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Numerical Study of The Mechanical Loss in Amorphous Oxides

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Title:
Numerical Study of The Mechanical Loss in Amorphous Oxides
Creator:
Hamdan, Rashid M
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (86 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
CHENG,HAI PING
Committee Co-Chair:
SABIN,JOHN R
Committee Members:
MERZ,KENNETH MALCOLM,JR
OBUKHOV,SERGEI
GOWER,LAURIE B
Graduation Date:
5/3/2014

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Subjects / Keywords:
Asymmetry ( jstor )
Atoms ( jstor )
Bromine ( jstor )
Doping ( jstor )
Energy ( jstor )
Graphite ( jstor )
Internal friction ( jstor )
Oxides ( jstor )
Simulations ( jstor )
Uniform Resource Locators ( jstor )
Physics -- Dissertations, Academic -- UF
amorphous -- dlpoly -- glasses -- ligo -- md -- moleculardynamics -- numerical -- oxides
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
Many high-precision optical measurement devices, including gravitational wave detectors, use layers of optical coatings to form highly reflective mirrors. These coating layers are usually dielectric amorphous oxides. However, the thermal noise associated with the mechanical dissipation in these coating materials is a major contributor to the total noise in the devices. This thermal noise is projected to be the limiting factor for the precision of the new generation of the Laser interferometer Gravitational Wave Observatory (LIGO). In this project, we investigate the sources of mechanical loss at the atomic level using numerical models for pure and doped amorphous oxides that are used for coating. We have implemented different numerical techniques including the trajectory bisection method and the non-local ridge method in the molecular dynamics simulation software DL-POLY. These techniques allow us to search the potential energy landscape for possible transitions between local consecutive energy minimums and to calculate the barrier height and other properties associated with each transition, including the energy asymmetry and the relaxation rate from saddle point to minimum. From distributions of these properties, we calculate the internal friction of pure and mixed amorphous oxides. We compare with experiment when possible and use the results of the numerical calculations to comment on the validity of the theoretical assumptions. In the future, we will use this method to propose new materials that should be better candidates to reduce thermal noise. In the last chapter, we present a first-principles study of the structure and functionality of stage two bromine doped of graphite, where an enhanced in-plane conductivity is reported experimentally. We study two forms of the bromine doping: molecular (Br$_2$) and atomic (Br). We compare their stability function of the interlayer separation. And in addition to the the charge transfer between the graphite layer and the bromine atoms, we calculate the density of state and the band structure for each form of doping to find their effect on the conductivity. Finally we investigate the effect of doping disorder on the out-of-plane band structure. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: CHENG,HAI PING.
Local:
Co-adviser: SABIN,JOHN R.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-05-31
Statement of Responsibility:
by Rashid M Hamdan.

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UFRGP
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Applicable rights reserved.
Embargo Date:
5/31/2016
Resource Identifier:
907294919 ( OCLC )
Classification:
LD1780 2014 ( lcc )

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NUMERICALSTUDYOFTHEMECHANICALLOSSINAMORPHOUSOXIDESByRASHIDMALEKHAMDANADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014RashidMalekHamdan 2

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ToYaraandJulia 3

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ACKNOWLEDGMENTS IwouldliketoexpressmydeepgratitudetomyadvisorProf.Hai-PingChengwhosesupervisionandsupportmadethisworkpossible.Prof.ChengbelievedinmeandwasalwaystherewithhersoundadviceandrecommendationsandthoughherguidanceIwasabletochooseandcompleteaprojectthatIfeelpassionateabout.Iwishtoacknowledgethecommitteememberswhotooktimefromtheirbusyschedulestoreadmydissertationandlistentomywork.MyverygreatappreciationareextendedtoallmyteachersandprofessorsattheUniversityofFloridaandelsewhere,fortheyhavesharpenedmyfascinationandpassiontowardphysicsandscienceingeneralandgavemetheknowledgeandtoolstobeinthispositiontoday.Iamparticularlygratefultoallmygroupmembersfortheirhelpandallourvaluablediscussionsandtheadvicetheygaveme.Finally,Iamindebttoallmyfriendsandfamilyfortheirendlessandunconditionalsupportandencouragement. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1OVERVIEW ...................................... 12 1.1Introduction ................................... 12 1.2Motivation .................................... 13 2THEORETICALBACKGROUND .......................... 15 2.1TheTwoLevelSystemsModel ........................ 15 2.2InteractionofTheTLS'swithPhonons .................... 16 2.3TheoreticalSimplication ........................... 20 3NUMERICALMETHOD ............................... 23 3.1MolecularDynamicsSimulationDetails ................... 24 3.2ForceField ................................... 26 3.3PreparingTheAmorphousStructures .................... 27 3.4TLS'sDistribution:IntervalBisection ..................... 29 3.5BarrierHeight:NonlocalRidgeMethod ................... 32 3.6TLS'sDensity .................................. 34 3.7TransitionRate ................................. 35 3.8DeformationPotential ............................. 37 3.9ElasticModulus ................................ 39 4INTERNALFRICTIONOFHOMOGENOUSANDHETEROGENOUSOXIDES 43 4.1BarrierHeightandAsymmetryDistributions ................. 44 4.2RelaxationTime ................................ 46 4.3DeformationPotential ............................. 49 4.4InternalFriction ................................. 50 5CONCLUSION .................................... 58 6STRUCTUREANDFUNCTIONALITYOFBROMINEDOPEDGRAPHITE ... 62 6.1Introduction ................................... 62 6.2MethodandComputationalDetails ...................... 65 6.3ResultsandDiscussion ............................ 67 5

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6.3.1CongurationsandEnergetics ..................... 68 6.3.2ElectronicStructure .......................... 70 6.3.3EffectsofDisorderandIntercalantConcentration .......... 74 6.4Conclusions ................................... 75 REFERENCES ....................................... 80 BIOGRAPHICALSKETCH ................................ 86 6

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LISTOFTABLES Table page 3-1PotentialParametersofthevariousoxides. .................... 27 4-1Thesetofcalculatedparametersnecessaryfortheinternalfrictionofpureanddopedsilica. ................................... 47 4-2Thesetofcalculatedparametersnecessaryfortheinternalfrictionofpureanddopedtantala. .................................. 48 6-1ThegroundstateenergyofdifferentthestartingcongurationasfunctionofinterlayerdistancecalculatedwithLDAfunctional. ................ 69 6-2ThegroundstateenergyofdifferentthestartingcongurationasfunctionoftheinterlayerdistancecalculatedwithvdW-DFfunctional. ............ 69 6-3ThechargetransferbetweenthecarbontheBratomsandthemagnetizationofthesystemperBr. ................................. 70 7

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LISTOFFIGURES Figure page 2-1Representationofthedouble-wellpotential. .................... 22 2-2Thedifferentrelaxationregimesatdifferenttemperatures. ............ 22 3-1Samplingoftheamorphousoxides. ........................ 30 3-2AschematicillustratingthealgorithmfortheTLSdistributionsearch. ...... 30 3-3AschematicoftheintervalbisectionoftheMDtrajectory. ............ 31 3-4Aschematicrepresentingthenon-localridgemethod. .............. 41 3-5AschematicillustratingthealgorithmfortheTLSdensitysearch. ........ 42 4-1Thecongurationofthelocalminimumsof2differentTLS'sofsilica. ...... 43 4-2Thecongurationofthelocalminimumsof2differentTLS'softantala. ..... 47 4-3Thenormalizedasymmetrydistributionf()=Nofthepuresilicaandsilicadopedwith50%zirconiaandhafnia. ........................ 48 4-4Thenormalizedasymmetrydistributionf()=Nofthepuretantalaandtantaladopedtitaniawiththeweakandstrongpotentials. ................ 53 4-5Thebarrierheightdistributiong(V)ofthepuresilicaandsilicadopedwithzirconiaandhafnia. ................................. 54 4-6Thebarrierheightdistributiong(V)ofthepuretantalaandtantaladopedtitaniawiththeweakandstrongpotential. ......................... 55 4-7Theinternalfrictionofsilicaanddopedsilica. ................... 56 4-8Theinternalfrictionoftantalaanddopedtantala. ................. 56 4-9Comparisonbetweenexperimentaldataofsilicaandthecomputationaldatafromthiswork. .................................... 57 4-10Theactivationenergyofpureanddopedtantala. ................. 57 6-1Representationofthesimulationsupercells. ................... 65 6-2Schematicrepresentationof4differentinitialpositionsoftheBr2moleculebetweenthegraphitesheets. ............................ 67 6-3Schematicrepresentationof3differentsiteoftheBratombetweenthegraphitesheets. ........................................ 68 8

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6-4Thetotalenergyofthedifferentatomicandmolecularcongurationsfunctionoftheinterlayerseparation. ............................. 69 6-5Iso-surfacesofthechargedensitydifferencebetweenthedopedsystemandisolatedBratomsandisolatedgraphitelayers. .................. 71 6-6Thetotalandprojecteddensityofstates. ..................... 72 6-7Bandstructurealonghighsymmetrylinesofthex-yplane. ........... 74 6-8Thenewsupercellmadeoftwooriginalsupercellsalongthec-direction. .... 77 6-9Bandstructurealonghighsymmetrylinesinthez-direction. ........... 78 6-10Bandstructurebetweenhighsymmetrylinesinthez-directionwiththelaterallocationoftheBratomschosenrandomlyrelativetothecarbonatom. ..... 79 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyNUMERICALSTUDYOFTHEMECHANICALLOSSINAMORPHOUSOXIDESByRashidMalekHamdanMay2014Chair:Hai-PingChengMajor:PhysicsManyhigh-precisionopticalmeasurementdevices,includinggravitationalwavedetectors,uselayersofopticalcoatingstoformhighlyreectivemirrors.Thesecoatinglayersareusuallydielectricamorphousoxides.However,thethermalnoiseassociatedwiththemechanicaldissipationinthesecoatingmaterialsisamajorcontributortothetotalnoiseinthedevices.ThisthermalnoiseisprojectedtobethelimitingfactorfortheprecisionofthenewgenerationoftheLaserinterferometerGravitationalWaveObservatory(LIGO).Inthisproject,weinvestigatethesourcesofmechanicallossattheatomiclevelusingnumericalmodelsforpureanddopedamorphousoxidesthatareusedforcoating.Wehaveimplementeddifferentnumericaltechniquesincludingthetrajectorybisectionmethodandthenon-localridgemethodinthemoleculardynamicssimulationsoftwareDL-POLY.Thesetechniquesallowustosearchthepotentialenergylandscapeforpossibletransitionsbetweenlocalconsecutiveenergyminimumsandtocalculatethebarrierheightandotherpropertiesassociatedwitheachtransition,includingtheenergyasymmetryandtherelaxationratefromsaddlepointtominimum.Fromdistributionsoftheseproperties,wecalculatetheinternalfrictionofpureandmixedamorphousoxides.Wecomparewithexperimentwhenpossibleandusetheresultsofthenumericalcalculationstocommentonthevalidityofthetheoreticalassumptions.Inthefuture,wewillusethismethodtoproposenewmaterialsthatshouldbebettercandidatestoreducethermalnoise. 10

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Inthelastchapter,wepresentarst-principlesstudyofthestructureandfunctionalityofstagetwobrominedopedofgraphite,whereanenhancedin-planeconductivityisreportedexperimentally.Westudytwoformsofthebrominedoping:molecular(Br2)andatomic(Br).Wecomparetheirstabilityfunctionoftheinterlayerseparation.Andinadditiontothethechargetransferbetweenthegraphitelayerandthebromineatoms,wecalculatethedensityofstateandthebandstructureforeachformofdopingtondtheireffectontheconductivity.Finallyweinvestigatetheeffectofdopingdisorderontheout-of-planebandstructure. 11

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CHAPTER1OVERVIEW 1.1IntroductionAmorphousanddisorderedsolidsshowanomalouspropertiesatlowtemperaturecomparedtowhatisknownforcrystallinematerials.Theseanomaliescanbelargelyexplainedbytheexistenceofabroaddistributionoflow-energyexcitationsinamorphousstructures.Theseexcitationsaredescribedastransitionsbetweentwolevelsystems(TLS)onthepotentialenergylandscapes(PEL),orasaparticlesmovinginadoublewellpotentialsinthehighdimensionalcongurationspace.Thetunnelingmodel,introducedbyPhilips[ 1 ]andAnderson[ 2 ]independentlyin1972,explainstheexperimentallyobserveddeviationsofthethermalpropertiesofamorphoussolidsatverylowtemperaturefromwhatispredictedbytheDebyemodel.Thestandardtunnelingmodule(STM)successfulpredictsthenearlylinear(T1:2)andquadratic(T1:8)dependenceoftheofthespecicheatthermalconductivityontemperature[ 3 ].Thetwolevelspictureofamorphoussolidscanbeextendedtoincludethermalrelaxationsbetweenthetwolevelsathighertemperatures(T5)Kandthusdescribestheelectricandacousticpropertiesinthesematerial.Specically,thepeaksinthedielectricdissipationandtheacousticattenuationattemperaturesbelowtheglasstransitiontemperature,inadditiontothevariationofthesoundvelocityasfunctionoftemperature[ 4 ].ThesuccessoftheSTMinexplainingthelowtemperaturepropertiesofdisorderedsolidshasbeenmainlyphenomenological[ 4 ].ItisbasedonthefactthatduetotheruggednessofthePELofamorphousmaterials,itispossibleforasingleatomoragroupatomoftomovebetweentwoormoreenergylevelsseparatedbyanenergybarrier.However,adescriptionofthenatureofthesetransitionsonanatomiclevelandthetruedistributionoftheirpropertiesisstilllacking.FewcomputationalstudieshavedealtwiththeissueoflocatingtheTLS's.DoliwaandHeuerusedmethodsbased 12

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onmoleculardynamicstolocateTLS'sandtheirenergybarriersofaLennard-Jonestypeglassmodel[ 5 ].ReinischandHeuerusedthesamemodelofglasssystematicallylocatingTLS'stoestimatetheirdensity[ 6 ].HeuerandSilbeyusedtheclassicalpotentialmodeltoestimatedistributionofthecouplingoftheTLS'swiththephonons[ 7 ].ThebroadvariationsfoundinthelocalenvironmentsofamorphoussolidsmeansthattheparametersofTLS'scomeinwiderangeofdistributions.TheparametersofTLS'sincludetheenergydifferencebetweenthetwolevelsthepotential,barrierbetweenthem,theircouplingtothephononsorconductingelectroninmetallicmaterials,andthetunnellingortransitionrate.Thedistributionsoftheseparametersarethebasesofthetunnelingmodel.Inourwork,wefocusspecicallyonthemechanicalloss(oracousticattenuationthatisproportionaltothelightscatteringintensity[ 8 ])ofpureandmixedamorphousoxide,whichcanbedescribedbytheinternalfriction.Theinternalfriction,ortheinverseofthequalityfactor,isaunitlessquantitydenedasQ)]TJ /F5 7.97 Tf 6.59 0 Td[(1(T)=v=![ 4 ],whereistheattenuationperunitlength,visthespeedofsoundinthesolid,and!isthefrequencyofthewave[ 4 ].isrelatedtotheenergypervolumelostinonecycleastheacousticwavepropagatesthroughthesolid:E=E0whereisthewavelengthofthewave.Computersimulationsbasedonmoleculardynamics(MD)tolocateTLS'swithintheamorphousoxidesandestimatethedistributionsandaveragevaluesoftheirparametersanddensity.Thenfromthesepropertieswecalculatetheinternalfrictionofthesilica(SiO2),silicamixedwithzirconiaandhana(SixZr1)]TJ /F7 7.97 Tf 6.58 0 Td[(xO2andSixHf1)]TJ /F7 7.97 Tf 6.59 0 Td[(xO2),tantala(Ta2O5),andtantalamixedwithtitina((Ta2O5)x(TiO2)y).WemodelthepotentialenergyofthesystemwithclassicalpairwiseatomicinteractionsbasedontheBuckinghampotential. 1.2MotivationThestudyofthemechanicallossismotivatedbytheeffortstoenhancetheprecisionofgravitationalwavedetectors.Interferometricgravitationalwavedetectors 13

