project3.docx
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project3.docx
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project3
Themental-InsulatorInterface
WendongWANG
Xi’anJiaotongUniversity
[ABSTRACT]Ahightensioncableconsistsofametalcoreandaninsulatingsheath.ItcanberegardasaMetal-InsulatorInterfaceSystem.Inthispaper,wewillusetheLanczosMethodtocreatea1dimensionalmodelsystemanduseittomimicthesystemandderivesomeimportantpropertiesbycalculatingtheenergylevelsforasingleexcesselectron.Andwediscusstherealisticofthe1dmodel,whytheexcitedstatewavefunctionssometimescontaindiscontinuities,andhowcanitbecorrected.
[KEYWORDS]:
Metal-Insulatorinterface,1dimensionalmodel,wavefunction,energylevel.
AccordingtotheSchottkymodel[1],whenamentalandadielectricformaninterface,thereisnochargetransferacrosstheinterfaceandthebarrierheightfortheelectronsisgivenbythedifferencebetweentheworkfunctionofthementalinthevacuumФm,vacandtheelectronaffinity.ButitwasobservedexperimentallythattheSchottkymodelisnotgenerallyobeyed.Thewell-knownsurfacestatemodelofBardeenexplainedthisobservation[2].Bardeen’smodelpostulatesahighdensityofsurfacestateoftheorder1persurfaceatom,andthesestatesacttopinthementalFermilevel.Heine[3]wasthefirsttopointoutthatthewavefunctionsofelectronsinthementaltailintothesemiconductorintheenergyrangewheretheconductionbandofthementaloverlapsthesemiconductorordielectricbandgap.Theseresultingstatesareknownasintrinsicstates.Theexistenceofmental-inducedgapstatesatthemental-dielectricinterfacewasalsopredictedbyCohen[4]usingself-consistentpseudopotentialcalculationsoftheelectronicstructureofmaterialinterfaces.ThesestatesarepredominantlydonorlikeclosetoEv,andmostlyacceptorlikenearEc,justlikethefigure2.TheenergylevelinthebandgapatwhichthedominantcharacteroftheinterfacestateschangesfromdonorliketoacceptorlikeiscalledthechargeneutralitylevelECNL.Chargetransfergenerallyoccursacrosstheinterfaceduetothepresenceofintrinsicinterfacestates.Figure2illustratesthecasewherethementalFermilevelisabovethechargeneutralitylevelinthedielectricECNL,d,creatingadipolethatischargednegativelyonthedielectricside.ThisinterfacedipoledrivesthebandalignmentsothatEF,mgoestowardECNL,d.
A.METHODS
●KrylovSpace
Itisonlyrecentlythatthemethodsdevelopedforlargesparselinearalgebraproblemshasbeenappliedtocontrolproblemsoflargedimensionality.MostofthemethodsarebasedontherecursivegenerationofKrylovspaces.Thesemethodsarebasedontheideaofprojectingtheproblemontoever-expandingsubspacesgeneratedbythematricesoccurringintheproblemitself.Ithasbeenshownthatformanylinearalgebraproblems,theconvergencerateismuchfasterthanwhatwouldbeexpectedfromarbitraryprojections.ThealgorithmsbasedonrecursivegenerationofKrylovspacescanbereferredtocollectivelyasLanczos-typealgorithms[5].ThesuccessofthesealgorithmscomesfromthefactthatastheKrylovspacedimensionincreases,thenewprojectioncanbeobtainedveryquicklyfromthepreviousprojectionontothenextlowerdimensionalKrylovspace.Now,IwanttostatehowtodefineaKrylovspaceanddescribesomeofthebasicalgorithmsforcomputingbasesfortheKrylovspace.
AKrylovsequenceisasequenceofvectorsgeneratedbyamatrixasfollows.Givenan(n*n)-matrixAandavectorx,thekthKrylovsequenceK(A,x,k)isasequenceofkcolumnvectors:
K(A,x,k)=(x,Ax,A2x,…,Ak-1x)
Askincreases,wegetasequenceofsequencesK(A,x,0)=x,K(A,x,1)=(x,Ax),….AblockKrylovsequenceisgeneratedbyamatrixAandan(n*p)-matrixXasfollows
K(A,X,k)=(X,AX,A2X,…,Ak-1X)
●Lanczosmethod
TheLanczosalgorithmwasoriginallyproposedbyLanczosasamethodforthecomputationofeigenvaluesofsymmetricmatrices.Theideawastoreduceageneralmatrixtotridiagonalform,fromwhichtheeigenvaluescouldbeeasilydetermined.ThemethodisbasedonaparticulartransformationwhichbringstheHamiltonianintoatridiagonalform.ThebasicideaoftheLanczosmethodisthefollowing:
Consideranarbitrarystate|ψ>anditsexpansionintermstheeigenstate|ψk>oftheHamiltonian.ThismethodamountstoconstructingorthogonalbasisstatesthatarelinearcombinationsoftheKrylovspacestatesinsuchawaythattheHamiltonianwritteninthisbasisistridiagonal.Thebasisconstructionstartsfromanarbitrarynormalizedstate|f0>,ofwhichisrequiredonlythatitisnotorthogonaltothegroundstateofH.Wecangettheotherstatesthroughthefollowingformula[6]:
|f1>=H|f0>-a0|f0>
a=H00/N0
|f2>=H|f1>-a1|f1>-b0|f0>
a1=H11/N1
b0=N1/N0
forgeneral,wecangeneralizeto
|fn+1>=H|fn>-an|fn>-bn-1|fn-1>
an=Hnn/Nn
bn-1=Nn/Nn-1
Finally,weobtainthenormalizedbasisstates:
|φn>=1*(Nn)-0.5|fn>
●Infinitesquarewell
Thebarriersoutsideaone-dimensionalboxhaveinfinitelylargepotential,whiletheinterioroftheboxhasaconstant,zeropotential.Inquantummechanics,theparticleinaboxmodel(alsoknownastheinfinitepotentialwellortheinfinitesquarewell)describesaparticlefreetomoveinasmallspacesurroundedbyimpenetrablebarriers.
