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First-Principles and Multi-Scale Modeling of Nano-Scale Systems

Permanent Link: http://ufdc.ufl.edu/UFE0022835/00001

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Title: First-Principles and Multi-Scale Modeling of Nano-Scale Systems
Physical Description: 1 online resource (143 p.)
Language: english
Creator: Cao, Chao
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: correlation, density, dynamics, function, functional, green, lda, molecular, nonequilibrium, theory, transport
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Development of nano-scale science and technologies has brought both great challenges and opportunities for the physics community. The study of nano-scale science blossomed over the last two decades because of the advances in experimental techniques, and it was further stimulated by the development of computer modeling methods. In this dissertation, I will start with demonstration of the current methods by applying them to realistic systems, and then development of the existing techniques. Firstly, we have employed classical and multi-scale simulations to study the mechanical properties of silica. We have identified the two-membered rings as the fingerprints of highly stressed structures, with their populations closely related with the fracture process. We have then studied the Pd-cluster-functionalized CNT systems. The conductance of metallic CNT based system increases upon hydrogen adsorption, and the conductance of semiconducting CNT based systems decreases upon hydrogen adsorption. The behavior is dominated by electron localization effects, and is related to charge transfer. The simulations show that both metallic and semiconducting CNTs should be better hydrogen-sensing materials individually than mixed ensemble CNTs. We have also demonstrated the importance of the on-site energy U in density functional Theory for a Ni-based single molecule magnet. Because of the strong correlation effects in this system, the DFT calculation fails. Thus the inclusion of a Hubbard-U like term is essential in order to obtain the correct ferromagnetic ground state and exchange-coupling constants. After taking the corrections into consideration, these properties were successfully reproduced by the calculation. For method development, we analyzed the shortcomings of the current implementation of multi-scale simulations, and introduced a new architecture. We have also employed maximally localized Wannier functions to obtain tight-binding model Hamiltonian for LaOFeAs. Finally we evaluated the approximations used in the start-of-art DFT+NEGF method. We have proven that under steady state condition, the system state is determined solely by its electron density profile, thus the effective Hamiltonian of the system is a functional of its electron density only. However, this functional is too complicated to obtain, and we propose a GW-type treatment as a first step.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chao Cao.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Cheng, Hai P.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-12-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022835:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022835/00001

Material Information

Title: First-Principles and Multi-Scale Modeling of Nano-Scale Systems
Physical Description: 1 online resource (143 p.)
Language: english
Creator: Cao, Chao
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: correlation, density, dynamics, function, functional, green, lda, molecular, nonequilibrium, theory, transport
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Development of nano-scale science and technologies has brought both great challenges and opportunities for the physics community. The study of nano-scale science blossomed over the last two decades because of the advances in experimental techniques, and it was further stimulated by the development of computer modeling methods. In this dissertation, I will start with demonstration of the current methods by applying them to realistic systems, and then development of the existing techniques. Firstly, we have employed classical and multi-scale simulations to study the mechanical properties of silica. We have identified the two-membered rings as the fingerprints of highly stressed structures, with their populations closely related with the fracture process. We have then studied the Pd-cluster-functionalized CNT systems. The conductance of metallic CNT based system increases upon hydrogen adsorption, and the conductance of semiconducting CNT based systems decreases upon hydrogen adsorption. The behavior is dominated by electron localization effects, and is related to charge transfer. The simulations show that both metallic and semiconducting CNTs should be better hydrogen-sensing materials individually than mixed ensemble CNTs. We have also demonstrated the importance of the on-site energy U in density functional Theory for a Ni-based single molecule magnet. Because of the strong correlation effects in this system, the DFT calculation fails. Thus the inclusion of a Hubbard-U like term is essential in order to obtain the correct ferromagnetic ground state and exchange-coupling constants. After taking the corrections into consideration, these properties were successfully reproduced by the calculation. For method development, we analyzed the shortcomings of the current implementation of multi-scale simulations, and introduced a new architecture. We have also employed maximally localized Wannier functions to obtain tight-binding model Hamiltonian for LaOFeAs. Finally we evaluated the approximations used in the start-of-art DFT+NEGF method. We have proven that under steady state condition, the system state is determined solely by its electron density profile, thus the effective Hamiltonian of the system is a functional of its electron density only. However, this functional is too complicated to obtain, and we propose a GW-type treatment as a first step.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chao Cao.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Cheng, Hai P.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022835:00001


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FIRST-PRINCIPLESANDMULTI-SCALEMODELINGOFNANO-SCALESYSTEMSByCHAOCAOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2008 1

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c2008ChaoCao 2

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Tomyparentsandmywife 3

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ACKNOWLEDGMENTS Iwouldliketorstthankmyadvisor,professorHai-PingCheng.Inmypast5yearsattheUniversityofFlorida,shetaughtmemuchmorethanscienticknowledge.Sheisnotonlyamentor,butalsoafriend.Noneofthisworkcouldhavebeendonewithoutherhelp.IwouldalsoliketothankProfessorsStephenHill,PeterHirschfeld,AndrewRinzler,ArthurHebard,andDimitriiMaslov.Itisgreatpleasureandhonortocollaboratewiththem.Theirsuggestionsanddiscussionsareveryimportantandvaluable.Allmycommitteemembers,includingProf.ArthurHebard,HenkJ.Monkhorst,SelmanHersheld,andBeverlySanders,haveusetheirvaluabletimetoreadmydissertationandimproveitsquality.AlthoughProfessorsSamuelB.Trickeyisnotonmycommittee,hehasalsogenerouslyoeredhelptocorrectmygrammaticandscienticimperfectionsinthisthesisuponmyrequest.HereIwouldliketoexpressmysincereappreciationtothemaswell.MythanksalsogotoDr.Xiao-GuangZhangatORNLandProfessorJing-GuangCheatFudanUniversity,aswellasDr.YaoHeandDr.ChunZhang.Ibenettedgreatlyfromtheirrigorousattitudesanduniquewaysofthinking.IwassponsoredbyNSFgrantNo.DMR-0325553andDMR-0218957,aswellasDOEgrantNo.DE-FG02-02ER45995throughoutmyPh.D.study.TheUF-HPCcenter,DOE/NERSCandORNLuserprogramhaveprovidedcomputerresourcesforalltheworkreported.Moreover,myfriendsinGainesvilledeservemysincereacknowledgementaswell.Theybroughthappinessintomylife.Withoutthem,lifeherewouldbedullandmiserable.Finally,Iwouldliketothankmyparentsandmywifeforsupportingmeduringthepast5years.Theyenduredtheextremehardshipofseparation,yettheyarealwaysatmyback.Theyaremyspiritualprop. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 LISTOFSYMBOLS .................................... 12 ABSTRACT ........................................ 13 CHAPTER 1SCIENTIFICBACKGROUND ........................... 15 2COMPUTATIONALMODELINGMETHODS .................. 20 2.1ClassicalMolecularDynamics ......................... 20 2.1.1BasicIdea ................................ 20 2.1.2EquationofMotion ........................... 21 2.1.3ExtendedHamiltonianandThermo/Barostats ............ 21 2.1.3.1Constanttemperature .................... 22 2.1.3.2Constantpressure ....................... 24 2.1.3.3Nose-Hooverthermo/barostat ................ 25 2.2DensityFunctionalTheoryandtheLocalDensityApproximation ..... 26 2.2.1BasicTheory ............................... 26 2.2.2SuccessandDrawbacks ......................... 29 2.3BeyondLDA:LDA+U ............................. 30 2.3.1HubbardModel ............................. 30 2.3.2LDA+UMethod ............................ 30 2.3.3LinearResponseTheorytoCalculateU ................ 32 2.3.4Self-ConsistentLDA+U ......................... 33 2.4BeyondDFT:GWMethod ........................... 34 2.4.1One-ShotGW(G0W0) ......................... 36 2.4.2Self-ConsistentGW ........................... 37 2.5Non-equilibriumGreen'sFunctionMethod .................. 38 3MECHANICALPROPERTIESOFSiO2NANO-WIRES ............. 42 3.1Introduction ................................... 42 3.2Method ..................................... 44 3.3ClassicalMDSimulationofAmorphousBulkandNano-wireSilicaFractureBehavior ..................................... 46 3.4Multi-ScaleSimulationofModelSiO2ChainInteractingwithH2OMolecules 50 3.5Conclusion .................................... 54 5

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4DFTSTUDYOFPd4-CLUSTER-FUNCTIONALIZEDCARBONNANOTUBESELECTRONICSTRUCTURE ............................ 56 4.1MethodsandCalculationalDetails ...................... 58 4.2ResultsandDiscussion ............................. 59 4.2.1PureCNTs ................................ 59 4.2.2Metallic(5,5)CNT ........................... 59 4.2.2.1Highcoverage ......................... 59 4.2.2.2Mediumcoverage ....................... 63 4.2.2.3Lowcoverage ......................... 66 4.2.3MediumCoverageSemiconducting(8,0)CNT ............ 68 4.3SummaryandConclusion ........................... 69 5NON-EQUILIBRIUMGREEN'SFUNCTIONSTUDYOFPd4-CLUSTER-FUNCTIONALIZEDCARBONNANOTUBESASHYDROGENSENSORS .. 71 5.1MethodandCalculationalDetails ....................... 71 5.2ResultsandDiscussion ............................. 73 5.2.1Structure ................................. 73 5.2.2ElectronicStructureAnalysis ...................... 74 5.2.2.1Transmissioncoecient ................... 74 5.2.2.2Densityofstates ....................... 76 5.2.2.3Chargetransferandhartreepotential ............ 78 5.2.2.4Localdensityofstates .................... 80 5.3SummaryandConclusion ........................... 81 6STRONGLYCORRELATEDELECTRONSINTHE[Ni(hmp)(ROH)Cl]4SINGLEMOLECULEMAGNET ............................... 82 6.1MethodandCalculationalDetails ....................... 84 6.2ResultsandDiscussion ............................. 87 7METHODDEVELOPMENT ............................ 93 7.1Multi-scaleSimulations ............................. 93 7.1.1DicultiesofCurrentMethods ..................... 93 7.1.2BasicIdea ................................ 94 7.1.3CodeDevelopement ........................... 96 7.1.4TestCase:NaClDissociationinWater ................ 98 7.1.4.1Staticquantumregionidentication ............ 99 7.1.4.2Dynamicquantumregionidentication ........... 101 7.2ConstructingEectiveModelHamiltoniansfromBandStructureCalculations:MaximallyLocalizedWannierFunction .................... 104 7.2.1Theory .................................. 104 7.2.2Application:Iron-BasedSuperconductorLaO1xFxFeAs ....... 105 7.3ElectronCorrelationsinTransportProblems ................. 114 7.3.1Background ............................... 114 6

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7.3.2IsaSingleElectronDensitySucient? ................ 116 7.3.3DirectiontoGo:GWCorrectedTransport .............. 118 8CONCLUSION .................................... 122 APPENDIX AEXAMPLEINPUTFILESFORPWSCF ..................... 125 A.1StructuralRelaxation .............................. 125 A.2Self-ConsistentCalculation ........................... 126 A.3BandStructure ................................. 127 A.4DOSandPDOSCalculations ......................... 128 BEXAMPLEINPUTFILESFORSMEAGOL .................... 130 B.1LeadsCalculation ................................ 130 B.2TransportCalculation ............................. 132 REFERENCES ....................................... 134 BIOGRAPHICALSKETCH ................................ 143 7

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LISTOFTABLES Table page 4-1Lowdinchargeanalysisofhigh-coveragePd4-CNTcomplex. ........... 62 5-1Excesschargeinthedeviceregionandoneachatomspecies. ........... 78 6-1TotalenergiesofNiSMMmagneticstatesrelativetoitsAFMstate. ....... 88 6-2MagneticmomentscapturedbyNi,O(1),Cl,NandO(2)atomsinNiSMM. .. 89 7-1ComparisonbetweenclassicalandquantumMDs. ................. 93 7-2Electronhoppingandon-siteenergiescalculatedfromMLWFs. ......... 111 8

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LISTOFFIGURES Figure page 1-1Scalingofphysicalquantitieswithsystemsize. ................... 16 1-2Quantizedconductanceobtainedinexperiments. .................. 18 2-1Totalenergyasafunctionoftheelectronoccupation. ............... 32 2-2LinearrelationshipbetweenUinandUoutinNi4singlemoleculemagnet. .... 34 2-3QPscGWcalculationofSiliconbandstructure. .................. 39 3-1Typicalstress-straincurve(schematicexample). .................. 42 3-2Ringdistributionanalysisforamorphoussilica. .................. 45 3-3Comparisonofstatisticallyaveragedstress-straindata. .............. 47 3-4Amorphoussilicafractureprocessringanalysis. .................. 48 3-5Silicanano-wirefracturewiththepresenceofwatermolecules. .......... 49 3-6Snapshotsoffracturepointforamorphoussilica. .................. 50 3-7Temperaturedependenceofamorphoussilicafracture. .............. 51 3-8Relaxedstructureof(H2O)n-silicachainattimet=0. .............. 52 3-9Fullquantumsimulationtypicaltrajectorysnapshots. ............... 53 3-10Multi-scalesimulationtypicaltrajectorysnapshots. ................ 54 3-11Defectsfoundin(H2O)160-SiO2chains. ....................... 55 4-1BandstructureandDOSofpure(5,5)and(8,0)CNT. .............. 59 4-2Structuresforhigh-coveragesystems. ........................ 60 4-3BandstructureandDOSforhigh-coveragesystems. ................ 60 4-4PDOSanalysisforhigh-coveragesystems. ..................... 61 4-5ElectronlocalizationinCNT+Pdsystem. ..................... 62 4-6Structuresformedium-coveragesystems. ...................... 64 4-7BandstructureandDOSformedium-coveragesystems. .............. 64 4-8PDOSanalysisformedium-coveragesystems. ................... 65 4-9Structuresforlow-coveragesystems. ........................ 66 9

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4-10BandstructureandDOSforlow-coveragesystems. ................ 67 4-11PDOSanalysisforlow-coveragesystems. ...................... 67 4-12Medium-coverage(8,0)CNTbandstructureandDOS. .............. 68 4-13Medium-coverage(8,0)CNTPDOS. ........................ 69 5-1Structureof(8,0)CNT-basedsystems. ....................... 72 5-2Structureof(5,5)CNT-basedsystems. ....................... 72 5-3TwoPd-clustersystemtransmissionspectrum. ................... 74 5-4SinglePd-clustertransmissionspectrum. ...................... 75 5-5PDOSofpureCNTsanddopedCNTswithouthydrogen. ............. 77 5-6ComparisonofdopedCNTPDOSbeforeandafterhydrogenadsorption. .... 77 5-7Hartreepotentialaveragedoverthexycrosssectionofthedeviceregion. .... 79 5-8LDOSof(5,5)CNT-basedsystems. ......................... 80 5-9LDOSof(8,0)CNT-basedsystems. ......................... 80 6-1Optimizedstructureof[Ni(hmp)(MeOH)Cl]4. ................... 83 6-2LinearresponseofnKSandnSCto. ........................ 85 6-3Calculationoftheself-consistentUparameter. ................... 86 6-4DFTcalculationoftotalandprojectedDOSfor[Ni(hmp)(MeOH)Cl]4. ..... 88 6-5DFT+UdcalculationoftotalandprojectedDOSfor[Ni(hmp)(MeOH)Cl]4. .. 89 6-6DFT+Up+dcalculationoftotalandprojectedDOSfor[Ni(hmp)(MeOH)Cl]4. 90 6-7VariationofJ1andJ2withrespecttodierentUpforO(1). ........... 91 7-1ForcemismatchfortheSi-ObondinSi(OH)4. ................... 94 7-2LogicalstructureanddataowofOPALarchitecture. .............. 96 7-3FlowchartoftheserverprocessintheOPALarchitecture. ............ 98 7-4OPALstaticquantumregiontestcasesnapshots. ................. 99 7-5OPALdynamicquantumregiontestcasesnapshots. ................ 103 7-6UndopedLaOFeAsbandstructure. ......................... 108 7-7DOSandPDOSofundopedLaOFeAs. ....................... 109 10

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7-8AtomicMLWFsandstrongesthoppinginthesystem. ............... 110 7-9DOScalculatedfromGGA+U. ........................... 112 7-10DOSofLaO1xFxFeAswithx=0.125projectedontoFeAslayers. ........ 112 11

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LISTOFSYMBOLS,NOMENCLATURE,ORABBREVIATIONS BOMDBorn-OppenheimerquantummoleculardynamicsCNTCarbonnano-tubeCORBACommonObjectRequestBrokerArchitectureDFTDensityfunctionaltheoryDIDomainidentication(inmulti-scalesimulations)DOSDensityofstatesGGAGeneralizedgradientapproximationHOMOHighestoccupiedmolecularorbitalsLDALocaldensityapproximationLDOSLocaldensityofstatesLUMOLowestunoccupiedmolecularorbitalsMDMoleculardynamicsMLWFMaximallylocalizedwannierfunctionsMPIMessagepassinginterfaceNEGFNon-equilibriumGreen'sfunctionPBEGeneralizedgradientapproximationofPerdew,BurkeandErnzerhofPDOSProjecteddensityofstatesSMMSingle-moleculemagnetSWCNTSingle-walledcarbonnano-tube 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyFIRST-PRINCIPLESANDMULTI-SCALEMODELINGOFNANO-SCALESYSTEMSByChaoCaoDecember2008Chair:Hai-PingChengMajor:PhysicsDevelopmentofnano-scalescienceandtechnologieshasbroughtbothgreatchallengesandopportunitiesforthephysicscommunity.Thestudyofnano-scalescienceblossomedoverthelasttwodecadesbecauseoftheadvancesinexperimentaltechniques,anditwasfurtherstimulatedbythedevelopmentofcomputermodelingmethods.Inthisdissertation,Iwillstartwithdemonstrationofthecurrentmethodsbyapplyingthemtorealisticsystems,andthendevelopmentoftheexistingtechniques.Firstly,wehaveemployedclassicalandmulti-scalesimulationstostudythemechanicalpropertiesofsilica.Wehaveidentiedthetwo-memberedringsasthengerprintsofhighlystressedstructures,withtheirpopulationcloselyrelatedwiththefractureprocess.WehavethenstudiedthePd-cluster-functionalizedCNTsystems.TheconductanceofmetallicCNTbasedsystemincreasesuponhydrogenadsorption,andtheconductanceofsemiconductingCNTbasedsystemdecreasesuponhydrogenadsorption.Thebehaviorisdominatedbyelectronlocalizationeects,andisrelatedtochargetransfer.ThesimulationsshowthatbothmetallicandsemiconductingCNTsshouldbebetterhydrogen-sensingmaterialsindividuallythanmixedensembleCNTs.Wehavealsodemonstratedtheimportanceoftheon-siteenergyUindensityfunctionaltheoryforaNi-basedsinglemoleculemagnet.Becauseofthestrongcorrelationeectsinthissystem,theDFTcalculationfails.ThustheinclusionofaHubbard-U 13

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liketermisessentialinordertoobtainthecorrectferromagneticgroundstateandexchange-couplingconstants.Aftertakingthecorrectionsintoconsideration,thesepropertiesweresuccessfullyreproducedbythecalculations.Formethoddevelopment,weanalyzedtheshortcomingsofthecurrentimplementationofmulti-scalesimulations,andintroducedanewarchitecture.WehavealsoemployedMLWFstoobtaintight-bindingmodelHamiltonianforLaOFeAs.Finally,weevaluatedtheapproximationsusedinthestate-of-artDFT+NEGFmethod.Wehaveproventhatundersteadystateconditions,thesystemstateisdeterminedsolelybyitselectrondensityprolen(r),thustheeectiveHamiltonianofthesystemHisafunctionalofn(r).However,thisfunctionalistoocomplicatedtoobtain,andweproposeaGW-liketreatmentasarststep. 14

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CHAPTER1SCIENTIFICBACKGROUNDTheconceptofnano-scienceandtechnologycanbetrackedbackatleastasfarasthetalk\ThereisPlentyRoomattheBottom"givenbyR.P.Feynmanin1959[ 1 ].Inithepredictedaprocessofminiaturizingmachinesanddevicesbymanipulatingindividualatomsandmolecules.Nearly50yearsafterthisfamoustalk,theareaofnano-scienceandtechnologyhasbecomeahotspotofscienticresearch.InadditiontotherealizationofFeynman'svisioninmodernelectronics,nano-structureshaveincreasinglywideapplications,especiallyinthechemicalindustryandinthebiosciences.Nano-scalematerialsturnedouttobenovelhigh-capacityanodematerials[ 2 3 ],bioactivematerialsthatcanpotentiallyserveasbonesubstituents[ 4 ],eectivecatalysts[ 5 { 10 ],hydrogenstoragematerials[ 11 { 16 ],etc.Inmodernelectronics,IBM,Intel,andAMDareusingstate-of-the-art45nmmanufacturingtechnologytoproducepowerfulCPUchips.Intheirlabs,10nmtechnologieshavebeenstudiedforapproximately10years.Asthedimensionofsinglefunctionaldevicesgoesdown,scalingeectsbecomemoreandmoreprominent.The10nmscaleisestimatedtobethelimitofconventionalcomplementarymetal-oxide-semiconductor(CMOS)integratedcircuits.Therefore,challengesarepresentforthephysicscommunitytoprovidenewideasandtheoriesfornewmaterialsandfuturetechnologies,e.g.electronics.Ingeneral,nano-scalesystemsarefundamentallydierentfrombulksystems(Figure 1-1 ).Thephysicalpropertyofasystemis,ingeneral,dependentonsomemeasureofsystemsizeR.WhenRisbigenoughtoreachthebulklimit(regionIII),(R)exhibitsasmoothscalablebehavior.WhileRiswithinthenano-scaleregime(regionI),(R)hascomplicatedbehaviorthatdependsstronglyondetailedinformationaboutsystemsize,structure,andchemicalcomposition,oringeneral,thecompletewavefunctionofthesystem(frig;fRIg),whereriaretheelectroncoordinates,andRIaretheioncoordinates. 15

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Figure1-1. Scalingofphysicalquantitieswithsystemsize(schematicrepresentation).RegionIisfocusofthenano-scalescience. IfweconsidertheHamiltonianofasystemofinterest,itcanbegenerallywrittenas: H=1 2MIXIr2I+1 2XI6=JZIZJ jRIRJj1 2Xir2iXi;IZI jriRIj+1 2Xi6=j1 jrirjj(1{1)Heretheatomicunit(~=1,me=1)isassumed.MIandZIrepresentsthemassandatomicindexofatomI,respectively.Butthetypicaltimescaleforelectronicmotionismuchsmallerthanthetypicaltimescaleforionicmotion,thusonecanintroducetheBorn-Oppenheimerapproximationtoseparatethetwomotions: (frig;fRIg)=(frig)(RI)(1{2)whereisthesolutionof He=1 2Xir2iXi;IZI jriRIj+1 2Xi6=j1 jrirjj(1{3)SincethedeBrogliewavelengthoftheionsisrelativelysmall,theionicmotionscanbeapproximatedwithNewtonianequationsverywell.ThusthetotalHamiltonian 1{1 becomes: H=P2I 2MI+1 2XI6=JZIZJ jRIRJj+He(1{4) 16

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Mostphysicalpropertiesofinterestdependoneithertheatomic(geometrical)structurefRIg(systemtemperature,pressure,volume,etc)oritselectronicstructure(frig)(polarizability,opticalproperties,magneticproperties,etc).AlthoughthesystemgeometryfRIgiscoupledwithitselectronicstructurethroughequation 1{4 ,itdoesnoharmtoviewthematdierentlevel,sinceonecanapplyvariousapproximationswhentreatingHe.ThedicultyofsolvingtheelectronicproblemHe=Eeliesintheelectron-electroncorrelations(whicharisefromthelastterminequation 1{3 )fromwhichallthecomplicatedmany-bodyeectsoriginated.In1965,W.KohnandL.-J.Shamproposedtheirfamousself-consistentproceduretocalculatee-ecorrelationsforagroundstateclosedsystem.[ 17 ]TheyarguedthatthegroundstateHamiltonianisafunctionalofnothingbuttheelectrondensity.Althoughtheyassumedthelocaldensityapproximation(LDA)intheiroriginalillustrationofthetheorems,theirmethodachievedgreatsuccess,andtodayisknownasKohn-ShamDFT.Nevertheless,LDAsometimescanyieldqualitativelywrongresults,sothatvariouseortstoimprovethisapproximationhavebeenproposedandstudied,includingGGA,LDA+U,andGGA+U[ 18 { 21 ].Whiletheseeortsareproventobeeectiveundercertaincircumstances,theyarenotuniversallyapplicablenorsystematicallybetterthanLDAand,forthe"+U"models,mostlysemi-empirical.Anothercompletelydierentapproach,withoriginsinquasi-particletheory,hasalsobeendevelopedforyearstotreate-ecorrelation,namelytheGWmethod.[ 22 { 25 ]Inthismethod,quasi-particlesinsteadofbareelectronsareconsideredinthetheory,providingamorerealisticphysicalpicture.OncethegroundstateofHeissolved,theclassicalionicmotioncanbeobtainedfromequation 1{4 .ThiscombinationofproceduresiscalledBorn-Oppenheimerquantummoleculardynamics(BOMD).Itprovidesagenerallyaccuratedescriptionofinter-atomicinteractions,andthususuallygivesexperimentallycomparablestructureandmechanicalproperties.Thereisanexperimentalchallenge,however.Themechanicalpropertiesof 17

