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Some Aspects of Electronic Transport in Graphite and Other Materials

Permanent Link: http://ufdc.ufl.edu/UFE0044451/00001

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Title: Some Aspects of Electronic Transport in Graphite and Other Materials
Physical Description: 1 online resource (112 p.)
Language: english
Creator: Pal, Hridis K
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: electrons -- graphite -- interaction -- magnetoresistance -- phonons -- transport
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We investigate several aspects of electronic transport in this work. First, we study the transport properties of graphite. We find that the experimentally observed unsaturating metallic behavior of resistivity at high temperatures may be explained by taking into account inter-valley scattering of electrons off hard optical phonons. Additionally, the presence of extremely light Dirac-like carriers at the $H$ and $H^{\prime}$ edges of the Brillouin zone is found to give rise to a regime where the in-plane transverse magnetoresistance is linear in the field, which explains, in part, some of the experimental observations. However, these carriers are found to have no effect on quantum oscillations. The out-of-plane transport, unlike in-plane, remains to be explained. Second, we look into the counterintuitive phenomenon of longitudinal magnetoresistance (LMR) (when current is applied parallel to magnetic field) and establish the minimal conditions for this effect to occur. We show that LMR can arise purely from band structure due to anisotropy of the Fermi surface (FS) provided, the FS is sufficiently anisotropic. We derive a condition on the anisotropy and prove that it is both necessary and sufficient. We also calculate LMR in graphite and compare it with the experimental data. Third, we discuss the effect of electron-electron interaction on the resistivity in a normal Fermi liquid. If Umklapp scattering is not allowed, unlike commonly accepted notion, the resistivity is not always guaranteed to show a $T^2$ behavior. When the FS is singly connected and two dimensional with a convex shape, there is no $T^2$ dependence. The leading dependence is $T^4$. We use this to predict how the surface resistivity in three dimensional Bi$_2$Te$_3$ family of topological insulators behaves as the Fermi energy is shifted since, with change of Fermi energy, the shape of the FS changes from convex to concave. We also derive a scaling form between resistivity and temperature in the vicinity of this convex-concave transition and show that the scaling form is universal, and therefore describes not only the Bi$_2$Te$_3$ family of topological insulators, but also any other material where the FS is two-dimensional and exhibits a similar convex-concave transition.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Hridis K Pal.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Maslov, Dmitrii.

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Permanent Link: http://ufdc.ufl.edu/UFE0044451/00001

Material Information

Title: Some Aspects of Electronic Transport in Graphite and Other Materials
Physical Description: 1 online resource (112 p.)
Language: english
Creator: Pal, Hridis K
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: electrons -- graphite -- interaction -- magnetoresistance -- phonons -- transport
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We investigate several aspects of electronic transport in this work. First, we study the transport properties of graphite. We find that the experimentally observed unsaturating metallic behavior of resistivity at high temperatures may be explained by taking into account inter-valley scattering of electrons off hard optical phonons. Additionally, the presence of extremely light Dirac-like carriers at the $H$ and $H^{\prime}$ edges of the Brillouin zone is found to give rise to a regime where the in-plane transverse magnetoresistance is linear in the field, which explains, in part, some of the experimental observations. However, these carriers are found to have no effect on quantum oscillations. The out-of-plane transport, unlike in-plane, remains to be explained. Second, we look into the counterintuitive phenomenon of longitudinal magnetoresistance (LMR) (when current is applied parallel to magnetic field) and establish the minimal conditions for this effect to occur. We show that LMR can arise purely from band structure due to anisotropy of the Fermi surface (FS) provided, the FS is sufficiently anisotropic. We derive a condition on the anisotropy and prove that it is both necessary and sufficient. We also calculate LMR in graphite and compare it with the experimental data. Third, we discuss the effect of electron-electron interaction on the resistivity in a normal Fermi liquid. If Umklapp scattering is not allowed, unlike commonly accepted notion, the resistivity is not always guaranteed to show a $T^2$ behavior. When the FS is singly connected and two dimensional with a convex shape, there is no $T^2$ dependence. The leading dependence is $T^4$. We use this to predict how the surface resistivity in three dimensional Bi$_2$Te$_3$ family of topological insulators behaves as the Fermi energy is shifted since, with change of Fermi energy, the shape of the FS changes from convex to concave. We also derive a scaling form between resistivity and temperature in the vicinity of this convex-concave transition and show that the scaling form is universal, and therefore describes not only the Bi$_2$Te$_3$ family of topological insulators, but also any other material where the FS is two-dimensional and exhibits a similar convex-concave transition.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Hridis K Pal.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Maslov, Dmitrii.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044451:00001


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SOMEASPECTSOFELECTRONICTRANSPORTINGRAPHITEANDOTHERMATERIALSByHRIDISKUMARPALADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012HridisKumarPal 2

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Tomyparents 3

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ACKNOWLEDGMENTS IamindebtedtomyadviserDr.DmitriiMaslovforhiscontinuedsupportandguidancethroughoutthiswork.Hisdepthofunderstandingandbreadthofknowledgeareexemplaryandtheyhaveinspiredmeimmenselyovertheyears.HehelpedmenavigatemywaythroughthemazeofscienticresearchandIowehimmuchforwhereIamnow.Withouthismentorship,thisdissertationwouldhaveneverseendaylight.Iamthankfultotheothermembersofmysupervisorycommittee-Dr.ArthurHebard,Dr.ChristopherStanton,Dr.Hai-PingCheng,andDr.CliffordBowers-forreadingthisdissertationcarefullyandprovidingmewithhelpfulsuggestions.PartsofthisworkhavebeendoneincollabortionwithotherresearchersandItakethisopportunitytothankthemallfortheirhelp.ThisincludesDr.ArthurHebardandDr.SefaattinTongay(Sef)whoprovidedmewiththeexperimentaldata;Dr.DmitriGutmanwhohelpedmewithsomeofthecalculationsintheprojectrelatedtotransportpropertiesofgraphite;andDr.VladimirYudsonwhoassistedmeintheprojectconcerningtheeffectofelectron-electroninteractionsontheresistivityofnon-Galileaninvariantsystems.Inthecourseofdoctoralwork,Ihavereceivedhelponcountlessoccasionsfromseveralofmyteachers,friendsandcolleagues.Iowethemallmysinceregratitude.Inparticular,IwouldliketomentionChungweiWangwhohashelpedmethroughoutmygraduatestudies,explainingtomeconceptsIcouldn'tunderstandmyself;TomoyukiNakayamaandLexKemperwhohavealwaysbeentheretoanswermyquestions;andKristinNicholaandPamMarlinwhoneverhesitatedtooffertheirsecretarialhelp.Thanksareduealsotoeveryonewhohashelpedmeinthepast,eitherdirectlyorindirectly,leadingtothesuccessfulcompletionofmydissertationwork.IwouldliketoacknowledgenancialsupportfromtheNationalScienceFoundationforpartofthisworkthroughgrantNSF-DMR-0908029. 4

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Lastbutnottheleast,Iwouldliketoexpressmyheartfeltappreciationtomyparentsandmysisterfortheirlove,supportandconstantencouragement.Withoutmyfather'szealforacademicresearchwhichheinfusedinmeatatenderageprovidingmetherequiredllip,mymother'sunconditionallovewhichhelpsmetrudgeforwardwhenthepathaheadseemsnarrowanddifcult,andmysister'spresencewhichconstantlyremindsmehowemptylifewouldhavebeenwithoutasibling,Iwouldhaveneverbeenabletoreachthisstageinlife.Iamgratefultothemall. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION .................................. 12 1.1Methodology .................................. 13 1.2ScopeandLayout ............................... 17 2ELECTRICALTRANSPORTPROPERTIESOFGRAPHITE .......... 20 2.1TheEnergySpectrumofElectronsandHolesinGraphite ......... 21 2.2DependenceofResistivityonTemperature ................. 28 2.2.1In-planeResistivity ........................... 28 2.2.1.1Modelofelectron-phononinteraction ........... 31 2.2.1.2Comparisontoexperiment ................. 35 2.2.1.3Discussion .......................... 36 2.2.1.4Summary ........................... 37 2.2.2Out-of-planeResistivity ........................ 38 2.3DependenceofResistivityonMagneticField ................ 41 2.3.1In-plane(Transverse)Magnetoresistance .............. 43 2.3.1.1Linearmagnetoresistance ................. 45 2.3.1.2Macroscopicinhomogeneities ............... 49 2.3.1.3Quantumoscillations .................... 51 2.3.1.4Summary ........................... 53 2.3.2Out-of-plane(Longitudinal)Magnetoresistance ........... 54 2.4ConcludingRemarks ............................. 55 3MINIMALCRITERIONFORLONGITUDINALMAGNETORESISTANCE:ANECESSARYANDSUFFICIENTCONDITION .................. 57 3.1Semi-classicalEquationsofMotion ..................... 59 3.2MinimalConditionsforLongitudinalMagnetoresistance .......... 61 3.2.1NecessaryCondition .......................... 61 3.2.2SufcientCondition ........................... 65 3.3Example:LongitudinalMagnetoresistanceinGraphite ........... 68 3.4ConcludingRemarks .............................. 70 4RESISTIVITYOFNON-GALILEANINVARIANTFERMILIQUIDSANDSURFACETRANSPORTINBi2Te3FAMILYOFTOPOLOGICALINSULATORS ...... 72 6

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4.1FormulationoftheProblem .......................... 74 4.2Electron-electronContributiontotheResistivity ............... 79 4.2.1LowTemperatures:PerturbationTheory ............... 79 4.2.2CaseswhentheLeadingTermVanishes ............... 83 4.2.2.1Isotropicsystemwithanarbitraryspectrum ........ 83 4.2.2.2Approximateintegrability:convexandsimplyconnectedFermisurfacein2D ..................... 84 4.2.2.3Subleadingcorrectionstotheresistivitywhentheleadingtermisabsent. ........................ 87 4.2.3Non-integrableCases ......................... 90 4.2.4Weakly-integrableCases ........................ 91 4.3EffectofeeInteractionsonSurfaceTransportinBi2Te3FamilyofTopologicalInsulators .................................... 93 4.3.1ConductivityNeartheConvex-concaveTransition .......... 94 4.3.2ProposalforExperimentalVerication ................ 99 4.4Discussion ................................... 100 4.5ConcludingRemarks ............................. 102 5CONCLUSIONS ................................... 104 REFERENCES ....................................... 107 BIOGRAPHICALSKETCH ................................ 112 7

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LISTOFFIGURES Figure page 2-1Unitcellsin(a)2Dgrahitewitha1anda2asthebasisvectorsand(b)3Dgraphite. 22 2-2TherstBrillouinzoneofgraphiteshowingthehighsymmetrypointsandtheFermisurfacesalongtheedges. .......................... 23 2-3Thebandstructure"vs.kofgraphiteneartheFermienergy. .......... 25 2-4AdiagramoftheFermisurfaceofgraphiteshowingtrigonalwarping(exaggerated). ............................................ 26 2-5Measuredin-planeresistivityofgraphite(squares)vsatheoreticalpredictioninthemodelwithscatteringrate1==1=0+T. ................ 30 2-6ExperimentaldataofabofgraphitettedwithEq. 2 ............. 35 2-7Experimentaldataofcvs.Tingraphite[ 10 ]. ................... 39 2-8Variationofhallresistancewithmagneticeldingraphite[ 10 ]. ......... 42 2-9Experimentaldataofabvs.Hat5Kingraphite[ 10 ]. ............... 44 2-10Calculateddependenceoftransversemagnetoresistanceonmagneticeldingraphiteinlowelds. ................................. 49 2-11CalculateddependenceofTMRonmagneticeldingraphiteoverawiderangeofeldvaluesbothinthepresenceandintheabsenceofinhomogeneities. .. 52 2-12Experimentaldataofcvs.Bat1.9Kingraphite[ 10 ]. .............. 55 3-1GeometricinterpretationofthenecessaryconditionforLMR. .......... 64 3-2CalculateddependenceofLMRonmagneticeldingraphite. ......... 70 4-1Isotropiccasein2D:Threepossiblescatteringprocessesnoneofwhichleadstocurrentrelaxation. ................................. 84 4-2Thenumberofself-intersectionpoints(markedbydots)dependsonwhetherthecontourisconvexorconcave. .......................... 87 4-3Thenumberofself-intersectionpointsdependsonthedimensionandthetopologyoftheFS. ....................................... 90 4-4IsoenergeticcontoursforthespectruminEq. 2 .Thedashedlinecorrespondstothecriticalenergyfortheconvex-concavetransition. .............. 94 4-5Factorsdeterminingthenumberofself-intersectionpointsin2Dcontours. ... 96 4-6Geometricinterpretationofaself-intersectingconcavecontour. ......... 98 8

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4-7DifferenttemperatueregimesforFermi-liquid(FL)andquantum-interference(QC)correctionstotheconductivity. ........................ 101 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySOMEASPECTSOFELECTRONICTRANSPORTINGRAPHITEANDOTHERMATERIALSByHridisKumarPalAugust2012Chair:DmitriiL.MaslovMajor:PhysicsWeinvestigateseveralaspectsofelectronictransportinthiswork.First,westudythetransportpropertiesofgraphite.Wendthattheexperimentallyobservedunsaturatingmetallicbehaviorofresistivityathightemperaturesmaybeexplainedbytakingintoaccountinter-valleyscatteringofelectronsoffhardopticalphonons.Additionally,thepresenceofextremelylightDirac-likecarriersattheHandH0edgesoftheBrillouinzoneisfoundtogiverisetoaregimewherethein-planetransversemagnetoresistanceislinearintheeld,whichexplains,inpart,someoftheexperimentalobservations.However,thesecarriersarefoundtohavenoeffectonquantumoscillations.Theout-of-planetransport,unlikein-plane,remainstobeexplained.Second,welookintothecounterintuitivephenomenonoflongitudinalmagnetoresistance(LMR)(whencurrentisappliedparalleltomagneticeld)andestablishtheminimalconditionsforthiseffecttooccur.WeshowthatLMRcanarisepurelyfrombandstructureduetoanisotropyoftheFermisurface(FS)provided,theFSissufcientlyanisotropic.Wederiveaconditionontheanisotropyandprovethatitisbothnecessaryandsufcient.WealsocalculateLMRingraphiteandcompareitwiththeexperimentaldata.Third,wediscusstheeffectofelectron-electroninteractionontheresistivityinanormalFermiliquid.IfUmklappscatteringisnotallowed,unlikecommonlyacceptednotion,theresistivityisnotalwaysguaranteedtoshowaT2behavior.WhentheFSissinglyconnectedandtwodimensionalwithaconvexshape, 10

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thereisnoT2dependence.TheleadingdependenceisT4.WeusethistopredicthowthesurfaceresistivityinthreedimensionalBi2Te3familyoftopologicalinsulatorsbehavesastheFermienergyisshiftedsince,withchangeofFermienergy,theshapeoftheFSchangesfromconvextoconcave.Wealsoderiveascalingformbetweenresistivityandtemperatureinthevicinityofthisconvex-concavetransitionandshowthatthescalingformisuniversal,andthereforedescribesnotonlytheBi2Te3familyoftopologicalinsulators,butalsoanyothermaterialwheretheFSistwo-dimensionalandexhibitsasimilarconvex-concavetransition. 11

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CHAPTER1INTRODUCTIONUnderstandingtransportpropertiesofsolidsinvolvesunderstandingtheowofelectricalorthermalcurrentsthroughthesystem.Currentsariseduetothetransferofchargeorenergybythecarriersfromoneendofthesystemtotheotherundertheinuenceofanexternalperturbation(drivingforce)suchasanelectriceldorathermalgradient.Ifboththedrivingforcesarepresent,onecanhavebothkindsofcurrentswithapossiblecouplingbetweenthem-aphenomenonknownasthermoelectriceffect.Thecarriersofthecurrentsareusuallychargedparticles(e.g.,electrons,holes,quasi-particles);however,theycouldbeunchargedtoo(e.g.,phonons).Inaddition,ifthereisanappropriateperturbationactingonsomeotherdegreeoffreedomofthecarrier(suchasspin),onecanhavecurrentsinthatchanneltoo(suchasspincurrent).Weare,however,interestedonlyintheelectricalaspectsoftransportinthiswork.Aknowledgeabouttheelectricalconductivityofthesystemisnotonlytechnologicallyrelevantbecauseofitspracticalutility,itisalsoindispensableinCondensedMatterPhysicsbecauseofitsroleinunderstandingthefundamentalpropertiesofsolids.Thevastplethoraofpropertiesobservedindifferentkindsofsolidsresultfrom,broadlystated,adelicateinterplayamongseveralfactors-theinteractionsamongtheparticlescomprisingthesystem(electron-electron,electron-phonon,electron-impurity,etc.),thedegreesoffreedomofeachofthoseparticlesinthesystem(charge,spin,pseudospin,etc.),andthedimensionalityandphysicalsizeofthesystem(one,two,orthreedimensionsandmesoscopicormacroscopic,etc.).Asthechargedparticlesmoveundertheapplicationofanexternalelectriceld,theirmotionisaffectedbyallthesefactors;thusitactsasaneffectivemirrortotheessentialunderlyingphysicalprocessesatplayinsidethesystem.Ourgoalistounderstandhowsomeofthesefactorsaffectelectronictransportindifferentcontexts. 12

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Dependingonwhethertheconductivityatzerotemperature(residualconductivity)isnon-zeroorzero,mostsolidsmaybeclassiedasmetalsandinsulatorsrespectively.Insolidswhichadmitabandpicture,thisdistinctionarisesfromthewaythebandsarelled-metalshavepartiallylledbandswhereasinsulatorshavebandswhicharecompletelylled(theremaybesolidswhichdonotadmitaband-picture,e.g.,Mottinsulator,wheretheunderlyingmechanismmaybedifferent).Further,ifinametallicsolid,partiallylledbandsoverlaptogiverisetoacarrierconcentrationsmallerthantypicalmetals,wecallitasemi-metal.Ontheotherhand,solidswhichareinsulatorsatzerotemperature,butwhoseenergygapsaresuchthatthereisanappreciableconductivityduetothermalexcitationattemperaturesbelowthemeltingpoint,arecalledsemi-conductors.Thisworkmainlydealswithmetalsandsemi-metalsandtheassociatedelectronictransport(althoughwediscusstopologicalinsulatorsinCh. 4 ,weareessentiallyinterestedinthemetallicsurfacestatespresentinthesesystems).Themechanismsofconductionandtheassociatedphysicalmanifestationsmaydifferdrasticallyfromonesystemtoanotherdependingontheidiosyncraciesoftheparticularsystem.However,therearealsosomefeatureswhichareeitheruniversalortrueforalargefamilyofsystemsthatowetheiroriginnottothespecicpropertiesofthesampleinquestionbutrathertosomegenralfeaturespresentinmanysystems.Therefore,tohaveareasonableundertandingofelectronictransportinagivencontext,itisnecessarytokeepinmindboththeuniversalfeaturesaswellasthefeaturesspecictothesysteminquestion.Inthecourseofthiswork,weexploreboththesesides-someofourresultswillbegeneral,applicabletoafamilyofsystems,whilesomeotherswillberelatedtoapropertythatisspecictoaparticularmaterial. 1.1MethodologySeveralmethodsexisttocalculatetheelectricalconductivityofasolid[ 1 2 ].Dependingonthesysteminquestionandtheeffectsbeingstudied,onemaybemoresuitedthantheothers.Inprinciple,ifonecouldalwaysstartatamicroscopiclevel 13

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andperformanexactquantumcalculationoftheconductivity,asinglemethodwouldsufce.However,moreoftenthannot,suchaprocedurebecomespracticallyformidableandapproximatemethodsthatarejustiedundergivenconditionsareimplemented.Throughoutthisworkweareinterestedinmacroscopicsystems(systemsizeisthelargestlengthscaleintheproblemasopposedtomesoscopicsornanosystems),inthelinearresponseregime,wheretheexternalelectriceldisconsideredsmallenoughsothatthecurrentdependslinearlyonit.Theproblemthenreducestondingtheconductivityfromtherelationj=E,wherejisthecurrentdensity,Eistheelectriceld,and,ingeneral,isatensor.Usually,however,theexperimentallymeasuredquantityistheresistivitywhichistheinverseoftheconductivity.Hence,onetypicallyinvertswhichisatensortondtheresistivitytocomparewithexperiments.Throughoutourworkweset~=kB=1.Thesimplestwaytocalculatetheconductivityofametal(considerzerofrequencyforsimplicity)istoassumeamodelwherethechargecarriersaretreatedasclassicalobjects.Ifnisthechargedensityandvisthevelocityofeachofthesecarrierswithmassm,theninthepresenceofE,theforceoneachofthecarriersiseE.Inthepresenceofimpurities,ifthecarrierschangetheirvelocitiesatanaverageintervalof,thentheaveragevelocityvavginthedirectionoftheeldiseE=m.Realizingthatthecurrentdensityisgivenby,j=nevavg,onegetstheconductivity=ne2=m.ThisistheDrudeformulaforthecondcutivity[ 3 ]which,inspiteofitssimplicity,workswellinmanycontexts.Onecanimproveuponthepreviousideabyfocussingonthedynamicsofnotjustonecarrier,butratheradistributionofcarriers.Thisisthemainstayofsemi-classicalmethodwhich,forthemostpart,willbeouradoptedmethodtocalculatetheconductivityinthiswork.Thesemi-classicalmethodisgovernedbytheBoltzmannequation(BE)whichpresupposesthatthereexistsadistributionfunctionf(r,k,t)whichisameasureofthenumberofchargecarrierswithcrystalmomentumk,atpositionr,andattimet.In 14

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thismethod,theeffectoflatticeisincludedquantum-mechanicallythorughtheinclusionofenergybands;however,thechargecarriersarestillconsideredtobeclassicalobjectssothatadistributionfunctionintermsofboththepostionandthemomentum,otherwiseforbiddenquantum-mechanically,isstillallowed.Tobemoreprecise,thisapproximationholdsaslongasthespreadoftheelectronicwavepacketissmallerthantheelasticmeanfreepath,i.e.,"F1,where"FistheFermienergy.Aswewillsee,thismethodreproducesthepreviousresultbutopensthedoortofurthercalculations.TheBEisbasicallyastatementthat,insteadystate,thereisnochangeinthedistributionfunction,i.e., df(r,k,t) dt=0.(1)ThelefthandsideofEq. 1 maybeexpandedas@f=@t+@r=@trrf+@k=@trkf@f=@t+vrrf+Frkfwhichcapturesthedifferentwaysbywhichthedistributionfunctionmaychangewithtime,vbeingthevelocityandFbeingtheforcearisingduetoexternalelds.Additionally,thecarriersscatteroffdifferentsourceswhichalsocontributeinchangingthedistributionfunction.AccountingforthescatteringbyacollisionintegralIc[f],Eq. 1 reducesto @f=@t+vrrf+Frkf=Ic[f].(1)ThisisthemostgeneralformoftheBoltzmannequation.Inourcase,eitherF=eE,ebeingtheelectroniccharge,or,inthepresenceofamagneticeldB,F=eE+e(vB).TheexactformofthecollisionintegralIc[f]dependsonthedetailsofthescatteringprocess,butinitsmostgeneralform,itmaybewrittenasIc[f]=Pk0Wk0!k)]TJ /F15 11.955 Tf 12.44 0 Td[(Wk!k0,whereWa!bdenotestheprobablityofscatteringprocessfromstateatostateb.NotethatWisafunctionofboththemicroscopicprobablitiesandthedistributionfunctions.Therefore,ingeneral,theBEisanintegro-differentialequationinf.Inalmostallcases,considerablesimplicationsneedtobemadeinordertohaveasolution.Inthecourseofthiswork,wewillmentionthemastheyareimplemented,onacase-by-casebasis. 15

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However,someofthesimplicationsarecommontoalltheresultsderivedinthiswork.Wementionthemnext.First,wewillmostlynotemployatime-dependentexternaleld.Therefore,thersttermdropsoutfromEq. 1 .Second,wewillalsonotconsideranyspatiallymodulatedexternaleldorsituationswherethedistributionfunctionisexplicitlyafunctionofposition.Therefore,thesecondtermdropsouttoo.Third,sinceweareinterestedinlinearresponsewheretheexternalelectriceldisassumedtobesmall,wecanlinearizethedistributionfunctionwithrespecttoitsequilibriumvalue.Towit,letf=f0+g(k),whereg(k)f0isthenon-equilibriumpartofthedistributionfunction.Irresepectiveoftheactualformofthecollisionintegral,onecanlinearizeitintermsofg(k).Andnally,inlinearresponse,theforcetermonthelefthandsideoftheBEinthepresenceofanelectriceldonly,canbere-writtenasevE@fo=@",wherev=rk"isthevelocityand"istheenergy.Ifthereisamagneticeld,fcannotbereplacedsimplybyf0intheforcetermarisingduetoBasitresultsinzero.Inthiscase,thetermisrewrittenase(vB)rkg.OncetheBEissolvedforthenon-equilibriumpartofthedistributionfunctiong,thecurrentiscalculatedbytheformulaj=2=(2)DRevgdDk,whereDisthenumberofdimensions.SincegisproportionaltoE,jisproportionaltoEtoo,andonecanextracttheconductivity.Asanexample,onecanre-derivetheDrudeformulafromthisapproach.Thesimplestmodelisthesocalledrelaxationtimeapproximation[ 3 ],wheremomentumrelaxationismimickedbyconsideringthecollisionintegralas,Ic[f]=)]TJ /F8 11.955 Tf 9.3 0 Td[(g=,whereissomephenomenologicalconstant.WritingtheBEasevE@fo=@"=)]TJ /F8 11.955 Tf 9.3 0 Td[(g=,wendg=)]TJ /F6 11.955 Tf 9.3 0 Td[(evE@fo=@".Usingthistondthecurrentinthesimplestcaseoffreeparticleswiththespectrum"=k2=2minthreedimensions,j=2=(2)3()]TJ /F6 11.955 Tf 9.3 0 Td[(e2)Rv(vE)@fo=@"dk,simplemanipulationsattemperatureT=0yieldtheresult=ne2=masabove,wheren=2=(2)3Rdk,theintegralrunningovertheentireBrillouinzone.Thisresultisthesameasbefore;however,notethatthis 16

