereapproximations(ASA),requiringthatthespaceisdividedintosphereswithinwhichtheKohn-Shampotentialissphericallysymmetric.Thisseriouslylimitstheapplicationofthelayer-KKRmethod,andsofarithasnotbeenusedinopenstructuresormolecularsystems. AdifferentlimitationexistsforthemethodemployedinPWCOND,whichisbasedontheformalismderivedinbyChoiandIhm,andwewillrefertoastheChoi-Ihmmethod.Thisderivationassumesthatthedirectionoftheelectriccurrent(transportdirection)isalongaparticularbaselatticevectora3,whichmustbeperpendiculartothebasalplaneformedbytwootherlatticevectorsa1anda2.Thisnecessarilyrequiresthattheunderlyinglatticeofthematerialmustbeorthogonal.Fornonorthogonallatticeswheretheredoesnotexistanya3perpendiculartothebasalplane,asshowninFigure 3-1 ,theexistingalgorithmbreaksdownandthePWCONDcodeproducesincorrectresults.Thusthismethodcannotbeappliedtosolidswithlowsymmetrytrigonalortricliniclattices.Evenforahighsymmetrylattice,inordertocomplywiththerequirementofthemethod,oftenamuchlargerorthogonalunitcellmustbeused.Forexample,tocalculatetransportpropertiesalongthe[111]directionofafacecenteredcubic(fcc)lattice,withtheexistingalgorithmanorthogonalunitcell,whichcontainsthreetimesmoreatomsthantheprimitivenonorthogonalunitcell,asisshowninFigure 3-1 ,mustbeused. 3.1GeneralizedMethodforPlaneWaveTransport Inourstudy,wehavegeneralizedtheplanewavescattering-basedmethodforballistictransporttolowsymmetrynonorthogonallatticesbasedonthree-dimensionalBloch'stheorem.InthissectionweoutlinethebasicdifferencesbetweenourmethodandChoi-Ihmmehod,andnecessarystepsrequiredforthegeneralization. 30

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Inacrystallinesolid,aneigenstateofaHamiltoniandescribingelectronunderperiodicpotentialobeysthecelebratedBloch'stheoremwhichiswrittenas k(r+R)=eikRk(r),(3) wherekisthewavevectorwhichdescribeseigenvectorsandeigenvaluesoftheHamiltonian.Risagenerallatticevectorthatcanbewrittenasalinearcombinationoflatticevectorscomprisingtheunitcellofthesolid,forexample,inathree-dimensionalsolidR=n1a1+n2a2+n3a3,wherea1,a2,anda3arethelatticevectorsoftheunitcell,andn1,n2,andn3areintegers.Correspondingly,ingeneralthephasefactoreikRinBloch'stheoremalsocontainsallcomponentsofthethree-dimensionalwavevectork.Incontrast,foraone-dimensionalsystemthephaseintheBloch'stheoremreducestoeikzd,wherewetakethedirectionofperiodicitytobeinzdirectionwithalatticevectord. IntheChoi-Ihmapproach,onlyone-dimensionalBloch'stheoremisapplied.ThisgreatlysimpliesthealgorithmanditsimplementationinthePWCONDcode[ 83 ].However,itimposesaseriousrestrictionwhenappliedtoathree-dimensionalsystem,sinceitrequiresanorthogonalunitcellfortheelectrode,inwhichthethirdlatticevectora3mustbeperpendiculartothebasalplaneformedbylatticevectorsa1anda2. Thethree-dimensionalBloch'stheoremalongthedirectionofthelatticevectora3foranonorthogonalunitcell,asshowninFigure 3-1 ,canbewrittenas k(r+a3)=eik.a3k(r).(3) Thewavefunctionsaresolvedfromthesingle-particleKohn-Shamequationinthetwo-dimensionalreciprocalspacecorrespondingtotheperiodicityofthebasalplaneofthelattice.Thusatwo-dimensionalFouriertransformationisperformedonthewavefunctionandpotential.UsingtheFouriertransformationofthewavefunction,asdenedinEquation 3 ,theaboveequationistransformedintothetwo-dimensionalreciprocalspace.Iftheunitcellisnonorthogonal,thephaseintheFouriertransformations, 31

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asshowninEquation 3 andEquation 3 ,includenon-zerocontributionscontainingG1zandG2z,whicharethezcomponentsofthebasalplanereciprocallatticevectorsG1andG2.ThesecontributionsareassumedtobezerointheChoi-Ihmformalism,andthecurrentimplementationofPWCONDalsoneglectsthesecontributions. ThederivationofthecorrectFouriertransformofthewavefunctionisgiveninsubsection3.2.1 .ItisconvenienttowritetheBlochtheoremintermsofthewavefunctionwithouttheextraphasefactor,ei(PG1z+QG2z)z,i.e.,thewavefunctionthatiscurrentlycalculatedbythePWCONDcode,whichwerefertoasCIk(P,Q,z).TheBloch'stheoremforthenonorthogonalunitcell,writtenintermsofCIk(P,Q,z),is, CIk(P,Q,z+d)=eikzdei(PG1?+QG2?)a3?CIk(P,Q,z),(3) wheredanda3?arethecomponentsofthelatticevectora3inthezdirectionandxybasalplane,andG1?andG2?arexycomponentsofreciprocalspacevectorsG1andG2,respectively.Theadditionalphasefactorei(PG1?+QG2?)a3?isneededforthenonorthogonallattice,andisthekeydifferencebetweenthispaperandtheChoi-Ihmmethod. Wealsoneedthetwo-dimensionallocalpotentialV(P,Q,z),whichiscalculatedfromthethree-dimensionallocalpotentialV(P,Q,R)bydividingtheunitcellintoslabsalongthezdirection,andassumingV(P,Q,z)ei(PG1z+QG2z)ztobeindependentofzwithineachslab.ThisconditionismathematicallymanifestedinEquation 3 .Incontrast,theChoi-IhmmethodassumesaconstantV(P,Q,z)withouttheadditionalphasefactorduetoG1zandG2z.Thesedifferencesleadtothephaseei(PG1z+QG2z)zinthenalequations,Equation 3 andEquation 3 ,forthewavefunctionscorrespondingtothelocalpotentialandnonlocalpartofpseudopotential. Boundaryconditionsneededtodeterminetheexpansioncoefcientsofthewavefunctionarebasedonthree-dimensionalBloch'stheorem,whichusestheperiodicityinthedirectionoflatticevectora3.Equationsresultingfromtheboundary 32

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conditions,asshowninthesubsection3.2.2 ,arearrangedinamatrixform AX=eikzdeik?a3BX,(3) whereXisacolumnvectorcontainingtheexpansioncoefcientsofthewavefunction,andAandBarecomplexgeneralmatrices.Wedeterminekzfromtheeigenvalueeikzdofthisgeneralizedeigenvalueproblem.kzcaningeneralbeacomplexnumberandconstitutesthecomplexbandstructureofthesystem.Again,thereisanadditionalphasefactoreik?a3absentintheChoi-Ihmformulation. 3.2ComplexBandStructurewithNonorthogonalUnitCell Beforesolvingthescatteringproblemwheretwosemi-inniteelectrodesareconnectedtothescatteringregion,wecalculatethescatteringstatewavefunctionsbysolvingsingle-particleKohn-Shamequationfortheelectrodes.Thedetailsaregiveninthefollowingsubsection3.2.1 andsubsection3.2.2 3.2.1CalculationofWavefunction WestartwithKohn-Shamequationforthescatteringstatewhichiswrittenas[ 80 83 ] E(r)=~2 2mr2(r)+V(r)(r)+XlmClmXR?eik?.R?Wlm(rR?).(3) IntheaboveequationV(r)isthelocalpotentialandWlm(r)istheprojectorfunctionofthenonlocalpartofthepseudopotential.isthepositionofthatomintheunitcellandR?,k?arethelatticevectorsandwavevectorsinthebasalplaneoftheunitcell.Clmisgivenby Clm=XpqDlm,pqZd3r0[Wpq(r0)](r0).(3) ThecoefcientsDlm,pqcharacterizetheultrasoftpseudopotential[ 83 ].WecanwritethegeneralsolutionofEquation 3 as (r)=Xnann(r)+XlmClmlm(r),(3) 33

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wheren(r)andlm(r)arethesolutionsofhomogeneousandinhomogeneouspartsofEquation 3 .Thehomogeneousequationinvolvingthelocalpotentialiswrittenas En(r)=~2 2mr2n(r)+V(r)n(r),(3) wherendenotesnumberoflinearlyindependentsolutionsobeyingtheperiodicityconditioninthebasalplanen(r+R?)=n(r)eik?.R?.Theinhomogeneousequationforthenonlocalpartofthepseudopotentialisgivenby Elm(r)=~2 2mr2lm(r)+V(r)lm(r)+XR?eik?.R?Wlm(rR?),(3) wherelmisaparticularsolutionforall(,l,m)obeyingtheconditionlm(r+R?)=lm(r)eik?.R?.UsingtheFouriertransformation,wecanwritethewavefunctionandthelocalpotentialinthereciprocalspaceas V(r)=XP,Q,RV(P,Q,R)ei(PG1+QG2+RG3).r,(3) n(r)=XP,Q,Rn(P,Q,R)ei(k+PG1+QG2+RG3).r.(3) Wesplittheabovetransformationforpotentialandwavefunctionas V(r)=XP,QV(P,Q,z)ei(PG1?+QG2?).r?ei(PG1z+QG2z)z,(3) V(P,Q,z)=XRV(P,Q,R)eiRG3zz,(3) n(r)=XP,Qn(P,Q,z)ei(k?+PG1?+QG2?).r?ei(PG1z+QG2z)z,(3) n(P,Q,z)=XRn(P,Q,R)ei(kz+RG3z)z,(3) where(k?+PG1?+QG2?).r?=(kx+PG1x+QG2x)x+(ky+PG1y+QG2y)y.InequationsoftheFouriertransformations,Equation 3 andEquation 3 ,thepresenceofaphaseei(PG1z+QG2z)zisinducedbythenonorthogonalunitcell,whichcausesthebasal 34

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planereciprocalvectorsG1andG2tohavenon-zerocomponentsG1zandG2z.ThisphaseisnotpresentinthecorrespondingequationsinChoi-Ihmmethod. UsingtheFouriertransformationsofpotentialandwavefunctioninEquation 3 ,weobtaintheKohn-Shamequationinthereciprocalspaceforthelocalpotentialas En(P,Q,z)=~2 2md2 dz22i(PG1z+QG2z)d dz+jk?+PG1?+QG2?j2+(PG1z+QG2z)2n(P,Q,z)+XP0,Q0V(PP0,QQ0,z)n(P0,Q0,z). (3) Thephaseei(PG1z+QG2z)zinEquation 3 leadstotwoextraterms2i(PG1z+QG2z)d dzn(P,Q,z)and(PG1z+QG2z)2n(P,Q,z)inEquation 3 ascomparedtothecorrespondingequationintheChoi-Ihmmethod. UpontheinversionofEquation 3 weget V(P,Q,R)=1 dZd0dzV(P,Q,z)eiRG3zz.(3) Thethree-dimensionallocalpseudopotentialV(P,Q,R)iscalculatedbyperformingaselfconsistenteld(SCF)calculationforthesystem.SimilartotheChoiandIhmmethodV(P,Q,z)isobtainedbydividingtheunitcellinthezdirectionintoNslabs,whereNisnumberofreal-spacegridpointsalongthezdirectionusedintheFastFourierTransform(FFT).But,insteadofassumingV(P,Q,z)tobeindependentofzineachslabasintheChoiandIhmmethod,hereweenforceV(P,Q,z)ei(PG1z+QG2z)ztobeindependentofzineveryslab.UsingtheseconditionsEquation 3 becomes V(P,Q,R)=NXp=11 dZzpzp1dzei(PG1z+QG2z+RG3z)zVp(P,Q),(3) whereVp(P,Q)isthelocalpotentialineachslab.Equation 3 containsoneofthemajordifferencesbetweenourmethodandtheonepresentedintheChoiandIhm 35