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humandentalenamelatnano-scalehavebeenstudiedwithatomicforcemicroscopy(AFM)[ 26 ].Experimentshaveexaminedthefractureofbonematerials[ 27 ],andimprovedmechanicalpropertiesbyincorporationofothermaterialstoformnanoceramics[ 28 29 ].Incontrast,computationalcostrestrictsstate-of-the-artDFT-basedBOMDtosmallclustersandcrystalswithperiodicboundaryconditions(PBC).Thereforenewmethodsneedtobedevelopedtostudythesemorecomplicated,nano-structuredmaterials. Figure1-2. Quantizedconductanceobtainedinexperiments.(FigurefromRef.[ 30 ]) Electronictransportpropertyofnano-scalestructuresisanotherhotspotofcurrentstudy.Becauseoftheirsmalldimensions,theconventionalBoltzmanntransportpictureisnolongerapplicableinthesesystems.Theoretically,R.LandauerrelatedtheconductivityofaballisticconductorGtoitstransmissioncoecientTbyG=MTG0(G0=2e2=histheconductancequanta),whereMisthenumberofconductionchannels[ 31 ].Buttikerextendedthistheorytoamulti-terminalcongurationandinamagneticeld[ 32 ],andderivedthefamousLandauer-Buttikerformulae.ThisformulaisthengeneralizedandextendedusingtheKeldyshGreen'sfunctions(non-equilibriumGreen'sfunctions),leadingtothecurrentlywidelyusedNEGFmethod.[ 33 { 35 ]Thestate-of-arttreatmentofthetransportproblemisbasedonHongGuo'sself-consistentprocedure.[ 36 37 ]ItusesDFTtotreattheelectron-electroncorrelations,andthusobtainasingle-electronHamiltonian,fromwhichtheNEGFisconstructed.TheelectrondensitycanbeobtainedbyintegratingtheNEGF,andthenusedtoconstructanewHamiltonianusingDFT.The 18

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cyclerepeatsuntiltheelectrondensitybecomesconverged.Experimentally,transportatnano-scalewasdemonstratedtobeballisticincharacterbyB.J.vanWeesetal.[ 30 ](Fig. 1-2 )andD.Wharametal.[ 38 ].Sincethedeviceissmall,broadenedbutdiscreteenergylevelsareexpectedfornano-scaleelectronics,leadingtointerestingbehaviorssuchasnegativedierentialresistance(NDR)[ 39 { 43 ]andCoulombblockades(CB)eects[ 44 { 47 ].Allthesephenomenashowthecrucialimportanceofelectron-electroncorrelationsinaballistictransportsystem.Althoughe-ecorrelationshavebeenstudiedextensivelyforequilibriumclosedsystems,thereissubstantiallylessworkonnon-equilibriumopensystems.Theremainingchaptersareorganizedasfollowing.Inthenextchapter,Ireviewthecurrentlywidelyusedcomputermodelingmethods,includingmoleculardynamics(MD),densityfunctionaltheory(DFT),theGWapproximation,andthenon-equilibriumGreen'sfunction(NEGF)method.Thesemethodsarethenappliedtomodelseveralinterestingnano-scalesystems.Inchapterthree,themechanicalpropertiesofSiO2nano-wiresarepresented.Chaptersfourandvediscussthechemical-electronicpropertiesofPd-cluster-functionalizedCNTs,andtheirapplicationinhydrogensensing.ChapterdiscussesaboutthestronglycorrelatedelectronsandmagneticpropertiesofNi4SMMs,aswellastheirpossibleapplicationinspin-tronics.Extensionofcurrentmethodsispresentedinchapterseven,andconclusionssummarizedinchaptereight. 19

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CHAPTER2COMPUTATIONALMODELINGMETHODS 2.1ClassicalMolecularDynamics 2.1.1BasicIdeaConsiderasystemconsistsofNclassicalparticles.ThecompleteHamiltonianofthesystemisthen: Hc=NXi=1p2 2m+U(r1;r2;r3;:::;rN)(2{1)ThersttermisthekineticenergyEk,whichisalsodenedtobethesystemtemperatureviaNfkBT=Ek,whereNfisthetotalnumberofdegreesoffreedomofthesystem,andkBistheBoltzmannconstant.Henceeventhecollectivemotioncontributestothesystemtemperatureinclassicalmoleculardynamics(MD)simulation.Theinteractionenergycanbewrittenas:U=XiUext(ri)+Xi
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thesystemHamiltonianiswell-dened,andhencegivenasetofinitialconditionri;vi,theNewtoniandynamicsofthesystemcanbedetermined. 2.1.2EquationofMotionBydiscretizingthetimevariable,onecanapproximatetheNewtonianequationsintoaseriesofequations:ai(t)=1 miriUvi(t)=vi(tt)+aitri(t+t)=vi(t)t+1 2ai(t)2t (2{3)Here,tisusuallyreferredtoastimestep.ThisparticularformwasrstproposedbyVerlet[ 48 ],andisoneofthemostwidelyusedmethodsinclassicalMD.Byeliminatingthevelocityterms,onecanobtainanexpressionwithoutv:ri(t+t)=2ri(t)ri(tt)+ai(t)2t.ThisapparentlyshowstheawkwardnessofthevelocityintheoriginalVerletalgorithm,andthatnumericalerrormayentersinceahigherordersmallterm2tisaddedtoa0tterm.Thehalf-stepleap-frogalgorithmwasintroducedtotacklethisproblem[ 49 ].Itsequationsetcanbewrittenasri(t+t)=ri(t)+vi(t+1 2t)tvi(t+1 2t)=vi(t1 2t)+ai(t)tvi(t)=1 2(vi(t+1 2t)+vi(t1 2t) (2{4)Hence,no2ttermisinvolvedinthealgorithmwhichreducesthepossibilityoflosingcomputationalprecision. 2.1.3ExtendedHamiltonianandThermo/BarostatsThusfar,thesystemunderstudyhasaconstanttotalenergyandconstanttotalnumberofparticlesbydenition.Itthereforecorrespondstoamicrocanonicalensemble. 21

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Incontrast,therealsystemsofinterestneverareisolatedfromtheexternalenvironment,andoftenhaveconstanttemperatureorpressure.Insuchacase,thecomputationalsystemhastobecoupledwithathermo/barostattomimictherealsystem. 2.1.3.1ConstanttemperatureSincethecomputationalsystemtemperatureisdenedtobethekineticenergyofthesysteminMD,itisverystraightforwardtouseadirectscalingmethodtokeepthesystemtemperatureconstant.Thatis,onerescalesalltheparticlevelocityto v0(t)=v(t)s T0 T(t)(2{5)wherev(t)istheoriginalvelocityattimet,T0isthetargettemperature,andT(t)isthetemperaturecalculatedfromoriginalvelocities.Despitethesimplicityofthisscheme,itisrarelyusedinrealisticsimulations.Therearetwodiculties.Thearticialscalingofvelocitydoesnotcorrespondtoanyphysicalprocessanditcompletelyruinsthedynamicsbyapplyingascalingfactortothewholesystem.Berendsen[ 50 ]introducedamorecomplicatedalgorithm,sothatateachtimestep,velocitiesarescaledby =s 1+t T0 T(t)1(2{6)Ifwetakethelimitoft!0+,equation 2{6 isinfact:T(t+t)=T(t)+t (T0T(t))dT dtjt=T0T jt (2{7)andthesolutionhasanexponentialformT(t)=T0+Cet=.Thus,thephysicalmeaningofisthesystemrelaxationtime.Bysettingtheproper,thesystemtemperaturegraduallygoestothetargettemperatureinamorephysicalwaythaninsimplerescaling.However,thearticialrescalingfactorisnoteliminated,andthustheresultingbehaviorisstillnotphysical. 22

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Inordertosimulateaconstanttemperaturesystem(acanonicalensemble),theHamiltonianhastoincludenotonlythedegreesoffreedomfromtheoriginalsystemofinterest,butatleastonedegreeoffreedomrepresentingtheenvironmentaswell.In1984,Noseintroducedthefollowingimplementationofthisapproach.[ 51 ]Theextradegreeoffreedomisdenotedass,withconjugatemomentumps.Thearticialvariablesplaystheroleofatime-scalingparameter,moreprecisely,thetimescaleintheextendedsystemisstretchedbythefactors.Anextrapotentialenergytermisassociatedwithsas: Vs=(f+1)kBT0lns(2{8)wherefisthenumberofdegreesoffreedomoftheoriginalsystemofinterest.Theenvironmentalkineticenergytermisalsospeciedas Ks=1 2Q_s2(2{9)whereQisthectitiousmasswhichcontrolstherateoftemperatureuctuations.ThesystemLagrangianisthen Lext=L0+KsVs(2{10)whereL0istheLagrangianoftheoriginalsystemofinterest.AsetofequationsofmotioncanbedeterminedfromLext,andtheextendedsystemHamiltonian, Hext=H0+Ks+Vs(2{11)istheconservedquantityofthesystem.HereH0denotesthetotalHamiltonianfortheoriginalsystemofinterest.Itisobviousthattheextendedsystemisamicrocanonicalensemble,anditcanbeproventhattheoriginalsystemofinterestisthentreatedinthecanonicalensemble.Inpractice,itiscrucialtochooseapropervalueofQfortheNosethermostat.IfQistoohigh,itwilltakeaverylongtimeforthesystemtorecoverthedesiredtemperature.InthelimitofQ!1theenergyowbetweenthesystemandenvironmentiszero, 23

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thusleadingtoaconventionalMD.IftheQistoolow,weakly-damped,signicantenergyoscillationswilloccur,resultinginaverypoorapproximationtoequilibrium.[ 51 ] 2.1.3.2ConstantpressureSimilartothermostats,algorithmshavebeendesignedtosimulatesystemdynamicsunderconstantpressureaswell.Berendsenpresentedaconstantpressuresimulationtechniqueinthesamespiritashisconstanttemperaturealgorithm.[ 50 ]Thesystemismadetoapproachthedesiredpressureaccordingto dP(t) dt=P(t)P0 P(2{12)whereP(t)istheinstantaneoussystempressure,P0isthetargetpressure,andPistherelaxationtime.Thesystemvolumeisthenscaledby P=1t P(P(t)P0)(2{13)andallparticleshavetheircoordinatesscaledby1=3P: r0=1=3Pr(2{14)ThistechniquesharesthedisadavantageastheBerendsenthermostat,and,aswell,hasthedeciencythatitdirectlymodiesthecoordinates.Similartoconstanttemperaturesimulations,aphysicalsolutionwouldhavetoincludeatleastonedegreeoffreedomfortheenvironment.AnaturalchoiceofsuchadegreeoffreedomwouldbethesystemvolumeV,asproposedbyAndersen[ 52 ].ThekineticenergyassociatedwithVturnsouttobe KV=1 2QP_V2(2{15)Here,QPisthecticiousmassofthebarostat.Ifweimaginethesystemofinterestiscontainedwithinaboxwithapiston,QPcanbethoughtofasthepistonmass.The 24

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potentialenergyis VV=P0V(2{16)P0isthetargetpressure.Tocoupletheparticleswiththesystemvolume,theircoordinatesandvelocitiesarerewrittenintoscaledformr=V1=3sv=V1=3_s (2{17)andthepotentialenergyV0andkineticenergyK0oftheoriginalsystemofinterestarerewrittenasfunctionsofsand_s:V0=V0(V1=3s)K0=1 2V2=3Ximi_si2 (2{18)TheequationsofmotionthuscanbederivedfromtheresultingLagrangian,LV=L0+KVVV,andtheextendedHamiltonian,HV=H0+KV+VV,istheconservedquantity. 2.1.3.3Nose-Hooverthermo/barostatIn1985,HoovercombinedtheAndersenbarostatandNosethermostattosimulatethenPTensemble.[ 53 ]Hederivedthefollowingequationsinthespiritofanextended 25

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Hamiltonian:fi=riU_si=pi mV1=3_pi=fi(P+T)pi_T=1 QXip2i mfkBT0!P=_V 3V_P=(PP0)V 2PkBT0 (2{19)Unlessotherwisespecied,thisisthethermo/barostatusedthroughoutthisthesis. 2.2DensityFunctionalTheoryandtheLocalDensityApproximation 2.2.1BasicTheoryByapplyingtheBorn-Oppenheimerapproximation,theelectronicmany-bodyHamiltoniancanbewrittenas: H=Xi(1 2r2i+vext(ri))+Xi6=j1 jrirjj(2{20)Here,vextistheexternaleld,includingtheioniceld.ThelasttermistheCoulombinteraction.HohenbergandKohn[ 54 ]managedtoprovethat: 1. Theexternalpotentialvextcanbecompletelydeterminedbythegroundstateelectronnumberdensityn(r)uptoanadditiveconstant. 2. Thegroundstateenergycanbeobtainedbyvariationalminimizationofasuitablefunctionalofthedensitywithrespecttotheelectronnumberdensity.ThereforethegroundstateHamiltonianisafunctionalofthegroundstateelectrondensity,i.e.:H=H[n]=T[n]+Vext[n]+Vee[n],whereTisthetotalelectronickineticenergy,Veeisthetotalinter-electronicenergy,andVextisthetotalexternalpotentialenergy. 26

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Intheoriginalproofofthesecondtheorem,HohenbergandKohnassumedthev-representabilityoftheelectrondensity,i.e.theelectrondensityisassociatedwithagroundstateHamiltonianwithcertainexternalpotentialVext.ThisrestrictionwaslaterliftedbyLevyandLiebintheirconstrainedsearchformalism.[ 55 56 ]Theydenedatwo-stepminimizationprocedure,i.e.: E0=minnmin!nhjT+Veeji+ZVext(r)n((r)d(r)(2{21)Theinnerminimizationisrestrictedtoallthatgiveaspecicn,andtheouterminimizationsearchesallreasonableelectrondensitiesthatintegratestoN.AlthoughtheHohenberg-KohntheoremsprovedthatthegroundstateHamiltonianisafunctionalofthegroundstateelectrondensity,theuniversalfunctionalcannotbedeterminedfromtheirproofs(whicharenotconstructive).In1965,KohnandSham[ 17 ]proposedaself-consistentmethodtocarrythroughthevariationalprocedure,whichisbasedontwoassumptions: 1. Theexactground-stateelectrondensitycanbereproducedbytheground-stateelectrondensityofanon-interactingreferrencesystem. 2. Thereferrencesystemisnon-interacting,whichmeanstheeectivepotentialofthereferencesystemislocal.NoticethedierencebetweentherstHohenberg-KohntheoremandtherstassumptionmadebyKohnandSham.Theformerprovedthatthegroundstateelectrondensitynisv-representable,butthelatterassumeditsnon-interactingv-representability,whichisneverrigorouslyproven.Fortunately,itseemsthatthenon-interactingv-representabilityconditionisalwaysmetbyanyphysicallyreasonableelectrondensity.ThetotalHamiltonianthereforecanbewrittenas H[n]=T[n]+Vext[n]+Vh[n]+Vxc[n](2{22)TheexternalpotentialfunctionalVext[n]=Rvext(r)n(r)drisratherstraightforward.Soistheclassicalpartoftheinter-electronicinteractions(theHartreepotential)Vh[n]= 27

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1 2Rn(r)n(r0) jrr0jdrdr0.However,explicitformsforthenon-classicalpartoftheinter-electronicinteractions(theexchange-correlationpotential)Vxc[n]andthekineticenergyfunctionalarenotsoobvious.ThepurposeoftheKohn-Shamprocedureistointroduce T0[n]=1 2NXi=1Zi(r)r2i(r)dr(2{23)andthedierencebetweenT0andTcanbecombinedwiththeunknownVxc,sothatequation 2{22 turnsinto H[n]=T0[n]+Vext[n]+Vh[n]+V0xc[n](2{24)where V0xc[n]=Vxc[n]+T[n]T0[n](2{25)TakingvariationalminimizationofH,undertherestrictionthatthetotalnumberofelectronsremainsunchanged:Rn(r)dr=0gives: n(r)(T0[n] n+vext(r)+vh(r)+V0xc[n] n)dr=0(2{26)where vh(r)=Vh[n] n=Zn(r0) jrr0jdr0(2{27)andwecandene: vxc(r)=V0xc[n] n(2{28)Thus,wehave: 1 2r2+veff(r)i(r)=ii(r)(2{29)where veff(r)=vext(r)+vh(r)+vxc(r)(2{30) n(r)=NXi=1ji(r)j2(2{31) 28

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whereiarethereferrenceorbitals.Equations 2{27 2{31 2{23 2{28 2{29 ,and 2{30 canbesolvedself-consistently,thusthegroundstateofmany-bodyHamiltonian 2{20 issolved. 2.2.2SuccessandDrawbacksWhiletheKohn-ShamDFTrecipeisexact,itisextremelydiculttoobtaintheexactexchange-correlationfunctional.TheoriginalKohn-ShampaperusedtheLDA,whichisbaseduponahomogeneouselectrongas.Surprisingly,LDAgivesfairlygoodresultsinmanysystems,includingveryinhomogeneousonessuchasatomsandmolecules.IntheKohn-Shampaper,theyalsoproposedagradientexpansionapproximation(GEA)thatincorporatesthegradientoftheelectrondensityjrnj,inthehopethatitcouldimprovetheperformanceforhighlyinhomogeneoussystems.However,theGEAoftenleadstopoorerresultsthanLDA,becauseitviolatessumrulesandotherrelevantconditions.[ 57 ]Variousgeneralizedgradientapproximations(GGAs)wereinvented.TheyretainGEAbehaviorasalimitintheweaklyinhomogeneouscasewhilepreservingthemoreimportantsumrules.Althoughitssuccessesarewide-spread,therearecasesthatcurrentimplementationsofKohn-ShamDFT(eitherLDAorGGA)giveverybadorevenqualitativelywrongresults.Itiswell-knownthatalltheexcitationenergiesexcepttherstionizationenergiesgivenbyLDAandGGAareverydierentfromexperimentalresults.Asforgroundstateproperties,LDAandGGAareknowntofailforsometransitionmetaloxidesandrare-earthelements(d-orf-electrons).TheexcitationenergyproblemisrelatedtothefactthatDFTisaground-statetheory,andalltheorbitalenergiesexceptthehighestoccupiedoneareofnodirectphysicalmeaning(evenitistroubledbyself-interactionerror).Thus,onehastoresorttoothertheoriesforthesekindofproperties,suchasTDDFT,GW.ThesecondproblemisbecauseLDAandGGAtendtodelocalizeelectronsexcessivelybecauseofincompleteaccountingforcoulombicself-interaction.Methodstoincorporatethesepropertiesalsohavebeendeveloped,includingLDA+U,GGA+U,andLDA+DMFT. 29

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2.3BeyondLDA:LDA+U 2.3.1HubbardModelTheoriginalHubbardmodelhastheHamiltonian HHub=tXj;;cj+ycj+UXjnjnj(2{32)wherejisthesiteindex,labelsthenearestneighbors,isthespin-index(withvalues,),nistheoccupationnumber,Uisthe"on-site"coulombrepulsion,andtisthehoppingamplitudebetweennearestneighbors. 2.3.2LDA+UMethodIn1991,V.I.Anisimovetal.rsttriedtoadaptHubbardmodelintoaDFTframework,acombinationnowcalledtheLDA+Umethod.[ 19 ]Init,thegroundstateenergyisgivenby ELDA+U[n(r)]=ELDA[n(r)]+EHub[fnImm0g]Edc[fnIg](2{33)whereEHubistheHubbard-Utermwithcorrecton-sitecorrelationincorporated;Edcisthedoublecountingterm,evaluatedasthemean-eldapproximationtotheon-sitecorrelation,Iisthesiteindex,isthespin-index(withvaluesand),andnistheoccupationnumber,denedas nImm0=Xk;ik;ijImIm0jk;i(2{34)Simplemanipulationsgive: ELDA+U[n(r)]=ELDA[n(r)]+XIU 2"Xm;6=m0;0nImnI0m0nI(nI1)#(2{35)Aftervariationwithrespecttohk;ij,wealsoobtainamodicationtotheeectivepotential Vjk;ii=VLDAjk;ii+XI;mU1 2nImjImihImjk;ii(2{36) 30

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Despiteitsstraightforwardphysicalexplanation,onecanseethatthemagneticquantumnumbersmandm0areinvolvedinequation 2{35 ,thusthetotalenergyisnotrotationallyinvariant.Inreality,itisofcourseirrelevanthowwechoosethereferenceaxisunlessanexternaleldpreselectsadirection.Toxthisproblem,V.I.Anisimovetal.derivedanewexpressionbydecomposingtheCoulombinteractionintoasummationoversphericalharmoniesYml.[ 20 ]TheresultingHamiltoniantermsareEHub[fnIg]=1 2Xm00;m000;;Ihm;m00jVeejm0;m000inImm0nIm00m000+1 2Xm00;m000;;I(hm;m00jVeejm0;m000ihm;m00jVeejm000;m0i)nImm0nIm00m000 (2{37)Edc[fnIg]=XIU 2nI(nI1)J 2nI(nI1)+nI(nI1)whereU=1 (2l+1)2Xm;m0hm;m0jVeejm;m0iJ=1 2l(2l+1)Xm;m0;m6=m0hm;m0jVeejm0;mi (2{38)Whileequation 2{38 isrotationallyinvariant,itisfairlycomplicatedanditsphysicalmeaningisnotstraightforward.M.Cococcionietal.studiedthisproblemandgaveaverysimplerotationally-invariantexpression[ 18 ]EHubEdc=U 2XIXm;(nImmXm0nImm0nIm0m)=U 2XITrnI(1nI) (2{39)ThisistheLDA+UmethodimplementedinPWSCF[ 58 ],andisusedthroughoutthisthesis. 31

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Figure2-1. Totalenergyasafunctionoftheelectronoccupation(schematicrepresentation).FigurefromRef.[ 18 ]. 2.3.3LinearResponseTheorytoCalculateUTheUinLDA+Uwasregardedasattingparameterorsemi-empiricalvalueforalongtime.In2005,Matteoetal.pointedoutthatUisrelatedtotheunphysicalcurvaturebehavioroftheLDA(andGGAaswell)totalenergyandissomehowrelatedwiththeself-interactioncorrection.[ 18 59 ]Consideraparticularatomwithinasolid.Theelectronoccupationn=N+(Nisinteger,06<1)onthisatomcanbefractional,whichmeansthatthelocalizeddensitymatrixforthisatomisnotapurestateofthisatom,butastatisticalmixtureofstateswithNandN+1electrons.Thus,thetotal(groundstate)energycontributedbythisatomisalinearcombinationoftheenergiesofthesestatesEn=(1)EN+EN+1(redcurve,gure 2-1 ).AlthoughthispiecewiselinearbehavioroccursproperlyforexactDFT,itismissingfromLDAandGGAcalculations,asshownviatheblackcurveingure 2-1 .Therefore,thephyscalmeaningoftheUcorrectionistoxtheunphysicalcurvaturebehavioroftheLDAenergy.Thus,theUparametercanbecalculatedfrom U=@2E[fqIg] @q2I@2EKS[fqIg] @q2I(2{40) 32

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whereqIisthechargeoccupationforsiteI,Eisthetotalenergy,anddEKSisatotalenergychangebecauseoforbitalrehybridizationinthenon-interactingsystemduetoanoccupationchange.Thisenergychangehasnothingtodowithon-sitecoulombinteractions,andthushastobesubtractedaway.Inpractice,itisverydiculttoxtheoccupationnumberforaspecicsite.Instead,whatisdoneisaconstrainedKohn-ShamDFTforthesystemusing E[fIg]=minn(r)(E[n(r)]+XIInI)(2{41)Thisleadsto,@2E @nI@nJ=@I @nJ@2E @I@J=@nI @J=IJ (2{42)weget @2E[fqIg] @q2I=()1II(2{43)whereistheresponsematrixdenedinequation 2{42 .ThereforeU=(KS)1II. 2.3.4Self-ConsistentLDA+UFromthediscussionabove,weknowthatUshouldxtheLDAorGGAunphysicalcurvaturebehavior.Infact,oncetheLDA+UorGGA+Umethodisemployed,thequadraticdependencyoftheHamiltoniancouldbewrittenasEquadratic=USCF 2XI"XiIiXjIj1!#Uin 2XIXiIiIi1 (2{44)whereIiistheelectronoccupationnumberfororbitaliofsiteI,USCFtermisthequadraticcontributionfromtheLDAorGGAfunctionalmodeledwithamean-eldexpression,andUinisthenormalUcontribution.AnaccuratechoiceofUrequiresUSCF=Uin.However,becauseoftheself-consistentprocedureinvolvedintheLDA+Ucalculation,theLDA+UgroundstatewouldbeverydierentfromtheLDAgroundstate, 33

