-1-1Lect

PhysicsSemiconductorBasic

Be

Mg

Zn

Cd

Hg

II

B

Al

Ga

In

Tl

III

C

Si

Ge

Sn

Pb

IV

N

P

As

Sb

Bi

V

O

S

Se

Te

Po

VI

4 5 6 7 89.012 10.81 12.01 14.01 16.00

12 13 14 15 1624.30 26.98 28.09 30.97 32.06

30 31 32 33 3465.38 69.72 72.59 74.92 78.96

65.38 69.72 72.59 74.92 78.9648 49 50 51 52

80 81 82 83 84200.6 204.4 207.2 209.0 210.0

beryllium

magnesium

zinc

cadmium

mercury

boron

aluminum

gallium

indium

thallium

carbon

silicon

germanium

tin

lead

nitrogen

phosphorus

arsenic

antimony

bismuth

oxygen

sulfur

selenium

tellurium

polonium

1.1Figure intocombinethatelementsshowingTablePeriodictheofPortion.chemicalitselement,theofnametheofconsistsentryEachsemiconductors.

byindicatedareTabletheofColumnsmass.andnumber,atomicsymbol,numerals.roman

-2-1Lect

SiSi

SiSi As

As

Ga

Ga

Se

Se

Cd

Cd

1.2Figure semiconductorsElementalcrystals.covalentindistributionCharge.areGe)(Si, covalentperfectly atomstwobetweensharedelectronssymmetryby;

Compoundatom.eachonprobabilityequalwithfoundbetoareofdegreesomehavealwayssemiconductors ionicity e.g.,compounds,III-VIn.

tonecessaryisthanchargemoreslightlyretainsatomAsfive-valenttheGaAs,Astheofchargepositivetheforcompensate 5 + onchargethewhilecore,ion

Gathe 3 + lessstilloccurselectronsofSharingcompensated.entirelynotisionCdionsthebetweenfairly 2 + Seand 6 + CdSe.compoundII-VIthein

Forsemiconductors.allforgoodreasonablyispicturebondcovalentThebynetworkbondingtherepresenttoconvenientoftenisitpurposeillustrative

thethatcourseofnotedbeshouldItelectrons).(sharedlinesand(ions)circles4hasindeedatomeachsemiconductorsmostFor3-dimensional.isnetwork

planesametheinnotlocatedarethesebutillustrated,asneighbors,nearestbelow.1.3Fig.seetetrahedron,aofverticestheatbut

-3-1Lect

θ

1.3Figure zincblendeordiamondainatomsofarrangementTetrahedral.anglebondtheanddiagonalscubethealongarebondsThestructure. θ is

cosbygiven −=θ 1/ regularadefinecornerstheinatomsFour.3centralthefromdifferentarefourthesestructurezincblendeaIntetrahedron.

same.theallaretheystructurediamondtheinatom,

-4-1Lect

3s

3p

Si atom3 s p2 2

Hybridized Bonded

Solid

Valenceband

Conductionband

sp3EG

bonding

antibonding

1.4Figure scheme.bondingcovalenttheininvolvedlevelsenergyElectronic.3thebetweenSeparation p 3and s 6aboutisorbitalsatomic eV ofMixing.

equivalent4intoorbitalsthese sp 3 oneofpromotionrequireshybrids s(levelsantibondingtheandbondingthebetweenSeparationelectron. "bare"

5aboutisbandgap) eV thealsoiswhich, "average" thebetweenseparationthesebetweenseparationminimumThestates.bandconductionandvalence

bandgap(thebands E G 1.2aboutisSiin) eV temperature.zeroat

-5-1Lect

1.5Figure Column-IVofcharacteristicstructure,diamondThe.Ge,Si,(diamond),Csemiconductors: α latticeBravaisanotisIttin).(grey-Sn

thealongaxeswithcoordinatesCartesianIncell.unitperatomstwohasandsideofcubeaofside a vectorabydisplacedisatomsecondthe

( /a ,4 /a ,4 /a isbasis)(theitselfunittwo-atomThefirst.thetorelative)4structureThelattice.(fcc)cubicface-centeredaformingperiodically,repeated

aisTheresublattices.fccinterpenetratingtwoasviewedbecan ofcenterinversion atoms.basistwothebetweenlinetheonhalfwaylocatedsymmetry,

allsooperation,inversionthebyinterchangedfullyaresublatticesfcctwoThearestructurediamondtheinatoms equivalentsymmetry doeshowever,This,.

ofsetachoosetowaynoistherebecauselattice,Bravaisadiamondmakenotsingleafromstructureentirethegeneratewouldthatvectorstranslationthree

atom.