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method.UsingEquation 3 ,wegettheKohn-Shamequationineachslabas Epn(P,Q,z)=~2 2md2 dz22i(PG1z+QG2z)d dz+jk?+PG1?+QG2?j2+(PG1z+QG2z)2pn(P,Q,z)+XP0,Q0eif(PP0)G1z+(QQ0)G2zgzVp(PP0,QQ0)pn(P0,Q0,z). (3) SolutionofEquation 3 iswrittenas pn(P,Q,z)=Xp(P,Q)ei(PG1z+QG2z)zfapneikp(zzp1)+bpneikp(zzp)g,(3) wherekp=p 2m(EEp)=~2,whichcouldberealorimaginarydependingonthegivenenergyEoftheelectronandp(P,Q)isthethsolutionofthefollowingequationinpthslab Epp(P,Q)=~2 2mjk?+PG1?+QG2?j2p(P,Q)+XP0,Q0Vp(PP0,QQ0)p(P0,Q0).(3) Theequationtodeterminep(P,Q)isidenticaltoChoi-Ihmmethod,howeverthewavefunctionpn(P,Q,z),asshowninEquation 3 ,againhasaphaseei(PG1z+QG2z)zwhichisabsentinChoi-Ihmmethod.Thus,generalizationtononorthogonalunitcellsleadstoaphaseintheequationofthewavefunctionforthelocalpotential.IfN2Disnumberoftwo-dimensionalreciprocalspacevectorsPandQ,thenthetotalnumberofunknowncoefcientsfapn,bpnginEquation 3 areN2N2D.Asthereare(N1)interfacesamongtheseNslabs,(N1)2N2Dcoefcientsaredeterminedbyusingthecontinuityofwavefunctionaswellasitsrstderivativeattheboundaryoftheslabs.Thephasefactorei(PG1z+QG2z)zfromtwoadjacentslabsattheboundarycancelsout,makingthedeterminationof(N1)2N2DcoefcientsidenticaltoChoi-Ihmmethod.Therest2N2DcoefcientsaredeterminedbyusingBloch'sequationwhichisdiscussedinthesubsection3.2.2 36

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AftersolvingEquation 3 containingthelocalpotential,nextweturntosolveEquation 3 ,whichcontainsthenonlocalpartofthepseudopotential.Similartothelocalpotential,wavefunctioninthereciprocalspaceforthenonlocalpotentialiswrittenas lm(r)=XP,Qlm(P,Q,z)ei(k?+PG1?+QG2?).r?ei(PG1z+QG2z)z.(3) LikeChoi-Ihmmethod,wedenetheFouriertransformationforthenonlocalpotentialas Wlm(k?,z)=1 SZd2r?Wlm(r)eik?.r?.(3) UsingEquation 3 andEquation 3 ,theKohn-Shamequationforthenonlocalpseudopotentialineachslabbecomes Eplm(P,Q,z)=~2 2md2 dz22i(PG1z+QG2z)d dz+jk?+PG1?+QG2?j2+(PG1z+QG2z)2plm(P,Q,z)+XP0,Q0eif(PP0)G1z+(QQ0)G2zgzVp(PP0,QQ0)plm(P0,Q0,z)+Wlm(k?+PG1?+QG2?,zz)ei(k?+PG1?+QG2?).?ei(PG1z+QG2z)z. (3) ComparingtheaboveequationtothecorrespondingequationintheChoi-Ihmmethod,wendextraterms2i(PG1z+QG2z)d dzplmand(PG1z+QG2z)2plminthekineticenergypart,eif(PP0)G1z+(QQ0)G2zgzinthelocalpotential,andei(PG1z+QG2z)zinthenonlocalpartofthepotential. ThesolutionofEquation 3 iswrittenas plm(P,Q,z)=Xp(P,Q)ei(PG1z+QG2z)zffplm(z)+aplmeikp(zzp1)+bplmeikp(zzp)g.(3)fplm(z)isgivenby fpm(z)=XP,Q[p(P,Q)]ei(k?+PG1?+QG2?).?Zzpzp1dz0fgp(zz0)Wlm(k?+PG1?+QG2?,z0z)g, (3) 37

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wheregp(z)=eikpz=2ikpforz>0andgp(z)=eikpz=2ikpforz<0.Theequationforthedeterminationoffpm(z)isidenticaltotheChoi-Ihmmethod,whilethewavefunctionforthenonlocalpartofpseudopotentialplm(P,Q,z)asshowninEquation 3 isdifferentfromthecorrespondingequationintheChoi-Ihmmethodbythephaseei(PG1z+QG2z)z.Similartothelocalpotential,the(N1)2N2Dunknowncoefcientsfaplm,bplmgcanbedeterminedbymatchingboundaryconditionsbetweenslabs,whichisthesameasintheChoi-Ihmmethod. Tosummarizethispart,generalizationoftheChoiandIhm'splanewavemethodforelectrontransportforhighsymmetryorthogonallattices[ 80 ]tolowsymmetrynonorthogonallatticesleadstoadditionalphaseei(PG1z+QG2z)zinthewavefunctionscorrespondingtothelocalpotentialandnonlocalpartofthepseudopotential.Thisphaseisalsorequiredtogettwo-dimensionallocalpotentialVp(P,Q)fromthethree-dimensionallocalpotentialV(P,Q,R). 3.2.2BoundaryConditions Wedeterminetheunknownexpansioncoefcientsofthewavefunctioncalculatedinsection 3.2.1 basedonthethree-dimensionalBlochtheorem.TosimplifytheimplementationofourmethodintheexistingPWCONDcode,hereafterwerewritethewavefunctionk(P,Q,z)intermsofthewavefunctionCIk(P,Q,z)ofChoi-Ihmmethodforawavevectork,as k(P,Q,z)=ei(PG1z+QG2z)zCIk(P,Q,z),(3) anduseCIk(P,Q,z)intheBlochequation.Thereare2N2D+Pl2nonlocal(2l+1)Naunknowncoefcients(an,Clm)inEquation 3 ,whereNaisthenumberofatomswhosenonlocalspheresexistinsidetheunitcell.2N2Dequationsinvolvingcoefcientsaredeterminedbyimposingthethree-dimensionalBlochtheoremalongthedirectionoflatticevectora3tothewavefunctionCIk(P,Q,z),whichisEquation 3 ,andtoits 38

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z-derivative: d dzCIk(P,Q,z+d)=eikzdei(PG1?+QG2?)a3?d dzCIk(P,Q,z). (3) ThePl2nonlocal(2l+1)NamoreequationscomefromEquation 3 .IfNcisthenumberofatomswhosenonlocalspherescrosseitherz=0orz=dplane,thenforNaNcatomswhosenonlocalspheresarecompletelyinsidetheunitcell,wehaveinthereciprocalspace Clm,k=Xu,vDlm,uvXP,Qei(k?+PG1?+QG2?)?Zd0dz[Wuv(k?+PG1?+QG2?,zz)]CIk(P,Q,z). (3) IntheotherNcatoms,forthosewhosenonlocalspherescrossthez=0plane,wehave Clm,k=Xu,vDlm,uvXP,Qei(k?+PG1?+QG2?)?Zd0dz[Wuv(k?+PG1?+QG2?,zz)]CIk(P,Q,z)+eikzdei(PG1?+QG2?)a3?Zd0dz[Wuv(k?+PG1?+QG2?,zzd)]CIk(P,Q,z) (3) andfortheotheratomswhosenonlocalspherescrossthez=dplane,wehave Clm,k=Xu,vDlm,uvXP,Qei(k?+PG1?+QG2?)?Zd0dz[Wuv(k?+PG1?+QG2?,zz)]CIk(P,Q,z)+eikzdei(PG1?+QG2?)a3?Zd0dz[Wuv(k?+PG1?+QG2?,zz+d)]CIk(P,Q,z) (3) TheseboundaryconditionequationscanbecastintoageneralizedeigenvalueproblemasshowninEquation 3 ,solutionofwhichgivesthecomplexbandstructure. 39

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3.3ScatteringProblem Weconsideratransportprobleminwhichtwosemi-inniteelectrodesareconnectedtotheleftandrightofascatteringregiondenedat0zl.Wecalculatecomplexbandstructureoftheelectrodesusingthemethoddenedintheprevioussection.Wealsocalculatethewavefunctionforthelocalpotentialandnonlocalpartofthepseudopotentialofthescatteringregion.TheBlochwavetravelingtowardsrightintheleftelectrodeortravelingtowardsleftintherightelectrodeisincidentonthescatteringregionandgetspartiallyreectedandtransmitted.IftheBlochwaveisincidentonthescatteringregionfromtheleftelectrode,wavefunctioninthreeregionswillbe[ 80 ] =8>>>>>>><>>>>>>>:k+Xk02Leftrkk0k0z0Xnann+XlmClmlm0zlXk02Righttkk0k0zl(3) Intheaboveequation,tkk0andrkk0arethetransmissionandreectioncoefcients,andLeftandRightcontainsallthestatesgoingtowardsleftandright,respectively.Thetotalnumberofunknowncoefcientsintheaboveequationsis4N2D+Pl2nonlocal(2l+1)(Ns+Ne),whereNsisthenumberofatomsinthescatteringregionandNeisthenumberofatomswhosenonlocalspherescrosseitherz=0orz=l.4N2Dboundaryequationsaredeterminedfromtheconditionthatthewavefunctionanditsderivativearecontinuousattheinterfacez=0andz=l.Equation 3 ofthesubsection3.2.1 givesPl2nonlocal(2l+1)Nsequations.WegetPl2nonlocal(2l+1)NeboundaryconditionsfrommatchingClmfromtheleftelectrodeandscatteringregionatz=0,andfromtherightelectrodeandscatteringregionatz=l.Thereectionandtransmissioncoefcientsaredeterminedbysolvingtheequationsobtainedfromcombiningalltheboundaryconditions.ThetransmissionandreectioncoefcientsforBloch'swaveincidentonthescatteringregionfromtherightelectrodeisdeterminedinsimilarfashion. 40

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3.4CalculationofVelocityofBloch'sState WecalculatethevelocityofaBloch'sstateintheelectrodebyusingthefactthatE(k)isananalyticalfunctionofk.Therefore,asmallimaginarypartEIaddedintheenergyEleadstoasmallimaginarypartkIinkz.vziscalculatedby[ 87 ] vz=1 ~@E @kz1 ~EI kIz.(3) Astheenergyisafunctionofthewavevector(kx,ky,kz),wecanwritethetotalchangeintheenergyas dE=@E @kxkx+@E @kyky+@E @kzkz.(3) FromtheEquation 3 ,wecancalculatevxbykeepingEandkyxed vx=vzkz kx.(3) InasimilarwayvycanbecalculatedbykeepingEandkxxed vy=vzkz ky.(3) 3.5ChoosingtheImaginaryPartofEnergy InordertocalculatevzusingEquation 3 ,weneedtochooseareasonablysmallvaluefortheimaginarypartoftheenergy.Thisvalueisdeterminedbyobtainingabalancebetweenapproximateerrorandroundofferrorinthecalculation.IfweaddasmallimaginarypartEI,totheFermienergyEFoftheelectron,usingTaylor'sexpansionwecanwrite kz(EF+EI)=kz(EF)+EI@kz @EE=EF+1 2(EI)2@2kz @2EE=EF+...(3) RearrangementofEquation 3 yields @kz @EE=EF=kIz EIEI 2@2kz @2EE=EF.(3) 41

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FromEquation 3 ,wendtheapproximateerrorvaezinthecalculationofvzis vaez=EI 2mv2z.(3) wheretheeffectivemassm=~2@2kz=@2E. Thecomputationallimitationgivesrisetoround-offerror.Iftheround-offerrorinthecalculationofkIziskroz,thenwecanwrite vz=1 ~EI kIz+kroz=1 ~EI kIz1kroz kIz, (3) whereonlytwoleadingtermsinTaylorexpansionaretakenbecauseofsmallkroz.AskIzisevaluatedatenergyEF+EI,TalorexpansionaroundEFgives kIz(EF+EI)=kIz(EF)+EI@kIz @EE=EF=EI@kIz @EE=EF (3) DuetoasmallEI,onlythersttwotermsaretakenintheexpansion.ForBloch'sstateattheFermienergytherewillbenoimaginarypartinkz,thuskIz(EF)=0.UsingEquation 3 inEquation 3 ,wegetround-offerrorinthecalculationofvzas vroz=kroz EIv2z(3) Inordertoobtainthebalancebetweenapproximateerrorandround-offerror,botherrorsshouldbeequaltoeachotherandweget EI=p 2mkroz,(3) wheremistheeffectivemassofelectrons.Foradouble-precisioncalculationkroz=1014.Thustheoptimumvalueofenergyforvzcalculationshouldbe EI=107p 2m(3) 42