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thusthescreeningeectfromLDA+UwouldbedierentfromtheLDAcalculation.Therefore,theUobtainedfromasimplelinear-responsetheorycalculationinfactmightnotxthecurvature.Instead,aself-consistentprocedurehastobeemployedtogetthecorrectUSCF. Figure2-2. LinearrelationshipbetweenUinandUoutinNi4singlemoleculemagnet.Thelinearregionisatleastfrom4.0eVto5.5eV. Fortunately,suchaprocedureispossible.IfoneweretocalculateUfromaLDA+Ugroundstateusingthelinear-responsetheorydescribedabove,theoutcomeUwouldbeUout=USCFUin mfromequation 2{44 .Althoughmis,ingeneral,dependentonUin,itisfoundtobeconstantoverawiderange(Figure 2-2 ).Therefore,USCFcouldbeobtainedbycalculatingUoutfromaseriesofLDA+Uingroundstates,andextrapolatingthelinearregiontoUin=0. 2.4BeyondDFT:GWMethodTheotherproblemwiththeKohn-ShamDFTansatzisthat,asaground-statetheory,itcannotdescribeexcitedstates,andthusexcitationenergies.However,theKohn-Shamorbitalenergiesoftenareregardedasappropriatezerothorderapproximation,becauseoftheformalresemblancebetweentheKohn-Shamequationsandquasi-particle 34

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equations.Therefore,othertheoriesarerequiredforthedescriptionofelectronexcitations.OneofthesetheoriesisdevelopedbyHedinetal.[ 22 ].Hedinrstderivedasetofself-consistentequationsbasedonthesingle-electronGreen'sfunctionpicture.Inthispicture,themany-bodyHamiltonianiswrittenas:H=1 2Xir2i+VH+Vext+XiZdr3(r;r0;Ei) (2{45)andtheself-energyisrelatedtothescreendCoulombpotentialvia(1;2)=iZG(1;4)W(1+;3)(4;2;3)d(3;4)W(1;2)=v(1;2)+ZW(1;3)P(3;4)v(4;2)d(3;4)P(1;2)=ZG(2;3)G(4;2)(3;4;1)d(3;4)(1;2;3)=(1;2)(1;3)+Z(1;2) G(4;5)G(4;6)G(7;5)(6;7;3)d(4;5;6;7) (2{46)wherethenotationfortheargumentsis1=(r1;t1),thespace-timecoordinateandd(1;2)denotesdr1dt1dr2dt2.Intheseexpressions,P(1;2)isthetime-orderedpolarizationoperator,W(1;2)isthedynamicallyscreenedinteraction,visthebareCoulombinteraction,and(1;2;3)isthevertexfunction.Hedin'sequationstogetherwiththeDysonequationG1=G10consistsaclosed,self-consistentset,whereG0istheGreen'sfunctionfornon-interactingsystem.ItisobviousthatthepolarizationfunctionPisinfactameasureofchargeresponsetotheexternaleldchange,andthevertexfunctionisaresponseoftheinverseGreen'sfunctiontotheexternaleldchange.Whileexact,theseequationsarecomputationallyextremelydemandingtosolve.Asaresult,certainapproximationshastobemade.ThefamousGWapproximationistoapproximatethevertexfunctionasadeltafunction,wherebyequations 2{46 arereduced 35

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to(1;2)=iG(1;2)W(1+;2)W(1;2)=v(1;2)+ZW(1;3)P(3;4)v(4;2)d(3;4)P(1;2)=iG(1;2)G(2;1) (2{47)Inpractice,theequationset 2{47 isstillcomputationallyintensive.Tofurthersimplifythecalculation,theinversedielectricfunctionisoftenusedinsteadofthepolarizabilityfunction,sothatW(r;r0;E)=Z1(r;r00;E)v(r00;r0)dr003 (2{48)Thephysicalinterpretationoftheseequationsisasfollowing.Consideranelectronwithinasolid.Becauseofthecoulombinteraction,itrepelsotherelectrons,thuscreatinganeectivepositivechargecloudaroundit.Theelectronplustheeectivepositivechargecloudisthencalledaquasi-particle,whichweaklyinteractswithothersduetothescreeningeect.Electronexcitationssuchasaddingorremovinganelectronthereforedonotsimplyaddorremovetheelectronitself,butalsomodifythepositivechargecloudaswell,thusisinfactcausingaquasi-particleexcitation.TheGWequationsreectthesephysicalargumentssincethebareCoulombinteractionvisreplacedwithscreenedinteractionW. 2.4.1One-ShotGW(G0W0)Becauseofitscomputationaldemands,theGWmethodsometimesisusedasapost-processingprocedureinsteadofaself-consistent.Thatis,oneobtainsthesingle-particleGreen'sfunctionanddielectricfunctionfromHartree-FockorDFTcalculations,thensimplyuseequation 2{48 and=iG0W0toobtainthedierencebetweenGWandDFTsingle-electronenergies.HereG0denotestheHForDFTGreen's 36

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functioncalculatedusing G0(r;r0;E)=XiDFT(HF)iy(r)DFT(HF)i(r0) EEDFT(HF)i+i(2{49)andW0denotesthescreenedpotentialobtainedfromequation 2{48 usingthedielectricfunctioncalculatedwithHForDFT. 2.4.2Self-ConsistentGWTheone-shotGWachievedgreatsuccessbycorrectlyreproducingarangeofexperimentalresults,mostnotably,semiconductorbandgaps.However,therearestillcasesforwhichitisfarfromsatisfactory.Also,theresultsofone-shotGWarestronglydependentonthequalityoftheinitialguess(G0)duetothelackofself-consistency.Twolevelsofself-consistencyarepossiblewithinGWcalculations.Atthelowerlevel,onlytheGreen'sfunctionGisupdatedself-consistentlywhilekeepingthescreenedinteractionWxed.Therefore,W0iscalculatedjustasinaone-shotGWrst,andself-consistencyisachievedbyusing (1;2)=iG(1;2)W0(1+;2)(2{50)andtheDysonequationintheform G(1;2)=G0(1;2)+ZG0(1;3)(3;4)G(4;2)d(3;4)(2{51)ThismethodisthereforealsocalledGW0.[ 60 ]Atthehigherlevelofself-consistency,theoriginalGWequations 2{47 areemployedsothatbothGandWareupdatedduringeachcycle.Thismethodiscalledthefullself-consistentGW.[ 61 ]Theintroductionofself-consistencyremovedtheinitial-guessdependency,butitdoesnotsystematicallyimprovethequalityoftheresults.Incertaincases,theself-consistentmethodsyieldresultspoorerthanG0W0.Subsequently,itwasfoundthatthemissingvertexfunctionandalackofself-consistencyactuallyadduptoanerrorcancellation,andsignicant 37

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improvementispossiblebyaddingvertexcorrectiontothecalculation.[ 62 ]However,thesecalculationsareevenmoredemandingtoperform.Thereisanotherwaytointroduceself-consistency.Sincetheone-shotGWdependssubstantiallyontheinitialguess,onecouldfocusonimprovingthequalityofthatinitialguessinsteadofGWresultitself.In2006,M.vanSchilfgaardeetal.proposedanewself-consistentprocedurecalledquasi-particleself-consistentGWorQPscGW.[ 23 { 25 ]TheymakeuseoftheGWself-energytoimprovetheLDAorGGAHamiltoniantoincludeasmanyofthemany-bodyeectsaspossible,andmeanwhilekeeptheHamiltonianhermitian.Duringtheprocess,theoriginalbareparticleswithintheDFTframeworkstarttogainmany-bodyeects,andarethuscalledbarequasi-particles.Theirself-consistentprocedurecanbewrittenas:W(r;r0;E)=Z1(r;r00;E)v(r00;r0dr003=iGWVxc=1 2Xi
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Figure2-3. QPscGWcalculationofSiliconbandstructure.Theexperimentsshowanindirectbandgapof1:12eV,whereastheLDAcalculationgivesabandgapof0:47eV,andQPscGWcalculationgivesabandgapof0:91eV. 39

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Hamiltonian.Wewillexaminetheformalismmorecarefullyinchapter7,sowejustbrieydescribethemethoditselfhere.WithintheDFT+NEGFformalism,thetransportsystemisdividedintothreepart,i.e.theleftlead,therightlead,andthedeviceregioninbetween.BecauseKohn-ShamDFTisusedforallthreeparts,itisimpliedthatelectronsarenon-interactingwithinbothleadsandthedeviceregion.Therefore,thesystemHamiltoniancanbewrittenas: H=Xk;kcykck+Xiidyidi+Xk;iTk;icyk;idi+h:c:(2{53)wherecandcyareannilation/creationoperatorsforleads;danddyarefordeviceregion;Tk;irepresentsthecouplingbetweendeviceandlead.Becausetheleadsarenon-interacting,MeirandWingreen[ 63 ]derivedthecurrentas J=ie 2~Zd 2TrLRG<+fLLfRRGRGA(2{54)wherewehaveL;R()=2TyL;R()fL;R()TL;R(),fL;R=1=e(L;R)=kBT+1aretheFermi-Diracdistributionfunctions(L;RaretheFermilevelsfortheleft/rightleadsrespectively),andGR,GA,andG
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usedtocalculatetheself-energies RI(E)=(+SDIHDI)GRII(E)(+SIDHID)(2{57)whereIiseitherLorR(leftorrightlead),+=E+i,GRIIistheretardedsurfaceGreen'sfunctionforleadI,andisaninnitesimalnumberinordertokeepcausality.TheeectiveGreen'sfunctionforthedeviceregionthencanbecalculatedas GRD(E)=+SDDHDDRL(E)RR(E)1(2{58)Jauhoetal.havederived[ 64 ]anexpressionfortheretardedself-energy <=fL(E)L+fR(E)R(2{59)whereiI=AIRI(I=L;R)isthecouplingstrengthbetweenleadIandthedeviceregion.TheDysonequationintheKeldyshGreen'sfunctionformalismis G<=G<0+GR0RG<+GR0
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CHAPTER3MECHANICALPROPERTIESOFSiO2NANO-WIRES 3.1IntroductionSilicondioxide(SiO2)isafascinatingmaterial,whichcomesinmanypolymorphicphasesthatexistoverawiderangeofpressuresandtemperatures.Thestructureandpropertiesofthesephasesareverydierentfromoneanother,makingSiO2oneofthemostscrutinizedmaterialsinsolid-statesciences[ 65 ].Themajorcrystallinephasesofsilicaare-and-quartz,-and-cristobalite,tridymite,coesite,andstishovite;itsamorphousstate(a-SiO2)isoftenreferredtoasvitreoussilica.Amorphoussilica(a-SiO2)isanimportanttechnologicalmaterial.Despiteitspoormechanicalproperties,itndsextensiveuseinthephotonicandelectronicindustriesduetoitsgoodopticalanddielectricproperties.Withtherecentemergenceofnano-technology,andtheconsequentdrasticreductioninthesizeofelectronicandopticaldevices,themechanicalpropertiesofa-SiO2havecomeunderscrutiny.Inaddition,therehasbeenafocusondevelopingandsynthesizingnewamorphous(andcrystalline)nano-silicamaterialsthatretainallpropertiesofthebulk,butwithbettermechanicalproperties. Figure3-1. Typicalstress-straincurve(schematicexample),notetheveryrichbehaviorandmechanicalproperties.Thegureistrievedfrominternet(URL: http://i240.photobucket.com/albums//uzidzit/img21.png ) 42

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Experimentally,themechanicalpropertiesofamaterialcanbecharacterizedbyitsstress-straincurve(atypicalstress-straincurveisshowninFig. 3-1 ).Mostoftheelasticregionischaracterizedwithalinearresponseofthestresstothestrain,withtheslopedenedastheYoung'smodulus.Theelasticregionisfollowedbyamuchlongernon-elasticorplasticregion,whichcanberoughlydividedintothreesubregions.Intheyieldingregion,thestressremainroughlyconstant(independentofthestrainapplied);theninthestrainhardeningregion,thestressincreasesinanonlinearmannerwhileadditionalstrainisapplied;afterahigheststress(theultimatestress)isreached,theneckingprocessoccurs,andthestressstartstodecrease;nallywhenitreachesthefracturepoint,thematerialfails,andthestressdroptozero.Thisfailurebehaviorhasbeenextensivelystudiedpreviously.Anumberofmoleculardynamicsstudiesfocusedoncrackpropagation,energyuxandtheRayleighwavespeed[ 66 { 70 ],andconrmedexperimentalndingsincludingthedynamicinstabilityofthecracktips.Insilicamaterials,MDsimulationshavefoundthatthebondbreakingprocessiscloselyrelatedtothecollectiveatomicmotions[ 71 72 ],andtheamorphousbulkmaterialexhibitsalargestrain-ratedependenceintheultimatestrainsustainedwhilethecrystalstructureexhibitsalmostnodependence[ 73 ].Itisalsofoundthatthebrittlefractureprocessinamorphoussilicaischaracterizedwiththeformationoftwo-memberedringstructuresinthebulkmaterials.[ 74 ]Allthesestudies,however,arebasedonthebrittlefractureprocessinthebulkmaterial.Fornano-phasematerials,intheSi3N4nano-phasestructure,ithasbeenfoundthatthemicrostructureplaysanimportantroleinthedynamicfractureprocess,andthatthefractureismuchmoreductilethaninthebulkmaterials.[ 75 ]Mostofthecurrenttheoreticalstudiesofsilicafracturedonotconsidertheinuenceoftheenvironment.However,thewater-silicainteractionisknowntobeoneofthemostimportantreasonsforsilicamaterialfailure,sinceH2OmoleculesreactwithSiO2tobreaktheSi-Obonds.Althoughthisreactionhasbeenknownforlongtime, 43

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microscopicmechanisticunderstandingofithascomeonlyratherrecentlyasaresultofadvancesincomputationalquantummechanicalmethodsandmoderncomputationalimplementations.Ithasbeenfoundthat,a)twowatermoleculesmustbeinvolvedintheSi-Obondbreaking,andb)two-memberedringsarethefavoredsites.Inthischapter,wepresentclassicalmoleculardynamics(MD)simulationstudiesofthemechanicalpropertiesofa-SiO2bulkandnano-wires,withthegoalofprovidingsomenewunderstandingoftheimportantdierencesbetweenbulkandnano-wiresilica.WealsopresenthereastudyoftheinteractionbetweenaSiO2nanochainandwatermolecules.ItsaimistoelucidatetheunderlyingmechanismofH2O-assistedSi-Obondbreaking. 3.2MethodWeusedclassicalMDintroducedin 2.1 forthesimulationofamorphoussilicafracture.Tobemorespecic,weusedtheDLPOLYMDsimulationpackage[ 76 ],togetherwiththeBKSpotentialforSiO2[ 77 ]andSPCforH2O[ 78 ].ToavoidunphysicalCoulombattractionsforhydrogenatomsinH2OandoxygenatomsinSiO2atsmalldistances,arepulsivetermwasaddedtotheoriginalforceelds.ForsiliconatomsinSiO2andoxygenatomsinH2O,aBKS-likeinteractionisassumed.ThebulkamorphousmaterialwasobtainedbyfollowingtheprocedureintroducedbyNormanT.Hu[ 79 ].Westartfromthesimple-crystobalitestructurewith3000atoms/cell,andheatitto9000Ktomeltit.Thistemperatureismuchhigherthantheexperimentalvaluebecausethecohesiveenergyfromtheclassicalforceeldismuchhigherthanthatfromexperiments.ThesamplethenwasgraduallyquenchedtoroomtemperatureintheNPTensemblebysettingthesystemtemperatureto8000K,6000K,4000K,2000K,1000K,and300Kinsequence.Theresultingmaterialhasadensityof2.48g/cm2,andsubsequentringanalysis(Fig. 3-2(a) )showsthatthemostabundantringstructureis7-membered.Thesearequitecomparabletotheexperimentalresults. 44

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(a)Bulk (b)WireFigure3-2. Ringdistributionanalysisfora)bulkandb)nano-wireamorphoussilica.Noticethatthenano-wiresamplehasamuchhigherabundanceforsmallringstructures. Fromthemodelbulkmaterial,werepeatthe3000atomunitcellthreetimesinthez-direction,andintroducealargevacuumregioninthex-andy-directions,sothatthenearestimagedistance(face-to-face)inthex-andy-directionswaslargerthan15A.Again,theresulting9000atomsupercellwasrelaxedunder300KintheNPTensemble.Theresultingnano-wirehasacross-sectionof30A30A.Similarringanalysisshowsthatthenumberoflow-coordinatedmemberedringsinthenano-wireismuchlargerthaninthebulkmaterial(Fig. 3-2(b) ).Sincelow-coordinatedringsareconsideredtobengerprints 45

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ofhighlystressedstructuresprimarilyfoundinsurfaces,theanalysisimpliesthatthesurfacemayplayanimportantroleinthenano-wire.Weusebothfullquantumandmulti-scalesimulationsforthestudyofwatermoleculesinteractingwithSiO2nanochain.Thefullquantumcalculationandthequantumpartofthemulti-scalesimulationsweredonewiththeSIESTApackage[ 80 ],whichimplementsKohn-ShamDFTinanumericallocalizedbasisset.WeusedthePBEexchange-correlationfunctional[ 81 ].Throughoutthischapter,weuseddouble-basispluspolarizationorbitals.Forthemultiscalesimulations,thequantumpartwasinterfacedwiththemultiscalesimulationpackagePUPILtocommunicatewiththeclassicalpart,whichisamodiedDLPOLYpackage.TheclassicalpartalsousedtheBKSforceeldforsilica.ThequantumregionconsistedofveSiO2unitsplusthewatermoleculeswiththeKohn-Shamorbitalsterminatedusingthelink-atomapproach[ 82 ],andtheinteractionsattheclassical/quantuminterfacewerealsomodeledusingBKSparameters. 3.3ClassicalMDSimulationofAmorphousBulkandNano-wireSilicaFractureBehaviorBoththebulkandnano-wiresampleswerestretchedwithconstantspeedinthesimulations,andtheirinternalstresswascalculatedtoobtaintheirfracturebehaviors.Thestretchingwasdonewithfollowingprocedure: Atcyclen,thesimulationunitcelliselongatedalongz-direction,sothatLn=Ln1+L.HereLnisthethecelllengthatcyclen,andL0istheoriginalcelllengthatequilibrium.Inpractice,wechoseL=L0=0:001. Thecoordinatesofalltheatomsarethenchangedto(xin=xfn1;yfn=yfn1;zin=zin1Ln=Ln1)toeliminatearticialdefects.Theirvelocitiesarekeptunchanged,i.e.vin=vfn1.Here,indenotestheinitialquantityofcyclenandfnisthenalvalueofcyclen. Thewholesystemisthenequilibratedfortime.Thepullingrateisthencalculatedas(L=L0)=.Threedierentpullingratesweresimulatedwithbothmaterials,eachwithvedierentinitialcongurationstocollectstatisticalinformation.Thestatisticalresults 46

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Figure3-3. Acomparisonbetweenstatisticallyaveragedstress-straindata.Theaveragesaretakenforbulkandnano-wireat0.1%/ps,0.2%/ps,and0.5%/pspullingrate,respectively.Noticethatthewiresamplesaremuchmoreductilethanthebulksamples. areshowningure 3-3 .Itisapparentthatbulkmaterialsareverybrittle,whilethenano-wiresareextremelyductile.Acomparisonofringanalysisonbothmaterialsateachstretchcycle(Fig 3-4 )showsthatthenumberoftwo-memberedringschangesdrasticallyduringthestretchingofabulkmaterial,especiallyatthefracturepoint,butthatthisnumberisrelativelystablewhenwestretchanano-wire.Thecomparisonalsoshowsthatthetwo-memberedringsaremostlylocatednearthesurfaces,andtheirpositionsarecontinuallychangingduringthestretch.Whenthebulkmaterialbreaks,themajorityofthetwo-memberedringsareproducedatthefracturesurface.Therefore,thetwo-memberedringsarethemajorsourceofthestressinsilicamaterials.Therelativelyconstantnumberoftwo-memberedringsinthenano-wirematerialandtheirhighmobilitymeanthatthesehighlystressedunstablestructurehelpedtheemissionofthestressduringthestretchingprocessinthenano-wires,andthusthenano-wiresbreaklikemetal.Snapshotsofthefracturepoint(Fig. 3-6 )showthatthenano-wirehasanobviousneckingprocess.Theforegoingsimulationsweredoneinvacuum,andthusnochemicalenvironmentwasinvolved.Toseeiftheenvironmenthasaninuenceonthesilicanano-wirefracture,wealsoperformedthesimulationwithnano-wireembeddedinsidewatermolecules. 47

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(a)Bulk (b)WireFigure3-4. Ringanalysisofamorphoussilicafractureprocess.Thex-axisistheappliedexternalstrain;theredcurveshowstheinternalstress;andthegreencurveshowsthenumberoftwo-memberedrings.Bothanalysisaredoneatpullingrate=0:1%/ps. Becausetheclassicalforceelds(BKSandSPC)weselectedarenotcapableofdescribingthechemicalinteractionsbetweenH2OandSiO2,thebondbreakingprocessisintrinsicallythesameasforthenano-wireinthevacuum.Therefore,thestress-straincurveweobtained(Fig 3-5(a) )doesnotlooknotablydierent.Nevertheless,asignicantdierenceinthelinearregionisobserved,showingthatthesilicamaterialhasweakerYoung'smodulusinthewetenvironment.Theringanalysis(Fig 3-5(b) )showsthatthenumberoftwo-memberedringsisdecreasedbyapproximatelyafactoroftwobecauseofthewater 48

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(a) (b)Figure3-5. Silicanano-wirefracturewiththepresenceofwatermolecules.Panela)isthecomparisonofthestress-straincurvesofthenano-wirewithandwithoutwatermolecules(drywireandwetwire);b)istheringanalysisforthenano-wirefracturewiththepresenceofwatermolecules.Bothdataareforapullingrate=0:1%/ps. molecules.Thiseectiscompletelyduetothestrongdipole-dipoleinteractionbetweenwatermoleculesandthesilicasurface,andshowsthatthewatermoleculeswillspeedtheprocessoftwo-memberedringbreakingevenwithoutchemicalinteractions.Thereductionoftwo-memberedringpopulationalsoexplainedthedecreaseofstressatthesamestraininawetenvironment. 49

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(a) (b)Figure3-6. Snapshotsoffracturepointfora)bulkandb)nano-wirea-SiO2.Noticetheobviousneckingprocessthatoccursforanano-wiresample. Thetemperaturedependenceofthefractureprocessisalsoexamined.AsshowninFig 3-7 ,theyieldstrengthdecreaseswhiletheyieldingregionextendsastheenvironmenttemperatureincreases.Asexpected,theincreasedkineticenergymakesthesilicamaterialseasiertorepairitsdefects,leadingtoamoreductilebehavior. 3.4Multi-ScaleSimulationofModelSiO2ChainInteractingwithH2OMolecules 1 Asdiscussedintheprevioussection,thetwo-memberedringpopulationonthesilicasurfaceisthengerprintinformationforitsmechanicalproperties.Theirinteractionwithwatermoleculesisofspecialinterest.Thereforeweundertookasystematiccomputationalstudyofthesilica-waterinteractioninthecaseofarepeatedtwo-memberedringsilicanano-chainwithbothfullquantumandmulti-scalesimulations.Thesilicanano-chainwasconstructedwith32SiO2unitsinsequentiallyorthogonaltwo-memberedringswith 1 ThisworkhasbeenpublishedasRef.[ 83 ] 50

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(a) (b)Figure3-7. Temperaturedependenceofamorphoussilicafracturefora)bulkmaterialandb)nano-wires.Dataaretakenatpullingrate=0:1%/ps. periodicboundaryconditions(PBC)assumed.Itsstructureandlatticeconstantswerefullyrelaxedat0K.Twodierentsituationswereinvestigated:alinearchainplustwowatermoleculesinthevicinity,thenthesameSiO2chaincoatedwithawaterlmwith160H2Omolecules(Fig. 3-8 ).Again,bothsystemsweretreatedunderPBC,andfullyrelaxedto0K.Wethenperformedsimulationsatroomtemperature(300K)withalatticeconstantequaltotheoneintheequilibriumstate,i.e.zeroexternalstress.Fig. 3-9 depictsatypicaltrajectorythatinvolvestheSi-Obondbreakingprocess.Attimestep1130(or339 51