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relyonthehighprecisionFabry-Perotlaserinterferometerstodetectdisplacementsofthesuspendedmassescoatedtoformhighlyreectivemirrors[ 9 10 ].Amorphousoxidesaremainlyusedforthemirrorcoating,whicharemadeofalternatingion-beamsputteredlayersofsilica(SiO2)andtantala(Ta2O5)inthecurrentdetectors[ 11 ].Afundamentallimitationtotheprecisionofsuchdevicesarisesduetothermalnoise,whichisproportionaltothemechanicaldissipationassociatedwiththesematerials[ 10 12 14 ].Innext-generationdetectors,itispredictedthatthethermalnoiseofthecoatingmaterialwillbethelimitingfactorforprecisionmeasurementsatlowfrequency[ 15 ].Thus,bothunderstandingtheoriginofmechanicallossandidentifyingmaterialsthatarecandidatestominimizethermalnoisearecentraltoimprovingthesemeasurements.Earlierexperimentalworksshowthatvariousglass-likesolidshaveawiderangeofacousticpropertiesatrelativelyhightemperatures[ 16 ]andlacksimple,universalwaystopredicttheseproperties.Therefore,itisimportanttodevelopnumericalmethodstopredicttheinternalfrictionfornewanddopedmaterials.Silica'sacousticpropertiesareextensivelystudiedexperimentally[ 16 26 ]andprovidesagoodmeasuretotestthenumericaltechniquepresentedhere.Thisthesisisorganisedasfollowing:Inchapter 2 wegiveabriefsummaryofthetunnelingmodelandthequantitiesinvolvedincalculatingtheinternalfriction.InChapter 3 wedescribethenumericalpotentialandcellusedtomodelthestudiedsystems.Thenwedetailthenumericaltechniquesthatweusetocalculateeachoftheparametersneededfortheestimationoftheinternalfriction.Inchapter 4 weanalyzetheresultsofthenumericalcalculations,wecomparetheseresultswithexperimentswhenpossibleandmakefewcommentsonthetheoreticalmodelandtting,followedbyaconclusionandnalcommentsinchapter 5 Finallyinchapter 6 wepresentadifferentprojectinwhichwestudytheeffectsofbrominedopingongraphiteusingrst-principlescalculations. 14

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CHAPTER2THEORETICALBACKGROUND 2.1TheTwoLevelSystemsModelAlthoughglassesandamorphousmaterialspossesslocalatomicorder,thelongrangeorderofcrystallinematerialsislost.Thelackoflongrangeorderallowsfordifferentgeometricalcongurationstobepossibleatthelocalscale.Transitionsbetweenthesedifferentcongurationsispossiblethroughtherearrangementofasingleatomorasmallgroupsofatoms,evenattemperaturesthatarewellbelowtheglasstransitiontemperature.Theselocaltransitionsgivetheglassestheiracousticandthermalproperties.Attemperaturesbelow1K,thepropertiesofglassesarewelldescribedbythequantumtunnelingmodel[ 1 2 ].Athighertemperatures,thermallyactivatedtransitionsdominateovertunnelingmechanisms[ 27 ],howeverthenatureofthestatesthemselvesisnotchanged.Thetunnelingmodelisbasedontheassumptionthatgroupsofatomsinamorphoussolidscanmovebetweentwoormorepreferredstatesbysimplechangesinbondlength,angle,orcombinationsofboth.Thesetransitionsarerepresentedasparticleswithtwopossibleenergylevelsoraparticlemovinginbetweentwoenergywellsinahigherdimensionalpotentialenergylandscape(thedimensionsaredeterminedbythenumberofatomsinvolvedinthetransition).Thesedoublewellsystemsarereferredtoastwo-levelsystems(TLS's)ortunnelingstates.Thusanamorphoussolidcanbedescribedasacollectionoranensembleofparticleswithtwoenergylevelortunnelingstates.EachtunnelingstateorTLSischaracterisedbytheenergyasymmetrybetweenthetwowells,theaveragebarrierheightV,andthecongurationaldistancedbetweenthetwostates[ 1 2 ]asrepresentedinFig. 2-1 .ThetwoenergylevelsoftheparticleortheTLSarereadilygiveninthewelllocalisedrepresentation.Ifthe1and2representthewavefunctionoftheparticlelocalizedintherstandthesecondwelloftheTLS,withgroundstateE1andE2 15

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respectively(correspondingtothelowestvibrationalfrequencyineachwell).AssumingthatE1=E2andthechoosingthezeroenergyasthemeanofthegroundstateenergies,theHamiltonianofthesystemiswrittenas: H=1 20B@00)]TJ /F4 11.955 Tf 9.3 0 Td[(1CA(2)0isthetunnelingsplittingortheoverlapoftheHamiltonian0=2h1jHj2i.ItcanbebeapproximatedusingtheWentzel-Kramers-Brillouinmethod: 0=E0e)]TJ /F7 7.97 Tf 6.58 0 Td[((2)isthetunnelingparameter: =d ~p 2mV(2)mistheparticleseffectivemassanddistheeffectivecongurationaldistancebetweenthetwowells.Theexactexpressionofthesetwoquantitiesishardtoexpressintermsoftherealsystemparametersoftheamorphoussolidsandmightbedeneddifferentlyindifferentsources.Mostimportantly,theenergysplittingbetweentheeigenstatesoftunnelingmatrixHis: E=q 2+20(2) 2.2InteractionofTheTLS'swithPhononsAnacousticwavewouldproducealocalstraininsidethesolidgivenbythestrainmatrixu: u=1 2(@u @x+@u @x)(2)uisthedisplacementoftheatomsinthedirection.TheTLS'scouplestostrainthroughtwodeformationalpotentials: D=@E @u= E@ @u+0 E@0 @u(2) 16

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M=1 2( E@0 @u)]TJ /F4 11.955 Tf 13.15 8.09 Td[(0 E@ @u)(2)DandMrepresenttheshiftoftheenergysplittingandthetheresonantinteractionbetweentheTLSandthephononsrespectively.Twomajorsimplicationcanbeintroducedtothedeformationpotentials.First,theeffectofstrainonthetunnelingsplitting0isnegligiblecomparedtotheeffectontheTLSasymmetry.Second,inisotropicsolids(whichisassumedfortheamorphoussolids)thelongitudinalandthetransversecomponentsofthedeformationpotentialscanbegroupedtogether.Hence,thesimplieddeformationpotentialsare: Dj=2 Ej(2) Mj=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(0 Ej(2) j=1 2@ @uj(2)whereujwithj=l;tarethelongitudinalandtransverseelementsofthestraintensor,andjwithj=l;tarethesimplieddeformationpotential.Thechoicebetweenlongitudinalortransversepolarizationdependsontheexperimentalsettings,andthefollowinganalysisapplyequallytoeitherpolarization.Theeffectofthestresscausedbyacousticwave(t)=0sin(!t)isexpressedasaperturbationH1addtothehamiltonianoftheTLS.Ineigenbasissetthisperturbationiswrittenas: H1=)]TJ /F4 11.955 Tf 10.5 8.08 Td[(1 20B@Dj2Mj2Mj)]TJ /F3 11.955 Tf 9.3 0 Td[(Dj1CA(t)(2)Theresemblancebetweenthissystemandthespin1=2systeminanoscillatingexternaleld[ 28 29 ]isusedtosolvetheBloch-likeequationfortheattenuation(internalfriction)anddissipationofsoundinamorphoussolidmodeledasanensembleofTLS's. Q)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2 EZ10Z10! 1+!22sech2( 2kBT)1 kBT( E)2P(E;)ddE(2) 17

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v v=2 2EZ10Z101 1+!22sech2( 2kBT)1 kBT( E)2P(E;)ddE(2)visthespeedofsoundinthemedium,Eistheappropriateelasticmodulus,istherelaxationtimeortheinverseofthetransitionrateinaTLSandP(E;)isthedistributionoftheenergysplittingandtherelaxationtimesofthesystem.Therelaxationoftheparticleinthedoublewellhavedifferentmechanismsandthedominantmechanismistemperaturedependant.Attemperaturebelow1Kthesinglephononprocessesdominateandtherelaxationrateisgivenby: )]TJ /F5 7.97 Tf 6.59 0 Td[(1=A0Ecoth[E 2kBT](2)whereAisaconstantthatdependsonboththecouplingoftheTLSwiththephononsandthespeedofsoundinthemedium[ 30 ].Othermechanismsincludetwo-phononprocessesorrstorderRamanprocessesatslightlyhighertemperature,electron-assistedtunnelinginconductingmetalsandincoherenttunneling.Athighertemperaturesthermallyactivatedrelaxation[ 4 ]becomesimportant.Ingeneraltheeffectiverateisthesumofthecontributionfromalltherelevantrelaxationprocesses.Thermalrelaxationbecomesimportantabove5Kanddominatesabove10Kwhichistheregioninwhichweareinterested.(seeFig. 2-2 ).TherateforthermallyactivatedrelaxationisgivenbyanArrheniusLaw.ThiscanbeseenbylookingatthesystemasanensembleofparticlesatthermalequilibriumattemperatureT.IfaspecicparticleintheensemblehasanenergyElargerthanthebarrierheightVtheparticlecancrossoverwithouttunneling.TheprobabilityofaparticletohaveanenergyEorlargerise)]TJ /F7 7.97 Tf 6.58 0 Td[(E=kBT.Therelaxationrateisgivenby[ 27 ]: )]TJ /F5 7.97 Tf 6.58 0 Td[(1=)]TJ /F5 7.97 Tf 6.59 0 Td[(10cosh[ 2kBT]e)]TJ /F7 7.97 Tf 6.59 0 Td[(V=kBT:(2)Theterm)]TJ /F5 7.97 Tf 6.59 0 Td[(10istheinverseoftheminimumrelaxationrate,ortherateatT=0.Itiscalledtheattemptfrequency[ 4 ]andcanbecalculatedfromthevibrationalfrequenciesatthebottomsofthepotentialwellsandthesaddlepoint.(Moreonthatinsection 3.7 .) 18

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TwoimportantobservationscanbemadetosimplifyEq. 2 .Firstduetotheterm(=E)2inEq. 2 onlyasymmetricstateswouldcontributetotheinternalfrictionandthustheenergysplittingEintheintegrationcanbereplacedwiththeasymmetry.SecondduetotheexponentialdependenceofonV,thedistributionofthebarrierheightisthemaincontributortothedistributionoftherelaxationtimeandthedistributionof0canbeignored.Hence,thedistributionsoverEandarebereplacedbyadistributionoverandVand0isbeapproximatedbyitsaveragevalue[ 27 ]. Q)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2 EZ10Z10! 1+!22sech2( 2kBT)(1 kBT)f()g(V)ddV:(2)Inthisformulationtheasymmetrydistributionf()hastheunitofenergy)]TJ /F5 7.97 Tf 6.59 0 Td[(1volume)]TJ /F5 7.97 Tf 6.59 0 Td[(1andcontainsinformationaboutthedensityNofTLS'spervolumeavailabletothesystem.Thebarrierheightdistributiong(V)isassumedtobeasmoothfunctionindependentoftemperature.Nosharppeaksing(V)areexpectedinglasses.Thistemperatureindependencestartstofailasthetemperatureapproachestheglasstransitionandthelocalpotentialuctuates,causingthedoublewellpicturetobreakdown[ 27 ].Eistheappropriateelasticmodulus.Gilroyetal.[ 27 ]haveusedtheYoung'smoduluswhereasotherauthorshaveusedthebulkmodulus[ 23 ].Inthiswork,itisfoundthattheYoung'smodulusismoreappropriatetoreproducetheinternalfrictioninsilicaandtantala(seeSec. 4.4 ).InthisworkEq. 2 isintegratednumericallytoestimatetheinternalfrictionofthevariousamorphousoxides.Alloftheingredientsinthatintegralarecalculatednumericallyontheatomiclevelusingaclassicalforceeldmodelfortheoxides.However,certainapproximationsareusuallyusedinliteratureandareimportanttocomparetoexperimentsandthusarediscussedbelow. 19

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2.3TheoreticalSimplicationSincetheintegralinEq. 2 hasnoanalyticalsolution,twomajorassumptionsabouttheasymmetryandthebarrierheightdistributionareusuallymadeinordertottheinternalfrictionexperimentaldatatothetheoreticalmodel. Wellbelowtheglasstransitiontemperature,f()isassumedtobeuniformf()=f0. Thedistributiong(V)decreasesrapidlyathighenergiesandisttedtoanexponentialdecay.Thetermsech2( 2kBT)inEq. 2 actsasaneffectivecutoffabove2kBTfortherangeofintegrationoftheasymmetry.AndwiththeuniformdistributionofthenumberofactiveTLS'sthatcontributetotheinternalfrictionvariesasf0kBT.Usingtheuniformdistributionofasymmetryandintegrationcutoff,theintegrationovercanbesolvedanalyticallyleadingto: Q)]TJ /F5 7.97 Tf 6.58 0 Td[(1=2f0 EZ10! 1+!22g(V)dV(2)with )]TJ /F5 7.97 Tf 6.59 0 Td[(1=)]TJ /F5 7.97 Tf 6.59 0 Td[(10e)]TJ /F7 7.97 Tf 6.59 0 Td[(V=kBT(2)Differentformswereproposedforthettingofthethebarrierheightdistributiong(V).EarliertheoreticalworkusedaGaussianfunctionwithasharplowenergycutofftottheasymmetrydistribution[ 31 ].Gilroyetal.[ 27 ]haveproposedanexponentialdecaywithnolowenergycutoffforthebarrierheightdistribution: g(V)=1 V0exp()]TJ /F3 11.955 Tf 9.3 0 Td[(V=V0)(2)V0isreferredtoastheactivationenergyandusuallyrelatedtotheglasstransitiontemperature. 20

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Withtheexponentialdecayofg(V)theintegralinEq. 2 canbesolvedanalyticallyandthenalformtheinternalfrictionis: Q)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2f0 E(!)(2)with=kBT=V0Inanalysisoftheexperimentaldata,itisusefultotthetheinternalfrictionmaximumandthetemperatureatthemaximumtothemeasuringfrequencies.FromthederivativeofequationEq. 2 ,Tpeakthetemperatureatwhichtheinternalfrictionismaximumis kBTpeak V0=)]TJ /F3 11.955 Tf 9.3 0 Td[(ln(!0)(2)andthemaximuminternalfrictionis Q)]TJ /F5 7.97 Tf 6.58 0 Td[(1max=2e)]TJ /F5 7.97 Tf 6.59 0 Td[(1f0kBTpeak EV0(2)Eq. 2 isusedtotfortheactivationenergyV0andthemaximumrelaxationtime0.Inthiswork,theinternalfrictioniscalculatednumericallyfromEq. 2 withthedistributionsfoundfromthemoleculardynamicssimulationswithoutanyassumptionsandthedistributionsoftheasymmetryandbarrierheightarecomparedtothettingequationstocommentonthevalidityoftheseassumptions.Insummary,Eq. 2 canbeusedtoestimatetheinternalfrictionofamorphousmaterialsattheatomiclevel.Thekeyquantitiesrequiredaretheasymmetrydistributionf(),thebarrierheightdistributiong(V)andthedensityoftheTLS's,inadditiontotheaveragedeformationpotential,theaveragemaximumrelaxationtime0,andtheYoung'smodulusEofeachsolid. 21

