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Resonant Surface Scattering on Nanowires

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Title:
Resonant Surface Scattering on Nanowires
Creator:
Roffman, David A
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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Language:
english
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1 online resource (112 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
HERSHFIELD,SELMAN PHILIP
Committee Co-Chair:
STEWART,GREGORY R
Committee Members:
MUTTALIB,KHANDKER A
CHENG,HAI PING
PHILLPOT,SIMON R
Graduation Date:
8/6/2016

Subjects

Subjects / Keywords:
Atomic interactions ( jstor )
Atoms ( jstor )
Conceptual lattices ( jstor )
Cubic lattices ( jstor )
Geometry ( jstor )
Mean free path ( jstor )
Phonons ( jstor )
Resonance scattering ( jstor )
Spring constant ( jstor )
Thermal conductivity ( jstor )
Physics -- Dissertations, Academic -- UF
thermal
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
For efficient thermoelectric devices there is a requirement for materials with low lattice thermal conductivity either as the active material generating the electrical current or as a material to stop the flow of heat from hot to cold regions. Skutterudites are one of the best thermoelectric materials not just because of their electrical properties, but also because of the low lattice thermal conductivity. Theoretical work on skutterudites indicates that it is the structure of the atoms in the cages which causes low thermal conductivity, sometimes referred to as resonant scattering. This is more effective than static disorder at scattering at low energy. The purpose of this thesis is to determine whether the same kind of reduced lattice thermal conductivity can be obtained by introducing resonant scattering on the surface of a wire. In part this means spraying the outside of the wire. The starting point for this work is a modeling of chains, cubic lattices, and skutterudites. This is then followed up by simple one dimensional scattering models that demonstrate the effectiveness of resonant scattering. Next is a study of uniform sprays which proved that ordered sprays do not reduce transmission as a function of sample length. Finally is a test of disordered surface resonant scattering, which did achieve the desired result of reduction in transmission and thermal conductivity as sample length increases. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2016.
Local:
Adviser: HERSHFIELD,SELMAN PHILIP.
Local:
Co-adviser: STEWART,GREGORY R.
Statement of Responsibility:
by David A Roffman.

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UFRGP
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Copyright Roffman, David A. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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LD1780 2016 ( lcc )

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RESONANTSURFACESCATTERINGONNANOWIRES By DAVIDALEXANDERROFFMAN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2016

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c 2016DavidAlexanderRoman

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ACKNOWLEDGMENTS ObviouslyImustrstthankmyadvisor,ProfessorSelmanPhilipHersheld.Hehas providedmanyexcellentexamplesofhowtousecodingtechniquestounlockmanyofthe mysteriesoftheuniverseatthenanoscale. Therewerepeopleatmyundergraduateschoolwhodidmuchtoprimemyinterest inscience,althoughmyphysicseldtherewasspacephysicswithafocusonMartian meteorology.ItwastherethatIrstlearnedaboutradioisotopethermoelectricgenerators RTGswhicharementionedbrieyinmythesisinconjunctionwithhowelectricalpower canbegeneratedduetoatemperaturedierence.InparticularIwanttosingleout Embry-RiddleAeronauticalUniversityProfessorsMichaelHickey,JohnOlivero,Jason Aufdenberg,andYonghoLee. MyinterestinthephysicsofRTGswasalsopeakedbyProfessorJamesTillman whoheadedtheVikingComputerFacilityforNASA.AlthoughMartianweatherisnot addressedinmythesis,Mr.TillmanandIhavediscussedoperationofinstrumentation onspacecraftenoughformetounderstandthattheknowledgeandskillsIacquiredwhile preparingmythesiscanhavepracticalapplication. Finally,Iwouldliketoacknowledgemyfatherforteachingmehowtoassert myselfwhennecessary,andtomymotherformakingsurethatIreceivedhomecooking throughoutmytimeattheUniversityofFlorida.Iamalsogratefultobothformaking surethatIgotthroughmyentireeducationwithouttheneedtotakeoutanyloans. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................3 LISTOFFIGURES....................................6 ABSTRACT........................................9 CHAPTER 1INTRODUCTION..................................10 1.1MotivationandBackground..........................10 1.2Objective....................................18 1.31-DCase.....................................23 1.42-Dand3-DCases...............................23 1.5SkutteruditeCases...............................24 1.61-DO"AtomCases.............................25 1.7SprayedSkutteruditeCases..........................25 1.8SprayedCubicLattices.............................25 21-DCHAINS,2-DAND3-DHARMONICRECTANGULARLATTICESWITH MASSDISORDER..................................28 2.11-DChains...................................28 2.22-Dand3-DHarmonicLattices........................31 3SKUTTERUDITESOFTHEFORMMX 3 .....................37 3.1AssumingSquareXGeometry.........................38 3.2RectangularXGeometry............................39 41-DOFF"ATOMSANALOG...........................48 5SPRAYINGSKUTTERUDITESOFTHEFORMMX 3 ..............60 5.1AssumingNoSprayed-XInteraction......................60 5.2AssumingSprayed-XInteraction........................62 6SPRAYEDCUBICLATTICES...........................81 6.1ScalingofTransmissioninaCleanCubicLattice..............81 6.2OrderedSpray..................................82 6.3SpringConstantDisorderedSpray.......................83 7NUMERICALMETHODS..............................96 7.1InitializationPartI:DeningParameters,ObtainingFrequencies,andWave Numbers.....................................96 4

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7.2InitializationPartII:Normalization......................97 7.3SettingUptheLoopandEOM........................97 7.4SimpleExampleofImplementingtheScatteringBoundaryCondition...99 7.5LUFactorization................................101 7.6ComputingtheVelocityofModes,OverallTransmissionandReection, andFinishingtheLoop.............................102 7.7ComputingThermalConductance.......................103 7.8ComputingMeanFreePath..........................104 8CONCLUSION....................................105 REFERENCES.......................................107 BIOGRAPHICALSKETCH................................112 5

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LISTOFFIGURES Figure page 1-1VoyagerRTGs.....................................26 1-2TheunitcellofCoSb 3 .................................27 2-1Transmissionthrougha1-DChain.........................35 2-2Transmissionthrougha1-DChainwithdisorder..................35 2-3Transmissionthroughacubiclatticewithdisorder.................36 2-4ModeNormalizedTransmissionthroughacubiclatticewithdisorder......36 3-1AtiltedofviewofaskutteruditeoftheformMX 3 .................40 3-2SkutteruditeInteractions...............................40 3-3ComparisonoftransmissionthroughM,Mwithdisorder,andMX 3 ........41 3-4ComparisonoftransmissionthroughMandMX 3 withvaryingk 1 ........42 3-5ComparisonoftransmissionthroughMandMX 3 withvaryingk 3 ........43 3-6ComparisonoftransmissionthroughMandMX 3 withvaryingk 5 ........44 3-7ComparisonoftransmissionthroughMandMX 3 withvaryingm Sb .......45 3-8Thermalconductanceofacubiclatticetoaskutterudite.............46 3-9TransmissionforsquareXgeometryvsrectangularXgeometry.........47 4-1CasesI-IVgeometries................................51 4-2TransmissionforCaseI...............................52 4-3TransmissionforCaseII...............................52 4-4Varyingo"atommassforCaseIII........................53 4-5Varyingthespringconstantoftheo"springforCaseIII............53 4-6Varyingthemassoftheo"atomforCaseIV..................54 4-7Varyingthespringconstantoftheo"springforCaseIV............54 4-8ComparisonoftransmissionforCasesI-IV.....................55 4-9CaseIIIdiptest...................................55 4-10CaseIIIFWHMtest1................................56 6

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4-11CaseIIIFWHMtest2................................57 4-12CaseIIIFWHMtest3................................57 4-13CaseIIIFWHMtest4................................58 4-14CasesVandVIgeometries..............................58 4-15TransmissionofCasesVandVI...........................59 5-1Sketchofasprayedskutterudite...........................64 5-2Cubicvssprayedcubictransmission........................65 5-3Weakskutteruditevssprayedweakskutteruditetransmission..........66 5-4Skutteruditevssprayedskutteruditetransmission.................67 5-5Skutteruditevssprayedskutteruditetransmissionforincreasedk ss .......68 5-6Skutteruditevssprayedskutteruditetransmissionforincreasedk sc ........69 5-7Skutteruditevssprayedskutteruditetransmissionforincreasedm So .......70 5-8Comparisonofvarioussprayedsetups........................71 5-9Thermalconductanceofacubicandasprayedcubiclattice...........72 5-10Thermalconductanceofaweakskutteruditeandasprayedweakskutterudite.73 5-11Thermalconductanceofaskutteruditeandasprayedskutterudite........74 5-12Thermalconductanceofaskutteruditeandasprayedskutteruditewithhigherk sc 75 5-13Thermalconductanceofaskutteruditeandasprayedskutteruditewithhigherk ss 76 5-14Thermalconductanceofaskutteruditeandasprayedskutteruditewithhigher m So ..........................................77 5-15Comparisonofconductanceforvarioussprayedsetups..............78 5-16EectofaSo-SbinteractionforN x =5.......................79 5-17EectofaSo-SbinteractionforN x =15......................80 6-1Transmissionwithcrosssectionvaryinginbetweentheplots............86 6-2Normalizedtransmissionwithlengthvaryinginbetweenplots...........87 6-3Sprayedgeometry1..................................88 6-4Sprayedgeometry2..................................88 7

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6-5Geometry2normalizedtransmissionwithplotsofvariouslengths.........89 6-6Geometry2transmissionofvaryingsamplelengthwithvariousplotsofdierent crosssections......................................90 6-7Sketchofsprayedgeometrywithdisorder......................91 6-8Obtainingpathlengthforlowerdisorder......................92 6-9ObtainingpathlengthforhigherdisorderI.....................92 6-10ObtainingpathlengthforhigherdisorderII....................93 6-11Transmissionasafunctionofdisorderandsamplelength.............94 6-12Meanfreepathforvaryingamountsofdisorder..................95 6-13Thermalconductanceforvaryinglengthsinthefulldisordercase........95 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy RESONANTSURFACESCATTERINGONNANOWIRES By DavidAlexanderRoman August2016 Chair:SelmanPhilipHersheld Major:Physics Forecientthermoelectricdevicesthereisarequirementformaterialswithlow latticethermalconductivityeitherastheactivematerialgeneratingtheelectricalcurrent orasamaterialtostoptheowofheatfromhottocoldregions.Skutteruditesareone ofthebestthermoelectricmaterialsnotjustbecauseoftheirelectricalproperties,but alsobecauseofthelowlatticethermalconductivity.Theoreticalworkonskutterudites indicatesthatitisthestructureoftheatomsinthecageswhichcauseslowthermal conductivity,sometimesreferredtoasresonantscattering.Thisismoreeectivethan staticdisorderatscatteringatlowenergy. Thepurposeofthisthesisistodeterminewhetherthesamekindofreducedlattice thermalconductivitycanbeobtainedbyintroducingresonantscatteringonthesurface ofawire.Inpartthismeanssprayingtheoutsideofthewire.Thestartingpointforthis workisamodelingofchains,cubiclattices,andskutterudites.Thisisthenfollowedupby simpleonedimensionalscatteringmodelsthatdemonstratetheeectivenessofresonant scattering.Nextisastudyofuniformsprayswhichprovedthatorderedspraysdonot reducetransmissionasafunctionofsamplelength.Finallyisatestofdisorderedsurface resonantscattering,whichdidachievethedesiredresultofreductionintransmissionand thermalconductivityassamplelengthincreases. 9

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CHAPTER1 INTRODUCTION 1.1MotivationandBackground Sincetheinventionofthelight-bulb,peoplehavecometovalueelectricity.The questionthatarisesfromthisneedis:howtogenerateit?Oneideaistouseatemperature dierencetogenerateelectricitydirectly.Devicesthatusethisconceptarecalled thermoelectricdevices.Theyhaveahotregion,asample"regionwiththermalinsulation aroundit,andacoldregion.Itisdesirabletohaveamaterialwithlowlatticethermal conductivity,sincetransferringelectricity,notheatistheobjective. Thestartingpointandmotivationforthisthesisareskutteruditematerials.Recently thesematerialshavebecomeapopulartopicinthecontextofthermoelectricity.A signicantpartofthisthesisreviewsthetheoreticalandexperimentalworkasitrelatesto thelatticethermalconductanceandconductivity. Thereareseveralkeyconceptsthatarefrequentlyreferredtointhepaperscited below.TherstisthatofaphononglasselectroncrystalPGEC.Tomaximizethe thermoelectricresponse,oneneedstominimizetheheatowviathelatticefromhotto coldregions.Aphononglasswoulddothat;however,theexperimentalevidenceisthat theskutteruditesarenotglassesinthesamemannerassilica.Rather,theyhavelow thermalconductivitieslikeglasses.Nolas,Morelli,andTrittgiveanexcellentreviewofthe propertiesofbulkskutteruditesbothlledandunlled.Thethermalconductivitygraphs showclearlytheroleofUmklappandboundaryscatteringintextbooklikeconductivity vs.temperaturecurves.Polycrystallineskutteruditeshavelowerthermalconductivities andareroughlytemperature-independentaboveabout50K.Theyarecloseinvalueto vitreoussilica[1]. Thesecondkeyconceptisthatofarattler,andmoregenerallymechanismstomodify thethermalconductivitywhileleavingtheelectronicpropertiesrelativelyunchanged.The skutteruditeshaveanaturalcagesintowhichextraatomscanbeplaced.Theseatomsare 10

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expectedtorattle"inweaklycoupledEinsteinphonons.Alargenumberofexperimental andtheoreticalworksexploretheroleofdierenttypesofrattleratoms.Theskutterudite materialsarebynomeanstheonlyclassofmaterialsbeingstudiedforthermoelectricity. GoncalvesandGodartreviewseveraldierentclassesofbulkthermoelectricsincluding theskutterudites,butotherclassesofmaterialsaswell[2].Foramoregeneraloverview ofrecentprogressinthermoelectrics,seetheMRSmedalpresentationbyMercuouri Kanatzidisforthediscoveryanddevelopmentofnanostructuredthermoelectricmaterials [3]. Theexperimentalaspectsofthesecondkeyconceptwillnowbeexplored.Instudying latticevibrationsthemostnaturalexperimentaltechniqueisinelasticneutronscattering. Viennoisetal.useinelasticneutronscatteringtodeterminethedierenceinthephonon densityofstatesbetweenalledandanunlledskutterudite.Subtractingthetwodensity ofstatesyieldsaclearpeakat7meVattributedtothelocalizedLa-atomsusedtoll theskutterudite[4].Thispeakisseenasevidencefortherattlingmodes.Dimitrovet al.performedinelastictimeofightscatteringandspecicheatmeasurements,andalso foundevidenceofanEinsteinoscillatororrattlerat5meVinthelledskutterudite Yb 0 : 2 Co 4 Sb 12 [5].However,whenYangetal.performedinelasticneutronscatteringon CeOs 4 Sb 12 todeterminethephonondispersioncurvesalongtwoprincipalaxes,they didnotdetectanyrattlingmodefortheCeatomalthoughtheyonlystudiedalongthe principleaxes[6]. Sergueevetal.performedhighpressurex-raydiractionandnuclearinelastic scatteringonalledskutterudite.ByexaminingthemodeandelementspecicGruneisen parameterstheyndthatthelargeanharmonicityoftherattlingmodeisreduced withpressure,suggestingthatanharmonicityplaysaroleinthelowlatticethermal conductivityofthesecompounds[7].Opticalspectroscopyhasalsobeenusedonthe skutteruditesmaterials.Kimetal.useddierentformsofX-rayscattering,transport measurements,andDFTcalculationsandconcludedthatthelowerthermalconductivity 11