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approachismorepowerfulbecause,rst,dependingonthescatteringprocess,onecanderiveanexpressionforthephenomenologicalconstant;andsecond,bywritingdownthecollisonintegralinfulldetail,wecanalsoincludeallotherformsofscattering,asnecessary,byincludingappropriatetermsinthecollisionintegralinsituationswheretherelaxationtimeapproximationfails.Indeed,todemonstratetheformeridea,assumingthatmomentumisrelaxedprimarilyduetostaticimpurites,startingfromthegeneralformforimpuritycollision,Iei[f]=Pk0Wk0,k(fk0)]TJ /F8 11.955 Tf 12.7 0 Td[(fk),onecanshowthat[ 4 ],1=(")=R(1)]TJ /F7 11.955 Tf 12.48 0 Td[(cos)S(",)d,whereS(",)isthedifferentialscatteringprobabilityandisdependentontheexactnatureoftheelectron-impurityinteraction,istheanglebetweenkandk0,andtheintegrationisdoneoverthesolidangle.InpassingwenotethattheBEequation,beingsemi-classical,missesalltheinterferenceeffectsinherentinthequantummechanicalnatureofelectrons.Tocapturethis,oneneedstoperfomquantummechanicalcalculationstypicallydoneinlinearresponsewiththehelpoftheKuboformula[ 2 ].Onecan,inprinciple,rederivealltheresultsobtainedviatheBEbythisformalism,includingtheDrudeformula,forexample[ 2 ].However,althoughthismethodismorerigorous,thecalculationsaremorecumbersomeandtherefore,unlessquantumcorrectionsarenecessaryfortheproblemathand,theBEisanexcellenttooltocalculatetheconductivity.Inmostofourwork,wewillbeinterestedinresultsthatareimportantonlyinthesemi-classicalregime.Wherevernecessary,wewillprovidethelimitswithinwhichourresultsmakesenseandhowtheycomparewitheffectsarisingduetoquantumcorrections. 1.2ScopeandLayoutInthiswork,weexplorequestionsthatarebothgeneralandmaterialspecic.Intermsofmaterials,graphiteisstudiedinsomedetail.Inregardtoquestionsthataremoregeneral,wediscusstwoareas:rst,therequirementtohavenon-zerolongitudinalmagnetoresistance(LMR)andsecond,thewayelectron-electroninteractionaffectstheresistivityinFermi-liquidsystemsthatarenon-Galileaninvariant.Inaddressingthislast 17

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question,wealsocomeupwithanovelresultthatisusefulinunderstandingthesurfaceresistivityinBi2Te3familyofthree-dimensionaltopologicalinsulators.Thematically,wecandividethisworkintotwoparts.Therstpart,consistingofChs. 2 and 3 ,dealwithtransportwithinthesingle-particleband-picture.Nointer-particleinteractionsamongthechargecarriersareconsidered,althoughwedoconsiderinteractionswithotherparticlessuchasimpuritiesandphonons.ThesecondhalfwhichcoversCh. 4 isgearedtowardunderstandingtransportwithinter-particleinteractionstakenintoaccount.Therestofthisworkisstructuredasfollows.InCh. 2 wediscussthetransportpropertiesofgraphite.Weexploreboththetemperaturedependenceofresistivityandthegalvanomagneticproperties.Whilemuchofthein-planetransportpropertiesisexplainedwithintheband-pictureatthesemi-classicallevel,theout-of-planepropertiesremainpoorlyunderstood.Inparticular,weshowthatthenon-saturatingmetallicbehaviorofthein-planeresistivitycanbewellexplainedathightemperatures(above300K)ifoneinvokesintervalleyscatteringofelectronsoffhardphonons.Asregardsin-planemagnetotransport,wendthatthepresenceofextremelylightDirac-likechargecarriersattheHandH0pointsoftheBrillouinzonegivesrisetoaregimewherethemagnetoresistance(transverse)isnon-analytic-itvarieslinearlywiththeeld.However,asweargue,theseDirac-likecarriershavenoeffectonquantumoscillations.Chapter 3 exploresthegeneralideaofLongitudinalMagnetoresistance(LMR),wheretheresistanceismeasuredalongthecurrentandinthesamedirectionastheeld.Thisisseeminglycontradictoryfor,accrordingtoLorentzlaw,thereisnoeffectofthemagneticeldonachargecarrieralongthedirectionofitsmotion.However,weshowthat,iftheFermisurface(FS)issufcientlyanisotropic,thiseffectcanarisepurelyasaresultanteffectoftheFSanisotropy.WederiveaconditionontheFSanisotropyforthistohappenandprovethatitisbothnecessaryandsufcient.Also,wecalculatetheLMRingraphiteasanexampleandshowwhytheeffectcannotarisesimplyfrom 18

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theband-structureandothermechanisms,outsidethescopeofthiswork,needtobeinvokedtoexplaintheexperimentalobservations.InCh. 4 weconcentrateontheeffectofelectron-electroninteractionsontheresistivityinnon-GalileaninvariantFermiliquids.WeshowthatthepresenceofaT2termintheresistivityisnotalwaysguaranteedwhenUmklappscatteringisnotallowed.Whetheritispresentorabsentdependson1)dimensionality(twovsthreedimensions),2)geometry(concavevsconvex),and3)topology(singlyvsmultiplyconnected)oftheFS.AnytwodimensionalsinglyconnectedconvexFSisfoundtohavenoT2term.Also,inmaterialssuchasBi2Te3familyofthreedimensionaltopologicalinsulators,wheretheFSofthetwo-dimensionalsurfacestatesundergoatransitionfromconvextoconcaveastheFermienergyischanged,theresistivityasafunctionoftemperaturefollowsauniversalscalingforminthevicinityoftheconvex-concavetransition.Finally,wesummarizethisworkinCh. 5 withafewconcludingremarks. 19

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CHAPTER2ELECTRICALTRANSPORTPROPERTIESOFGRAPHITEGraphiteisoneoftheallotropesofcarbonfoundinnature.Itisclassiedasasemimetalsinceitsvalenceandconductionbandsoverlapleadingtoasmallconcentrationofchargecarriers-considerablysmallerthantypicalmetals-whichhelpsinconductingcurrentatzerotemperature.Itsharessomegeneralfeaturescommontoallsemimetals(someotherexamplesare:As,Bi,Sb,GeTe),suchashighvaluesofmagnetoresistanceandthermalemf,inadditiontohavingsmallcarrierconcentrationandFermienergy.Besides,lightcyclotronmassesinthesematerialsenablethestudyofquantummagneto-oscillationsatmoderatemagneticeldsandtemperatures.Inaddition,graphitealsoexhibitsastrikinganisotropyinthephysicalpropertiesowingtoitslayeredcrystalstructure.Hencegraphitehasbeenaconstantsourceofinterestforresearchersinthepastandcontinuestobesotoday.Inspiteofavastamountofworkdonebackduringtheperiodbetween50'sand80's,somebasicquestionsremainunansweredwhichwewishtoexploreinthischapter.Tentativeobservationsinthelasttenyearsrelatingtometal-insulatortransition[ 5 ],quantumHall-effect[ 6 ],supermetallicconductivityinintercalatedgraphite[ 7 ],andevenintrinsicsuperconductivity[ 8 ]haveusherednewexcitementintheeld.Therecentisolationofsinglelayergraphitecalledgraphenewithremarkableelectronicpropertieshasonlyfurtheredtheinterestinthismaterial.Inthischapter,however,wefocusentirelyontheelectricaltransportpropertiesofgraphite.Allourresultsarederivedwithinthesingle-particlepicturearisingfromthebandstructure.Thischapterisnotintendedtobeanexhaustiveaccountoftheworkdonealreadyinthiseld.Althoughwehaveincludedsomeresultsfromexistingliteratureinthecourseofderivingourresults,wehavedonesoonlytoputourresultsincontext.Foramoredetailedreviewonthistopic,thereaderisreferredtoRef.[ 9 ].Thetheoreticalresultsderivedinthischapteraremainlyinspiredbyexperimental 20

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observations[ 10 ].MostoftheexperimentaldatareferredtointhisChapterrelatetoexperimentsonHighlyOrientedPyroliticGraphite(HOPG)-akindofgraphitesynthesizedarticiallyinthelaboratory.Anyreferencestoexperimentaldataonotherkindsofgraphite,suchasKishandnaturalgraphite,arementionedexplicitly.Therestofthechapterisstructuredasfollows.InSection 2.1 wediscussthebandstructureofgraphitepointingout,inshort,theideasinvolvedinderivingtheenergyspectrum.Section 2.2 describesthedependenceofresistivityontemperatureprovidingsometheoreticalunderstandingbehindtheobservedbehaviors.Thegalvanomagneticpropertiesofgraphite,specicallythemagnetoresistance,arepresentedinSection 2.3 .Finally,weendthischapterinSection 2.4 withsomeconcludingremarks. 2.1TheEnergySpectrumofElectronsandHolesinGraphiteAnyseriousattemptatexplainingthetransportpropertiesofanymaterialnecessarilybeginswithanunderstandingoftheunderlyingatomicstructureandenergyspectrum.Therefore,itisimperativethatwestartwiththecrystalstructureofgraphiteanditsresultingbandstructure.Ourpresentationhereisfarfromcomprehensivefor,thedetailsandintricaciesinvolvedinbandstructurecalculationsdemandaseparatediscussionontheirownright.Instead,inthissectionwepresentasynopsisoftheessentialideasgoverningtheenergyspectrumofcarriersandpresentsomesimpleexpressionsthatmakethecalculationoftransportpropertieslateranalyticallytractable.ForamoredetaileddescriptionofthebandstructurethereaderisreferredtoRef.[ 9 ].Graphiteisalayeredcrystal.Itiscomposedoflayersofcarbonatomsarrangedontopofeachother.Carbonatomsineachlayerarearrangedinahexagonalfashion.Thedistancebetweennearestneighboratomsinalayeris1.42Awhiletheinterplanardistanceis3.35A.Thisdifferenceintheinteratomicseparationintwodifferentdirectionsisresponsibleforthelargeanisotropyinmostofthephysicalpropertiesofgraphite.SincehexagonsdonotformaBravaislattice[ 3 ],todescribetheperiodicstructureinasinglelayer,oneusuallyconsidersarhombusasaunitcellwithatwo-pointbasis:Aand 21

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Figure2-1. Unitcellsin(a)2Dgrahitewitha1anda2asthebasisvectorsand(b)3Dgraphite(ReprintedbypermissionfromM.S.DresselhausandG.Dresselhaus,Adv.Phys.30,139(1981)). B,asshowninFig. 2-1 (a).Theprimitivetranslationvectorsa1,a2inplane(ab-direction)havethesameabsolutevaluesequaltoja1j=ja2j=a=p 31.42A=2.46A.Theperiodicityintheperpendiculardirection(c-direction)istwicetheinterplanardistance,viz.,c=23.35A=6.71AowingtothefactthatthesinglelayersingraphitearestackedupinanABABABfashion,usuallyreferredtoasBernalstacking.TheresultingunitcellisshowninFig. 2-1 (b).Thereciprocallatticesharesthesamesymmetryasthedirectlattice.TherstBrillouinzoneisshowninFig. 2-2 .Thehighsymmetrypointsare:thecenter-)]TJ /F1 11.955 Tf 6.77 0 Td[(,thecorners-HandH0,thecentersofthetopandbottomfaces-A,thecentersofthesidefaces-M,thecentersofthesideedges-KandK0,andthecentersofthetopandbottomedges-L.Carbonatomsingraphitearesp2hybridizedwhichmeansthreeoutofthefourvalenceelectronsofeachcarbonatomparticipateintheformationofbondswhichlieintheplaneofthelayer.Theseareconstructedfromthewavefunctions: 2s, 2pxand 2py(xandyaxeslieintheplaneofthelayer)andtheyultimatelycontributeto 22

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Figure2-2. TherstBrillouinzoneofgraphiteshowingthehighsymmetrypointsandtheFermisurfacesalongtheedges(ReprintedbypermissionfromM.S.DresselhausandG.Dresselhaus,Adv.Phys.30,139(1981)). bandswhicharecompletelylledanddonotcontributetotransport.Thefourthelectron,describedby 2pzstickingoutoftheplaneofCatoms(inthezdirection),takespartintheformationofbonds.Thevalenceandconductionbandsingraphitewhichhaveasmalloverlap,contributingtotheobservedtransportproperties,arisefromtheseorbitals.ThemostgenerallyacceptedmodelfortheelectronicspectrumwasdevelopedbySlonzewski,WeissandMcClure[ 12 13 ],commonlyreferredtoastheSWMcmodel.Thismodeldescribesthedependenceofenergyonthewave-vectorinthekpmethod,basedontheunderlyingsymmetryofthelattice.Themethodemployssevenparametersi(1=1,...,7)capturingthehoppingtermsbetweendifferentcarbonatoms.Theycanbeenlistedasfollows: 0:hoppingbetweennearestneighborsinalayer(3.2eV). 1:hoppingbetweentwonearestAtypeatomsfromtwoconsecutivelayers(0.4eV). 2:hoppingbetweentwoBtypeatomsfromtwonearestequivalentlayers()]TJ /F7 11.955 Tf 9.3 0 Td[(0.02eV)(notethateachBatomfollowsanotherBatomonlyineveryalternatelayer,resultingfromBernalstacking). 3:hoppingbetweenanAatominonelayerandthenearestBatomintheadjacentlayer(0.3eV). 23

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4:hoppingbetweenanAatominonelayerandthenextnearestAatomintheconsecutivelayer(0.1eV). 5:hoppingbetweentwoAtypeatomsfromtwonearestequivalentlayers(0.01eV). 6(alsocalled):on-siteenergydifferencebetweenthetwoinequivalentCatomsina2Dunitcellinalayerarisingoutofadifferenceinthechemicalenvironmentfacedbythetwoatoms(0.01eV).TheenergybandscanbefoundoutbysolvingfortheeigenvaluesoftheHamiltonian H=0BBBBBBB@"010H13H130"02H23)]TJ /F8 11.955 Tf 9.3 0 Td[(H23H13H23"03H33H13)]TJ /F8 11.955 Tf 9.3 0 Td[(H23H33"031CCCCCCCA,(2)where"01=+1)-216(+1 25)]TJ /F3 7.97 Tf 6.78 4.34 Td[(2,"02=)]TJ /F6 11.955 Tf 11.89 0 Td[(1)-217(+1 25)]TJ /F3 7.97 Tf 6.78 4.34 Td[(2,"03=1 22)]TJ /F3 7.97 Tf 6.78 4.34 Td[(2,H13=1 p 2()]TJ /F6 11.955 Tf 9.3 0 Td[(0+4\ei,H23=1 p 2(0+4\eiandH33=3)]TJ /F8 11.955 Tf 6.77 0 Td[(ei.Hereistheazimuthalanglemeasuredbetweenkandthe)]TJ /F8 11.955 Tf 6.77 0 Td[(KdirectionintheBrillouinzone,)-365(=2cos,=1 2kzc,=p 3 2ak,withkzmeasuredfromtheKpointandkmeasuredfromtheHKHedge.ThesecularequationfortheaboveHamiltonianisafourthorderequationwhichingeneralcannotbesolvedanalytically.However,togainsomeunderstandingofthebandstructure,ifweassumeforsimplicity,3=0,thisfourthorderequationdegeneratesintotwosecondorderequationswhichcanthenbesolvedexactly.Underthisassumption,thefourbranchesoftheenergyspectrumcanbewrittenas "1=1 2("01+"03)f1 4("01)]TJ /F6 11.955 Tf 11.95 0 Td[("03)2+v2k2[1)]TJ /F6 11.955 Tf 13.15 8.08 Td[(4 0]2g1 2,"2=1 2("02+"03)f1 4("02)]TJ /F6 11.955 Tf 11.95 0 Td[("03)2+v2k2[1+4 0]2g1 2, (2) wherev=p 3 20a.Fig. 2-3 showsthevariationofenergyofthefourbranchesasafunctionofthewavevectoralongdifferentdirections.ThedottedlinereferstotheFermienergywhichiscalculatedbyenforcingtheconditionofelectricalneutrality,i.e.,thenumberdensitiesofelectronsandholesareequalinpristinegraphite.Branches 24

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"+2and")]TJ /F3 7.97 Tf 0 -7.97 Td[(1correspondtothemajorityelectronsandholesrespectively,while")]TJ /F3 7.97 Tf 0 -7.97 Td[(2and"+1,respectively,aretheminorityholesandelectrons.SurfacesofconstantenergycorrespondingtotheenergyspectrumcanbedrawnaroundtheHKH(orH0K0H0)edge.Thesearesmallspindle-shapedsurfaceswhichareextendedalongtheout-of-planec-directionbutwithverylittleextentalongtheradialin-planeab-direction.Indeed,attheFermienergy,thesesurfacesoccupylessthan1%oftheBrillouinzoneradially.Fig. 2-2 showstheFermisurfacearoundtheedgesoftheBrillouinzone.Itisworthnotingthatwithintheapproximation3=0,Eqs. 2 resultinacylindricallysymmetricFermisurface.Thishoweverbreaksdownwhen3isnottakentobezero.AnexpandeddiagramoftheFermisurfaceasshowninFig. 2-4 corroboratesthisfact. Figure2-3. Thebandstructure"vs.kofgraphiteneartheFermienergy.Here=p 3=2ak,E1andE2refertotheminorityelectronandholebranchesrespectively("+1and")]TJ /F3 7.97 Tf 0 -7.98 Td[(2respectively),andE3referstothemajorityelectronandholebrancheswhicharedegeneratealongthekzdirection(ReprintedbypermissionfromM.S.DresselhausandG.Dresselhaus,Adv.Phys.30,139(1981)). Inthecourseofunderstandingthetransportpropertiesofgraphite,wewilloftenresorttoapproximationsbyneglectingsomeoftheparametersthatappearinthespectrumsothatwehavesimpleexpressionswhichwecanworkwithanalytically.Indeed,historicallytherstspectrumofgraphitewasproposedbyWallace[ 14 ]whodid 25

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Figure2-4. AdiagramoftheFermisurfaceofgraphiteshowingtrigonalwarping(exaggerated)(ReprintedbypermissionfromM.S.DresselhausandG.Dresselhaus,Adv.Phys.30,139(1981)). asimpletight-bindingcalculationwithnearestneighborinteractionbothinplaneandoutofplane.Thespectrumthatresultsfromthistwo-parametermodel(hereaftercalledtheWallacespectrum)matcheswiththeSWMcresultinthelimitweputallparametersi,i=2to6equaltozeroandleaveonly0and1non-zero.Itiseasytoseethattheresultingspectrumbecomes "=1 21)]TJ /F2 11.955 Tf 9.43 0 Td[(f1 421)]TJ /F3 7.97 Tf 6.77 4.94 Td[(2+v2k2g1 2(2)Inaddition,notethattheWallacespectrumcanbefurthersimpliedbyexpandingEq. 2 inkandkeepingonlytwodegeneratebranchesofelectronsandholes(hereaftercalledsimpliedspectrum): "k=k2 2m(kz),(2)wherem(kz)=)]TJ /F6 11.955 Tf 19.63 0 Td[(1=v2isthekz-dependentmassofthein-planemotion.ThisspectrumisvalidaslongasweareawayfromtheH(H0)points.Inpassing,wenotethat,if1 26

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iszerowhichphysicallymeansthatthelayersaredecoupled,Eq. 2 reducesto"=vkwhichisthediracspectrumofgraphene,somuchinvoguethesedays!Beforeconcludingthissection,letusdiscuss,inshort,howeachofthesevenparametersconsideredintheSWMcmodelaffectsthespectrumqualitatively.Tointroduceanyapproximationinthespectrum,suchknowledgeiscrucialfor,itallowsonetohavecontrolovertheapproximations.0isthelargestenergyscaleintheproblemaffectingthein-planevelocityofthecarriers.Thisdemarcatesthebandfromthebands.1denotestheenergyscalethatseparatesthequadraticpartfromthelinearpart(Dirac-like)ofthespectrumasseeninEq. 2 .Thus,forenergiesabove1,graphitelosesits3Dstructureandbecomesabunchofalmostnon-interactinggraphenesheets.2,althoughverysmall,servesaveryimportanttask:theFermienergyderivesitsvalue(250K)fromthisparameter.Hence,niteoverlapoftheconductionandvalencebands,nitenumberofchargecarriersandnon-zeroconductivityatT=0K,andtheverynotionthatgraphiteisasemimetalowetheiroriginsto2.Thenon-mirrorcharacterofthetwobandsresultsfromtheparameter4,whichfollowseasilyfrominspectionofEqs. 2 .Thepresenceofboth5and6mainlyaffectsthespectruminitsquantitivedetailsalthoughmentionmustbemadeofthesmallrelativeshiftofthebandboundariesdueto5.Andnally3,aswasmentionedbefore,breaksthecylindricalsymmetryoftheFermisurface.ItintroducesthesocalledtrigonalwarpingontheFermisurfacewherebythesurfacegetsconvolutedalongtheazimuthalanglewithaperiodicityof2=3radians.ThiscannotbeveriedfromEqs. 2 as3wasassumedtobeidenticallyzeroinderivingthoseequations.Theeffectof3canbeincludedperturbativelytoyieldthefollowingexpressionfortheenergyspectrumofmajorityholesandelctronsinthevicinityoftheHKH(orH0K0H0)edge(i.e.,forsmallk)[ 13 ]: "="03+A2[B24+2B3)]TJ /F6 11.955 Tf 6.77 0 Td[(3cos(3)+23)]TJ /F3 7.97 Tf 6.78 4.94 Td[(22]1 2,(2) 27

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where=p 3 2ak,A=1 2[(0)]TJ /F18 7.97 Tf 6.58 0 Td[(4\2 "03)]TJ /F18 7.97 Tf 6.59 0 Td[("01+(0+4\2 "03)]TJ /F18 7.97 Tf 6.59 0 Td[("02],andB=1 2[(0+4\2 "03)]TJ /F18 7.97 Tf 6.58 0 Td[("02)]TJ /F3 7.97 Tf 13.47 5.47 Td[((0)]TJ /F18 7.97 Tf 6.59 0 Td[(4\2 "03)]TJ /F18 7.97 Tf 6.59 0 Td[("01].Thetermcontainingcos(3)capturesthesaidtrigonalwarpingeffect. 2.2DependenceofResistivityonTemperatureWiththeknowledgeofthebandstructureathand,wenowwishtoexplorethevarioustransportpropertiesofgraphite.Webeginwiththedependenceofresistivityontemperature.Ingeneral,thereisstrikinganisotropyofthemagnitudeofresistivitybetweentheab-andc-directions(zz=xx104).Addedtothat,whilethein-planeresistivityshowsametallicbehaviorforalltemperatures,thec-axisresistivityisanon-monotoniccurvewithbothmetallicandinsulatingparts.Thephysicsofthein-planetransport,aswewillsee,canbeunderstoodbyconsideringscatteringofelectronsfromhardorsoftphonons,dependingonthetemperatureweareinterestedin.Thec-axistransport,ontheotherhand,seemstodefyacommonband-picturedescriptionandisyettobeexplainedinitsentirety.Nonethelesswewillmakesomegeneralcomments. 2.2.1In-planeresistivity1Experimentsdatingbacktothe1950s[ 15 ]showthatthein-planeresistivityrisesmonotonicallywithincreaseintemperature.Thesemeasurementsweremainlycarriedoutuptotheroomtemperature,belowwhichthechargecarriersingraphitearemoreorlessinthedegenerateregime(Fermi-statistics).ThephysicsbehindthisbehavioriswellunderstoodandcanbeaccommodatedwithinaDrude-likepicturewherethetemperaturedependenceofresistivityarisesoutofacompetitionbetweenthecarrierconcentrationwhichriseswithT,andthescatteringtimewhichdecreaseswithT.Forexample,OnoandSugihara[ 16 ]calculatedtheconductivityusingspectrum(Eq. 2 )andthescatteringtimeduetoelectron-phononinteraction,anditagreedwellwiththe 1Theresults,gures,andmostofthetextinthissectionarefromourarticle:D.B.Gutman,S.Tongay,H.K.Pal,D.L.Maslov,andA.F.Hebard,Phys.Rev.B80,045418(2009).Copyright(2009)bytheAmericanPhysicalSociety. 28

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experimentaldata.Thisideaisalsosupportedin[ 5 ]where,amongotherthings,theauthorsshowedthattheresistivitycurvecanbeverywellunderstoodusingathree-bandmodeleachobeyingaDrude-likeformula.Inthissection,however,weareinterestedtoseehowtheresistivitybehavesfortemperaturesgreaterthantheroomtemperature,somethingnotexploredbefore.Inthesetemperatures,allcouplingsusedasparametersintheSWMcspectrumexcept0and1becomeirrelevantbecausetheyareall.300K.Withonly0and1asnon-zerocouplings,graphitecanbeviewedasastackofgraphenebilayers,whichwehenceforthrefertoas`Bi-layergraphite'(BLGT).ThesimpliedWallacespectrum,describedbeforeinEq. 2 ,ishenceagooddescriptionofBLGT.NotethatBLGThasbeenreducedtoazerogapsemiconductorastheFermienergywithinthismodeliszero.Thisresultsinanenergy-independentdensityofstates,=161=v2cuptoO("=1)andthenumberdensityofchargecarriersn=2Rd!f0(!),wheref0(!)=1=(exp(!=T)+1)istheFermifunctionwithFermienergy"F=0,increasinglinearlywithT.BecauseofthelinearincreaseofthecarrierdensitywithT,thein-planeconductivityscalesas[ 14 ] xx=4ln2 e2 cT,(2)againuptoO(T=1).Consideringscatteringtobepredominantlyduetophonons,attemperatureswheretheelectron-phononscatteringisinthequasi-elasticregime,oneexpects1=toscalelinearlywithTandconsequently,xxtobeTindependent,meaningthattheresistivityisexpectedtosaturateattemperatureshigherthanroomtemperature.However,averydifferentbehaviorisobservedexperimentallyforT>300K.Fig. 2-5 showsmeasurementsupto900Kindicatingthattheresistivitycontinuestoincreaseunabated.Thesolidcurveshowsthetheoreticalpredictionifxxiscalculatednumericallyusing1==1=0+Tandarealisticspectrumofgraphite.Ascanbeseen,itaccountsonlyfor12%increaseinresistivityfrom300Kto900Kasopposed 29

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to100%increaseexperimentally.Evenwithamoredetailedspectrumofgraphite,hence,onecannotaccountfortheincreaseintheresistivity.Clearlyweneedtoinvokeanewscatteringmechanismwith1=growingfasterthenTwhichwillexplaintheobservation. Figure2-5. Measuredin-planeresistivityofgraphite(squares)vsatheoreticalpredictioninthemodelwithscatteringrate1==1=0+Tandrealisticspectrumofgraphite. Beforeproceedingwiththisgoalinmind,recallEq. 2 whereweexpandedtheWallacespectrumtofurthersimplifythespectrumas "k=k2 2m(kz),(2)wherem(kz)=)]TJ /F6 11.955 Tf 19.41 0 Td[(1=v2isthekz-dependentmassofthein-planemotion.Evaluatingthein-planeconductivityas xx(T)=4e2Zd3k()]TJ /F6 11.955 Tf 9.3 0 Td[(@fT=@"k)v2("k,T)=(2)3(2) 30

