Citation
Thermal and Electrical Nonlinear Currents Driven through a Nano-Bridge Bearing an Electron-Phonon Interaction

Material Information

Title:
Thermal and Electrical Nonlinear Currents Driven through a Nano-Bridge Bearing an Electron-Phonon Interaction
Creator:
Nartowt, Bradley J
Publisher:
University of Florida
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
MUTTALIB,KHANDKER A
Committee Co-Chair:
HERSHFIELD,SELMAN PHILIP
Committee Members:
HEBARD,ARTHUR F
MASLOV,DMITRII
PHILLPOT,SIMON R
Graduation Date:
12/17/2017

Subjects

Subjects / Keywords:
dyson
efficiency
electron
electron-phonon
feynman
interaction
many-body
nanoelectronics
nonequilibrium
nonlinear
peltier
perturbation
phonon
resonant
seebeck
thermoelectricity

Notes

General Note:
Motivated by the goal of studying thermoelectric phenomena in systems of low dimensionality, electrical and thermal currents through a nano-bridge with an electron/phonon interaction are calculated in a toy model using the non-equilibrium Green function (NEGF) technique. Thus, the goal of this work is to determine the effect of (localized) interactions of energy $\omega$ electrons and energy $\Omega$ phonons in a junction between left ($L$) and right($L$) leads at disparate potentials $\mu_L \ne \mu_R$ and temperatures $T_L>T_R$. The bias $\mu_R-\mu_L$ drives an electrical current $I$ at a (thermal) voltage $V^T$, and thus an exerted power $P^T$, upon its surroundings. A thermal-energy current $Q$ accompanies this $P$, and the positive quantity $\eta=P/Q \le 1-T_{cold}/T_{hot}$ is how efficiently (which can be no greater than the Carnot efficiency) this power $P$ is generated by a quantity of $Q$. The junction is a nano-bridge which is electronically and mechanically coupled to the leads.

Record Information

Source Institution:
UFRGP
Rights Management:
All applicable rights reserved by the source institution and holding location.
Embargo Date:
12/31/2019

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THERMALANDELECTRICALNONLINEARCURRENTSDRIVENTHROUGHA NANO-BRIDGEBEARINGANELECTRON-PHONONINTERACTION By BRADLEYJOSEPHNARTOWT ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2017

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c 2017BradleyJosephNartowt

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ToErinandourchildren

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ACKNOWLEDGMENTS IacknowledgethelifeandworkofthelateProf.MildredDresselhaus,whodirected research-eortstowardsthermoelectricityinsystemsoflowdimensionality,givingme somethingtowritemydoctoraldissertationabout. IthankthewomenandmenresponsibleforlandingApolloXIontheMoon{a tremendousengineeringfeatthat,whenmyworkwaslledwithyet-to-be-detectedbugs, encouragedmethatitwasinfactpossibletorealizeaworkingandcorrectcomputercode. IthankMr.MarkLanari,whoisresponsibleforasignicantamountofwhatIknow about Mathematica Ithankmyco-supervisorProf.SelmanPhilipHersheldforteachingmehow importantproblem-solvingskillsare,andtounderstandeveryobjectenteringintoa calculation.Ialsothankhimforprovidinghisexpertiseonnon-equilibriumthermodynamics. IalsothankmysupervisorProf.KhandkerAbdulMuttalibforallowingmecountless hoursofdiscussionatthechalkboardinhisoce,wherethedirectionofmyresearch-eorts wasoftensteeredbackoncoursesometimesagainstmuchinertia.Profs.Hersheldand Muttalibcontinuouslyandgenerouslyinvestedtheirtimeinme.Itshouldbefurthermore saidthatthereisarathernelinebetweenbeinghelpfulandguidingtothepointof micromanaging.Somehow,despitealltheguidancetheyprovided,Profs.Hersheld andMuttalibrespectedmyautonomyandallowedmetoguidethecourseofthiswork, providinganotherwayinwhichIlearnedmuch.Ithankthemforskillfullyholdingthe delicatebalanceofsuchamodeofinstruction. Finally,IthankmywifeErinforhersaintlypatience,unwaveringdevotion,quiet strength,brightbeauty,andshiningtalent.Herenthusiasmforworldhistory,theuniverse ofJ.R.R.Tolkien,StarTrek,cooking,architecture,theVictorianAgeoer,andmany moresuchfascinationsoercountlesstopicsofdiscussionwhicharerefreshingafterlong daysofthinkingonlyaboutphysics. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFFIGURES....................................11 ABSTRACT........................................15 CHAPTER 1INTRODUCTION..................................16 1.1AbridgedHistoryofThermoelectricity....................17 1.1.1TheBeginningoftheFieldofThermoelectricity...........17 1.1.1.1Avoltage-gradientresultingfromatemperature-gradient -theSeebeckeect......................17 1.1.1.2Aheatpumpfromelectricalow-thetime-reversedSeebeck eect..............................18 1.1.1.3Athermoelectriccurrentfromatemperature-gradientwithin asinglematerial.......................18 1.1.1.4TheWiedemann-Franzlawofcorrelationbetweenelectrical andthermalconductivity..................18 1.1.1.5Asummaryofthephysicallawsgoverningthethermoelectric responseoffermionicchargecarriers............19 1.1.1.6ThethermogalvanomagneticeectsofNernst,Ettingshausen, Righi,andLeducinanalogytotheHalleect.......20 1.1.2TheStudyofThermoelectricityinBulkMaterialsintheMid-20th Century.................................20 1.1.2.1ThethermoelectricgureofmeritofIoe.........20 1.1.2.2ThethermoelectriceectduringtheWorldWarsandthe ColdWar...........................21 1.1.2.3TheSeebeckeectaftertheSpaceAge...........21 1.1.3TheoreticalandExperimentalMaturationoftheFieldofNanoelectronics andThermoelectricity..........................22 1.1.3.1Thermoelectricityinsystemsofreduceddimensionality..22 1.1.3.2Conductioninsystemsofreduceddimensionality.....23 1.2OverviewofthePhysicalProblemofThermoelectricity...........24 1.2.1MotivationandScope..........................24 1.2.1.1Separatemanipulationofthermalandelectricaltransport24 1.2.2UseofLinearvs.NonlinearResponse.................25 1.2.2.1Conservednonlinearcurrentsasthechargecarriersinteract withothercurrents.....................25 1.2.2.2Thephysicsinthefullnonlinearcurrentandtheproblem ofthermoelectricity......................26 1.2.2.3Saturationofthemaximumchemicalpotential.......26 5

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1.2.2.4Theproclivityoflinearresponseintheundriventhermoelectric device.............................26 1.2.2.5Thedriventhermoelectricdevice..............27 1.2.3ThermodynamicEciencyofaDrivenSystem............28 1.2.4TheSemiconductingHeterostructureModel..............29 2THEORETICALTOOLS..............................30 2.1Thermodynamics................................30 2.1.1TheTransport-CoecientDescriptionofTransmissionDrivenby ArbitrarilySmallThermoelectricGradients:TheLinear"Response30 2.1.1.1Theelectricalandthermalconductivities..........32 2.1.1.2TheSeebeckandPeltiertransportcoecients.......33 2.1.1.3Transportcoecientsimpliedbyagivenbandstructure andspectraldensity.....................34 2.1.1.4TheCutler-MottformulafortheSeebeckcoecient....37 2.1.2Undrivenvs.DrivenThermoelectricCircuits.............39 2.1.2.1Thethermoelectric,theload,andthedriving-agent....40 2.1.2.2Thethermoelectricdevice V T asanimperfectbattery...42 2.1.2.3ModeloftheOhmicandnon-Ohmicloads.........42 2.1.3TheFirstLaw-TheCircuit,theCurrents,andtheThermoelectric BlackBox................................44 2.1.3.1System-subdivisionsgovernedbytheFirstLaw......44 2.1.3.2Theoutowofpowerfromthetheromelectricgenerator..45 2.1.3.3Theloweredpower-inputtoaheatpumpduetoatemperature gradient:anonlinearthermoelectriceect.........45 2.1.3.4Constancyof P A 0 onapplicationof T ...........46 2.1.3.5EciencyofthethermalassistwithintheFirstLaw...50 2.1.4TheSecondLawandEciency-Bounds................50 2.1.4.1Theimpossibilityofreversibledriving...........51 2.1.4.2Impossibilityofsimultaneousoperationasaheatpump..51 2.1.4.3EciencyboundsfromtheSecondLaw...........52 2.1.4.4Eciencyasafunctionof ................53 2.2ModeloftheThermoelectricNano-Device..................54 2.2.1FermionicElectronsandBosonicPhonons..............54 2.2.1.1Thesimplicityofrepresentingparticlesasoperators....54 2.2.1.2Thealgebraoftheparticleoperators............55 2.2.1.3Highercommutatorsofparticleoperators..........56 2.2.2Two-Lead/Center-RegionSystemofElectrons............58 2.2.2.1TheelectronicquadraticHamiltonian............58 2.2.2.2Conservationintheelectronnumber-operator.......60 2.2.2.3Theenergycurrentoperator.................62 2.2.3Two-Lead/Center-RegionSystemofPhonons.............65 2.2.3.1TheAnderson-FanoquadraticHamiltonian.........65 2.2.3.2TransportintheFanoHamiltonian.............66 6

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2.2.4TheElectron/PhononInteraction...................67 2.2.4.1Thecontributionoftheelectron/phononcouplingtothe otherwise-quadraticHamiltonian..............67 2.2.4.2TheA-typevs.B-typephononsandcomputationalcost.68 2.2.4.3Scatteringofelectronsoofphonons............68 2.2.4.4Excitationsofcharge-neutralelectron/holepairsbyphonons70 2.2.4.5TheHamiltonianofthemolecularbridge..........71 2.3TheNon-EquilibriumGreenFunctionNEGFMethod...........71 2.3.1TheEquilibriumDistributions.....................72 2.3.1.1Thequantum-statisticalexpectation............72 2.3.1.2Theequilibriumdistributionoffermionandbosonensembles72 2.3.1.3Theeectivefermionicandbosonicthermalfunctions...72 2.3.1.4TheSommerfeldexpansioninaniteband.........73 2.3.2TheDenitionsoftheGreenfunctions................76 2.3.2.1TheGreenfunctionasaspectraldecomposition......76 2.3.2.2Single-fermionGreenfunctionsforelectronicdensity...77 2.3.2.3BosonGreenfunctionsformechanicalamplitude......78 2.3.2.4Two-fermionGreenfunctionsforparticle/holeexcitations78 2.3.2.5Single-bosonGreenfunctionsforatheoryofbosonscontaining number-preservingprocesses.................78 2.3.2.6TheexistenceofexactpropertiesimpliedbytheGreen functiondenitions......................79 2.3.3ExactPropertiesoftheGreenFunctions...............79 2.3.3.1Therelationamongthetime-orderingsoftheGreenfunctions andself-energies.......................79 2.3.3.2TheLehmannrepesentationoftheGreenfunctionsinthe domain...........................80 2.3.3.3TheGreenfunctionrepresentationoftheuctuation-dissipation theorem............................81 2.3.3.4Thefermionicspectralfunctionsumruleatequilibrium..82 2.3.3.5Thebosonicspectralfunctionsumruleatequilibrium...83 2.3.3.6TheKramers-Kronigidentity................83 2.3.4QuadraticTemperature-IndependentGreenFunctions........83 2.3.4.1Occupationoperatortimedependenceinastationarystate83 2.3.4.2ThestationarystateretardedandadvancedelectronicGreen functionsinpositionspace..................85 2.3.4.3ThestationarystateretardedandadvancedphononicGreen functionsinmodespace...................85 2.3.4.4RetardedsurfaceGreenfunctionandenergy-momentum relationdispersionofthe th lead'selectrons.......87 2.3.4.5Energy-momentumrelationdispersionofthe th lead's phonons............................90 2.3.4.6RetardedelectronicsurfaceGreenfunctionbyaspectral decomposition.........................93 7

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2.3.4.7RetardedphononicsurfaceGreenfunctionbyaspectral decomposition.........................96 2.3.5Dyson'sEquation............................97 2.3.5.1TheGreenfunctionsolutiontoordinarydierentialequations andtheSchrodingervs.Heisenbergpictures........97 2.3.5.2Theinniteperturbativesum-theDysonequationfor theGreenfunction......................98 2.3.5.3InversesandanalyticityforGreenfunctions........99 2.3.5.4ThedressingofaGreenfunctionbyaperturbation....99 2.3.6LangrethRules.............................100 2.3.6.1Thecontourideaforstreamlininganalyticcontinuationof Greenfunctions........................100 2.3.6.2TheLangrethorderingsofaconvolution..........101 2.3.6.3TheLangrethorderingsofamultiplication.........102 2.3.7PropertiesoftheDysonEquationUndertheLangrethRules....103 2.3.7.1Theretardedandthermal 7 Dysonequationsandadditivity oftheself-energy.......................103 2.3.7.2Independenceofthe 7 electronicGreenfunctionsfroma center-sitetemperatureforan N C -sitecenter-region....104 2.3.7.3TheLangreth-orderingsoftheDysonequationinaperturbative seriesform...........................108 2.3.8RewritingLead/Center-RegionGreenFunctionsinTermsofCenter Regionand/orSurfaceGreenFunctions................109 2.3.8.1Greenfunctionindicesbelongingtothecenterregionand totheadjoiningreservoirs.................109 2.3.8.2De-mixinglead/reservoirGreenfunctionswithoutLangreth ordering............................110 2.3.8.3TheorderoftheGreenfunctionpoleforball-and-spring Greenfunctions........................111 2.3.8.4ApplicationofLangrethorderingtotheGreenfunctions.111 2.3.8.5Mixed-indexelectronicGreenfunctions...........112 2.3.8.6Analogousindexde-mixingforball-and-springphonons..114 2.3.9GreenFunctionswithExplicitTemperatureandPotentialDependence115 2.3.9.1ElectronicthermalGreenfunctions.............115 2.3.9.2PhononicthermalGreenfunctions.............116 2.3.10Self-EnergiesoftheElectron/PhononInteractionandtheSelf-Consistent BornApproximation..........................116 2.3.10.1Mathinterlude:theFouriertransformandconvolution theorems............................116 2.3.10.2Theelectronicself-energyduetoanelectroncollidingwith aphonon...........................118 2.3.10.3Thephononicself-energyduetoaphononexcitingacharge neutralelectron/holepairfromtheFermisea.......120 2.3.10.4Mathinterlude:principle-valueintegralsbyaniteinterval method............................120 8

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2.3.10.5Probability-theoryandexplanationoftheappearanceof theconvolutionoperation..................122 2.3.10.6AnalyticrstBornapproximationtotheselfenergy by Sommerfeldexpansionfor N C =1..............124 2.4ObservablesintheAbsenceofanAppliedMagneticField..........134 2.4.1TheContinuousElectricalandNumberCurrent...........134 2.4.1.1ThematerialcurrentasaLandauer-Buttikertermplusa correction...........................135 2.4.1.2Thematerialcurrentasasumofelastic"andinelastic" terms.............................138 2.4.1.3Interpretationsofthetransmissionpluscorrectionvs.elastic plusinelasticdecompositions................138 2.4.1.4Sommerfeldexpansionoftheinelasticmaterialcurrent..139 2.4.1.5Thenon-interactinglimit..................141 2.4.2TheDiscontinuousEnergyCurrentintheElectrons.........142 2.4.3TheDiscontinuousEnergyCurrentinthePhonons..........143 2.4.4Thermodynamicsinterlude:theLandauer-Buttikercurrentandthe thermalstateofthereservoirs.....................144 3NONLINEARTHERMOELECTRICRESPONSEINRESONANTTUNNELING147 3.1TheoryoftheResonantRegime........................147 3.1.1Spectral-densityofacollectionofinter-coupledlong-livedquantum levels...................................147 3.1.2Thespectraldensityofasinglelevel..................148 3.2TheThermoelectricResonantGenerator...................149 3.2.1Currentwithanammeterasload...................149 3.2.2Voltagewithavoltmeterasload....................150 3.3TheNonlinearMaterialCurrent........................151 3.3.1Thelargegradientsaccessiblebythedriventhermoelectric.....151 3.3.2Eectofelectron-phononcouplinguponnonlinearnumbercurrent.154 3.3.2.1Lowtemperature.......................154 3.3.2.2Hightemperature.......................155 3.3.3Divergenceofthenonlinearmaterialcurrent.............157 3.4TheNonlinearEnergyCurrent.........................159 3.4.1Lackofcontinuityofthecurrent E e )]TJ0 g 0 G/F15 11.9552 Tf 15.96 -4.338 Td [(.................160 3.4.1.1Divergenceof E e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(andelectron-phononcoupling g ....162 3.4.1.2Divergenceof E e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(andgradients T and .......162 3.4.2 E e )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(asafunctionof ........................163 3.5TotalElectronicPlusPhononicThermalCurrent.............164 3.6TheApproximately-ElectronicThermalCurrent...............166 3.7Temperatureofthecenter-regionphonons..................169 3.8OperationoftheThermoelectrically-AssistedCircuit............170 3.8.1Settingloadresistanceequaltoreciprocal-dierential-conductance formaximumpower-hysteresis....................170 9

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4CONCLUSION....................................173 APPENDIX ADRIVENVS.NON-DRIVENEFFICIENCYFORANINTEGRABLERESONANT LIKE"TOYSPECTRALFUNCTION.......................177 BBOXCARSPECTRALFUNCTIONOFACHAINCENTERREGION.....186 REFERENCES.......................................187 BIOGRAPHICALSKETCH................................191 10

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LISTOFFIGURES Figure page 1-1Densitiesofstatesorspectralfunctionsinzero,one,two,andthreedimensions allowingcircumvention[15,16,18]oftheWiedemann-Franzlaw..........25 2-1Drivingagentinserieswiththermoelectricblackboxallowingaccesstoregions beyondthegenerator-regimewhereatemperature-gradientincreasesthecurrent.41 2-2Inputof P T 0 tothermoelectricdevicethateitherpumpsthermalenergyfrom coldtohot,orscattersuselesslyandoperatesintandemwiththespontaneous owofhottocold...................................51 2-3SchematicofthequadraticHamiltonian2{35and2{39for N C =2......59 2-4Atype-A"phonononlyhasanonzerointeraction-vertex M Aii 0 for i = i 0 .For acentralsitewith N C sites,therearethus N C possibleverticesandthus N 2 C possibleFock-diagrams................................68 2-5Atype-B"phonononlyhasanonzerointeraction-vertex M Bii 0 for i 6 = i 0 .Fora centralsitewith N C sites,therearethus N C N C )]TJ/F15 11.9552 Tf 12.091 0 Td [(1possibleverticesandthus N 2 C N C )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 possibleFock-diagrams.........................69 2-6Atype-A"phonononlyhasanonzerointeraction-vertex M Aii 0 for i = i 0 .For acentralsitewith N C sites,therearethus N C possibleverticesandthus N 2 C possible -independentHartree-diagrams......................69 2-7Atype-B"phonononlyhasanonzerointeraction-vertex M Bii 0 for i 6 = i 0 .Fora centralsitewith N C sites,therearethus N C N C )]TJ/F15 11.9552 Tf 12.091 0 Td [(1possibleverticesandthus N 2 C N C )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 possible -independentHartree-diagrams...............69 2-8Aphononoftype A or B interactswiththeFermiseaandexcitesanelectron/hole pair,withself-energy.Iftheoutgoing/ingoingphononisoftype A or B the vertexhastheconditions i = i 0 vs. i 6 = i 0 ,and j = j 0 vs. j 6 = j 0 ..........71 2-9Bandvelocityandenergyasafunctionofmomentumforadiatomicchain....89 2-10Opticalandacousticphononbranchesinadiatomicchainforvariousmass-ratios.92 2-11Poleplotfortheintegrandofthecontourintegraldening g r 0 ` ` 0 withexaggerated inntesimal.......................................95 2-12PhononicpoleplotfortheretardedphononicGreenfunctionshowingtheimportance ofusingthesignedinntesimal =0 + sgn .Themagnitudeof isexaggerated.96 2-13Convolutionsofuniform,exponential,andGaussiandistributions.........124 2-14ComparisonoftheanalyticandnumericalrstBornapproximatedelectron/phonon collisionalimaginaryself-energies 7 F 1 forpositiveresonance-energy........131 11

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2-15ComparisonoftheanalyticandnumericalrstBornapproximatedelectron/phonon collisionalimaginaryself-energies 7 F 1 fornegativeresonance-energy.......131 2-16ComparisonoftheanalyticandnumericalrstBornapproximatedelectron/phonon collisionalimaginaryself-energies 7 F 1 forpositiveresonance-energy........133 2-17ComparisonoftheanalyticandnumericalrstBornapproximatedelectron/phonon collisionalimaginaryself-energies 7 F 1 forpositiveresonance-energy........133 3-1DepthoftheFermiseaforaresonanttunnelingdevice...............149 3-2Eectofhot-leadtemperture T L Numbercurrentatzerochemicalpotentialdierence. ShallowFermiseaisinfaintdottedlineforreference,whilethedeepFermisea isshownasasolidline.................................150 3-3Maximumthermalbiasatzerochemicalpotentialdierence.ShallowFermisea isinfaintdottedlineforreference,whilethedeepFermiseaisshownasasolid line...........................................151 3-4Numbercurrent N solidandwithoutelectron/phononinteraction N 0 dashed vs.potential-gradient V T acrossthethermoelectricblackboxat T L =T r =3 ; 5 ; 7 ; 9 blue,green,purple,andredsolidlines,respectivelyand T R =0accentuating quantumfeaturesinashallowFermiseaand g =0 : 15...............152 3-5Numbercurrentmagnitudeasafunctionofeachpossiblegradient V T and foradeepFermiseashowingtheevolutionofsaturationof E th tothevalues j U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( j ,hot/coldasymmetrywithrespectto T ,andasingleundulation inthecurrentduetophononassistedtunneling...................153 3-6Numbercurrentmagnitudeasafunctionofeachpossiblegradient V T and forashallowFermisea3{5withoutelectron/phononcoupling g =0.....154 3-7Numbercurrentasafunctionofpotential-gradientacrossathermoelectricblack boxforathickFermisea,3{4withafaint-dashedthinFermisea3{5for reference........................................155 3-8Numbercurrentasafunctionofpotential-gradientacrossathermoelectricblack boxforathick[23]Fermisea.............................156 3-9Numbercurrentmagnitudeasafunctionofeachpossiblegradient V T and foradeepFermiseashowingtheevolutionofsaturationof E th tothevalues j U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( j ,hot/coldasymmetrywithrespectto T ,andasingleundulation inthecurrentduetophononassistedtunneling...................156 3-10Number-current-discontinuity N = N L )]TJ/F22 11.9552 Tf 12.079 0 Td [(N R inunitsof T room asafunctionof V T ,whichunexpectedlylessensforashallowFermisea..............157 3-11Number-current-discontinuity N = N L )]TJ/F22 11.9552 Tf 12.74 0 Td [(N R asafunctionof V T ,exceptat highertemperature..................................158 12

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3-12Numbercurrentdivergencemagnitudeinunitsof T room asafunctionofboth gradients T and for g =0 : 15andashallowFermisea............159 3-13Thenumbercurrentdivergencemagnitudeinunitsof T room asafunctionofboth gradients T and for g =0andashallowFermiseaisontheorderofmachine epsilon10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(16 ,andshowsupasnumericalnoise"..................160 3-14Electronicenergycurrentdiscontinuityplottedvs.absolute-band-densityphononic energycurrentdiscontinuity..............................161 3-15Electronicenergycurrentdiscontinuityplottedvs.potentialgradient,exceptat highertemperature, T L =10 T room ..........................161 3-16Divergenceof E e )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(asafunctionofbothgradients T and thelatterin unitsof U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( forashallowFermiseafor g =0 : 15................163 3-17Divergenceof E e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(asafunctionofbothgradients T and forashallowFermi sea,ontheorderof10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(14 atmostfor g =0.....................163 3-18Energycurrentasafunctionofpotential-gradientacrossathermoelectricblack boxforathick[23]Fermiseafor T L =2 T 2 room ....................164 3-19Energycurrentasafunctionofpotential-gradientacrossathermoelectricblack boxforathickFermiseathinFermiseashownasdashedforreferenceinunits of T 2 room .Highertemperature.............................165 3-20Energycurrentasafunctionofpotential-gradientacrossathermoelectricblack boxforathinFermiseaat T L =2 T room .......................165 3-21Eectofelectron-phononcoupling g uponenergycurrentasafunctionofpotential gradientacrossathermoelectricblackboxforathinFermiseaat T L =10 T room .166 3-22Totalthermalcurrent Q = Q e )]TJ/F15 11.9552 Tf 10.038 -4.338 Td [(+ Q ph .Thefeaturesof Q e )]TJ/F15 11.9552 Tf 10.986 -4.338 Td [(stronglydepending upon arelargelydrownedoutbythemonotonicincreaseof Q ph with T lendingmerittotheideaofregarding Q e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(asaseparateentity...........167 3-23Heatplotofthequantity j Q e )]TJ/F19 11.9552 Tf 7.085 -4.339 Td [(j =T 2 room asafunctionof and T for g =0 : 15 inashallowFermisea.................................167 3-24Heatplotoftheelectronicthermalcurrent j Q e )]TJ/F19 11.9552 Tf 7.085 -4.338 Td [(j =T 2 room asafunctionof and T for g =0inashallowFermisea.........................168 3-25Dierence-magnitude T ph )]TJ/F22 11.9552 Tf 11.956 0 Td [(T =T incenter-regionphonontemperature T ph fromtheaveragetemperture T = T L + T R = 2forvarious T and .For T 0and attheresonance,thereisalargedeviationof T ph from T .For T '1 ,thecentersitephonontemperaturequicklysaturatestotheaverage T andbecomesindependentof ..........................170 13

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3-26Electricalcurrent I = )]TJ/F22 11.9552 Tf 9.298 0 Td [(N vs.appliedbias E a foranOhmicloadofunitresistance R L =1,forthinFermisea,withthickFermiseashownindashedlineforreference. Hotleadtemperatureof10 T room used........................171 3-27Electricalcurrentvs.appliedbiasforvariouselectron-phononcoupling g ,thick Fermisea,forloadresistance R L = j dN=dV T j )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 at T L =2 T room ..........171 A-1Comparisonofcalculatedgenerator-regimepowerasafunctionofmaximum U i andspread)]TJ/F23 7.9701 Tf 67.21 -1.793 Td [(i ofaLorentzianwhosespectralfunction )]TJ/F23 7.9701 Tf 5.289 0 Td [(= )]TJ/F23 7.9701 Tf 6.587 0 Td [(U 1 +)]TJ/F18 5.9776 Tf 15.168 2.269 Td [(2 comesfrom theHamiltonianofaresonance U 1 withescaperates)]TJ/F23 7.9701 Tf 99.466 -1.794 Td [(! toaleftandaright leadvs.theexponentiatednegativeabsolutevaluewhichdoesnotcomefrom aHamiltonianbutintegratestoclosedformwithaFermi-Diracdistribution 2{45.Thetwodierenttransmissionfunctionsproducevaluesofthermoelectric generatorpowerwhicharequalitativelysimilar...................180 A-2Comparisonofcalculatedgenerator-regimeutilitycoecienteciencydivided bythemaximumCarnoteciencyasafunctionofmaximum U i andspread )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(i ofaLorentzianvs.theexponentiatednegativeabsolutevalue.Aswiththe powerinFig.A-1,thereisqualitativesimilarity...................181 A-3Left:featuresof N and Q forA{1asafunctionof = E th .Right:onsetof numericalbreakdownfortoosmallavalueof W ..................182 A-4Numberandthermalcurrents:ordersofmagnitudeandseparatebehaviours...182 A-5 P A vs. ,witharrowsshowingtherootnding-processthatsolves P A =0 withrespectto andndsthechemicalpotentialgradient = P A =0 Shadedboxesindicatethatagiven and T isnotgauranteedtoyielda P A =0 thatsolves P A =0............................183 A-6 P A =0 and P L vs. ...............................183 A-7Eciencyandutilityasafunctionof ,withanadded independent Q to obey2{26.ShallowFermisea.Eciencyisundenediftheredoesnotexist satisfying P A =0................................184 A-8EciencyandutilityasinFig.A-7,exceptforadeepFermisea.Eciencyappears todropduetothedecreasedeectof T upon P A asinFig.A-5.......184 14

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy THERMALANDELECTRICALNONLINEARCURRENTSDRIVENTHROUGHA NANO-BRIDGEBEARINGANELECTRON-PHONONINTERACTION By BradleyJosephNartowt December2017 Chair:KhandkerA.Muttalib Cochair:SelmanP.Hersheld Major:Physics Motivatedbythegoalofstudyingthermoelectricphenomenainsystemsoflow dimensionality,electricalandthermalcurrentsthroughanano-bridgewithanelectron/phonon interactionarecalculatedinatoymodelusingthenon-equilibriumGreenfunction NEGFtechnique.Thus,thegoalofthisdissertationistodeterminetheeectof localizedinteractionsofelectronsandphononsinajunctionbetweenleft L and right L leadsatdisparatepotentialsandtemperatures.Thepotentialgradientacrossthe thermoelectricdevicedrivesanelectricalcurrent I = )]TJ/F22 11.9552 Tf 9.298 0 Td [(eN atthevoltage V T aboutthe thermoelectricdevice,andthusexertapower P T uponitssurroundings.Athermal-energy current Q accompaniesthis P T .Thepositivequantity = P=Q 1 )]TJ/F22 11.9552 Tf 12.462 0 Td [(T cold =T hot isthe mostecientlythispower P L canbedeliveredtoaloadbyaquantityof Q Trulynonlinearfeaturesoftheelectroniccurrentremaininaccessibletoathermoelectric devicefunctioningsolelyasageneratornetpoweroutow,ratherthaninow.The thermoelectricdeviceisalsoexternallydrivenbyapower P A ,andthechangeinpower deliveredtoaloadduetoapplicationofthetemperaturegradientwhileholdingthe appliedpowerconstantisstudied,andaneciencyisproposedforthis.Whilethereis stillapower-inowtothethermoelectricdevice,itneverthelessremainsthatadditional powerisdeliveredtotheloadasadirectresultofapplyingthetemperaturegradient. 15

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CHAPTER1 INTRODUCTION Theaimofthisdissertationistogeneralizeawell-knownspherical-cow"model popularsincethe1980sofasemiconductingheterostructuretonitetemperature anditinerantphonons.Anelectron/phononinteractionlocalizedtoacenter-regionis introduced.Thenumber,electricalandthermal nonlinear steady-statecurrentsofcharge carriersandphononsforbothadriventhermoelectricandathermoelectricgeneratorare thecentralresultsofthisdissertation. Useofthefullnonlinearcurrentsasopposedtojusttransportcoecientslike electrical/thermalconductivity,Seebeckcoecient,Peltiercoecient,Thomsoncoecient etc.allowsincorporationoftheeectsofnitetemperatureuponfar-out-of-equilibrium 1 featuresofcurrentssuchasnegativedierentialresistanceandresonancefornite-width spectralresonance.Onewhocalculatesthenonlinearcurrentscanalsodeterminethe eectofpower-delivery 2 tosomeloadonapplicationofatemperature-gradientof arbitrarymagnitude. However,applicationofatemperaturegradienttoathermoelectricdeviceisbarely abletogenerateachemicalpotentialgradient thatreachestheresonanceofaspectral function.Hence,toaccesstrulynonlinearfeaturesofathermoelectricdevice,onemust drivethethermoelectricdeviceusinganexternaldrivingagent.Forsomechemical potentialgradients ,applicationofatemperaturegradient T increasesdeliveryof powertotheload.Itispossibletothenadjust suchthatthedrivingagentapplies thesameamountofpower.Thisbringsupthepossibilityofaneciency d ofadriven system,andsuchan d isproposedinthisdissertation. 1 Suchnonlinearfeaturesappearwithrespecttothe th lead'schemicalpotential .In asemiconductingheterostructure,anexternalagentbringsthisheterostructurefaroutof equilibriumassuch. 2 Theratesofpower-deliverytovarioussystemsaresubjecttoenergy-conservation. 16

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Throughoutthisdissertation,equation-numberingisusedassparinglyaspossible.An equationisonlynumberedifitisreferredtolater.Therefore,un-numberedequationsare designatedasjustshowingintermediaryexplicitstepsofaderivation. 1.1AbridgedHistoryofThermoelectricity Althoughunrelatedtothetechnicalcomplexitiesoftheresultsofthiswork,an abridgedhistoryofthermoelectricityispresented. 1.1.1TheBeginningoftheFieldofThermoelectricity ThermoelectricitybeganwithSeebeck'sdiscoverythatathermalcurrentcancarry electricalcurrent,andwithPeltier'sdiscoverythatanelectricalcurrentcancarrythermal energyagainstatemperaturegradient.LordKelvinestablishedtheoreticalrelation betweenthesetwophenomenaimpressivelyabout10yearsbeforeMaxwell'slawsof electromagnetismwerepublished,whentheSeebeckeectwasstillmistakenforbeing magnetic ratherthan electric 1.1.1.1Avoltage-gradientresultingfromatemperature-gradient-the Seebeckeect Thedrivingofanelectricalcurrentbyatemperature-gradientwasdiscoveredwhen, in1821,ThomasJohannSeebeckfashionedaclosedloopformedbytwosemicirclesof disparate metalsatdisparatetemperatures.Seebeckreportedthatthisloopcarriedcurrent whenheobservedanearbycompassneedlealigningitselfperpendicularto 3 theloop's plane.Assuggestedbythetitleofhismonograph[1],Seebeckincorrectly 4 concludedthe responsetothetemperature-gradientwas magnetic ,ratherthanelectrical. 3 Theloop'smagneticdipolemoment ~ loop alignedwiththecompass'smagneticmoment ~compass duetotheloopbeingofarea ~ A loop andcarryingcurrent I loop .Obviously,the loop'scurrentissustainedbyanEMF. 4 InSeebeck'sdefense,Maxwell'sequationswerepublishedbetween1861and1862 40yearsafterSeebeck'smonograph. 17

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1.1.1.2Aheatpumpfromelectricalow-thetime-reversedSeebeckeect Arelatedpre-Maxwellthermoelectriceectwasthetemperature-gradientbetween two disparate metalsinelectricalcontactbearingthesamecurrentanddevelopinga temperaturegradientacrossthesiteofcontact,whichwasPeltier's[2]observationabout 10yearslater.Peltier'sexperimentwastorunanelectricalcurrentbetweentodespairat materialsandthenobserveatemperaturedierencebuildingupbetweenthosetwomatter Sinceseabeck'sworkwasthoughttobetheresultofmagneticphenomenonitwasnot realizebyPeltierthattheSeebeckeectwasthetimereversedeectofthePeltiereect. 1.1.1.3Athermoelectriccurrentfromatemperature-gradientwithinasingle material Athermoelectriceectwithin one material,theelectricalcurrentowinginany conductorbearinganon-uniform 5 temperature-prole,wasmadeknown[3]byLord Kelvinin1854.LordKelvin'scontributionwasthatatemperaturegradientwithinone materialresultedinapotentialgradientwithinthatsamematerial.Thismeantthatthe SeebeckandPeltiereectsweretime-reversedintrinsicpropertiesofagivenmaterial. Kelvinwentontointroducethequantity K @ T )]TJ/F22 11.9552 Tf 12.553 0 Td [(S e foramaterialattemperature T ,Seebeckcoecient S e ,andPeltiercoecient.Kelvinalsoproducedtherelation = TS e ofthesame,indicatingtime-reversal-symmetrybetweentheSeebeckandPeltier eects. 1.1.1.4TheWiedemann-Franzlawofcorrelationbetweenelectricaland thermalconductivity GermanphysicistsGustavHeinrichWiedemannandRudolphFranzreportedin1853 anapproximatecorrelationbetweenthe unitsofinntesimalcurrent-densityperunitof appliedinntesimalelectriceldelectricalconductivityandthe unitsofinntesimal thermal-uxperunitofinntesimalappliedtemperaturegradientthermalconductivity 5 Kelvin'sthermoelectriceectallowsonetoviewasampleofanymaterialconsidered tobeablackboxasaheat-enginewhoseeciencyisboundedbytheCarnoteciency. 18

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formetals.In1900,PaulKarlLudwigDrudeprovidedamodel[4{6]for and ofa metalattemperature T ,oneoftheconsequencesofwhich 6 wasWiedemannandFranz's reportedcorrelation, = T = 2 k B =e 2 k B = e = ~ =1 = 3 = 2{1 1.1.1.5Asummaryofthephysicallawsgoverningthethermoelectricresponseoffermionicchargecarriers Whenoneignoresthermaltransportbybackground"elementsphonons,photons, andmaterialcontactandconsidersonlythatduetoelectrons,thethermoelectriceects areduetothermalenergytransportedbyanychargecarriers.Thatis,beginwiththelaws thatchargecarrierscancarrythermalenergy,thermalenergyowsspontaneously fromahotregiontoacoldregion,andthermalenergymaybecreatedordestroyed, andisanon-conservedinexactdierential.Becauseofthis,onecanhavethefollowing eectsintransport: SeebeckEect: Generateanelectricaleldpotentialgradientonapplicationofa temperaturegradientsoastodeliveraowofpowertoaloadandgenerateuseful energy. PeltierEect: Generateatemperaturegradientonapplicationofanelectricalpotential gradient.Thermodynamically,theregionbeingcooledbecomestheload. ThomsonEect: Createordestroythermalenergyatsomepointinamaterialasa consequenceoftheSeebeckeect'stemperature-dependencelocalizedtothatpoint ofthermalenergycreation/destruction. 6 Tobeprecise,1{1isimpliedbytheDrudemodelonlybecausethemodel mis-estimates and byacommonfactorthatcancelsintheratioof1{1.Furthermore, theDrudemodeltreatedthe duesolelytotheelectrons. 19

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Thenonlinearcurrentsthatcontainthistransport-coecientdescriptionaresimilarly governedbytheabovementionedlawsgoverningthermalenergycarriedbymaterial charge-carriers. 1.1.1.6ThethermogalvanomagneticeectsofNernst,Ettingshausen,Righi, andLeducinanalogytotheHalleect Thedevelopmentofapotential-gradient eV transversetoaspontaneousow Q ofheat,atemperature-gradient T transversetoaspontaneousowa I of electricalcurrentorb Q ofthermalcurrenteachinthepresenceofamagneticeld B ,arethethermogalvanomagneticeects.TheyarerespectivelynamedtheNernst[7][8] ,Ettingshausen,andRighi-Leduceects[9].Alloftheseeects postdateMaxwell'slawsofelectromagneticelds. 1.1.2TheStudyofThermoelectricityinBulkMaterialsintheMid-20th Century Oncethermoelectricphenomenawereobservedinbulkmaterialsamples,theobvious goalbecameoptimizationoftheeectusefulenergyobtainablefromatemperature gradientinanappreciablequantitybeingahighlydesirableengineeringgoal.Becauseof theWiedemann-Franzlaw,thisgoalunfortunatelystagnateduntilthestudyofelectrical responseinsystemsofreduceddimensionalityinthe1980s. 1.1.2.1ThethermoelectricgureofmeritofIoe In1949,RussianphysicistAbramFedorovichIoemadetheproblemofgood thermoelectric"well-posedandquantiedbyintroducingasinglequantity ZT that ishigh/lowforahigh/lowthermoelectricresponsefromagiventemperaturegradient inthelinearresponse.Namely,consideranymaterial.Letthatmaterialsustain S e unitsofinntesimalvoltage-responseperunitofinntesimalappliedtemperature gradientSeebeckcoecient.Letthematerialletpassthroughit unitsofinntesimal current-densityperunitofappliedinntesimalelectriceldelectricalconductivity. Similarly,letthematerialletpassthroughit unitsofinntesimalthermal-uxper unitofinntesimalappliedtemperaturegradientthermalconductivity.Finally,letthe 20

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materialbeoftemperature T .Then,Ioedenedsuchamaterial'sthermoelectricgure ofmerit Z T = ZT as, ZT S 2 e = {2 Theproblemofoptimizing ZT requiresindependentcontrollingof S e e ,and whichcannotbedoneaccordingtotheWiedemann-Franzlaw.However,becausethe Wiedemann-Franzlawassumes[5][6]abulksystem,reductionofthedimensionalityofthe materialtothenanolevelinoneormoredimenisonsisaworkaroundtothisobstacle. 1.1.2.2ThethermoelectriceectduringtheWorldWarsandtheColdWar AfterIoeinventedthegureofmerit ZT inthelate1950s,theSpaceRacebegan. Clearly,whenthegoalwasbringinghumanlifetotheextremely-coldreachesofspacea naturaltemperature-gradientexisted.Oneonlyneededsomethinghotandlightweight 7 to bringaboardtheshuttle,becausethecoldreachesouter-spaceservedasacold-reservoir thatcouldgenerateelectricitywhichisscarceinouterspace.Thus,ifnotinthe infrastructureofEarth,thermoelectricshadatleastfoundanicheapplicationinecient astheywereatthattimeinspacetravel. 1.1.2.3TheSeebeckeectaftertheSpaceAge Perhapsbuoyedbytheoptimismfromhavinglandedonthemoon,workersquickly triedtomakemoreandmoreecientthermoelectricmaterials.Attemptssaturatedat agureofmeritofaround1or2,andsoupperlimitswhichwereinitiallyconjecture 8 begantobetakenmoreseriously.Manybegantodoubtthatthermoelectricitywould becomepartofourinfrastructure.Thematterbegintolullfromthe1970stothe 7 Thereisacostof$18,000perkilogramofshuttlepayload.Thus,radioactivematerials suchasplutoniumwereusedasthehotreservoirs. 8 Inthe1960s,anupper-limitof ZT 1uponthegureofmeritwashypothesized. 21

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1990s.Yet,thermoelectricsremainedessential,forinstanceinouter-space 9 andinthe environmentalconcerns 10 ofcushioningtheenergy-crisisandminimizingpollution. 1.1.3TheoreticalandExperimentalMaturationoftheFieldofNanoelectronicsandThermoelectricity Afteraperiodofdisinterestinthermoelectricityinthe1970s,electricalcurrent andconductivityinnano-systemsbecameexperimentallyaccessibleinthe1980s.This promptedaurryoftheoreticalresultsintwoandonedimensionalmodels[10][11][12][13][14], butthisworkalmostinvariablyassumedzero-temperature.Thelullofinterestin thermoelectricitycontinuedduringthistimeuntilalargegureofmeritwasmeasured inananisotropicquantum-wellsuperlatticebyHicksandDresselhaus[15],anda diverging gureofmeritwascalculatedbyMahanandSofo[16]forthezerocurrentthrougha perfectspectral-resonance. 1.1.3.1Thermoelectricityinsystemsofreduceddimensionality Theideatostudyandproducethermoelectricityinsystemsofreduceddimensionality beganin1993,whenHicksandDresselhausshowed[15]thatthethermoelectricgureof merit ZT oftwo-dimensionalanisotropicquantumwellsuperlatticecouldbemadetobe aslargeas Z 2 D T 10.G.D.Mahansubsequentlypointedout[16]thatathermoelectric materialwithaperfectspectralresonance 11 givingwell-studied[17]resonanttunneling 9 Whereoneisfarawayfromanyconvenientpower-source,andwithalarge temperature-gradientreadilyavailableouterspacehavinganabsolutetemperatureof =3 : 11 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 W ;here, W setsthescaleforallquantitieshavingdimensionsofenergyin thiswork,thoughitistobeintroducedlaterthough,shortly. 10 Ifanecientthermoelectricdeviceisrealized enmasse ininfrastructure,power sourcesmightbequiet,havenomovingpartsandthuswouldrequirelittlerepair, andweputoutlowtonopollution.Infact,theadvantagesanddisadvantagesof thermoelectricitybecomingamajorpartofinfrastructurehasanalogiestotheadvantages anddisadvantagesofstandardharddrivesorHDD"soversolidstatedrivesorSSD"s asstorageandretrievaldevicesincomputers. 11 AperfectspectralfunctionisaDirac-deltafunction,whichallows zero currenttopass through. 22

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throughastationarystatehadanarbitrarily-largegureofmerit.Sometimelaterin2007 Dresselhausandotherspointedout[15]thatthethermoelectriceectwouldbeaccentuated insystemsofreduceddimensionalityduetoqualitativedierencesinthedensityofstates thatisinzero,one,andtwodimensionalsystems. Ingeneral,theeectsofconnementtolowerdimensionalityisincreasedboundary-scatteing ofphononsandqualitativechangeintheelectronicdensityofstates.Nano-electronicswas birthedinthe1980s,andmaturedinthe1990sandearly2000s.Oncethismaturation completed,previously-inaccessiblemeansoftuningmaterialpropertiesavailedthemselves toIoe'sproblemofindependentlycontrolling S e e ,and 1.1.3.2Conductioninsystemsofreduceddimensionality Thus,thehighgureofmeritthatwassoughtafterinvainbeforetheadventof nano-electronicshadbeenobserved.Abrieftechnicaldescriptionofconductioninsystems ofreduceddimensionality,asaninterludetothishistoricaloverview,isinorder.The electricalresistance R encounteredbyanelectricalcurrent I runningparalleltoalength L andthroughacross-sectional-area A ofconducting-materialofresistivity is L=A Thetheoryofelectricalconductioninnano-andmeso-systemsbeganbyquestioningat whatpointinthelimits L 0and A 0thisformulafortheelectricalresistancebreaks down.ItshouldbesaidthatDirac'smonographonquantumtheorywaspublishedin 1930,andsuggestedthetheoreticalanswerof L;A = h=p forelectronscomprising I ofmomentum p = mv = mI= nAe implyingadeBrogliewavelength for n electrons perunitvolume,asindicated.Furthermore,Drude'stheoryrelatingthedriftvelocity tothecurrentarrivedin1900,acoupleofdecadesbeforethebirthofquantumtheory suggestingthetheoreticalanswerof L;A v = I= nAe foranaveragetimeof betweenlattice-collisions.Inthesameyear1900,Planckpublishedhisdiscoveryofthe constant h fromhisexplanationofblackbodyradiationandtheultravioletcatastrophe. Nevertheless,thefabricationofnano-devicesinwhichthislimitwouldplayoutremained unavailableuntilthe1980s,revealingthemeanfreepath v ,thespatial-separation 23

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betweenlattice-collisions,asoneofthreerelevantlength-scales.Theothertwolength scalesaretheabovementioneddeBrogliewavelengthandthephase-relaxation-lengththe characteristiclengthoverwhichthephaseofanelectronicwavefunctionse.g.,inaWKB semiclassicalwavefunctionsignicantlychanges. 1.2OverviewofthePhysicalProblemofThermoelectricity Havingestablishedahistoricaloverviewofthermoelectricity,anoverviewofthe benetsofoptimizingthethermoelectriceectandthefundamentalphysicsisinorder. Theseserveasmotivationforthetechnicalitiestobesluggedthroughlater. 1.2.1MotivationandScope Theoreticalstudyofathermoelectricdeviceisnotonlydesirableformanypractical reasons,butalsoaddressesafundamentaltheoreticalproblem.Inpracticalconsiderations, thethermoelectricengineoperateswithoutproducingpossibly-toxicemissions,withonly atemperaturegradienttodriveit.Likethesolidstatehard-drivesthatarereplacing thosewithspinningdisks,thethermoelectricdeviceissuperiortothosedevicesthathave movingpartsrequiringlittlemechanicalmaintenance. 1.2.1.1Separatemanipulationofthermalandelectricaltransport Studyofthethermoelectriceectisalsoafundamentalphysicalproblem.Theaim istoindependentlycontrolelectricalandthermaltransport.Pursuitoftheproblemof thermoelectrictransportpursuesthesefundamentaltopicsclearlyasamatterofcourse. Thepropensityofabulkthree-dimensionalmacroscopicmaterial'sconduction-electrons totransportthermalenergyandelectricalcurrentarecorrelatedbytheWiedemann-Franz 12 law[4{6].Toindependentlycontrolthesetransportcoecientsandthenonlinear responsebeyond,thedimensionalityofthesystemisreduced.Thishastheeectof 12 TheWiedemann-Franzlawgovernstheelectricalconductivityandthethermal conductivityofelectrons,whicharetransportcoecients.Thisdissertationtreatsthe nonlinearresponse. 24

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Figure1-1.Densitiesofstatesorspectralfunctionsinzero,one,two,andthree dimensionsallowingcircumvention[15,16,18]oftheWiedemann-Franzlaw. sharpeningthetransport-governingspectralfunctionintoatunablepeak,permittingthe desiredindependentcontrol. Thethermoelectricdeviceinthisdissertationisasinglequantumleveloccupiableby twospin-opposedelectrons,andthushasthezero-dimensionalspectralfunctionofFig. 1-1. 1.2.2UseofLinearvs.NonlinearResponse 1.2.2.1Conservednonlinearcurrentsasthechargecarriersinteractwith othercurrents Thenonlinearcurrentstobecalculatedarethenumbercurrent N ofcharge-carriers andthetotalenergycurrent E .Thisdissertationhasfoundnotonly N butalso E in theconservingapproximations 13 ituses.Iftheenergycurrent E e )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(ofcharge-carriers is interacting withanothercurrent E other i.e.,anelectron/phononinteraction,an electron/photoninteraction,etc.then E e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(and E other bythemselves arenotconserved, buttheirsum E = E e )]TJ/F15 11.9552 Tf 10.205 -4.338 Td [(+ E other is.Thenonlinearthermalcurrent Q = Q e )]TJ/F15 11.9552 Tf 10.205 -4.338 Td [(+ Q other is obtainedfrom E = Q + hot N .Ifthereiselectron/other"interaction,thentotheextent ofthestrengthoftheinteraction Q e )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [(is not abletobeseparatelycalculatedfrom Q other .If thestrengthoftheinteractionisparametrizedby g ,theninthelimitof g 0, Q e )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [(indeed isseparatefrom Q other .Thisdissertationspecializestothecaseofother"beingphonons. 13 Self-energiesofaninteractionarecalculatedusingtheself-consistentBorn approximation. 25

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1.2.2.2Thephysicsinthefullnonlinearcurrentandtheproblemofthermoelectricity Therehasbeendebateamongtheoreticians[19][20]astowhetheralinear-response transport-coecientdescriptionofthethermoelectriceectissucient,orifa nonlinear-responsecurrentsandthermodynamiceciencyisrequired.Sincethe linear-responsedescriptionistheTaylor-expansionofthenonlinearcurrentstruncated atlinearorder,theargumentforalinear-responsedescriptionisinpursuitofsimplicity withoutneglectingphysics.Itisthendingsofthisdissertationthattherearemany interestingnonlinearphenomenaresonance,negativedierentialresistance,phonon assistedtunnelinglyingbeyondthislineargenerator-regime,but only ifthethermoelectric deviceisexternallydriven.Thethermoelectriceectinthetheoryofnonlinearcurrent responseisdescribableasatemperature-gradient T sustainingachemicalpotential gradient .If T alonesustains ,then foralargerangeofparameters[21]failsto growlargeenoughtoaccesstheusualnonlinearfeaturesofresonance,negativedierential resistance,andphonon-assistedtunneling. 1.2.2.3Saturationofthemaximumchemicalpotential Whathappensifonemakes T arbitrarilylarge?Forasteady-state[19]number-current N describedbyaLandauer-Buttikerform,thereisamaximum = U )]TJ/F22 11.9552 Tf 12.471 0 Td [( setbythe distanceoftheresonance U fromtheequilibriumchemicalpotential .However,sucha large T willcausetheFermi-DiracdistributionstobecomeBoltzmann-liketheclassical high-temperturelimitofboththeFermi-DiracandBose-Einsteindistributions,thus spoilingthesharpnesstypifyingmostdistributionsofaquantumsystem. 1.2.2.4Theproclivityoflinearresponseintheundriventhermoelectric device Thus,inwhatisknownsofar,authorsarejustiedinclaimingthenonlinearresponse isunnecessary[22].Infact,thisiscorroborratedbyrandomlychoosing[21]parameters andleast-squares-ttingastraightlineto I vs. L )]TJ/F22 11.9552 Tf 12.895 0 Td [( R data,citingtheexcellentt R 2 0 : 92ofalinearmodel.Otherspointoutthatthenonlinearresponseisrequired 26