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3.6Summary Wehavegeneralizedtheplanewavescattering-basedtransportmethodofChoiandIhmtolowsymmetrynonorthogonallattices,wherethe3rdlatticevectora3isnotperpendiculartothebasalplane.Thefullthree-dimensionalBlochtheorem,ratherthantheone-dimensionalBlochtheorem,isappliedinthetransportdirection.Thisrequirestheinclusionofanadditionalphasefactorei(PG1z+QG2z)zinthescatteringstatewavefunctionwhichhasbeenneglectedintheChoi-Ihmmethod.WehavealsoenrichedthePWCONDcodewiththeabilitytocalculateBloch'sstatevelocity.OurgeneralizationoftheChoi-Ihmmethodallowsthestudyoftransportpropertiesinabroadrangeofstructureswithlowsymmetrylattices. 43

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Figure3-1. Unitcell.a)AnonorthogonalunitcellusedingeneralizedPWCOND.Latticevectora3isnotperpendiculartobasalplanevectorsa1anda2.Thezdirectionisthetransportdirectionperpendiculartothebasalplane.b)AnorthogonalunitcellinanfcclatticeusedinoldPWCONDcalculation.Thetransportisalong[111]latticedirection. 44

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CHAPTER4ELECTRONTRANSPORTINMATERIALSWITHDEFECTS WeapplythemethoddevelopedinthisworktocalculateinterfaceresistivityofvariousGBsincrystallinecopper.Presenceofcrystallographicdefectssuchasgrainboundaries(GBs)inasolidreducesthesymmetryofanotherwisehighsymmetrylattice.GBscanalterthecharacteristicsofmaterialsbymodifyingtheirphysicalpropertiessuchasmechanicalstrength,thermalconductivity,andelectricalconductivity.IntroducingGBsinasolidincreasesthescatteringoftheelectronwavefunctions,whichinturnleadstoadecreasedelectricalconductivity.DifferentGBsscatterdifferently.Inameasurementonnanocrystallinecopper,Luetal.[ 90 ]ndthattwinboundariesaffectelectricalconductivitylesssignicantlythanothertypesofGBs.Thereisarecentrst-principlescalculationoftheGBresistivityforcopperusingthelayer-KKRapproach[ 91 ].However,becauseoftheatomicsphereapproximationnecessaryfortheKKRmethod,theresultofthatworkmaybeaccurateonlyforthetwinboundarybutlessaccurateforothertypesofGBs. 4.1TestCasesfortheGeneralizedMethod BeforewestarttoapplyourmethodtostudyGBs,wetestourmethodbycalculatingcomplexbandstructureofcopperin[001]and[111]directions.IntheoriginalPWCOND,orthogonalunitcellscontainingtwoandthreeatomsarerequiredtocorrectlycalculatethecomplexbandstructurein[001]and[111]directions,respectively.However,withourgeneralizedmethod,wecangetthebandstructurebyusingoneatomnonorthogonalunitcellsinbothcases. Inordertoobtaincomplexbandsin[001]direction,aunitcellwithsquarebasalplanehavinglatticevectorsa1=a0(1=2,1=2,0)anda2=a0(1=2,1=2,0),andthethirdlatticevectora3=a0(1=2,0,1=2)ischosen.Latticeconstanta0isfoundtobe PendingpublicationinPhysicalReviewBbytheAmericanPhysicalSociety. 45

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6.71Bohr.Aunitcellwithhexagonalbasalplanewithlatticevectorsa1=a0(1,0,0),a2=a0(1=2,p 3=2,0),anda3=a0(1=2,1=(2p 3),p 2=3)isusedtogetthecomplexbandsinthe[111]direction.ResultsareshowninFigure 4-1 andFigure 4-2 .RedpointsarerealpartofthecomplexbandstructurecorrespondingtotheBloch'sstates,whilethegreenpointsareimaginarypartofcomplexbandstructurerepresentingevanescentstates. 4.2GrainBoundaries 4.2.1Geometries AlltheGBsunderourstudyaretwistboundaries,whicharedescribedbythecoincidentsitelattice(CSL)model[ 40 ].CoincidenceGBsarecreatedupontherotationoftwohalvesofthecrystalrelativetoeachotheraboutacommonaxis,leavingsomeofthelatticepointscommonbetweenthetwolattices.HereweconsidervarioustypesofGBs,including(twin),,,andGBsformedon(001)and(111)planesofcopper.GeometriesofthesefourtypesofGBsareshowninFig. 4-3 .Forthetwinboundary,thestackingsequenceoflatticeplanesinthe[111]directionchangesfrom`..ABCABCABCABC..'offcccopperto`..ABCABCBACBA..'.Althoughformationofthetwinboundaryrequiresacommonplanebetweenthetwohalvesofthecrystal,otherGBsconsideredheredonothavethisconstraint.GBiscreateduponrotationofthetwohalvesofthecrystalrelativetoeachotherby38.21aboutthe[111]direction,whileandGBsareformedbyrotationof36.87and22.62aboutthe[001]direction.OutofallthepossiblestructuresforGB,wehavechosentype-2inourcalculation.Intype-2,inadditiontotherotation,atomsofthetwohalvesofthecrystalarealsodisplacedby(0.1,0.1)a0relativetoeachotherinthedirectionsofthebasalplanelatticevectors,wherea0isthelatticeconstantoftheunitcelldescribingtheCSL. 4.2.2ModelandCalculationDetails Wehaveimplementedourgeneralizedplanewavescattering-basedtransportmethodinthePWCONDcodeoftheQuantumEspressopackage.Calculationof 46

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transmissionandreectioncoefcientsoftheBlochstateincidentonthescatteringregionfollowsasimilarprocedureadoptedbytheChoi-Ihmmethod[ 80 ].Werstperformtheelectronicstructurecalculationsoftheelectrodesandthescatteringregionseparatelytoobtaintheself-consistentpotential.Thenthetransmissionandreectioncoefcientsareobtainedbysolvingtheequationsthatmatchtheboundaryconditionsforthewavefunctionanditsderivativeattheinterfacesbetweentheelectrodesandthescatteringregion,andforthecoefcientsClm,asdenedinEquation 3 ,thatareusedtoexpandthenonlocalpartofthesolution.ThegroupvelocitiesoftheBlochstatesintheelectrodesarecalculatedwiththeproceduredescribedinsection 3.4 Thecalculatedtransmissionandreectionprobabilities,andvelocitiesofBlochstatesarethenusedasinputintheBoltzmanntransportequation[ 66 67 68 ]tosolvefortheresistivityoftheGBs.TondtheresistivityoftheGBs,werstusetheBoltzmannequationtocalculatetheresistivityofasolidofcertainthicknesscontainingtheGBasRA(Bulk+GB)withanassumedscatteringlifetimethroughoutthesolid.Thedirectionofelectrontransportissettobeperpendiculartotheinterface.Thevoltagedropasafunctionofpositioninthecrystaliscalculatedwithavoltagedifferenceof1Voltacrossthissegmentofthesolid.TheinterfaceresistivityofGB(RA(GB))isgivenby RA(GB)=RA(Bulk+GB)VIF=1Volt,(4) whereVIFisthediscontinuityinthevoltagedropattheinterface. Forthetwinboundary,theunitcellofeachelectrodecontainsonlyoneatom.For,,andboundaries,theunitcellhas5,7,and13atoms,respectively.Thescatteringregion,asshowninFig. 4-3 ,contains11and12(111)atomicplanesfortwinand,whileeight(001)planesareusedtomodeland,respectively.Ineachcase,extraatomicplanesadjacenttothescatteringregionateachsideareincludedasbufferlayersintheself-consistentsupercellcalculation. 47

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TheCuatomisdescribedbytheVanderbiltultrasoftpseudopotential[ 92 ]withexchangeandcorrelationmodeledbylocaldensityapproximationofPerdew-Zunger[ 93 ].Weuseenergycutoffsof40Ryand600Ryforthewavefunctionandchargedensity,respectively.Gaussiansmearingwithawidthof0.01eVisusedfortheoccupationofelectroniclevels.272727k-pointgridisusedtosampletheBrillouinzone(BZ)oftheelectrodesofthetwinboundaryinMonkhorst-Pack'sscheme[ 94 ].ForotherGBs,k-pointgridsarescaledininverseproportiontothelatticevectorsdescribingtheirunitcells.Thenumberofk?pointsinthetwo-dimensionalBZcorrespondingtothebasalplanefortransmissionandvelocitycalculationisdeterminedthroughconvergencecheckofinterfaceresistivity.Convergenceisachievedbyusing2071,361,343,and169k?pointsfortwin,,,andGBs,respectively. 4.2.3Results InFig. 4-4 ,weshowthetotaltransmissionprobabilityT(k?)attheFermienergyinthetwo-dimensionalBrillouinzoneoftheGBsunderstudy.T(k?)iscalculatedbyaddingthetransmissionprobabilitiesforalltheBlochstatesateachk?.WenotethatthedistributionofT(k?)reectsthesymmetryofthelatticeaboutthetransportdirection.FortwinandGBswhosebasalplanesare(111),T(k?)has6-foldsymmetry.Thetransmissionhas4-foldsymmetryinandGBsduetotheir(001)basalplanes. InTable. 4-1 ,welistthecalculatedinterfaceresistivityofallfourtypesofGBs.TheresistivityofthetwinboundaryisthesmallestamongalltheGBs.Thoseof,,andGBsarelargerandveryclosetoeachother.Resistivityofthetwinboundaryobtainedinourcalculationisingoodagreementwithboththeexperimentalvalueof1.71017-m2reportedbyLuetal.[ 90 ],aswellasarecentstudy[ 91 ]ofcopperGBresistivityusingthelayer-KKRmethodwhichfoundthetwinresistivityas0.2021016-m2.ThelatterstudyalsofoundthatthetwinboundaryhasthesmallestresistivityamongallfourtypesofGBs.Thelayer-KKRmethod,however,yieldslargervaluesofresistivityforandGBsascomparedtoourresultsfromtheplanewavemethod. 48

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4.3Summary UsingthegeneralizedPWCOND,wehavestudiedtwin,,,andgrainboundariesincrystallinecopper.Thetotaltransmissionprobabilityhasthesamesymmetryasthatofthelatticeaboutthetransportdirection.AmongalltheGBs,twinhasbeenfoundtohavetheleastresistivity.ResistivityofotherGBsarehigherandclosetoeachother.CalculatedvaluesofGBresistivityareingoodagreementwithavailableexperimentaldata,aswellasapreviouscalculationusingthelayer-KKRmethod. 49

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Table4-1. Calculatedinterfaceresistivity(resistance-areaproduct)ofvariouscoppergrainboundaries.Unitfortheinterfaceresistivityis1016m2. TypeofGBThisstudyLayer-KKRExperiment TwinRelaxed0.2080.2020.170 Unrelaxed0.5850.807 Unrelaxed0.5420.532 Unrelaxed0.4650.863 50

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Figure4-1. Complexbandstructureofbulkcopperin[001]direction.RedandGreenpointsrepresentrealandimaginarypartofcomplexbands.TheFermienergyisshownbyahorizontaldottedblackline. Figure4-2. Complexbandstructureofbulkcopperin[111]direction.Realandimaginarypartofcomplexbandsareshownwithredandgreenpoints.AHorizontaldottedblacklinerepresentstheFermienergy. 51

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Figure4-3. Topandsideviewsofgeometriesofa)twin,b),c),andd)grainboundaries.Thedashedboxrepresentsthescatteringregioninthetransmissioncalculation.Yellowandbluespheresarecopperatomsoftwohalvesofthelattice.Forthetwinboundary,redspheresrepresentthecommonplaneattheinterfaceandaperspectivewayisusedforthetopview. 52

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Figure4-4. Totaltransmissionprobabilityintwo-dimensionalBZofa)twin,b),c),andd)grainboundaries.Theunitsofkxandkyare2=a,whereaisthelengthofthelatticevectorinthebasalplane. 53