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withspectrum(Eq. 2 )and=const,wereproduceEq. 2 .Typicalmomentacontributingtoxxarekz1=candkkTp 2mT,wheremm(kz=0)=1=v2.AlthoughexpansioninkbreaksdownneartheHpoints(kz==c),where)]TJ /F1 11.955 Tf 10.1 0 Td[(vanishesandthespectrumisDirac-like,thecontributionofDiracFermionstoxxissmallinproportiontothevolumeoftheBrillouinzonetheyoccupyandleadstocorrectionsoforderO("=1)toEq. 2 .Therefore,atypicalcarrierinBLGTismassiveratherthanDirac-like;theisoenergeticsurfacesarecorrugatedcylinderscenteredneartheKpoints,andthedensityofstatesisenergy-independent.Hence,itisjustiedtousethesimpliedspectrum. 2.2.1.1Modelofelectron-phononinteractionFacedwiththetaskofndingamechanismthatproduces1=growingfasterthanT,letusrevisittheideaofelectron-phononscatteringingraphite[ 17 ].With4atomsperunitcell,graphitehas3acousticand9opticalphononmodes.Thesecanberegroupedintohardandsoftphonons[ 18 ].Thecharacteristicenergyscaleofhardmodes,0.1eV,correspondstoDebyeenergiesofthein-planeacousticphononsandfrequenciesofthein-planeopticalphononswithin-phasedisplacementsofatomsinadjacentgraphenesheets.Softmodes,withtypicalenergiesoforder10meV,arisefromweakcouplingbetweengraphenesheetsaswellasfromout-of-phasedisplacementsofatomsinadjacentsheets.Forthetemperaturesofinterest(T>300K),allsoftmodesareintheclassicalregime,inwhichtheoccupationnumberand,correspondingly,thescatteringratescaleslinearlywithT.AlthoughhardacousticmodesarestillbelowtheirDebyetemperatures,theyarealsointheclassicalregime.Forexample,typicalin-planephononmomentainvolvedinscatteringatahard,graphene-likeacousticmodewithdispersion!A=sabqareqk.Thecorrespondingfrequencies!AsabkaresmallerthanTaslongasTms2ab1K.Therefore,allsoftmodesandhardacousticmodesleadtolinearscalingofthescatteringratewithtemperature.Theremaininghardmodesareopticalin-planephonons,i.e.,the 31

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longitudinalopticalmodewithfrequency!00.17eVatthe)]TJ /F1 11.955 Tf 10.1 0 Td[(point[ 18 19 ].ForT.!0,scatteringatthesemodesleadstoanexponentialgrowthoftheresistivitywithtemperaturewhichexplainstheexperimentalobservation.Weconstructasimpliedmodel[ 20 21 ]fortheinteractionbetweenelectronsandthelongitudinalopticalphonons.Letusstartbywritingthetight-bindingHamiltonianas Hgraphite=0B@HkH?Hy?Hk1CA.(2)Here,HkandH?describehoppingwithinandoutofgraphenesheetsrespectivelyandcanbewrittenas Hk=0B@00~Sk0Sk0Sk00~Sk1CAandH?=0B@21cos(kzc)0001CA.(2)Above,Sk=eikxa=p 3+2e)]TJ /F9 7.97 Tf 6.58 0 Td[(ikxa=2p 3cos(aky=2)and~S=4cos)]TJ /F2 11.955 Tf 5.48 .27 Td[(p 3kxa=2cos(kya=2)+2cos(kya)arethestructurefactorsforin-planehoppingbetweennearestneighbor(AB)andnext-to-nearestneighbors(AAandBB,denotedby00).Thereasonforincludingthenext-to-nearestneighborinteractioninwritingtheHamiltonianwillbecomeclearshortly.However,oneshouldnotethatitsintroductionintheproblemdoesnotchangethespectruminanyappreciableway.NeartheKandK0points,~S)]TJ /F7 11.955 Tf 23.48 0 Td[(3+3k2a2=4.;sothemomentum-dependentpartof~Schangesthespectrumonlyatlarge(1=a)kand,therefore,canbeneglected.Ontheotherhand,0Skvkv(kx+iky),wherevistheDiracvelocityofasinglegraphenelayer.Uptoanadditiveconstant,theeigenvalues"koftheHamiltonianabovearegivenbythesameWallacespectrumEq. 2 whiletheeigenvectorsarerepresentedbyaspinork=C("k=k,1,"k=k,1)neartheKpointandbykneartheK0point.Theelementsofkaretheamplitudesofndingachargecarrierononeofthefouratoms(A,B,~A,~B)ofthegraphiteunitcell.Iftheopticalphononfrequencyisstillmuchsmallerthanthe1hoppingelement,bothinitialandnalstatesofascatteringprocessaredescribedbythespectrumin 32

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Eq. 2 .Forthisspectrum,theamplitudesonAand~Aatomsaresmallasj"kj=jkjk=mv(T=1)1=21.Neglectingtheseamplitudes,wereplacetheeigenvectorsby0k=(0,1,0,1)=p 2.Inreality,!0=10.5andcorrectionstotheresultsfollowingfromthisapproximationwouldbeimportantinamoredetailedtheory.Inthesimplestmodel,phononsmodulatehoppingamplitudesbystretchingthecorrespondingbonds.TheHamiltonianofthisinteractionhasthesamestructureasthatforanideallattice He)]TJ /F3 7.97 Tf 6.59 0 Td[(ph=0B@~Hk~H?~Hy?~Hk1CA,(2)wheretildedenotecorrectionstohoppingmatrixelementsduetolatticedistortions.Itisinterestingtonotethatif~Hkdoesnothavediagonalelements,thematrixelementofHe)]TJ /F3 7.97 Tf 6.59 0 Td[(phbetweenthespinors0kvanishes.Therefore,theelectron-phononinteractionappearsonlyinthediagonalelements,whichexplainstheintroductionof00intheHamiltonian..ExpandingHamiltonian(Eq. 2 )inatomicdisplacements,onends HAAe)]TJ /F3 7.97 Tf 6.59 0 Td[(ph=i@00 @aXqF^uA,q;Fx=sinaqx+2sinaqx 2cosp 3 2aqy;Fy=2p 3sinap 3 2qycos1 2aqx, (2) where^uA,qistheoperatorofthedisplacementofAatomsinthemomentumspace.Displacementsinthec-direction,beingperpendiculartotheplane,donotchange(tolinearorder)thedistancebetweenadjacentAatoms,andarethereforeuncoupledfromfermions.SincetheformfactorsFxandFyaresmallforsmallq,processeswithlargeqhavehigherprobabilities.Forthisreason,wewillfocusoninter-valleyscatteringbetweenKandK0points,correspondingtoq=q02=a)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F7 11.955 Tf 9.3 0 Td[(1=p 3,1=3.Employing 33

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Eq. 2 ,wendthematrixelement Mqk1!k2=k2,k1+q1 L3=2s 1 2!qq,(2)whereq=(@00=@a)dqFisthedeformationpotentialforopticalphonons,dqisthedimensionlesspolarizationvector,Lisasystemsize,andisthemassdensity.Thecorrespondingscatteringtimecanbeevaluatedas[ 22 ] )]TJ /F3 7.97 Tf 6.59 0 Td[(1e-ph(k1)=Xk2Wqk1!k21)]TJ /F8 11.955 Tf 11.96 0 Td[(f0(k2) 1)]TJ /F8 11.955 Tf 11.96 0 Td[(f0(k1),(2)whereWqk1!k2=W+qk1!k2+W)]TJ /F3 7.97 Tf 6.59 0 Td[(qk1!k2isthetotalscatteringrate,Warethetransitionrateswithemissionandabsorptionofphonons,respectively, Wqk1!k2=2jMqk1!k2j2Nq+1 21 2("k1)]TJ /F6 11.955 Tf 9.96 0 Td[("k2!q),(2)andNqistheBosefunction.Neglectingthedispersionoftheopticalmodeandusingsimpliedelectronspectrum(Eq. 2 ),wendfortheinter-valleyscatteringrate )]TJ /F3 7.97 Tf 6.59 0 Td[(1iv("k,T)=)]TJ /F3 7.97 Tf 6.59 0 Td[(1coth(!0=2T)cosh2("k=2T) cosh2("k=2T)+sinh2(!0=2T),(2)whereallmodel-dependentdetailsoftheelectron-phononinteractionareincorporatedintoanominalscatteringrate)]TJ /F3 7.97 Tf 6.59 0 Td[(1.Forathermalelectron("kT)thescatteringratebehavesasexp()]TJ /F6 11.955 Tf 9.3 0 Td[(!0=T)forT!0.ForT!0,thismodegoesintotheclassicalregimeand)]TJ /F3 7.97 Tf 6.59 0 Td[(1iv/T.Atsufcientlyhightemperatures,whereintervalleyscatteringisthedominantmechanism,andtheconductivityisobtainedbysubstitutingiv("k,T)intoEq. 2 (iv)ab=4ln2)]TJ /F7 11.955 Tf 11.95 0 Td[(1 3e2 cTexp!0 T.(2) 34

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Figure2-6. ExperimentaldataofabofgraphitettedwithEq. 2 2.2.1.2ComparisontoexperimentBasedontheresultfortheconductivityforintervalleyscattering(Eq. 2 ),wettheobservedab(T)bythefollowingformula: ab=c e21 0+T1 "+c e21 a0Texp)]TJ /F6 11.955 Tf 10.5 8.09 Td[(!0 T,(2)wherea0=2(4ln2)]TJ /F7 11.955 Tf 11.96 0 Td[(1)=30.376and"cRd3kv2jj)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F6 11.955 Tf 9.3 0 Td[(@f0k=@"k=23.Whencalculating",weinclude20.02eVinthespectrumasnon-zero2resultsinniteFermienergyleadingtonon-zeroconductivityatT=0K,whichisnecessarytottheexperimentalcurve.TherstterminEq. 2 accountsmostlyforthelow-Tbehaviorofab,whenscatteringatimpurities(1=0)andsoftphonons(T)dominatestransport.Thesecondtermisduetointervalleyscattering.Equation( 2 )containsfourttingparameters:0,,,and!0.TheresultsofthetareshowninFig. 2-6 .Thettingparametersare0=6.2910)]TJ /F3 7.97 Tf 6.59 0 Td[(12s,=0.09,=1.410)]TJ /F3 7.97 Tf 6.59 0 Td[(14s,and!0=0.22eV.Thevaluesof0andareinreasonableagreementwithvaluesfoundinpreviousstudies[ 5 ].Thefrequency!0issomewhathigherbutstillclosetothefrequencyoftheE2gmode 35

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[ 18 19 ].Arathershortnominaltimeindicatesstrongcouplingbetweenelectronsandopticalphononsingraphite. 2.2.1.3DiscussionThescatteringmechanismdescribedaboveisnottheonlysourceofscatteringpresentinthesystem.Thereareothermechanismsatworktoo.However,theyfailtoproducetheexpectedbehavior.Inpassing,webrieyremarkonacoupleoftheseotherscatteringmechanismsforcompleteness.First,westartwiththeelectron-holeinteraction.Incontrasttotheelectron-electroninteraction,thismechanismgivesrisetoniteresistivityinacompensatedsemi-metalevenintheabsenceofUmklappprocesses[ 22 23 ].ForTEF,wehaveausualFermi-liquidbehavior1=e-h/T2.TheT2-behaviorofabisindeedobservedingraphitebelow5K[ 5 24 ].However,BLGTisnotaFermiliquidbutratheranon-degenerateelectron-holeplasmawithxed(andequaltozero)chemicalpotential.ToestimatethestrengthofCoulombinteraction,wecalculatethersparameter,i.e.theratiooftypicalpotentialandkineticenergies rs(T)=e2 1lEkin,(2)where1isthebackgrounddielectricconstantofgraphiteandl=(4=3)1=3n)]TJ /F3 7.97 Tf 6.59 0 Td[(1=3/T)]TJ /F3 7.97 Tf 6.58 0 Td[(1=3isthetypicalinter-carrierdistance.Usingtheexperimentallymeasuredvaluesofnumberdensityn1019cm)]TJ /F3 7.97 Tf 6.58 0 Td[(3atT=300K(Ref.[ 25 ])and15(Ref.[ 26 ]),andevaluatingthekineticenergy2asEkin=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(2=12ln2T1.2T,weobtainrs(T=300K)1.2.Astemperatureincreases,rs(T)decreasesasT)]TJ /F3 7.97 Tf 6.59 0 Td[(2=3.Therefore,theperturbationtheoryfortheelectron-holeinteractionisreasonablyaccuratealreadyatT'300KandbecomesevenbetterathigherT.IntheThomas-Fermimodel,the 2Theequipartitiontheoremoftheclassicalstatisticalphysics,whichstatesthatthekineticenergyassociatedwitheverydegreeoffreedomisequaltoT=2,isnotapplicabletoacompensatedsemimetal(evenifthespectrummassive),becausethedistributionfunctionisnotMaxwellianbutratheraFermi-Diraconewithzerochemicalpotential. 36

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screenedCoulombpotentialU(q)=4e2=1q2jj+q2z+2isisotropiceveniftheelectronspectrumisnot;alldetailsofthespectrumareencapsulatedinthe(squareof)screeningwavenumber2=4e2=1proportionaltothedensityofstates.InBLGT,2isT-independentand,sinceisproportionaltothe(small)in-planemass,c)]TJ /F3 7.97 Tf 6.59 0 Td[(1.Also,atnottoohightemperatures,kT/p T.Sinceqjjcannotexceedthetypicalelectronmomentum,qjj.kTandU(q)almostdoesnotdependonqjj;therefore,U(q)4e2=1)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(q2z+2.UsingthisformofU(q)andthespectrumfromEq. 2 ,weobtainfromtheFermiGoldenRule 1 e-he21 v2T.(2)Thislinearscalingpersistsevenathighertemperatures(T&1),inthesingle-layerlimit[ 27 ].Therefore,electron-holeinteractiondoesnotprovideanexplanationoftheexperiment.Also,wenotethatforTabovetheDebyefrequency,themulti-phononprocessesmodifythescatteringrateas1==1=0+)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(T+(T=T2)2+(T=T3)3+O(T4),whereT2,3correspondtoanenergyscaleatwhichanharmonicitybecomesstrong.ThedatacanbettedbyT2T3103K,whichiswellbelowthescaleof104Katwhichanharmonicitybecomesstronginotherphysicalproperties,suchasthec-axisthermalexpansioncoefcientandelasticmoduli[ 28 ].ThisreinforcesourconclusionthatanharmonicityisnotimportantintransportforT<1000K. 2.2.1.4SummaryTosummarize,wehavestudiedin-planeresistivityofHOPG.Wefoundthatitstemperaturedependenceisdeterminedbyacompetitionbetweenthoseofthecarriernumberdensityn(T)andofthescatteringrate1=.Attemperaturesbelow50K,thenumberdensityispracticallyindependentofthetemperature,whilethethescatteringrateincreaseswiththetemperature;asaresult,theresistivityincreaseswithT.AttemperaturescomparabletotheFermienergy,theincreaseinn(T)almost 37

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compensatesforthatin1=,leadingtoaquasi-saturationofxxatT300K.However,fullsaturationneveroccursbecause,asthetemperatureincreasesfurther,scatteringoffhardopticalphonons,characterizedbyanexponentialincreaseof1=withT,becomesimportant.ThisresultsinafurtherincreaseofxxwithT.However,weshouldpointoutthatalthoughtheresistivitycurveshowsnosignofsaturationupto900K,ourtheorypredictsthatitshouldeventuallysaturatefortemperatureshigherthan!0.Thisregimeishoweverdifculttoattainexperimentally. 2.2.2Out-of-planeResistivityThec-axisresistivitybehavesinamannerwhichisstrikinglydifferentfromtheab-resistivity,bothqualitativelyaswellasquantitatively.Fig. 2-7 presentstheexperimentalmeasurementsfortemperaturesupto900K.Therearethreeaspectsofthecurvethatcaptureattention.First,atzerotemperature,zzisapproximately104higherthanxxinmagnitude(compareFigs. 2-5 and 2-7 ).Second,unliketheab-resistivity,thec-axisresistivityisnon-monotonic.IthasametallicpartatlowtemperaturesandaninsulatingpartathighertemperatureswiththemaximumoccurringatT50K.Andthird,thetailofthecurveatthehighesttemperaturemeasureddoesnotsaturate.Unfortunately,acompletetheoreticalunderstandingofthebehaviorisstillmissing.Unlikein-planetransportwhichcanbedescribedquitewellatalltemperaturesbyasimpleBoltzmannpicturewithappropriatelychosenscatteringprocesses,thec-axistransportseemstobegovernedbysomeothermechanism.AcalculationbasedontheBoltzmannequationwiththebandpicturetakenintoaccountcannotexplainallpartsofthec-axisresistivitycurve.Belowwediscusssomeoftheseissues.First,westartwiththelargeanisotropyintheresistivities.Foraroughestimate,ifweassumethatzzhasaDrude-likeformula,i.e.,=m ne21 ,wherenistheelectronnumberdensity,mistheeffectivemassderivedfromtheenergyspectrumandisthescatteringtime,thenzz xx=(mzz mxx)(xx zz),subscriptszzandxxdenoteout-of-planeandin-planerespectively.Assuminganellipsoidalmodelfortheenergyspectrumin 38

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Figure2-7. Experimentaldataofcvs.Tingraphite[ 10 ]. graphite,mzz4m0andmxx0.03m0,wherem0isthebareelectronicmass[ 9 ].Tomeettheratioof104betweenthetworesistivities,weneedxx zz102.UsingamorerealisticspectrumforgraphiteandcalculatingtheresistivityexactlyforT=0K(assumingtobeindependentofkor"),yieldstheratioxx zztobe75whichisisnotasignicantimprovement.Ordinarilyifweassumethat1=0arisesduetostaticimpurities,itisnotclearimmediatelywhythetwoscatteringtimesarisingoutofimpurityscatteringshouldbesodifferent,unlesssomeotherexoticprocessistakingplace.Arecentsuggestionthatstackingfaultscouldleadtotheanisotropy[ 29 ]maybetheanswer.Next,inordertoexplainthepeakintheresistivitycurve,wenotethatwithinourmodelthiscanariseonlyfromacompetitionbetweenandn:asTincreases,nincreasesanddecreases.Sincezz1=zz n,forsmalltemperatures,1=zzhastogrowfasterthannandeventuallylosetonthusbringingthecurvedown.Ifweassumethesamescatteringprocessesasinthein-planecasefromtheprevioussection,attemperatureswherethepeakoccurs,1=hastheform1=0+T.Since1=0isbiggerthanitscounterpartinthein-planecase,thetemperaturedependentpartof1=zz 39

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needstobelargeratleastbyoneorderfor1=zztohaveanyappreciabletemperaturedependencewhichcouldwinovern.However,theoreticallyifweconsiderquasi-elasticscatteringofelectronsandsoftphonons,itisdifculttomeettherequirementof1=zztobegreaterthan1=xx[ 22 ].Finally,wecometothehightemperaturetailintheresistivitycurve.Atthistemperature,followingourargumentofthein-planeresistivity,wecanviewgraphiteasastackofgraphenebi-layers.Asinthepreviouscase,wearethenallowedtousethesimpliedspectrum(Eq. 2 )andcalculatetheconductivityusinganexpressionsimilartothein-planecase.ItcanbeeasilyshownthattheintegralscalesasT2ln(1=T),whichmeansnincreasesslightlyfasterthanT2.Atthistemperature,thein-planetransportisdominatedbyscatteringofelectronsandhardphononswhichresultin1=toincreaseexponentiallywithT.Ifweapplythesameargumentinthecaseofc-axis,thentheresistivitycurveshouldbeanincreasingoneinsteadofdecreasingbecausetheexponentialfunctionin1=willalwayswinovertheT2ln(1=T)term.Evenifoneneglectsthetemperaturedependencein1=assumingthelargephenomenologicalvalueof1=0,andallowsforthewholedependenceoftheresistivitycurveinthehightemperatureregimetocomefromn/T2ln(1=T)alone,itdoesn'texplainthecurveinitsentirety.Indeed,attingoftheexperimentaldataonresistivityagainstthefunctionalformT2ln(1=T)(notshowninthegure)showsthattheytverywellandthevalueof1=0thatcomesoutturnsouttobeclosetothatextractedfromthezerotemperaturevalueindependently.This,therefore,wouldseempromisingwereitnotfortheresultingproblemthatwewillfaceintryingtoexplainthemetallicpartofthecurvefromthesamemodel.Withinthismodel,sincetheentiretemperaturedependenceofzzarisesfromnalone,themetallicparttooshouldbedueton.Unfortunately,thisisnotborneoutbyactualcalculation.Usingthesamesimpliedmodel,butnowintroducinganiteFermienergy,ifwedoaSommerfeldexpansioninT="F,thecoefcientfortheT2terminncomesouttobepositive,meaning1=nshouldconsistentlydecreasewith 40

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temperaturestartingfromT=0Kuptothehighesttemperature.Specically,asimplecalculationyields zz("2F+2 3T2)ln(1 "F),(2)fortemperaturesbelow"F.Improvinguponthespectrumdoesnotyieldanynewresult.ExactcalculationoftheintegralnumericallyusingamorerenedspectrumasinEq. 2 demonstratesasimilarmonotonicincreaseinn.Thuswendthatitisdifculttoexplainthedependenceofc-axisresistivityontemperaturebasedononlythebandpicture.Toachieveacompleteunderstandingofthebehavior,probablyweneedtoinvokesomeothermechanismofconductioninthec-direction. 2.3DependenceofResistivityonMagneticFieldItiswell-knownthatthetransportpropertiesinanymaterialcanbegreatlyaffectedbythepresenceofamagneticeld.This,infact,providesaveryusefultooltounderstanddifferentpropertiesofthematerial,e.g.,theelectronicstructure.Whenconsideringtheelectricaltransportproperties,theadditionalpresenceofthemagneticeldgivesrisetomagnetoresistanceandHalleffectand,forhighquantizingelds,magneto-oscillations.Also,therehavebeenrecentreports[ 30 ]arguingthatgraphiteexhibitsbothintegerQHEaswellasfractionalQHEinhighenoughelds,whichopensupnewpossibilities.Inthepresentcase,however,wewillbecontentwithmainlynon-quantizingelds;although,towardstheend,wewillcommentoncertainaspectsofquantumoscillations.Sincen=1LandaulevelcrossingtakesplaceatB7T,thismeansmostofourdiscussionspertaintoeldslessthan7T.Theoretically,magnetotransportinnon-quantizingeldsisdescribedbytheBoltzmannequationwhichneedstobesolvedtocalculatetheconductivitytensor.Usuallyoneinvertsthistensortogettheresistivitytensorwhosecomponentscanthenbedirectlycomparedwithexperimentaldata.Galvanomagneticpropertiesofgraphitehavebeenstudiedinthepast[ 31 ].Semi-classicalexpressionsforthemagnetic 41

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Figure2-8. Variationofhallresistancewithmagneticeldingraphite[ 10 ]. conductivitytensorcomponentswerederivedbysolvingtheBoltzmannequationintherelaxationtimeapproximationwithconstant.Theseauthorsconsideredasimplespectrumwithamulti-bandmodelformajorityandminoritycarriers.ForexamplethehallresistanceplottedagainstmagneticeldinFig. 2-8 canbequalitativelyunderstoodbyusingasimple2-bandmodelwithparabolicspectrum: Hall=(R122+R221)B+R1R2(R1+R2)B3 (1+2)2+(R1+R2)2B2.(2)wherethesubscripts1and2refertotherstandsecondbandrespectively,Rreferstothehallcoefcient,andreferstotheresistivity.Ascanbeseen,thepresenceofalinearandacubicterminthenumeratorpredictsaminimumpointinthegraph,whiletheasymptoticbehaviorforveryhighorloweldsshouldbelinear,asseenintheexperimentaldata.Thereare,ofcourse,subtletiesinvolvedwhichneedmorerenedtreatment,butonthewholeEq. 2 doesexplainthedata.Inthissectionhowever,wewillnotdealwiththesesubtletieswhichhavebeendealtwithinthepast.Instead,we 42

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focusonthemagnetoresistancewherewendinterestingresults.Inwhatfollows,wewillalwaystakeBtobeinthezdirection,i.e.,perpendiculartothegraphitelayers. 2.3.1In-plane(transverse)magnetoresistance3Despitethefactthatthedependenceofin-planeconductivityontemperatureiswellunderstoodusingthesemi-classicalBoltzmannequation(cf.,Sec. 2.2.1 ),thedependenceofin-planemagnetoresistanceoneldpresentssomedifculty.Experimentally,in-planetransversemagnetoresistance(TMR)(Bkc),isfoundtodependonthetheeldinawaythatisessentiallysampledependent.Whilethebehaviorinsomesamplesisfoundtomatchwell,atleastqualitatively,withtheexpectedbehaviorcalculatedsemi-classicallybytheBoltzmannequationusingamulti-bandmodel[ 16 32 ],inothercasesitdeviates[ 30 33 ].Interestingly,intheselattercases,TMRisfoundtovarylinearlywiththeeld,thelinearityspanningawiderangeofeldvalues,beginningfromverysmallBandpersistinguptotheultra-quantumregimeandbeyond[ 9 ].Fig. 2-9 showstheexperimentallyobservedlineardependence.Althoughthelinearityintheultra-quantumregimecanbeaccountedforbyintroducingelddependenceinthescatteringtime[ 9 ],thelinearbehaviorinsemi-classicaleldsstilllacksaproperunderstanding.Itisusuallythoughttobearisingduetoextrinsicreasonssuchasmacroscopicinhomogeneitiesbuttheissueisfarfrombeingsettled[ 30 ].Fromsimplesymmetryarguments,sincetheeldisavector,themagnetoresistancecanonlybeafunctionofB2.Infact,onecanshowverygenerallythatforanyspectrum,intheasymptoticlimit,TMRbehavesquadraticallyforverysmalleldsandeithersaturates(Fermisurfacewithclosedorbits)orgrowsquadratically(Fermisurfacewithopenorbits)forveryhighelds[ 4 34 ].Alineardependenceontheeld,thus,istheoreticallyinterestingasthefunctionaldependenceisessentiallynon-analytic. 3Theresults,gures,andmostofthetextinthissectionarefromourarticle:H.K.PalandD.L.Maslov,tobesubmittedtoPhys.Rev.B. 43

