Lecture,Mar.1
ThermalConductivityduetoConductionElectronsinMetals
OnMonday,Iderivedκ=(1/3)<v2>τcVfromtheDrudemodel.
YoumightobjectthatIshouldhavetakenintoaccountthatthevelocityofelectrons
comingfromthehighertemperaturesideshouldbehigherthanthatofelectrons
comingfromthelowerTside,onaverage.Thereare2reasonswhythisisn’ta
problem:
1) OnlyelectronsclosetotheFermienergycontributetothethermal
conductivityandthevariationintheirvelocitiesissmall
2) Thermalconductivitymeasurementsareusuallydoneon“opencircuit”
samplesinwhichnoelectricalcurrentisflowing.Iftherewasanon-zero
averagevelocityfromthehighertolowerTend,therewouldbeanelectrical
current.Whenthetemperaturegradientisfirstappliedsomeelectrical
currentdoesflowbutthenchargebuildsuponthesurfacesofthesample,
creatingauniformelectricfieldinthesamplewhichcancelsthiselectrical
currentsothat,inthesteadystatethereisnoelectricalcurrent.Thisiscalled
athermoelectricfieldandtheSeebeckeffect.Thisleadstoacancellation
whichallowsustoignorethevariationofvx.
AnotherthingIhavebeenignoringisthermalexpansionofthesolid.Thiscould
maketheelectrondensityvaryslightlywithtemperature.Thiscouldmakeτvary
also.Thisisgenerallyleadstoonlysmallcorrectionstothethermalconductivity.
Ihavegivenarathersketchy,non-rigorousderivationofthermalconductivity.Fora
morerigorousone,seeAshcroftandMermin,chapter13.
ThermalConductivityduetoPhonons
Vibrationofionsalsocontributestothethermalconductivityinmetals,andisthe
onlysourceininsulators.Althoughtheionsdon’thaveanynetmotionthroughthe
crystal,wecanthinkofthephonons,correspondingtoquantizedionvibrationsas
transportingenergythroughthecrystalinasimilarwaytotheconductionelectrons.
Phononsalsosuffer“collisions”.Thesecaninvolvescatteringoffimpurities,the
surfacesofthesampleandinteractionwithotherphonons(andelectrons).
Impuritiescanincludeamixtureofdifferentisotopesofatomsinthecrystal,since
theyhavedifferentmasses,affectingtheclassicalvibrationmodes.Inalargepure
samplephonon-phononinteractionsusuallyplaythedominantrole.Ifweinclude
cubictermsintheion-ioninteractionpotential,thisleadstoprocesseswheretwo
phononscollideandturnintoasinglephonon,andasinglephonondecaysintoa
pairofphonons.Thisisanalogoustohighenergyphysicsprocesseswhereunstable
particlescanbeproducedandcanannihilate.Wecanagainwritethesameformula
asforthermalconductivityduetoconductionelectrons:κ=(1/3)<v2>τcVbutnowall
parametersrefertophonons,notelectrons,includingthecollisiontime,τ.The
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phonon-phononcontributiontoτturnsouttohaveastrongtemperature
dependence,becomingverysmallatT<<TDwheresurfacescatteringdominatesand
growingatlargerTastheaveragenumberofphononsgrows.Iwillreturntothis
later.
ThermalconductivityofisotopicallypureLiF.AtlowT,itisdominatedbyscattering
offthesurfaces,asindicatedbythestrongdependenceonthecross-sectionalareaof
thesampe.AthighTitisdominatedbyphonon-phononscatteringandthe
dependenceoncross-sectionalareadisappears.
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Semiconductors
Metalshavepartiallyfilledbandswhereasinsulatorsusuallyhaveonlyfilledor
emptybandsinthegroundstate(T=0).However,notallinsulatorsarethesame.
