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AdvancedTextsin Physics This program of advancedtexts covers a broad spectrumof topic,swhich are of cczOt=demerghgHtereMhphysiO.EaGbookproddesacompzGensive=d introductioato a âeldat the forelont of modernresearch.As such, yet accessible Gesetexts are intendedfor serliozundernduate aadgraduatestnrlents atthe MS and PhDlevekhowever,researc,h scientjstsseeklngan introductionto particalar areas of physicswill also bemeât from thetitles in tlzis collection.
Springer Berïin Heidelberg N'ewYork HongKong London
ISBN3-540-00635-4Springer-verlag Berl'inHeidelbergNewYork whethe.rthe whole or part of the matedal Tlziswork Lssubjed to cepydght.M right.sace reservedy recitatior,breadk concttrned,swcifcatlythezigiztsof Mnslation,reprintinpreuse ef illustrationsp casting,reproductionon znicroGlmor ia any other myb aad storagein data bnnkx Duplicatioa of tlzispublicationor partstâeaeof is perrnlttedonlyunderthe prodsionsof ttte GermanCopyrightLaw of September py*65, in its mtrrent version,andpermirqionfor use mmt alwap 5e obtainedâom tmdertheGermanCopyrightLaw. Spzinger-versap Viohdonsare liable for prosecudon
BerlinHcidelberg NcwwYork Speger-vcrlag a memberof Bemelsmnnnspringer Sdenceeusiness MediaGmbH
h'inted'm Germany etc. hzt.hspublicadoadeesnot Theuse of genemldesczipdve nnmes, zegistered namesy trademnrksa implykeven in tùeabsence of a speec stateme'nttlzatsuchnaraes are exemptfromtàerelevaatproandthereforefreefor genetxd use. tectivelawsaad regalations bytiœauthorusinga SpdagerT> maco package Typesetrlng: Dataprepared leutersitausea Finallamut FmnkHeawegy GmblfyHeiddberg Cove.rdaslgnt#e.J;> +production Printed on add-fzeepaper SPD:k0855596 57/514:1.*a5 4 3 z I o
Friedhelm Bechstedt
Principles of SurfacePhysics WiG zo7 Figttres
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Springer
To Andreas, Slzsanne,and UV
M the rivers raa iato the sea; yet the sea is not h111:unto the place from whencethe rivezs come,
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Ecclezias'te (or,The Preadzea'l
Preface
'
J.nrecent decades,surfaceatd intezfacephysicshas becomeall increasiugly important subdiscipve withic the physicsof condensed matte,r as well as a'a interdisciplico feld betweenphysics, c-z:ystallography,cltemsKtzy,biology, a'admaterials science.There are severaldriving forcu for the developmentof the âeld, among them semiconductor tenBnology,new materials, epitaxy and chemical catalysis. The (alectzlcaland optical properties of nanostructmes basedon diferen.tseMconductorsaze governedby the interfacesor, at least, by the presence of ictezfaces. A microscopic kmdemtandingof the growth procœses zequiresthe investigation of the surface processes at an atomic level. Elementary processeson surfaces,such as adsorptioa ar.d desorption, play a key role in the understandingof heterogeneo)ls catiysis. During the course of the surfacehwestigations, it has bee'npossible to observe a dzomn.tic progzessi'a tile abttit.gto stad.y surfaces of materials in generi, aa.don a microscopic scale in particlzlar. There are t'wo mai'n reasons for this provess. Fcom the expem'mentalpoiat of view it is largely due to the developmentam.dawdlability of aew types of powtkrhtl microscopes.Spectaclzlar advancesiu techniques suck ms srltnns'ngtl:nmeling microscopy now allow us to observeindividual atoms on sarfaces,and to follow tlb.e,irpatlus with a clarity qlnl'rnaginablea few years ago. bnromthe theoretical point of view(orrathez the viewpoint of simulation) progress is related to the wide àvailability of computers and tEe c'lrnmn.ticinczeazeof their power. Todayj early methodological developmentssuc,has dezusityftmctional theory allow a full Trnunttlm-mechanicaltreatment of electronsia materiab. I'a the futm'e, compate,r experiments will be able to simtzlatethe beavior of surfacesaad processE'son s'prTncesat the level of individual atomic corœ and their smzotmdingelectronswith high accmacy and Temarlçable predic-tivepower. TMSenormous progress in surfacesciencehas been documentedi.c rnn.ny excellent boolcson surfacestructmw, surfacepzocesses,theoretkal modeling of surfaces,a'adsurfacesacd interfacesof particular solids Likesemiconductors. Howevea', only very few books try to treat thc subjecttn a lpnifed and compreheasiveway, This holds true in particulaz for the ex-persrnentalaud theoreticalmethodstzsedin surface physicsand, most of a11,fo< the principles an.dcoacepts. Hence7 perceivedthe acsedfor a book dealing wxith.smface physics at the level of aa advancedtextbook. The n'l= hereis to descdbethe
V1I1
Preface
ftmdaentals of the seld and to provide a lamework for the cliscussionof sttrfacephenomenain a siugletext. Examplesof pazticulazsuzface.s of materiaks suG as sczniconductoz's or metals aze only discussedas a means of illus'tzating the fundamentalsor pzinciples. Specialtheoretical or mcperimental methods of suzfacestudies are mentioned but not describedin detail. Particulaz attention ks paid to physical approachesthat c-an be applied to the discoveryand dismlssioaof nove,lsurfacephenomena. Among them are symmetzy azguments, energetics,driving forces and elementsof gcometrick chauges,elementazyexdtationsl and other charadezistic propertics. These elementsshouldlzelpto clazsifysrface pzoblemsand to facilitate their kmderstanding.The only prior knowledgeassumedis tmdewaduate phMcsand mathematics course material. Mp.imlytextbook quantum meianics and geometrical arguments az'e used to discussand desczibesarfacesand surface processes. Graduate-leveltopics such as second qxpxntization are avoideê. Whenevezmacy-body argttments are needed,a bzief (more phenomenological)introduction is Tven.Greeu'sffmctions are intzoducedby using their' relationshipto obsezvablequautities. The use of gzoap titeor.r is restricted to geometricalazgumentsand its notations. Feynmaudiagrnml aze only shown to illustrate interactions betweenpazticlœ on s:Arfnces. An Hended mzbjxt iade,xwill help s'tudeatsand scientixs to use the book for referenceand dmu iag their every-dayscieutisc wozk. To keepformulasto a mnrageablelength, thty are written in tke lamework of cgsuaits. In Hdttion, use is vnnzieof the fad that the energiesof valenceelectronsare of the order of electron volts and atomic distancesare of the order of angstiozns. The book is basedoc lectmesgiven at the Humboldt-universttâtz'u Berlin and the nieth4ch-snhlller-univemRâtJenn.and on s'tudeatsernsnars. I would like to nrlmowledgemaay discussions with colleaguesaround the world, I aksothnunk my cozeaguesazd studentsfor their critical reading of paz'ts of the manuscript. Among others 1 am iadebted to R. Del Sole, N. Esser, J. Fnrkhrniiller, S. Glutsch, P. Kratze, J. Neugebauer,G. Onida, M. Ro1)lfmg, A. Se.himdlrnnzyz, W.G. Sn%ml'dt,azzdJ.-M. Wagner. The typing of the m=uscdpt was achievedwith competencesnd inBnlte patience by my sem'etanr Sylda Hofmnnn. Coordinationand production of the book were tmdertaken by Petra 'mdber and Angeh Lahee 1om SptingerVerlag. .
Jena, Ma.rc.h2003
'
FriedhelmSec/udeff
1.1 t%lodelS =es......................................... 1.1.1 S ace V=c B14IV..........-.......m.......-... 1.1.2 The S ace ms a Physick Objed . . . 1.2 TvmpDimRnql'on/Crys .-.-.-...............-......... 1.2-1 Lattice Planes of BT7lk-Crys-tals . 1.2.2 Om'ented Slabs---..............--..............-. S'arface. Pln.nn.rPom-t Gzoups 1.2.3 Id 1.2.4 R Smfaces:Reconstruction xnd Relaxation -
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1.2.5 Superlatticesat S'urfa' ces 1.2.6 XvoodNotatl'oa 1.2.7 S etry t'Xs:l-qx-s6cat-on 1.3 Reciprocf Space 1.3-1 D1rect n.ndRem-pro Lat-tices Zones 1.3-2 zBri'llo-l'lt-n 1.3-3 Projectionof 317Onto 2D Bm-llonain Zones *pro Space :..3.4 S etzy of Points aad Lm-es in .
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*cS Thermod Te'cl 2.1 Km-etz-cProcesesnrd S aces m- Equilibrz-' 2.2 ThermodynzmlcRelatio'nqfor S aces 2,2.1 Thermody-aAmicPotentim.lA L 2.2.2 S c.e Moda-qcatz'on of Thermodynxmic Potentin.lq 2.2.3 S ace Tensiona'ndSTq'rface Stress 2.3 E -brixlm Sbnpeof Smnall als 2.3.1 à'nsqotropyof S ace Bnergy 2.2.2 AbsoluteValuesfor S ace Energie.s 2.7.3 'W:t1@ Clonntruction 2.4 S ace Elergy n.md lTorphology 2.4.1 Facett' and itoughe 2.4.2 3D Versu 2D Gro....-.......
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3O3 306 310 312 312 313
1. Symmetry
1.1
Vodel Smfaces
1.1.1 surface vez'sus Bulk
'
.
Evcry re,alsoEdis boundedby surfaces.Nonetheless,the model of ac in6n7'te solid which neglect,sthe presence of sucfees works very welt in the casq of m=y physical properties. TNe remsoa is, Astly, that one usually deals with properties, suc,has traztsport, optical, magnetic, menhxnscalor thermaz properties, to wllic.h all the atoms of the soliclcontribute more or 1- to the same extent, and, seconiy, that there are many more atoms f.n the buik of a solid samplethan at its surfRe, prokided t'he solid is of macrœcopic size. Irz the case of a srdiconcube of 1 =3, for e-x=ple, one has 5 x 1022bulk atolns and 4 x 1015surfaceatoms. The surface atozcs are only visible in smfacesensitive experimentaltecy niques or by studying propertio or processeswif.ch are determinedby sttrface atoms only. Among them ate phenomenalike crystal growth, adsorption,oxjdation, etching or catalysis. They cnmnot be describedby the mode) of aa l'nflnlte solid. However, there aze also eseds wlf.c,hare detnrminedby the For interplay of bu.lk and surface (or,more stridly spexlaang)the hterface). i'astance, the clmnne-lof the cnrrler transport in âeld-efect traaetors is dcutermnenedby the slxrfnœ (interface) staœ ws well ms tb.e bulk dopbzg.H one appliedthe of tlze ftrst theoretical approaees to the feld esect,Bardeen (1.1) prtamiseof chargeneutrality at the sT:vhces/intcrfaces. TMs conditiön meaus that in thermal equilibzium the surfacebard bending adjustsin such a way that the net chargein s'nrfnne states is balancedby a space nbxrge below tahe suxfve of the semlconductorfomningthe maH paz't of 'the electzicaldevice. 1.1.2 The Sttrface
as a
Physical
Object
Under normAl conditions,i.e., atmosphericpressurean.droom temperatuke, the zeal smfaceSf a solid is far removedfrom the ideal sylems desirablein physical Gvestigations. A freshly prepared smface of a material Ls normally of vezy readive toward atoms a'ad.moleculesîn the enviro=ent. M kn'mds particle adsorptioa from strong chnmt'sorptionto weak physisorption give rise to an adlayeron the topmist atomic layers of the solid. Oneevxmple -
-
:. Symmetry
is the l'rnrnediateformation of an extremely thin otde layer o:a a frebhzy deaved silicon crystal. Usually the chemicazcomposition aud the geometrëcal strudm-e of such a contmsnation adlayerare not well defned. A.saz objectof physical izwestigations a welydeMed soace kas to 5e preparedon a particular soEd,in a special preparâtion procces, under wem deG'ned extez'nalconditions.Such a solid colzldbe a czysinlli'ne material, a singlecrystal or a mystplline filrn depositedby epitax'y in a well-controlled of suG a crystnlline sygtemmight alsobe prepared my. A rathe,rcleaaslxrfar..ta as an electrodesurfazein an elec-trrïz-ynn,l'ci cell, or a somiconductorsnHnce iu a reactor where vapor phase epîtaxy (V.PB) Ls performed at s'tandard prvttre conditions and at elevated temperature. Howeverzthe processesof the tmderlyingmethodsand the results are l'athe.r comple,xacd dsocult to characterize.The simplestways to prepare a sozd surfaceshotlld happea in altrœgh vacuum (UHV1, i.e.) at ambient pressurelower than 10-S Pa (about 10-10 to=l. There are essentiallythree ways to mauufacture cleaa suzfaces tmder UHV conditiozzs: .
-
i. Cleamgeof brittle materials in UHV. Of course, only snlrênreswhicz are cleavageplanes of the czystal nnn be made in this way. ii. Treatmeat of imperfect and contnmsnatedsurface of arbitrary orieatation by ion bombardment and thermal n.nnenling (YA), generallyin severalcycles.There are no lsrnstationsio certain materials and to certain tallographicorientations. 1. Epitaxial growth of crystal layers (oroverlayers) by mexns of emporation or molecular beam epitax'y (MRE). '
Obvioasly,a smooth and cleansurfacenn.nnot be realized i.c the ideal form, bui rathe,r oxulyto some approxsmation. My real smface wi:tl exhibit irregl'lnr deviations from perfed smoothnessand pttrity despitetEe care taken Snits preparation. A.zl illustration of such a surface is given in Fig. 1.1. J.n reality a surface conskstsof a nltmbez of irregular portions of parallel sum facelattice planes whic,hare displacedvertinn.lly by one or more lattice plaae sepratlons 'VM respec't to ev% other. Atomic steps TERRACE
xjxx
occlzr
at the botmd-
STEP
ADATOM
STEP-ADATOM
VACANGY
Jllustratîon of structural imperfeotiomsof crystal sudaces. Atoms aï.d thekrelcctronshel)sare indicated by little cubu.
Fip. 1.1.