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Figure2-1. Representationofthedouble-wellpotential.disthecongurationaldistancebetweenthetwowells,istheenergyasymmetry,E1andE2arethegroundstateenergiesinthetworepectivewells,whileVistheaverageofbarrierheightmeasuredfromthebottomeachwelltothesaddlepoint. Figure2-2. Thedifferentrelaxationregimesatdifferenttemperatures. 22

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CHAPTER3NUMERICALMETHODWhenmodelingamorphoussolids,afewthingsneedtobetakenintoconsideration.First,amorphousstructurelacktheperiodicitythatisfoundincrystallinesolids.Henceawiderangeofdifferentlocalstructurescanbefoundinanexperimentalsampleofamorphoussolidthatareoftheorderof10nm[ 11 ].Second,thetransitionsbetweenthelocalminimumsonthepotentialenergylandscape,eventhoughlocal,involvetensofatomsandextendoverdistancesofmorethan10A.Theseissuesimposealowerlimitonthesizeofthesimulationboxthatcanbeusedtomodelanamorphousoxide.Whereas,thecomputationaltechniqueandcomputertimeavailableimposeanupperlimitonthesizeofthesamplethatcanbeused.Inthiswork,asimulationboxwithsidesbetween20and26Aischosenwith648to1008atomsineachboxwithperiodicboundaryconditionstosimulateabulkmaterialwithnosurfaces(Fig. 3-1 ).Thisislargeenoughtocapturethetransitionsinthepotentialenergylandscapewithoutanysignicantnitesizeeffects.However,forcapturingthebroadspectrumofstructuresandtransitionsfoundinanexperimentalsampleofamorphousoxides,insteadofonehugesample,asetofthesmallersamplesdescribedaboveisindependentlypreparedandusedasthestartingcongurationsforthesimulations.Thisstrategyensuresthatmostofthepossiblecongurationscanbesampled.Thenumericalmethodusedinthisworkarebasedonmoleculardynamics(MD)trajectoriesandenergyminimization.TheatomicinteractionsaremodeledwithaclassicalpotentialcontainingCoulombandvanderWaalsinteractions.ThemoleculardynamicsimulationpackageDL POLY[ 32 ]isusedtogeneratetheneededtrajectoriesandtheallthedifferenttechniquesthataregoingtobedescribedinthischapterwereimplementedinDL POLY2.20,tomakethesimulationsmoreautomatedwithintheMDsimulationsandtoreducethememoryrequirementofsavinglongtrajectories. 23

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Inthischapter,ashortdescriptionofthedetailsoftheMDsimulationsusedinDL POLYisgivenandtheclassicalforceeldsusedtomodeltheinteractionsbetweentheatomsoftheamorphousoxidesisdescribed.Then,asetofnumericaltechniquesusedtocollectatheTLS'sandtheirpropertiesarediscussed.ThedistributionsoftheasymmetriesandthebarrierheightsarefoundwithonemethodandtheTLS'sdensitieswithanother.ThenthecollectedTLS'sisusedforcalculatingtheparametersthatarerelevantfortheinternalfrictionoftheamorphousoxides. 3.1MolecularDynamicsSimulationDetailsTheperiodicboundaryconditionspermitstheuseofEwaldsummation[ 33 ]tocalculatethelongrangeCoulombinteractions.Ewaldsummationisusedtoaccountfortheinteractionofeachparticlewiththeotherparticlesinthesimulationcellandtheirimagesininnitearrayofperiodiccells.Thetrickistoaddaneutralizingchargedistributionofequalandoppositechargeontopofeachchargeinthesimulationcell.Thepotentialsumoverthepointchargesandtheirneutralizingshellsconvergesrapidlyasanerrorfunction.Thenasecondchargedistributionoppositetotheoriginaloneisaddedontop.ThepotentialsumoverthisnewdistributionisperformedinFourierspaceandconvergesveryrapidly.Theequationsofmotionsaresolvedusingthevelocityverletalgorithm[ 34 ].Inthisalgorithmthepositions,velocities,andforcesareallfoundateachfulltimestep.thecalculationsaredoneintwosteps.Firstthevelocitiesattimet+t=2arecalculatedfromtheforcesandvelocitiesattimet.Secondthefulltimesteppositionsareevaluatedfromthehalftimestepvelocitiesandthentheforceatthenewpositionsareupdated. v(t+1 2t)=v(t)+1 2tF m (3a)x(t+t)=x(t)+tv(t+1 2t) (3b) 24

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IncollectingtheTLSdistributionandpreparingthesamples,theMDsimulationsarerunwithinthreeensembles:microcanonicalensemblewithconstantenergyandcellvolume(NVE),canonicalensemblewithconstantvolumeandtemperature(NVT),andaconstantpressureconstanttemperatureensembleinwhichthevolumeisallowedtovary.IntheNVEensemble,theconservedquantityisthetotalenergyH=KE+U.FortheNVTensemblethetemperatureiscontrolledbytheNose-Hooverthermostat[ 35 ].Africtiontermisaddedtotheequationofmotionwiththecoefcient(t),whichiscontrolledbytheequation: d(t) dt=NkB Q(T(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Text)(3)whereNisthedegreesoffreedom,QistheeffectivemassandT(t)istheinstantaneoustemperatureattimet.Theconservedquantitybecomes: H=KE+U+1 2Q(t)2+Q TZt0(s)ds(3)Tisthetimeconstantusuallyintheorderof0:5)]TJ /F4 11.955 Tf 11.96 0 Td[(2ps.FortheNPTensembletheHooveralgorithm[ 36 ]isusedwhichcouplesabarostatwithNose-Hooverthermostatbyaddingapressurefrictiontermtothetheequationsofmotion.Theconservedquantityinthisalgorithmis: H=KE+U+V(t)Pext+1 2Q(t)2+1 2W(t)2+Q TZt0(Q T(s)+kbText)ds(3)wherePextistheexternalpressure,V(t)istheinstantaneouscellvolumeattimet,Wisthepressureeffectivemassand(t)isthetimedependentpressurefrictioncoefcient.ThelocalpotentialenergyminimumcongurationsarefoundusingconjugategradientenergyminimizationmethodstartingfromcorrespondingMDcongurations.Theconjugategradientmethodchangesthecoordinatesoftheatomsgraduallymovingthemclosertothebottomoftheenergywellineachstepuntiltheconvergencecriteriaisreached.Intheconjugategradientmethodthedirectioninwhichtheatomsmovesat 25

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everyiterationshkisconjugatetothedirectioninthepreviousstephk)]TJ /F5 7.97 Tf 6.58 0 Td[(1: hk=)]TJ /F12 11.955 Tf 9.3 0 Td[(gk+khk)]TJ /F5 7.97 Tf 6.59 0 Td[(1 (3a)k=(xk)]TJ /F12 11.955 Tf 11.96 0 Td[(gk)]TJ /F5 7.97 Tf 6.59 0 Td[(1)xk gk)]TJ /F5 7.97 Tf 6.59 0 Td[(1gk)]TJ /F5 7.97 Tf 6.59 0 Td[(1 (3b)wheregk=rUkisthepotentialenergygradientatthecongurationxk.Theconjugateddirectionsarenotorthogonaltoeachotherwhilethegradientsareorthogonalateachstep.Incontrasttothesteepestdescentmethodinwhichthedirectionofmotionalwaysfollowsthegradient,theconjugategradientmethodavoidstheoscillatorymotioninnarrowvalleys.ThisversionoftheconjugategradientwasimplementedbytheauthorinDL POLYandusedinallthetechniquesthataredescribedbelow. 3.2ForceFieldTheinteratomicinteractionsintheamorphousoxidesaremodeledwithapotentialbasedontheBuckinghampotential[ 37 ].Thispotentialisreliableandtransferable.TheinteractionenergyBKSijbetweenatomsiandjisgivenby: BKSij=qiqj=rij+Aijexp(rij=ij))]TJ /F3 11.955 Tf 11.95 0 Td[(Cij=r6ij(3)whererijistheinteratomicdistancebetweenthetwoatoms,qiqj=rijistheCoulombinteractionwithqiandqjbeingthecharges.Aijexp(rij=ij)andCij=r6ijarethePaulirepulsionandvanderWaalsattractiveterms,respectively.ThePaulirepulsionandthevanderWaalsattractivetermsareonlyincludedforcation-anionandanion-anioninteractions.Thecation-cationinteractionisdescribedbytheCoulombstermalone.Thisprovideshightransferabilityfortheforceeldandallowsformixingofthedifferentoxideswithoutchangingtheparameters.Experimentsshowthatthecation-anionbondsinHfO2andTiO2possesssomecovalentfeatures,whichisnotcapturedbytheBKSalone.Therefore,anextraMorsetermisaddedtotheinteractiontocontrolthedegreeofcovalence[ 38 ].Furthermore, 26

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amorphousTiO2commonlyexistintwoformwithfourorsixcoordinationoftheTication,dependingonthemethodofamorphization.Thestrongandweakcoordinationformsrequiretwodifferentsformsofthecation-anioninteraction.ThenalinteratomicpotentialenergyisaMorse-BKS(M)]TJ /F7 7.97 Tf 6.59 0 Td[(BKSij): M)]TJ /F7 7.97 Tf 6.58 0 Td[(BKSij=BKSij+Dij(1)]TJ /F3 11.955 Tf 11.95 0 Td[(exp()]TJ /F3 11.955 Tf 9.29 0 Td[(aij(rij)]TJ /F3 11.955 Tf 11.96 0 Td[(re))2(3)theparametersofthepotentialforthevariousoxidesareprovidedinTable 3-1 Table3-1. PotentialParametersofthevariousoxides. InteractionsAij(eV)ij(A)Cij(A6)Dij(eV)aij(A)re(A)Charge(e) O-O[ 37 ]1388.7000.36232175.0000.000000.00000.0000O:-1.2Si-O[ 37 ]18003.7000.20520133.5300.000000.00000.0000Si:+2.4Zr-O[ 39 ]17243.3940.22650128.3500.000000.00000.0000Zr:+2.4Hf-O[ 38 ]12372.0000.2285981.3500.327401.62302.0480Hf:+2.4Ta-O[ 38 ]100067.0000.131916.0480.378981.62532.5445Ta:+3.0Tiweak-O[ 38 ]5105.1202.2531020.0000.347771.90001.9600Ta:+2.4Tistrong-O[ 38 ]5505.1202.2531020.0000.547761.90001.9600Ta:+2.4 3.3PreparingTheAmorphousStructuresThecongurationsofthesimulationsofpureandmixedamorphousoxidesarepreparedviaannealing,startingfromthecrystallinestructureforthepuresamplesorfrompureamorphousstructuresforthedopedones.Tendifferentamorphousoxidesareexaminedinthisstudy: puresilicaSiO2 25%and50%zirconiaandhanadopedsilica Zr0:25Si0:75O2 Zr0:5Si0:5O2 Hf0:25Si0:75O2 Hf0:5Si0:5O2 puretantalaTa2O5 20:4%and53:8%cationdopedtantalawiththestronglyandweaklycoordinatedtitania (Ta2O5)0:398(TiweakO2)0:204 27

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(Ta2O5)0:231(TiweakO2)0:538 (Ta2O5)0:398(TistrongO2)0:204 (Ta2O5)0:231(TistrongO2)0:538-quartzisthemostcommonlowtemperaturecrystallinestructureofsilica.IthasatrigonalstructureandbelongstothespacegroupP3221(No.154)withlatticevectorsa=4:9Aandc=5:4Aandcontains3SiO2units.AmorphousSiO2ispreparedbyannealingasupercrystallinecellformedof72unitcellsof-quartz,arrangedascloseaspossibletocubicshape.Theresultisanamorphousstructurewith648atoms.Anoutlineoftheannealingprocessisasfollows: (a) Equilibratethecrystallinestructureat300KwithaNPTensembleMDsimulationforshorttimeintervalt (b) IncreasethetemperaturebyanincrementofTandre-equilibrateusinganNPTensembleuntilthemaximumtemperatureisreached. (c) Adjustthelatticevectortoformaperfectcubiccellkeepingthevolumeconstant.AndperformMDsimulationwithaNVTensembleforalongerintervaloftime. (d) reducethetemperaturebyanincrementofTandre-equilibrateusinganNPTensembleuntiltheroomtemperatureisreached (e) AnalNVTsimulationisperformedatroomtemperature.Themaximumtemperatureusedfortheannealingprocessis6000K,thetemperatureincrementsrangebetween50and400Kandtheequilibrationtimeintervalsbetweenthetemperatureschangesisontheorderof10ps.AfterannealingtheamorphousstructureisrelaxedtooneofthelocalpotentialenergyminimumtobeusedinthenextpartoftheTLSsearch.Thecrystallinestructurechosenasastartingpointfortantalais-Ti2O5.IthashexagonalstructureandbelongstothespacegroupP6=mmm.Theunitcellcontains4tantalumand10oxygenatomswithlatticevectorsa=7:61A,b=7:66Aandc=3:8A.thesupercellusedtogeneratetheamorphousstructureismadeof72unitcellscontains1008atomstotal.Forthemixedsilicastructuresthestartingpointistheannealedsilicasupercellwhererandomatomsofsiliconarereplacedwithatomofzirconiumandhafniumand 28

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thewholestructureisannealedagainusingtheproceduredescribedabove.Forthedopedtantasomeofthetantalumatomsaredirectlyreplacedwithtitaniumhoweverextratitaniumatomsreplacesomeoxygenatomstokeepthetotalnumberofatomsconstantandthestoichiometricratioscorrect.Thenannealingisperformedwitheitherthestronglyortheweaklycoordinatedforceeld. 3.4TLS'sDistribution:IntervalBisectionAftereachamorphoussampleisannealed,ThestartingcongurationsfortheTLS'ssearcharepreparedbyrunninganMDsimulationwithinacanonicalensembleatarelativelyhightemperaturearound1200K.Thenafteracertaintimeintervalasnapshotofthetrajectoryistakenandquenchedtothelocalenergyminimumusingenergyminimization[ 6 ].ThesequenchedstructuredareusedasstartingcongurationsinthenextstepinthesearchedforthelocalTLS'sdistribution.Thetimeintervalischosentobelargeenough(10)]TJ /F4 11.955 Tf 11.95 0 Td[(20ps)toallowfordiversityintheinitialstructureschosen.EachinitialcongurationisbroughttoarunningtemperatureTrunusingashortNVTequilibrationMDrun.AfterequilibrationatTruntheMDrunisswitchedtomicrocanonical(NVE)ensembleandtheMDtrajectoryx(t)iscollected.FromtheMDtrajectoryadiscontinuoushoppingtrajectorynisfound,byquenchingtheMDtrajectorytolocalminimumofthepotentialenergyatequaltimeintervalst.Aftereachtimeintervalthequenchedcongurationniscomparedwiththepreviousonen)]TJ /F5 7.97 Tf 6.59 0 Td[(1.IfthetwolocalminimumsareidenticalthenitisassumedthatnohoppinghaveoccurredduringthattimeintervalandtheMDtrajectoryx(t)betweenthesetwopointsisdiscarded.However,ifthetwominimumsaredistinguishablethereisnoreasontosupposethathoppingfromthebasinofn)]TJ /F5 7.97 Tf 6.59 0 Td[(1tonisadirectone.Thesystemcouldhavevisitedoneormorebasinsontheway.InordertoresolvethelocalminimumsthataredirectlyconnectedfurtherenergyminimizationontheMDtrajectoryisneed.Forthat,theintervalbisectionmethodisused[ 5 ].Fig. 3-2 showsaschematicofthiscontinuoussearchofTLS'swiththentheMDrun. 29