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(a) (b)Figure3-8. Relaxedstructureof(H2O)n-silicachainattimet=0.Panela)depictsawaterdimer(n=2)withthechainandb)amonolayerwaterlmwiththechain(n=160). fs,Fig. 3-9(a) ),ave-coordinatedSiisformed,followedbyacollectiveprotontransferprocessinvolvingbothwatermolecules.Asaresult,onewaterdissociatesandformstwoSi-O-Hgroups,whiletheotherplaystheroleofcatalyst.Attimestep4300(1.3ps,Fig. 3-9(e) ),aSi-Obondbreaks.Withinonly50fsfromtheformationoftheve-foldSiunit,theprotontransferprocessisdone.ThelifetimeoftheH3O+transientingure 3-9(c) isonlyabout20fs.Othertrajectorieswithdierentinitialcongurationsdemonstratethesamefeaturesandtimescaleforthereactionprocess.CollectiveprotontransfermotionofthissortwasobservedbeforeintheSiO2-waterinteractionbytracingtheenergylandscapeorbyraisingthetemperature.Here,wehavewitnesseddirectlyafast,spontaneousbondbreakingandformationprocessatroomtemperature.Thisresultimpliesthattheseparticularsilicachainsshouldbeveryunstableinhumidenvironment.Tocollectstatistics,thezerostrainsimulationsofasilicachainwithtwoH2O'swererepeatedusingmulti-scalesimulations.Atypicaltrajectoryofamulti-scalesimulationisshowninFig. 3-10 .Justasinthefullquantumsimulation,ave-coordinatedSiisformedat345fs;andtheSi-Obondbreaksat1.5fs.Acollectiveprotontransferprocesssimilartothefullquantumsimulationalsocanbeidentied,andtheevolutiontimeisfoundtobe 52

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(a)nt=1130 (b)nt=1190 (c)nt=1250 (d)nt=1300 (e)nt=4300 (f)nt=4800Figure3-9. Snapshotsduringatypicaltrajectoryofafullquantumsimulation.ntdenotesthetimestepsatwhichthesnapshotistaken.Panela)showstheformationofave-coordinatedsilica,whichistheprecursorofareaction;b)-d)depictthewaterprotontransferthatleadstotwoSiOHgroups;e)andf)showaSi-Obondbreakingandstabilizationofthebrokenbond.Abrokenbondisdenedasoccurringwhentheinteratomicdistanceismorethan20%ofthecorrespondingT=0Kequilibriumlength. ofthesamescale.ThetotalCPUtimerequired,however,isreducedbyafactorof102comparingtofullquantumcalculations.Thesecondsystemwassimulatedunderthesamephysicalconditions.Withmanymorewatermoleculesinvolved,thephysicalbehaviorismuchmorecomplicatedthanfortherstsystem.Again,theprotontransferprocessisobservedduringthesimulation,accompaniedwiththeformationofmanylocaldefectstructures.WehereplottheabundanceofthesedefectsinFig. 3-11 .ItisremarkablethatbothH3O+andSiOHhavehighabundance,suggestingthatthesystemisveryecientatdissociatingwatermolecules.Also,thenumberofthesedefectsreachstablevalueswithin400-500fsofsystemevolutiontime.Bothdefects,butespeciallythehydroniumionH3O+,arequitemobileinthesystem. 53

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(a)nt=1500 (b)nt=3800 (c)nt=4755 (d)nt=5055Figure3-10. Snapshotsduringatypicaltrajectoryofamulti-scalesimulation.Aprocesssimilartothatfoundinafullquantumsimulationcanbeidentied.ThedenitionofbondbreakingisthesameasinFig. 3-9 Bothfullquantumandmulti-scalesimulationsshowthatthetwo-membered-ringsilicachainisveryreactivewithwater.Evenatzeroexternalstress,theintrinsicstraininthechainstructureissucienttodissociatewatermoleculesspontaneouslyinsubpicosecondtimeatroomtemperature.Additionofexternalstressmakestheprocessevenfasterbutdoesnotchangeitsessentialcharacter.Duringthedissociation,watermoleculesactasbothreagentsandcatalyst,andthereactionisacollectiveeortofallwateronthesurfaceofthesilicachain. 3.5ConclusionViaclassicalMDsimulation,thetwo-memberedringsareidentiedasthengerprintsofhighlystressedstructures,andtheirpopulationiscloselyrelatedwiththefractureprocess.Thenano-wirea-SiO2canreleaseinternalstressbybreakingexistingtwo-memberedringsandformingnewones,andthusismuchmoreductilethanthebulkmaterial.Hightemperaturehelpstheevolutionoftwo-memberedringsandthereforealsoleadstomoreductilebehaviors.H2Omoleculescanreactwiththetwo-memberedringsevenatzeroexternalstress,andtheH2Odipoleassiststhebreakingofthehighlystressedtwo-memberedrings.Thusthesilicamaterialisexpectedtobreakmoreeasilyinahumidenvironment.Bothfullquantumandmulti-scalesimulationsyieldedverysimilartrajectoriesforamodelSiO2chainfractureinthepresenceofH2Omolecules,and 54

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(a) (b)Figure3-11. Varioustypesofdefectsfoundin(H2O)160-SiO2chains.a)Evolutionofthedefectpopulationasafunctionoftime;b)Snapshotscontainingthesedefects. signaturestructuressuchasve-coordinatedSiatomsandH3O+groupswerefoundinbothsimulations,showingtherobustnessofthecurrentimplementationofmulti-scalesimulations. 55

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CHAPTER4DFTSTUDYOFPd4-CLUSTER-FUNCTIONALIZEDCARBONNANOTUBESELECTRONICSTRUCTURE 1 Carbonnanotubes(CNTs),especiallysinglewalledcarbonnanotubes(SWCNT),havedrawngreatattentionsincetheirdiscoveryin1991.ThedependenceofCNTbandstructureonthechiralityanddiameterwererstexplainedbyNoriakiHamadaetal.[ 85 ]bycalculatinggraphenebandstructurewiththetight-bindingmodel.CNTsformconductors,narrowormoderategapsemi-conductorsaccordingtotheirhelicalindices(n,m).HencetheCNTsareconsideredtobepromisingmaterialsandbuildingblocksforfutureelectronicdeviceapplications.[ 86 { 88 ]Furthermore,functionalizedCNTshavecontrollablebandstructuresarisingfromthechemicalpropertiesofdierentsidegroups,andarethereforethoroughlystudiedfordierentpurposes,e.g.,chemicalsensors,nano-catalysts,andmolecularswitches.[ 87 { 90 ]Thestudyofthemetal-CNTinteraction,includingmetal-CNTcontacts,metal-doped,andmetal-coatedSWCNTshasthereforedrawnmuchattentioninthelastdecades.IthasbeendemonstratedthatCNTscanbeusedaschemicalsensorscapableofdetectingsmallconcentrationsofmoleculeswithhighsensitivityunderambientconditions[ 89 { 98 ],andthusCNT-basedsensorshavepotentialimpactonawiderangeofhumanactivities,fromdomesticgasalarms,spacemissions,agriculturalandmedicaldiagnosticapparatus,tochemicalplantinstrumentsforsafetycontrol[ 99 ].Traditionalgassensors,suchassemiconductormetaloxides,silicon,organicmaterialsandcarbonblackpolymercomposites,areknowntobeoperableonlyathightemperatures(200C600C)becauseofthechemicalreactionbarrierbetweenthesensingmaterialsandthegasmolecules.Hole-dopedsemiconductorCNTs,however,havebeenproventohavesubstantial 1 ThisworkhasbeenpublishedasRef.[ 84 ]. 56

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conductancechangeuponexposuretoNO2andNH3moleculesatroomtemperature[ 90 ],andthusshedlightonroomtemperatureelectricalsensing.AmongallCNT-basedsensors,palladiumdopedCNTshavedrawnspecialattention.Palladium(Pd)andPd-clustersareknowntobereactivewithhydrogenmolecules,andthePdbulkandsurfaceshavebeenstudiedascatalystsandhydrogenstoragematerials.ThereactionpathsforhydrogendissociationonextendedPdsurfaceshavebeenexaminedingreatdetail[ 100 { 108 ].Bothbarrier-lessandnite-barrierdissociationofhydrogenhasbeenfoundonallthePdsurfacesstudied,andthebarrierheightisquitesensitivetotheadsorptionsiteandtheorientationofthehydrogenmolecule.Inspiteofthevastamountofworkonsurfaces,thereisrelativelylittleworkonhydrogenadsorptionontoPdclusters.In2001,ErnstD.Germanetal.[ 109 ]studiedhydrogeninteractionswithPd4-clustersusingspin-polarizeddensityfunctionaltheorycalculations,andobtainedresultssimilartothosefromsurfacestudies.HydrogendissociationonPdclustersalsowasdemonstratedexperimentallytobedependentonthesizeofthecluster[ 110 ].Recently,experimentsdemonstratedthatPddopedand/orcoatedCNTcanbeusedasahydrogensensor[ 91 { 98 ].Intheseexperiments,ensemble(mixed)SWCNTsshowlimitedhydrogensensitivity,whilesemiconductingSWCNTsshowanearly50%conductancechangeinthepresenceofhydrogen.Experimentalistswerealsoabletoidentifythatthepalladiumformsnon-continuous,closelydeposited,cluster-likestructure[ 91 93 ].Theoretically,S.Dagetal.[ 111 ]comparedtheadsorptionanddissociationofhydrogenmoleculesonbareandfunctionalizedCNTs,withtheconclusionthatPddopingpromotesthechemisorptionofhydrogenmoleculessignicantlycomparedtobareCNTs.L.Miaoetal.[ 112 ]haveemployedacontinuousatomicchainmodelandstudiedthebandstructureofPdandPd/NialloychaindopedCNTsusingdensityfunctionaltheory.TheirresultsshowthathydrogenadsorptionmodiestheDOSaroundEFdramaticallyinbothsemiconductingandmetallicCNTs,andthereforecausestheconductancechangeobservedexperimentally. 57

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ThischapterreportstheoreticalresultsonhydrogenadsorptionontoPd4-cluster-functionalizedCNTs.WefocusontheeectofthehydrogenadsorptiononthetransportpropertiesofthePd4-cluster-functionalized(5,5)CNTs.Weexaminebothdissociativeandmolecularadsorption,withhigh-,medium-andlow-densityPd4-clustercoverage.Therestofthischapterisorganizedasfollows:insection2,webrieydiscussthecalculationdetails;insection3,wepresentandanalyzeourresults;nallyinsection4,wesummarizethischapter. 4.1MethodsandCalculationalDetailsTheelectronicstructureandstructuralrelaxationcalculationswereperformedwithinDensityFunctionalTheory(DFT)(pleasereferto 2.2 )[ 17 ]frameworkusingthequantumespressopackage[ 58 ].Inthesecalculations,aplanewavebasisset,thePBEexchange-correlationfunctional[ 81 ]andRRKJultrasoftpseudopotentials[ 113 ]wereemployed.ForPdatoms,weusednonlinearcorecorrections,andthesemicored-statewastreatedasavalencestate.Theuseofultrasoftpseudopotentialsenabledustouseanenergycut-oassmallas32Ryasbasiswhilethedensitycut-owastakentobe400Ry.TheBrillouinzonesweresampledby1132,1116or118specialk-pointsusingtheMonkhorst-Packscheme[ 114 ]inthecaseofhigh,mediumandlowcoveragerespectively.ForDOScalculations,thek-pointswerechosentobe11512forhighcoverage,and11256fortherest.TospeeduptheKohn-Shamconvergence,Gaussiansmearingofwidth=0:001Rywasusedforelectronoccupations,andthenextrapolatedto=0.Forthechargeanalysis,Lowdinchargesonindividualatomswereobtainedbyprojectingthetotalwavefunctionontotheatomicorbitals.TomodelindividualCNTs,a40Asuper-cellinthexandydirectionswasintroducedsothatinteractionsbetweentwoneighboringunitcellscouldbeneglected.Accordingtoreference[ 109 ],thebond-topandbond-bondapproachcongurationsgivethemoststablephysisorptionandchemisorptioncomplexstructureforhydrogenadsorptiononPd4clusters,respectively.Therefore,weusedthesameapproachcongurationsfor 58

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physisorptionandchemisorptionrespectivelyinourcalculations.Weobtainedallthestructuresbyfullrelaxationwithoutanyrestrictiononsymmetry.Itiswellknown,however,thatphysisorptionexistsonlyatlowtemperatures,sincethebarrierfromphysisorptiontochemisorptionisverysmall.[ 107 108 ] 4.2ResultsandDiscussion 4.2.1PureCNTsThepure(5,5)and(8,0)CNTswererstmodeledforcomparisonpurposes(Fig. 4-1 ).Afteroptimization,weobtainedalatticeconstantof2.46Aand4.26Aforpure(5,5)and(8,0)CNTrespectively,whichcorrespondstoaC-Cbondlengthequalto1.42AinbothCNTs.Thebandstructureanddensityofstates(DOS)indicatethatthepure(5,5)CNTisaconductor(Fig. 4-1(a) )andthepure(8,0)CNTisasemiconductorwithabandgapof0.6eV(Fig. 4-1(b) ). (a)(5,5)Metallic (b)(8,0)SemiconductingFigure4-1. BandstructureandDOSofapure(a)(5,5)metallicand(b)(8,0)semiconductingCNT.Inallgures,EFisshiftedto0.0eV. 4.2.2Metallic(5,5)CNT 4.2.2.1HighcoverageHighcoveragewassimulatedwithtwounit(5,5)CNTdopedwithonePd4-clusterinonesuper-cell.ThefullyrelaxedstructureisshowninFig. 4-2(a) .TheaveragedPd-Pdbondlengthwithinonesuper-cellis2.67A,andthePd-Pdbondsacrosstwosuper-cells(3-20and4-20inFig. 4-2(a) ,i0referstothei-thPdatomintherightnearestneighbor) 59

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(a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-2. Structuresforhigh-coveragesystems.Unlessotherwisespecied,greenatomsarecarbon,whiteatomsarehydrogen,andgoldatomsarepalladiuminallgures.Right-topinsertsarethesamestructuresfromfrontview. (a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-3. BandstructureandDOSforhigh-coveragesystems.Inallgures,EFisshiftedto0.0eV. 60

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(a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-4. PDOSanalysisforhigh-coveragesystems.TheredlinesrepresentPDOSoncarbonatoms;thebluelinesrepresentPDOSonpalladiumatoms.ThecontributionsfromhydrogenatomsarenegligiblecomparedtothosefromCandPd.Inallgures,EFisshiftedto0.0eV. haveabondlengthof2.86A.SinceCNTisaquasi-1Dsystem,wedenethepalladiumcoverageintermsoflinearcoverage,i.e. coverage=lengthcoveredbyPd4cluster TotallengthofCNT(4{1)throughoutthispaper.Therefore,thecoverageisapproximately100%forthiscase.Inexperiments[ 91 92 ],thenumberofpalladiumatomsperlengthisoftenevenhigher.Becauseofdoping,aradialdistortionoftheCNTtakesplace,andthecross-sectionoftheCNTelongatesinthedirectionwheretheclustersweredeposited.Thewholesystem,however,becomesasemi-conductorwithabandgapofEg=128meV(Fig. 4-3(a) ).Projecteddensityofstates(PDOS)analysis(Fig. 4-4(a) )showsthatclosetotheFermienergylevel,thePdclustersdominatethehighestoccupiedband,whilePdclustersandtheCNTcontributeroughlythesametothelowestunoccupiedband.InTable 4-1 ,weshowtheLowdinchargesforthissystem.ThePd4-clustertransferredabout0.1electron 61

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PureCNTWithoutHWith-H2With-2H TotalC 160.00160.09160.12160.08C 136.00135.81135.85135.89TotalPd 39.9139.9039.83TotalH 1.982.08 Table4-1. Lowdinchargeanalysisofhigh-coveragesystems.Thelistednumbersarescaledbytheratioofthetheoreticaltotalchargetothecalculatedtotalcharge.TotalCreferstoallcarbonatoms;Ccarbontoatomsnotbondedwithpalladiumatoms(inthepureCNTcase,itisthetotalchargeof34carbonatoms);WithoutHreferstotheCNT+Pdsystemwithouthydrogenadsorption;With-H2tothehydrogenphysisorbedsystem;With-2Htothehydrogenchemisorbedsystem. totheCNT,butthetotalnumberofelectronsonthecarbonatomsawayfromthePdcluster(i.e.,thecarbonatomthatisnotbondedwithPdcluster)wasactuallyreducedbyabout0.2.Infact,thisindicatesthatduetothestrongbondingbetweenthecarbonandpalladiumatoms,localizedstatesformattheinterfacebetweentheclusterandtheCNT.WeshowthiseectbyplottingthechargedierenceofCNT+PdCNTPdinFig. 4-5 .ElectronlocalizationopensthegapattheFermilevel. Figure4-5. ElectronlocalizationinCNT+Pdsystem.CalculatedfromCNT+PdCNTPd;theredareaindicateschargeaccumulation(electronslocalizedinthisarea),andtheblueareaindicatesholeaccumulation.ElectronlocalizationwasobservedbetweentheCNTandPd4-cluster.Inthisgure,yellowatomsarecarbon,andwhiteatomsarepalladium. Fig. 4-2(b) and 4-2(c) illustratethestablecongurationofdissociativeandmolecularH2adsorptiononthisdenselycoveredsystem,respectively.Forbothchemisorptionand 62

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physisorption,thePd-Hbondlengthsare1.85A.TheH-Hdistancesinchemisorptionandphysisorptionare2.09Aand0.81ArespectivelycomparingtotheequilibriumH-Hbondlengthof0.75Ainhydrogenmolecule.ThebindingenergyisthencalculatedfromEb=ECNT+Pd+EH2ECNT+Pd+H,whichturnsouttobe1.06eVforchemisorptionand0.37eVforphysisorption.ThemolecularbindingofH2doesnotsignicantlychangethebandstructurenortheDOS,butthebandgapisreducedfrom128meVto52.1meV(Fig. 4-3(b) ).Inthedissociationcase,however,thesystemwasfoundtobeagoodconductor(Fig. 4-3(c) ).Thisiscausedbythemajorstructuralchangeinducedbythechemisorptionofhydrogen.Asaconsequence,thePdclustersformanatomicwire.Inthissituation,thePd-Pdbondlengthscanbeclassiedintotwocategories:(1)thePd-Pdbondswithinthesamex-yplane(aprincipallayer)(1-4,4-3and3-1inFig. 4-2(c) ),whichhavebondlengthsof2.64A;and(2)theinterlayerPd-Pdbonds(2-1,3,4and20-1,3,4inFig. 4-2(c) ),whichare2.93A.Totestthestabilityofthehigh-coveragesystembeforehydrogenadsorption,weremovedthehydrogenatomsfromthechemisorbedcomplex,inwhichPd4clustersaredistortedandformawire,andthenrelaxedthegeometry.Remarkably,thestructurerecoverstobethesamebeforehydrogenadsorption.Thisreversibilityshowsthatthestructurebeforeadsorptionishighlystableandthesensingmechanismisrobust. 4.2.2.2MediumcoverageThemediumcoveragewassimulatedwithonePdclusteron3unitcellsofCNT,whichresultsina66%coverage.TherelaxedPdclusterisaperfecttetrahedronwithanedge(Pd-Pdbond)lengthof2.68A(Fig. 4-6(a) ).RadialdistortionoftheCNT,whichismuchlessthaninthehighcoveragecase,isalsoobserved.SincethepercentageofthecarbonatomsbondedwiththePdcluster(andhencethenumberoflocalizedelectrons)issignicantlydecreasedfromthehighcoveragecase,itisexpectedthatthemediumcoveragesystemhasabetterconductivity.ThebandstructureandDOScalculations(Fig. 4-7(a) )conrmthisspeculation,andtheniteDOSattheFermienergylevelindicates 63

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(a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-6. Structuresformedium-coveragesystems.Right-topinsertsarethesamestructuresfromfrontview. (a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-7. BandstructureandDOSformedium-coveragesystems.Inallgures,EFisshiftedto0.0eV 64

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(a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-8. PDOSanalysisformedium-coveragesystems.TheredlinesrepresentthePDOSoncarbonatoms;thebluelinesrepresentthePDOSonpalladiumatoms.ThecontributionsfromhydrogenatomsarenegligiblecomparedtothosefromCandPd.Inallgures,EFisshiftedto0.0eV. thatthesystemismetallic.ThePDOSanalysis(Fig. 4-8(a) )showsthataroundtheFermilevel,thePdclusterscontributemorethan60%tothetotalDOS,almosttwiceasmuchastheCNTdoes.ThebandstructurealsoshowsthatthestatecontributedbythePdclusteraroundtheFermilevelisactuallyaatband(vF=3:3104m/s.,theotherbandis1:1105m/s),whichissemi-localized;thus,theconductivityislowforthisband.ThePd-Hbondlengthsforthemolecularanddissociativeadsorptionsare1.79Aand1.82A,respectively,andthecorrespondingbindingenergiesare0.45eVand1.30eV,respectively.Thetetrahedronisexpandedbecauseofhydrogenadsorption.Inmolecularadsorption,theresultingPd-Pdbondlengthsare2.71A,whileinchemisorption,theyareelongatedto2.81A(Fig. 4-6(c) ).However,thedistancebetweentwoneighboringtetrahedronsisnowmuchlongerthaninthehighcoveragecase;hence,thedrasticstructuralchangeobservedinthehighcoveragecasenolongerexists.Nevertheless,hydrogendissociationstronglyinuencestheelectronicstructureofthesystem.TheFermi 65

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velocitiescalculatedfromthebandstructure(Fig. 4-7(c) )arenow2:1105m/sand2:0105m/s,suggestingamuchhigherconductanceinthediusiveconductingregion.Also,80%oftheDOSaroundEFiscontributedbytheCNT(Fig. 4-8(c) ),indicatingthathydrogendissociationonthePdclusterreducedthelocalizationofelectronsbetweenCNTandPd4clusters.Thiseectisalsopresentinphysisorption,whichleadstoapproximatelyequalcontributionstothetotalDOSfromthePdclustersandtheCNT(Fig. 4-8(b) ).TheFermivelocitiesinphysisorptionare2:2105m/sand8:7104m/s(Fig. 4-7(b) ),indicatingaconductancebetweenthesystemwithouthydrogenandchemisorption. 4.2.2.3Lowcoverage (a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-9. Structuresforlow-coveragesystems.Left-topinsertsarethesamestructuresfromfrontview. WealsodepositedonePd4clusterper4unit-cellCNTstosimulatethelowcoveragesystem.TheCNTisapproximately50%coveredbythePdcluster.Inthissituation,thelocalgeometryofthePd4clusterisalmostidenticaltothatforthemediumcoveragecase.However,withevenanlowerpercentageoflocalizedelectronsthanthatinthemediumcoveragecase,thesystemismoreconducting(vF=4:4104m/sand1:6105m/s).Similartothemedium-coveragecase,hydrogenphysisorptionpreservestheoverallshapeofthebandstructurearoundtheFermilevel.ItwascalculatedfromthebandstructurethattheFermivelocitiesofthephysisorbedsystemare3:4105m/sand5:8105 66

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(a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-10. BandstructureandDOSforlow-coveragesystems.Inallgures,EFisshiftedto0.0eV (a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-11. PDOSanalysisforlow-coveragesystems.TheredlinesrepresentthePDOSoncarbonatoms;thebluelinesrepresentthePDOSonpalladiumatoms.ThecontributionsfromhydrogenatomsarenegligiblecomparedtothosefromCandPd.Inallgures,EFisshiftedto0.0eV. 67

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m/s(Fig. 4-11(b) ),indicatingamuchlargerelectronmobility.Withchemisorption,thismobilityisfurtherincreasedto6:5105m/sand6:7105m/s(Fig. 4-11(c) ).PDOSanalysisrevealsasimilarmechanismbehindthisconductancechange,i.e.,thebondingbetweenhydrogenandpalladiumreducesthelocalizationofelectrons. 4.2.3MediumCoverageSemiconducting(8,0)CNTInexperiments,thePdlayersonthePd-dopedCNTsformanon-continuous,closelydeposited,cluster-likestructure[ 91 93 ],whichismostclosetothemediumcoveragecaseinourmodel.Thuswehereonlysimulatethemedium-coveragecaseforthesemiconducting(8,0)CNTbasedsystems. (a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-12. BandstructureandDOSformedium-coverage(8,0)CNTsystems.Inallgures,EFisshiftedto0.0eV ThePd-dopedsemiconductingCNTsshowcompletelydierentbehavior.FromthebandstructureandDOScalculations,oncethesemiconductingCNTisdopedwithPdclusters,anewstateformsaroundtheFermilevelEF 4-12(a) .ThePDOScalculationrevealsthatthestateisduetoahybridizationbetweenPdandCorbitals,andisprimarilycontributedbyPd.Althoughthesebandsaresemi-localized,niteFermivelocitiesvFarestillpresent,leadingtoniteconductance.Also,thebandgap(betweentheconductionbandandlowestunoccupiedband)isgreatlyreducedbyabouthalf.Onceahydrogenmoleculephysisorbsonthesystem,itreopensthebandgapto0.5eV.TheimpuritystateinducedbythePddopingispushedtohigherenergyandbecomescompletelylocalized.Whenthehydrogenmoleculedissociates,theimpuritystate 68