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inlledCoSb 3 isduetophonon-localizedresonantscatteringbytheIn-llerrattling [8].Heetal.usedanopticalpumpprobetechniquewhichallowsthemtomeasurethe longitudinalsoundvelocityandrelaxationofphonons.Theyfoundthatthehighfrequency acousticphononmodesarestronglydampedastheYbllingfractioninYb x Co 4 Sb 12 increases[9].Bothoftheseresultsaregoodevidenceforrattlingmodes. Luetal.performedRamanscatteringonlledandunlledCoSb 3 andfoundthat lowerfrequenciesshifteddownandhigherfrequenciesshiftedupbroadening[10].This doesnotshowtheeectofrattlersdirectly,butitdoeshighlighttheimportanceof disorder.Gueetal.doultrafastspectroscopyonskutteruditespartiallylledwithBa,Yb, andLa,respectively,andthenafourthsamplewithallthreeelements.Theirresultsshow thatdierentguestelementscauseresonantoscillationswithdierentfrequencies.This demonstratesthatmultipleguestelementscanscatterawiderspectrumofphononsthan asingleguest[11].Neartheendofthisthesiswewillagainseetheimportanceofdisorder withdierentfrequencyscales. Finally,thereisabodyofexperimentalworkwhichexaminestheeectofdierent kindsofmodicationstoskutteruditesandtheireectonthermalproperties.Thelist ofexperimentsthatfollowindicatethatllingthevoidsintheskutteruditedecreases thermalconductivity.Yangetal.measurethelowtemperaturetransportproperties ofBa x R y Co 4 Sb 12 R=La,Ce,andSr,andndthatusingllerelementsofdierent chemicalnaturesrareearths,alkalineearths,andalkalinesprovidesstrongerphonon scatteringoverabroaderrangeinfrequencies[12].Prytzetal.addedLarattlers inallavailablevoidsofCo 8 P 24 andreplacedCobyFetoformLa 2 Fe 8 P 24 .Using electronenergylossspectroscopytheyfoundthatelectronicstructureoftheparent andmodiedcompoundarequitesimilar[13].Harnwunggmoungetal.reportthatthe lledskutteruditeTl 0 : 1 In x Co 4 Sb 12 showsadramaticreductioninthelatticethermal conductivityduetotherattlingofTlandIn,andalsonaturallyformIn 2 O 3 particles [14].Salesetal.measurethelatticethermalconductivityofR 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(y Fe 4 )]TJ/F22 7.9701 Tf 6.586 0 Td [(x Co x Sb 12 R=La, 12

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Ce,orThandndgreatlyreducedlatticethermalconductivity.Thelatticethermal conductivityisquitesmallcomparableatroomtemperaturetothatofvitreoussilica [15].Fuetal.exploretheeectsofthermaldiusionofNinanoparticlesinaskutterudite material,ndingalargereductionofboththeelectricalconductivityandthelattice thermalconductivity[16].Ohtakietal.putatomsinacagelikestructureincertain oxidesandndexperimentallygreatlyreducedthermalconductivity[17].Thompsonet al.measurethecharacteristicsofCaFe x Co 4 )]TJ/F22 7.9701 Tf 6.586 0 Td [(x Sb 12 andndevidenceforphononmodesat 7meVinadditiiontothepreviouslyobserved17meVmode[18].Kimetal.synthesizedan SmlledCoSb 3 withoutchargecompensationandstronglyreducedthephononthermal conductivityandenhancedthethermoelectricperformance[8].Thegistofthisresearchis thatmoremodicationsanddisorderdecreasesthethermalconductivityandatleastin somecasesseemstoleavetheelectricalpropertiesrelativelyunchanged. Theskutteruditeclassofmaterialshavealsobeenstudiedextensivelytheoretically. Sometheoreticalworksfocusonndingparametersformodelstfromexperimental data.FeldmanandSinghperformadetailedtofdierentlatticemodelstoobserved Ramanscatteringspectra[19].Morerecentlytheyusedinelasticneutronscatteringdata todeterminelatticemodelsinbothdopedandundopedskutteruditecompounds[20]. Othertheoreticalworksfocusonndingtheunderlyingstructureofthephononsandtheir scattering.Kozaetal.useDFTandlattice-dynamicscalculationstostudythevibrational dynamicsofvariousdopedskutterudites.Theirresultsaresupportedbyinelasticneutron scatteringandheat-capacitymeasurements.Thetheoreticalandexperimentaldensity ofphononstatesareingoodagreementwithpeaksat4.9and5.7meV[21][22].More recently,Kozaetal.compareinelasticneutronscatteringresultsfromLaFe 4 Sb 12 toresults fromabinitiocalculationsandndenergyeigenmodeswithhighamplitudesoflanthanum vibrations[23].Qiuetal.examinethesimulationsofthevibrationsinlledskutterudites andshowthatpartsofthecrystalareweeklycoupledandpartsarestronglycoupled leadingthemtointroducetheconceptofapart-crystallinepart-liquidstate.Theyobserve 13

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thatthelowtemperatureconductivityclosetoamorphouslimitmustbedescribedbythe rattling-likedampingasaresonantphononscattering[24].GhosezandVeithenperform DFTcalculationsonCoSb 3 andTlFeCo 3 Sb 12 andalsondalowenergymodedueto thecoupledvibrationsofTlandSbwhichleadstopeakinthephonondensityofstates [25].Theabovecalculationsallpointtoaroleofsomekindoflocalizedmodeinphonon scattering. Othercalculationsndeectsbesidesscatteringooflocalizedmodesmaybe important.Chenetal.dorstprinciplescalculationsofthephonondispersionand showhowitisstronglyanisotropic,whichcanleadtoincreasedscatteringatnanograin boundaries[26].Lietal.performabinitiocalculationsonCoSb 3 andBaCo 4 Sb 12 .Upon llingtheyndthatthethermalconductivityismainlyduetothereductionofphonon lifetimeslimitedbyanharmonicscattering-notrattlingmodes[27]. Manyofthecalculationsfocusontheroleofllingtomakemodicationstothe skutterudite.Thisisinthephilosophyofcomputationaltechniquesusedtodesign materials.Allenoetal.examinetheeectsofllingskutterudites,ndinginsomecases stronglyreducedphononthermalconductivity.TheseareantimonidesA y M 4 Sb 12 withA=Ca,Sr,BaLa-Sm,Eu,Yb,Tl,M=Fe,Co,Ru,Rh,Os,Ir[28].Weeetal. useabinitiocomputationstoinvestigatetheeectofllerionsonthepropertiesof Ba-lledCoSb 3 skutterudites.Theyareabletoidentifytheller-dominatedmodes, whichhaveweakdispersionindicatingtheyarelocalized,andalsoshowthatthemodes arestronglycoupledtonearbySbatoms[29].UsingmoleculardynamicsYangetal. examinedtheeectsofporesinCoSb 3 ,ndingthatreducingtheporediameterorthe porosityreducesthethermalconductivity[30].HuangandKavianyperformedabinitio andmoleculardynamicssimulationsinanorderedcrystal,skutteruditeandanordered lledskutterudite,andthenexplainedsmallerphononconductivityofpartiallylled skutteruditesasthesolidsolutionofemptyandlledstructures.Theyndthatthe couplingbetweenthellerandthehostisstrongwithminoranharmonicity[31].Zebarjadi 14

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etal.considerusea2-Dtoymodelwithatomsinacageanddoequilibriummolecular dynamicstocomputethethermalconductivity.Theyexaminetheeectschangingthe meanllerdisplacementandthemassoftheller,claimingtofollowtrendsseeninthe experiment[32].UsingabinitiocalculationsWilliamsonetal.studyarangeoflled skutterudites.Theyndthatthethermalconductivityiscontrolledbytheacoustic modes,andtheoptimalllingisapartiallydopedsystem[33].Theideathatonewants disorder,hereintheformofpartiallling,willappearlaterinthethesisaswell. Mostoftheabovecalculationsfocusedontheroleofthellerorextraatoms placedintheskutteruditecages.Othermodicationstotheskutteruditeshavealso beenexamined.UsingabinitiocalculationsandlatticedynamicsChi.etal.foundthat replacingtwooftheatomsinthepnicogenringsSbinCoSb 3 withasingleatomchosen soastomaintainchargeneutralityresultedinlowthermalconductivity[34].Liand MingoperformabinitiocalculationsonCoSb 3 andIrSb 3 tounderstandthedierences inthelatticethermalconductivitybetweenthematerials.Theyndthattheincreasein massfromCotoIrisultimatelythedominantfactor,leadingtoadecreaseinacoustic phononfrequenciesandtheDebyetemperature[35].Jietal.usemoleculardynamicsand abinitiocalculationstoinvestigatetheroleofvacanciesinIn 4 Se 3 )]TJ/F22 7.9701 Tf 6.587 0 Td [(x .Sevacanciesstrongly suppressphononpropagationinonedirectionwithlittleeectinotherdirections.The latticethermalconductivityisreduced[36].Kimetal.performabinitiocalculationson Ba x Co 4 Sb 12 ndingseveralstablecongurationsoftheBaorderedintheintrinsicvoids. Thepredictedphononconductivityshowaminimuminatwophaseregime[8].While manydierentskutteruditemodicationshavebeenconsidered,thegeneraltrendisthat moredisorderincreasesscattering. Theabovetheoreticalworkwaslargelyrstprinciplescalculations.Thereisanother groupoftheoreticalworkwhichfocusesonmoremodelsystems.Someofthisworkfocuses onderivinggeneralresults.MinandRowepointoutthatwhilemoderateimprovement inthethermoelectricgureofmeritmaybeobtainedbyreducingthelatticethermal 15

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conductivity,togobeyondthattheelectricalpropertiesneedtobeimprovedaswell[37]. ChenandPodlouckyusingDFTandBoltzmanntransporttheorytoshowthatsimply separating K ph = K )]TJ/F21 11.9552 Tf 12.143 0 Td [(K el doesnotworkwhentheSeebeckcoecientbecomeslarge[38]. Theconclusionsofbothofthesepapersseemquitereasonable. Thereisanotherclassoftheoreticalpaperswhichusesphenomenologicalmodels forphononscatteringmechanismsandtexistingdata.Thefourpaperslistedbelow allclaimtotthedata;however,theyareallusingdierentscatteringmechanisms. Thisillustratestheperilofjustifyingatheorybyitsttothedata.Nolasetal.use asimplephenomenologicalphonon-scatteringrelaxationtimetomodelLa-andYblledskutteruditeantimonidesandcomparetoexperimental.Theirtincludesthe relativeimportanceofboundary,pointdefect,andresonantscattering[39].Baiet al.useaphenomenologicalmodelforthephononlifetimethatincludephononpoint defectscatteringandphononresonantscatteringbyllers.Theyalsondgoodtto experiments[40].Zebarjadietal.alsousephenomenologicalrelaxationtimestto experimentaldatatostudythemeanpathsinlledandunlledskutterudites.They needtoincludethefullphonondispersioninordertotthedata[41].Gengetal.usea phenomenologicalmodelusinglatticeanharmonicitytotthehightemperaturelattice thermalconductivityofsomeskutteruditeallows.Basedonthesuccessoftheirt,they concludethatphononresonantscatteringisnotneeded[42]. Finally,wenotethattheremaybeotherphysicsgoingonintheskutteruditesbesides justtheroleofphononscatteringbyextrallersordisorderinduced.Mocheletal.use resonantultrasoundspectroscopytostudythelledskutteruditeYbFe 4 Sb 12 andnd ananomaloussofteningoftheelasticconstantsat50Kwhichcannotbeexplainedby thedynamicsoftheller[43].Viennoisetal.showthatLaFe 4 Sb 1 2isaferromagnetic quantumcriticalpointsystemandCeFe 4 Sb 12 isamoderateheavyfermioncompound[44]. Whilethisreferstotheelectronicproperties,itmayverywelleectthelatticeproperties aswell. 16

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Inthisthesiswestartoutcomputingthethermallatticetransportofsimple modelsbasedontheskutteruditestructureandtrytounderstandthebasicmechanism forreductionofthephononthermalconductancebasedonourmodel.Basedon thisunderstanding,wethentrytoinducethesamekindofextrascatteringina non-skutteruditematerialwithsurfacescattering.Theroleofphononsscatteringo ofatomsabsorbedonasurfacehasrecentlybeenstudied.BabaeiandWilmerperform moleculardynamicscalculationsonagasabsorbedontoaporouscrystalstructure.They showthatthedecreasedconductivityassociatedwithincreasedgasconcentrationisto phononscatteringothegasmoleculeswitharateproportionaltothenumberofgas moleculesinapore[45]. Wewillnowconsiderafewexamplesofthepracticalusesofthermoelectricdevices. Salvadoretal.indicateaveryusefulworkinprogressinvolvesthermoelectricgenerators forcarexhaustwasteheatrecovery[46].Theheatthatisrecoveredisusedtoimprove automotivefueleconomybecauseitisusedtoproduceelectricityfortheon-board electricalsystem.Thisnewsourceofpowersupplantselectricityproducedbythe alternator.Theloadontheenginerequiredtodrivethealternatoristhusreducedor eveneliminatedifthethermoelectricgeneratorpoweroutputishighenough. Thermoelectricdevicesarealsooftenemployedinspaceprobes,suchastheVoyager missions[47],MarsLanders,etc.Forspaceprobes,thesedevicesarecalledradioisotope thermoelectricgeneratorsRTGs.RTGsarenecessarybecausebatteriesdiefairlyfast andsolarpanelsareincreasinglylesseectiveastheygetfurtherfromthesunVoyager missions. Thoughnotthetopicofthisreport,itishelpfultoshowaschematicofathermoelectric device.InthecaseoftheVoyagermissions,thesetupisshowninFig.1-1. Nowthatanexampledevicehasbeenshown,thisthesiswillnowmoveontoselection ofthermoelectricmaterials.OnepossibilityisskutteruditesoftheformMX 3 ,astheyare cheaptomake,andhavemuchuntappedpotentialforusesinindustry.Guoetal.indicate 17

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theirpropertiesareeasytomodify[48].Hencedetermininghowtoreducetheirlattice thermalconductivityisofimportance. MuchoftheoriginalmodelingworkforthesesubstanceswasdonebyLutzand Kliche,howeverseveraloftheirpapersaren'tavailableinEnglish.Asidefromthelowest frequencymodewhenappliedtoCoSb 3 ,itisveryaccurate.TheirLKmodelismodied inmanypapersaswithFeldmanandSingh[19].Itappearsthatthecauseforreduced transmissionandthermalconductivityisthattherearecagedatoms. Thenextquestionis:howisitpossibletocreateamaterialwithlowthermal conductivity,withoutsimplyusinganexistingmaterial?Toanswerthis,othermechanisms aresoughtforreducingtransmissionandthermalconductivityintheformofsprays,etc. Thistopicisthethrustofthisthesis. 1.2Objective Theobjectiveistoreducethermalconductivity,whichmeansthereisaneedtolessen heattransfer.Sinceheatisrelatedtomotionofatoms,obtainingthepositionofthese atomsasafunctionoftimeforgiveninputstoasystemiscritical. Atomsinalatticecaninteractthroughavarietyofpotentials.Forargument'ssake, taketheLennard-Jonespotential[49].ByTaylorSeriesexpandingthispotentialabout theatoms'equilibriumpositionsandkeepingonlytermsupto1storder,onewillarrive atspringenergyinteractionsquadraticinposition.ThisisknownastheHarmonic approximation. U harm = X R;R 0 u R D R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R 0 u R 0 2 {1 Heretheu'sarethedisplacementsoftheatomsvectorobjects,Risthesiteofatom,and theR'areallthesitescorrectedtositeR.TheDisthedynamicmatrix,whichgenerates thespringconstantsfromtheLennard-Jonespotential.Thedynamicmatrixelementsare obtainedbytakingsecondpositionalderivativesofthepotential. Thismodelingeneralallowsformultiplespringconstantsforeachdirection's equationofmotionEOM.MeaningifonesimplywritesNewton'ssecondlaw,skipping 18