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Figure3-1. Samplingoftheamorphousoxides.Multiplesmallercomputationalcellsareusedtocoverthediverselocalenvironmentsofthesematerials. Figure3-2. AschematicillustratingthealgorithmfortheTLSdistributionsearch.InthismethodtheMDsimulationisposedatspecictimeintervalstandrelaxationisperformedinordertolookfortransitions.ThentheMDsimulationiscontinuedfromthelastconguration.Thisallowsforthesamplingofdifferentlocalenvironmentsoftheamorphoussolid. 30

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Theintervalbisectionisimplementedasfollowing: (a) Sett1![n)]TJ /F4 11.955 Tf 11.96 0 Td[(1]tandt2![n]t (b) Set1!n)]TJ /F5 7.97 Tf 6.59 0 Td[(1and2!n (c) Sett!(t1+t2)=2 (d) QuenchtheMDcongurationx(t)togetthelocalminimum (e) If1thent1!tand1!,elset2!tand2!. (f) repeatsteps(c)through(e)untilt2)]TJ /F3 11.955 Tf 11.95 0 Td[(t1<2MDsteps.AschematicoftheintervalbisectionisshowninFig. 3-3 .Theintervalbisectionwillpinpointthetransitionupto1MDstepwiththetwodistinguishablebasinsthataredirectlyconnectedbythisMDstep.Withthismethodonlyonetransitionislocatedintheintervaltsoalltheotherbasinsvisitedbysystemandthebackandforthmotionsarelost. Figure3-3. AschematicoftheintervalbisectionoftheMDtrajectory.thismethodisusedtolocatetheexacttransitionpointbetweentwolocalminimums.Thecongurationatthetimehalfwaybetweenthestartingcongurationsistestedtoseetowhichbasinitbelongs.Thenthecongurationcorrespondingtothatminimumisreplacedwiththenewoneandthenewhalfwaycongurationistaken. Tojudgeiftwocongurationsaredistinguishableornotthemassweightedaveragedistancedmwbetweenthetwocongurationsiscalculated.Ifdmwislessthanacriticaldistancedc,thetwocongurationsaredeemedidentical.Otherwise,theybelongtotwo 31

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differentbasins. dmw=PNi=1midi PNi=1mi(3)diisdistancebetweentheithatomintherstcongurationandthecorrespondingatomintheconguration.TheparametersrelevantforthesimulationaretherunningtemperatureTrunandthetthetimeintervalbetweenthesuccessivequenching.Therightchoiceofparametersareimportanttoinsurethehoppingoccursfrequentlyenoughtooccurinthetimeinterval,butnottoofrequentthatthemultipletransitionswithinonetimeintervalaremissed.InthefollowingworkTrunischosenbetween400and800KandTbetween0:5103and2103MDsteps,wheretheMDstepis0:5fs.Andthecriticaldistancefordistinguishingbetweencongurationsisdc=0:2A.TheintervalbisectionhasbeenimplementedbytheauthorswithintheMDsimulationsusingtheDL-POLYpackage.ThisallowstheidenticationofthetransitionswithoutthestoppingthethesimulationandwithouttheneedoflargememoryspaceforthefullMDtrajectory.Onlytrajectoriesinonetimeintervaltisneededatatime.ThissearchmethodiscontinuouswithintheMDrun.Thestartingpointofthenewsearchistheendpointofthepreviousone,thusthecongurationofamorphousstructureisdynamicandthesystemdoesntretainitsinitialstructure(Fig. 3-2 ).Thismethodallowsforthesamplingawiderrangeoflocalenvironmentswithoutbreakingthesearch.However,sincetheinitialstructureisnotpreservedthismethoddoesnotallowustoestimatethetotalnumberofTLSpervolume. 3.5BarrierHeight:NonlocalRidgeMethodAfterlocatingthetwoadjacentlocalminimums1and2andthetwoMDcongurationsx(t1)andx(t2)thatbelongtothe1andxi2respectively,thetransitionstate(xTS)connectingthetwominimumsisfoundusingthenonlocalridgemethod.ThetransitionstatebetweentwolocalminimumsisastationarypointinthePELF(x)=0.Itisaminimuminalldirectionsexceptone.ThusxTSisanenergyminimum 32

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ontheborderseparatingthetwobasinsoftheconnectedminimums.Inotherwords,ifonedrawsaridgeseparatingthetwobasinsonthePEL,wherecongurationsononesiderelaxesto1andthoseontheothersiderelaxesto2whenenergyminimizationisapplied,xTSwouldbethethelowestpointonthatridge.Hence,asteepestdescentminimizationofanypointexactlyonthebordershouldleadtoxTS.Nevertheless,thisisnotapracticalprocedurebecauseeveniflocatingacongurationperfectlyontheridgeispossiblefromtheMDtrajectory,theslightestnumericalerrorsduringminimizationareenoughtoknockthesystemofftheridgeandbringittooneofthelocalminimums.Insteadthetwocongurationsx(t1)andx(t2),oneoneachsideoftheborder,arerelaxedinparallelandperiodicallybroughtbacktotheridgebylinearintervalbisection[ 5 ].Theprocedureisasfollow: (a) Setx1!x(t1)andx2!x(t2) (b) Bringx1andx2closertotheridgeusingintervalbisection (i) Findthecongurationmidwaybetweenx1andx2,x=(x1+x2)=2 (ii) Quenchxtothelocalminimum. (iii) If1thent1!tand1!,elset2!tand2!. (iv) Repeatsteps(i)-(ii)3or4times. (c) Performapresetnumber(M)ofenergyminimizationssteps (d) Repeatsteps(b)through(c)untilmaximumforcecoordinatemax(jFi;j)Fcut (e) PerformashortenergyminimizationontheauxiliarypotentialeV(x)=1=2jF(x)j2.ThecombinationofrelaxationstepsandintervalbisectionwillbringthesystemtothevicinityofxTSinfewiterations.Inmostcaseseitherx1orx2convergetoxTSwithF(x)=0.However,ingeneraltheiterationisstoppedatmax(jFi;j)FcutwithFcut=10)]TJ /F5 7.97 Tf 6.58 0 Td[(3eV=A.TheauxiliarypotentialeV(x)=1=2jF(x)j2isapositivefunctionandonlynullatthestationarypoints.Thus,minimizingeV(x)inthevicinityofxTSshouldguaranteeconvergencetothetransitionstate.Fig. 3-4 showsaillustrationofthemethodonthePEL. 33

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Insomecasetheintervalbisectionentersanewbasin.Andtheenergyminimizationleadstoalocalminimumdifferentthan1and2.TomakesurethatxTSfoundisthetransitionstateconnecting1and2insteadofanewminimumasimpletestisperformed.xTSisslightlymovedalongthetwovectors1)]TJ /F12 11.955 Tf 12.85 0 Td[(xTSand2)]TJ /F12 11.955 Tf 12.85 0 Td[(xTS.Theenergyminimizationoftheresultingcongurationshouldleadto1and2respectively.OtherwisethecongurationxTSisdeemeduntasatransitionpointbetweenthesetwominimums.TheHessianmatrixofthetransitionstatexTSshouldhaveonlyonenegativeeigenvalue.However,thecalculationofthehessianmatrixiscomputationallyexpensiveandleftforlatercalculations.TheproceduredescribedaboveisallimplementedwithintheTLSsearch.Thusbynding1,2andxTSandtheircorrespondingenergy,theasymmetry(=jU(1))]TJ /F3 11.955 Tf 12.98 0 Td[(U(1)j)andtheaveragebarrierheight(V=U(xTS))]TJ /F4 11.955 Tf -422.3 -23.91 Td[((U(1)+U(1))=2)foreachTLSisfound.AlltheTLS'sfromthedifferentinitialstartingcongurationsareputtogethertoformtheenergydistributionoftheasymmetryandbarrierheightsf()=Nandg(V). 3.6TLS'sDensityTheasymmetrydistributionf()hasthedimensionsofenergy)]TJ /F5 7.97 Tf 6.58 0 Td[(1volume)]TJ /F5 7.97 Tf 6.58 0 Td[(1,henceontopoftheasymmetrydistributionoverenergywhichisestimatedbythedynamicbisectionmethod,thetotalnumberoftheTLSspervolumeNisrequired.ThedynamicbisectionmethodimplementedabovedoesnotprovidethisinformationbecausetheMDsimulationiscontinuedfromthenalcongurationandtheinitialcongurationisnotxed.Forthat,aseparateprocedureisusedtoestimatethedensityNforeachstartingcongurationandtheaverageoverallthedifferentstartingcongurationsistaken.Afterthesampleisbroughttotherunningtemperature(Trun),thecongurationx0issavedandquenchedtothelocalminimum,0.0servesastherstminimumforallthedoublewellsystemsthatneedtobefound.Inthesecondstepthevelocitiesofx0areresettorandomnumberswithaGaussiandistributionaroundTrunandthetrajectoryof 34

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anMDsimulationiscollectedforasettimet.Afterthisthesystemisquenchedtothelocalminimum.Ifisdistinguishablefrom0accordingthemassweighteddistancedmw,intervalbisectionisusedtondthelocalminimumnthatisconnectedto0with1MDstep.Thenthecongurationofthenewminimumisstored.Ifthenewminimumisequivalentto0,eitherthevalueofMisincreasedandtheMDruniscontinuedorthevelocitiesareresetandtheprocedurerestarted.Thisprocedureisrepeateduntilallthenewminimumsarefoundatleastontheorderof10timesandnonewminimumscanbefound.AschematicofthismethodisshowninFig. 3-5 .Allthenewminimumsfoundthatareconnectedtotheinitiallocalminimum0arecomparedtoeachothertoestimatethenumberofthedistinguishabletransitionsthatthesystemcanundergo.Duetothelargenumberoflocalminimumscollectedthedirectcongurationdistancebetweeneachoneofthemisnotcalculated;Howevertheyareclassiedaccordingtofourmaincriteria: (i) Thenumberofatomsinvolvedinthetransition (ii) theasymmetry (iii) themassweighteddistancedmw (iv) Theeffectivemassp=PNi=1d2i di;max.Allthesequantitiesarecalculatedduringtheintervalbisectionmethodperformedwiththeoriginalminimum0. 3.7TransitionRateInthethermalregimetherelaxationtimeisgivenbyequationEq. 2 .Asdiscussedinchapter 2 duetotheexponentialdependenceofonthebarrierheightV,asmallrangeofVwillresultinawilddistributionoftherelaxationtime;Therefore,thedistributionoftheprefactor0canbeignoredanditsaveragevalueoveralltheTLS'scanbeused.ThiswouldallowfortheintegrationovertherelaxationtimeinEq. 2 tobereplacedbyanintegrationoverthebarrierheightleadingtoEq. 2 .Theprefactor 35

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0isthemaximumrelaxationtimeforeachTLSortheinverseoftheminimumtransitionrate.ThetransitionratefromeachwellwithintheTLSisdenedseparately,Thus0foreachTLSistakenastheminimumoftheinverseofthetworates(k10andk20)associatedwiththeTLS0=min[(k10))]TJ /F5 7.97 Tf 6.59 0 Td[(1;(k20))]TJ /F5 7.97 Tf 6.59 0 Td[(1].Fromeachwelltheminimumtransitionratek1;20isalsoknownastheattemptfrequencyorthefrequencyfromthebottomofthewellinthedirectionofthesaddlepoint.Itisdenedintermsofthenormalmodesfrequencies[ 40 ]: k1;20=QNi=11;2i QN)]TJ /F5 7.97 Tf 14.71 0 Td[(1i=1si(3)1;2iarethenormalmodefrequenciesatthebottomofthecorrespondingwellandsiarethefrequenciesatthesaddlepoint.Nisthenumberofdegreesoffreedominthesystem.Inthedenitionoftherelaxationtime,itismoreappropriatetousethefreeenergybarrier,F=V)]TJ /F3 11.955 Tf 12.05 0 Td[(TS,insteadofthepotentialenergyone[ 41 ].Thisintroducesanentropytermtothemaximumrelaxationtime: )]TJ /F5 7.97 Tf 6.59 0 Td[(10=min[(k10))]TJ /F5 7.97 Tf 6.58 0 Td[(1;(k20))]TJ /F5 7.97 Tf 6.58 0 Td[(1]eS=kB:(3)Fortwo-levelsystems,S=kBisexpectedtobeln2;However,inamorphousoxidesthetransitionsystemsmaybemorecomplexandformedfrommorethan2connectedwells.AshasbeenreportedbyTielburger[ 23 ]andconrmedinthiswork,S=kB=ln4isoftenabettertfortheexperimentaldataonsilica(seeSec. 4.2 ).ThenormalmodefrequenciesarecalculatedusingtheHessianmatrixatthesaddlepointandatthewells'bottom. i=1 cp i:(3)iarethepositivenonzeroeigenvaluesofthemassweightedHessianmatrix:H;i;j: H;i;;j=1 p mimj@2U @xi;@xj;(3) 36

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xi;alphaisthecoordinateoftheithinthedirection.AndmiisthemassoftheithatomandUisthetotalpotentialenergy.TheHessianmatrixhasN=3Ndegreesoffreedom,whereNisthenumberofatomsinthesimulationbox.Itissymmetric(@2U @xi;@xj;=@2U @xj;@xi;),thereforethereisN2+N=2independentelements.Inpractice,theHessianmatrixelementsarecalculatedastherstderivativeoftheforcescomponentsassincewithinDL-POLYtheforcesarecalculatedfromananalyticalformdirectly.Thederivativeiscalculatedbysmalldisplacementoftheatomsaroundtheirequilibriumpositions. @2U @xi;@xj;=@Fi; @xj;=Fi;(:::;xj;+;:::))]TJ /F3 11.955 Tf 11.95 0 Td[(Fi;(:::;xj;)]TJ /F3 11.955 Tf 11.95 0 Td[(;:::) 2(3)whereFi;istheforceontheithatominthedirectionandisasmallnumber.Theperiodicboundaryconditionsofthesimulationboxresultsintranslationalinvariance,whichreducesthenumberofdegreesoffreedomofthesystembythree.ThusthreeoftheeigenvaluesofthemassweightedHessianmatrixarezeroandneedtoberemovedbeforetakingtheproduct.ThecalculationoftheHessianmatrixisthemostcomputationallyexpensivepartofallthetechniquesneededtoestimatetheinternalfriction.TheprocedureofcalculatingthetheHessianmatrixanditseigenvalueswasparallelizedandimplementedinDL-POLY2.20[ 32 ],whereeachelementofthematrixcanbecalculatedonaseparateprocessorsimultaneously.Therefore,nocommunicationbetweenthecpu'sisneededandthetimerequiredscalesalmostperfectlywiththenumberofprocessors.TheeigenvaluesarecalculatedusingtheLapackroutines[ 42 ]. 3.8DeformationPotentialThedeformationpotentialisameasureofthecouplingoftheasymmetryoftheTLStothestrainappliedonthesystem.Thestraintensorisaunitlesssymmetric3by3 37