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(a)Withouthydrogen (b)Physisorption (c)ChemisorptionFigure4-13. PDOSanalysisformedium-coverage(8,0)CNTsystems.TheredlinesrepresentthePDOSoncarbonatoms;thebluelinesrepresentthePDOSonpalladiumatoms.ThecontributionsfromhydrogenatomsarenegligiblecomparedtothosefromCandPd.Inallgures,EFisshiftedto0.0eV. getsfurtherpushedupduetotherehybridization.Itisthusexpectedthatthehydrogenadsorptionwillsuppresstheconductivityinbothcases. 4.3SummaryandConclusionInsummary,weperformedrst-principlessimulationsontheeectsofhydrogenadsorptionontheelectronicandtransportpropertiesofPd4-cluster-functionalizedCNTs.Asystematicchangeintheband-gapwidthversusthecoveragepercentagewasobserved,whichsuggeststhepossibilityofmanipulatingtheCNTbandstructurebyappropriatepalladiumdoping.With100%palladiumcoveredmetallic(5,5)CNTs,thesystemturnsouttobesemiconductingbecauseofelectronlocalizationeect;andforthesemiconducting(8,0)CNTs,thePddopingwillturnitintoaconductor.Allthe 69

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investigated(5,5)CNTbasedsystemsshowedanincreaseinconductanceuponhydrogenchemisorptionbutthemechanismsvary:athighcoverage,theadsorptionofhydrogeninducedasubstantialstructuralchange,andtheclustersformedanatomicwire;atmediumandlowcoverage,thedissociationofhydrogengreatlyreducedthelocalizationofthebindingelectronsbetweenCNTandPd4-clusters.Butforthe(8,0)CNTbasedsystems,thehydrogenadsorptionwillturnthesystembackintosemiconductors,andthussuppressestheconductivity.TheconductancechangeinbothsystemssuggeststhatthePd-clusterdopedCNTcanpotentiallyserveasahydrogensensor. 70

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CHAPTER5NON-EQUILIBRIUMGREEN'SFUNCTIONSTUDYOFPd4-CLUSTER-FUNCTIONALIZEDCARBONNANOTUBESASHYDROGENSENSORSIntheChapter 4 ,westudiedthebandstructureofPd4-clusterfunctionalizedCNTs.FormetallicCNT-basedsystems,weconcludedthatthehydrogenadsorptionreducestheelectronlocalizationeect,andhencetheconductanceofPd4-clusterfunctionalizedmetallicCNTsisexpectedtoincreaseuponhydrogenadsorption.ForsemiconductingCNT-basedsystems,electronlocalizationeectsreopensagapatthefermi-level,andthusthesystemconductanceisexpectedtodropuponhydrogenadsorption.However,withoutdirecttransportcalculations,itisstillnotclearwhatthesensingmechanismisandwhythereissuchalargedierencebetweensemiconductingandmetallicSWCNTs.Inthischapter,wepresentarst-principlesinvestigationoftheconductanceresponseofPd4-functionalizedSWCNTstohydrogenenvironment.WeexamineandcontrastsemiconductingandmetallicSWCNTsystems,andrevealthephysicsbehindPd-dopedSWCNThydrogensensing.Therestofthechapterisorganizedasfollows:inthefollowingsection,webrieyreviewthemethodsandcalculationdetails,thenwepresentourresultsforsemiconducting(8,0)SWCNTsinparallelwithmetallic(5,5)SWCNTs,combinedwithdiscussion;nallywesummarizethischapteranddrawconclusions. 5.1MethodandCalculationalDetailsAsbefore,wechosea(5,5)SWCNTasthemetallicmodelsystemanda(8,0)SWCNTasthesemiconductingmodelsystem.TheirtransportpropertieswerecalculatedusingtheDFT-combinedNEGFmethodintroducedinSection 2.5 .ThetransportisalongtheaxisoftheCNT(z-axis),andalargeinter-celldistancewasintroducedinthex-andy-directionstoeliminatetheinteractionsbetweentwoneighboringimages.TwotetrahedralPd4clustersweredepositedontopoftheCNTtoformthescattering/reactioncenter.Inordertotestthestabilityofourresults,wealsousedone-orthree-clusterdepositedsystemstorepeattheconductancecalculations.Theleadswerechosentobe 71

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(a) (b) (c)Figure5-1. Geometricstructureofthesemiconducting(8,0)CNT-basedsystems(a)dopedonlywithpalladium,(b)hydrogenmoleculephysisorbedonpalladium,and(c)hydrogenmoleculechemisorbedonpalladium.YellowatomsareAuintheleads;greenatomsarecarbon;magentaatomsarepalladium;whiteatomsarehydrogen.Theright-topinsertsineachpanelshowthelocalstructureofthePd-clusterfromfront-view.ThedottedlineshowstheH-Hbond. (a) (b) (c)Figure5-2. Geometricstructureofthemetallic(5,5)CNT-basedsystems(a)dopedonlywithpalladium,(b)hydrogenmoleculephysisorbedonpalladium,and(c)hydrogenmoleculechemisorbedonpalladium.YellowatomsareAuintheleads;greenatomsarecarbon;magentaatomsarepalladium;whiteatomsarehydrogen.Theright-topinsertsineachpanelshowthelocalstructureofthePd-clusterfromfront-view.ThedottedlineshowstheH-Hbond. 72

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Auwiresinthe(111)direction,andthelead-moleculedistancewasoptimizedtoensurestability.However,theCNTstructureandAusurfacestructureattheinterfacewerekeptrigidrespectivelyduringtheoptimization.Wearguethattheimportantphysicsoccursatthedeviceregion,andtheinterfaceintroducesnothingbutaxedinterfaceresistance.Therefore,thedetailedinterfacestructureisnotimportanttoourresults.Structuresotherthanthedevice/leadinterfacewerefullyrelaxeduntiltheforceoneachatomwassmallerthan0.01eV/A.Weusedthedouble-zetasplit-valencenumericalbasissets[ 115 ]andtheminimalbasissetsforthestructuraloptimizationandthetransportcalculations,respectively.MinimalbasissetswereusedfortransportcalculationsbecauseonlythestatesaroundEFarecrucialtotransportproperties,andthusaminimalbasissetisgoodenoughforthispurpose.Also,redundantbasisfunctionswillleadtosingularmatrixproblemswhencalculatingtheGreen'sfunction.Inallcalculations,thelocaldensityapproximation(LDA)inPerdew-Zungerform[ 116 ]wasincorporatedtodescribetheexchange-correlationsofthesystem.Theequivalentplane-waveenergycutoforthereal-spacegridwastakentobe200Ry.Fortheleads,weuseda1132Monkhorstschemek-grid[ 114 ]tosampletheBrillouinzone,whileforthedeviceregionwetakeonlythegammapointintoconsideration.Finally,weenforcechargeneutralitycriteriontobe0.001ebymatchingtheHartreepotentialsatleads-deviceregioninterface. 5.2ResultsandDiscussion 5.2.1StructureFigures 5-1 and 5-2 showtheoptimizedgeometriesforthedeviceregionsofthesemiconductingandmetallicCNTs,respectively.EventhoughhydrogenmoleculescanbeeitherphysisorbedorchemisorbedonPd-clusters,physisorptionisrarelyobservedexperimentallyduetothesmallbarrierbetweenthetwostates.[ 107 108 ]Wesimulatebothstatesforcompletenessandcomparisonpurposes.ThePd4clusterformsadistorted(slightlyelongatedalongthez-axis)tetrahedronontheCNTsduetostronginteractionsbetweenthecarbonandpalladiumatoms.ThePd-Hbondlengthsinthephysisorbed 73

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andchemisorbedcomplexareapproximately1.67Aand1.80A,respectively.TheH-Hdistanceisincreasedfrom0.75A(inaH2molecule)to1.00Ainphysisorption,andisfurtherincreasedto2.38Ainchemisorption. 5.2.2ElectronicStructureAnalysis 5.2.2.1Transmissioncoecient (a)PureCNTs (b)MetallicCNTbasedsystem:TwoClusters (c)SemiconductingCNT-basedsystem:TwoClustersFigure5-3. Transmissionspectrumof(a)pureCNTswithingoldleads(withoutanyPdorhydrogendoping),(b)metallic(5,5)CNT-basedsystemswithtwoPd-clusters,and(c)semiconducting(8,0)CNT-basedsystemswithtwoPd-clusters.\Pdonly"indicatesaPd-cluster-dopedCNTsystemwithouthydrogenadsorption;\Physi"indicatesahydrogenphysisorbedsystem;and\Chemi"indicatesahydrogenchemisorbedsystem.EFisalignedat0.0eVinallgures. WeshowtheNEGFcalculationoftheconductanceingure 5-3 .ThesmallbiasconductancenearEFcorrespondstothequantitymeasuredinmostexperiments.Theconductanceofpure(5,5)and(8,0)CNTwithoutPddopingis0.62G0and0.23G0,whereG0=2e2=hdenotestheconductancequanta(gure 5-3(a) ),respectively.Note 74

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thatinFig. 5-3(a) thereisaniteconductanceatEFforthesemiconductingCNTduetothetunnelingcurrent.ThechangesintheconductanceafterPddopingandhydrogenabsorptionareverydierentforthesemiconductingandthemetallicCNTs.AfterPddoping,theconductanceofthemetallicCNTsdecreasesto0.28G0(Fig. 5-3(b) ),andthatofthesemiconductingCNTsincreasesto0.54G0(Fig. 5-3(c) ).Withtheadsorptionofhydrogen,theconductancechangesarereversed.ForthesemiconductingCNT-basedsystems,theconductanceisreducedto0.27G0forphysisorptionand0.24G0forchemisorption(gure 5-3(c) ),respectively.Forthemetallicsystem,hydrogenadsorptionenhancestheconductanceto0.36G0forphysisorptionand0.53G0forchemisorption(gure 5-3(b) ). (a)(5,5)CNT-basedsystem:OneCluster (b)(8,0)CNT-basedsystem:OneClusterFigure5-4. Transmissionspectrumof(a)metallic(5,5)CNT-basedsystemswithonePd-cluster,and(b)semiconducting(8,0)CNT-basedsystemswithonePd-cluster.\Pdonly",\Physi",and\Chemi"havethesamemeaningasinFig. 5-3 .EFisalignedat0.0eVinallgures. Tocomparewithexperimentalresults,wedenethesensitivityas: sensitivity=maxGf Gi;Gi Gf(5{1)whereGfisthepalladiumdopedsystemconductanceafterhydrogenadsorption,andGiisthepalladiumdopedsystemconductancebeforehydrogenadsorption.Fromtheconductancescalculatedabove,apure(8,0)semiconductingsensingsystemhasasensitivityof2.Foranmixedensemblesystemconsistingof1=3metallicCNTsand2=3 75

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semiconductingCNTs,ifweassumethatallCNTsareinaparallelcongurationandusetheconductancecalculatedforbothsystems,ourmodelsystemgivesasensitivityof1.35becausethemetallicsystemconductancevariesintheoppositedirectionfromthatofthesemiconductingsystems.IfaswearguedthattheconductanceofthemetallicsystemsvariesmuchlessthanthezerobiasconductancecalculatedatEF,thenthesensitivityofthemixedensembleisincreasedslightlyabove1.49butstillsignicantlybelow2.TheseresultsperfectlymatchtheexperimentsperformedbyKong,etal.[ 91 ],whichmeasuredsensitivitiesof2and1.36respectivelyforthesemiconductingbasedsystemsandthemixedensemblesystems.TheconductanceincreaseinthemetallicCNT-basedsystemsisalsoconsistentwithourpreviousbandstructurecalculationof(5,5)CNT-basedsystemsaspresentedinchapter4.Totestthestabilityoftheseresults,wealsorepeatedtheconductancecalculationsforone-andthree-cluster-dopedsystems.TheresultsareshowninFig. 5-4 .Intheone-cluster-dopedsystems,theconductancesforthemetallicandsemiconductingCNT-basedsystemswithouthydrogenadsorptionare0.57and0.076G0,respectively.Oncehydrogenisphysisorbed,theybecome0.64and0.028G0,respectively.Withhydrogenchemisorption,thesenumbersare0.60and0.034G0,respectively.Inthiscase,thesensitivityofthesemiconductingCNT-basedsystemisstillaround2;butthesensitivityofthemixedensemblesystemdroppedtoalmost1(nosensitivityatall)sincethemetallicsystemconductanceismuchlargerthanthesemiconductingones. 5.2.2.2DensityofstatesToanalyzethephysicsbehinddierentbehaviorsinducedbyPd4-clusterdoping,werstcalculatedtheelectronicdensityofstates(DOS)andtheDOSprojectedontotheCNTbackboneforbothmetallicandsemiconductingsystemswithouthydrogenadsorptionasshowninFigs. 5-5(a) and 5-5(b) .ForthemetallicCNTs(gure 5-5(a) ),theresultconrmsthatobtainedinthepreviouschapter.BondingwiththePdclustercauseslocalizedstatestoformattheCNT/Pd-clusterinterface.Theseshouldaddconsiderable 76

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(a)Metallic(5,5) (b)Semiconducting(8,0)Figure5-5. DOSaroundEFprojectedontotheCNTbackboneforthepureandPd-clusterdopedCNTsystemswithoutgoldleads.EFisalignedat0.0eV.Panel(a)isthemetallic(5,5)basedsystem,and(b)isthesemiconducting(8,0)basedsystem. (a)(5,5)Metallic (b)(8,0)SemiconductingFigure5-6. DOSofmodelsystemswithoutgoldleadsprojectedontotheCNTbackbone.(a)isthemetallic(5,5)CNT-basedsystem;(b)isthesemiconducting(8,0)CNT-basedsystem.EFisalignedat0.0eV. scatteringtotheconductionchannels,and,aswewillshowbelow,shouldsignicantlyreducetheconductance.ForthesemiconductingCNTs,theprojectedDOSofpureCNTandCNTwithPddoping(gure 5-5(b) )showsthatthepresenceofPdclustersgreatlyreducesthebandgap,from0.8eVto0.4eV.Thisisexpectedtosignicantlyincreasetheconductance.RealisticCNT-basedsystemsareusuallymuchlongerthanourmodelsystems,butsimulatingsystemswithsuchmuchlongerlengthisnotcomputationallyfeasible.Therefore,toseepossibleeectsofPddopingandhydrogenadsorptiononlongerCNTs, 77

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(5,5)(8,0) PdonlyPhysiChemi PdonlyPhysiChemi C 9:859:729:65 8:928:888:79Pd 5:655:785:78 4:634:764:74Au 4:204:666:21 4:294:926:37H N/A0:722:34 N/A0:802:32Table5-1. Excesschargeinthedeviceregionandoneachatomspecies(numberofelectrons).\Pdonly"denotesthePd-functionalizedCNTwithouthydrogenadsorption;\Physi"denoteshydrogenphysisorbedsystem;\Chemi"denoteshydrogenchemisorbedsystem. wepresenttheprojectedDOScalculationofmodelsystemswithoutthegoldleads(gure 5-6 ).Byeliminatingthegoldleads,wecanfocusonthechangewithinthescatteringcenterareathatmorecloselymimicsthepartoftheCNTsfarawayfromtheelectrodesinarealisticsystem.ThemetallicCNT-basedsystemshowsasmiliarreducedelectronlocalizationeectwithhydrogenchemisorption,sincethestatearoundEFinthetoppanelismuchmoreextendedthaninthebottompanel.ThesemiconductingCNT-basedsystemexhibitsanevenlargerlocalizationeectsuchthatthebandgapatEFisagainenlarged. 5.2.2.3ChargetransferandhartreepotentialTofurtherunderstandthesensingmechanism,weperformedaMullikenpopulationanalysis(table 5-1 )onourmodelsystems.ForpureCNTs,thePddopingcauses5.6eand4.6etobetransferredtothemetallicandthesemiconductingsystemsfromthethePdclusters,respectively.Thesechargetransferscausedierenteectsinthetwotypesofsystems.Forthesemiconductingsystem,thechargetransfercausesasignicantshiftintheconductancepeaksandbringsoneofthepeaksveryclosetoEF,thusleadingtoalargeincreaseintheconductance.Forthemetallicsystem,however,theshiftintheconductancepeaknearEFismuchsmaller,lessthan0.01eV.Thelargechangeintheconductanceismainlyduetothesharpnessofthepeak.Whenintegratedoveratypicalbiasvoltagewindow,theeectofthechargetransferontheconductanceinthemetallicsystemshouldbegreatlyreduced. 78

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Forhydrogenadsorbedsystems,thechargeanalysisshowsthatbothhydrogenphysisorptionandchemisorptionwillfurtherdepletechargeonPd-clustersby0.1e.Interestingly,nosignicantdierencebetweenphysi-andchemisorptionisobservedoneitherthePd-clusterortheCNTbackbone.However,sincethehydrogenatomhasmuchhigherelectronanitythanthePd-clusterafterhydrogendissociation,approximately1.6moreelectronswillbetransferedontohydrogentoformPd-Hchemicalbondinginchemisorptionthanphysisorption.AsimilaramountofelectronsthenwastransferredfromtheAuleadsviatheCNTbackboneontothePd-clustertoneutralizetheclusters. (a)Metallic(5,5) (b)Semiconducting(8,0)Figure5-7. Hartreepotentialaveragedoverthexycrosssectionofthedeviceregionalongthez-axisfor(a)(5,5)basedmodelsystems,and(b)(8,0)basedmodelsystems.Thetwopanelsshouldnotbecompareddirectlyagainsteachother.Thecurveswithineachpanel,however,aredirectlycomparable.Thelabels\Pdonly,"\Physi,"and\Chemi"havethesamemeaningasinFig. 5-3 ThelargeeectofhydrogenadsorptioninthesemiconductingsystemsisagainduetothesignicantshiftinthetransmissionpeakrelativetoEF,thistimeintheoppositedirectionthaninthecaseofPddoping.Forthemetallicsystems,thereisstillverylittleshiftinthetransmissionpeak.Thelargechangesintheconductanceareagainduetothetinyshiftofaverysharppeak,andareexpectedtodiminishonceintegratedoverasmallbiaswindow.OnewaytovisualizetheeectofchargetransfersisthroughtheplotoftheHartreepotentialwithinthedeviceregion,asshowningure 5-7 .Bothplotsindicatethateitherhydrogenphysisorptionorchemisorptionsubstantiallymodiestheelectrostaticpotential 79

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withinthedeviceregion.Yet,nobigdierencesareseenattheAu-lead/CNTinterfaces(around12A).Thisfurthersupportsourinitialassumptionthatthedetailsoftheinterfacearenotimportantfactorsinunderstandingtheeectofhydrogenadsorption. 5.2.2.4Localdensityofstates (a)Pdonly (b)ChemiFigure5-8. LDOSofmetallic(5,5)CNT-basedsystems.ThecontouristakenattheLDOSvalue1:0104.ThegureshowsonlyoneofthetwoCNT-clusterunitsforsimplicityandclarity.(a)isPd-clusterdopedCNTwithouthydrogenadsorption;(b)isPd-clusterdopedCNTwithhydrogenchemisorption. (a)Pdonly (b)ChemiFigure5-9. LDOSofsemiconducting(8,0)CNT-basedsystems.ThecontouristakenatLDOSof1:0104.ThegureshowsonlyoneofthetwoCNT-clusterunitsforsimplicityandclearnesspurposes.(a)isPd-clusterdopedCNTwithouthydrogenadsorption;(b)isPd-clusterdopedCNTwithhydrogenchemisorption. Wenallyanalyzehowthelocaldensityofstates(LDOS)(gure 5-8 and 5-9 )respondstohydrogenadsorption.TheLDOSisobtainedbyintegratingthereal-space 80

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Green'sfunctionforthedeviceregion: (r)=1 ZEF+EFIm[G(r;r;E)]dE;(5{2)wherewechoose=0:05eV.TheguresclearlyshowthatthehydrogenadsorptioninducedelectronlocalizationonCNTbackboneforsemiconductingCNTsystems,whichreducestheconductivityapproximatelybyhalf(gure 5-9 ).FormetallicCNT-basedsystems,littledierenceisobservedontheCNTbackbone.HoweverthestatesattheCNT/Pd-clusterinterfacearegreatlyreducedbyhydrogenchemisorption.ThesestatesareactuallystrongchemicalbondingbetweenCNT/Pd-cluster[ 84 ],whichservesasascatteringcenterindeviceregion.Thus,thehydrogenchemisorptionenhancstheelectrontransport. 5.3SummaryandConclusionInconclusion,wehaveperformedDFT-combinedNEGFcalculationsonPd4clusterfunctionalized(5,5)and(8,0)CNTmodelsystems.UponPddoping,theconductanceofmetallicCNTsdecreasesduetoelectronlocalizationeects,andtheconductanceofthesemiconductingCNTsincreasessinceitcreatesnewstatesthatreducesthebandgap.Formetallic(5,5)system,theconductanceincreasesby90%withhydrogenchemisorption;whileforsemiconducting(8,0)system,theadsorptionsupressestheconductanceby60%.Thesebehaviorisdominatedbyelectronlocalizationeectuponhydrogenadsorption,andisrelatedtochargetransfer.ThereforebothmetallicandsemiconductingCNTsaremuchbetterhydrogensensingmaterialsindividuallythanmixedensembleCNTs. 81

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CHAPTER6STRONGLYCORRELATEDELECTRONSINTHE[Ni(hmp)(ROH)Cl]4SINGLEMOLECULEMAGNET 1 Single-moleculemagnets(SMMs)havedrawnmuchattentionsincetheirdiscoveryin1991[ 118 { 120 ].SMMcrystalscontainorderedarraysofmolecularnanomagnets,eachpossessingalargespingroundstate(S=10forMn12-Ac)andasignicantuniaxialmagneto-anisotropy(DS2z,withD<0).Thesetwoingredientsgiverisetoamagneticspectrumforanisolatedmoleculeinwhichthelowestlyinglevelscorrespondtothe`spin-up'and`spin-down'states(ms=S),separatedbyanenergybarrieroforderDS2.Thisbarrierresultsinmagneticbistabilityandhysteresisatlowtemperatures(kBT<
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densityfunctionaltheory(DFT)calculations(includingon-siteCoulombenergies)whichprovidecrucialinsightsintotheoriginofthisferromagneticstate. (a) (b)Figure6-1. Optimizedstructureof[Ni(hmp)(MeOH)Cl]4.a)Thecompletestructurewithouthydrogenatoms;b)theexchange-couplingschemeofthe4Niatomsfromthesameangle.MagentaatomsareNi;redatomsO(1);yellowatomsCl;orangeatomsotheroxygens(i.e.O(2));blueatomsN;andgrayatomsC.TheNiandO(1)atomsformaslightlydistortedcube. WhileDFT[ 17 ]hassuccessfullyexplainedthepropertiesofavarietyofSMMs,includingMn12,Mn4,Co4,Fe4[ 127 { 131 ],andevensomeothernickelbasedSMMs[ 132 ],ithassofarfailedmiserablyforNi4.Notonlyweretheearlytheoreticalattemptsunabletoreproducethecorrectgroundstate,buttheresultingcouplingconstantswerealsofoundtobeantiferromagnetic,andordersofmagnitudehigherthantheexperimentalvalues[ 133 ].Itwasalsofoundthatthecalculatedspindensityisnotquitelocalizedaroundthenickelatoms,asexpected.Thus,ithasbeensuggestedthatthediscrepancybetweentheoryandexperimentsmightariseduetothesmall\spindensityleakage"inthissystem,resultinginspindelocalization.Thereisanotherpossibility.Duetothelocalizednatureof3delectrons,transitionmetaldioxides,includingnickeloxides,areknowntobestronglycorrelatedmaterials.Thefunctioningcoreof[Ni(hmp)(ROH)Cl]4,ontheotherhand,isacubictetra-nickeloxide(Ni4O4),whichisstructurallyveryclosetothenickeloxidecomplex.Therefore,itismoreplausiblethatthelackofstrongcorrelationinDFTwithapproximateexchange-correlation 83