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overthedynamicmatrixequation,theallowedextraspringconstantswouldlikelyhave beenoverlooked.LieghtonindicatesthatwecanusetheRosenstock-Newellmodel[50]. TheaboveequationwhenappliedtothesetupsinthisthesisrevealsthattheX,Y,andZ EOMareindependent.ThismeansthatXdoesn'tappearintheYandZEOM,etc. Themostimportantfeatureofthisapproximationisthatityieldsananalytic solution.FortheworkIdid,thesolutionsareplanewavesinthebaths.Thismodelwas implementedinmyvariouscodestoobtaintheresultsinthisthesis.Itwillbeshown belowhowtousetheU harm formula.Letthelatticespacingbea"intheXdirection, b"intheYdirection,andc"intheZdirection.Forthisderivationa3-Drectangular harmoniclatticewithnearestneighboronlyinteractionsisassumed. FirstcomputeD.Asthisisavector,IwillturnitintomatrixelementformD QW , whereQandWtakeonthedirectionsX,Y,andZ. D QW R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R 0 = R;R 0 X R 00 E QW R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R 00 )]TJ/F21 11.9552 Tf 11.955 0 Td [(E QW R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R 0 {2 LetEbethepotentialenergyandE QW be: E QW R )]TJ/F21 11.9552 Tf 11.956 0 Td [(R 0 = @ 2 E @Q@W {3 Nowlettheatom'sindices'sx,y,zbeintegersn,m,l.Positionisthenthese integersmultipliedbythelatticespacing.IntheequationforD QW ,RandR'canbe: na,mb,lc,na-a,mb,lc,na,mb-b,lc,andna,mb,lc-c,na+a,mb,lc,na,mb+b,lc,and na,mb,lc+c. ForthecaseoftheLennard-Jonespotentialtheinteractionenergyis: E =4 12 r 12 )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( 6 r 6 {4 UsingtherulescitedinAshcroftandMermin[49],manyoftheD qw arethesame: D QW a; 0 ; 0= D QW )]TJ/F21 11.9552 Tf 9.298 0 Td [(a; 0 ; 0= )]TJ/F21 11.9552 Tf 9.299 0 Td [(E QW a {5 19

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D QW ;b; 0= D QW ; )]TJ/F21 11.9552 Tf 9.299 0 Td [(b; 0= )]TJ/F21 11.9552 Tf 9.299 0 Td [(E QW b {6 D QW ; 0 ;c = D QW ; 0 ; )]TJ/F21 11.9552 Tf 9.298 0 Td [(c = )]TJ/F21 11.9552 Tf 9.298 0 Td [(E QW c {7 D QW = E QW a + E QW b + E QW c {8 TheseequationsareinsertedintoU harm : U harm = 1 2 X QW Q n;m;l D QW W n;m;l + Q n;m;l D QW a W n +1 ;m;l + Q n;m;l D QW b W n;m +1 ;l + Q n;m;l D QW c W n;m;l +1 + Q n +1 ;m;l D QW W n +1 ;m;l + Q n +1 ;m;l D QW a W n;m;l + Q n;m +1 ;l D QW b W n;m;l + Q n;m;l +1 D QW c W n;m;l + Q n;m +1 ;l D QW W n;m +1 ;l + Q n;m;l +1 D QW W n;m;l +1+{9 Q n;m;l D QW a W n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ;m;l + Q n;m;l D QW b W n;m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ;l + Q n;m;l D QW c W n;m;l )]TJ/F15 11.9552 Tf 11.955 0 Td [(1+ Q n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ;m;l D QW W n )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ;m;l + Q n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ;m;l D QW a W n;m;l + Q n;m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ;l D QW b W n;m;l + Q n;m;l )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 D QW c W n;m;l + Q n;m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ;l D QW W n;m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ;l + Q n;m;l )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 D QW W n;m;l )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ApplytheRosenstock-Newellmodel[50],meaningQ=W,completingthesquaresandthen simplifyingreducesU harm to: U harm = 1 2 X n;m;l )]TJ/F21 11.9552 Tf 11.955 0 Td [(X n +1 ;m;l 2 + X n;m;l )]TJ/F21 11.9552 Tf 11.955 0 Td [(X n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ;m;l 2 E xx a {10 + Y n;m;l )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y n;m +1 ;l 2 + Y n;m;l )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y n;m )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ;l 2 E yy b + Z n;m;l )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z n;m;l +1 2 + Z n;m;l )]TJ/F21 11.9552 Tf 11.955 0 Td [(Z n;m;l )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 E zz c Thenegativeofthepotentialisdierentiatedtwicewithrespecttooneofthe coordinatesandinsertedintowhat'sessentiallyNewton's2ndLaw.Thisyieldsthe equationsofmotion,whicharefoundlaterinthisthesis.Thediversityofspringconstants 20

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mustbeinvestigated.ReturningtotheLennard-JonespotentialandevaluatingE QW : E QQ R )]TJ/F21 11.9552 Tf 11.955 0 Td [(R 0 =24 6 28 Q 2 r 16 )]TJ/F15 11.9552 Tf 17.508 8.088 Td [(2 r 14 6 + 1 r 8 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(8 Q 2 r 10 {11 Thereare6uniquespringconstantsthatcomefromtheequationabove: k x = E xx a =24 6 26 6 a 14 )]TJ/F15 11.9552 Tf 15.662 8.088 Td [(7 a 8 {12 k y = E yy b =24 6 26 6 b 14 )]TJ/F15 11.9552 Tf 15.079 8.088 Td [(7 b 8 {13 k z = E zz c =24 6 26 6 c 14 )]TJ/F15 11.9552 Tf 15.109 8.088 Td [(7 c 8 {14 k a = E yy a = E zz a =24 6 )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 6 a 14 + 1 a 8 {15 k b = E xx b = E zz b =24 6 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 6 b 14 + 1 b 8 {16 k c = E xx c = E yy c =24 6 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 6 c 14 + 1 c 8 {17 Sincemymodelsstudyparametersonthebasisofspringconstantsandnot and ,Ijust assumethese6constantscanbearbitrarilyaltered. NowthattheEOMandspringconstantshavebeendetermined,computethe transmissionfunction.Thisisnecessarytocomputethethermalconductivityandmean freepath.Transmissioninthisthesisdiersabitfromthestandarddenitionwherethe maximumvalueoftransmissioncanassumeisunity.Transmissioninthisdocumenthas amaximumtheoreticalvalueequaltothenumberofincidentmodesthatareallowedfor agivenfrequency.Thetransmissionandreectionarecomputedforeachincidentmode, whichmeansthatsumoftransmissionandreectionforeachincidentmodeisunity, howeverthesumofthesequantitiesforallincidentsmodesequalsthetotalnumberof modes. In1-D,thetransmissionandreectionare: T = j x N +1 j 2 j A j 2 {18 21

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R = j x 0 j 2 j A j 2 {19 Wherethesubscripts0andN+1meanthepartofthesolutiontotheEOMthatisthe reectedortransmittedpartofthewavesinthebaths.In2-Dorhigherdimensionsitis irrelevantwhichatomsinagivenlayerofthebathareselected,asthequantitiesx 0 and x N +1 willbedividedbytherelevanttrigonometricfunctionfortheatom'spositionwithin alayerseelaterchaptersforfurtherdetails. Asstatedearlier,theobjectiveofthisthesisistoreducethermalconductivity.This hasunitsof[51]: K = rateofheatflow [ W ] objectlength [ m ] crosssection [ m 2 ] temperaturedrop [ K ] {20 Foraboutacentury,thermalconductivitywasconsideredamaterialpropertywith somesmallchangesfortemperature,andhencethisthesiswouldhavebeenconsidered irrelevant.Thegeometrywasassumedtosomehowcancelout.Nowthatnanoscale fabricationisareality,itispossibletocreatematerialsinwhichgeometrysizeandshape ofsampledoesaectthermalconductivity.Hencethisthesisisrelevant. Thelastquantitycomputedisthemeanfreepathl.Atarudimentarylevel,the meanfreepathistheaveragedistanceanobjectcantravelbeforeacollisionoccurs.In thisproblem,thetravelingobjectsarephononsandthecollisionsrefertoscatteringo anatom'spotential.Thereasonforcomputingmeanfreepathistobetterunderstand thephysicsofthesituationdescribedinthisthesis.Onecancreateanexpressionforthis quantitywithoutevenunderstandingthephysicsbehinditbythefollowingreasoning:In adisorderedsample,asthelengthgoestoinnitythetransmissionshouldgotozero.The reasonisthatwithinaninnitesamplelength,acollisionshouldoccurandhence: T l L {21 Theproblemwiththisrelationshipisthatasthesamplelengthgoestozero,the transmissionbecomeinniteinsteadofapproachingunity.Anadjustmentisneededtothe 22

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aboveformula: T = l L + l {22 BytakingthelimitasL !1 ,T ! 0andasL ! 0,T ! 1. 1.31-DCase Inthiscasetherearemonatomicinniteheatbathsontheleftandrightendsofthe sample.ThesamplehasNatoms,andallthemassescanbedierent,howeverallthe springconstantsarethesameintheprogramIwrote.Thereisanincidentplanewave fromtheleftandnoincidentwavefromtheright.Theheatbathsarelocatedatthe layerscorrespondingtox=0and x = N x +1 a ,where a isthelatticespacing.The transmission,reection,andtimeaveragedpoweroftheleadswillbecomputed.Since thereisonlyonemodeintheheatbathsperincidentangularfrequency,thephysicsis quiteabitdierentthanthe2-Dand3-Dcases. 1.42-Dand3-DCases Therearemonatomicsemi-inniteheatbathsontheleftandrightendsofthe sample.Thismeansin2-DthatthereisanitenumberofatomsrunningintheY direction,butthatthesebathsextendinnitelyintheXdirectionofeithersideofthe sample.For3-Dbaths,thereareanitenumberofatomsintheY-Zplane,butit'sstill inniteintheXdirectiononeithersideofthesample.Inthe2-Dand3-Dcases,Iusethe scatteringboundaryconditionfortheleads,openboundaryconditionsfortheedges,and theRosenstock-NewellmodelfortheEOM.The1stoftheseallowsformodetransitions. In2-Dand3-Dtherearemodetransitionsifthereisdisorderinthelattice;this,and thefactthattherearemultipleallowedincidentmodesforonefrequency,arethemajor physicaldierenceswhencomparedtothe1-Dcase. Theopenboundaryconditionmeanstheperpendicularwavenumbersisare quantizedexplainedlater.Edge"meansy=borN y borz=corN z c.Open"means thattheedgeatomsarefreetovibrate.Thisisdierentfromaxed"endatoms' boundaryconditionortheverycommonlyusedperiodicboundarycondition. 23

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AsforthethirdconditionRosenstock-Newellmodel,itmeansthatthemotionin theX,Y,andZdirectionsaren'tcoupled;however,therearemultiplespringconstants. SpecicallyfortheX-EOMtherearetwospringconstantsin2-Dand3springconstants in3-D.Onespringconstantisforcouplingtheatomswhoserestsitesareleftandright Xdirectionoftheatomofinterest.Theotherspringscorrespondtoatomswhoserest sitesareupanddownYdirection,oraboveandbelowZdirection.TheY-EOMand Z-EOMaresimilarinthattherearetwoadditionalspringconstantsfor2-Dandthree constantsin3-D.In3-D,thereareingeneralatotalof6springconstants. Forthesample,thereareN x atomsintheXdirection,N y atomsintheYdirection, andN z atomsintheZdirection.Theheatbathsarelocatedatthelayerscorresponding tox=0andx= N x +1 a ,where a isthelatticespacingintheXdirection.TheEOM fortheatomsontheedgesarethesameastheinterior,excepttheyhavefewerthan4 -Dor6-Dotheratomstointeractwith;thesetermsareremovedfromtheEOM. 1.5SkutteruditeCases TheunitcellofagenericskutteruditeoftheformMX 3 contains32atoms.Mis typicallyCoandXisusuallyAsorSb.Thismodelisanexpandedversionoftheonein thelastchapter-Drectangularlattice,withevenmorespringsaddedtoreectthefact thattherearerectanglesinsidethecubes.Inreality,theserectanglesarealmostsquare. ForCoSb 3 thesidesare2.85and2.97angstroms[19].IconnectedtheMcubiclatticeto thecagedXatomswithspringconstantk 1 .TheXatomsareconnectedtotheirnearest neighborXatomswithspringconstantsk 3 ork 4 .Alsothereisaninteractionbetween thenearestplanesofXatoms.Thatis,anXatominagivenrectangleinteractswiththe nearestXatomsinaanotherrectanglewithspringconstantk 5 ork 6 .Thismodeldoesn't consider2ndnearestneighborXinteractions,eventhoughthisdistanceismuchlessthan thedistanceassociatedwiththek 5 andk 6 parameters. 24

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1.61-DO"AtomCases Thepurposeofthissectionistoillustratewhyaspraycouldreducetransmission.It alsopredictsatwhichfrequencythetransmissiondropso.Fromthis,onecanextrapolate tohigherdimensions. Inthiscasetherearemonatomicinniteheatbathsontheleftandrightendsofthe sample.Thesamplehasonetothreeatomsinachainalongwithoneortwoo"atoms. Deneo"atomstomeanatomsthatcoupledtotheatomsinthechainandothero" atoms,butnottotheheatbaths.Thereisanincidentplanewavefromtheleftandno incidentwavefromtheright.Theheatbathsarelocatedatthelayercorrespondingtox= 0and x = N x +1 a ,where a isthelatticespacing. 1.7SprayedSkutteruditeCases Takethesituationforskutteruditesandthenplaceatomsontheoutsideofthefour sidesofthesurfaceoftheskutteruditethataren'tattachedtotheheatbaths.These atomsaretobeplacedadistancez"fromthesurfaceofthecentersofthecubicpartof theskutterudite.ImagineanFCClatticewherethecubicpartrepresentsthecubicpart oftheskutteruditeandwherethefacedcenteredatomssprayed-onatomsaredragged outwardsadistancez"insteadofbeingleftonthesurfacesofthecubes. Therearetwoextrainteractionsconsideredwhencomparedtoanordinaryskutterudite: k ss whichistheinteractionbetweensprayed-onatomsandk sc whichistheinteraction betweenthesprayed-onatomsandcubiclattice.Brieytestedaretheinteractions betweenthesprayed-onatomsandtherectanglescagedXatoms. 1.8SprayedCubicLattices Atthispoint,themotivationforsprayingandtheeectsofauniformsprayhave alreadybeenexplored.Thissectiondiscusseshowtransmissionscaleswithsamplelength andcrosssection.Thenadisorderedsprayistestedtoseeiftransmissionandthermal conductancewilldropasafunctionoflength.Lastly,themeanfreepathisobtainedfor thedisorderedspraycase. 25

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Figure1-1.VoyagerRTGs.TheRTGsusedintheVoyagermissions[47].Thehotregionis madeof24100Wceramicspheresofplutoniumoxideandthecoldregionisa convertercasemadeofBerylliumwithanirontitanatecoating.What's consideredthesampleisfairlycomplicatedasitsmadeofmultiplematerials. 26

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Figure1-2.Theunitcell[52]ofCoSb 3 . 27

