Aquantumdot(QD)confineselectronsinallthreedimensions.Inthisway,QDsarejust likeatoms or moleculeswithquantum states; somepeople call quantum dots“artificial atoms.”QDscanbefabricatedinnumerousways.Onecanstartwithaquantumwelland etchittoleavesmallpillarsthatareafewnanometersacross.Anothermethodthatworks surprisinglywellistosimply depositontothesurfaceofasolidanumberofatomsless thanenoughtocoverthewholesurfacewithoneatomthickness.Inthiscase,theatomswill tendtoformsmallbeads,justlikewaterbeadinguponasurface.Thesebeadscanthenbe coveredbyabarrierlayerwithadifferentbandgap.Oneadvantageofthismethodisthat itdoesnotrequireetching;ingeneralitisdifficulttogrownewlayerswithtechniqueslike molecularbeamepitaxyafter etchinghasbeendone,sincetheetchingaltersthesurface andtypicallyintroducesoxygen.Quantumdotscanalsobefabricatedbywetchemistry.
TheQDsstartoutassedimentparticlesafewnanometersacross,whichareallowedtodrop fromaliquidsolutionontoasurface.Theliquidisthenremoved,leavingthedotsonthe surface.Insomecases,abarriermaterialmaybeaddedatthispointtocoverallthedots;
inothercasesthesedimentparticlesmaybecoatedwithabarrier materialviachemical reactionwhilestillinsolution.Finally,organicmoleculesafewnanometersacrossmaybe treatedasQDs.
±Fig.2.34 Aquantumdotwithtunnelingbarriersintoandoutofthedot,andagatecontacttochangeitspotential.FromA.G.
Sheer,UniversityofKonstanz.
SinceQDsarezero-dimensional,thereisnocurrentconductiontospeakofwithinadot.
TherearetwowaystomakeQDsactiveindevices,however.Onewayistoallowtunneling ofchargeintoandoutofaQDviathinbarriers.Anotherwayistosendphotonsintothe dot,whichexciteelectronsfromthevalencebandtotheconductionband.
Exercise2.8.6 Quantumdotsarerarelyexactlysymmetric.SupposethataQDisrectangu- larwithalengthof7.5nm,widthof8nm,andheightof6nm.Computethelowest fiveconfinedstateenergiesofanelectroninthisdot,foranelectroneffectivemass of0.1timesthevacuumelectronmass,assuminginfinitebarrierheightaroundthe dot.
Coulomb blockade.Figure2.34showsanexampleofQDstructurefabricatedbyetch- ingawaymaterialtoleavebehindaverysmallmetaldot(thecentralcircleintheimage) witharadiusofafewnanometers,andthreemetalcontactsseparatedfromthedotbythin tunnelingbarriers.
Figure2.35(a)showsanexampleoftheintrinsicconductionbandstructureforthedot withthetwotunnelingbarriersandtwosurroundingmetalcontacts.Anelectroninthedot hasdiscrete energy levelsbecause of thequantum confinement inallthree dimensions.
Thebandsaretiltedduetothevoltagedifferenceofthecontactontheleftandthecontact ontheright.Inthisconfigurationofthebands,anelectroncantunnelfromtheleft-hand contactintotheupper confinedstateofthequantumdot.Weassumethatthelowerstate isfilled,beingbelowbothcontactFermienergies,andthereforeitdoesnotparticipatein conduction.
InFigure2.35(b),thequantizedenergylevelsofthedotareshiftedupwardduetoone additionalelectronbeinginthedot.TheenergylevelsareshiftedupbecausetheCoulomb repulsionoftheelectronsgivesthemhigherpotentialenergy.Inthiscase,nonewelectron cantunnelfromtheleftcontactintothedot,sinceitwouldhavetogainenergytogoup
141 2.8 QuantumConfinement
barrier
barrier
EF(R)
EF(R) EF(L)
EF(L)
dot (a)
(b)
±Fig.2.35 (a)Bandstructureforaquantumdotwithtunnelingbarriersbetweentwoelectronreservoirs.(b)Thealteredband structurewhenanelectroninthedotraisesthequantizedenergylevelsinthedot.
tothe upper quantized state. (Weassume that thesystem is at lowenough temperature thattheelectronshavesmallprobabilityofjumpingtohigherlevels.)Theelectroninthe upperlevelcanstilltunnelouttotherightinthiscase.However,ifthetunnelingbarrier ishighenough,thismaytakesometime,sothedotcanbeleftinametastablestatewith asingleelectron intheupper quantized level. Theconductionthrough thedot therefore correspondstoaseriesofsingleelectrons, tunnelingone atatimeintothedotandthen out.Aslongasoneelectronisinthedot,nomorecancomein.
Thisisknownasa Coulomb blockade.Asingle electronchangestheeffectiveresis- tance of the current path through the dot by orders of magnitude, since the tunneling currentdepends very sensitively onthe relativeenergies ofthe states. This canalso be usedforasingle-electrontransistor,sinceoneelectroninthedotcontrolswhetherthedot conducts or not. There are many different versions of single-electron transistors, all of whichusethefactthatasingleelectroncansignificantlyshiftthequantizedenergylevels inadot.
The sensitivity of the states of a dot to single electron charge can also lead to an unwanted effect. Sometimes impurities in a solid can lead to trapped single electrons or holes. If one of these is near a quantum dot, it can substantially shift the states of thedot.
Exercise2.8.7 Calculate the Coulomb potential energy of two electrons separated by 10nm, in a solid with dielectric constant of 10. How does this energy compare tothetypicalenergylevelspacingof10–100meVinnanostructures?