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duetomanifestly-largegradientstypifyingananosystem[21].Thisdissertationclaims thenonlinearresponseisrequiredforthecaseofthethermalassist.Nonlinearelectrical currentshavebeencalculatedforalongtime[13,23],anddeliberatelyatzerotemperature notonlytoaccentuatequantumfeaturesbutalsoinpursuitofmathematicalsimplicity. Obviously,temperaturemustbenonzerointhethermoelectriceect,andmoreover non-inntesimaltoachieveanitethermoelectricpower.Quantumfeaturessuchas resonantpeaksmayliedeepwithintheFermisea,whilethemaximumchemicalpotential dierence E th intheun-assisted"caseisontheorderofthetemperature-gradient, T ; thatis E th T Asapuregeneratori.e.,withoutbeingdrivenbyavoltage-supply,thethermoelectric generatorpowersaload.Iftheloadisanammeterwhichdropszerovoltage,thecurrent isamaximum.Iftheloadisavoltmeterwhichdropsmaximumvoltage,thecurrentis zero.Intermediatetothesetwocasesisthedeliveryofthemaximumpossiblepowerto theload.Itisfoundthatthegeneratorregimeislargelylinear,owingtothefactthatthe maximumvoltageisapproximatelyequaltotheappliedtemperaturegradient.Thisisthe functionofthethermoelectricthatistraditionallystudied. Duetobeingconnedtothisintervalofvoltage,thethermoelectricislargelylinear. Theonlynonlineareectistheonsetofresonant-currentgivingthedivergingMahan-Sofo gureofmerit,whichmayevenbetoodeepintheFermisea.Atleasttwononlinear eects,negativedierentialresistancewhichgivesbothhysteresisanddipsin Q e )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [([and thusspikesineciency]andphonon-assisted-tunnelingneededfortheelectron/phonon interactiontogeneratethermoelectricpower,liewelloutsideofthisgenerator-regime". 1.2.2.5Thedriventhermoelectricdevice Thenaturalsolutionwouldbetodrivethethermoelectricdeviceasasemiconducting heterostructureisdriven.Theloadthethermoelectricdevicealone would powerwould alsobedrivenbythedriving-agent.Theload,thethermoelectricdevice,andthevoltage supplywouldallbeinserieswitheachother.Aparallelcircuitwouldaddtheresistance 27

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duetothedrivingagent'sinternalimperfectionsasanextraparametertotheproblem[to avoidshortingcurrentawayfromtheload],andthusisasuperuouscomplexitylacking newphysics.SeeFigure3-4,wherethereisacircuit-insert. Perhapsaconcreteexperimentmightbeinorder.Althoughtheauthorofthis dissertationisunfamiliarwiththedicultiesanddetailsofscanning-tunnelingmicroscopy inwhichoneobservesanonlinearcurrent,hesuggeststhatthescanning-tunneling microscope'sreceiverofthenonlinearcurrentberegardedastheload,andpower-delivery totheloadbyapplicationofatemperature-gradientbecalculatedwhileholdingapplied powerconstanti.e.,constantbeforeandaftertheapplicationofthetemperature-gradient. Althoughthevoltageofthedrivingagentwouldbedroppedacrossthethermoelectric device'sinternalimperfectionssinceitisnowaquantumscattererwithadierential-conductance fromthevoltage-supply'spointofview,andthusdissipatespower,oneisstillfreeto considerthe change inpower-deliverytoallcircuitelementsuponapplicationof T .The casecouldbemadethatthisisanotherclassofthermoelectriceects,whichseemtobe unexplored. 1.2.3ThermodynamicEciencyofaDrivenSystem Eciency isdenedasworkoutputconsideredtobeuseful"dividedbywork consideredtobeinput".Itshouldbenotedthatthereissubjectivitytothedenitionof eciencye.g.,numberofbananasperunitfertilizermaybeafruit-farmer'sdenition. However,thereisobjectivitytotheupperlimitoneciencygivenbythethermodynamics-laws ofconservationandentropy-increasee.g.,massconservationplacesanupperlimitonhow manybananasmaybegrownfromagivenunitquantityoffertilizer. Thus,oneisfreetodeneaneciencythatisafunctionofhowmuch extra power issuppliedtoanalready-drivenloade.g.,byavoltagesupplyonapplicationofa temperature-gradient,andsubsequentlyusethermodynamicstoboundthiseciencyto anupperlimit.Itturnsoutthattherstandsecondlawsofthermodynamicsrequire adriventhermoelectrictonotsimultaneouslybeactingasathermalpumpcarrying 28

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thermalenergyoutofacoldreservoirintoahotreservoirwithitspower-inputtoprevent aperpetualmotionmachineofthesecondkindi.e.,onewhichisothermallyconverts thermalenergydirectlyintousefulwork. 1.2.4TheSemiconductingHeterostructureModel Thegeneralitiesofthethermodynamicsofthepartially-driventhermoelectric arerealizedinawell-knownspherical-cow"modelpopularsincethe1980sofa semiconductingheterostructureadaptedtonitetemperatureanditinerantnon-Einstein freephonons.Anelectron/phononinteractionlocalizedtoacenter-regionisintroduced. Ofprimaryinterestarethenonlinearsteady-statenumber,electricalandthermalcurrents ofelectronsandphononsforbothadriventhermoelectricandathermoelectricgenerator. Previousworksincorporatingsuchalocalizedelectron/phononinteractionhave usedanreservoiressentiallyathirdleadofEinsteinsingle-modedphononsata xedtemperature.Thisworkcannotusethissincethenonlinearthermalcurrent Q is calculatedfromenergy-conservation.Instead,eachmonatomicleadispopulatedwith phononsthatdonotinteractwithelectrons,andalocalizedmodebetweenthetwoleads ofalongbutnitelifetimeinteractswithsimilarly-localizedelectrons.Withinthismodel, theelectronsandphononsformaclosedsystem,energy-conservationisrestored,andthe nonlinearthermalcurrent Q isyielded. UseofanEinsteinreservoirofphononsalsorequiresspecifyingsaidreservoir's temperature.Often,theaverageoftheleftandrightleads'temperaturesisused.Inthis dissertation,athirdleadwithitsowntemperature T t ,andphononicthermalcurrent Q ph t isintroducedasathermometer.Thecenter-sitephonontemperature T ph couldbe takentobethenumerically-determinedsolutionto Q ph t =0thehotandcoldleads' respectivetemperaturesnaturallylendthemselvestobeingupperandlowerlimitsin thebisection-rootndingmethod.Thistemperature T ph isfoundtobe not equaltothe averageofthetemperaturesoftheleftandtherightlead. 29

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CHAPTER2 THEORETICALTOOLS 2.1Thermodynamics AlthoughaspecicHamiltonianisstudiedinthisdissertation,theimplicationsof thermodynamicsholdingregardlessofhowcomplexorsimpleaHamiltonianisused aretoberstestablished.Therstarethelawsofconservationgoverningthesystemor anyclosedsubdivisionofthesystem.TherearethreeusefulinstancesoftheFirstLaw energy-conservation:theentirecircuit,theelectronicandphononiccurrentswith aninteractionlocalizedtothemolecularbridge,andthethermoelectricdeviceitself. Eciencyisthenintroduced,andaneciencyofthedriventhermoelectricissuggested. Thedrivenjunctionactingaseitherathermalpumpwithacoecientofperformanceor anonlinearscattererirreversiblydissipatingpower-inputisnotedtosatisfytheFirstLaw. TheecienciespossibleundertheFirstLawarestated. TheSecondLawisthenintroduced.Traditionally,theSecondLawgivesanupper boundontheeciencyofadevice,andthiscaseisnoexception.However,theSecond Lawalsoforbidsthedriventhermoelectricdevicefromsimultaneouslyactingasathermal pumpandanenginewhosemaximumeciencyistheCarnoteciency.Thus,theSecond Lawrequiresthedriventhermoelectricdevicetobeanonlinearscatterer. Finally,thefollowingsystemofunitsareused, k B = e = ~ =1{1 2.1.1TheTransport-CoecientDescriptionofTransmissionDrivenby ArbitrarilySmallThermoelectricGradients:TheLinear"Response Thetemperature-independentretardedandadvancedGreenfunctionsobtained throughoutSec.2.3.4aresucientfordescriptionofnonlineartransportwithout interactions.Moreover,nonlineartransportsubsumeslineartransport,wherecurrentsare directlyproportionaltothegradientsthatelicitthemandtheconstantsofproportionality themselvesofgreattheoreticalinterestarecalledtransportcoecients".Forthermoelectric 30

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transportinvolvinganelectricalcurrent I = )]TJ/F22 11.9552 Tf 9.298 0 Td [(eN duetoamaterialcurrent N of charge-carriersofcharge )]TJ/F22 11.9552 Tf 9.298 0 Td [(e andathermalcurrent Q simultaneouslyowing,the transportcoecientsofinterestaretheelectricalandthermalconductivities and andtheSeebeckandPeltiercoecients S e and,respectively. Althoughthisdissertationconcernsitselfalmostexclusivelywiththenonlinear currents I and Q ,thetransportcoecientsareadirectdescriptionofwhathappenswhen measurementsoftransportphenomenaaretakene.g.,theSeebeckcoecient S e isthe unit-responseofvoltage-gradientperunit-applicationoftemperature-gradient.Hence,a theoryofthemshouldbeestablished. Thelinear-response/transport-coecientdescriptiontreatsagivenmaterialandthe thermalization-processes 1 asacontinuum.Therefore,thesystemofunits2{1requires electricalconductancetobedimensionless. Onenalgeneralityconcerningtransportcoecientsisthattheyaretensorial quantitiesingeneral.Forinstance,the i;j th elementoftheelectricalconductivity tensor ij istheunit-response J i ofcurrentdensityinthe i th directionduetothe unit-applicationofanelectriceld E j inthe j th direction.Thoughamentionworthy generalization,nowheredoesthisdissertationconcernitselfwiththreedimensional transportmuchlessan i th -directioncurrentelicitedbya j th -directiongradientfor i 6 = j .Hence,thisdissertationshallregard asascalar.However,thisspecialcase ubiquitoustoone-dimensionaltransport cannot beusedwhenamagneticeldisapplied. Onapplicationofamagneticeld,threedimensionalityandtheconsequentre-castingof thetransport-coecientsastensorsevenifdiagonalismandatory. 1 Alsocalledrelaxation-processes,asintherelaxationtime assumedbytheDrude model. 31

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2.1.1.1Theelectricalandthermalconductivities Consideranexperimentalistwhoappliesapotential V = EL duetoauniform electriceld E droppingalongthelength L ofasampleofcross-sectionalarea A and whothenmeasuresthecurrent I = JA = qnvA constitutedbyacharge-density nq ofdrift-velocity v .Theywouldthenreportanelectricalconductivityof = IL= AV forthatsample.Similarly,anexperimentalistwhoappliesatemperature-gradient r T acrossthesamesampleoflength L andcross-sectionalarea A andobservesathermal current Q inresponsetoatemperaturegradient 2 r T wouldthenreportathermal conductivityof = j Q= r T j .Itshouldbenotedthatmeasurementofthermalcurrentcan bedoneonlyindirectly.Forinstance,consider[24]themeasurementofthermalcurrentin atwo-dimensionalsample.Theon-diagonaltransportcoecients and makereference tothesample'slength L andcross-sectionalarea A ,andsothereistacitassertionof three-dimensionalityofasample.Thisstillallowsforthepossibilityof p A L ,where thedesirablefeaturesoftheelectronicspectralfunctioninFig.1-1remainaccessible. Onebeginswithasampleofmaterialandremovesahollowofmaterialtoconstitutetwo separatedsubstratesreservoirsjoinedbyalength 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 m andthickness 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 m of thinlm.Next,anelectrical-insulatorofwidth 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(5 m isdepositeduponthecenterof thelm,leavingequaldistancesbetweentheedgesoftheinsulatorandthesubstrates. Immediatelyabovetheelectrical-insulatoraheaterisdepositedatthecenterofthelm. Thetemperatureriseoftheheaterisdeterminedbymeasuringthechangeinitselectrical resistance.Thesubstratetemperatureisassumedtobeuniform.Theexperimenter thensendsanamountofpower P totheheaterattemperature T where T isalsothe 2 Notetheuseof r T ratherthan T .Forabulksampleoflength L andcross-sectional area A possiblywith p A L havingdimensionality D =1 ; 2 ; 3ofFig.1-1,a displacement-dierential d ~ L overwhichthetemperature-dierential dT occursmustbe referredto.Fordimensionality D =0ofFig.1-1,thereisnolengthscaleandgradients are T and instead. 32

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temperatureofthesubstratesandofheat-capacity C V togiveatemperature-riseof T> 0.Anamountofthermal-current Q = 1 2 P=A traversesthedistances L throughthe cross-sectionalarea A = td .Atsteady-state,where T = T x isalinearfunctionin x thetemperatureofthesubstrateis T ,andthetemperatureoftheheateris T + T ,and T x = )]TJ/F21 7.9701 Tf 6.586 0 Td [( T L x + T + T ,whichsatisestheboundary-conditions T = T + T ,and T L = T givingathermalconductivityof = PL= A T .Hence,measurementof is moreindirectthanmeasurementof 2.1.1.2TheSeebeckandPeltiertransportcoecients Anexperimentalistwhosweeps[25]aseriesoftemperaturegradients r T that produceacorrespondingseriesofvoltage 3 gradients V maytaslope 4 ofmagnitude S e tothisdata.This S e wouldthenbethereportedSeebeckcoecient.Similarly,an experimentalistwhomeasuresathermalcurrent Q solelyinresponseto 5 acurrent I will reportaPeltiercoecientof.Intheabsenceofamagneticeld,and S e forasample oftemperature T aresubjecttotheThomsonrelation= TS e .Whenmeasuringthe diagonal"transportcoecients and ofSec.2.1.1.1,gradients r T and V = EL were 3 Thisisoftencalledtheopencircuit"voltage,sonamedbecauseitisthevoltage thatdevelopsacrossavoltage-measuringdevicewhoseresistanceismuchgreaterthan theresistanceoeredbytheinternalimperfectionsofthethermoelectricmaterial.Such alargeresistancepermitszerocurrent,andthusthereiszerodeliveryofpowertothe voltage-probe,asignatureofthelinearresponse. 4 Here,itisassumedthattheresponding V islinearin T .Infact,an experimentalistattemptingtofurnishatransport-coecientdescriptionofamaterial wouldconsidernonlinearitytobetheresultofpoorelectricalcontact[25]forwhich resonant-tunnelingwouldbethedominantcurrent-producingmechanism. 5 SeparationofthethermalenergiesduetoJouleheatingandduetothePeltiereect isdicult.Eachmechanismisanuisancetomeasurementoftheother'scorresponding transportcoeciente.g.,thePeltiereectisanuisancetoresistivitymeasurements. Reductionofthecorrespondingerrorscanbereduced[25]withalternating-currentor pulsed-direct-currentwhosecurrent-amplitudesarelowenoughtorenderJouleheating insignicant.Properheat-sinkingofthesampleisalsoameansoferror-reduction. 33

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presentseparately.Whenpresentsimultaneouslythetemperaturegradient r T remains asitis,butthepotentialgradientmustbegeneralizedsuchthatavoltmeterwitha reading[5]of )]TJ/F27 11.9552 Tf 11.291 9.631 Td [(H V E obs d ~ ` duetothetotaldropofpotentialovertheinternalresistance d ~ R = =A d ~ ` overtypicallength-element d ~ ` duetothecomposition E obs = E )-223(r = )]TJ/F22 11.9552 Tf 9.298 0 Td [(e Dependingupon E obs ? 0,thethermoelectriceecteithersustainsanelectriceldableto deliverpowerintoitssurroundingsuppersignordissipatesapower-inputsuppliedby itssurroudingslowersignbybothirreversibleJouleheatingandthereversible 6 Peltier eect.Itshouldbenotedthat E obs isanalogoustothedecomposition E = E a + E th that shallbeeectedlater. 2.1.1.3Transportcoecientsimpliedbyagivenbandstructureandspectral density Electronsdocarrythermalenergythoughnotasmuchasphononsdo,andsoone considersthisnow.Firstconsiderasmallsub-regionofthesolidwithnotemperature-gradients. Next,oneconsidersadierentialofthermalenergy, dQ = T dS .Considering S = S U;N forasampleofconstantvolume,andusing @S @U N =1 =T and @S @N U = )]TJ/F22 11.9552 Tf 9.298 0 Td [(=T ,one computes dQ as, dS = T dS U;N = T @S @U dU + @S @N dN = T 1 T dU + )]TJ/F23 7.9701 Tf 6.586 0 Td [( T dN = dU )]TJ/F22 11.9552 Tf 11.956 0 Td [( dN = dQ=T Theaboveistrueatalltimesandspaces.Therefore,dividingthroughby dA dierentialofbounding-areaofcontrol-volumeandatime-dierential dt ,onegets current-densities, j q i = dQ dA i dt = T dS dA i dt = T j S i = dU )]TJ/F22 11.9552 Tf 11.956 0 Td [( dN dA i dt = j i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j n i j q i = Tj S i = j i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j n i 6 Inthenonlinear"caseofnitepowerdeliverybetweentheexperimentalsetupused asathermoelectricgeneratoranditssurroundings,theSecondLawthereforeplacesa lowerlimitupontheproclivityofJouleheating. 34

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Fordimensionalities D =1 ; 2 ; 3,momentum k isagoodquantumlabelforthe charge-carriers.Letthe n th bandofallowedenergies n k ofmomentum k charge-carriers haveanonequilibriumdistributionfunctionof g n k momentum-statesperunitenergyand band-velocity v i n k inthe i th direction 7 .Theenergy E andnumber N currentdensities aregivenbythefollowinginwhichtheband-index n isomitted, j = X n 2 Z 1 D d D k n k 1 v n k g n k j n = X n 2 Z 1 D d D k n k 0 v n k g n k Puttingalltheseelementstogether,onehasthethermalcurrentcarriedbythe fermioniccharge-carriers, j q i = j i )]TJ/F22 11.9552 Tf 11.955 0 Td [(j N i = Z 1 D d D k 2 k 1 v i k g k )]TJ/F22 11.9552 Tf 11.956 0 Td [( Z 1 D d D k 2 k 0 v i k g k = Z 1 D d D k 2 k )]TJ/F22 11.9552 Tf 11.956 0 Td [( v i k g k Havingestablishedexpressionsforthecurrentasfunctionsofthenonequilibrium distributionfunction g n k andband-structure v i n k ;" n k ,theassumptionoflinearresponse tothegradients E obs and r T isused, j N i = L 11 ij E obs j + L 12 ij r r j T j q i = L 21 ij E obs j + L 22 ij r r j T Theabovesetofcoecients L 11 ij ;L 12 ij ;L 21 ij ;L 22 ij isthenrewrittenintermsofthe function L ij .Thenonequilibriumdistributionfunction g isalsosubjecttotherelaxation-time 7 Whileitwasstatedearlierthattensorialtransportcoecientswouldnotbepursued, itishelpfultoseehowtheynaturallyarise. 35

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approximation 8 givingthefollowinginwhich k k )]TJ/F22 11.9552 Tf 11.955 0 Td [( L 11 ij = J n i E obs j = e 2 Z n k n k =0 v i n k v j n k )]TJ/F22 11.9552 Tf 9.298 0 Td [(df d! = L =0 ij L 21 ij = L 12 ij = L 1 ij = )]TJ/F22 11.9552 Tf 9.299 0 Td [(e L 22 ij = L 1 ij = e 2 T Consideringthesetransportcoecientstonolongerbetensors,onecannowwrite theelectricalconductivity ,theiso-electronici.e., E obs = 0 thermalconductivity e thenon-materiali.e., j N =0thermalconductivity e ,theSeebeckcoecient S e ,the Peltiercoecient,andthethermoelectricgureofmerit ZT impliedforcharge-carriers ofcharge )]TJ/F22 11.9552 Tf 9.298 0 Td [(e andtemperature T = qj N = E obs r T = 0 = qj N =E = e 2 L 11 =T e = j j q j = jr T j E obs = 0 = )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 T 0 L 21 + r 1 T L 22 = jr T j = j L 22 j = e 2 T e = j j q j j N =0 = jr T j = )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 T r e L 21 + r 1 T L 22 = jr T j = L 11 L 22 )]TJ/F22 11.9552 Tf 11.955 0 Td [(L 12 L 21 = L 11 T 2 S e 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(e jr j = jr T j = 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(e T L 12 L 11 r T )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = jr T j = 1 eT L 12 =L 11 = )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 e )]TJ/F22 11.9552 Tf 5.479 -9.683 Td [(j q =j N r T = 0 = 1 e )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 T r e L 21 + 0 L 22 = L 12 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(L 11 r e T )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = 1 e L 12 =L 11 ZT = S e 2 T e = 1 eT L 12 =L 11 2 e 2 L 11 =T L 11 L 22 )]TJ/F22 11.9552 Tf 11.955 0 Td [(L 12 L 21 = L 11 T 2 T = 1 L 11 L 22 L 12 L 21 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Itremainstospecializetoaspecicbandstructureandspectraldensitytoobtain specicresultsfortheabovetransportcoecients. Itisre-emphasizedthattheSeebeckcoecient S e istheunit open-circuit voltage perunittemperaturegradient,andsodescribesthecaseofzeropower-delivery.Power 8 Thetransportcoecientdescriptionofresponsetheorybecomesexactwhen theappliedgradientsareinntesimal.Similarly,therelaxation-timeapproximation approximatesthedistributionofparticlesasanequilibrium-distributionwith spatially-dependentpotentialandtemperature.Thesystemdepartsfromequilibrium correspondinglyinntesimally. 36

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deliveryiszerounderconditionsofmeasurementof S e owingnotonlytotheopen-circuit conditionbutalsotheby-denitionexactnessofthelinearresponseonlyfor inntesimal gradients.Thecaseofnitepower-deliveryandwhyitisimportantisdetailedinthe followingsections,andsotheactualpower-deliveringornonlinear"currents N and Q willbethefocusofthisdissertation. 2.1.1.4TheCutler-MottformulafortheSeebeckcoecient ThefollowingshallbedoneinSIunits,whicharemoreappropriateforbulkand continuoussystems.Consideramaterialoftemperature T andchemicalpotential .Let itstransportpropertiesbeabletobedescribedbyatransmission-probability T = T = T assignedtoallparticlesofenergy partakinginsaidtransport.Letthecharge-carrier energy beconnedtoaverywideband 2 [ 1 ;! 2 ]ofenergies.TheSeebeckcoecient S e iscalculatedundertheseconditionsas, L m = 2 Z 1 T ! )]TJ/F22 11.9552 Tf 11.955 0 Td [( m )]TJ/F22 11.9552 Tf 9.298 0 Td [(df d! d! 2 Z 1 T m )]TJ/F22 11.9552 Tf 9.298 0 Td [(df d! d!S e L 1 eTL 0 2 6 k B e k B T d ln T d! {2 Theresult2{2iscalledtheCutler-Mottthermopower,anditshallnowbederived. Theaimistoevaluatethelinear-responseintegrals L m .Thisbeginsbythefollowing TaylorexpansionwhichissimilartotheideabehindtheSommerfeldexpansion,owingto k B T inmostmaterialsaboutthematerial'schemicalpotential assumingthatthe transmissionprobabilityisasmooth 9 functionof L m = 2 Z 1 1 X n =0 )]TJ/F22 11.9552 Tf 11.955 0 Td [( n n @ n T m @! n )]TJ/F22 11.9552 Tf 9.299 0 Td [(df d! d! = 1 X n =0 1 n @ n T m @! n 2 Z 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( n )]TJ/F22 11.9552 Tf 9.299 0 Td [(df d! d! 1 X n =0 1 n @ n T m @! n I n 9 Apointofcautionregardingthispremiseofthesmoothnessof transmission-probabilityisasfollows.Oneshouldnotethataconsequenceofthe Cutler-Mottformula2{2isthatanarbitrarily-largeSeebeckcoecient S e followsfrom anarbitrarily-sharply-varyingtransmission-probability T intheneighborhoodofthe chemicalpotential 37

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Takingthewide-bandlimit,theintegral I n havingunitsofenergytothepower n vanishesforodd n andisevaluatedas, I n = 2 Z 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [( n )]TJ/F22 11.9552 Tf 9.298 0 Td [(df d! d! = )]TJ/F23 7.9701 Tf 6.587 0 Td [(n 2 + Z 1 + )]TJ/F22 11.9552 Tf 11.955 0 Td [( k B T n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 x 4 sech 2 x 2 dx k B T n 1 4 + 1 Z x n sech 2 x 2 dx Truncatingtheseriesatfourth-orderandbeyond,thelinear-responseintegral L m simpliesto, L m 'T m I 0 + 1 1! @ T m @! I 1 + 1 2! @ 2 T m @! 2 I 2 = T m 1+ @ T m @! 0+ 1 2 @ 2 T m @! 2 k B T 2 2 3 Hence,thelinear-responseintegrals L 0 and L 1 thatenterintotheSeebeckcoecient S e areevaluatedas, L 0 = 2 Z 1 T ! )]TJ/F22 11.9552 Tf 11.955 0 Td [( 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(df d! d! = 2 Z 1 T )]TJ/F22 11.9552 Tf 9.298 0 Td [(df d! d! 2 Z 1 T ! )]TJ/F22 11.9552 Tf 11.955 0 Td [( d! = T L 1 = 2 Z 1 T ! )]TJ/F22 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(df d! d! = 1 X n =0 1 n @ n T 1 @! n I n 0+0+ 1 2! 2 3 T )]TJ/F22 11.9552 Tf 11.955 0 Td [( = 00 k B T 2 Onethenworksoutthelogarithmicderivativeofthetransmissionprobability appearingin2{2asfollows, S e = 1 eT L 1 L 0 = 1 eT 2 6 @ 2 @! 2 T )]TJ/F22 11.9552 Tf 11.955 0 Td [( k B T 2 T = 2 6 k B e k B T 1 T @ 2 @! 2 T ! )]TJ/F22 11.9552 Tf 11.955 0 Td [( = = 2 6 k B e k B T 1 T 0 B @ T 00 ! )]TJ/F22 11.9552 Tf 11.955 0 Td [( + T 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(0+ T 0 1 C A = = 2 6 k B e k B T d ln T d! Forthesystemthatisofreduceddimensionalityandnitelyoutofequilibriumdue tonitegradients T and ,theequilibriumchemicalpotential shouldbeplaced withinaneighborhood)-327(oftheresonance U 1 .Indeed,forathermoelectricwithnonlinear current2 R d! 2 f
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areservoiratabsolutezero f
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loadedwithadevicewhoseimperfections 12 scatterfar fewer charge-carrierscomparedto theinternalimperfectionsofthegeneratori.e.,aperfect ammeter ,thenthemaximum current runsthroughtheloop-circuitformed.Allvoltageisdroppedbytheinternal imperfectionsofthegeneratorandsozeropowerisdeliveredtotheload.Conversely,ifthe generatorisloadedwithadevicewhoseimperfectionsscatterfar greater charge-carriers comparedtotheinternalimperfectionsofthegeneratori.e.,aperfect voltmeter ,then themaximum voltage isdroppedbytheload.Zerocurrentrunsthroughtheloopcircuit formed 13 andsozeropowerisdeliveredtotheloadinthiscaseaswell. Momentarilyreturningtoatransport-coecientdescription,oneshouldnotethatto obtainmaximumexternalpowerfromasourcee.g.,athermoelectricvoltagegenerator withsomeinternalresistance,theresistanceoftheloadmustequaltheresistanceofthe sourcethetheoremofelectricalengineeringgivingtheconditionofmaximumpower transfer.Notingthatloadpowerisloadvoltagetimescurrent,thechemical potentialgradient isloadvoltage,circuitcurrentdecreasesas increases,one seesthattheabovedescriptionsatisestheseconditions,andishowthegenerationofa thermalvoltageinresponsetoatemperaturegradient T manifestsitselfinanelectrical circuit. 2.1.2.1Thethermoelectric,theload,andthedriving-agent Athermoelectricdevicedropsavoltage V T givenby, V T = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(e =+ {3 12 Thetermimperfections"isusedinplaceofthetermresistance",becauseresistance isatransportcoecientthatcarriesconnotationsofthelinearresponse. 13 Thisistheopencircuit"conditionfeaturingzeropowerdeliverythattheSeebeck coecientisdenedby. 40

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Figure2-1.Drivingagentinserieswiththermoelectricblackboxallowingaccesstoregions beyondthegenerator-regimewhereatemperature-gradientincreasesthe current. This V T whichissink-sensewhetherthethermoelectricdeviceisageneratororis beingdrivenisinserieswithasource-sensedriving-agent E a andasink-senseload V T to formasingleloop.Thevoltagelawforthisloopis, 0= X i E i + X i V i = E a + V T + V L {4 Throughouttheloop,thereiscontinuouscurrent I = )]TJ/F22 11.9552 Tf 9.298 0 Td [(eN ,whichthusimpliesenergy conservationthesumofthepowers[currenttimesvoltage]dissipatedandsuppliedis zero, 0= X i P i = P A + P T + P L {5 Because directlydetermines I fromthespecicHamiltonianofthesystem,itis mostconvenienttoputinbyhandadesired andlet2{4determine E a .Thus,simple modelsfor V L and V T arenowtobeintroduced. 41

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2.1.2.2Thethermoelectricdevice V T asanimperfectbattery Considerthecircuitofsection2.1.2.1with E a =0.Because T istakentobethe sourceof forathermoelectricfunctioningpurelyasagenerator,thefollowingbounds existon intheabsenceofanappliedbias, 0 > > )]TJ/F22 11.9552 Tf 9.299 0 Td [(e E th )]TJ/F15 11.9552 Tf 21.918 0 Td [( T ; E a =0;{6 Let E th of2{6beasource-sensevoltageinserieswithinternalimperfections droppingsink-sensevoltage V th .Thus, V T = E th + V th .Thisthermally-sustainedbias E th mustbefoundbyndingthechemicalpotentialdierence atwhichthecurrent N iszerobynumericalbisection[26].Thisisat = E th 7 0for T ? 0,atwhich N approaches0fromthe side, T ? 0 lim V T 'E th N V T =0 {7 Thethermalbiasmodeledasasource E th inthecircuitofFigure3-4isoneoftwo placeswherethespecicsofanyHamiltonianforinstance,2{43ofthisdissertation entersintothiscircuitmodel,theotherbeingthespecicsofanycurrent I forinstance 2{36ofthisdissertation. 2.1.2.3ModeloftheOhmicandnon-Ohmicloads Twosimplemodelsoftheloadarenowtobegiven.By2{4,theseseparatemodels implyseparateformulaefor E a :onewithexplicitdependenceupon E th andtheotherupon N Thevoltagemodelisasfollows.Forloadswhichdropavoltagethatisnotsimply proportionaltotheloop-current )]TJ/F22 11.9552 Tf 9.298 0 Td [(N ,theloadvoltagedrop V L isassertedtobeafraction ofthesum E th + E a ofsource-sensevoltages, V L = )]TJ/F22 11.9552 Tf 9.298 0 Td [( E = )]TJ/F22 11.9552 Tf 9.298 0 Td [( E a + E th ;0 < 1;{8 42

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Thismodel2{8requirestheappliedbias E a tobecalculatedfromKircho'svoltage law2{4as, E a = E th )]TJ/F15 11.9552 Tf 11.955 0 Td [( = )]TJ/F22 11.9552 Tf 11.955 0 Td [( ; = 2 [ E th ; 0];{9 As approachesarbitrarilycloseto1,theloadbecomesofarbitrarilylargeresistance, andthustheappliedbias E a inthemodel2{9becomesarbitrarilylargeforaxed current I .Thedisadvantageofthismodelisthatthespeciccurrent-voltagepropertiesof theloadcomparedtothethermoelectric'sinternal-imperfectionsarerequiredtoobtain anexactnumericalvalueof .Thatis, isaphenomenologicalparameter,ratherthan aphysicalonelikeresistanceiswhosescaleisphysicallysetbytheunit-voltageper unit-nonlinear-current.By2{9, E a isdominatedwithinthegenerator-regime2{6by E th for 1andby inthedrivenregimefor 0. Although =0correspondstoashortcircuitand 1anopencircuit,thereis notananalyticconditionfor P L tobemaximized.Thatis,thereisnota whichsolves @ P L =0,whichcanbeseenas, @ P L = I@ )]TJ/F22 11.9552 Tf 9.299 0 Td [( E = )-222(E th = )]TJ/F22 11.9552 Tf 11.955 0 Td [( 2 {10 TheOhmiccurrentmodelshallnowbedescribed,andhysteresisoccursinthismodel. Thecounterpartoftheabovemodelisfor V L tobeafunctionofthetotalcurrent I ratherthanthetotalsource-voltage E .ThesimplestmodelfeaturingthisisOhm'sLaw, V L = )]TJ/F22 11.9552 Tf 9.299 0 Td [(IR L = )-222()]TJ/F22 11.9552 Tf 23.911 0 Td [(eNR L .Notethatby2{4the E a itselfisafunctionof I E a = )]TJ/F22 11.9552 Tf 9.298 0 Td [(NR L )]TJ/F15 11.9552 Tf 11.955 0 Td [( ; = 2 [ E th ; 0];{11 Oneseeshysteresisinthecurrent I asafunctionof E a for R L V T =N ,meaning NR L dominates E a in2{11.Thehysteresisundersuchaconditionistraceabletothe decreaseof N as increasesthisiscallednegativedierentialresistance,andisseen laterinFig.3-4.Byhysteresis",itismeantthat I asafunctionof E a ,aparametricplot 43

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byEq.2{11,canhave2valuesof I toagiven E a ,indicatingsystemmemorydespite lateruseoftheadiabaticapproximation. Itisimportanttonotethatthechemicalpotentialdierence takesonadierent physicalmeaningforthedrivensystem2{9and2{11comparedtotheundrivenone 2{6.Intheundrivencase,thebounds2{6of characterizetheloadrelativetothe thermoelectric.Themaximumcurrentoccursat =0asiftheloadwasanammeter, andthemaximumvoltageoccursat = E th asiftheloadwasavoltmeter.In contrast,thedriventhermoelectric2{9and2{11 xes theloadasbeingdescribedby anindependentparameterratherthan :asdroppingafraction of E ,orwithan Ohmicresistance R L .Thisisconsistentwiththefactthatintheinterval2{6there may beadrivenagent E a ,whileintherangesgivenin2{9and2{11therenecessarily is a drivingagent. 2.1.3TheFirstLaw-TheCircuit,theCurrents,andtheThermoelectric BlackBox ThethreerepresentationsoftheFirstLawareasfollows.Therstistheabovementioned 2{5,acorollarayof2{4,whenthecircuitisthesystem.Thesecondtakesthe electricalandphononicenergycurrentswiththespecicnanoscopicmodel2{36and 2{41,specializedtolatertobethesystem.Thethird,essentially dU = TdS )]TJ/F22 11.9552 Tf 12.254 0 Td [(PdV + dN at dV =0constantvolume,takesthethermoelectricdeviceitselftobethesystem havinganinputofenergy P T andows Q h ;Q c ofthermalcurrent,giving2{13and 2{15. 2.1.3.1System-subdivisionsgovernedbytheFirstLaw TheFirstLawpositsconservationacrosstheboundaryofwhateverisconsidered tobethesystem,whichmaybeasubdivisionofalargersystem.Consideringthe circuitintroducedinSection2.1.2.1asasystem,thelawofenergyconservationis2{5. Consideringthethermoelectricblackboxasasystemwithaowofenergydueto hot N electronic-energycurrent E e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(andphononicenergycurrent J ,energyconservationis 44

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requiredtoobtain Q andwiththeSecondLawtoboundeciency, E = E e )]TJ/F15 11.9552 Tf 9.741 -4.936 Td [(+ J = hot N + Q ; Q = Q ph + Q e )]TJ/F15 11.9552 Tf 7.085 -4.936 Td [(;{12 Itisimpossibletoseparatelycalculate Q ph and Q e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(inthepresenceofanelectron/phonon interaction,althoughapproximationstothesamecanbemadetotheextentthatthe interactionissmalle.g.,withinaperturbationtheoryoftheinteractionasdoneinthis dissertation. 2.1.3.2Theoutowofpowerfromthetheromelectricgenerator Withinthewell-studiedgenerator-regime2{6,thethermoelectricdevicenomatter howintricateaHamiltonianmaydescribeitistheheatenginestudiedinintroductory thermodynamics.Onethereforeconsidersahotreservoirhavingthermalenergy )]TJ/F22 11.9552 Tf 9.299 0 Td [(Q h leavingandbecomingeitherthermalenergy+ Q c owingintoacoldreservoiroran outow+ P T ofenergycurrentacrossthethermoelectric'sbounding-surfacewhich,by 2{5with P A =0,isdirectlydeliveredtotheload. 2.1.3.3Theloweredpower-inputtoaheatpumpduetoatemperature gradient:anonlinearthermoelectriceect Ofinteresttothestudyofspecicallynonlinearthermoelectricityisusing P A to delivernotonlythepower )]TJ/F22 11.9552 Tf 9.298 0 Td [(P L butalso )]TJ/F22 11.9552 Tf 9.299 0 Td [(P T .BytheFirstLaw,thepowerinput )]TJ/F22 11.9552 Tf 9.298 0 Td [(P T is allowedtoeitherpumpsheatfromcoldtohotorbedissipated.Thepower-input P A allowsthechemicalpotential togooutsidetheboundsin2{6. TheFirstLawisnowappliedtothesetwocasesofthermalpumpingandenergy dissipation.Athermoelectricblackboxattemperature T C 0 ;T C hasenergy+ P T 0 ;P T owinginacrossitsboundary,thermalenergy Q h 0 ;Q h owing in out fromahotreservoir attemperature T h 0 ;T h ,and Q c 0 ;Q c out in toacoldreservoir, P T Q h = Q c ; P T 0 Q h 0 = Q c 0 ;{13 45

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Thereisacommontemperature T = T c 0 = T C 0 = T h 0 beforetheapplicationofa temperature-gradient T ,andtemperatures T h = T c + T and T h T C T c afterthe applicationofatemperaturegradient.Thus,thesetemperaturesareparametrizedas, T c h 0 = T 1 2 dT ; T c h = T 1 2 T ; dT; T> 0;{14 Thus,intheapplicationofonlyaninnitesimaltemperature-gradient,onereservoir maybecalledhot",andonecold",preservingadirectionoftimeevenifaninnite amountoftimeisrequiredforanyniteamountoftransport. Ofinterestisthecasewhereapplicationof T increasesthepower-inputtotheload. Onapplicationof T ,let P T 0 beloweredto P T duetothethermalcurrentsnowbeing )]TJ/F22 11.9552 Tf 9.299 0 Td [(Q h and+ Q c .At T = dT ,thesameare Q h 0 ;Q c 0 ;P T 0 .Thatis,applying T givesa loweredenergyinputtothethermoelectricdevice.Theformof2{13isnoted,andone cansubtracttheseandintroducethenotations Q = Q )]TJ/F22 11.9552 Tf 11.93 0 Td [(Q 0 and P = P )]TJ/F22 11.9552 Tf 11.93 0 Td [(P 0 toobtain, ~ P T ~ Q h = ~ Q c {15 Thetildein2{15refersto2{5whichis P T = )]TJ/F22 11.9552 Tf 9.298 0 Td [(P L fortheundriventhermoelectric. Tobeobtainedis2{15 without thetilde,whichdenotestheimportantspecialcasewhere power-inputis P A 0 ,butdoesnotchangeafterapplicationof T sothat P A 0 = P A Implementingthisattheproof-of-conceptlevelisnowtobediscussed. 2.1.3.4Constancyof P A 0 onapplicationof T By2{5,someof P T arrivesattheloadandconstitutes )]TJ/F22 11.9552 Tf 9.299 0 Td [(P L ,andsomeconstitutes )]TJ/F22 11.9552 Tf 9.298 0 Td [(P A .Theaimistocausetheeectof T appliedtothethermoelectrictobedirectly deliveredtotheload. Consideracalorimetristwhocanmeasureanychangeinenergyofanyofthe abovementionedsystems.Thecalorimetriststartswithtworeservoirsatequaltemperatures, andappliesavoltageacrossthereservoirstodrivecurrent.Perhapsthisvoltageistobe largeenoughtoaccessthephonon-assisted-tunnelingregioninFig.3-4.Then,the 46

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calorimetristbringsonereservoirtoanelevatedtemperature.Knowingexactlyhowmuch powertheirvoltagesupplyisputtingout,thecalorimetristadjuststhevoltage-supply suchthatequalpowerisputout beforeandafter thewarmingofthereservoir.The calorimetristthusiscontrolling P T ,andtheyadjusttheirapparatussoastohave P T = P T 0 .Thisgives P L =0,wheretheabsenceofthetildeof2{15indicates thiscondition.By2{5and2{15, P T Q h = Q c ; P T + P L =0;{16 Thus,theFirstLawforthedrivensystemispreciselyanalogoustotheFirstLawfor theundrivensystemgivenacalorimetristwhocaneect P A =0,causingtheoutput" )]TJ/F15 11.9552 Tf 9.298 0 Td [( P T > 0ofthethermoelectrictobedirectlyrealizedas+ P L > 0. Thetranscendentalequation ~ P A =0requiresNewton'smethodtodetermine V T ,andthusrequiresevaluationofthemodel'scurrentsovereachiterationoften computationally-expensive,especiallyifaself-consistencycalculationisrequiredforeach valueofthecurrent.Yet,thedriveneciency d obtainedlaterin2{26iscontingent uponobtainingthecondition ~ P A =0.Thealternativeisforthecalorimetristto continuouslyadjustadialontheloadi.e.,adjust R L or to R L 0 or 0 respectivelysuch that ~ P A =0 P A .Thisgives, R L 0 = R L N 2 + V T N =N 0 2 {17 0 = V T )-222(E th )]TJ/F15 11.9552 Tf 11.955 0 Td [( N 0 =N V T )]TJ/F22 11.9552 Tf 11.955 0 Td [( V T )-222(E th + N 0 =N )]TJ/F22 11.9552 Tf 11.955 0 Td [( {18 Using2{17and2{18isfarlesscomputationallyexpensive.Furthermore,changing R L ; to R L 0 ; 0 leaves unchanged.However,thecondition ~ P A =0mayyieldan analyticexpressionrequiring R L 0 tobenegativeorfor 0 = 2 [0 ; 1eachareunphysical. Sincethecurrent N andthermalbias E th mayfollowfromanarbitrarily-intricatemodel, suchanalyticconditionsforthephysicalityof and R L areoflittleuse. 47

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Ontheotherhand,onemaybeunconcernedaboutchoosingaspecic R L or at T =0.Indeed,intheproblemofattaininganecientandpowerfulthermoelectric device,theparametersofinterestarethoseofthemodelHamiltonian,notthecircuit. Dependingonthecases N 0 7 0for R L ,oneneeds, N 0 ? 0 R L ? )]TJ/F22 11.9552 Tf 9.298 0 Td [(V T N=N 2 {19 For 0 2 [0 ; 1,itiseasiesttoselectadesired 0 withinthisinterval,andthen choosinga givenby, = )]TJ/F22 11.9552 Tf 9.298 0 Td [(NV T )]TJ/F22 11.9552 Tf 11.955 0 Td [( 0 + N 0 V T )-222(E th 0 N 0 V T )-222(E th N )]TJ/F22 11.9552 Tf 11.955 0 Td [( 0 + N 0 0 {20 However,the resultingfromtheaboveexpressionmusthavetheproperty 2 [0 ; 1. ThisisanalogoustotheconstraintthattheOhmic R L mustbepositive-denite. Thismethodisdesignedaroundthefollowingnaturalsequenceofcomputationalsteps calculatecurrents N;Q whichtakesthemostcomputationaltime,so N;Q shouldbe storedonthecomputer'shard-disk,andthencalculatetheimplied P T ;P L ;P A ;Q::: whichwouldretrievethe N;Q storedonhard-disk.Thus,forgivenstored N;Q ,many dierent R L ; andthe P T ;P L ;P A ;Q::: theyimplycouldbetriedandstudiedwithout spendingthelargeamountoftimerequiredtocalculate N;Q Complicatedmodelscanhavelargeparameter-poolswithsmallornegligibleislands" wherethe R L and determinedbythecondition P A =0areactuallyphysical.This callsforusingrootndinge.g.,theabovementionedNewton'smethodtocalculate atwhich P A =0.Dueto N beingdirectlysensitiveto ,avalueof making P A =0isalmostalwayssuretoexistperhapsthiscouldbemathematicallyproven. Implementing P A =0withrespectto callsforthesimplestpossiblemodel, includingtoy"modelsnotnecessarilycorrespondingtoaparticularHamiltonianyet whichhavematchingmathemticalpropertiese.g.,atoymodelwithapeak,spread, andnormalizationlikethatofaLorentzian.Suchtoymodelsexistforwhich N is 48

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determinabletoclosedformatnitetemperatureofcourse,withoutelectron/phonon interaction.Thisistosaythattheband-density N mayactuallybeintegratedto closed-form. Implementing P A =0withrespectto hasthefurtheradvantageofleaving R L and freetovaryforpurposesofoptimization.Sincechanging changes N N varies withrespectto R L .Thus,oneshouldimmediatelyoptimize P L withrespectto R L to determinethe R L atwhich P L isanextreme-value, 0= @ P L @R L P A =0 R L = @ @R L ln 1 N 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(N 0 2 P A =0 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 {21 Evenifaclosedformexistsfor N ,thederivative2{21dependson R L throughthe condition P A =0asindicatedanexampleofthisdependenceisoeredby2{17. Thus,2{21mustbesolvedasaself-consistencyconditionwiththederivativenumerically approximated. Anupperlimitissetupontheload-resistance R L bythemaximumapplicablevoltage setbytheequipmentonehasattheirdisposalandtheminimumresultingcurrentset bythethresholdbelowwhichthedetectedcurrentisconsideredtobenoise,whichinthis theoreticalcalculationistheband-slice4 W=N band forthedivisionoftheband4 W into N band elementalslices.Inthesystemofunits2{1,thismaximumloadresistanceis, max R L =max V L j I j = max V L min j I j max V T min j j max j j 4 W=N band N band {22 Sincethecritical R L of2{21istobedeterminedbyarootndingmethod,having bothanupperlimit2{22andanobviouslowerlimitof R L =0upon R L opensupthe optionsofclosed-intervalmethodswhichhavegauranteedalbeit,slow 14 convergence. 14 Forinstance,thebisectionmethodconvergesatwhatcomputerscientistscallalinear rateandisoflogarithmiccomplexity. 49

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2.1.3.5EciencyofthethermalassistwithintheFirstLaw Theaimnowistoassociateadrivensystem"eciency d withthethermalassist describedinSection2.1.3.4thatwouldbeanalogoustothefamiliarundrivensystem eciency u = P L =Q h inwhich P T T 2 ofthethermoelectricgeneratorwithin2{6. Notingthat P T T 2 ,itseemssensibletodene d = P L =Q h d = P L Q h = )]TJ/F15 11.9552 Tf 9.299 0 Td [( P T Q h = P T 0 Q h 1 )]TJ/F22 11.9552 Tf 13.757 8.088 Td [(Q c Q h {23 Itisseenthatthedriven-systemeciency2{23isadimensionlessmeasure P T 0 =Q h ofhowfaroutofequilibriumthethermoelectricisdrivenby alonebythedriving agentplusthequantity )]TJ/F22 11.9552 Tf 12.573 0 Td [(Q c =Q h ,theuppersignofwhichisexactlytheundriven engine'seciency.Thereisnorelationbetween P T 0 and Q h withoutconsultingaspecic modeli.e.,goingbeyondthermodynamics,sothisisthemostsimplicationthatis possible.Incontrast,itiswell-knownthatecienciesoftheengineandthermalpump havemaximumvaluesthatarefunctionsexclusivelyoftemperature. OnusingtheSecondLaw,itshallbeseenthatthetwosignsin2{23correspondto twoseparatewaysinwhichitisimpossibletobuildaperpetualmotionmachineofthe secondkindonewhichisothermallyconvertsthermalenergyintoanequalquantityof usefulwork. 2.1.4TheSecondLawandEciency-Bounds Aboundupontheeciency2{23governingtheincreaseinpower-deliverytothe load P L dueto T impliedbytheSecondLawisthegoalofthissection.Thesecond lawsofthermodynamicsforaninntesimalandanitetemperaturegradientare, Q c 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(Q h 0 0 S c )]TJ/F22 11.9552 Tf 11.956 0 Td [(S h = Q c =T c )]TJ/F22 11.9552 Tf 11.955 0 Td [(Q h =T h 0{24 Itshallnowbeshownthat2{24impliesthefollowing:intheabsenceofanite temperaturegradient T theremustbenon-zeroentropyproductionifthejunctionis beingdrivenby P T 0 ,thejunctionisnotallowedtosimultaneouslyoperateasaheat 50

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Figure2-2.Inputof P T 0 tothermoelectricdevicethateitherpumpsthermalenergyfrom coldtohot,orscattersuselesslyandoperatesintandemwiththespontaneous owofhottocold. pumptopreventtheconstructionofaperpetualmotionmachineofthesecondkind.That is,thedrivingpower P T 0 mustbedissipated,andtheuppersignin2{13andonwardmust beselected,andtheeciency2{23isboundedbetween0andtheCarnoteciency inthenon-driven P T 0 0limit. 2.1.4.1Theimpossibilityofreversibledriving Theuppersignin2{24correspondstothepower-inputs P T ;P T 0 beinguselessly scattered,whilethelowersigncorrespondsto P T ;P T 0 drivingathermalpumpwarming T h with Q h comprisedpartiallyof Q c pumpedfrom T c .Combiningtherstlaw2{13 with2{24,itisseenthat P T 0 0,indicatingthatitisimpossibletoreversiblydrive thesystemwith P T 0 intheabsenceofatemperaturegradient T .Acorollaryofthisis thatthepower-input P T 0 mustpumpthermalenergyinthedirectionof T ,ratherthan against T duetothisopeningthepossibilitythat P T 0 ispoweringathermalpump. Onapplicationofatemperature-gradient T whilekeepingconstantthepower suppliedbyadrivingagentinFig.2-2,thedevicemayonlyactasanengineifitisnot simultaneouslypumpingthermalenergy. 2.1.4.2Impossibilityofsimultaneousoperationasaheatpump Applicationofthetemperaturegradient T givesingeneralanincreaseordecrease inentropyproduction,notstrictlyanincrease.Thus,thegoalnowistocalculate P L rev = )]TJ/F15 11.9552 Tf 9.299 0 Td [( P T rev = P T 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(P T rev ;thatis, P L atthecondition S c )]TJ/F22 11.9552 Tf 11.955 0 Td [(S h =0, P L rev = P T 0 Q h rev )]TJ/F22 11.9552 Tf 11.955 0 Td [(Q c rev = P T 0 )]TJ/F23 7.9701 Tf 13.649 4.707 Td [(T c T h Q h {25 51

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Theupperlimituponeciencyintheabsenceofdrivingis P T 0 0in2{25,which wouldjustgivetheCarnoteciencyintakingthepositivesign.However,whatthesetwo signsphysicallycorrespondtomustbeconsidered,regardlessofwhetheronechoiceofsign givesadesiredlimit. Dissipationofthepowerinputresultsingreater entropyproduction.Forthe positive-signedCarnoteciencyin2{25,thepower-inputs P T ;P T 0 aredissipated. Theapplied T workswithwhatthe P T ;P T 0 aredoing.Letrespectiveamounts S + 0 ;S + of entropybecreatedinthepresenceof dT; T Powerinputattemptedtobeusedasaheat-pumpresultsinlesser entropy production.Forthenegative-signedCarnoteciencyin2{25,thepower-inputs P T ;P T 0 operateathermalpump.Theapplied T worksagainstwhatthe P T ;P T 0 aredoing. Ifrespectiveamounts S )]TJ/F21 7.9701 Tf -0.696 -7.879 Td [(0 ;S )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(ofentropyarecreatedinthepresenceof dT; T ,then S )]TJ/F21 7.9701 Tf -0.696 -7.879 Td [(0 ;S )]TJ/F22 11.9552 Tf 10.406 -4.338 Td [( 0at whichthereisthermalcurrent Q h fromthehotreservoirhasthefollowingboundsuponits 52