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CHAPTER5STUDYOFADSORPTIONOFSMALLMETALCLUSTERSONGRAPHENE Graphene,asingletwodimensionalsheetofcarbonatomsformingahoneycomblatticehasdrawnwideinteresttheoreticallyaswellasexperimentallybecauseofitsinterestingelectronicandmechanicalproperties.SinglelayergrapheneisazerobandgapsemiconductorwithalineardispersionrelationaroundFermienergynearthecornersofthehexagonalBrillouinzone[ 12 95 ].Metalsadsorbedongraphenecanformdifferenttypesofstructuresandchangegraphene'selectronicbehaviorgivingrisetointerestingphysicalproperties.Theapplicationofgrapheneinelectronicdevicescanberealizedifonecancontrolthecarrierdensityaswellascarriertype.InanexperimentbyGierzetal.,holedopingofgraphenewasaccomplishedbyadsorptionofgold,bismuth,orantimonyonitssurface[ 96 ].AsreportedbyChenetal.,potassiumwasalsousedtochemicallydopegraphene[ 14 ].Changeofcarrierconcentrationwithadsorptionofmoleculesonthesurfaceofgraphenemakesthegrapheneanexcellentcandidateforgassensors[ 23 ].Inordertounderstandhowadsorptionofatomsaffectstheelectronicstructureofgraphene,theoreticalcalculationsusingDensityFunctionalTheory(DFT)havebeenperformed.Duffyetal.studiedadsorptionoftransitionmetaladatomsanddimersongraphite[ 97 ].Chanetal.performedDFTcalculationformetaladatomsadsorptionongraphene[ 33 ].First-principlesstudiesoftransitionmetaladatomsanddimersongraphenewerecarriedoutbyCaoetal.andJohlletal.[ 98 99 ]. Althoughperfectgrapheneisnon-magnetic,thepresenceofimpuritiesordefectscangiverisetomagnetism[ 100 101 102 103 104 ].Inanexperimentwheregraphitewasirradiatedwithprotons,amagneticsignalwasobserved[ 105 106 ].Whiletheaboveexperimentsdemonstratethestatisticalpropertiesoftherandomcollectionofvacanciescreatedduringirradiation,controlledexperimentshavebeenrecentlyperformed.Ugeda PendingpublicationinPhysicalReviewBbytheAmericanPhysicalSociety. 54

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etal.generatedsinglevacanciesongraphiteandusedlowtemperaturescanningtunnelingmicroscopytostudytheeffectofeachvacanciesontheelectronicandmagneticpropertiesofgraphite[ 107 ].Inatheoreticalapproachtounderstandthisphenomenon,spinpolarizedDFTstudieswereperformedbyLehtinenetal.andYazyevetal.,inwhichthemagneticsignalwasexplainedbythepresenceofavacancyingraphitestructure[ 102 29 ].Irradiationofgraphenewithhighenergyparticlescanknockanatomoutandgiverisetoavacancy.Thepresenceofavacancycausesonesp2danglingbondineachofthreeneighboringcarbonatoms.Uponatomicrelaxation,twooftheneighboringatomsreconstructacovalentbondwhilethethirdatomisleftwithadanglingbond,leadingtolocalmagnetism.Inadditiontodefectsandvacancies,adatomsadsorbedondefectedgraphenecangiverisetoratherunusualandexcitingproperties.Krasheninnikovetal.performedaDFTstudyoftransitionmetalatomsinteractingwithgraphenecontainingsingleanddoublevacancies[ 108 ].Theyreportedthatanironadatomadsorbedongraphenewithsinglevacancytobenonmagnetic,whileagoldadatomontopofthevacancyhadamagneticmomentof1B.Thestrongsensitivityofgraphene'sresponsetothecharacteroftheadatomisaninterestingeffectthatmeritsfurtherstudyoftheexactnatureoftheinteraction. DFTwithlocalandsemilocalexchangecorrelationfunctionalshasbeenverysuccessfulindescribingthephysicalpropertiesofmaterialswhichhavestronglocalbonds.However,materialswhichhaveregionsoflargeinteratomicdistanceandthuslowelectrondensitycannotbehandledaccuratelywithDFTalone,andlongrangevanderWaals(vdW)interactionsareneededtobetakenintoaccount.Graphite,softmatter,raregases,andgraphenewithmetalsareafewexamplesofthesystemswhereinclusionofvdWinteractionsinDFTisnecessary[ 109 110 111 ].Forinstance,ithasbeenfoundthatthegeneralizedgradientapproximation(GGA)givesverylowornobindingatallforgrapheneadsorbedon(111)surfaceofnoblemetals,andvdWinteractionswereincludedtoimprovethebinding[ 112 113 ].Adsorptionofnoblemetal 55

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adatomsongraphenewasstudiedbyAmftetal.,wheretheyfoundanincreaseinthebindingenergiesoftheadatomswiththepresenceofvdWinteractions[ 114 ].OutofmanyschemesthathavebeenusedtoincorporatelongrangevdWinteractionsintoDFT,vanderWaalsdensityfunctional(vdW-DF)byDionetal.,andsemi-empiricalmethod(DFT-D2)byS.Grimmearethemostwidelyused[ 111 115 ].TheresultofvdW-DFmethoddependsgreatlyonthechoiceofexchangefunctionalswhichisshownbyKlimesetal.[ 116 117 ].OriginalimplementationofvdW-DFbyDionetal.,whichusesrevPBEexchangefunctional[ 118 ],overestimatesbondlenghandunderestimatesbindingenergy.ComparedtotherevPBEexchangefunctional,Klimesetal.foundresultsofadvancedexchangefunctionalssuchasoptB88-vdWandoptB86-vdWtobeclosertoexperiments. Motivatedbytheinterestingresultsofadatomanddimeradsorptionongraphene,wepushtheissuefurtherandstudymetalclusteradsorptiononbothperfectanddefectedgraphene.Asasubjectoffundamentalinterest,metalclustershavebeenproducedandstudiedexperimentallyandtheoreticallyfordecades[ 119 ].Itisknownthatdelocalizedselectronsformtheso-calledshellstructureandphysicalpropertiesofsmallclustersvaryatom-by-atomwhichenrichesthebuildingblocksfornanostructuredsystems.Theprocessbywhichelectronicpropertiesofgraphenechangewithanincreaseinnumberofadsorbedatoms,i.e.adsorptionofsmallclustersisthesubjectofourinterest. Inourstudy,wereportDFTcalculationsofAuandFeclusterscontaining1-to5atomsadsorbedonperfectgrapheneaswellasdefectedgraphenecontainingasinglevacancy.WendthatFeclustersbindstrongertographenethanAuclusters,independentofclustersizeandwhetheravacancyispresent.InclusionofvdWinteractionsincreasesbindingbetweenAuclustersandperfectgraphenesubstantially.WeshowthatFeclustersadsorbedonperfectgrapheneleadstodopingofgrapheneforallclustersizesconsidered,whereasAuclustersshowanodd-evenoscillationinthe 56

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dopingofgraphenewiththenumberofatomsintheclusters.DefectedgraphenewithAuclustersismagneticforodd-numberedclustersandnon-magneticforeven-numberedclusters,whilethewholesystemismagneticforFeclusterswithn>1. 5.1Method Weusedensity-functionaltheory,asimplementedinViennaab-initiosimulationpackage(VASP)[ 120 121 ],toperformourcalculations.Projectoraugmentedwave(PAW)potentials[ 123 122 ]areusedtodescribetheioncores,andexchangeandcorrelationaremodeledbygeneralizedgradientapproximationofPerdew,Burke,andErnzerhof(PBE)[ 124 ].Allthesimulationsareperformedusing66supercellofgraphenewhichcontains72Catoms.Intergraphenelayerdistanceischosentobe25Atoavoidinteractionsbetweenperiodicimages.Kineticenergycutoffforthewavefunctionsinallthecalculationsis400eV.K-pointsaregeneratedusingMonkhorst-Pack'sscheme[ 94 ]and551-centeredgridisusedtosampleBrillouinzone.WidthofGaussiansmearingfortheoccupationofelectroniclevelsis0.2eV.Toobtainaccuratemagneticmoment,calculationsinvolvingAuclustersaredonewithsmallerwidthof0.01eV.Structuresinthisstudyarerelaxedtilltheforceoneachatomissmallerthan0.025eV/A.ThevdWinteractionsareincorporatedinDFTthroughvdW-DFaswellasDFT-D2methods[ 111 115 ].CalculationsinvolvingvdW-DFmethodaredonewithtwoexchangefunctionals,namelyrevPBEandoptB88-vdW[ 118 116 ].ParametersforDFT-D2aretakenfromSawinskaetal.whostudiedtheinteractionofgraphenewithAu(111)surface[ 113 ].DispersioncoefcientC6andvdWradiusRoforgoldare40.62J-nm6/moland1.772A,respectively.Pairinteractionsuptoaradiusof12Aaretakenintoaccountandglobalscalingfactors6ischosentobe0.75. Tounderstandthebindingoftheclusterwithgraphene,wedenethreetypesofbindingenergies BE1=(EG+MnEGnEM1)=n,(5) BE2=EG+MnEG+Mn1EM1,(5) 57

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BE3=EG+MnE0GEMn,(5) whereGrepresentseitherperfectgrapheneordefectedgraphene.MisAu/Feandnisthenumberofatomsinthecluster.IntheequationforBE1,EGistheenergyoffreestandinggraphene,whileinBE3,E0Gistheenergyofthegrapheneobtainedafteroptimizationoftheclusteronitbutwithoutrelaxingitafterwards.BE1containsinformationofcohesiveenergyoftheclusterinthepresenceofgraphene,whileBE2signiestheenergygainedbythesystemuponaddingonemoreatomtothealreadyexistingcluster.TheenergygainedthroughtheinteractionbetweengrapheneandclusteriscontainedinBE3.Inordertovisualizechargetransferbetweenclusterandgraphene,weplotchargedensitydifferencen(r),whichisdenedas n(r)=nG+Mn(r)nG(r)nMn(r)(5) wherenG+Mn(r),nG(r),andnMn(r)arechargedensitiesofcombinedsystem,graphene,andmetalcluster,respectively. 5.2ClustersonPerfectGraphene Twometalclusters,AunandFen(n=1-5)onaatgraphenesheetarestudied.Theenergeticsandelectronicstructuresofisolatedclustersandisolatedgrapheneareobtainedasreferencesystemsforinvestigatingthecluster-grapheneinteraction.Theunitcellcontains72Catomsandthecluster-clusterdistanceinthein-planeadjacentcellisabout15A. 5.2.1GroundStateGeometryandEnergeticsofGoldClusters Forasingleadatomonperfectgraphene,therearethreepossibilitiesoftheadsorptionsite,namelytheatopsite,whichisdirectlyaboveoneofthecarbonatoms,thebridgesite,whichisontopofoneofthecarbon-carbonbonds,andthehollowsite,whichisinthecenterofthehexagon.WendthemoststablepositionfortheAuadatomongraphenetobetheatopsite.ThedistancebetweentheAuadatomandCatomis2.82A.ComparisonofourresultwiththepreviouscalculationsbyJalkanenetal.[ 125 ] 58

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andChanetal.[ 33 ],whoreportedthedistanceoftheAuatomabovetheCatomtobe2.62Aand2.55A,respectively,showsagoodagreement,withabout10%deviationisbecauseofdifferentparameterschosenforenergycutoff,K-points,andforcethresholdintheoptimization. TheinitialgeometriesofthegoldclustersAunforn>2istakenfromtheworkofHakkinenetal.[ 126 ]andWangetal.[ 127 ].GoldclustersAunforn>3areplanar,andwendthattheyretaintheirplanarpropertywhenadsorbedonthegraphene.Thepositionoftheclusterwithnnumberofatomsongrapheneisachievedbybringingoneatomtothealreadyrelaxedclusterwith(n1)atomsandthenrelaxingthewholestructure.SeveralstructuresbasedonvariouspositionsoftheaddedatomareoptimizedandweshowthemoststablestructuresamongtheminFigure 5-1 .Wendthatthegeometriesoftheclustersontopoftheperfectgraphenearesimilartotheirfreestandingstructures[ 126 127 ].Everyclusterhasoneatomwhichisclosertothegraphenelayerandtherestoftheclusterisbondedtothegraphenethroughthatatom.Hereafter,wewillrefertothatatomas`anchoratom'.Theheightoftheanchoratomabovethegrapheneplane(ha)anditsdistancefromitsnearest-neighbor(NN)Catom(dac)aregiveninTable 5-1 .LengthsofAu-Aubondsincreasewithclustersizefrom2.53AforAu2to2.67AforAu3andnallysaturateto2.70AforAu4andAu5clusters.WendthattheanchoratominthecaseofAu2isatadistanceof2.32AabovetheCatominthegraphenelayer,whichisabout20%shorterthandacoftheadatom.ThesecalculationsareinexcellentagreementwiththoseofJalkanen'setal.,whofoundthedimerbondlengthtobe2.52Aanddactobe2.32A.TheanchoratominAu3clustermovesawayfromthetopoftheCatomby0.57Atoanunsymmetricalpositionbetweenthehollowandbondsites. ThebindingenergiesoftheAuclustersadsorbedontheperfectgrapheneareshowninTable 5-1 .WendthatthecalculatedBE1,asdenedinEquation 5 ,ofAu2clusterisclosetothatofAu3cluster.SimilarvaluesofBE1arealsoobservedforAu4 59