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dimensionalmatrix.Thestraintensorisdenedby: u=1 2(@u @x+@u @x)(3)whereuisthedeformationoftheboxinthedirection.Thussixindependentdeformationsareneededtoestimatethecouplingconstant.Ifthesimulationcellisdenedbythreelatticevectorfakgthenthecelldeformationsthatproducethesixindependentelementsofthestraintensorinthecartesiancoordinatesare: 1. ak1=ak1ak1;(k=1;2;3) 2. ak2=ak2ak2;(k=1;2;3) 3. ak3=ak3ak3;(k=1;2;3) 4. ak1=ak1 2ak2&ak2=ak2 2ak1;(k=1;2;3) 5. ak1=ak1 2ak3&ak3=ak3 2ak1;(k=1;2;3) 6. ak2=ak2 2ak3&ak3=ak3 2ak2;(k=1;2;3)isasmallnumber.Therstthreedeformationscorrespondtolongitudinaldeformationandthelastthreetotransversedeformations.Thenthesixelementsofthecouplingaregivenby: =1 2@ @u(3)Thusforeachdeformation,isttedtoastraightlineasfunctionofwhoseslopeis.Thecouplingconstantisverysensitivetothedeformationsandisimportantforthesystemtobeatthelocalminimumenergyasafunctionofthecellshapeandsize(whichmightnotbethesameforeachTLSduetothenitesizeofthebox)forthesystemtobeinthelinearresponseregime.ThiscriteriawasnotveryimportantintheprevioussearchesforTLS.Therefore,therestrictionontheshapeofthecell(cubicshape)isabandonedandtheenergiesofasetofdifferentstartingpointsareminimizedasafunctionofthelatticevectorcoordinatesandshortsearchesfornewTLS'sare 38

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conductedtoinsurethatthesystemisstillinthelinearregime.ThenthecouplingconstantfromalltheTLS'sareaveragedgeometricallyandthethreelongitudinalandthreetransversedirectionsareaveragedtogethertondlandt. 3.9ElasticModulusInordertocalculatetheelasticpropertiesbycalculatingtheelastictensorckl.crelatesthesixindependentelementsofthestresstensorktothesixindependentelementsofthestraintensoruk: k=Xl=1;6cklul(3)Thedeformationsusedtoreproducethestrainelementsaredescribedinsection 3.8 .Thestresstensorelementsarecalculated =1 V )]TJ /F9 11.955 Tf 11.29 11.36 Td[(Ximivi;vi;+1 2Xi;j6=i(xi;)]TJ /F3 11.955 Tf 11.95 0 Td[(xi;)fij;!(3)miisthemassofoftheithatom,vi;andxi;isthecomponentofthevelocityandpositionofthetheithatomrespectively,andfij;isthecomponentoftheinteractionforcebetweentheithandjthatoms.Twodifferentapproximationscanbeusedtocalculatethebulkmodulus(B)andtheshearmodulus(G)directlyfromtheelastictensorelements:theVoigtapproximation[ 43 ]andtheReussapproximation[ 44 ].Heretheaverageofthetwomethodsisused. BR=1 (s11+s22+s33)+2(s12+s13+s23) (3a)GR=15 4(s11+s22+s33))]TJ /F4 11.955 Tf 11.95 0 Td[(4(s12+s13+s23)+3(s44+s55+s66) (3b)BV=1 9(c11+c22+c33)+2 9(c12+c13+c23) (3c)GV=1 15(c11+c22+c33)]TJ /F3 11.955 Tf 11.96 0 Td[(c12)]TJ /F3 11.955 Tf 11.96 0 Td[(c13)]TJ /F3 11.955 Tf 11.95 0 Td[(c23)+1 5(c44+c55+c66) (3d)BRandGRarethebulkandtheshearmodulicalculatedusingtheReussapproximationrespectivelyandBVandGVaretheVoigtapproximationmoduli.sklaretheelementsof 39

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inverseoftheelastictensor.Theaveragemoduliare G=1 2(GR+GV) (3a)B=1 2(GR+GV) (3b)TheaverageYoungsmodulusYisgivenby: Y=9BG 3B+G(3) 40

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Figure3-4. Aschematicrepresentingthenon-localridgemethod.TheblacklinerepresentstheMDtrajectoryandthereddashedlineistheridgebetweenthetwobasins.Themethodusesasuccessionofenergyminimizationstepsandintervalbisectiontondthetransitionstatebetweentwolocalminimums. 41

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Figure3-5. AschematicillustratingthealgorithmfortheTLSdensitysearch.InthismethodtheMDrunisrestartedfromtheoriginalcongurationsaftereachtransitionisfound.Thusallthepossibletransitionsfromthisoriginalcongurationscanbefound. 42

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CHAPTER4INTERNALFRICTIONOFHOMOGENOUSANDHETEROGENOUSOXIDESTheTLS'sobservedmayinvolveaslowas5atomsandashighas150atoms.Fig. 4-1 andFig. 4-2 illustratetheatomiccongurationoftwolocalminimainvolvedintwotypicalTLS'sofsilicaandtantalarespectively.Theatomsdisplacedinthetransitionareshowninbrightcolorsandtheonesthatdon'tsignicantlymovefromoneminimumtotheotherareingray.Thetypicaltransitioninthesesystemsmayinvolvearotationofoneortwotetrahedralstructureoraipofabondfromonetetrahedrontoanother.Inadditiontothesemajormovementsthereisarepeleffectwhereanextendednumberofatomthatadjusttheirpositionsinthenewminimum. Figure4-1. Thecongurationofthelocalminimumsof2differentTLS'sofsilica.TherstTLS(panelsA,BandC)involves21atomsandthesecond(panelsD,EandF)involve59atoms.Theatomsingrayaretheatomsnotinvolvedinthetransitionwhiletheatomsparticipatingarecolored.A)representsthetherstpotentialenergyminimumoftherstTLSandC)representsthesecondminimum.inB)the2minimumsareimposedontopofeachothertohighlightthedifference.D),E)andF)arethesameforthesecondTLS 43

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Inthefollowing,wepresenttheasymmetry(f())andbarrierheight(g(V))distributionsofamorphoussilicaandsilicadopedwithoxidesofgroup4:puresilica(SiO2),hafnia-dopedsilica(Hf0:25Si0:75O2andHf0:50Si0:50O2),zirconia-dopedsilica(Zr0:25Si0:75O2andZr0:50Si0:50O2),puretantalaandtantaladopedwithtitaniamodeledwithaweakandstrongbondingpotential:Ta2O5,(Ta2O5)0:398(TiweakO2)0:204,(Ta2O5)0:231(TiweakO2)0:538,(Ta2O5)0:398(TistrongO2)0:204and(Ta2O5)0:231(TistrongO2)0:538.Afterdiscussingthesedistributions,wepresentthemechanicallossofthesesystemsandstudytheeffectofdopingonthedifferentelementsofEq. 2 .Whereasthemechanicallossinsilicaisextensivelystudiedexperimentallyandtheoreticallyandwillprovideagoodconrmationfortheaccuracyofournumericalmethods,experimentsandcalculationsonthedopedsystemsarequiterareandourcalculationswillprovideimportantbenchmarksforfuturestudies. 4.1BarrierHeightandAsymmetryDistributionsAsmentionedinsection 2.3 ,inordertottoexperimentaldatatwosimplicationsaremadetotheasymmetryandbarrierheightdistributionsinordertondananalyticalsolutionfortheintegralinEq. 2 .First,f()isassumedtobeuniformf()=f0.Second,thatdistributiong(V)followsanexponentialdecayg(V)=1 V0exp()]TJ /F3 11.955 Tf 9.3 0 Td[(V=V0)whereV0istheactivationenergy.Usingthesetwoassumptions,Eq. 2 simpliesintoEq. 2 .TheTLS'sdistributionsareestimatedinexperimentallyasf0g(V)byttingthemeasuredinternalfrictionusingthesimpliedformofQ)]TJ /F5 7.97 Tf 6.59 0 Td[(1inEq. 2 .However,thetheoreticalfoundationfortheseassumptionsaboutf()andg(V)thatleadtothisequationarenotfullyjustiedorexperimentallyveried.Numerically,thedistributionsarecalculateddirectlybyfollowingthesystemduringtransitionsattheatomiclevelusingthemethodsdescribedin.Thus,thenumericallycalculateddistributionsprovideamethodtotesttheassumptionsmadeinthederivationofEq. 2 44

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Theasymmetrydistributionisrstcalculatedasafunctionofenergy,f(V),usingthebisectionmethoddescribedabove,startingfrom5to10differentcongurationalsamples.ThisdistributionisnormalizedtothetotalTLS'sdensityincludingasymmetriesbelowa0.1eVenergycutoff.ThiscutoffisjustiedbytherapiddecayoftheintegrandinEq. 2 asexceeds2kBT.Fig. 4-3 andFig. 4-4 representthenormalizedasymmetrydistributionsf()=Nforsilicadopedmaterialsandtantaladopedmaterialsrespectively.Thenormalizedasymmetrydistributionsarenotuniformovertheintegrationrangeoftheasymmetry.However,theTLS'swithsmallerasymmetryaremoreabundantthanthosewithlargerasymmetry.TheearlytheoreticaljusticationforauniformdistributionisgivenbyPhilips[ 1 ]onlyforthetunnelingregimebelow1K.Philipsarguesbasedonexperimentaldataoftheinteractionofthetunnelingstateswithstrainthatthedistributionf()canbedescribedbyawidenormaldistributioncenteredatzero.Therefore,dataclosetothecenterofthedistributioncanbeapproximatedasaconstant.Later,thisapproximationisextendedtothethermalregimebelowtheglasstransitiontemperaturebutwithoutjustication.[ 27 ]Thenumericallycalculateddistributionsuggeststhatthisassumptionisnotvalid.AlthoughonecouldarguethatthesearchmethodusedheremightfavorlowerasymmetryTLS's,thecalculationshavebeenrepeatedatdifferentrunningtemperaturestoensureconvergenceanddifferentcriteriaweresettoeliminateanyrepeatedTLS'sasdiscussedinsection 3.6 .Thebarrierheightdistributiong(V)ofthedifferentsystemsareshowninFig. 4-5 andFig. 4-6 .Inconstructingg(V),onlyTLS'swithlessthanthe0.1eVasymmetrycutoffareconsidered.Theoretically,g(V)isassumedindependentoftheobservationtemperatureandthusoftheasymmetrycutoff.Fromthelogarithmicscale,itisclearthedecayofthebarrierheightdistributiondoesnotfollowtheexactexponentialdecayproposedandthedecayoflog(g(V))changesslopeatoneortwopoints.Keilet.al.foundthatamodiedGaussianoftheformVe)]TJ /F7 7.97 Tf 6.59 0 Td[(V2=V20isabettertforthebarrier 45

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distributionwith=0:27foramorphousSiO2[ 45 ].Nonetheless,theexponentialdecaystillprovidesareasonableapproximatetforg(V).TheTLSdensity,N,belowtheasymmetrycutoffof0:1eVisgiveninTable 4-1 (Nisreportedperparticle).Nforsilicaisfoundtobe2:6TLS/1000particlewhileforpuretantalatheTLS'sdensityis1:3TLS/1000particle.TheTLSdensityofsilicaisslightlyaffectedbydoping,aszirconia-dopedsilicahasahigherdensityofTLS'sandhafniadopingslightlyreducesthenumberofTLS.Silica,zirconiaandhafniahavethesametetrahedralstructureandcoordinationnumber,hencelargechangesinthenumberoftheTLS'sshouldnotbeexpected.ThemajorcontributortotheTLSinsilicaaretherotationsandelongationofSi-ObondsaswellastherotationorshiftoftheSiO4tetrahedral[ 6 21 ].However,shiftsintheatomicpositionareextendedandeachtransitionmightinvolvebetween10and100atoms(Fig. 4-1 andFig. 4-2 ).Thesetransitionsshouldstillbeexpectedinthedopedsystems.ThedopingoftantalawithtitaniaalsoreducesthetotalnumberofTLS's;However,theeffectofthestronglycoordinatedtitaniaishigherthanthatoftheweaklycoordinated.TheweaklycoordinatedtitaniahasthesamecoordinationnumberastantalaandthusitseffectonthenumberofTLS'sisnotbig.Whereas,thestronglybondingtitaniahasacoordinationof6,thuslargerdisturbanceisexpectedinthestructure.Thedopingalsocausesashiftinthebarrierheightpeaks(Table 4-1 ),buttheeffectisnotlinear.Itisclearthatdopedsystemshavealowerbarrierheightdistributionmaximumthanpuresilica.Thiscouldbeduetodisruptionofthetetrahedralstructureresultingfromthedifferenceinbondlengthbetweenthecationsandoxygenthatallowsforsmallerenergybarriers. 4.2RelaxationTimeIfEq. 3 isusedforcalculatingthemaximumrelaxationtimewithoutincludingtheentropyfactor,thevalueof0forpuresilicais1:35ps,whichislargerthanthetheexperimentalvalueof0:18[ 24 ]to0:2ps[ 25 ].ThissuggeststhatEq. 3 ismore 46

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Figure4-2. Thecongurationofthelocalminimumsof2differentTLS'softantala.TherstTLS(panelsA,BandC)involves11atomsandthesecond(panelsD,EandF)involve39atoms.Theatomsingrayaretheatomsnotinvolvedinthetransitionwhiletheatomsparticipatingarecolored.A)representsthetherstpotentialenergyminimumoftherstTLSandC)representsthesecondminimum.inB)the2minimumsareimposedontopofeachothertohighlightthedifference.D),E)andF)arethesameforthesecondTLS. Table4-1. Thesetofcalculatedparametersnecessaryfortheinternalfrictionofpureanddopedsilica. f00tlY[ 38 ]TLS/100particlepseVeVGpa SiO20.260.341.322.2896Si0:75Hf0:25O20.230.351.402.47119Si0:50Hf0:50O20.240.411.152.04138Si0:75Zr0:25O20.280.361.462.69114Si0:50Zr0:50O20.350.371.361.86138 47

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Figure4-3. Thenormalizedasymmetrydistributionf()=Nofthepuresilicaandsilicadopedwith50%zirconiaandhafnia.Onlythe50%isshownsincetheotherpercentagesshowthesamepattern. Table4-2. Thesetofcalculatedparametersnecessaryfortheinternalfrictionofpureanddopedtantala. f00tlY[ 38 ]TLS/1000particlepseVeVGpa Ta2O21.90.521.82.4145(Ta2O5)0:398(TiweakO2)0:2041.50.451.42.2139(Ta2O5)0:231(TiweakO2)0:5382.00.441.32.2121(Ta2O5)0:398(TistrongO2)0:2041.70.731.72.6143(Ta2O5)0:231(TistrongO2)0:5381.20.291.42.4163 48

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appropriateforthecalculation)]TJ /F5 7.97 Tf 6.58 0 Td[(10,whichincludesanentropy-dependentterm.UsingS=KB=ln4[ 23 ].ThisvalueofS=kBsuggestthatparticlesthatformthemovingunitineachtransitionhaveonaveragefourstablelocalminimathataredirectlyconnectedbyarstordersaddlepoint.Thisisareasonabledeductionfortheconnectedtetrahedronsstructureoftheseoxides.UsingS=KB=ln4forsilica0=0:34ps,thisisamuchbettercomparisonwithexperiments.Thecollectedtransitionsindicatethatmorethanoneminimumareconnectedbymovementcenteredaroundthesamesetofatoms.Theexactnumberoftheseconnectedpotentialwellsisbeyondthecurrentsearchmethod.Therefore,thesamevalueofS=kBisusedforallofthemixedoxidesandtheresultsarepresentedinTable 4-1 andTable 4-2 4.3DeformationPotentialThetransverseandlongitudinaldeformationpotentialforsilicaare1:32eVand2:28eVrespectively.Theexperimentalvalueoftis0:9)]TJ /F4 11.955 Tf 12.55 0 Td[(1:1eV[ 26 ]with2l=2t=2:5[ 17 ].Thecalculatedratiofromourcalculationsis2l=2t=3:0.Theeffectofmixingonthedeformationpotentialisnotdirect,inthat25%ofzirconiaandhafniaincreasethedeformationpotentialofsilica,forthe50%dopedsystemsislessthanforpuresilica.Althoughthereislittleexperimentalevidenceforcomparison,previousstudieshavefoundtfor(ZrO2)0:89(CaO)0:11tobe1eV,thesameaspuresilica[ 16 ].Dopingalsoaffectstheratio2l=2t,for50%ZrO2dopedSiO2theratioisthelowestat1.9.Thecalculateddeformationpotentialofpuretantalais1:8eVand2:4eVforthetransverseandlongitudinalpolarizationrespectively.Theratioof2l=2tis1.8lowerthanthevaluepredictedbyexperimentforsilicaandotheramorphousmaterials.Thedopingoftantalawithtitaniaresultinthereductioninthedeformationpotentialforbothpolarizations.Theratio2l=2tis2:46and2:86forthe20:4%andthe53:8%weaklycoordinatedtitaniadopingand2:33and2:94forthe20:4%andthe53:8%stronglycoordinatedtitaniadoping. 49