1.2 Two-DimensionatCzystals
3
ade'sof thesc httice-plane portions whic,hin this coatext are.calledterraa. The steps may exhibit lrsnk-q.1.naddition to teztaces, steps and ksnkq, other stmmtmal irregularities may ocmzr wiziclzcaa be substtmedunder the temn isurface roughness'.Adatoms aud vacancies bdong to tMs categozy, as do complexœof thœe m'rnpleddects. ln the case of slzrfnrzxsof compotmdezystals qaite often atoms of one of the contributiag elementsare dcpleted more thn.n those of the other which ruults in an emrinbrnentof the latte,r and in a noa-stoiYometzy at the sllrfnre. The most signiâcaut fo= of chemical disttlrbauceof surfaces)whic,happliesboth to compotmdaad elementalcnrstals, is the contaminxtioa by impuritics or adatoms of anothes spœies. The impurity atoms or adatoms may be situated at regular or nomegular sites of the surfacelattice plane: at locations above and slightly below'it.
1.2 Two-Dimensional
'
Cl-ys'tnlA
A completeGaradezization of a solid surfacerequires Howledgeof not only atoms of Ewhatspecies'aze present but lwhere' they are. Just as in the bttk it Lsnot that the atoec coordinatesas suchare of pucb direct interest. Rather, besidesthe clmmicalnatme of the atoms their geometricalarrangememtgoverns tke electronic; magnetichoptical, and otNe.rpropeztiesof surfaees. 1.2.1 Lattice Plnnes of BI:11zCrystals
A geometrical constzucdonwhich Lsof particular signifcxnce in describiug czystal surfacesis that of a lattia pltme. Ipatticeplan.e,sare uslpnl'y deuoted by MCJerindicos (W) where h, k, î aie the iutege,rredpzoca)axis intezvals given by the i'atersectiomsof the lattice planeswith the three crjrstallographic axes. They have a simple men.nl'ng izzthe czaseof zectangalarcrystal systems, for example,denoteslattice plaau e.g., the cubk system. The smbol (100), perpendicularto the cubic a-axis, (111) means lattice planespezzlealc'tictllnr to the body diagonaliu the ftrst odant of the cubic unit cell, aud (110) denotes the lattice plan.e.sperpendiclzlar to the face diagonal izl the fzst (padran.t of the o-plane of the cubic lmii cell. Ustlnlly, the collection of suchplanœ t'hat are eqttivale'nt by symmetry is labelH (à,k!J. 'lnhlzs(100J ld>nds for the collection(100), aad (OOï), if these planes are (010),(001) (1.00)r (010)) equivalent. The bar aotatlon ï ldicates the correspondiugnegstive coeëciext. J.nthe case of trigonal and hexagonallatiic%, four crys-tazlographic &es are conside-red, tlzree bxsteadof two perpendicular to the c-A='q. The lattiœ instcad of three- The &st plres are then cltazacterizedby fom indices (h,kïI) three, However,are not independentof eackother. Iu fact h + k + ï = 0. The fourth n='q (correspondbzg to t;e iude,x2)is perpenclicular to the he-xagonat aze sometimestermed Bvavaisï'rlJces. basalplatae.The Lhkélj A particular geometzical plane 'can also be characterized by its normal diredion
ê. Symmetr,g
4 n
.;V1v5,1.
=
J.nthe ca-se of latticeplanesit is convenien.tto relateit to a lincar combination 1
'jg gez+ k%+ !;a)
6, =
(1.1)
of the primitive vectors b.f (/ = 1, 2, 3) of the reciprocal httice with the iuteger coescients h., k, a'n.dJ. The vectoz's bj are directly related to the prlrnl'tive lattice vectors aé (ï = 1, 2, 3)by the relation (zç . '
.
bj
=
zmAj.
(1.2)
Apart Fom the cmxe of priml'tive Bravaislattices) they are Hq'Ferentfrom the crystallograpilic axes. M:yway, a lattiee plane can be charaderized by the Miller indices(i,kJ) anGhence,a normal parallc) to the vedor Ghkt = hbï + k%+ Iba of the reciprocallattice. Eowever,as a con' ! v- I
/
.f. q
%
& N &
+
> ., *
v
#.
r= :N1
-
v.
IW1
'f
.,.
i
-10E1 (:1
-
D1n:l
suzfaceof diamond1.15. Two diferent 2x1 recoctructions of the (111) stmcttlre crystaks.(a) ideal mtdnne; (b) 2xl recons-tnzcted=z#v..e due to rthxl'u
Fk.
formationiand (c)2x1 reconstrudH mtrfnne due to m'rfnz,e atoms. Dots: nominal fust-hyer atozns; clple
an heqtlivalent bunlrMngof nomloxl second-layer atoms.
certai'a recorsstruction denoted by an oression of the type ca,t be realizedby rls/erent atomic coegurations. This is dezronstrated (1.14) t'a Fip 1.15 for diferent arrangements of fzrst- and second-layeratoms oî a surface.The easiestway to reconstmzc'tthe ideal diamond-likematerial (111) The daaatoms (Fig. s':rixz!e (Fig. is by bunkling the frrst-laye,r 1.15c). 1.15a) ne accompanyingdeerent becomeineqzzivalent. gling bondsparaDelto (11lJ ElJJ.Mwith electronssupports the tendencyto an inert surface.However,the bonds can alqo be rebondedif they occur at atoms which are 62stThe sarfaceatoms may be the bulk point of view). zteaœestneighbors(from a possibleexamlying nex't to eac,hothe,r (see arrangedin the form of c-lznsrtq A11atoms of a càain are coupledtogethe,r by 'r-bonds of p1eia Fig. 1.15b). paralle:danglingorbitals. The gcceration of daqglingbondsat the formerly second-layeratoms icclude zemarlcablechaazgesof the bondingtopoloa beaeath.The atoms in ballt custals of the diamoad structme aze bonded in however,fvefold azd sevenfold sidbld zings.In a rr-bondedc'hainmodel I1.9J, Hngs are formed. Fourth, the Wood noiation e>n also be used for sllrfxvte overlayerstmlc1-%% dae to adsorbatœ. A periodk arrangement of adatoms or adsorbed c'tureawhich caa be classife accordmolemtlesA.1qngivœ rise to a su a ttRmn-pA to the Wood Eowever,one usqlnlly NICIS ing to exprëssion (1.14). notation. The chemical stoichiometzyof the atomic or molecular overlayer is given by X, and T is the nllnnberof adspedesi.n tba overlaye.runit cell. susrface at a h'actional For example, CO adsorbedmolecularlyon the Ni(100) surfacecoveage of one 'Axlf formqan overlayershown by the dots in Fig. 1,13. in this case the suzfacedeaotation bcxcomes According to oression (1.14),
'PMrd,
a
'
or Ni(10O)(UfxVl)A45O-CO. Ni(1ûO)c(2x2)-CO
1.2 l'wo-DimensionalCrystals
25
1.2.7 Symmetry Classecation
In the commensurate case all the reconstractedsltrrn.ca with the tmderlying bttlk halfspacespossessa translational symmetzy nyxracterized by the four Bramis classE'swith group elements (slA). The corresponding 2D Bravais Fig. :.8), lattice tramsformsaccording to a certain planar point group (sce the so-calledholohedral pott groap, with elememts(aI0J,The 2D crystal, the atoms in the surfaceand the bttlk below, transform according to a suN group?its poict group. 'I'he combination of tNe translatiomal and point voup smmetrie,s gives the spRe group. There are 17 plnmxrspacegroups. Latliceg with a corr-onding spnmetry are shown in Fig.1.16. It is evidemtthat fur.h of the 10'point p'otlps oi equivckleatdirectior.s combiuedwith the conuponding aodated lattice gives rise dizedly t,o a so-enlled symmorphic space group with ele-meats(tx(A).The spee groups p1, p2l1, J1m1, pzrnm, p4, fmrn., p3, p3r11, #, and fmm orioate in tids mn.nner. Since tke point groups of the rectangular crystal s'ys-temare eachassociatcdwitk t'wo Bravais lattsces,prlmltive or ceatered,we 6nd two f:IVGe.r spae,evoups, elrp,l sad czmm. 1xïthe ctkse of the point group 3,m, there exist t'wo dslerent possibilitiœ of positioning t'wo reNedionlinesrelative to the hengonal lattice vectors, elther through the vertice of the equilateral hexagon of the Wigner-seitz cell as assumedin the case of p3mI, or such that they bisect its edges.J.cthe latte,r c%e oae has, as the thirteenth s'pace group, the voup p31m. The point grou.p rfAmn.l'nR uncha'agedif in its space group a glide refecvtionliue Lssubstituted foz a.c ordinary re:ection liae. One musi thereforee-xarnine the 13 spacegroupsalzeadyestablishedto determine wheGe.rthe mzbstitutioa of a reNectionline r?z by a glide reflec-tioali'ae g (i.e., with a trnrmlaf-ion'v by half of thE shortet a zefectioa irt rrù in conjtmdion lattëce vector parallel to m) 1(.)* to a new space group. Oae easily 61.1.* that th.is is not the cmse for the hexagonalcrys'tal system. Ia the quadkatic crys'tal sys'te,mit is possibleto s'ubstitaztea s'yste,mof glide reEectioa liaes for one but rot b0th. of the non-eqttivalezttrefedion lin.e systems. Tizis yields the additional space goup pngrrt. The rematniag space groups .p1.gl (lom acd pzmn, p2.:.: (lomp2mrzlocctzr in 213crystals with a prlmstive .y1m1) Bravakslatuce. They contain elements of taheform (aIA+'rJwith :rectangulaz :i- ks a Factional latdc,e translation. The ceateredrectnmotlxr aad oblique .'iotal systems do not =ve rkse to additional space voum. Cocmuently, ''four of the 17 2D space groups hwolve glide rezections,i.e., they m'e aon-
'.b#zdmorphic groups. ',
Ia Table 1.3 we sllrnrnnrize the symmetr.g classïcation of 2D crystals. ;',ïVJnternational notation is used.Despitethe fact that the atomic basksis parallel to tite negative sttrface normk direction, we use a plane Elékvnded '' è'ç8Angularcoordsnxte system with uzlit vectors ex, ev. The ozigin of the co:?. 'érdihatesys'temLspositionedon the rotation x='q if one ests. rne prlmitive ', .bzsisvectoz'sof the 2D Bravais lattice are az, Q. The secondbar inclicat'
Egg. 7..16. The 17 phnxr space groups representedms parts of lat-tice,ssatisfying the synzmetriesof thosespacevoups. Tîe sma:tldotsare at the corne> of the unit cells, or at their midpoints, for reference.Onelarge dot is positioned at a.c arbitrary lecation within the tmit cell, and khe other large dots are obuined non-synjmetrical 1om thts one by appl>g a:llthe relevantsyrnrnetry operatioc. ARe.r(1.7).
l.2 Twceimensional CUS'i?IS
27
Table 1.3. Symmetzyclassifcation of 'twmdîmensionalcrystals (1.2, 1.10j. Crystal Bravais system
class
Point
Space Symmetry
group
group
elements
pl pall
(cLA) Le.a), (4Ix) (s &, tznogxl
oblique DobDque 1 2 c rec.t-
p-rect=rllpr
r?z
plrzl
p1J1 2mm
pzmm
pz''n'l,p
(s1A),tznvr'r+Al,= = Ye. (cIx), (JIIAJ,(NIAJ, (m.1AJ (4IAJ,'(mxl'r+ A)','tmvrr+ A), (c1AJ, v. = 1e. (FIAJ,(JilA), frrul'r+ AJ,(mvl'r+ aJ, m .ve.+ ve. (sIR),('maIA) ..
ing the reconstmzction ksdropped ic the followiqg. The oorrespondicgplaae lattice constaats aze tzz acd ca. The generalclassiîcationof 2D cz-ys-tals izzTable 1.3caa be usedto elhn.racterize the symmetry of mlrface of 1.ea1 Fxxamples are given in Table 1.4. Tllis table iudicates the space groups of low-index surfacesof tgpkal s/micondactozscrystnl7n'zingin diamond?zinc-blende,or wurtzite structuzes. Ideal and reconstructedmlrface are consideredand the reslzlting groups are discussedfor diferent nlzmbezsof atomic layers below the uppermost one. Table 1.4 clealy showsthat the reskfting spacegroup dependson the czystal .
1.3 ReciprocalSpace
29
orientation, the nlzmbe,rof atomib laycs takeninto accotmt, and the model of reconstntction. That means, the space voup of a 217Orstem depends on all of the above-mentionH details.
Space
1.3 Reciprocal
1.3.1 Direct and Reciprocal
Lattices
A two-dîmensionalsolid smface is chazacterizedby a 2D Bra,vaishttice (cf. Table 1.3)'ivith prirnitive basis vedors 41 and Ga. We use the vectors with only one bar independent of whether or not a reconstruction ks pre-sOt. Ia coorrlirate system with l:mit vectors termq of a rectangtzlarpla'aa.rCartesslm cx an.dev, the bmsisvectozs read as J,z = Azze. + Azzew, 5,2 = Azzew+
(1,16)
Jbzev.
The determsmantdet .j. of the 2x2 matrh
./i
-4.1:Jkz .,121.d%2
=
givesthe area W of the nlanl'tcell of the Bravais lattice. 1.nfact A = 'n' (&1x&z) = det aiwith n as the surfacenormal. A correspondsngreciprocalIctâcc in Fourier space is mssociatecl with the Bravaislattice in real space.TEe redprocatlattice, as we shallsee throughout the book, is ex-tremelyuseful an.d pertinent in all dlfrraction methods, i.u pwrticularin tlle case of low-energ,geledron doaction (EEED). h.sin threedimezssionalspace, the prt'mstive bmsisvectors 5zacd bl of the 2D reciprocal lattlceare desnedaccordîngto the orthogonalityrelatior ,
% 8J= With '
'zz as
(ï,#
2rctjk
-
=
1, 2)
(1.18)
.
the Alnit vector normal to the surface, solutions of the relatioa
are (1.18) -.
bt
27
=
J.a x .p, j
Ië1 :,21 x
%
=
2x
n x ,
a1
l41) 4
p4mm p4mg
4
Qnm
4=
,4
m
Az
J
2r=
2mm
Z
m
m
X'
4mm
4rzr4
m
m
or
Ltae
X
.D Irzedudble part of BZ
-
44
:. Symmetry
Tbble 1.6.
(conthued)
e)He-mgortallattice Srmmetrz
Spacegroup
.
points or lines ,3
p31m
p3m1 .3% fmm
JN
3m
3m
3
6
6mm
zs',!;œ
m
m
4z-/'
3m
3a
ZZ'
m
m
##'
m
2mm
AA'
-
-
z?s
-
m
Heduc3ble pat't of BZ
The Heducible repruentations of these point voups are gvenin (1.10) for the vaziousspacegroups of a, corresponds'ngBravais httice. TEe dimensionof the Heducible representation deterrnsnesthe degeneraeyof an eigenvalueat a given:. For the systtamKtmder considerationonly Heduciblereprœeutations with fîsmensîons equal to 1 or 2 will appear.