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andAu5clusters.ThisisbecauseBE1,asdened,containscohesiveenergiesoftheclusters,whichlieclosetoeachotheraswell.Wendcohesiveenergiestobe-1.14eVand-1.16eVfortheAu2andAu3clusters,and-1.50eVand-1.65eVfortheAu4andAu5clusters,respectively.BindingenergyBE2,asdenedinEquation 5 ,arelargerforAu2andAu4clustersascomparedtootherclusters.SinceAuhasonlyonevalenceelectron,Au2andAu4clustershaveclosedshellelectroniccongurationsandthustheyaremorestableandhaverelativelylargerBE2.Interactionbetweengrapheneandclustersismainlythroughtheanchoratomanditsstrengthdependsonha,andthusondac.BecauseofthesimilardacforAu2-Au4clusters,bindingenergyBE3,asdenedinEquation 5 ,areaboutthesameforthoseclusters.However,forAu5cluster,dacis2.45A,whichislargerthanthoseoftheAu2-Au4clustersbyabout0.12A,explainingthesmallerBE3.ThebindingenergyBE3oftheAuadatom-0.12eVisincloseagreementwiththeresultsofcalculationsbyJalkanenetal.andChanetal..OurcalculatedBE3oftheAudimeralsoagreeswellwiththereportedvalueof-0.53eVbyJalkanenetal.. AsevidentfromthebindingenergyBE3,theAuclustersareweaklybondedtothegraphene,andlongrangevdWdispersionforcesmustbetakenintoaccountinthecalculationtocorrectlydescribetheinteractionbetweentheAuclustersandthegraphene.WeemployrecentlydevelopedvdW-DFmethod[ 111 ]aswellasDFT-D2method[ 115 ]toincludevdWinteractionsinDFT.BE3andhaobtainedbythevdW-DFmethodusingrevPBE[ 118 ]andoptB88exchangefunctionals[ 116 ]aswellastheDFT-D2methodareshowninTable 5-2 .ComparedtothePBEcalculationswhichdonotincludevdWinteractions,wendanincreaseinthebindingenergyBE3ofalltheclustersadsorbedonthegrapheneineverymethodused.TheBE3andhaofalltheclustersobtainedbythevdW-DFandtheDFT-D2methodsareinexcellentqualitativeagreement.Inallthemethods,wendBE3oftheAuadatomsmallestandhalargest.BE3increaseswiththeclustersizeanditsmaximumvalueisobservedfortheAu4cluster.BE3ofAu2andAu5clustersaresimilarinmagnitudeandthesameholdstrue 60

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forAu3andAu4clusters.InthevdW-DFmethod,wendhatobelargerandBE3tobesmallerfortherevPBEfunctionalascomparedtotheoptB88functionalforalltheclusters.OverestimationofthebondlengthandunderestimationofthebindingenergyisawelldocumentedfeatureoftherevPBEfunctional[ 116 ]anditisalsoobservedinthepresentwork.BE3andhaobtainedbytheoptB88functionalofthevdW-DFmethodandthesemiempiricalDFT-D2methodareingoodagreementwitheachother. AstheresultsofthevdW-DFcalculationswiththeoptB88exchangefunctionalandtheDFT-D2calculationsareclosetoeachother,furtherquantitiessuchasBE1,BE2,anddacarecalculatedusingonlythevdW-DFmethodandshowninTable 5-3 .OtherthantheAuadatom,whichmovesclosertothegraphenelayerby0.3A,dacoftheclustersaresimilartothePBEcalculations.Thedistancebetweencentersofmassoftheclustersandthegraphenelayer(zCM)aswellashaaresmallerthanthoseofthePBEcalculations,indicatingthattheclustersmoveclosertothegraphenewiththeinclusionofvdWinteractions.TheinteractionbetweenthegrapheneandtheclustersiscontainedinthebindingenergyBE3,whichchangesmorethanBE1orBE2.AsdacforAu2-Au5clustersaresimilartothePBEcalculations,theincreaseinBE3isattributedtothelongrangevdWinteractionsbetweenthegrapheneandtheatomsoftheclustersotherthantheanchoratom.TheeffectofthelongrangeinteractionsismanifestedinthevariationofBE3withtheclustersize.WendthatAu2andAu3clustershaveidenticaldacbutmarkedlydifferentBE3,whiletheoppositeistrueforAu2andAu5clusters.IfweconsiderthechangeBE3(BE3vdWBE3PBE),wendthatBE3increasesmonotonicallywiththeclustersize,forexample,BE3ofAu2clusterincreasesby-0.36eV,whilethechangeinBE3ofAu5clusteris-0.66eV.Thechangeinthecluster'scenterofmasszCMalsoincreaseswiththeclustersize,forexample,zCMofAu2clusterdecreasesby0.02Aascomparedto0.08AofAu5cluster.Becauseofthelongrangeinteractionsbetweenthegrapheneandtheclusters,thedecreaseinzCMandtheincreaseinthenumberofatomsoftheclustersgiverisetolargerBE3astheclustersizeincreases. 61

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ThisexplainsthevariationofBE3withtheclustersize.BE1andBE2oftheclusterswiththevdWinteractionsareonlyslightlyincreasedascomparedtothePBEcalculations.BE1andBE2dependmostlyonthecohesiveenergiesoftheclusterswhichdonotchangesubstantiallywhenvdWinteractionareincluded.WiththevdW-DFmethod,wendcohesiveenergisofAu2,Au3,Au4,andAu5clusterstobe-1.17eV,-1.19eV,-1.56eV,and-1.71eV,respectively,whichareveryclosetothecohesiveenergiesoftheclustersobtainedwiththePBEcalculations. Itshouldbenotedthat,withtheinclusionofvdWinteractions,thelowestenergyadsorptioncongurationfortheAu5clusterislyingparalleltothegraphenesheet(Figure 5-1 ),andhereafterwewillrefertothisasAu5(P).ThecalculatedtotalenergyoftheAu5(P)clusterongrapheneisloweredby287meVfromthecongurationinwhichtheclusterisbondedtothegrapheneonlyviaananchoratom,primarilybecauseofthelargercontactareawhentheclusterislyingdown,resultinginstrongervdWinteractionsbetweenthevegoldatomsandthegraphene. OurresultsfortheadatomareconsistentwiththevdW-DFandthesemiempiricalcalculationsofAmftetal.[ 114 ],whoalsoobservedanincreaseinBE3withvdWinteractionsinbothmethods.TheirresultofBE3oftheadatom-0.39eVwiththevdW-DFmethodusingtherevPBEfunctionalliesclosetoourresultoftherevPBEcalculation.WiththesemiempiricalmethodAmftetal.foundBE3tobe-0.89eV,whichishigherthanourresult.Intheircalculation,haisaboutthesameinbothmethodsascomparedtothePBEcalculations,whichisdifferentfromourresultswhereasignicantchangeinthepositionoftheanchoratomisobserved.Thereasonforthesedifferencesislikelyduetotheirprocessofrelaxation.IntheircalculationtheadatomwaskeptxedandonlytheCatomsofthegraphenewereallowedtorelaxwhilewerelaxthewholesystem,i.e.,theadatomaswellastheCatomsofthegraphene.ThesemiempiricalcalculationsperformedbyJalkanenetal.[ 125 ]yieldBE3oftheadatomanddimertobe-0.61eVand-0.74eV,respectively.Intheirstudy,duetothedifferentchoice 62

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ofparameters,onlytherelativeenergiesaresuggestedtobetrusted,andweareinagreementwiththefactthatBE3oftheAudimerislargerthanthatoftheAuadatom. 5.2.2GroundStateGeometryandEnergeticsofIronClusters TheFeadatomfavorsthehollowsiteasitsbondingpositionwithhaanddacof1.55Aand2.11A,respectively.ThisresultisincloseagreementwithJohlletal.[ 99 ],whoreportedthehollowsiteasthebondingsitewithhatobe1.54A.Ourresultisabout8%largerthanCaoetal.[ 98 ],whofoundhatobe1.44A.Weprovideexplanationforthedifferenceinsubsection5.2.4 .ThestartinggeometriesofironclustersFenforn>2areobtainedfromthepreviousworksofCastroetal.[ 128 ],Maetal.[ 129 ],andBalloneetal.[ 130 ].UnliketheAuclusters,Feclusterswithn>3arethreedimensional.OptimizedgeometriesoftheclustersadsorbedonthegrapheneareshowninFigure 5-2 .LiketheAuclusters,geometriesoftheFeclustersontopofthegraphenearesimilartothecorrespondingfreestandingstructureswiththeexceptionofFe5cluster.Although,thegroundstatestructureofisolatedFe5clusteristrigonalbipyramidal,onthetopofgraphene,squarepyramidalstructureisfoundtobemorestable.Theaveragebondlengthsoftheclustersincreasemonotonicallywiththenumberofatoms,thesmallestbeing2.07Aforthedimerand2.20A,2.29A,and2.33AforFe3,Fe4,andFe5clusters,respectively.IntheFedimer,theanchoratomisnotexactlyatthemostsymmetrichollowpositionbutitslightlymovesawayfromthecenter.WendtheFedimerbondlength2.07Aandhatobe1.89A.TheseresultsagreewellwithJohlletal.,whoreportedthedimerbondlengthof2.08Aandhatobe1.86A.OurresultsaresmallerthanCaoetal.,whofoundthedimerbondlengthandhatobe2.21Aand2.02A,respectively. ThebindingenergiesoftheFeclustersadsorbedonthegraphenearelistedinTable 5-4 .WendthatthecalculatedBE1increasesmonotonicallywiththeclustersize.However,BE3doesnotfollowthistrend.BE3ofFe2clusterisclosetothatofFe5cluster,whileforFe3cluster,ithassimilarvalueasofFe4cluster.ThevariationofBE1 63

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canbeexplainedviathecohesiveenergiesofthefreestandingclusters,whichalsoincreasemonotonicallywiththeclustersize[ 129 ].InthecaseofFe3andFe4clusters,theanchoratomisclosertothegraphene,whichinturnmakesdacsmallerandthusresultsinhigherBE3,whereastherelativelylowervaluesofBE3fortheFe2andFe5clustersarebecauseoflargerdac.ComparedtotheAuclusters,BE3oftheFeclustersdecreaseslowlywithha.BE2increaseswiththenumberofatomsintheclusters,indicatingthatFeatomsprefertoformclustersonthegraphene.AsindicatedbyhigherBE3andsmallerhaforalltheclusters,interactionbetweengrapheneandFeclustersisstrong.ThephysicalpropertiesofstronglybondedsystemsarewelldescribedbyPBEfunctionalandthelongrangevdWinteractionsarenotneededtobetakenintoaccount. Recently,Johlletal.[ 131 ]reportedadsorptionofFe3andFe4clustersongraphene.ThegeometriesoftheclustersontopofthegrapheneobtainedinourworkaredifferentthanthosereportedbyJohlletal..UponcomparingthetotalenergiesofthemoststablestructuresofFe3andFe4clustersobtainedintheirworkwithourresults,wendthatthestructuresreportedintheirworkarenotthegroundstatestructuresoftheclustersonthegraphene.TheFe3andFe4clustersadsorbedonthegrapheneinourstudyare0.09eVand0.24eVmorestablethanthosereportedbyJohlletal.. 5.2.3ElectronicStructureandMagnetizationofGoldClusters WeperformBaderanalysis[ 132 ]toquantifythechargetransferbetweenthegrapheneandtheAuclusters.Visualizationofthechargetransferisdonebyplottingchargedensitydifferencen(r),asdenedinEquation 5 ,andisshowninFigure 5-3 .WendthatthechargetransferbetweentheAuclustersandthegrapheneissmall.Interestingly,thechargetransfer,ascalculatedfromBaderanalysis,isfromthegraphenetotheAuclustersforalltheclustersize,withlargestchargetransferbeinginthecaseoftheadatom.InclusionofvdWinteractionsincreasesthechargetransferbyaverysmallamountwhichisrelatedtosmallerzCMoftheclusters.ThetotalmagnetizationofthesystemcomesfromtheAuclusters;thecontributionofC 64