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eciency d denedas P T =Q h forwhich P T = )]TJ/F15 11.9552 Tf 9.298 0 Td [( P L underthecondition2{16of constantpowerinput P T = P T 0 0 d P L =Q h P T 0 =Q h + )]TJ/F22 11.9552 Tf 11.955 0 Td [(T c =T h {26 Themotivationforobtaining2{26wastoboundtheeciencyofthedelivery ofpowertoaloadbyadevicedriventoachemicalpotentialdierence V T = atwhichtheremaybeinterestingquantumfeatures.Inthecaseofthisdissertation, thoseinterestingfeaturesareresonantcurrentresponsibleforMahanandSofo's perfect-resonancethermoelectricanditsinnitegureofmerit,negativedierential resistance,andphonon-assistedtunneling.Ingeneral,2{26couldbeusefulindetermining thelimitsupontheeciencyofanynano-devicegeneratingusefulenergyfroma temperaturegradient. 2.1.4.4Eciencyasafunctionof Theeciency2{26issimilartotheconventionaleciencies u )]TJ/F22 11.9552 Tf 22.75 0 Td [(V T N=Q ofthe undriventhermoelectric,orthatofaheatpump t ,whichare, u = P L Q 1 )]TJ/F22 11.9552 Tf 13.756 8.088 Td [(T c T h ; t = Q h + Q c P T T h + T c T h )]TJ/F22 11.9552 Tf 11.955 0 Td [(T c ;{27 Intheundrivencase,thesource-sensevoltage V T acrossthethermoelectric blackboxwasthenegativeofthevoltageacrosstheloadseenbyremovingthe drivingagentfromtheschematicinFig.3-4,andtheamountofvoltageacross thethermoelectric/loaddependedontherelativevoltage-dropsacrossthisandthe thermoelectric'sinternal-imperfectionsliketheinternalresistanceofanimperfect battery.However,fromthethermoelectricdevice'spointofview,thevoltage V T evenifwithinthegeneratorregime,maypartiallycomefromadrivingagent.Thatis,a givenvoltagedropacrossthethermoelectricblackboxmaybeasumofcontributionsfrom E th andfrom E ap 53

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Facedwiththisambiguity,lettheeciencyasafunctionof V T bedenedas, V T = u )]TJ/F23 7.9701 Tf 6.587 0 Td [(V T E th + V T + d V T + E th )]TJ/F23 7.9701 Tf 6.587 0 Td [(V T {28 Thishastheadvantageofallowingcomparisonofthedriven-junction'seciency d takingadvantageofallnonlinerfeaturesforinstance,inFig.3-4withtheconventional undriveneciency u 2.2ModeloftheThermoelectricNano-Device 2.2.1FermionicElectronsandBosonicPhonons 2.2.1.1Thesimplicityofrepresentingparticlesasoperators Asystemoffermions/bosonsrequiresitsstatetobeantisymmetricorsymmetric underparticle-interchange: j 1 2 i = j 2 1 i .Thisantisymmetry-requirementis identicallysatisedbyacumbersomeSlaterdeterminantorpermanentdet i q j = 1 q 1 2 q 2 2 q 1 1 q 2 inthe2 2caserequiredfortwoparticles.Ifoneeschewsstatesofthe SchrodingerpictureinfavorofoperatorsoftheHeisenbergpicturetheantisymmetric/symmetric requirementisidenticallysatisedbyannon-anticommuting/non-commutingalgebraof operatorsrepresentingtransitionsbetweenoccupanciescolloquiallycalledladder" operators.Thisalgebrabecomesespeciallysimpleifthetransitionsduesolelytothe perturbationsincetheunperturbedHamiltonianisdiagonalarebetweenoccupancies ofstatesthatare stationary ,andthisholdsalwaysiftheperturbationisintroduced adiabatically.Thus,particlestatisticsarebosonicandfermionic,andthefermionic statisticsimplyPauliexclusion.Theseparticlestatisticsarerootedinthesymmetrization SorantisymmetrizationArequirementoftheidentical-particle-wavefunction,and thisS/A-requirementistidilybookkeptbythedevicesofsecond-quantization,which positstheanticommutation-relationsmentionedearlierforbosonic-phononsand fermionic-electrons. 54

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2.2.1.2Thealgebraoftheparticleoperators Thenano-systembeingconsideredinthisworkismadeoffermionicelectronsand bosonicphonons.Theoccupancyofthe j th siteinthe th leadattime t isassignedthe dually-conjugate[27]pairofoperators c j t $ c y j t .Theoccupancyofthe j th sitein thecenter-regionattime t isassigned c j t $ c y j t .Similarly,theoccupancyofthe th leadbyaphononofmomentum q attime t isassigned b y q t $ b q t .However,the center-sitephononshaveneitherwell-resolvedpositione.g.,site-index j ormomentum e.g.,momentum-index q ,onlyamode-index 15 m = A;B independentof t ,sotheir operatorsare b y m t $ b m t .Thealgebraoftheseoperatorsforalltime-independent 16 occupanciesis, f c j ;c 0 j 0 g =0=0 y = f c y j ;c y 0 j 0 g ;[ b q ;b q 0 0 ]=0=0 y =[ b y q ;b y q 0 0 ]; [ b m ;b y m 0 ]= mm 0 ; f c j ;c y j 0 g = jj 0 ; [ b q ;b y q 0 0 ]= qq 0 0 ; f c j ;c y 0 j 0 g = jj 0 0 ;{29 Electronsandbosonsitinerantthroughoutthethermoelectricdeviceinthisdocument aredescribedbythisalgebra2{29.Electricalenergyshallbecarriedbyelectronsand thermalenergyshallbecarriedbybothelectronsandphonons. 15 Themotivationforusingthelabel A;B isexplainedinFigures2-6,2-7,2-4,and2-5, wherediagramsdescribingtheinteractionbetweentheabovementionedelectronsand phononsisshown. 16 Aswouldbeiftheelectronsandphononswerealldescribedbyatime-independent Hamiltonian,oranytime-dependentterminsaidHamiltonianisintroduced adiabatically 55

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2.2.1.3Highercommutatorsofparticleoperators Considerparticleswithaquantumlabel k withanalgebradenedbyKronecker-delta functionsforcommutatorsoranticommutators, [ ^ A y k ; ^ A y k 0 ] =0=0 y =[ ^ A k ; ^ A k 0 ] ;[ ^ A k ; ^ A y k 0 ] = k 0 k ;[ A;B ] AB BA ;{30 Commutatorsbetweentransitionoperatorslike ^ A y k ^ A k 0 frequentlyoccurwhen calculatingtime-evolution.Thus,oneneedsthefollowingidentitywhichcanbeveried bydirectsubstitution, [ AB;CD ] )]TJ/F15 11.9552 Tf 10.405 1.794 Td [(= A [ B;C ] D +[ A;C ] DB C [ A;D ] B AC [ B;D ] {31 Followingfromthiscommutator-identity2{31arethefollowingparticle-operator identities, [ ^ b y ^ b ; ^ b 0 ] )]TJ/F15 11.9552 Tf 10.406 1.793 Td [(= )]TJ/F22 11.9552 Tf 9.299 0 Td [( 0 ^ b $ [ ^ b y ^ b ; ^ b y 0 ] )]TJ/F15 11.9552 Tf 10.405 1.793 Td [(=+ 0 ^ b y ;[^ c y ^ c ; ^ c 0 ] )]TJ/F15 11.9552 Tf 10.405 1.793 Td [(= )]TJ/F22 11.9552 Tf 9.298 0 Td [( 0 ^ c $ [^ c y ^ c ; ^ c y 0 ] + =+ 0 ^ c y ; Directcomputationiseectedfrom2{31tofurtherobtainafour-particlecommutation identity, [ ^ A y k 1 ^ A k 2 ; ^ A y k 3 ^ A k 4 ] )]TJ/F15 11.9552 Tf 10.405 1.793 Td [(= ^ A y k 1 [ ^ A k 2 ; ^ A y k 3 ] ^ A k 4 +[ ^ A y k 1 ; ^ A y k 3 ] ^ A k 4 ^ A k 2 ^ A y k 3 [ ^ A y k 1 ; ^ A k 4 ] ^ A k 2 ^ A y k 1 ^ A y k 3 [ ^ A k 2 ; ^ A k 4 ] = ^ A y k 1 k 2 k 3 ^ A k 4 +0 ^ A k 4 ^ A k 2 ^ A y k 3 k 1 k 4 ^ A k 2 ^ A y k 1 ^ A y k 3 0 Thus,oneobtainsthefollowingidentityforthealgebraofparticle-operators,able tobeusedforcommutationbetweenparticle-densityoperatorsaswellascommutation betweentunneling-operators, [ ^ A y k 1 ^ A k 2 ; ^ A y k 3 ^ A k 4 ] )]TJ/F15 11.9552 Tf 10.405 1.793 Td [(= ^ A y k 1 ^ A k 4 k 2 k 3 )]TJ/F15 11.9552 Tf 15.042 3.022 Td [(^ A y k 3 ^ A k 2 k 1 k 4 ; ^ A k $ ^ A y k = bosons fermions ;{32 Theabovecommutatorhasmanyimportantcorollariesforthestudyofanano-bridge describedbythetight-bindingmodel.Forinstance,oneoftenencountersdensity 56

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commutedwithdensity,whichisdemonstratedbywriting[^ n k ; ^ n k 0 ] )]TJ/F15 11.9552 Tf 12.022 1.793 Td [(=[^ c y k ^ c k ; ^ c y k 0 ^ c k 0 ] )]TJ/F15 11.9552 Tf 12.022 1.793 Td [(= ^ c y k ^ c k 0 kk 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y k 0 ^ c k kk 0 =^ c y k ^ c k 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y k 0 ^ c k kk 0 ,andthusconcluding, X k [^ n k ; ^ n k 0 ] )]TJ/F15 11.9552 Tf 10.406 3.487 Td [(=0{33 Higherpowersofdensityvanishonlyifcertainindicesaresummedover,whichis demonstratedbywriting, [^ c y k 00 ^ c k 00 ; ^ c y k ^ c k ^ c y k 0 ^ c k 0 ] )]TJ/F15 11.9552 Tf 10.405 1.793 Td [(=^ c y k 00 ^ c k )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y k ^ c k 00 ^ c y k 0 ^ c k 0 k 00 k +^ c y k ^ c k ^ c y k 00 ^ c k 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y k 0 ^ c k 00 k 00 k 0 {34 Thedensity-hoppingcommutator: Suppose ^ A y k 1 ^ A k 2 =^ c y j ^ c j =^ n j andthiswas commutedwiththehoppingoperator^ n hop j 0 0 =^ c y j 0 0 ^ c j 0 +1 ; 0 +^ c y j 0 +1 ; 0 ^ c j 0 0 .Twoinstancesof 2{32yields, [^ n j ; ^ n hop j 0 0 ]=[^ c y j ^ c j ; ^ c y j 0 0 ^ c j 0 +1 ; 0 +^ c y j 0 +1 ; 0 ^ c j 0 0 ] = ^ c y j ^ c j 0 +1 ; 0 jj 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 0 ^ c j j;j 0 +1 +^ c y j ^ c j 0 0 j;j 0 +1 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 +1 ; 0 ^ c j jj 0 0 [^ n j ; ^ n hop j 0 0 ]= ^ c y j ^ c j 0 +1 ; 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 +1 ; 0 ^ c j jj 0 +^ c y j ^ c j 0 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 0 ^ c j j;j 0 +1 0 Thehopping-hoppingcommutator: Thenexttaskistocalculatethecommutator between^ n hop j =^ c y j ^ c j +1 ; +^ c y j +1 ; ^ c j andanindependenthopping^ n hop j 0 0 =^ c y j 0 0 ^ c j 0 +1 ; 0 + ^ c y j 0 +1 ; 0 ^ c j 0 0 ,as, [^ n hop j ; ^ n hop j 0 0 ]= [^ c y j ^ c j +1 ; ; ^ c y j 0 0 ^ c j 0 +1 ; 0 ]+[^ c y j +1 ; ^ c j ; ^ c y j 0 +1 ; 0 ^ c j 0 0 ] +[^ c y j ^ c j +1 ; ; ^ c y j 0 +1 ; 0 ^ c j 0 0 ]+[^ c y j +1 ; ^ c j ; ^ c y j 0 0 ^ c j 0 +1 ; 0 ] = 0 B @ ^ c y j ^ c j 0 +1 j +1 ;j 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 ^ c j +1 j;j 0 +1 +^ c y j +1 ^ c j 0 j;j 0 +1 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 +1 ^ c j j +1 ;j 0 +^ c y j ^ c j 0 j +1 ;j 0 +1 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 +1 ^ c j +1 jj 0 +^ c y j +1 ^ c j 0 +1 j;j 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 ^ c j j +1 ;j 0 +1 1 C A [^ n hop j ; ^ n hop j 0 0 ]= 0 B @ ^ c y j +1 ^ c j 0 +1 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 +1 ^ c j +1 jj 0 +^ c y j ^ c j 0 +1 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 +1 ^ c j j +1 ;j 0 + ^ c y j +1 ^ c j 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 ^ c j )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 j;j 0 +1 +^ c y j ^ c j 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 ^ c j j +1 ;j 0 +1 1 C A 0 57

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Thedensityanti-hopping"commutator: Considertheanti-Hermitianoperator ^ m hop j =^ c y j ^ c j +1 ; )]TJ/F15 11.9552 Tf 13.005 0 Td [(^ c y j +1 ; ^ c j ;whileitneverappearsinanyHamiltonian,itappearsin commutatorsresultingfromcanonicaltransformation, [^ n hop j ; ^ m hop j 0 0 ]= [^ c y j ^ c j +1 ; ; ^ c y j 0 0 ^ c j 0 +1 ; 0 ]+[^ c y j +1 ; ^ c j ; ^ c y j 0 0 ^ c j 0 +1 ; 0 ] )]TJ/F15 11.9552 Tf 9.299 0 Td [([^ c y j +1 ; ^ c j ; ^ c y j 0 +1 ; 0 ^ c j 0 0 ] )]TJ/F15 11.9552 Tf 11.955 0 Td [([^ c y j ^ c j +1 ; ; ^ c y j 0 +1 ; 0 ^ c j 0 0 ] = 0 B @ ^ c y j ^ c j 0 +1 j +1 ;j 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 ^ c j +1 j;j 0 +1 +^ c y j 0 +1 ^ c j j +1 ;j 0 )]TJ/F22 11.9552 Tf 11.956 0 Td [( 0 ^ c y j +1 ^ c j 0 j;j 0 +1 + ^ c y j 0 +1 ^ c j +1 jj 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j ^ c j 0 j +1 ;j 0 +1 +^ c y j +1 ^ c j 0 +1 j;j 0 )]TJ/F15 11.9552 Tf 12.198 0 Td [(^ c y j 0 ^ c j j +1 ;j 0 +1 1 C A 0 [^ n hop j ; ^ m hop j 0 0 ]= 0 B @ ^ c y j ^ c j 0 +1 +^ c y j 0 +1 ^ c j j +1 ;j 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(^ c y j 0 ^ c j +1 +^ c y j +1 ^ c j 0 j;j 0 +1 + ^ c y j 0 +1 ^ c j +1 +^ c y j +1 ^ c j 0 +1 jj 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(^ c y j 0 ^ c j +^ c y j ^ c j 0 j +1 ;j 0 +1 1 C A 0 2.2.2Two-Lead/Center-RegionSystemofElectrons TheportionoftheHamiltonianthatdescribestheelectronsandwhichisquadratic isnowintroduced.AsmuchinformationaspossibleisextractedfromthisHamiltonian withouttakingitsquantum-statistical-expectationthisisdoneafterintroducingGreen functions,whichisinsection2.3. 2.2.2.1TheelectronicquadraticHamiltonian Fortheelectrons,thereisacentralregionmadeof N C siteswhichcostanenergy U i i =1 ; 2 ;:::;N C tooccupy.The i th and j th center-regionsiteshavethenegative energetically-favorableoverlap-energy )]TJ/F22 11.9552 Tf 9.299 0 Td [(W ij )]TJ/F22 11.9552 Tf 11.956 0 Td [( i;j inwhich i;j istheKroneckerdelta. H C = N C X j =1 U j c y j c j ; H CC = N C X j = i +1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(W ij c y i c j + h:c: ; Thisoccupancy-energyandoverlap-energy 17 togetherconstitutethetight-binding" quadraticmodeloftheelectrons.Eachleadisaone-dimensionalsemi-innitechainof 17 Theoverlapenergyissometimescalledthehopping"energy,butagiven tight-bindingHamiltonianisquadraticandthusstationary. 58

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Figure2-3.SchematicofthequadraticHamiltonian2{35and2{39for N C =2. siteswhichcost U = tooccupyandwhichareinterconnectedbyoverlap-matrix-element )]TJ/F22 11.9552 Tf 9.298 0 Td [(W ,so, H = 1 X j =1 U c y j c j ; H = 1 X j =1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(W c y j c j +1 + h:c: ; Withoutlossofgenerality,onecantakesite-1tobecoupledby )]TJ/F22 11.9552 Tf 9.299 0 Td [(W tothe th lead inwhich = L;R isaleftLorrightRlead-indexatsite1 withinthelead, H C = )]TJ/F22 11.9552 Tf 9.298 0 Td [(W 1 c y 1 c 1 + h:c; ThisconstitutesthequadraticportionoftheHamiltonianoftheelectrons,as, H 0 e )]TJ/F15 11.9552 Tf 10.405 2.955 Td [(= H C + H CC + X H + H + H C {35 ThisHamiltonian2{35isschematcallyillustratedforthecase N C =2inFigure2-3, alongsidetheparametersbelongingtothephononicHamiltonian2{39. InFig2-3,themode-index m = A;B forthephononsisnotaposition-label,meaning that,e.g.,site-1inthecenterregionisnotthelocationofmodeA.Rather,eachofthetwo modesrattlesaboutanarbitrarily-complicatedcenter-region.Thereisonlyonemodefor N C =1. Aleftandrightlead,whicharebothmonatomicandsemi-innite,areatdisparate temperaturesandpotentials.Theyhaveelectronicenergiesfromoccupancyandfrom intersite-tunneling.Thereisalsotunnelingtoacentral-region-inhomogeneity.This 59

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left/centerandcenter/righttunnelingaloneisresponsibleforthenonlinearcurrents ofprimaryinteresttothiswork.Thisisthestandardandpopulartight-binding" Hamiltonianforelectrons.Anelectronofmomentum k inalatticewithintersitespacing a ,occupancyenergy U ,andintersitetunneling )]TJ/F22 11.9552 Tf 9.299 0 Td [(W hasanenergy k of k = U )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 W cos k a 2.2.2.2Conservationintheelectronnumber-operator Itisusefultoimmediatelycalculatetime-derivativesofoccupation.Let N = 1 i =1 c y i c i bethetotalnumberdensityinthe th leadandsimilarly N c N C i =1 c y i c i .With these,itisausefulchecktoverifythe operator formofthestatementofthecontinuityof thenumbercurrent.Todoso,onerequires, i c 1 =[ c 1 ;H ]= U c 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(Wc 2 )]TJ/F27 11.9552 Tf 11.955 11.358 Td [(X i W i c i ; i c j 2 = U c j )]TJ/F22 11.9552 Tf 11.955 0 Td [(W c j +1 + c j )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 ; 60

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Thus,bydirectcomputationandextensiontohighertermsinthesum,onecalculates thetime-evolution i N ofelectrondensityinthe th leadas, i N = 1 X i =1 i c y i c i + c y i i c i = c y 1 U c 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(Wc 2 )]TJ/F27 11.9552 Tf 11.955 11.358 Td [(X i W i c i + 1 X i =2 c y i )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(U c i )]TJ/F22 11.9552 Tf 11.955 0 Td [(W c i +1 + c i )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(h:c: = 0 B @ U c y 1 c 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(Wc y 1 c 2 )]TJ/F27 11.9552 Tf 11.956 8.967 Td [(P i W i c y 1 c i )]TJ/F15 11.9552 Tf 9.298 0 Td [( U c y 1 c 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(Wc y 2 c 1 )]TJ/F27 11.9552 Tf 11.955 8.967 Td [(P i W i c y i c 1 1 C A + 1 X i =2 0 B @ U c y i c i )]TJ/F22 11.9552 Tf 11.956 0 Td [(W c y i c i +1 + c y i c i )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 )]TJ/F15 11.9552 Tf 9.298 0 Td [( U c y i c i )]TJ/F22 11.9552 Tf 11.955 0 Td [(W c y i +1 c i + c y i )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 c i 1 C A = Wc y 2 c 1 + X i W i c y i c 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(Wc y 1 c 2 )]TJ/F27 11.9552 Tf 11.956 11.357 Td [(X i W i c y 1 c i + 1 X i =2 W c y i +1 c i + c y i )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 c i )]TJ/F22 11.9552 Tf 11.956 0 Td [(c y i c i +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(c y i c i )]TJ/F18 5.9776 Tf 5.757 0 Td [(1 = 0 B @ Wc y 2 c 1 + P i W i c y i c 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(Wc y 1 c 2 )]TJ/F27 11.9552 Tf 11.955 8.966 Td [(P i W i c y 1 c i 1 C A + W 0 B @ c y 3 c 2 + c y 1 c 2 + c y 4 c 3 + c y 2 c 3 + ::: )]TJ/F22 11.9552 Tf 9.298 0 Td [(c y 2 c 3 )]TJ/F22 11.9552 Tf 11.955 0 Td [(c y 2 c 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(c y 3 c 4 )]TJ/F22 11.9552 Tf 11.955 0 Td [(c y 3 c 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(::: 1 C A i N = N C X i =1 0 B @ W i c y i c 1 )]TJ/F22 11.9552 Tf -44.298 -23.908 Td [(W i c y 1 c i 1 C A Onehasthuscalculatedthetime-derivativeofthenumberdensityinthe th lead, i N ,whichisdirectlyproportionaltotheoverlap-energy W j withthe j th site, i N =[ N ;H 0 e )]TJ/F15 11.9552 Tf 7.085 2.956 Td [(]= X N C j =1 [ N ;H j ]= W j c y j c 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(W j c y 1 c j = W j c y j t y a c j t a )]TJ/F22 11.9552 Tf 11.955 0 Td [(h:c: {36 61

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Thisresult2{36bearssimilaritytotheoperatorforspatialtranslation 18 bythe latticeconstantbetweenthe th leadsurfacesiteandthe j th centersite.Indeed,thisisthe formexpectedbyvirtueofthecurrent-operatorfromelementary 19 quantummechanics. Thetime-derivativeofthecenter-sitecentersiteoccupancygivesthesum i N ,giving an operator statementofmass-conservationforthesystem H 0 e )]TJ/F15 11.9552 Tf 7.085 2.955 Td [(, i N c = X N C j =1 [ c y j c j ;H 0 e )]TJ/F15 11.9552 Tf 7.084 2.955 Td [(]= X j; [ c y j c j ;H C ]= )]TJ/F27 11.9552 Tf 11.291 11.358 Td [(X N {37 Itshouldalsobenotedthatthisconservationlaw2{36holdsforacenterregionof arbitrarily-manysites. 2.2.2.3Theenergycurrentoperator Becauseanelectronicenergycurrent E e )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [(,anelectronic-thermalcurrent Q e )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [(,anda numbercurrent N arerelatedbytherstlawofthermodynamics 20 as E e )]TJ/F15 11.9552 Tf 10.406 -4.339 Td [(= Q e )]TJ/F15 11.9552 Tf 8.073 -4.339 Td [(+ hot N onerequirestheenergycurrenttocalculate Q e )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [(.Now,incontrasttothenumber-current, energyconservationonlyappearswhenonetakestheexpectationvalueof iE .Theenergy intheleadisgivenby H + H .Onerstnotesthatthetime-derivativeoftheoccupancy energy H = U N isalreadyknownduetohavingworkedout i N soitremainsonlyto 18 Recallthatthecontinuous-position-spacerepresentationofthetranslationoperatoris exp x p i ~ ,andthecurrentofthestateofmass m particlesis ~ 2 mi r )]TJ/F22 11.9552 Tf 11.955 0 Td [(h:c: 19 Let[28] P V t = R V x t x t d x betheprobabilityofaparticleinstatesatisfying i@ t x t = ^ H x t x t beingmeasuredwithinthevolume V attime t .Letthisvolume V be boundedbytheclosedsurface S .Then,thecurrent J x t totaledover S isinterpretable asaprobabilitycurrent J x t because P V t = R V x t d x = R V x t x t + x t x t d x = i ~ 2 m H S r )]TJ/F15 11.9552 Tf 11.955 0 Td [( r d ~ S H S J x t d ~ S .Byconstruction, J x t satises_ x t = r J x t ,in which x t isclearlytheprobabilityperunitvolume-dierential d V ofndingtheparticle attime t 20 Thisassumestheabsenceofacouplingoftheelectronstoanyothersystem,e.g.a phononensemble.Inthepresenceof,e.g.,electron/phononcoupling,thephononicenergy current J isrequired.Then, E e )]TJ/F15 11.9552 Tf 11.352 -4.339 Td [(+ J = Q + hot N ,inwhich Q isthetotalheatcurrent. Inthiscase, Q canbeonlyapproximatelybrokenupinto Q e )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(and Q ph becausethereisno longeranindividualrstlawfortheelectronsandthephononsasindependentsystems. 62

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calculatethekineticenergy H .Proceedingdirectly, i H )]TJ/F22 11.9552 Tf 9.299 0 Td [(W = 1 X i =1 i c y i c i +1 + c y i i c i +1 + i c y i +1 c i + c y i +1 i c i = 0 B @ )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(U c y 2 c 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(Wc y 2 c 2 )]TJ/F27 11.9552 Tf 11.955 8.966 Td [(P i W i c y 2 c i )]TJ/F27 11.9552 Tf -190.654 -10.637 Td [( U c y 1 c 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(Wc y 2 c 2 )]TJ/F27 11.9552 Tf 11.955 8.967 Td [(P i W i c y i c 2 1 C A + 1 X i =2 0 B @ i c y i c i +1 + c y i +1 i c i 1 C A + 1 X i =1 0 B @ c y i i c i +1 + i c y i +1 c i 1 C A = U c y 2 c 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(c y 1 c 2 + N C X i =1 W i c y i c 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(W i c y 2 c i + 1 X i =1 0 B @ c y i + c y i +2 i c i +1 + i c y i +1 c i +2 + c i 1 C A i H )]TJ/F22 11.9552 Tf 9.299 0 Td [(W = U 0 B @ c y 2 c 1 )]TJ/F22 11.9552 Tf -30.253 -23.908 Td [(c y 1 c 2 1 C A + N C X i =1 0 B @ W i c y i c 2 )]TJ/F22 11.9552 Tf -44.298 -23.908 Td [(W i c y 2 c i 1 C A + 1 X i =1 0 B @ c y i +2 + c y i U c i +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(W c i +2 + c i )]TJ/F15 11.9552 Tf -195.74 -23.908 Td [( U c y i +1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(W c y i +2 + c y i c i +2 + c i 1 C A 63

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Writingoutexplicitlysometermsoftheinnitesum,itisseenthatterm-by-term cancellationsoccur,anditsumstozero, [ sum ]= 1 X i =1 0 B @ c y i +2 + c y i U c i +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(W c i +2 + c i )]TJ/F15 11.9552 Tf -195.74 -23.908 Td [( U c y i +1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(W c y i +2 + c y i c i +2 + c i 1 C A = U 0 B @ c y 2 c 1 )]TJ/F22 11.9552 Tf -30.252 -23.908 Td [(c y 1 c 2 1 C A + 0 B @ c y 3 + c y 1 U c 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(W c 3 + c 1 )]TJ/F15 11.9552 Tf -172.957 -23.908 Td [( U c y 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(W c y 3 + c y 1 c 3 + c 1 1 C A + 0 B @ c y 4 + c y 2 U c 3 )]TJ/F22 11.9552 Tf 11.955 0 Td [(W c 4 + c 2 )]TJ/F15 11.9552 Tf -172.957 -23.908 Td [( U c y 3 )]TJ/F22 11.9552 Tf 11.955 0 Td [(W c y 4 + c y 2 c 4 + c 2 1 C A + ::: = U 0 B @ c y 3 c 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(c y 2 c 3 + c y 4 c 3 + c y 2 c 3 )]TJ/F22 11.9552 Tf 9.298 0 Td [(c y 3 c 4 )]TJ/F22 11.9552 Tf 11.955 0 Td [(c y 3 c 2 + ::: 1 C A + W 0 B @ c y 3 + c y 1 c 3 + c 1 + c y 4 + c y 2 c 4 + c 2 + ::: )]TJ/F15 11.9552 Tf 9.298 0 Td [( c y 3 + c y 1 c 3 + c 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( c y 4 + c y 2 c 4 + c 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(::: 1 C A =0 Thus,onehasthefollowing,inwhich,inthenalstep,asumoverallelementsofan antisymmetricrank-2tensor W i W j c y i c j )]TJ/F22 11.9552 Tf 11.026 0 Td [(W j W i c y j c i occursandthuscontributesnothing evenifatime-reversal-symmetry-breakingmagneticeldisapplied, iE = i H + i H = U N C X i =1 0 B @ W i c y i c 1 )]TJ/F22 11.9552 Tf -44.298 -23.907 Td [(W i c y 1 c i 1 C A + )]TJ/F22 11.9552 Tf 9.298 0 Td [(W N C X i =1 0 B @ W i c y i c 2 )]TJ/F22 11.9552 Tf -44.297 -23.907 Td [(W i c y 2 c i 1 C A = N C X i =1 0 B @ U 0 B @ W i c y i c 1 )]TJ/F22 11.9552 Tf -44.297 -23.908 Td [(W i c y 1 c i 1 C A )]TJ/F27 11.9552 Tf 11.956 27.617 Td [(0 B @ W i c y i )]TJ/F22 11.9552 Tf 9.298 0 Td [(i c 1 )]TJ/F27 11.9552 Tf 11.955 8.967 Td [(P j W j c j + U c 1 )]TJ/F22 11.9552 Tf -174.024 -23.908 Td [(W i i c y 1 )]TJ/F27 11.9552 Tf 11.955 8.967 Td [(P j W j c y j + U c y 1 c i 1 C A 1 C A = N C X i =1 i W i c y i c 1 + W i c y 1 c i + X j W i W j c y i c j )]TJ/F22 11.9552 Tf 11.955 0 Td [(W j W i c y j c i iE = i N C X i =1 W i c y i c 1 + W i c y 1 c i +0 Hence,thetime-derivativeof H + H ,theenergycurrentoperator E ,inthe domainof t;t 0 whichbecomesthe t )]TJ/F22 11.9552 Tf 13.02 0 Td [(t 0 domainduetothetunneling W i $ W i 64

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ofelectrons c y $ c betweenthecentralregionandthe th reservoiristhefollowing Hermitianoperator, E = E t )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 = H + H = X N C i =1 W i c y i t 0 c 1 t + W i c y 1 t 0 c i t {38 Itisnotedthattime-derivativesremainin2{38. Anafterwordconcerninglaterworkisinorder.Thetermsin2{38become i! onre-expressioninterms < ofGreenfunctionsandsubsequentFouriertransformation. Thisexpressionintermsofthe < Greenfunctionsofsection2.3isdoneinsection2.4.2. Furthermore,thetime-derivative i H center shallbecalculatedinthelatersection2.2.4when aperturbation H 0 A + H 0 B isadded. Thereisnotanoperator-statementofenergyconservation,astherewasfornumber conservationinSec.2.2.2.2.Rather,energy-conservationappearsaftertakingthe expectation.Thismightindicatetherelevanceofuctuationsandnoisetopicswhich arenotstudiedinthiswork,butarepursuedin,e.g.,[29]. 2.2.3Two-Lead/Center-RegionSystemofPhonons TheportionoftheHamiltonianthatdescribesthephononsandwhichisquadratic isnowintroduced.AsdoneinSection2.2.2,asmuchinformationaspossibleisextracted fromthisHamiltonianwithouttakingitsquantum-statistical-expectation. 2.2.3.1TheAnderson-FanoquadraticHamiltonian Phononsareintroducedusingwhatisamongthesimplermodelsstillfeaturing transport:theFano-AndersonHamiltonian.Lettherebephonons b y q $ b q ofmomentum q andenergydispersion q inlead= L;R ,andlettherebelocalized m th -mode Einsteinphonons b y m $ b m ofenergy m H 0 leads = X q; q b y q b q ; H 0 center = X m = A;B m b y m b m 65

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Thereiscoupling q m betweenthemomentum q phononsandthelocalized m th phononmode, H 0 coupling = X m;q; m; )]TJ/F23 7.9701 Tf 6.586 0 Td [(q b y )]TJ/F23 7.9701 Tf 6.586 0 Td [(q b m + m; + q b y m b q ThisconstitutesthefollowingHamiltonianproposedsimulataneouslybyU.Fano[30] andP.W.Anderson[31]in1961, H 0 ph = H 0 leads + H 0 center + H 0 coupling {39 Thedispersion q ofamomentum q phononisanyresultingfromadiagnolizable Hamiltonianforphononsinthe th lead.Inthisdissertation,thedispersionofphononsin amonatomicchainisused.Itisalsopossiblethoughnotdoneinthisdissertationtouse adiatomicchainHamiltonian,givingatwo-brancheddispersionwithagap. 2.2.3.2TransportintheFanoHamiltonian Theenergydensity H ph ofthephotonsinthe th leadinthephononHamiltonian isnowcalculatedattheoperatorlevel.Here,takingthequantumstatisticalexpectation ofthe th lead'senergydensityisavoidedasitwasinSections2.2.2.2and2.2.2.3. Theresultingenergycurrentappearsvirtuallyidenticaltotheenergydensity2{38 oftheelectronicHamiltonian.Thetimeevolutiongeneratedbytheabove-mentioned FanoHamiltonianofamomentum q phononinthe th leadandthe m th phononmode localizedtothecentersiteeachare, i b m t =[ b m t ;H ph 0 ]= b m m + X q q m b q ; i b q t = q b q + X m m; )]TJ/F23 7.9701 Tf 6.586 0 Td [(q b m ;{40 This2{40isthenusedtocalculatethetime-evolutionofthein-leadphononsas, i H ph = X q 0 B @ q b y q b q + P m m b y q b m )]TJ/F22 11.9552 Tf 9.299 0 Td [(h:c: 1 C A = X qm q m 0 B @ b y q b m )]TJ/F22 11.9552 Tf -27.529 -23.908 Td [(b y m b q 1 C A {41 In2{41,itshouldbenotedthat,unlikethe electronic energy k = U )]TJ/F15 11.9552 Tf -416.184 -23.908 Td [(2 W cos k a 2 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 W; 2 W ],thephononicenergyrangesas q = 0 p 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos q a 2 66

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[0 ; p 2 0 ].Thus,integralsofphononicband-densitieslike2{41areovertheappropriate range. 2.2.4TheElectron/PhononInteraction TheHamiltoniandevelopedsofarhasbeenquadratici.e.,itadmitsclosed-form eigen-energiesandGreenfunctionsinthelatersection2.3.Here,aHolsteinelectron/phonon interactionisintroduced,whichgivesself-energiesduetoelectronscollidingwithphonons andphononsexcitingcharge-neutralelectron/holepairs.Incorporatingbothofthese processeswillbeessentialforcalculatingthethermalcurrent. 2.2.4.1Thecontributionoftheelectron/phononcouplingtotheotherwisequadraticHamiltonian Anelectron/phononcouplingshallbeincorporatedperturbativelytotheotherwise-quadratic Hamiltonianintroducedsofar.Thefollowingbearsemphasizing:theenergyi phonons, althoughlocalizedtoacentral-region,donotcarryaquantum-labelofposition,butrather amode-index 21 i = A;B asstatedinFig.2-3. Specically,acouplingofenergy M Aii existingbetweenamodeA phonon's mechanicalamplitude b y A + b A interactingwithadensity c y i c i ofelectronsatthe i th siteofthecenterregionshallhavetheHamiltonian H 0 Aii ,givingtheterm H 0 A = N C i =1 H 0 Aii Similarly,acouplingofenergy M Bi 6 = j existingbetweenamodeB phonon'smechanical amplitude b y B + b B interactingwithanelectronicoverlapamplitude c y i c j + c y j c i between the i th and j th ofthecenterregionshallhavetheHamiltonian H 0 Bi 6 = j ,givingthe term H 0 B = N C i;j =1 H 0 Bij )]TJ/F22 11.9552 Tf 13.037 0 Td [( ij .Thisconstitutesthefollowinggeneralizationofthe Holstein-Hamiltonianwiththeindicatedcoupling M ijk betweenthephononsandthe 21 Inacenter-regionwithmorethan1site,themode-indx i wastakento belongtophononsallowableatelectron/phonon-interactionverticeswith quantum-selection-rule-likestipulationsontheGreenfunctionindicesatthesamei.e., i = A for j = k i = B for j 6 = k ,andsoon. 67

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Figure2-4.Atype-A"phonononlyhasanonzerointeraction-vertex M Aii 0 for i = i 0 .For acentralsitewith N C sites,therearethus N C possibleverticesandthus N 2 C possibleFock-diagrams. electrons, H 0 = X ijk M ijk b y i + b i c y j c k = X j M Ajj b y A + b A c y j c j + X jk )]TJ/F22 11.9552 Tf 11.955 0 Td [( jk M Bjk b y B + b B c y j c k {42 Theelectron/phononinteractionaddsnotermstothetime-dependence2{38. 2.2.4.2TheA-typevs.B-typephononsandcomputationalcost TheseparationofphononsintoA-type"vs.B-type"phononsmatchingvs. mis-matchingelectron-indicesattheelectron/phononverticesintheusualFeynman diagrammaticrepresentationoftheinteractiongreatlycutsdownthecomputation-timeof thecalculation.Theperturbation2{42whosediagrammaticrepresentationisshownin Figs.2-6,2-7,2-4,and2-5has2 N 2 C +2 N 2 C N C )]TJ/F15 11.9552 Tf 12.241 0 Td [(1 2 HartreeplusFockdiagramsinstead ofthe2 N 4 C thatwouldbewithouttheselectionrules.Thisgivesafractional-decreaseof + N C )]TJ/F15 11.9552 Tf 11.504 0 Td [(1 2 =N 2 C inthenumberofdiagrams,whichequals1for N C =1,hasaminimum of1 = 2at N C =2thet-stub[32]andmonotonicallytendstounityinthelimit N C '1 2.2.4.3Scatteringofelectronsoofphonons Onehas ,theself-energyofanelectroncollidingwithaphonon.Anelectron-density iscoupledtothemechanicalamplitudeofaharmoniclocalizedphonon.Duetothis coupling,aphononmaybeemittedandre-absorbedbyanelectron,constituting aFeynmandiagramwithaFockshapetoitfromthetheoryofelectron/electron interactions. 68

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Figure2-5.Atype-B"phonononlyhasanonzerointeraction-vertex M Bii 0 for i 6 = i 0 .For acentralsitewith N C sites,therearethus N C N C )]TJ/F15 11.9552 Tf 11.955 0 Td [(1possibleverticesand thus N 2 C N C )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 possibleFock-diagrams. Figure2-6.Atype-A"phonononlyhasanonzerointeraction-vertex M Aii 0 for i = i 0 .For acentralsitewith N C sites,therearethus N C possibleverticesandthus N 2 C possible -independentHartree-diagrams. Theelectron/phononcouplingmayalsomediateaninteractionbetweenanitinerant electronandalocalizedelectron,constitutingaFeynmandiagramwithaHartree-like diagram. Figure2-7.Atype-B"phonononlyhasanonzerointeraction-vertex M Bii 0 for i 6 = i 0 .For acentralsitewith N C sites,therearethus N C N C )]TJ/F15 11.9552 Tf 11.955 0 Td [(1possibleverticesand thus N 2 C N C )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 possible -independentHartree-diagrams. 69

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Thediagrams2-4,2-5,2-6,and2-7givecollisionalself-energieswhichdressthe electron. 2.2.4.4Excitationsofcharge-neutralelectron/holepairsbyphonons Alternatively,aphononmaybeabsorbedbyFermiseaandexciteacharge-neutral electron/holepairwhichrecombineandre-emitthephonon,givingaself-energywhich dressesthephonon. Thepolarizationbubbleself-energiesaredenedinSection2.3.2.4asexpectations ofamplitudesofoverlapbetweenseparateelectronicdensities.Becausetheseparate electronicdensitiescarryseparatespinindices,thesepolarizationbubblesareofspin1, andthusarebosons.Correspondingly,the 7 Langrethorderings[33]respectivelycarrya signs. InstudiesusingareservoirofEinsteinphononandtwoelectronreservoirsatabsolute zerotemperature,thepolarizationbubblemerelyrenormalizes[23]thephononenergy to1partin10 3 andsowereneglected.However,thisworkcalculatestheinteracting thermalcurrentduetothephononsandthatduetotheelectronswithanelectron/phonon interaction.Thisthermalcurrentiscalculatedfromanenergycurrentwhichmustbe conserved.Withaninteraction,electron-energyandphonon-energyarenotseparately conserved;onlytheirsumis.Thus,thepolarizationbubbleisrequiredsothatthephonons aredressedwithinteractionscausinginow/outowofenergyfromtheelectronsjustas theelectronsaredressedwithinteractionscausingthesameinow/outowofenergyfrom thephonons. InFig.2-8,iftheoutgoingphononisoftype A andtheincomingphononisoftype B ,thispolarizationisawayforatypeA /typeB phononinteraction.Phonon/phonon interactionsarenotconsideredinthisdissertation. 70

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Figure2-8.Aphononoftype A or B interactswiththeFermiseaandexcitesan electron/holepair,withself-energy.Iftheoutgoing/ingoingphononisof type A or B thevertexhastheconditions i = i 0 vs. i 6 = i 0 ,and j = j 0 vs. j 6 = j 0 2.2.4.5TheHamiltonianofthemolecularbridge Hence,theHamiltonians H 0 e )]TJ/F15 11.9552 Tf 10.986 2.955 Td [(and H 0 ph ofitinerantelectronsandphononswith localizedinteraction H 0 isgivenas, H = H 0 e )]TJ/F15 11.9552 Tf 9.741 2.955 Td [(+ H 0 ph + H 0 {43 2.3TheNon-EquilibriumGreenFunctionNEGFMethod Thisworkshallnowdetailthemethodtobeusedtostudythemodel-Hamiltonian H = H 0 ph + H 0 e )]TJ/F15 11.9552 Tf 10.549 2.956 Td [(+ H 0 ;thatis,thequadraticsystemofFig.2-3perturbedby2{42. Thismethodistowritedowntime-dependentquantum-statisticalexpectationsof particle-amplitudes,orGreenfunctions"forshort.Onebeginsbywritingdown Schrodinger'sequation,andwritingdownanansatztoitinvolvingoperators G 0 ;G somethingcalledaGreenfunction.Inanalogytostudyingoperatorsratherthanstates asonedoeswhenstudyingtheHeisenbergpictureratherthan 22 theSchrodingerpicture, onecalculates G 0 ;G andeschewsreferencetothestatestheyareequivalentto. 22 Thetwopicturesareequivalentbecausetheyleadtothesameobservables.Operators, states,andwhateverotherobjectsonemaydreamofareonlyconceptualcrutchesalong thewaytothesingulargoalofcalculatingtheresultsofexperiments. 71

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2.3.1TheEquilibriumDistributions 2.3.1.1Thequantum-statisticalexpectation Thequantum-statisticalexpectationofanoperator X inagrand-canonical-ensemble ofpotential inwhichthe i th stationarystate j i i hasenergy E i duetobeingoccupiedby N i particlesisthefollowing[34]weightedsumthissumisalsoexpressibleasatrace, h X i = P i h i j X j i i e )]TJ/F23 7.9701 Tf 6.586 0 Td [( E i )]TJ/F23 7.9701 Tf 6.586 0 Td [(N i Z = Tr Xe )]TJ/F23 7.9701 Tf 6.587 0 Td [( H )]TJ/F23 7.9701 Tf 6.587 0 Td [(N Z Z = X i e )]TJ/F23 7.9701 Tf 6.587 0 Td [( E i )]TJ/F23 7.9701 Tf 6.586 0 Td [(N i {44 2.3.1.2Theequilibriumdistributionoffermionandbosonensembles Theoccupation-probabilities f < and b < ofastateofenergy byaparticlefrom agrandcanonicalensembleofchemicalpotential andtemperature T isgivenbythe followingnon-standardbutconvenientsumsoverboundquantizedequilibriumstatesof energy E andthuseachofBoltzmannprobabilities e )]TJ/F23 7.9701 Tf 6.586 0 Td [(E f < = P 1 n =0 ne )]TJ/F23 7.9701 Tf 6.587 0 Td [(n )]TJ/F23 7.9701 Tf 6.586 0 Td [( P 1 n =0 e )]TJ/F23 7.9701 Tf 6.587 0 Td [(n )]TJ/F23 7.9701 Tf 6.586 0 Td [( = 1 e )]TJ/F23 7.9701 Tf 6.586 0 Td [( +1 =1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(f > ; 1 e x +1 =1 )]TJ/F15 11.9552 Tf 28.998 8.088 Td [(1 e )]TJ/F23 7.9701 Tf 6.587 0 Td [(x +1 ; b < = P 1 n =0 ne )]TJ/F23 7.9701 Tf 6.586 0 Td [(n! P 1 n =0 e )]TJ/F23 7.9701 Tf 6.586 0 Td [(n! = 1 e )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 = b > )]TJ/F15 11.9552 Tf 11.955 0 Td [(1= )]TJ/F22 11.9552 Tf 9.299 0 Td [(b > )]TJ/F23 7.9701 Tf 6.586 0 Td [(! {45 TheaboveareknownastheFermi-DiracandBose-Einsteinfunctionswhich,forany systemofidenticalfermionsorbosonsinequilibrium,aretherespectiveprobabilitiesthat aquantumstateofenergy isoccupied[35].Itisacknowledgedthatthenotations f 7 and b 7 arenon-standard.However,theexpressionsresultingfromusingthemarecompactand symmetric. 2.3.1.3Theeectivefermionicandbosonicthermalfunctions Theeective"versionsoftheequilibriumfunctions2{45occurringinthe 7 Green functionsofacenter-regioncoupledby)]TJ/F23 7.9701 Tf 209.958 4.339 Td [( tothe th reservoirwithoccupancyenergy U ,inter-siteoverlap-energy )]TJ/F22 11.9552 Tf 9.298 0 Td [(W ,and / C -tunneling )]TJ/F22 11.9552 Tf 9.298 0 Td [(W forelectronsorinter-site 72

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oscillation-frequency 0 = p k=m and / C -mechanical-coupling are, f 7 eff = P )]TJ/F23 7.9701 Tf 7.314 4.339 Td [( f 7 P )]TJ/F23 7.9701 Tf 7.314 3.454 Td [( ;)]TJ/F23 7.9701 Tf 19.2 4.936 Td [( = W 2 W Re p 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(" 2 ; = )]TJ/F22 11.9552 Tf 11.955 0 Td [(U )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 W ; b 7 eff = P b 7 P ; = 2 0 ;{46 2.3.1.4TheSommerfeldexpansioninaniteband Onebeginswithanintegral I overaniteband 2 [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 W; 2 W ]whichisoftheform ofaband-density H = H localizedthatitsleadingdependenceupon isthe n th power ofthesameattheband-edges 2 W timesafermionicthermalfunction f 7 I = Z 2 W )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W H f 7 d" ; f < = )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(e )]TJ/F23 7.9701 Tf 6.587 0 Td [( = k B T +1 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 =1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f > ;lim 2 W H n ; Onethenusesthefundamentaltheoremofcalculustodenetheantiderivativeof H ,whichis K ;i.e., K Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W H 0 d" 0 H K 0 = @ K Onesubsequentlyintegratesbypartsi.e.,integratebothsidesof Hf = Kf 0 )]TJ/F15 11.9552 Tf -432.582 -23.908 Td [( Kf 0 totransferthederivative d=d" ontothethermalfunction, I = Z 2 W )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W K f 7 0 )]TJ/F27 11.9552 Tf 11.955 13.27 Td [( K f 0 7 d" =0 )]TJ/F27 11.9552 Tf 9.298 16.272 Td [(Z 2 W )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W K f 0 7 d" = )]TJ/F27 11.9552 Tf 11.291 16.272 Td [(Z 2 W )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W K f 0 7 d" Eachthermalfunction'sexplicitderivative,usingtheabbreviation x = )]TJ/F22 11.9552 Tf 11.897 0 Td [( =k B T ,is ahyperbolic-secant-squaredfunctionwhichisstrongly-peakedaradiusofthetemperature k B T aboutthechemicalpotential df 7 d" = )]TJ/F15 11.9552 Tf 9.298 0 Td [( 1 k B T e )]TJ/F23 7.9701 Tf 6.587 0 Td [( = k B T +0 e )]TJ/F23 7.9701 Tf 6.587 0 Td [( = k B T +1 2 = )]TJ/F21 7.9701 Tf 16.796 4.707 Td [(1 k B T e x= 2 + e )]TJ/F23 7.9701 Tf 6.587 0 Td [(x= 2 2 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 k B T sech x= 2 2 2 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 4 k B T sech 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [( 2 k B T Becausethethermalfunctionderivative f 0 7 isstronglypeakedabout ,the integral I musttakeitsgreatestsupportfromitsintegrandbeingsampledat 2 73

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[ )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 W; 2 W ].OnethusTaylor-expandstheantiderivative K about = I = +2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W 1 X n =0 )]TJ/F22 11.9552 Tf 11.955 0 Td [( n n d n K d" n )]TJ/F22 11.9552 Tf 9.298 0 Td [(df 7 d" d" = +2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W K |{z} @ 0 K )]TJ/F22 11.9552 Tf 9.299 0 Td [(df 7 d" d" + 1 X n =1 1 n d n K d" n +2 W Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W )]TJ/F22 11.9552 Tf 11.955 0 Td [( n )]TJ/F22 11.9552 Tf 9.298 0 Td [(df 7 d" d" Hence,byuseoftheantiderivative K 0 = H andTaylor-expansion,thelimit T 0isbuiltintotheleadingtermwhichsomemightsayisthegeniusofthe Sommerfeldexpansion.Ascendingpowersoftheseriesarecorrectionsduetononzero temperature.Thus,thequantity I whichistheintegralofathermalfunction f 7 anda band-density H whichislocalizedtoaband 2 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 W; 2 W ]suchthatithas[atmost] power-lawbehaviorattheband-edgesduetoSommerfeld[5]mayberepresentedasa zero-temperatureleadingtermpluscorrections I = 2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W H f 7 d" = Z 2 W H d" 1 4 1 X n =1 1 n d n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 H d" n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k B T n +2 W + =T Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W + =T x n sech 2 x 2 dx {47 ConvolutionintegralsbySommerfeldexpansion: Aquantity Q whichisa convolution-integralof H = H andFermifunction f 7 overaniteenergy-band 2 [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 W; 2 W ]inasystemhavingchemicalpotential andtemperature T isrepresentableas azero-temperaturetermpluscorrections,as, Q 7 = 2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d! 0 f 7 0 + H 0 = )]TJ/F23 7.9701 Tf 6.587 0 Td [(! 2 W )]TJ/F24 5.9776 Tf 5.756 0 Td [(! Z 2 W H 0 d! 0 + 1 X n =1 T n H n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(! b n {48 b n T n 4 n ! +2 W )]TJ/F23 7.9701 Tf 6.586 0 Td [( =T Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W )]TJ/F23 7.9701 Tf 6.587 0 Td [( =T dx x n sech 2 x 2 Thelimitsofintegrationhavestep-functionsbecausethefreely-varyingindependent variable intheconvolutionvariesover 2 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 W; 2 W ].Accordingly,theupperlimit ofintegrationmustvanishiftheintegraltakesitssupportfromapeakedfunctionthatis outsideofthestepped-thermal-function. 74