2. Thermodynamics
Processes and Surfaces in Equilibrium
2.1 Kinetic
The real surfaceof a solid tmder atmosphericpressureis ver.v diFerent hom alt ideal system desirable in surface physics.Therefore, surfus as objKts of physical studles are usually prepared in UlIV. Besidescluavagea'ad the combination of jon bombardment and n.nn-b'ng, acother chssical method to prepare a fresh, clean surfaceis thc euporation and condensationof an overlayer on a substrate, e.g., within a:ct MBB procedure. 'ne resulting s'urface system, more predsely tke solid-vacu7nminterface, is no static system. SeveralMnetic processesocmtr to a,cez'tainextent dependingoa the mzbstrate temperatm'e T.
A seledion of such processes is iudicated scornxticaxy in Fig. 2.1. Tltey represent elemeatary events whic.hhappen daring epiteal growth, e.g., NIBE. Hbwcver,such processesalso oc= at the substrate-vacuttm interorl a terrhaoe (a)deposition(aisorptson)
(b)depositionon an island
.
q
'
(c)
on on a
ce
.q
. .
(d)daorption
.
.
(e)cucleation
.
. .
. ..
.
,.
w
...
.
.
! .
.
.
r
.
.
'
I .
.
.
.-.
.
.
.
.
. .
'
'
'
'.
.(..
.
'..
'
-
. .
.
.
z'
vh'
.5 J ' ..'., ., . .
..
... .-.
..
' .
.. ...
..,
'
'
,
.
..
.
.. . .
'
,
..
.
. .
. .
'''F
'''
'
.
. .
..
.
.
'
7 .::4
(exchange) ($knterdlfrusioc (g)atlcmont
at an island
(h)de/chment from an is:and (i)attachmentat a step F5g. 2.1. Schfemxtic representationof R'nziame-ntal atomic procassesoccurriztgduring epitaxizl vowth.
46
2. Thnrmodynamics
face,though with
reducedprobability due to the lower tamperattlre aud the smxller number of atoms ia tike gas surrotmdiug the s:arlhce.'Ia any case, the vacaum Lsnot perfect, rather a reatgmsoccms. The atoms and molecula in the restgas interRt with the Rmrfaem.Adxrption and desorptionof atoms are observed.Bœidesthe temperatme and t'hc cknmicalnattzre of tlm atoms, the strength of 1%- proces- also depemdson the atomic sites on terrazes, happ= on or in the surfaze, e.g., at islandsor near steps. Other proca
dsfFnqionon terraces or along stem and jnterdlelnion due to exchangereo tions. Onepossibleconxqueace LsnucleatiomAaotheaone is the atttement at stem or isLandsgiviag rise to a layer and/or isl=d growth. The elementary ldnetic processesare also infuealced by the sudaze morpkologyard geometrya,s the presenceof tzenehesand stepswhich detmmnlne the local suzfaceemergy.The spatial vadation of the surfaceenergy deoes the adsorptionsites ard the difhcqion barriers. The clhn.razteristicbondhlg sites enea.a of an atom at a specialsite a'adthe enerr barriers betweea.such the probabity for adsomtion, desorptiou and difhlqson of atoms detèrrnsne at a given temperatuze. In the thmrrnodpmrnic equilibrittm al1surfaceprocesserproceedin t'wo opposîte diredions at equal rates accoreg to the Sprincipleof detailedbala'ace'. Detailed balance means that the rate constacts for the forward direction, vç, a'nd.tke back-wa'rddirection, /b, of a process satLsfythe rehtion where AE is the energy dsFerence betweenthe vtjrs = exp(-1S/kBT), iaitial an.dftnal states. Processessuch as adsorption and desorption, decay a'adformation of 11a41*,etc-, must obey the detailed balance.The equali'ty of forward azd baclcwazdrates in equilibrium is incompatible with the net epitaxx growth of a surfaceoverlayer. Crystal growth is clerly relate-d to non-equilibrium ldnetic processes.Yet the 'pzincipleof detiled balance'is still 'h11671ed. However,closiagthe shutteo of the emlsion cells, Le.t intenmptholding the system at not too high ir.g the molecularor atomic beamszua'nd temperatuzes, a ratb.ers-tatic sttrfac'e'caa be prepared.,which doesnot change the principal strudme a,s is indicatcd, e.g.: by the conselwation of a cez'tatu LEED pattern. Such a surface can be consideredto be in timmnodynnmt'c equilibri'tzm with the substrate and the surroundings, e.g., the restgas.
2.2 Thermodynnmic 2.2.1 Thermodyzmmic
Relations for Surfaces Potentials
The eqttilibxium state of a one-component system consisting of N pxriucle at a %cd temperatmv T and pressurep is the one wit,h the mn'nimum Gibbs 1ee enthalpy G(Ttp,N) (2.1), G = F + p:r
whereFLT,Jr A'),
(2.1)
2.2 ThermodynxmscRelations for Sttrfacœ
F
U
=
-
ST,
47
(2.2)
is the Eelmboltz Fee enera. It is related to the intelmal energy U = tJ(S,J,rN) and the entropy S by a Legendretraasformation. The energy conservatioalaw atd the relatioizshipbetweenheat and work n>.n be written ia the fo= KU = TdS
pdF + JzdN
-
(2.3)
for azt kTIIIiteSiIIIAIchangeof the internal energy. Variation of the numbe,rof pmicles N Lsalloweddue to pazticleKxchaagewith a reservoir characterized by the cEemicalpotentiat Jz of the particle. For an isolated system with no
heat Gchange(0 is
=
0)a'n.dparticle Gchange(dN= 0)the intcrnal energy tdy
constant at cozkstant volume
0).The corresponeg
micascopic distribution is the micronltnonicaleasembleof statistical met+n.nics. a
=
The thermodynmp'c potential G (orF) can be used to deriw the thew modynnmlcquantities of the consideredsystem at consta'at temperatm'e T, paztide aumber N and pressur'e p (orvolt'tme F). Tn6mitemnl changesof the three vadablesgive rl'- to l'n6nitesimal s of tlze poteatial, so that dG = -S&1' + V'd.p+ MI:'L?V or
(lF
=
-5'dT
-
pd7 + Jl KN.
(2.5)
1xlequillbriAlrn, F is a ml'nimllm wtth respec'tto the Her variablesat constant T, 7, and N, whcreasG is a nninirnllm at constant T', p, aud N. The correspondingmicroscopic distribution is the rltnonicazensemble.The Gemical potentia,lJzic (2.3), (2.4),or (2.5)is given by .-
(w'&)(w'') (N,x) -
S,V
-
:r'I5J
.
r,a
Under normal pzessm'eof abou.t 1 atmosphere,the dllerence betweeathe Helmholtz 1ee energy F and the Gibbs free energy G, J?= F
-
G = -pF)
(2.7)
is insîgniftcan.tfor a bnlllcsoDdor liquid. TMs holds'i.apazticular for volllrneinduced changcs-pdF. 'nus, it is mlmcient to 'ttse F for most cases in solid state physics.Thc difrerence (2.7) is Kramer'sgzandpotential J2= .O(T, F)Jz) Despiteits smxl:ness, more preciselyits vaaishingMuence on changes (2.22. irt bulk systems includizlgphmse trnnsitions, the potettial is convemiOtto use for system transformxtions that occur at a constrt temperature T9 vob:rne F, and chemicalpotential y. This may be of pazticular iuterest for tke surface
region of the estems under consideration.Togetherwith tKe Gibbs-Duhem
48
2, Thermodyaamics
eqaation, SdT
-
Vdp+Nd;&
dfz = -SdT -Jd7-
=
0, inflnitesimal nhxnge of the variablœ result
Ndp.
(2.8)
In this cmse the thermodrmrnic properties of a syswte,m are governedby the graadcanonicalstatistical operator. Comparisonof (2.5)a'ad(2.8)indicates a tqansforrnlttion law J) t= .F' /zN.
(2.9)
-
This is a consequeaceof the fact that the Gibbs free enthalpy varies lin-rly wit,h the zplmbe.rof particles (cf.(2.7)and (2.$), G = P'N vrith the proportionality factor /.: =
(2.10) for eaG homogeneous phaze(2.1j. gLT,p)
2.2.2 Sarface Modmcation of Thermodynnmlc
Potentials
A stlrfnne of az'e,aA infaencesnot oaly the spectroscopicproperties of a solid but also its thermodynnaicpzoperties. Tn order to discassthe Muence of a h'eesllzûce of a hnlfqpacc, we follow Gibbslideaof the dteqllsrnolarl dividing smfaceh(2.2,2.3). This is illustrated in Fig.2.2 in terms of tke particle demsitjr n = NjV as a fnnction of the distaacenormnl to the sltrfnce. It es gradually from its solid to its vapor vaâue.In Fig. 2.2 ihe vertical lines incticatea partition of khe total space into a bulk solid vobnrneVi, a bulk mpor vol=e 'Ui,an.da volame W of the trxnRition region,the surface.The correspondkg nz and 'pu characteriz,e dwlnm'tjes the llnlform bulk phases,the l-nal'-in6nl'tel solid and its vapor with whic,hthe solid coexists azd which occupiesthe othe,r halfspace.The sllrfnre region, whose spatial eoctentis of atomic dimeasions is thus a strongly tnhomogeneousregion sulu 10 or less atomicylayers), (about rotmdedby t'wo homogeneous phases,the solid and the vapor. Unforttmately, the partition in Fig. 2.2 is aot uitîque, siace the numbe,rof particles in eac,h
n g. 2 v c . Parucle density'of a onmcomponent system near a
surfacearotmd z
=
0.
2.2 ThermodynamicRelatiomsfor Surfaces
phase 1 or 2 dependson thc n=ber
of particles Ns in the s'arfacetransition
regitm.The same holdsfoTthe voblmes.TMSuncertatnty is of the same order of martude as the surface eFectitself. Howeve-rjiu the framework of a the partition is madeunique macroscopictheozy kere the titrqrrnodyztpml'œ, by applying the n.ataralconditions (2.1) N
ï,l = J4.+ Vi, 'nzh + nzFz.
=
(2.11)
J.ncompnriKonto the total nllmber of particles the nllrnber of partëclesNs in the surfaceregion is assumed to be negligibie. In thc macroscopic limjt one has N :2:0. The extensive thermodyzmrnic potentl'n.lq tmde.r consideratîon,the free energy F azzdthe freeenthalpy % can be written as contributioztslom phases 1 and 2 plus a surfaceterm. The ozigia of the surfacet-rrn rltn be discussed 2.2,2.41We follow tke derivation of tssingat least t'wo eqaivalemtvîews (2.1, a'adSmmjaard as we)l as that of Desjonqubes 12.2J Landau aad Twifshitz(2.14 and considerKramer's g'razd potential. Sinceiu thlrrnodpmmic equslibrillm the pressare p is the same in the tvo homogeneousbulk phazes,at least for eqaatioa (2.7)can thus be written in the fonn a plaue surface g2.l), '
.
-p(M.+ vi) +
J) =
fls.
The surfacecontTibutionJ?sto the grand potGtial should be proportionalto the surfacearea A
q
(
'
wîth y
=
(2.13)
'yA
ms the'surfacee-xce,ssdensity of
.0.
2.2;3 Sttrface Tension a'ad Smface Stress E
czcexss be identfed as the s'arjace cc energymr zlrzif area or s'udace .freee'rternyfor short (butimprecisely) Lh-' $bra oae-componentsystem the chemîcalpotential p is equi in b0th phases Witlz ihe total nttrnbe,zof particles accordiug tn thermodynxmlc eql'phbrinpnn. From our approximate description is st2l A11611ed.. G = yN (2.40) to (2.11), of the tclividiug surface' with Ns = 0, one readily obtaius Gs = O for the contribution. Tlms, M'itll (2.13) : sizrface
Tlle proporttonality factor ''f in
can (2.13)
.
:
I
Fs = f)s = JX
(2.14)
fensftm, holds. The surfacee'xcess h'eeenezr 'y is sometsmescalledthe snrj'aco althoughthis term is somewhatconhlsing despitethe corresponding common t'nst of measmement. Using the Glbbs-Duhemrehtion it cau be shownthat all thermodylmrnîcquactities of a s'arfacecan be expressedjn termK of J. The
2. Thermodynnm''cs
50
of ''f also allows the de6nition of the s'srfqce temperatme depezdence e'rcas,s entropy Ss (2.14,
'
'
Ss =
0J2s
-
OT
=
,z1,/$
-.A
X
&T
.
Aj/z
.
The surface9ee ene'rgy ,(2.14) contnalnstwo physically diFerent contributioas. Au l'n6mitesimalchangeis given by dl's = y ILA+ A d'y. The flrst term y dx represents the reversiblework doneto cllange the surface =e,a. by dA. This cllangemay happen by incremsingthe n'lmber of atoms ic 'ihesurfaceat a Sxed averagedarea per surfaceatom. lt ksthereforerelated to an ideally plastic deformation. Ic contrast, the srcondcontribution A d'y is related to a,n ideally e'lastic defo=atiom The zu:rnberof suzfaceatoras remnsnq constant but their interatomic distancesvazy. This is accompanied by a vadation of the =e,a, per atom. Sach a stretched surfacehas a modifed surfaceenergy y-j-k, whie,hleadsto azl additional contribution to the chaage of Fs The c'hacgeof the smfacearea may be interpreted in a rnnnroscopicsense as the consequenceof a (biaxial) strain ,ac'tjng on the smface atoms. With the components%j of the adual strain tensor azd assllrnl'aglinear elasticity the work done i'a the surface region can be descibed as the c,hangeof the elastic enerr .
dFs = X Voudqj
(2.17)
.iIJ'
wlth (7vJas the componentsof the s'ur.fatte siress tensor (2.54. They are deoed as the spatiatly averaged deviations o'u = J dzko(z) cz?j) of the local stress (qjtz)from the stress âeld deepin the btlllc ct. Accortiimgly,considera plane norrnnl to the sudacearld labe,l the norrnxl to the plaue as the dimction j, Ju îs the force pe,r Ilnnit length whicll the atoms exez't across the line of intersection of the plnnaewith the smface iu the Ftk. dtrection.The dements of the strain ttmsor qJ are dqflned in direct analogywith the corruponding bllllc quantities. With d.â = A E: dqç, comparison of expre-ssions(2.16) a'ad -
leads to (2.17) *3i = '/V +
(2.18)
.