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CHAPTER2 1-DCHAINS,2-DAND3-DHARMONICRECTANGULARLATTICESWITHMASS DISORDER Inthischapterwepresentresultsforrectangularlatticeswithmassdisordersothat wecanlatercomparetheseresultstothecaseofresonantscattering.Wealsointroducein somedetailthemethodologyusedforcomputingthetransmissionprobabilities.Westart withthesimplestcaseofaonedimensionalchainandthenproceedtotwodimensional andthreedimensionallattices. 2.11-DChains LeftandrightbathsareassumedtobeattemperaturesT L andT R .Becausethe systemisharmonic,themodesareindependent.Hereweshowhowtondamodeand computeitstransmissionprobability. Assumethatthereisanincidentplanewaveintheleftbathgivenby:X i x,t= Ae iqx )]TJ/F22 7.9701 Tf 6.586 0 Td [(iwt ,where q isthewavenumber, w istheincidentangularfrequency,and x isthe positionoftheatom'srestsite.Forthisonedimensionalcasethepositionis x = na , wherenistheintegersitelocation.Toobtainthedispersionrelation,substitutetheplane waveX i x,tintotheequationofmotionforthebaths m B d 2 X n dt 2 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 kX n + kX n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + kX n +1 ; {1 wherethespringconstantisk,andm B isthemassofaheatbathatom.Thisresultsin thefollowingequation: )]TJ/F21 11.9552 Tf 11.955 0 Td [(m B w 2 e iqna )]TJ/F22 7.9701 Tf 6.586 0 Td [(iwt A = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 ke iqna )]TJ/F22 7.9701 Tf 6.587 0 Td [(iwt A + ke iq n +1 a )]TJ/F22 7.9701 Tf 6.586 0 Td [(iwt A + ke iq n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 a )]TJ/F22 7.9701 Tf 6.587 0 Td [(iwt A {2 Dividingthroughby Ae iqna )]TJ/F22 7.9701 Tf 6.587 0 Td [(iwt yields: )]TJ/F21 11.9552 Tf 11.955 0 Td [(m B w 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 k + ke iqa + ke )]TJ/F22 7.9701 Tf 6.587 0 Td [(iqa : {3 28

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Simplifyingtheaboveequationresultsin: 1 )]TJ/F21 11.9552 Tf 13.15 8.088 Td [(m B w 2 2 k = e iqa + ke )]TJ/F22 7.9701 Tf 6.586 0 Td [(iqa 2 {4 Noticethattheright-hand-sideisacosineandthensolveforq: q = arccos )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(m B w 2 2 k : {5 Fortheincidentwavesonlyrealwavevectorsareconsidered.Complexwavevectors willlaterbeconsideredfortheoutgoingwaves,creatingevanescentdecayingmodes.The reectedwavehastheform: X r x;t = Be )]TJ/F22 7.9701 Tf 6.586 0 Td [(iqx )]TJ/F22 7.9701 Tf 6.587 0 Td [(iwt {6 andthetransmittedwavehastheform: X t x;t = Ee iqx )]TJ/F22 7.9701 Tf 6.586 0 Td [(iwt {7 TheunknownreectedandtransmittedamplitudesareBandE.Tondthesewewill havetousetheequationsofmotionfortheatomsinthesample: m n d 2 X n dt 2 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 kX n + kX n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + kX n +1 ; {8 where n inthesampleregiongoesfrom1toN.Thus,thereareNequationshere;however, thereare N +2unknownsbecauseinadditionto X 1 , ::: , X N ,thereareBandEforthe reectedandtransmittedwaves,respectively. TwomoreequationscanbegeneratedbywritingX 1 x,tintermsofBandX n x,t intermsofE.Thiscanbedonebywritingtheequationofmotionforatoms0andN+ 1andsolvingforatoms1andN.Wewillseethatthisisverysimilartothescattering boundaryconditionusedlaterinhigherdimensions. Lettingtheincidentamplitudebeunityforthissimpleonedimensionalcase,the transmissionprobabilityis T = j E j 2 andthereectionprobabilityis R = j B j 2 .BandE arecomputedindependentlyi.e.notsimplysettingR=1 )-224(j E j 2 .Ifthesamplehasmass 29

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disorder,thetransmissionwilltypicallybelowerthanforanorderedcase.Disordercan beviewedasachangeinmediumeverytimeanatomofadierentmassisencountered. ShowninFig.2-1isaplotwithoutdisorder,andthenthenextinFig.2-2isaplotwith disorder. Withthetransmissionobtained,aformulaforathermalconductancemustnowbe derived.WewillusetheLandauerformalism,meaningthestartingpointistheLandauer equationforthermalcurrent[51].Thisthermalcurrentisequalto: I = Z W 0 ~ wT w h n s )]TJ/F21 11.9552 Tf 11.955 0 Td [(n d ; d ~ w {9 wheren s andn d aretheBose-Einsteinfunctionsofthebaths, ~ isPlanck'sconstant,and W isthecut-ophononbandwidth.Nowmultiplythenumeratoranddenominatorby T,whichisthetemperaturedierenceinbetweenthebaths.Letthedierenceofthe Bose-Einsteinfunctionsdividedbythetemperaturedierencebecomeapartialderivative. RecallthattheBose-Einsteinfunctionis: n = 1 exp ~ w k B T L )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 {10 whereT L istemperatureoftheheatbathsandk B isBoltzmann'sconstant.Nowthe thermalcurrentequals: I = Z W 0 ~ 2 w 2 T w hk B exp ~ w k B T L T 2 L exp ~ w k B T L )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 ; d ~ w {11 Thenusethefollowingbasicrelationshipbetweenthermalcurrent,conductance,and temperaturedierence: I = )]TJ/F21 11.9552 Tf 9.298 0 Td [(J T {12 SolvingfortheconductanceJyields: J = Z W 0 ~ 2 w 2 T w 2 k B exp ~ w k B T L T 2 L exp ~ w k B T L )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 d w {13 30

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2.22-Dand3-DHarmonicLattices Inthissectionwewillanalyzeathreedimensionalharmoniclatticewithmass disorder,generalizingtheonedimensionalcasedescribedintheprevioussection.Thecase ofatwodimensionallatticeisobtainedinthesameway. TheequationsofmotionfortheXdisplacementwithinaleadis m B d 2 X n;m;l dt 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 k x + k b + k c X n;m;l + k x X n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;m;l + k x X n +1 ;m;l + k b X n;m )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;l + k b X n;m +1 ;l + k c X n;m;l )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + k c X n;m;l +1 {14 andsimilarlyfortheYandZdisplacements m B d 2 Y n;m;l dt 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 k a + k y + k c Y n;m;l + k a Y n )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;m;l + k a Y n +1 ;m;l + k y Y n;m )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;l + k y Y n;m +1 ;l + k c Y n;m;l )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 + k c Y n;m;l +1 {15 m B d 2 Z n;m;l dt 2 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 k a + k b + k z Z n;m;l + k a Z n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;m;l + k a Z n +1 ;m;l + k b Z n;m )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;l + k b Z n;m +1 ;l + k z Z n;m;l )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + k z Z n;m;l +1 {16 Rememberthesearedecoupledastheequationsarederivedinchapter1.2.Hereasite n,m,lindicatestheX,Y,Zatomnumber.Thesitemcanvarybetween1andN y ,and thelsitevariesbetween1andN z .Therearesixspringconstants.The1stspringconstant k x [k a ]isusedintheX[YandZ]equationofmotionthatisforatomswhoserestsites areleftorrightoftheatomofinterestXdirection.Thenextspringconstantisk b [k y ], thespringconstantusedintheXandZ[Y]equationofmotionthatisforatomswhose restsitesareaboveorbelowoftheatomofinterestYdirection.Thenalsetofspring constantsarek c [k z ],thespringconstantusedintheXandY[Z]equationofmotionthat isforatomswhoserestsitesareaboveorbelowoftheatomofinterestZdirection. Thesolutionsinthetransversedirections,yandz,aredeterminedbythefactthat sitemvariesbetween1andN y andsitelvariesbetween1andN z .Therearenoatomsfor m =0andfor m = N y +1,andconsequentlytherearenospringsconnecting m =0and 31

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m =1orconnecting m = N y and m = N y +1.Thesolutionsinthetransverseyandz directionshavetheformcos q y m )]TJ/F15 11.9552 Tf 12.454 0 Td [(1 = 2 b andcos q z l )]TJ/F15 11.9552 Tf 12.455 0 Td [(1 = 2 c ,where b and c arethe latticespacingsinthetransversedirectionsand q y = n N y b {17 q z = n N z c {18 wheren=0:N y -1forq y andn=0:N z -1forq z .Thisiscalledanopenboundary condition. Asintheonedimensionalcasethewavefunctionparalleltothewirehastheform e iq x x ,where q x isagainobtainedbysubstitutingtheplanesolutionintotheequationsof motionforthebaths. Theresultingwavevectorq x isgivenby q x = acos )]TJ/F22 7.9701 Tf 13.151 5.045 Td [(m B w 2 2 k x + k b )]TJ/F22 7.9701 Tf 6.586 0 Td [(cos q y b k x + k c )]TJ/F22 7.9701 Tf 6.587 0 Td [(cos q z c k x 3 a + acos )]TJ/F22 7.9701 Tf 13.151 5.045 Td [(m B w 2 2 k a + k y )]TJ/F22 7.9701 Tf 6.586 0 Td [(cos q y b k a + k c )]TJ/F22 7.9701 Tf 6.587 0 Td [(cos q z c k a 3 a {19 + acos )]TJ/F22 7.9701 Tf 13.15 5.046 Td [(m B w 2 2 k a + k b )]TJ/F22 7.9701 Tf 6.586 0 Td [(cos q y b k a + k z )]TJ/F22 7.9701 Tf 6.587 0 Td [(cos q z c k a 3 a Thefactorof3arisesfromthefactthatthereare3EOMthatneedtobesatisedfor eachatomoneperdimension.Ifk a =k x ,thenthersttwoacostermsandthefactor of3canbeomitted.Thestrategynowissimilartotheonedimensionalcase.Intheleft leadwehaveanincomingwaveandreectedwave: X n;m;l = A q cos q y m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 = 2 b cos q z l )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 = 2 c e iq x na + q X r =1 R q cos q y m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 = 2 b cos q z l )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 = 2 c e )]TJ/F22 7.9701 Tf 6.586 0 Td [(iq x na {20 Intherightleadwehaveatransmittedwave X n;m;l = q X r =1 T q cos q y m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 = 2 b cos q z l )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 = 2 c e iq x na {21 32

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Thesummationisovereverysetq x ,q y ,q z thatisaplanewavesolutioninthebaths. The R q andthe T q arethegeneralizedreectionandtransmissioncoecients.TheA q isnotsummedoverbecauseitrepresentstheincomingwave.Whensolvingthematrix equationforallthereectionandtransmissioncoecients,onlyoneincidentmodeata timeisconsidered.Howeverthismodecantransitionintoothertransversemodes. Wenowcountthenumberofunknownsandthenumberofequations.Thereader mightthinkthatthereareN x N y N z amplitudesX n;m;l inthesample,N y N z reection coecients,andN y N z transmissioncoecients.HoweverthereareonlyN x N y N z equations ofmotionforthesampleregion.Theresolutionofthisissueisthatthereare2N y N z equationsrelatingtheamplitudesofthex=aandx=N x alayerstothereectionand transmissioncoecientsviatheEOMofthebathatoms'layersx=0andx=N x +1a. Thesetofequationsisalsoreferredtoasthescatteringboundarycondition.Withthe scatteringboundaryconditionthetotalnumberofequationsisN x N y N z ,whichisequalto thetotalnumberofunknowns. Thetotaltransmissionandreectionprobabilitiesforasingleincidentmodeisgiven by: T = N X q =1 V q T q V i {22 R = N X q =1 V q R q V i {23 V q = k x asin q x a m B w {24 Here,Nisthenumberofmodeswitharealq x , V q isthevelocityofamodethathasa realq x ,T q isthetransmissioncoecientofthatmode,R q isthetransmissioncoecient ofthatmode,andV i isthevelocityoftheincidentmode.Thesumof T and R isone forasingleincidentmode.Formultipleincidentmodeswesumoverthetransmission probabilitiesforeachmode,meaningTisingeneralgreaterthanone.Forexample,if thereare5incidentmodes,themaximumallowedtransmissionis5. 33

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Fig.2-3showstheeectsofmassdisorderinthesample.Ifthereismassdisorder, modetransitionswilloccur.Roughlyspeaking,themodechangesallowresonant scatteringdescribedlater,whichwilldecreasetransmission.Ifcheckingthelimiting caseastheamountofdisordergoestozero,theaectontransmissionwillbecome negligibledonetocheckthenumericalstabilityofmycode.Theamountofdisorderis denedasthevariationinthebasesamplemass.Ifthebasesamplemassis1.3,Disorder =0.3"referstomassesinthesamplerandomlydistributedbetween1.3-1.6andDisorder =1.0"referstomassesinthesamplerandomlydistributedbetween1.3-2.3. Lastly,takethedatashownFig.2-3anddividethetransmissionbythenumberof incidentmodesseeFig.2-4.Thisquantitywillbecalledmodenormalizedtransmission. Moredisorderstillmeanslowermodenormalizedtransmission.Howeveratlowenergy, thereismoreofanimpactwithdisorderthanisapparentfromjustaplotoftransmission. Note:ifonelooksaheadattheskutteruditechapteritshouldbeapparentthatcubic massdisorderisfarlesseectiveatsuppressingtransmissionatlowerenergiesthancaged atoms. 34

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Figure2-1.Transmissionthrougha1-DChain.Thisisatransmissionplotfornodisorder inthe1-Dsample.IntheunitsIuse,N=15,m B =1,basemassofallchain atomsis1.3,k=12.2624,a=1,andA=0.03*i+0.1. Figure2-2.Transmissionthrougha1-DChainwithdisorder.Thesameastheprevious gure,exceptthebasemassnowhasdisorder.Thesampleatomshave randomlydistributedmassesbetween1.3and1.6.Disorderissupposedto decreasetransmission,whichforthemostpartisthecaseinthisplot. 35

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Figure2-3.Transmissionthroughacubiclatticewithdisorder.Thisisatransmissionplot varyingamountsofdisorderrandomnessintermsofthemassesinthe3-D sample.Intheunitschosen,N x =N y =N z =5,m B =1,basemassofall sampleatomsis1.3,k x =k a =12.2624,k b =k y =k c =k z =5.7425,a=b= c=1. Figure2-4.ModeNormalizedTransmissionthroughacubiclatticewithdisorder.Inthe unitschosen,N x =N y =N z =5,m B =1,basemassofallsampleatomsis 1.3,k x =k a =12.2624,k b =k y =k c =k z =5.7425,a=b=c=1. 36

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CHAPTER3 SKUTTERUDITESOFTHEFORMMX 3 Inthissectionwestudytransmissionthroughaskutteruditeregionplacedbetween twoperfectcubiclatticeleads.Theresultsherewillbecomparedandcontrastedwith thosefromthemassdisordercase.ThegeometryisshowninFig.3-1[53]. Thenameskutteruditeoriginatesfromthefactthattheywerefoundasanaturally formingmineralCoAs 3 inSkutterud,Norway[1].Itwasdiscoveredthatthisparticular substancehasahighgureofmeritZT,meaningithasahighthermoelectriceciency. Whyisthisthecase?SlackindicatesthatthecriteriaforahighZTvalueare[54]:a sizableunitcellinthiscase32atoms,largeconstituentatommasses.93amuforCo and74.92amuforAs[55],smallelectronegativitydierencesbetweentheatoms,and largecarriermobility. Substancesofthisformalsohavevoids,ascanbeseeninFig.3-1,inwhich2of the8cagesareemptyinsteadofbeinglledinwithXatomslikeAs.Bernsteinetal. statethatthesevoidscanbelledbyavarietyofatoms,suchasLaandFe[56].The reasonforwantingtollthesevoidsistoreducethemeanfreepathviaintroductionof phonon-scatteringcenters[57].Fillingskutteruditecagestoreducethemeanfreepathis notexploredinthisthesis,howeveradisorderedsurfacesprayisinvestigateditachievesa similarresultintermsofreducingthermalconductivity. NotethatthisthesisonlyconsidersbinaryskutteruditesoftheformMX 3 .This familyofskutteruditesinadditiontocontainingCoAs 3 ,alsoincludes:CoP 3 ,CoSb 3 ,RhP 3 , RhAs 3 ,RhSb 3 ,IrP 3 ,IrAs 3 ,andIrSb 3 .AsthereassomanysubstancesoftheformMX 3 , itishelpfultoconsiderwhathappenswhenarbitrarilyalteringthebondstrengthsofthe M-M,M-X,andX-Xbondsexploredinthisthesis. Now,someofthechemicalpropertieswillbediscussed.Uponearlyinspection ofsubstancesoftheformMX 3 ,itwasfoundthatthesematerialsarediamagnetic semiconductorsnounpairedspins.TheexceptionisNiP 3 ,inwhichmetallicconductivity 37