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Thewide-bandlimit: Muchsimplicationoftheaboveresultoccursforan innitely-wideband.Thesecorrectiontermsareonlythosewithevenpowersof )]TJ/F22 11.9552 Tf 12.627 0 Td [( because f 0 isevenabout and )]TJ/F22 11.9552 Tf 11.955 0 Td [( n iseven/oddalternatinglyin n I 0 + 1 Z 1 X n =1 )]TJ/F22 11.9552 Tf 11.956 0 Td [( n n d n H d" n )]TJ/F22 11.9552 Tf 9.298 0 Td [(@f 7 @" d" = 1 X n =1 0 @ 1 n d n H d" n 2 4 + 1 Z )]TJ/F22 11.9552 Tf 11.955 0 Td [( n )]TJ/F22 11.9552 Tf 9.298 0 Td [(@f 7 @" d" 3 5 1 A = 1 X n =1 1 n d n H d" n + 1 Z 2 6 4 n odd odd n even even 3 7 5 [even] | {z } parity d" = 1 X n =1 1 n d 2 n H d" 2 n + 1 Z )]TJ/F22 11.9552 Tf 11.955 0 Td [( 2 n )]TJ/F22 11.9552 Tf 9.298 0 Td [(@f 7 @" Onethengetsridof K infavouroftheoriginal H ,andalsointroducesthe substitution x )]TJ/F23 7.9701 Tf 6.587 0 Td [( k B T .Onethengets, I 0 = 1 X n =1 0 @ 1 n d 2 n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 H d" 2 n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 + 1 Z k B Tx 2 n )]TJ/F22 11.9552 Tf 9.299 0 Td [(df 7 x d k B Tx + 1 A d k B Tx + = 1 X n =1 2 4 + 1 Z x 2 n n )]TJ/F22 11.9552 Tf 9.298 0 Td [(df 7 x dx dx 3 5 k B T 2 n d 2 n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 H d" 2 n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 X n =1 [ a n ] k B T 2 n d 2 n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 H d" 2 n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 Thedimensionlessnumbersintroducedabovecanbewrittenintermsofthe Riemann-zeta-function, a n 1 Z x 2 n n )]TJ/F22 11.9552 Tf 9.299 0 Td [(@f 7 x @x dx = 1 Z x 2 n 4 n sech 2 x 2 dx = 1 X m =1 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 m +1 4 n 8 n m 2 n =2 1 X m =1 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 m +1 m 2 n 2 )]TJ/F15 11.9552 Tf 26.79 8.088 Td [(1 2 2 n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 n ThisgivesthefamiliarformoftheSommerfeldexpansion, I = + 1 Z H f 7 d" = Z H d" 1 X n =1 a n k B T 2 n H n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 ; a n = 2 )]TJ/F15 11.9552 Tf 26.79 8.088 Td [(1 2 2 n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 n ;{49 75

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2.3.2TheDenitionsoftheGreenfunctions 2.3.2.1TheGreenfunctionasaspectraldecomposition Thespectraldecomposition[28]ofanoperatorshallnowbederived.Letanarbitrary quantumstatebe j i ,andletacompletesetofstatesi.e.,stationarystates j n i havethe resolutionoftheidentity ^ 1= P n j n ih n j .Then,theactionoftheoperator ^ H upon j i for ^ H j n i = E n j n i hasarepresentationasthefollowingsummation,whichiscalledthe eigenfunction-expansion, ^ H j i = ^ H X n j n ih n j i = X n E n j n ih n j i = X n E n j n ih n j j i! ^ H = X n E n j n ih n j TheoperatorusedaboveneednotbetheHamiltonian.Carryingoutthesame machinationsfortheoperatorbeing ^ G R A z ^ 1 )]TJ/F15 11.9552 Tf 14.991 3.022 Td [(^ H )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 withtheaddedrequirementthat z becomplexsothattheoperator-polesarenonsingularontherealaxis,guaranteeingthe convergenceoftheresultingsummationeigenfunctionexpansionandthesum-representation 23 of ^ G R A ^ Q = z ^ 1 )]TJ/F15 11.9552 Tf 14.99 3.022 Td [(^ H )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = X n z ^ 1 )]TJ/F15 11.9552 Tf 14.99 3.022 Td [(^ H )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 j E n ih E n j = X n j E n ih E n j z )]TJ/F22 11.9552 Tf 11.955 0 Td [(E n ^ G ; z = E + i 0 + ;{50 Whatmakesthisresult2{50sousefulisthatforagivenregione.g.,innite potentialwellofnite,semi-innite,orinnitewidthinwhichonewishestosolvethe Schrodingerequationin,say, -spacee.g., beingposition,momentum,spin,etc.,one needsonlytoapplythebrah j and-ket j 0 i totheappropriatesidesof2{50anduse the -spacestates h j E n i and h 0 j E n i obeyingtheboundary-conditionsofthatgiven region.Onethenhasa ^ G thatcontainsthesameinformationasthewavefunction,with theaddedconveniencethat ^ G isaconvenientformfortheperturbativetheoryofthe Dysonequation. 23 Onerequiresconvergenceof ^ G = z ^ 1 )]TJ/F15 11.9552 Tf 14.99 3.022 Td [(^ H )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = P 1 n =0 C n 1 n z ^ 1 )]TJ/F15 11.9552 Tf 14.99 3.022 Td [(^ H n sothat ^ G j n i = z ^ 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(E n j n i ,requiring z tobecomplex. 76

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Now,suppose ^ Q = z ^ 1 )]TJ/F15 11.9552 Tf 14.99 3.022 Td [(^ H )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 inwhich ^ H j E n i = E n j E n i andthepowerseries ^ Q = P 1 n =0 C n 1 n ^ Q n n existswhichrequiresthat z becomplex.Then, ^ Q = z ^ 1 )]TJ/F15 11.9552 Tf 14.99 3.022 Td [(^ H )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = X n z ^ 1 )]TJ/F15 11.9552 Tf 14.99 3.022 Td [(^ H )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 j E n ih E n j = X n j E n ih E n j z )]TJ/F22 11.9552 Tf 11.955 0 Td [(E n ^ G {51 ThisisaGreenfunctionofa N -stationary-statesystemin N N matrix-form. For N '1 ,thesumin2{51maybere-castasanintegral.Anexampleofwhenthis continuum-limitmaybetakeniswhena D =1 ; 2 ; 3dimensionallattice 24 ofboundstates extendstoinnity,andtheallowedenergylevelscoalesceintoacontinuousniteband. ThisGreenfunctionmatrixinaposition x ormomentum p basisis, G x ; x 0 = h x j ^ Q j x 0 i = X n h x j E n ih E n j x 0 i z )]TJ/F22 11.9552 Tf 11.955 0 Td [(E n ; G p ; p 0 = h p j ^ Q j p 0 i = X n h p j E n ih E n j p 0 i z )]TJ/F22 11.9552 Tf 11.956 0 Td [(E n 2.3.2.2Single-fermionGreenfunctionsforelectronicdensity Thefermionic 1-bodyelectron-Greenfunctionsarethusdenedasthetime-dependent expectationsofthehole-propagation 25 operator c y 0 t 0 c t andtheelectron-propagation 26 operator c t c y 0 t 0 .Summingtheseelectron/holeabsorptionsandemissionssoastoforman anticommutatoralsoyieldsaquantityobeyingan -domainsum-rule,soaGreenfunction isreservedforthisaswell. =1 ; 2 ; 3 ; 5 ;::: g < t 0 t 0 = D c y 0 t 0 c t E ; g > t 0 t 0 = D c t c y 0 t 0 E ; ig r a t 0 t 0 = t )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 D f c t ;c y 0 t 0 g E ;{52 24 Thislatticemaybeineitherpositionormomentumspace;notethattheFermisphere ofaSommerfeldmetal,forinstance,residesinmomentumspace. 25 Aholeiscreatedwithposition/momentum,spin,etc. attime t andisabsorbedat t 0 7 t ,and g < 0 t;t 0 istheamplitudeforthis. 26 Similarly,anelectroniscreatedwithposition/momentum,spin,etc. 0 attime t 0 and isabsorbedat t 7 t 0 ,and g > 0 t;t 0 issimilarlydesignated. 77

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2.3.2.3BosonGreenfunctionsformechanicalamplitude Thebosonic 1-bodyphonon-Greenfunctionsaredenedasthetime-dependent expectationsoftheabsorption 27 operator )]TJ/F22 11.9552 Tf 9.298 0 Td [( 0 t 0 t andoftheemission 28 operator + t 0 t 0 .Introducing t = 1 p 2 b t + b y t d < t 0 t 0 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 h 0 t 0 t i ; d > t 0 t 0 =+2 h t 0 t 0 i ; id r a 0 t;t 0 = 2 t )]TJ/F23 7.9701 Tf 6.587 0 Td [(t 0 h [ t ; 0 t 0 ] i ;{53 2.3.2.4Two-fermionGreenfunctionsforparticle/holeexcitations Thebosonictwo-electron Greenfunctionsaredenedasthetime-dependent expectationsoftheoverlapofanamplitudeofelectronswithquantumnumber attime t havingdensity n t withelectronswith 0 t 0 andthusdensity n t 0 0 .Thus,electron-density n t playstherolethatmechanicalamplitude t = p 2didintheboson-Greenfunctions 2{53mechanicalamplitude andelectronicdensity n bothbeingHermitianoperators, inwhich n t c y t c t < t 0 t 0 = )-167(h n 0 t 0 n t i ; > t 0 t 0 =+ h n t n 0 t 0 i ; i r a t 0 t 0 = t )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 h [ n t ;n 0 t 0 ] i ;{54 2.3.2.5Single-bosonGreenfunctionsforatheoryofbosonscontaining number-preservingprocesses Itispossible[36]tointroduceelectron-like"butstillbosonicdensity-overlapGreen functionsforphonons.Since b m t b y m t representsabsorptionemissionattime t ofonemode m phononwithstationary-statealgebra[ b m ;b y m 0 ]= mm 0 b y m 0 t 0 b m t isa number-preservingoperator,andsoonecanmakeGreenfunctionsoutofthesephysical 27 Absorption ofamechanical-amplitude t occurringwithmodeormomentum at time t correlatedwithanamplitude 0 occurringwithmodeormomentum 0 attime t 0 7 t 28 Emission ofamechanical-amplitude 0 occurringwithmodeormomentum 0 attime t 0 correlatedwithanamplitude occurringwithmodeormomentum attime t 7 t 0 78

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objects, i d r a mm 0 = t )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 D [ b mt ;b y m 0 t 0 ] E ; d < mm 0 = )]TJ/F27 11.9552 Tf 11.291 13.27 Td [(D b y m 0 t 0 b mt E ; d > mm 0 =+ D b mt b y m 0 t 0 E ;{55 TheseGreenfunctions2{55areexactinatheoryignoringanharmoniccontributions e.g.,termswithtwo-phonon-emission b y m t b y m 0 t 0 ,three-phonon-emission b y m t b y m 0 t 0 b y m 00 t 00 andtheirabsorptiontime-reversedcounterpartstothelatticepotentialthatthe phononsareitinerantin.TakingtheFouriertransformof2{53andneglecting b y b y $ b b andhigher-order n -foldemission/absorption, d r a = d r a + )]TJ/F23 7.9701 Tf 6.587 0 Td [(! d a r )]TJ/F23 7.9701 Tf 6.587 0 Td [(! = d a r y ; d 7 = d 7 )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(! d ? )]TJ/F23 7.9701 Tf 6.586 0 Td [(! ;{56 Itshouldbenotedthatthestep-functionsin2{56arerequiredforthe -domain retardedandadvancedGreenfunctionstobetime-reversalsofeachotherthatis,the relation d r a = d a r y issatised. 2.3.2.6TheexistenceofexactpropertiesimpliedbytheGreenfunction denitions Thedenitions2{52,2{53,2{54,and2{55implysomeexactpropetiesvaluable bothforcheckingoldresultsandobtainingnewresults. 2.3.3ExactPropertiesoftheGreenFunctions 2.3.3.1Therelationamongthetime-orderingsoftheGreenfunctionsand self-energies Thereisavaluablerelationamongthetime-orderingsoftheGreenfunctionsand self-energies.TheGreenfunction g subjecttotheLangrethruleshasthevarious time-orderings g r ;g a ;g < ;g > ,whichobeythefollowingidentityinanydomaine.g., t )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 ;!;::: anddeneafunction N obeyingasum-ruleinthe -domain, i g r )]TJ/F22 11.9552 Tf 11.955 0 Td [(g a = g < + g > 2 N {57 79

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Giventheexistenceoftheobject obeying 29 g )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 11.9552 Tf 12.282 0 Td [(G )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 ,theidentity2{57is preserved,andoccursagainforboth and G G )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = g )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 11.9552 Tf 11.956 0 Td [( ; i G r )]TJ/F22 11.9552 Tf 11.955 0 Td [(G a = G < + G > ; i r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a = < + > ;{58 2.3.3.2TheLehmannrepesentationoftheGreenfunctionsinthe domain NowtobeobtainedistheLehmannrepesentationoftheGreenfunctionsinthe domain.TheLehmannrepresentationoftheGreenfunctions2{52and2{53occurafter Fouriertransformationusingkernel e i! t )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 ofthesametothe -domain.Thesystem H musthavetime-independentenergy,so H j n i = E n j n i ,andtime-independentoccupation ^ N ,so ^ N j n i = N n j n i .Thesestates h n j withstationaryenergyandparticle-numberare complete,soapossibleresolutionoftheidentity-operator ^ 1is ^ 1= P n j n ih n j Thus,forparticles a y $ a withanticommutationupper/positivesignorcommutation lower/negativesign,itisfoundthat, g < jj 0 = 1 Z X nn 0 2 + E n + N n )]TJ/F15 11.9552 Tf 11.955 0 Td [( E n 0 + N n 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! h n 0 j a y j 0 j n ih n j a j j n 0 i e )]TJ/F23 7.9701 Tf 6.587 0 Td [( E n 0 + N n 0 g > jj 0 =+ 1 Z X nn 0 2 )]TJ/F15 11.9552 Tf 9.298 0 Td [( E n + N n + E n 0 + N n 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! h n j a j j n 0 ih n 0 j a y j 0 j n i e )]TJ/F23 7.9701 Tf 6.586 0 Td [( E n + N n g r a jj 0 = 1 Z X nn 0 1 + )]TJ/F15 11.9552 Tf 9.299 0 Td [( i + n )]TJ/F22 11.9552 Tf 11.955 0 Td [(! n 0 h n j a j j n 0 ih n 0 j a y j 0 j n i )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(e )]TJ/F23 7.9701 Tf 6.586 0 Td [(! n e )]TJ/F23 7.9701 Tf 6.587 0 Td [(! n 0 {59 TheLehmannrepresentation2{59existsbecausetheensembleofparticles associatedwiththeseGreenfunctions,thoughnumerous,areallinstationarytime-independent states.Therefore,itisappropriatetorepresentthecorrespondingparticle-operatorsas theirmatrix-elementsinthebasisofstationarystates.Specically,manyproperties 29 Later,thisisshowntobetheself-energythatissummedtoinniteorderinDyson's equation.Here,assertingthisdenitionimplies2{58. 80

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importantandusefulforbothcomputingnewresultsanddetectingerrorsinoldresults followfromthisrepresentation. 2.3.3.3TheGreenfunctionrepresentationoftheuctuation-dissipation theorem AnotherresultistheGreenfunctionrepresentationoftheuctuation-dissipation theorem.Thisgivesanadditionalanalyticconditionupontheimaginarypartofthe retarded/advancedGreenfunctionsbeyond2{57.Throughoutthiswork,thiscondition shallbecalledsimplytheuctuation-dissipationtheorem".Thisuctuation-dissipation theoremisstatedinthe -domainforstationarystatesi.e.,stateswithtime-independent energyofacanonicalorgrandcanonicalensemble,andfollowsfromtheLehmann representation2{59.ItstatestheequilibriumGreenfunction g eq 7 intermsofthe spectralfunction A andthemostprobableequilibriumdistribution f 7 impliedbya temperature =1 =T andfermionicpotential .Abosonicanaloguealsoexists.The fermionicandbosonicuctuation-dissipationtheoremsare, g eq 7 = f 7 A fermions eq ; d eq 7 = b 7 A bosons eq ;{60 In2{60, N isthe -domainfunctionsshownin2{57,and2 A = d < + d > istheanalogueof N forphononsfrom2{53.The eq appearingassuperscriptor subscriptindicatesthat2{60holdsonlyforGreenfunctionsofasysteminequilibrium thoughthisisarguablyredundantduetotheoccurrenceoftheequilibriumdistributions f 7 and b 7 .Examplesofasystem in equilibriumwouldbecanonicalorgrandcanonical ensembles.Anexampleofasystem outof equilibriumwouldbeacentralregiondriven out ofequilibriumbycouplinge.g.,quantumtunnelingtooneormoregrandcanonical ensembles. Acorollaryof2{60:supposetherewerequantities r and a suchthat r = [ g r 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 11.9552 Tf 13.349 0 Td [([ g r ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 =[ a ] y =[ g r 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 11.9552 Tf 11.956 0 Td [([ g r ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 y ,andquantities 7 suchthat 7 = [ g r ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g 7 [ g a ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 .Later,thesearecalledself-energies".Furthermore,let[ g r 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 11.9552 Tf 11.984 0 Td [([ g a 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 / 81

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0 + .Then,usingthedenitionofthespectralfunction, g r 7 g a eq = g 7 eq = f 7 2 A e )]TJ/F23 7.9701 Tf -3.929 -7.892 Td [(! = f 7 i g r )]TJ/F22 11.9552 Tf 11.955 0 Td [(g a eq = f 7 g r;eq i r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a g a;eq Clearly,theequilibriumGreenfunctions g r;eq and g a;eq dropoutfrombothsidesofthe aboveequation.Furthermore,thesamesetofstepsmaybeeasilyrepeatedforthebosonic case,forwhichonenamestheself-energies r ; a ; 7 .Thisgives, 7 eq = f 7 i r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a eq ; 7 eq = b 7 i r )]TJ/F15 11.9552 Tf 11.955 0 Td [( a eq ;{61 Afurthercorollaryisobtainedfrom2{60and2{61bydirectlycalculatingthe quantity g > < )]TJ/F22 11.9552 Tf 11.955 0 Td [(g < > g > < )]TJ/F22 11.9552 Tf 11.955 0 Td [(g < > = i g r )]TJ/F22 11.9552 Tf 11.955 0 Td [(g a f > f < )]TJ/F22 11.9552 Tf 11.955 0 Td [(f < f > i r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a = i g r )]TJ/F22 11.9552 Tf 11.955 0 Td [(g a 0 i r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a =0 Again,thebosoniccasereadilyfollowsfromthefermioniccase,andonehas, N f in + N f out = g > < + )]TJ/F22 11.9552 Tf 9.299 0 Td [(g < > =0; N b in + N b out = d > < )]TJ/F22 11.9552 Tf 11.955 0 Td [(d < > =0;{62 Hence, ? g 7 forthecaseoffermionsareinterpretable[23]astheratesofmaterial-scattering ofparticles out in ofthecenterregion.Inanumber-conservingtheoryofphononssuchas 2{55asimilarinterpretationholds. 2.3.3.4Thefermionicspectralfunctionsumruleatequilibrium Asecondpropertyusefulasaconsistency-checkgovernsthefermionicspectral function2 A = g < + g > c.f.,2{58.Atequilibrium,thisfermionicspectral functionobeysasumrule.Atequilibriumwhere V T = T =0,onehas2 A =2 A eq = g eq ,yieldinginthe domain, 1 Z d! 2 A eq jj 0 = 1 Z X n e )]TJ/F23 7.9701 Tf 6.586 0 Td [(! n h n j a j a y j 0 + a y j 0 a j j n i = D f a j ;a y j 0 g E = jj 0 {63 Thiscanbeusedasacheck.Thereferencetothepartitionfunction Z = Tre )]TJ/F23 7.9701 Tf 6.586 0 Td [( E )]TJ/F23 7.9701 Tf 6.586 0 Td [(N iswhatrequiresequilibrium-conditionsin2{63. 82

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2.3.3.5Thebosonicspectralfunctionsumruleatequilibrium Thereisabosonicspectralfunctionsumruleatequilibrium,althoughtheruleisthat thespectralfunctionintegratestozeroandthusisnon-normalizable.Nevertheless,sucha sumrulecanalsobeusedacheckofconsistencyofthebosonicGreenfunctions, 1 Z d! 2 D eq mm 0 = 1 N X n e )]TJ/F23 7.9701 Tf 6.587 0 Td [(! n h n j 2 m y m 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 y m 0 m j n i = D [ m ; y m 0 ] E =0{64 Thestructureofthisspectralfunctionisduetotheoppositelysignedprocesses ofemissionandabsorption,whichareequalinequilibriumandthusgiveafunction integratingtozero. 2.3.3.6TheKramers-Kronigidentity ThelastpropertyistheKramers-KronigidentityforGreenfunctionsandself-energies. ItfollowsfromtheLehmannrepresentation2{59, g r = 1 Z d! 0 2 g < 0 + g > 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 + i ; r = 1 Z d! 0 2 < 0 + > 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 + i ;{65 Takingtherealpartofbothsidesof2{65givesaresultthatcanbeusedfor computationofanyretardedself-energy.Theaccuracyof2{65incalculatingaretarded self-energyisinfactsuperiortothecorrespondingLangrethrulefromwhichanyofthe time-orderings <;>;r;a follow. 2.3.4QuadraticTemperature-IndependentGreenFunctions GreenfunctionsforthequadraticHamiltoniansofSections2.2.2.1and2.2.3.1are obtained.Therststepindoingthisisobtainingthetime-dependenceoftheoperators obeying2{29. 2.3.4.1Occupationoperatortimedependenceinastationarystate Considerparticles a y t $ a t withquantumlabel whichmaybeposition, momentum,spin,etc.attime t .Theparticleshavethestandardanticommutation relationsiftheparticlesarefermionsbosons.Thefollowingisrequired:theunitary 83

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time-evolutionofanoperator,andaproductruleforanticommutators, [ AB;C ] )]TJ/F15 11.9552 Tf 10.406 1.793 Td [(= A [ B;C ] [ A;C ] B ; i O t [ O t ;H ] $ O t = e iHt ^ Oe )]TJ/F23 7.9701 Tf 6.586 0 Td [(iHt ; Bytheaboveexpressionforoperator-time-dependenceandoperatoralgebra,the Hamiltonian H whichhastheindicatedquadraticforminsomerepresentationgenerates thefollowingtime-evolution, i a t [ a t ;H ]=[ e iHt a e )]TJ/F23 7.9701 Tf 6.586 0 Td [(iHt ; X a y 0 t a 0 t ]= e iHt X 0 [ a ;" 0 e iHt a y 0 e )]TJ/F23 7.9701 Tf 6.587 0 Td [(iHt e iHt a 0 e )]TJ/F23 7.9701 Tf 6.586 0 Td [(iHt ] e )]TJ/F23 7.9701 Tf 6.587 0 Td [(iHt = X 0 0 e iHt [ a ;a y 0 a 0 ] )]TJ/F22 11.9552 Tf 7.084 3.884 Td [(e )]TJ/F23 7.9701 Tf 6.586 0 Td [(iHt = )]TJ/F27 11.9552 Tf 11.291 11.357 Td [(X 0 0 e iHt a y 0 [ a 0 ;a ] [ a y 0 ;a ] a 0 e )]TJ/F23 7.9701 Tf 6.586 0 Td [(iHt Thisgivesarst-orderdierentialequationintimefortheoccupation a t ofastate withquantumlabel position,spin,momentum,etc.attime t foraHamiltonian H thatinsomerepresentationisquadraticas H = P a y 0 t a 0 t .Thisdierentialequation isimmediatelysolvablewiththeinitialconditionthattheexpectationof a y t a t ,in accordancewiththiswork'suseoftheadiabaticregime,istheappropriatethermal function2{45, i a t = )]TJ/F27 11.9552 Tf 11.291 11.357 Td [(X 0 0 e iHt 0 a 0 e )]TJ/F23 7.9701 Tf 6.586 0 Td [(iHt = a a t = e )]TJ/F23 7.9701 Tf 6.586 0 Td [(i" t a {66 Astatewithsuchunitarytime-dependenceisastationarystate,whichdescribesa quantumsystematequilibrium.Linearcombinationsofsuchstateswithdisparateenergies time-evolve,andcorrespondtostatesoutofequilibrium.Non-equilibriumstatesmay bebuiltoutoftheseequilibriumstateswhentheagentdrivinganotherwise-stationary systemisintroducedadiabaticallyi.e.,overatime-scale )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 '1 .Notethatthisis perhapsinanalogytothefactthatstateswithpossiblyelaboratetime-dependenceare representableaslinearcombinationsoftime-independentstates. 84

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2.3.4.2ThestationarystateretardedandadvancedelectronicGreenfunctionsinpositionspace Combining2{52and2{66,andsubsequentlyFourier-transformingtothe -domain usingthekernelexp+ iz TinwhichT= t )]TJ/F22 11.9552 Tf 12.616 0 Td [(t 0 and z = + i ,theretardedGreen functionfortheoverlap-amplitudebetweenstatesofenergy U and U 0 isobtained, ig r 000 0 = 1 Z ig r 0 0 T e iz T d T= 1 Z 0 c y 0 c + c c y 0 e )]TJ/F23 7.9701 Tf 6.587 0 Td [(iU T e iz T = 1 Z 0 0 e )]TJ/F23 7.9701 Tf 6.586 0 Td [(iU T e iz T Adelta-functionresultsaftercarryingouttheintegration.Dividingbothsidesby i oneobtainsthe ; 0 th matrix-element g r 000 0 oftheretardedGreenfunction, g r 000 0 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(i 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(iU + iz = 0 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U = g a 0 0 y ; z = + i 0 + ;{67 Forthenano-junctionofFig.2-3,thequantumnumbers ; 0 includelabelsof positionadditionallabelswouldbe,e.g.,spin. 2.3.4.3ThestationarystateretardedandadvancedphononicGreenfunctions inmodespace TheretardedGreenfunction2{55ofthe m th phononmode'sdensity, d ro ,istobe obtainedforamodeofenergy m withcoupling tothe th lead.Phononsinthesystem Fig.2-3areassumedtohavetheirnumberconserved,allowing2{56tobeusedtoobtain 2{53.However,becauseof2{41,time-evolutionoftheboson-operatorsgenerated bytheHamiltonian2{39isnotthesameas2{66.Itisstillabletobebroughttoa quadraticform.Thecoupledequationsof2{41maybedecoupledas, 2 6 4 b m z )]TJ/F15 11.9552 Tf 13.256 3.022 Td [(~ m b q z )]TJ/F22 11.9552 Tf 11.955 0 Td [(! q 3 7 5 = 2 6 4 P q q m b q P m m; )]TJ/F23 7.9701 Tf 6.587 0 Td [(q b m 3 7 5 $ 2 6 4 b q b m 3 7 5 = 2 6 4 1 z )]TJ/F23 7.9701 Tf 6.587 0 Td [(! q 0 P m mq b m 1 z )]TJ/F21 7.9701 Tf 7.527 2.015 Td [(~ m P q q m b q 3 7 5 Fortheisolatedmode b m ,anembeddingself-energy[37]witharealandimaginary partresults.Althoughthecoupling q m betweenthe m th localizedmodeanda momentumq phononinthe th leadmayhaveascomplicatedadependenceupon q asdesired,thismomentum q issummedout.Hence,aphenomenologicalparameter 85

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theimaginarypartoftheembeddingself-energy,results, b m = 1 z )]TJ/F15 11.9552 Tf 13.256 3.022 Td [(~ m X q q m q m z )]TJ/F22 11.9552 Tf 11.955 0 Td [(! q b m z )]TJ/F15 11.9552 Tf 13.256 3.022 Td [(~ m = X q j q m j 2 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(! q r 0 m =Re r 0 m + i m b q = 1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(! q X m mq q m z )]TJ/F15 11.9552 Tf 13.256 3.022 Td [(~ m b q z )]TJ/F22 11.9552 Tf 11.955 0 Td [(! q = X m j q m j 2 z )]TJ/F15 11.9552 Tf 13.255 3.022 Td [(~ m = X m 0 m Re r 0 m + i m = X m 0 m m + i m Above,amissinglabelupon denotessummationi.e., = m m = m; m = .Therealpartoftheembeddingself-energyisabsorbedinto ~ m toform m takentobeasecondphenomenologicalparameter.Asimilarsetofphenomenological parameters,aresonance-energyandanescape-rategivingbroadeningintunneling spectroscopy[38],existsforelectronstunnelingthroughaninsulatingregionfromone metallicleadtoanother.Thus,thetime-evolutionofthephononoperatorsiscompletely determined.Abbreviating z m = m + i m and z = q + m 0 m m + i m ,oneobtains, b m t = e )]TJ/F23 7.9701 Tf 6.586 0 Td [(iz m t b m $ b y m t = e + iz m t b y m ; b q t = e )]TJ/F23 7.9701 Tf 6.586 0 Td [(iz t b q $ b y q t = e + iz t b y q ;2{68 Thislocalizedphononmodeofenergy m havingthemostspectralweightaradius m about m shallinteractby2{42withalocalizedelectronhavingaresonance at U havingthemostspectralweightaradius)]TJ/F23 7.9701 Tf 224.396 -1.793 Td [(U about U .Thisinteractionis similarlygivenasinglephenomenologicalparameterthatignoresthepossibly-complicated dependenceuponphonon-momentum q andelectron-momentum k NowonemayfollowthesamesetofstepsasinSection2.3.4.2usingthetime-dependence 2{68intheGreenfunctions2{55assumingconservationofphononnumberintheform 2{56toobtain, d r 0 m = d r 0 m ! + d a 0 m )]TJ/F22 11.9552 Tf 9.298 0 Td [(! )]TJ/F23 7.9701 Tf 6.587 0 Td [(! = + + )]TJ/F15 11.9552 Tf 11.955 0 Td [( m + i m + )]TJ/F23 7.9701 Tf 6.587 0 Td [(! )]TJ/F22 11.9552 Tf 9.299 0 Td [(! )]TJ/F15 11.9552 Tf 11.956 0 Td [( m )]TJ/F22 11.9552 Tf 11.955 0 Td [(i m =[ d r 0 m ] y {69 Theisolated th leadGreenfunctionshallalsoberequired,anditisobtainedin analogyto2{69, d r 0 = + + )]TJ/F22 11.9552 Tf 11.955 0 Td [(! q + i P m 0 m m + i m + )]TJ/F23 7.9701 Tf 6.587 0 Td [(! )]TJ/F22 11.9552 Tf 9.298 0 Td [(! )]TJ/F22 11.9552 Tf 11.955 0 Td [(! q )]TJ/F22 11.9552 Tf 11.955 0 Td [(i P m 0 m m )]TJ/F23 7.9701 Tf 6.587 0 Td [(i m 86

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Itis independentbecause 0 ischosenlaterinSec.2.3.4.5togivethephononsthe samebandwidthastheelectrons,giving q = j j .Thus, d r 0 isexpressibleasfollows, inwhichanorder-of-magnitudeestimatetoitsrealandimaginarypartsinwhichthe estimates 0 10 0 W m 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W ,and m 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 W aremadeisincluded d r 0 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(i + =! 0 P m m m + i m + i )]TJ/F23 7.9701 Tf 6.587 0 Td [(! =! 0 P m m m )]TJ/F23 7.9701 Tf 6.587 0 Td [(i m = )]TJ/F22 11.9552 Tf 9.299 0 Td [(i sgn !=! 0 P m m m + i m sgn 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(i 10 1 sgn W {70 Itshouldbenotedthat2{60allowscalculationofthedisconnected 7 phononic surfaceGreenfunctions,subsequentlyyielding d 7 0 m .Thecalculated 7 Greenfunctions, whichordinarilymakereferencetothesystem'stemperature,isdesiredtobeindependent ofreferencetoacenter-sitetemperature.ThesegoalsrequiretheLangrethrules,which aredevelopedlaterinSec.2.3.6. 2.3.4.4RetardedsurfaceGreenfunctionandenergy-momentumrelation dispersionofthe th lead'selectrons Consideraone-dimensionalchainofsitesofalternatingtype A and B withrespective occupancy-energies U A and U B ,inter-siteenergy )]TJ/F22 11.9552 Tf 9.299 0 Td [(W duetoanamplitudeoftunneling, andinter-sitespacing a= 2implyingalatticeconstantof a foraunitcellcontaining sites A and B ;thisisconvenientlater.Duetothischainbeinghomogeneousinthe limit U A U B ,thismodelsananowireintheappropriatelimit,achainofperiodic copolymersformingaconductingcovalent-nanowire,andevena GaAs wellsandwiched betweentwo Al x Ga 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(x As barriers.Whetherthenanowireiscovalentormetallicis determinedbythesingleparameter W ,whichinturndeterminesthebandwidth. Recallthattheoperator T = T x = T x eectingatranslationby x ofthe-ket j r i to yield j r + x i suchthat ^x j r + x i = r + x j r + x i isgivenasfollows, T x j r i = j r + x i$h r j T y x = h r + x j ; T x = e )]TJ/F23 7.9701 Tf 6.586 0 Td [(i x k $ T y x = e + i x k = T )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 x = T )]TJ/F39 7.9701 Tf 6.587 0 Td [(x ; k = p = ~ = )]TJ/F22 11.9552 Tf 9.298 0 Td [(i r x ; 87

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Duetothepropertiesofthistranslationoperator,theretardedGreenfunctionsinthe leadare, g r 0 mn = e ikja= 2 g r 0 m + j;n + ` e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ik`a= 2 = e ik j )]TJ/F23 7.9701 Tf 6.587 0 Td [(` a= 2 g r 0 m + j;n + ` ; m;n 2;{71 Thegoalnowistoobtainthedispersion-relationinthisdiatomicchain.Todo this,theretardedDyson'sequationisusedtocalculatenon-surfaceGreenfunctionsof neighboringsite-overlap;say, g r 0 23 and g r 0 34 .Thus,tothematrix-multiplicationinDyson's equation2{79applyretardedLangrethordering2{81oftheconvolution 30 .Then,use thetranslation-operatortoobtaininstancesof g r 0 23 and g r 0 34 soastopursueahomogeneous 2 2matrixsystem, 2 6 4 g r 0 23 g r 0 34 3 7 5 = 2 6 4 )]TJ/F23 7.9701 Tf 6.587 0 Td [(W z )]TJ/F23 7.9701 Tf 6.587 0 Td [(U B g 33 + g 13 )]TJ/F23 7.9701 Tf 6.587 0 Td [(W z )]TJ/F23 7.9701 Tf 6.586 0 Td [(U A g 44 + g 24 3 7 5 = 2 6 4 )]TJ/F23 7.9701 Tf 6.586 0 Td [(W z )]TJ/F23 7.9701 Tf 6.587 0 Td [(U B g r 0 34 e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ika= 2 + e + ika g r 0 34 e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ika= 2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(W z )]TJ/F23 7.9701 Tf 6.586 0 Td [(U A e )]TJ/F23 7.9701 Tf 6.586 0 Td [(ika g r 0 23 e + ika= 2 + g r 0 23 e + ika= 2 3 7 5 = 2 6 4 )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W cos ka 2 z )]TJ/F23 7.9701 Tf 6.586 0 Td [(U B g r 0 34 )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W cos ka 2 z )]TJ/F23 7.9701 Tf 6.587 0 Td [(U A g 23 3 7 5 Becausetheabovesystemishomogeneous,nontrivial g r 0 23 and g r 0 34 resultfora vanishingdeterminant,whichgivesthedispersion, 1 )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W cos ka 2 z )]TJ/F23 7.9701 Tf 6.586 0 Td [(U B )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W cos ka 2 z )]TJ/F23 7.9701 Tf 6.586 0 Td [(U A 1 =0 $ z = k + i k = U q U 2 + W cos ka 2 2 | {z } U = U A + U B = 2; U = U A )]TJ/F23 7.9701 Tf 6.587 0 Td [(U B = 2; {72 Thedispersion2{72canbeseentohaveaband-gapifplottedasinFig.2-9. TheDysonequationisrequiredtocalculate g r 0 11 ;g r 0 22 whichareconnectedbythe translationoperator,buthaveanoverallnormalizationthatisunknown,inwhichthe dispersion2{72intheform2 W cos ka 2 = p z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U A z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U B recallthehalved lattice-constantisusedthetwo-signedfunctioncos ka 2 musthavebothsquareroots. Beginningwith g r 0 11 = 1 z )]TJ/F23 7.9701 Tf 6.587 0 Td [(U A )]TJ/F22 11.9552 Tf 12.471 0 Td [(Wg r 0 21 = 1 z )]TJ/F23 7.9701 Tf 6.587 0 Td [(U A )]TJ/F22 11.9552 Tf 12.471 0 Td [(We )]TJ/F23 7.9701 Tf 6.587 0 Td [(ika= 2 g r 0 11 ,onesolvesfor g r 0 11 and 30 Specically,thesumoverinternalindicesdeningmatrix-multiplicationisa convolution 88

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Figure2-9.Bandvelocityandenergyasafunctionofmomentumforadiatomicchain. obtains, g r 0 11 = 1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U A + We )]TJ/F23 7.9701 Tf 6.587 0 Td [(ika= 2 = 1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U A + W q z )]TJ/F23 7.9701 Tf 6.586 0 Td [(U B z )]TJ/F23 7.9701 Tf 6.587 0 Td [(U A 4 W 2 i q 1 )]TJ/F21 7.9701 Tf 13.151 5.699 Td [( z )]TJ/F23 7.9701 Tf 6.586 0 Td [(U B z )]TJ/F23 7.9701 Tf 6.587 0 Td [(U A 4 W 2 {73 Requiringthelimitingcaseofthemonatomicchaintooccurfor U B U A = U resolvesthe ambiguityfromthesquare-rootoperation.Ontakingthislimit,itisseen thatonemusttakethenegativeroot, lim U B U A g r 0 11 =lim U B U A 1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U A + W )]TJ/F27 11.9552 Tf 9.298 14.596 Td [(q z )]TJ/F23 7.9701 Tf 6.586 0 Td [(U B z )]TJ/F23 7.9701 Tf 6.587 0 Td [(U A 4 W 2 + i q 1 )]TJ/F21 7.9701 Tf 13.15 5.699 Td [( z )]TJ/F23 7.9701 Tf 6.587 0 Td [(U B z )]TJ/F23 7.9701 Tf 6.586 0 Td [(U A 4 W 2 = 1 W 1 z )]TJ/F23 7.9701 Tf 6.586 0 Td [(U A 2 W + i q 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( z )]TJ/F23 7.9701 Tf 6.587 0 Td [(U A 2 W 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 W e ika= 2 TousetheGreenfunction2{73forthe th lead,changethenotationwhichup untilnowhasbeenlocaltothissectionas g r 0 ij g r 0 i j .ThesurfaceGreenfunction g r 0 1 1 89

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maybeusedtocalculatetheretarded -domainself-energy r 0 dressingthecenter-site Greenfunction g r 00 ij into g r 0 ij r 0 ij = r 0 i 1 g r 0 1 1 r 0 1 j = )]TJ/F22 11.9552 Tf 9.298 0 Td [(W i g r 0 1 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(W j = W i W j g r 0 1 1 Re r 0 ij + i )]TJ/F23 7.9701 Tf 7.314 4.936 Td [( ij {74 Theimaginarypartofthisself-energy)]TJ/F23 7.9701 Tf 205.796 4.338 Td [( ij iscalledanescaperate",andiswhat occursinthethermalfunctions2{46andistheanalogueofthephonon-parameter in2{69.Itisalsothesine-functionplottedinFig.2-9,whichvanishesatthe band-edgesandthusisresponsiblefornegativedierentialresistanceinacurrent driventhroughajunction.Furthermore,intheabsenceofanappliedeldthatbreaks time-reversal-symmetry,onehas W i = W i = W i 2.3.4.5Energy-momentumrelationdispersionofthe th lead'sphonons Thedispersion = k ofamomentum k phononshallbecalculated.Consider[5] alinearchainwithinter-sitespacing a inwhichalternatesiteshavemass M 1 and M 2 andonlynearestneighborsinteractbyaspringconstant C .Letthe n th site-1have displacement u 1 na fromequilibrium,andthe n th site-2havedisplacement u 2 na from equilibrium.Then,the n th site-1hasenergy 31 1 2 U 1 duetodisplacement u 1 na )]TJ/F22 11.9552 Tf 12.627 0 Td [(u 2 na and duetodisplacement u 1 na )]TJ/F22 11.9552 Tf 12.05 0 Td [(u 2 n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 a ,whilethe n th site-2hasenergy 1 2 U 2 duetodisplacement u 2 na )]TJ/F22 11.9552 Tf 12.376 0 Td [(u 1 na andduetodisplacement u 2 na )]TJ/F22 11.9552 Tf 12.376 0 Td [(u 1 n +1 a .Thetotalenergyofalatticeofsimilar suchsitesisfoundfromdirectcalculationas, U harm = 1 2 U 1 + 1 2 U 2 = 1 2 1 2 C X n 0 B @ u 1 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 2 na 2 + u 1 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 2 n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 a 2 + u 2 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 1 na 2 + u 2 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 1 n +1 a 2 1 C A 31 Theextrafactorof1 = 2istoavoiddouble-counting. 90

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Correspondingly,theforce F = )]TJ/F23 7.9701 Tf 10.494 4.707 Td [(dU dx canbecalculatedfromthispotentialenergy,for useinNewton's2ndlawoneinvocationforeachsite.Carryingthisoutfor M 1 M 1 u 1 n 0 a = )]TJ/F22 11.9552 Tf 10.494 8.088 Td [(@U harm @u 1 n 0 a = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 4 C X n @ @u 1 n 0 a 0 B @ u 1 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 2 na 2 + u 1 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 2 n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 a 2 + u 2 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 1 na 2 + u 2 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 1 n +1 a 2 1 C A = )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 C X n 0 B @ u 1 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 2 na nn 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(0+ u 1 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 2 n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 a nn 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(0+ u 2 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 1 na )]TJ/F22 11.9552 Tf 11.955 0 Td [( nn 0 + u 2 na )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 1 n +1 a )]TJ/F22 11.9552 Tf 11.955 0 Td [( n +1 ;n 0 1 C A M 1 u 1 n 0 a = )]TJ/F21 7.9701 Tf 10.494 4.707 Td [(1 2 C )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( u 1 n 0 a )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 2 n 0 a + u 1 n 0 a )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 2 n 0 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 a )]TJ/F15 11.9552 Tf 11.955 0 Td [( u 2 n 0 a )]TJ/F22 11.9552 Tf 11.956 0 Td [(u 1 n 0 a )]TJ/F15 11.9552 Tf 11.955 0 Td [( u 2 n 0 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ;a )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 1 n 0 a Hence,Newton'sLawsappearas, M 1 u 1 n 0 a = )]TJ/F22 11.9552 Tf 9.299 0 Td [(C )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 u 1 n 0 a )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 2 n 0 a )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 2 n 0 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ;a ; M 2 u 2 n 0 a = )]TJ/F22 11.9552 Tf 9.298 0 Td [(C )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 u 2 n 0 a )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 1 n 0 a )]TJ/F22 11.9552 Tf 11.955 0 Td [(u 1 n 0 +1 ;a ; Usinganoscillatoryansatz u i x t = A i e i kx )]TJ/F23 7.9701 Tf 6.586 0 Td [(!t where i =1 ; 2, x = na ,and disparateamplitudes A 1 6 = A 2 makestheaboveintoahomogeneoussystemofequations. Becausethesystemishomogeneous,theconditionforanon-trivialsolutionisavanishing determinant, 2 6 4 M 1 2 A 1 M 2 2 A 2 3 7 5 = 2 6 4 C A 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(A 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(A 2 e )]TJ/F23 7.9701 Tf 6.586 0 Td [(ika C A 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(A 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(A 1 e ika 3 7 5 2 6 4 2 C M 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 2 )]TJ/F23 7.9701 Tf 6.586 0 Td [(C M 1 + e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ika )]TJ/F23 7.9701 Tf 6.586 0 Td [(C M 2 + e ika 2 C M 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 2 3 7 5 2 6 4 A 1 A 2 3 7 5 = 2 6 4 0 0 3 7 5 Settingthedeterminantequaltozeroandsolvingfor ,ofwhichtherearetworoots correspondingtoseparateacousticlow-frequencyandopticalhigh-frequencybands ofpropagatingphononenergies, 2 C M 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(C M 1 + e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ikx )]TJ/F23 7.9701 Tf 6.587 0 Td [(C M 2 + e ikx 2 C M 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 2 =0 ! p C = r M 1 + M 2 M 1 M 2 q 1 M 1 2 + 1 M 2 2 +2 1 M 1 M 2 cos ka Simplifyingthisresultyields, p C = 1 p M 1 M 2 q M 1 + M 2 p M 1 2 + M 2 2 +2 M 1 M 2 cos ka {75 91

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Figure2-10.Opticalandacousticphononbranchesinadiatomicchainforvarious mass-ratios. Aformmoresuitableforplottingis =! 1 where 1 p C=M 1 vs. ka= ,and introducingthesingleparameter m M 2 =M 1 ,onecanplotthefollowingforvarious mass-ratios,asdoneinFig.2-10, =! 1 = r 1 m + m q 1+ 1 m 2 +2 1 m cos ka Thedispersion2{75isahomogeneous-quadraticforlow-momentumphonons ka= 0andforphononsneartheband-edge ka= 1.PerformingaTaylor expansionof2{75forlowphonon-momentum, =! 1 ka= 0 = q 1 m + m 1 m +1 1 = 4 + m q 1 m + m 1 m +1 ka 2 = c 0 + c 2 ka 2 Onemayalsoexpandtheopticalbranchinthemassratio m as, + =! 1 m 0 = p 2 =m + p m= 2 1 2 cos ka + O 3 = 2 m FortheopticalbranchinFig.2-10,thebottomoftheopticalbranchisdominatedby thereciprocal-reduced-mass, m 1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 + m 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 ,anditsedgesbythelightermass m 1 .Forthe acousticbranch,theedgesaredominatedbytheheaviermass m 2 Thegapbetweentheopticalandacousticbranchcouldbeusefulforcontrollingthe appearanceofphononicabsorptionsandemissionscontributingtotheelectron/phonon interactionself-energytobeincorporatedlater.Thisworkusesamonatomicchainin 92

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whichphononspropagate,butfutureworkcouldstudytheeectofgapsinthephonon dispersionuponthethermoelectricresponse. Inthelimitof m 1,thesystemwhosedeterminantwassettozerobecomeslinearly dependent,andthushasatrivially-vanishingdeterminant.Thus,onemustrepeatthe calculationtoobtainthemonatomicdispersion, M K x n = )]TJ/F15 11.9552 Tf 9.298 0 Td [([ )]TJ/F15 11.9552 Tf 9.299 0 Td [( x n +1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(x n + x n )]TJ/F22 11.9552 Tf 11.955 0 Td [(x n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ] $ M K 2 n =2 n )]TJ/F22 11.9552 Tf 11.955 0 Td [( n +1 e ika )]TJ/F22 11.9552 Tf 11.955 0 Td [( n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ika Thisgivesthemonatomicdispersion,whichisparametrizedas, q M K 2 = p 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2cos ka ! = 0 p 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos ka {76 2.3.4.6RetardedelectronicsurfaceGreenfunctionbyaspectraldecomposition Onebeginsbyprojectingthematrix-elementsof2{50intheposition-basisofthe ` th siteinthe th lead.Lettheleadbeoflength L whichcancelsinthecontinuum-limit. Duetotheboundary-conditionthatthe ` th siteeigenstate h ` j k i ofmomentum k mustvanishatthebeginningofthesemi-innite th lead,theeigenstateis h ` j k i = q 2 L sin k a` .Onplacingthis2{50andnon-dimensionalizingtheintegrand, g r 0 ` ` 0 = X k h ` j k ih k j ` 0 i )]TJ/F22 11.9552 Tf 11.956 0 Td [(! k + i + a =a Z )]TJ/F23 7.9701 Tf 6.586 0 Td [(a =a L d k a 2 q 2 L sin k a` q 2 L sin k a` 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [( U )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 W cos ka + i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 2 W + Z )]TJ/F23 7.9701 Tf 6.586 0 Td [( d sin ` sin ` 0 )]TJ/F23 7.9701 Tf 6.587 0 Td [(U )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos )]TJ/F22 11.9552 Tf 11.955 0 Td [(i Onethenusesstandardtechniques[39]forintegratingarationalfunctionwhose numeratoriscomprisedoftrigonometricfunctionstosomepower,andwhosedenominator 93

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containstworst-orderpolesandan ` th -orderpole, g r 0 ` ` 0 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 2 W + Z )]TJ/F23 7.9701 Tf 6.586 0 Td [( d 1 2 i e i` )]TJ/F22 11.9552 Tf 11.955 0 Td [(e )]TJ/F23 7.9701 Tf 6.586 0 Td [(i` 1 2 i e i` 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(e )]TJ/F23 7.9701 Tf 6.587 0 Td [(i` 0 )]TJ/F23 7.9701 Tf 6.587 0 Td [(U )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W )]TJ/F21 7.9701 Tf 13.151 4.707 Td [(1 2 e i + e )]TJ/F23 7.9701 Tf 6.586 0 Td [(i )]TJ/F22 11.9552 Tf 11.955 0 Td [(i = 1 8 W I ccw dz iz z ` )]TJ/F22 11.9552 Tf 11.955 0 Td [(z )]TJ/F23 7.9701 Tf 6.586 0 Td [(` z ` 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(z )]TJ/F23 7.9701 Tf 6.587 0 Td [(` 0 )]TJ/F21 7.9701 Tf 13.151 4.707 Td [(1 2 z + z )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(i z ` + ` 0 z ` + ` 0 = i 4 W I ccw dz z 2 ` )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 z 2 ` 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 z 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(i z +1 z ` + ` 0 = i 4 W I ccw dz z 2 ` )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 z 2 ` 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(z + z )]TJ/F22 11.9552 Tf 11.956 0 Td [(z )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( z ` + ` 0 Above,thefollowingstandardchangeofintegrationvariableswaseected,which mapstherstBrillouinzone )]TJ/F22 11.9552 Tf 9.299 0 Td [(< = k a totheunitcircle j z j =1,as, z = e ix dz = ie ix dx dx = ie ix dz = i dz=z = i d ln z Thefollowingfactorizationwasalsoused,givingrootswhoseproductisunity 32 and arethusconjugate,normalized,andrepresentableasexponentials.Dening " )]TJ/F22 11.9552 Tf 12.182 0 Td [(i onehasthefollowingunitary 33 representation, z 2 )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 z +1= z )]TJ/F22 11.9552 Tf 9.298 0 Td [(z + z )]TJ/F22 11.9552 Tf 9.299 0 Td [(z )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( z = p 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1= i p 1 )]TJ/F15 11.9552 Tf 12.743 0 Td [( 2 = e ik a e sgnIm z e ik a e Onethensumsoverresidues,inwhichthe ` + ` 0 orderpolemustbewrittenasanite sumwhoseclosed-formsumisguessedbyautomatedtrialanderrorusingMathematica totestaguessedformulafor1 `;` 0 60, g r 0 ` ` 0 = i 4 W 2 i z + 2 ` )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 z + 2 ` 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 z + )]TJ/F22 11.9552 Tf 11.955 0 Td [(z )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( z + ` + ` 0 + 1 ` + ` 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1! lim z 0 d ` + ` 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 dz ` + ` 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 z 2 ` )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 z 2 ` 0 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 z )]TJ/F22 11.9552 Tf 11.956 0 Td [(z + z )]TJ/F22 11.9552 Tf 11.955 0 Td [(z )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( 1 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 2 W 0 @ z + 2 ` )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 z + 2 ` 0 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 z + )]TJ/F22 11.9552 Tf 11.955 0 Td [(z )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( z + ` + ` 0 + ` + ` 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 X k =0 z )]TJ/F23 7.9701 Tf 7.085 6.73 Td [(k )]TJ/F23 7.9701 Tf 6.587 0 Td [(` )]TJ/F23 7.9701 Tf 6.587 0 Td [(` 0 z + )]TJ/F23 7.9701 Tf 6.586 0 Td [(k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F25 7.9701 Tf 11.956 15.874 Td [(j ` )]TJ/F23 7.9701 Tf 6.587 0 Td [(` 0 j)]TJ/F21 7.9701 Tf 8.939 0 Td [(1 X k =0 z )]TJ 7.084 6.73 Td [()]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(k z + k j ` )]TJ/F23 7.9701 Tf 6.586 0 Td [(` 0 j 1 A 32 Proof:notethat z + z )]TJ/F15 11.9552 Tf 18.978 1.793 Td [(= + j 1 )]TJ/F15 11.9552 Tf 12.744 0 Td [( 2 j ,forwhichlim 0 Re[ + j 1 )]TJ/F15 11.9552 Tf 12.743 0 Td [( 2 j ]=1and lim 0 Im[ + j 1 )]TJ/F15 11.9552 Tf 12.743 0 Td [( 2 j ]=0,meaningtheirproductisasiftheywereunitaryinthe 0 limit. 33 Intermsoftrigonometricfunctions,andwithouttheinntesimal ,onehas z = i p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(" 2 =cos k a i p 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos 2 k a =cos k a i sin k a = e ik a 94