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atomsofthegraphenesheetisclosetozeroforalltheclusters,whichisshowninthemagnetizationdensityinFigure 5-3 .Fromtheprojecteddensityofstates(PDOS)ontotheclusters,wendthattheelectronsoftheoutermostsorbitalsofatomsofAuclustersmainlygiverisetothemagnetization,butacontributionfromtheelectronsofdorbitalstothetotalmagneticmomentisalsopresentinsmallamountinthecaseofAu3andAu5clusters.Thetotalmagnetizationofthegraphenewithclusterscontaininganoddnumberofatoms(Au1,Au3,Au5,andAu5(P))iscloseto1B=cellandforanevennumberofatoms(Au2andAu4)iszero. Weshowthetotaldensityofstates(DOS)andPDOSontotheporbitalsofthegrapheneinFigure 5-4 .ThetotaldensityofstatesisniteattheFermienergy(EF)ofthegraphenewiththeclusterscontaininganoddnumberofatomsandzerowiththosecontaininganevennumberofatoms.Thiscanbeexplainedbyelectronpairingeffect.Auclusterswithanoddnumberofatomshaveopenshellelectroniccongurations,whilethosewithanevennumberofatomshaveclosedshellcongurations.TheHOMO-LUMOenergygapfortheclosedshellislargerthanthatfortheopenshell,makingtheclusterswithaclosedshelllesschemicallyreactivethanthosewithanopenshell.ThepeakinthetotalDOSatEFfortheclusterswithanoddnumberofatomsisduetotheelectronsofsorbitalsoftheatomsoftheclusters.ForAuadatom,thespin-upspeakisbelowEF,whilethespin-downspeakispartiallyoccupiedatEF,indicatingthechargetransferfromthegraphenetotheadatom.AsimilarobservationisalsotrueforAu5andAu5(P)cluster.However,inthecaseofAu3cluster,thespin-uppeakispartiallyemptyatEFandthespin-downpeakliesaboveEF,whichisapossibleindicationofthechargetransferfromtheAu3clustertothegraphene.ForAu2andAu4clusters,thespin-upandspin-downsstatesarelocatedabout1.0eVand0.6eVbelowEF. FromthePDOSasshowninFigure 5-4 ,weobservethattheporbitalsofthegrapheneareonlyslightlyperturbedbythepresenceoftheclustersandDiracconeofthegrapheneispreserved.ThePDOSshowslittlehybridizationofgraphenep 65

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orbitalswithAusorbitals.TheDiracpoint(ED)ofthegraphene,wherethespin-upandspin-downDOSarezero,shiftsawayfromEFbythepresenceofsomeoftheclusters.Dopingcanbeidentiedasn-typeorp-type,dependingonwhetherEDislowerorhigherthanEF.Interestingly,thePDOSsuggeststhatthegrapheneisdopedbyodd-numberedclustersbutremainsundopedinthepresenceofeven-numberedclusters.WehavecheckedthisbehavioruptoAu7clusterandconrmedthisodd-evenoscillation.WiththeshiftofED,wenotethep-typedopinginthecaseofAu1,Au5,andAu5(P)clusters,andthen-typedopinginthecaseofAu3cluster,consistentwithourtotalDOSobservation.AsthereisnoshiftofEDfromEF,thegrapheneisundopedinthecaseofAu2andAu4clusters.Odd-evenoscillationintheDOShasalsobeenobservedingraphenewithopenshellNO2andclosedshellN2O4molecules[ 133 ].Baderanalysis,whichdemonstratesthep-typedopingofthegrapheneforalltheclusters,isnotingoodagreementwiththeodd-evenoscillationofdopingpredictedbythePDOS.WendthatthetotalDOSandthePDOSremainunaffectedwithvdWinteractions. 5.2.4ElectronicStructureandMagnetizationofIronClusters Wendthatforalltheclustersonthegraphene,thechargetransferisfromtheclusterstothegraphene,andtheamountofthechargetransferismuchlargerthanthatfortheAuclusters.Chargedensitydifferencen(r)isshowninFigure 5-5 .Foralltheclusterswithsizen>1,theanchoratomlosesabout0.4e,andtherestamountofthechargetransfercomesfromotheratomsintheclusters.Theamountofchargetransferfromtheotheratomsintheclusterstothegraphenedependsonthepositionoftheanchoratom.Wendthatsmallerdacfacilitatesmorechargetransferfromtheotheratoms,whichexplainsthelargerchargetransferinthecaseofFe3andFe4clustersascomparedtotheotherclusters. UnliketheAuclusters,graphenewiththeFeclustersismagneticforallclustersizes.ThetotalmagnetizationofthesystemmainlycomesfromtheFeclustersasweshowinthemagnetizationdensityinFigure 5-5 .ThePDOSrevealsthatthe 66

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magnetizationiscontributedfromtheelectronsofdorbitalsoftheclusteratoms.InthecaseoftheFeadatom-graphenesystem,oneelectronfromthesorbitalofFeistransferredintodorbitals,causingthetotalmagnetizationtobeabout2B=cell.Theindividualmagneticmomentsoftheatomsotherthantheanchoratomintheclustersareabout3.2B=cell,whilethemomentoftheanchoratomvarieswithdac.InthecaseofFe3andFe4clusters,theanchoratomhasasmallermagneticmomentof1.74B=celland1.80B=cell,respectively,whileforFe2andFe5clustersithasalargermagneticmomentof2.99B=celland2.61B=cell,respectively.AsdacofFe3andFe4clusterslieclosetothatoftheFeadatom,theeffectofthegrapheneontheorbitalsoftheiranchoratomisalsosimilartothatoftheadatom.Thus,liketheFeadatom,electrontransferfromthesorbitalintothedorbitalscausesthemagneticmomentoftheanchoratomtobeabout2B=cellinthecaseofFe3andFe4clusters.However,duetolargerdacforFe2andFe5clusters,thepresenceofgraphenedoesnotcauseelectrontransferbetweensanddorbitals.Theanchoratompossessesabout7electronsinthedorbitals,yieldingthemagneticmomenttobecloseto3B=cell. WecompareourresultsfortheadatomandthedimerwiththoseofCaoetal.andJohlletal..OurcalculationsyieldBE3oftheadatomtobe-1.19eV,whichisincloseragreementwiththeresultsofCaoetal..ThemagneticmomentsreportedbyCaoetal.andJohlletal.fortheadatomwere2.13B=celland2.0B=cell,respectively,whilechargetransferreportedbybothgroupswas0.69e.FortheFedimerthebindingenergyreportedbybothgroupswascloseto-0.75eV,howeverourvalueofthebindingenergy-0.92eVisabout20%larger.Themagneticmomentofthedimerobtainedinourcase6.16B=cellissmallerthanthereportedvalueof6.46B=cellbyCaoetal,butisalmostthesameas6.15B=cellobtainedbyJohlletal..ThechargetransferreportedbyCaoetal.andJohlletal.were0.24eand0.37e,respectively,whicharesmallerthanourvalueof0.42e.Ourresultsforbondlengths,bindingenergies,magneticmoments,andchargetransferqualitativelyagreeandthesmallquantitativedifferences 67

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couldbecomingfromthedifferentchoiceofpseudopotentials.WhileCaoetal.andJohlletal.usedPWSCF[ 82 ]andultrasoftpseudopotentials[ 92 ],ourcalculationsaredoneusingVASPandPAWpotentials,whichgivemagneticmomentsclosertoexperiments[ 134 ]. InFigure 5-6 ,weshowthetotalDOSandthePDOSontotheporbitalsofgraphene.AtEF,thetotalDOSforthespin-upchannelofsmallerclusters(Fe1-Fe3)isclosetozero,whilethatoftheFe4andFe5clustersisnite.ThetransferofelectronfromthesorbitaltodorbitalsinthecaseofFeadatomisconrmedwiththetotalDOSwhichshowsthespin-downdstateatEF-0.25eV.ForFe2cluster,thespin-downdorbitalsfrombothFeatomsareatEF,givingrisetoalargermagnetization.TheinteractionofthegraphenewiththeFeclusterschangesitselectronicbehaviorbuttheDiracconeofthegraphene,asseeninthePDOS,isstillpreservedwithsomedeviationduetothehybridizationoftheporbitalsofthegraphenewiththesanddorbitalsoftheatomsoftheFeclusters.InthecaseofFe2,Fe4,andFe5clusters,EDshiftsbelowEFindicatingthen-typedopingofthegraphene.TheshiftofEDwithrespecttoEFisnotasignalofdopingwhenthereissubstantialhybridization.InthecaseofFeadatom,althoughEDliesclosetoEF,grapheneisfoundtobedoped.DuetothehybridizationoftheporbitalsofgraphenewiththedorbitalsoftheFeadatom,theDiracconeofthegraphenegetsmodiedwithapeakatEF-0.25eVinthespin-downDOSwhichleadstothedopingofthegraphene.AsimilaranalysisisalsotrueforFe3cluster.ThePDOSresultofthegraphenebeingn-typedopedforalltheclustersisinagreementwiththeBaderanalysis. 5.3ClustersonDefectedGraphene Wefocusonlyongraphenewithsinglevacancydefect,whichgreatlydistortsthelatticeandleadstoa1.29B=celllocalmomentafterafullrelaxation.Witha15A15Asupercell,thecalculatedformationenergyofsinglevacancyconvergesto7.88eV,whichisinagreementwiththeresultsofMaetal.[ 135 ]. 68

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5.3.1GroundStateGeometryandEnergeticsofGoldClusters InFigure 5-7 ,weshowoptimizedgeometriesofAuclustersondefectedgraphene.WendthemoststablepositionoftheAuadatomonthedefectedgraphenetobeabovethevacancyatadistanceof1.75A.TheNNCatomssurroundingthedefectsitearenotinthesameplaneasotherCatomsofthedefectedgrapheneandmoveupfromthegrapheneplane.WeobservethatthepositionsoftheNNCatomsdonothavethreefoldsymmetry.TwooftheCatomsmoveupfromthegrapheneplaneby0.42A,whilethethirdCatomis0.30Aawayfromtheplane.AverageC-CdistanceofNNCatomsofthedefectedgrapheneinthepresenceoftheAuadatomisabout0.18Amorethanthatofthedefectedgraphenewiththevacancyonly.OurresultsareincloseagreementwithKrasheninnikovetal.[ 108 ],whoreportedhaanddactobe1.85Aand2.10A,respectively.WendthegeometriesandorientationsoftheAunclustersforn>1ontopofthedefectedgraphenetobesimilartothoseontheperfectgraphene.Clustersretaintheirplanarpropertyandbondtothedefectedgraphenethroughtheanchoratom.AverageAu-Aubondlengthsofalltheclustersareaboutthesameasthoseofthecorrespondingclustersintheperfectgraphenecase. WeshowthebindingenergiesalongwithhaanddacoftheAuclustersontopofthedefectedgrapheneinTable 5-5 .BE1oftheadatomisfoundtobe-2.46eV,whichindicatesastrongbindingoftheadatomwiththevacancy.ThisresultagreeswithKrasheninnikovetal.,whoreportedthebindingenergytobe-2.0eV.Fortheclusterswithn>1,BE1decreasesmonotonicallywiththenumberofatoms.However,BE3increasesfromtheadatomreachingitsmaximumvaluefortheAu3clusterandthendecreasesonwards.Althoughtheclustersarebondedtothedefectedgraphenethroughtheanchoratom,thevariationofBE3withtheclustersizedoesnotdependondacwhichisaboutthesameforalltheclusters.BE1oftheAudimerislargerthanthatoftheadatom,whichindicatesthebindingbetweentwoAuatomsofthedimerisstrongerthanthatbetweentheanchorAuatomandthevacancy.ThiscanbeveriedbyBE2,which 69