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Figure2-9. Experimentaldataofabvs.Hat5Kingraphite[ 10 ]. Itshouldhoweverbenotedthatexperimentally,linearmagnetoresitanceisfoundtooccurinmanymaterialsandgraphiteiscertainlynotanexception[ 35 ].Whileanunderstandingexistsonlyinahandfulofsituations[ 36 ],mostcasesingeneraldefyexplanations.Nevertheless,researchershaveshownthatnon-analyticdependenceontheeldincertaincasescanariseduetospecialfeaturesoftheFermisurface[ 37 38 ].Therefore,itisworthwhietorevisitthesemi-classicalcalculationsdoneusingtheBoltzmannequationandtrytoseeifalineardependencecouldresultfromanysuchpeculiaritiesingraphite.Surprisinglyenough,asweshowinthissection,theredoesexistaregimeofeldswhereTMRscaleslinearlywiththeeldwhichresultsentirelyfromthebandstrucutreofgraphite.ThelinearityisfoundtooriginatefromthepresenceofextremelylightcarriersneartheH-pointsintheBZofgraphite.ThisseemspromisinginlightofthefactthatitistheH-pointswhichharbortheDiracFermionsingraphiteandalotofefforthasbeenmadeintherecentpasttoseeiftheseDiracFermionscanhaveanysignatureintransportpropertiesingraphite.However,beyondthesaidregime,TMR,asseeninourcalculations,isnolongerfoundtovarylinearlywiththeeld,in 44

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contrasttothatobservedexperimentally,whichthereforeprobablyarisesduetoextrinsicreasonssuchasthepresenceofmacroscopicinhomogeneities.ApartfromthelinearbehaviorofTMR,quantumoscillationswhichappearontopofthelinearbackgroundinTMRhavealsodrawnsomeattentioninrecentyears.Therehasbeenclaimsbasedonrecentexperiments[ 39 ]thatthesequantumoscillationscarrysignaturesofDiracFermionsingraphite,whichhoweverseemstobedisputable[ 40 ].InspiredbytheeffectofextremelylightcarriersneartheH-pointonTMR,weaddressthisissuetheoreticallyandndthat,withinthepresentlyacceptedbandstructure,itisnotpossibletohaveanysignatureofDiracFermionsonquantumoscillationsingraphite.Thereasonisinfactsubtle.Aswepointoutlaterinthechapter,itisthetypeofstacking(Bernal:ABAB..)inconventionalgraphitewhichpreventsonefromseeinganyeffectofDiracFermionsinquantumoscillations.ArticialgraphitewithAAAA..typeofstackingwillexhibitquantumoscillationsarisingsolelyfromDiracFermionscomingfromH-points. 2.3.1.1LinearmagnetoresistanceInordertocalculatethedependenceofin-planetransversemagnetoresistanceoneld,recallthatgraphiteiswelldescribedbythesimpliedformofthespectrum:"k=k2 2m(kz),wherem(kz)=)]TJ /F6 11.955 Tf 28.52 0 Td[(1=v2isthekzdependentmassofthein-planemotion,aslongasweareawayfromtheH-points(kz==c)becauseattheH-points)]TJ /F1 11.955 Tf -425.54 -23.91 Td[(vanishesandtheexpansioninkwhichledtothisformisnolongervalid.Note,aspointedoutbefore,theoverlapoftheconductionandvalencebandwhichgivesrisetoasmall(31018cm)]TJ /F3 7.97 Tf 6.59 0 Td[(3atT!0)butnon-zerocarrierconcentrationinthecompletemodelisduetohoppingbetweennext-to-nearestplanesandtheFermienergyisalsoontheorderofthismatrixelement.Therefore,strictlyspeaking,theFermienergyiszerointhissimpliedmodel.However,allourensuingcalculationsonmagnetotransportwillbedoneatzerotemperature,wherenon-zeroconcentrationofcarriersandhencea 45

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non-zeroFermienergyisrequired.Therefore,althoughwekeeponlynearestneighborinteractionsinthespectrum,weincludeanon-zeroFermienergyinourcalculations.Wenowcalculatethesemi-classicalTMR.Thestandardwayistoinverttheelementsoftheconductivitytensorfoundfromtheexpressionforcurrent:j=2e=(2)3Rv(k)g(k)dk,whereg(k)isthenon-equilibriumpartofthedistributionfunctionobtainedbysolvingthelinearizedBoltzmannequationwhich,intherelaxationtimeapproximation,reads: )]TJ /F8 11.955 Tf 11.96 0 Td[(eEv@f0 @"=1 +e(vB)@ @kg(k).(2)Ingeneral,itisnoteasytosolvetheBoltzmannequationforanarbitraryspectrum(evenwithinisotropicrelaxation-timeapproximation).However,owingtothesimplicityofourspectrum,wecanarriveatanexactsolutioneasily.Assumeg(k)=vA,whereAisanunknownvectortobedeterminedandisallowedtobeafunctionofkzalone.Substitutingg(k)intoEq. 2 andrepresentingAintermsofatriad[ 4 ]composedofB,E,andBE,wegetaftersomealgebra, g(k)=)]TJ /F6 11.955 Tf 9.3 0 Td[(e@f0 @"vE+(e=m(kz))2BBE)]TJ /F7 11.955 Tf 11.96 0 Td[((e=m(kz))BE 1+(e=m(kz))2B2.(2)SinceBinthezdirection,tounderstandTMRwewefocusonxx.UsinggasobtainedinEq. 2 intheexpressionforcurrent,wehaveforeachband, xx(B)=4e2 (2)3Zv2x 1+!c(kz)22)]TJ /F6 11.955 Tf 10.5 8.08 Td[(@f0 @"dk,(2)wherethefactorof4hasbeenincludedtoaccountforspinandvalleydegeneracy,!c(kz)=eB=m(kz)andisassumedtobeindependentofk.Substitutingvxintheintegrandandsimplifying,weget: xx(B)=4e2 (2)3"F"2 c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(42 cZ=201 cos2+2d#,(2) 46

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where=kzc=2and=!c(kz=0)=eB=m(0),m(0)beingthethevalueofthein-planemassatkz=0.Clearly,for1,xx(B)goestozeroas1=B2asexpected.Ontheotherhand,for1,onecannotexpandthedenominatorin,becausetheresultingintegraldivergesasR=20d cos2at==2.Itmeansthatthedominantcontributiontotheintegralcomesfromasmallregionnear==2,i.e.,fromthevicinityoftheH-point.Thisallowsustoreplacecos2by(=2)]TJ /F6 11.955 Tf 12.45 0 Td[()2,shiftthevariableas0==2)]TJ /F6 11.955 Tf 12.22 0 Td[(,andextendtheupperlimitofintegrationto1,uponwhichtheintegralreducestoR10d0 (0)2+2==(2jj).SubstitutingthisbackinEq. 2 ,weseethatxx/jBj.Thislinearityisreectedalsointheresistivityasidealgraphiteisaperfectlycompensatedmaterialandtherefore,theresistivityxx(B)issimply1=xx(B).InpassingwenotethattheintegralinEq. 2 isactuallysimpleenoughtobecalculatedexactly.Carryingouttheintegrationatzerotemperatureexactlyonends xx(B)=xx(0)1)]TJ 27.72 8.08 Td[(jj p 2+1.(2)Theabovedescribedbehaviorisselfevident.Thefactthatxx(B)isgovernedbyextremelylightcarriersresidingneartheH-pointsintheBZisinteresting;butitalsoservesasacaveatthatthepictureisnotcomplete.Allthecalculationspresentedsofarhavebeendoneusingthesimpliedspectrum,whichwasderivedbyexpandingtheWallacespectrum.TheapproximationisvalidawayfromtheH-points.Intheregimeof1,thelowertheeldtheclosertotheH-pointsarethosecarriersthatmostlycontributetotheconductivity.Atsomelowenoughvalueoftheeld,theexpansionoftheWallacespectrumitselfbreaksdownandthesimpliedspectrumisnolongervalid.Tondthescaleatwhichthishappens,werecalltheWallacespectrumfromEq. 2 :"=[)]TJ /F6 11.955 Tf 9.3 0 Td[(1)-228(+(21)]TJ /F3 7.97 Tf 6.77 4.34 Td[(2+v2k2)1=2],)-310(=cos(kzc=2).Clearlytheexpansioninkbreaksdownwhen1)]TJ /F2 11.955 Tf 10.48 0 Td[(vk.AttheFermienergy,vk"F,whichmeansthetwotermsbecomecomparablewhen)]TJ /F2 11.955 Tf 11.42 0 Td[("F=1.SinceinEq. 2 thedominantcontributionintheintegralcomesfrom)]TJ /F2 11.955 Tf 10.76 0 Td[(,itmeans 47

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theresultpredictingthelinearbehaviorforxx(B)isvalidonlyfor"F=11.Below"F=1,oneneedstoconsiderthefullWallacespectrumandrecalculatethedependenceoftheconductivityonthemagneticeld.Goingthroughsimilarstepsasbefore,itisstraightforwardtoshowthatinthislimit,xx(B)21="FandtheresultingresistivityalsoscalesasB2,asinconventionalmaterials.Tosummarize,wehave,theoretically,twoscalesinidealgraphite,correspondingtowhichtherearethreeregimeswhereTMRbehavesdifferentlywithrespecttotheeld.For,"F=1,thedependenceisquadratic,for"F=11,thedependenceislinear,andfor1TMRagainvariesquadratically.Interestinglyenough,thelinearityinthemagnetoresistancewhichisthecentralresultofthissection,isaclassicallylow-eldphenomenon.Thisshouldbecomparedtotheotherexistingmechanisms(eitherclassicalorquantum)intheliteraturewherelinearmagnetoresistanceresultsinasymptoticallyhighelds[ 41 ].Sinceinrealsamplesimpuritiesalwaysgiverisetosomedecompensationandhencetoanon-zeroHallcomponentintheconductivitytensor,itisinstructivetocalculatetheHallpartoftheconductivityandseeifthelinearityisstillpreserved.Forthesimpliedspectrum, xy(B)=4e2 (2)3Zv2x(!c) 1+!2c2@f0 @"dk=xx(0)2 tanh)]TJ /F3 7.97 Tf 6.58 0 Td[(11 p 2+1 p 2+1. (2) Note,duetocylindricalsymmetryofthespectruminthismodel,xx=yyandalltheoff-diagonalcomponentsoftheconductivityvanish,i.e.,z=z=0,for=x,y.Thustheconductivitytensorhasablock-diagonalformleadingtoxx=xx=(2xx+2xy).Consideringboththeelectronandholebandsandintroducingasmalldecompensationdenedbyn=ne)]TJ /F8 11.955 Tf 13.2 0 Td[(nh(neandnharetheconcentrationofelectronsandholesrespectively;nenh=n,say),weextracttheasymptoticlimitsofEqs. 2 and 48

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2 forsmalleldstoobtainxx(B)=xx(0)(1)-297(jj)=((1)-297(jj)2+42 2(n n)2),for1.Thisimpliesthat,inthislimit,xx(B)=xx(0)/jBj.Thelinearitythereforeisnotdestroyedbydecompensation.Itiseasytoshowthatintheoppositelimitoflarge,xx(B)continuestogrowquadraticallyasbeforebutnotindenitely.Asexpected,anewscale1=(n=n)emergesabovewhichthemagnetoresistanceeventuallysaturatesasxx(B)/1=(n=n)2.WenotethattheHallresistancewithinthismodel,however,showsnoanomalousbehavior. Figure2-10. Calculateddependenceoftransversemagnetoresistanceonmagneticeldingraphiteinlowelds.Inset:Ablown-upportionofthesamegraphshowingthelineardependenceonmagneticeld.Here=eB=m(0). 2.3.1.2MacroscopicinhomogeneitiesAlthoughtheoreticallywehavealinearregimeofmagnetoresistanceingraphite,itdoesnotcompletelyexplaintheexperimentallyobservedlinearity.Indeed,usingtypicalvaluesforandbandparameters[ 5 9 ],wendthattherange"F=11translatesintoeldrangeof100gaussto1000gauss.Inexperimentshowever,linearityisobservedatmuchhigherelds[ 30 33 ](cf. 2-9 ).Linearmagnetoresistanceinclassicallyhighelds(!c1)issometimesascribedtomacroscopicinhomogeneitiesinthesample.Theideaoflinearmagnetoresistancearisingfromaninhomogeneoussampleisnotnew-severalresearchershaveworkedonthisproblembefore[ 42 ],andrecentobservationsoflinearmagnetoresistanceinsome 49

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othermaterialssuchassilverchalcogenides,havebeenattributedtothepresenceofsuchinhomogeneities[ 43 44 ].Theessenceoftheideaisasfollows:Supposethatthechargecarriersinthematerial,insteadofbeinguniformlydistributed,formpuddles,eachofwhichisrichineithertypeofchargecarriers.Theentirematerialisstillelectricallyneutralowingtothebackgroundpositivecharges.Thesepuddlesaremacroscopicallylargeinthesensethattheirtypicallengthscaleislargerthanthemeanfreepathbutsmallerthanthesystemsize.Therefore,onecandeneaconductivitytensorforeachofthesepuddles.Physically,puddlescanariseduetoseveralreasons,e.g.,grainsinthesample;however,werefrainfromspecifyinganyparticularmechanism.Thequestionthatonewouldliketoansweris:whatistheeffectiveconductivityoftheentiresample,giventhatweknowtheconductivityofeachpuddle?Asolutiontothisproblemforthegeneralscenarioexistsintheliteratureandinvolvesnumercialcalculationsatthemean-eldlevel.However,theonlysituationwhereananalyticalsolutionexistsisthesimplercaseofatwodimensionalmaterialwithtwocomponentswithequalpartialvolumefractions[ 45 46 ](strictlyspeakingthematerialcanstillbethreedimensional,onlytheinhomogeneityhastobespreadoutinaplane).Sinceweareinterestedingainingabasicunderstandingoftheeffectsofsuchinhomogeneitiesingraphite,weadoptthissimpliedmodelfortherestofthissection.Consideraninhomogeneousheterophasegraphitecrystalcomposedoftwophasesofequalfractionsinvolumeanddenedbytheirownconductivitietensorsi,i=1,2,whosecomponentsaredescribedbythesameexpressionsasinEqs. 2 and 2 .Forsimplicity,weassumethattheonlydifferencebetweenthetwophasesisthedecompensationfractions:1,2=n1,2=n1,2()]TJ /F7 11.955 Tf 9.3 0 Td[(11,21).Thuswehaveid=xxandit=ixy,wheredandtinthesubscriptsstandforthediagonalandtransversecomponentsoftheconductivitytensor,andxxandxyareobtainedfromthesameequationsasbefore,i.e.,Eqs. 2 and 2 .UsingtheexpressionsobtainedbyBulgadaevandKusmartsev[ 45 ],onecanndthecomponentsoftheeffective 50

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conductivityeas ed=p 1d2dA,whereA="1+1t)]TJ /F6 11.955 Tf 11.96 0 Td[(2t 1d+2d2#1=2, (2) et=2t1d+1t2d 1d+2t. (2) Forthecaseofgraphite,usingEqs. 2 and 2 ,itisstraightforwardtoshowthatforelds1thebehaviorofTMRremainsunchanged,i.e.,magnetoresistanceofinhomogeneouscrystalsdoesnotdifferfromthatofhomogeneousonesintermsoftheelddependence(however,theexactexpxressionformagnetoresistancenowisofcoursedifferentandinvolves1,2).However,forelds1,thebehaviorchanges.Itstartsoffquadratically,asinthehomogeneouscase,butbecomeslinear,asopposedtosaturating,athighelds.Thislinearityhappensfor1=min(12)andasymptoticallythelinearslopeof isj1)]TJ /F18 7.97 Tf 6.58 0 Td[(2j 2(21+22).TheHallresistivityalsoscaleslinearlyinthisregimewithaslopeof1+2 2(21+22)(inunitsof0,thezeroeldresistivity).Wenotethat,withinthismodel,itispossibletondanotherscaleforeldsbetween1and1=min(12),namely1=max(12).Theasymptoticdependenceoneldintheregion1=max(12)1=min(12)dependscruciallyonthemutualsignsof1,2:ifsgn1=sgn2,meaningthatboththephasesarericherbythesametypeofcarriers,themagnetoresistancetendstosaturatefromitsinitialquadraticdependencebeforeincreasinglinearlyforhigherelds;ifontheotherhand,sgn1=)]TJ /F8 11.955 Tf 9.3 0 Td[(sgn2,meaningthatoneofthephasesisrichinelectronswhiletheotherisricherinholes,themagnetoresistancechangesintoB3fromB2beforebecominglinear.Fig. 2-11 showsanillustrativecurveintheinhomogeneouscasecomparedtothesameinthehomogeneouscase. 2.3.1.3QuantumoscillationsWithinthesemi-classicalregime,athighelds(!c1),onendstheusualquantumoscillationssuperimposedonthebackgroundmagnetoresistanceingraphiteascanbeseeninFig. 2-9 .ItiswellknownthatextremalcrosssectionsintheFermi 51

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Figure2-11. CalculateddependenceofTMRonmagneticeldingraphiteoverawiderangeofeldvalues:solid(red)forasinglecrystalwithoutanyinhomogeneitiesanddashed(blue)forsamplewithinhomogeneities.Here=eB=m(0),1=0.09,and2=)]TJ /F7 11.955 Tf 9.3 0 Td[(0.01. surfaceperpendiculartotheeldgiverisetothesemagneto-oscillations.Withintheacceptedbandpicture,ifminoritycarriersareneglected,graphitethereforeisexpectedtoshowtwosetsofoscillationsarisingfromelectronandholepockets.Themaximalcross-sectionsinboththeelectronandholeFermisurfacesoccuratpointsawayfromtheH-pointswhereonendsnormalmassivecarriers(c.f.,Eq. 2 ).However,attheH-pointswheretheBZendsinthez-direction,theFermisurfaceremainsopenleadingtoatinybutnon-zerocross-section.Therefore,itisnaturaltoaskifquantumoscillationscanalsoariseduetothesecross-sectionsharboringDiracFermions.Opinionsdifferonthisissueontheexperimentalsideasbothclaimsandcounterclaimstothiseffecthavebeenmaderecently[ 39 40 ].Inthissectionweshowthattheoretically,withintheacceptedbandstructure,itisnotpossbiletohavesignaturesofDiracFermionsinquantumoscillationsingraphite.Toestablishourclaim,werecallthattheoscillatorypartoftheconductivityleadingtoShubnikov-deHaasoscillations(ormagnetizationincaseofDeHaas-vanAlpheneffect)isgivenby Xq6=0ZdkzIn(kz)ei2qn(kz)(2) 52

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whereIn(kz)intheintegrandisacombinationofthesingle-particleGreen'sfunctionsdependingonthequantitybeingcalculated,andn(kz)istheLandauindexasafunctionofkz.InthelimitoflargenumberofLandaulevels(n1),itiscustomarytocomputetheintegralbymeansofasaddlepointapproximation,wherethemaincontributiontotheresultcomesfromthosevaluesofkzwheren(kz)hasanextremum.Forgraphite,thenearlyDiracfermionsneartheH-pointsaredescribedby,"=vk)]TJ /F6 11.955 Tf 12.19 0 Td[(1cos(kzc=2).TheLandaulevelscanbeeasilyfoundleadingtothefollowingexpressionforn: n=1 2ev2B"n+1coskzc 22.(2)ItisobviousthatthisexpressiondoesnotadmitanextremumvalueattheH-pointswherekz==c.Therefore,noquantumoscillationsareexpectedtoresultfromthecarriersneartheH-points.Notethatthesolereasonforthisnegativeresultistheextrafactorof2inthedenominatorwithinthecosineterminEq. 2 .Thisfactorof2resultsfromBernal-stackingofthegraphenelayersingraphite,whichmakestheperiodicityinthez-directiontobetwolatticeplanesinsteadofone.IfoneweretoconstructanarticialgraphitebyplacinggraphenelayersdirectlyontopofeachotherinAAAA..fashion,DiracFermionsattheH-pointswouldindeedgiverisetoitsownsetofquantumoscillations. 2.3.1.4SummaryInsummary,wendthatthereexistsaregimeofmagneticeldwherethein-planetransversemagnetoresistanceingraphite(eldparalleltoc-axiswhilethecurrentisintheab-plane)scaleslinearlywiththeeld.ExtremelylightcarriersneartheH-points,wherethespectrumisDirac-like,contributestosuchanon-analyticbehavior.Interestingly,thelinearityinthiscaseappearsinthelow-eldregion,unlikeotherknownmechanismswheresuchabehaviorresultsonlyinasymptoticallyhighelds.However,theobservedlinearityinactualsamplessometimesspanstheentirerange-uptoclassicallyhigheldswhichcannotbeaccountedforwithinasimplesemi-classical 53

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treatementbasedonthebandpicture.Anargumentbasedonthepresenceofmacroscopicinhomogneitiesinthesesampleshasbeenusedtounderstandtheobserveddependencesemi-quantitatively.Inaddition,wealsolookintothepossbilityofndinganysignatureofDiracFermionsresidingneartheH-pointsinquantumoscillationsandconludeonthenegative.Thereasonfortheabsenceofsuchaneffectisfoundtobeadirectconsequenceofthestackingstyleingraphite. 2.3.2Out-of-plane(Longitudinal)MagnetoresistanceWhenthecurrentowsinthesamedirectionasthemagneticeld,theresistancemeasurediscalledthelongitudinalmagnetoresistance(LMR).LMRingraphiteexhibitsastrikingdependenceonthemagneticeld.Atsmalltemperatures,therearetwopointstonote:rst,thevalueofLMRishuge,andsecond,zz(B)=zz(0)showsanon-analyticp Bbehavior.Fig. 2-12 showsthevariationofmeasuredzzwithBat1.9K,whereitcanbeseenthatthecurvefollowsap Bdependence.Onceagain,onsymmetryarguments,thisisnotordinarilyexpected.Oneexpectsaquadraticbehavioratsmallelds(!c1)andsaturationathigherelds.Theotherfeature,similartotheTMRcase,isthatthecurveshowsnosignofsaturationatthehighesteldmeasured(7T)whichiswellintothequantumregimeforgraphite.Beyondthiseldgraphiteenterstheultra-quantumregimewhereonlythelowestLandaulevelisoccupied.WewouldliketopointoutthatsimilardependenceoneldwasobservedbySpain[ 47 ],althoughinhiscasethemagnetoresistancesaturatesforhigherelds(8T)asexpectedfromtheory[ 34 ].Wedonothaveaconclusiveexplanationforthenon-analyticbehaviorofthelongitudinalmagnetoresistanceingraphiteatthispoint.Thisshouldcomeasnosurpriseaswearestilllackingacompleteunderstandingofthedependenceofc-axisresistivityontemperatureintherstplace.Arguably,c-axisconductivityisnotgovernedbyasimpleBEtypetransportarisingpurelyoutofsingle-particlebandpicture.Nevertheless,inourquestforanexplanationforthebehaviorofLMRingraphite,we 54

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Figure2-12. Experimentaldataofcvs.Bat1.9Kingraphite[ 10 ]. encounteredadifferentproblem-onethatismorefundamentalandpressing:Whydoesgraphite(orforthatmatteranymaterial)shownon-zerolongitudinalmagnetoresistanceintherstplace,whenconventionalwisdomwouldsuggesttothecontrary?TheLorentzforceactingonachargecarrieractsinadirectionperpendiculartothedirectionofthemagneticeldandhenceshouldnotaffectthemotionofthecarrierinthedirectionoftheeld,resultinginzeroLMR.Experimentally,however,LMRisobservedtobenon-zeronotonlyingraphitebutinahostofothermaterials.Itshouldthereforebeagoodexercisetondtheminimumconditionrequiredfornon-zeroLMR.Thisformsthebasisofthenextchapter,wherewederivetheminimalrequirementsaFSshouldsatisfyinorderforthiseffecttoarisepurelyoutoftheband-picture. 2.4ConcludingRemarksWehavestudiedtheelectricaltransportpropertiesofgraphiteinthischapter.Evenwithinthesemi-classicalregime,wefoundthatmuchremainstobeunderstoodinthismaterialinspiteofitsrichhistoryofresearch.Startingwithashortaccountof 55

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thebandstructure,weexploredin-planetransportingraphitewhichcanbeexplainedverywellwithintheBoltzmannpicture.Fromthetemperaturedependenceofthein-planeresisitivitywefoundthatlowtemperaturetransportisdominatedbyscatteringofelectronsandsoftphononswhilehightemperaturedataconrmsintervalleyscatteringofelectronsandhardphonons,leadingtoasteadyincreaseintheresistivitywithtemperature.Thevariationofthein-planetransversemagnetoresistancewithmagneticeldcanbeexplainedwithinthesameBoltzmannpicture.Calculationrevealsthreescalesfor!cwheretransversemagnetoresistancemanifestsdifferentdependencies:below"F=1ithasquadraticbehavior,between"F=1and1ithasalinearbehavior,between1andn=nitisquadratic,andnallyitsaturatesfor!cgreaterthann=n,wherenrepresentsthedecompensation.However,thelinearityobservedinexperimentsevenathighereldscannotbeexplainedwiththispicturearisingpurelyoutoftheband-structure.Weconjecturedthepresenceofmacroscopicinhomogeneitiestobethecausebehindthis.Also,becauseoftheBernalstackingingraphite,thelightDirac-likecarriersattheHandH0edgeswerefoundtohavenosignatureinquantumoscillations. 56

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CHAPTER3MINIMALCRITERIONFORLONGITUDINALMAGNETORESISTANCE:ANECESSARYANDSUFFICIENTCONDITION1Magnetoresistance,asdescribedinthelastchapter,canbedistinguishedintotwotypesdependingonthemutualorientationofthecurrentandthemagneticeld:transverse(TMR)andlongitudinal(LMR).AlthoughachangeinthetransverseresistanceduetoamagneticeldisnaturalbecauseelectronsexperienceLorentzforceinthatdirection,theveryexistenceofLMRissomewhatsurprising,atleastatrstglance.Indeed,sinceLorentzforceisperpendiculartotheeld,onedoesnotexpectthemotionofelectronsalongtheeldtobeaffected.Aweakpointofthisargumentisthatitapplies,strictlyspeaking,onlytofreeelectronsbutnottoelectronsinmetals.Moreover,LMRisabsentinamorerealistic(yetstillincomplete)dampedBlochelectronsmodel(DBEM),inwhichaphenomenologicaldampingtermisintroducedintothesemi-classicalequationsofmotionforanarbitraryelectronspectrum[ 48 50 ].However,wewillargueinthispaperthatthedampedBlochelectronsmodelisnotequivalenttotheBoltzmannequation(BE),whichprovidestheonlycompletesemi-classicaldescriptionofsemi-classicaldynamicsofelectronsinsolidsinthepresenceofscattering.Therefore,absenceofLMRinDBEMdoesnotimplyitsabsenceinreality.Experimentally,LMRhasbeenobservednotonlyingraphiteasseenbefore,butalsoinmanyothermaterials[ 51 52 ].Theoretically,ageneralsolutionoftheBEinthemagneticelddoesnotexcludeLMR[ 53 ];calculationsperformedforparticularmetals,e.g.,copper,doyieldniteLMR[ 51 52 ].However,itisnotclearfromthisgeneralsolutionwhichsymmetriesmustbebroken,i.e.,howanisotropictheelectronspectrum 1Theresults,gures,andmostofthetextinthischapterarefromourarticle:H.K.PalandD.L.Maslov,Phys.Rev.B81,214438(2010).Copyright(2010)bytheAmericanPhysicalSociety. 57