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functionalsisresponsibleforthisfailure,andthe\spindensityleakage"isjustanartifact.Tojustifythisspeculation,wecalculatedtheelectronicstructureof[Ni(hmp)(MeOH)Cl]4usingboththeDFTandDFT+Umethods.ThelatterwasintroducedbyV.I.Anisimovetal.[ 19 ]andsimpliedbyM.Cococcionietal.[ 18 ]asdescribedin 2.3.2 6.1MethodandCalculationalDetailsAllthereportedcalculationsweredoneusingthePWSCFpackage[ 58 ],whichutilizesPBEexchange-correlationfunctionals[ 81 ],ultrasoftpseudopotentials[ 134 ],andaplane-wavebasis-set.Werespectivelychoseenergycut-osforthewavefunctionsandchargedensitiestobe40Ryand400Rytoensuretotalenergyconvergence.ThestructureofthemoleculewasoptimizedwithaxedtotalspinS=4untiltheforceoneachatomwassmallerthan0.01eV/A_Therelaxedstructureisingoodagreementwithexperimentalresults.ThesamestructurewasthenusedforsomeoftheAFMstates(S=0)andS=2,aswellasforalloftheDFT+Ucalculations.Duetosymmetryrestrictions,weonlysimulatedtheS=4,S=2andAFM(S=0)states.FortheDFT+Ucalculations,theself-consistentLDA+Umethoddescribedin 2.3.4 [ 18 ]wasincorporatedtodeterminetheUvalueforNi.StartingwithUin=4:0eV,bysettingdierentvalues(-0.05to0.05eV)intheinputle,weobtainasetofnKSsintheoutputleintherstKohn-Shamiterations:(exampleisfor=0:01eV) Parametersofthelda+Ucalculation:Numberofiterationwithfixedns=0StartingnsandHubbardU:enterwrite_nsU(1)=4.0000U(2)=4.0000U(3)=0.0000U(4)=0.0000U(5)=0.0000U(6)=0.0000U(7)=0.0000alpha(1)=0.0100alpha(2)=0.0000alpha(3)=0.0000alpha(4)=0.0000alpha(5)=0.0000alpha(6)=0.0000alpha(7)=0.0000atom1Tr[ns(na)]=8.4953856<---......atom2Tr[ns(na)]=8.4953889<---......atom3Tr[ns(na)]=8.4953856<---......atom4Tr[ns(na)]=8.4953889<---...... 84

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ThearrowsindicatesthenKSinformation.Similarly,nSCcanalsobefoundinthelastiterations(exampleisalso=0:01eV): Parametersofthelda+Ucalculation:Numberofiterationwithfixedns=0StartingnsandHubbardU:enterwrite_nsU(1)=4.0000U(2)=4.0000U(3)=0.0000U(4)=0.0000U(5)=0.0000U(6)=0.0000U(7)=0.0000alpha(1)=0.0100alpha(2)=0.0000alpha(3)=0.0000alpha(4)=0.0000alpha(5)=0.0000alpha(6)=0.0000alpha(7)=0.0000atom1Tr[ns(na)]=8.4944046<---......atom2Tr[ns(na)]=8.4953875<---......atom3Tr[ns(na)]=8.4953892<---......atom4Tr[ns(na)]=8.4953873<---...... (a)nKSvs. (b)nSCvs.Figure6-2. ThelinearresponseofnKSandnSCto.Accordingtolinearresponsetheory,theslopeoftheselinearrelationshipsaretheelementsoftheresponsematricesKSand,andcanbeusedtocalculatetheon-siteenergyU. TheresultingnKSsandnSCsarelinearwithrespecttos(Fig. 6-2 ).Byttingthemtolines,wehavetherstrowelementsoftheresponsematricesKSIJandIJ.Theoretically,weshouldrepeatthisprocedureforallfourNiatomstoobtaintheremainingthreerows,however,thankstothehighsymmetryoftheS=4groundstate,wedonotneedtodothatbecausealltheatomsareequivalent,andalltheremainingrows 85

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arejustpermutationsoftherstrows:KS=0BBBBBBB@0:146460:001490:001800:001490:001490:146460:001490:001800:001800:001490:146460:001490:001490:001800:001490:146461CCCCCCCA=0BBBBBBB@0:0995900:0004290:0004110:0004290:0004290:0995900:0004290:0004110:0004110:0004290:0995900:0004290:0004290:0004110:0004290:0995901CCCCCCCA Figure6-3. Calculationoftheself-consistentUparameterforNi.TheredlineisthecalculatedUout(from4.0eVto5.5eV),andthebluelineisttedtothecalculatedvalues. ThisyieldsUout=(KS)1II=3:21eVforUin=4:0eV.RepeatingthisprocedureforU=4.0,4.5,4.6,5.0,5.4,5.5,and6.0eV,weobtainalinearrelationshipbetweenUinandUoutasshowninFig. 6-3 .Accordingtotheself-consistentUproceduredescribedinchapter2,theself-consistentUparametercanbeobtainedbyextrapolatingthelinearregiontoUin=0:0eV,whichturnsouttogive6.20eVforthissystem.Foroxygen,wetookthewell-establishedUvalueof5.90eV[ 21 ].FortheDOSandprojectedDOS,weused0.1eVgaussiansmearingtosmooththeresults. 86

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6.2ResultsandDiscussionTheDFTcalculationsconrmedndingsofParketal.[ 133 ].TheoptimizedstructureisshowninFigure 6-1(a) .Inordertobetterdisplaythegeometry,wehideallhydrogenatoms.Thefournickelatomsandfouroxygenatomsonthehmpgroup(wecallthemO(1)fromnowon)deneaslightlydistortedcube.TheAFMstate(S=0)turnsouttobethegroundstate,whichis14.8meVlowerthantheS=2stateand35.1meVlowerthantheS=4state(table 6-1 ,columnDFT).UsingaLowdinchargeanalysis,onecanseethatabouta1.48{1.50Bmagneticmomentisfoundoneachnickelatom,andeachofthefourO(1)atomscontributesa0.1{0.26Bmagneticmoment(table 6-2 ,columnDFT).TheHeisenbergHamiltonianingeneralcanbewrittenas: Hex=Xi
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Figure6-4. TotalDOSandPDOSonthe2pand3dorbitalsof[Ni(hmp)(MeOH)Cl]4fortheS=4stateusingDFT.Theredlinesrepresent-spinandbluelinesrepresent-spin.ThegreenlineistheFermilevel,EF.AllDOSdrawntothesamescaleforcomparisonpurposes. DFTDFT+UdDFT+Up+d AFM(S=0) 0.00000.000000.000000S=2 0.00110.000120:000069S=4 0.00260.000190:000368 Table6-1. TotalenergiesinRydbergs.AllnumbersarerelativetotheAFMstate(S=0) weturnedontheUparameterforboththenickel3dandO(1)2porbitals.Infact,itisknownthatCoulombinteractionsbetweenoxygen2pelectronsarecomparabletothosebetweendelectrons[ 135 136 ],andshouldhencebetakenintoconsiderationaswell.However,sinceoxygenusuallybaresafullyoccupiedp-shell,thiscorrelationeectisoftenthoughttobeneglegible.Therefore,inmostcases,DFT+Udcanalreadyyieldasatisfactorydescriptionoftheground-statewithoutoxygen2p-electroncorrections.Nevertheless,DFT+Uhastobetakenintoconsiderationexplicitlyhereforboththe3dandoxygen2pelectronsinordertoobtainthecorrectgroundstateforthismolecule. 88

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DFTDFT+UdDFT+Up+d AFMS=2S=4 AFMS=2S=4 AFMS=2S=4 Ni 1.491.491.50 1.681.681.68 1.741.741.74O(1) 0.110.130.26 0.060.080.16 0.040.060.11Cl 0.090.090.09 0.060.060.06 0.060.060.06N 0.080.080.08 0.060.050.05 0.050.050.05O(2) 0.060.060.06 0.030.030.03 0.030.030.03 Table6-2. Magneticmoments(inB)capturedbyNi,O(1),Cl,NandO(2)atoms.AFMindicatestheantiferromagneticstate(S=0).Allnumbersareaveragedoverthesamespecies. Figure6-5. TotalDOSandPDOSonthe2pand3dorbitalsof[Ni(hmp)(MeOH)Cl]4fortheS=4stateusingDFT+Ud.Redlinesrepresent-spinandbluelinesrepresent-spin.ThegreenlineistheFermilevel,EF.AllDOSdrawntothesamescaleforcomparisonpurposes. TheDFT+UenergiesareshowninTable 6-1 .ByturningonDFT+Uforthenickelatomsonly,theenergydierencesbetweendierentspinstateswerereducedgreatly,hencegivingmuchsmallervaluesfortheexchangecouplingconstants.However,thegroundstatehereisstillAFM(S=0),andtheenergiesfortheS=4andS=2statesrelativetotheAFMstateare2.61meVand1.60meV,respectively.ButoncewetakeintoconsiderationthestrongCoulombinteractionsforboththeNiandO(1)atoms,theorderisreversed, 89

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Figure6-6. TotalDOSandPDOSonthe2pand3dorbitalsof[Ni(hmp)(MeOH)Cl]4fortheS=4ground-stateusingDFT+Up+d.Redlinesrepresent-spinandbluelinesrepresent-spin.ThegreenlineistheFermilevel,EF.AllDOSdrawntothesamescaleforcomparisonpurposes. yieldingcorrectlyaS=4groundstate.Aspin-unrestrictedcalculationalsoconrmedthisdiscovery.TheS=2stateisnow0.94meVlowerthantheAFMstate,andtheS=4groundstateis5.00meVlower.Usingthesevalues,weobtainedferromagneticexchange-couplingconstantsfortheDFT+Up+dcalculationfromattoequation2,i.e.J1=0:50meVandJ2=0:68meV 6{2 .Theseresultsmatchexperimentreasonablywell(0:68and2:28meV)[ 137 ].TounderstandthecontributionoftheHubbard-Uliketermbetter,werstperformedaLowdinchargeanalysistocalculatethemagneticmomentscapturedbytheNi,O(1),Cl,NandO(2)atoms(table 6-2 ).TheresultsshowthatthespindensityismorelocalizedintheDFT+Up+dcalculations(1.74B)thaninnormalDFT(around1.50B),andthatthemagneticmomentsfoundontheO(1)atomsaregreatlyreduced.Thisisbecausethestrongon-siteCoulombinteractionpreventshybridizationbetweenthenickel3dandoxygen2porbitals,thuspreventingtheunphysical\spin-leakage".TheunmodiedDFT 90

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Figure6-7. VariationofJ1andJ2withrespecttodierentUpforO(1). calculationsfavortheAFMgroundstatebecausethelackofanon-siteenergytendstocoupleelectronswithoppositespinprojections,andthusleadtotheincorrectgroundstate.ThelocalmagneticmomentonanindividualatomisameasurablequantityusingNMR,andthuscanbeusedtovalidatethesetheoreticalpredictions.ThetotalDOSandPDOSintheDFT+Udmethod(Figure 6-5 )andDFT+Up+dmethod(Figure 6-6 )werealsocalculated.IncontrasttotheDFTresults,thedominantcontributiontotheHOMOandLUMOstatesinbothDFT+Ucalculationsisnowfromthe2p(ClandO)and3d(Ni)orbitals,respectively(Figure 6-6 ).Wedonothavedirectexperimentalresultstocomparewiththisfeature,however,fornickeloxideitiswellknownthatDFTwithLDAorGGAgivesincorrectPDOScontributions.Earlyexperimentsandcalculations[ 138 { 142 ]showthatinsteadofad-dgapgivenbyDFT,nickeloxidesactuallyhaveap-dgap.Thelargestspindensitycontribution,ofcourse,isstillfromtheNi3delectrons.FortheS=4groundstate,theDFT+Up+dcalculationyieldsaLUMO-HOMOgapof2.95eV,whichisfrommajorityspintominorityspin.IntheDFT+UdandDFTcalculations,thesenumbersare2.56eVand1.09eV,respectively.SinceDFThasbeenknowntounderestimateenergygapsandexcitationstates,DFT+UcalculationshaveprovennecessaryinordertoobtainagreementwithexperimentssuchasresonantinelasticX-rayscattering(RIXS)andXPS[ 143 ].Thepresentstudyalsolikely 91

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callsintoquestionrecentreportsofHOMO-LUMOgapsoffractionsofaneVintheFe8SMM[ 144 ].Finally,wehavealsorepeatedDFT+UdandDFT+Up+dcalculationswithUdfornickelrangingfrom4.58to6.20eVandUpforoxygenrangingfrom1.0to8.0eV.Theenergeticorderofspinstatesremainsthesame,andthemagnitudesoftheenergydierencesbetweenspinstatesaremore-or-lessinsensitivetovariationsofUpfrom3.0to8.0eV,asseeninFigure 6-7 .Thisclearlydemonstratesthereliabilityandrobustnessofourresults.Inconclusion,wehaveperformedDFTandDFT+Ucalculationsfor[Ni(hmp)(MeOH)Cl]4.Becauseofthestrongcorrelationeectsinthissystem,theDFTcalculationfailsduetothefactthatthelackofon-siteenergyunphysicallyencouragesthehybridizationoforbitals,leadingtoAFMcoupling.TheinclusionofaHubbard-UliketermforboththeNi3dandO(1)2pelectronsgreatlyenhancesthelocalizationforbothstates,andisessentialinordertoobtainthecorrectferromagneticgroundstateandexchange-couplingconstants.Aftertakingbothcorrectionsintoconsideration,thesepropertiesweresuccessfullyreproducedbythecalculations.WethenanalyzedtheDOSandprojectedDOSofthesystem,andthecalculationpredictsthattheopticaltransitionfromHOMOtoLUMOisp-dlike,andthegapis2.95eV. 92

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CHAPTER7METHODDEVELOPMENT 7.1Multi-scaleSimulations 7.1.1DicultiesofCurrentMethodsClassicalMDhasachievedgreatsuccessinqualitativelyexplainingmanyinterestingphenomenainliquidsandamorphoussolids(uptomillionsofatoms),butduetotheuseofclassicalforceelds,itisdiculttodescribeachemicalprocessofasystemofinterestwithsatisfactoryquantitativeaccuracy.Itisalsopracticallyimpossibletoobtainasetofuniversallyapplicableforce-eldparametersforalltheelementsofinterest.Onthecontrary,quantumMDcaneasilyxtheseshortcomings,butusageofsuchmethodscurrentlyislimitedtosmallclustersorcrystallinesolids(1023atoms)becauseofthecomputationaleortitrequires.Therearemanysituationsthatneedcalculationsonalargesystemwithchemicalaccuracy.Biochemistryandmaterial-failurestudiesaresuchtypicalexamples.Amongtheseproblems,acertainsubsetdoesnotactuallyrequirechemicalaccuracythroughoutthesystem,becausethechemicalprocesstakesplaceonlyinaspecicregion.Therestofthesystemsimplyplaystheroleofasuitableenvironment,sothathighlyaccuratestructuredoesnotaectthechemicalprocessmuch.Therefore,thesystemnaturallycanbedividedintotworegions:thecoreareawherechemicalprocesstakesplace,andtheenvironment.Multiscalesimulationisacombinedclassical/quantumMDtechniquetotreatsuchasystem. Method ClassicalMD QuantumMD Algorithm O(N) O(N2lnN)SystemSize 109atoms 103electronsAccuracy Empirical,mechanicalaccuracy First-principle,chemicalaccuracyUniversality Parametersarespecicto Universal,noparameterisneeded materialsandchemicalenvironment Table7-1. ComparisonbetweenclassicalandquantumMDs. 93

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7.1.2BasicIdeaInmultiscalesimulations,thecoreregionistreatedwithquantumMD(andisthereforealsoreferredasquantumregion),andtheenvironmentistreatedwithclassicalMD(andisthereforealsoreferredasclassicalregion).Bothregionsareinterfacedproperlysuchthattheinuenceoftheclassicalregionisproperlyrelayedtothequantumregion.Inaotherword,thesystemHamiltoniancanbewrittenas: Htotal=K+VQQM+VCCM+VI(7{1)whereHtotalisthetotalHamiltonian;Kisthekineticenergyofthewholesystem;VQQMisthequantumregionpotentialenergycalculatedwithquantummechanics;VCCMistheclassicalregionpotentialenergycalculatedwithclassicalmechanics;VIisthepotentialenergyfortheinterface. Figure7-1. TheforcemismatchfortheSi-ObondinSi(OH)4.ThequantumcalculationisdonewithSIESTAusingdouble-orbitalsandthePBEexchange-correlationfunctional. Bycomparingequation 7{1 withthefullquantumHamiltonianHtotal=K+VQ,themulti-scalesimulationwillbeexactif VI=VtotalQMVQQMVCCM(7{2)Achievingequalityispracticallyimpossible,thusonealwaysmakescertainapproximationstoequation 7{2 .Themostcommontreatmentoftheinterfaceis: 94

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Terminateunsaturatedbondwithlinkatomsinthequantumregion. SetVI=VtotalCMVQCMVCCMsothatHtotal=K+VQQM+VtotalCMVQCM.Thisisactuallytosubtractthedoublecountingofpotentialenergywithinquantumregions.Therefore,inthecaseofmultiplequantumregions,theclassicalpotentialinvolvingatomsfromdierentquantumregionsshouldbecounted,andtheclassicalpotentialinvolvingtwodierentimagesinPBCshouldbecountedaswell.Whileseeminglyquitereasonable,thisprocedureisaverybigapproximation.Becauseofthesimplesubtraction,thetotalHamiltonianisnolongeraconservedquantity.AndthevariationofthetotalHamiltonianwithrespecttotimedependsmuchonthequalityoftheclassicalforceeld.Inthecasethatclassicalforceelddiersgreatlyfromthequantumforceeld,thetotalHamiltonianiscompletelyruined,leadingtounreliabledynamics.Thisistheforcemismatchproblem.WehereillustratethisproblemwithatypicalexampleofSi-ObondstrengthinSi(OH)4Fig. 7-1 .ItshowsthatwhiletheBKSforceeldtsthequantumforceeldfrom1.3to1.7A,itiscompletelyooutsidethisrange.Moreover,thetwoyielddierentequilibriumbondlengthsandtheBKSpotentialismuchsofterinthestretchedregion.Thusamulti-scalesimulationofsilicafractureusingBKSastheclassicalforceeldwillencounteraseriousprobleminthatthequantumregionwilltendtorepairitselfandtransferthestressintotheclassicalregion.Asamatteroffact,wehavetriedtoapplystressinthemodelSiO2chainsimulationdiscussedinchapter3,anditturnsoutthatwithaxedquantumregion,themodelchainalwaysbreaksoutsidethequantumregion.Toxthisproblem,eitherabetterapproximationfortheinterfaceorabetterclassicalforceeldisneeded.Itisstillanopenproblemtoimprovetheapproximation,anditseemsmorestraightforwardtoimprovetheclassicalforceeld.Previously,wehaveimplementedmulti-scalesimulationwiththePUPILcode([ 145 ]),andwehavedemonstrateditsusageinchapter3.However,thePUPILimplementation,becauseofitsarchitecturalrestriction,hasthefollowingdisadvantages: ItisnotMPI-capable,andthereforeallexecutionisserial. 95

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Itishardtoextendtheservercapabilitytosupportmultiplequantumregionsormultiplelevelsofsimulation(largerthan2). TheusageofJAVAandCORBAgreatlyreducedtheeciencyoftheserverexecution.Tosolvetheseproblems,wehavere-evaluatedthemodel,andhereweproposeanewimplementation.ItiscompletelybasedonMPI-2protocols,andthecodeiswritteninC(itcanbeeasilyportedtoC++orFortran).Therefore,boththeserverexecutionandthecommunicationeciencyismaximized. 7.1.3CodeDevelopement Figure7-2. LogicalstructureanddataowofOPALarchitecture. AschematicdemonstrationofthenewarchitectureisshowninFig. 7-2 .SinceallMD/DFT/DIcodesaretreatedatthesamelevelinthisarchitecture,adatastructuremss subjobthatrepresentsthesejobsisabstracted,whichcontainsthejobtype,theglobalMPIcommunicatorusedtocommunicatewiththeserver,thepathtotheexecutable,theargumenttostarttheprogram,itsworkingdirectory,etc.Itisdenedas: typedefstruct__mss_subjob{inttype;//JobType:0:MD,1:DFT,2:DIMPI_Commcomm;//Globalcommunicatorintsize;//Numberofprocessorsintionode;//IOnodeglobalindexcharexec[MAXLNLEN];//Pathtoexecutablechar*argv[MAXARGC];//Executionargumentcharworkdir[MAXLNLEN];//Workingdirectoryintnatms;//Numberofatoms}mss_subjob; 96

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Anotherimportantdatastructureisthemss infowhichcontainsalltheinformationrequiredbyOPALserver.Itisdenedas: typedefstruct__mss_info{MPI_Commworld;//Globalcommunicatorintsize;//Totalnumberofprocessorsintmaxqmregsize;//MaximumnumberofQMregionsintqmregsize;//CurrentQMregionsmss_subjob*md;//MDjobmss_subjob**dft;//DFTjobsmss_subjob*qmid;//DIjobintnatms;//totalnumberofatoms}mss_info; Theowchartoftheserverprocessisshowningure 7-3 .Atthebeginningoftheexecution,theserverwillinitializethesestructuresbyreadingtheserverinputle.ItisinprincipleevenpossibletoimplementdynamicallocationofprocessorsinthisarchitecturesothatalltheinformationneededisthetotalnumberofprocessorsandDIalgorithm.Aftertheserverprocessinitiatescorrectly,itwilllocatetheMD/DFT/DIexecutableles,andspawnthemusingMPIcallMPI Comm spawnaccordingly: ierr=MPI_Comm_spawn(job->exec,job->argv,job->size,MPI_INFO_NULL,0,MPI_COMM_SELF,&(job->comm),MPI_ERRCODES_IGNORE); Duringthesimulation,allpossiblecommunicationsaredoneviatheOPALserverusingMPIprotocols(MPI SendandMPI Recv),andtheserverprocesscontrolsthedata 97

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Figure7-3. FlowchartoftheserverprocessintheOPALarchitecture. ow(andthustheexecution).Whenthesimulationisnished,theMDcodewillsetitsvalueofnatmsto0,andsendthatbacktotheserver.Theserverwillthensendmessagetoallrunningsubjobs,makethemexit,cleanup,andexititself. 7.1.4TestCase:NaClDissociationinWaterTodemonstratethepowerofOPALarchitecture,wesimulatedasimpletestcaseofNaCldissociationinwater.ItisimportanttopointoutthatthesedemonstrationsareintendedtoshowtheideaofOPALarchitectureaswellasitsimplementation,insteadofthephysicalprocessitself. 98

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7.1.4.1StaticquantumregionidenticationWerstsimulatethetestcasewithastaticquantumregionidenticator.Bydenition,thereisnoparticalexchangebetweenthequantumregionandclassicalregioninatestrun.Thequantumatomsarespeciedbysettinguptheinputles. (a)InitialFigure7-4. Severalsnapshotsofthetestcase(staticquantumregion).TheredatomisNa+,theyellowatomisCl,thegreyatomsareOinclassicalregion,theblueatomsareOinquantumregion,andthewhiteatomsareH.TheinitialseparationbetweenNaandClis0.97A(panel"Initial").After0.06ps(panel"Critical"),theseparationbecomes9.73A,andtwoseparatequantumregionformsafterthat.Thenalseparationis10.3A(panel"Final"). WechoseNa+andClionsandtheH2Omoleculesaroundthemasthequantumregion.ThewatermoleculesaretreatedwithTIP4Pforceeldintheclassicalregion. 99

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(b)CriticalFigure7-4. Continued. Periodicboundaryconditionsareassumed.Becauseoftheshortsimulationtime(0:03ps),onecanassumethattheparticleswithinthequantumregiondoesnotaltermuch,andthusthestaticquantumregionapproximationcouldbetaken.TheH2Omoleculesinitiallywithin9.7Aawayfromeitherionswerechosentobethequantumregion.Thereareinall34H2Omoleculesplusthetwoionsincludedinthequantumregion.Initially,theNa+andClionswereputveryclosetoeachother(only0.97Aaway),sothattheywouldintensivelyrepeleachother.WeshowtheinitialconditionsinFig. 7-4(a) .After200steps(0.06ps),theybecomeveryfaraway(9.73A)(Fig. 7-4(b) ),andshouldbetreatedastwoseparatequantumregions.Thuswe 100

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(c)FinalFigure7-4. Continued. re-evaluatedthedistancebetweenwatermoleculesandeachion,andchosetheH2Omoleculeswithin4.9AawayfromNa+orCliontobetherstorsecondquantumregion,respectively.Thesimulationkeepsrunningforanother13timesteps,givingannalseparationof10.3A. 7.1.4.2DynamicquantumregionidenticationInthedynamicquantumregionidenticatortest,theidenticatorwillcollectinformationthatneededtoidentifythequantumatomsfromtheinputlepreparedtheuser.Forexample,inthistestcase,theinformationneededinclude: thecenter(s)ofthequantumregion(s)(Na+andClionsinthetestcase); 101