-6-1Lect

1.6Figure tosimilarisfigureTheGaAs.e.g.,structure,zincblendeThe.Galliumdifferently.coloredareplanesalternateinatomsthatexcept1.5Fig.

sublatticeidenticalAnlattice.cubiccenteredfaceaformatoms – shiftedbut1cubetheofdiagonalbodythealong / lengthitsofth4 – atomsbyformedis

inversionanhavenotdoesstructurezincblendethediamond,UnlikeAs.ofVolumesymmetry. a 3 cellunitprimitivethetimesfourisshowncubeof

volume v c 8thusiscrystaltheofvolumeunitperatomsofnumberThe. a/ 3.

-7-1Lect

A

A

c

a

B

b

1.7Figure aformatomsCadmiumCdSe.e.g.,structure,wurtziteThe.atomsCdofplanesecondThecell.unitperatomstwowithlatticehexagonal

BTheatoms.Aofplanestwothebetweenmiddletheinexactlylocatedis(B)Atheasatomsofarrangementhexagonal2-dimensionalsamethehasplaneisAtorelativeatomBtheofPositionlaterally.shiftedisbutplane

a 1 /3 + a 2 /3 + a 3 / .2

bydenotedisneighborsAin-planebetweenDistance a A-Averticaltheandbydistance c abyatomscadmiumfromdisplacedareatomsSelenium.

distance b wurtzitegeneralaThusaxis.hexagonaltheofdirectiontheinconstantlatticetheparameters:3bydescribedisstructure a twoand

parametersdimensionless =γ a/c and =µ a/b .

thebycharacterizedaresemiconductorswurtziteMost "ideal" values=γ (8/3)1/2 andlattice)packedclose(hexagonal =µ (3/8)1/2 arebonds(all

blockstetrahedralundistortedwithbuiltwhenobtainsstructureThisequal).arekindoppositeofatomsonlystructure,zincblendetheinLike1.3).(Fig.

bonded.

-8-1Lect

A

A

A A

A

A

A

B B

BC

C

C

1.8Figure ofAtomsaxis.hexagonalthealongviewedstructurewurtziteThe.formedtrianglesequilateralsix)of(outthreeofmiddletheinprojectBtypespeciesatomicsamethetocorrespondBandAboththatNoteA.atomsby

andCd)(e.g. together Seof(Atomslattice.packedclosehexagonalaformtheof3/8byapproximatelypagetheintodisplacedlatticehcpanotherform

distance c planes.)Atwobetween

(planetheintouchtheythatsoAspherestheexpandingImagine "closepacking" cannotwelevel,nexttheonspheressimilarstackingthatclearisIt).

togowewhenHowever,short).tooisBC(distancepositionsCandBbothfillis(aspositionsAthreetheeitherfilltofreeareweagain,levelnextthe

arrangementlatter(thepositionsCthreetheorlattice)hcpforappropriateatomicsecondtheofsublatticesimilaraAddinglattice).fcctocorresponds

thatNotestructure.diamondtheorwurtzitetheeitherobtainwespeciesandwurtzitespheres,ofpackingdenseatocorrespondfccandhcpwhile

loose.veryarediamond

-9-1Lect

1.9Figure IV-VItheofSomestructure.chloride)(sodiumrocksaltThe.ascrystallizePbTe)PbSe,PbS,family:lead-saltthe(mainlysemiconductors

isstructureThisrocksalt. octahedrally (coordinated z = ofions)(orAtoms).6thatwayasuchinlatticecubicsimpleaofsitesalternateoccupySandPb

zincblende,theLikekind.othertheofneighborsnearestsixhasatomeachhasandbasistwo-atomawithcubicface-centeredisstructurerocksaltthe

8 a/ 3 shadedthethatNotevolume.unitperatoms "atoms" inpositionedare1.6Fig.inaswaysamethe – Thearrangement.face-centeredtheexhibiting

shadedtheandwhite "atoms" sublattices.fccinterpenetratingtwoform

-10-1Lect

1.10Figure parallelogramAnylattice.honeycombaofcellsunitPrimitive.latticeothernoprovidedserve,willverticesaspointslatticefourbyformed

betohavenotdocelltheofBoundariesinside.orboundaryitsonfallspointofpairabyreplacedbemaysidesoppositethelines:straightofmade

area.samethehavecellsprimitiveAllcurves.congruent

itsNotecell.primitiveWigner-SeitztherepresentshexagonshadedThetheobserveandlattice)honeycombtheofsymmetry(pointsymmetry

pointcentralthefromdrawnwereLinesconstruction.itsforusedscaffoldinginbutsufficient,wereneighborsnearestonlycasethis(inneighborsitsto

waslineeachandrequired)bemayneighborsofshellsseveralgeneralline.perpendicularawithbisected

-11-1Lect

1.11Figure rhombicalattice,cubicface-centeredtheforcellWigner-Seitz.volumeofdodecahedron v c = a 3 / faces.12andedges24vertices,14hasIt4.

