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Semiconductor

Asemiconductorisasolidmaterialthathaselectricalconductivitybetweenthoseofaconductorandaninsulator;

itcanvaryoverthatwiderangeeitherpermanentlyordynamically.[1]

Semiconductorsareimportantinelectronictechnology.Semiconductordevices,electroniccomponentsmadeofsemiconductormaterials,areessentialinmodernconsumerelectronics,includingcomputers,mobilephones,anddigitalaudioplayers.Siliconisusedtocreatemostsemiconductorscommercially,butdozensofothermaterialsareused.

Braggreflectioninadiffuselattice

Asecondwaystartswithfreeelectronswaves.Whenfadinginanelectrostaticpotentialduetothecores,duetoBraggreflectionsomewavesarereflectedandcannotpenetratethebulk,thatisabandgapopens.Inthisdescriptionitisnotclear,whilethenumberofelectronsfillsupexactlyallstatesbelowthegap.

EnergylevelsplittingduetospinstatePauliexclusion

Athirddescriptionstartswithtwoatoms.Thesplitstatesformacovalentbondwheretwoelectronswithspinupandspindownaremostlyinbetweenthetwoatoms.Addingmoreatomsnowissupposednottoleadtosplitting,buttomorebonds.Thisisthewaysiliconistypicallydrawn.Thebandgapisnowformedbyliftingoneelectronfromthelowerelectronlevelintotheupperlevel.Thislevelisknowntobeanti-bonding,butbulksiliconhasnotbeenseentoloseatomsaseasyaselectronsarewanderingthroughit.Alsothismodelismostunsuitabletoexplainhowingradedhetero-junctionthebandgapcanvarysmoothly.

Energybandsandelectricalconduction

Likeinothersolids,theelectronsinsemiconductorscanhaveenergiesonlywithincertainbands（ie.rangesoflevelsofenergy）betweentheenergyofthegroundstate,correspondingtoelectronstightlyboundtotheatomicnucleiofthematerial,andthefreeelectronenergy,whichistheenergyrequiredforanelectrontoescapeentirelyfromthematerial.Theenergybandseachcorrespondtoalargenumberofdiscretequantumstatesoftheelectrons,andmostofthestateswithlowenergy（closertothenucleus）arefull,uptoaparticularbandcalledthevalenceband.Semiconductorsandinsulatorsaredistinguishedfrommetalsbecausethevalencebandinthesemiconductormaterialsisverynearlyfullunderusualoperatingconditions,thuscausingmoreelectronstobeavailableintheconductionband.

Theeasewithwhichelectronsinasemiconductorcanbeexcitedfromthevalencebandtotheconductionbanddependsonthebandgapbetweenthebands,anditisthesizeofthisenergybandgapthatservesasanarbitrarydividingline（roughly4eV）betweensemiconductorsandinsulators.

Inthepictureofcovalentbonds,anelectronmovesbyhoppingtoaneighboringbond.BecauseofthePauliexclusionprincipleithastobeliftedintothehigheranti-bondingstateofthatbond.Inthepictureofdelocalizedstates,forexampleinonedimensionthatisinawire,foreveryenergythereisastatewithelectronsflowinginonedirectionandonestatefortheelectronsflowingintheother.Foranetcurrenttoflowsomemorestatesforonedirectionthanfortheotherdirectionhavetobeoccupiedandforthisenergyisneeded.Forametalthiscanbeaverysmallenergyinthesemiconductorthenexthigherstateslieabovethebandgap.Oftenthisisstatedas:

fullbandsdonotcontributetotheelectricalconductivity.However,asthetemperatureofasemiconductorrisesaboveabsolutezero,thereismoreenergyinthesemiconductortospendonlatticevibrationand—moreimportantlyforus—onliftingsomeelectronsintoanenergystatesoftheconductionband,whichisthebandimmediatelyabovethevalenceband.Thecurrent-carryingelectronsintheconductionbandareknownas"

freeelectrons"

althoughtheyareoftensimplycalled"

electrons"

ifcontextallowsthisusagetobeclear.

Electronsexcitedtotheconductionbandalsoleavebehindelectronholes,orunoccupiedstatesinthevalenceband.Boththeconductionbandelectronsandthevalencebandholescontributetoelectricalconductivity.Theholesthemselvesdon'

tactuallymove,butaneighboringelectroncanmovetofillthehole,leavingaholeattheplaceithasjustcomefrom,andinthiswaytheholesappeartomove,andtheholesbehaveasiftheywereactualpositivelychargedparticles.

Onecovalentbondbetweenneighboringatomsinthesolidistentimesstrongerthanthebindingofthesingleelectrontotheatom,sofreeingtheelectrondoesnotimplydestructionofthecrystalstructure.