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islargerfortheAudimerascomparedtothatoftheAuadatom.FurtherdecreaseintheBE1withtheclustersizecanalsobeexplainedbyBE2.BE2ofAu3-Au5clustersaresimilarinmagnitudewhichcausethedecreaseinBE1astheclustersizeincreases. AstheclustersonthedefectedgraphenehavesmallerhaandsubstantiallylargerBE3,inclusionofvdWinteractionsshouldnotcauseanysignicantchanges.WeverifythisbyincludingvdWinteractionsintheadatom-defectedgraphenecase.WithvdWinteractions,weobtainBE3andhatobe-3.19eVand1.77A,respectively.TheseresultsaresimilartothoseobtainedinthePBEcalculations.Asexpected,differencescausedbyvdWinteractionsaresmallandPBEfunctionalissufcienttodescribetheclustersonthedefectedgraphene. 5.3.2GroundStateGeometryandEnergeticsofIronClusters TheoptimizedgeometriesofFeclustersondefectedgrapheneareshowninFigure 5-8 .AdsorptionsiteoftheFeadatomonthedefectedgrapheneissimilartotheAuadatom,andhaanddacarefoundtobe1.23Aand1.77A,respectively.WeareincloseagreementwithKrasheninnikovetal.,whofoundhaanddactobe1.30Aand1.76A,respectively.UnliketheAuadatomcase,eachNNCatomsbondedtotheFeadatommoveupfromthegrapheneplaneby0.35A,preservingthethree-foldsymmetry.InthepresenceoftheFeadatom,averageC-CdistanceofNNCatomssurroundingthedefectsiteisabout0.12Amorethanthatofthedefectedgraphenewiththevacancyonly.Duetoitssmallersize,theFeadatomcauseslessdistortionnearthevacancyinthedefectedgraphenethantheAuadatom.UnliketheAuclusters,orientationsoftheFeclustersontopofthedefectedgraphenearedifferentfromthoseontopoftheperfectgraphene.Forexample,bondaxisofthedimerismoreparalleltographeneplaneratherthanbeingperpendiculartoit.Asaresultoftheorientation,clustersarenotbondedtothedefectedgraphenethroughtheanchoratomonly.ThemoststablestructureoftheFe5clusteristrigonalbipyramidalwhichisdifferentfromthesquarepyramidalstructureinthecaseoftheperfectgraphene.AscomparedtotheAuclusterswheretheaverage 70

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Au-Aubondlengthsoftheclustersareaboutthesamefortheperfectanddefectedgraphene,averagebondlengthsoftheFeclustersonthedefectedgrapheneareabout0.1Amorethanthecorrespondingclustersinthecaseoftheperfectgraphene.WendthatunliketheAuclusters,haanddacincreasemonotonicallywiththeclustersizewhichareshownintheTable 5-6 WendBE1oftheadatomtobe-7.62eV,whichisinagreementwiththereportedresultof-7.3eVforthebindingenergybyKrasheninnikovetal..ComparedtotheAuadatom,theFeadatomisabout0.5Aclosertothevacancywhichgivesrisetostrongerbinding.ThebindingenergiesoftheFeclustersontopofthedefectedgraphenearelistedinTable 5-6 .BE1decreaseswiththeclustersize,whileBE3decreasesfromFe1toFe2clusterandthenincreasesfortherestoftheclusters.Theinteractionoftheclusterswiththedefectedgrapheneisnotonlythroughtheanchoratom,butalsothroughotheratomsintheclusterswhichlieclosetothegrapheneplaneandcontributetothebinding.Thus,similartotheAuclustersonthedefectedgraphene,thevariationofBE3withtheclustersizecannotbesimplyexplainedbydacalone.LiketheAuclusters,thevariationofBE1withtheclustersizecanbeunderstoodthroughBE2. 5.3.3ElectronicStructureandMagnetizationofGoldClusters Baderanalysisisperformedtoestimatethechargetransferandthechargedensitydifferencen(r)isshowninFigure 5-9 .Comparedtotheclustersontheperfectgraphene,thedirectionofthechargetransferreverses.Weobservethechargetransferfromtheclusterstothedefectedgraphenewhosemagnitudeislargerthanthatintheperfectgraphenecase.ForalltheclustersAunwithn>1,theanchoratomlosesabout0.4eandtherestamountofthechargetransferiscontributedfromotheratomsintheclusters.Thetotalmagnetizationofthedefectedgraphenewiththeclusterscontaininganoddnumberofatomsiscloseto1.0B=cell,whilewiththeclusterscontaininganevennumberofatomsiszero.Themagnetizationoftheentiresystemisprovidedbytheclustersaswellasthedefectedgraphene.Thecontribution 71

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oftheclustertothetotalmagnetizationofthesystemisabout35%fortheAu1aswellasAu5clusters,and50%fortheAu3cluster.FromthePDOS,wendthatthemagnetizationinthecaseofAuadatomonthedefectedgraphenedoesnotcomefromtheelectronofthesorbital,insteadfromtheelectronsofthepanddorbitalsoftheadatom.HybridizationwiththedefectedgraphenegivesrisetoelectronswiththepcharacteristicsintheAuadatom.Electronsfromthepanddorbitalscontribute0.13B=celland0.19B=cell,respectively,tothetotalmagnetizationoftheadatom.ForAu3andAu5clusters,contributionsoftheelectronsoftheporbitalsoftheclusteratomstothemagneticmomentarenegligible,andtheelectronsofthesanddorbitalsoftheclusteratomsprovidethemagnetization.AlthoughthemagneticmomentofthedefectedgrapheneislocalizedontheCatomsadjacenttothevacancyfortheAuadatom,itbecomesdelocalizedforAu3andAu5clusters.AsshowninthemagnetizationdensityinFigure 5-9 ,theextentofthedelocalizationincreaseswiththeclustersize.Intheadatomcase,Auonlycontributesabout0.32B=cell,whiletwooutofthreeNNCatomswhichareabout0.42Aabovethegrapheneplanehavemagneticmoments0.35B=celleach,andthethirdCatomwhichisabout0.30Aabovethegrapheneplanehasamagneticmomentof-0.19B=cell.ThisresultagreeswithKrasheninnikovetal.,whoreportedthemagneticmomentoftheadatomonthedefectedgraphenetobe1.0B=cell. Togainafurtherunderstandingoftheelectronicstructure,weshowthetotalDOSandthePDOSontotheporbitalsofthedefectedgrapheneinFigure 5-10 .WendthattheclustersmodifytheDOSofthedefectedgraphenesubstantially.HybridizationbetweentheporbitalsofthedefectedgrapheneandthesanddorbitalsoftheAuclustersissolargethatthePDOSofthedefectedgraphenealmostresemblesthetotalDOSnearEF.ThePDOSshowsoscillationatEFasafunctionoftheclustersize.InFigure 5-10 ,itisshownthatthedefectedgraphenehasanitedensityofstateswiththeclusterscontaininganoddnumberofatomsandzerodensityofstateswiththeclusterscontaininganevennumberofatomsatEF. 72

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5.3.4ElectronicStructureandMagnetizationofIronClusters Weshowthechargedensitydifferencen(r)inFigure 5-11 .UponperformingtheBaderanalysis,wendthatthechargetransferisfromtheclusterstothedefectedgraphene.TheamountofthechargetransferislargerascomparedtotheperfectgrapheneaswellastotheAuclustersadsorbedonthedefectedgraphene.FortheclustersFenwithn>1,theanchoratomlosesabout0.9eandtherestamountofthechargetransferiscontributedfromotheratomsintheclusters.ThemagnetizationofthesystemisonlyduetotheclustersasshowninthemagnetizationdensityinFigure 5-11 .Liketheperfectgraphene,themagneticmomentofthesystemmainlycomesfromthedorbitalsoftheFeatomsintheclusters.UnliketheoscillationofthemagnetizationintheAuclusters,wendthatthesystemismagneticfortheironclustersFenwithn>1.ThemagneticmomentoftheFeadatomonthedefectedgrapheneisfoundtobe0.14B=cell,whichisclosetotheresultreportedbyKrasheninnikovetal.wherethesystemwasfoundtobenonmagnetic.AsexplainedbyKrasheninnikovetal.,threeoutofeightvalenceelectronsoftheFeadatomformsigmabondswiththeNNCatoms.Oneelectronformsoutofplanebondandthespinoftheremainingfourelectronspairuptogiverisetozeromagnetization. Thevariationofthemagnetizationofthedefectedgraphenewiththeclustersforn>1canbeexplainedthroughthemagneticmomentoftheanchoratom.ThemagneticmomentoftheanchoratomdependsonthelocalgeometriesoftheclustersaroundthevacancyanditsvalueisdeterminedfromthefactthattheelectronsoftheanchoratombindwiththedanglingbondsoftheNNCatoms.WendthatintheisolatedFedimer,eachoftheFeatomshavemagneticmomentscloseto3.5B=cell.AstheanchoratomoftheFedimeronthedefectedgrapheneissurroundedbythreeCatomswhichhaveonedanglingbondeach,threeelectronsoftheanchoratombindwiththeNNCatomsandgiverisetothemagneticmomentoftheanchoratomcloseto0.5B=cell.TheotheratomintheFedimeronthedefectedgraphenehasamagnticmomentof3.3B=cell, 73

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thuswegetthetotalmagnetizationofthesystemcloseto4B=cell.InthecaseofFe3cluster,whosegeometryisshowninFigure 5-8 ,thirdatomdoesnotchangethelocalgeometryaroundthedefectsite,andthusthemagneticmomentoftheanchoratominthiscaseisalsocloseto0.5B=cell.Otheratomsintheclusterhavemagneticmomentsof2.76B=celland2.99B=cell,whichmakesthetotalmagnetizationoftheFe3clusteronthedefectedgraphenecloseto6B=cell.AlthoughthemagneticmomentoftheanchoratominthecaseofFe2andFe3clustersiscloseto0.5B=cell,itincreasesto1.0B=cellforFe4andFe5clustersduetothedifferentlocalgeometryaroundthevacancy.AsshowninFigure 5-8 ,thegeometryoftheFe4clusterisconstitutedoftwoplanarFe3triangleswhichhavetheanchoratomincommon.Bothtrianglescontributeabout0.5B=celleach,whichmakestotalmagneticmomentoftheanchoratomcloseto1.0B=cell.FortheFe5cluster,thelocalgeometryaroundthevacancyissimilartotheFe4cluster,henceitsanchoratomalsohasthesamemagneticmomentasthatoftheFe4cluster.Otherthantheanchoratomoftheclusters,therestatomshavemagneticmomentsofabout3.0B=cell,thereforethetotalmagnetizationoftheFe4andFe5clustersonthedefectedgraphenearecloseto10.0B=celland13.0B=cell,respectively. WeshowthetotalDOSandthePDOSontotheporbitalsofthedefectedgrapheneinFigure 5-12 .LiketheAuclustersonthedefectedgraphene,hybridizationbetweentheporbitalsofthegrapheneandsanddorbitalsoftheFeclustersislarge.WendthatatEF,thetotalDOSforthebothspinchannelsoftheFe1andFe2clustersisclosetozero,whileniteforthoseoftheFe4andFe5clusters.FortheFe3cluster,thespin-downDOSisniteandthespin-upDOSisclosetozeroatEF.ThePDOSshowssimilarbehaviorasthetotalDOS,butunliketheAuclustersthereisnoodd-evenoscillationatEF. 5.4Summary WehaveperformedDFTcalculationsofgoldandironclusters(AunandFen,n=1,5)adsorbedonperfectanddefectedgrapheneinordertostudysize-dependentstructures, 74

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energetics,andelectronicandmagneticproperties.Wesummarizeourconclusionsasfollows(1)Alltheclustersarebondedtographenethroughananchoratom,withtheexceptionofFeclustersondefectedgraphene.ThegeometriesofAuandFeclustersadsorbedonperfectanddefectedgraphene,exceptforFe5clusterontopofperfectgraphene,aresimilartothecorrespondingfreestandingclusters. (2)BindingenergyandchargetransferofAuclustersaresmallerthanthoseofFeclustersinthecaseofbothperfectanddefectedgraphene. (3)InclusionofvanderWaalsinteractionsinAu-graphenesystemsuggeststhatlongrangeinteractionsarecrucialinthebindingofAuclusterstoperfectgraphene,andtheAu5clusterbecomesparalleltothegraphene.WendexcellentagreementbetweenvdW-DFmethodwithoptB88functionalandDFT-D2method. (4)Anodd-evenoscillationinthedopingofperfectgraphenewithAuclustersisobserved.Perfectgrapheneisfoundtobedopedforclusterswithanoddnumberofatomsandundopedforclusterswithanevennumberofatoms.PerfectgraphenewithFeclustersisdopedinallcases.RealspaceBaderanalysisdoesnotexplainthedopingcorrectlyinsystemswherechargetransferissmall. (5)MagnetizationofFeclustersadsorbedondefectedgrapheneisonlycausedbytheclusters.Bycontrast,whenAuclustersareadsorbedondefectedgraphene,magnetizationisduetotheclustersaswellasthedefectedgraphene.Magnetizationofthedefectedgraphenebecomesmoredelocalizedastheclustersizeincreases.Anodd-evenoscillationinthemagneticmomentofAuclustersonbothperfectanddefectedgrapheneisobserved.Thesystemisfoundtobemagneticwithamagneticmomentofapproximately1.0B=cellforanodd-numberedclustersandnonmagneticforaneven-numberedclusters.PerfectgraphenewithFeclustersismagneticinallcases,whiledefectedgrapheneismagneticforclusterswithn>1. 75