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Figure2-11.Poleplotfortheintegrandofthecontourintegraldening g r 0 ` ` 0 with exaggeratedinntesimal. Theinntesimal controlswhichpolecontributestotheresidue-sum,soitsrole hasbeenservedandthelimit 0maybeexplicitlyeected.Onethenreferstothe consequentfunctionalform z e ik a oftheroots z .For )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 < Re < 1,theroots areparametrizablebytrigonometricfunctionsgivingaunitarycomplexexponential,as indicated.Becausetherootsarereciprocalsandcomplex-conjugatesofeachother,their productisunity,andthesum s = disappears,leavingonlytheorder` pole, g r 0 ` ` 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 2 W e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ik a j ` )]TJ/F23 7.9701 Tf 6.587 0 Td [(` 0 j )]TJ/F22 11.9552 Tf 11.955 0 Td [(e ik a j ` )]TJ/F23 7.9701 Tf 6.586 0 Td [(` 0 j 2 i sin k a )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ik a ` + ` 0 )]TJ/F22 11.9552 Tf 11.956 0 Td [(e ik a ` + ` 0 2 i sin k a + e ik a` )]TJ/F22 11.9552 Tf 11.955 0 Td [(e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ik a` e ik a` 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(e )]TJ/F23 7.9701 Tf 6.586 0 Td [(ik a` 0 2 i sin k a Hence,the th leadGreenfunction g r 0 ` ` 0 inaone-dimensionalsemi-innitelatticeof momentum k electronswithdispersion k = U )]TJ/F15 11.9552 Tf 10.682 0 Td [(2 W cos k a interspacedbyadistance a andwithinter-sitetunneling )]TJ/F22 11.9552 Tf 9.298 0 Td [(W andoccupancy-energy U fortransitionbetweensites ` and ` 0 isgivenas, g r 0 ` ` 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = W 2 i sin k a 0 B @ e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ik a ` + ` 0 +2 e ik a ` + ` 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(e ik a j ` )]TJ/F23 7.9701 Tf 6.587 0 Td [(` 0 j )]TJ/F22 11.9552 Tf 11.955 0 Td [(e ik a ` )]TJ/F23 7.9701 Tf 6.586 0 Td [(` 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(e )]TJ/F23 7.9701 Tf 6.587 0 Td [(ik a ` )]TJ/F23 7.9701 Tf 6.587 0 Td [(` 0 1 C A ` = ` 0 =1 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 W e iak {77 95

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Figure2-12.PhononicpoleplotfortheretardedphononicGreenfunctionshowingthe importanceofusingthesignedinntesimal =0 + sgn .Themagnitudeof isexaggerated. 2.3.4.7RetardedphononicsurfaceGreenfunctionbyaspectraldecomposition ItshallalsobeusefultoobtaintheretardedphononsurfaceGreenfunctions,because theyareinvolvedinthecalculationofthephononicenergycurrent.Usingthespectral decomposition,butthistimeusingtheinntesimal =0 + sgn toavoidabranch-cuton therealaxis, d r 0 q q = X q h q j ih j q i )]TJ/F22 11.9552 Tf 11.955 0 Td [(! q + i 0 + + a =a Z )]TJ/F23 7.9701 Tf 6.587 0 Td [(a =a Ld q a 2 q 2 L sin q a q 2 L sin q a )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 p 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos q a + i 0 + = 1 I j z j =1 dz iz [ 1 2 i z )]TJ/F22 11.9552 Tf 11.955 0 Td [(z )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 ] 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 q 1 )]TJ/F21 7.9701 Tf 13.15 4.707 Td [(1 2 z + z )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 + i 0 + = i 4 I j z j =1 dz z 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 z 3 )]TJ/F22 11.9552 Tf 11.956 0 Td [(! 0 q 1 )]TJ/F21 7.9701 Tf 13.151 4.707 Td [(1 2 z + z )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 + i 0 + Clearly,thereisathird-orderpoleat z =0.Theunitaryandconjugatepoles z = z inthedenominator'squadraticdeterminethebounds0 0 p 2 W upon 0 Abbreviating = + i 0 + =! 0 ,noting 0 = p 2 W and )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 W
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ThebehaviorofthepolesinFig.2-12givesthefollowingresidue-sum,inwhich therepresentation[27] z _= e iqa oftheroots 34 z greatlysimpliesthetermduetothe third-orderpole, d r 0 q q =2 i i 4 1 0 X Res z 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 2 z 3 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(z + z )]TJ/F22 11.9552 Tf 11.955 0 Td [(z )]TJ/F15 11.9552 Tf 7.085 1.794 Td [( = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 2 0 R + + 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1! lim z 0 d 3 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 dz 3 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 z 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(z + z )]TJ/F22 11.9552 Tf 11.955 0 Td [(z )]TJ/F15 11.9552 Tf 7.084 1.794 Td [( = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 2 0 z + 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 z + 3 z + )]TJ/F22 11.9552 Tf 11.956 0 Td [(z )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( + 1 2! 2 z )]TJ/F22 11.9552 Tf 7.085 1.793 Td [(z + + z + 2 + z )]TJ/F21 7.9701 Tf 7.085 6.132 Td [(2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 z + 2 z )]TJ/F21 7.9701 Tf 7.085 5.247 Td [(3 z + 3 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 2 0 z + 2 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 2 0 e i 2 q a Hence,theretardedGreenfunction d r 0 q q ofmomentum q phononswithdispersion q = 0 p 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos q a inthemonatomic th leadwhoseatomshavenaturalmechanical vibrationalfrequency 0 isgivenas, d r 0 q q = )]TJ/F15 11.9552 Tf 9.299 0 Td [( 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 e 2 iq a q = 0 p 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos q a {78 2.3.5Dyson'sEquation 2.3.5.1TheGreenfunctionsolutiontoordinarydierentialequationsandthe Schrodingervs.Heisenbergpictures TheGreenfunctionmethodisusedtoobtainthesolutiontoaninhomogeneous linearordinarydierentialequation[40]withaknownhomogeneoussolution 0 by introducingafunction G suchthatequals 0 plusanindex-contraction 35 of 0 with theinhomogeneity V .Theaforementionedquantitiescouldbeoperators,makingthe Greenfunctionmethodnaturallytintothemethodsofquantummechanics.Oneneeds onlytorerealizethehomogeneoussolution 0 asaDirac-ket j 0 i ,andlikewisethe paricularsolution.Theindex-contractionintheansatzistheinner-productonthe 34 Asintheelectroniccase,theroleoftheinntesimalistopickthepolethat contributestotheresidue-sum.Onceithasservedthatpurpose,itmaybediscarded, leavingthevery-handypropertiesofunitaryoftheroots. 35 Atthispoint,onemustmakereferencetoapossiblywell-resolvedcoordinate: position,momentum,spin,etc.;callthiscoordinate q 97

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solution-space,andsothedualoftheDirac-ketthatismanifestlyanindex-contraction i.e.,aninstructiontointegrate"inthecontinuous-case,thebrah j comesintoplay. 2.3.5.2Theinniteperturbativesum-theDysonequationfortheGreen function Thus,consideraquantumsystem H = H 0 + V with H 0 diagonalizedtoyield thestationary-statesandtheirenergies U .Theintroductionof V givestheSchrodinger equation i@ t j i = H j i anditsindex-contractionansatzas, j i = 0 + GV 0 ; G = i@ t )]TJ/F22 11.9552 Tf 11.955 0 Td [(H )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ; j i = 0 + G 0 V j i ; G 0 = )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(i@ t )]TJ/F22 11.9552 Tf 11.955 0 Td [(H 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 ; FollowingtheHeisenbergpicture,onemayconcernthemselvesonlywiththeoperators H;V;G andtheiranalyticity,andmakereferencetothestates ; 0 onlyasneeded.The questionastowhethertheinverses G;G 0 existhasnotbeenaddressedyet,andforthat onetemporarilyreturnstotheSchrodingerpictureofstatesandtheirlifetimes. Theansatzinvolving G 0 proposedabove,whensubstitutedintoitselfandarranged asaperturbativeseriesofascendingorders,immediatelyyieldsthefollowingsummable series, j i = 0 + G 0 V )]TJ 5.48 0.478 Td [( 0 + G 0 V )]TJ 5.48 0.478 Td [( 0 + G 0 V ::: = 1 X n =0 G 0 V 0 = )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(G 0 V )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 0 =+ GV 0 Combiningtheaboveresultwiththeansatzinvolving G ,oneobtainsanequationfor twooperatorseachoperatingupon j 0 i ,soonethenworksonlywithoperators G;G 0 ;V andcontinuestoassumetheydonotcommute.Oncarefuluseoftheshoesandsocks" theorem[41]fortheinverse-operationoneimmediatelyobtainstheDysonequationafter solvingfor G in )]TJ/F22 11.9552 Tf 11.955 0 Td [(G 0 V )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 j 0 i =+ GV j 0 i G = )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(G 0 V )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 V )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = G 0 V )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(G 0 V )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 V )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = G 0 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(V )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 98

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2.3.5.3InversesandanalyticityforGreenfunctions Anobjectiononemightraiseisthequestionoftheexistenceof G 0 ,whichisan inverse.Becausethestationary-stateenergies U areknown,thestationary-stateis determinedforalltime,andthetime-derivativeactsuponthestatetakentotime-evolve to t from t 0 forwhich t 0 7 t as, i@ t 0 t = i@ t e )]TJ/F23 7.9701 Tf 6.587 0 Td [(iH 0 t 0 t 0 = i@ t 1 X n =0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(it n n H 0 n 0 t 0 = i@ t 1 X n =0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(it n n U n 0 t 0 = i@ t e )]TJ/F23 7.9701 Tf 6.587 0 Td [(iUt 0 t 0 = U 0 t Physicallythisisatautologicalstatement 36 ,butmathematicallyitrequiresreference totheindicatedTaylor-seriesrepresentationofexp )]TJ/F22 11.9552 Tf 9.298 0 Td [(iH 0 t .Onethenassertsthatthe eectoftheinhomogeneity V istogivethestationarystateanitebutlargelifetime 1 = adecayrate .Thus,onFourier-transformingtothe -domain,oneusesthe kernelexp )]TJ/F22 11.9552 Tf 9.299 0 Td [(i i t andthetime-derivativebecomes + i .For nonzero,butthe smallestenergy-scaleintheproblem,theinverse G 0 existsandisequaltoasadvertised, butwiththenewparametrization i@ t + i .Whentheinteraction V isintroduced andsubsequentlytransformedtothe + i -domain,itseectistomaketheenergy U stationary-system H 0 = i@ t )]TJ/F15 11.9552 Tf 12.006 0 Td [( G 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 haveanon-inntesimaldecay-rateIm V intowhich i isabsorbed. 2.3.5.4ThedressingofaGreenfunctionbyaperturbation Summarizingtheseresults,theansatz j i = j 0 i + GV j 0 i = j 0 i + G 0 V j i inthe Schrodingerequation H j i = i@ t j i givestheDysonequationfor G ,whichcanbeused tointroduce V toallorders, G = G 0 1 X n =0 VG 0 n = G 0 + G 0 VG = i@ t )]TJ/F22 11.9552 Tf 11.955 0 Td [(H )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 36 Onerealizes,specically,thatof course astatethatisstationary"isdeterminedfor alltime. 99

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G 0 = i@ t )]TJ/F22 11.9552 Tf 11.955 0 Td [(H 0 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ; G 0 = + i )]TJ/F22 11.9552 Tf 11.955 0 Td [(H 0 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ;{79 2.3.6LangrethRules InSec.2.3.5,itwasobservedthatfortheinverse G 0 toexistinthedomain where time t )]TJ/F22 11.9552 Tf 12.694 0 Td [(t 0 issummedoutbyFouriertransform,theparameter wasrequiredtobe extendedtothecomplexplane i ,where wasinterpretableasadecay-rateofan otherwise-stationarystateandthusisthesmallestenergy-scaleintheproblem.The ideaoftheLangrethrulesistoextendthetime-variablefromtheone-dimensionalinterval t;t 0 2 [ ; 1 ]toadeformablecontourinthecomplex-planethathasarealpart extendingoverthesame t;t 0 2 [ ; 1 ],butanimaginarypartthatcharacterizesthe Greenfunctions2{52,2{53,andothers. 2.3.6.1ThecontourideaforstreamlininganalyticcontinuationofGreen functions Intheperturbationtheorytobeused,onewishestostreamlinetheprocessofanalytic continuationintheconvolutionsandproductsthatself-energiesofaninteractione.g., electron/phononinteractionshallbe.Therefore,consideracontour-domainparticle amplitude iG asfollows, iG t;t 0 D T C [ a t a y t 0 ] E = g > t;t 0 c t )]TJ/F23 7.9701 Tf 6.587 0 Td [(t 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g < t;t 0 c t 0 )]TJ/F23 7.9701 Tf 6.587 0 Td [(t = D a t a y t 0 E c t )]TJ/F23 7.9701 Tf 6.587 0 Td [(t 0 )]TJ/F27 11.9552 Tf 11.955 13.271 Td [(D a y t 0 a t E c t 0 )]TJ/F23 7.9701 Tf 6.586 0 Td [(t Above,theHeavisidestep-functionhasbeengeneralizedtohavinganon-contour argument.Thepropertiesoftheo-contourhorizontal-domainstepfunctionsare familiar,andtheyaresummarizedasfollows, t )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 = + ; t 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(t = )]TJ/F15 11.9552 Tf 7.084 1.794 Td [(; = ; =0;{80 100

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2.3.6.2TheLangrethorderingsofaconvolution Bydirectcomputationfrom iG andusingthepropertiesoflinearityoftheconvolution operationandthebreakingupofanintegrationinterval,theless/greaterLangreth rules[33]foraseries"/convolution D t;t 0 = A B t;t 0 ,using2{80,hasthefollowing 7 Langrethordering, d 7 = d 7 t;t 0 = A t; t B t;t 0 7 = A t; b 7 ;t 0 + a 7 t; B ;t 0 = I C 7 t 0 d A t; b 7 ;t 0 + I C xt d a 7 t; B ;t 0 = t Z d )]TJ/F22 11.9552 Tf 9.298 0 Td [(ia > t; b 7 ;t 0 + t Z d )]TJ/F22 11.9552 Tf 9.299 0 Td [(ia < t; b 7 ;t 0 + t 0 Z d a 7 t; + ib > ;t 0 + t 0 Z d a 7 t; + ib < ;t 0 = t Z d )]TJ/F19 11.9552 Tf 5.479 -9.683 Td [()]TJ/F22 11.9552 Tf 9.299 0 Td [(i t )]TJ/F23 7.9701 Tf 6.586 0 Td [( a > t; + a < t; b 7 ;t 0 + t 0 Z d )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(+ i t 0 )]TJ/F23 7.9701 Tf 6.587 0 Td [( a 7 t; b > ;t 0 + b < ;t 0 d 7 = 1 Z )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(a r t; b 7 ;t 0 + a x t; b a ;t 0 d = a r t; b 7 ;t 0 + a 7 t; b a ;t 0 = d 7 t;t 0 Theretardedseries"/convolution D t;t 0 = A t; t B t;t 0 followsfromusingthe obtainedrule AB 7 = a r b 7 + a 7 b a uponthedenitionoftheon-contouramplitude iG Thisproceedsas, id r t;t 0 = d > t;t 0 + d < t;t 0 + = a r t; t b > t;t 0 + a > t; t b a t;t 0 + a r t; t b < t;t 0 + a < t; t b a t;t 0 + = a r t; t b > t;t 0 + b < t;t 0 + a > t; t + a < t; t b a t;t 0 + = )]TJ/F22 11.9552 Tf 9.299 0 Td [(i a > t; t + a < t; t + b > t;t 0 + b < t;t 0 + a > t; t + a < t; t + i b > t;t 0 + b < t;t 0 )]TJ/F27 11.9552 Tf 7.084 15.063 Td [( + id r t;t 0 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(i a > t; t + a < t; t + b > t;t 0 + b < t;t 0 + +0= )]TJ/F22 11.9552 Tf 9.298 0 Td [(i + ia r t; t + ib r t;t 0 = )-222()]TJ/F22 11.9552 Tf 23.91 0 Td [(ia r t; t b r t;t 0 = ia r t; t b r t;t 0 Thenextstepistocheckiftheaboveproperty d r = A B r = a r b r oftheadvanced orderingofaconvolutionsatisestheproperty d r y = d a )]TJ/F22 11.9552 Tf 9.298 0 Td [(id a t;t 0 = d > t;t 0 + d < t;t 0 )]TJ/F15 11.9552 Tf 10.406 1.793 Td [(= )]TJ/F22 11.9552 Tf 9.298 0 Td [(i a > t; t + a < t; t + b > t;t 0 + b < t;t 0 + a > t; t + a < t; t + i b > t;t 0 + b < t;t 0 )]TJ/F27 11.9552 Tf 7.085 15.063 Td [( )]TJ/F15 11.9552 Tf -439.235 -26.698 Td [(=0+ i a > t; t + a < t; t )]TJ/F15 11.9552 Tf 7.085 1.794 Td [( b > t;t 0 + b < t;t 0 )]TJ/F15 11.9552 Tf 10.406 1.794 Td [(=+ i )]TJ/F22 11.9552 Tf 9.299 0 Td [(ia a t; t )]TJ/F22 11.9552 Tf 9.299 0 Td [(ib a t;t 0 = )-222()-222()]TJ/F22 11.9552 Tf 33.209 0 Td [(ia a t; t b a t;t 0 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(ia a t; t b a t;t 0 101

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Thus,theLangrethrules[33]governingthe 7 andretarded/advancedorderingofa convolution D t;t 0 = A B t;t 0 areasfollows, d 7 t;t 0 = A t; t B t;t 0 7 = a r t; b 7 ;t 0 + a 7 t; b a ;t 0 ; d r t;t 0 = A t; t B t;t 0 r = a r t; t b r t;t 0 =[ d a t;t 0 ] y ;{81 2.3.6.3TheLangrethorderingsofamultiplication BecausetheconvolutiontheoremstatesthattheFouriertransformofaproduct/convolution isaconvolution/product,thenextnaturalstepistoobtainthesamerulesfora multiplication D t;t 0 = A t;t 0 B t;t 0 .The 7 parallel"multiplicativeLangreth rulesare, F t;t 0 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(if > t;t 0 c + + if < t;t 0 c )]TJ/F15 11.9552 Tf 10.405 2.955 Td [(= )]TJ/F22 11.9552 Tf 9.298 0 Td [(iA t;t 0 B t 0 ;t = )]TJ/F22 11.9552 Tf 9.299 0 Td [(i )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [()]TJ/F22 11.9552 Tf 9.299 0 Td [(ia > t;t 0 c + + ia < t;t 0 c )]TJ/F27 11.9552 Tf 7.085 12.64 Td [()]TJ/F19 11.9552 Tf 12.951 -9.684 Td [()]TJ/F22 11.9552 Tf 9.299 0 Td [(ib > t 0 ;t c )]TJ/F15 11.9552 Tf 9.741 2.956 Td [(+ ib < t 0 ;t c + = i ia > t;t 0 ib < t 0 ;t c + + i ia < t;t 0 ib > t 0 ;t c )]TJ/F15 11.9552 Tf 10.405 2.955 Td [(= )]TJ/F22 11.9552 Tf 9.298 0 Td [(i a > t;t 0 b < t 0 ;t c + + i )]TJ/F22 11.9552 Tf 9.299 0 Td [(a < t;t 0 b > t 0 ;t c )]TJ/F19 11.9552 Tf 10.406 2.955 Td [(! f 7 t;t 0 = a 7 t;t 0 b ? t 0 ;t Theretardedparallel"/multiplicative D t;t 0 = A t;t 0 B t;t 0 followsfromusingthe obtainedrule f 7 t;t 0 = a 7 t;t 0 b ? t 0 ;t uponthedenitionoftheon-contouramplitude iG ,inexact analogywithwhatwasdonefortheconvolution.Thisproceedsas, f r t;t 0 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(i t )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 f > t;t 0 + f < t;t 0 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(i t )]TJ/F23 7.9701 Tf 6.587 0 Td [(t 0 )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [( a > t;t 0 + a < t;t 0 b 7 t 0 ;t )]TJ/F22 11.9552 Tf 11.955 0 Td [(a 7 t;t 0 b > t 0 ;t + b < t 0 ;t = a ? t;t 0 b a t 0 ;t + a r t 0 ;t b ? t 0 ;t )]TJ/F22 11.9552 Tf 9.299 0 Td [(iA t;t 0 B t 0 ;t r Thus,theLangrethrulesgoverningthe 7 andretarded/advancedorderingofa product D t;t 0 = A t;t 0 B t;t 0 areasfollows, F 7 t;t 0 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(iA t;t 0 B t 0 ;t 7 = a 7 t;t 0 b ? t 0 ;t F r;a t;t 0 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(iA t;t 0 B t 0 ;t r;a = a r;a t 0 ;t b 7 t 0 ;t + a 7 t;t 0 b a;r t 0 ;t {82 Thefollowingimmediatecorollaryisnoted:theless,greater,retarded,andadvanced orderingsmayberemovedfromaGreenfunction. 102

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2.3.7PropertiesoftheDysonEquationUndertheLangrethRules InwhichtheDysonequationispresented,andsomeofitspropertiesunderthe Langrethrulesareobtained. 2.3.7.1Theretardedandthermal 7 Dysonequationsandadditivityofthe self-energy Onebeginswiththe 7 equationinmatrixform,andeectsahandyrearrangement whichisdonewithoutassumingcommutationafterapplyingtheLangrethrule2{81 andsolvingfor g 7 g 7 = g 7 0 + g r 0 r g 7 + g r 0 k g a + g 7 0 a g a $ ^ 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r 0 r g 7 = g 7 0 ^ 1+ a g a + g r 0 k g a Onethenusestheretarded 37 andadvancedDysonequations,whichrespectivelyare rearrangeableas, g r = g r 0 + g r 0 r g r $ ^ 1 )]TJ/F22 11.9552 Tf 10.927 0 Td [(g r 0 r = g r 0 [ g r ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ; g a = g a 0 + g a 0 a g a $ ^ 1+ a g a =[ g a 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g a ; Combiningtheseresults,andsolvingforthekineticGreenfunction, g r 0 [ g r ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g 7 = g 7 0 [ g a 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g a + g r 0 k g a $ g 7 = g r [ g r 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g 7 0 [ g a 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g a + g r k g a Theretarded,advanced,andkineticDysonequationsallfeatureadditivityof self-energies.IllustratingthisintheretardedDysonequationisstraightforward, g r = [ g r 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( r )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = [ g r 00 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 11.9552 Tf 11.956 0 Td [( r 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( r )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = [ g r 000 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( r 00 )]TJ/F22 11.9552 Tf 11.956 0 Td [( r 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [( r )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 37 Theretardedequationisconventionally g r = g r 0 + g r 0 r g r ,buttheDyson renormalization-seriescanbesummedintheotherdirection"toobtain g r = g r 0 + g r r g r 0 Proof :bydirectlysolvingfor g r ,bothhavethesolution g r =[ g r 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 11.9552 Tf 11.956 0 Td [( r )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 103

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Illustratingthisforthekineticandtherefore 7 Dysonequationappearscumbersome, butisstillstraightforward, g 7 = g r 7 +[ g r 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(g 7 0 [ g a 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g a = g r 7 +[ g r 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g r 0 7 0 +[ g r 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g 7 00 [ g a 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g a 0 [ g a 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g a = g r 7 + 7 0 +[ g r 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g 7 00 [ g a 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g a g 7 = g r 7 + 7 0 + 7 00 +[ g r 00 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g 7 000 [ g a 00 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g a Thus,ifdressingGreenfunction g r 00 ;g 7 00 withself-energy r 0 ; 7 0 bytheDyson equationtoconstituteGreenfunction g r 0 ;g 7 0 ,andthenagainwithself-energy r ; 7 to constitute g r ;g 7 ,onecanjustusetheDysonequationwiththeself-energies r 0 + r and 7 0 + 7 onceintheDysonequationas, g r = )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [([ g r 00 ] )]TJ/F15 11.9552 Tf 11.956 0 Td [( r + r 0 + r 00 + ::: )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 {83 g 7 = g r 7 + 7 0 + ::: +[ g r 000 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g 7 000 [ g a 000 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g a g r 7 + 7 0 + 7 00 + ::: g a {84 Thisisconvenientdueto 7 beingabletoplaytheroleoftheself-energyofboththe leadsandthemany-body-perturbation,with[ g r 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g 7 0 [ g a 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 beingatermdueentirelyto theisolatedcentralsite. 2.3.7.2Independenceofthe 7 electronicGreenfunctionsfromacenter-site temperatureforan N C -sitecenter-region Consideracollectionof N C isolatedlattice-siteseachbearinganinnite-lifetime state.ThesesitesaredescribedbyGreenfunction g 000 .Lettheselatticesitesthenbe coupledbyaself-energyof 00 .Next,letthecenterregionthiswouldconstitutebe joinedtoanynumberofreservoirsbyaself-energy 0 ,givingtheGreenfunction g 0 .It iswell-knownthatfor N C =1forwhichonewouldobviouslyhave 00 =0,reference 104

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toacentersitetemperature T C appearsin g 7 000 butnotin g 7 0 becausethequantity 38 7 000 [ g r 000 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g 7 000 [ g a 000 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 isdirectlyproportionaltotheinntesimal0 + .Thisresultis nowtobegeneralizedto N C > 1sitesadjoinedbyanynumberofreservoirs,forwhichall Greenfunctionsare N C N C matrix-functionswithadomainof .Onebeginsbynoting that 7 000 isdirectlyproportionaltotheinntesimal0 + for N C > 1.Thisisbecause, [ g r 000 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 =diag 1 i N C z )]TJ/F22 11.9552 Tf 11.956 0 Td [(U i _= 2 6 6 6 6 6 6 6 4 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 0 0 0 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 2 . . . . . . . 0 0 0 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U N C 3 7 7 7 7 7 7 7 5 =[ g a 000 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 y ; z = + i 0 + ; Carryingoutthematrix-multiplicationbetweendiagonalmatricesimmediatelyyields, 7 000 =diag 1 i N C z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i diag 1 i N C 2 f 7 C 0 + z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i z )]TJ/F22 11.9552 Tf 11.956 0 Td [(U i diag 1 i N C z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i =2 ^ 1 f 7 C 0 + 0 Thenexttaskisobviouslytoshowthatthequantity 39 g r 0 7 00 g a 0 issimilarly independentof T C ,where g r 0 isaGreenfunctionassumedtohaveanon-inntesimal imaginarypart 40 initsdenominator.Inthe 7 Dysonequations,thisevaluatesto, g r 0 7 00 g a 0 = g r 0 ^ 1 7 00 ^ 1 g a 0 = g r 0 h [ g r 000 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g r 000 det g r 000 det g r 000 i 7 00 h g a 000 [ g a 000 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 det g a 000 det g a 000 i g a 0 = det g r 000 g r 0 [ g r 000 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 det g r 000 g r 000 7 00 g a 000 1 det g a 000 det g a 000 [ g a 000 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g a 0 38 Thisquantity 7 000 isnotaself-energy,buthasdimensionsofoneandfurthermore entersadditivelytothe 7 Dysonequations. 39 Inreferringtothequantity g r 0 ,theadditionoftheself-energy 0 oftheleadshasbeen presupposed. 40 Thus,thelimit 0 0isoneforwhich 0 isassumedtobemuchlessthanthe inntesimal0 + thrownawayin,e.g., 7 000 .Thisgivesalower-limitupon 0 whichmay placearestrictionuponamany-bodyselfenergy beyondthatrequiredforconvergence ofthegeometricDysonseries.Forinstance,Hyldgaardet.al.[23]introduce= g = )]TJ -460.686 -14.445 Td [(foranelectron-phononinteractionwithvertex M = p g introducedtoacenterregion withcouplingIm r 0 =)-326(toaleftandarightlead. 105

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Forconvenience,require~ g r 0 ~ 7 00 ~ g a 0 andthusintroducenewtilde"Greenfunction withnewdimensionsi.e., )]TJ/F23 7.9701 Tf 6.587 0 Td [(N C ,whichmayhavelatent 41 Diracdeltafunctions", ~ g r 0 det g r 000 g r 0 [ g r 000 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 $ ~ g a 0 det g a 000 [ g a 000 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g a 0 =[~ g r 0 ] y Now,itmustbeshownthatthequantity~ 7 00 1 det g r 000 g r 000 7 00 g a 000 1 det g a 000 havingunitsof 2 N C )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 isindependentof T C .Thisproceedsbyusingthefactthatthe determinantofadiagonalmatrixisjusttheproductofthediagonalelements.Oncethis determinantisintroduced,theobject 7 00 becomesadjoinedbymatricesofproductswith the i th poledueto z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i fromdet g r 000 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = N C i =1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i missing, ~ 7 00 = g r 000 7 00 g a 000 det g r 000 det g a 000 = N C Y i =1 z )]TJ/F22 11.9552 Tf 11.956 0 Td [(U i diag 1 i N C 1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i r 00 g 7 000 a 00 diag 1 i N C 1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i N C Y i =1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i Althoughthepolesdueto z )]TJ/F22 11.9552 Tf 13.004 0 Td [(U i and z )]TJ/F22 11.9552 Tf 13.004 0 Td [(U i wereeliminatedbytherespective determinants,anotherpairofpolesresultsonintroducingtheexplicitformof g 7 000 ~ 7 00 = diag 1 i N C Y i 0 6 = i z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i 0 r 00 diag 1 i N C 2 + f 7 C z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i a 00 diag 1 i N C Y i 0 6 = i z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U i 0 Itisseenthatonecaneasilywriteanexpressionforthe i;j th elementsofthematrix below,insteadoftheentirematrix.Oneistoswitchtosuchanindex-representation, andmanipulatesoastoentirelyeliminatetheinntesimal i 0 + bycancellingitfromall denominators.Introducetheself-energy r 00 mn = )]TJ/F22 11.9552 Tf 9.298 0 Td [(W mn mn inwhichthecomplementary" deltafunction ij =1 )]TJ/F22 11.9552 Tf 9.896 0 Td [( ij isintroducedtorequire m 6 = n .Ineectingthisswitchtoindex notation,oneisabletocarryoutthemultiplicationofthediagonalmatricesbymoving their i;j th elementthrough r a 00 mn ,causingthelastpoletodisappear.Thiseliminatesany 41 Thatis,Diracdeltafunctionsthatwouldresultifthealgebraicformof~ g r 0 were manipulatedsoastofavoroccurrencesof 1 0 + = x 2 + + 2 x 106

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latent"Diracdeltafunctions 42 ,yieldingthefollowinggeneralizationassuming, ~ 7 00 !; = a N C Y p =1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U p ab z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U b )]TJ/F22 11.9552 Tf 9.299 0 Td [(W bc bc cd 2 + f 7 C z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U c z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U c )]TJ/F22 11.9552 Tf 9.299 0 Td [(W de de ef z )]TJ/F22 11.9552 Tf 11.956 0 Td [(U f N C Y p =1 z )]TJ/F22 11.9552 Tf 11.956 0 Td [(U p f =2 + f 7 C N C Y p =1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U p N C X c =1 W c c W c c z )]TJ/F22 11.9552 Tf 11.956 0 Td [(U z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U c z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U c z )]TJ/F22 11.9552 Tf 11.956 0 Td [(U N C Y p =1 z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U p ~ 7 00 !; =2 + f 7 C Y p 6 = ;p 6 = c z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U p X c 6 = ;c 6 = W c c W c c Y p 6 = ;p 6 = c z )]TJ/F22 11.9552 Tf 11.955 0 Td [(U p 0 Hence,noreferencetothecenter-sitetemperatureintheformofcenter-regionFermi functionsisrequired,nomatterhowmanysitesacenterregionmaybemadeof.Thus, theless/greaterDysonequationstakeonthefamiliarformthatavoidsreferencetoa center-regiontemperaturespecicallyinresponsetotheadditionofcoupling 0 tooneor morereservoirs, g 7 0 = g r 0 7 00 + 7 0 g a 0 = g r 0 + 7 0 g a 0 = g r 0 7 0 g a 0 {85 ThepricetopayinusingthisformoftheDysonequationappearstobethe occurrenceoftheinverse"termshown,forinstance,intheadditivity-property2{84 whenanotherself-energy,say 7 ,isintroducede.g.,oneduetointeractions.However, thistooisavoided,andtheless/greaterDysonequationscontinuetoretaintheform 2{84thatavoidsreferencetoacenter-region-temperature.Directlycalculating g 7 and introducingthedenitionof g 7 0 from2{85tohandletheterm[ g r 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g 7 0 [ g a 0 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 ,one obtainsaresultthatisinkeepingwiththeDysonequationfeaturingadditivityofany numberofself-energies, g 7 = g r 7 +[ g r 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g r 0 7 0 g a 0 [ g a 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g a = g r 7 + 7 0 g a {86 42 Moreprecisely:whatremainsissomethingproportionalto0 + ,withnodenominators bearinginntesimalanalyticcontinuationsthatcouldpossiblyyieldDiracdeltafunctions. 107

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Avoidanceofacenter-regiontemperatureisanenormouslyfortunatesimplication foronewishingtostudymorecomplicatedcentralregions.Considercenter-regionkinetic Greenfunctions g 7 ii and g 7 jj .Thetwofunctionsoughttohavethesamethermalfunction withthesametemperatureforacommoncenter-regiontemperature.However,ifthe centerregioncanbemadearbitrarily-large,andifthecenterregionisdrivenoutof equilibriumbytwoormorereservoirsatwell-denedtemperaturebyvirtueofthe reservoirsbeinganinnitechain,area,orvolumeofhomogeneousoverlap-energies )]TJ/F22 11.9552 Tf 9.299 0 Td [(W andoccupancy-energies U ,alatticesitespatiallyclosertoonereservoirwith g 7 ii ought tohaveacorrespondinglyclosetemperaturetothenearbyreservoir.Anothersitewith g 7 jj wouldhaveyetanothertemperature.Ingeneral,onewouldhaveatemperature-prole ofthecenterregion.Tocalculatethetemperatureprole,onemustsolvetheequation wherezeroheatcurrentisowinginandoutofthesiteintoathirdreservoir[42]actingas athermometer,andthisiscomputationallyexpensive. Theresultofthissectionadmitsaphysically-intuitiveinterpretation.Consider theoccupancyofthecenterregionbyaparticle,phononmode,magneticspin,orin generalaquantumstationarystate.Thelifetimeofthisoccupancyisinniteifthere iszerocoupling 0 tothe th lead.Thisrequiresthestationarystateequilibratenot withthe th leadbutwiththecenterregion.Bythezerothlawofthermodynamics,this requiresthestatetohavethecenterregion'stemperature.Foranonzerocoupling 0 tothe th leadwhichistheonsetoftransportifanygradientsareapplied,asseenin theexpressionsforcurrentobtainedinSection2.4,thestationarystateisnolonger stationary,andthustheinntesimaldecay-ratetheinnite-lifetimestationarystatewas assignedisdwarfedbytheimaginarypartof 0 2.3.7.3TheLangreth-orderingsoftheDysonequationinaperturbativeseries form Forthecomputer-implementedself-consistencycalculationwhichobtainsthe self-energies,theDysonequationisbestwrittenas2{84.Foranalyticworktostudy 108

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thefunctionalformoftheGreenfunction,theself-energiesthatdressit,andthe conservation-lawstheircorrespondingobservablese.g.,numberandenergycurrents mayobey,itisbettertowritetheDysonequationasseriesofascendingperturbative orders.ThisiseasilydonefortheretardedDysonequationof2{84, g r = [ g r 0 ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( r )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = g r 0 X 1 n =0 g r 0 r n = g r 0 1+ r g r 0 + r g r 0 2 + ::: {87 The 7 Greenfunctionof2{84requiresmorecare.Duringthefollowingalgebraic steps,onecanrecalltheLangrethrule2{81,whichprescribestheformthatoughtto occur.Using2Re z z + z y bothtocompactifyandtocheckRe g 7 =0with2{52, g 7 = g r 7 0 + 7 g a g r 7 g a = g r 0 + r g r 0 + r g r 0 2 + ::: 7 + g a 0 a + g a 0 a 2 + ::: g a 0 = g r 0 0 B @ 2Re r g r 0 k +2Re r g r 0 2 7 +2Re r g r 0 2 k g a 0 a + k + r g r 0 k g a 0 a + r g r 0 2 k g a 0 a 2 + O 5 g + ::: 1 C A g a 0 Onecanbeginfromtheabovetoobtainhigherorders.Below,anoccurrenceof g 7 0 = g r 0 7 0 g a 0 istobeworkedinbyintroducing 7 = 7 0 + 7 ,yieldingthefollowing whichislowest-orderin 7 r ,and a g 7 = g r 0 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( 7 +2Re r g r 0 7 + ::: g a 0 = g 7 0 + g r 0 )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [( 7 +2Re r g r 0 7 + ::: g a 0 {88 2.3.8RewritingLead/Center-RegionGreenFunctionsinTermsofCenter Regionand/orSurfaceGreenFunctions 2.3.8.1Greenfunctionindicesbelongingtothecenterregionandtothe adjoiningreservoirs ConsidertheGreenfunction g dressedwithaninteractionlocalizedtoaregion withRomanindices i;j andattachedatsite k byoverlap-energy 1 k toreservoir at reservoir-site 1 .ReservoirshaveGreek-letterswithRomansubscripts,i.e., 5 wouldbe the th reservoir's5 th siteawayfromitssite 1 coupledtothecenterregion's j th siteby 1 j .TheGreenfunction g 0 describesparticle-amplitude-overlapsinthesamesystemas 109

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g exceptwithoutthelocalizedinteraction.Finally,theGreenfunction g 00 isthesame as g 0 exceptwithoutcouplingtoanyreservoirs.Finally, g 000 isthesameas g 00 except withoutanyoftheregion'sinterconnectionsinthetightbindingmodel,thiswouldbean overlap-energy )]TJ/F22 11.9552 Tf 9.298 0 Td [(W ij inthecenter-regionand )]TJ/F22 11.9552 Tf 9.298 0 Td [(W inanyoftheleads.LetGreekindices without subscripts,like ,varyover both reservoir-indiceslike 5 andcenter-region indiceslike j =2,becauseindicesaretobesummedoverallnonvanishingself-energies. 2.3.8.2De-mixinglead/reservoirGreenfunctionswithoutLangrethordering Usingtheadditivity-propertyofself-energyintheDysonequation,onedressesthe isolated-regionGreenfunction g 00 withtheself-energy= 0 + whichisthetotal self-energyduetointer-regioncoupling 0 andduetoaninteraction localizedtothe centerregion.TheinternalsumoftheDysonequationisoverbothlead-indicesand center-regionindices,soaGreekletter isused.Thus,thecenter/leadfullGreenfunction g j ` isrstexpressedintermsofnon-mixed-indexGreenfunctionsas, g i ` = g 00 i ` + g i 0 g 00 0 ` =0+ g i 0 k j + 0 j k + jj 0 g 00 0 ` = g i k 0 k j 0+ g ij 0 j k g 0 k ` k 1 + g ji jj 0 0= X j g ij 0 j 1 g 0 1 ` Thetransposeoftheaboveresultrequiresthatonerepeatthederivationexceptwith g expressedas g 0 + g 0 g insteadof g 0 + gg 0 asdoneabove, g ` i = g 00 ` i + g 00 ` 0 g 0 i =0+ g 00 ` 0 k j + 0 j k + jj 0 g 0 i = X j g 0 ` 1 0 1 j g ji Althoughanexplicitsumisindicatedovercenter-regionindex j ,thissumshallbe omittedforcompactness.Next,themixed-indexGreenfunction g 1 ` istobesimilarly re-expressed.Lettingdoubly-occurringindicesimplysummation,theGreenfunction g 1 ` forwhich ` =1 ; 2 ;::: is, g 1 ` = g 0 1 ` + g 00 1 0 g 0 ` = g 0 1 ` + g 0 1 1 0 1 i +0 0 i 1 +0 ij g i ` = g 0 1 ` + g 0 1 1 0 1 i g ij 0 j 1 g 00 1 ` 110

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2.3.8.3TheorderoftheGreenfunctionpoleforball-and-springGreen functions Itisimportanttocountthenumberofisolated-leadGreenfunctions g i j thathave occurreduptothisstep.Thisisbecausethephononic g 1 1 issingularas 0,and mustoccurwithapower n 2 tobenite.So,thecenter-leadhybridGreenfunction g j 1 = g ji 0 i 1 g 00 1 1 hasonesurfaceGreenfunctionrequiring n 2 ,whilethefullsurface Green 43 function g 1 1 = g 00 1 1 + g 00 1 1 0 1 i g ij 0 j 1 g 00 1 1 hastworequiring n 4 2.3.8.4ApplicationofLangrethorderingtotheGreenfunctions Thenexttaskistoapplythe 7 Langrethseries"rule2{81overindex-contractions i.e.,convolutionstotheaboveresults.Beginningbycalculating g 7 j ` and g 7 ` j asfollows, g 7 i ` = g ij 0 j 1 g 0 1 ` 7 = g r ij r 0 j 1 g 7 0 1 ` + g 7 ij a 0 j 1 g a 00 1 ` + g r ij 7 0 j 1 g a 0 1 ` = g r ij r 0 j g 7 0 1 ` + g 7 ij a 0 j g a 0 1 ` g 7 ` i = g 0 ` 1 0 1 j g ji 7 = g r 0 ` 1 r 0 1 j g 7 ji + g 7 0 ` 1 a 0 1 j g a ji + g r 0 ` 1 7 0 1 j g a ji = g r 0 ` 1 r 0 1 j g 7 ji + g 7 0 ` 1 a 0 1 j g a ji Thus,the 7 tunnelingGreenfunctionsforeitherbosonsorfermionsforcoupling betweenthe th leadandacenterregionare, g 7 i ` = g r ij r 0 j g 7 0 1 ` + g 7 ij a 0 j g a 0 1 ` ; g 7 ` i = g r 0 ` 1 r 0 1 j g 7 ji + g 7 0 ` 1 a 0 1 j g a ji {89 Itismorelabor-intensivebutasstraightforwardasthepreviouscalculationto calculate g 7 1 ` .Forgreatersimplicity,calculate g 7 1 ` ,becauseitturnsoutthat g 7 1 ` is 43 Notethatthe ij 0limitof g j 1 is g 0 j 1 = g 0 ji 0 i 1 g 00 1 1 .Meanwhile,the ij 0 limitof g 1 1 is g 0 1 1 = g 00 1 1 + g 00 1 1 0 1 i g 0 ij 0 j 1 g 00 1 1 with g 0 ij behavingasasurfaceGreen functionfromthepointofviewof g 0 1 1 ,andthus 0 1 i g 0 ij 0 j 1 istheself-energy. 111

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nonzeroonlyif = theGreenfunctionisbetweentwoleads, g 7 1 ` =+ g 00 1 1 0 1 i g ij 0 j 1 7 g a 00 1 ` ++ g 00 1 1 0 1 i g ij 0 j 1 r g 7 00 1 ` =+ g 7 00 1 1 a 0 1 i g a ij a 0 j 1 +0+ g r 00 1 1 r 0 1 i g 7 ij a 0 j 1 +0 g a 00 1 ` ++ g r 00 1 1 r 0 1 i g r ij r 0 j 1 g 7 00 1 ` g 7 1 ` =+ W i W j g r 00 1 1 g 7 ij g a 00 1 ` ++ W i W j g 7 00 1 1 g 7 00 1 ` g a ij g a 00 1 ` + W i W j g r 00 1 1 g r ij g 7 00 1 ` Itisnotedthattheabovesimplieswhenusedtoevaluate g 7 1 1 ,sincethe2 nd parenthesizedtermisoftheform z + z y 2Re z forcomplex z ,holdingeveninthe presenceofavectorpotential, g 7 1 1 = g a 00 1 1 + g 7 00 1 1 + W i W j g r 00 1 1 2 g 7 ij +2 g 7 00 1 1 Re[ W i W j g r 00 1 1 g r ij ] Furthermore,termsweregatheredunder g 7 00 1 ` 2 f 7 Im g a 00 1 ` ,preventingsymbolic clutteronuseofthefermionicuctuation-dissipationtheorem. Lastly,itisconvenientandmucheasiertocalculate g r 1 ` ,the th -leadsite-1/site ` overlapdressedwithoverlap-energy 0 toacenter-regionandtheenergy r ofan interactionlocalizedtosaidcenter-region.Thisiseectedasfollows, g r 1 ` = g 00 1 1 + 0 1 i g ij 0 j 1 g 00 1 1 r = g r 00 1 1 + W i W j g r ij g r 00 1 1 {90 Itshouldbementionedthat,despitethenotation,noneoftheaboveGreenfunctions wereassumedtobeexclusivelybosonicorfermionic. 2.3.8.5Mixed-indexelectronicGreenfunctions Here,onespecializesthegeneralGreenfunctionsobtainedinSec.2.3.8.4toelectrons. TheresultsoftheaboveseriesofcalculationsaretheGreenfunctions g 7 j ` and g 7 1 ` in termsof g ij i.e.,theinteracting g withonlycenter-siteindicesor g 00 1 ` i.e.,theisolated g 00 withonlyreservoirindices,atleastoneofwhichisthereservoir-sitecoupledtothe center-region.TheGreenfunctions g r 0 ` 1 and g 7 0 ` 1 areoftheunconnected th leadat 112

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equilibrium,sotheuctuation-dissipationtheorem2{60maybeusedtorelatethese, g 7 ` i = g r 0 ` 1 r 0 1 j g 7 ji + A e )]TJ/F23 7.9701 Tf -3.928 -7.892 Td [(! f 7 a 0 1 j g a ji ; g r 0 ` 1 = g r 0 ` 1 eq ; g 7 0 ` 1 = g 7 0 ` 1 eq ;{91 Itshallbeconvenienttohavecalculatedthequantity N 7 i W i g 7 i ` )]TJ/F22 11.9552 Tf 11.124 0 Td [(W i g 7 ` i asthe imaginary-partofthefollowingdierence, N 7 i = W i g 7 ij a 0 j 1 g a 0 1 ` + g r ij r 0 j 1 g 7 0 1 ` )]TJ/F22 11.9552 Tf 11.955 0 Td [(W i g 7 0 ` 1 a 0 1 j g a ji + g r 0 ` 1 r 0 1 j g 7 ji = W j g 7 ij W i g r 0 ` 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(W i g a 0 1 ` )]TJ/F22 11.9552 Tf 11.955 0 Td [(W j g 7 0 ` 1 W i g r ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(W i g a ji N 7 i =2 i Im )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(W j g 7 ij W i g r 0 ` 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(W j g 7 0 ` 1 W i g r ij =2 i )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(g 7 ij Im[ W j W i g r 0 ` 1 ] )]TJ/F22 11.9552 Tf 11.956 0 Td [(g 7 0 ` 1 Im[ W j W i g r ij ] Onethenusestheuctuation-dissipationtheorem2{60toobtainthesubtraction betweentwo < Greenfunctionsoccurringlaterinthecalculationofthenumbercurrent N ,clearlyyieldingaquantitythatisthesumofatermwith g 7 ij plusatermthatisan interactingspectralfunctiondueto2Im g r ij forreal-denitetunnelingmatrixelements timesan th leadthermalfunction f 7 N 7 i =2 i g 7 ij )]TJ/F23 7.9701 Tf 7.314 4.936 Td [(` 0 ji )]TJ/F15 11.9552 Tf 11.955 0 Td [(2)]TJ/F23 7.9701 Tf 13.168 4.936 Td [(` i Im g a 0 ` 1 ;)]TJ/F23 7.9701 Tf 23.752 4.936 Td [(` 0 ji Im W j W i g r 0 ` 1 ;)]TJ/F23 7.9701 Tf 19.2 4.936 Td [(` i Im W j W i g r ij ;{92 Somelimitsoccurringfor2{92are ` =1whichallowsuseof2{74tointroduce theescaperates,zeroappliedmagneticeld A = 0 ,andasingle-siteorhomogeneous centralregion N C =1, N 7 i ` =1 = 2 i g 7 ij )]TJ/F23 7.9701 Tf 7.314 4.936 Td [( 0 ji )]TJ/F15 11.9552 Tf 11.955 0 Td [(2)]TJ/F23 7.9701 Tf 13.167 4.936 Td [( i Im g a 0 1 1 A = 0 = 2 i )]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [( ij )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(g 7 ij )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 A e )]TJ/F23 7.9701 Tf -3.929 -8.012 Td [(ij f 7 N C =1 = 2 i )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [( )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 A e )]TJ/F22 11.9552 Tf 7.085 -4.936 Td [(f 7 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 7 Thisresult2{91isused,forinstance,incalculatingthequantum-statistical -domainexpectationofthecommutators[ N ;H ]and[ H + H ;H ].Itshouldbenoted thattheelectronGreenfunction g r 00 i j requiresevaluationofacontour-integralfollowing fromitseigenfunction-expansion.Theresultofdoingso[43]followsfromchoosingonesign 113

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ofthemomentum k from2{77;choosingthepositiveone, g r 0 j j 0 = 1 2 iW sin k a exp ik j j )]TJ/F22 11.9552 Tf 11.956 0 Td [(j 0 j +exp ik j + j 0 j = j 0 =1 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 W e ik a {93 ThisGreenfunction2{93satisesthesumrule2{63,whichiseasilyveried analyticallyifonepicksoutspecicvaluesof j and j 0 togetridoftheexponentiated absolutevalue j j )]TJ/F22 11.9552 Tf 11.955 0 Td [(j 0 j .Thecase j = j 0 =1isneededtoconnecttheGreenfunctions. Abosonicanalogueto2{93isrequiredforball-and-springphonons,butnotforthe Fano-Andersonmodeloftheopticalphononsconsideredinthiswork.Thisisseenlaterin Section2.3.9.2,wherethe 7 non-interactingphononGreenfunctionsforthephonons' m th opticalmodearecalculated. 2.3.8.6Analogousindexde-mixingforball-and-springphonons JustaselectronicGreenfunctionswithmixedspatialregionindicesarecritical fornon-equilibriumexpectations,transport,andinteractionsarecritical,sotoowould phononicGreenfunctionswithmixedmodeindicesbecriticalforthesameinphonons.A modelofphonon/phononinteractionsmightbemassesonspringswithposition-labels insteadofmodelablels.Theresult2{91isgeneralizabletoaregionoflocalized phonons,the m th modeofwhichhastunneling-matrix-element-likecoupling m tothe th reservoir.Theonlydierenceisanextra correspondingto 7 uponall equilibrium-distributionsduetothebosonicuctuation-dissipationtheorem d 7 00 2 b 7 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2Im d r 00 ,yieldingthefollowing, d 7 1 ` =+ m n d r 00 1 1 d 7 ij d a 00 1 ` + m n d 7 00 1 1 d 7 00 1 ` d a ij d a 00 1 ` + d r 00 1 1 d r ij b 7 2Im d a 00 1 ` {94 Inanalogytothefermionicelectronic2{91,thisresult2{94isrequiredfor calculatingthequantum-statistical -domainexpectationofthecommutator[ H ph ;H ]. LiketheelectronGreenfunctions, d r 00 i j requiresevaluationofacontour-integralresulting fromitseigenfunction-expansion. 114