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andparamagnetismhasapermanentmagneticmomentoccur[1].EachXatomhas5 valenceelectrons.ThisisobviousforsomeoftheXatomchoices;P,As,Sbareinthe samecolumnoftheperiodictable.HulligerindicateswithinthesquaresofX,thebonds are typeusesup2valenceelectronsandtheM-Xbondstakeuptheremainingvalence electrons[58].AllofthemetalatomsareoctahedrallycoordinatedbytheXatoms. 3.1AssumingSquareXGeometry ThespringinteractionsareshownonapictureoftheskutteruditeinFig.3-2.Recall thatthissectionisasimpliedversionofthenextsectionlessspringtypes. Thebasictechniqueemployedinthischapteristhesameaspreviouschapterfor the3Dlattice.AssumesquaresofXareinthecubicMlattice.Thismeansk 3 =k 4 and k 5 =k 6 .Theequationsaresimilartothosefoundinthe3-Dlatticechapter.Justadd theappropriatetermsfornearbyXatomsfortheMatoms'EOM.Iavoidwritingdown equationsbecausewhenIcodedthis,thenotationisdiculttoexpress.Icreatedtwo matriceswhichstoretherestsitesoftheMandXatoms.ThenIcreatedanothermatrix foralltheEOMoftheMandXatoms,whichtookatomsthatwereseparatedbyspecic distancesandinsertedtheappropriatespringconstants.Atomsnumbersarechosenbased ontherestsitesstoragematrices. Therstquestionposedis,howisthetransmissionthroughaskutteruditedierent fromstaticdisorder?Cubicdisorderjustsuppressesthehigherendofthefrequency spectrum.Ontheotherhand,skutteruditescandrasticallydecreasetransmissionatlower frequenciesandactuallyincreasetransmissionathigherenergyvalues.Thespectrumof theskutteruditelooksshiftedtotherightwhencomparedtothecleancubiclattice. Fig3-4throughFig.3-7changeonepropertyoftheskutteruditeatatimeandthen seewhatchangesoccurinthetransmissionfunction.Itturnsoutthatalteringk 1 achieves themostsignicantdecreaseintransmission.Increasingk 3 ,ork 5 onlyseemstoalterthe transmissionfunctionatlowfrequencies,ratherthanadjustingthewholespectrum.In addition,increasingm Sb onlyseemstohaveasmalleect. 38

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Finally,comparethethermalconductanceofacleancubiclatticetoskutteruditeFig. 3-8.Asareminder,thermalconductanceisdenedas: J = Z W 0 ~ 2 w 2 T w 2 k B exp ~ w k B T L T 2 L exp ~ w k B T L )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 d w {1 WhereT L istemperatureoftheheatbaths,Wisthecut-ophononbandwidth, k B isBoltzmann'sconstant,and ~ isPlanck'sconstant.Integrateoverallprovided frequenciesusingthetrapezoidapproximation.Thepresenceofthesquaresinthe skutteruditereducesthethermalconductance. 3.2RectangularXGeometry Byusingrectanglesinsteadofsquares,therearetwoadditionalspringconstants,k 4 associatedwiththelongerbondandk 6 associatedwiththelongerbond.Thisisa morerealisticLKmodel,asalltherelevantspringconstantsarenowincluded.Thereare dierencesinthetransmissionfunctionatlowerfrequencies,howeverthelast40percentof thespectrumappearsidentical,withmuchoftherestofthespectrumbeingsimilar.This comparisonisshowninFig.3-9. 39

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Figure3-1.AtiltedofviewofaskutteruditeoftheformMX 3 . Figure3-2.SkutteruditeInteractions.Thebluelineisk 1 ,theredlineisk 3 ,andthegreen lineisk 5 . 40

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Figure3-3.ComparisonoftransmissionthroughM,Mwithdisorder,andMX 3 .Thisisa transmissionplotforMwithandwithoutmassdisorderandMX 3 without disorder.Intheunitschosen,N x =N y =N z =5,m B =1,basemassofallM atomsis1.3,basemassofallXatomsis1.7,k x =k a =12.2624,k b =k y =k c =k z =5.7425,a=b=c=1.TheMandXinteractionparametersarek 1 = 3.0,k 3 =0.7andk 5 =0.3. 41

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Figure3-4.ComparisonoftransmissionthroughMandMX 3 withvaryingk 1 .Thisisa transmissionplotforMandMX 3 .Intheunitschosen,N x =N y =N z =5,m B =1,basemassofallMatomsis1.3,basemassofallXatomsis1.7,k x =k a =12.2624,k b =k y =k c =k z =5.7425,a=b=c=1.TheMandX interactionparametersarek 3 =0.7andk 5 =0.3. 42

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Figure3-5.ComparisonoftransmissionthroughMandMX 3 withvaryingk 3 .Thisisa transmissionplotforMandMX 3 .Intheunitschosen,N x =N y =N z =5,m B =1,basemassofallMatomsis1.3,k x =k a =12.2624,k b =k y =k c =k z = 5.7425,a=b=c=1.TheMandXinteractionparametersarek 1 =3.0and k 5 =0.3. 43

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Figure3-6.ComparisonoftransmissionthroughMandMX 3 withvaryingk 5 .Thisisa transmissionplotforMandMX 3 .Intheunitschosen,N x =N y =N z =5,m B =1,basemassofallMatomsis1.3,k x =k a =12.2624,k b =k y =k c =k z = 5.7425,a=b=c=1.TheMandXinteractionparametersarek 1 =3.0and k 3 =0.7. 44

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Figure3-7.ComparisonoftransmissionthroughMandMX 3 withvaryingm Sb .Thisisa transmissionplotforMandMX 3 .Intheunitschosen,N x =N y =N z =5,m B =1,basemassofallMatomsis1.3,k x =k a =12.2624,k b =k y =k c =k z = 5.7425,a=b=c=1.TheMandXinteractionparametersarek 1 =3.0,k 3 =0.7,andk 5 =0.3. 45

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Figure3-8.Thermalconductanceofacubiclatticetoaskutterudite.Thebluecurveis thecleancubiclatticeandtheredcurveistheskutterudite.Notethatthe latticethermalconductanceoftheskutteruditeislessthanthecubiclattice. Intheunitschosen,N x =N y =N z =5,m B =1,basemassofallMatomsis 1.3,k x =k a =12.2624,k b =k y =k c =k z =5.7425,a=b=c=1.TheM andXinteractionparametersarek 1 =3.0,k 3 =0.7,andk 5 =0.3.Themass oftheXatomsis4.Bothcurvesaremultipliedbyaconstant,whichisthe sameforbothdatasets. 46

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Figure3-9.TransmissionforsquareXgeometryvsrectangularXgeometry.Thisisa transmissionplotforMX 3 withsquareandrectangularX.Intheunitschosen, N x =N y =N z =5,m B =1,basemassofallMatomsis1.3,basemassofall Xatomsis1.7,k x =k a =12.2624,k b =k y =k c =k z =5.7425,k 1 =3,a=b =c=1.TheMandXinteractionparametersarek 3 =0.7,k 4 =0.5,k 5 =0.3 andk 6 =0.1. 47

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CHAPTER4 1-DOFF"ATOMSANALOG Inordertounderstandwhyphonontransmissionthroughaskutteruditeisnotthe sameasacubiclattice,lookato"atomsinsimple1-Dsetups.Therearefourcasessee Fig.4-1.CaseIisasingleatominsertedinbetweentwoinniteheatbaths.Thisatom hasamassdierentfromthebathatommasses.CaseIIusesathreeatomsample,where oneofthespringsinthesampleisn'tthesameasthoseintheheatbaths.Thesersttwo casesarelikenedtomodifyingacubiclattice.CaseIIIhasathreeatomsamplewithone ofthoseatomsbeingano"atom.Thiso"atomisn'tcoupledtothebaths,butis coupledtotheotheratomsinthesample.CaseIVhasatwoatomsample,buttheo" atomiscoupledtotheotheratominthesampleandnottotheheatbaths.CaseIIIis mostanalogoustotheskutterudite. ForCaseI,thetransmissioncoecientisgivenby: T = 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 cos qa 2 k x )]TJ/F22 7.9701 Tf 6.587 0 Td [(m C w 2 k x 2 +4 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 2 k x )]TJ/F22 7.9701 Tf 6.586 0 Td [(m C w 2 k x cos qa {1 NoticethatthereisnodipinthetransmissionpredictedshowninFig.4-2.Thismeans thatthereisnopointinthetransmissioncurveatwhichthetransmissiondropstozero asidefromwhenthereisnoallowedincidentmode.WhyistherenodipinCaseI? Apotentialdipshouldoccurwhenthenumeratoroftheanalyticexpressionisequalto zero.Thersttimesuchadipcouldoccurisatthemaximumallowedfrequencyinthe bath.Thismeansthedipoccursatthemaximumfrequency,andishencenotadipsince transmissioniszeroatanyfrequencygreaterthanthisvalue. TheformulasforCasesIIandIIIaretoolengthyevenifoneassumesthemasses ofthesampleatomsareidenticaltothoseinthebathsexcepto"atoms.Testedare varyingk 1 forCaseIIFig.4-3andm P Fig.4-4andk P Fig.4-5forcaseIII.Notice thatforCaseIIIthatthereisadip. 48

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TheCaseIVformulausedbelowallowsforchainando"atommassestobe dierentfromthemassesinthebaths.NoticethatthereisadipseeFig.4-6andFig. 4-7. T = 4 k 2 x )]TJ/F21 11.9552 Tf 9.299 0 Td [(m P w 2 + k P 2 sin qa 2 A 11 + A 22 {2 Intheaboveformula,A 11 andA 22 aregivenby: A 11 =4 k 2 x k P )]TJ/F21 11.9552 Tf 11.955 0 Td [(m B w 2 2 + )]TJ/F21 11.9552 Tf 9.298 0 Td [(k 2 P + )]TJ/F21 11.9552 Tf 9.298 0 Td [(m C w 2 +2 k x + k P k P )]TJ/F21 11.9552 Tf 11.955 0 Td [(m P w 2 2 {3 A 22 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(4 k x k P )]TJ/F21 11.9552 Tf 11.956 0 Td [(m P w 2 )]TJ/F21 11.9552 Tf 9.299 0 Td [(k 2 P + )]TJ/F21 11.9552 Tf 9.298 0 Td [(m C w 2 +2 k x + k P k P )]TJ/F21 11.9552 Tf 11.955 0 Td [(m P w 2 cos qa {4 WhileCasesIandIIseemdierent,theybothshowageneraldecreaseintransmission withincreaseinfrequency.InCasesIIIandIVthereisadip. ThedipsforCasesIIIandIVarefoundbysettingthenumeratorsintheanalytic expressionsforthetransmissioncoecientsequaltozero.Notethatthebathmassesdon't determinethelocationofthedipsineithercase.Thebathspringconstant,o"atom springconstantsando"atommassareallthatmatters.Thedipsarelocatedat: w CaseIII = s k 2 p +2 k p k x m P k x {5 w CaseIV = r k p m P {6 Forthefullwidthhalfmaximum,thereisanapproximaterelationthatcanbeused ifk p ismuchlessthank x .Thiswasfoundempirically.Otherwise,anexpressionwasnot readilyapparent. w FWHM = 3 : 4328 k p k x {7 MycodeistestedagainsttheseexpressionsinFig.4-9throughFig.4-13. Afterseeingthesefourcases,it'snecessarytoseeifit'spossibletocreatemorethan onedipinthetransmission.CaseVhasthreeordinarysampleatomsandtwoo"atoms. 49

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CaseVIisthesame,exceptthatthetwoo"atomscaninteractwitheachother.The geometryisshowninFig.4-14. ForCaseV,therearetwodipsifeithertheredspringsisn'tthesameaseitherof thedarkgreenspringsoriftheyellowmassisn'tthesameasthepurpleone.Ifneitherof theseconditionsissatised,thereisonlyonedip. w CaseV = s k 2 1 +2 k 1 k x m P 1 k x and s k 2 2 +2 k 2 k x m P 2 k x {8 Thesubscripts1indicatesthelefto"atomandcorrespondingsetofsprings,whereas 2isfortheonesontheright.CaseVIhasaverycomplicatedformulaforthedipsin transmission.Byaddinganinteractionbetweentheo"atoms,evenifthetwosetsof o"atomsandinteractionsarethesame,thetwodipsoccuratdierentfrequencies. ThisisshowninFig.5-15. Extrapolatingtotheskutterudite,thediplocationisapproximatedby: w skutterudite = r 2 k 1 m P {9 Intheformulaabove,k 1 isthespringconstantconnectingthecubeatomstothesquare atoms,andm P isthemassoftheatomsinthesquares.Thisformulaonlyworksifk 1 is muchlessthank x .Ifthisconditiondoesn'thold,thenthereisnosimpleformula. 50

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Figure4-1.CasesI-IVgeometries.Theblackatomsarethebathatoms,theblueatoms arethesampleatomsnoto",andthepurpleatomsaretheo"atoms. Blacklinesarebathconstantspringconstantsandredspringsareanyother springconstant. 51

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Figure4-2.TransmissionforCaseI.VaryingmassesforCaseI.Theparametersusedin thisplotare:m B =1andk x =12.26. Figure4-3.TransmissionforCaseII.Varyingthespringconstantinbetweenchainatoms 1and2forCaseII.Theparametersusedinthisplotare:m B =1andk x = 12.26. 52

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Figure4-4.Varyingo"atommassforCaseIII.Varyingthemassoftheo"atomfor CaseIII.Theparametersusedinthisplotare:m B =1,k x =12.26,andk p = 12.26. Figure4-5.Varyingthespringconstantoftheo"springforCaseIII.Notethatforthe k p =18case,thedipintransmissionshouldoccuroutsideoftheallowed frequencies,sothereisnodip.Theparametersusedinthisplotare:m B =1, m P =1,andk x =12.26. 53

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Figure4-6.Varyingthemassoftheo"atomforCaseIV.Theparametersusedinthis plotare:m B =1,k x =12.26,andk p =12.26. Figure4-7.Varyingthespringconstantoftheo"springforCaseIV.Theparameters usedinthisplotare:m B =1,m P =1,andk x =12.26. 54

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Figure4-8.ComparisonoftransmissionforCasesI-IV.DarkblueisCaseI,darkgreenis CaseII,redisCaseIII,andcyanisCaseIV.Theparametersusedinthisplot are:m B =1,m C =1.3,m P =1.8,k x =12.26,k 1 =3,andk p =12.26. Figure4-9.CaseIIIdiptest.ForCaseIIIthedipoccursatw=2.274andthepredicted isatw=2.2736.ForCaseIVthedipoccursatw=1.491andthepredicted isatw=1.4907.Thebinsizeforwis0.001.Theparametersusedinthisplot are:m B =1,m C =1,m P =1.8,k x =12.26,k 1 =3,andk p =4. 55

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Figure4-10.CaseIIIFWHMtest1.ForCaseIIIthedipoccursatw=0.236andthe predictedisatw=0.2359.TheactualFWHMis0.2359andthepredicted factoristhesame.ForCaseIVthedipoccursatw=0.167andthe predictedisatw=0.1667.Thebinsizeforwis0.001.Theparametersused inthisplotare:m B =1,m C =1,m P =1.8,k x =12.26,k 1 =3,andk p = 0.05. 56