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4.4InternalFrictionInthefollowingcalculations,themechanicallossiscalculatednumericallyfromEq. 2 usingtheYoung'smodulusandtheaverageofthelongitudinalandthetransversecouplingconstants.TheinternalfrictionofsilicaasafunctionoftemperatureisshowninFig. 4-7 .Acomputedat100Hz,10kHz,and1MHz.Fig. 4-7 .Brepresentstheinternalfrictionofsilicaandpureandmixedwithzirconiaandhanaat100Hz.Fig. 4-8 .Arepresentstheinternalfrictionofamorphoustantalacomputedat100Hz,10kHzand1MHz.AndthatofdopedtantalawithboththeweakandthestrongcoordinatedtitaniaareshowninFig. 4-8 .B.Inordertocomparewithexperimenttheinverseofthetemperatureatwhichtheinternalfrictionismaximum1=Tpeakisplottedversusthelogarithmofthemeasuringfrequency.IfthetheoreticalassumptiondiscussedinSec. 2.3 abouttheshapeofasymmetryandbarrierheightdistributionsareused,thefollowintegralinEq. 2 simplifytoEq. 2 .FromthederivativeofEq. 2 ,ln(!)islinearfunctionof1=Tpeakwithaslope)]TJ /F3 11.955 Tf 9.29 0 Td[(V0theactivationenergyandinterceptln(0)(seeEq. 2 ),andthethemaximumoftheinternalfrictionQ)]TJ /F5 7.97 Tf 6.59 0 Td[(1maxisalinearasfunctionofTpeak.Fig. 4-9 .Arepresentsln(!)versus1=TpeakforamorphoussilicaascalculatedfromEq. 2 incomparisonwithvaluescollectedfromdifferentexperimentalwork.Thelineartofoftheexperimentaldata,V0=540KandthatofthecomputeddataisV0=580K.Thus,thismethodprovidesamoreprecisedenitionoftheactivationenergythanthedirectttingofg(V).Theinterceptofthislineisproportionaltothe)]TJ /F3 11.955 Tf 9.29 0 Td[(ln(0),whichisoverestimatedinourcalculations,andexplainsthesmalldiscrepancybetweenthenumericalandexperimentaldata.InFig. 4-9 .B,themaximuminternalfrictionofsilicaQ)]TJ /F5 7.97 Tf 6.58 0 Td[(1maxatdifferentfrequenciesisplottedasafunctionofTpeak.Thecalculatedinternalfrictionislargerthantheexperimentalvaluesduetotheoverestimationofthedeformationpotential.Moreover,thenumericaldatadonotttoastraightlineaspredictedinEq. 2 .Thissuggests 50

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thattheapproximationsresultinginEq. 2 arenotvalid.Similaranalysiscouldnotbemadewiththeexperimentaldataduetothelackofdataatintermediatefrequencies.Thisistherstndingthatdemonstratesthattheexponentialtforg(V)maynotbeaccurate,andindicatesthenecessityoffurthercomputationalstudytounderstandthedetailsofbarrierdistributionsinotheramorphoussystems.Fig. 4-10 representstheln(!)asfunctionof1=Tpeakwiththelinearttingforthetantalaandtantaladopedwith20:4%ofweaklyandstronglycoordinatedtitania.Fromtheslopeofthelinearttingofthetantaladatatheactivationenergyis580K.ExperimentsbyMartinet.al.[ 11 ]omion-beamsputteringthinlmoftantalaanddopedtantalafoundanactivationenergyforthepuretantalaof330Kandamaximumrelaxation5:9ps.Thedifferencebetweentheexperimentalvaluesandtheonescalculatedheremaybeaduetotheforceeldusedtorepresentthematerialortothesamplepreparationmethodandeffectofsurfacesorduetoafailinginthetheoryitself.Thisisdiscussedfurtherinchapter 5 .Theexperimentalvalueoftheactivationenergyoftantaladopedwithtitania(14:5%Tication)thinlmisreportedas510K[ 46 ]and460K[ 11 ],whilethecalculatedvaluesare616and690fortheweaklyandstronglycoordinated20:4%dopingrespectively.Fig. 4-7 .Brepresentstheinternalfrictionofthedopedsilicasamplescomputedat100Hzincomparisonwiththepuresilicainternalfrictioncalculatedatthesametemperature.Theinternalfrictionofsilicadopedwith25%zirconiaandhafniaishigherthanthatofpuresilica,whereas50%dopingwitheitheroxideresultsinthereductionofQ)]TJ /F5 7.97 Tf 6.59 0 Td[(1;However,inallthedopedsystemsQ)]TJ /F5 7.97 Tf 6.58 0 Td[(1ofthedecreasesfasterathightemperaturethanpuresilica.Theparametersthatcontributetothevalueofthemechanicallossincludethedeformationpotential,theTLSdensity,theYoung'smodulusandtheactivationenergyV0whichdescribesthedecayofthebarrierheightdistribution.Theeffectoftitaniadopingontantalaat1000HzisshowninFig. 4-8 .Ingeneraldopingbytitaniareducesthemagnitudeoftheinternalfrictionoftantala.Thiswas 51

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observedexperimentally.However,againaswithsilicadopingtheeffectisnotstraightforward,whilethe20:4%dopingwiththestronglycoordinatedtitaniaincreasestheinternalfriction,theincreaseddopingwiththesamecoordinationresultsinreducedQ)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Andtheweaklycoordinatedtitaniadecreasestheinternalfrictionforbothlevelofdoping.ThedopingoftantalawiththestronglycoordinatedtitaniaincreasesitsYoung'smodulus,howevertheweaklycoordinatedtitaniareducedtheYoung'smodulusoftantala.TheotherfactorthatstronglycontributetothemagnitudeoftheinternalfrictionistheTLS'sdensity.ThestronglycoordinateddopanthaveastrongereffectontheTLS'sdensitythantheweaklycoordinated.Thevastnumberofdependenciesmakeithardtopredictpatternsintheeffectofdopingontheinternalfriction.Thisistherststudytoourknowledgethathasdeterminedtheinternalfrictionforvariousdopinglevelsinamorphousoxides.Thendingsindicatethatpreciselevelsofdopingcanhaveverydifferenteffectsonthermalnoiseinamaterial,emphasizingtheimportanceoffurtherdopingstudiestodetermineoptimallevelsfornoisereduction. 52

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Figure4-4. Thenormalizedasymmetrydistributionf()=Nofthepuretantalaandtantaladopedtitaniawiththeweakandstrongpotentials. 53

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Figure4-5. Thebarrierheightdistributiong(V)ofthepuresilicaandsilicadopedwithzirconiaandhafnia.Therawdistributionsareshownasbluebarsandthelogarithmofg(V)arerepresentedasgreenlines.Thesolidredlinesrepresentthetofthedecayofg(V)toEq. 2 54

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Figure4-6. Thebarrierheightdistributiong(V)ofthepuretantalaandtantaladopedtitaniawiththeweakandstrongpotential.Therawdistributionsareshownasbluebarsandthelogarithmofg(V)arerepresentedasgreenlines.Thesolidredlinesrepresentthetofthedecayofg(V)toEq. 2 55

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Figure4-7. Theinternalfrictionofsilicaanddopedsilica.PanelAistheinternalfrictionofsilicacomputedat3differentfrequencies102,104,and106Hz.PanelBistheinternalfrictionofsilicadopedwithzirconiaandhanacomputedat100Hz Figure4-8. Theinternalfrictionoftantalaanddopedtantala.PanelAistheinternalfrictionofpuretantalacomputedat3differentfrequencies102,104,and106Hz.PanelBistheinternalfrictionoftantaladopedwithtitaniacomputedat1000Hz 56

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Figure4-9. Comparisonbetweenexperimentaldataofsilicaandthecomputationaldatafromthiswork.theexperimentaldataiscollectedfromvariousreferences[ 18 23 ].PanelA.representsthelogarithmofthemeasuringfrequencyfunctionoftheinverseofthepeaktemperatureoftheinternalfriction.PanelB.showsthepeakoftheinternalfrictionasfunctionofthetemperatureatthepeak.Theexperimentaldataareshownasbluecircles,thesolidlinesrepresentthelinearttingoftheexperimentaldata,theblacksquaresarethenumericaldatawiththelinearttingshownasgreendashedlines. Figure4-10. Theactivationenergyofpureanddopedtantala.Thelogarithmofthemeasuringfrequencyfunctionoftheinverseofthepeaktemperatureoftheinternalfrictionofpuretantalaandfortantala20:4%cationdopedwiththestronglyandtheweaklycoordinatedtitania. 57

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CHAPTER5CONCLUSIONInthisthesisasetofnumericalmethodswereemployedwithinclassicalmoleculardynamicssimulationsintheDL POLYpackagetoconstructthedistributionoftwolevelsystems(TLS's)inamorphousoxidesandotherparametersimportanttocalculatetheinternalfrictionsorthemechanicallossinthesematerials.Thecalculationsoftheinternalfrictionarecarriedoutforpuresilicaandpuretantalaaswellaszirconia-andhafnia-dopedsilicaandtitania-dopedtantala.Thechoiceofmaterialsstudied(thedopedandpuresilicaandtantala)wasinspiredbythecombinedtheoreticalandexperimentaleffortstoenhancethesensitivityofthelasergravitationalwavedetectors(LIGO)byreducingthethermalnoiseassociatedwiththemirrorcoatingsofthedetector.ThecurrentdetectorsuseslayersofSiO2andTa2O5fortheircoatingthustheeffortstondtherightdopantthatreducesthethermalnoiseofthesematerialthatisassociatedbythemechanicallossofacousticwavesinthesematerials.Inadditiontothecalculationofmechanicallossandotherlowtemperatureproperties,Thedistributionsofthetwolevelsystemsinamorphousmaterialsandthemechanismofthetransitionsbetweenthemgivesagreatinsighttothenatureofthesematerialsandthedynamicsbelowtheglasstransitiontemperature.Firstthingtonoteisthenumberofatomsinvolvedineachtransition.Duetothetetrahedralstructureoftheseoxides,itwasassumedthatthetransitionsarelocalizedtoonetetrahedronincludingoneorfewatoms,whereasthetransitionsfoundnumericallyarenotrestrictedtoonebondoronetetrahedron,butincludeseveralconnectedtetrahedronsandspreadoutovermanyatomsfurtherwithsmallshiftsintheequilibriumpositionsofseveralatomstoaccommodateforthecentralstructurechange.Atypicaltransitionmayengagebetween10and150atoms,inarangeofupto10A.Thecenterofthetransitionmainlyinvolvearotationofthetetrahedralstructurearoundacertainbond,aippingofanoxygenbondfromonestructuretotheotheroracollectiveshiftinconnectedstructures. 58

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Afurthercategorizationofthesetransitionsmightbeinterestinginmanylevelsbutwashotsubjectofthecurrentwork.(SeeFig. 4-1 andFig. 4-2 )Twooftheimportantfactorsthatcontributetotheinternalfrictionare:1)thedeformationpotentialorcouplingoftheTLStotheappliedstrain(),and2)theaveragemaximumrelaxationtime().Theestimatedvalueofthetransversedeformationpotential,t,and0ofsilicaare1:32eVand0:34psrespectivelyandtheirexperimentalvaluesare1eVand0:18to0:2ps.Theforceeldsthatareusedinthisworkarebuilttoreproducetheelasticpropertiesoftheoxides,whichareanimportantfactorforaccurateQ)]TJ /F5 7.97 Tf 6.59 0 Td[(1calculations.However,theforceeldsarenotexpectedtoexactlyreproducetheexperimentalvaluesofand0,whichrequirettingtosecondderivativesoftheHamiltonian.Nonetheless,thesecalculationswouldprovideagreatcomparisonbetweentheinternalfrictionofdifferentmaterials.Fittingtheforceeldstoexperimentalvaluesofand0isaverychallengingtask,sincethesequantitiesareaveragedquantitiesthatcanonlybeestimatedafterndingtheTLSdistributionofaverylargeamorphoussampleoracrossmultiplesamples.Theinternalfrictioncalculationsinthisworkcorrespondtobulksystems.Theshapeofthecalculatedinternalfrictionofsilica,thetemperatureofthepeakandtheactivationenergycomparewellwithexperimentaldata.CalculatedactivationenergyisV0=580KandtheexperimentalvalueisV0=540K.However;Martinet.al.[ 24 ]measuredtheinternalfrictionofion-sputteredthinlmofsilicaandreportedanactivationenergyof370K.Theyacknowledgethatthisvalueoftheactivationenergyislowerthanwhatmeasuredforbulksilica.Similarlythecalculatedactivationenergyoftantalaanddopedtantalainthisworkislargerthanthatoftheactivationenergymeasuredfortheion-sputteredthinlmofthesamematerial[ 11 46 ].Therefore;thesurfaceeffectandthesamplepreparationsmethodseemstobeimportant.Pennet.al.[ 47 ]measuredtheinternalfrictionforarodoffusedsilicatobeontheorderof10)]TJ /F5 7.97 Tf 6.59 0 Td[(7atroomtemperature. 59

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Thisisalsoisnotobservedforthebulksilicainthesecalculation.Hence,furthercalculationswheresurfacesareincludedarethenextstepinthisproject.Otherimportantobservationsfromexperimentaldataare:First,themagnitudeoftheinternalfrictionatthepeakisnotamonotonicfunctionofthemeasuringfrequency[ 11 24 ].ThisisincontradictiontowhatisexpectedfromEq. 2 .Second,forthesilicathinlm,anotherpeakintheinternalfrictionathighertemperatureisobserved[ 24 ].Eventhoughthecalculationsheredonotincludesilicalms,fromEq. 2 asecondpeakinQ)]TJ /F5 7.97 Tf 6.58 0 Td[(1isnotexpected.ThirdthemeasuredQ)]TJ /F5 7.97 Tf 6.59 0 Td[(1ofthinlmtantalashowsaplateformwithconstantvalueathightemperatureabove100K.Thecalculatedinternalfrictiondecreasesmonotonicallyathighertemperature.Whetherthesediscrepanciesareduetothesurfaceandinterfaceeffectintheexperimentorafundamentalprobleminthemodelisnotfullyclear.However;itisimportanttoexploresomepossiblemodicationsinthetheorylikedifferentrelaxationproceduresoratemperaturedependenceofthebarrierheightdistributions.Thenumericalresultsprovideaninsightonthevalidityofthetheoreticalassumptionsusedinthettingoftheexperimentalvalues,especiallytheshapeofthedistributions.Therstpointworthemphasizingisthatf()isnotuniformovertheenergyrangeanddecreaseswithincreasing.Second,thebarrierdistributiong(V)doesnotttoaperfectexponentialdecay,howeverthenotionoftheactivationenergyisstillusefulandcanbecalculatedfromthelinearttingofexperimentalornumericaldatatoEq. 2 (Fig. 4-9 .A).Theeffectoftheseimperfectassumptionsareshown(Fig. 4-9 .B)wherethenumericalvaluesofQ)]TJ /F5 7.97 Tf 6.58 0 Td[(1maxdonotttoEq. 2 .Theinuenceofdopingontheinternalfrictionofsilicaisnotstraightforward.Whileasmallamountofdopingmightincreaseacertainfactor,suchasthedeformationpotential,increasingtheamountofdopingmightreversethiseffect.TheTLSdensity,deformationpotential,andtheYoung'smodulusarethecontributingfactorstothechangesinQ)]TJ /F5 7.97 Tf 6.59 0 Td[(1ofthedopingsystems.WhereastheYoung'smodulusincreaseswith 60