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2.3.9GreenFunctionswithExplicitTemperatureandPotentialDependence HavingdevelopedtheLangrethrules,thequadraticGreenfunctionswithexplicit dependenceuponthethermodynamicparameterstemperatureandchemicalpotential ofthereservoirsmaybeobtained.Theisolated-center-regionGreenfunctionsofthe electronsareobtained,followedbytheseparatesurfaceGreenfunctionsthatareusedto calculatetheconnected"center-regionGreenfunctionswhich,by2{85,isindependent oftemperature.Thoseofthephonons,ontheotherhand,arenotseparatelyanisolated centerGreenfunctionandasurfaceGreenfunctionasin2{69. 2.3.9.1ElectronicthermalGreenfunctions The 7 thermalGreenfunctionsoftheelectronsarecalculatedfromtheDyson equation2{84withtheproperty2{85.Followingthese,theuctuation-dissipation theorem2{60andthedenition2{74oftheescape-rateareused.Followingthesesteps overthecourseofadirectcalculation, g 7 0 ij = g r 0 ii 0 7 0 i 0 j 0 g a 0 j 0 j = g r 0 ii 0 r 0 i 0 g 7 0 a 0 j 0 g a 0 j 0 j = g r 0 ii 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(W i 0 f 7 )]TJ/F15 11.9552 Tf 9.298 0 Td [(2Im g r 0 )]TJ/F22 11.9552 Tf 9.299 0 Td [(W i 0 r 0 j 0 g a 0 j 0 j OnintroducingtheimaginarypartofthesurfaceGreenfunction,andgeneralizingthe denitionoftheeectiveFermifunctionwhichisasumoverthelead-index g 7 0 ij = g r 0 ii 0 )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [( W i 0 W j 0 Resin k a=W f 7 g a 0 j 0 j g r 0 ii 0 )]TJ/F23 7.9701 Tf 17.72 4.936 Td [( i 0 j 0 f 7 g a 0 j 0 j g r 0 ii 0 )]TJ/F23 7.9701 Tf 17.719 -1.794 Td [(i 0 j 0 f 7 eff g a 0 j 0 j {95 115

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2.3.9.2PhononicthermalGreenfunctions The 7 phonon-densityGreenfunctionfollowsfrom2{85holdingforbothfermions andbosonsandsubsequentexpansionofthetermsintomatrix-multiplications, d 7 0 mn = d 7 0 m = X m 0 m 00 0 q q 0 d r 0 mm 0 r 0 m 0 q d 7 0 q q 0 r 0 q 0 m 00 d a 0 m 00 m = X q d r 0 mm mq b 7 q 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(! q q m d a 0 mm d 7 0 mn = X d r 0 m 2 b 7 = d r 0 m 2 b eff 7 Animportantpropertyofthisresulting d 7 mn isthatitisindependentofcenter-region temperature. Asdonethroughoutthiswork,lettherebenotransitionsbetweenthe m th and n th modes.Following2{56,onemayobtainthe 7 phononicmechanical-amplitudeGreen functionsfromtheabovephonon-densityGreenfunctionsinwhichoccur d r 0 givenby 2{69, d 7 = d r 2 b eff 7 )]TJ/F27 11.9552 Tf 11.955 10.162 Td [( d r )]TJ/F23 7.9701 Tf 6.587 0 Td [(! 2 b eff ? )]TJ/F23 7.9701 Tf 6.586 0 Td [(! = b eff 7 2 ++ )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F22 11.9552 Tf 27.737 8.087 Td [( b eff 7 2 + )]TJ/F22 11.9552 Tf 9.298 0 Td [(! )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 = 4 !b eff 7 ! )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 + 2 + 2 + 2 2.3.10Self-EnergiesoftheElectron/PhononInteractionandtheSelfConsistentBornApproximation 2.3.10.1Mathinterlude:theFouriertransformandconvolutiontheorems Here,theoremsgoverningtheinterplayofFourier-transformation,convolution,and multiplicationareobtained.OfconcernistheresultofapplyingaFouriertransform withkernel e i!t toaproduct f t g t of t -domainfunctions f = f t = f t and g = g t = g t ,aconvolution f g t = f g t = R dt 0 f t 0 g t )]TJ/F23 7.9701 Tf 6.587 0 Td [(t 0 ofthesame,andnallya reverse-convolution" f g t = f g t = R dt 0 f t 0 g t + t 0 ofthesame. 116

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First,convolutionandreverse-convolutionarecommutative,whichisprovedbythe followingdirectcalculations, Z 1 dt 0 f t 0 g t t 0 = Z d t t 0 f t 0 g t t 0 = Z d t f t )]TJ/F21 7.9701 Tf 6.704 1.471 Td [( t g t = Z 1 d t f t )]TJ/F21 7.9701 Tf 6.704 1.471 Td [( t g t Takingtheupperandlowersigns,oneobtainsthefollowingcommutation-propertyof theconvolutionandreverse-convolutionoftwofunctions f and g f g = Z 1 dt 0 f t 0 g t )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 = g f ; f g = Z 1 dt 0 f t 0 g t + t 0 = g f ;{96 Next,theeectofFouriertransformingovervariousmultiplicativeoperationsistobe determined.TheFouriertransformofaconvolutionisamultiplication,whichisseenby directcalculationusingkernel e i!t asfollows, h = F [ f g t ]= Z 1 dt e i!t Z 1 dt 0 f t 0 g t )]TJ/F23 7.9701 Tf 6.587 0 Td [(t 0 = Z 1 dt Z 1 dt 0 e i!t 0 f t 0 e i! t )]TJ/F23 7.9701 Tf 6.587 0 Td [(t 0 g t )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 = Z 1 dt 0 e i!t 0 f t 0 Z 1 dt e i! t )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 g t )]TJ/F23 7.9701 Tf 6.587 0 Td [(t 0 = Z 1 dt 0 e i!t 0 f t 0 ~ g = ~ f ~ g Now,theeectofFouriertransformingareverse-convolutionisdetermined.This proceedsthesamewayastheabovecalculationoftheFouriertransformoftheforward" Fouriertransform, h = F [ f g t ]= Z 1 dt e i!t Z 1 dt 0 f t 0 g t + t 0 = Z 1 dt Z 1 dt 0 e i! )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 f t 0 e i! t + t 0 g t + t 0 = Z 1 dt 0 e i! )]TJ/F23 7.9701 Tf 6.587 0 Td [(t 0 f )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 Z 1 d t + t 0 e i! t + t 0 g t + t 0 h = g Z 1 dt 0 e i! )]TJ/F23 7.9701 Tf 6.587 0 Td [(t 0 f )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 = g f = f g Theoverbaronthefunction f indicatesthefunctionisevaluatedatitsreversed argument.Thatis, f x f )]TJ/F23 7.9701 Tf 6.587 0 Td [(x .Thisiswhy f g hasbeencalledareverse"convolution. Summarizingtheseresults, h = Z 1 dte i!t Z 1 dt 0 f t 0 g t t 0 = f g ; h = Z 1 dte i!t f t g t = Z 1 d! 0 2 f 0 g ! 0 ;{97 117

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Later,thetheorems2{96and2{97willbeusedtonumericallyevaluateconvolution integralsbyfast-Fourier-transformwhichwouldotherwiserequirethecomputationally expensiveendeavorofusingquadratureover 0 foreveryoneofthe N 10 4 slicesof thattheband 2 [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 W; 2 W ]isdividedinto.Thisisdoneas, Z 1 d! 0 2 f 0 g ! 0 = Z 1 dte i!t f t g t = F F )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 [ f ] F )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 [ g ] Carryingthisoutnumericallyrequiresincorporationofthefactthatinthetheorems 2{96and2{97,thedomains t and areassumedtobeinnite.Aprocesscalled zero-padding"isimplemented,whereanypairoffunctions f and g tobeconvolvedby fastFouriertransformhavetheirdomains 2 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 W; 2 W ]extended,andthefunctionsare takentobezerointhisdomain-extension.Lastly,forthefastFouriertransformtorun thefastest, N shouldbepickedtobeanintegerpowerof2i.e., N =2 ; 4 ; 8 ; 16 ; 32 ;::: inadditiontohavingaband-slice thatissuciently-smallerthanthesmallestenergy scaleintheproblem. 2.3.10.2Theelectronicself-energyduetoanelectroncollidingwithaphonon TheFockself-energies 7 F ij fromtheFeynmandiagramsofFigs.2-4and2-5 foraphononofenergy 0 scatteringanelectronofenergy togoing out in ofanyofthe centerregion'ssitesindices i and j beingsummedoverwhentheDysonequation g ij = g 0 ij + g 0 ii 0 i 0 j 0 g j 0 j isused,isgivenasfollows, 7 F !ij = Z d T e i! T M mii 0 g 7 T i 0 j 0 d ? )]TJ/F21 7.9701 Tf 6.587 0 Td [(T m M mj 0 j = M mii 0 M mj 0 j Z d! 0 2 g 7 0 + !;i 0 j 0 d ? 0 m {98 Theself-energies 7 F ij thendeterminetheself-energies rF ij =[ aF ij ] y = rF ji bythe morenumericallyaccurateKramers-Kronigidentity2{65,orlessaccuratelybutuseful asacheckbytheretardedLangrethruleof2{82, r !ij = Z d! 0 2 > 0 ij + < 0 ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 + i 0 + = M mii 0 M mj 0 j Z d! 0 2 g r 0 + !;ii 0 d 7 m! 0 + g 7 0 + !;ii 0 d a m! 0 {99 118

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TousetheKramers-Kronigidentity2{65,adroitevaluationoftheprinciple-value occurringintheCauchy-identity x + i 0 + )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = P x )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 11.9552 Tf 12.212 0 Td [(i x isrequired.Namely,one mustusethemethodsinSec.2.3.10.4twosubsectionslater. TheHartreeself-energy rF ij isreal-denite,requiring 7 F ij =0.SinceitsFeynman diagrams,Figs.2-6and2-7,aretime-independent,onemustevaluatetheself-energy directlyfromthecontour-integraldenitionusingmachinationssimilartothoseusedto derive 44 2{81and2{82.First,obtaintheself-energyusingFeynmanrules,making suretoappendafactorof )]TJ/F15 11.9552 Tf 9.298 0 Td [(1duetothefermion-loop, H ij t; t =lim t t H ij t; t = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 n S I C dt 0 M mij d m t;t 0 M mi 0 j 0 g i 0 j 0 t 0 ;t 0 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(M mij M mi 0 j 0 2 I C i 0 j 0 Thecontour-integral I C mij aboveisthenevaluatedas, I C mij = I C dt 0 d m t;t 0 g ij t 0 ;t 0 = Z t dt 0 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( )]TJ/F22 11.9552 Tf 9.298 0 Td [(i d > m t;t 0 ig < ij t 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 + Z t 2 dt 0 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(+ i d < p t;t 0 ig < ij t 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 = i 2 Z t 2 dt 3 [ )]TJ/F22 11.9552 Tf 9.298 0 Td [(d > m t;t 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(d < m t;t 0 t )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 ] g < ij = )]TJ/F22 11.9552 Tf 9.299 0 Td [(i 2 Z + 1 dt 3 [ d r m t;t 0 ] g < ij I C mij = ~ d r m g < ij =lim 0 d r m g < ij Writingthistime-independentfunctionasaninverseFouriertransformusingthe kernel e )]TJ/F23 7.9701 Tf 6.587 0 Td [(i! t )]TJ/F23 7.9701 Tf 6.586 0 Td [(t 0 = 2 t t 0 = )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 rH ij = F )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 [ )]TJ/F22 11.9552 Tf 9.298 0 Td [(M mij M mi 0 j 0 2 d r m g < i 0 j 0 ]= )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 M mij M mi 0 j 0 d r m Z 2 W )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d! 0 2 g < i 0 j 0 0 {100 44 Itisinobtainingthistime-independentHartreeself-energythattheLangrethrules revealtheirusefulness.Thereasonwhythecontour-integraldenitionisrequiredisthat thederivationsusedtoobtain2{81and2{82werepredicateduponthepropertiesof theon-contourstep-function,whichbecomesill-denedatzeroargument. 119

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2.3.10.3Thephononicself-energyduetoaphononexcitingachargeneutral electron/holepairfromtheFermisea InanalogywithSec.2.3.10.2,theself-energies 7 !m foramodem localizedphononof energy impingingupontheFermiseaandexcitingfromitwithanenergy-inputof 0 a charge-neutralelectron/holepairare, 7 !m = Z d T e i! T M mii 0 g 7 T ij M mjj 0 g ? )]TJ/F21 7.9701 Tf 6.587 0 Td [(T ;j 0 i 0 = M mii 0 M mjj 0 Z d! 0 2 g 7 0 + !;ij g ? 0 j 0 i 0 {101 Theself-energies 7 !m thendeterminetheself-energies r m =[ a m ] y = r m ,whichmust beconsistentwiththeLangrethrulesaswell, r !m = Z d! 0 2 > 0 m + < 0 m )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 + i 0 + = M mii 0 M mjj 0 Z d! 0 2 g r 0 + !;ij g 7 0 j 0 i 0 + g 7 0 ij g a 0 + !;j 0 i 0 {102 Inzero-temperatureworkthatisdisconcernedwithanythermalcurrents,the polarization-bubble m isadiscardablerenormalizationofthefrequency m ofamode m localizedphonon.However,whenthethermalcurrentcarriedbythephononsand electronsisdesiredasinthiswork,thephononsandtheelectronsmustbeaclosed system,becausethethermalcurrentiscalculatedfromtheFirstLaw.Thus,theenergy owin/outoftheelectronsmustbeequalandoppositeoftheenergyowin/outofthe phonons.Iftherearetobecollisionalself-energies whichdresstheelectrons,theremust alsobepolarization-self-energieswhichdressthephonons. 2.3.10.4Mathinterlude:principle-valueintegralsbyaniteintervalmethod Considertheprinciplevalue P integral I ofafunction f = f x dividedby x )]TJ/F22 11.9552 Tf 12.009 0 Td [( over theinterval x 2 [ a;b ]forwhich 2 [ a;b ].Theprinciplevaluegivesanitevaluedespite non-analyticityatthepole x = I = P Z b a f x x )]TJ/F22 11.9552 Tf 11.956 0 Td [( dx =lim 0 Z )]TJ/F23 7.9701 Tf 6.586 0 Td [(" a f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx +lim 0 Z b + f x x )]TJ/F22 11.9552 Tf 11.956 0 Td [( dx Theproblemisthatonecannotcomputationallyeectthislimit-variable for computationalproblemswithnite-elements.Letsuchaniteelement x havean 120

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associatedparameter= x ,where x canbemadearbitrarilysmallbutnot innitesimalorsmallerthanamachine-epsilon;forinstance,thatofMATLABis2 : 22 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(16 .Then,onemightdivideuptheintervalofintegrationanitedierenceabout thetroublesomepole x = as x 2 [ a; )]TJ/F15 11.9552 Tf 12.504 0 Td [(], x 2 [ )]TJ/F15 11.9552 Tf 12.503 0 Td [( ; +],and x 2 [ + ;b ], insteadofinto x 2 [ a; )]TJ/F22 11.9552 Tf 12.072 0 Td [(" ]and x 2 [ + ";b ]asdonebythelimitingprocedureindicated above.Thesearecontrastingmethodsofinterval-divisionsi.e.,withnite-dierences,vs. inntesimals,whichcanbeshowngraphically. AsolutionproposedbyThompson[44]istosplittheintegral I intoasingularterm I "; containingthelimitvariable ,andanon-singularterm I whichbyitselfonly approximates I I = P b Z a f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx =lim 0 0 @ )]TJ/F23 7.9701 Tf 6.586 0 Td [(" Z )]TJ/F21 7.9701 Tf 6.587 0 Td [( f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx + + Z + f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx 1 A + )]TJ/F21 7.9701 Tf 6.587 0 Td [( Z a f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx + b Z + f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx lim 0 I "; + I So,thetaskisnowtocalculate I 0 ; ,ifitexists.ByTaylor'stheorem,itdoes, I 0 ; =lim 0 0 @ )]TJ/F23 7.9701 Tf 6.587 0 Td [(" Z )]TJ/F21 7.9701 Tf 6.586 0 Td [( f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx + + Z + f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx 1 A =lim 0 1 X n =0 0 @ )]TJ/F23 7.9701 Tf 6.587 0 Td [(" Z )]TJ/F21 7.9701 Tf 6.586 0 Td [( dx x )]TJ/F22 11.9552 Tf 11.955 0 Td [( x )]TJ/F22 11.9552 Tf 11.955 0 Td [( n f n n + + Z + dx x )]TJ/F22 11.9552 Tf 11.955 0 Td [( x )]TJ/F22 11.9552 Tf 11.955 0 Td [( n f n n 1 A = 1 X n =0 f n n lim 0 0 @ )]TJ/F23 7.9701 Tf 6.587 0 Td [(" Z )]TJ/F21 7.9701 Tf 6.586 0 Td [( x )]TJ/F22 11.9552 Tf 11.956 0 Td [( n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 dx + + Z + x )]TJ/F22 11.9552 Tf 11.955 0 Td [( n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 dx 1 A = 1 X n =0 f n n + Z )]TJ/F21 7.9701 Tf 6.587 0 Td [( x )]TJ/F22 11.9552 Tf 11.955 0 Td [( n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 dx I 0 ; = 1 X n =0 f n n n )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F15 11.9552 Tf 9.299 0 Td [( n n = 1 X n =0 f n n n 1+ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 n +1 n = 1 X n =0 f n +1 n +1! 2 2 n +1 2 n +1 Hence,theexactvalueofaprinciplevalueintegralcalculatedusingnite-dierences, byTaylor'sTheorem,iswhatthefollowinginniteseriesconvergesto, I = P b Z a f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx = )]TJ/F21 7.9701 Tf 6.587 0 Td [( Z a f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx + b Z + f x x )]TJ/F22 11.9552 Tf 11.955 0 Td [( dx + 1 X n =0 f n +1 n +1! 2 2 n +1 2 n +1 {103 121

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Asanexampleofusing2{103,onecouldtaketheprinciplevalueofaKramers-Kronig integralfrom2{65over 0 calculatedwithband-element whoseintegrand,having thenumerator 0 < 0 + > 0 ,involvesapole 0 = Re r = P 2 W Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W d! 0 2 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 = )]TJ/F23 7.9701 Tf 6.587 0 Td [(! Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W d! 0 2 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 + +2 W Z + d! 0 2 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 + 1 X n =0 n +1 n +1! 2 2 n +1 2 n +1 Aquestionofhowtouse2{103inprogramming-practicemayremain,evenafter presentationofthisexample.Implementingthecut"ofthepole = 0 thatbeginsafter = 0 )]TJ/F22 11.9552 Tf 12.187 0 Td [(! andendsbefore = 0 + ,thersttwoterms,isimplementedinpractice bycarryingouttheconvolutionandsettingtheresultingcolumnvectorarray'svalueat = 0 equaltothethirdterm'sscalarvalue. 2.3.10.5Probability-theoryandexplanationoftheappearanceoftheconvolutionoperation Theself-energiesintheprevioustwosections2.3.10.2and2.3.10.3followfromthe Feynman-rules,whichthemselves,roughlyspeaking,arebaseduponBorn'sprobabilistic interpretation[45]ofthequantumstatewavefunctionofasystemandconservation ofenergy.Abriefargumentisnowgivenfortheappearanceofconvolutiondueto conservationofenergyandprobability.Conservation-principlesarejustsuperpositions. Letthetotalenergy z ofasystem Z maybethesumofenergies x and y ofsubsystems X and Y .However,thetotalenergies x and y mayhaveanamountofenergy drawn 45 from X and Y ,whichvariescontinuously.Thus,letthesesub-parts X and Y have correspondingsub-distributions 46 g x = g x and g y = g y ,whicharetheprobabilites thatanamountofenergy isdrawnfromtherespectivesystems.Ifthesystemsare 45 Thewordsdrawn",drawing",etc.,inreferencetotheamountofenergy ,refers toperforminganexperiment/measurementwherearesultconsistentwiththeenergy is observed. 46 Here,theprobabilisticyetfundamentalBorninterpretationisinvoked. 122

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independent 47 ,thentheprobability g = g ofdrawinganamountofenergy fromthe twosystemstakenasawholeisgivenbytheconvolution g = R 0 g x 0 g y )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 d! 0 Theproofisasfollows.Let X and Y beindependentreal-valuedrandomvariables withrespectivedensityfunctions f X x and f Y y .Furthermore,let X X;Y ,sothat usingthesamenotation f X X;Y = f X x f Y y bytheproduct-ruleforindependent probabilities.OnethencalculatesPr X + Y z where z = x + y toobtainthecumulative distributionfunctionof Z = X + Y ,whichthenshallbedierentiatedwithrespectto z Pr=Pr X + Y z = ZZ X + Y z d 2 x f X X;Y = 8 > > < > > : 1 R z )]TJ/F23 7.9701 Tf 6.586 0 Td [(x R dxdy f X x f Y y = 1 R dx f X x z )]TJ/F23 7.9701 Tf 6.586 0 Td [(x R dy f Y y z )]TJ/F23 7.9701 Tf 6.586 0 Td [(y R 1 R dxdy f X x f Y y = 1 R dy f Y y z )]TJ/F23 7.9701 Tf 6.586 0 Td [(y R dx f X x 9 > > = > > ; Pr= f Y F X z = 1 Z dz 0 f Y z 0 F X z )]TJ/F22 11.9552 Tf 11.956 0 Td [(z 0 Dierentiatingbothsidesoftheaboveimmediatelygivestheprobabilitydistribution f X fromthecummulativedistribution F X ,yieldingtheconvolutionadvertisedearlier, d Pr dz = 1 Z dz 0 f Y z 0 d dz F X z )]TJ/F22 11.9552 Tf 11.955 0 Td [(z 0 = 1 Z dz 0 f Y z 0 f X z )]TJ/F22 11.9552 Tf 11.955 0 Td [(z 0 = f Y f X z = f Z z Thus,theconvolution-operationnaturallyappearsinquantitieswhicharerandomly distributed,conserved,anddrawnfromindependentsystems,asisoftenthecasein quantummany-bodysystems. Onecansketchconvolutionsofuniform,exponential,andGaussiandistributions,as doneinFig.2.3.10.5. 47 Notethattheself-consistentBornapproximationtothesedistributionswhich are -domainGreenfunctionsisinitializedwithdistributions g x and g y whichare independentnon-interacting. 123

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Figure2-13.Convolutionsofuniform,exponential,andGaussiandistributions. 2.3.10.6AnalyticrstBornapproximationtotheselfenergy bySommerfeldexpansionfor N C =1 Here,theself-energies electron-phononcollisionsandphonon-stimulated electron/holepolarizationcalculatedtotherstBornapproximationarepresented. Toincorporatetheeectofnitefermiontemperature,theSommerfeld-expansionshall beused.InthecaseoftheBose-Einsteindistribution,itwillbenotedthat x 2 )]TJ/F23 7.9701 Tf 6.587 0 Td [(d dx b 7 x 2 3 x 2 + 2 ,where isattingparameter.Thisbecomesexactinthelimitof T 0the expansioniszeroeverywhere,andshouldhavenoleadingtemperature-dependentterm. Withthe th leadatnitetemperatures T andconstantescape-rates)]TJ/F23 7.9701 Tf 142.778 4.338 Td [( )]TJ/F23 7.9701 Tf 7.314 4.338 Td [( U 1 = W 2 =W q 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( U 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(U )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W 2 1 )]TJ/F28 7.9701 Tf 6.587 11.557 Td [( U 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(U )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W ,thenon-many-bodyelectronicGreenfunctionsare, g r 0 = 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 + i )]TJ/F19 11.9552 Tf 11.83 8.201 Td [(! g r 0 2 = 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 2 +)]TJ/F21 7.9701 Tf 19.076 3.454 Td [(2 g 7 0 = 2)]TJ/F22 11.9552 Tf 13.167 0 Td [(f 7 eff ! )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 2 +)]TJ/F21 7.9701 Tf 19.075 3.454 Td [(2 = X = L;R 2)]TJ/F23 7.9701 Tf 13.168 -1.793 Td [( 2 W j )]TJ/F23 7.9701 Tf 6.586 0 Td [(U j f 7 ! )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 2 +)]TJ/F21 7.9701 Tf 19.075 3.454 Td [(2 Similarly,theretardedphononicGreenfunctionsisasfollowsexceptthistimethe analogueoftheescaperate alreadyisconstant, d r 0 = )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F22 11.9552 Tf 11.955 0 Td [(i )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 + 2 )]TJ/F15 11.9552 Tf 15.789 8.087 Td [( + )]TJ/F22 11.9552 Tf 11.955 0 Td [(i + 2 + 2 = X s = 1 s )]TJ/F22 11.9552 Tf 11.956 0 Td [(s )]TJ/F22 11.9552 Tf 11.955 0 Td [(i )]TJ/F22 11.9552 Tf 11.955 0 Td [(s 2 + 2 = 2 + i 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 The 7 phononicGreenfunctionscanberearrangedinamannersothattheirroots aremoreobviousthisisconvenientforpartial-fractionsdecompositionthatistobe 124

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eectedforintegration, d 7 0 = X = L;R 2 b 7 ! )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 + 2 + 2 b ? )]TJ/F23 7.9701 Tf 6.586 0 Td [(! + 2 + 2 = X s = 1 X = L;R 2 s b 7 ! )]TJ/F22 11.9552 Tf 11.956 0 Td [(s 2 + 2 = X = L;R 8 b 7 ! )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 + 2 + 2 + 2 Theelectron-phononself-energiesattheHartree-Focklevelare, 7 F M 2 = Z d! 0 2 g 7 + 0 d ? 0 rF = Z d! 0 2 > 0 + < 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! 0 + i 0 + rH M 2 =2 Z d! 0 2 )]TJ/F22 11.9552 Tf 9.299 0 Td [(g < 0 d r 0 Onealsorequires N -foldpartialfractionsdecomposition.Aformulaforpartial fractiondecompositionwithoutdegenerate/repeatedrootsamathematicalidentity,for N 1is, f N 1 Y i =1 1 y )]TJ/F22 11.9552 Tf 11.955 0 Td [(z i N X j =1 1 Q N i =1 z j )]TJ/F22 11.9552 Tf 11.955 0 Td [(z i + ij 1 y )]TJ/F22 11.9552 Tf 11.955 0 Td [(z j z i 6 = z j {104 The n +1 th derivativeoftheabovepartialfractionsdecompositionmaybe immediatelycalculatedusingmathematicalinduction, d n +1 f dy n +1 = n X k =0 n k n )]TJ/F22 11.9552 Tf 11.955 0 Td [(k d k f dy k N 1 X j =1 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 y )]TJ/F22 11.9552 Tf 11.955 0 Td [(z j n )]TJ/F23 7.9701 Tf 6.587 0 Td [(k +1 = N X j =1 1 Q N i =1 z j )]TJ/F22 11.9552 Tf 11.955 0 Td [(z i + ij )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 n +1 y )]TJ/F22 11.9552 Tf 11.955 0 Td [(z j n +2 Thethermalstatesoffermionic-electronsandbosonicphononsandtheirrespective approximantsarenowtobeobtained.TheFermifunctions f 7 andtheBosefunctions b 7 ,andtheirzero-temperaturelimit,are, f < = e )]TJ/F23 7.9701 Tf 6.586 0 Td [( =T +1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 =1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f > b < = e !=T )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = b > )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 lim T 0 f 7 = )]TJ/F23 7.9701 Tf 6.587 0 Td [(! lim T 0 b 7 = TheFermifunction f 7 canbehandledbySommerfeldexpansion2{48orby ttinganapproximantthisdissertationelectstodotheformer,whiletheBosefunction 125

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b 7 occurringas !b 7 whichisfreeofsingularitiescanbehandledbytting 48 an approximant.SuchanapproximanttotheBose-Einsteindistributioncouldbe, !b 7 ! !e !=T + T e 3 != T + ! + e !=T + T e 3 != T {105 TheSommerfeldexpansionisnowusedforthiscaseofnitefermiontemperature.In Sec.2.3.1.4,itwasfoundthataquantity Q whichisthereverseconvolutionintegralof H andFermifunction f 7 i.e.,integrand f 7 0 + H 0 overanenergy-band 2 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 W; 2 W ]ina systemhavingchemicalpotential andtemperature T isapproximatedbythefollowing seriesinwhichoddtermssurviveduetotheasymmetricboundsandtheabbreviation = )]TJ/F22 11.9552 Tf 11.956 0 Td [(! ismade, Q 7 = 2 W Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W d! 0 f 7 0 + H 0 = 2 W Z 2 W H 0 d! 0 1 X n =1 T n H n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 b n b n +2 W )]TJ/F23 7.9701 Tf 6.587 0 Td [( =T Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W )]TJ/F23 7.9701 Tf 6.586 0 Td [( =T dx T n x n sech 2 x= 2 4 n Beforeconsideringthecaseofnitetemperature,theless/greaterself-energiesat zero temperatureshallbedeveloped.IntherstBornapproximationandatzerotemperature, onecalculates 7 F usingfourfold-partial-fractions-decompositioni.e.,usingtheidentity 2{104with N =4.ThisisstillnovelandimportantbecauseGreenfunctionsof itinerant phononsareused,anddeterminingtheperturbation-parameterenteringasa premultipleoftheself-energyisnecessarytodeterminetheconditionsunderwhich2{42 48 Oneshouldnoticethatthisapproximantgives R 1 0 y 3 dy e y )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = 4 15 R 1 0 y 2 ye )]TJ/F23 7.9701 Tf 6.587 0 Td [(y + e )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 y= 2 dy = 178 27 =1 : 01519 4 15 ,i.e.,a1 : 52percenterrorinthe now-pedagogicalcalculationoftheheat-capacityofaphononorphotonensemble. 126

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isabletobeincorporatedperturbatively, 7 F 1 M 2 = 2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d! 0 2 g 7 0 + 0 d ? 0 0 = X 0 2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d! 0 2 2)]TJ/F23 7.9701 Tf 13.167 -1.793 Td [( + 0 )]TJ/F23 7.9701 Tf 6.587 0 Td [( 0 + )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 2 +)]TJ/F21 7.9701 Tf 19.075 3.454 Td [(2 8 0 0 0 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 + 2 0 + 2 + 2 = X 0 16 0 )]TJ/F24 5.9776 Tf 5.289 -0.996 Td [( 2 2 W= Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W= du 0 0 u 0 0 u 0 0 )]TJ/F23 7.9701 Tf 6.952 5.255 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(! u 0 0 6 Q 6 i =1 u 0 0 )]TJ/F23 7.9701 Tf 6.586 0 Td [(z 0 i = X 0 6 X j =1 0 )]TJ/F24 5.9776 Tf 5.289 -0.996 Td [( = 3 Q 6 i =1 z 0 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(z 0 i + ij )]TJ/F23 7.9701 Tf 6.587 0 Td [( 0 0 Z 0 u 0 0 du 0 0 u 0 0 )]TJ/F23 7.9701 Tf 6.586 0 Td [(z 0 j Above,thechangesofintegrationvariable u 0 0 = 0 u 0 = ,and 0 = )]TJ/F22 11.9552 Tf 11.975 0 Td [( were used.Thisgivesthezero-temperature 7 self-energies.Introducing g = M 2 = 2 ,itisseen thatthedimensionlessquantity g )]TJ/F22 11.9552 Tf 7.314 0 Td [(= 2 controlstheimaginarypartoftheself-energy 7 F if 7 F ismeasuredinunitsof, 7 F 1 = 6 X j =1 1 Q 6 i =1 z 0 j )]TJ/F22 11.9552 Tf 11.956 0 Td [(z 0 i + ij X ; 0 g 0 )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [( 2 z 0 j ln 0 0 + z 0 j z 0 j )]TJ/F22 11.9552 Tf 11.955 0 Td [( 0 0 {106 Above,thefollowingscaledrootsfrompartial-fractionsdecomposition2{104are introducedandmusthaveanonzeroimaginarypart 49 toavoiddegeneracy, z 0 1 2 = U 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(! i )-556( z 0 3 4 =+ i z 0 5 6 = )]TJ/F15 11.9552 Tf 9.299 0 Td [( i Theself-energiesofHyldgaardet.al.[23]maybeobtainedbyreturningtobeforethe integraliscarriedout,eecting 0,introducingtheresultingDirac-deltafunctionsin placeoftheLorentzians,andintegrating. Nowthegeneralizationtonitetemperatureshallbemadeforthe 7 Fockself-energies. Atnitetemperature,onemayscaletheintegration-variabletothe th lead'schemical potential )]TJ/F22 11.9552 Tf 22.195 0 Td [(W phonon-temperature T 0 insteadofthemode-energyasdoneforthe 49 Herethepremisethattheimaginaryparts ; )-326(arenonzeroisused. 127

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z 0 i .Introducingtheabbreviation H 7 P 0 8 M 2 0 )]TJ/F24 5.9776 Tf 5.289 -0.996 Td [( !b ? Q 6 i =1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(z 0 i ,subsequentlyyields, 7 F 1 = X ; 0 6 X j =1 = M 2 )]TJ/F24 5.9776 Tf 5.289 -0.996 Td [( 0 T 0 2 =T 0 6 Q 6 i =1 z 0 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(z 0 i + ij y Z )]TJ/F23 7.9701 Tf 6.586 0 Td [(y du 0 f 7 u + u 0 u 0 b ? u 0 u 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(z 0 j 7 F 0 1 + 1 X n =1 1 n T n H 7 ; n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(! b n {107 Thezero-temperaturelimitof2{107is2{106.However,thelimit 0 + at nonzerotemperaturemustbeeectedatthestepsinSections2.3.7.2and2.3.9.2. Thisisbecauseinthe 7 Dysonequation2{84itisassumedthat 0 + ,andthe term 50 [ g r 000 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g 7 000 [ g a 000 ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 isdiscardedbecauseofthis.Thereasonwhythe 0 + limitissoeasilyeectedforallsystemsatabsolutezerotemperatureisthezerothlawof thermodynamics. Above,thenotationisusedthat,duetotheself-consistentstructureoftheself-energies ofSec.2.3.10.2,aself-energyispartofaperturbativeseriesthe n th termofwhich itself canbeSommerfeld-expanded.Hence,onemustbeawarethattheyhaveaserieswithina series, F = F 1 + F 2 + ::: F 1 = F 0 1 + X 1 n =1 1 n T n H n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(! b n Twocommentsareinorderhere.First,2{107isanimportantresulti.e.,assigned anequationnumberbecauseitshowsthepremultiplethatcontrolswhetherornotthe perturbationtheoryconvergesregardlessofwhatthedimensionlessintegralevaluates 51 to. Second,notethattheleadingterm 7 F 0 1 is not thezero-temperatureself-energyobtained 50 Referencetoacenter-regiontemperatureonlyoccursin g 7 000 ,andfornon-inntesimal tunnelingtothegrandcanonicalresevoirsthisisdrownedout". 51 TheapproximateevaluationoftheintegralrequiresSommerfeldexpansionwhose leadingtermofzeroelectrontemperatureiscumbersomeduetonitephonontemperature andthepiecewisevalidityoftheapproximant2{105.Theonlyadvantageofcarrying outthisintegrationisasaquantitativecheckoftherstiterateoftheself-consistentBorn approximationthatneedsacomputertocarryout. 128

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earlier.In 7 F 0 1 ,the fermion electrontemperatureiszero,butthe boson phonon temperatureisnonzero,andtheantiderivativeisevaluatedusingtheapproximant2{105. Incontrasttothezero-temperaturecalculationof 7 F 1 ,therootsintheterm 7 F 0 1 are thistimescaledtothelead0 phonon-temperature T 0 T 0 z 0 1 2 = U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! i )]TJ/F22 11.9552 Tf 13.956 0 Td [(T 0 z 0 3 4 =+ iT 0 z 0 5 6 = )]TJ/F15 11.9552 Tf 9.298 0 Td [( i Consider,now,theleadingzero-electron-temperaturetermoftheSommerfeld expansion 7 F 0 1 .Ironically,the0 th -ordertermresultingfromtheSommerfeldexpansion .7requiresfarmoreeort.Thechangesofvariables T 0 u 0 = 0 T 0 u = T 0 = )]TJ/F22 11.9552 Tf -451.004 -23.908 Td [( ,and 0 T 0 = T shallalsobeneededtoworkoutthecaseofnitetemperature.One requirestheapproximant2{105,notingtheboundsofintegration u 0 7 2 [ f )]TJ/F23 7.9701 Tf 6.587 0 Td [(y )]TJ/F23 7.9701 Tf 6.587 0 Td [( g ; f )]TJ/F23 7.9701 Tf 6.587 0 Td [( y g ] breakupoverthepiecewisedomainoftheapproximant2{105asfollows, w ? j = )]TJ/F23 7.9701 Tf 6.586 0 Td [( Z y du 0 u 0 e u 0 + e 3 u 0 = 2 u 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(z 0 j w ? j = 0 Z y du 0 u 0 e u 0 + e 3 u 0 = 2 u 0 )]TJ/F22 11.9552 Tf 11.956 0 Td [(z 0 j + )]TJ/F23 7.9701 Tf 6.587 0 Td [( Z 0 du 0 u 0 + e u 0 + e 3 u 0 = 2 u 0 )]TJ/F22 11.9552 Tf 11.956 0 Td [(z 0 j Thus,the0 th -Sommerfeld-ordertermisdeterminedbythefollowingintegrals I 7 0 j whichrequiretheantiderivatives 52 R e sx x )]TJ/F23 7.9701 Tf 6.586 0 Td [(a dx e as Ei sx )]TJ/F22 11.9552 Tf 12.144 0 Td [(as and R xe sx x )]TJ/F23 7.9701 Tf 6.587 0 Td [(a dx ae as Ei sx )]TJ0 g 0 GETq1 0 0 1 72 234.108 cm[]0 d 0 J 0.398 w 0 0 m 144 0 l SQBT/F21 7.9701 Tf 80.202 224.053 Td [(52 Duetothesixpolesforthecaseofnitetemperature,theEifunctionsare calledabout30times.The analytic rstBornapproximationisironicallymuchmore computationallyexpensiveandcorrespondinglyslowertocalculatethanthe numerical self-consistentBornapproximation.Specically,MATLABtookabout30secondsto evaluatethisanalyticrstBornapproximationandabouthalfasecondtoevaluatethe self-energies bythenumericalself-consistentBornapproximation. 129

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as + e sx =s inwhichEiistheexponential-integralfunction, I 7 0 j = Z y )]TJ/F23 7.9701 Tf 6.587 0 Td [( du 0 u 0 b ? u 0 u 0 )]TJ/F23 7.9701 Tf 6.586 0 Td [(z j Z y )]TJ/F23 7.9701 Tf 6.587 0 Td [( du 0 u 0 e u 0 + e 3 u 0 = 2 u 0 )]TJ/F23 7.9701 Tf 6.586 0 Td [(z j 0 B @ R y 0 du 0 u 0 e u 0 + e 3 u 0 = 2 u 0 )]TJ/F23 7.9701 Tf 6.586 0 Td [(z j + R 0 )]TJ/F23 7.9701 Tf 6.586 0 Td [( u 0 + e u 0 + e 3 u 0 = 2 u 0 )]TJ/F23 7.9701 Tf 6.587 0 Td [(z j 1 C A Returningtothenite-temperatureroots,considerthepre-multiple.Itistobe determinediftemperaturethatistoohighcausesaself-energytobearbitrarilylarge.One carriesoutthemachinations, 7 F 1 0 j = 6 X j =1 = M 2 )]TJ/F24 5.9776 Tf 5.289 -0.997 Td [( 0 T 0 4 Q 6 i =1 z 0 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(z 0 i + ij I 7 j =2Re X j =1 ; 3 ; 5 = M 2 )]TJ/F24 5.9776 Tf 5.288 -0.997 Td [( 0 T 0 4 Q i =1 ; 3 ; 5 j z 0 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(z 0 i + ij j 2 I 7 j = 16 M 2 )]TJ/F24 5.9776 Tf 5.289 -0.996 Td [( 0 T 0 4 I 7 1 j z 0 3 )]TJ/F23 7.9701 Tf 6.587 0 Td [(z 0 1 j 2 j z 0 3 )]TJ/F23 7.9701 Tf 6.586 0 Td [(z 0 5 j 2 + I 7 3 j z 0 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(z 0 1 j 2 j z 0 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(z 0 3 j 2 + I 7 5 j z 0 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(z 0 3 j 2 j z 0 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(z 0 5 j 2 7 F 1 0 j = 4 g )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [( 0 4 2 I 7 1 )]TJ/F25 7.9701 Tf 11.875 0 Td [()]TJ/F23 7.9701 Tf 6.586 0 Td [( 2 + U 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(! )]TJ/F21 7.9701 Tf 6.586 0 Td [( 2 )]TJ/F25 7.9701 Tf 15.169 0 Td [()]TJ/F23 7.9701 Tf 6.586 0 Td [( 2 + U 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(! + 2 + I 7 3 )]TJ/F21 7.9701 Tf 6.586 0 Td [(\051 2 + )]TJ/F23 7.9701 Tf 6.587 0 Td [(U 1 + + 2 + I 7 5 )]TJ/F21 7.9701 Tf 6.586 0 Td [(\051 2 + )]TJ/F23 7.9701 Tf 6.586 0 Td [(U 1 + )]TJ/F21 7.9701 Tf 6.586 0 Td [( 2 Thus,the th lead'stemperaturedropsoutaltogetherfromthepre-multipleofthe imaginarypartoftheself-energy,anddoesnotenterinasaperturbativeparameter. Thisisin contrast totheuseofEinsteinphononsofimaginaryself-energies 7 ;ein =M 2 b ? g 7 + b 7 g 7 evaluatedatacenter-regiontemperature T C whoseperturbative contributiongrowsmonotonicallyastheBosefunctions b 7 dowith T C .Thisisafeature ofusingitinerantratherthanstationary-statephonons. Asdetailedinanearlierfootnote,therstBornapproximationisusefulforchecking thatonehascorrectlyprogrammedtheself-consistentBornapproximationandnotmuch beyondthat.JuxtaposingtheoutputoftheMATLABcodewiththeevaluationofthese analyticexpressionsfor 7 F 1 of2{107isdoneinFigs.2-14and2{107isdoneinFigs. 2-15. Todeterminetheconditionsunderwhichtherealpartoftheself-energyavoids adivergingperturbativeseries,onecalculatestheretardedFockself-energyrstat zerotemperature,andthenatnitetemperature.Todoso,onecoulduseanyofthe thefollowingformulae:oneisfromtheKramers-Kronigrelation2{65whichrequires 130

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Figure2-14.ComparisonoftheanalyticandnumericalrstBornapproximated electron/phononcollisionalimaginaryself-energies 7 F 1 forpositive resonance-energy. Figure2-15.ComparisonoftheanalyticandnumericalrstBornapproximated electron/phononcollisionalimaginaryself-energies 7 F 1 fornegative resonance-energy. evaluationofaprinciplevalueandtheotherisfromtheLangrethrule2{82, rF 1 = Z d! 0 2 >F 1 0 +
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obeytheconditions 0 7 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! ? 2 W ,yielding, rF 1 M 2 = 2 W Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W d! 0 2 0 B @ g r; 0 0 + d 7 0 0 + g 7 0 0 + d a 0 0 1 C A = X s = 1 X = L;R 2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d! 0 2 0 B @ 1 0 + )]TJ/F23 7.9701 Tf 6.586 0 Td [(U 1 + i )]TJ/F25 7.9701 Tf 7.68 10.951 Td [( 2 s 0 0 )]TJ/F23 7.9701 Tf 6.586 0 Td [(s 2 + 2 + 2)]TJ/F24 5.9776 Tf 9.523 -0.996 Td [( 0 + )]TJ/F24 5.9776 Tf 5.756 0 Td [( 0 + )]TJ/F23 7.9701 Tf 6.586 0 Td [(U 1 2 +)]TJ/F18 5.9776 Tf 11.875 2.269 Td [(2 s 0 )]TJ/F23 7.9701 Tf 6.586 0 Td [(s )]TJ/F23 7.9701 Tf 6.587 0 Td [(i 1 C A = X s; s 2 W Z 0 d! 0 0 )]TJ/F21 7.9701 Tf 6.587 0 Td [( U 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(! )]TJ/F23 7.9701 Tf 6.587 0 Td [(i \051 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [( s + i 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [( s )]TJ/F23 7.9701 Tf 6.586 0 Td [(i + X s; )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [( s 2 W Z )]TJ/F23 7.9701 Tf 6.586 0 Td [(! )]TJ/F24 5.9776 Tf 5.756 0 Td [(! +2 W d! 0 0 )]TJ/F21 7.9701 Tf 6.587 0 Td [( U 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(! )]TJ/F23 7.9701 Tf 6.587 0 Td [(i \051 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [( U 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(! + i \051 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [( s + i rF 1 M 2 = X s; s 3 X j =1 ln 2 W )]TJ/F23 7.9701 Tf 6.586 0 Td [(z r 0 j )]TJ/F21 7.9701 Tf 6.586 0 Td [(ln )]TJ/F23 7.9701 Tf 6.586 0 Td [(z r 0 j Q 3 i =1 z r 0 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(z r 0 i + ij + X s; )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [( s 3 X j =1 ln 2 W )]TJ/F23 7.9701 Tf 6.586 0 Td [(z r 0 j )]TJ/F21 7.9701 Tf 6.587 0 Td [(ln )]TJ/F23 7.9701 Tf 6.586 0 Td [(! )]TJ/F24 5.9776 Tf 5.756 0 Td [( +2 W )]TJ/F23 7.9701 Tf 6.586 0 Td [(z r 0 j Q 3 i =1 z r 0 j )]TJ/F21 7.9701 Tf 7.078 0 Td [( z r 0 i + ij Thenon-degenerateroots z r 0 1 ;z r 0 2 ;z r 0 3 andtheircomplexconjugates z r 0 1 ; z r 0 2 ; z r 0 3 appearingintheabovepartial-fractionsdecompositionare, z r 0 1 = U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! )]TJ/F22 11.9552 Tf 11.955 0 Td [(i )]TJ/F22 11.9552 Tf 13.956 0 Td [(z r 0 2 = s + i = z r 0 1 z r 0 3 = s )]TJ/F22 11.9552 Tf 11.955 0 Td [(i Hence,thenextnaturalstepistocalculatetheretardedfunctionatnitetemperature, andthisiseectedas, rF 1 M 2 = 2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d! 0 2 0 B @ g r 0 0 + d 7 0 0 + g 7 0 0 + d a 0 0 1 C A = 2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d! 0 2 8 0 b 7 0 0 )]TJ/F21 7.9701 Tf 6.587 0 Td [( 2 + 2 0 + 2 + 2 0 + )]TJ/F23 7.9701 Tf 6.586 0 Td [(U 1 + i \051 + 2)]TJ/F24 5.9776 Tf 9.523 -0.996 Td [( f 7 0 + 2 0 + )]TJ/F23 7.9701 Tf 6.586 0 Td [(U 1 2 +)]TJ/F18 5.9776 Tf 11.875 2.813 Td [(2 0 )]TJ/F23 7.9701 Tf 6.587 0 Td [(i 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [( 2 = X 0 B B B B @ 5 P j =1 4 T = Q 5 i =1 z r j )]TJ/F23 7.9701 Tf 6.587 0 Td [(z r i + ij 0 @ f 2 W 0 g R f 0 )]TJ/F24 5.9776 Tf 5.756 0 Td [( 2 W g du e 3 u= 2 ue u u )]TJ/F21 7.9701 Tf 7.078 0 Td [(~ z r j + f 0 2 W g R f )]TJ/F24 5.9776 Tf 5.756 0 Td [( 2 W 0 g du e 3 u= 2 u + e u u )]TJ/F21 7.9701 Tf 7.078 0 Td [(~ z r j 1 A + 4 P j =1 2)]TJ/F24 5.9776 Tf 9.523 -0.997 Td [( = Q 4 i =1 z a j )]TJ/F23 7.9701 Tf 6.587 0 Td [(z a i + ij 2 W R )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W f 7 0 + d! 0 0 )]TJ/F23 7.9701 Tf 6.587 0 Td [(z a j 1 C C C C A rF 1 M 2 X 5 X j =1 4 T = Q 5 i =1 z r j )]TJ/F23 7.9701 Tf 6.587 0 Td [(z r i + ij w r 7 2 W ; ~ z r j + 4 X j =1 2)]TJ/F24 5.9776 Tf 9.523 -0.996 Td [( = Q 4 i =1 z a j )]TJ/F23 7.9701 Tf 6.586 0 Td [(z a i + ij I 7 ! Above,thefunction w r 7 y;z withaccompanyingroots z r j andtheSommerfeld-expandable function I 7 withaccompanyingroots z a j eachwereintroduced, w r 7 y;z f y 0 g Z f 0 )]TJ/F24 5.9776 Tf 5.756 0 Td [(y g du e 3 u= 2 ue u u )]TJ/F23 7.9701 Tf 6.587 0 Td [(z + f 0 y g Z f )]TJ/F24 5.9776 Tf 5.756 0 Td [(y 0 g du e 3 u= 2 u + e u u )]TJ/F23 7.9701 Tf 6.586 0 Td [(z = y Z 0 du e )]TJ/F18 5.9776 Tf 5.756 0 Td [(3 u= 2 + ue )]TJ/F24 5.9776 Tf 5.756 0 Td [(u u )]TJ/F23 7.9701 Tf 6.586 0 Td [(z + e )]TJ/F18 5.9776 Tf 5.756 0 Td [(3 u= 2 + u + e )]TJ/F24 5.9776 Tf 5.756 0 Td [(u u )]TJ/F23 7.9701 Tf 6.587 0 Td [(z 132

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Figure2-16.ComparisonoftheanalyticandnumericalrstBornapproximated electron/phononcollisionalimaginaryself-energies 7 F 1 forpositive resonance-energy. Figure2-17.ComparisonoftheanalyticandnumericalrstBornapproximated electron/phononcollisionalimaginaryself-energies 7 F 1 forpositive resonance-energy. I 7 = ln )]TJ/F22 11.9552 Tf 11.955 0 Td [(! )]TJ/F23 7.9701 Tf 6.587 0 Td [(! +2 W )]TJ/F22 11.9552 Tf 11.955 0 Td [(z r j 2 W )]TJ/F22 11.9552 Tf 11.955 0 Td [(z r j 2 6 )]TJ/F22 11.9552 Tf 9.299 0 Td [(T 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(z j 2 + ::: z r 1 = U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(! )]TJ/F22 11.9552 Tf 11.955 0 Td [(i )]TJ/F22 11.9552 Tf 13.955 0 Td [(z r 2 3 =+ iz r 4 5 = )]TJ/F15 11.9552 Tf 9.298 0 Td [( iz r j = T ~ z r j z a 1 2 = U 1 i )]TJ/F22 11.9552 Tf 13.956 0 Td [(z a 3 4 = + iz a j =~ z a j T Acomparisonoftheanalyticandnumericalrstiterateoftherealandimaginary partsof rF 1 isdoneinFigs.2-16and2-17. 133