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shouldbeforLMRtooccur.ThisisprobablythereasonwhyLMRissometimesviewedasakindofsurprise[ 54 55 ].Inadditiontoanisotropicspectrum,severalmorespecialmodelshavebeeninvokedtoexplainLMR.Itwasshown,forexample,thatLMRcanariseduetoanisotropicscattering[ 56 ],macroscopicinhomegeneities[ 57 ],includingbarrierinhomogeneitiesinsuperlattices[ 55 ],aswellasduetoamodicationofthedensityofstatesbythemagneticeldintheultra-quantumregime,whenallbutthelowestLandaulevelsaredepopulated[ 58 ].WhereasobservedLMRinmanycasesislikelytobecausedbythesemoreevolvedmechanisms,itisstillnecessarytoexplorewhetherLMRcanarisesimplyduetoanisotropyoftheFermisurface(FS)andtoformulateaminimalconditionforLMRtooccur.Magnetotransportinnon-quantizingeldsisdescribedbytheBoltzmannequationwhichgivestheconductivitytensor.Tondmagnetoresistance,oneinvertsthistensor.Itiswellknownthatforanyisotropicspectrumandalsoforisotropicscattering,themagneticelddependencesofthediagonalandoff-diagonalconductivitiescancelout,sothatbothTMRandLMRareabsent.WhileTMRcanbemadenitebyeitherinvokinganykindofanisotropyoftheFermisurfaceorintroducingamultibandpicturewhilekeepingthespectrumisotropic,thestorywithLMRisnotsosimple.Asisshowninthischapter,notalltypesofanisotropygiverisetoLMR,e.g.,deformingasphericalFermisurfaceintoanellipsoidaloneisnotenough.WederivethenecessaryandsufcientconditionthespectrummustsatisfyforLMRtooccuranddiscusstheimplicationsofthisconditionforseveraltypesofband-structure.Forexample,metalswithface-centeredcubic(FCC)andbody-centeredcubic(BCC)latticessatisfythenecessaryandsufcientconditionevenifonlynearest-neighborhoppingistakenintoaccount,whereasasimplecubic(SC)latticehasLMRonlyduetohoppingbetweennext-tonearestneighbors.Thesameistrueforlayeredstructures,suchashexagonalplanesstackedontopofeachother,whereonehastoincludeout-of-planenext-nearest-neighborinteractionstoseetheeffect.Notethattheconditionservesasaminimalconditioninthesensethat 58

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itdemonstrateshowLMRcanarisepurelyduetoFermisurfacegeometryevenifscatteringisisotropic.Therestofthischapterisorganizedasfollows.InSec. 3.1 weshowthatLMRisabsentinDBEMandanalyzethedifferencesbetweenthisandBoltzmann-equationapproach.InSec. 3.2 ,wederivethenecessaryandsufcientconditionforLMRintheBoltzmann-equationformalismanddiscusstheimplicationsofthiscondition.Asaparticularexample,weconsiderthecaseofBernal-stackedgraphiteinSec. 3.3 .Ingraphite,thenecessaryandsufcientconditionissatisedduetotrigonalwarpingoftheFermisurface.Wend,however,thatthestrongnon-parabolicLMRobservedinhighlyorientedpyrolyticgraphite(HOPG)samples[ 9 ]cannotbeaccountedforbysimplyconsideringtheanisotropyoftheFermisurface.OurconclusionsaregiveninSec. 3.4 3.1Semi-classicalEquationsofMotionTheeffectofweakelectricandmagneticeldsonelectronsinsolidscanbedescribedbythesemi-classicalequationsofmotion[ 3 ] v=@"k @k, (3) dk dt=e(E+vB), (3) whereeistheelectronchargeandweset~=1.Weneglectheretheanomaloustermsinthevelocitywhich,evenifpresent,aresmallinweakmagneticelds[ 59 60 ].Intheabsenceofscattering,Eqs. 3 and 3 arevalidforanarbitraryspectrum"(k)andprovideaninvaluabletoolforanalyzingcollisionlessdynamicsofelectronsinsolids.Toaccountforscatteringofelectronsbyimpurities,phonons,etc.,itiscustomarytoreplaceEqs. 3 and 3 byaphenomenologicaldampedBlochelectronmodel(DBEM)withadampingterm)]TJ /F5 11.955 Tf 9.3 0 Td[(k=insertedintotheright-handsideofEq. 3 [ 48 50 ].Insteady-state,DBEMreducesto k =e(E+vB).(3) 59

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WearenowgoingtoshowthatthisapproacheliminatesLMRnotonlyforanisotropicbutalsoforanarbitraryspectrum.TondLMR,weassumethatthecurrentj=encv,wherencisthenumberdensityofconductionelectrons,isalongBchosenasthez-axis.Then, vz(kx,ky,kz)=jz nce, (3) vx(kx,ky,kz)=0, (3) vy(kx,ky,kz)=0. (3) Furthermore,theequationofmotion(Eq. 3 )forthekzcomponentgives kz=eEz.(3)ThesetoffourequationsEqs. 3 3 )denesaninhomogeneoussystemofequationsforfourunknowns:kx,ky,kz,andEz.Ingeneral,suchasystemhasauniquesolution.Therefore,EzcanbefoundasafunctionofjzusingonlyEqs. 3 3 .Sincenoneoftheseequationsinvolvethemagneticeld,thelongitudinalresistivityzz=jz=EzdoesnotdependonBeither,whichimpliesthatLMRisabsentforanarbitraryspectrum.Ontheotherhand,componentsExandEyhavetobefoundfromtheequationsofmotionforkxandkywhichdoinvolveB,andhence,TMRisnotzeroforanarbitraryspectrum.Iftheaboveconclusionwerecorrect,itwouldbeinvariancewithexperimentalobservations.Aswewillshowshortly,non-zeroLMRcanbeunderstoodonlybyusingtheBEasdescribedearlierinCh. 1 .AlthoughtheBEisasemi-classicaldescriptionjustlikethepreviousmethod,thereissomeconceptualdifferencebetweenthetwoapproaches.Theproblemisthat,whiletheequationsofmotionsintheabsenceofscatteringcanbederivedfromtheSchroedingerequation,theDBEMdoesnotfollowfromanymicroscopicapproach.Indeed,themomentumkintheabsenceofscatteringstillhasthemeaningofthequantumnumberparameterizingtheBlochstate k(r). 60

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Hencea(slow)evolutionofkwithtimeinthepresenceofelectricandmagneticeldsdescribestheevolutionof k(r).Inthepresenceofscattering,e.g.,bydisorder, k(r)becomesarandomquantitywhoseaverageoverdisorderrealizationsdoesnothaveaparticularmeaning.Therefore,itisnotsurprisingthatanadhocinsertionofthedampingtermintotheequationofmotiondoesnotcaptureessentialphysics.Evenintheabsenceofthemagneticeldtheshortcomingsofthisprocedurebecomeobvious-e.g.,itiseasytoshowthattheconductivitycalculatedbyDBEMmethoddoesnotcoincidewiththatcalculatedfromtheBEforageneralbandstructure,theexceptionbeingonlyinthecaseofasimpleisotropicspectrum. 3.2MinimalConditionsforLongitudinalMagnetoresistanceHavingdealtwiththeinconsistenciesofthedampedBlochelectronsmodel,wenowreturntotheoriginalproblemofndingtheminimumrequirementfornon-zeroLMRforanarbitraryspectrum"="(k).WenowderivetherequiredconditionbasedonsolvingtheBEandshowthatasingleconditionwhichisbothnecessaryandsufcientdoesexisttoserveasaminimumrequirementtohavenon-zeroLMR[ 61 ]. 3.2.1NecessaryConditionInthelinear-responseregime,onecanrewritetheBEforthenon-equilibriumpartofthedistributionfunctiong(k)=fk)]TJ /F8 11.955 Tf 11.96 0 Td[(f0kas(cf.Ch. 1 ) 1+^g(k)1+e(vB)@ @kg(k)=)]TJ /F6 11.955 Tf 9.3 0 Td[(eEv@f0k @"k, (3) wherewehavealsoadoptedtherelaxation-timeapproximation(whichisexactforisotropicimpurityscattering).Sinceweareinterestedintheminimalcondition,weallowtodependonlyon"butnotonthedirectionofkandassumethatallcomponentsofkrelaxatthesamerate,i.e.,1=isascalarratherthanatensor.Wewillcomebackto 61

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thispointlaterinthechapter.ForBjj^z, ^=e(vB)@ @k=eBvy@ @kx)]TJ /F8 11.955 Tf 11.96 0 Td[(vx@ @ky.(3)FollowingtheZener-Jonesmethod[ 4 ].weexpressg(k)viaaninniteseriesintheoperator^: g(k)=(1+^))]TJ /F3 7.97 Tf 6.58 0 Td[(1)]TJ /F6 11.955 Tf 9.29 0 Td[(eEv@f0k @"k=1Xn=0()]TJ /F7 11.955 Tf 10.69 2.66 Td[(^)n)]TJ /F6 11.955 Tf 9.3 0 Td[(eEv@f0k @"k. (3) Notethattheoperator^alwaysyieldszerowhenitactsonanyfunctionthatdependson"konly.Hence,inEq. 3 ,^actsonlyonv.SubstitutingEq. 3 intothecurrentj=2eRd3kvg(k)=(2)3,wendtheconductivityas =2e2Zd3k (2)3)]TJ /F6 11.955 Tf 10.5 8.09 Td[(@f0k @"kv1Xn=0)]TJ /F7 11.955 Tf 10.69 2.66 Td[(^nv.(3)IntheLMRgeometry,EjjBjj^z.If^vz=0,allbutthen=0terminEq. 3 areequaltozero.Therefore,anecessaryconditionforzztodependonthemagneticeldis ^vz6=0.(3)Rewriting^incylindricalcoordinates,thecondition 3 canbere-expressedas: @" @@ @k)]TJ /F6 11.955 Tf 15.75 8.09 Td[(@" @k@ @vz6=0,(3)or @ @kz@"=@ @"=@k6=0.(3)Ontheotherhand,Eq. 3 isnotasufcientconditionbecauseevenif^nvz6=0forthenthtermintheseries,thecontributionofthistermtozzmayvanishuponintegratingovertheFermisurface.Forexample,sincezzmustbeanevenfunctionofB,alloddtermsintheseriesmustvanish. 62

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Equation 3 impliesthattheminimumconditiononthespectrumisthattheratioof@"=@and@"=@k(equaltokv=v)mustdependonkz.Geometrically,thismeansthattheanglebetweenthecomponentofvelocityperpendiculartotheeldandtheradialdirectionatagivenpointontheFermisurfacemustvarywithkz.Itcanbeeasilyseenthatifthespectrumdoesnotdependon,condition 3 istriviallyviolatedandthereisnoLMR.Therefore,angularanisotropyoftheFSaboutthemagnetic-elddirectionisaprerequisite.However,anisotropymustbeofaspecialkind.Forexample,spectrasuchas"k="1(k,)+"2(kz)and"k="1(k,)"2(kz),whicharearbitrarilyanisotropicinthedirectionbutseparableinkz,violatecondition 3 andthusdonotleadtoLMR.Asanexample,letusconsideraSClatticewithlatticeparametera.Inthetight-bindingmodelwithnearest-neighborhopping(parameterizedbycouplingt1),theenergyspectrumisgivenby"k=)]TJ /F7 11.955 Tf 9.3 0 Td[(2t1[cos(kxa)+cos(kya)+cos(kza)]which,beingseparableinallthreecoordinates,clearlyviolatestheLMRcondition.Ifnext-to-nearest-neighborhopping(parameterizedbycouplingt2)istakenintoaccount,additionalterms)]TJ /F7 11.955 Tf 9.3 0 Td[(4t2[cos(kxa)cos(kya)+cos(kya)cos(kza)+cos(kza)cos(kxa)]occurinthespectrum,whichnomoreviolatestheLMRcondition.Thus,theeffectcomesonlyfromnext-to-nearest-neighborhoppingforaSClattice.Ontheotherhand,anFCClatticesatisestheconditionalreadyatthenearest-neighborlevelbecausethespectruminthiscase"k=)]TJ /F7 11.955 Tf 9.3 0 Td[(4t1[cos(kxa=2)cos(kya=2)+cos(kya=2)cos(kza=2)+cos(kza=2)cos(kxa=2)]isnon-separable;thesameistrueforaBCClattice.Ontheotherhand,layered,e.g,hexagonal,structureswillrequirecouplingbetweenanatomlocatedinoneplaneandanotheratomintheadjacentplanebutsituatedobliquelyfromtheformer,ifthemagneticeldisperpendiculartotheplanes(moreonthislaterforthespeciccaseofgraphite).Aquantitymeasuredinatypicalexperimentisnottheconductivitybuttheresistivity.Generallyspeaking,thedependenceoftheconductivityonthemagneticelddoesnotautomaticallyimplyadependenceoftheresistivityontheeldawellknowncaseis 63

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Figure3-1. Geometricinterpretationofthenecessaryconditionforlongitudinalmagnetoresistance.Here,v?isthecomponentoftheelectronvelocityperpendiculartotheeld. theisotropicspectrum,whenthe(transverse)diagonalcomponentsoftheconductivitydependonBbutthediagonalcomponentsoftheresistivitydonot.Itisnecessary,therefore,tomakesurethatEq. 3 isnotonlyanecessaryconditionforlongitudinalmagnetoconductancebutalsoformagnetoresistance.Itisdifculttoprovethatnon-zeromagnetoconductanceimpliesnon-zeroLMRforanarbitraryspectrum.Toproceedfurther,werelaxaconditionontheenergyspectrum,assumingthatBisperpendiculartotheplaneofsymmetry,i.e.,that"(kx,ky,kz)="(kx,ky,)]TJ /F8 11.955 Tf 9.3 0 Td[(kz).Thisconstraintisstrongerthanthatimposedbytimereversalsymmetry(intheabsenceofthespin-orbitinteractionandmagneticstructure),i.e.,inthiscase,vzisoddwhilevxandvyareeveninkz,andtheoff-diagonalcomponentsz(6=z)vanishbothinzeroandnitemagneticelds.Forexample,alltermsintheexpressionforxzvanishuponintegrationoverkz: xz=2e21Xn=0()]TJ /F8 11.955 Tf 9.3 0 Td[(eB)nZd3k (2)3vxvy@ @kx)]TJ /F8 11.955 Tf 11.96 0 Td[(vx@ @kynvz@f0k @"k=0. (3) 64

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BytheOnsagerprinciple,z=aswell.Therefore,thematrixofisblock-diagonalandzz=1=zz.ThusEq. 3 isanecessaryconditionfornon-zeroLMRaswell,providedthatthespectrumissymmetriconinversionofkz. 3.2.2SufcientConditionTheconditionpresentedinEq. 3 isonlyanecessaryconditionforLMR,astheintegralinEq.( 3 )maystillvanishduetosomesymmetryeveniftheintegrandsatisesEq. 3 .Toformulateasufcientcondition,weapproachtheproblemfromthestrong-magneticeldlimit.Inthislimit,itisconvenienttousethemethodofLifshitz,Azbel'andKaganov[ 34 53 62 ],inwhichthek-spaceismappedontoaspacedenedbythesetofvariables""k,kzandt1,wheret1,denedbytheequation dk dt1=evB,(3)isthetimespentbyanelectronontheorbitinthek-spaceinthepresenceofthemagneticeldonly.Accordingly,theintegrationmeasureistransformedas ZZZdkxdkydkz=eBZZZdt1d"dkz.(3)Thenon-equilibriumcorrectiontothedistributionfunctioncanbewrittenas g=e@f0 @"Es,(3)wheressatises @s @t1=Ic[s]+v.(3)Adoptingtherelaxation-timeapproximationforIcandkeepingonlytheleadingtermin1=B,itiseasytoseethat[ 53 ] sz=hvzi,(3)wherehvzi=1 TRvzdt1withTbeingeithertheperiodofanorbit(forclosedorbits)orthetimeoverwhichanorbitreachestheboundaryoftheBrillouinzone(foropenorbits). 65

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Thezzcomponentoftheconductivitytensorinthislimitisthenequalto zz(1)=2e2 (2)3eBZZZd"dkzdt1vzhvzi)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@f0 @"=2e2 (2)3Id`ZdkzZd" v?vzhvzi)]TJ /F6 11.955 Tf 10.5 8.09 Td[(@f0 @", (3) whered`isalineelementalongtheorbitandv?=p v2x+v2y.Obviously,zz(1)doesnotdependonB.Ontheotherhand,thezero-eldvalueofzzis zz(0)=2e2 (2)3Zd3kv2z)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@f0 @".(3)Pippard[ 52 ]suggestedthattheratiozz(1)=zz(0)maybeusedtogetinformationaboutthescatteringmechanismsontheFS.We,however,usethisratiotoconstructasufcientconditionforLMR.Keepingthesameconstraintontheenergyspectrum"(kx,ky,kz)="(kx,ky,)]TJ /F8 11.955 Tf 9.3 0 Td[(kz)sothatzz=1=zz,thesufcientconditionforLMRcannowbeformulatedasfollows:ifzz(1)6=zz(0),wehavenon-zeroLMR.Itisonlyasufcientconditionbecause,evenifitisviolated,LMRcanstillexist.Indeed,evenifasymptoticlimitsofthefunctionzz(B)coincide,itisnotnecessarilyaconstant.Toformulatethesufcientconditioninmoretransparentterms,weusethefollowingtrick.Theintegrationmeasureintheexpression(Eq. 3 )forthezero-eldconductivitycanformallyberewrittenintermsofthevariables",kzandt1,asspeciedbytransformation(Eq. 3 ).Sincetheresultdoesnotdependonthemagneticeld,thistransformationcanbeappliedforanyvalueoftheeld;but,tocomparethezero-andstrong-eldvalues,wechoosethesameBasintherstlineofEq. 3 .Then, zz(0)=2e2 (2)3eBZZZd"dkzdt1v2z)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@f0 @".(3) 66

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ComparingthisequationwiththerstlineofEq. 3 ,weseethatthesufcientconditionisequivalentto ZZZd"dkzdt1)]TJ /F6 11.955 Tf 10.49 8.09 Td[(@f0 @")]TJ /F8 11.955 Tf 5.48 -9.69 Td[(vzhvzi)]TJ /F8 11.955 Tf 19.26 0 Td[(v2z6=0.(3)Integratingovert1,werewritethelastequationasZZd"dkz)]TJ /F6 11.955 Tf 10.5 8.09 Td[(@f0 @")]TJ /F2 11.955 Tf 5.48 -9.68 Td[(hv2zi)-222(hvzi2=ZZd"dkz)]TJ /F6 11.955 Tf 10.5 8.09 Td[(@f0 @"h(vz)-222(hvzi)2i6=0.Sincetheintegrandisnon-negative,theintegralcanvanishonlyifvz=hvzi,whichisthecaseifvzdoesnotdependont1.Hence,thesufcientconditionisequivalenttotherequirementthat @vz @t1=@vz @kx@kx @t1+@vz @ky@ky @t1+@vz @kz@kz @t16=0.(3)RecallingthatksatisesEq. 3 ,were-writethelastequationas vy@ @kx)]TJ /F8 11.955 Tf 11.96 0 Td[(vx@ @kyvz6=0,(3)or,recallingthedenitionoftheoperator^inEq. 3 ,as ^vz6=0.(3)SincethesufcientconditioninEq. 3 coincideswiththenecessaryconditioninEq. 3 ,weconcludethatEq. 3 isbothnecessaryandsufcientforLMR.Asacorollary,italsofollowsthatthestrong-eldvaluezz(1)isalwayssmallerthanorequaltozz(0),implyingthatifLMRisnon-zero,itispositive.Beforeconcludingthissection,wewouldliketocommentthatouraimwastoestablishaminimalconditionfortheappearanceofLMRinmaterials.Specically,wewantedtoexplorewhether,inthesimplestmodelforscattering,anisotropyofthebandstructurealonecangiverisetoLMR;theanswerturnsouttobeintheafrmative. 67

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ItshouldbepointedoutthatLMRcanalsooccurduetoanisotropicscattering.Indeed,aswasshownbyJonesandSondheimer[ 63 ]whochoseaspecialformofthescatteringprobabilitytosolvetheBEexactly,non-zeroLMRcanoccurevenforanisotropicspectrum,ifthescatteringprobabilityisappropriatelyanisotropic.Ingeneral,scatteringofBlochelectronsistobedescribedbyatensorofrelaxationtimes,becausedifferentcomponentsofmomentumrelaxatdifferentrates.Inlieuofafullymicroscopicdescription,weadopthereanheuristicmodel,inwhichtherelaxationtime,beingstillascalar,dependsonthepointinthekspace,=(k).Itiseasytoseethatthenecessaryconditionfornon-zeroLMRinthiscaseismodiedto: ^(vz)6=0,.(3)Thatmeansthatevenifthespectrumaloneviolatesourpreviouscondition( 3 ),i.e.,^vz=0,Eq. 3 maystillbesatisedbecause^maybenon-zero.Ifthisisthecase,LMRisniteaswell.Ontheotherhand,anattempttoprovethatEq. 3 isalsoasufcientconditioninthiscasefailsbecauseofthefollowingreason.With=(k),expressionsforthehigh-eldandzero-eldlongitudinalconductivitiesarestillgivenbyEqs. 3 and 3 ,exceptthatnowisinsidetheintegrals.Followingsamereasoningasbefore,asufcientconditionfornon-zeroLMRwouldbezz(B=1)6=zz(B=0),whichnowimpliesthatRR()]TJ /F18 7.97 Tf 6.59 0 Td[(@f0 @")(hv2zi)-235(hvzihvzi)d"dkz6=0.Unlikethepreviouscase,however,theintegrandcannotbeproventobeapositivefunction;therefore,anon-zerointegranddoesnotguaranteethattheintegralisalsonon-zero.Therefore,thesufcientconditioncanonlybeformulatedintheintegralform,asgivenabove. 3.3Example:LongitudinalMagnetoresistanceinGraphiteAsaparticularexampleofamaterialwithsignicantLMR,weconsiderthecaseofgraphite,whereahuge-uptothreeordersofmagnitude-LMReffectisobservedwhenboththecurrentandmagneticeldarealongthecaxis[ 9 ].RecallthattheenergyspectrumofgraphiteiswelldescribedbytheSlonczewskiWeissMcClure(SWMc) 68

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model(seeSec. 2.1 )whichinvolves7parameters0,...,6describingdifferentkindsofinteractionsbetweenlatticepoints.Parameter3playsaspecialroleasitbreaksrotationalsymmetryoftheFS.Without3theFSiscylindricallysymmetricabouttheBrillouinzoneedge.Therefore,theLMRconditionisclearlyviolated.However,nite3leadstotrigonalwarping,i.e.,athree-folddeformationoftheFS.Aspointedoutbefore,anexpressionforenergyspectrumofelectronsandholeswith3includedinaperturbativewaycanbewrittenas[ 13 ] "="03+A2[B24+2B3)]TJ /F6 11.955 Tf 6.77 0 Td[(3cos(3)+23)]TJ /F3 7.97 Tf 6.77 4.94 Td[(22]1=2,(3)where=p 3ak=2,)-523(=cos(kzc=2),and==2+,withaandcbeingthein-planeandout-of-planelatticeconstants,respectively.AlsoinEq. 3 ,"03,AandBareallfunctionsofkzandcontainotherinteractionparameters.Neglectingallthenext-to-nearestneighborcouplingsexceptfor3inthespectrum,wehave"03=A=0andB=20=1)]TJ /F1 11.955 Tf 10.1 0 Td[(.Withthisapproximation,Eq. 3 canberewritten[uptoO(23)]as "=k2 2m(kz)+3)]TJ /F6 11.955 Tf 6.78 0 Td[(cos(3)+123)]TJ /F3 7.97 Tf 6.77 4.33 Td[(3 220sin2(3),(3)wherem(kz)=21)]TJ /F6 11.955 Tf 6.77 0 Td[(=3a220.AsisobviousfromEq. 3 ,thetermscontainingintroducethetrigonalwarpingeffectinthespectrum.Duetothepresenceoftheseterms,theconditionfornon-zeroLMRissatised.Fig. 3-2 showsthecalculateddependenceofLMRonthemagneticeldinunitsofof!c,where!c=3eB=m0withm0m(kz=0)ingraphiteatzerotemperature(for0=3.16eV,1=0.39eV,and3=0.315eV)[ 9 ].Asexpected,LMRisquadraticatsmalleldsandeventuallysaturatesatlargeelds.However,wenotethatalthoughthisexplainsqualitativelywhygraphitehasnon-zeroLMRintherstplace,thecurvedoesnotnearlymatchtheexperimentquantitatively.Namely,wendthatrelativemagnetoresistancezz(B)=zz(0)(zz(B))]TJ /F6 11.955 Tf 11.95 0 Td[(zz(0))=zz(0)saturatesapproximatelyatavalueof0.2.However,observedvalueofthisratioishigherbyordersofmagnitude[ 47 ].Thisimplies 69

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thatthemechanismofLMRinrealgraphite(asopposedtoidealgraphitedescribedbytheSWMcmodel)isnotsimplyanisotropyoftheFS.Thedisagreementisnotsurprisinginlightofthefactthatthemechanismofc-axistransportingraphite(notonlyinnitebutalsoinzeromagneticeld)isstillnotcompletelyunderstoodandgenerallybelievedtobeduetoprocessesnotdescribedbythestandardBE,e.g.,phonon-assistedresonanttunnelingthroughmacroscopicdefects,e.g.,stackingfaults[ 64 66 ],ordisorder-assisteddelocalization[ 29 ]. Figure3-2. CalculateddependenceofLMRonmagneticeldingraphite. 3.4ConcludingRemarksToconclude,wederivedanecessaryandsufcientconditionthatanelectronicspectrumshouldsatisfyinordertoshownon-zerolongitudinalmagnetoresistancewithinthesemi-classicalregimeofelectrontransport.Wefoundthatanisotropyisessentialfornon-zeroLMRalthoughthisanisotropyhastobeofaspecialkind,namely,thespectrummustsatisfyaparticularnon-separabilityconditiongivenbyEq. 3 .WealsoshowedthataphenomenologicaldampedBlochelectronsmodeldoesnotcaptureessentialphysicsofsemi-classicaltransportinanisotropicmaterials.Inparticular,thismodelpredictsthatLMRisabsentnotonlyforisotropicbutalsoanisotropictransport, 70

