Theabilityto makevarious semiconductor heterostructuresmeansthatwe canmakeall types of band structures analogous to the one-dimensional Kronig–Penney model we studiedinSection1.2.
133 2.8 QuantumConfinement
As we discussedabove, a semiconductor layer sandwiched between two othersemi- conductorlayerswithlargerbandgapsmakesaquantumwellwhichhasapotentialenergy profileinonedimensionthatisexactlythesameastheone-dimensionalsquarewellstudied inSection1.1.Inthesameway,ifwehaveaseriesoflayerssandwichedbetweenotherlay- erswithhigherbandgaps,thenthepotentialenergyprofileinthedirectionperpendicularto theplanesisaseriesofcoupledquantumwellsjustlikethatoftheKronig–Penneymodel studiedinSection1.2.Thisisknownasa superlattice.Inthiscase,justaswehaveseenin numerousexamples,theoverlapofthewavefunctionscanleadtotheformationofenergy bands.We nowhave thepossibilityofbands withinbands,or mini-bands, asshown in Figure2.30.
If the layers are madein a periodic pattern with the same thicknesses, then Bloch’s theoremapplies again. The periodiclayer structure acts asa set ofBragg reflectorsfor theelectronwavesjust asatomiclayersdo. Wecanthereforetreattheelectronsorholes as free particles and redraw the electron bands in a smaller zone, as shown in Figure 2.30.Sincethe electronsand holes inthe effective massapproximationact as freepar- ticles,wecanusethenearlyfreeelectronapproximationofSection1.9.3todeterminethe bandgaps.
Therearenumerouspossibilities,therefore,ofalteringthebandstructureofmaterialsby theappropriatechoiceofmaterialthickness.Asdiscussedabove,wehavethepossibility ofbandstructureengineering,designingwhateverbandstructureisneededforaparticular device.
Exercise2.8.4 InmanysemiconductorssuchasGaAs,theconductionbandhasaconduc- tionbandminimumatzonecenter,andanindirectgapwithhigherenergyatanother minimum,atacriticalpointonthezoneboundary.Thezone-centerminimumistyp- icallycalledthe ³-valley,whiletheother,indirect minimaarecalledthe X-valley andthe L-valley.
Supposethat wemake a superlatticewithwellmaterialGaAs, whichhasa ³- valley1.5eVabovethevalencebandmaximum,andbarriermaterialAlAs,which hasa ³-valley 2.6 eV higherthan theGaAs valenceband maximum, butan [X- valley]only 1.7 eV higher than the GaAs valence band maximum. If the GaAs quantumwellwidthissmallenough,theconfinedstateofthe ³-valleycanbepushed higherthantheAlAs X-valley,asshowninFigure2.31.Inthiscase,electronswill fallfromtheGaAs ³-valleyintothe X-valleyofthebarriers,sothattheAlAsformsa quantumwellfortheelectrons,withtheGaAslayersactingasbarriers.Thevalence- bandholeswillremainconfinedintheGaAslayers,however.Thisiscalleda Type II superlattice.Astructure in whichboththe electronsand holesare inthesame layeriscalleda Type Isuperlattice.
Assumingthatthereisnegligiblepenetrationofthewavefunctionofthe ³-valley electronsintothebarriers,estimatetheGaAsquantumwellthicknessatwhichthe abovestructure willconvertfrom aTypeIto aTypeIIstructure. Theconduction electroneffectivemassisapproximately0.06m0forGaAs.
(a)
(b) (c)
z
d Ec
Ev
E E
0
k a
d
– a 0
k d – d
±Fig.2.30 (a)Spatialbandprofileofasuperlatticestructure.(b)Theoriginalconductionandvalencebands,foraquantumwell materialwithlatticeconstanta.(c)Mini-bandsarisingfromthequantumwellstates,inthereducedzone.The additionalperiodicityofthesystemleadstonewBlochstates.
L
2.6
1.7 1.5 X
Γ
AlAsGaAs
±Fig.2.31 ConductionbandenergiesforaTypeIIsuperlattice,discussedinExercise2.8.3.
135 2.8 QuantumConfinement
Bloch oscillations.Superlatticeshavebeenusedtodemonstrate(Waschke etal. 1993) thebasiceffectof Bloch oscillations,inwhichaDCelectricfieldgeneratesACelectro- magneticradiation dueto thereflectionsofthe electronwavesin theperiodicpotential.
Thiseffectisagenericpropertyofallperiodicsystems(seeKriegerandIafrate1986),but isnotobservedinstandardbulksolidsbecausetheoscillationfrequencyinbulkmaterials istoolow,forreasonswewillseebelow.
The basic effect can be understood as follows. As an electric field accelerates the electronsin aperiodic solid, their quasimomentum ±k increases. If k increasespastthe boundaryof the first Brillouin zone, itwill be wrappedaround to the oppositeside of theBrillouinzone.Assumingthattheelectronremainsinthesameband,itwilltherefore undergoperiodicmotion.(Transitionstootherbands,whichcanoccurinhighfields,are discussedbyKriegerandIafrate1986.)Toputitanotherway,astheelectronsaccelerate, theirwavelengthbecomesshorter, untiltheyhavewavelengthsoshortthattheyundergo Braggreflection(discussedinSection1.2)fromtheperiodicstructureandthusarereflected totheoppositedirection.Theythenacceleratebacktheotherwayagainduetothestatic electricfield,andsoon.