'
witk %g =
ru
,w
.
tlevLt
(2.19)
TMS Lsthe venerable Shuttlewoz'th equation (2.6). Sinze in a liquid there is no resistaace to plutic deformn.tion,the second termq in (2.16) and (2.18) vanish aud the surfacee'nergyand sudacestress becomeequ.at(2.5).
2.3 Eqailibdum Shapeof SmallCzystals
51
It1 the case of surfacœof czystalsthis is usually not the case. The deviatiozusvij play an important role for the reconstruction or relaxation of metaz smfaces (2.74. The stabitity of the 7x7 reconstruction of snrniconductorsurfacessuchas Si(111) sçemsto be a consequenceof stressesin cllfbrent atomic tensor '3'dependson the surface symmetry. If a layers (2.$. The secozd-rn.nk sllr'hce has tikreefoldor lzigherrotational s-immetz'y, then the tensor becomes is So the smfacestress (2.18) diagonz with equal components, no' = 'ré%. isotropic in a surfaceplane. Anlsotropic smfacestressesmay haveau Muence on the step geneaatioa on surfaces.For instance, the 2x1 spnmetrsc tasy'mhas only ml'=or symmetry with the metric)dimer reconstmtdion on Si(10O) point group 2mm (m)(cf.Tables 1.3'azd 1.4)=dl Nence)the surface stress with a height of one atomic teasor is anisotropic. Single-hyers'teps(steps layer)may occur witlz dlserent orientations with respect to the Hsmers.The o:a dsferent tezraces formatioa of b0th 2x 1 and 1x2 domaias (cf.Fîg. 1.14) is likely, since tEe sarface becomesisotropic 9om a macroscopic point of view. For semiconductorssmface stresses have beea direcvtlymeasured (2.9) or calculated(2.10-2.12). 'A liquid Lschaœaderized by a vauishingresistanceto a îow of atoms 9om the bulk to the surface,a'ad'dce versa. For solid surfacesone may therefore coasiderthc tenmr '? as a driving therrnodpmmscforce to move atoms h'om the balk iato tlze stu'facelayez. The signs of the diagonal elements deterrnsne the strain stste of a surface.When 'lnrê> 0, then the smfaceteads to accw mulate more atoms. When Trê < 0, the opposite tendency sho'aldoccur. As a consequcncethe surfaceatoms prefe,rsmaller or lrger lattice spaciagsthan in the bulk. This arzangementis accompaied by compressiveor tensile stzrtinq. In the cmse Tr'? < 0, a'aothe,rcozzsequencecokfd be a (srnzlletr.r-brenb'ng) smface reconstruction of the type schematicmllyiudicated in Fig. 1.9b. Stuw facerearrangements like atomic dislocations and elastic buckling of a sulface colzld also be possible.ln a metal surfaceatoms might be placed in unfavorablebonding posltions with respect to tàe next laye,rof the substrate. shouldbe large enoughto compe=ate for this enerr expense. Thereby,1Trê1 The important role of tke Hi/ereace be>een sllrfnzte'energyand surfaces'tress for the actual smfacestracture is generallyaccepted.Howeve'r,there is a contmveray about this dl'ference az a drivi'ag force for the recoas-truction of at surface (2.11, 2.13,2.14J. leazt metal surfaces,e.g., of the Aa(110)
2.3 Equilibrium 2.3.1 Anisotropy
Shape of Small Crystals of Stu-face Enerr
The smface free e'nerg.rper llnit area 'y of a cez'taincrystal surfacevaries with or the its czystallograpkicorientation characterizedby the surfacepla'aeLhkLt izl general.A plot of tMs energy surfacenormal zz, i.e., 'y = 'y(h,kJ) or y = 7('n,) plot (2.15!, plays azl importa'at role in versus orientation, a so-called W'IzlF
2. Tkermodynmics
52
Un! ro1(1
(ln)
Ez-g.2-3- A
ce,
whic.h -:s slightly
mus-om-ented Fom
the
(01)s
ace.
the theory of the eqllsFlbri:'mshape of crystals and morphological stabillt.y a two-dimensional soltd To illustrate this orientation dependence, (2.16,2.174. 1om the gO1) direction is shown in Fig.2.3 for a squaze which is mirw-a'gcted surface(1>> 1)represents a lattice with lattice comstant a. Its nomsnal (1n,) 'dcsnal (01)stzrface.It consists of a Mgh n'lmber of (01)ten'aces separated by atomic steps of height c. With f 'w A as 'kke azlgleof orientation of (1p1 orientation, the step density is given ms tan ela.If Js is the agnsnqtthe (01) emergyper step aud g(0)is the energy of a (01)face,the mndhce energyof a (1n)surfaceis
'y(8) =
,
cos
9
g(0)+
taa 9
ps
(2.20)
.
c
The prefador cos 0 g'uarantee that the relative amount of the (01)terraces to the total surface=e,3, reduceswith hcrewsing azzgleJ. The htcaction of steps IhRAbeeaneglectedin (2.20). Asmlrnsngthat tbe s'tep mode: also holds for large.rangles8, tlze reslzltîng siu 81is plotted 'm Fig. 2.4 for angles # in h'ndion .,7/) = g(0) Icos :1+ J.r-I the fokzrquadzantsof the plane. Suc,ha polar plot of àhesurfaceenerr cac the above be dz'awnfor 0e,1:place 'wit%x'na 6n5te crystal. Even msmlma'ng trivial dependence of 'y on 3, it gives a WG plot of the surfaceenerg.g(2.15J. Y z
Z
z'
> > - e
.- -
.--w
N
/ î
N
N
N
-
!
''k
$. N.
( (X
/ l
z
z
j
0
hx .
t
'> >.
N N.
!
' e'
N
N ..ee
N
N
y;
...
-A
A
Z
/
1 l
X
Fig. 2.4. Polar plot of the sarface ezkergyfor a vinlns'l s'urfRe augle 8. The with a 'mzsalignment simple energy exwession (2.Q0) has been USCXCI for several ratios
J%/ h(0)4q= 1 lsolidlinel, Aa (dashed linel,and ) (dotted linel.
2.3 Equilibrsum Shapeof Small Crystals
Polar (W:)1'FF) plots of the smface auerg.gare possible fol. arbitrazy plres in the crystal. lt is obvious that the variation of the surfaceenergy with orientation is of great Lmport=ce for a a:lmber of pzobleznsrelated to sttrface inhomogeneitiesor surfaee morpholoor. Notice thst the surface energy in Lsa conthmous gmction but that it hmsdiscontinuousderimtes' Fig.2.4, 'y(J), at angleslimiting the quadrants,e.g., b = 0. J.nfact
-.2
'dù
dê
:=+0
z
=
-
*=-0
.
c
That means, there is a cusp at 8 = 0. The increase of the acgle 9om : = 0 iè large vazuesis azcompazkiedby an increaze of the s'tep density. A proper ixpression%r 'y(û)must hencei'ncludethe iateraction betweensteps. Ia this èase it has been sho'wnthat
'y(p)lzasa cusp at
eveo- angle wilic,h corresponds
to a set of Msller indices whoseratios are rational Iplmbers(2.181. The shat'p.
npessof the cusp Ls a rapidly decreasingfnrnction of the index
mw
Yig.2.3).
1/a4 tsee
rone exnmple, whiclz can be interpreted as a renlization of the model above: could be a surfaceof a diltmond-stzudmem'ystal, e.g., Si, EètsEctuqsed àfiented between (001j an.d (111). As the sxmple orientation is tilted 1om j1'j 01)to g111j, i.e., û = 0 to 54.7o , the smfacemorpholor varies h'om (001) .7g r
.
''.
' .
.
.''':''' :':''' .' .' '' .'7* .'.' '' .' '' .'
('1*:':':'5* :' i':' :' :''' E'
''
''
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.
r
)..t... .,.j.
'
.. .
j..(:è.t'Ii 1 ' .: . .
1.
()a,!
. .
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el-ls
.
-112
,
r3 :
:
i:('
11S
:1112
337
r5
55 12
.
..
el.:$-1
'
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-!19
114
335
.
117
.
..' .. ' '
.. . g
plsnes. The and (7.7.1) #lkkE i''!7 'a'à sideview of a disrnond lattice betveen the (001) of the lattice li'aesrepresentprojections of atomk positions onto the jkflsèçtions ;E iï'd''' ;,E(. .)..pl%e The connections of the dots with tbe circle indicate the surfaceplpne àtï' oiedby (hk2). After (2.l9J. t@'' p
.....
.
-
-
'
-
Thermodneœ
54
(011) (111)
+
(137)
(11i) (113) (114)
(001)
Pig. 2.6. Stereovaphic trirmrlè for cubic-czys'talsurfnrm ahîbiting the low-index corner points (111), (011), and (001).In addition, one hig,hindex smface.is indicated in the i'atedor.
to (113) to (5512)to (111) as has been mexqlpred for stli.con (2.19j. (114) Possibleimk-mmediate surfu orienvtions are shown in Rg.2.5 irt a (lï0) to
.
plane. The Mlller indicesof these snxrfacesare indicated. Given a set of Mnmer then tke polar aagle8 can be obteed 1om tan 9 = Wvk. iatlices Lltkl), The sllrêxce orieatation r-qn atso1yeVaHH Fom (oO1q in anothe,rdirection thnan(111). Arbikary orientations of high-index surfacesof cubic crystals are visaalizediu a sterovaphic tzi=gle. The construction of the stereovaphic tregle ks based on three-dsrnensional consîderatiozus. M crystallovaplkic cnn be given by poàt.s on a sphere. If one connects all points of direc-tiozus the upper (north) half vith Ge south pole, eve.ry direction can be marked by the pokt of inteoection in the meridiaa plre. Tile aazest are'a of noneqdvalent points is givea by the s'tereopaplzictriangle plotted in Fig. 2.6. TEe three corners are givea by the three low-inde.xsurfacw. The surfMes whlch Iie on the co=ection lines be-een the low-index eoz'ae,r points are formally lin a bulk-tmlncatedview)composedout of the corner planes.Any plane in the inteior of the trinmrle kscrmposedin the same sense out of the three come,r place's.TMS shows that the compleYty of the diferent planes increases 9om the corners to the connection lines to the interior. The variation of the sllrfnne energy betweenthe low-hdex surhces (1.11), acd (311) , (110), may also be representedversus a polar angle 0 iu a (100) plane as izl Fig. 2.7. Witk a surfaze normal E!l0q for $ = 0 one observ' e's (110) the variation 1om (ï12(,(I13q, (11:35, E1ï0), (1ï15, (001)) (1îl),(1IZ!,(00i1: Axvllmsng thn: the (110) and (001) surfae are b0th mirror gïllj to LI1J). ezm be represemted plauœthe measuredvalues'y(#)(2.29) as a relative slrrênrza Wlnlfrplots have aksobeen energyverstus the polar angle as in Fig.2.7. Sn'ml'lnr ex-tractH by other groups (2.21).
2.3 Equllibrium Shapeof S=xll Czystals :.20
glch
4. W
.
ve w
G
1.-1.0a
r
w
'-N
':. ns & %-'' >'
*
pp s o
Z
o
s
5 u
;
*
t (lot)l k
z
#
120
*
wg .
p
N
) )
= o
>-
(3.11) (a:1)
=
:.1.j
r:l
55
au an
p
:
:
m
=
':
.
u
$f
:
e
J (4(* I
r
Z
% /
()
90
'I8O
040)
u u a
p
%
t t ! (A11h (117.)(I:.'lJ
:-95
r u
t
(1111 167û
,36:
Fig.
Slnrfnzteenera plot 7L9j.The surface energy ratios havebeenextracted by a reverse Wtzl'm constnction fxm voids 2.7.
(2.20J .
2.3.2 Absolute Values for Surface Energies
The absolute'valueof the sarface9ee euergy of solid materials is a fn'ndamentally importan.t energetic quantity which is neededfor the tmderstandsngof a large numbe,rof basican.dappliedpheaomemwsuc,has crystal growth, smfwze fafettklg, growth and stabllity of thln fzlms,the shapeof sma:tlmystctes in a suppoded catalys't, and ma'ay generazmaterials sdence applications. Dœpîte its wemrecogOed sigadcance) there are relatively little reliable primazy data of ecperimentalsurfacefree energiesbecausethey are vezy diëcult to memsme. k. 1u.contrast to ftzid interfaces,where the surfaceenerr or tension r-an usually be obtaizzedquite emsilyby capillar.y and similar eyperimental tech-
zliques)the determsnation of g(a) for solid-vapor inte'rfaces(i.e.1 surfaces) is eAremely diec-ult. Thereforc, at present not much reliable expem'rnental information cau be fotmd about J in the literatme. TMS is in parrticulartrue for the Anssotropy of the surfaceenergy'.Only a few ttvthnl'ques,such as zeroaud cleavagetechnsques(2.244, havebeen USHrepeatedlyin camep(2.22, 2.23) the past to obtain (pln.ntitative valuesfor a, lirnited nlzmberof solids,mostly metals. Irt other exmerimentsthe equilibrium shapesof voids izt crystals, suc,h as Si (2.201, have been measmed.The void shape, to a good approximntioa a tetmknldecahedronlsee also Fig. 2.10a), is related to the eqtlilibrblm czystal shape, and the surfaceFeeenergy is extracted via the W111fF construction (see next sedion) of suchau equilibrittm shape.Resultsare lis'tedfor Si sarfaces i'a Table 2.1. Absolute surface 9ee eaea'giese-q,n also be calrnllltted..However, ârstprinciples calculations are dlëc'alt for numerical and methodological reain detail ia Chap. 3. The slab coogarations usually used sons as dkscussed
56
2. Thermodynamic.s
Table a.1- calculated slxrfnce energie.sa' (1 J/m2)of low-isdex surfacœof fcc semiconductors cmtnlllzing in the diamond structure (C,S$ Ge)or zinc-blende and of bcc (Mo,W)and fcc (,M,Au)metaks.Reconstraetedlrestaxldttre (.fazAsl Iu the (311) column (metals). laxcdlsnrfxces lzaveyyens'tutliM for sezniconductmrs of TnAn Mo, W) Ls listed. ln t'he the value for tke (ï11)surface 1(211) Se'rGne) compotmdcaa the aaion cllemic.alpotemtialis fLXI.dat Jzas= Jolk 0.2 eV. In G't.'XIvaltlt.s (2.204 are also given in. parectkeses. the cmse of Si, expe-ram' -
possesstwo sarfaces.Izl the cmse of crystals consistingof only (2.12,2.25-2.27) one element the use of symmetric slabsinducesconvergenceproblennq(2.25).