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dopingwithzirconiaandhafnia,andNarelesspredictable,astheychangenonmonotonicallydependingonthepercentageofdoping.Therefore,inordertoreduceinternalfriction,oneshouldbalancetheeffectsofthedecreaseofthedeformationpotentialwiththeeffectoftheincreaseoftheTLSdensity.Hence,thenumericalcalculationsprovidecrucialinformationaboutpropertiesofdopedamorphousoxidestoassistexperimentalstudiesinpredictingnewandbettermaterials. 61

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CHAPTER6STRUCTUREANDFUNCTIONALITYOFBROMINEDOPEDGRAPHITE 6.1IntroductionGraphiteintercalationcompounds(GIC)areformedbyinsertingdifferentchemicalspecies(atomicormolecular),calledtheintercalant,betweenlayersofthegraphitehost.yTheintercalationprocessisfacilitatedbythehighlyanisotropiclayeredstructureofgraphite[ 48 ].TheperiodicoccurrenceoftheintercalatelayersinbetweenthegraphitelayergivesrisetoanimportantorderingpropertyoftheGIC'sknownasthestagingphenomenon;hence,theGIC'scanbeclassiedusingastagingindexthatrepresentsthenumberofgraphitelayersbetweensuccessiveintercalatelayers[ 48 ].Therstpaperaboutgraphiteintercalationcompoundsdatesbackto1841whenC.SchafhautlreportedaboutH2SO4-GIC's[ 49 ].Morethan100reagentscanbeintercalatedintographite,rangingfromalkalimetalstohalogenmixturesandmetalhalidestooxyhalidesandoxides.GIC'shaveunusualproperties[ 48 ],andawiderangeofapplications,suchashighlyconductivematerials,batteryelectrodes,catalystsfororganicsynthesis,andagentsforstorageandseparationofhydrogenisotopes.[ 49 ].Graphitehasalowdensityofchargecarriers,ontheorderof10)]TJ /F5 7.97 Tf 6.59 0 Td[(4percarbonatom,withequalnumbersofelectronsandholes.Therefore,asmallchargeexchangewiththeintercalantwillresultinsignicantincreaseinthefreecarrierdensity[ 48 ].Undercertainconditions,graphiteintercalatedwithacceptorcompoundslikethestrongacidicuorides(AsF5andSbF5)werereportedtohavehighconductivityatroomtemperature[ 50 ].In-planesuperconductivitywasrstreportedbyHannayetal.[ 51 ]inrststagealkaliintercalantcompoundsC8M(M:K,Rb,Cs)[ 51 52 ]withtransitiontemperaturesupto0:55KforC8K.InthemetastablehighconcentrationLi-intercalantC2Li,the yReproducedwithpermissionfromJ.Chem.Phys.138,164702(2013).Copyright2013,AIPPublishingLLC. http://dx.doi.org/10.1063/1.4801786 62

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superconductingtransitiontemperatureis1:9K[ 53 ].MostrecentlysuperconductivitywasreportedinC6YbandC6Cawithtransitiontemperaturesof6:5Kand11:5K,respectively[ 54 ].Thephysicalmechanismunderlyingthechangesinconductivityuponintercalationisthechargetransferbetweengraphiteandtheintercalant,whichisthesubjectofmanyexperimentalandtheoreticalstudies.Dependingontheintercalate,changetransfercanoccurinbothdirections.DiCenzoetal.[ 55 ],usingX-RayPhotoelectronSpectroscopy,foundthattheC-1sspectrarevealsanelectronicchargedistributionwhichisstronglylocalizedonthecarbonatomsofC24Kthatarenearesttotheintercalantions.Holzwarthetal.[ 56 ]foundsimilarconclusionsforC6Li,C12LiandC18LiusedasmodelsforGIC'sinrst-principleselectronicstructurecalculations.Howeverinaself-consistenteffectivemasstheoryD.P.DiVincenzoetal.[ 57 ]predictedthatforGIC'swithdensitybetween1 6and1 12ofintercalantperCatom,thechargetransferredisquitehomogeneouslydistributedwithsmallenhancementofchargeneartheintercalantsandasmalldepletionawayfromthem.Thestudyofbromineintercalatedgraphitedatesbackto1979,whenRosenmanetal.[ 58 ]studiedquantummagnetothermaloscillationsandexplainedtheresultsviaamodulatedelectronicstructurecomposedofanalternatesequenceoftwozones;onezoneispuregraphite,theotherisatwo-dimensionalmetallicsandwichmadeofthebrominelayerandthetwoadjacentgraphitelayers.Tongayetal.[ 59 ]foundthatexposureofhighlyorientedpyrolyticgraphitetobrominegasresultsinamonotonicincreaseofthein-planeconductivitywiththeintercalationtime.Forintercalantdensityof6%thein-planeconductivityreachesvaluesthataresignicantlyhigherthanthatofcopperoverawidetemperaturerange(300K>T>1:7K).HallandX-RayphotoelectronspectroscopymeasurementsshowthatBrisanelectronacceptordopant;thusholedopingthegraphenesheets.Theincreaseofthein-planeconductivity 63

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isaccompaniedbyanincreaseoftheinter-layerseparationandadecreaseoftheout-of-planeconductivity.Itshouldbementionedthatsincetheexperimentaldiscoveryofgraphenein2004,andwiththefascinationwithitsexceptionallyhighcrystalandelectronicquality,GIC'sareusedasintermediatesfortheexfoliationofgraphitetoformgrapheneinsolutions[ 60 62 ],wheretheintercalantpushesthegraphitelayersapartfromeachotherandsignicantlyreducestheinter-layerbindingenergy[ 60 ].Boththeoreticalandcomputationalcommunitiesshowgreatinterestinsingle-layerandfew-layergraphenesystemsbecauseoftheirexoticelectronicstructure.Recently,theadsorptionofhalogendiatomicmoleculesongraphenewasstudiedusingdensityfunctionaltheory(DFT)withtheself-consistentinclusionofvanderWaalsinteractions.Theorientationofthemoleculesarefoundtobeparalleltothegraphenelayerandthechargetransferleadstotheformationofaconductingbandwiththeincreaseoftheimpurityconcentration[ 63 ].OtherDFTinvestigationsofBr2intercalatedgraphitefoundthatintercalationreducestheinterlayerbindingenergyinC32Br2toalmostonetenthofitsvalueinpuregraphite.Thechargetransferfromthebrominemoleculetothecarbonatomsisreportedtorangefrom0.01to0.08electronsfordifferentintercalationconcentration[ 60 ].ThechargetransferfromgraphenetoasingleBratomisfoundtobemuchmoresubstantialcomparedwiththatofaBr2molecule[ 58 ].Inthisstudy,weuserstprinciplesnumericalsimulationstoinvestigatethepropertiesoflow-andhigh-concentrationbrominedopedgraphite.Whilerecenttheoreticalstudiesonbrominedopedsystems[ 63 65 ]havebeenfocusedonpropertiesofgrapheneorultra-thingraphitelmswithC32Br2[ 64 ]andgraphitewith(C18)n(Br2)m,(n=1,m=1,2)[ 65 ],theemphasisofourworkistomodelbulkGICatmuchlowerconcentrations(suchasC196Br2)andtorelatethecalculationstoexperimentalobservationsinsuchbulksystems.WeexaminethephysicalconditionsforwhichatomicBrbecomesenergeticallyfavoredovermolecularBr2betweengraphenesheets. 64

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WecalculatethechargetransferbetweencarbonatomsandBratomsandthebandstructureofthedopedsystem.Effectsofdisorderareinvestigatedandthebandmodications(bothin-planeandalongthec-axis)areusedtoexplaintheexperimentallyobservedchangeinconductivity.Therestofthispaperisorganizedasfollows:SectionIIdiscussestheoreticalmodelsandcomputationaldetails,SectionIIIpresentsresultsandcalculations,andSectionIVconcludesourinvestigation. Figure6-1. Representationofthesimulationsupercells:(A)Twosupercellsstagetwointercalationofbromineingraphite.EachcellcontainstwolayersofgrapheneandtwoBratomsplacedalternatelybetweenthelayers.C196Br2.(B)TwosupercellsofstageoneintercalationC196Br9. 6.2MethodandComputationalDetailsLowconcentrationBromineintercalationismodeledbyasupercellmadeof77ABAB-graphiteunitcellsrepeatedinthexy-planewithstagetwointercalationoftwobromineatoms;i.e.196carbonatomsarrangedintwographenesheetswithtwobromineatomsinbetween(C196Br2).ThetwosheetsbetweenwhichtheBratomsareplaced,arepushedawayfromeachotheralongthec-axistoadistanced,whilethedistancebetweenthesecondlayerandtheperiodicimageoftherstlayeriskeptat 65

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itsvalueinundopedgraphiteFigure 6-1 A.ThetwoBratomsareeitherinacovalentlyboundstatetomodelmoleculardoping,orfarfromeachothertomodelatomicdoping.Inthecaseofatomicdoping,theeffectsofdisorderandintercalationconcentrationontheout-of-planeconductivityareinvestigatedusingstagetwodopedC196Br8andC196Br4,plusstageonedopedC196Br9andC196Br18(onegraphitelayerbetweenintercalatelayers).InthesecongurationstheBratomsaredistributedrandomly.Figure 6-1 BThestructural,energeticandelectroniccalculationsareperformedusingdensityfunctionaltheory[ 66 ]asimplementedintheQuantumEspressopackage[ 67 ].ThesingleparticleKohn-Shamwavefunctionsareexpandedinaplane-wavebasiswithcutoffenergyof40Ry.Theaugmentedchargedensityenergycutoffischosentobe400Ry.TwomethodsfortheExchangeCorrelationeffectsarecontrastedandcompared.TherstisthelocaldensityapproximationLDAasprescribedbyCeperlyandAlder[ 68 ]andparameterizedbyPerdew-Zunger[ 69 ];thesecondmethodisthevanderWaalsdensityfunctionalvdW-DF[ 70 71 ]asimplementedbyRoman-Perezetal.[ 72 ].TheirreducibleBrillouinzoneissampledby223specialk-pointsusingtheMonkhorst-Packscheme[ 73 ].ThechargedistributionontheindividualatomsisdeterminedusingtheBaderanalysismethod[ 74 ].ThecombinationofGGAparameterizedbyPerdew,Burke,andErnzerhof[ 75 ]andanempiricalvanderWaalsinteractionasimplementedintheQuantumEspressocode(DFT-D)[ 76 ]isalsoinvestigatedfortheprimarysystemsandtheresultsarecomparedwiththosefromtheLDAandvdW-DFcalculations.Theintercalationcongurationsareobtainedusingconstrainedoptimizationbyallowingallcarbonandbromineatomstorelaxtoalocallyminimumenergy,whilethesizeofthesupercelliskeptxed.Theheightcofthesupercellisvariedmanuallyholdingthedistancebetweentheundopedgraphitesheetsatitspuregraphitevalue. 66

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6.3ResultsandDiscussionBeforepresentingthemainresults,fewsimplesystemsareinvestigatedusingLDA,GGA,DFT-DandvdW-DF.ThebindingenergyandbondlengthofBr2calculatedwithLDAare2:90eVand2:30Arespectively,and2:40eVand2:30A,usingGGA.Thecorrespondingexperimentaldataofthetwovaluesare2:00eVand2:28A.ThusitisclearthatLDAoverestimatesthebindingoftheBratom.Forbulkgraphite,theexperimentalinterlayerdistanceis(3:35Aat300K)[ 77 ].TheinterlayerdistancecomputedwithLDAis3:25AandwhileGGAalonefailstoreproducetheinterlayerbindingofgraphite,theinterlayerdistancecomputedwithDFT-Dis3:16AandthatcomputedwithvdW-DFis3:6A[ 70 ],[ 78 ].LDAyieldsagoodestimatefortheinterlayerdistanceofgraphiteandbindingenergy,butitfallsshortwhentheintermoleculardistanceincreases[ 79 ],anditfailstoreproducetheexperimentalcompressibilityofgraphite[ 80 ].TheseshortoutcomesofLDAareaddressedbyemployingthevdW-DFmethodwhichisdevelopedtoreproducethepropertiesofexpandedsystemslikegraphite.[ 81 ] (A) (B) (C) (D) Figure6-2. Schematicrepresentationof4differentinitialpositionsoftheBr2moleculebetweenthegraphitesheets.Inconguration(A),theBratomsarealignedwithahexagoncenterfromonesheetandacarbonatomfromtheotheralternatively;in(B),theBratomsarealignedwithacarbonatomsfrombothsheets;in(C),bothatomsarealignedwiththehexagoncenterofthesamesheetandacarbonatomfromtheother;whilein(D),bothatomsarealignedwiththemidpointofaC-Cbondofonesheet. 67

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6.3.1CongurationsandEnergeticsIntheequilibriumcongurationsBr2andBrsitinthemiddlebetweentwoadjacentsheets.Withoutexternalstresstheequilibriumdistancebetweentheintercalatedlayersofgraphiteis6:86AascalculatedwithLDAand5:84Aascalculatedwithvdw-DF.However,consideringthatingeneralthesystemwillmostlikelyhavelocalstress,theinterlayerdistancewilldifferfromtheequilibriumvalue.Theselocaldifferencesaremodeledbyvaryingthelatticeconstantcinthedirectionperpendiculartothegraphitelayersasdescribedintheprevioussection.Foreachlatticeconstant,wefoundthefavorablecongurationoftheBratomandtheBr2molecule,startingfromfourdifferentcongurationsoftheBr2moleculeandthreecongurationsoftheBratomsasshowninFigure 6-2 andFigure 6-3 .Afteroptimizationtheinterlayerdistancebetweentheintercalatedlayersisfoundasd=ZC2)]TJ /F4 11.955 Tf 14.65 3.03 Td[(ZC1,whereZC1istheheightaveragedoverallthecarbonatomsintherstlayerandZC2isaveragedoveratomsofthesecondlayer.Table 6-1 andTable 6-2 listthegroundstateenergyofallthedifferentcongurationsasfunctionoftheinterlayerdistanced,calculatedusingbothLDAandvdW-DF.ThepreferredcongurationofmolecularbrominepredictedbyLDAiswhentheBratomsalignwiththemidpointoftheC-CbondsfromthesamesheetFigure 6-2 .D.ThesecondmoststablemolecularcongurationsareFigure 6-2 .CandFigure 6-2 .A,wherethetwoBratomsalignwiththecentersoftwoadjacenthexagonsfromthesamelayerorwithwithcenteroftwohexagonsfromtheoppositelayers.vdW-DFpredictsFigure (A) (B) (C) Figure6-3. Schematicrepresentationof3differentsiteoftheBratombetweenthegraphitesheets.Inconguration(A),theBratomisalignedwithonehexagoncenterofonesheetandacarbonatomoftheother;in(B),theBratomisalignedwithacarbonatomfromeachsheets;whilein(C)BrisalignedwiththemidpointofaC-Cbondofonesheet. 68