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Finally,theHartreeretardedself-energycanbeimmediatelydoneatnitetemperature sincetheintegralisastandardarctangentfunction.Avoidingthewide-bandlimit 53 one obtains, rH 1 M 2 =2 d r 0 0 2 W Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W d! 0 2 g < 0 0 =2 )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 2 + 2 X = L;R 2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d! 0 2 2)]TJ/F23 7.9701 Tf 13.167 -1.794 Td [( f < 0 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 2 +)]TJ/F21 7.9701 Tf 19.075 3.454 Td [(2 = X = L;R )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 )]TJ/F23 7.9701 Tf 15.768 -1.793 Td [( 2 + 2 1 )]TJ/F27 11.9552 Tf 10.502 25.058 Td [( tan )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 )]TJ 27.647 8.201 Td [(+tan )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 W + U 1 )]TJ/F27 11.9552 Tf 27.668 25.058 Td [( )]TJ/F22 11.9552 Tf 13.151 8.088 Td [( 2 3 T 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 )]TJ/F21 7.9701 Tf 11.867 3.453 Td [(2 + )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 2 2 + ::: 2.4ObservablesintheAbsenceofanAppliedMagneticField 2.4.1TheContinuousElectricalandNumberCurrent Onebeginwiththecontinuous/incompressiblenumbercurrent N = N L + N R = )]TJ/F22 11.9552 Tf 9.299 0 Td [(I BytheHeisenberg-Diracequationofmotion,thenumbercurrent N = )]TJ/F22 11.9552 Tf 9.298 0 Td [(I inthe th leadtimes i equalstheexpectationofthenumberdensityinthe th leadandits, commutatorwiththeglobalHamiltonian H = H 0 + H 0 .Takingtheleft-leadtobe thesourcepositively-signedandtheright-leadtobethesinknegatively-signed,and beginningfrom2{36andusing2{92with ` =1and A = 0 ,onegets, D N i E = )]TJ/F22 11.9552 Tf 9.298 0 Td [(iW i g < i 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g < 1 i = X N C j =1 2)]TJ/F23 7.9701 Tf 13.167 4.936 Td [( ij A e )]TJ/F23 7.9701 Tf -3.928 -8.012 Td [(ij f < )]TJ/F22 11.9552 Tf 11.956 0 Td [(g < ij {108 Lettherebeleftandrightnumbercurrentband-densities N L R whichabsorbafactor of1 = 2sometimesexplicitlyshowntoindicateanaverageof N L and N R duetotunneling )]TJ/F22 11.9552 Tf 9.298 0 Td [(W i betweenthe i th center-regionsiteandthesurface-siteofthe th leadwithretarded, advanced,and 7 overlap g ij betweenthe i th and j th sitesofthecenterregionarethe following,bothwithandwithoutanappliedtime-reversal-breakingeld, 53 Oneisfreetoapproximatetan )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 W + U 1 )]TJ/F19 11.9552 Tf 22.094 4.123 Td [(' = 2theabovementionedwide-bandlimit. 134

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N = N L + N R = N A + N C = N el + N in 1 2 X N C i =1 D N L i + N R i E {109 Theterms N i havetwoterms.Bothtermshaveacommonmultipleof)]TJ/F23 7.9701 Tf 297.978 4.339 Td [( ij builtofthe product W i W j inter-regiontunneling-amplitudes.Thersttermisthespectral-density 2 A e )]TJ/F23 7.9701 Tf -3.928 -8.012 Td [(ij = g < ij + g > ij acharge-neutralsumofelectronandholetransitionsbetweenthe i th and j th center-regionsitesandthesecondtermistheamplitude g < ij ofelectronictransitions betweenthe i th and j th center-regionsites.Both A ij and g < ij arecenter-regionGreen functionswiththedetailsofwhathappensatthecenter-regione.g.,simpleinter-site tunneling )]TJ/F22 11.9552 Tf 9.298 0 Td [(W ij ,orthe[lesssimple] hole particle lifetimes 7 [whichareself-energies]buried inside".Buryingthecenter-region'sdetailsintotheGreenfunctionallowsexpressionof thecurrentinaLandauer-Buttikerform. 2.4.1.1ThematerialcurrentasaLandauer-Buttikertermplusacorrection Here,thekineticorKeldyshGreenfunction ig k g < )]TJ/F22 11.9552 Tf 12.711 0 Td [(g > isintroduced,aswell asthecorrespondingkineticequilibriumdistribution f k f < )]TJ/F22 11.9552 Tf 12.525 0 Td [(f > =2 f < )]TJ/F15 11.9552 Tf 12.525 0 Td [(1andits eective"counterpart f k e P f k )]TJ/F23 7.9701 Tf 7.314 4.338 Td [( = P )]TJ/F23 7.9701 Tf 7.314 4.338 Td [( .Thecurrent2{109isrewritteninterms of g k ,whichsymmetrizesthecurrent'sreferencetoparticle-ow g < andhole-ow g > Switchingfromindex-notationtonon-commutingmatricesinthisnotation,multiplicative inversesiseasiertoeect,anddeningthesuminthedenominatorof)]TJ/F23 7.9701 Tf 375.402 4.339 Td [( = P )]TJ/F23 7.9701 Tf 7.314 4.339 Td [( notas 135

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matrix-multiplicationbutratherasaHadamard 54 product. N = N L + N R = 1 2 i )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [(L )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r ^ 1 f k L )]TJ/F22 11.9552 Tf 11.955 0 Td [(g k )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [(R )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r ^ 1 f k R )]TJ/F22 11.9552 Tf 11.955 0 Td [(g k = 1 2 i )]TJ/F23 7.9701 Tf 7.315 4.936 Td [(L g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r [ f k e + )]TJ/F24 5.9776 Tf 5.289 2.813 Td [(R )]TJ/F24 5.9776 Tf 5.289 2.269 Td [(L +)]TJ/F24 5.9776 Tf 11.875 2.269 Td [(R f k L )]TJ/F22 11.9552 Tf 11.956 0 Td [(f k R ] )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [(L g k )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [(R g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r [ f k e )]TJ/F21 7.9701 Tf 22.104 4.707 Td [()]TJ/F24 5.9776 Tf 5.289 2.813 Td [(L )]TJ/F24 5.9776 Tf 5.288 2.269 Td [(L +)]TJ/F24 5.9776 Tf 11.875 2.269 Td [(R f k L )]TJ/F22 11.9552 Tf 11.955 0 Td [(f k R ]+)]TJ/F23 7.9701 Tf 29.536 4.936 Td [(R g k = 1 2 i 0 B @ +)]TJ/F23 7.9701 Tf 16.419 4.338 Td [(L g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r f k e + )]TJ/F24 5.9776 Tf 5.289 2.813 Td [(L g a )]TJ/F23 7.9701 Tf 6.587 0 Td [(g r )]TJ/F24 5.9776 Tf 8.581 2.813 Td [(R )]TJ/F24 5.9776 Tf 5.288 2.269 Td [(L +)]TJ/F24 5.9776 Tf 11.875 2.269 Td [(R f k L )]TJ/F22 11.9552 Tf 11.955 0 Td [(f k R )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 7.9701 Tf 7.314 4.338 Td [(L g k )]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F23 7.9701 Tf 7.314 4.338 Td [(R g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r f k e + )]TJ/F24 5.9776 Tf 5.289 2.813 Td [(R g a )]TJ/F23 7.9701 Tf 6.586 0 Td [(g r )]TJ/F24 5.9776 Tf 8.582 2.813 Td [(L )]TJ/F24 5.9776 Tf 5.288 2.269 Td [(L +)]TJ/F24 5.9776 Tf 11.875 2.269 Td [(R f k L )]TJ/F22 11.9552 Tf 11.956 0 Td [(f k R +)]TJ/F23 7.9701 Tf 26.284 4.338 Td [(R g k 1 C A N = 1 2 i )]TJ/F23 7.9701 Tf 11.867 4.936 Td [(L )]TJ/F15 11.9552 Tf 11.956 0 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [(R [ g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r f k e )]TJ/F22 11.9552 Tf 11.955 0 Td [(g k ]+ )]TJ/F23 7.9701 Tf 7.314 4.339 Td [(L g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r )]TJ/F23 7.9701 Tf 11.867 4.339 Td [(R +)]TJ/F23 7.9701 Tf 19.075 4.339 Td [(R g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r )]TJ/F23 7.9701 Tf 11.867 4.339 Td [(L )]TJ/F23 7.9701 Tf 7.314 3.454 Td [(L +)]TJ/F23 7.9701 Tf 19.075 3.454 Td [(R f k L )]TJ/F22 11.9552 Tf 11.955 0 Td [(f k R ThesecondtermisobviouslyoftheLandauer-Buttikerform,whiletherst termisacorrection.Thecorrectioncanbeshowntohaveatermofthenetin/out particle-scatteringrate g > < )]TJ/F22 11.9552 Tf 11.955 0 Td [(g < > whichisrequiredtovanishfor N L )]TJ/F22 11.9552 Tf 12.851 0 Td [(N R =0 times )]TJ/F24 5.9776 Tf 5.289 2.813 Td [(L )]TJ/F21 7.9701 Tf 6.586 0 Td [()]TJ/F24 5.9776 Tf 5.289 2.813 Td [(R )]TJ/F24 5.9776 Tf 5.289 2.27 Td [(L +)]TJ/F24 5.9776 Tf 11.875 2.27 Td [(R whichidenticallyvanishesforasymmetricjunctioninthelinearresponse plussomeextraterms.Letting r ; a ; k betheself-energiessolelyduetoamany-body interactioni.e.,theself-energy r 0 =Re r 0 + i )-326(isnotincludedintheseterms, g k )]TJ/F15 11.9552 Tf 11.955 0 Td [( g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r f k e = g r )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [( k )]TJ/F15 11.9552 Tf 11.955 0 Td [( a )]TJ/F22 11.9552 Tf 11.955 0 Td [( r +2 i \051 f k e g a = g r )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [( k )]TJ/F15 11.9552 Tf 11.955 0 Td [( r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a f k e )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 i )]TJ/F22 11.9552 Tf 7.314 0 Td [(f k e g a =2 ig r )]TJ/F21 7.9701 Tf 8.116 -4.977 Td [(1 2 i k )]TJ/F22 11.9552 Tf 11.955 0 Td [(i r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a 1 2 f k e )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F22 11.9552 Tf 7.314 0 Td [(f k e g a 54 Thisiselementwisemultiplicationanddivision,whichiscommutativeandfrequently foundinsoftwarepackagesratherthaninanalyticwork,withherebeinganexception. 136

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Theabovetermsmayberearrangedas, g k )]TJ/F15 11.9552 Tf 11.955 0 Td [( g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r f k e = 1 2 i )]TJ/F27 11.9552 Tf 10.502 35.818 Td [(0 B @ )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r [ g a ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k g a + g r r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a [ g r ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g k 1 C A = 1 2 i )]TJ/F27 11.9552 Tf 10.502 35.817 Td [(0 B @ )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r [ g a ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 11.9552 Tf 12.952 -9.684 Td [([ k ;g a ]+ g a k + [ g r ; r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a ]+ r )]TJ/F22 11.9552 Tf 11.956 0 Td [( a g r [ g r ] )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g k 1 C A = 1 2 i )]TJ/F27 11.9552 Tf 10.502 35.817 Td [(0 B @ )]TJ/F22 11.9552 Tf 9.299 0 Td [(i g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r i k )]TJ/F22 11.9552 Tf 11.956 0 Td [(i r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a ig k + )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r [ g a ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 [ k ;g a ]+[ g r ; r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a ][ g r ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g k 1 C A g k )]TJ/F15 11.9552 Tf 11.955 0 Td [( g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r f k e g k )]TJ/F15 11.9552 Tf 11.955 0 Td [( g a )]TJ/F22 11.9552 Tf 11.955 0 Td [(g r f k e = 1 i )]TJ/F27 11.9552 Tf 8.476 13.806 Td [()]TJ/F22 11.9552 Tf 5.48 -9.683 Td [(g > < )]TJ/F22 11.9552 Tf 11.956 0 Td [(g < > + 1 2 )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(g r [ g a ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 [ k ;g a ]+ 1 2 [ g r ; r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a ][ g r ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g k Therefore,thematerialcurrent N mayberewritten[13][12]as N A plusacorrection N C ,whichare, N A = )]TJ/F23 7.9701 Tf 7.315 4.338 Td [(L i g r )]TJ/F22 11.9552 Tf 11.955 0 Td [(g a )]TJ/F23 7.9701 Tf 11.866 4.338 Td [(R +)]TJ/F23 7.9701 Tf 19.076 4.338 Td [(R i g r )]TJ/F22 11.9552 Tf 11.955 0 Td [(g a )]TJ/F23 7.9701 Tf 11.867 4.338 Td [(L )]TJ/F23 7.9701 Tf 7.314 3.454 Td [(L +)]TJ/F23 7.9701 Tf 19.076 3.454 Td [(R f < L )]TJ/F22 11.9552 Tf 11.955 0 Td [(f < R {110 N C = 1 2 )]TJ/F23 7.9701 Tf 7.314 4.338 Td [(L )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 7.9701 Tf 7.314 4.338 Td [(R )]TJ/F23 7.9701 Tf 7.314 3.453 Td [(L +)]TJ/F23 7.9701 Tf 19.075 3.453 Td [(R 0 B @ g < > )]TJ/F22 11.9552 Tf 11.955 0 Td [(g > < )]TJ/F21 7.9701 Tf 13.151 4.707 Td [(1 2 )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(g r [ g a ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k ;g a )]TJ/F21 7.9701 Tf 10.494 4.707 Td [(1 2 [ g r ; r )]TJ/F22 11.9552 Tf 11.955 0 Td [( a ][ g r ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g k 1 C A {111 Thus,thecorrection-termsmaybecomearbitrarilyimportantforarbitrary N C correspondingtoanynumberofsitesatwhichamany-bodyinteractionmayimpactthe totalcurrent.Specically,anycurrentinaLandauer-Buttikerformmakesreferenceonly totheleftandrightreservoirs,theleft/centerandcenter/rightcoupling,andthedensity ofstatesateachsitewhichonlyobeysasum-ruleatequilibrium.Theideabehindthe formofthecorrection"2{111isthatthematerialcurrentisincompressible/continuous asshownin2{37OnecouldconjecturethatachainofsitesbeginningatreservoirL andendingatreservoirR shouldhavedominancebyintermediaryscatteringstothe extentthatthejunctionisasymmetric. Therefore,itismoreaccurateaswellasmorephysicallyillustrativeandanalytically easiertowritethetotalcurrentasthesumoftwoterms,theformerhavingnoexplicit occurrencesofamany-bodyself-energytheelastic"currentandthelatterhaving 137

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explicitmany-bodyself-energiestheinelastic"current,witheachofthesetermshaving acommonoverallleft-multipleof g r andaright-multipleof g a 2.4.1.2Thematerialcurrentasasumofelastic"andinelastic"terms favor 7 objects,inwhichasumover isimplied, N =)]TJ/F23 7.9701 Tf 19.74 4.936 Td [( ij )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [( g < ij + g > ij f < )]TJ/F22 11.9552 Tf 11.955 0 Td [(g < ij =)]TJ/F23 7.9701 Tf 19.74 4.936 Td [( ij g r ii 0 < i 0 j 0 + > i 0 j 0 +2)]TJ/F23 7.9701 Tf 24.928 -1.794 Td [(i 0 j 0 f < )]TJ/F22 11.9552 Tf 11.956 0 Td [( < i 0 j 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2)]TJ/F23 7.9701 Tf 13.167 5.667 Td [( i 0 j 0 f < g a j 0 j =)]TJ/F23 7.9701 Tf 19.74 4.937 Td [( ij g r ii 0 )]TJ/F22 11.9552 Tf 9.299 0 Td [( < i 0 j 0 f > + > i 0 j 0 f < +2)]TJ/F23 7.9701 Tf 29.481 -1.793 Td [(i 0 j 0 f < )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 7.9701 Tf 7.315 5.668 Td [( i 0 j 0 f < g a j 0 j N =)]TJ/F23 7.9701 Tf 19.74 4.936 Td [( ij g r ii 0 > i 0 j 0 f < )]TJ/F22 11.9552 Tf 11.955 0 Td [( < i 0 j 0 f > +2 X = L;R )]TJ/F23 7.9701 Tf 7.314 5.668 Td [( i 0 j 0 f < )]TJ/F22 11.9552 Tf 11.955 0 Td [(f < g a j 0 j sumtoconstitutethecurrent,inwhichthekineticfermidistributionsmakean appearance, N =)]TJ/F23 7.9701 Tf 19.74 4.936 Td [(L ij g r ii 0 )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [( > i 0 j 0 f < L )]TJ/F22 11.9552 Tf 11.955 0 Td [( < i 0 j 0 f > L +2)]TJ/F23 7.9701 Tf 24.929 4.936 Td [(R i 0 j 0 f < L )]TJ/F22 11.9552 Tf 11.955 0 Td [(f < R g a j 0 j )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [(R ij g r ii 0 )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [( > i 0 j 0 f < R )]TJ/F22 11.9552 Tf 11.955 0 Td [( < i 0 j 0 f > R +2)]TJ/F23 7.9701 Tf 24.929 4.936 Td [(L i 0 j 0 f < R )]TJ/F22 11.9552 Tf 11.955 0 Td [(f < L g a j 0 j = 2 6 4 )]TJ/F23 7.9701 Tf 7.314 4.338 Td [(L ij g r ii 0 )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [( > i 0 j 0 f < L )]TJ/F22 11.9552 Tf 11.955 0 Td [( < i 0 j 0 f > L g a j 0 j )]TJ/F15 11.9552 Tf -142.083 -23.908 Td [()]TJ/F23 7.9701 Tf 7.314 4.339 Td [(R ij g r ii 0 )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [( > i 0 j 0 f < R )]TJ/F22 11.9552 Tf 11.955 0 Td [( < i 0 j 0 f > R g a j 0 j 3 7 5 + 2 6 4 0 B @ )]TJ/F23 7.9701 Tf 7.314 4.338 Td [(L ij g r ii 0 )]TJ/F23 7.9701 Tf 7.314 4.338 Td [(R i 0 j 0 g a j 0 j + )]TJ/F23 7.9701 Tf 7.314 4.338 Td [(R ij g r ii 0 )]TJ/F23 7.9701 Tf 7.314 4.338 Td [(L i 0 j 0 g a j 0 j 1 C A 2 f < L )]TJ/F22 11.9552 Tf 11.955 0 Td [(f < R | {z } f k L )]TJ/F23 7.9701 Tf 6.586 0 Td [(f k R 3 7 5 N in + N el Therstbracketedtermistheinelasticcurrent,andthesecondwhichhasbotha Landauer-ButtikerandFisher-Leeformistheelasticcurrent, N in = )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [(L ij g r ii 0 )]TJ/F22 11.9552 Tf 5.48 -9.683 Td [( > i 0 j 0 f < L )]TJ/F22 11.9552 Tf 11.955 0 Td [( < i 0 j 0 f > L )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [(R ij g r ii 0 )]TJ/F22 11.9552 Tf 5.48 -9.683 Td [( > i 0 j 0 f < R )]TJ/F22 11.9552 Tf 11.955 0 Td [( < i 0 j 0 f > R g a j 0 j {112 N el = )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [(L ij g r ii 0 )]TJ/F23 7.9701 Tf 7.314 4.936 Td [(R i 0 j 0 g a j 0 j +)]TJ/F23 7.9701 Tf 19.076 4.936 Td [(R ij g r ii 0 )]TJ/F23 7.9701 Tf 7.314 4.936 Td [(L i 0 j 0 g a j 0 j 2 f < L )]TJ/F22 11.9552 Tf 11.955 0 Td [(f < R {113 TheLandauer-Buttikerformof2{113impliesthatthereisatransmission function"-likeobject. 2.4.1.3Interpretationsofthetransmissionpluscorrectionvs.elasticplus inelasticdecompositions TheLandauer-Buttikerformof2{113impliesthatthereisatransmission function"-likeobject,astherewasin2{110.Bothofthesearesimilarinthatthey aresymmetricundertheinterchange L R ,butdierinthatneitherareinterpretable astheprobabilityoftransmissionofa single energy charge-carrier.Inthepresenceof 138

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interparticle-coupling,thelabel nolongercorrespondstoasingleparticle,butrather justtheenergyatwhichtheremaybeanyproclivityofexchangesofenergybetween particles.Inthecaseoftheelectron/phononinteractionself-energiesinSections2.3.10.2 and2.3.10.3, 2.4.1.4Sommerfeldexpansionoftheinelasticmaterialcurrent Thereisthermal-suppression[23]oftheleft-current-inscatteringandright-current-outscattering contributionstotheinelasticmaterialcurrent2{112.Hence,itisnaturaltoeecta Sommerfeldexpansion2{47ofthesametoisolatetheeectofanonzerotemperature. Approximatingthe 7 phononicGreenfunctionsasfree 55 andtakingtheirsupport[46] entirelyfrom givingreadily-integrabledeltafunctions, 7 F !mij / Z d! 0 2 g 7 0 + !;i 0 j 0 d ? 0 m Z d! 0 2 g 7 0 + !;i 0 j 0 b ? e;! 0 X s = s 0 )]TJ/F23 7.9701 Tf 6.587 0 Td [(s m = 1 2 g 7 + m ;i 0 j 0 b ? e; m + g 7 )]TJ/F21 7.9701 Tf 6.587 0 Td [( ;i 0 j 0 b 7 e; m 55 AlthoughacasewasmadeinSection2.3.10.3fortheessentialityofthe polarization-bubbletothepnononicenergycurrent,aqualitativeunderstandingof theinelasticcurrent2{112ispursuedhere.Thus,assimpleaspossibleaformfor convolving d ? 0 m issought,andthisisadelta-function,whichassumesthephononshave arbitrarily-smalldecay-rate duetothe th lead.Thiselicitsaseriouscaution:this meansthemode m localizedtothecenterregionisofinnitelifetime,requiringitto equilibratenotwiththe th lead'stemperature,butwiththecenter-regiontemperature. Mathematically,thisisrealizedastheimportanceoftheinntesimalthatwasdiscarded becauseitwasdwarfedbythedecay-rate duetothe th leadtoobtain2{86. 139

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Bytheaboveformfor 7 F !ij ,oneofthetermsintheinelasticcurrentdropsout,as follows, lim 0 lim T L 0 < !mij f > L = M mii 0 M mj 0 j 1 2 g < + m ;i 0 j 0 lim T L 0 f > L / M mii 0 M mj 0 j 1 2 L )]TJ/F21 7.9701 Tf 6.587 0 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(! )]TJ/F23 7.9701 Tf 6.587 0 Td [( L = M mii 0 M mj 0 j 1 2 0 lim 0 lim T R 0 > !mij f < R = M mii 0 M mj 0 j 1 2 g > )]TJ/F21 7.9701 Tf 6.586 0 Td [( m ;i 0 j 0 lim T R 0 f > R / M mii 0 M mj 0 j 1 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [( R + R )]TJ/F23 7.9701 Tf 6.586 0 Td [(! = M mii 0 M mj 0 j 1 2 0 Oneisthenleftwiththefollowingsimpliedinelasticcontributiontothematerial current'sband-density,whichwouldconstitutetheleadingzerotemperatureterminthe Sommerfeldexpansion2{47onewouldeectupontheintegralgivingthecurrent, lim T ; N in =)]TJ/F23 7.9701 Tf 19.74 4.937 Td [(L ij g r ii 0 > i 0 j 0 f < L g a j 0 j +)]TJ/F23 7.9701 Tf 19.075 4.937 Td [(R ij g r ii 0 < i 0 j 0 f > R g a j 0 j {114 Ofcourse,tohavethethermoelectriceect,atleastoneleadmustbeatan above-zerotemperature,givinganasymmetrytotheproblem.Choosingthe right left lead tobeofzerotemperaturegivespartialsuppressionto out in -scatteringofelectronsi.e.,the self-energy ? .Boththeleftandtherightinelasticcurrents2{112haveanegativeterm thatvanishesduetothesetwopossiblezero-temperaturesuppressions,givingincreases inthesecurrents.Intheresonant-regimethatis,)]TJ/F22 11.9552 Tf 265.427 0 Td [(; beingthesmallestenergy-scalesin theproblem,onecaninterprettheserespectiveincreasesasfollows.Anincreasein N in;L correspondstoanincreaseofcharge-carriersofenergy enteringfromlead L occupation oftheresonance,andsubsequentlyleavingitatenergy )]TJ/F15 11.9552 Tf 11.963 0 Td [( m ,havinggivenenergy m to exciteamodem phonon,inwhichcaseonlycharge-carriersscatteredoutby > .Similarly, anincreasein N in;R correspondstoanincreaseofcharge-carriersofenergy + m entering occupationoftheresonance,andsubsequentlyleavingitintolead R atenergy ,having givenenergy m toexciteamodem phonon,inwhichcaseonlycharge-carriersscattered outby < 140

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Althoughthisisnotanewinterpretataion[23],itisimportanttomentionthis interpretationinthecontextofthermoelectricity.Takingthewide-band-limitandthus using2{49,thequantity N in =2 R 2 W )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d! 2 N in isexplicitlycalculatedas, N in = L Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W )]TJ/F23 7.9701 Tf 7.314 4.937 Td [(L ij g r ii 0 > i 0 j 0 g a j 0 j d! + 2 W Z R )]TJ/F23 7.9701 Tf 7.315 4.937 Td [(R ij g r ii 0 < i 0 j 0 g a j 0 j d! + 6 0 B @ T L 2 )]TJ/F23 7.9701 Tf 11.866 4.338 Td [(L ij g r ii 0 > i 0 j 0 g a j 0 j 0 = L )]TJ/F22 11.9552 Tf -115.22 -23.908 Td [(T R 2 )]TJ/F23 7.9701 Tf 11.866 4.338 Td [(R ij g r ii 0 < i 0 j 0 g a j 0 j 0 = R 1 C A Analytically,itisnotworthwhiletocarryouttheProductRuleuponthefourfold-product above.Innumericalpractice,oneeectsnumericalevaluationofthederivativeand evaluatesitattherespectivechemicalpotentials.Furthermore, g r ij and g a ij constitute anoverallmultipleuponthetermintheexpansionanditsderivativetakessupport fromevaluationattherenormalizedresonance ~ U i ,whilethe th lead'sescape-rates)]TJ/F23 7.9701 Tf 105.384 4.339 Td [( ij arerelativelyconstant.Thus,thederivativetakesitssupportfromoutscattering > ij of electronsofenergy L andinscattering < ij ofelectronsofenergy R 2.4.1.5Thenon-interactinglimit Thenon-interactinglimitofeither2{110or2{113isreadilyobtained,andbearsa Landauer-Buttikerform, N 0 = )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 7.9701 Tf 7.314 4.936 Td [(L ij g r 0 ii 0 )]TJ/F23 7.9701 Tf 7.314 4.936 Td [(R i 0 j 0 g a 0 j 0 j +)]TJ/F23 7.9701 Tf 19.076 4.936 Td [(R ij g r 0 ii 0 )]TJ/F23 7.9701 Tf 7.315 4.936 Td [(L i 0 j 0 g a 0 j 0 j 2 f < L )]TJ/F22 11.9552 Tf 11.955 0 Td [(f < R 2 T 0 f < L )]TJ/F22 11.9552 Tf 11.955 0 Td [(f < R = E 0 =! {115 Thefunction T istheprobabilityoftransmissionofanelectronofenergy from theleftreservoirtotherightreservoir,andthisform2{115forthematerialcurrentis afunctionaleof T alone.Althoughthecenter-regionHamiltonian2{35determines T usually T isunabletobeintegratedtoclosedformwiththeFermifunctiondierence f < L )]TJ/F22 11.9552 Tf 12.026 0 Td [(f < R .However,approximationsarepossible.Forinstance,acenter-regionHamiltonian intheresonant-regimeyieldsaLorentziantransmissionfunction.Mathematically,a Lorentzian = = x )]TJ/F22 11.9552 Tf 12.567 0 Td [(x 0 2 + 2 ischaracterizablebythehorizontalcoordinate x 0 at whichittakesitsmaximum/peakvalue,ameasureofspread aboutthispeakvalue, hasaxedunit-amountofareai.e.,theLorentzianobeysasumrule,anddecaysas 141

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x )]TJ/F22 11.9552 Tf 12.416 0 Td [(x 0 2 for x )]TJ/F22 11.9552 Tf 12.415 0 Td [(x 0 .AfunctionwhichintegrateswithaFermifunctionandhas theseexactattributesexceptitdecaysexponentiallyinsteadofreciprocal-quadratically isthefunction N 0 e j x )]TJ/F23 7.9701 Tf 6.587 0 Td [(x 0 j = whichobeysasumruleforasuitablenormalization N 0 withthedisadvantagethattoosmallavalueof willcauseoating-pointoverowin itsnumericalevaluationduetotheexponentialproducingsubtractionsbetweentwenty digitnumericalquantitieswherethecurrentisoforder N 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 ,meaningthe signicantdigitslietwenty-twodecimalplacesdeepintothemantissa.Another possibleregimeiswherethecenter-regionisalongstraightchainof N C siteswith tunneling-energies )]TJ/F22 11.9552 Tf 9.299 0 Td [(W L ; )]TJ/F22 11.9552 Tf 9.298 0 Td [(W 12 ; )]TJ/F22 11.9552 Tf 9.299 0 Td [(W 23 neartheleftmost-sites1 ; 2 ; 3 ;::: andthetunneling energies )]TJ/F22 11.9552 Tf 9.298 0 Td [(W R ; )]TJ/F22 11.9552 Tf 9.299 0 Td [(W N C ;N C )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 ; )]TJ/F22 11.9552 Tf 9.298 0 Td [(W N C )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ;N C )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 neartherightmost-sites N C ;N C )]TJ/F15 11.9552 Tf 12.217 0 Td [(1 ;N C )]TJ/F15 11.9552 Tf 12.217 0 Td [(2 ;::: ontheorderofin-reservoirtunnelingenergies )]TJ/F22 11.9552 Tf 9.298 0 Td [(W ,andnearlyzeroresonantnearthe chain'scentergraduallylocalizinganite-widthwindowofoccupancy-energies,where WhitneyandButtiker[47]nd T tobewell-describedbyaboxcar-function. 2.4.2TheDiscontinuousEnergyCurrentintheElectrons Takingthequantum-statisticalexpectationof2{38,introducingtheGreenfunctions 2{52,andusing2{91,andFouriertransformingwithrespecttoT t )]TJ/F22 11.9552 Tf 12.118 0 Td [(t 0 togotothe domainortakingashortcutandjustnotingthat,aftersaidFouriertransformation,the setofstepswillexactlymatchthosetakenbeforeobtaining2{108, iE L R = N C X i =1 1 Z e i! T D W i c y it 0 c 1 t + W i c y 1 t 0 c it E d T = N C X i =1 )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(W i + i! g < i 1 + W i )]TJ/F22 11.9552 Tf 9.298 0 Td [(i! g < 1 i = i!N L R AsclaimedinSection2.2.2.2where2{36wasobtained,energyisconservedonly whenonetakesthequantum-statisticalexpectation.Hence,theenergy-current-band-density E atelectron-energy 2 [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 W; 2 W ]issimplyrelataedtotheelectron-number-current's 142

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band-density2{108as, E L R = !N L R ! 2 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 W; 2 W ]; E L R =2 Z 2 W )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d! 2 E L R {116 2.4.3TheDiscontinuousEnergyCurrentinthePhonons Takingthequantum-statistical-expectationof2{41andintroducingoccurrencesof thephonon-densityGreenfunctions2{55, i D H ph E = X qm q m b y q b m )]TJ/F22 11.9552 Tf 11.955 0 Td [(b y m b q = X qm q m d < q m )]TJ/F15 11.9552 Tf 14.021 3.154 Td [( d < mq Itisnotedthattheaboveisthepreciseanalogueofthecalculatedexpectations 2{108andof2{116,howeverwiththe barred Greenfunctionsforwhichthis dissertationdoesnottreatFeynmandiagramsof!of2{55.Thisrequirestheinverse transform"of2{56,butthistransformdoesnotexist.Oneseesthisbywriting, d < + = 1 2 d < + + d > )]TJ/F23 7.9701 Tf 6.586 0 Td [(! + d > )]TJ/F23 7.9701 Tf 6.587 0 Td [(! )]TJ/F22 11.9552 Tf 11.955 0 Td [(d > )]TJ/F23 7.9701 Tf 6.587 0 Td [(! = 1 2 d < + )]TJ/F22 11.9552 Tf 11.955 0 Td [(d > )]TJ/F23 7.9701 Tf 6.586 0 Td [(! + d > )]TJ/F23 7.9701 Tf 6.586 0 Td [(! ; d r + = 1 2 d r + + d a )]TJ/F23 7.9701 Tf 6.586 0 Td [(! )]TJ/F15 11.9552 Tf 14.021 3.155 Td [( d a )]TJ/F23 7.9701 Tf 6.586 0 Td [(! ; d > )]TJ/F23 7.9701 Tf 6.587 0 Td [(! = 1 2 d < + )]TJ/F22 11.9552 Tf 11.955 0 Td [(d < + + d > )]TJ/F23 7.9701 Tf 6.587 0 Td [(! + d < + = 1 2 d > )]TJ/F23 7.9701 Tf 6.587 0 Td [(! )]TJ/F22 11.9552 Tf 11.955 0 Td [(d < + + d < + ; d a )]TJ/F23 7.9701 Tf 6.587 0 Td [(! = 1 2 d a )]TJ/F23 7.9701 Tf 6.587 0 Td [(! + d r + )]TJ/F15 11.9552 Tf 14.021 3.155 Td [( d r + ; Aworkaroundistointergratethebanddensity J tocalculatethephononiccurrent J = R d! 2 J .Onemustusethefactthatnotonly J = R 2 W )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W d! 2 J ,butalsothechange ofvariablesholdingforanydeniteintegrationi.e.,thefollowingisamathematical identity, J = )]TJ/F27 11.9552 Tf 11.291 16.273 Td [(Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W 2 W d )]TJ/F23 7.9701 Tf 6.587 0 Td [(! 2 J )]TJ/F23 7.9701 Tf 6.587 0 Td [(! = Z 2 W )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W dy 2 J y = Z 2 W )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W dy 2 J )]TJ/F23 7.9701 Tf 6.587 0 Td [(! Onethenobtainsthebanddensity J ofphononicenergycurrentas, iJ = i J + J )]TJ/F24 5.9776 Tf 5.756 0 Td [(! 2 = 1 2 X qm ! q d < q m )]TJ/F15 11.9552 Tf 14.021 3.154 Td [( d < mq + d < q 0 m )]TJ/F22 11.9552 Tf 9.298 0 Td [(! )]TJ/F15 11.9552 Tf 14.021 3.155 Td [( d < mq )]TJ/F22 11.9552 Tf 9.299 0 Td [(! = 1 2 X qm q d < q m )]TJ/F22 11.9552 Tf 11.955 0 Td [(d < mq Thisresultagainbearsresemblancetoitselectroniccounterpart2{116.The dierence,however,entersinthedensityofstatesoverthe electronic energyband 2 143

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[ )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 W; 2 W ]andthe phononic energyband 2 [0 ; p 2 0 ], J L R = Z 0 p 2 0 d! 2 J L R = 1 2 X qm Z 0 p 2 0 d! 2 j j d < q m )]TJ/F22 11.9552 Tf 11.955 0 Td [(d < mq ; 2 [0 ; p 2 0 ];{117 2.4.4Thermodynamicsinterlude:theLandauer-Buttikercurrentandthe thermalstateofthereservoirs Consider[48]tworeservoirs,left L andright R ,attemperatures T L and T R Theyareofxedvolumes V L and V R sopressuredoesnotenterthecalculation.Their respectiveparticle-populations N L and N R allowedtovaryi.e.,byanelectricalcurrent duetorespectivedisparatepotentials L and R .Theirrespectiveenergeticcontents E L and E R arealsoallowedtovarydueto dE S;N = T dS + dN inwhich T dS = dQ Theconservationofmassandenergyimply, dN =0= dN L + dN R dN L = )]TJ/F22 11.9552 Tf 9.299 0 Td [(dN R ; dE =0= dE L + dE R dE L = )]TJ/F22 11.9552 Tf 9.299 0 Td [(dE R ; Letthe th reservoirhaveanenergy E dependinguponitsparticle-content N and entropy S .Then,acalorimetristeectingachangeofenergy dE tothe th reservoir,for = L;R ,wouldobservethat dE R isexpressibleas, dE L = T L dS L + L dN L ; dE R = T R dS R + R dN R = )]TJ/F22 11.9552 Tf 9.298 0 Td [(dE L = T R dS )]TJ/F22 11.9552 Tf 11.668 0 Td [(dS L + R )]TJ/F22 11.9552 Tf 9.299 0 Td [(dN L ; Thissystemofequationsmaybesolvedforthequantity dE L =dN L .Notingthatthe calorimetristwouldnditeasiesttovary T and insteadof N R and S ,theabove systemofequationsmaybesolvedtoobtain T L R dN L )]TJ/F22 11.9552 Tf 12.399 0 Td [(T L T R dS )]TJ/F22 11.9552 Tf 12.399 0 Td [(T R L dN L = dE L T L )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R .Thisyieldsa dE L =dN L of, dE L dN L = T L R )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R L T L )]TJ/F22 11.9552 Tf 11.956 0 Td [(T R )]TJ/F22 11.9552 Tf 17.643 8.087 Td [(dS dN L T L T R T L )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R = dE dN {118 Inthelaststepof2{118,itwasnotedthattheexpressionissymmetricunder interchangeofreservoir-labels, L R ,because dN L = )]TJ/F22 11.9552 Tf 9.298 0 Td [(dN R and T L )]TJ/F22 11.9552 Tf 11.271 0 Td [(T R = )]TJ/F15 11.9552 Tf 9.299 0 Td [( T R )]TJ/F22 11.9552 Tf 11.271 0 Td [(T L producingtwocancellingminus-signs. 144

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Betweenthereservoirs L;R andtheuniverse,thereservoirs'entropymustalways remainconstantorincrease, dS 0.Thus,thecalorimetristmustconsiderthecases T L 7 T R separately.Thecase T L T R givesalowerboundupon dE L =dN L T L 7 T R dE L dN L = T L R )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R L T L )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R )]TJ/F22 11.9552 Tf 17.643 8.088 Td [(dS dN L T L T R T L )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R Q T L R )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R L T L )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R Now,supposethecalorimetristreversibly dS =0= dS L + dS R eectedthis energy-exchangebetweenreservoirs.Theywouldseethatthequantity dE L =dN L is boundedinsuchamannerthatapplyingtheminimizing-operationtobothsidesofits denitionyields, min dE dN =min T L R )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R L T L )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R )]TJ/F22 11.9552 Tf 17.643 8.088 Td [(dS dN L T L T R T L )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R = T L R )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R L T L )]TJ/F22 11.9552 Tf 11.955 0 Td [(T R {119 Now,supposethetworeservoirswerenanowires,andthecalorimetristwasinterested inthethermoelectriceect,whereatemperature-gradient T L )]TJ/F22 11.9552 Tf 9.298 0 Td [(T R sustainsapotential-gradient R )]TJ/F22 11.9552 Tf 10.929 0 Td [( L andthusamaterialcurrent N = N L + N R .Iftheconservationofmassandenergy simultaneouslyholdtrueacrosstheregionbetween 56 thereservoirs,thecurrentadmits descriptionbyaLandauer-Buttikerformula.Thismeansonecanassignaprobability T = T toacharge-carrierofenergy ,amost-probabledistribution t < = t < ofallcharge-carriers[35]ofenergy inthe th reservoir,givingamaterialcurrentof N / R d! T t
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suchthat t
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CHAPTER3 NONLINEARTHERMOELECTRICRESPONSEINRESONANTTUNNELING 3.1TheoryoftheResonantRegime Thefeaturesoftheresonant-regimearetobeobtainedfromtheGreenfunction denitions.Theresonant"regimeisdenedashavingtunnelingmatrixelements )]TJ/F22 11.9552 Tf 9.298 0 Td [(W c.f.,theHamiltonian H C in2{35betweenthecenter-regionthatareamongthe smallestenergy-scalesintheproblem.Specicallythe )]TJ/F22 11.9552 Tf 9.299 0 Td [(W 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 W constituteanescape rate)]TJ/F23 7.9701 Tf 31.376 -1.793 Td [(! whichisontheorderofroomtemperatureinbulkcopper[50], )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(! = X )]TJ/F23 7.9701 Tf 7.314 4.936 Td [( = X W 2 =W 1 )]TJ/F28 7.9701 Tf 6.586 11.557 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(U 2 W q 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(U )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W 2 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W X 1 T room {1 Tunnelingthroughalong-livedquantumlevelbecomesthedominantmodeof transportintheregimecharacterizedby3{1,andthespectral-densityisapproximately theLorentzianofFig.1-1. Anotherfeatureoftheregimedenedby3{1isthefactthattheescaperates)]TJ/F23 7.9701 Tf 416.889 4.338 Td [( taketheirsupportfromtheresonance U 1 .Therefore,nophysicalcontentofthemodelis lostbyeectingthefollowingapproximation, W W )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [( )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [( U 1 = q 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( U 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(U )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W 2 {2 3.1.1Spectral-densityofacollectionofinter-coupledlong-livedquantum levels Consideracollectionoflong-livedquantumlevels U i inwhichthereareinter-level tunnelingmatrix-elements )]TJ/F22 11.9552 Tf 9.299 0 Td [(W ij $)]TJ/F22 11.9552 Tf 26.406 0 Td [(W ji fortunnelingbetweenthe i th and j th levels. Agivenquantumlevelcouldbethatofasingleatomorofanentirebuckyball:bothare roughlydescribableashavinganenergy-costof U i tooccupy.Thusitisthequalitative shapeofthenanoscalecenter-regionthatdetermineshowmanyisolatedlevelsthereare. Twoisolatedlevelsareformedbyanano-systemcalledat-stub[32]andasingleisolated levelisaqualitativemodelofscanningtunnelingmicroscopy[51].Suchasystemgivesa spectraldensitythatisaseriesofapproximately-Lorentzianpeaks. 147

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3.1.2Thespectraldensityofasinglelevel Thesinglesite N C =1center-regionspectraldensityhasgeneralform[34]when dressedbyaself-energy r =Re r + i )]TJ/F23 7.9701 Tf 11.867 -1.793 Td [(! +Im rF whoseimaginarypartincludesa coupling)]TJ/F23 7.9701 Tf 54.138 -1.794 Td [(! = P )]TJ/F23 7.9701 Tf 7.314 4.338 Td [( duetoacollectionofreservoirs.Omittingsite-indices,onegets, A e )]TJ/F23 7.9701 Tf -3.928 -7.892 Td [(! = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2Im g r = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2Im 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( r = 2)]TJ/F23 7.9701 Tf 17.72 -1.793 Td [(! +Im rF ! )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(Re r 2 +)]TJ/F23 7.9701 Tf 23.628 -1.793 Td [(! +Im rF 2 {3 Atequilibrium,thisspectraldensity3{3followsthesum-rule2{63andentersinto thecontributions2{110and2{113tothecurrent. Theparameterregimesofasinglespectrallevelforthecharacteristicsofcurrent vs.voltagearedeterminedbythedistancetheequilibriumchemicalpotential = L + R = 2isfromthenegativebandedge )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 W comparedtootherenergyscalesinthe problemparticularlytheenergy m ofthe m th localizedphononmode.Thisiscalledthe depthoftheFermisea.AFermiseathatissaid[23]tobedeep"fortheproblemofthe electron/phononinteractionisoneforwhich j +2 W j m ,andthisissatisedby, U 1 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 7 W = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 9 WW L = W R =0 : 1 W =0 : 2 Wg = M 2 = 2 =0 : 1{4 Conversely,onecouldhave j +2 W j & ,inwhichcasetheFermiseaissaid[23]to beshallow"comparedtothephononmodeenergy.Thisissatisedby, U 1 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 1 W = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 5 WW L = W R =0 : 15 W =0 : 7 Wg = M 2 = 2 =0 : 1{5 TherenormalizationofthespectraldensitybyRe hasarst-ordercontribution proportionaltotheperturbativeparameter g = )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(U 1 ,meaningthat)]TJ/F23 7.9701 Tf 86.655 -1.793 Td [(U 1 cannotbemade arbitrarilysmall,givingalocalizedenergylevelofinnitelifetimeandalowerlimitupon )]TJ/F23 7.9701 Tf 7.314 -1.794 Td [(U 1 foragiven g .TheseregimesareillustratedinFig.3-1. 148

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Figure3-1.DepthoftheFermiseaforaresonanttunnelingdevice. 3.2TheThermoelectricResonantGenerator 3.2.1Currentwithanammeterasload Totheextentthatthethermoelectricgeneratorislinearfortheambienttemperatures ofplanetEarth T room 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 ,itis,itscurrent I = )]TJ/F22 11.9552 Tf 9.298 0 Td [(e N asafunctionofvoltage V T is characterizablebyaxisintercepts:thevertical-interceptmax I andthehorizontal-intercept E th .Themaximumcurrentandzerodroppedvoltageloadisaperfectammeteroccursat V T =0iscalculatedinFig.3-2asafunctionofelectron/phononcoupling g A 11 Theeectofamany-bodyinteractionself-energy =Re + i Im isfor i Imto broadenandforRe toshifttheresonantspectralfunction3{3.Ifthethermoelectric currentisdrawnintoanexperimentalistperformingscanningtunnelingspectroscopy, theeectofthemany-bodyinteractionwouldshowupasashiftinthedierential conductancei.e.,thequantity dN=dV T .Fig.3-2,conrmsthis:thedependenceof N V T =0upon g A 11 iswithoutabruptdipsorspikes;thedependenceisstrictly linear.Theelectron/phononinteractionisknowntoproduceaseriesofdelta-function spikestimesaPoissondistribution[52]inthespectralfunctionofanisolatedelectronic levelinteractingwithabathofEinsteinphonons,andthesespikesarebroadenedinto approximatelyLorentzianswhenthisisolatedlevelisbroughtintocontact[23]witha leftandarightreservoir.Theswitch-onoftheinelasticcontributiontothecurrentgives 149

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Figure3-2.Eectofhot-leadtemperture T L Numbercurrentatzerochemicalpotential dierence.ShallowFermiseaisinfaintdottedlineforreference,whilethe deepFermiseaisshownasasolidline. risetocurrent-spikesatpotential-gradients V T whicharewell-beyondthelimitedinterval 2{6. 3.2.2Voltagewithavoltmeterasload Theminimumzerocurrentandmaximumdroppedvoltageoccursat V T = E th theloadisaperfectvoltmeter.Forthesameregimes3{4and3{5,theeectofthe many-bodycoupling g andhot-leadtemperature T L aredetermined,andshowninFig. 3-3. Clearly,fortheshallowFermisea3{5andwiderresonance)]TJ/F23 7.9701 Tf 321.13 -1.793 Td [(U 1 theresonantspike incurrentisaccessedmorereadily,andinsteadofalineardependenceupon g thereis curvature.However,thiscurvatureisthermallysmearedoutathighertemperatures meaningthatthereisacompetitionbetweenincreasingthethermoelectriceectby increasing T andsharpeningtheresonancetoachievetheMahan-Sofo[16]divergent 150

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Figure3-3.Maximumthermalbiasatzerochemicalpotentialdierence.ShallowFermi seaisinfaintdottedlineforreference,whilethedeepFermiseaisshownasa solidline. gureofmerit.Consequently,thereisatransitionfromtheeectof g increasing E th = E th 0 atlower T L todecreasing E th = E th 0 athigher T L ThedeepFermisea3{4featuresanunremarkablylineardecreasein E th = E th 0 inFig. 3-3duesolelytotheshiftingandbroadeningofthespectralfunction. 3.3TheNonlinearMaterialCurrent Thenonlinearmaterialcurrent N ispresented.Incontrasttothelinearregimewhich requiredthedepthoftheFermiseatobeshallowenoughtoaccessonlytherstvestiges oftheresonantcurrent,thenonlinearregimefeaturesinascendingorderof V T inwhich theyappearthewholeresonance,theresultingcurrent-saturation,thefollowingnegative dierentialresistance,andthephonon-assistedtunneling. 3.3.1Thelargegradientsaccessiblebythedriventhermoelectric Thematerialcurrentofelectrons N iscalculatedasafunctionof = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 V T = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1= L )]TJ/F22 11.9552 Tf 12.489 0 Td [( R inFig.3-4.Theonsetofresonant-current,negativedierentialresistance,and 151

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Figure3-4.Numbercurrent N solidandwithoutelectron/phononinteraction N 0 dashedvs.potential-gradient V T acrossthethermoelectricblackboxat T L =T r =3 ; 5 ; 7 ; 9blue,green,purple,andredsolidlines,respectivelyand T R =0accentuatingquantumfeaturesinashallowFermiseaand g =0 : 15. phonon-assistedtunnelingarenonlinearfeaturesthatareomittedwhenconsideringonly thelinearresponse. InFig.3-4, N 7 and T 7 0correspondstonumbercurrentpumpedfromhotto coldspontaneouspluspumpedowfromhottocold.Onlyspontaneousowof Q e )]TJ/F15 11.9552 Tf -432.989 -28.246 Td [(fromhottocoldoccursinthesecondquadrant, ,givingathermoelectricgenerator.For referencethenon-interactingcurrent N 0 atthesametemperaturesarelabeledwiththe samecolorsexceptwithfaintdottedlines.Thefeaturesoftheplot,labeled A;B;C;D;E areasfollows: 152

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Figure3-5.Numbercurrentmagnitudeasafunctionofeachpossiblegradient V T and foradeepFermiseashowingtheevolutionofsaturationof E th tothe values j U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( j ,hot/coldasymmetrywithrespectto T ,anda singleundulationinthecurrentduetophononassistedtunneling. AThermoelectricgeneratorregioninthesecondquadrant,andblownupinthe insert-plot.Here, N> 0and V T < 0sothatpower P L = V T )]TJ/F22 11.9552 Tf 9.298 0 Td [(N = V L I L deliveredtotheloadispositivethenonlinearSeebeckeect.Atlowtemperataures bluelines N;N 0 areOhmic/linear.For T L '1 ,thenon-interactingthermal voltageis E th 0 = 2 )]TJ/F22 11.9552 Tf 12.784 0 Td [(U .Thereisalmostneveranynonlinearfeaturesinthis generatorregimeunlessthetemperatureoftheleft-leadismadewellinexcessofthe already-depressedmelting-temperatureofanynano-system.FeaturesBthrough Eareonlyaccessiblewhenoneaddsthedrivingagentapplyingbias e E ap ,asshown inthecircuit. BOnsetofresonant-currentsoccurringbeyond )-326(of N V T = E th =0,whichinthe limitof T L '1 isapproximatelyanarctangentfunction. CElectron/phononinteractiontransitionsbetweenaugmentingandreducingthe numbercurrent. DNegativedierentialresistancecausedpartlybytheescaperates p 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( V T = 2 W 2 approachingzero. EPhonon-assisted[23]tunneling. 153

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Figure3-6.Numbercurrentmagnitudeasafunctionofeachpossiblegradient V T and forashallowFermisea3{5withoutelectron/phononcoupling g =0. ConsiderFig.3-6,whichisaplotofnumbercurrent0 : 8 j N j =T room 0asafunctionof eachpossiblegradient V T and forzeroelectron/phononcoupling g =0andashallow Fermisea. Fig.3-6hasthesamequalitiesasFig.3-9evenwithdisparateFermiseadepths exceptwiththeabsenceofphonon-assistedtunneling. 3.3.2Eectofelectron-phononcouplinguponnonlinearnumbercurrent 3.3.2.1Lowtemperature ConsiderFig.3-7,aplotofnumbercurrent N asafunctionofpotential-gradient V T acrossathermoelectricblackboxforathickFermisea3{4,withthinFermisea3{5 shownasdashedforreference.Variouselectron/phononcouplings g areused,andthecold leadisheldatabsolutezero T R =0toaccentuatequantumfeatures.Thehotleftleadis heldatareasonableandmoderatetemperature T L =2 T room .Anypartofthecurrentwith aLandauer-expressionisanoddfunctionof V T if T L = T R becauseatzerotemperature theFermifunctiondierence f < L )]TJ/F22 11.9552 Tf 11.955 0 Td [(f < R iscorrespondinglyanoddfunctionof V T 154

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Figure3-7.Numbercurrentasafunctionofpotential-gradientacrossathermoelectric blackboxforathickFermisea,3{4withafaint-dashedthinFermisea3{5 forreference. ForboththeshallowanddeepFermisea,theeectoftheelectron/phononinteraction istobroadenandshiftthespectralfunction,yieldingasmearingoffeatures A;B;C;D ItisnotuntiltheFermifunctionsin2{113switchonthecollisionalself-energies 7 subsequentlyyieldingphonon-assistedtunneling. 3.3.2.2Hightemperature Next,considerFig.3-8,whichisexactlythesameasFig.3-7exceptatmuch highertemperature T L =10 T room .Thefeatures A;B;C;D;E areasbefore.Inthe phonon-assistedtunneling E ,thereissmearingofthehumpfortheowof N fromhotto cold,butnotforcoldtohot. Themaximumthermalbias E th infeature A isgreatlydecreasedby g inthiscase. Thenon-interactingmaximumthermalbias E th 0 canalsobeseentosaturatetothevalue )-167(j U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( j atsuchhightemperture.Thissaturationisshownintheheat-plotFig.3-9. 155