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Figure4-11.CaseIIIFWHMtest2.ForCaseIIIthedipoccursatw=0.334andthe predictedisalsoatw=0.334.TheactualFWHMis0.028andthepredicted factoristhesame.ForCaseIVthedipoccursatw=0.236andthe predictedisatw=0.2357.Thebinsizeforwis0.001.Theparametersused inthisplotare:m B =1,m C =1,m P =1.8,k x =12.26,k 1 =3,andk p = 0.1. Figure4-12.CaseIIIFWHMtest3.ForCaseIIIthedipoccursatw=0.581andthe predictedisatw=0.5809.TheactualFWHMis0.085andthepredictedis 0.084.ForCaseIVthedipoccursatw=0.408andthepredictedisatw= 0.4082.Thebinsizeforwis0.001.Theparametersusedinthisplotare:m B =1,m C =1,m P =1.8,k x =12.26,k 1 =3,andk p =0.3. 57

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Figure4-13.CaseIIIFWHMtest4.ForCaseIIIthedipoccursatw=0.894andthe predictedisatw=0.8944.TheactualFWHMis0.196andthepredicted factoristhesame.ForCaseIVthedipoccursatw=0.624andthe predictedisatw=0.6236.Thebinsizeforwis0.001.Theparametersused inthisplotare:m B =1,m C =1,m P =1.8,k x =12.26,k 1 =3,andk p = 0.7. Figure4-14.CasesVandVIgeometries.Theblackatomsarethebathatoms,theblue atomsarethesampleatomsnoto",andthepurpleandyellowatomsare theo"atomswhichcanbedierentmasses.Blacklinesarebath constantspringconstantsandredanddarkgreenspringsarespringsthat coupletheo"atomstotheordinarysampleatoms.Thelimecoloredlineis aspringthatcouplesthetwoo"atomstoeachother. 58

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Figure4-15.TransmissionofCasesVandVI.Duetotheverysmallbinsizeusedto createthisplot,thedipspredictedforCaseVexactlycorrespondtowhere theyareontheplot.Thesearelocatedatw=2.411568andw=3.05536. Noticethatthetransmissionneverdipsto0forthek P 3 =2.Thismaybe becauseit'spossibletohaveimaginaryrootsinthenumeratoroftheanalytic expressionfortransmission.Theparametersusedinthisplotare:m B =1, m C =1,m P 1 =1.6,m P 2 =1.6,k x =12.26,k P 1 =4,andk P 2 =6. 59

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CHAPTER5 SPRAYINGSKUTTERUDITESOFTHEFORMMX 3 5.1AssumingNoSprayed-XInteraction Basedontheresultsoftheo"atomsinthe1-Dchainsstudiedearlier,my hypothesisisthatbysprayingatomsaroundtheoutsideofaskutterudite,thatthe latticethermalconductancewilldropconsiderablyformuchofthefrequencyspectrum. Placeatomsontheoutsideofthefoursidesofthesurfaceoftheskutteruditethataren't attachedtotheheatbaths.Theseatomsaretobeplacedadistancez"fromthesurface ofthecentersofthecubicpartoftheskutterudite.ImagineanFCClatticewherethe cubicpartrepresentsthecubicpartoftheskutteruditeandwherethefacedcentered atomssprayedonatomsaredraggedoutwardsadistancez"insteadofbeingleftonthe surfacesofthecubes. Therearetwoextrainteractionsconsideredwhencomparedtoanordinaryskutterudite: k ss whichistheinteractionbetweensprayedonatomsandk sc whichistheinteraction betweenthesprayedonatomsandcubiclattice.Inthissection,thereiscurrentlyno interactionbetweenthesprayedonatomsandtherectanglesofXatoms.SeetheFig.5-1 foradiagram. Theresultsareasexpected,meaningthatthelatticethermalconductanceisgenerally lessforthesprayedsystem.Alsotestedwastheeectofsprayingasimplecubiclattice inthesamemannerasthesprayedskutterudite.Thelatticethermalconductanceofthe sprayedcubiclatticeislessthantheordinarycubiclattice.Aftertherstplot,allofthe otherplotschangeoneparameteroftheskutteruditeandseetheeectontransmission. Outofthese,itappearsthatk sc hasthemostsignicantimpactonreducingtransmission. Theanalogoussituationistheun-sprayedskutteruditehavingk 1 asthemostimportant parameter.Intheanalog,thecagetocagedatominteractionisk 1 andinthesprayed setupthecageisthesprayandthecagedpartisthecubicpartofthesample. 60

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StartinFig.5-2withacubiclatticeandsprayedcubiclatticeforcomparison.Wesee thatthesprayedatomsreducethetransmissionatlowenergy,butthatthetransmission isactuallyenhancedalittleathigherenergies.Notethisisnotsimplyashiftinthe transmissionsincethereissignicantreductionatlowenergy. ForcomparisoninFig.5-3wehaveaskutteruditeandasprayedskutterudite; however,theskutteruditeonlyhasaweakcouplingbetweenthecageandthecagedatoms. Thiscouplingisonetotwoordersofmagnitudelowerthantheotherinteractions.The eectisthattheskutteruditetransmissionslookquitesimilartothatforthecubiclattice forthisweakcoupling.Whenthesprayedatomsareaddedthetransmissionplotsalso looksimilartothesprayedcubicatomcasewiththeexceptionthatthereisasmalldip wherethetransmissionalmostgoestozeroatlowfrequencyinthiscase. InFig.5-4weuseamorerealistick 1 ,whichiscomparabletothespringsinthe transversedirections.Heretheskutteruditetransmissionisclearlydierentfromthecubic transmissioninFig.5-2andtheweakskutteruditeinFig.5-3withlowtransmissionat moderatelylowenergy.Addingthesprayedatomssignicantlyreducedthetransmission inthemiddleofthephononspectrum,andweseesharpresonantfeaturesinthelowerhalf ofthespectrum.Overallthetransmissionisreducedbythesprayexceptatthehighest energieswhereitisslightlyenhanced. InFig.5-5weusethesamebaseskutteruditeasinFig.5-4andincreasethe sprayed-sprayedinteraction.Theoverallstructureofthetransmissionasafunction ofenergyisthesame,buttherearesomeadditionalresonances.Thisissimilarto whatwefoundintheonedimensionalmodelswherethestructurealsochangedwith increasedinteractionbetweenthesprayedatomsFig.4-15.Theinteractionmodiesthe vibrationalmodesofthesprayedatoms. WhileinFig.5-5wemodiedthesprayed-sprayedinteraction,inFig.5-6we modiedthesprayedatomstotheskutteruditecageinteraction.Hereweseealarge changeinthetransmission.Thereisactuallyanenhancementofthetransmissionaround 61

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the2to3valuesofthefrequencies,whileforlargerenergiesthereisadramaticdecrease inthetransmission.Asnotedearlier,thisisnotsurprisingbecausefortheunsprayed skutteruditek 1 isthemostimportantinteraction.Theanaloghereofk1isthesprayed atom-skutteruditecageinteraction. Finally,inFig.5-7westartagainwiththeparametersofFig.5-4,andthistime changethesprayedatommass,increasingitbyalmostafactoroftwo.Thetransmission looksquitesimilartoFig.5-4exceptthatthereisasmallatregionbetween3and4 ,whileinFig.5-4therewereresonancesthere.Thisisnotsurprisingbecausewefound thatthemassesoftheatomseectsthepositionsofresonancesinouronedimensional models. Insummary,inFig.5-8wecomparethecubicsprayedcaseofFig.5-2,the skutteruditesprayedcaseofFig.5-4,andtheskutteruditesprayedcaseofFig.5-6which hadlargespraytocageinteraction.Bothoftheskutteruditecaseshavelowertransmission atlowerenergy,meaningthatwewouldbeeectiveatreducingthethermalconductance atlowertemperatures.Forthetwosprayedskutteruditecasestherearetrade-os.Ifone wantstoreducethetransmissionatlowenergies,thenjustthesprayedskutteruditeis thebest.Ontheotherhand,ifonewantstoreducethetransmissionoverawiderangeof temperatures,thenoneshouldincreasethesprayedatom-skutteruditecageinteraction. Fig.5-9onwardsareplotsofthermalconductanceforthedatausedtogeneratethe transmissionplotsinthesameorder.Sprayingdecreasesthermalconductanceasdoes usingcagedatoms. 5.2AssumingSprayed-XInteraction Nowthatthesimplermodelhasbeenstudied,considerthesamemodelwithsprayed toXatominteractionsk sr 1 ,k sr 2 ,andk sr 3 .Therearethreespringconstants,asthere arethreepossibledistancesconsideredbetweenthesprayedandXatoms.Qualitatively, theredoesnotappeartobemuchofadierence.IntherstplotFig.5-16,SKUis thesameasthesprayseeninFig.5-4.ThesecondplotFig.5-17changesthelengthof 62

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thesampletoseeifthatmakesthesenewspringconstantshavemoreofaneect.There isnotmuchofaneectfromusingthesethreespringconstants,sotheyarenotusedin futuresections. 63

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Figure5-1.Sketchofasprayedskutterudite.Aviewfromaboveofthesprayed skutterudite.TheshadedgrayareasindicatethattherearerectanglesofSb undertheCoatoms.TheredatomsareSo,andblueareCo.Asforsprings, bluelinesarek ss andredlinesarek sc notallofthesearelabeled. 64

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Figure5-2.Cubicvssprayedcubictransmission.Atransmissionplotcomparingacubic latticetoasprayedcubiclattice.Forlowerfrequencies,thesprayedcubic latticehaslowertransmissioncomparedtothecubiclattice.Thereare200 binsusedinthisplot.Theparametersare:N x =N y =N z =5,k x =k a = 12.262494774575245,k y =k b =k z =k c =5.7425,k ss =5,k sc =2,m B =1, m Co =1.3,m So =1.4,m Sb =1.7. 65

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Figure5-3.Weakskutteruditevssprayedweakskutteruditetransmission.Atransmission plotcomparingaskutteruditetoasprayedskutteruditewithacubeto rectangleinteraction,k 1 =0.1.Thereare200binsusedinthisplot.Therest oftheparametersare:N x =N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c =5.7425,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k ss =5,k sc = 2,m B =1,m Co =1.3,m So =1.4,m Sb =1.7. 66

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Figure5-4.Skutteruditevssprayedskutteruditetransmission.Atransmissionplot comparingaskutteruditetoasprayedskutteruditewithacubetorectangle interaction,k 1 =3.Thereare200binsusedinthisplot.Therestofthe parametersare:N x =N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c =5.7425,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k ss =5,k sc =2, m B =1,m Co =1.3,m So =1.4,m Sb =1.7. 67

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Figure5-5.Skutteruditevssprayedskutteruditetransmissionforincreasedk ss .A transmissionplotcomparingaskutteruditetoasprayedskutteruditewitha sprayedtosprayedatominteractionofk ss =8.Thesprayedskutteruditehas lowertransmissioncomparedtotheordinaryskutteruditeformostfrequencies. Thereare200binsusedinthisplot.Therestoftheparametersare:N x =N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c =5.7425,k 1 = 3,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k sc =2,m B =1,m Co =1.3,m So = 1.4,m Sb =1.7. 68

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Figure5-6.Skutteruditevssprayedskutteruditetransmissionforincreasedk sc .A transmissionplotcomparingaskutteruditetoasprayedskutteruditewitha sprayedtocubicatominteractionofk sc =7.Forlowerfrequenciesthesprayed skutteruditehaslowertransmissioncomparedtotheordinaryskutterudite. Thereare200binsusedinthisplot.Therestoftheparametersare:N x =N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c =5.7425,k 1 = 3,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k ss =5,m B =1,m Co =1.3,m So = 1.4,m Sb =1.7. 69

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Figure5-7.Skutteruditevssprayedskutteruditetransmissionforincreasedm So .A transmissionplotcomparingasprayedskutteruditewithasprayedonatom massofm So =1.4toonewithm So =2.5.Forlowerfrequenciesthesprayed skutteruditehaslowertransmissioncomparedtotheordinaryskutterudite. Thereare200binsusedinthisplot.Therestoftheparametersare:N x =N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c =5.7425,k 1 = 3,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k ss =5,m B =1,m Co =1.3,m Sb = 1.7. 70

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Figure5-8.Comparisonofvarioussprayedsetups.Atransmissionplotcomparingvarious sprayedsetups.Thereare200binsusedinthisplothigherk sc =7andhigher k 1 =3.Otherwise,therestoftheparametersare:N x =7,N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c =5.7425,k 1 =3,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k ss =5,m B =1,m Co =1.3,m So =1.4,m Sb =1.7. 71

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Figure5-9.Thermalconductanceofacubicandasprayedcubiclattice.Noticethatthe latticethermalconductanceislowerforthesprayedcubiclattice.The parametersare:N x =N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c =5.7425,k ss =5,k sc =2,m B =1,m Co =1.3,m So =1.4,m Sb = 1.7. 72

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Figure5-10.Thermalconductanceofaweakskutteruditeandasprayedweak skutterudite.Thelatticethermalconductanceofaskutteruditeandsprayed skutteruditewithk 1 =0.1.Thelatticethermalconductanceisnotas predictableastheotherplots.Theparametersare:N x =N y =N z =5,k x = k a =12.262494774575245,k y =k b =k z =k c =5.7425,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k ss =5,k sc =2,m B =1,m Co =1.3,m So =1.4,m Sb =1.7. 73

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Figure5-11.Thermalconductanceofaskutteruditeandasprayedskutterudite.The latticethermalconductanceofaskutteruditeandsprayedskutteruditewith k 1 =3.Noticethatthelatticethermalconductanceislowerforthesprayed skutterudite.Theparametersare:N x =N y =N z =5,k x =k a = 12.262494774575245,k y =k b =k z =k c =5.7425,k 3 =0.7,k 4 =0.5,k 5 = 0.3,k 6 =0.1,k ss =5,k sc =2,m B =1,m Co =1.3,m So =1.4,m Sb =1.7. 74

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Figure5-12.Thermalconductanceofaskutteruditeandasprayedskutteruditewith higherk sc .Thelatticethermalconductanceofaskutteruditeandsprayed skutteruditewithk sc =7.Noticethatthelatticethermalconductanceis lowerforthesprayedskutterudite.Therestoftheparametersare:N x =N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c =5.7425,k 1 = 3,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k ss =5,m B =1,m Co =1.3,m So = 1.4,m Sb =1.7. 75

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Figure5-13.Thermalconductanceofaskutteruditeandasprayedskutteruditewith higherk ss .Thelatticethermalconductanceofaskutteruditeandsprayed skutteruditewithk ss =8.Noticethatthelatticethermalconductanceis lowerforthesprayedskutterudite.Therestoftheparametersare:N x =N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c =5.7425,k 1 = 3,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k sc =2,m B =1,m Co =1.3,m So = 1.4,m Sb =1.7. 76

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Figure5-14.Alatticethermalconductanceplotcomparingasprayedskutteruditewitha sprayedonatommassofm So =1.4toonewithm So =2.5.Noticethatthe latticethermalconductanceislowerforthesprayedskutterudite.Therestof theparametersare:N x =N y =N z =5,k x =k a =12.262494774575245,k y = k b =k z =k c =5.7425,k 1 =3,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k sc = 2,k ss =5,m B =1,m Co =1.3,m Sb =1.7. 77

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Figure5-15.Comparisonofconductanceforvarioussprayedsetups.Alatticethermal conductanceplotcomparingvarioussprayedsetupshigherk sc =7and standard"k 1 =3.Otherwise,therestoftheparametersare:N x =7,N y = N z =5,k x =k aandhigher =12.262494774575245,k y =k b =k z =k c =5.7425, k 1 =3,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k sc =2,k ss =5,m B =1,m Co =1.3,m So =1.4,m Sb =1.7. 78

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Figure5-16.EectofaSo-SbinteractionforN x =5.Atransmissionplotcomparinga sprayedskutteruditewithoutSo-Sbinteractiontoonewiththoseinteractions. Thereare200binsusedinthisplot.Therestoftheparametersare:N x =N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c =5.7425,k 1 = 3,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k ss =5,k sc =2,k sr 1 =1.5,k sr 2 = 1.4,k sr 3 =1.3,m B =1,m Co =1.3,m So =1.4,m Sb =1.7. 79