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Table6-1. ThegroundstateenergyofdifferentthestartingcongurationasfunctionofinterlayerdistancecalculatedwithLDAfunctional. d(A)moleAmoleBmoleCmoleDd(A)atomicAatomicBatomicC 4.752.492.562.322.324.802.182.012.075.720.480.540.440.515.720.810.750.806.21--0.10-6.180.70(0.67)0.686.690.010.010.01(0.00)6.680.800.790.80 EnergymeasuredineV.Theenergiesofthemoststableatomicandmolecularcongurationsareshowninparantheses. 6-2 .CtobethemoststableandFigure 6-2 .Athesecondmost.Inthecaseofatomicdoping,bothmethodsagreethattheBratomspreferapositioninwhichtheyalignwiththecarbonatomsfrombothlayersFigure 6-3 .B. Figure6-4. Thetotalenergyofthedifferentatomicandmolecularcongurationsfunctionoftheinterlayerseparation.Panel(A)iscalculatedusingLDAfunctionalandpanel(B)usingvdw-DF. Table6-2. ThegroundstateenergyofdifferentthestartingcongurationasfunctionoftheinterlayerdistancecalculatedwithvdW-DFfunctional. d(A)moleAmoleBmoleCmoleDd(A)atomicAatomicB 5.370.2070.2030.2010.2195.430.1420.1555.87-0.073--5.93(0.000)0.0176.370.0180.042(0.017)0.0306.430.0510.0617.370.2430.2500.2440.2466.930.2130.213 EnergymeasuredineV.Theenergiesofthemoststableatomicandmolecularcongurationsareshowninparantheses. 69

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ThemajordiscrepancybetweentheLDAandvdW-DFpredictionisintheglobalgroundstateofthesystem.LDApredicatesthatwithoutanystress,bromine,intercalatedbetweenthegraphitelayers,favorsthemolecularformBr2,andtheequilibriuminterlayerdistanceis>6:7A;However,undercompressionthepreferenceisshiftedtowardatomicBr,andbelowaninterlayerseparationof5:2AatomicBrbecomesmorestable.Ontheotherhand,usingvdW-DFthegroundstatecorrespondstotheatomicBrwithinterlayerdistanceof5:8AFigure 6-4 .Nonetheless,withlargerinterlayerseparationmoleculardopingisstillmorestablethanatomicdoping.ThedifferenceinthegroundstateenergybetweentheatomicandmolecularintercalationismoreprominentinLDAthanvdW-DF.Thequasi-bondstrengthEbdenedasdifferenceinthetotalenergyofthegroundstateofmolecularandatomicdopingis+0:67eVforLDAcalculationand)]TJ /F4 11.955 Tf 9.3 0 Td[(0:017eVwithvdW-DF. 6.3.2ElectronicStructureComputationswithLDAandvdw-DF,yieldsimilarelectronicstructuresofgraphiteintercalatedbyatomicandmolecularbromine.Herewepresentbothresultstogether. Table6-3. ThechargetransferbetweenthecarbontheBratomsandthemagnetizationofthesystemperBr. molecularCmolecularDatomicBatomicCd(A)n(e)]TJ /F1 11.955 Tf 7.08 -4.34 Td[()n(e)]TJ /F1 11.955 Tf 7.09 -4.34 Td[()d(A)n(e)]TJ /F1 11.955 Tf 7.08 -4.34 Td[()M(B)n(e)]TJ /F1 11.955 Tf 7.09 -4.34 Td[()M(B) 4.750.050.114.800.330.000.340.005.720.060.125.720.500.380.490.426.210.05-6.180.530.41-0.426.69--6.680.540.400.540.36 ThemoststablecongurationfordifferentlatticeconstantcalculatedusingLDAfunctionalisused. Thechargetransferbetweenthecarbonandbromineatomsdependsontheinterlayerseparation,andmostimportantly,ontheformofthedopantbromine.FortheBr2moleculethechargetransferissmall.IncongurationFigure 6-2 .Cthechargetransferfromthegraphitetothebromineis0:06e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(and0:08e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(perBratomusingLDAandvdW-DF,respectively.IncongurationFigure 6-2 .Dthechargecapturedby 70

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eachBratomis0:11e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(and0:13e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(forLDAandvdW-DF,respectively.Inthecaseofatomicdoping,thechargetransferfromthegraphitetobromineisbetween0:34and0:54e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(perBratomTable 6-3 .Thesevaluesareofthesameorderofmagnitudeasthevalues(0:5e)]TJ /F1 11.955 Tf 7.09 -4.34 Td[()reportedbyexperiment[ 58 ].Figure 6-5 showstheiso-surfaceofthechargedensitydifferencebetweenthedopedsystemanditsindividualcomponents. n=ntotal)]TJ /F3 11.955 Tf 11.96 0 Td[(nbromine)]TJ /F3 11.955 Tf 11.96 0 Td[(ncarbon;(6)wherentotalisthetotalchargedensityofthedopedsystemwhilenbromineisthatofbromineatomsaloneandncarbonisthechargedensityofthegraphenesheetsalone. Figure6-5. Iso-surfacesofthechargedensitydifferencebetweenthedopedsystemandisolatedBratomsandisolatedgraphitelayers.n=ntotal)]TJ /F3 11.955 Tf 11.13 0 Td[(nbromine)]TJ /F3 11.955 Tf 11.12 0 Td[(ncarbon.Redandblueshowiso-valuesof0:0067e=A3,redpositiveandbluenegative.Panel(A)representsmoleculardopingincongurationFigure 6-2 .C,andpanel(B)representsatomicdopingincongurationFigure 6-3 .B. ThedifferenceinthechargetransferbetweenatomicandmoleculardopingisclearfromtheprojecteddensityofstatesFigure 6-6 .Inthecaseofmoleculardoping,theemptyanti-bondingpporbitaloftheBr2moleculeislocatedattheFermisurface.Inthecaseofatomicdoping,thebrominestateoriginatingfromthehalf-lledpzorbitaloftheBratomsislocatedwaybelowtheFermisurfaceandthestatesattheFermisurfacearegraphitestates. 71

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Figure6-6. Thetotalandprojecteddensityofstates:ThedashedbluelinerepresentsthetotaldensityofstatesandsolidredlinerepresentstheDOSprojectedontheBratomsscaledupbyafactorofve.TheFermienergyissettozero.Panel(A)representsatomicdoping;panel(B)representsmoleculardoping.Inthecaseofmoleculardoping,thestatecorrespondingtotheppofBr2isrightattheFermisurface,whiletheatomicpzorbitaliswellbelowtheFermienergy. Thecarbonatomsingraphitehavesp2hybridization,formingstrongbondsintheplaneofthegraphenesheets,whiletheweaklycoupledpzatomicorbitalsgiverisetobondsthatlierightaboveandbelowtheplane.Thebondingandanti-bondingbandsarefarbeloworabovetheFermisurface;thus,onlytheelectronscontributetochargetransferwiththeintercalantbromine.ThisisseenfromgureFigure 6-5 ,where 72

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thechargedifferenceexistsonlyaboveorbelowthegraphiteplanes,butnotexactlyintheplane.TheamountofchargedonatedbyeachCatomdependsonitsdistancefromtheclosestBrdopant.Whilethenearestcarbonatomscontributethelargestnumberofelectrons,evenatlargedistances,thechargetransfernevergoesexactlytozero.AtomicBrhasaonehalf-lledporbital(s2p5)andisexpectedtobemagnetic.Whenintercalatedingraphite,attheequilibriumseparationBrretainsitsmagnetizationandthetotalmagnetizationofthesystemcalculatedwithLDAisabout0:41BperBratom.Butwithcompression,thesystemlosesitsmagnetizationandforinterlayerseparationbelow4:75Aitbecomeszero-Table 6-3 .Tofurtherinvestigatetheelectronicpropertiesofthedopedsystem,wecalculatethebandstructurealongthehighsymmetrylinesinthex-yplaneoftherstBrillouinzoneandcomparethemwiththatofpuregraphite,asshowninFigure 6-7 (Themainfeaturesofthebandstructurefordifferentatomicandmolecularcongurationsarethesame;therefore,weonlypresentthemoststablecongurationsofatomicandmolecular).Inthecaseofatomicdoping,theoriginallyfullyoccupiedbandsofgraphitearepartiallyoccupiedanddeformedtointersecttheFermisurfaceatmorethanonepoint.ThesteepbandscrossingtheFermisurfacecorrespondtonon-localizedelectrons,suggestingahighconductivityinthex-yplane.Inthecaseofmoleculardoping,bandscrossingtheFermisurfacearelesssteepandcorrespondtothelocalizedmolecularorbitalsofBr2thusdonotcontributetotheconductivity.AlsotheoverlapofthegraphitebandsattheKandHpointsisreplacedbyabandgap.Therefore,weexpectatomicBrdopedgraphitetohavehigherconductivityinthex-yplanethanmolecularBr2dopedgraphiteandpuregraphite. 73

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Figure6-7. Bandstructurealonghighsymmetrylinesofthex-yplane.Panel(A)representspuregraphite;panel(B)representsgraphitedopedwithtwoseparateBratoms;panel(C)representsgraphitedopedwithaBr2molecule;andpanel(D)representsstageoneintercalationC196Br9 6.3.3EffectsofDisorderandIntercalantConcentrationThebandstructurealongthehighsymmetrylinesinthekzdirectioncontainsbandsthatintersecttheFermienergy.Thismayindicateanincreaseinthebandconductivityalongthec-directionofgraphite,whichisnotobservedexperimentally[ 59 ].ThisbandconductivityisduetothealignmentoftheBratomsalongthec-direction,whicharisesfromtheperiodicityofourmodel.Forthat,webuiltalargersupercellcontainingtwooftheoriginalsupercellsrepeatedinthec-direction.Theatomsinonepartofthenewsupercellareshiftedinthex-yplanewithrespecttotheotherFigure 6-8 toremovethealignmentoftheBratoms.Figure 6-9 showsacomparisonbetweenthebandstructureoftheoriginalsupercellandthenewdoubledone.Asexpected,Thelargesupercellcontainstwiceasmanybandsastheoriginal.Butmoreimportantly,thebandsareat 74

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andnolongercrosstheFermisurface.Thisisthepropertyoflocalizedstatesthatdonotresultinbandconductivity.TheeffectofthebromineconcentrationisillustratedbythebandstructureofstageonedopedgraphitewithnineBratoms(C196Br9)Figure 6-7 .D.MoreofgraphitebandsareabovetheFermienergy,duetothehigherchargetransfertotheBr,leavinghighernumberofholesavailableonthegraphenesheetsforelectricconductivity.WealsoinvestigatetheeffectofintercalationstagingbycomparingC196Br2andC196Br8stagetwodopingwithC196Br4,C196Br9andC196Br18stageonedopingFigure 6-1 .B.ThechargecapturedbyeachBratom0:45e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(anditsmagnetization0:4Bisthesameinbothstages.ThebandstructurealongthekzdirectionFigure 6-10 showsanincreaseinthenumberofbandsneartheFermisurface.InstagetwodopingsomebandshaveasmallslopeacrosstheFermisurfaceduetothealignmentoftheBratomsinthatdirection.Yet,thebandsinstageonedopingareatandparalleltotheFermisurface.Thusonlylocalizedstatesexistsasaresultofthefurtherseparationofthegraphitelayers.Thiseliminatesthebandconductivityinthec-directionbutdoesnoteliminatethepossibilityofelectrontunnelingconductivity.However,thisisbeyondthescopeofourwork. 6.4ConclusionsLowconcentrationstagetwobromineintercalationofgraphitewasstudiedusingrstprinciplesDFTcalculationswithbothLDAandvdw-DFexchange-correlationfunctionals.Whengraphiteisintercalatedwithbrominethecarbonsheetsarepushedawayfromeachothertoaseparationthatdependsonthebromineform.Theinterlayerseparationforatomicdopingissmallerthanthatformoleculardoping.DespitethatLDAresultsindicatethatthegroundstatecorrespondstomolecularBr2doping,theenergydifferencebetweenmolecularandatomicbromineis0:65eVwhichis4timessmallerthanbondstrengthoffreeBr2molecule(2:40eV).vdW-DFshowsthatintercalatedbrominepreferstheatomicform,andenergydifferencebetweenatomicand 75

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molecularbromineis0:017eV.BothLDAandvdw-DFagreethatthemoleculardopingismorefavorablewithlargeinterlayerdistancesandatomicdopingisfavorableforsmallerdistances.TheenergyofthepporbitalofmolecularBr2isclosetoFermienergy,whereasthatofthepzorbitalisbelowtheFermienergy.Hence,thechargetransferredfromgraphitetotheBr2moleculeislessthanthattransferredtoBratoms.Thechargetransferdependsontheinterlayerseparation,andrangesbetween0:35and0:55e)]TJ /F1 11.955 Tf -430.56 -28.25 Td[(perBratominthecaseofatomicdopingandlessthan0:12e)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(inthecaseofmoleculardoping.Thischargetransferincreasesthenumberofholecarriersinthegraphenesheetsleadingtoanincreaseinthein-planeconductivity.Further,thebandstructureshowsthatthebandsofatomicdopedgraphitehavelargerslopeneartheFermilevelthuslargerFermivelocityandhigherin-planeconductivity.Thesmallenergydifferencebetweenthegroundstateofatomicandmoleculardopingsuggestthatbothformwillcoexistatroomtemperature,incontrasttopreviousstudieswhereonlyBr2intercalationformwasconsidered[ 60 ].ThepresenceofatomicBrplustheevidencefromthebandstructureprovidetheexplanationofthesupermetallicin-planeconductivityobservedinexperiment[ 59 ].Whereas,theincreaseintheresistivityalongthec-direction[ 59 ]canbeexplainedbytheincreaseintheseparationbetweentheintercalatedlayersofgraphiteandtheatbandsalongthekzdirectionofthedisorderedsupercellFigure 6-9 .D. 76

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Figure6-8. Thenewsupercellmadeoftwooriginalsupercellsalongthec-direction.TheBratomsofonecellisshiftedrelativetotheotherinordertobreakthealignmentalongthec-direction. 77

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Figure6-9. Bandstructurealonghighsymmetrylinesinthez-direction.Panel(A)representspuregraphite;panel(B)representsatomicdopedgraphitewiththeoriginalsupercelloftwolayers;panel(C)representsmoleculardoped;andpanel(D)representsthedoublesupercellwiththeBratomsshiftedtoeachother.NotethatthebandsneartheFermilevelthathadahighslopeinpanel(B)areatintheexpandedcell. 78

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Figure6-10. Bandstructurebetweenhighsymmetrylinesinthez-directionwiththelaterallocationoftheBratomschosenrandomlyrelativetothecarbonatom.Panel(A)representsstagetwodopingofC196Br8;panel(B)representsstageonedopingC196Br8;andpanel(C)representsstageonedopingofC196Br18. 79

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BIOGRAPHICALSKETCH Dr.RashidHamdangrowupinatypicaltowninthemountainsofLebanonandgraduatedfromhighschoolin1998withadreamtobecomeascientist.RashidjoinedthecollegeofscienceintheLebaneseUniversityfromwhichhereceivedaBachelorofSciencewithaTeachingDiplomainchemistryin2002.RashidleftthetheLebaneseUniversitywithagreatenthusiasmforscienceandafascinationwithquantumphysics.ForthatheswitchedhismajorandenrolledintheAmericanUniversityofBeirutayearlater.HereceivedaMasterofScienceinphysicsinsummer2007.Thesubjectofhismastersthesiswastheinteractionofmulti-blockcopolymerswithliquid-liquidinterfaces.AttheUniversityofFlorida,RashidJoinedProf.Hai-PingChengsgroupinwhichheworkedonseveralprojects,buthefoundhisinterestinstudyingtransitionsandlocatingofsaddlepoints.HedefendedthisdissertationandreceivedaDoctorateofPhilosophyinMay2014.Rashidhasatruepassionforteaching,asthroughouthisgraduatestudieshewasalsoateacherofhighschoolandcollegelevelphysics. 86


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