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Table5-1. Anchoratom'spositionabovegrapheneplane(ha),distancefromNNCatom(dac),bindingenergies(BE1-BE3),chargetransfer(q),andtotalmagnetization(mcell)ofAuclustersadsorbedonperfectgraphene.`-'signinchargetransfercolumnindicateschargetransferfromthegraphenetothecluster.Unitsfordac,BE1-BE3,q,andmcellareA,eV,e,andB,respectively. SystemhadacBE1BE2BE3mcellq Au12.892.82-0.107-0.107-0.1220.91-0.10 Au22.452.32-1.373-2.639-0.5260.00-0.09 Au32.432.33-1.345-1.288-0.6540.91-0.03 Au42.492.34-1.608-2.397-0.5150.00-0.08 Au52.572.45-1.681-1.975-0.2181.00-0.07 Table5-2. haandBE3ofAuclustersadsorbedonperfectgraphenewithinclusionofvanderWaalsinteractionsusingdifferentmethods. vdW-DF(revPBE)vdW-DF(optB88)DFT-D2 SystemhaBE3haBE3haBE3 Au13.40-0.3292.59-0.5062.50-0.583 Au22.56-0.5502.43-0.8872.35-1.009 Au32.51-0.7272.37-1.1222.30-1.270 Au42.57-0.7902.43-1.1512.30-1.316 Au52.73-0.5772.52-0.8822.35-1.097 Table5-3. WithvdWinteractions(optB88),ha,dac,BE1-BE3,q,andmcellofAuclustersadsorbedonperfectgraphene.SignconventionforthechargetransferissameasthatinTable 5-1 SystemhadacBE1BE2BE3mcellq Au12.592.48-0.451-0.451-0.5060.91-0.12 Au22.432.32-1.588-2.725-0.8870.00-0.10 Au32.372.32-1.545-1.458-1.1220.91-0.06 Au42.432.31-1.828-2.678-1.1510.00-0.11 Au52.522.42-1.874-2.059-0.8820.92-0.12 Au5(P)---1.932--1.1200.83-0.15 76

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Table5-4. ha,dac,BE1-BE3,q,andmcellofFeclustersadsorbedonperfectgraphene.`+'signinthechargetransfercolumnindicateschargetransferfromtheclustertothegraphene. SystemhadacBE1BE2BE3mcellq Fe11.552.11-1.126-1.126-1.1942.06+0.84 Fe21.892.37-1.960-2.698-0.9226.16+0.42 Fe31.662.20-2.447-3.435-1.3048.09+0.63 Fe41.742.24-2.705-3.478-1.20911.57+0.63 Fe51.942.41-2.868-3.519-0.83315.68+0.46 Table5-5. ha,dac,BE1-BE3,q,andmcellofAuclustersadsorbedondefectedgraphene.SignconventionforthechargetransferissameasthatinTable 5-4 SystemhadacBE1BE2BE3mcellq Au11.752.09-2.463-2.463-2.9971.02+0.29 Au21.702.10-2.549-2.635-3.2160.00+0.11 Au31.792.08-2.331-1.894-4.0860.84+0.13 Au41.782.11-2.257-2.037-3.7010.00+0.23 Au51.792.09-2.194-1.940-3.4840.94+0.12 Table5-6. ha,dac,BE1-BE3,q,andmcellofFeclustersadsorbedondefectedgraphene.SignconventionforthechargetransferissameasthatinTable 5-4 SystemhadacBE1BE2BE3mcellq Fe11.231.77-7.621-7.621-8.3890.14+1.08 Fe21.311.79-4.900-2.179-7.7124.03+1.42 Fe31.381.80-4.423-3.470-8.0025.99+1.36 Fe41.451.82-4.300-3.930-8.4549.81+1.74 Fe51.571.85-4.115-3.373-8.77713.53+1.85 77

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Figure5-1. OptimizedgeometriesofAuclustersadsorbedonperfectgraphene.Au1-Au5,andAu5(P)clustersareshownfromtoptobottomrow.Leftandrightcolumnsshowtopandsideviews. 78

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Figure5-2. OptimizedgeometriesofFeclustersadsorbedonperfectgraphene.Fe1-Fe5clustersareshownfromtoptobottomrow.Leftandrightcolumnsshowtopandsideviews. 79

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Figure5-3. Chargedensitydifferencen(r)andmagnetizationdensityofAuclustersadsorbedonperfectgraphene.Au1-Au5,andAu5(P)clustersareshownfromtoptobottomrow.Leftandrighttwocolumnsshowtopandsideviewsofchargedensitydifferenceandmagnetizationdensity,respectively.Inthechargedensitydifferencesubgures,redandlightblueareasrepresentpositiveandnegativechargedifference,whereasinthemagnetizationdensitysubgures,redandlightblueareasrepresentthespin-upandspin-downdensity.Allthesubgureshaveisosurfacevalueof0.01. 80

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Figure5-4. TotaldensityofstatesofperfectgraphenewithgoldclustersAun(n=1-5),andAu5(P)onleftandprojecteddensityofstatesontotheporbitalsofgrapheneonrightcolumn.Solidredandbluelinesrepresentthespin-upandspin-downdensityofstatesofgraphenewithclusters.Inordertoseetheeffectoftheclustersonthegraphene,densityofstatesofthefreestandinggrapheneisalsoshown.Dottedredandbluelinesrepresentthespin-upandspin-downdensityofstatesoftheisolatedgraphene.Fermienergyissetatzeroandshownbyverticalblackdottedline. 81

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Figure5-5. Chargedensitydifferencen(r)andmagnetizationdensityofFeclustersadsorbedonperfectgraphene.Fe1-Fe5clustersareshownfromtoptobottomrow.FormatsofthesubguresaresameasthoseofFigure 5-3 82

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Figure5-6. TotaldensityofstatesofperfectgraphenewithironclustersFen(n=1-5)onleftandprojecteddensityofstatesontotheporbitalsofgrapheneonrightcolumn.FormatsofthesubguresaresameasthoseofFigure 5-4 83

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Figure5-7. OptimizedgeometriesofAuclustersadsorbedondefectedgraphene.Au1-Au5clustersareshownfromtoptobottomrow.Leftandrightcolumnsshowtopandsideviews. 84

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Figure5-8. OptimizedgeometriesofFeclustersadsorbedondefectedgraphene.Fe1-Fe5clustersareshownfromtoptobottomrow.Leftandrightcolumnsshowtopandsideviews. 85

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Figure5-9. Chargedensitydifferencen(r)andmagnetizationdensityofAuclustersadsorbedondefectedgraphene.Au1-Au5clustersareshownfromtoptobottomrow.FormatsofthesubguresaresameasthoseofFigure 5-3 86

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Figure5-10. TotaldensityofstatesofdefectedgraphenewithgoldclustersAun(n=1-5)onleftandprojecteddensityofstatesontotheporbitalsofdefectedgrapheneonrightcolumn.Solidredandbluelinesrepresentthespin-upandspin-downdensityofstatesofdefectedgraphenewithclusters.Effectoftheclustersonthedefectedgrapheneisassessedbyshowingdensityofstatesofthefreestandingdefectedgrapheneaswell.Dottedredandbluelinesrepresentthespin-upandspin-downdensityofstatesoftheisolateddefectedgraphene.Fermienergyissetatzeroandshownbyverticalblackdottedline. 87

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Figure5-11. Chargedensitydifferencen(r)andmagnetizationdensityofFeclustersadsorbedondefectedgraphene.Fe1-Fe5clustersareshownfromtoptobottomrow.FormatsofthesubguresaresameasthoseofFigure 5-3 88

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Figure5-12. TotaldensityofstatesofdefectedgraphenewithironclustersFen(n=1-5)onleftandprojecteddensityofstatesontotheporbitalsofdefectedgrapheneonrightcolumn.FormatsofthesubguresaresameasthoseofFigure 5-10 89

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CHAPTER6CONCLUSION Usingrst-principlemethods,wehaveperformedtransportandelectronicstructurestudiesofmaterialswithdefectsandimpurities.Wehavedevelopedaplanewavescattering-basedtransportmethodforlowsymmetrynonorthogonallattices,wherethe3rdlatticevectora3isnotperpendiculartothebasalplane.Thisisdonebygeneralizingtheplanewavescattering-basedtransportmethodofChoiandIhm.AsopposedtoChoiandIhm'smethod,whichemploysonedimensionalBlochtheorem,weusethreedimensionalBlochtheorem.SolutionoftheKohn-Sham'sequationforthewavefunctionforlowsymmetrylatticesrequirestheinclusionofanadditionalphasefactorei(PG1z+QG2z)z.Theequationtoobtaintwodimensionalpotentialalsogetsmodiedwiththesamephasefactor.Wehaveappliedourgeneralizedmethodtocalculateinterfaceresistivityoftwin,,,andGBsincrystallinecopper.OurresultsshowtwinboundarytohavetheleastresistanceamongalltheGBsandinterfaceresistanceofotherGBstobeaboutthesame.OurgeneralizationoftheChoi-Ihmmethodhaspavedthewayforthestudyoftransportpropertiesofawiderangeofmaterialswithlowsymmetrylattices. Inadditiontoplanardefects,wehavealsostudiedpointdefects.Wehaveperformedelectronicstructurecalculationsofgoldandironclusters(AunandFen,n=1,5)adsorbedonperfectgrapheneanddefectedgraphenewithasinglevacancy.Wehavefoundthatalltheclustersarebondedtographenethroughananchoratom,exceptforFeclustersondefectedgraphene.WiththeexceptionoftheFe5clusteronperfectgraphene,thegeometriesofAuandFeclustersadsorbedonperfectanddefectedgraphenearesimilartothecorrespondingfreestandingclusters.FeclustersbindmorestronglythanAuclusterstobothperfectanddefectedgraphene.AlthoughperfectgraphenewithFeclustersisdopedforallclustersizes,anodd-evenoscillationinthedopingofperfectgraphenewiththesizeofAuclustersisobserved.Perfect 90

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grapheneisfoundtobedopedforclusterswithanoddnumberofatomsandundopedforclusterswithanevennumberofatoms.TheoriginofmagnetizationofFeclustersandAuclustersondefectedgrapheneisdifferent.WhilemagnetizationofFeclustersadsorbedondefectedgrapheneisonlycausedbytheclusters,inthecaseofAuclustersmagnetizationisduetotheclustersaswellasthedefectedgraphene.ForAuclusters,magnetizationofthedefectedgraphenebecomesmoredelocalizedastheclustersizeincreases.Anodd-evenoscillationinthemagneticmomentofAuclustersonbothperfectanddefectedgrapheneisobserved.Thesystemisfoundtobemagneticforodd-numberedclustersandnonmagneticforeven-numberedclusters.Thebindingenergyofmetalclustertodefectsitescanbeusedtocreatemetalclusterarrays,orquantumdotarrays.Basedontheinterestingsize-dependentelectronicandmagneticpropertiesofsmallclusters,optical,magneticandelectrontransportpropertiesofthesystemcanbemanipulatedbycontrollingtheclustersize. 91

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BIOGRAPHICALSKETCH ManojSrivastavawasborninasmalltownnamedBallia,150KmeastofVaranasi,India.Sincehischildhoodhewasinterestedinscience.AftermovingtoDelhiforhisundergraduatedegree,hisinterestinphysicsdeepened,anduponcompletionofhisdegreehedecidedtocometoUSAforhigherstudies.HewantstoapplytheknowledgehegainedduringPh.D.forthebettermentofthesociety.Otherthanphysics,Manojisinterestedinsports,music,andtravelling. 99