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Figure5-17.EectofaSo-SbinteractionforN x =15.Atransmissionplotcomparinga sprayedskutteruditewithoutSo-Sbinteractiontoonewiththoseinteractions. Thereare200binsusedinthisplot.Therestoftheparametersare:N x = 15,N y =N z =5,k x =k a =12.262494774575245,k y =k b =k z =k c = 5.7425,k 1 =3,k 3 =0.7,k 4 =0.5,k 5 =0.3,k 6 =0.1,k ss =5,k sc =2,k sr 1 = 1.5,k sr 2 =1.4,k sr 3 =1.3,m B =1,m Co =1.3,m So =1.4,m Sb =1.7. 80

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CHAPTER6 SPRAYEDCUBICLATTICES 6.1ScalingofTransmissioninaCleanCubicLattice Inordertounderstandhowtoreducetransmissioninasprayedlattice,itisrst necessarytoseehowthetransmissionscalesasafunctionofsamplelengthandcross section.Forthevariationinsamplelength,thewiggles"onthetransmissionplotare aected,buttheoverallshapeandmagnitudeofthetransmissionfunctionaren't. Crosssectionvariationisalittletrickierastherewillbemoreincidentmodesfor largercrosssections.Thexistodividethetransmissionbythecrosssectiontogeta normalizedtransmission.Forexample,ifthesamplecrosssectionis5X5atoms,divide thetransmissionby25togetthenormalizedtransmission.Theresultisthesameas thesamplelengthvariation,thewiggles"change,buttheoverallshapeandmagnitude doesn't. Fig.6-1containsplotswherethecrosssectionistheonlythingchangedbetween plots.Eachplotcontainsavarietyofsamplelengthstoprovethatthesamplelength doesn'tseemhaveasignicanteectontransmissionforacleancubiclattice.Thevariety ofcrosssectionsistoshowthatthisfactisn'tduetojustincludingonlysmallorlarge samplecrosssections.Note:thedatashowninthissectionassumesthatallatomsare connectedbyaspringconstantofunity,bathmassofone,andbasecubicmassof1.3. Fig.6-2hasplotswithnormalizedtransmission.Inthese,theonlythingtochange betweenplotsisthesamplelength.Likethevaryingcrosssectiontests,thisvaryinglength testrevealsthatthetransmissionfunctionisn'treallyalteredsignicantly.Thereisone dierencethough,thesmallercrosssectionsseemtobehaveabitdierentlyatlowenergy, reassemblingastepfunction. 81

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6.2OrderedSpray Weconsidertwoorderedspraygeometries.Forgeometry1imagineanFCClattice withthemiddleatompulledoutwardashortdistance.Ifviewedfromaboveitlookslike Fig.6-3. Next,geometry2isdenedasaspraythatperfectlyenclosesthesample.For example,ifthebathhasa5X5crosssection,thebasicsamplehasa5X5crosssection thatissurroundedbyspray,meaningthesampleis7X7asawholeinneratomsand 24sprayedonatoms.Asliceinthedirectionparalleltotheinniteheatbathcoordinate isshowninFig.6-4. Notshownineitheroftheguresarethebondsbetweenthesprayed-sprayednearest neighboratoms.Theyareusedinthedatainthissection;howeversincetheydon'tseem tobeparticularlyimportantandmakethingsunnecessarilymorecomplicated,theyare assumedtohaveavalueofzerointhenextsection. Inmymodel,atrstglance,onewouldthinkthatsprayingacubiclatticewould lowertransmissioninameaningfulway.Meaningthatirregardlessofthesamplecross section,therewouldbeapronouncedreductionintransmission.Undermorecareful inspectiontherearetwopointsthatshootdownthisidea.Auniformsprayessentially createsanewlattice,sothereisnochangeinmediabetweenlayers"onthiscrystal.The secondpointisthatthereismismatchcreatedbetweenthecrosssectionofthebathand thatofthesample. Eventhoughthereisareductionintransmission,itisasurfaceeect,andthelarger crosssectionofthesample,thelesspronouncedthereductionintransmission.Thiscan alsobeseenbycomputinganormalizedtransmissionviadividingthetransmissionbythe crosssectioninthesecondgeometrytheperfectlyenclosedsprayvariant. Ivariedthesamplelengthtoseeiftherewasanynoticeablechangeintransmission foraxedcrosssection.Theansweristhatthewiggles"inthefrequencyvstransmission change,buttheoverallshapeandmagnitudeofthetransmissionfunctiondoesn'tchange 82

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appreciably.Ofcoursechangingthebasemassoftheinnerpartofthenewcrystalwill aecttransmission,butitwillonceagainnotmatterifImakethesamplelonger. Whilenotshownhere,withtheskutterudite,thepointsabovestillholdtrue.The shapeofthetransmissionfunctionisdierentfromthecubiclatticeduetotheinternal structureoftheskutterudite.Note:thedatashowninthissectionassumesthatallatoms areconnectedbyaspringconstantofunity,bathmassofone,andbasecubicmassof1.3, andsprayedatommassof1.4. Normalizedtransmissionisthetransmissiondividedbythetotalareaofasprayed sampleofcrosssection.Smallsamplelengthsandsmallcrosssectionsdeviatefromthe pointsmadeabovethatnormalizedtransmissionisn'teectedbysamplelengthandcross section.Fig.6-5isofnormalizedtransmissionwithlengthbeingtheonlyparameterto varyinbetweentheplots.Fig.6-6isoftransmissionwithcrosssectionvaryinginbetween theplots.Geometry1isnotshownhereastheresultsfortransmissionareverysimilarto geometry2. 6.3SpringConstantDisorderedSpray Fromtheresultsofthecaseforcubiclatticemassdisorder,onecanseethereisa fundamentaldierencefromauniformcrystalevensprayedones,specicallyinthe transmissionfunction.Withthismind,Imodiedsprayedgeometry1toincluderandom cubictosprayedinteractions.Thismeansthatwhilethesprayedmassesareuniform,the springconstantsbetweenthemandtheinnersamplearedierentseeFig.6-7.Most sprayedsitesareleftdisconnectedfromtheinnersample.Notethatunliketheprevious sections,thereisnosprayed-sprayedbond. Inthiscasethetransmissionfunctionchangesbetweenrunsforsamplesofthesame size,somultiplerunsarerequiredforaxedsamplesize.Averagingiseasy:addthe transmissionfunctionstogetheranddividebythenumberofrunsforthatsamplesize. Herewefoundthatthetransmissionwilldecreaseasthesamplegetslonger. 83

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Sincethetransmissionnowdecreaseswithincreasinglengthunliketheuniform spray,onecancomputethemeanfreepathofphononsinthesample.Arudimentary formulais[51]: T = 1 1+ L l {1 Wherethesamplelengthis L "andthemeanfreepathis l ." Thismustbemodiedasmymodelhasmultipleincidentmodes.Multiplybythe numberofmodesexcitedN m inthebathandournalexpressionis: T = N m 1+ L l {2 Igeneratedseveralplotsofsamplelengthvs1/Transmissionatxedfrequencyfor mymodelFig.6-8throughFig.6-13.Inthemthereisalinearregion.Whydoesthis matter?Ifonerearrangesthelastequationfor1/Transmissionandtstheplotswitha line,itbecomesapparentthat: slopeoftheplots = 1 lN {3 Solvingforthepathlengthisthentrivial. Ingeneralasthetransmissionincreases,thepathlengthgoesup.Thisisbecause moreofthestu"incidentfromtheleftbathgetsthroughthesample.However,ifone normalizeswithrespecttonumberofmodesmultiplybynumberofmodesthepath lengthasafunctionofnumberofmodesisfairlyat.Notethatbasecrosssectionmeans thesamplesizewithoutthespray.Forexample,a5X5basecrosssectionmeansthereare atomssprayedaroundthebase25atomsinalayer. Thenextquestionis,howdoesincreasingdisordereecttransmission?Let'sassume thatoneiscomparingalittledisorderwithtwiceasmuch.Higherdisorderisdenedas havingsprayedspringswithvaluesbetweenzeroand0.6.Theansweristhattransmission dropsinanevenmoreprofoundway.It'sfarmoreobviousforalargerportionofthe 84

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frequencyspectrum.Inadditionthepathlengthseemstobecutbyincreaseddisorder sincethetransmissionisreduced.Recallthatintheseunitsthex-directionlatticespacing isunity.ThisquestionisaddressedinFig.6-14. Now,whathappenstothemeanfreepath?Theanswerisitappearsthatincreasing theamountofdisorderinthesprayreducesthemeanfreepathFig.6-15. Finally,howdoesthethermalconductancevariesasafunctionoflengthforthe maximumdisordercase.Unsurprisingly,theconductancedecreaseswithsamplelength Fig.6-16. 85

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Figure6-1.Transmissionwithcrosssectionvaryinginbetweentheplots. 86

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Figure6-2.Normalizedtransmissionwithlengthvaryinginbetweenplots. 87

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Figure6-3.Sprayedgeometry1.Fromaboveitlookslikethis.Theblackatomsarethe cubicpart,andtheredarethesprayed-onatoms.Theblacklinesarethe springsconnectingthecubicatomsandtheredlinesarethesprings connectingthecubicatomstothesprayedones. Figure6-4.Sprayedgeometry2.Fromaboveitlookslikethis.Theblackatomsarethe cubicpart,andtheredarethesprayed-onatoms.Theblacklinesarethe springsconnectingthecubicatomsandtheredlinesarethesprings connectingthecubicatomstothesprayedones. 88

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Figure6-5.Geometry2normalizedtransmissionwithplotsofvariouslengths. 89

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Figure6-6.Geometry2transmissionofvaryingsamplelengthwithvariousplotsof dierentcrosssections. 90

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Figure6-7.Sketchofsprayedgeometrywithdisorder.Fromaboveitlookslikethis.The blackatomsarethecubicpart,andtheredarethesprayed-onatoms.The blacklinesarethespringsconnectingthecubicatomsandallothercolored linesarethespringsconnectingthecubicatomstothesprayedones.Each colorisadierentvalue.Noticehowsomesprayedatomsaren'tusedno spring. 91

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Figure6-8.Obtainingmeanfreepathforlowerdisorder.Thisgurecontainsvariousplots of1/Transmissionasafunctionoflength.Eachplothasadierentenergy. Figure6-9.ObtainingpathlengthforhigherdisorderI.Aplotof1/Transmissionasa functionoflengthforhigherdisorderandconstantdrivingfrequencysquared of2.228.Thepathlengthis6.4767. 92

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Figure6-10.ObtainingpathlengthforhigherdisorderII.Aplotof1/Transmissionasa functionoflengthforhigherdisorderandconstantdrivingfrequencysquared of3.280.Thepathlengthis15.3846. 93

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Figure6-11.Plotsoftransmissionfortwodierentamountsofspringconstantdisorder withaxedbasesamplecrosssectionof5X5.Thesamplelengthvaries betweeneachplot. 94

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Figure6-12.Meanfreepathforvaryingamountsofdisorder.Notethatforincreasing disorderthatmeanfreepathdecreases.Thesamplelengthsusedare7,13, 21,27,31,41,51,71atoms.Thebasesamplecrosssectionis5X5. Figure6-13.Thermalconductanceforvaryinglengthsinthefulldisordercase.Thisplot isforthecaseoffulldisorder,andasthesamplelengthincreases,thethermal conductancedecreases. 95

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CHAPTER7 NUMERICALMETHODS 7.1InitializationPartI:DeningParameters,ObtainingFrequencies,and WaveNumbers Thissectionisdedicatedtohowmycodeworksandthemathinvolved.Thesections areplacedintheorderinwhichthecodeiswrittenandtheproblemsthatarose.Forthis section,letCobethecubicsampleatoms,Sbbethecagedatoms,andSobethesprayed onatoms.Allconstantparametersaredenedatthestartofthecode:springconstants ortheamountofdisorder,masses,samplesize,numberoffrequencybinsetc. NextIsimultaneouslyndthemaximumallowedfrequencyandassignthefrequencies forthepredenednumberofbins.TodothisIneeddwchangeinfrequencybetween binsasthenumberofbinsisxed.Assumeaverysmalldwwhichwillbechangedlater andastartingfrequencynearzeroex.10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 .Referringbacktotheequationscitedearlier inthisthesis,calculatethetransversewave-numbers: q y = n y N y b {1 q z = n z N z c {2 Recallthatn y andn z areintegersvaryfrom0toN y -1andN z -1respectively.Tomake thissimpler,letk x =k a .Fromthese,onecanobtaintheparallelwavenumberforeach frequency: q x = acos )]TJ/F22 7.9701 Tf 13.151 5.046 Td [(m B w 2 2 k x + k b )]TJ/F22 7.9701 Tf 6.587 0 Td [(cos q y b k x + k c )]TJ/F22 7.9701 Tf 6.587 0 Td [(cos q z c k x a {3 Iterateoverfrequencies,whichmeansincreasethefrequencybyasmallamounteach loop.Ifthereisnoallowedq x foragivenfrequency,thenthemaximumfrequencyisthe currentvalueminusthestartingdw.Thensetdwtobethemaximumfrequencydivided bythepredenednumberofbins.Thefrequencybinsarethenlledin. 96

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Nowdenethematricesthatareofdimension3XNumberofRelevantAtomsto holdthethreecoordinatesofeachatominthesample.Rowoneisforx,rowtwoisfory, androw3isforz.TherelevantnumberofatomsisthetotalnumberofCoorSborSo atoms. 7.2InitializationPartII:Normalization DenearowvectortoholdthenormalizationconstantsN c forplanewavesolutions totheEOMlikeso:Ifn y =0andn z =0then N c = s 1 N y N z {4 Ifn y =0andn z 6 =0then N c = s 2 N y N z {5 Ifn y 6 =0andn z =0then N c = s 2 N y N z {6 Ifn y 6 =0andn z 6 =0then N c =2 s 1 N y N z {7 Thisrowvectorislledinthesameorderasabove.Asforwhythenormalization constantsarewhattheyare,thesolutionstotheEOMhaveapart: cos n y N y y b )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 = 2 cos n z N z z c )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 = 2{8 Squarethispartandaverageovereachofthesecosines.Theresultforeachtermis1/4 unlessn y =0orn z =0,inwhichcasetheresultis1/2or1.Asthenumberofmodesis equaltothecrosssectionofthebaths,thesumissupposedtoequalN y N z . 7.3SettingUptheLoopandEOM Nowtothebulkofthecode.Setupaloopoverthefrequenciesandatthestartof thisloop,calculatetheincidentallowedwave-numbersthatareallowedaswellascreate theemptymatrixandvectorthatarefortheEOM.Theunknownssolvedforarethe 97

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reected,transmitted,andsampleamplitudesthislastoneisafairlyloosedenition. Thedimensionsofthismatrixarethenumberofatomsinthesamplebythenumberof atomsinthesampleandthevector'sdimensionisjustthenumberofatomsinthesample. DuringthisstepIobtainthenumberofallowedincidentmodesforagivenfrequency.This isveryimportantasitsavesmealotofcomputationaltime. Thecodeisdividedintosubsectionsthatconnecttheatoms.Forexample,thereis asectionwhichconnectsthecubicsampleatomstothesprayed-onatomsandaseparate sectionforconnectingcubicsampleatomstocagedatoms.Todetermineifatomsare connected,cyclethroughalltheatoms'positionsinthepositionmatrices.Takethe absolutevalueofthedierenceinpositionsbetweentheatomsandifitislessthanthe tolerableerrorof10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 ,thenconnecttheatomsbyenteringtheappropriatespringsinthe correctslotofthematrix. TheequationsenteredintothematrixaretheEOM.Forthispartofthediscussion, assumethatasprayedcubiclatticeisbeingreferenced.Letthenumberofsprayed connectedatomsconnectedtoagivencubicatombeN con .Assumingoneistryingto obtaintheEOMforsampleatomswhicharemorethantwolayersinwardx > 3aandx < N x a-a,whereaisthelatticespacing,theyareoftheform: m Co d 2 X n;m;l dt 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 k x + k b + k c + k sc N con 2 X n;m;l + k x X n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;m;l + k x X n +1 ;m;l + k b X n;m )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;l + k b X n;m +1 ;l + k c X n;m;l )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 + k c X n;m;l +1 + k sc SprayedAtoms {9 Thewaythematrixisarrangedfromlefttorightis:reectedamplitudes,cubic atoms,transmittedamplitudes,cagedatomsifapplicableandsprayedatoms.Ofcourse ifatomsaremissingforagivenEOM,thentheyarecutoutofwhat'sseenabove.Let thenumberofsprayedatomsconnectedtoothersprayedatomsbeN ss andthenumberof cubicatomsattachedtoasprayedatombeN sc .Forsprayedatomsthatareinteriorlikein 98