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thecut-oradiusrcutofthequantumregion(s)(5.0Ainthetestcase); theintendednumberofatomsNqminthequantumregion(s)(104atomsinthetestcase).Theidenticatorwillrsttestthevalidityoftheseinputparameters.Incasethattheyaretoofarapart(2rcut)butwithinthesamequantumregion,itwillprintoutawarningtonotifytheuserthatthesecentersshouldbeseparatedintodierentquantumregions.Onthecontrary,iftheyaretooclose(<2rcut)andwithindierentquantumregion,itwillalsoproduceawarningbecausethesetwoquantumregionsshouldbemergedintoalargerone.Iftheinputparametersarecorrect,theidenticatorwilltrytogroupalltheatomsintomoleculesaccordingtotheirinteratomicdistances.Inthesimpletestcase,itisactuallyachievedbyreadingtheinputlefromDLPOLY,whichalreadyspeciedtheinformation.Thecoordinateofeachmoleculeisthencalculated,whichisactuallythecoordinateoftheoxygeninH2Omoleculesinthetestcase.Theinformationisthenusedtocalculatethedistancebetweeneachmoleculeandthequantumcenters.Sincethequantumcentersmightcorrespondtodierentquantumregions,eachmoleculeisthenpre-assignedtoitsclosestquantumregion,andthisdistanceisrecorded(willbereferredasthecharacteristicdistance).Theidenticatorwillthensortthepre-assignedmoleculesineachquantumregionaccordingtotheircharacteristicdistance,andtaketherstNqmatomsfromthesortedlistasthequantumatoms.IncasethattheNqm+1thatomhasacharacteristicdistanceshorterthanrcut,thenanotherwarningwillbeprintedtoinformtheuserthatNqmissettooloworrcutissettoohigh.Thesimulationusesthesameinitialcongurationasintheprevioustest.However,thedynamicidenticationisobviouslymuchmorephysical,asallthesurroundingwatermoleculesaretreatedwithquantummechanics.Onecanactuallyobservetheevolutionofthequantumregionasthesimulationcontinues.After0.0645ps,whentheseparationbetweenNaandClis10.00A,thecodecorrectlyprintsoutawarning.Fromthesnapshot 102

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(a)CriticalFigure7-5. Snapshotsofthetestcase(dynamicquantumregion).(a)The"critical"timestep,whentheseparationbetweenNaandClis10.00A(timestep215,or0.0645ps);(b)the"nal"timestep.TheredatomisNa+,theyellowatomisCl,thewhiteatomsareH,thegrayatomsareOintheclassicalregion,andtheblue/greenatomsareOintherst/secondquantumregion,respectively.Theinitialcongurationisthesameasthestaticquantumregiontest(Fig. 7-4(a) ). atthattimestep(Fig. 7-5(a) ),onecanclearlyseetheseparationthequantumregion.Thesimulationcontinueswithtwoseparatequantumregions.After20moresteps(0.006ps),theseparationbecomes10.61A(Fig. 7-5(b) ). 103

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(b)FinalFigure7-5. Continued 7.2ConstructingEectiveModelHamiltoniansfromBandStructureCalculations:MaximallyLocalizedWannierFunction 7.2.1TheoryConsiderasetof(N)Blochfunctionsnk(r).Asetof(N)WannierorbitalswnR(r)=wn(rR)canbeconstructedthrough: jwnRi=V (2)3ZBZ"NXm=1Ukmnjmki#eikRdk(7{3)whereksamplestherstBrillouinzone(BZ),RrepresentstheBravaislatticevector,andUkareasetofunitarytransformationmatrices.FordierentselectionofUks,dierentwnRscanform.ForeachwnR,wecandeneitsspread: !n=hwn0jr2jwn0ihwn0jrjwn0i2(7{4) 104

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ThemaximallylocalizedWannierfunctions(MLWF)arewnRsobtainedfromasetofspecicUkssuchthatthesumoftheirspread=PNn=1!nisminimized.[ 146 ]Itcanbeproven[ 146 147 ]thatthespreadfunctionalcanbedecomposedintoagaugeinvariantpartIandavariantpart~.Thersttermcanbewrittenas: I=Xn"hwn0jr2jwn0iXm;RjhwmRjrjwn0ij2#(7{5)leavingthesecondtermtobe: ~=XnXmR6=n0jhwmRjrjwn0ij2(7{6)Theminimizationthereforeisdoneon~only.AftertheMLWFsareobtained,theunitarytransformationmatricesUksarealsodened.ApplyingtheunitarytransformationtotheHamiltonianmatrix,wethenhave: Hw(k)=(Uk)yH(k)Uk(7{7)whereHnm(k)=nknm.Thereal-spaceHamiltonianHwnm(R)canthenbecalculatedbyFouriertransformingHw(k): Hwnm(R)=1 NkXkeikRHwnm(k)(7{8)whereNkisthenumberofunitcellsrepresentedbythek-mesh. 7.2.2Application:Iron-BasedSuperconductorLaO1xFxFeAs 1 Therecentdiscoveryofsuperconductivitywithonsettemperatureof26KinLaO1xFxFeAs[ 149 ]hasgeneratedconsiderableinterestbecauseofanumberofunusualaspectsofthismaterial.First,withtheexceptionofsomeoftheA-15materials,Feisneverfoundinsuperconductorsatzeropressure(althoughFeitselfsuperconductsat 1 ThisworkhasbeenpublishedasRef.[ 148 ]. 105

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10GPa).Second,bothferromagneticandantiferromagneticuctuationsareapparentlypresentinthematerial,suggestingpossibleanalogiestoternaryrareearth,heavyfermion,borocarbide,ruthenateandcupratesuperconductors.Finally,thediscoveryofsuperconductivityattherelativelyhighcriticaltemperatureof26Kimpliesthatanewpairingmechanismmaybeinplay.TheanalogsystemLaO1xFxFePhasacriticaltemperatureof7K,sothereappearstobeanewclassofsuperconductingmaterialswithnoobviouslimitonTc.Naively,ferromagneticorderisinimicaltosuperconductivitysincetheexchangeeldoftheferromagneticionbreakssingletpairs.FerromagneticorderwasthereforefoundonlyveryrecentlytocoexistwithsuperconductivityinUGe2[ 150 ]andURhGe[ 151 ],insituationswheretheorderisquiteweak.Coexistenceofantiferromagneticorder,ontheotherhand,islesspairbreakingifthecoherencelengthismuchlargerthanthewavelengthofthemagneticmodulation,whichisgenerallythecase.Thusmanyexamplesofsuperconductorscoexistingwithantiferromagneticorderareknown,andhavebeenrecentlyreviewed[ 152 ].ThestructureofthenewmaterialconsistsofLaOlayerssandwichingalayerofFeAs,anddopingwithFappearstooccurontheOsites.SincetheFeisarrangedinasimplesquarelattice,theanalogywiththecuprates,whereelectronshoponasquarelatticeanddopingoccursviaanearbyoxidechargereservoirlayer,istemptingtodraw.EarlyelectronicstructurecalculationsforboththeP[ 153 ]andthenewAsmaterials[ 154 { 156 ]havepresentedasomewhatdierentpicture,however.Whileexperimentallyalowchargedensitywasmeasuredforthismaterial[ 157 ],thesecalculationssuggestthatvebands,ofprimarilyFe-Ascharacter,crosstheFermilevelandgiverisetoamulti-sheeted,quasi-2Dmaterial.Noevidenceforlong-rangeorderwasfoundinthesestudies,althoughproximitytobothantiferromagneticandferromagneticorderedstateswasnoted.TheweakcouplingoftheLaOlayerstotheFeAslayersfoundhereandinpreviousworksalsosuggeststhatinsightmaybegainedbyexaminingironmonoarsenideFeAs,a 106

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layeredmetallichelimagnet[ 158 ]withlargepitchangleinthezincblendestructurewithlow-temperatureeectivemoment0:5B.Theelectronicstructureandspinmomentforthiscompoundhavebeencalculatedbydensityfunctionaltheory[ 159 ].Whilethetheoryissuccessfulinthesensethatanantiferromagneticgroundstateisfound,itdoesnotdistinguishthecomplicatedmagneticstructureandndsamomentoforder2B.TounderstandtheelectronicstructurepropertiesofthenewFe-basedsuperconductorsandtheinterplaywithstructureandmagneticstates,wehaveperformedrst-principlesdensityfunctionaltheorysimulationsontheundopedandx=0.0625aswellasx=0.125dopedLaO1xFxFeAs.MostofthereportedresultsonelectronicstructureandmodelHamiltonianhavebeenobtainedusingthePWSCFpackage[ 58 ],whichemploysaplane-wavebasissetandultrasoftpseudopotentials[ 134 ].WehavealsousedVASP[ 160 161 ]toconrmourcalculationswhenthesamecalculationscanbeperformed,asdescribedinthefollowingsectionsindetail.Thelocalspindensityapproximation(LSDA)andgeneralizedgradientapproximation(GGA)ofPerdew,BurkeandErnzerhof(PBE)[ 81 ]potentialshavebeenincorporatedforthesimulation.Fordensityofstates(DOS)calculations,wehaveuseda16168Monkhorst-Packdensegrid[ 114 ]tosampletheBrillouinzone;whileforstructuralrelaxationandself-consistentcalculations,a884Monkhorst-Packgridhasbeenused.Allstructureshavebeenfullyoptimizeduntilinternalstressandforcesoneachatomarenegligible.OurGGA+UcalculationshavebeenperformedviatheVASPcode.Theexistingliteratureusesanon-siteColumbenergythatvariesfrom4.0eVto6.9eV[ 19 139 140 ],andwehaveexploredtheparameterspacewithintherangeU=2.0-5.0eVandJ=0.89eVonFeinourcalculations.Ourcalculationsshowanunambiguousantiferromagnetic(AFM)groundstatewithstaggeredmoment2.3BforundopedLaOFeAs,whichis84meVperFelowerthanparamagnetic(PM)andferromagnetic(FM)states.Theenergydierencebetweenthelattertwoisfoundtobenegligible.Infact,theFMstatehasaverysmallmagneticmoment(0.05perFe);thereforeitcanberegardedasaPMstate.TheAFMground 107

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(a) (b)Figure7-6. UndopedLaOFeAsbandstructureof(a)theAFMstateand(b)thePMstate.RedlinesrepresentDFTcalculationresults;bluelinesarebandstructuresreconstructedfromthetight-bindingmodelusingmaximallylocalizedWannierfunctions(MLWF).Sincethespin-upandspin-downbandsaredegeneratefortheAFMstate,weplotonlyspin-upbandshere.Forbothgures,Fermienergiesareindicatedbythegreenlineat0.0eV. statehasbeenconrmedbyindependentVASPcalculationsusingtheprojectoraugmentedwave(PAW)method[ 162 ].Theoptimizedstructurehasalatticeconstantofa=4.0200Aandc=8.7394A;andthebondlengthsforFe-AsandLa-Oare2.35Aand2.40Arespectively.Forreference,theparamagneticstatehasanoptimizedlatticeconstantofa=3.9899Aandc=8.6119A,whilethebondlengthsforFe-AsandLa-Oare2.34Aand2.33A,respectively.BoththeAFMandPMbandstructuresareshowninFig. 7-6 withredcurves.Inbothstates,asmalldispersionalongthec-axis(fromtoZandfromAtoM)indicatesinteractionsbetweenlayersareweak,andthustheseparationofthestructureintoLaOandFeAslayersispossible.ThePMstatebandstructurereproducespreviousDFTcalculationresults[ 153 154 ],exhibiting5bandsacrosstheFermilevel.TheAFMstatebandstructureisqualitativelydierent,exhibitingonly3bandsacrossEF.InVASPcalculations,AFMstatesare14meVperFelowerthanPMandFMstates.SimilartoPWSCFcalculations,thebandstructuresoftheAFMstateareverydierentfromthePMstate.Theseresultsindicatethedelicacyofthemagneticstatesinthissystem,andthatthemagnetismstronglyaectstheelectronicstructure. 108

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Figure7-7. DOS(toppanel)andPDOS(FeAsandLaOplanes:middlepanelandbottompanel,respectively)ofundopedLaOFeAs.Sincethespin-upandspin-downstatesaredegenerateforAFMstate,weplotspin-upstatesonly.TheFermilevelisalignedto0.0eV. Tofurtherexamineandconrmthesendings,wehaveperformedtwoseriesofadditionalcalculations.First,wehaveperformedGGA+UcalculationsusingVASP.Hauleet.alusedadynamicalmeaneldtheory(DMFT)-LDAapproach,andfoundthatacriticalvalueofU=4.5eVledtoaMotttransitionwithagapattheFermisurface.WehavecalculatedtheelectronicstructureusingVASP'simplementationofGGA+U,andalsondaMotttransitionfortheLaOFeAssystematacriticalU3eVforFe.NotethatlowerboundoftheempiricalvalueofUchosenincalculationis3.5-4.0eVforFedorbitals([ 19 ]).ThegroundstateisfoundtobealwaysAFMforalltestedUwithin0.0-5.0eVinourcalculations,buttheDOSchangesdramatically(Fig. 7-9 ).AMottgapofabout1.0eVisobservedintheGGA+UcalculationatU=4:5eV.Experimentally,itisobservedthatbelow100K,theresistivityofundopedLaOFeAsincreaseswhentemperaturedecreases,butappearstoremainmetallic[ 149 ],suggestingthatthesystemisinfactontheedgeofaMotttransition.[ 156 ].Second,wehaveinvestigatedbulkFeAs.TheAFMstateisagainfoundtobethegroundstate,inagreementwithexperiment[ 158 ],aswellaswithpreviouscalculationsbasedonfull-potentiallinearizedaugmentedplanewave(FLAPW)method[ 159 ].Inaddition,anisolatedlayerofFeAshasalsobeensimulated,andthesystemagainhasan 109

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AFMgroundstate.ThecalculatedAFMstatesofbulkandtheisolatedlayerare53meVand408meVperFeatomlowerthantheirPMstates,respectively.ComparedtoFLAPWcalculations,ourresultshaveshownasmallerenergydierencebetweenAFMandPMstate,indicatingthatourcalculationsdonothaveanarticialbiasfortheAFMstate.TheelectronDOSderivedfromtheAFMbandstructureandthecorrespondingprojectedDOS(PDOS)ontoLaOandFeAsplanesarepresentedinFig. 7-7 .ItisclearfromthePDOSanalysisthatFe3dorbitalsdominatetheDOSaroundEF(-2to2eVrelativetoEF)andfrom-12to-10eV;whereasDOSbelow-13eVisalmostcompletelyderivedfromLaOlayers.ThisclearseparationintheDOSconrmsthatthismaterialcanbeviewedasalayeredstructure.However,bothLaOandFeAslayerscontributeapproximatelythesamefrom-6to-2eV,suggestingahybridizationbetweenlayerswithinthisenergywindow.ForAFMstate,thecalculationsgivea2.30BlocalmagneticmomentonFebyintegratingthePDOStoEF,whichissimilartothevalueobtainedfromcalculationsforthebulkFeAscrystal. (a) (b)Figure7-8. Diagramsfor(a)atomic-typeMLWFsand(b)strongesthoppinginthesystem.Thediagramispresentedonthex-yplanewiththemostimportanthoppingslabeled.Ironatomsandarsenicatomsarelocatedattheverticesandcentersofthesquarelattice,respectively.Notethat(b)depictsanirreduciblesubsetofhoppings,andthez-displacementofAsatomsisnotshown. TofurtherunderstandthephysicswithintheFeAslayersandconnectourcalculationstomodelcalculations,wehaveusedthemaximallylocalizedWannierfunctions(MLWF)method[ 146 147 ]toanalyzetheatomicorbitalswhichdominatetheelectronicstructureneartheFermisurface.SixteenMLWFs,including10d-typeMLWFsonFeand6 110

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type PM AFM on-site 3dx2y2 11.35 11.679.85 3dx(y)z 11.26 11.569.76 3dxy 11.18 11.499.60 3dz2 11.14 11.9610.06 4px(y) 9.97 9.339.33 4pz 6.35 3.173.17 hoppings 1 3dxy-4py 0.79 0.800.622 3dxz-4px0 0.60 0.670.483 3dxz-4px 0.81 0.790.744 3dz21-4pz 1.02 0.830.465 3dx2y2-4pz 1.26 1.211.006 4px-4px 0.68 0.680.687 3dxz-4pz 0.49 <0:10.688 4pz-4pz 0.17 0.550.55Table7-2. Electronhoppingtijandon-siteenergiesi(ineV)matrixelementscalculatedfromMLWFs.IntheAFMstate,hoppingwillbedierentforironatomsondierentsub-lattices. p-typeMLWFsonAs,havebeenusedtoobtainatight-bindingeectiveHamiltonianHeff=Piicyicyi+Pi;jtijcyicjbyttingthebandstructurearoundEF.TheseMLWFsarethenusedtoconstructamodelHamiltonianmatrix,fromwhichwecanregeneratethebandstructureusingatight-bindingframework(bluecurvesinFig. 7-6 ).InbothAFMandPMcases,thetight-bindingbandstructuretstheDFTbandstructurewell,showingthevalidityofourmodelHamiltonian.WeshowthemostimportanthoppingtermsinFig. 7-8 ,andthecorrespondingvaluesarelistedintable 7-2 togetherwiththeon-siteenergies.DuetotheS4symmetryoftheFeAstetrahedra,theFe3dorbitalssplitinto3non-degenerate(3dx2y2,3dxy,3dz2)and1doublydegenerateenergystate(3dx(y)z)inbothPMandAFMstates.Interestingly,thelowestlying3dz2stateinPMisthehighestinAFMstate,leavingtheorderofother3statesunchanged.Theenergydierencebetweenlowest3dstatesandhighest4porbitals(4px(y))are1.2eVand1.6eVinPMandAFMstate,respectively,whichisaboutthesamemagnitudeastheCu-Osplittingincuprates.InthePMstate,thestrongesthoppingscomefromFe3dx2y2,3dz2andAs 111

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4pzorbitals,followedbythecouplingfromFe3dxz,3dxyandAs4pxorbitals.Remarkably,thedirecthoppingbetweenAs4px(y)inthesamex-yplaneareaslargeastheFe-Ashopping.IntheAFMstate,sincethemirrorsymmetrywithintheunitcellisremoved,thesehoppingmatrixelementssplitintotwogroupsforspin-upelectronsandspin-downelectronsonFesites,respectively.Furthermore,thehoppingbetweentwoneighboringAs4pzand4pzorbitalsaregreatlyenhancedinAFMstates.ThedirecthoppingsbetweenFe3dorbitalsarenite,butmuchsmallercomparedtoFe-AsandAs-Ashoppinginbothcases. Figure7-9. DOScalculatedfromGGA+UwithdierentUvalues,2.0,3.0,and4.5eV.EFisalignedto0.0eVinallcases.Becauseofdegeneracy,onlythe-spinDOSisplotted. Figure7-10. DOSofLaO1xFxFeAswithx=0.125projectedontoFeAslayers.EFisalignedto0.0eV. WenallypresentourcalculationsondopedLaOFeAs.Thedopingissimulatedbysubstitutinganoxygenatomin8primitivecellswithauorineatom,andthenfullyrelaxingthestructurewiththeoptimizedlatticeconstantsoftheundopedsystem.Since 112

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eachprimitivecellcontains2oxygenatoms,thedopingcorrespondstoLaO1xFxFeAswithx=0:0625.TheresultingmaterialturnsouttobestillAFM,butwithanextratotalspin.Withx=0.125doping,wereducedthenumberofprimitivecellsinvolvedbyhalf,sothatoneoxygenatominevery4primitivecellswasreplacedwithauorineatom.TheuorinesubstitutionhasaneectonelectronicstructurewhichisprimarilyconcentratedintheFeAslayersatenergiesnearEF,plusanimpuritystateat7.5eVbelowEF.Weshowthecomparisonofundopedandx=0.125dopedsysteminFig. 7-10 .Thex=0.0625dopedsystemhasasimilardopingeect,butsmallerinmagnitude.Theoverallmagneticoderingremainsthesame,butthemagneticmomentisalteredby6%.Inconclusion,wehaveperformedrst-principlescalculationsforLaOFeAsandLaO1xFxFeAssystems.AnAFMgroundstatehasbeenfoundforundopedLaOFeAsviaDFTcalculations.Wendthatthegeometry,electronicstructureandthemagneticstateofthissystemarestronglyrelated.InbothAFMandPMstates,thebandstructuresaroundFermilevelarederivedfromFe3dandAs4porbitals,andwehavettedbandscrossingtheFermisurfacetotight-bindingHamiltoniansusingMLWFs.TheparametersforthemodelHamiltoniansfromtherst-principlescalculationscanbeusedformodellingtransport,magneticandsuperconductingphenomenaassociatedwithstronglycorrelatedelectronsinthesystemunderinvestigation.WhilethesystemexhibitsmetallicbehaviorinDFTcalculations,aninclusionofanon-siteenergyof4.5eVonFeturnsitintoasemiconductorwithagapof1.0eV,whichimpliesthatthesystemisclosetoaMott-typeinsulator.DuetotheevidentproximityofsuperconductivitytoantiferromagnetismandtheMotttransition,wesuggestthatthesystemmaybealarge-spinanalogoftheelectron-dopedcuprates,whereAFMandsuperconductivitycoexist.Asamatteroffact,shortlyafterthisworkwasdone,alinearspindensitywavestatewaspredictedinanelectronicstructurecalculation[ 163 ],anddiscoveredinneutronscatteringexperiments[ 164 ],thenextensivelystudiedin[ 165 ].WehavecomparedtheenergyofsuchamagneticstatewiththesublatticetypeAFMstatediscussedinthis 113

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work,andcanconrmthataccordingtoPWSCFcalculationsitis109meVperFelowerinenergy.WehavealsorepeatedthesecalculationswithPAWmethodimplementedinthenewestPWSCFversion.Contrarytothepseudopotentialresults,thesublatticetypeAFMstructureisnotstableinthePAWcalculations,butthestripe-likeAFM(linearspindensitywavestate)remainsthegroundstate.However,themagneticmomentperFeisreducedto1:4B,muchlowerthantheoneobtainedwithpseudopotentialmethod.Theenergydierencebetweenthestripe-likeAFMstateandPMstateisonly0:01eV,whichisalsomuchlowerthanthepseudopotentialcalculations.Aftertheourinedoping,themagneticmomentperFeisfurtherreducedto0:7B,showingthereductionofthemagneticordering.ThePAWcalculationsttoexperimentalresultsmuchbetter,meaningthatthemagneticstructureofthissystemstronglydependsonthequalityofthepseudopotential/PAWdataset.Duringoursimulation,wenoticedthattheultrasoftpseudopotentialsgeneratedusingVanderbiltschemewillyieldincorrectchargestates(Featomsgainedelectrons).ThisartifactisduetothelargercutchosenbytheVanderbiltpseudopotentials(inordertolowertheenergycut-osrequiredbythecalculations).TheRRKJpseudopotentialsuserelativelysmallerrcut,thustheyarecapableofcorrectlydescribethechargestates. 7.3ElectronCorrelationsinTransportProblems 7.3.1BackgroundWeintroducedthestate-of-the-artDFTcombinedwithNEGFtransportcalculationmethodinchapter2,anddemonstrateditsuseinchapter5.Despitetheapparentsuccessofthismethod,largediscrepanciesbetweenthesecalculationsandexperimentalresultscanbeseen.Variouseortstoimprovethequalityofthesecalculationshavebeendonebyintroducingself-interactioncorrections(SIC)[ 166 { 168 ]orHubbard-Uliketerms(NEGF+LDA+U)[ 169 ].However,theproblemhasmuchdeeperrootsthansimplyabetterexchange-correlationfunctional. 114

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LetusrstbrieyreviewthecurrentDFT+NEGFmethod.Severalimportantapproximationsareinvolved: 1. Atthelevelweareinterested(DFT),bothleadsandthedeviceregionarenon-interacting. 2. Theelectrondensitiesandpotentialsretaintheirgroundstatebulkvaluesforbothleftandrightleads. 3. ThesystemHamiltonianisstillafunctionalofonlytheelectrondensity.Moreover,thecalculationisdoneinthreemajorsteps: A ConstructaHamiltonianHfromtheelectrondensitynusingDFT. B ConstructtheGreen'sfunctionsGR,GA,andG
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theelectronscanberegardedasnon-interactinginsideleads,onecannotassumethesamethingforthedeviceregion. 7.3.2IsaSingleElectronDensitySucient?Inapreviouswork[ 170 ],wehadbelievedthatasingleelectrondensityisnotsucienttodeterminethestateofanon-equilibriumopensystem.The(incorrect)argumentisasfollows:Onaccountofapproximation2in 7.3.1 ,bothleadshavethebulkgroundstateelectrondensity.BecauseoftherstHohenberg-Kohntheorem,theexternalpotentialoftheleftleadisdetermineduptoanarbitraryconstant,thereforetheFermilevelLoftheleftleadisalsodetermineduptoanarbitraryconstantVL.SoistheFermilevelRoftherightlead.LetthisconstantbeVR.Sincebothleadsareconnectedtoadrain/source,thetwoconstantsVLandVRarenotindependent.However,accordingtothegroundstateHohenberg-Kohntheorem,ifweshifttheFermienergiesoftheleftorrightleads,theelectrondensitiesremainunchanged.Thisleadstoacontradiction.Themajorawinthisargumentliesintheapproximationittakes.Inordertoseehowbigtheapproximationis,letusexamineasimplemodelsystem:freeelectronowscatteredoastep-function-likepotential(in1-Dforsimplicity): U(x)=8>><>>:0;(x<0)U0;(x>0)(7{9)Itisobviousthatitssolutionhasformof: (x)=8>><>>:Aleiklx+Bleiklx;(x<0)Areikrx;(x>0)(7{10)wherekl=p 2E,kr=p 2(EU0).(Here,atomicunitsareassumed,i.e.me=1,~=1) 116