To prove this, we assume a one-dimensional system, and write the time-dependent Schrửdingerequation,
− ±2 2m
∂2
∂x2ψ + U(x)ψ = i±∂
∂tψ. (2.8.9)
Weassumethatthepotential U(x)in(2.8.9)hasthesameperiodicityasthemedium.Now we guessthe solutionof the form ofa Bloch state ψk = uk(x)ei(kx−ωt), where uk(x) is aBlochcellfunctionwiththeperiodicityofthelattice.Substitutingthisinto(2.8.9),we have
±2 2m
´
−∂2uk
∂x2 − 2ik∂uk
∂x + k2uk
à
+ U(x)uk= (±ω) uk. (2.8.10) Thisisadifferentialequationthatwecan,inprinciple,solvefortheperiodicfunction uk(x) foranygivenvalueof k.
Nowletusaddatermtothepotentialenergyforaconstantelectricfield,namely −qEx.
Weguessnowthatthesolutionofthetime-dependentSchrửdingerequationwillhavethe formofaBlochstate ψk= uk(t)(x)ei(k(t)x−ω(t)t),withanexplicittimedependencefor k and ω.SubstitutingthisintotheSchrửdingerequation(2.8.9),wehave
±2 2m
´
−∂2uk
∂x2 − 2ik∂uk
∂x + k2uk
à
+ U(x)uk − qExuk
= i±∂uk
∂k
∂k
∂ t + i±
´ ix∂k
∂t − iω − it∂ω
∂t à
uk (2.8.11)
=−x∂(±k)
∂t uk+ i∂uk
∂k
∂(±k)
∂t +
´
±ω + t∂(±ω)
∂t à
uk. Wecansolvethisequationbyfirstsetting
∂(±k)
∂t = qE = F, (2.8.12)
whichisjust whatwewouldexpect forthechangeofmomentumwithaforce.Wethen have
i∂uk
∂k
∂(±k)
∂t + (±ω)uk+ t∂ω
∂k
∂(±k)
∂t uk
= ±2 2m
´
−∂2uk
∂x2 − 2ik∂uk
∂ x + k2uk
à
+ U(x)uk. (2.8.13) Weassumethat k = 0at t = 0,whichgives ±k = qEt,or t = ±k/qE.Usingthisin(2.8.13), andrearranging,wehave
(±ω)uk= ±2 2m
´
−∂2uk
∂x2 − 2ik∂uk
∂ x + k2uk
à
+
´
U(x) + i uk
∂uk
∂k qE − k∂ (±ω)
∂k à
uk. (2.8.14) Thishasexactlythesameformas(2.8.10),butinsteadofjust U(x),wehavethelastterm inparenthesesontheright-handside,withtwoextraterms.However,eachoftheseterms is also periodic in x, since uk(x) hasthe periodicity of U(x), and the third term in the parenthesishasno x-dependence atall. Thus,wehaveanequationforaBloch function with analtered periodic potential, which canbe solved self-consistentlyfor uk(x). The overallwave functionis theproduct ofthis periodic functiontimesaplanewave factor eikx,with k = qEt/±.When k equals π/a,Braggreflectionwilloccur,taking k from π/a to −π/a.Then k willcontinuetoincreaseatthesamerate.
Thetotaltimetogothroughthewholeallowedrangeof k valuesisfoundbytakingthe totalrangeof k,equalto2π/a,anddividingbytherateofadvance of k,whichfromthe aboveis dk/dt = qE/±.Wethenhavetheperiod T = (2π/a)/(qE/±),or,using ω = 2π/T, theoscillationfrequency
ωB = a|q|E
± , (2.8.15)
whichisproportionaltotheelectricfield E andthelatticeconstant a.
Thelargerthelatticeconstant,thelesstimeittakesfortheelectronstospeeduptohave wavelengthcomparabletoit;thereforelargerlatticeconstantcorrespondstoshorteroscil- lationperiod.Latticeconstantsoftensofnanometers,whicharetypicalforsemiconductor superlattices,giveoscillationsintheTHzfrequencyrangeforDCelectricfieldsofkV/cm, thatis,afewvoltsacrosstensofmicrons;thishastechnologicalrelevanceasasourceof THzradiation.Inprinciple,Blochoscillationsexistinanyperiodicsystem,includingbulk crystals withperiodic atomiclattices; the atomic spacing ofangstromsin typical solids impliesfrequenciesintheGHzrangeforcomparableelectricfields.Fortheselongerperi- ods,however,scatteringoftheelectronsbreaksupthecoherenceoftheoscillations.Aswe willdiscussinChapter5,typicalscatteringtimesforelectronsinsolidsareoftheorderof tensofpicoseconds.
Accordingtotheaboveanalysis,iftherewerenoscatteringoftheelectrons,theelectrons inacrystal wouldnevercarryDCcurrent,andinsteadsimplyoscillatebackand forthin responsetoaDCfield.Scatteringthereforeplaysanessentialroleinelectricalconduction.
137 2.8 QuantumConfinement
Exercise2.8.5 Showthat inthecasewhen U(x) = 0 and uk = 1,that is,the statesare planewaves ψ = ei(kx−ωt)inavacuum,andboth k and ω aretime-dependent,the solutionof(2.8.14)for k = 0at t = 0implies k = qEt/± and ω = (qEt)2/6±m.
Showthatthisimpliesthattheaveragevalueoftheenergy,definedby
² ∞
−∞dx ψ∗Hψ =
² ∞
−∞dx ψ∗i±∂ψ
∂t , (2.8.16)
isequalto ±2k2/2m.Inotherwords,thekineticenergygrowsintimeinthiscase, andthereisnoBraggreflection.