of slab 111the case of compotmdswith partially ionic bonds,i'a the malorit.y the elecorientatiozssone buasto deal with t'wo inequivalent surfacesto A11611 Moreover, in th.is case the surtrostatic neutratity coztditlontc.f. Sec't.3.4.3). facefree energiesdependon the preparation..conditioas or, more precisely,on the chernical potentials of the constîtuents (cf.Table 2.1).Neverthelv, suck calculationsare =ow possible.Resultsfor covaleatand ionlc semiconductozs with a'a fcc Bravais lattice as well ms for bcc and fcc metals are listed in Table 2.1. A rath.er compléte colledion of data for the llnrelaxed surfacesof 60 metals Lslisted in (2.27). The absolute valuœ aze of the order of 1 J/m2. Eowever,the values vazy with the surfaceorientation az.d the bondingbehavior. For smicondudors a surfacereconstrudioa may considerablylogez the smfacefree enera. .An intuitively reasonable,but rough estternxtionrelatc the surface energy 'y of Camaterial to the cohesiveeaergy per bond? Le., apprnvfrnately half of the value given in Table 2.2. Together with an sttrfacœof area of about c2/2per atom (avalue whic,his correct for (100) ziuc-blende/diamond ) one Mds for econ 2.5 J/m2withou.t reconstmzctioninto accm'nt. The absolutevaluœ of the surface9ee ener#es dependon the dnn-qity,in particular on the elmtron density n, of the m= terîxh. This is cl-ly demonstmted in Fig. 2.8 fqr snbonded metals. With the sttrfRe eneir decreasingdis-tanceof the eledrons rs = (3/4m)w (2.28) iucreasesan.dvice versa- The behaviori'a the tow-densit'ylimit rs k 4 follo= tlle predictton of smface energieswitlnsn the jel'tiummodel of a metal surface(2.291. '
2.3 Eq:lsllbrblmShapeof Sma'tlCastals 1.4
57
AI
1.2
D
E 1.o
R
: >-
> 0.8
Ga egeds ; rf rnnnroscopic (orat least mesoscopic) ;yL c%t the eqttilibriïlm 9k1.4 (2.8)ar.d (2.13), t keglected.Accordingto expressioms shape (scs) at constaat temperature 7. witu fuxedvoblrne >- and '$'f'! ( kt'û; Vkak'' 'èxl potential p, is determc'nedby the zairsvnxl excess suzface9ee eneroI.l i'' Liîzukebhèct to ths surfacea4.
'
'
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-.
.
.
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..j.,,...
..
.. .::..' .
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.
,.
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@' q.( '
--.
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:.
u jiiyi #,
,
.
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....
..
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jjz 'y('n,)
.
.A(F)
(; gg; .
58
2. Thermodynaaics
subjectto thE constraint of âxed volttme kr
=
Yd'Z.The WIO
theorem
v'(z)
the ECS is not nec-arily that of tEe min:'mll= smface area. It may be a complex polyhedron witll the lowest total surfaceenerrr for a given volume. A minimal suzfaceonly occurs for a perfectly sphericalWhtlfr plot, i.c., an. isotropic excess sarfacefree qnergp The corzepponding ECS is a sphere.Tikis hasbeenexperimentallyshowafor ia the absenceof gravity. ln the case of czystals the variatjon water dzoftets of y with the normal .rt will produze, on eac,hsurface elemènt dA, a foree proportional to àyjonwhie,hwi.ll tend to alter its direction a,t the same time as ''/ tends to shtr'tuk5ts area. Consequently,the ECS can no loager be a sphea'e. Figure 2.9 sehematicallyshows a res'alt i'n.two dlmensiorus.It indicates shapeard the morphologithat the WttlF plot c('n,) governstke eqlTilsbrblrn In the case of real three-dimensionalC'Iy'SC;aLS cal stability of a c'zystal(2.20j. the sîtuation ks more compncated.Besidesthe variation of 'y with the surface normalhalso the strength of the vaziation plays a role. This is demozustraîed in Fig. 2.10. The ECSis construded for silicon tytking mto account the surface and (111). In Fig.2.10a energiesof fom' orientations, namely (0011, (1131, E011), but < 'y(100) < 'y(110) < 'y(113) and b the same eneageticordorlngof 'y(111) dl'ferent values (see Table 2.1)have been ased. The ftrst paramete,r set is derived 1om meaaurementswhile the secondone %nAbeen calculated by azl the shape,skz thc t'wo âgtu'es(a)and (b) ab initio method. Qaalitatively aze the same. However)the relative areaz of the cnrstal facets vaor with the absolutenllmbe'rsof the various energles. Tn.kinginto accout only the two orientations, the cazbicsyma'n.d(100) lowest surface energiesfor the (111) facets ocwtalledron with (111) shapeis a regnzla,r metzy suggeststhat the W111fF states that 2.16,2.30) (2.22)(2.15,
.
Fig. 2.9. A polar plot of the stau face free energy for a 2D mystal (solidlinel and the ECS based construcion (dotted on the W111fF in (2.30J. Seealsof1g7.u.% Linel.
59
2.4 SurfaceBneav and Morphology
(b)
(a)
.41% o'w. ''q t.
.
.
.
jh
.
s .h. ;. ykayyjo
... . .
, ,
.
.
.
.
.
-
% # ..
l's .
y
,. ,.j
.
.k. .
'
.v . 4 ;
.
j;.. N.-K
'
.
.
'ê
x. . . .
.
.
.
x :fk
.i; j. .
! ,,,.J ï. >
), b
.
1 'Il; ?* Wr' (' .
. dbài. .
i.
r
..
.
,
...
-
t &.g'
.
v q. +i ! .
)k4',''',
'è, () the fat surfaceis stable (or + (d2'y/d82):so whereasfor 'y(0) at least metaztablc), + (dW'-t/d92)#=a < 0 the bucktedstlrface
is more si'able. Izza lzighly nansqotropic crystal with a strong variation of 'y with the smface orientation the buclclimgesed dlscussedaboveindicates that such a czystal will ml'nlml'ze the surfaceenergy for a given Itornsmal normal by the formation phenomenonwaz Emt discussedby Herriug (2.16,2.301 of facds. This jacetti'ng constntctiozu with the hèlp of a geomctzsc.al However: at :5Up to noo tempeaxture efec'tshave not beenconsidea'ed. aite temperatures the discllmion must be supplemented'by the Melusion of emtropy esects. At low tempezature a stretched surface Ls:at on the micl'oscopic scale. Whea the temperatme incremses,thermal factuations appear: tlle surfaceis no longer Qat and may buclde. These thermal âuctuations are related to the mean square deviatioas of the atomic positioas with respect to the averageposîtions. Their strengths dependo:a the actual temperature. The fuduations may remsn'n Gnsteor diverge.J.nthe fzrstcase one speaksabout a sm00th surface.J.nthe secondll'rnstk one says that the smfaceis rough. Above transition. a criticat temperatare the sarfaceundergoesa roughœnén.g 2.4.2 aD Versus 213 Growth
I'n Sec't.2.1 we stated that, i.a thermodrmrnl'c equitibril:m, there is no net growth. All elementar.yprocesse.sproceed in two opposite directioas according to tEe priaciple of detailedbal=ce. The czystal growth must be a nonequilibriclm kinetic process. The xesulting macroscopk state of the system dependson the reaction paths in the covguration space ms tnclicatedin tlle obtained state is not Fig. 2.1. Siacethe result is ldnetiem,llydetmmnsned, pal'ts of the overall processmay necessarilythe most s'tableone. Neverthelcxss,
2.4 SurfaceEzlergyand Morphology
61
'be ldnetically forbsdden,whereasothers may be in loci thermodynazrdc equilibrillc). Consequently,equilibrblm arrlments may be applied locally, even though the total growtiz processis a non-eqz:c'l'lbzlphrn process. Exactly these ideas are used in modern Emt-priaciplesstudies of epitaxia1growth (2.32-2.341. The activation energiesof the processesocclzmqngon the growing surface,such as diffusion, nucleation,and attaeAment ot detachfrom ab i'aitio : ment of adatomsto existing islands and steps, a-rc ezxtracted total-energy calculations, The correspondiug atornic processeshappe!l in the length and time domain of 0.1-1 mm and femto- to picoseconds.Hencethe èazclllntions for modeting atomistic aspects of growth have been hampered : by thc needto bridge length an.d time scalesby many orders of mn.gnitude. First-principlesmoleculazstudies, while being powerful tn the irwestigation ipri)findividual events on a time scale shorter tlza'a 100 ps, a're not suitable Yraccessingthe time scalesi'nvolvedin epiteal growth, nor cnn they tackle i,lq.r',iih: statisticalinterplay of the nttrnerous processesthat are respoMiblefor tke ë.: Tùtitcome of a growth expersrnent.Howcver, ldnetic Monté Carlo simulations g,tfger an emcien.taad accurate way to cope with tMs dilctzlts J;).y rz Viteadof following such atomisttc approaches one may consider the ) of an overlayer more pheuomenologically. ln the cmse of heterocpikrsàrii$th lsfiky,the overlayer grows on a substrate in one of thTeedilerent growth i:'illbdes as indicated in Fig.2.12. In tlle bwnk-va.n de.r Mezve mode (2.35q, :i>ibzizs or moleculesgrow layer-by-laye,r in asequentialfashion.Jn contrastj the è:''V' yliher-weber mecûanism(2.36) leads to the fomnntion of indMdual threih.EE..)L, irtéljliéùsional Lslands.The third mode is a combinationof tke two an initial sltkisdiiensional process creates afew monouyezs, the so-canedwetticg layez, ;'':' tttu it t. thèn udivi4ual islazïdsform'wsmore matarial is deposited. This process i r,.tkskàkuws ws t:e stranslu-xrastauov modeg2.aq. It migbz 'beinduce: by sew ij'''''? 7tjktE'sfacsors. , (i) A cezta- lattice msmatcz betveen the substrate aud the Efi'' :v cnmnqtcompletely be accomodatedin an elastic mauer as vithin j;Jjrsktzlayer q.2.j J'ljhëkck-va,n der Merve mode. (ii) Alternatively, the dieezencein the czysEs'p)j;t1k. ji liltjj lrE)i or the orientation of the overlayerwith resped to the subs-trate :((12 (( ' ( jinzfzttzy )!).i. ?(!!FEVd2'' tt'sjucll a growth mode. As we will see below, the Strxnski-Krastanov ''' @t'y: gj.gts! (jjuckatsr tue selpassembledgrowth of an azray of quantnnrndots on Pr)P '
'
'
. ..
..
.
.
-
'
'
'
. .
.
''
'
.
..
'
:. '
'
.
.
.. .
..
'.
.
,
'
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.
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.'
.
.
.
li(#.jtt' àitiutb.
h',)ip?'k of the loc.alsurface1ee energy per t'nit )j.. jjgtilryEfjuckssioc jj.E....L-.,.ijjE. .j
area eyalsoazlowsa kk !(:;di'' 'àizin iti6n of the growth modes. One has to introducethree dsFerent energiœ, : .t,.!11.' .tf.t1(; ''a:ljj.!j jgu s:o energ.gof the sttbstrate-vamr:rn intezface,x, that of the ' ''
. '
: . . : i. . .. ' .::..
.
.
,
kEtt )kJ lav(tijjk jjg9uyym yatesaae,yo , aud.tkat of the substrate-overlayer interfacc, :(yï't)i#)) jiyl'''k.. (;;..'-i!. tgkdjkg to the discussionin Sect. 2.7 the quantîty'.'y ca,n also be ,l(t' .-rrr! 'kël.-2kq,.j.:;ïj(q',L; '.,;rj' -)tëi.. .k :.2.: a force per Trnlt lengt.h of bolmdary. Disregakrztmga possible bt,è884tks .t j: .: 'J one may considera. situation as showaia Fig. 2.13, where the tiltkitt/kllayvr, :,.. Jfllfijtp) #,. k't jj .m uuuu u assn,medto be depositedintla.eform of i'adividuilislands. s:'r.I ; rE.h.jp'r'yttu ë toucz? the equilibrium of rj rj ujore substratej isln.ndj and vacuum )! ..:..jzjgEEg(l iï. j''') j. E
rllmerent modes of epitxxial growth tseailezztatie--a.llyl Fig. 2.12. T for dilerent coverages:below one monolayer,betweenone and two mono ezs, above t'wo moztolayers.In tzhe217Fknmlc-vander Meme mode, layers of mxtfameAlgrow on top of eachothet'.In the Volmeer-Weber mode, separate3D Is'hnds form on the substrate. 1.nthe Strsknmld-nutanov mode one or two monolayers(wetting form fzzst, laym') followedby m' dividual islands.