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Figure3-8.Numbercurrentasafunctionofpotential-gradientacrossathermoelectric blackboxforathick[23]Fermisea. Figure3-9.Numbercurrentmagnitudeasafunctionofeachpossiblegradient V T and foradeepFermiseashowingtheevolutionofsaturationof E th tothe values j U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( j ,hot/coldasymmetrywithrespectto T ,anda singleundulationinthecurrentduetophononassistedtunneling. ConsiderFig.3-9,whichisaheatplotoftheabsolutevalueoftheinteracting dimensionlessnumbercurrent0 : 7 j N j =T room 0asafunctionofeachpossiblegradient V T and foramoderateelectron/phononcoupling g =0 : 15. 156

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Figure3-10.Number-current-discontinuity N = N L )]TJ/F22 11.9552 Tf 11.956 0 Td [(N R inunitsof T room asafunction of V T ,whichunexpectedlylessensforashallowFermisea. Theheatplotisabletoshowthecontinuousevolutionofcertainfeatureswith respectto T .Thehorizontalaxisputsthegradient inunitsof U 1 )]TJ/F22 11.9552 Tf 12.499 0 Td [( =0 : 2 W to showthesaturationof E th tothisvaluethatoccursforaninnitely-hotleadsendinga thermoelectriccurrentintoaleadatabsolutezero.Thereisalsoclearhot/coldasymmetry thatevolvesin T .Finally,onecanseethatthephonon-assistedtunneling,afainttrace ofblueathighbiases,undergoesaslightundulationwithrespectto T 3.3.3Divergenceofthenonlinearmaterialcurrent Beforeplottinganycurrents,onemustverifythattheyarecontinuousandhavezero divergence.Ifthereisanydiscontinuity,itmustevolvefrombeingzeroat g =0tosome nitevalue.Furthermore,recallingthattheself-energies ; arecalculatedbyanumber ofiterationsoftheself-consistentBornapproximationaconservingapproximation, thediscontinuity N mustlessenandatworstsaturateatsomelowervalueasthe numberofiterationsincreases.TheserequirementsareveriedinFig.3-10,wherethe number-currentdivergence N = N L )]TJ/F22 11.9552 Tf 11.955 0 Td [(N R isplottedasafunctionof = V T 157

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Figure3-11.Number-current-discontinuity N = N L )]TJ/F22 11.9552 Tf 11.956 0 Td [(N R asafunctionof V T ,exceptat highertemperature Somefeaturesoftheplotofthisdivergencearelettered A and B inFig.3-10,and theyaredescribedasfollows: AAlow-couplingspikeoccursat V T )]TJ/F22 11.9552 Tf 21.918 0 Td [(W oncoldside" N owsfromcoldtohot. BLargerdiscontinuityforphonon-assistedcurrents. Hence,thenumbercurrentthroughthenano-junctionindeedisconserved.The numbercurrentdivergencecanbeplottedathighertemperature,andthisisdonein Fig.3-11.ThefeaturesandtrendsofFig.3-10areveriedtobepresent.Onemay alsoverifythesmallnessofthemagnitudeofthedivergenceofthenumbercurrentasa functionofbothgradients T and ,andthisisdoneinFig.3-12. Indeed, N=T room remainsontheorderof10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 asitdidinFigs.3-11and3-10.One canalsoseethat N=T room becomeslargeratlarger T thatareintheresonantportion of V T forthecaseofthethermoelectricdevicebeing driven correspondingtoroughlythe rstandthirdquadrants.Forathermoelectricdevicedrivingitssurroundingsthelinear regimeineitherthesecondorfourthquadrants,thedivergenceissignicantlysmaller. 158

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Figure3-12.Numbercurrentdivergencemagnitudeinunitsof T room asafunctionofboth gradients T and for g =0 : 15andashallowFermisea. Tore-verifythatthenumberdivergencedecreaseswith g ,aplotofthisdivergenceatzero electron/phononcouplingmaybeconstructed,andthisisdoneinFig.3-13. Indeed,onecanseethatthedivergenceisontheorderof10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(16 ,whichinturnison theorderofmachine-epsilon. 3.4TheNonlinearEnergyCurrent Thenonlinearelectroniccurrent E e )]TJ/F15 11.9552 Tf 7.084 -4.338 Td [(,whichhasdimensionsofsquaredenergy,is presented.However,unlikethisdissertation'spresentationoftheelectronicnumbercurrent N ,thenonzerodivergence 1 E = E L )]TJ/F22 11.9552 Tf 12.239 0 Td [(E R of E e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(ismentionedstraightaway.Because thishasnonzerodivergence,it cannot becalledanenergycurrentenergyonlygainsthe meaningthatcomeswithbeingcalledenergy"insofarasitisconserved.Nevertheless, thedivergenceof E e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(doesapproachzeroas g similarlyapproacheszero.Thus,itisnot thedivergenceof E e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(whichiszero,butthedivergenceof E = E e )]TJ/F15 11.9552 Tf 9.762 -4.338 Td [(+ J thatis:theenergy 1 Hereandonwards,divergence"meanstheoutow,notamathematicaltendancy towardsaninnitemagnitude. 159

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Figure3-13.Thenumbercurrentdivergencemagnitudeinunitsof T room asafunctionof bothgradients T and for g =0andashallowFermiseaisontheorder ofmachineepsilon10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(16 ,andshowsupasnumericalnoise". currentoftheelectronicenergycurrentandthephononicenergycurrentsummedmeaning theelectroniccurrentandphononiccurrenttakentogetherasoneclosedthermodynamic system.Since g isincorporatedperturbatively,andsincethedivergenceof E = E e )]TJ/F15 11.9552 Tf 10.098 -4.339 Td [(+ J iszero, E e )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [(gainsmeaningasthe approximate contributionoftheelectronstotheenergy current.Thisrestoresthisdissertation'sinterestincalculating E e )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(alone. Fortheproblemofthermoelectricity,theconservationof E isrequiredtocalculatethe thermalcurrent Q appearinginthelawofthermodynamics dE = dQ + hot dN governing thetransportofanyincrements dE;dQ;dN ofenergy,thermalenergy,andnumbercurrent respectivelyinasystemofconstantvolumeandconstantpotential hot 3.4.1Lackofcontinuityofthecurrent E e )]TJ/F15 11.9552 Tf -227.637 -28.761 Td [(Straightaway,oneshoulddeterminehowbadly E e )]TJ/F15 11.9552 Tf 10.986 -4.338 Td [(failstohavezerodivergence,and so E iscalculatedasafunctionofcouplingandoftemperature.Zeroelectron-phonon coupling g =0isveriedtogivezerodivergence. 160

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Figure3-14.Electronicenergycurrentdiscontinuityplottedvs.absolute-band-density phononicenergycurrentdiscontinuity. Figure3-15.Electronicenergycurrentdiscontinuityplottedvs.potentialgradient,except athighertemperature, T L =10 T room 161

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3.4.1.1Divergenceof E e )]TJ/F38 11.9552 Tf 11.568 -4.338 Td [(andelectron-phononcoupling g Thedivergenceof E e )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(iscalculatedforahotleadofveryhightemperature10 T room in Fig.3-15.ThefeaturesofFig.3-15arequalitativelythesameasthoseofFig.3-14.There isnooverallincreaseofthedivergenceof E e )]TJ/F15 11.9552 Tf 10.986 -4.338 Td [(withtemperature. Thedivergenceof E e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(iscalculatedforahotleadoftemperature2 T room inFig.3-14. ThefeaturesofFig.3-14.areasfollows: ANeargenerator-regime,electronsloseenergytophononsindeepFermisea. BElectronsgainenergyfromphononsinresonant-regime j V T j 2 W ,andthisregionof energy-absorptionsplitsintotwonarrowerpeaksonthinningtheFermisea. CPeakheightincreasesat j V T j 2 : 5 W ,whichisthelocationofthephonon-assisted numbercurrent. Interestingly,thedivergenceof E e )]TJ/F15 11.9552 Tf 10.986 -4.338 Td [(isnotaslargenearthe V T ofphonon-assisted tunnelingasitisnearthe V T oftheresonantcurrent.Furthermore,theabsolutevalue ofthedivergenceof E e )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [(hasnotbeentaken,anditisseenthatthisdivergenceislargely positive.Apositivedivergenceof E e )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [(meansthereisalargeoutow/smallinowof electronenergyfromthehot/coldlead,meaningthattheelectronsaresendingenergyinto thephonons. 3.4.1.2Divergenceof E e )]TJ/F38 11.9552 Tf 11.568 -4.338 Td [(andgradients T and ConsiderFig.3-16.Fortheelectron-phononcoupling g =0 : 15,theamountofenergy transferredbetweentheelectronsandthephononsisabout0 : 25 T 2 room atmost,which occursintheresonance.Largertemperatureincreasestheamountofelectron/phonon energyexchange.Thehighestamountofelectron-phononenergyexchangeoccursinthe rstandthirdquadrantsataround = U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( 5. ConsiderFig.3-17.Itshowsthatthedivergenceof E e )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(isontheorderof10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(14 whichis10 2 timesmachineepsilonfor g =0,andthus E e )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(infactisthe conserved energycurrent E .Becausetheenergycurrent E isconserved,itappearsinthelawof 162

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Figure3-16.Divergenceof E e )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [(asafunctionofbothgradients T and thelatterin unitsof U 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( forashallowFermiseafor g =0 : 15. Figure3-17.Divergenceof E e )]TJ/F15 11.9552 Tf 10.986 -4.338 Td [(asafunctionofbothgradients T and forashallow Fermisea,ontheorderof10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(14 atmostfor g =0. thermodynamics dE = dQ + hot dN ,andthuscanbeusedtocalculatethethermalcurrent dQ 3.4.2 E e )]TJ/F38 11.9552 Tf 11.568 -4.338 Td [(asafunctionof Asanintermediarytothegoalofcalculating Q e )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [(,oneshoulddeterminethe magnitudeof E e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(todeterminecomparisontothedivergence E oforder E 2 T 2 room ConsiderFig.3-18.Itshowsenergycurrentasafunctionofpotential-gradientacross athermoelectricblackboxforathick[23]FermiseathinFermiseashownasdashedfor 163

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Figure3-18.Energycurrentasafunctionofpotential-gradientacrossathermoelectric blackboxforathick[23]Fermiseafor T L =2 T 2 room referenceinunitsof T 2 room for T L =2 T 2 room .AAlmosteverywheretheincreaseofthe electron/phononcouplingcausesmoreenergytobedumpedintotheelectrons,although N decreasesslightlywith g inFigure3-7.For g =0,energy-conservationfortheelectrons asaclosedsystemis E e )]TJ/F15 11.9552 Tf 11.506 -4.339 Td [(= hot N + Q e )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [(,andsotheonlyotherplaceforthisenergyto bedumpedintowouldbeintothermalcurrent Q e )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [(.Qualitatively,onecanexpect Q to increasewith g .BEnergycurrentincreaseswith g atthesatellitepeaks. ConsiderFig.3-20.Itshowsenergycurrentasafunctionofpotential-gradientacross athermoelectricblackboxforathinFermisea.Energycurrentsharplyincreasesinthe resonantregimewhentheFermi-seathins,butdecreasesslightlyandshiftswithcoupling g .Incontrastto3-18,theenergycurrentdecreasesslightlywith g ,whichmaymeanthat Q issimilarlydecreased.A E saturatesandshiftswith g ,incontrasttoB E increasing with g 3.5TotalElectronicPlusPhononicThermalCurrent Thetotalthermalcurrent Q = Q e )]TJ/F15 11.9552 Tf 10.266 -4.339 Td [(+ Q ph shallnowbedetermined.Thephononic current Q ph isonlydependentupon throughtheimpingementofphononsuponand 164

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Figure3-19.Energycurrentasafunctionofpotential-gradientacrossathermoelectric blackboxforathickFermiseathinFermiseashownasdashedfor referenceinunitsof T 2 room .Highertemperature. Figure3-20.Energycurrentasafunctionofpotential-gradientacrossathermoelectric blackboxforathinFermiseaat T L =2 T room 165

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Figure3-21.Eectofelectron-phononcoupling g uponenergycurrentasafunctionof potentialgradientacrossathermoelectricblackboxforathinFermiseaat T L =10 T room re-emissionfromtheFermiseaexcitingcharge-neutralelectron/holepairs.Thisprocess isrepresentedbythepolarizationbubbleofFig.2-8.Theresultingdependenceupon isontheorderofeachofthedivergences E e )]TJ/F19 11.9552 Tf 12.833 -4.339 Td [()]TJ/F15 11.9552 Tf 24.346 0 Td [( J ,whichistosayweakly dependent.Indeed,ithasbeenfoundthatthepolarizationbubblerenormalizesthe phononmodefrequencytoonepartin1000.Thesolepointofretainingthepolarization bubbleofFig.2-8istorestorecontinuityto E = E e )]TJ/F15 11.9552 Tf 9.742 -4.338 Td [(+ J 3.6TheApproximately-ElectronicThermalCurrent AlthoughitwasemphasizedatthebeginningofSec.3.4that E e )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [(isnotanenergy currentinthepresenceofinteractions,itremainsthatthequantity E e )]TJ/F19 11.9552 Tf 10.155 -4.339 Td [()]TJ/F22 11.9552 Tf 12.369 0 Td [( hot N Q e )]TJ/F15 11.9552 Tf -446.322 -28.246 Td [(isafunctionaleexplicitlyofelectronicGreenfunctionsandthusadjustmentofelectronic parameters U 1 ;W ; moststronglyinuences Q e )]TJ/F15 11.9552 Tf 7.085 -4.338 Td [(.Though Q e )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(strictly isn't thethermal currentcarriedbyelectronsalonethisseparationisimpossible,adjustmentofthe abovementionedelectronicparametersdirectlyinuences Q e )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [(.Thisdissertationwouldbe remissinfailingtopointouttheinterestingfeaturesof Q e )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [(. 166

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Figure3-22.Totalthermalcurrent Q = Q e )]TJ/F15 11.9552 Tf 9.741 -4.339 Td [(+ Q ph .Thefeaturesof Q e )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [(strongly dependingupon arelargelydrownedoutbythemonotonicincreaseof Q ph with T ,lendingmerittotheideaofregarding Q e )]TJ/F15 11.9552 Tf 10.986 -4.338 Td [(asaseparateentity. Figure3-23.Heatplotofthequantity j Q e )]TJ/F19 11.9552 Tf 7.084 -4.338 Td [(j =T 2 room asafunctionof and T for g =0 : 15inashallowFermisea. InFig.3.6,aheatplotoftheapproximate 2 j Q e )]TJ/F19 11.9552 Tf 7.085 -4.338 Td [(j =T 2 room asafunctionofgradients and T atcoupling g =0 : 1inashallowFermiseaisshown. 2 Inthepresenceof2{42itisimpossibletoformallyseparatetheelectronic contributiontothethermalcurrentfromthatofthephonons.Suchaseparationisstrictly anapproximationwithinperturbationtheory. 167

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Figure3-24.Heatplotoftheelectronicthermalcurrent j Q e )]TJ/F19 11.9552 Tf 7.084 -4.339 Td [(j =T 2 room asafunctionof and T for g =0inashallowFermisea. InthisplotFig.3.6, T =0wouldcorrespondtoaleft/rightsymmetricelectronic thermalcurrent.Featuresofthisplotareasfollows. AResonantthermalcurrentconcurrentwiththeresonancein N inFig.3-9. BMonotonicincreaseof Q e )]TJ/F15 11.9552 Tf 10.986 -4.338 Td [(with T CThermalsuppressionoftheresonant-spikein Q e )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(inthethermalpumpquadrant, oppositeofthethermal-suppressionsthathappenin3-9. DThermalsuppressionofthephonon-assisted-tunnelingspikesin Q e )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(inthe generator/dissipaterquadrants. EZeroelectronicthermalcurrentinthegenerator/dissipaterquadrants. InFig.3.6,aheatplotoftheexact j Q e )]TJ/F19 11.9552 Tf 7.085 -4.338 Td [(j =T 2 room asafunctionofgradients and T atelectron-phononcoupling g =0inashallowFermiseaisshown. 168

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Clearly,thereisanabsenceofphonon-assistedtunneling,andtheresonancelastsinto higher T .Hence,theelectron-phononinteractioncouldactuallygiveamoreecient thermoelectricdevice.However,thethermoelectricdevicewouldneedtobeexternally driveninordertoaccessthisresonance,andtheunconventionaleciency2{26inthis dissertationwouldbeused.Becausethiseciency2{26isunconventionalassuch,there isnocomparisontobemadetothewell-establishedbodyofresultsforthethermodynamic eciencyofadevicedrivingitssurroundingswithatemperaturegradient T .Itremains toimplementtheuseof2{26inasystembeyondjustatoymodelwhichproducedthe ecienciesofFigs.A-7andA-8. 3.7Temperatureofthecenter-regionphonons Nowtobepresentedisthetemperatureofathirdleadintroducedtothecenter regionasathermometer.Ifthethirdlead'sphonons'temperature T ph issuchthatthere iszeronetthermalcurrentowinginoroutofit.Thatis, T ph isthesolutionto,byway ofnumericalbisection,thecondition J T net = J T )]TJ/F22 11.9552 Tf 12.703 0 Td [(J L + J T )]TJ/F22 11.9552 Tf 12.703 0 Td [(J R =0.Thus, T ph couldbeinterpretedasthecenterregion'sphonons'temperatureinsofarastheleftand rightleadsareabletoberegardedasasinglesystem 3 theirseparatedbyacentralregion notwithstanding. OneseesinFig.3.7that T ph isonephononenergyabovetheaveragetemperature T oftheleftandrightleads.Itisthusincorrecttoappealtointuitionandassertthatthe centersitephonons'temperatureisjust T = T L + T R = 2. 3 Thenetthermometercurrent J T net mayfeatureaowofphononicthermalenergy J T )]TJ/F22 11.9552 Tf -458.702 -14.446 Td [(J L in/outoftheleftleadbut T ph isthetemperatureatwhich J T )]TJ/F22 11.9552 Tf 14.276 0 Td [(J L iscompensatedby J T )]TJ/F22 11.9552 Tf 15.259 0 Td [(J R towithinthetoleranceofthebisectionalgorithm.Suchasystem,the L and R takenasawhole,exhibitsHenryLouisLeChatelier's dynamic equilibrium. 169

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Figure3-25.Dierence-magnitude T ph )]TJ/F22 11.9552 Tf 11.955 0 Td [(T =T incenter-regionphonontemperature T ph fromtheaveragetemperture T = T L + T R = 2forvarious T and .For T 0and attheresonance,thereisalargedeviationof T ph from T For T '1 ,thecentersitephonontemperaturequicklysaturatestothe average T andbecomesindependentof 3.8OperationoftheThermoelectrically-AssistedCircuit 3.8.1Settingloadresistanceequaltoreciprocal-dierential-conductancefor maximumpower-hysteresis Finally,theoperationofthethermoelectriccircuitistobecalculated.Instead ofplottingasafunctionofthetheorist'sthermodynamicgradients and T ,this dissertationshallpresentelectricalcurrentasafunctionofthebiasappliedbyan experimentalist'sinstrument.Onebeginsbyplottingtheelectricalcurrent I = )]TJ/F22 11.9552 Tf 9.299 0 Td [(eN asafunctionofappliedbias E a ,asinFig.3-26. Becausetheelectricalcurrentisthenegativeofthenumbercurrentandtheapplied biassimilarlyhasanegativesign,thegenerator-regimeissentfromthesecondquadrant tothefourthquadrant.However,itremainsindependentofthe chargeconvention thatdrivigofthethermoelectricgivespowerintherstandthirdquadrants,andthe thermoelectricdrivingitssurroundingsgivespowerinthesecondandfourthquadrant. 170

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Figure3-26.Electricalcurrent I = )]TJ/F22 11.9552 Tf 9.299 0 Td [(N vs.appliedbias E a foranOhmicloadofunit resistance R L =1,forthinFermisea,withthickFermiseashownindashed lineforreference.Hotleadtemperatureof10 T room used. Figure3-27.Electricalcurrentvs.appliedbiasforvariouselectron-phononcoupling g thickFermisea,forloadresistance R L = j dN=dV T j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 at T L =2 T room 171

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Fig.3-27isofelectricalcurrentvs.appliedbias,thick Fermisea,forloadresistance R L = j dN=dV T j )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 loadresistanceequalsthereciprocaloftheabsolutevalueofthe thermoelectricblackbox'sdierentialconductivity,ofnoteduetothemaximum-power-transfer theorem[53]. AHysteresisisachievablenearthegenerator-regime. BHysteresisfor E ap > 0,butnotfor E ap < 0,whichisatemperature/interaction-eect. CSaturationof N for E ap > 0to 0 : 5,butnotfor E ap < 0. DFornegativedierentialresistancein N ,the E ap becomesarbitrarily-largein magnitudetoattainagiven I ,correspondingtoinniteload-resistance. EHysteretic-deformationofthephonon-assistedcurrent. Resistanceofamacro-systemisaphysicalquantitythatiseasilyvariedovermany ordersofmagnitude,sotheselatentfeaturesarereadilyaccessiblewhenperforming measurements. 172

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CHAPTER4 CONCLUSION Inconclusion,thethermoelectriccurrents N and Q throughananojunctioninthe resonantregimehavebeencalculatedforuseinpredictingthequalitativemeasurementsof nonlinertunnelingcurrente.g.,thoseinscanningtunnelingmicroscopyandthrough double-potential-barrierjunctionspreparedbymolecular-beamepitaxyatnite temperaturefortheproblemofthermoelectricity.Anelectron-phononinteractionhas beenintroduced.Thus,themodelstudiedbythisdissertationissimilartothatstudiedby Hyldgaardet.al.[23].However,therearesomecriticaldierences. Therstdierenceisthatthecalculationisgeneralizedtonitetemperaturean obviousgeneralizationforthestudyofnonlinearthermoelectricity.Asecondsharper dierenceisthatthephononthatislocalizedtothecenterregionis itinerant insteadof static becauseitsdecay-rateisnottheinntesimal0 + likethatofHyldgaardet.al.'s Einsteinphononsbutratheranite .Thetworeasonswhyitinerantratherthanstatic Einsteinphononswereusedareasfollows.First,temperature-eectsaresovitaltothe problemofthermoelectricitybutthestaticEinsteinphononswouldbeofthecenter region 1 temperatureadicultquantitytocalculate.Thatis,onewishingtomakethe generalizationofHyldgaardet.al.'smodeltonitetemperatureis forced toeitherconsider itinerantphononsorintroduceacenter-regiontemperatureasanadditionalindependent parameter. Thelatteroftheseoptionsturnsouttobenotsuchagoodideaduetothesecond reasonforconsideringitinerantphonons.Thissecondreasonisthatthethermalcurrent Q iscalculatedfromtheenergycurrentwhichisconservedbyvirtueofbeinganenergy" 1 Referencetoacenter-regiontemperatureiswell-knowntodisappearfromthe 7 Dyson equationsintheelectroniccase,andthisdissertationshowsthatthisdroppingoutalso happensinthe 7 Dysonequationsphononiccase.Thereasonwhythe 0 + limitisso easilyeectedforallsystemsatabsolutezerotemperaturewasfoundtobethezerothlaw ofthermodynamics. 173

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current.Theelectronandphononenergycurrentsalonearenotseparatelyconservedin thepresenceofanelectron-phononinteractionsotheyarenotreallyenergy"currents assuch.Onlythetotalenergycurrent{thatconstitutedbytheelectronsandphonons together{isconserved,andthisdissertationpresentsasimplemodelinwhichthisholds. ThissimplemodelisoneinwhichthephononsaremodeledbytheFano-Anderson Hamiltonian 2 andapproximatedasbeingnethercreatednordestroyedi.e.,atheoryof phononswhose m th modehasno b y m b y m $ b m b m processesresultinginthenetcreationor destructionofapairofphonons. Typically,thermoelectricmaterialsaredescribedwithinatransportcoecient" description,whereacurrentislinearlyproportionaltoagradient.Amajorshortcomingof thelinearresponseregimeindescribingthermoelectrics.Bydenition,thelinearresponse considerszeropowerdeliverybyathermoelectric,owingtotheopen-circuit-voltage conditionunderwhichtheSeebeckcoecientisdened/measuredandtheexactnessof thelinearresponseonlyforinntesimalgradients.Therefore,deliveryofanon-inntesimal quantityofpowerbyathermoelectricmaterialismanifestlynonlinear,andthusonemerit ofthisdissertationistheinclusionofthermoelectricpowerdelivery. Inadditiontothefeatureofnitepower-delivery,workbyHyldgaardet.al.[23]and Hersheld[21]serveasthemotivationforcalculatingthenonlinearthermoelectriccurrents. Hence,beyondthemeritofnitepowerdeliveryisalsothenoveltyofinclusion ofnonlinearfeaturesofelectricalcurrentobservedinscanningtunnelingmicroscopy andthedrivendouble-barrierjunctionfrommolecularbeamepitaxy.Thenonlinear currentfromaleftgrandcanonicalensembleintoarightgrandcanonicalensemblefor thecaseofthoseensemblesseparatedbyasinglequantumleveltowhichtheyareweakly coupledfeaturesaresonantspikeincurrent.Withinthetransport-coecientdescriptiona 2 Whenusedforelectrons,theFano-Andersonmodelisasingleelectroniclevelcoupled toacontinuumof k statessuchacontinuumisfoundinametal. 174

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thermoelectricgureofmeritthatwasarbitrarilylargeinsofarasthelead/levelcoupling wasarbitrarilyweakwasfoundduetotheresonantspikeincurrent.Thisresonantspike incurrentisanexclusively-nonlinearfeatureifthecouplingislargerthantheopencircuit voltage.Negativedierentialresistanceandphonon-assistedtunnelingarealsoexclusively nonlinearphenomena. Thus,anaimofthisdissertationistodeterminehowathermoelectricdevicecould partakeinthesenonlinearphenomena.Indeed,itwasshownthatadrivingagent's externalbias E a wasrequiredtoaccesstheexplicitlynonlinearportionsofthecurrentasa functionofchemicalpotentialdierence.Resonance,negativedierentialresistance,and phononassistedtunnelingarethemainnonlinearfeatures.Thedierencebetweenthe currentatzeroandnitetemperaturedierenceforthefullgamutofchemicalpotential gradient issurmisedinthisdissertationtoresultingreaterpowerdeliverytoaloadin someportionsofthecurrent-voltagecharacteristicofthethermoelectricdevice. Itwassubsequentlywonderedifsomethingthatactslikeeciencycouldbedeveloped foradevicedespiteitshavingnetpower input .Thismotivateddevelopedtheproposal ofaneciencyofasystemwithexternaldriving,andatoymodelofthiswasdeveloped byintroducingaspectralfunctionthathadapeakandaspreadaboutthispeaklike theLorentzianspectralfunctionsoftheresonantlevelwhileatthesametimebeing closed-formintegrablewiththeFermi-Diracfunctiongivingaclosed-formcurrent.Results forthisdriveneciencyinthepresenceofachemical-potential-gradientindependent phononicthermalcurrentwhichwasfoundtoberequiredbytheSecondLawwere presented. Currently,itisofinteresttominimizethedeleteriousnessofthepresenceofphonons tothethermoelectriceect.Amongotherthings,phonon-assistedtunnelingisone possibilitytodothis.Howeversincephononassistedtunnelingoccursforbiaseswell beyondtheopen-circuitmaximumvoltage,thedriventhermoelectricappearsto betheonlywaytoharnessphonon-assistedtunnelingtocompelphononstohelpthe 175

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thermoelectriceect.Whilethisdissertationdoesnotpresentresultsfortherisein eciency 3 thiswouldbestowrather,onlyatoymodelwithoutinteractions,thiswould beinterestingfuturework. Fundamentalphysicalproblemsarealsotreatedinthisdissertation.Aconserved energycurrentforsimultaneouselectronicandphononictransportwasdeveloped. Thetempertureofalocalizedphononmodedrivenfaroutofequilibriumbyadjoining reservoirswasalsosolvedbycalculatingtheBose-Einsteindistributiontemperatureat whichthethermalcurrentin/outofthelocalizedphononmodewaszero. 3 Thatthethermoelectricdeviceisdrivenhaspowerinputratherthanoutputupends thenotionofeciency,andanewdenitionofthesamewasdevelopedinSec.2.1.4.4. 176

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APPENDIXA DRIVENVS.NON-DRIVENEFFICIENCYFORANINTEGRABLERESONANT LIKE"TOYSPECTRALFUNCTION Computationalcostintheformofcomputingtimeisfrequentlythebulkofthe workenteringintoevaluationofnonlinearcurrents.Onecansignicantlycutdownon thecomputationaltimerequiredintheprocessoftryingoutalargevarietyofchoices ofparametersiftheyndatoy"spectralfunctionobeyingcertainessentialproperties. Suchcost-cuttingoeredbyagiventoyspectralfunctionisessentialtoimplementingthe condition P A =0withrespectto forproof-of-concept 1 work.Onesuchtoy"spectral functionisproposedhere:anexponentiatednegativeabsolutevalue.Althoughchosen forthemathematicalreasonthatitintegratestoclosedformwithathermalfunction,it alsoarisesinthephysicalproblemofstudyingthequantumstationaryboundstates boundbyaDirac-deltapotentialwell 2 ofarbitrarily-smallwidth.Anexponentiated negative-absolute-valuespectralfunction N 0 e j )]TJ/F23 7.9701 Tf 6.586 0 Td [(U i j = )]TJ/F24 5.9776 Tf 5.289 -1.216 Td [(i waschosenbecauseitactually piecewiseintegratestoaclosedformwhenmultipliedbytheFermi-Diracdistribution 1 Theproof-of-conceptworktobecarriedoutcouldbethetestingofthedriven eciency2{26. 2 Consider[28]theone-dimensionalevenpotential V x = )]TJ/F22 11.9552 Tf 9.299 0 Td [( x e.g.,duetoan attractivepoint-defectalongsomelineofstrength thattrapsasetofstationarystates ofevenoroddparity.Atleastoneoccupiablestateofevenparityisboundbythis potential.Inpositionspace,thispotentialwouldlocalizeastate h x j i = x .For x 6 =0, thisstateis x 6 =0 = 0 e )]TJ/F23 7.9701 Tf 6.586 0 Td [( j x j andobeys )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 m @ x 2 x 6 =0 = )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 m 2 x 6 =0 = E x 6 =0 .Integratingboth sidesoftheSchrodingerequationabout x =0proceedsas R 0 + 0 )]TJ/F15 11.9552 Tf 10.903 1.881 Td [( )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 m @ x 2 x )]TJ/F22 11.9552 Tf 11.955 0 Td [( x x dx = R 0 + 0 )]TJ/F22 11.9552 Tf 10.903 1.882 Td [(E dx ,andgivesthenormalization 0 = p = m .Hence,amass m particle mayoccupyabound-state h x j i ofenergy E = )]TJ/F21 7.9701 Tf 14.239 4.707 Td [(1 2 m 2 = )]TJ/F21 7.9701 Tf 10.494 4.707 Td [(1 2 m 2 inaDirac-delta potentialofstrengthgivenbytheexponentiatednegativeabsolutevalue stateis h x j i = p me )]TJ/F23 7.9701 Tf 6.587 0 Td [(m j x j .Notethatthemomentum-spacerepresentationofthisstatefrom itsposition-spacerepresentationis h k j i = 1 2 p R 1 h x j i e ikx dx = = 2 + k 2 ,whichisa Lorenziancenteredatzeromomentumandofspread wheretheprefactor 1 2 p onthe Fouriertransformwaschosentopreservenormalizationtounitywhenintegratedoverthe dimensionless k= domain. 177

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2{45.LiketheLorentzianspectralfunction A i = )]TJ/F24 5.9776 Tf 5.288 -1.215 Td [(i = )]TJ/F23 7.9701 Tf 6.587 0 Td [(U i 2 +)]TJ/F24 5.9776 Tf 11.875 -1.215 Td [(i 2 whosepowerand currentsmustbecalculatedbynumericalintegration,ithasapeaklocatedat U i ,a dispersionaboutthispeakof)]TJ/F23 7.9701 Tf 164.564 -1.793 Td [(i ,andthespectral-functionsum-rule2{63realizable 3 asanormalization, A 0 = Z 2 W )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W e j )]TJ/F23 7.9701 Tf 6.587 0 Td [(U i j = )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i d! )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = 1 )]TJ/F23 7.9701 Tf 7.314 -1.794 Td [(i 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2e )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W= )]TJ/F24 5.9776 Tf 5.289 -1.216 Td [(i cosh U i )]TJ/F23 7.9701 Tf 7.315 -1.794 Td [(i )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 W )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i 1 2)]TJ/F23 7.9701 Tf 13.167 -1.794 Td [(i Using2{115,andapproximatingtheescape-ratesas)]TJ/F23 7.9701 Tf 286.563 4.339 Td [( )]TJ/F23 7.9701 Tf 7.314 4.339 Td [( U i i.e.,theescape ratestaketheirsupportfrombeingevaluatedattheresonant-energy U i ,sothisestimate becomesasymptoticallyequal insofarasoneisintheresonant-regime)]TJ/F23 7.9701 Tf 211.778 -1.794 Td [(i 0. Abbreviating F < = f L< )]TJ/F22 11.9552 Tf 11.956 0 Td [(f R< N 0 4)]TJ/F23 7.9701 Tf 13.167 4.339 Td [(L U i )]TJ/F23 7.9701 Tf 7.314 4.339 Td [(R U i )]TJ/F23 7.9701 Tf 7.314 4.118 Td [(L U i +)]TJ/F23 7.9701 Tf 19.076 4.118 Td [(R U i e U i = )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i 2)]TJ/F23 7.9701 Tf 13.167 -1.794 Td [(i csch 2 W )]TJ/F23 7.9701 Tf 7.314 -1.794 Td [(i 0 @ U i Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W d!e + )]TJ/F23 7.9701 Tf 6.586 0 Td [(U i = )]TJ/F24 5.9776 Tf 5.288 -1.215 Td [(i F < | {z } closed )]TJ/F21 7.9701 Tf 6.586 0 Td [(form )]TJ/F21 7.9701 Tf 6.587 0 Td [(integrable + 2 W Z U i d!e )]TJ/F21 7.9701 Tf 6.587 0 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(U i = )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i F < | {z } closed )]TJ/F21 7.9701 Tf 6.587 0 Td [(form )]TJ/F21 7.9701 Tf 6.586 0 Td [(integrable 1 A To tidily integrate 4 theaboveexpressionfor N 0 andalso E 0 = !N 0 for E 0 ,onemust eectachangeofvariables,namely = )]TJ/F22 11.9552 Tf 11.955 0 Td [( N = b Z a e )]TJ/F23 7.9701 Tf 6.587 0 Td [(U i = )]TJ/F24 5.9776 Tf 5.289 -1.216 Td [(i e )]TJ/F23 7.9701 Tf 6.586 0 Td [( +1 d! = b )]TJ/F23 7.9701 Tf 6.587 0 Td [( Z a )]TJ/F23 7.9701 Tf 6.587 0 Td [( e T + )]TJ/F23 7.9701 Tf 6.586 0 Td [(U i = )]TJ/F24 5.9776 Tf 5.289 -1.216 Td [(i e )]TJ/F23 7.9701 Tf 6.587 0 Td [( +1 d )]TJ/F22 11.9552 Tf 11.955 0 Td [( = T e )]TJ/F23 7.9701 Tf 6.586 0 Td [(U i = )]TJ/F24 5.9776 Tf 5.289 -1.216 Td [(i b Z a e T = )]TJ/F24 5.9776 Tf 5.288 -1.216 Td [(i e +1 d 3 Caution:thenormalization A 0 onlyexistsifthewide-band-limiti.e.,)]TJ/F23 7.9701 Tf 207.421 -1.793 Td [(i 0isnot taken. 4 Bytidily",itismeantthatifthischangeofvariablesisnoteected, Mathematica willyieldabigmess. 178

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E = b Z a !e )]TJ/F23 7.9701 Tf 6.586 0 Td [(U i = )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i e )]TJ/F23 7.9701 Tf 6.587 0 Td [( +1 d! = e )]TJ/F23 7.9701 Tf 6.587 0 Td [(U i = )]TJ/F24 5.9776 Tf 5.288 -1.215 Td [(i b Z a T + e T = )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i e +1 d = I N + T 2 e )]TJ/F23 7.9701 Tf 6.587 0 Td [(U i = )]TJ/F24 5.9776 Tf 5.288 -1.215 Td [(i b Z a e T = )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i d e +1 Above,onehasthefollowingantiderivativesinwhichtheconstantofintegrationis omittedinwhichthenotationforthehypergeometricfunctions 2 F 1 F 2 1 and p F q F p q thelatterhavingvectorargumetsof Mathematica isused, I 0 = Z e T = )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i e +1 d = )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(i T e T = )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i F 2 1 ; T )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i ; 1+ T )]TJ/F24 5.9776 Tf 5.288 -1.215 Td [(i ; )]TJ/F22 11.9552 Tf 9.299 0 Td [(e I 1 = Z e T = )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i e +1 d = )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(i T 2 e T = )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i T F 2 1 ; T )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i ; 1+ T )]TJ/F24 5.9776 Tf 5.288 -1.215 Td [(i ; )]TJ/F22 11.9552 Tf 9.299 0 Td [(e )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 7.9701 Tf 7.314 -1.794 Td [(i F p q f 1 ; T )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i ; T )]TJ/F24 5.9776 Tf 5.288 -1.215 Td [(i g ; f 1+ T )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i ; 1+ T )]TJ/F24 5.9776 Tf 5.289 -1.215 Td [(i g ; )]TJ/F22 11.9552 Tf 9.299 0 Td [(e Onethenevaluatestheantiderivativesattheendpointsandsubtracts 5 the left-currentfromtheright-current,giving, N 0 = N L + )]TJ/F22 11.9552 Tf 11.956 0 Td [(N R + + N L )]TJ/F19 11.9552 Tf 10.163 2.956 Td [()]TJ/F22 11.9552 Tf 11.955 0 Td [(N R )]TJ/F15 11.9552 Tf 8.174 2.956 Td [(; E 0 = E L + )]TJ/F22 11.9552 Tf 11.955 0 Td [(E R + + E L )]TJ/F19 11.9552 Tf 9.741 2.956 Td [()]TJ/F22 11.9552 Tf 11.955 0 Td [(E R )]TJ/F15 11.9552 Tf 7.626 2.956 Td [(= hot N 0 + Q 0 ;A{1 Intheresonant-regime,thetransmissionfunctionisdescribablebyapeak-location U i andaspread)]TJ/F23 7.9701 Tf 76.965 -1.793 Td [(i aboutthatpeak.Hence,onecaningeneraldescribetheeectof U i and )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(i uponpowerandeciency.Theapplicabilityofthisisextendabletotheself-energy ofanymany-bodyinteraction ,if theeectof ismerelytherenormalizations U i ~ U i and)]TJ/F23 7.9701 Tf 30.076 -1.794 Td [(i ~ )]TJ/F23 7.9701 Tf 7.315 -1.794 Td [(i .Iftheserenormalizationsare,themselves,afunctionofanyparameter p 5 Thisdiersfromtheconventionusedlater,wherethe L numbercurrentisassigneda positivesense,whilethe R numbercurrentisassignedanegativesense. 179

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FigureA-1.Comparisonofcalculatedgenerator-regimepowerasafunctionofmaximum U i andspread)]TJ/F23 7.9701 Tf 67.21 -1.793 Td [(i ofaLorentzianwhosespectralfunction )]TJ/F23 7.9701 Tf 5.289 0 Td [(= )]TJ/F23 7.9701 Tf 6.586 0 Td [(U 1 +)]TJ/F18 5.9776 Tf 15.168 2.269 Td [(2 comes fromtheHamiltonianofaresonance U 1 withescaperates)]TJ/F23 7.9701 Tf 99.466 -1.794 Td [(! toaleftanda rightleadvs.theexponentiatednegativeabsolutevaluewhichdoesnot comefromaHamiltonianbutintegratestoclosedformwithaFermi-Dirac distribution2{45.Thetwodierenttransmissionfunctionsproducevalues ofthermoelectricgeneratorpowerwhicharequalitativelysimilar. otherthan ,implicitdependenceupon p willberealizedin ~ U i and ~ )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(i inallclosed-form expressions. Asamatterofcourse,oneinvestigatesthethermoelectricresponsepowerand eciencyover^ U i ^ +10 T andcomparetotheLorentzianresponseRecallthat ^ = T L R )]TJ/F22 11.9552 Tf 11.998 0 Td [(T R L = T L )]TJ/F22 11.9552 Tf 11.998 0 Td [(T R = + T= T V T ,whichistheequilibrium-energybetween tworeservoirsofdisparatetemperaturesandpotentials.Onendsthattheresultsare qualitativelysimilar,andwithinanorderofmagnitude. Ahighvalueof W mustbeusedtoavoidnumericaloverowintheoating-point storageofanynumericalquantity.Thisactuallyhighlightstheadvantageofthenumerical calculation:closed-formantiderivativesintheresonant-regime W 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 W inthespecic caseoftheexponentiated-negativespectralfunctionareexponentialsorhypergeometric 180

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FigureA-2.Comparisonofcalculatedgenerator-regimeutilitycoecienteciency dividedbythemaximumCarnoteciencyasafunctionofmaximum U i and spread)]TJ/F23 7.9701 Tf 44.448 -1.793 Td [(i ofaLorentzianvs.theexponentiatednegativeabsolutevalue.As withthepowerinFig.A-1,thereisqualitativesimilarity. functionsevaluatedatthe bandedges thelargestenergyscaleintheproblemover theenergyscaleof\050 U i amongthesmallestenergyscalesintheproblemifinthe resonant-regime.Therequirementofthearbitrarily-largedimensionlessquantity 2 W= \050 U i fromevaluationoftheantiderivativeatthelimitsofintegration placesalowerlimitupon\050 U i ,evensotheresonantregimeitselfrequiresanupper limit 6 upon\050 U i .Numericalintegrationquadrature,trapezoidalrule,convolutionby fastFouriertransform,etc.,ontheotherhand,avoidtheevaluationofsuchrunaway functionsofsuchlargequantities.Integralsaredonebyquadratureandinsteadconvolve byfastFouriertransform.Theresultofdoingsoproducesnumericalquantitieswhichare notrunawaytranscendentalfunctions. 6 Inthissituationwherenumericalcalculationisactuallysuperiortoanalytic calculation,thedierencebetweenevaluatingadeniteintegralbyniteRiemannsum vs.byantiderivativeisfelt.Onemightbeinclinedtotakethefundamentaltheoremof calculuslessforgranted. 181

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FigureA-3.Left:featuresof N and Q forA{1asafunctionof = E th .Right:onsetof numericalbreakdownfortoosmallavalueof W FigureA-4.Numberandthermalcurrents:ordersofmagnitudeandseparatebehaviours. InFig.A-3,oneseesthenumbercurrent N andthermalcurrent Q forthetoy integrablespectralfunction. InFig.A-4,onecanseethatthethermalcurrentbecomespositivefor T .For sgn T =sgn Q ,theowofthermalenergyisspontaneousi.e.,fromthehotleadtothe coldlead,and P T 0 =Q contributespositivelytotheCarnotupperlimit1 )]TJ/F15 11.9552 Tf 12.237 0 Td [( T cold =T hot of eciency.Thisisinterpretableas Q performingusefulworkintheformof P L which cannotsimultaneouslypumpthermalenergyuphill"thegradient T ,thatis,fromcold tohot. InFig.A-5,aplotof P A at T =0comparedtothesame P A atnonzero T illustratessolving P A =0withrespectto toyield = P A =0 .If P A T at some = a isgreaterthanthecorresponding P A T =0 overtheentire ,thenasolution 182

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FigureA-5. P A vs. ,witharrowsshowingtherootnding-processthatsolves P A =0 withrespectto andndsthechemicalpotentialgradient = P A =0 Shadedboxesindicatethatagiven and T isnotgauranteedtoyielda P A =0 thatsolves P A =0. FigureA-6. P A =0 and P L vs. P A =0 to P A =0doesnotexistforthat a .Moreover,thereisnothingtostopa rootndingalgorithmfromndinga P A =0 oneitherthepositivesideornegativeside, asinFig.A-5. InFig.A-6,thecriticalchemicalpotentalgradient P A =0 giving P A =0is plottedasafunctionofthezero-temperaturegradient T =0 .Sincethisquantityis calculatedbyarootndingalgorithm,itisnotgauranteedtobecontinuousduetothe possibilityofmultipleroots.Indeed,onecanseethat2{11forthecaseofanOhmicload givesaconditionfor P A =0thatisatleastlinearin 0= P A = E a I )]TJ/F22 11.9552 Tf 11.955 0 Td [(E a 0 I 0 = E a )]TJ/F22 11.9552 Tf 9.298 0 Td [(N )]TJ/F22 11.9552 Tf 11.955 0 Td [(E a 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(N 0 = N 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(N 2 0 R L + N P A =0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(N 0 183

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FigureA-7.Eciencyandutilityasafunctionof ,withanadded independent Q toobey2{26.ShallowFermisea.Eciencyisundenediftheredoesnot exist satisfying P A =0. FigureA-8.EciencyandutilityasinFig.A-7,exceptforadeepFermisea.Eciency appearstodropduetothedecreasedeectof T upon P A asinFig.A-5. Because N;N 0 areeachnon-monotonicfunctionsduetonegativedierentialresistance, multiplesolutionscanexist.If P A istoolarge,a P A =0 maynotexistatall. Armedwiththisunderstanding,onethencalculatestheeciencyatwhichpoweris deliveredtoaloadbyathermoelectricgeneratorwithpoweroutputduetoapplication of T andcomparesittoathermoelectricgenertorwithpower input thatis lessened byapplicationof T givingincreasedpower-deliverytotheload.Thisisdonefora shallowFermiseainFig.A-7,andthenforadeepFermiseainFig.A-8. Tunneling-spectroscopy[38]couldserveasapracticeinwhichtheaboveresultscould playout.Anonlinearcurrentwouldbegeneratedbythespectrometerandsubsequently 184

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usedtodriveanano-load.Thespectrometerpoweredexternallywouldserveasboththe thermoelectricdeviceaswellasthedriving-agent.Hence,settingsonthespectrometer wouldbeadjustedtokeepthepower-deliveryconstantoverthecourseofapplyingthe temperature-gradient T .MeasurementofcurrentthroughandvoltageacrossanOhmic loadwouldgivedatayieldinginformationaboutthepower-delivery. 185

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APPENDIXB BOXCARSPECTRALFUNCTIONOFACHAINCENTERREGION Theboxcarspectralfunctionis, A = + )]TJ/F22 11.9552 Tf 11.955 0 Td [(! )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(! )]TJ/F15 11.9552 Tf 6.753 2.867 Td [( + )]TJ/F23 7.9701 Tf 6.587 0 Td [(! Thenumbercurrentfortheboxcarspectralfunctionis, N = 4)]TJ/F23 7.9701 Tf 13.167 4.339 Td [(L U i )]TJ/F23 7.9701 Tf 7.314 4.339 Td [(R U i )]TJ/F23 7.9701 Tf 7.314 4.118 Td [(L U i +)]TJ/F23 7.9701 Tf 19.075 4.118 Td [(R U i 2 W Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W d!A F = 4)]TJ/F23 7.9701 Tf 13.167 4.339 Td [(L U i )]TJ/F23 7.9701 Tf 7.314 4.339 Td [(R U i )]TJ/F23 7.9701 Tf 7.315 4.118 Td [(L U i +)]TJ/F23 7.9701 Tf 19.076 4.118 Td [(R U i 0 B @ F +2 W +[+2 W ] )]TJ/F23 7.9701 Tf 6.586 0 Td [(! + + F + + + )]TJ/F21 7.9701 Tf 6.587 0 Td [([+2 W ] )]TJ/F15 11.9552 Tf -201.829 -23.907 Td [( F )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W )]TJ/F21 7.9701 Tf 6.587 0 Td [([ )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W ] )]TJ/F23 7.9701 Tf 6.586 0 Td [(! )]TJ/F21 7.9701 Tf 6.254 1.074 Td [( + F )]TJ/F15 11.9552 Tf 6.752 2.867 Td [( )]TJ/F21 7.9701 Tf 6.586 0 Td [( )]TJ/F25 7.9701 Tf 6.255 1.074 Td [()]TJ/F21 7.9701 Tf 6.586 0 Td [([ )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W ] 1 C A Theenergycurrentfortheboxcarspectralfunctionis, E = 4)]TJ/F23 7.9701 Tf 13.168 4.339 Td [(L U i )]TJ/F23 7.9701 Tf 7.315 4.339 Td [(R U i )]TJ/F23 7.9701 Tf 7.314 4.118 Td [(L U i +)]TJ/F23 7.9701 Tf 19.076 4.118 Td [(R U i +2 W Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 W d!A F ! = 4)]TJ/F23 7.9701 Tf 13.167 4.339 Td [(L U i )]TJ/F23 7.9701 Tf 7.314 4.339 Td [(R U i )]TJ/F23 7.9701 Tf 7.314 4.118 Td [(L U i +)]TJ/F23 7.9701 Tf 19.075 4.118 Td [(R U i 0 B @ G L +[+2 W ] )]TJ/F23 7.9701 Tf 6.587 0 Td [(! + + G + + + )]TJ/F21 7.9701 Tf 6.587 0 Td [([+2 W ] )]TJ/F15 11.9552 Tf -195.806 -23.908 Td [( G R )]TJ/F21 7.9701 Tf 6.586 0 Td [([ )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W ] )]TJ/F23 7.9701 Tf 6.587 0 Td [(! )]TJ/F21 7.9701 Tf 6.255 1.074 Td [( + G )]TJ/F15 11.9552 Tf 6.753 2.867 Td [( )]TJ/F21 7.9701 Tf 6.587 0 Td [( )]TJ/F25 7.9701 Tf 6.254 1.074 Td [()]TJ/F21 7.9701 Tf 6.586 0 Td [([ )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 W ] 1 C A Above,thefollowingantiderivativesappearusinglocalnotation, F T R ln+ R )]TJ/F22 11.9552 Tf 10.447 0 Td [(T L ln+ L ; G = F )]TJ/F22 11.9552 Tf 10.447 0 Td [(T L 2 Li 2 )]TJ/F22 11.9552 Tf 9.298 0 Td [(" L + T R 2 Li 2 )]TJ/F22 11.9552 Tf 9.299 0 Td [(" R ; e )]TJ/F23 7.9701 Tf 6.587 0 Td [( =T ; 186

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BIOGRAPHICALSKETCH BradleyJosephNartowtwasborninWorcester,MA,andgrewupinOxford,MA.He attendedSt.Peter-MarianJr./Sr.HighSchoolinWorcester,MA,whereheparticipatedin football,track-and-eld,andcrosscountry.HewasawardedtheGilreinFamilyMemorial Scholarshipongraduation. DuringhisundergraduateprogramatFranciscanUniversityofSteubenville,in Steubenville,OH,BradleyearnedrepeatedplacementontheDean'sList,andinJuneof 2006receivedhisBachelorofSciencewithhonorsinMathematicalScience,withminors inchemistryandengineeringscience.Afterworkinginametallographylabandoperating furnaceswhiletakinggraduateclassesforayear,hewastheninvitedintothegraduate programinmaterialsscienceandengineeringattheUniversityofFlorida,wherehe receivedhisMasterofScienceinDecemberof2008.Hethenbecameagraduateresearcher andteacherattheUniversityofMinnesota,Duluth,from2009to2011,earningaMaster ofScienceinphysicsinMayof2011. BradleyreturnedtotheUniversityofFloridainAugustof2011tobeginaPhDin physics.Duringthisprogram,heresearchedthethermoelectriceect,andinventeda methodattheproof-of-conceptleveltoaccessthenonlinearregimeofathermoelectric device. Bradleyplanstocontinueworkingasapostdoctoralresearcherintheeldof transportinnano-devices. 191