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whichisnotconsistenteitherwiththepredictionsoftheBoltzmann-equationtheoryorexperiment.Ingeneral,thelimitingvaluesofthelongitudinalconductivitiesinthezero-andhigh-eldlimitsdifferonlyinhowthesquareofthez-componentoftheelectronvelocityisaveragedovertheFS.Excludingsomepathologicalsituations,thesetwoaveragescanonlydifferbyanumericalcoefcientontheorderofunity.Therefore,anLMReffectcan,inprinciple,resultfromFSanisotropyifitsmagnitudedoesnotexceedoriscomparableto100%.If,inaddition,thelatticestructureissuchthatLMRispossibleonlyduenearest-neighbor-hopping,oneshouldexpectevensmallervaluesofLMR.Inmanymaterials,e.g.,copper[ 51 ]andSr2RuO4[ 54 ],theobservedLMReffectisontheorderof10%,whichiswellwithintheanisotropic-FSmechanism.However,giganticLMReffects,suchastheoneobservedingraphite,requireexplanationswhichmightinvolvemacroscopicinhomogeneitiesofthesample. 71

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CHAPTER4RESISTIVITYOFNON-GALILEANINVARIANTFERMILIQUIDSANDSURFACETRANSPORTINBi2Te3FAMILYOFTOPOLOGICALINSULATORS1AT2scalingoftheresistivitywithtemperature(T)isconsideredasanarchetypalsignatureoftheFermi-liquid(FL)behaviorinmetals.ThisresultowesitsorigintothePauliexclusionprinciplewhichdictatesthat,atlowtemperatures,onlythosequasiparticlesthatresidewithinawidthoforderTneartheFermienergyparticipateinbinarycollisions.Thisargument,however,appliesonlytotheinverseofthequasiparticlerelaxationtime1=eebutnottotheresistivity,,perse-theT2scalingoftheformerdoesnotnecessarilyimplythatofthelatter.AverysimpleexampleisaGalilean-invariantFL,wheretheelectron-electron(ee)interactiondoesnotaffecttheresistivity,although1=ee,asmeasured,e.g.,bythermalconductivity,doesscaleasT2.Thereasonisthat,sincevelocitiesofelectronsareproportionaltotheirrespectivemomenta,conservationofmomentumautomaticallyimpliesconservationoftheelectriccurrent.Inordertoachieveasteady-statecurrentundertheeffectofanexternalelectriceld,amomentumrelaxationmechanismisneeded.Ofcourse,theFLofelectronsinametalisnotGalilean-invariant.Inthepresenceoflattice,thecurrentmayberelaxedbyUmklappcollisions[ 67 ],whichconservethequasimomentumuptoareciprocallatticevector:k+p=k0+p0+b.Umklappprocessesareallowed,however,iftheincomingelectronmomentakandpaswellasthemomentumtransferq=k)]TJ /F5 11.955 Tf 12.72 0 Td[(k0=p0)]TJ /F5 11.955 Tf 12.72 0 Td[(parealloforderb.Theserequirementsaresatisedif1)theFermisurfaceislargeenough,e.g.,atleastquarter-lledinthetight-bindingcase[ 68 ],and2)theinteractionissufcientlyshort-ranged.Inconventional 1Theresults,gures,andmostofthetextinthischapterarefromourarticles:(1)H.K.Pal,V.I.Yudson,andD.L.Maslov,Phys.Rev.B85,085439(2012).Copyright(2012)bytheAmericanPhysicalSociety.(2)H.K.Pal,V.I.Yudson,andD.L.Maslov,arXiv:1204.3591(SubmittedtoaspecialissueoftheLithuanianJournalofPhysicsdedicatedtothememoryofY.B.Levinson). 72

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metals,thesetwoconditionsareeasilymetduetoalargenumberofcarriersandeffectivescreeningoftheCoulombinteraction;thusUmklappcollisionsoccurataratecomparableto1=ee,and/T2.However,therearesituationswhentheseconditionsarenotmet;e.g.,therstconditionisviolatedinsystemswithlowcarrierconcentration,suchasdegeneratesemiconductors,semimetals,surfacestatesofthreedimensionaltopologicalinsulators,etc.,andthesecondconditionisviolatedwhenametalistunedtothevicinityofaPomeranchuk-typequantumphasetransition(QPT)[ 69 ],e.g,aferromagneticQPT.Inthischapter,wefocusontheformercase,i.e.,onsystemswithlowcarrierconcentration.Foradiscussionoftheothercase,thereaderisreferredto[ 70 71 ].IfUmklappsaresuppressed(andthetemperatureistoolowfortheelectron-phononinteractiontobeeffective),currentcanberelaxedonlyviaelectron-impurity(ei)collisions.Still,thenormal,i.e.,momentum-conserving,eecollisionscanaffecttheresistivity,ifcertainconditionsaremet.ThecombinedeffectofnormaleeandeiinteractionsdoesnotnecessarilyleadtotheT2dependenceoftheresistivity.Whetherthishappensdependsonthreefactors:1)dimensionality(twodimensions(2D)vsthreedimensions(3D)),2)topology(simplyvsmultiplyconnected),and3)shape(convexvsconcave)oftheFermisurface(FS).TheT2termisabsentnotonlyforaGalilean-invariantbut,moregenerally,foranisotropicFLwithanon-parabolicspectrum,aswellasforanisotropicbutquadraticspectrum.In2D,theconditionsaremorestringent.Inadditiontocasesmentionedabove,theT2termisabsentforasimply-connectedandconvexbutotherwisearbitrarilyanisotropicFS.ThereasonbehindthisisthattheT2termarisesfromelectronsconnedtomovealongtheFScontoursuchthat,fortheconvexcase,momentumandenergyconservationsaresimilartothe1Dcase,wherenorelaxationispossible.Thegoalofthischapter,inpart,istoelucidatethesepoints.Additionally,wewishtoapplytheresultstosurfacetransportinBi2Te3familyof3Dtopologicalinsulators(TI).The2Dsufacestatesinthesematerials 73

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havetheuniquepropertythat,astheFermienergyincreases,the2DFSchangesrapidlyfromacircletoahexagonandthentoahexagram,changingitsshapefromconvextoconcave.Therefore,thesematerialscanserveasatestinggroundforthetheoreticalresultsoutlinedabove.Further,wederivethedependenceofresistivityontemperatureneartheconvex-concavetransition,and,asweshow,theresistivityisfoundtoobeyauniversalscalingformapplicablenotonlytothesesystems,butalsotoanyothersystemexhibitingsimilarchangeofFSshape.Therestofthechapterisorganizedasfollows.WebeginbyformulatingtheproblemintermsoftheBoltzmannequation(BE)inSec. 4.1 .InSec. 4.2 ,wesolvetheBEperturbativelywithrespecttoeescattering,whichisanadequateapproximationatlowenoughtemperatures,andanalyzevariousstituationsmentionedabove.InSec. 4.3 ,wediscussthecaseofBi2Te3familyof3DTIsandderivethescalingformoftheresistivityneartheconvex-concavetransition.InSec. 4.4 ,wediscussseveralissuessuchaswhathappensinthehightemperaturelimitandwhatarethelimitationsofourresults.Finally,ourconcludingremarksarepresentedinSec. 4.5 4.1FormulationoftheProblemThemoststraightforwardwaytondtheeffectoftheeeinteractionontheconductivityinthesemi-classicalregimeisviatheBoltzmannequation(BE)which,forthecaseofatime-independentandspatially-uniformexternalelectriceldE,reads(cf.,Ch. 1 ) eE@fk @k=Iei[fk]+Iee[fk],(4)whereeistheelectronchargeandfkisthedistributionfunction.ThecollisionintegralsIeeandIeiontheright-handsidedescribetheeffectsoftheeeandeiinteractions,respectively.Explicitly, Iei=Zk0wk0k(fk0)]TJ /F8 11.955 Tf 11.96 0 Td[(fk)("k)]TJ /F6 11.955 Tf 11.95 0 Td[("k0),(4) 74

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and Iee=ZpZp0Zk0Wk,p!k0p0("k+"p)]TJ /F6 11.955 Tf 11.96 0 Td[("k0)]TJ /F6 11.955 Tf 11.95 0 Td[("p0))]TJ /F5 11.955 Tf 5.48 -9.69 Td[(k+p)]TJ /F5 11.955 Tf 11.95 0 Td[(k0)]TJ /F5 11.955 Tf 11.95 0 Td[(p0[fk0fp0(1)]TJ /F8 11.955 Tf 11.95 0 Td[(fk)(1)]TJ /F8 11.955 Tf 11.96 0 Td[(fp))]TJ /F8 11.955 Tf 11.96 0 Td[(fkfp(1)]TJ /F8 11.955 Tf 11.96 0 Td[(fk0)(1)]TJ /F8 11.955 Tf 11.95 0 Td[(fp0)], (4) whereRkisashort-handnotationforRdk (2)D,andwk,k0andWk,p!k0p0aretheeiandeescatteringprobabilities,correspondingly.Foraweakelectriceld,theleft-handsideoftheBEreducestoevkEn0k,wherevkistheelectrongroupvelocityandnkn("k)istheequilibriumdistributionfunction,withprimedenotingaderivativewithrespecttotheelectronenergy,"k(measuredfromtheFermienergy).Linearizingtheeecollisionintegralontheright-handsidewithrespecttothenon-equilibriumcorrectiontonk,denedas fk=nk)]TJ /F8 11.955 Tf 11.95 0 Td[(Tn0kgk=nk+nk(1)]TJ /F8 11.955 Tf 11.96 0 Td[(nk)gk,(4)oneobtains[ 68 ] Iee=ZpZp0Zk0Wk,p!k0p0(gk0+gp0)]TJ /F8 11.955 Tf 11.96 0 Td[(gk)]TJ /F8 11.955 Tf 11.95 0 Td[(gp)nknp(1)]TJ /F8 11.955 Tf 11.95 0 Td[(nk0)(1)]TJ /F8 11.955 Tf 11.95 0 Td[(np0))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(k+p)]TJ /F5 11.955 Tf 11.95 0 Td[(k0)]TJ /F5 11.955 Tf 11.95 0 Td[(p0("k+"p)]TJ /F6 11.955 Tf 11.96 0 Td[("k0)]TJ /F6 11.955 Tf 11.96 0 Td[("p0). (4) BeforeproceedingwiththeacutalsolutionoftheBE,wemakethefollowingobservations. Hiddenphonons.Thelinearizedformofthesteady-stateBEassumesimplicitlythattheelectron-phononinteractionisalsopresentinthesystem;otherwise,thetotalelectronenergywillincreaseindenitelyduetotheworkdonebytheelectriceld.Asusual(see,e.g.,Ref.[ 22 ]),weassumethatthetemperatureislowenoughsothatonecanneglectadirectelectron-phononcontributiontotheresistivity(whichrequiresthatephee,ei,wheresarethetransportscatteringtimesforcorrespondingprocesses)buthighenoughsothat,foraxedelectriceld,theelectron-phononinteractioncanstill 75

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equalizetheelectronandlatticetemperatures(whichrequiresthattheworkdonebytheelectriceldontheenergyrelaxationlengthismuchsmallerthanthetemperature). Parityofanon-equilibriumpartofthedistributionfunction.AlinearinEterminfkcanbewrittenas fkfk)]TJ /F8 11.955 Tf 11.96 0 Td[(nk=)]TJ /F5 11.955 Tf 9.3 0 Td[(AkETn0k,(4)whereAkcontainsexplicitlyonlytheeffectsoftheeeandeiinteractions.Atlowenoughtemperatures(asspeciedinthepreviousparagraph),theelectron-phononinteractionshowsuponlyinthenextquadratictermanddoesnotaffecttheresistivitydirectly.Atevenlowertemperatures,wheneei,eescatteringcanbetreatedasaperturbationtoeiscattering.Inthiscase,Akisdeterminedbythecrystalsymmetryandbytheeiscatteringprobabilityand,withoutspecifyingbothofthem,nopropertiesofAkcanbefurtherinferred.However,iftheeiscatteringprobabilitysatisesthemicroreversibilitycondition[ 22 ],i.e.,wk,k0=wk0,k,thenAkisoddink.Indeed,reversingthesignofkintheBEandrelabelingk0!)]TJ /F5 11.955 Tf 24.58 0 Td[(k0,weobtain evkEn0k=Zk0w)]TJ /F3 7.97 Tf 6.59 0 Td[(k0,)]TJ /F3 7.97 Tf 6.59 0 Td[(k(f)]TJ /F3 7.97 Tf 6.59 0 Td[(k)]TJ /F8 11.955 Tf 11.95 0 Td[(f)]TJ /F3 7.97 Tf 6.58 0 Td[(k0)("k)]TJ /F6 11.955 Tf 11.96 0 Td[("k0).(4)Usingtime-reversalsymmetrywk,k0=w)]TJ /F3 7.97 Tf 6.59 0 Td[(k0,)]TJ /F3 7.97 Tf 6.59 0 Td[(k(whichisguaranteedintheabsenceofthemagneticeldandmagneticorder)andmicroreversibility,weseethatA)]TJ /F3 7.97 Tf 6.59 0 Td[(k=)]TJ /F5 11.955 Tf 9.3 0 Td[(Ak.Thisisthepropertyofthenon-equilibriumdistributionfunctionwewillbeusinglateron.Tosimplifythepresentation,wewillrstuseamodelformoftheeicollisionintegral,namely,arelaxation-timeapproximation2Iei=)]TJ /F7 11.955 Tf 11.29 -.16 Td[((fk)]TJ /F8 11.955 Tf 11.96 0 Td[(nk)=i,whichallowsforaclosed-formsolution,andthenextendtheproofforthegeneralformoffkgivenbyEq. 4 2Strictlyspeaking,thesecondterminthecollisionintegralshouldcontainthedistributionfunctionaveragedoverthedirectionsofkratherthantheequilibriumdistributionnk.However,thisdifferenceisimmaterialtolinearorderinE. 76

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However,onehastokeepinmindthatmicroreversibilityisnotageneralprinciplebeyondtheBornapproximation.Rather,itisaconsequenceoftwomicroscopicsymmetries,i.e.,symmetrieswithrespecttotime-andspace-inversions,andisthusabsentinsystemsthatarenon-centrosymmetric[ 72 ]. Nodisordernosteady-statelinear-responseregime.Sincethemomentumisconservedinnormalcollisions,thecollisionintegral 4 isnulliedbyacombinationBk,whereBisk-independentbutotherwisearbitrary.Thismeansthatthereisnouniquesteady-statesolutioninthelinear-responseregime.Obviously,thesteady-statesolutionisabsentbecausethetotalmomentumoftheelectronsystem(perunitvolume),K=Rkkfk,increaseswithtime.Indeed,restoringthetimeandspatialderivativesintheBE,multiplyingitbykandintegratingoverkweobtain @Ki @t+@ij @xj=)]TJ /F8 11.955 Tf 9.3 0 Td[(eZkki@fk @kjEj,(4)whereij=Rkkivjfk.Integratingbypartsintheright-handsideandtakingintoaccountthatthenumberdensityN=Rkfk,weobtain @Ki @t+@ij @xj=eNEi(4)Theleft-handsideisjustthecontinuityequationwhiletheright-handsideisthetotalforceperunitvolume.Therefore,althoughtheelectronliquidisnot,generallyspeaking,Galilean-invariant,itisacceleratedasawholebytheelectriceld(inacrystal,anincreaseofthemomentumintimeleadstoBlochoscillationsofthecurrent;thecurrentaveragedovertimeisequaltozero).Therefore,oneneedstoinvokeimpurityscatteringinordertorendertheproblemwell-dened3. 3Awell-knowncasewhentheresistivityisniteonlyduetonormaleecollisionsisaperfectlycompensatedsemi-metal[ 22 ].UndopedgrapheneisaspecialcaseofacompensatedsystemwithzeroFermienergy,wheretheconductivityisnite(anduniversal)atT=0evenintheabsenceofanyscattering[ 73 ]. 77

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NolatticenoT2termintheresistivity.AddingjustdisorderbutnolatticedoesnotgiverisetoaT2termintheresistivity.Noticethatthisstatementisweakerthantheeeinteractiondoesnoteffecttheresistivityatall,whichistrueifwk0,kdependsonlyonthescatteringanglebutnotontheelectronenergy.Thesimplestcaseisthatofpoint-likeimpurities,whenwk,k0=1=("k)i,where("k)isthedensityofstates(peronespincomponent)andiisaconstant.Inthiscase,theBEreducesto evkEn0k=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(fk)]TJ /F7 11.955 Tf 12.05 2.66 Td[(f i+Iee(4)withfisanaverageoffkthedirectionsofk.Intheabsenceoflattice,k=mvandhencetheelectriccurrentj=2eRkvfk=(2e=m)Rkkfk.NowonecanmultiplytheBEequationbyvandintegrateoverk,uponwhichIeedropsout,andobtainarelationbetweenjandEdirectly,withoutsolvingforfk: eZkvk(vkE)n0k=m 2eij(4)Theresulingconductivity=ne2i=mdoesnotcontainanyeffectsoftheeeinteraction,exceptforpossiblyFLrenormalizationsofmandi.Thesameistrueifthescatteringprobabilitydependsonlyontheanglebetweenkandk0.Parameterizingwk,k0as wk,k0=w"k,^k^k0=("k),(4)weexpandfkandw"k,^k^k0overacompletebasissetof,e.g.,Legendrepolynomialsin3D: fk=X`ff`g("k)P`(cos);w=X`wf`g("k)P`(cos)P`(cos0)+Wo, (4) where(0)istheanglebetweenEandk(k0),andWoisanoddfunctionofthepolaranglesthatvanisheswhensubstitutedintothecollisionintegral.Intermsofangular 78

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harmonics,theBEreducesto evkEn0k=)]TJ /F13 11.955 Tf 11.29 11.36 Td[(X`ff`g f`g("k)P(cos`)+Iee (4) with 1 f`gi("k)wf0g("k))]TJ /F7 11.955 Tf 26.05 8.08 Td[(1 2`+1wf`g("k).(4)Iff`gi("k)doesnotdependontheelectronenergy,oneproceedsinthesamewayasforpoint-likeimpurities,i.e.,oneobtainsadirectrelationbetweenjandEbymultiplyingEq. 4 byvandintegratingoverk.Theresulting(Drude)conductivity=ne2tri=mcontainsthetransporttimetrif1gibutnoeffectsoftheeeinteraction(again,uptoFLrenormalizations).Ifwdoesdependontheelectronenergy,asisoftenthecaseforsemiconductors,theintegralRkkIeidoesnotreducetotheelectriccurrent,andoneneedstosolveforfkinordertondtheconductivity.Sincetheeeinteractionaffectsfk,theconductivityisalsoaffected.However,asweshowlater,theeffectstartsonlyfromtheT4termintheresistivity(orT4lnTin2D). 4.2Electron-electronContributiontotheResistivityItmayseemthat,oncelatticeanddisorderispresentinthesystem,aT2termintheresistivityisguaranteed.Whiledisordertakescareofmomentumrelaxation,latticebreakstheGalileaninvariance.Asaresult,vk=@"k=@k6=k=m,whichmeansmomentumconservationdoesnotimplycurrentconservation,andonecannotobtainarelationbetweenthecurrentandtheelectriceldwithoutactuallysolvingtheBE.Ingeneral,therefore,oneshouldindeedexpectaT2termintheresistivity.Whilethisisreadilythecasein3D,itturnsoutthattheconservationlawsin2DforbidtheT2terminseveralcaseswhichwewishtoexplorenext. 4.2.1LowTemperatures:PerturbationTheoryAtlowtemperatures,whentheeecollisionsarelessfrequentthantheeiones,theeecontributioncanbefoundviatheperturbationtheorywithrespecttoIeeasin[ 70 ] 79

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wherethisparticularsituationwasinvestigatedindetail.Webeginwiththesimplestisotropicmodelforelectron-impurityscattering,whentheBEisgivenbyEq. 4 .However,wekeepthedependenceoftheeirelaxationtimeontheelectronenergyforthetimebeing.AttherststepwesolveEq. 4 withIee=0,whichyields g(1)k=)]TJ /F8 11.955 Tf 9.3 0 Td[(ei("k)vkE=T.(4)Next,wesubstituteg(1)kbackintoEq. 4 andndacorrectionduetoIee g(2)k=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(i("k) Tn0kIeehg(1)ki(4)Thecorrespondingcorrectiontotheijcomponentoftheconductivitytensorisgivenby ij=)]TJ /F7 11.955 Tf 9.3 0 Td[(2e2 TZdDq (2)DZZZd!d"kd"pIIdak vkdap vpWk,p(q,!)`ik`jn("k)n("p)[1)]TJ /F8 11.955 Tf 11.96 0 Td[(n("k)]TJ /F6 11.955 Tf 11.96 0 Td[(!)][1)]TJ /F8 11.955 Tf 11.96 0 Td[(n("p+!)]("k)]TJ /F6 11.955 Tf 11.96 0 Td[("k)]TJ /F3 7.97 Tf 6.59 0 Td[(q)]TJ /F6 11.955 Tf 11.96 0 Td[(!)("p)]TJ /F6 11.955 Tf 11.96 0 Td[("p+q+!). (4) Here,qk)]TJ /F5 11.955 Tf 12.47 0 Td[(k0=p0)]TJ /F5 11.955 Tf 12.47 0 Td[(pisthemomentumtransfer,dakisthesurface(line)elementofanisoenergeticsurface(contour)atenergy"kin3D(2D),and`i("k)vk+i("p)vp)]TJ /F6 11.955 Tf 13.01 0 Td[(i("k)]TJ /F6 11.955 Tf 13.01 0 Td[(!)vk)]TJ /F3 7.97 Tf 6.59 0 Td[(q)]TJ /F6 11.955 Tf 13 0 Td[(i("p+!)vp+qisavectormeasuringthechangeinthetotalvectormeanfreepath`kvki("k)duetoeecollisions.Theenergytransferwasintroducedbyre-writingtheenergyconservationlawas("k+"p)]TJ /F6 11.955 Tf 11.95 0 Td[("k0)]TJ /F6 11.955 Tf 11.95 0 Td[("p0)=Rd!("k)]TJ /F6 11.955 Tf 11.96 0 Td[("k0)]TJ /F6 11.955 Tf 11.96 0 Td[(!)("p)]TJ /F6 11.955 Tf 11.96 0 Td[("p0+!).ThescatteringprobabilityWk,p(q,!)Wk,p!k)]TJ /F3 7.97 Tf 6.58 0 Td[(q,p+qisnowallowedtodependon!.UsingthesymmetrypropertiesofWk,p(q,!),onecancastEq. 4 intoamoresymmetricform ij=)]TJ /F8 11.955 Tf 13.12 8.09 Td[(e2 2TZdDq (2)DZZZd!d"kd"pIIdak jvkjdap jvpjWk,p(q,!)`i`jn("k)np("p)[1)]TJ /F8 11.955 Tf 11.95 0 Td[(n("k)]TJ /F6 11.955 Tf 11.95 0 Td[(!)][1)]TJ /F8 11.955 Tf 11.95 0 Td[(n("p+!)]("k)]TJ /F6 11.955 Tf 11.95 0 Td[("k)]TJ /F3 7.97 Tf 6.59 0 Td[(q)]TJ /F6 11.955 Tf 11.95 0 Td[(!)("p)]TJ /F6 11.955 Tf 11.95 0 Td[("p+q+!). (4) 80

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NowletuscountthepowersofTinEq. 4 .Eachofthethreeenergyintegrals(over!,"k,and"p)givesafactorofTwhich,inacombinationwiththeoverall1=Tfactor,alreadygivesaT2dependence,asexpectedforaFL.TheT2resultholdsaslongastheintegraloverqdoesnotintroduceadditionalTdependence.ThisisthecaseintheFLregime,whentypicalqareoforderoftheultravioletcutoffoftheproblem,i.e.,thesmallestofthethreequantities:thereciprocallatticevector,atypicalsizeoftheFS,andtheinverseradiusoftheeeinteraction.Inthiscase,the!dependenceofWk,p(q,!)canbeneglected.TheenergydependenceoficontributesonlytohigherordertermsinTandweneglectitforthetimebeingaswell,sothat`=ivwith v=vk+vp)]TJ /F5 11.955 Tf 11.95 0 Td[(vk)]TJ /F3 7.97 Tf 6.58 0 Td[(q)]TJ /F5 11.955 Tf 11.95 0 Td[(vp+q(4)beingthechangeintheelectroncurrentduetoeecollisions.Finally,sincetheintegralsofthecombinatonoftheFermifunctionsoverenergiesalreadyproduceafactorofT2,electronscanbeprojectedontotheFSintherestoftheformula.Thismeansthatonecandrop!inboth-functionsandperformthesurfaceintegralsovertheFS.Wepauseheretoremarkthatneglecting!inthe-functionsdoesnotmeanperforminganexpansionin!="k,!="p,etc.Infact,allquasiparticlesenergies("k,"k)]TJ /F3 7.97 Tf 6.59 0 Td[(q,etc.)areequaltozerobecausetheelectronswereprojectedontotheFS!Whatitreallymeansisthatthe-functionsimposeconstraintsontheanglesbetweenkandq(andpandq)withelectrons'momentabeingontheFS.Typicalvaluesoftheseanglesaredeterminedbytheratiooftypicalq(q)tokF.Inasystemwithashort-rangeinteraction,qminfkF,1=a0g,wherea0isthelatticespacing;therefore,typicalanglesareoforderunity.Ontheotherhand,typical!(!)areoforderT,andcorrectionstoanglesduetonite!aresmallaslongasTminf"F,Wg,whereWisthebandwidth,whichweneedtoassumeanyhowtobeintheFLregime.Iftheinteractionradius,r0,ismuchlongerthanboththelatticespacingandtheFermiwavelength,qissmallbutinproportionto1=r0ratherthantoT,while!isstilloforderT.Thismeansthateffective 81