Js = Js/o + 7o cos
(2.25)
$
the angle betweenthe overlayer-vacuumfaceand the substvatevacuam hce. The two lsmiting growth modes,the Frank-vaa d. Merve mechxniRm with / = 0 azd the Volmer-Webermode with 4 > 0, can be charac-
4 Eig. 2.13. Faquil*lbrbtm of forces (schematically) at substrate, depositedislazïd and vactmm. (overlayer)
2.4 Surface Baergy aud Morp/ology
63
terizedby ï'y f 0 or Ay > 0 with 17 = yo + wo %. For the intermediate Siranski-Kraàtanovgrowth; A.f S Ofor the flrst atomic layers (wetthg layex -
and Arf > 0 for the islands. 2.4.3 Formntion of
Dots Qv3Arttaarn
long time, it was believed that the of 3D kslaudsin b0th the mode and Strnmld-Krnstanov modeLsrcompanied by plasVolmer-Webe.r tic rehvxtion, for instancq by the formation of dislocalons n.ear the island base.In recent years, it 'hxq bee,afotmd for severil heteroepitaMalsystems islaudsform' vith a lattice miqf.t 2: 2%that dislocation-hv(i.e.,coheremt) in Stransld-Krastacovgrowth on a wettiug layer. Suc.hsysttarnqare Ge on Si Tn As on GAAKwith 7%(2.39,2.40): and1nP oa JAGa,P with 4% with 4%(2.382, akisst.Eventhe system Ge on SiC with a norninally mueh high.e,rlattice mlpreûevemucph showssuc.ha growth mode. Thœe 1s1a11(1s ft of aboat 23%(2.41) of the czisst-inducedelasticenerr by ckaqgingthe implane lattice const=t in the island for layers away from the intezfacè.Tllese naztoscalecoherent islands,which are often fouud to have a narrow size distribution a'adto be are accompaGedby a spatial quanarrangedin a regular array (2.39,2.40j, tization of electrons aud holes i'n three dirnensions. Hence,the islandsbaaed aze promisicg for on direct-semicondudor combinationssuc,has Tnls/Gn.A.K diodes (LEDs) azd lmsers.lmages of such use in quantum dot light-vitting whicà havebeenobteed by sc-annîng ttmqu-tum dots or nanoczystaltites, neling Mcroscopy (STM), are shown in Figs.2.14 and 2.15 together with a schvxtical represOtation of their facets.The examples are an TnAs pyrxrnld surface(2.43J on GAAq(OO1.) and a Genanoczystalon a 4H-SiC(0001) (2.421 For
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W W,''k. $A% > :%'&'% é.: :u ' 'k qY l e %.' L..xs;: z--.w kiB'?: -. m. V ''+ ' ' . . val w . . 'e'zt : x#&. '
4':t .;T'.%2; k' .%!' (.(. ' u?B. , ' ' ' Q'. iy o . . . 9 .2' w &' ' , , r ' ' h . 'Tsh k' 'f é J ' ''rf' rk ï'. EF E ..L;t k $)' f '' . ' t! fd:'k';ï' Tt: k .1 $2 hilk k4f. .LL''k.Eëçl L(l? i;r . '. '5;' b'''6ç''7' lh w) u;i. owV/kyuz W qg ' ' 7 ;: hkl.hl i-l '-1i-l i'': 7'' 'b.bLkê'Lt-A' ikkkikh îikii'ik.ui/' lkuz./dskik)' sLzil'qs W '
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chemicalvapor deposition(CVD)
.;
yy yy v. .
xo
qs
yy yy yy 1+' '' 4
Ge $ u-s-y,y
hh-rs-hh-r--sh-slg
molecular beam epitaxy (MBE)
Pig. 2.18. Schematicrepresentationof two diserent epstmdaltecAnl'ques,CACD aud MBE, tzsedto prepaz'ea mrr'ace or to deposite au overhyer, here Ge on Si.
The actual chemic,alpoteatials dependon the surfaceprepazatype ï (2.56) tion conditionsor the kn'ndof epitaxy usedto deposttea surfaceoverhyer.The latter is schematscallyindicated in Flg. 2.18 for the depositioa of aa nzlqor'bate (here: Ge)on a subarate (here: Si)using clsleren.t epiteal tenhnreques. ln the fozowing we excludeadsorption,rather we study thc preparation inâueaceon the eqlzilibdnlmsmface of an AB compound ms a'a exnmple. The relevant chemicalpotentials are p.x and #s. Eowever,i'a addition the clzemical potential Vxkô of the corresponcllngcrystal bulk also has to be take.ainto accotmt: sirze the b'll1r solid Ls a reervoir whic,he-q.n e,x e atoms with the surface (2.541. For condsmqed states, e.g., the GM.AS sudaces,bllllc Gadts,or blll1 0)staudsfor layer gzowth in the Frank-va,uder Merve mode tisla'ad growth in the Volmer-Webermode). The Garacteristic quantity .4J* for growth of a ceztaizbmatcrial on a substrateis no longer a, constant material parameter, but caa be c-hanged with temperattu.e and pzasm'e. The use of relations of thc type (2.48) allows one to relate the prepara-
-
tion conditions to pat-tial press'tzresand substrate temperatmes. In principle, the mse of theseqppnrtities also allows the determination of the surfaceemrgy and,hence)the stable surfaceph%e for certain preparatioa conditions. (2:39) Examplesare given ia Fig.2.20 for the MBE preparation of GaA,s(O01) (2.67j and 1z1P(00:) (2.6$surfaces.Htead of a partial pressurea beamequivalent js ttsedto account- for the presemceof molecttlar beams.lt is pressure (BEP) a pressttre whic,his equivalent to the ftux of molecules or atoms impinging on the suzface.Togetherwith the measurement of the s'arface reconstruction by refledion high-energyelectron rllFraction IR.H%EDI at a given substrate temperatme, the beam eqaivalen.tpresstzresallow the construction of BEP-T p'hnxediagra'ms.Jn tke cmse of GaAs(0O1) the BEP ratio for As molectzles aud Ga atoms is varied, wheremsfor ToP(001) tEe ûux of Pa moleculesis varied in the presenceof an atmostzero ln ûttx. Iu the languageof the theoretical p'hnse cliagrn.rnqin Fig. 2.19 an increasiug BEPASa/BEPCa ratio (increasing BEPPZ ûuxlas well as decreasingsubstratetemperatme T correspondsto a tendency toward more As-rich (P-rich) preparation conditions. The opposite tendency, decreasingBEPAw/BEPOa (BEPP ) and incremsingtemperature, describes ? Ga-rich tln-z'ichl preparation condstioms.The meastkredphmsediarytrn!x in Fig. 2.20 and the calctfated surfaceenergies versus the cation chemicalpoteatial i'a Fig. 2.19showthe same trends for the most stablereconstmzctions. For example,in the GaAscase the stable4x2 reconstmzctions in Ga-rieh conreco' (2t i ons changeover into a c(4x4) nstruction undezAS-HC,h preparation
z z; zxs'.%v.1 '''VAJ*'%*'T T'=rl-'71r4h1'û;m, Ga dro pIel:s a.q 6 pe'lgW.v :kN%.). rpliu' X1) a'4xl 4x 4x2 ( .1 %%%s%+
.
.--.
.
1I I
-x sttrfnce. Howeve'r,tlze interwat of the variation of Jzc has to be remn.rkably
modoed if s'table compoundsAC, BC, or ARC (vith porssibleadclitional variations of the stoickiometry) exist. One evmple could be tke adsorption of C = Lî on a zinc-blende crystal Znse (A = Zn, B = Se).An' upper botmd on the cxemicalpotential of the adsorbateis fouad by eloring tlle vaziouscompoundsthat the adatom nltn form in its interactioa with the sys-tem. For Li, a possibletppe.r botmd on ;tu is of couzse imposedby Li metal. Howevu, the most stringent constraint arises from the com(bt17k) pound Lizse;whie,h leads to the constraint jm the chernicalpotential of Li, b lk = 2 b!znc 2 6gj 2/:L2+ #se = Jilse Jzl,z + p's E Bjk zxnaase f For a certain smfacestructure or, more precisely, for a cez'tain adsorbaàe Thus, structure, the ntlmbe.rsNx, NB, and Nc are fzxedin expression (2.49). f)s only dependson l/za aad zlpsc. Two adsorbatestmzctlzres are in ,eqlzilibrbtm for equal %. TMs condition allows the coastmzctionof the phase boudarie. ln a region of Zï/IA and Apn wbic,h Lsbouuded by such botmdaries, the adsorbatephasewith the lowestsurfaceenergy f)s Lstlle most stable -
'
'
One.
In order to iliustrate the stability of adsorbatestructm'es on compolmd thc cleamge face of Iztlsis suzfaces,the deposition of As atoms on TnP(l10), studied. Foz simplidty the translational symmetry is pstricted to the smallschematicallyi.a Fig. 2.21. They
contûmed layer stntcture Le = 1),-(d)ewxck=ge-reacted surface 'wath a;a extra -&s oveau layer Le= 2z) P (In,Asj atoms are denoted by full tempty, smbols (2.70j. shaded)
ial
.
coverage0- = 1 conwponds to one monolayer of arsemic.1c the case of a (110) cleavageface e = 1 3srelated to t'wo adatomsper surfacetmit cell. The clean stufacewith O = 0 is describedby a reiaxedzig-zag cha,instrudttre. The &st s'tep of art adsorbatewith a coverage @ = àzrepreseat,sa'a exchange-reacted geometzy The upplrmost phosphorusatoms are replacedby arsenic adatoms resalting in an ï= AA monolayer. In the nex-t step of coverage,0- = 1, an As smface.Severalstructuzal modelshave monolaye,roccurs on top the ïnP(110)
been suggested.The most stable one is a so-calledeyïfczriclcontinzedzcper The largest coverage@ = .j is representedby an stractur6 (ECLS) g2.702. geometrs ECSL on top of a,n exchacge-reacted of the fottr stradmes tmder coasiderThe results for the surfaceener#es in a m/z:n,-Ayxs phasediagram with two triple points ation are slzrnrnarized i'n the allowedregion in Fig. 2.22 g2.N.Despite the existemzeof the stable
80
2. Thermodpmrnics
> S
1 '
ex oange-readed
=.
exch.reacted
SUOCO
' t of tile electron: ùal energy (or èlectrozz interaeion. In generi, besidesthe efect of the core and the Hartree due to tite clmssicalsnteraction with the otlzer electrons, it coatxsnq Sotential $ certa in contdbutions of exchangeand correlatiou. The spin varisble of the that it is inLelectronis not e'xgocitly consideredhere. Rather, we msmnrnç cluded in the space coordinate œ. The spinor character of tke wave hlnctions e' : i s thereforealso not xplicitly tikea into accotmt. ln cases where the spin importaat, if e.g., the spG-orbit hteraction strongly modifes the becomes electrozlicstmzeure, the spin variable and the spin quartttlm nllrnber will be explicitly memtioned. .
82
3. Bonding and Bnergetics
of an orderedaad comrnensttrate sttdacesystem the total poteutial in (3.2) obeysthe trn.nqlatîonal symmetry of the halfspacewith the surface.Then J.1lthe
case
+ .&)= F@) V'(œ
with R beiug a vector of the 2D Bravais lattce of the suzface.As > conse+ A) = ebuR#Lœ), quence the eigenslnctions obey the Bloch theorem, '$Lm and an eigemstate ca.a be classiâedi:a terzns of a 217wave vector k &om acd a b=d index v. The the corresponasng2D Brillouin zone (cf.Fig. 1.22) scbrHnger-like equation (3.2)takes the fo= '
(3'4) JfW:@) fy(i)#y:@) with 2D Bloc.hA'anctions#vx(z)a'adcorrespondingBloe,hbands zv(:) de=
vector i. Onehmsto mention that the theory presented in (3.3)and (3.4)ks also valid for a system with 3D translationat symmetzy and.wave vedors k from a 3D Brillolnl'n zone.
pendikgon the 217wave
..
3-1.2 Tight-lBinding
.
Approac,h
I.zlorder to tmderstaudthe bondmgbehaviorin the smfaceon a length scaleof the order of neazest-neighbordistance-s,it is convenient to expand the' singlepazticle wave Alndions #@)in (3.1)izt termq of orbitals 4.(z) centeredat aa atomic core at the origin of coordinates. The inde.xa labels the kiud Tizis approacll to the electronic of atom azd the atomic quantntm nthmbez's. structure requires knowledge of the atomic positions A... The solution of i'a Sect.3.3. Eere we adopt a locnlivedthis strucvtmalproblem Lsdkscussed orbital basis set (4o(z .&..))and assn'rne that their locnlization centers Rs, are lmowzuWithn'n the so-called linear combination of atomk orbitals method.,the following equali'ty hole for an arbitrazy polyatomic (1uCAO) system u'ade,rconsideratioa '
-
#(=)=
V %i$=LT.&)' -
(3.5)
l,$
where in explicit calculazionsatomic mMe fn'nctions, hybrids, or bonding and as /.(œ).ln fzrst-priuciple or semi-empiric,al antibonding orbitals are IZSC.CI electronic stracttu'e methods the hmdions should be lmown a'ad may even be adaptediu self-consistentc'ycles.In more or lesse'mpirscalapproaGesthe =alytic fo= of the basis Gtmctionse-qn remain n''nk-nown. Mther, one introducesthe matrix elemeats of the Hxrnsltonian H and the overlap integrals and tries to S'adexplicit expressionsfor thesequ=tities. With the a'asatz (3.5)the Schrödinger-likeequation (3.1)tmnqforrrminto an imfnite system of homogeneous algebraicequations. The expznqion coef-
fcients restttt from tlze eigenvalueproblem
3.1 Orbitals and Bond'mg
?pj .RJ)! R,.) sksabLltz' (J2u(a., I'''.2 -
-
b,j
(3-6)
0,
=
83
'
where
HzsLmt .7) are
J
d3tr/o'(œ m'IHSb# AJ)
=
-
-
the matrh elements of the one-electronHn.rniltonian S
tem, and
(3.2)of the
s'ys-
'
%sL.% S.J) =
-
j
d.3z4.* (œ m.I6LT A#) -
-
The intra-atomic iâtegralsare Krodenotethe intenatomicoverlapi'ategraks. of the oae-cente,r necker smbols becauseof the asgprnedoz-tbonormnlbguation orbitals. I'n.ab izlitio calculations, the multicenter iutevals in (3.7)and (3.8)are Apart evaluatedexplidtly for a givea S and a given set of %Lm .Rz+). requests suck a procedarerequires heav,gnuh'om possible self-conskstence -
merical calculations because,in general,the matrix elements do not converge rapiclly iu spsce. Therefore,m=y empideal and semi-empiricaltecbnlques have been developedto reduce the tplrnerkal efort (3.1, 3.2).For eAample, assllrnes that HC.bLm', the extended Hfic'keltheory (EHT)(3.31 .Rj) is proportiona) to S=SLK.&) if i # j. Besidesthe enormous' reduction izl the computatioaal esoz't,empirical methods give muc,hinsight iu'to the chnmical bonding processe involvedan.da deeperunde>tandi'ngof the trends .in properties 9om oue system to azmther.A majordevelopmenti.n thq history waswthework by Slate.raud of the empirical tight-binding method (ETBM) who suggesteda new and waiuablerole foz the LCAO method, Koste,r (3.4j, that of azl intezpohtion scheme.First, they suggestedto treat the Hn.m51tozzia'amatrix elements ILVL.%, .aj) (3.7)ms pazameters aztdto ft tkem to one-electroneaeates.Second:they invoked lœown tmeasttred or caklzlated) accordingto whic,ha,n orthonormal a theorem,ftrst proven by Löwdia (3.5), set of LCAO orbitals could be de6medrigorously suc,hthat -
XbLm XJ)= Qbéa.p -
.