Holes:

electronabsenceasachargecarrier

Thenotionofholes,whichwasintroducedforsemiconductors,canalsobeappliedtometals,wheretheFermilevellieswithintheconductionband.WithmostmetalstheHalleffectrevealselectronstobethechargecarriers,butsomemetalshaveamostlyfilledconductionband,andtheHalleffectrevealspositivechargecarriers,whicharenottheion-cores,butholes.Contrastthistosomeconductorslikesolutionsofsalts,orplasma.Inthecaseofametal,onlyasmallamountofenergyisneededfortheelectronstofindotherunoccupiedstatestomoveinto,andhenceforcurrenttoflow.Sometimeseveninthiscaseitmaybesaidthataholewasleftbehind,toexplainwhytheelectrondoesnotfallbacktolowerenergies:

Itcannotfindahole.Intheendinbothmaterialselectron-phononscatteringanddefectsarethedominantcausesforresistance.

Fermi-Diracdistribution.StateswithenergyεbelowtheFermienergy,hereμ,havehigherprobabilityntobeoccupied,andthoseabovearelesslikelytobeoccupied.Smearingofthedistributionincreaseswithtemperature.

Theenergydistributionoftheelectronsdetermineswhichofthestatesarefilledandwhichareempty.ThisdistributionisdescribedbyFermi-Diracstatistics.Thedistributionischaracterizedbythetemperatureoftheelectrons,andtheFermienergyorFermilevel.UnderabsolutezeroconditionstheFermienergycanbethoughtofastheenergyuptowhichavailableelectronstatesareoccupied.Athighertemperatures,theFermienergyistheenergyatwhichtheprobabilityofastatebeingoccupiedhasfallento0.5.

Thedependenceoftheelectronenergydistributionontemperaturealsoexplainswhytheconductivityofasemiconductorhasastrongtemperaturedependency,asasemiconductoroperatingatlowertemperatureswillhavefeweravailablefreeelectronsandholesabletodothework.

Energy–momentumdispersion

Intheprecedingdescriptionanimportantfactisignoredforthesakeofsimplicity:

thedispersionoftheenergy.Thereasonthattheenergiesofthestatesarebroadenedintoabandisthattheenergydependsonthevalueofthewavevector,ork-vector,oftheelectron.Thek-vector,inquantummechanics,istherepresentationofthemomentumofaparticle.

Thedispersionrelationshipdeterminestheeffectivemass,m*,ofelectronsorholesinthesemiconductor,accordingtotheformula:

Theeffectivemassisimportantasitaffectsmanyoftheelectricalpropertiesofthesemiconductor,suchastheelectronorholemobility,whichinturninfluencesthediffusivityofthechargecarriersandtheelectricalconductivityofthesemiconductor.

Typicallytheeffectivemassofelectronsandholesaredifferent.Thisaffectstherelativeperformanceofp-channelandn-channelIGFETs,forexample（Muller&

Kamins1986:

427）.

Thetopofthevalencebandandthebottomoftheconductionbandmightnotoccuratthatsamevalueofk.Materialswiththissituation,suchassiliconandgermanium,areknownasindirectbandgapmaterials.Materialsinwhichthebandextremaarealignedink,forexamplegalliumarsenide,arecalleddirectbandgapsemiconductors.Directgapsemiconductorsareparticularlyimportantinoptoelectronicsbecausetheyaremuchmoreefficientaslightemittersthanindirectgapmaterials.

Carriergenerationandrecombination

Whenionizingradiationstrikesasemiconductor,itmayexciteanelectronoutofitsenergylevelandconsequentlyleaveahole.Thisprocessisknownaselectron–holepairgeneration.Electron-holepairsareconstantlygeneratedfromthermalenergyaswell,intheabsenceofanyexternalenergysource.

Electron-holepairsarealsoapttorecombine.Conservationofenergydemandsthattheserecombinationevents,inwhichanelectronlosesanamountofenergylargerthanthebandgap,beaccompaniedbytheemissionofthermalenergy（intheformofphonons）orradiation（intheformofphotons）.

Insomestates,thegenerationandrecombinationofelectron–holepairsareinequipoise.Thenumberofelectron-holepairsinthesteadystateatagiventemperatureisdeterminedbyquantumstatisticalmechanics.Theprecisequantummechanicalmechanismsofgenerationandrecombinationaregovernedbyconservationofenergyandconservationofmomentum.

Astheprobabilitythatelectronsandholesmeettogetherisproportionaltotheproductoftheiramounts,theproductisinsteadystatenearlyconstantatagiventemperature,providingthatthereisnosignificantelectricfield（whichmight"

flush"

carriersofbothtypes,ormovethemfromneighbourregionscontainingmoreofthemtomeettogether）orexternallydrivenpairgeneration.Theproductisafunctionofthetemperature,astheprobabilityofgettingenoughthermalenergytoproduceapairincreaseswithtemperature,beingapproximately1×

exp（−EG/kT）,wherekisBoltzmann'

sconstant,TisabsolutetemperatureandEGisbandgap.

Theprobabilityofmeetingisincreasedbycarriertraps–impuritiesordislocationswhichcantrapanelectronorholeandholdituntilapairiscompleted.Suchcarriertrapsaresometimespurposelyaddedtoreducet