Thesizeoftheenergygapbetweenthehighestfilledband(valenceband)and
lowestemptyband(conductionband)isextremelyimportant.Ifthisgapisnottoo
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large,asignificantnumberofelectronswillbeexcitedfromthevalencebandtothe
conductionbandatroomtemperature.Thustheconductivitycanbesignificantat
roomtemperature.Suchmaterialsarecalledsemiconductors.ThegapinSiis1.12
eV,itis.37eVinPbS.ItcanbeseveraleVingoodinsulators.Thebehaviordepends
9. Semiconductors
exponentiallyonthegap,EGsince,asweshallsee,theconcentrationofelectrons
andholesis~exp[-EG/(2kBT)].Furthermore,suchmaterialsareextremely
sensitivetorelativelysmallconcentrationsofimpurities,whichcaneffectivelyadd
Intrinsic semiconductors are materials where the valence band is full
electronsorholes.Interfacesbetweenregionswithdifferentimpurity
and the conduction band is empty at T=0. However the band gap is
concentrations(suchasdiodes)areamajoringredientofthesemi-conductor
so small that there is a significant concentration [n] of thermally
industry.
excited electrons in the conduction band and a concentration [p] of
Gapsinsemiconductorsareoftenintheopticalphotonrange.Thisisalsoimportant
“holes” in the valence band near room temperature where [n]=[p].
forapplications.So,adsorbingaphotoncanexciteanelectronfromthevalenceto
The “n” and “p” refer to the “negative” and “positive” charge of the
conductionband,correspondingtocreatinganelectron-holepair.Thisplaysa
carriers respectively. This leads to very T dependent conductivities
crucialroleinsolarcellsandotherapplications.Suchprocessesmustconserve
which are intermediate between a metal and an insulator, hence
crystalmomentumaswellasenergy.Visiblelighthasawavelengtharound500nm
7m-1.(Thecorrespondingenergyisabout
called semiconductors.
andawave-vectoraround2π/λ=1.3x10
hc/λ~2.4eV.)Thiswave-vectorisusuallyverysmallcomparedtothewidthofthe
Semiconductors are distinguished by whether the energy gap is
Brillouinzone,2π/asinceaistypically<1nm.So,directexcitationofanelectron
fromvalencetoconductionbandmustessentiallypreservethewave-vector.
direct or indirect. In a direct gap semiconductor such as GaAs an
Semiconductorswherethetopofthevalencebandandbottomoftheconduction
electron may be excited from the top of the valence band (valence
bandoccuratthesamewave-vectorsaresaidtohaveadirectgap.Otherwise,they
band edge) to the bottom of the conduction band (conduction band
aresaidtohaveanindirectgap.Aphotoncanexciteanelectronfromvalenceto
edge) without change in the crystal momentum. i.e. the maximum in
conductionbandinasemiconductorwithanindirectgap,providedthataphononis
the valence band and minimum in conduction band occur at the value
alsoproducedtoconservemomentum(andenergy).Theadsorptionintensityis
of crystal momentum. In an indirect gap material such as Si these
weakerinthiscaseandtheadsorptionintensityversusenergyofthephotonis
occur at different values of k.
broadernearthreshold(bandgap).
Fig.1 (a) Direct gap material where the minimum in conduction band
(CB) occurs at the same k value as the maximum in the valence
band(VB). (b) Indirect gap material where there is some offset in k3
between the CB minimum and VB maximum
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!
Nearthetoporbottomofaband, ε ( k ) canbeapproximatedasquadratic.(Linear
termsvanishsinceweareexpandingaroundastationarypoint.)Mostgenerally,the
!
k
minimum/maximumoccursatsomenon-zero
andtheshapeismorecomplicated
!
k fromthebandminimum/maximumand
thanasimpleparaboloid.Ifwemeasure
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assume,forsimplicitythatitisasimpleparaboloid,wecanapproximate
!
!
"2k 2
"2k 2 €
ε c (k ) ≈ ε c +
,
ε
(
k
)
≈
ε
+
v
2mc * v
2m€v *
butnotethattheeffectivemassesobeymc*>0andmv*<0.Theycanbeverydifferent
inmagnitudethanthefreeelectronmass,iftheperiodicpotentialislarge.In
dealingwiththevalencebanditismoreconvenienttousetheconceptofholes,
discussedinmyJan.27lectureandinKittel,chapter8.Theenergytocreateahole
!
inthevalencebandatwave-vector kh ,correspondingtoremovinganelectronwith
!
wave-vector −k h is
!
"2k 2
ε h ( k ) ≈ −ε v +
wheremh*=-mv*>0.
2mh *
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Itismoreconvenienttouseholes,partlybecausethevalencebandstaysmainly
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filled,sowehavemanyvalenceelectronsbutfewholes.
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