(3.9)
The Löwdl'n theorem statez that a set of non-orthogonal orbitals located at dferent atoms ca'a be trxnKformedinto â.new set of orbitals whicit are othe,raad preserve the atomic geometzy orthogonal to e'a.C'N The advantagesof the empirical methods becomeobvious when considericg an isolated bond formedby two orbitals c = 1, 2 localizedat neigbboring .atoms. Examplescodd be simple diatozaicmolemzles,e.g., the hydxogen moleculcH: with a pare colminntbonda'nda I;iI'I molecalewith a, heteronoIJr (orsimply, polcr)bond, for which, howevez,the consideredorbitals are restricted to the most importau.t s-like ones of the valence electroils.Without the dgenvalueproblem.readsas the neglect of ovezlap (3.8),
3, Bondingand Bnergetics
84
antibonding ca
+(AE)2 IH12l2
Ha
AE AE
1-11$
IH1zl2 +(A:)2 Eb
bonding Eig- 3.1- Formationof bonding and antibondingmoleculeleveb fromatomic level.
Szz e Nlz czsa S-lc sS* 12 Sa2 e -
-
-
-
-
:1
Q
=
(3.10)
0.
Neglectiugthe overlapof the t'wo orbitals forrnlng the bond, i.e,, S1c= û,
12+ the eigenwlues sap = I ::i: are the eneates of the antibondingan.dbonding orbital combinations, respectively. They are schematm ically reprœented i'a Fig. 3.1. The abbreviations ë' = (fftz+ Szz)/2and zâe= (JA2-Szl)/2 defmethe averageof the two atomic ener#es ard one-half of their dileaence, respectively.Tn the case of diferent atoms in the molecule with Jêza> Joz, the atom 1 (2)represents an aalion (cation). Co=equently, ls n-xn alqo be called the yolar cncrr g3.6) , in contzwt to 'the colmlenie'rJern 1Sz2 1,whic,h îs also nonzero in the lt'rnl't of equal atoms and orbitals. The polr energy determinesto what extent the electron densit.yalong the bond is deformedtoward the anion. Iu. the point-nhnrgep6dure, zîc indicates how many eledrons are traasferredbetweO the catioa and anion alongthe bond. A quaatitative me=zre is given by the bond,polczwzp (3.6)
glsza(zk)j
Gp
=
Iffzzlz+
(3.11)
.
(2!42
àz = It is related to the eigenveeors by IQ12 j2= lz(),::F:aplfor (1+ap)ar.d 1?2 the bonding/anbibondzng statc. Ia the votmd (bondsng) state of the molecule one Ods the rœtllt that the probability of an electron appering ou atom 1 The is (1+ (zp)/2 and the probability of Mdtengit on atom 2 is (1 &p)/2. = ap. Sinceihe coYent dipole of the bond is proportîomnl to (1ê1I2rJaJ2) enea'r is a, Alrctiozz of the distanced = lRz Az I of the t'wo atoms, the polarit.y aud the ezmrg.ysplitting of the molectklelevels dependon d. The bond length deqfollows fz'omthe condition of mypvlrnl:m enera ecltsqsbrihtm -
-
-
3.1 Orbitals acd Bonding
85
gain of the t'wo electronsdueto bonding. The enero- (.FJ1z12 + (1s)2shotzld be a mprrim:lm. However, to perform sach an optlmlmation e-xplicitly, the overlap interadion ($n general,a short-raagerepulsiveiateraction) has to be
takeu into aecout (3.1,3.64. I'n the case of trn.nslationallyinvariant systems of the type defned by (3.3) and (3.4)with atomic positions m' = .R + rz (Bravais lattice vector A aud atomic basisvector .rv)the LCAO exnmnsion(3.5) can be speaifed as = '?Jb:(z) 'h7cvc,ililxt@l
(3.12)
c,ï
with Bloch
sllrnq
1
s:.)z$
:kf@) Gn J((2 /.(z A;) =
e
-
constmzctedfz'om the locatized basis ozbitals. The R-sllm exteads over N elementa'cycells of tbe hlndamemtalregkn of the 2D (or3D, theaythwave vector k) crystal. The Bloc.hbands ss,(k) at a.a azbitrazy poiut k az'e then obtainedby solvingthe eigenvalueproblem
c,aj.(:) gl:l'oçe''t:l J-'-J ,u(i)JoyJvq =
-
0,
by.#
wherethe lbrnsltozsianmatrix reads ms Jfoïby(i) EE:
=
1
y
c -i:(a-.a,.).s ab (a. .%) V AyR/ .
,
of the orbit/ a localized at m. = A + 'but t:e overlap (3.9) ;. itocnlized at Rj = R' + o is neglected.
Ju
aad orbital b
é' '
j
E.321.3Atomic ;
'
;
.::.':. :
.
Orbltals aud Thelr Interactcon
.
'!Eè
'Let us considerelamcnts A an.dB belongingto colllmns Nx=N aad Nz = (8'':,X) (N = 1)2, 3, 4) of the Periodic Table and formi'ng compolmdsA.NBs-x.
Vsuallythey
crystnlllze within the cubic crystal syste,m with two atoms A i and B s:a the primitive cell. The occurring crys'tals posses the dsamond, qzinc-blende, roclcsalt,or cesblrn-chloridestructure. Sometimesthe resulting LïKNBZ-Ncompound crystals belong to the hGagonal crystal system, e.g., i ïhey possessthe wurtzite structure with %ur atoms pe,r'lmst celt. Th.ebondiug Tbfsuch compozmdsis governedby the N and (8-N) valenceelectrons of the Ao atoms of a cation-azzion pair. However, L'V-m compolmds,e.g., the lead I'ubiltsPbS) Pbse, and Pbrl'e: cau be descaibedi'n a &'=1'1n.rmxnne,r allowiz.g Lïheformation of a lone pair of electronsat the grottp-Rq atom. 7 Tlle s andp atozaic orbitals (or,more strictly, the Löwdin orbitals) of the !:: oute,r electronicshell with orbital euergiescs, cp acd radial ftlnctions Ra(r), .
.
.'
86
3. Bondinz and Bnergetîcs
coordinxtes m = polq Usiug bondiag. the clmmical most to contxibute .Rp/) the Arll wave cos 0)and sphedcal hnmnonic,s w) 8 sinw, cos siu ybmtp, $ p, rtsin rephced by tlle,ir combinations are umzally fllndions 4c(m)
1
1J) X04:,W)&(r) .zw.&(r)' =
=
-1
= J'zz (p, pj 4&-z(p,p)) a,(r) ( 1yu) W
f
yp) w -
-
yi z (9,w)+s-,(:,.,)q as(r)E
W)&(r) lpa) Fzûlû, =
=
Z-J.w(z.;, .
=
4x
v
-tX.w(r), 4.
p
(3.16)
t3 z
s-c&('r)'
where t:e corresponeg atom is asstanedto be sitaated as the origin of tb.e coozdinatesystem. Suc,horbitals are represeatedin Fig. J.2.The new p n='R. orbitals are directedalong a Cartesia'a fktledd la more complicatcdcompouudsor even in mtztals,inzompletelf and I shellsmay contribute to the cheraicalbondi'ag.Jmportaatexamples three''series in the Peziodic itwolve t'ransition metal atozas. These belong to to of 3d (Ti to Ni),4: (Za. Table whic.hcorrcspoudto the progresive 611n'ug Pd) and 5d (Hfto Pt) sbells. Real combinations of the d orbitals with a'a enexgyed and a radial part &(r) aœe z
Pz
S
Py
9
Px x
Fig. 3.2. Scàemltticrepresentation of s and p valemceorbitals.
3.t Orbîtals tmd Boadinz
87
15 Az
ltfwv) y,,.g 2Q(r), Ta =
IGz)=
15.:, s r a&(r), 15 zz
Idam) spadtr), =
z2 v2 15 = Idz2-#a) 16zr r z .!Q(r), 3z2 r2 15 = Edaxu-ral .TQ(r). 16c r a The atoznic orbitals givea ict (3.16) nnn be usedto calculate the and (3.17) matzix elements.J.ILyIJk., at lemst,to calcalate their smmetly and Aj) (3.7), geometzicdependemce. We illuswtratesuc,ha calculation of the Hxrniltozzian matrix in a more generalFamework.The following assumptions are made: -
-
i. The basis set is restricted to one s orbital and the tkee p., pu, and pz ozbitals of the mlence shell of eachatom tsee Fig. 3.$. 5i. Orbitals on Werent atoms are assumed to be orthogonalso that (3.9) is flgfllled. iii. Only the nearest-neighboror second-nearest-neilbor iuteractions S0:(.&,Ro'4 (3.7)are retainetwith the additional asslfmptiontha,t the Hn=sltonia'a lnAAlocal cylindricazsymmetry around the n.='K couecting a pa,ir of interacti'ngatozns, so that J.Qs(A,., Rj) H IILSLR..Rj). -
The adequacyof the tuse of s and p orbitals of the highest (partially) occupied shell (i) seems to be quite clear from cizemical azgamertts,Le., that only valenceelectronsessKtially participate in chemicalbonding. This is obvious for tetrahedrally coordinatedhNBs-x compolmdsbut also for simple metals suchaz: e.g.j A1. J.naddition) for trnmqition petals or transition-metal compotmdsthe d orbitals (3.17) have to be taken iato accolmt. Of coul-se, all thesebmsissets are incomplete. NevertheHs, electrordc stnzctmes axd total eaergiescazz be deduced,at least, usi!zgthe Harniltonian matrix elementsas parameters to be ftted.. Foz iustance, the restdction to s a'ad p orbitals is qlzite accurate for the desciption of both valencea'ad.conduction bands of s/miconductors. Howeverjbaides the nearest-neighborinteraction (3.7) also the second-neazet-nejghborinte-raction has to be taken into account, Only by the inclttsioa of suc.hmatrix elements beyoad the fzst-nearut-neëghbor interaction, cau the Dlike condudion b=d msmima of the i'adired materials Si, Ge aud GaP be obtained near the X and L poiats of tNe Brilloqlln zone (3.8) The improvement of tike condtzctionbandsca,'n alsobe acllievedby an enlvgement of the b%is set wilbsn the fzstmearest-neighborapprcfm'rnatioa. For instance, the periplleral atomic states o-xn be approximatedby e.n excited s state (denoted The additional s* statc couples with e*)(3.9, 3.10). .
88
3. Bondingand Bnergetks
the aatibonding p-like conductionstates near X or L and pushesthesestates towazdlower energy. The neglect of overlap (ii) has often been viewed aa a seziousdrawbac,k of the ETBM, since simple estimates fnd SabLm.Aj) betweeaatomic orbitals on nearest neighbors to be of the orde.rof 0.5. However, according in au.y case a'a ortkonormat set of basis Atnctioms may be to Löwdt'n (3.5)) constructed without loss of local symmetry. The restrictson to frst- ar.d perhaps second-neazest-neighborinteracvtiozls(iii) in the Hamiltoaiaa (3.7)is also aa mssllmption. A justiâcationcltn be given by viewing tlle JQ:(.R., .&j) efecvtiveaveragehteradions Jueto the Mterpolation scheme.The îttîng procedure of known electronicbands or othe.rquazttitîes lduces the partial Snclusionof longer-rangedinteractions izl the Xective quaatitites. The smmetr.g restridiol in (iii)means that the intra-atomic matzix elementscan be replacedby one-center integrals wheremsthe interatomic rnntrix elementscan be treated as t'wo-cente.rintegrals (2A. or Witk localized wave faactions /atœ Az.)of the type given in (3.16) can the intra-atomic (i m j) matrix elememts of the J'hmiltonian (3.7) (3.17) be written with the symmetr.v restdction in the fo= -
'
-
= Hab(A4 , K' )
Xb
eu +
j
d3z2) (ls) G =
-
(3.30)
for the two sp hybrids. They are plotted in Fig. 3.4c. 3.1.5 Bonds and B=ds
The consequencesof the hybridization stage (Fig.3.4)and tke interactioa cff hybrids (Fig.3.5)for tEe azlowedelectronic states can be eazitydemonstrated for tetrahedrally coordsnatedsystems, e.g., covateator partially ionic spmicozductors c'oestn.lll'zingin the dinmond structurre or the zinc-blendestrac= follows with the fcc Bravais ture. The lbmiltonian matrix Iïlàbllk)(3.15) lattice po>ts (A) and neighborsof an atom along the positive or negative Siucet'wo atoms A and B are i:a one J'n:'t cell, tetrAedron vectol's dj (3.26). one ftnds an 8x8 matrix (3.7)
3-1 Orbitn.7qaad.Bon
el
0 O
o
SA
Egagz(%) Eapn
0 () c
-
(k) Bvpgz(A)Espgh(1)
Aaagz (J4.E'irs4l/o) Erugzqkj -.Esp:z(%) o .-àspgzlkt.Ebvg4(%) ,EbzJz(J4 EnvgnLkj o o o .E)z.vgatpl scvgzllèz'==.vz(k) s) .-k.pg.iLkt eBO (1 0 0 .*l.,#2(b)-Xap>;(%)-Xop#1(A)-Xx,p9:(1) &Bp 0 O 0 BspgqlktSwzp''z(h) BzwgL (1) Ecug-o(D) cpB o 0 c Bnpgllhsfwvp't(h) Em=g*z(â') fa,yv;(=) cD 0 Exp.q.z(k) E=ugq 0 0 (*) .%og*L (%) .FQp,1(k) P c c
S'A >
.