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ultravioletenergyscaleisreducedtovF=r0,andtheFLdescriptionisvalidonlyatlowenergies,wheretheeffectofaniteenergytransferonthekinematicsofcollisionsisnegligible.Thiscanbeillustratedforasimpleexampleofthequadraticspectrum,whentheanglebetween,e.g.,kandq,satisescosk,q=(q2=2m+!)=vFq.Neglecting!isjustiedaslongasTq2=2m.Asanotherremark,typicalqmaybedifferentfordifferentobservables.WhatwesaidaboveistruefortheleadingtermintheeecontributiontotheelectricalconductivityinalldimensionsD>1,becausethesmallqbehavioroftheintegrandinEq. 4 isregularizedbythe`ifactorsthatvanishinthelimitofq!0.(Thisisanalogoustotheregularizingeffectofthe1)]TJ /F7 11.955 Tf 12.62 0 Td[(cosfactorinatransportcross-sectionforelasticscattering.However,whencalculatingthesingle-particlelifetime(theimaginarypartoftheself-energy)[ 74 ]andthermalconductivity[ 75 ]in2D,onerunsintoinfraredlogarithmicdivergences,whichmeansthattheinfraredregionofthemomentumtransfers(qT=vF)doescontributetotheresult.Inthosecases,neglecting!inthe-functionsisnotjustied.(Thesubleadingtermsintheconductivityalsorequiremorecare;seeSec. 4.2.2.3 below.)Comingbacktomaintheme,weneglect!inthescatteringprobability,whichisallowedinthecaseofaFL.Afterallthesesimplications,thediagonalcomponentoftheconductivityreducesto ii=)]TJ /F8 11.955 Tf 13.12 8.09 Td[(e2 2T2iZdDq (2)DZZZd!d"kd"pIIdaFk vFkdaFp vFpWk,p(q,0)(vi)2n("k)n("p)[1)]TJ /F8 11.955 Tf 11.95 0 Td[(n("k)]TJ /F6 11.955 Tf 11.95 0 Td[(!)][1)]TJ /F8 11.955 Tf 11.95 0 Td[(n("p+!)]("k)]TJ /F6 11.955 Tf 11.96 0 Td[("k)]TJ /F3 7.97 Tf 6.59 0 Td[(q)j"k=0("p)]TJ /F6 11.955 Tf 11.95 0 Td[("p+q)j"p=0, (4) wheresuperscriptFindicatesthatthecorrespondingquantityisevaluatedattheFS.Nowtheintegralsoverallenergiescanbeperformedwiththehelpofanidentity 1 TZd"1Zd"2Zd!n("1)n("2)[1)]TJ /F8 11.955 Tf 11.95 0 Td[(n("1)]TJ /F6 11.955 Tf 11.96 0 Td[(!)][1)]TJ /F8 11.955 Tf 11.96 0 Td[(n("2+!)]=22 3T2,(4) 82

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andweobtainaT2termintheconductivitywithaprefactorgivenbyacertainaverageovertheFSii=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(2 3e22iT2ZdDq (2)DIIdaFk vFkdaFp vFpWk,p(q,0)(vi)2("k)]TJ /F6 11.955 Tf 11.96 0 Td[("k)]TJ /F3 7.97 Tf 6.59 0 Td[(q)j"k=0("p)]TJ /F6 11.955 Tf 11.95 0 Td[("p+q)j"p=0. (4) Clearly,whethertheleadingcorrectiontotheresidualconductivityindeedscalesasT2dependsonwhethertheintegralovertheFSisnonzero.Sincetheintegrandispositive,theintegralmayvanishonlyifv=0undertheenergyconservationconstraintsimposedbythe-functions.Asasimplecheck,weapplyEq. 4 fortheGalilean-invariantcase,whenvk=k=m.Inthiscase,v=0,asitshouldbe. 4.2.2CaseswhentheLeadingTermVanishesWenowanalyzeseveralcaseswherev=0leadingtotheabsenceoftheT2term. 4.2.2.1IsotropicsystemwithanarbitraryspectrumTherstcaseisthatofanisotropicbutotherwisearbitraryenergyspectrum.Suchasituationmayariseduetorelativisticeffects.Another(pseudo-relativistic)exampleisweaklydopedgraphenewithanegligiblysmalltrigonalwarpingoftheFS.Since"kisafunctionofjkjonly,the-functionconstraintsinEq. 4 implythatjkj=jk)]TJ /F5 11.955 Tf 12.56 0 Td[(qjandjpj=jp+qj.Then, vjk=2@"k @(k2)kj=(k)kj;vjk)]TJ /F3 7.97 Tf 6.59 0 Td[(q=2@"k @(k2)jk)]TJ /F3 7.97 Tf 6.58 0 Td[(qj=jkj(kj)]TJ /F8 11.955 Tf 11.96 0 Td[(qj)=(k)(kj)]TJ /F8 11.955 Tf 11.95 0 Td[(qj), (4) where(k)vk=k.NoticethatthesecondlineinEq. 4 isnotanexpansioninsmallqbutanexactrelation.SubstitutingEq. 4 (andsimilarexpressionsforvjpandvjp+q) 83

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intov,itiseasytoseethatvvanishesidentically.Thus,thereisnoT2correctiontotheresistivityofanon-Galilean-invariantbutisotropicsystem.Thisresultalsoholdsforageneralquadraticspectrum"k=kikj=2mij,inwhichcasevj=ki=mjiandvj=0.Noticethat,incontrasttotheGalilean-invariantcase(with"k=k2=2m)]TJ /F6 11.955 Tf 12.22 0 Td[(F),whennotonlytheT2termbutallhigherordertermsareabsent,higherorder(T4,etc.)termsarenon-zeroforanon-parabolicspectrum. Figure4-1. Isotropiccasein2D:threepossiblescatteringprocessesnoneofwhichleadstocurrentrelaxation. 4.2.2.2Approximateintegrability:convexandsimplyconnectedFermisurfacein2DThefactthattheT2terminresistivityisabsentforanisotropicFSdoesnotmeanthatitisnecessarilypresentforananisotropicFS.Infact,theT2termisalsoabsentforasimply-connectedandconvex4butotherwisearbitraryFSin2D[ 70 76 77 ].Beforeconsideringthegeneralcase,however,letusstudythesimplestexampleofsuchaFS,i.e.,a2DcircularFSwithquadraticspectrum.SincethisisjustaGalilean-invariant 4Agureiscalledconvexifalinesegmentdrawntoconnectanytwopointsonthegurewillhavenopointsoutsidethegure(e.g.,acircle).Incontrary,ifatleastonelinesegmentconnectingtwopointsonthegureliespartlyoutsidethegure,thegureiscalledconcave(e.g.,ahexagram). 84

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case,wealreadyknowthattheeeinteractionhasnoeffectontheresistivity.However,itisinstructivetoseeinageometricalwayhowtheT2termvanishesthiswillbeusefulforthesubsequentanalysisofthegeneralcasein2D.Geometrically,oneneedstondthetwoinitialmomenta,kandp,belongingtotheFS,suchthatthenalmomenta,k)]TJ /F5 11.955 Tf 12.31 0 Td[(qandp+q,alsobelongtotheFS.AsshowninFig. 4-1 ,onlythreesituationsarepossible[ 78 ]:(1)Cooperchannel,whenthetotalinitialand,therefore,thetotalnalmomentaareequaltozero;(2)swappingofvelocities,whentheinitialmomentumofonetheelectronscoincideswiththenalmomentumofanotherelectronandviceversa,i.e.,p=k)]TJ /F5 11.955 Tf 12.57 0 Td[(q;and(3)noscattering-thisisthetrivialcasewheretheinitialandnalmomentaofindividualelectronsarethesame.Forallofthesecases,v=0andthustheT2termisabsent.Toseethatthesesituationsindeedexhaustallthepossibilities,onecansolvethemomentumandenergyconservationequations,i.e.,k)]TJ /F5 11.955 Tf 10.37 0 Td[(k0=p0)]TJ /F5 11.955 Tf 10.36 0 Td[(p=qandk2=k02;p2=p02,subjecttotheadditionalconstraintk=p=k0=p0=kF.Theseleadtotwoequations:q2)]TJ /F7 11.955 Tf 12.43 0 Td[(2kqcoskq=0andq2+2pqcospq=0,whereijdenotestheanglebetweenthevectorsiandj.Thethreepossiblesolutionsare:kq)]TJ /F6 11.955 Tf 12.37 0 Td[(pq=,correspondingtocase(1);kq+pq=,correspondingtocase(2);andq=0forarbitrarykqandpq,correspondingtocase(3).ThesituationdescribedaboveisnotspecictoacircularFSin2DbutoccursalsoforagenericconvexFS,seeFig. 4-2 (a).Indeed,introducinganewvariablep=)]TJ /F5 11.955 Tf 9.3 0 Td[(pinEq. 4 andusingthetime-reveralsymmetry(")]TJ /F3 7.97 Tf 6.59 0 Td[(p="p)andsymmetriesofthescatteringprobability,weobtain ii=)]TJ /F6 11.955 Tf 10.5 8.08 Td[(2 3e2T22iZd2q (2)2IIdaFk vkdaFp vpWk,p(q,0)vik)]TJ /F5 11.955 Tf 11.96 0 Td[(vip)]TJ /F5 11.955 Tf 11.96 0 Td[(vik)]TJ /F3 7.97 Tf 6.59 0 Td[(q+vip)]TJ /F3 7.97 Tf 6.59 0 Td[(q2("k)]TJ /F6 11.955 Tf 11.95 0 Td[("k)]TJ /F3 7.97 Tf 6.58 0 Td[(q)("p)]TJ /F6 11.955 Tf 11.95 0 Td[("p)]TJ /F3 7.97 Tf 6.58 0 Td[(q). (4) Foragivenq,wemustndtwomomentasatisfyingtherelations"k="k)]TJ /F3 7.97 Tf 6.59 0 Td[(qand"p="p)]TJ /F3 7.97 Tf 6.59 0 Td[(q.Geometrically,ndingthesolutiontothesetwoequationsisequivalenttoshiftingtheFSbyq,andndingthepointsofintersectionbetweentheoriginalandtheshiftedFSs.A 85

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convexFShasatmosttwoself-intersectionpoints.Therefore,theequation"k="k)]TJ /F3 7.97 Tf 6.59 0 Td[(qhasonlytwosolutions.Inaddition,ifkisasolution,then)]TJ /F5 11.955 Tf 9.3 0 Td[(k+qisalsoasolutionsothattherootsoftherstequationformasetfk,)]TJ /F5 11.955 Tf 9.3 0 Td[(k+qg.Sincethesecondequationisthesame,itstworootsfp,)]TJ /F7 11.955 Tf 9.51 0 Td[(p+qg=f)]TJ /F5 11.955 Tf 15.28 0 Td[(p,p+qgmustcoincidewiththerootsoftherstequation.Thiscanhappenif1)k=)]TJ /F5 11.955 Tf 9.29 0 Td[(p,whichgivestheCooperchannelorif2)k=p+qwhichgivesswapping.Thesituationwithq=0,whennoscatteringoccurs,istriviallypossible.Forallthescatteringprocesseslistedabove,v=0andtheT2termvanishes.AlthoughtheanalysisabovewasbasedonEq. 4 ,obtainedintherelaxation-timeapproximationforeiscattering,itcanbereadilyextendedforthegeneralformoftheeicollisionintegralinEq. 4 .Thenon-equilibirumpartofthedistributionfunctioninthepresenceofeiscatteringaloneisgivenbyEq. 4 ,whichimpliesthatg(1)kinEq. 4 isreplacedbyg(1)k=AkE.Thelowest-orderiterationineescatteringistobefoundfromanintegralequation Iei[g(2)k]=1 Tn0kIee[AkEn0k].(4)Theeicollisionintegralcanbeviewedasaintegraloperator,theinverseofwhichisdenedby ^I)]TJ /F3 7.97 Tf 6.59 0 Td[(1ei[fk]X^kOk,k0fk0,(4)where^kk=k.Thankstomicroreversibility,Ok,k0=Ok0,k.AformalsolutionofEq. 4 is: g(2)k=1 Tn0k^I)]TJ /F3 7.97 Tf 6.59 0 Td[(1eiIee[AkE].(4)UsingthemicroreversibilitypropertyofOk,k0andthefactthatAkisoddink,itiseasytoseethatg(2)kisoddinkaswell.Thisisallonereallyneedstorepeatthestepsofthepreviousanalysis.ThecorrectiontotheconductivitynowcontainsacombinationviAj,whereAAk+Ap)]TJ /F5 11.955 Tf 12.05 0 Td[(Ak0)]TJ /F5 11.955 Tf 12.05 0 Td[(Ap0.Beingoddinallmomenta,Abehavesinthesamewayasvuponthechangep!)]TJ /F5 11.955 Tf 25.63 0 Td[(p.Thescatteringprocessesareclassiedin 86

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thesamewayasbefore,andthevanishingoftheT2termfollowsfromthevanishingofv.Alimitednumberofpossibleoutcomesoftheeecollisionsmeansthatour2Dsystembehavessimilartoa1Dsystem,wherebinarycollisionsdonotleadtorelaxation.Theanalogyworksbecause,tondtheleading(T2)termintheconductivity,itsufcestoprojectelectronsontotheFS,whichisalinein2D.Therefore,kinematicseffectivelybecomes1Dand,althoughthisisa2Dcase,wehaveanintegrablesystem.However,thisanalogyhascertainlimitations.First,the2Dcaseisintegrableonlywithrespecttochargebutnotthermalcurrentrelaxation;thismaybecomparedtothe1Dcasewherenorelaxationcanhappeninanyofthequantities.Second,evenwhenitcomestothechargecurrent,relaxationisabsentonlyuptonext-order-termsinT="F(seeSec. 4.2.2.3 ).Third,notanyFSlinein2Disintegrable:concaveandmultiply-connectedcontoursbehaveinanon-integrableway.Withalltheselimitationsinmind,wewillrefertothe2Dconvexcaseasapproximateintegrability. Figure4-2. Thenumberofself-intersectionpoints(markedbydots)dependsongeometry:(a)Aconvexcontourhasatmosttwoself-intersectionpointsand(b)aconcavecontourcanhavemorethantwoself-intersectionpoints(sixintheexampleshown). 4.2.2.3Subleadingcorrectionstotheresistivitywhentheleadingtermisabsent.Tondthesubleadingcorrectionforthecaseconsideredintheprevioussection,wegobacktoEq. 4 ,replaceagainibyaconstantinthescatteringprobability,butnow,insteadofneglecting!inthefunctions,expandtheproductofthefunctionsto 87

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secondorderin!.Thezeroth-orderterm,("k)]TJ /F6 11.955 Tf 11.95 0 Td[("k)]TJ /F3 7.97 Tf 6.58 0 Td[(q)("p)]TJ /F6 11.955 Tf 11.96 0 Td[("p+q),nulliesvi.Theoddin!termsvanishuponintegrationover"k,"p,and!.IntheFLcasethisgives ii=1 2e2 T2iZdDq (2)DZZZd!!2d"kd"pIIdak vkdap vpWk,p(q,0)vi2n("k)n("p)[1)]TJ /F8 11.955 Tf 11.96 0 Td[(n("k)]TJ /F6 11.955 Tf 11.96 0 Td[(!)][1)]TJ /F8 11.955 Tf 11.96 0 Td[(n("p+!)]0("k)]TJ /F6 11.955 Tf 11.95 0 Td[("k)]TJ /F3 7.97 Tf 6.58 0 Td[(q)0("p)]TJ /F6 11.955 Tf 11.95 0 Td[("p+q))]TJ /F7 11.955 Tf 10.5 8.09 Td[(1 2f00("k)]TJ /F6 11.955 Tf 11.96 0 Td[("k)]TJ /F3 7.97 Tf 6.59 0 Td[(q)("p)]TJ /F6 11.955 Tf 11.95 0 Td[("p+q)+("k)]TJ /F6 11.955 Tf 11.95 0 Td[("k)]TJ /F3 7.97 Tf 6.58 0 Td[(q)00("p)]TJ /F6 11.955 Tf 11.96 0 Td[("p+q)g. (4) Thederivativesofthe-functionsproducethesamerootsforkandpasthe-functionsthemselves.However,integratingbyparts,wemakethederivativesactonvi2.Althoughvi2vanishesforkandpsatisfyingenergyandmomentumconservations,itsderivativesdonot.Thismaketheintegralnon-zero.Sincewenowhavetwomorefactorsof!thecorrectiontotheconductivityscalesas ii/T4.(4)Inmoredetail,letk0beoneoftherootsoftheequation"k="k)]TJ /F3 7.97 Tf 6.59 0 Td[(q.Thecorrespondingrootforpisthenp0=k0)]TJ /F5 11.955 Tf 11.96 0 Td[(q.Expandingvaroundtherootsgives vi=([k)]TJ /F6 11.955 Tf 9.3 0 Td[(p]r))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(vik0)]TJ /F5 11.955 Tf 11.95 0 Td[(vik0)]TJ /F3 7.97 Tf 6.58 0 Td[(q, (4) wherekk)]TJ /F5 11.955 Tf 12.58 0 Td[(k0andpp)]TJ /F5 11.955 Tf 12.59 0 Td[(k0+q.Subsequentintegrationproceedsasintheintegral ZZdxdy0(x)0(y)1 2(x)]TJ /F8 11.955 Tf 11.95 0 Td[(y)2=)]TJ /F13 11.955 Tf 11.29 16.27 Td[(Zdx0(x)x=1,(4)wherekandpplaytherolesofxandy(andsimilarlyforanintegralwithaproduct00(...)(...)).FurthercancelationsforaparticularFSmaymaketheTdependenceevenweakerbutthegenericanswerisT4.Inadditiontothemechanismdescribedabove,thereareothersourcesofhigherthanT2-ordercorrectionstotheconductivity;oneofthemistheenergydependenceofiwhichwehaveneglectedsofar.ThismechanismoperateseveninaGalilean-invariant 88

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system:althougheecollisionsconservethemomentum,theyredistributeelectronsintheenergyspaceandthusaffecttheconductivity,ifidependsontheenergy[ 22 79 ].Toestimatethemagnitudeofthiseffect,weapplyEq. 4 totheGalilean-invariantcase(v=k=m)andexpandtheimpurityrelaxationtimesenteringthevectormeanfreepathasi("l)=i(0)+0i"l,where0i@i("l)=@"lj"l=0.Thisyields `=0i! m(k)]TJ /F5 11.955 Tf 11.96 0 Td[(p)]TJ /F7 11.955 Tf 11.96 0 Td[(2q). (4) SinceEq. 4 containstwofactorsof`,andeachofthemisproportionalto!,wehaveanextra!2factorintheintegrand.In3D,thisimmediatelygivesaT4term ii3D/(0i)2T4.(4)In2D,thesituationismoredelicatebecausethepartoftheintegrandassociatedwiththek)]TJ /F5 11.955 Tf 13.07 0 Td[(pterminEq. 4 islogarithmicallydivergent.Thisawell-knownDlogsingularitythatoccurs,onamoregenerallevel,asthemass-shellsingularityoftheself-energy(seeRef.[ 74 ]andreferencestherein).Thisisalsothesamesingularitythatoneencounterswhencalculatingthethermalconductivityin2D(intheabsenceofimpurityscattering)[ 75 ].Indeed,ourproblembearsaformalsimilaritytothatofthethermalconductivitybecausethechangeinthethermalcurrentjTk=vk"kduetoeecollisions, jTk+jTp)]TJ /F5 11.955 Tf 11.95 0 Td[(jTk)]TJ /F3 7.97 Tf 6.58 0 Td[(q)]TJ /F5 11.955 Tf 11.96 0 Td[(jTp+q=(k)]TJ /F5 11.955 Tf 11.96 0 Td[(p)]TJ /F7 11.955 Tf 11.96 0 Td[(2q)!+q("k)]TJ /F6 11.955 Tf 11.95 0 Td[("p) m, (4) containsthesametermas`inEq. 4 .ThesingularitycanberesolvedbythesamemethodasinRef.[ 75 ].i.e.,byconsideringadynamicallyscreenedCoulombinteraction.Theresultisthat,similartothethermalconductivity,theconductivitycontainsanextra 89

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logfactorascomparedtothe3Dcase: ii2D/(0i)2T4ln("F=T).(4)TheDlogdoesnotoccurintheT2termintheconductivity,ifthelatterisniteduetobrokenintegrability,whichisthesubjectofthenextsection. 4.2.3Non-integrableCasesItfollowsfromthepreviousdiscussionthatwhethertheT2termisabsentorpresentdependsentirelyontheFShavingtwoormorethantwoself-intersectionpoints.AconcaveFSin2Dcanhavemorethantwoself-intersectionpoints(cf.Fig. 4-2 (b),thereforetherearemorethantwosolutionsfortheinitialmomentaforgivenq.SomeofthesesolutionsstillcorrespondtointegrableprocessesencounteredalreadyforaconvexFS,buttheremainingonesdorelaxthecurrent.Therefore,aT2termsurvivesinthiscase. Figure4-3. Thenumberofself-intersectionpointsdependsonthedimensionandthetopology:(a)A3DFShasaninnitenumberofself-intersectionpoints(aline)and(b)amultiplyconnectedFSalsohasmorethantwoself-intersectionpoints. Similarly,in3D,themanifoldofintersectionbetweentheoriginalandshiftedFSsisaline,seeFig. 4-3 (a).Therefore,theequation"k="k)]TJ /F3 7.97 Tf 6.59 0 Td[(qhasinnitelymanyroots.Thereisnocorrelationbetweentherootsoftheequations"k="k)]TJ /F3 7.97 Tf 6.59 0 Td[(qand"p="p+q.Geometrically,thismeansthattheinitialmomenta,kandp,donothavetobeinthe 90

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sameplaneasthenalones,k0andp0.Therefore,ananisotropic(butnotquadratic)FSin3DallowsforaT2correctiontotheresistivity.Also,iftheFSismultiplyconnected,aT2termintheresistivityispresent,eveniftheindividualFSsheetsdonotallowforaT2termontheirown.Evenmoreso,theindividualsheetscanevenbeisotropic.ThereasonisobviousfromFig. 4-3 (b)whichshowsanexampleoftwocircularFSsin2D.Clearly,theequation"k="k)]TJ /F3 7.97 Tf 6.58 0 Td[(qhasmorethantworootseveninthiscase.Thus,accordingtoourpreviousarguments,thereisnogeneralreasonforthevanishingoftheT2terminsuchasituation. 4.2.4Weakly-integrableCasesSincethequestionofintegrabilitydependsonthedimensionalityandtheshapeoftheFS,onecanexploresituationsofweaklybrokenintegrabilitybymanipulatingthesetwoparameters.Forexample,inthecaseofaquasi-2Dmetalwithveryweaklycoupledplanes,onecananticipatetheonsetoftheT2terminproportiontotheinterplanarcoupling.Similarly,ifoneconsidersasystemwith2DFSwhichisweaklyconcave,theT2termisexpectedtogrowatapacedecidedbythestrengthoftheconcavity.Infact,thelattersituationisphysicallyrealizedinthe2DsurfacestatesoftheBi2Te3familyof3Dtopologicalinsulators.Wethereforeexploreonlytheformercaseinthissection,leavingthelattercasetobestudiedinmoredetailinthenextsection,inthecontextofthesetopologicalinsulators.Consideralayeredmetalwithaquasi-2Dspectrumwhich,forsimplicity,weassumetobeseparableintothein-andout-of-planepartsas "k="jjkjj+"zkz,(4)wherekjjandkzarethein-planeandout-of-planecomponentsofthemomentum,correspondingly.Inthetight-bindingmodelwithnearest-neighborhopping,"zkz=t?[1)]TJ /F7 11.955 Tf 11.96 0 Td[(cos(kzc)],wherecisthelatticespacinginthez-direction.Themetalisinaquasi-2Dregimewhent?"F.Inregardtothein-planepartofthespectrum,"jjkjj,we 91

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assumethatthecorrespondingenergycontoursareanisotropicbutconvexsothat,intheabsenceoftheinter-planehopping,theT2-terminthein-planeconductivitywouldbeabsent.(IftheplanesareassumedtobeGalilean-invariant,i.e.,"jjkjj=k2jj=2mjj,asinacorrugatedcylindermodel,theT2-termistriviallyzerobecausethein-andout-of-planecomponentsofthemomentumareconservedindependently,andhencevjjkjj+vjjpjj)]TJ /F5 11.955 Tf 12.57 0 Td[(vjjk0jj)]TJ /F5 11.955 Tf 12.57 0 Td[(vjjp0jj=0.)TondtheT2-terminthein-planeconductivity,weuseamethodsimilartothatinSec. 4.2.2.3 ,i.e.,weexpandthe-functions,exceptforthatnowweexpandbothin!and"zkz.AsweexplainedinSec. 4.2.1 ,theexpansionin!isreallyanexpansionin!normalizedbytheappropriateultravioletenergyscaleoftheproblem.Likewise,theexpansionin"zkzisreallyanexpansionint?=F,whichisanaturalsmallparameterforaquasi-2Dsystem.Thezeroth-orderterm(!=0,"zkz=0)nulliesvjj.Therst-ordertermsalsovanish:theones,proportionalto!,dosobyparity,andtheonesproportionalto"zkz,dosobecausetherst-orderderivativesofthe-functionsnullify(vjj)2afterasingleintegrationbyparts.Finally,thecrossproductsinsecond-orderterms,beingoddin!,alsovanish.Therefore,theonlysurvivingsecond-ordertermis "jjkjj)]TJ /F3 7.97 Tf 6.59 0 Td[(qjj)]TJ /F6 11.955 Tf 11.96 0 Td[("jjkjj+"zkz)]TJ /F9 7.97 Tf 6.59 0 Td[(qz)]TJ /F6 11.955 Tf 11.96 0 Td[("zkz)]TJ /F6 11.955 Tf 11.95 0 Td[(!"jjpjj+qjj)]TJ /F6 11.955 Tf 11.95 0 Td[("jjpjj+"zpz+qz)]TJ /F6 11.955 Tf 11.95 0 Td[("zpz+!=1 2h)]TJ /F6 11.955 Tf 5.48 -9.68 Td[("zkz)]TJ /F9 7.97 Tf 6.58 0 Td[(qz)]TJ /F6 11.955 Tf 11.96 0 Td[("zkz2+!2i00"jjkjj)]TJ /F3 7.97 Tf 6.59 0 Td[(qjj)]TJ /F6 11.955 Tf 11.96 0 Td[("jjkjj"jjpjj+qjj)]TJ /F6 11.955 Tf 11.96 0 Td[("jjpjj+1 2h)]TJ /F6 11.955 Tf 5.48 -9.68 Td[("zpz+qz)]TJ /F6 11.955 Tf 11.95 0 Td[("zpz2+!2i"jjkjj)]TJ /F3 7.97 Tf 6.59 0 Td[(qjj)]TJ /F6 11.955 Tf 11.96 0 Td[("jjkjj00"jjpjj+qjj)]TJ /F6 11.955 Tf 11.96 0 Td[("jjpjj+)]TJ /F6 11.955 Tf 10.46 -9.68 Td[("zkz)]TJ /F9 7.97 Tf 6.59 0 Td[(qz)]TJ /F6 11.955 Tf 11.96 0 Td[("zkz)]TJ /F6 11.955 Tf 12.95 -9.68 Td[("zpz+qz)]TJ /F6 11.955 Tf 11.96 0 Td[("zpz)]TJ /F6 11.955 Tf 11.96 0 Td[(!20"jjkjj)]TJ /F3 7.97 Tf 6.59 0 Td[(qjj)]TJ /F6 11.955 Tf 11.95 0 Td[("jjkjj0"jjpjj+qjj)]TJ /F6 11.955 Tf 11.95 0 Td[("jjpjj. (4) Equation 4 containstwoindependentcorrections.Alltermsproportionalto!2produceaT4correctiontotheconductivitythatexistseveninapurely2Dsystem.Alltermscontainingthesquaresoftheout-of-planedispersionsproduceaT2correction[ 76 ].Therefore, ii=A4T4+A2t2?T2,(4) 92

