Greenfunctionapproachtointerfacestatesinband-inverted
junctions
FranciscoDom´ınguez-Adame
DepartamentodeF´ısicadeMateriales,FacultaddeF´ısicas,UniversidadComplutense,E-28040
Madrid,Spain
arXiv:cond-mat/9410098v1 26 Oct 1994TypesetusingREVTEX
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Narrow-gapsemiconductorcompoundslikePb1−xSnxTeandHg1−xCdxTepresentbandinversionundercompositionalvariation.Inaband-invertedheterojunctionthefundamentalgap,definedasthedifferencebetweenΓ6andΓ8energies,hasoppositesignsoneachside[1].TypeIIIsuperlatticeswithbandinversionofCdTe/HgTeandPbTe/SnTehasbeensuccessfullygrowninthepast[2,3].Oneofthemostconspicuouscharacteristicofband-invertedheterojunctionsistheexistenceofinterfacestateslyingwithinthefundamentalgap,providedthatthetwogapsoverlap[4–8].InIV-VIcompoundsthoseinterfacestatesareproperlydecribedbymeansofatwo-bandmodelusingtheeffectivek·papproximation.Onthecontrary,theanalysisismorecomplexinII-VIcompoundsduetomixingwithheavy-holestatessincenoncentro-symmetryeffectsarenotnegligibleinthiscase.Theequationgoverningconduction-andvalence-bandenvelope-functionsinasimpletwo-bandmodel,neglectingfar-bandcorrections,isaDirac-likeequation.Exactsolutionscanthenbefoundinviewofthisanalogybecauseonecanuseelaboratedtechniqueslikethoserelatedtosupersymmetricquantummechanics[7].Theaimofthispaperistopresentanalternativewayofsolutionbasedonthesocalledpointinteractionpotentials[9,10](anyarbitrarysharplypeakedpotentialapproachingtheδ-functionlimit)alongwithaGreenfunctionmethod.Webelievethatourtreatmentgivesaveryintuitiveexplanationoftheoriginofinterfacesstateswhileotherapproachesmayobscurethewayhowthosestatesarise.Moreover,theeffectsofexternalelectricandmagneticfieldscanbeincludedinastraightforwardfashion,aswewillshowlater.
Intheeffective-massapproximationtheelectronicwavefunctionisasumofproductsofBlochfunctionsattheband-edgewithslowlyvaryingenvelope-functions.Thetwo-bandmodelHamiltonianintheabsenceofexternalfieldsisoftheform
H=v⊥αypy+vzαzpz+
1
elementsareconstantthroughthewholeheterostructureduetothesimilarityofthezonecentreinbothsemiconductors.Sincethegapdependsonlyuponz,thetransversalmomen-tumisaconstantofmotionandwecansettheYaxisparalleltothiscomponent.Inthetwo-bandcasetherearefourenvelope-functionsincludingspinandwearrangetheminafourcomponentvectorF(r).Thisvectorsatisfiestheequation
HF(r)=[E−V(z)]F(r),
(2)
whereV(z)givesthepositionofthegapcentre.Itisunderstoodthatthegrowthdirectionis[111].ThewayV(z)changesfromonelayertoanotherisnotwellunderstoodbut,assumingthattheinterfacestatesspreadoverdistancesmuchlargerthantheinterfaceregion,wecanconfidentlyconsideritasastep-likefunction.Accordinglywetake
EG(z)=EGLθ(−z)+EGRθ(z),V(z)=VLθ(−z)+VRθ(z),
(3a)(3b)
θbeingtheHeavisidestepfunction.Here,thesubscriptsLandRmeanleftandrightsidesoftheheterojunction,respectively.
Aswehavealreadymentionedabove,themomentumperpendiculartotheinterfaceisconserved,andthereforewelookforsolutionsoftheform
F(r)=F(z)exp
i
2
βEG(z)−E+V(z)F(z)=0.
(5)
AsimplewaytosolvethisequationistheFeynman-Gell-Mannansatz[11]
F(z)=αyv⊥p⊥+αzvzpz+
1
dz2
+
1
4
22
EG(z)2−[E−V(z)]2+v⊥p⊥−i∆αz(β−λ)δ(z)χ(z)=0.
(7)
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Forbrevitywehavedefined
EGR−EGL
∆=
E,
(8b)
GR−EGL
andwehaveusedtherelationshipsdθ(±z)/dz=±δ(z).Notethatinthecaseofband-invertedheterojunctionEGREGL<0.