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Therefore,wehaveAl=kl+kr 2klAr (7{11)Bl=klkr 2klAr (7{12)thustheelectrondensityn(x)is: n(x)=j(x)j2=8>><>>:(1U0 Esin2klx)n0;(x<0)n0;(x>0)(7{13)Theleftandrightleadsinthiscasearesimplyemptyspace.Accordingtoapproximation2in 7.3.1 ,theyshouldhavethesameelectrondensity.ThisapproximationisgoodwhenU0Ebecauseofequation 7{13 .Inthismodelsystem,EFEandVbU0,thusapproximation1isvalidwhenVbEF.Ifwelookatthismodelsystemfromanotherangle,bydeningx00)tobethedeviceregion,wecanseethattheelectrondensityn(x)somehowcontainsthebiasinformationU0.Thusaone-to-onemappingbetweentheelectrondensitynandthebiasinformationU0exists.Itisquiteobviousinthissimplecase,becauseiftwoelectrondensitiesn1andn2areexactlythesame,k1lhastobeequaltok2l,whichmeansE1=E2.ThereforeU10hastobeequaltoU20.Thisprovideuswithahintthatasingleelectrondensitymightstilldeterminethesystemstateinthecaseofsteadystatecurrent.Nowwetrytoprovethis.Westartwiththestandardthree-partsystem,i.e.theleftlead,thedeviceregion,andtherightlead.However,wewillintroduceacomplexpotentialVsthatrepresentsthesourceanddrainsothatthesystemisnowclosed.Thecomplexpotentialhastobecarefullychosensothatitwillnotintroduceextrascatteringwithinthesystem.Thatistosay,attheboundaryofthesystem,thepropagatingelectronstatesareeigenstatesofthebulkleads.Suchchoiceispossible,accordingto[ 171 172 ].So,wehaveamappingfromanon-equilibriumopensystemtoanon-equilibriumclosedsystembytheintroductionofcomplexpotentials.Weconsiderthetime-dependent 117

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processthatturnsanequilibriumclosedsystemtoanon-equilibriumclosedsystem,orintermsofcomplexpotentials,weconsideratimedependentcomplexpotential: Vs(t)=V0s 1+et=(7{14)Sothatatthelimitt!Vs()=0,andt!1Vs(1)=V0s.Thereforethesystemisinitiallyatitsgroundstateatt=.Theparameterhastobesucientlylargesuchthatatanyspecictimet,thereisnoelectronaccumulationwithinthesystem.Thatcorrespondstoagradualincreaseofthebiasvoltageofthesystemsothatthesystemcanberegardedasinaquasi-steadystateatanytimet,andtheelectronnetowiszero.AccordingtoRungeandGross[ 173 ],theinstantaneousstateofthesystemiscompletelydeterminedbyitscurrentelectrondensityn(t).Takingthelimitthatt!1,thesystem'seectiveHamiltonianH(1)hasaone-to-onemappingwithitselectrondensityn(1).Thereforeinsteadystate,theeectiveHamiltonianHisafunctionaloftheelectrondensityn(r)only. 7.3.3DirectiontoGo:GWCorrectedTransportAlthoughwehaveproventheone-to-onemappingbetweentheeectiveHamiltonianHandtheelectrondensityn(r)understeadystatecondition,itdoesnotimplythatthefunctionalwilltakeexactlythesameformastheoneinthegroundstateDFT.Infact,oneshouldexpectthatthegroundstateDFTbecomesaspecialcaseofthismoregeneralsteadystateDFT.Consideringthedicultiesonehastogetanaccurateexchange-correlationfunctionalforevengroundstateDFT,theexactfunctionalforthegeneralsteady-stateDFTmightremaininthedarkforcenturies.TheapproximationhencehastobemadethatthegroundstateDFTfunctionalisareasonablezerothorderapproximationtothesteady-stateDFTfunctional.Secondly,evenifoneweretoobtaintheexactsteady-stateDFTfunctional,theHamiltonianobtainedusingthatfunctionalstillwouldnotbesuitableforuseintransportcalculation.Aswestatedbefore,theelectrontransportprobleminvolves 118

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electronexcitations,andtheexcitationsarerelatednottothebareelectrons,buttothequasi-particles.HencetheDFTfunctionalswillnotyieldtheimportantexcitationenergiesrequiredbytransportcalculations.Variouseortstopolishthisfunctionalhavebeendoneundercertainobviouscircumstances.Forexample,C.ToherandK.Burkeetal.havepointedouttheimportanceofself-interactioncorrection(SIC)intransportcalculation[ 166 { 168 ];A.R.Rochaetal.haveincorporatedtheLDA+Uschemeintotransportcalculationstoaccountfortheincompleteon-sitecoulombinteractions[ 169 ].BothmethodsimprovetheLDAorGGAresultsincertaincases,buttheimprovementisnotsystematicjustasexpected.BothmethodsareinfactcalculatinganinteractingdeviceregionbeyondLDAlevel.Thesuccessofthesemethodsdemonstratestheimportanceofincorporatinge-einteractionswithinthedeviceregion.Wehereproposetoincorporatethee-einteractionsintotheNEGFmethodbyintroducingaGW-likecorrection.WerstwritedownthetotalHamiltonianofaninteractingdeviceregioncoupledwithnon-interactingleads: H=Xk;kcykck+Xiidyidi+XU(dy;d)+Xk;iTk;icyk;idi+h:c:(7{15)wherecandcyareannilationandcreationoperatorsfortheleads,danddythecorrespondingoperatorsforthedeviceregion.Tk;irepresentsthecouplingbetweendeviceandleads,andUrepresentstheelectron-electroninteractionswithindeviceregion.Sincetheleadsarestillnon-interacting,thecurrentformuladerivedbyMeirandWingreen[ 63 ]stillapplies,i.e.: J=ie 2~Zd 2TrLRG<+fLLfRRGRGA(7{16)wherewehaveL;R()=2TyL;R()fL;R()TL;R(). 119

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TheHamiltonian 7{15 canbewrittenintoanequivalentform: H=h+int+tr(7{17)wherehistheisolatednon-interactingdeviceregionHamiltonian,intrepresentstheelectron-electroninteractionswithinthedeviceregion,andtrrepresentsthecouplingwithleads.WeadopttheassumptionthatthedierencebetweenthegroundstateDFTfunctionalandthesteady-stateDFTfunctionalissmall,andthusonedonothavetoconsiderextraself-energytermsduetothenon-equilibriumcondition.Thatistosay,weassumethatee=0.Basedontheseapproximations,wecanintroduceg(!)=(!h)1(theGreen'sfunctionforthenon-interactinguncoupleddeviceregion),G0(!)=(!hint)1(theGreen'sfunctionoftheinteractinguncoupleddeviceregion),andG(!)=(!hinttr)1(thecompleteGreen'sfunction).Theseparationallowsustocalculatetheelectron-electroninteractionself-energyintfromtheGWapproximation,andobtainabetteruncoupledGreen'sfunctionG0thantheonedirectlyfromDFTcalculationfortransportcalculations.WethenhavethefollowingDysonequations:G<=G<0+G<0AtrGA+GR0
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Wealsohave GR;A=gR;A+gR;A(R;Atr+R;Aint)GR;A(7{21)Therefore,wecandesignanewself-consistentprocedureasfollowing: 1. Thedeviceregionelectrondensitynisusedtoconstructanewnon-interactingHamiltonianh,whichisusedtocalculategR;A. 2. Thee-einteractionself-energycanthenbeevaluatedusingGWapproximation,i.e.Rint(!)=i 2RgR(!+!0)W(!0)d!0. 3. ApplytherstDysonequation 7{21 toobtainGR;A. 4. ApplythesecondDysonequation 7{20 togetG<. 5. Usingn=RG<(r;r;)d,we'llgetanewdeviceregiondensityn. 6. Repeattheprocedureuntilself-consistencyisreached.Althoughthisprocedurelooksquitefascinating,ithasoneanotherapproximationinitstheory.Inpractice,thedeviceregionalwayshastoincorporatepartoftheleadssothatthewavefunctionsandelectrondensitiesattheleft/rightboundariesofthedeviceregionwillbeexactlythesameastheleft/rightleads.Thatis,toensurethatallscatteringprocessoccurwithinthedeviceregion.Inthenewproceduredescribedabove,theleadsremainnon-interactingwhilewehaveaninteractingdeviceregion.Thisinevitablyintroduceextrascatteringpotentialsatbothboundaries.Onemightaskifitispossibletointroduceinteractingleadsintothecalculationaswell.Inthatcase,theMeirandWingreencurrentfomula 7{16 nolongerholds,anditisdenitelynottrivialtoderivethecurrentformulaforinteractingleads.Thus,itisthebesttoregardthisasanapproximationaswellatthecurrentstage.Thisapproximationisexpectedtoholdwellforlargeleadsmadeofgoodmetal,becauseDFTisexpectedtoworkforsuchsystems.Fortunately,mostexperimentalset-upsuselarge,goodmetalsasleads. 121

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CHAPTER8CONCLUSIONWehaveemployedclassicalMDandmulti-scalesimulationstostudythemechanicalpropertiesofSiO2.ViaclassicalMDsimulation,thetwo-memberedringsareidentiedasthengerprintsofhighlystressedstructures,withtheirnumbercloselyrelatedwiththefractureprocess.Wefoundthatthenano-wirea-SiO2ismuchmoreductilethanthebulkmaterialbecauseitcanemitinternalstressbybreakingexistingtwo-memberedringsandformingnewones.Hightemperaturehelpstheevolutionofthetwo-memberedringsandthereforealsoleadstomoreductilebehaviors.H2Omoleculescanreactwiththetwomemberedringsevenatzeroexternalstress,andtheH2Odipoleassiststhebreakingofthehighlystressedtwo-memberedrings.Thusthesilicamaterialisexpectedtobreakmoreeasilyinahumidenvironmentthanadryone.BothfullquantumandmultiscalesimulationsyieldverysimilartrajectoriesforamodelSiO2chainfracturewithexistenceofH2Omolecules.Signaturestructuressuchasve-coordinatedSiatomsandH3O+groupsarefoundinbothsimulations,showingthevalidityofthecurrentimplementationofmultiscalesimulations.WethenappliedDFTandDFT+NEGFmethodstomodelPd-cluster-functionalizedCNTsystems.TheDFTcalculationsshowasystematicchangeintheband-gapwidthversuscoveragepercentage,whichsuggeststhepossibilityofmanipulatingtheCNTbandstructurebyappropriatepalladiumdoping.With100%palladium-coveredmetallic(5,5)CNTs,thesystemturnsouttobesemiconductingduetoelectronlocalizationeect;andforthesemiconducting(8,0)CNTs,Pddopingturnsitintoaconductor.Allthe(5,5)CNTbasedsystemsinvestigatedshowedanincreaseinconductanceuponhydrogenchemisorptionbutthemechanismsvary.Athighcoverage,theadsorptionofhydrogeninducedasubstantialstructuralchange,andtheclustersformedanatomicwire.Atmediumandlowcoverage,thedissociationofhydrogengreatlyreducedthelocalizationofthebindingelectronsbetweenCNTandPd4-clusters.Butforthe(8,0) 122

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CNTbasedsystems,hydrogenadsorptionturnsthesystembackintosemiconductors,andthussuppressestheconductivity.Todeterminetheactualconductancechangeinthesystem,asetofDFT+NEGFcalculationswasperformedaswell.Thesecalculationsalsoshowthat,uponPddoping,theconductanceofmetallicCNTsdecreasesduetoelectronlocalizationeects,andtheconductanceofthesemiconductingCNTsincreasessincePddopingcreatesnewstatesthatreducethebandgap.Forthemetallic(5,5)system,theconductanceincreasesby90%withhydrogenchemisorption,whileforthesemiconducting(8,0)system,theadsorptionsupressestheconductanceby60%.Thisbehaviorisdominatedbyelectronlocalizationeectsuponhydrogenadsorption,andisrelatedtochargetransfer.ThereforebothmetallicandsemiconductingCNTsaremuchbetterhydrogensensingmaterialsindividuallythanmixedensembleCNTs.Wehavealsodemonstratedtheimportanceoftheon-siteenergyUinLDAandGGAbyapplyingGGA+UtoaNi4singlemoleculemagnet.Becauseofthestrongcorrelationeectsinthissystem,thesimpleDFTcalculationbecausethelackofon-siteenergyunphysicallyencouragesthehybridizationoforbitals,leadingtoAFMcoupling.TheinclusionofaHubbard-UliketermforboththeNi3dandO(1)2pelectronsgreatlyenhancesthelocalizationforbothstates,andisessentialinordertoobtainthecorrectferromagneticgroundstateandexchange-couplingconstants.Aftertakingbothcorrectionsintoconsideration,thesepropertiesweresuccessfullyreproducedbythecalculations.WethenanalyzedtheDOSandprojectedDOSofthesystem,andthecalculationpredictsthattheopticaltransitionfromHOMOtoLUMOisp-dlike,andthegapis2.95eV.Wethenanalyzedtheshortcomingsofthecurrentimplementationofmultiscalesimulation.AnewmultiscalesimulationarchitectureOPAL,whichiscompletelybasedontheMPIprotocolswasintroduced.ByneglectingtheCORBAinterfaces,theeciencyofthecommunications,andthustheexecutionoftheserverprocessisgreatlyenhanced.Thenewarchitectureiscapableofadvancedfeaturesincludingparallelexecution,multiple 123

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centersimulations,anddynamicprocessorallocation.WedemonstrateitspowerbyapplyingtoasimpletestingcaseofNaCldissociationinwater.WehavealsoemployedMLWFstoanalyzetheelectronicstructureofrecentlydiscoveredironbasedsuperconductorLaO1xFxFeAs.AnAFMgroundstatehasbeenfoundfortheundopedparentcompound,andaMott-typetransitionhasbeendiscoveredwithaninclusionofanon-siteenergyof4.5eVonFe.Asetoftight-bindingmodelHamiltonianparametershasbeenobtainedfortheundopedparentcompound.DuetotheevidentproximityofsuperconductivitytoantiferromagnetismandtheMotttransition,wesuggestthatthesystemmaybealarge-spinanalogoftheelectron-dopedcuprates,whereAFMandsuperconductivitycoexist.Finally,weexaminedthestate-of-artDFT+NEGFmethod,andevaluatedtheapproximationsusedinthismethod.Wehaveproventhatundersteadystateconditions,thesystemstateissolelydeterminedbyitselectrondensityprolen(r),thustheeectiveHamiltonianofthesystemHisafunctionalofn(r),andthegroundstateDFTfunctionalisaspecialcaseofthissteady-stateDFTfunctional.However,thisfunctionalistoocomplicatedtoobtain,andweproposeaGW-liketreatmentasarststep. 124

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APPENDIXAEXAMPLEINPUTFILESFORPWSCFPWSCFisaplanewavedensityfunctionaltheorycodethatusespseudopotentialsorprojectedaugmentedwave(PAW)method.Itisopensource,freelydistributableunderGNUPublicLicense(GPL).Inthelatestversion(4.0.1),itcontainsrichfeaturesincludingCar-Parinello-likemeta-dynamics,phonon-electroncoupling,linear-responsecalculations,VanderWaalscorrections,LDA+UandGGA+U,etc.AevenlargercommunityissupportingPWSCFbyreleasingevenmorepost-processingcodes,includingtheGWmethod(codeSAX),maximallylocalizedwannierfunctions(wannier90),ballistictransport(WanT),etc.AlltheDFTcalculationsreportedinthisthesis,unlessspecied,weredoneusingthiscode.ThisappendixsketchesuseofPWSCFwiththeLaOFeAscalculationinputles. A.1StructuralRelaxationAtypicalstructuralrelaxationinputlelookslikethis: &controlcalculation='vc-relax'\\Variablecellrelaxationcalculationprefix='lofa'/&systemibrav=0,celldm(1)=1.88972687777\\Structuralparameters.nat=8,ntyp=4\\8atomsof4typesnbnd=64,ecutwfc=40,ecutrho=400\\Energycutoff(s)occupations='smearing'\\Metallicsystemdegauss=0.001\\ElectrontemperatureinRy/&electrons/&ionsion_dynamics='damp'\\Minimizationalgorithm/&cellcell_dynamics='damp-pr'\\Minimizationalgorithm/ATOMIC_SPECIESO1.599943O.pbe-paw_kj.UPF\\PseudopotentialtouseLa1.389055La.pbe-sp-paw.UPF 125

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Fe1.558452Fe.pbe-paw.UPFAs1.7492160As.pbe-paw.UPFCELL_PARAMETERS(alat)\\latticeconstants4.0233089200.0000000000.0000000000.0000000004.0233089010.0000000000.0000000000.0000000008.580818525ATOMIC_POSITIONS(crystal)O0.0000000000.0000000000.000000000O0.5000000000.5000000000.000000000La0.5000000000.0000000000.148273878La0.0000000000.500000000-0.148273878Fe0.0000000000.0000000000.500000000Fe0.5000000000.5000000000.500000000As0.0000000000.5000000000.372844853As0.5000000000.0000000000.627155147K_POINTSautomatic16,16,8,1,1,1 A.2Self-ConsistentCalculationThemainpurposeofaself-consistentcalculationistogetareliable,well-convergedelectrondensityforlateruse. &controlcalculation='scf'\\Self-consistentCalculationprefix='lofa'tprnfor=.true.\\Printforcetstress=.true.\\Printstress/&systemibrav=0,celldm(1)=1.88972687777nat=8,ntyp=4nbnd=64,ecutwfc=40,ecutrho=400occupations='smearing'degauss=0.001/&electrons/ATOMIC_SPECIESO1.599943O.pbe-paw_kj.UPFLa1.389055La.pbe-sp-paw.UPFFe1.558452Fe.pbe-paw.UPFAs1.7492160As.pbe-paw.UPFCELL_PARAMETERS(alat)4.0232572070.0000000000.000000000 126

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0.0000000004.0232572260.0000000000.0000000000.0000000008.580781298ATOMIC_POSITIONS(crystal)O0.0000000000.0000000000.000000000............As0.5000000000.0000000000.627157033K_POINTSautomatic16,16,8,1,1,1Ittypicallyusetheparametersusedtorelaxthestructure.Bysettingtprnforandtstress,onecanseeiftherelaxationhasbeendonecorrectly. A.3BandStructureUsingtheelectrondensityobtainedinthelaststep,onecanproceedwiththebandstructurecalculation. &controlcalculation='bands'\\Bandstructurecalculationprefix='lofa'/&systemibrav=0,celldm(1)=1.88972687777,nat=8,ntyp=4nbnd=64,ecutwfc=40,ecutrho=400/&electrons/ATOMIC_SPECIESO1.599943O.pbe-paw_kj.UPFLa1.389055La.pbe-sp-paw.UPFFe1.558452Fe.pbe-paw.UPFAs1.7492160As.pbe-paw.UPFCELL_PARAMETERS(alat)4.0232572070.0000000000.0000000000.0000000004.0232572260.0000000000.0000000000.0000000008.580781298ATOMIC_POSITIONS(crystal)O0.0000000000.0000000000.000000000............As0.5000000000.0000000000.627157033K_POINTScrystal1410.0000000.0000000.00000010.0250000.0000000.0000001 127

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............0.5000000.5000000.02500010.5000000.5000000.0000001Thenoticeabledierenceisthatoccupationsshouldnotbespeciedinbandstructurecalculations,sinceonedoesnotneedtolltheelectronDOSinband-structurecalculations.Also,onehastoexplicitlylistthespecialk-pointsalongthehigh-symmetrylinesinsteadofusingautomaticMonkhorst-Packgrids. A.4DOSandPDOSCalculations &controlcalculation='nscf'\\DOSCalculationprefix='lofa'/&systemibrav=0,celldm(1)=1.88972687777nat=8,ntyp=4nbnd=64,ecutwfc=40,ecutrho=400occupations='tetrahedra'/&electrons/ATOMIC_SPECIESO1.599943O.pbe-paw_kj.UPFLa1.389055La.pbe-sp-paw.UPFFe1.558452Fe.pbe-paw.UPFAs1.7492160As.pbe-paw.UPFCELL_PARAMETERS(alat)4.0232572070.0000000000.0000000000.0000000004.0232572260.0000000000.0000000000.0000000008.580781298ATOMIC_POSITIONS(crystal)O0.0000000000.0000000000.000000000............As0.5000000000.0000000000.627157033K_POINTSautomatic32,32,16,1,1,1InaccurateDOScalculations,onehastomakesurethatoccupationsissettobetetrahedra,anduseamuchdenserK-grid. 128

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Oncethecalculationisdone,onecanusetheprojwfc.xpostprocessingcodetogetPDOS.Thiscodewillprojectthetotalwavefunctionontotheatomicorbitalsofeachatom,andthereforegetthePDOSandchargeonindividualatoms. &inputppprefix='lofa'Emin=-25.0\\CalculatePDOSfromthisenergyEmax=25.0\\tothisenergy/ 129

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APPENDIXBEXAMPLEINPUTFILESFORSMEAGOLSMEAGOL[ 174 ]isaDFT+NEGFtransportcodebasedonSIESTA[ 80 ].Itusesnorm-conservingpseudopotentialmethodsandnumericallocalizedbasissets.AllthetransportcalculationsinthisthesisaredoneusingSMEAGOL.AtypicalSMEAGOLcalculationconsistsoftwosteps: 1. AleadcalculationtoobtainthesurfaceGreen'sfunctionoftheleads,whichhastobeaserialcalculation.Theresultisusedtoconstructtheself-energiesL=R. 2. Transportcalculationofthedeviceregion,whichcanbedoneineitherserialorparallelexecution. B.1LeadsCalculationTypicalinputlefortheleadcalculationreads: SystemName(111)GoldLeadSystemLabelAu.leadNumberOfSpecies1NumberOfAtoms43%blockChemicalSpeciesLabel179Au%endblockChemicalSpeciesLabelPAO.BasisSizeSZ%blockPS.lmaxAu2%endblockPS.lmaxLatticeConstant1.0Ang%blockLatticeVectors22.7329070.0000000.0000000.00000022.7329070.0000000.0000000.0000007.045380%endblockLatticeVectorsAtomicCoordinatesFormatScaledCartesian 130

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%blockAtomicCoordinatesAndAtomicSpecies
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B.2TransportCalculationAfterthecalculationfortheleadsisdone,oneproceedtocalculateforthetransportinthedeviceregion: SystemNameGoldLeads+(8,0)CNTSystemLabelcntNumberOfSpecies2NumberOfAtoms367%blockChemicalSpeciesLabel16C279Au%endblockChemicalSpeciesLabelPAO.BasisSizeSZ%blockPS.lmaxAu2%endblockLatticeConstant1.420806667Ang%blockLatticeVectors16.00.000.000.0016.00.000.000.0038.081613253015514%endblockLatticeVectorsAtomicCoordinatesFormatAng%blockAtomicCoordinatesAndAtomicSpecies
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ElectronicTemperature300KSolutionMethoddiagonDM.UseSaveDMTEMTransportT\\TransportcalculationVInitial0.0eV\\ThefirstbiasvoltageVFinal0.0eV\\ThelastbiasvoltageNIVPoints0\\#ofbiasvoltagestobecalculatedDelta1.0d-5\\TheinfinitesmalusedtocalculateG^RTrCoefficientsT\\CalculatethetransmissioncoefficientT(E)InitTransmRange-3.0eV\\FromE=thisvalueFinalTransmRange-1.0eV\\ToE=thisvalueNTransmPoints500\\NumberofmeshpointswetakeHartreeLeadsLeft-21.074143Ang\\MatchtheHartreepotentialsHartreeLeadsRight25.987087Ang\\oftheleadsanddeviceregionHartreeLeadsBottom-2.054361760eV\\toenforcechargeneutralityForthesamereasonsasstatedpreviously,onlysingle-basissetisusedforthecalculation.Forthetransportcalculations,onehastomakesurethatDeltaissmallenoughtoensureconvergence.Also,itispreferedtoperformanormalDFTcalculationofthedeviceregiontoobtaintheconvergedelectrondensity,andusethatdensityasthestartingpointtospeedtheself-consistentprocedure. 133

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BIOGRAPHICALSKETCH ChaoCaowasborninHangzhou,P.R.China,in1981.Hehasbeenfascinatedbystoriesoffamousscientistssincehewasalittleboy.In1999,heenteredthePhysicsDepartmentofFudanUniversityinShanghai.TherehemetProf.JingguangChe,anddecidedtodevotehimselftocomputationalcondensedmatterphysics.In2003,ChaoacceptedanoerfromtheUniversityofFlorida.TherehebeganworkingforProf.Hai-PingCheng.Hisinitialprojectwasmultiscalesimulationsandtheinterfaceproblem.Duringtheprocess,hebecamemoreandmoreinterestedinstronglycorrelatedelectronsandtransporttheory.Heiscurrentlyworkingtoincorporatetheelectroncorrelationeectsintotransportcalculations. 143