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thecaseabove: m So d 2 X So dt 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [( k sc N sc + k ss N ss X So + k ss N ss X r =1 X SoConnected + k sc N sc X r =1 X CoConnected {10 Thesummationsabovearetosumoveralltheadjacentatoms.Thereisnoeasysetof incidencesforthesprayedatoms. 7.4SimpleExampleofImplementingtheScatteringBoundaryCondition Fortherstlayersofthesamplex=aandN x a,thescatteringboundarycondition mustbeused.Asthisisexcessivelycomplicatedin3-DbutwhatIimplementedinmy code,a1-Dsampleproblemisusedtoshowhowthisisimplemented.Letthesamplebe 3atomslongwithdierentmasses.Atrstglancethereare5unknowns:thereected amplitude,3sampleatomamplitudes,andatransmittedamplitude.Thescattering boundaryconditioneliminatestwoofthesampleamplitudesinfavorofreectedand transmittedamplitude. Theplanewavesolutionsinthebathsare: X n t = Ae iqna + Be )]TJ/F22 7.9701 Tf 6.586 0 Td [(iqna e )]TJ/F22 7.9701 Tf 6.587 0 Td [(iwt {11 X n t = Ee iqna e )]TJ/F22 7.9701 Tf 6.587 0 Td [(iwt {12 NextwritethesolutionsfortheX )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ,X 0 ,X 4 ,andX 5 bathatoms: X )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 t = Ae )]TJ/F22 7.9701 Tf 6.587 0 Td [(iqa + Be iqa e )]TJ/F22 7.9701 Tf 6.587 0 Td [(iwt {13 X 0 t = A + B e )]TJ/F22 7.9701 Tf 6.587 0 Td [(iwt {14 X 4 t = Ee 4 iqa e )]TJ/F22 7.9701 Tf 6.586 0 Td [(iwt {15 X 5 t = Ee 5 iqa e )]TJ/F22 7.9701 Tf 6.586 0 Td [(iwt {16 99

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NowwritetheX 0 andX 4 EOMtosolveforX 1 andX 3 intermsofA,B,andE.Thisis thescatteringboundarycondition.TheX 0 EOM: m B d 2 X 0 dt 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 kX 0 + kX )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + kX 1 {17 TheX 4 EOM: m B d 2 X 4 dt 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 kX 4 + kX 3 + kX 5 {18 Whichsimpliesto: )]TJ/F21 11.9552 Tf 11.955 0 Td [(m B w 2 A + B = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 k A + B + k Ae )]TJ/F22 7.9701 Tf 6.587 0 Td [(iqa + Be iqa + kX 1 {19 )]TJ/F21 11.9552 Tf 11.956 0 Td [(m B w 2 Ee 4 iqa = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 kEe 4 iqa + kX 3 + kEe 5 iqa {20 SolvethesetwoequationsforX 1 andX 3 : X 1 = k )]TJ/F21 11.9552 Tf 11.956 0 Td [(m B w 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(ke )]TJ/F22 7.9701 Tf 6.586 0 Td [(iqa k A + k )]TJ/F21 11.9552 Tf 11.956 0 Td [(m B w 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(ke iqa k B e )]TJ/F22 7.9701 Tf 6.586 0 Td [(iwt {21 X 3 = k )]TJ/F21 11.9552 Tf 11.955 0 Td [(m B w 2 e 4 iqa )]TJ/F21 11.9552 Tf 11.955 0 Td [(ke 5 iqa k Ee )]TJ/F22 7.9701 Tf 6.586 0 Td [(iwt {22 Nextwritedowntherearrangedequationforthewavenumber: k )]TJ/F21 11.9552 Tf 11.956 0 Td [(m B w 2 = k e iqa + e )]TJ/F22 7.9701 Tf 6.587 0 Td [(iqa {23 SubstitutethisintotheequationsforX 1 andX 3 : X 1 = Ae iqa + Be )]TJ/F22 7.9701 Tf 6.586 0 Td [(iqa e )]TJ/F22 7.9701 Tf 6.587 0 Td [(iwt {24 X 3 = Ee 3 iqa e )]TJ/F22 7.9701 Tf 6.587 0 Td [(iwt {25 WritetheEOMforX 1 ,X 2 andX 3 : m 1 d 2 X 1 dt 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 kX 1 + kX 0 + kX 2 {26 m 2 d 2 X 2 dt 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 kX 2 + kX 1 + kX 3 {27 100

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m 3 d 2 X 3 dt 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 kX 3 + kX 2 + kX 4 {28 NowthatthesolutionstoX 1 andX 3 havebeenfoundintermsofA,B,andE,rearrange andplugtheresultsintotwooftheequationsabove: k )]TJ/F21 11.9552 Tf 11.955 0 Td [(m 1 w 2 Ae iqa + Be )]TJ/F22 7.9701 Tf 6.586 0 Td [(iqa = k A + B + kX 2 {29 k )]TJ/F21 11.9552 Tf 11.955 0 Td [(m 3 w 2 Ee 3 iqa = kX 2 + ke 4 iqa E {30 Removethetimedependencebyassumingaharmonicsolutionandrewritingall3EOM: k )]TJ/F15 11.9552 Tf 11.955 0 Td [( k )]TJ/F21 11.9552 Tf 11.955 0 Td [(m 1 w 2 e )]TJ/F22 7.9701 Tf 6.587 0 Td [(iqa B + kX 2 = k )]TJ/F21 11.9552 Tf 11.955 0 Td [(m 1 w 2 e iqa )]TJ/F21 11.9552 Tf 11.955 0 Td [(k A {31 ke )]TJ/F22 7.9701 Tf 6.587 0 Td [(iqa B + m 2 w 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k X 2 + ke 3 iqa E = )]TJ/F21 11.9552 Tf 9.299 0 Td [(ke iqa A {32 kX 2 + ke 4 iqa )]TJ/F15 11.9552 Tf 11.955 0 Td [( k )]TJ/F21 11.9552 Tf 11.955 0 Td [(m 3 w 2 e 3 iqa E =0{33 Let 1 =k-k-m 1 ! 2 e )]TJ/F22 7.9701 Tf 6.587 0 Td [(iqa and 3 =ke 4 iqa -k-m 3 ! 2 e 3 iqa .Writethisasamatrix equation: 0 B B B B @ k )]TJ/F21 11.9552 Tf 11.955 0 Td [(m 1 w 2 e iqa )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F21 11.9552 Tf 9.298 0 Td [(ke iqa 0 1 C C C C A A = 0 B B B B @ 1 k 0 ke )]TJ/F22 7.9701 Tf 6.587 0 Td [(iqa m 2 w 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 kke 3 iqa 0 k 3 1 C C C C A 0 B B B B @ B X 2 E 1 C C C C A {34 7.5LUFactorization NormallyonecansimplyinvertthematrixequationfortheEOMtosolveforthe unknowns.However,ifthesystemissucientlylargetherewillbenumericalerrorswith MATLAB'sinverseoperator.ThesolutiontothisproblemistouseLUfactorization whereLUstandsforlowerupper[59]. Thisprocedureisusedtosolvematrixequationsoftheform: Ax = b {35 101

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WhereAisamatrix,xisthecolumnvectorofunknowns,andbistheinhomogeneous partofthesystemofequations.LUfactorizationtakesthematrixAanddismantlesit intoaproductofalowertriangularmatrixLandanuppertriangularmatrixU. Thisistheprocedure.Startwith LUx = b {36 andpre-multiplybothsidesbyL )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 . Ux = L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 b {37 Deneanewvector b 0 : b 0 = L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 b {38 MultiplyingtheaboveexpressionbyLyieldsandsubstitutingitintotheonetwo equationsabovethelastone: Lb 0 = b {39 Ux = b 0 {40 AtthispointsincetheLandUmatricesaretriangular,onecaneasilyobtainb'fromthe topequation.Nowthebottomequationcanbesolvedviabacksubstitutionatriangular matrixbasicallygivestheanswertooneoftheunknowns.Theonlyquestionleftishow doesonegetLandU?OftentheseareobtainedviaGaussianeliminationbytargeting elementsaboveorbelowthediagonaldependingonwhichmatrixisdesired. 7.6ComputingtheVelocityofModes,OverallTransmissionandReection, andFinishingtheLoop Nowthattheamplitudesforeachtransitionedmodehavebeenfoundassuming2-D or3-D,theoveralltransmissionforagivenincidentmodeandfrequencymustbefound. Steponeistocomputethevelocityofeachmode: V q = k x asin q x a m B w {41 102

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Thetransmissionisthen T = N X q =1 V q T q V i {42 Foreachincidentmode,thereareatotalofN y N z possiblemodesitcanmorphinto oneofthemisthesameastheincidentmode.V i isthevelocityoftheincidentmode andT q isthetransmittedamplitudeforagivenmode.Sumoveralltheallowedmode transitionstogettheoveralltransmission.Thereectedamplitudeis: R = q X r =1 V q R q V i {43 Thereasonforcomputingthereectionandtransmissionseparatelyistoensurethat foreachincidentmodethatT+Risequaltounity.Thisisgreatindicatorofwhetheror notmycodeisworkingproperly,butbynomeansisitaawlesscheck.Next,repeatthis procedureforeachincidentmode.Then,T+RwillequalN y N z . Atthispoint,theloopoverfrequenciesisclosed.Thismeansrepeatallthesteps aboveuntilw max isreached.Nowthethermalconductancecanbecomputed. 7.7ComputingThermalConductance Createavectoroftemperaturesandanemptyvectorfortheconductance.Thereis adoubleloop,withtheouterloopbeingovertemperatureandtheinnerloopbeingover theoveralltransmission.Thelengthofthetransmissionvectoristhenumberoffrequency bins.Theconductanceis: J = Z W 0 ~ 2 w 2 T w 2 k B exp ~ w k B T L T 2 L exp ~ w k B T L )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 d w {44 Thereisnotananalyticsolutionforthis,soIevaluatethisintegralnumerically,using themidpointrule: F x = Z X 0 f x d x x f x i +1 + f x i {45 Thismeansthatforaxedtemperature: J ~ 2 w 2 i +1 T i +1 w i +1 2 k B exp ~ w i +1 k B T L T 2 L exp ~ w i +1 k B T L )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 + ~ 2 w 2 i T i w i 2 k B exp ~ w i k B T L T 2 L exp ~ w i k B T L )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 dw {46 103

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Onceiteratedovertemperature,thedesiredresultisobtained. 7.8ComputingMeanFreePath Therearesomeplotsthathavemeanfreepath,butthecomputationofthisquantity isnotpartofthemaincode.ThisisbecauseIhavetoaverageoverdisorderinthoseplots domultiplerunsforthesamesamplesize.ForthisItakethetransmissiondataforthese runsstoredintxtlesandaverageforverunspersamplesize.Avectorforthesample lengthsiscreatedandthenwithinaloopoverfrequenciesItalinetothelengthvs1/T plot.Thisyieldstheslope,whichinturngivesthemeanfreepathforeachfrequency. 104

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CHAPTER8 CONCLUSION Inthisthesiswehavestudiedthethermalconductanceandconductivityinwireswith resonantscattering:skutterudites,resonantsurfacescattering,andskutteruditeswith resonantsurfacescattering.Thegoalwastondandunderstandmechanismsforreducing latticethermalconductance. Ouroriginalmotivationforlookingatthesesystemswasthewellestablishedlow latticethermalconductivityinskutterudites.InChapter3whenwecomparedsmall wiresoftheskutteruditestothosewithmassdisorderwefoundthattheskutterdites wereparticularlyeectiveinreducingthetransmissionofthelowenergyphonons.Mass disorderwasmoreeectiveforhighenergyphonons. ThisledusinChapter4toconsidersomesimpleonedimensionalmodelswhichwe couldsolveanalytically.Theonedimensionalmodelswhichmimickedmassdisorderwere againineectiveatreducingthetransmissionatlowenergies.Ontheotherhandtwo resonantscatteringmodelsproducedlargereductionsinthescatteringandtransmissionat lowenergy.Sincewewereabletosolvethesemodelsanalytically,wewereabletoidentify thesourceofthisreductionunequivocallyasbeingduetoresonantscattering,andwe wereabletoanalyticallydeterminethezerosofthetransmission.Eachresonantscatterer hasitsowncharacteristicfrequency.Havingmorethanoneresonantscattererproduces multiplezerosinthetransmission;however,whentheresonantscatterersarecoupled stronglyenoughsomeofthezerosareeliminated. ThisledusinChapter5toaddresonantsurfacescatteringtoourwires.Firstwe addedtheresonantscatteringtoaskutteruditewire.Thisdidreducetransmissionand thermalconductance.Howeverincreasingsamplelengthforasprayedskutteruditedidn't reducetransmissionandconductance.Nextweaddedresonantsurfacescatteringtojust ansimplecubiclattice.Forauniformspray,thetransmissionandconductancedecreased. Thewaythatthetransmissiondecreasedwasdierentfromtheskutterudite,howeveronce 105

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again,increasingthelengthdidnotfurtherdecreasetransmissionorconductance.Two dierentgeometriesofsprayswereusedandbothhadsimilarresults. Alloftheseresultswereforthethermalconductance{notthethermalconductivity, whichisusuallythemostrelevantforexperiments.Thethermalconductivityisthe thermalconductancetimesthelengthofthesampleanddividedbythearea.Inthe rsttwosectionsofChapter6weexaminedthescalingofthethermalconductancewith lengthandcrosssectionalareaandfoundthattheearlierresultsdidnotscaleasthe thermalconductivity.Weconcludethatthescatteringandthermalconductancewasdue tophononmodemismatchbetweentheleadsandthesampleintheearlierexamples, eventhoughresonantscatteringisaneectivewaytoreducethelowenergythermal conductance. InthelastsectionofChapter6weactuallycalculatethethermalconductivityby consideringamodelofdisorderedresonantsurfacescattering.Thisproducesathermal conductancewhichdecreaseswithsamplelengthandallowsustocalculatenotonlythe thermalconductivity,butalsothemeanfreepathafunctionenergy.Resonantsurface scatteringisaneectivemeansofreducingthethermalconductivity,buttheremustbe somedisorderwithinourmodelorelsetheresonantscattererswillformphononbands themselves. Thisleavesuswithseveralimportantopenquestions.First,canoneexperimentally addresonantscattererstosurfacesofnanowiressoastoreducethethermalconductivity? Thismightbedone,forexample,bydepositingamoleculeonasurfacewithawelltuned vibrationalmodel.Second,sinceweseewithinourmodeltheimportantroleofdisorder, thenaturalquestioniswhatistheamountofdisorderinatypicalskutteruditewire.Can thisbecharacterized?Withinourharmonicmodelwecaneasilyincludeotherkindsof disorder. 106

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BIOGRAPHICALSKETCH DavidAlexanderRomanwasborninOakland,CAin1993.However,mostofhislife wasspentinsouthernFlorida.In2011hereceivedhisbachelor'sdegreeinspacephysics fromEmbry-RiddleAeronauticalUniversity,DaytonaBeach,Florida.Thenin2013he graduatedwithM.Sc.inphysicsfromUniversityofFlorida.Hethencontinuedwithhis PhDprogramatUniversityofFlorida,nishinghisPhDinthesummerof2016. 112