Themodelismainlyusedasahypotheticalexampletoillustratethedifferencesbetweenclassicalandquantumsystems.
Theparticleinaboxmodelprovidesoneoftheveryfewproblemsinquantummechanicswhichcanbesolvedanalytically,withoutapproximations.Thismeansthattheobservablepropertiesoftheparticle(suchasitsenergyandposition)arerelatedtothemassoftheparticleandthewidthofthewellbysimplemathematicalexpressions.Duetoitssimplicity,themodelallowsinsightintoquantumeffectswithouttheneedforcomplicatedmathematics.
●Finitesquarewell
Thefinitepotentialwell(alsoknownasthefinitesquarewell)isaconceptfromquantummechanics.Itisanextensionoftheinfinitesquarewell,inwhichaparticleisconfinedtoabox,butonewhichhasfinitepotentialwalls.Unliketheinfinitesquarewell,thereisaprobabilityassociatedwiththeparticlebeingfoundoutsidethebox.Inthequantuminterpretation,thereisanon-zeroprobabilityoftheparticlebeingoutsidetheboxevenwhentheenergyoftheparticleislessthanthepotentialenergybarrierofthewalls.
B.RESULTANDDISCUSSION
●Partone:
WesolvedtheSchrödingerequationfortheinfinitewallpotentialsquarewellandfoundtheenergyofquantumstatentobe
eV,whereqistheelectroniccharge,mthemassandLthewidthofthewell.WewillconsiderthesquarewellasrepresentingthepseudopotentialofanexcessN’thelectronwiththeremainingN-1electronsandthenucleus.Wewillrepresentametal(A)byaninfinitesquarewell(ISW)ofwidthLa=1.0nmandpotentialwellat-2.0eV(withcodeparameterwallev=-2.0eV),andaninsulator(B)byanISWofwidthLb=0.5nmandapotentialwellat0eV(withwallev=0.0eV)
Theinfinitesquarewellfor‘insulator’(B),V(x)=00 a)Thefirst5energylevelsineachsystemare: Mental: -1.6230-0.49201.39304.03207.4250 Insulator: 0.37701.50803.39306.03209.4250 b)GiventhatourresultsareforasingleexcesselectronandthatweassumeourenergiesarerepresentativeofaninfiniteN+1electronsystemwherewouldyouplacetheFermilevelinthemetal(highestoccupiedelectronenergy)andwheretheconductionbandintheinsulator? ANSWER: AccordingtothePauli-exclusionprincipleandprincipleofminimumenergy,electronsoccupythelowenergyorbitsfirstandeachorbitcanoccupytwoelectrons.So,N+1electronsfillthe(N+1)/2states.Thefermilevelis +V(x) Intheinsulator,theconductionbandisthenextenergylevelofthehighestoccupiedelectronenergy c)Theelectronaffinityisdefinedas where isthevacuumenergy,=0.eV,whataretheelectronaffinitiesforthetwopurematerials? Commentontheresult. ANSWER: Evac=0 So,EEA=-EC. Forthemental, -V(x) Fortheinsulator, d)Plotthefirst5eigenfunctionsforeachsystem. ●Parttwo(usingtheLanzcoscodewithnf0=0) WewillnowputthesematerialstogetherandcreateaninterfacewiththeinsulatorbychangingtheISWintoaFSWwithpotential=wallev=-2eVevforx<1.0nm,wallev=10eVand1nm Changethecodetocalculate: ! usetoputFSWpotentialtozeroinsidewell if(x.ge.b.and.x.le.1)then wall=dfac*evtoscale*wallev1 elseif(x.ge.1.and.x.le.1.5)then wall=dfac*evtoscale*wallev2 else wall=dfac*evtoscale*wallev3 endif ChangetheFILE=’lanINfswthree’: 2.,1.,1.,0.001,1.,1.,2500,0.01,-2,10,0,1.0e25,0, a)Comparetheenergylevelsinthesystemabovewiththosejustintheisolatedmetalandinsulator(assumeISW).Fromfigure1thereareintrinsicsurfacestates.Assumingourjunctionmodelisrepresentativeoftheinterfacewhataretheseenergies.WhatistheEAofthejunctionandhowdoesitcomparewithbulkPE? ANSWER: Thefirstsixenergiesare: -1.66-0.651.011.183.324.656.25. Mental: -1.6230-0.49201.39304.03207.4250 Insulator: 0.37701.50803.39306.03209.4250 Comparedtothepurementalorinsulator,theenergyoftheinterfaceislower. TheEAofthejunctionisbiggerthanmentalortheinsulator.PEisaninsulator.Asaresult,EAofthejunctionisbiggerthanthebulkPE. b)Plotwavefunctionsforthejunction c)Intheexcitedstatescharge
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