ItisworthmentioningthatEq.(7)isnothingbutaKlein-Gordonequationwithscalar-likeandelectrostatic-liketermsdependingonposition(likearelativisticspinlessparticlewithaposition-dependentmassinanelectricfieldasoccursinQED)plusapointinteractionpotentialarisingfromthediscontinuityofthegapandthegapcentre.Theoccurrenceofthisshort-rangepotentialmakesitpossibletheexistenceofboundstatesdeepinthegap.InordertofindtheboundstatesweuseaGreenfunctionformalism,similartopreviouslyusedinthecaseoftheDiracequationwithpointinteractionpotentials[12].Tothisend,letusconsidertheGreenfunctionassociatedtoEq.(7)withoutthepointinteractionpotential
−
∂2
1
h¯2v2z
Diracequationforpointintereactionpotentials(seeRef.[9]andreferencestherein).OncetheGreenfunctionisknown,the4-vectorχ(z)canbeobtainandusingEqs.(4)and(6)theenvelope-functionsarefinallydetermined.Boundstatelevelscanbecomputedtakingthelimitz→0inEq.(11).Toobtainnontrivialsolutionswerequirethe4×4determinanttovanish.Thus,usingthedefinitionsof∆andλgivenin(8)weobtain
1
4
(EGR−EGL)2−(VR−VL)2
=
1
W[u+,u−]
,(13)
whereW[u+,u−]istheWronskianofthetwosolutions.Definingtworealparameters
K1
L=
1
h¯v
EGR2−(E−VR)2+v2⊥p2⊥,
z4
(14)
thetwoindependentsolutionsareu+=exp(−KRz)andu−=exp(KLz)sothatg(0,0;E)=2/(KR+KL).UsingEq.(12)onefinallyobtains
K1
1
R+KL=
potentials.Nevertheless,itisclearthatappliedelectricormagneticfieldscanbeeasilyhandledwithminormodificationsoftheequations.NotethatthecrucialpointisthatoneassumesthattheKlein-GordonequationwithoutthepointinteractionpotentialarisingfromtheabruptinterfacecanbesolvedexactlyandthecorrespondingGreenfunctionsisexplicitlywrittenout.Thisissoforalargevarietyofelectricandmagneticfieldconfiguration,aspointedoutinRef.[13].Thus,forinstance,itispossibletoinvestigateLandaulevelsinband-invertedheterojunctionsinarathersimpleway,insteadofusingmoreelaboratedmathematicaltreatments,asthoserecentlycarriedoutbyAggasi[14].Inaddition,itisalsopossibletostudyconfinedStarkeffect,atopicwhichremainsopenintheliterature.Thesecondaspectweremarkisthefactthatthereisnoneedtouseanabruptheterojunction
−1−1
model,simulatedbyasteppotential.TheonlyrequirementisthatKRandKLmust
bemuchlargerthantheinterfaceitself,animplicitassumptionwhenusingtheenvelope-functionformalism.Qualitativelytheprofileoftheheterojunctionissoliton-like[7]and,asaconsequence,itderivativeisasharplypeakedfunction.Thustheintegralequation(11)canbesolvedbyalimitingprocess,inanalogouswaytotheDiracequationforsharplypeakedfunctionsapproachingtheδ-functionlimit[12].Toconclude,wefeelthattheapproachwedevelopedholdsinalargevarietyofcasesofpracticalinterestanditmayhelpinabetterunderstandingofinterfacestatesinband-invertedheterojunctions.
ACKNOWLEDGMENTS
ThisworkissupportedbyUCMthroughprojectPR161/93-4811.
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REFERENCES
[1]K.Seeger,SemiconductorPhyscis,(SpringerVerlag,Berlin,1991).[2]J.P.Faurie,IEEEJ.QuantumElectron.QE-22,1656(1986).[3]A.Ishida,M.Aoki,andH.Fujiyasu,J.Appl.Phys.58,1901(1985).[4]B.A.VolkovandO.A.Pankratov,JETPLett.42,178(1985).[5]V.KorenmanandH.D.Drew,Phys.Rev.B35,6446(1987).[6]D.AggasiandV.Korenman,Phys.Rev.B37,10095(1988).[7]O.A.Pankratov,Semicond.Sci.Technol.5,S204(1990).
[8]V.I.LitvinovandM.Oszwaldowski,Semicond.Sci.Technol.5,S364(1990).[9]F.Dom´ınguez-AdameandE.Maci´a,J.Phys.A:Math.Gen.22,L419(1989).[10]E.Maci´aandF.Dom´ınguez-Adame,J.Phys.A:Math.Gen.24,59(1991).[11]R.P.FeynmanandM.Gell-Mann,Phys.Rev.109,193(1958).
[12]B.Mend´ezandF.Dom´ınguez-Adame,J.Phys.A:Math.Gen.25,2065(1992).[13]B.M´endezandF.Dom!nguez-Adame,IlNuovoCimentoB107,489(1992).[14]D.Agassi,Phys.Rev.B49,10393(1994).
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