equivalent:allnotareverticesTherhombi.equaltwelvearecelltheofFacesrhombi,fourtocommonarecubebigtheofcentersfacethetouchthatsixthe

andObserverhombi.threetocommonareverticeseightotherthewhilerocksaltandzincblende,diamond,Insymmetry.cellthecontemplate

atoms.twoaccommodatescellprimitivethestructures

Exercise whiteoflatticefccthefromStartgeometrically.cellWigner-SeitzfcctheConstruct:"atoms" cubesmallaDrawcube).theofcentertheatsiteonehasit(conveniently,1.9Fig.in(edge /a ofreplicasMake1.11.Fig.inlinedashedthebyshownasatom,centralthearound2)

replicatheofdiagonalsbodytheUsechessboard).three-dimensionala(likecubesmallthepyramids6identifytocubes – pyramidsThesecube.centraltheoffaceonecoveringeach

theofverticesSixdodecahedron.rhombictheformcubecentralthewithtogethertheeach,edgesfourtobelongpyramids)ofapicesthewithcoincidethat(thosedodecahedron

edges.threeofintersectionsarecorners)cubesmallthewith(coincidingverticeseightother

-12-1Lect

W

kx

kz

ky

L

K

UΛ

Σ

∆

Γ

Q

Z

X

W

U

X

1.12Figure alattice,cubiccenteredfacetheofzoneBrillouinfirstThe." octahedrontruncated " (24volumeof π a/ )3 alongsquaressixfaces:14hasIt.the <100> thealonghexagonsregulareightanddirections <111> directions.

thatNoteprimes).the(withoutconventionallylabeledarepointsSymmetryUandKpoints ′ are identical aresoandvector)latticereciprocalaby(different

WandWpoints ′ – XandXnotbut ′ stateselectronicofnumberThe.8iszoneBrillouintheinspin)(including a/ 3 volume.unitper

-13-1Lect

kx

ky

ΓU

R

KΛ Σ∆

kz

H

M

P

L

K

A

T

A

H

M

1.13Figure theshowinglattice,hexagonalaofzoneBrillouinfirstThe.00.dsvolumeofprismhexagonalaisItaxes.symmetryandpointssymmetry

√ (2π)3 v/ c where, v c = a 2 /c prismsimilaraofvolumetheis230byrotatedareprismstwoThesecell.Wigner-Seitzthetocorresponding ˚

spin)(includingstateselectronicofnumberTheanother.onetorespectwith00.ds √ 4iszoneBrillouinthein /(a 2c volume.unitper)

-14-1Lect

E ( )k1

E ( )k2

E ( )k3

E ( )k4

Pseudo-momentum

Ene

rgy

1.14Figure respectwitharrangedbemaybandsdifferenthowofIllustration.(likebandstwoofOverlapanother.oneto E 2 and E 3 theirimplynotdoes)

inpointsofpairsbetweenenergiesofCoincidencestouching.orintersectionBandsdegeneracy.accidentalofexampleanisbandsthese E 1 and E 2 touch

gapEnergydegeneracy.thisofcauselikelytheissymmetrywhosepoint,aatbandsseparates E 3 and E 4.

-15-1Lect

k 1 k 2

F1 F2

q̂

1.15Figure tonormalplanetheinzoneBrillouinaofcross-sectionSchematic.FFacesline).dashedtheby(shownplanesymmetrycrystala 1 Fand 2 are

directionatoperpendicularandplanesymmetrythetoparallel q̂ Opposite.F(symbolically,identicalarezoneBrillouintheoffaces 1 = F2 ≡ Therefore,).F

pointsrelatedsymmetry k1 and k2 Formomentum.crystalsametherepresentbandenergyevery En (k ofareaentiretheonvanishesderivativenormalits),

F.polygonthe

-16-1Lect

ΓΛ

L∆

X

ELEXEΓ

Esohh

lh

so

1.16Figure cubicofstructurebandtheinpointsextremalImportant.forappropriateisscale)tonot(drawnpictureschematicThesemiconductors.

indicatedthetemperatureroomatGaAsForsemiconductor.III-Vdirect-gapaare:energies

E Γ = 1.42 eV, E L = 1.71 eV, E X = 1.90 eV , E so = 0.34 eV .

theinispointbandconductionlowestthesiliconIn ∆ theof85%direction,are:K300atsiliconforenergiesindicatedThepoint.Xtoway

E Γ = 4.08 eV, E L = 1.87 eV, E ∆ = 1.13 eV , E so = 0.04 eV .

thebutLatispointbandconductionlowesttheGeIn Γ away:farnotispointE Γ = 0.89 eV, E L = 0.76 eV, E ∆ = 0.96 eV , E so = 0.29 eV .