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wherei=x,y,andconstantsA4andA2dependondetailsofthein-planespectrum;generically,A4A2.Equation 4 describesadimensionalcrossoverfromthe2D-likeregime(ii/T4)atTt?tothe3D-likeregime(ii/T2)forTt?).Noticethat,inthe3Dregime,theT2-terminthein-planeconductivitydependsontheout-of-planehopping. 4.3EffectofeeInteractionsonSurfaceTransportinBi2Te3FamilyofTopologicalInsulatorsTheclassofmaterialsBi2Te3,Bi2Se3,andSb2Te3(Bi2Te3family),knownpreviouslyduetoitsthermoelectricproperties,havereceivedrenewedinterestascandidatesforthree-dimensionaltopologicalinsulators(TI)[ 80 ].TIsarecharacterizedbyagappedbulkspectrumwithconductingsurfacestatesextendingacrosstheentiregap.ThesurfacestatescontainanoddnumberofDiracconesandareprotectedagainstanyperturbationthatpreservestime-reversalsymmetry.PhotoemissionshowsthatthesurfacestatesoftheBi2Te3familyofcompoundshaveasmall,singly-connectedFSatthecenteroftheBrillouinzone(BZ).FollowingFu[ 81 ],theelectronicdispersioninthesesystemscanbedescribedby k=p v2k2+2k6cos2(3),(4)whereistheazimuthalangle,vistheDiracvelocity,andisaconstant.CorrespondingisoenergeticcontoursarepresentedinFig. 4-4 .AstheFermienergyincreases,theFSchangesrapidlyfromacircletoahexagonandthentoahexagram.AtsomecriticalvalueoftheFermienergyF=c(=0.16eVforBi2Te3,forexample[ 81 ]),theshapechangesfromconvextoconcave.Basedonthediscussionintheprevioussection(cf.,Sec 4.2 ,onecanstraightforwardlypredict,therefore,thatthee-econtributiontotheresistivityscalesasT4ontheconvexsideandasT2ontheconcaveside.However,thisconclusionisvalideitherwellbeloworwellabovetheconvex-concavetransition,i.e.,whenjjF,where="F)]TJ /F6 11.955 Tf 12.58 0 Td[("c.Onecanaskthequestion:whathappensin 93

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thevicinityofthetransitionwhenjjc,whenitisweaklyintegrable?Thissectionisgearedtowardsansweringthisquestion.Indeed,thesematerialsareuniqueinthesensethat,bychangingtheFermienergy(bydopingorgating),onecantunethesystemcontinuouslyfromanintegrabletoanon-integrablesituationpassinginbetweenthroughaweakly-integrablestate.Thus,weproposethesurfacestatesoftheBi2Te3familyof3DTIsasatestinggroundforthetheoreticalresultsoutlinedintheprevioussectionandinthesectiontofollow. Figure4-4. IsoenergeticcontoursforthespectruminEq. 2 .Thedashedlinecorrespondstothecriticalenergyfortheconvex-concavetransition. 4.3.1ConductivityNeartheConvex-concaveTransitionWerstfocusonthemostinterestingcaseofT,whentheisoenergeticcontoursneartheFermienergyareconcave,andthendiscussthecaseofjj.T,whenbothconvexandconcavecontoursneartheFSarethermallypopulated5.Nearthetransition,severalquantitiesinEq. 4 exhibitacriticaldependenceon.Tobeginwith,wehavev,whichiszeroontheconvexsideandnon-zeroontheconcaveside.Additionally,therearetwootherquantitieswhichalsoshowacriticalbehavior.AsFigs. 4-5 (c)and 4-5 (d)illustrate,eveniftheFSisconcave,ithasmorethantwo 5Although,inprincipleweshouldusespinorswhilesolvingtheBE,wecanstillusethescalarBEwithoutaffectinganyconclusionsAwayfromtheDiracpoint,theeffectofthephasefactorsinthespinorwavefunctionsisonlytosuppressbackscatteringprobabilityofallprocessesconsidered. 94

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self-intersectionpointsonlyifitisshiftedalongoneofthespecialdirectionsandthemagnitudeoftheshiftissufcientlysmall.Thesespecialdirectionsarehigh-symmetryaxesthatintersecttheFSatpointswithpositivecurvature,asinFig.2(b).Therefore,thewidthoftheangularintervalnearaspecialdirection(q)andthemaximumvalueofq,qmax,alsodependoninacriticalmanner.Notethat,morethantwointersectionpointsisalsopossibleforlargemomentumtransferofq2kF,kFbeingtheFermimomentum.However,backscatteringisnotallowedinthecaseofTIsduetothehelicalnatureoftheelectronstates.And,inthecaseofanon-helicalmetal,thiseffectcanbeshowntogiverisetoonlyhigherordertermsin.ApproximatingRd2qbyqq2max,resolvingthefunctions,andintegratingoverallenergies,weobtain jj=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(e22iT2 12Xl,mqjMkl,pm(qmax)j2[vj]2lmkl vkl^klpm vpm^pm1 jv0kl^qj1 jv0pm^qj,(4)wherethesumrunsoverallintersectionpoints,theprimedenotesaderivativewithrespecttotheazimuthalangle,and^ll=jlj.Noticethatalthoughafactorofq2maxfromthephasespaceofintegrationcancelswiththesamefactorfromthefunctions,itwillreappearinthecalculationofvj.Hence,weneedtocalculateitsdependenceonaswell.Wearenowgoingtoshowthat q/3=2,qmax/1=2,andvj/3=2.(4)Webeginwithq.Underanassumption(tobejustiedlater)ofsmallq,theequationk)]TJ /F6 11.955 Tf 12.23 0 Td[(k)]TJ /F3 7.97 Tf 6.58 0 Td[(q=0reducestovkq=0,whichimpliesthatqisatangenttotheFSattheintersectionpoints[cf.Fig. 4-6 (a)].DeningasananglebetweenthenormaltotheFSatanygivenpointandq,weplotasafunctionoftheazimuthalangle.Figure 4-6 (b)clearlydemonstratesadistinguishingfeaturebetweentheconvexandconcavecontours:()ismonotonicfortheformerandnon-monotonicforthelatter.Thenon-monotonicpartiscenteredaroundcertaininvariantpoints,i.e.,commonpoints 95

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Figure4-5. Factorsdeterminingthenumberofself-intersectionpointsinthe2Dcontours:(a)Aconvexcontourhasnomorethantwoself-intersectionpoints,(b)forqalongaspecialdirection,aconcavecontourhasmorethantwoself-intersectionpoints,(c)evenintheconcavecase,thenumberofself-intersectionpointsislessthanthemaximalnumberallowedbysymmetry,ifqisnotalongaspecialdirection,and(d)forqlargerthanacriticalvalue,thenumberoftheself-intersectionpointsislessthanthemaximumnumberallowedbysymmetry. forallcontours.TheoscillationsreecttherotationalsymmetrysixfoldinourcaseoftheFS.Fromsymmetry,ifisasolution,sois+;wethusconsideronlythedomain2[0,].Wenowneedtondtheangularintervalofqaboutaspecialdirectioninwhichtheequation=q+=2hasthreeroots-theT2termisnon-zeroonlyinthiscase.Sincethereisaone-to-onecorrespondencebetweentheanglesandq,wecanndthecorrespondingintervalinsteadofq.Clearly,theregionsonthecurvewhereitisnon-monotonicareresponsibleforthemultipleroots6.Redeningvariables 6Inlightofthis,itisnowclearwhyaconvexcontourdoesnotallowformorethantwosolutionsandwhyaconcavecontourallowsformorethantwosolutionsonlyforspecialdirectionsofq. 96

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andasmeasuredfromtheinvariantpoints,thenon-monotonicpartofthecurvecanbeconjecturedtoobeyacubicequation[cf.Fig. 4-6 (c)]: =b3)]TJ /F8 11.955 Tf 11.96 0 Td[(a(),(4)wherea()/andb>0isaconstant.Indeed,weneedatleastacubicequationtoprovideforthreerealroots;whetherthereisoneorthreerootsdependsonthesignofa()whichmustbenegative/positiveintheconvex-concaveregimes,correspondingly.ForthemodelspectrumofEq. 4 ,wendb=2anda()=16 9q 7 61=2( v3).Thequantityistheverticaldistancebetweenthemaximumandminimumofthiscurvewhich,accordingtoEq. 4 ,scalesas3=2.Nextinlineisvj,whichweexpandinsmallqasvj[@ @kj(qv)]jk2)]TJ /F7 11.955 Tf 10.69 0 Td[([@ @kj(qv)]jk1=[@ @kj(qv)]jk2k1,wherek2andk1areanytwosolutionsoftheequationqvk=0.ReferringtothegeometryofFig. 4-6 (d),wendvj/q[@ @()]j21,whereuseofEq. 4 yieldsvj/qqmax.Finally,tondqmax,werelaxtheassumptionofsmallqandsolvetheequationk)]TJ /F3 7.97 Tf 6.59 0 Td[(q=qforarbitraryq.ItiseasiertodothisbycastingEq. 4 intoanequationforthecontourintermsoflocalcartesiancoordinates.Tothiseffect,weapproximatetandky=dkxandtan)]TJ /F8 11.955 Tf 23.73 0 Td[(kx=k0F,withk0FbeingtheFermimomentumattheinvariantpoint[cf.Fig. 4-6 (d)]andsubstituteintoEq. 4 togetthefollowingcontourequation:ky=)]TJ /F8 11.955 Tf 9.3 0 Td[(bk4x=4+a()k2x=2,wherekx,yarethemomentameasuredfromtheinvariantpointsandnormalizedbyk0F.Usingthisexpressiontosolvefortherootsofk)]TJ /F3 7.97 Tf 6.58 0 Td[(q=q,onearrivesatacubicequationinkxwhichhasthreedistinctrealrootsifq2p a()=b.Thismeansthatqmax/1=2(andthusvj/3=2).This,inhindsight,validatestheassumptionofsmallq.Substitutingtheseresultsintotheexpressionfortheconductivity,wendthatq[vj]2/9=2,whichmeanstheprefactoroftheT2termintheresistivityscalesas9=2.TheT4termisalwayspresent,asdiscussedbefore.Hence,theresistivityhasthe 97

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Figure4-6. Geometricinterpretationofaself-intersectingconcavecontour:(a)Forsmallq,pointswherethenormaltotheFSisperpendiculartoqarethepointsofself-intersection(blackdots),(b)vs[asdenedinpanel(a)].Dotted:c,(c)azoomofthenon-monotonicpartofthegraphinpanel(b),and(d)aportionoftheFScontourshowingthegeometricconstructionforthederivationoftheequationforthecontour. form =0+A "F9=2()T2+BT4 "2F,(4)where0istheresidualresistivity,(x)isthestepfunction,andAandBarematerial-dependentparameters(generically,AB).AcrossoverbetweentheT4andT2regimesoccursatT"F(="F)9=4"F.Returningtothecaseof.T,whenbothconvexandconcavecontoursarepopulated,itiseasytoseethatthe9=2prefactorisreplacedbyT9=2,leadingtoaT13=2termin.Thisterm,however,issubleadingtotheT4one.Therefore,Eq. 4 describestheleadingT-dependenceoftheresistivityinbothsituations(jjTandjj.T)nearthetransition.Notethattheexponentsof2,4,and9=2inEq. 4 are 98

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universal,i.e.,theyarethesameforanarbitrary2DFermisurfacewithanon-quadraticenergyspectrumnearaconvex-concavetransition. 4.3.2ProposalforExperimentalVericationWenowdiscussthefeasibilityofobservingthesepredictionsinanexperiment.First,ourresultsareapplicableatrelativelyhightemperatures,whenquantumcorrectionstotheconductivitymaybeneglected.AswasshowninRef.[ 82 ],thisispossibleintheballisticquantumregime,whereaquantumcorrectionscalesasT(Ref.[ 83 ]),whiletheFLcorrectionalwaysscalesasT2.ComparingtheprefactorsonearrivesattheconditionT1=iwhich,evidently,dependsonthepurityofthesample.InBi2Se3thinlms(Ref.[ 84 ]b)thelogarithmicdownturnoftheconductivity,indicatingthedominanceofthediffusivequantumcorrection,startsatabout5KtheFLcontributiontotheconductivityshouldthereforebesearchedattemperaturesabove5Kinthesesamples.Next,oneneedstoaskifthee-phcontributiontotheresistivitymasksthee-eone.Ingeneral,thee-phcontribution,whichscalesasT5atT
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conclusionsonthetemperaturedependenceatdifferentvaluesofFcarryovertothefrequencydependence.Theadvantageofanopticalmeasurementisthatthee-phpartof\()saturates[ 22 ]forTBGF,whilethee-epartcontinuestogroweitheras2or4dependingonthesignof.Thisadvantagewasusedinthepasttodetectthee-econtributionto\()innoblemetals[ 22 ],andweproposetoapplythesametechniquetoTIs.Inpassingwenote,althoughphotoemissionandtunnelingmicroscopy[ 89 ]haveconvincinglyestablishedthepresenceofsuchsurfacestatesinthesematerials,signaturesofthesestatesintransportmeasurementsaremoredifculttoobserve,mainlybecauseofstrongconductioninthebulk[ 90 ].Withrecentexperimentalprogress,however,intheabilitytotunethenumberofsurfacechargecarriers[ 91 ],itisnowpossibletoseemoreclearlyevidenceofsurfacetransport.Indeed,includinge-einteractionturnedouttobecrucialforexplainingtheobservedeldandtemperaturedependencesinquantummagnetotransport[ 84 ].Inlightofthisprogress,wehopeourpredictionsaremeaningfulandwillbeputtotestinrecentfuture. 4.4DiscussionInthischapter,wehavefocusedourattentionsofarontheconventionalFLT2term;ourgoalhasbeentoexploresituationswherethismaybeabsent.Thisisessentiallyalow-temperatureresultderivedwithinthesemiclassicalregime.Toputourndingsintoperspective,severalcommentsareinorder.First,fromtheexperimentalpointofview,itmaybeimportanttounderstandwhethertheeecontributionmaybecomelargerthantheeioneathighertemperatures,andifso,whathappenstotheresistivityinthislimit.InthecaseforUmklappscattering,theeecontributiongrowsunabateduptothetemperaturescomparabletotheFermienergy.Thenormalcontribution,however,isdifferent:itsaturatesinthelimitwhentheeerelaxationtimebecomesshorterthentheeione.Theeffectofsaturationwasunderstoodalreadyintheearlierdaysoftheelectrontransporttheory[ 92 93 ]: 100

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veryfrequenteecollisionestablishaquasi-equilibriumstatewiththedriftvelocityxedbyeiscattering.Thepreviousanalysiswas,however,limitedtothecasewhennormaleecollisionsaffecttheresistivityviatheenergydependenceoftheeirelaxationtime[ 22 94 ].However,onecanshowthatthesaturationoccurseveniftheeirelaxationtimedoesnotdependonenergy[ 71 ].Insuchacase,theresidualresistivityandthesaturatedvalueoftheresistivityaretypicallyofthesameorder,thusthereisnotruescalingregimefortheT2(orT4)term.Thischanges,however,ifoneconsidersatwobandmodelwithverydifferentbandmasses,inwhichcaseatruescalingregimeemerges.Formoredetails,thereaderisreferredto[ 70 71 ]. Figure4-7. DifferenttemperatueregimesforFermi-liquid(FL)andquantum-interference(QC)correctionstotheconductivity.TheshadedregiononthetemperaturescaleistheregimewheretheFL(T2)correctionisdominant. Second,recallthatallourresultssofararebasedontheBE.Effectsarisingduetoquantuminterferencearemissinginthisapproach,whicharealsoimportantatlowtemperatures.Therefore,itisessentialtoknowthatlimitsofvalidityofouraforestatedresults.Recall,theFL-likecontributiontotheresistivitydiscussedsofarbehavesasT2(orT4,ifthereisapproximateintegrability)inthelow-temperatureregime,denedby1=ee1=i,andsaturatesinthehigh-temperatureregime,denedby1=ee1=i.Innon-integrablesystems,1=ee=gT2="F,wheregisthedimensionlesscouplingconstant.InagenericFL,g1andthecrossoverbetweenthetwolimitsoccursatT?=p "F=i.Foragoodmetal,"Fi1sothat1=iT?"F.Incaseofquantum 101

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corrections(QC),thescalethatdifferentiatesbetweenlowandhightemperatures,i.e.,betweenthediffusiveandballisticregimes,isTDB=1=i.ForTTDB,oneisinthediffusivelimit,characterizedbyalogarithmicallydivergentAltshuler-Aronovcorrection[ 95 ];withallcouplingconstantsbeingoforderone,jj=Dln(1=Ti)="Fi,whereD=e2"Fi=istheDrudeconductivity.ForTTDB,oneisintheballisticlimit,wherethecorrectionscaleslinearlywithT:jj=DT="F[ 83 ].Apartfromtheinteractioncorrection,thereisaalsoaweak-localizationcorrection,)]TJ /F6 11.955 Tf 9.3 0 Td[(WL=Dln(=i)="Fi,whereisthephase-breakingtime,apreciseformofwhichdependsonwhetheroneisinthediffusiveorballisticlimits:intheformer,1=Tln("Fi)="Fi;inthelatter,1=1=ee.Inthediffusivelimit,theweak-localizationcorrectionissimilartotheAltshuler-Aronovresult,differingonlyintheprefactor.Intheballisticlimit,theweaklocalizationcorrectionissmallerthantheinteractioncorrectionbyafactorofln(T=T)=Ti.Therefore,thecorrectorderofmagnitudeforthequantum-interferencecorrectionisstillgivenbytheinteractioncorrectionbothinthediffusiveandballisticlimits.ComparingtheFL-contribution)]TJ /F6 11.955 Tf 9.3 0 Td[(FL=DT2i="Ftothequantumcorrections,wendthatjQC=FLjln(1=Ti)=T22i1andjFL=QCjTi1inthediffusiveandballisticlimits,correspondingly.Therefore,itismeaningfultoconsidertheFLcontributionandneglectquantum-interfenceprocessesintheballisticbutnotinthediffusivelimit.TheinterplayofdifferentmechanismsisshownschematicallyinFig. 4-7 .Intheintegrablecase,theT2termintheresistivityvanishesandtheFLcorrectionscalesasjFLj=DT4i="3F.Inthiscase,theFLcorrectiondominatesoverthequantumoneonlyattemperatureswellabovethediffusion-ballisticcrossover:T("Fi)2=3TDBTDB. 4.5ConcludingRemarksThepurposeofthischapterhasbeentwofold.First,wewantedtoanalyzetheeffectofeeinteractionsontheresistivityofFLsinthesituationwhenUmklappscatteringofelectronscanbeneglected.Suchasituationarises,e.g.,inlow-carrierdensity 102

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materials,wheretheFSistoosmalltoallowUmklaps.Insuchcases,theconventionalT2dependenceoftheresistivityontemperatureisnotguaranteed.Whetheritispresentdependson1)dimensionality,2)shape,and3)topologyoftheFS.IftheFSisquadraticorisotropic,thereisnoT2contributiontotheresistivity.However,anisotropyisnotsufcienttoguaranteetheT2dependence.InthecaseofaconvexandsimplyconnectedFSin2D,thereisnoT2dependenceeither.Insuchcases,theleadingtemperaturedependenceonresistivityduetoeeinteractionsisT4.Inallothercases,theT2behaviorisallowed,asacorrectiontotheDruderesistivity.Second,weappliedtheseresultstopredictthebehaviorofsurfaceresistivityintheBi2Te3familyof3Dtopologicalinsulators.Inparticular,wehaveshownthat,whentheFSchangesitsshapefromconvextoconcaveasafunctionofthellingfraction,asisthecaseinthesesystems,theresistivityfollowsauniversalscalingformneartheconvex-concavetransition.WehavealsosuggestedapossibleexperimentalwaytoconrmourpredictionsintheseTIs. 103

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CHAPTER5CONCLUSIONSThegoalofthisworkhasbeentoinvestigatetheoreticallydifferentaspectsofelectricaltransport.Inourattempttounderstandthetransportproperties,wehaveaddressedquestionsthatarebothuniversalandmaterialspecic.MostofourresultshavebeenderivedfromsolvingtheBoltzmannequationunderdifferentcircumstances.Therefore,mostofourresultsarevalidinthesemi-classicalregime.Whereverneeded,wehaveprovidedthejustcationforusingthesemi-classicalmethodandwhyquantumeffectswerenotimportant.Belowwelistourconclusionsandfuturedirectionsofresearch.First,weexploredthetransportpropertiesofgraphite.Ourtheoreticalworkherewaslargelyinspiredbyseveralinterestingexperimentalobservations.Tobeginwith,thein-planeresistivityincreaseswithtemperatureinametallicwaywithoutshowinganysignsofsaturationincontradictionwithpredictionsofmodelsthatwereusedinthepasttosuccessfullyexplainthebehaviorofresistivityatlowertemperatures.Wefoundthatinvokingtheideaofelectronsscatteringoffhardopticalphononscouldexplaintheobservedbehavior.Intermsofthein-planetransversemagnetoresistance,experimentsshowthatmanysamplesexhibitlinearmagnetoresistance.Westudiedthisaspectwithintheacceptedband-structureofgraphiteandarrivedattheresultthat,duetopresenceofextremelylightcarrierswhichareDirac-likeattheHandH0pointsintheBZofgraphite,thereexistsanintermediateregimewherethein-planetransversemagnetoresistanceindeedbehaveslinearly.However,itwasalsoshownthatthiscouldn'taccountfortheexperimentallyobservedlinearitywhichextendsuptoquantizingelds.Thepresenceoflinearityathighereldsisthoughttobeduetothepresenceofmacroscopicinhomogeneities.Besides,itwasalsoshownwhytheselightcarrierscannothaveanysignatureinquantumoscillations,althoughtheycontributetotheclassicalmagnetoresistance.Theexperimentallyobservedout-of-planeproperties 104

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werefoundtobewantinginexplanation.Itwasconjecturedthatadifferentmechanismwasattheheartofout-of-planetransportsinceitcouldn'tbeexplainedbysimplebandpictureattheBoltzmannlevel.Thec-axistransportingraphiteisstillpoorlyunderstoodandisanopenareaofresearch,somethingwhichmaybepursuedinfuture.Second,weinvestigatedthephenomenonoflongitudinalmagnetoresistancewheretheeffectofmagneticeldonresistanceismeasuredinacongurationwhenboththecurrentandtheeldareinthesamedirection.Since,duetoLorentzlaw,thereshouldn'tbeanyeffectoftheeldonthecurrent,suchaphenomenonissurprising,atleastatarstglance.Whileseveralexoticmechanismshavebeenintroducedintheliteraturetoexplainthiseffect,weshowedthatthiseffectcouldarisesimplyfromtheband-structure.Ourgoalwastoexaminetheminimalmodelrequiredtohavethiseffect.Andaswedemonstrated,suchaneffectcouldresultsimplyfromtheFSanisotropyprovided,theanisotropysatisescertaincriterion.Wedervivedthiscriterionandprovedthatitisbothnecessaryandsufcient.Also,weusedthisconditiontocalculatetheLMRingraphite.SincethetheoreticallycalculatedLMRdidnotmatchwiththeexperimentalobservations,wearrived,oncemore,attheconclusionthatout-of-planetransportingraphiteisprobablygovernedbysomeothermechanism.Finally,westudiedtheeffectofelectron-electroninteactionsontransportinnon-GalileaninvariantsystemsandusedtheresulttopredicthowthesurfaceresistivityoftheBi2Te3familyofthreedimensionaltopologicalinsulatorswouldbehave.TheacceptednotionthataFLatlowtemperaturesalwaysdemonstratesaT2dependenceisinfactnotuniversallytrue-ifUmklappscatteringofelectronsisnotallowed,thereexistsaclassofsystemswheretheT2termisabsentandtheleadingdependencegoesasT4.Whetherthishappensornotdependson1)dimensionality(twovsthreedimensions),2)geometry(concavevsconvex),and3)topology(singlyvsmultiplyconnected)oftheFS.Inparticular,asystemwitha2DFSwhichissinglyconnectedandconvexinshapewillhavenoT2dependence.Thishelpeduspredicthowthesurface 105

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resistivityinthreedimensionalBi2Te3familyoftopologicalinsulatorswouldbehaveastheFermienergyisshifted,sincewithchangeofFermienergytheshapeoftheFSchangesfromconvextoconcave.Wealsoderivedascalingformbetweenresistivityandtemperatureinthevicinityofthisconvex-concavetransitionandshowedthatthescalingformisuniversal,andthereforedescribesnotonlythethreedimensionalBi2Te3familyoftopologicalinsulators,butalsoanyothermaterialwheretheFSistwodimensionalandexhibitsasimilarconvex-concavetransition.FortheparticularcaseofBi2Te3familyoftopologicalinsulators,suggestionsweremadeonhowtotestourresultsexperimentallybymeasuringtheopticalconductivityinthesesystems. 106

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BIOGRAPHICALSKETCH HridisPalwasborninthecityofBerhampurinWestBengal,India.AfterspendinghisearlyyearsatNaihati,hemovedwithhisfamilytoKalyanitotheirpermanentresidencewherehegrewupalongwithhissister.AftergraduatingfromJulienDaySchoolin2001,hewenttoJadavpurUniversity(JU),Kolkata,fromwhereheobtainedtheB.Scdegreein2004.FollowingthishejoinedIndianInstituteofTechnologyBombay(IITB),Mumbaiasastudentinthe2yearM.Sccourseandreceivedthedegreein2006.WithanaimtocontinueresearchinphysicsandtoobtainthePh.Ddegree,hearrivedattheUniversityofFlorida(UF),USAinfall2006,wherehehasbeenstudyingsince.EnroutetoobtainingthePh.Ddegree,hereceivedtheM.Sdegreeinsummer2009fromUF.HeisexpectedtoreceivethePh.DdegreeinAugust,2012fromUF. 112