book.theofendtheatTablesinfoundbecandatastructurebandAdditional

-17-1Lect

1.17Figure bandconductiontheofvicinitytheinenergyconstantofSurface.alongextendedrevolution,ofellipsoidssixrepresentssiliconinedge <100>

(heavydirectionlongitudinaltheinleastiscurvaturebandThedirections.masseffectiveThemass)(lightdirectiontransversetheinhighestandmass)

ratio ml m/ t ∼∼ .5

-18-1Lect

1.18Figure bandconductiontheofvicinitytheinenergyconstantofSurface.alongextendedrevolution,ofellipsoidsfourrepresentsgermaniuminedge

<111> large,veryisratiomasseffectiveThedirections. ml m/ t = thatso,20sausage.alikelooksreallyellipsoideach

forcellprimitiveachosetohavewouldweellipsoidfullaexhibittoorderIneachpicture,zoneBrillouintheIninternal.bewouldpointLsomewhich

thetoshiftedishalfequivalenttheandboundarythebytwoincutisellipsoidvector.latticereciprocalabyfaceopposite

-19-1Lect

kx

ky

1.19Figure theinsurfaceconstant-energyaofplane(001)theonProjection.diamond-structureinedgebandvalencedegeneratethree-foldtheofvicinity

warpedareenergyconstantofsurfacesThree-dimensionalsemiconductors.exaggerated.muchisfiguretheinwarpingThespheres.

-20-1Lect

EG

Eso hh

lh

so

s1/2s

p3/2p

p1/2

(1)

(3)

(2)

(4)

(2)

(a) (b)

1.20Figure cubicinzoneBrillouintheofcentertheatstatesofGenesis.spinelectronNeglectingsemiconductors. (a) formedistopbandvalencethe,

bondingfourtheofcombinationslinearthreeby sp 3 1.4).Fig.(cf.hybridspredominantlyareThese p combinationindependentlinearly4thTheorbitals.

(predominantly s shownnotisandbandvalencetheofbottomtheatis-type)banddirect(inbandconductiontheofbottomThefigure.thein

predominantlyissemiconductors) s hybridsantibondingotherthewhile-type,(p inNumbersband.conductiontheofrangeuppertheinare-type)

atdegeneraciestheindicateparentheses k = weIf0. "include" (butspinthedouble.degeneraciestheinteraction),spin-orbitthenot

interactionspin-orbittheofInclusion (b) twothesplits-off p-statesmomentumangulartotalthetocorresponding jm > =

21__ ±

21__ > The.

statesfourremaining 23__ m > lightbands,degeneratedoublytwotorisegive

(holes 23__ ±

21__ > (holesheavyand)

23__ ±

23__ > isbandvalencetheoftopThe).

31__ E so bandsplit-offtheoftop(andabove

32__ E so unperturbedthebelow)

coupling.spin-orbitofabsencetheinedgebandvalence

-21-1Lect

EG

Eso s1/2

p3/2

p1/2

(4)

(2)

(2)

m = 1/2 +_

m = 3/2 +_

1.21Figure Intin).(greysemiconductorsgaplessindiagrambandSchematic.changingbyobtainedformallyisstructurebandgaplessthemodel,Kanethe

ofsignthe E G The(1.78).Eq.cf., s 1/2 energyinlowerbecomesthenbandthethan p 3/2 theinbranchlight-holeofcurvaturetheandband p 3/2 isband

isbandgapTheelectrons).intotransformedareholes(lightreversedthedegeneratemakesthatsymmetrysametheofvirtueinzeroidentically

Thissilicon.andgermaniuminbranchesheavy-holetheandlight-holetoleadswhichfield,magneticaofapplicationthebybrokenbecansymmetryabyaccomplishedbecaneffectsimilargap;forbiddenaofemergencethe

strain.uniaxial

-22-1Lect

E

EG

k

e

lh

1.22Figure andconductiontheinrelationenergy-quasimomentumThe.linesthinrelation,exactthe(schematically)indicatelinesFatbands.light-hole

thegiveslinedashedtheofSlopemodel.bandtwothebygivenrelationthevelocitymaximum v max tocorrespondswhichelectrons,bandof c thein

approximatelyismodeltwo-bandthethatevidentisItanalogy.relativisticcurve.realtheonpointinflectionthetoupvalid

-23-1Lect

(a)

(b)

(c)

1.23Figure functions.waveenvelopeandBlochtheofillustrationSchematic.

(a functionBloch) u (r at) k0 = cell.unitcrystalthewithinrapidlyvaries0

(b functionEnvelope) F (r scale.cellunittheonlittlevaries)

(c functionwaveComplete) ψ (r masseffectivetheinelectronlocalizedaof)approximation.