(3.31) hteractions with the composite matrix elemeatsof the ftrstrmearest-netghbor
(3.2:) zas = 4vuv, Eap= 4K,./7i
2ap=
-4!V./V%
.pu
=
En
=
4(k;p.+ 27,.,.)/3, 4(vr,w *9p=)/3. of the phasefactors gjLkj(J=
(3.32)
-
The stlmq Bloc,hwave vedor k, 1
ipa,
are
+ e kud, +
.çt(k) .,.jge 1 spaz .ç2(k) '; Ee + =
=
::
ipaz
J3(k) 'k Ee 1 l/od.z :4(2) 4 (e =
=
-
dto-fnedby
-
-
eipaa
-
eiaas
es/oax
eioaa
aaa
eipla
e
:aaa
+
e
-
1,2, 3, 4),which dependon the 3D
.j- (/Tea.j ,
ppwj
-
,
eipaij
-
(:.a3)
,
+
emaxj .
ca'a TMs LCAO I'Iackiltoaian matrix for the zinc-blende structure (3.31) be easily diagonnlizedfor certnsmhigh-symmetry points and directions i'a the bz,lk fcc Brilloplin zone. For instaace, for the direction k = k(1,0, 0) Fig. 1.25) along a cubic nan's betweenthe Mgh-symmetry poizztsF and X (see one obtnsnsin the lirnit of a dbmond-stract'are crystal (A= B) 'with lattice
(3.7)
cozustant tu
c1/2/3/4(k)ep =
:i:
c
2
-% *1
(f.2.cos2 1-Jj + Qv sin (k< )j 2
(3.34)
for the two pg/z-lilte bandswhich are twofold degenerate,azzd
) (s-
ss,e?,,,s(p) -
1
jp -
:
+sp +
'+' (,-,-û )q 4 l sin2 + .ss2p (k-% 4 )j tk-t)
(s-
+
s-)
cos
,
sa
+
(s..- sss)cos
a
notation is tha,t fo'ar for the Btr spz-like bands. Here the meaning of the *:':F They are + + +: + +, + -, combiuationsof the two signs n.='.qe in (3.35) for the bandsu = 5, 6, 7, 8. an.d .
-
-
-
-
-
3. Bondingand Bnergetics
> > U1 u X c
UJ
Fig. 3.6. Ba'adstmzctme of a sinconczystal witk.in the d:st-nearœt-nerghborapprovt'mationand mstngan ap3 basis set. The valenceband mnr'm= (r1s)is taken 1uszero energp Tke re#onof the Aamdamental gap ls shnzled.
There
are
m=y
sets of frst-nearesfmneighbor tight-binding parameters.
one fnds describedin (3.22) Withsmthe tplmsvezsal' moêel of Harrison g3.1! Sss = -7.28 eV, Bsp = 4.52eV, Szz = 1.76eV, az.dEn = 5.24ev for silicon bond length dqq= 2.35A. Usingsa = -14.79 ev and sp = -7.59 with a b177k an.d(3.35) are plotted in Fig. 3.6. The uppermost vatence ev the bands (3.34) b=d varying Som J'cs,to ,X4 symmetry charader is twofold degenerate.Togethe.rwith the lowest conductionband vary'ingfrom Jlz to X1 it surrounds the fl:ndn.rnezttaâ gap region. Wheremsthe valence bands are reasonably desczibed by tlle Ast-nearest-neighborapprovsrnation and the spn bmsisset: the cpnductionbands show features it disagxeemeutwith experimental observations. Thls holds in particular for the position of the conduction b=d minirnttrn in the BZ, which is expeded to be situated at about 0.85TW.J.ustead, the lowest conduction band îs înevitably predicted to zise in going frozzz .;' to X. However, with the inclusion of a peripheral .s* state in the bae set interactions tke dispersionof the lowest or of tlle second-nearyst-neighbor couductionband can be describedcorredly. '
2.2 DanglingBonds
3.2 Dangling 3.2.1 Formation
There is
97
Bonds of Dangling Hybrids
natlzral quasi-chemical view of the creation of a surface, as a, by brexkeirgof interatomic bonds. Such a simpMed process accompstnied quasi-cimmlcalpictme can conveniently be discussedin the fmmework of the tight-binding appronn'h described izl Sec't.3.1.2 =d, hemce,for homopo1a.rand heteropolar semiconducztom with n.7=os%directional bonds.' I'a sue.h crystals the eledronic structuze is deterrnînedby hybrid ozbstakswhidz are directed toward eacz other and fo= bonding orbitals as sEownin Fig. 3.5 for the t'wo hybrids coupled by %. When creating a surface with a certain orjcntation suchbonds and bondir.gorbitaksare truncated. J.nthe ideal cmse there (withbl:lk-terrninated atomic positions a'cdno electron redistributîon), appear hybrid orbitals which are directedout of the stufacear.d remaiu ttzlbonded. These orbitals are to.lled danglLnghpérrld.v(Fequeutly, but less apFor the discusslonof these dangling hybrids propziately,also danglingbonds). or bonds, prototypical systemq are the (111) a'ad.(100) surfacesof din.mondtype crystals. TMs is illustrated in Fig. 3.7 for silicon. The âgure showsthe a
(a) .'
.,.-.-.'
,'
.'..', '--.,-.--..'....'-'' ...-,'
'---------''''''''''''''
1.( E1
t
VV X# .%py,; .-----... -----,,,,,,,. .y. -----
-
C
------.
,
.'
-,-/
..
E.l
,
)
y ygjyy, j
Fig.3.7. Contour plot of the total electron density itz a (110)plane intersecting l (111) plazzeat zight a'ngles:(a)b4'lk-siticon; (b)ideal Si(111) sttrface.A contour ppacing of 0.10 1/A?is used.
3. Bondlng and Energetics
98
(b) e' t' Tb
#' #
e
2 e
e
1 '
1 .
-&'
-
A
*
2e .------mw-
I
------
:'
l 1 1
k
3
1
. '
I '.
I
1
,
.7..i.
j j
e
?' #' l1 e' # I 2e I
1
e
1
1
1.
I t
.p
1
1 z
1 1
)
.F
I I
1
I
I
I 1
j
1 l
'
2
1
I
'
1
.*
t
v..
4
e
e
'
'
ç A#'
,
J 2 sp3 hybrids in the Pig. 3-8- mustration of bozd cutting durin.zsmfaceformation. four bonds surrouztdingan atom in a diamond-or zhc-blende-sirudme cryst/ are sarface.The surface planœ are ùatchedwithin surface; (b) (100) shown: (a)(111) the little cubes.
total electron denzity in
a
(1ï())plane for a
btzlk crystal and an ideal
(111)
shlr6nce.
The cutting of bonds darizzgsurface formation is d=onstrated in more detai) in Fig. 3.8. It showsa little cube with edge lengkh co/2that contains orientation iu a climond or the four bonds around one atom. For the (111) strudme, two types of surfacesaz'e possible.Oneof thesehwsone zinc-blende dangling orbital per surfaceatom while tile othe,r haz three daagtinghybrids. here only the fzst type, the singledacgliag-bondsmface,whic.k We conside,r is thQnatural cleavageplaae of diamoud-structmeczystals.ln ma'ay cases it surface,sinceonly one bondhmsto betruncatedper seems to be the favorable 1x1 surfaceuzzit cell. Bac,hsmfaceatom hmsone dangliug hybrid perpemdicu1arto the surfaceand three bac,kbondswith atoms i.a the furstuuderlayer.In. tEesecondcase, the formation of a triple daugling-bondsurfacewoltld reqttire tHe cuttmg of thzee bonds. Three danglinghybrids would occar per smface atom which Ls bonded to att atom in the naxt atomic layer by one back bond pmlle,l to the sztrfnne normal. Por diamondtllllsuc,htriple êangling-bond m,rfnzveshave indeedb*n dlrœulr.d?d i3.13!. cax two bondsbaveto be tmmcated dlxring the formntion of ln the (100) the stxrfn.ce.Thts is reprxted in Fig. 3.8b. Each sarface atom has twolsp3 y bulk-lilcedangling hybrids.It is bondedto atoms in the aext atozniclayerY. two back boads. 3.2.2 soduence
on
Electronic States
The allowedeledxonic states of a, hzlfspaceare ve,rysensitive to the presence of dangling azd bac,kbonds.This can be demonstrateêby tbe investigation
3.2 Dangling Eonds
99
of the band structuze due'to the presenceof an ideal sttrof the c'hanges face. For two orientations the projectedbulk bacd strtzctures are plotted izz Fig. 3.9 for a, silicon crystal. Accordlng to the procedre describedin Sect. 1.3.3the alloled ban.dsof a Si crystal are show.afor wave vectoz'svacyingizl the correponding surfaceBrillo':sn zone. Besidesthe Alndamental gap regiou between the occupiedvalenceba'adsand.empty conductionbands,one n.lrxn bands ms a consequenceof the forbiddea observespockets in the projecvted regions îzzthe bulk b=d structme. M1 theseregiozzsbeing forbidden for bulk states. states allow a, clear identlcation of surfaceuderived 1 surfacewith half-fttledda'n.In spite of the faet that the ideal Si(111)1'x gling boads has never been observed,we discussa Si imlfqpace vith sucb a surfacefor the purpose of illustratiom Tke llnit cell and the BZ are sbown in Fig. 1.6 and Fig. 1.22ej zespectively.The irreduciblepart of the BZ for the space group p3m1 (Table 1.4)Lsindicatedin Table 1.6e. The most prominent efect of the smfaceis the occuzrence of a band td' in the A'ndamental band structme sllown in Fig. 3.9a. The corresponding gap of the projected electronicstates are rnninly due to the drgling sp3 hybrid orbitals as shown i.n the backin Fig. 3.10. Only a small part of the wave hlncwtionksloenllzeè (a)
Si(111)1x1
(b)
Si(1O0)1x1
10
217wave vedor
2Dwave vedor
Fig. 3-9- Band stmzctures of the ideal, bulk-termsnated Si(111)1x1 (a) and are calculatedby means of the empMcal tightSi(100)1x1 (b) surfaces.They Tile probinrling method inclualng second-nearest-neighboriuteraeions (3.141. aud the bar.dsof botmd surfacestatœ jeded bulk band Gtructure (shaded rebozbs) lînes)of types $d'1 tbr' ; and %' are shown. Surfacc resonxnces and antizescs(soDd nances in the shadedretonsare not plotted.
100
3. Bonding and Bnergetëcs
((l11J
l
Y
3.10. Contour plot of the wave-Gpnction squarefor the td' state at the # point in t>eBZ of the Si(111)lxlsurface(cZFig. 1.22). A (110) plane tntersecting the (111) surfaceis shown. A contou.r spacing of 0.031/â3is uged. Fig.
The dangling hybrid energetically resides between the bonding bands.Its energy ccT= npx (3.25) is a'adantiborzdt-mg (valence) (coaduction) vet'y closeto the top of the bl:lk rca, valencebands at T' vith energy sp -S= svith the three hybrids in the back bonds The interaction 'yc (3.28) (3.34). Fig. 3.5)shifi;sthe dangling-bondband slightly to higher energies.Siuce (see the point of view the (111) surfaceatoms are secozd-neal'estneighbors(from tlle interadion of the dacgling hybrids at an ideal surface of the blllli-czystal), is weak, =d, tims, the dispersion of the sttrfaceband td' is fairly smaz. TMs bazldpins the Fermsenerr. It is partially Oed, and therefore the ideal (111) Tn .adclition surfacesof group-W crys-talsshould have a metxlllc charactea'. to the dangling-bond-derivedband i.n the fhindamentalgap, the surfacealso rise to b=ds %' in pockets of the projected baad stracture arcund the gjve,s K point in Fjg. 3.9a. They are mn.l'nly related to bonding a'aê antibonding combimlntionsof the t'wo hybrids fomninga b>ck bond. The bandsdb'related to the bondr'ng (antibonding) combinatioM occuz i:a the projectedvalence iateractions bands.Sincethe Emt- azd secoad-nearest-nekghbor (cozduction) of the bnnt- bonds with the environrnent are ed in compa'dsonto the situatson of bonds in the bu)k crystal, such an energetic splitting between blk7k'statœ ar.d ba'ek-bondstat% Lsunderstazdable. 1.n.the cmse of the (100) s'arface(cf.Fip3.9b)tEe suzfacemodiâcatioâ of the electronic structme is more drmsticcompared with that of the (111) smface.First, i:a the (100) cmsbtwo bondshave to be brokec per surface atom (cf.Fig. 3.8).This resdts ia two danglingsp' hybrids.Sineenow thesehybrids are localized at the sn.rne surface site, for symmetzy reasons they dehybridize tuto bridge-bond (tbr')orbitals which are pazalle:to the surfaceand danglinp As'a bond (td')orbitals whic.hare pezwndiculazto the surfaceplane (3.15j. resalt, the orbital chazacterof the correpondsng smfacestates strongly difers from tha,tof the origiaal sp3hybdds.A possibledebybzidizationof the tvo sp3 bond
a'rea,.
3
i otq
3
S7
Sp .u -
De hybn-dQatiOn
..z.
Eo1 14 Bulk
ldeal surface
of the surfacesp3 dattgling hybrids on Eig. 3.11- De-hybridizatioxp (schematic)
a
(â00)suzface.
hybrids into a p orbitaèacd an sp orbital is Mdicated in Fig. 3.11. Formxlly, and (3.27) the zzew orbîtals can be written as with the de6nîtions (3.16) autisymmetric and symmetric linea,rcombinations 1
:$
:$
1
+ pa)) 2)) .a (1pv) 1 1 a s + I.pz)1 + 1.) i'S.p 2)1 vg EIs) 1d) v,j. ((.4:.'P
lbr) xgj (ldp1) =
-
sp
=
=
=
,
(3.36)
of the danglinghybzids, whea'etke surfaceatom is asstlmedto be located'at tite origin. The two states with eaegies =
Sapz -'- ,72,
d;d =
hox + 'ya
cbr
(3.37)
by the interadsotl malonger degenerate.The splitting is determn'ned also Fig. 3.5). of two sp3 hybrids at the sn.rne atom (see trix elements'ya (3.28) The strongest iuteradions be>een bridge-bondaad dangling-bond orbitals cemteredon dz'fbrentsites of the square lattice of s'arfaceatoms mxsnly happen via the substrate and lead to a broadfmingof the two levelssbr a'cdgd (3.37) i'nto surfacebands in Fig. 3.9b. Sincethe smface atoms are second-nerestthe broaderiing neighbor atoms (fromthe poi'n.tof view of the blzlk crystal), comes essentiallyfrom indirect iateractions betweensurfaceatoms locatedat They are due to the coupling of tke surface the same atomic row aloztg(0ï1). of the srst underlayer atoms pointaad.the sp3hybrid.s (3.27) orbitals (3.36) ing along a bond. Nevertheless,the Id' band i'n Fig. 3.9b showsoruly a weak dispersionbecauseof the small efective rr-like interaction of the (d)orbita,ls The esective interaction of the lbr)orbitals is much stronger, iu par(3.36). dtredioa. TV correspondingbaad therefore developsa tictz:ar in tke g()llJ quasi-one-dimensional character.Accordsngly,the dispersionof the (br' band directions. The interaction of the bridge and .1Re is strong along thc 13..11 aze no
102
3. Bondtng asd energetic,s
is the is of r type. The consequence bondsalongthe chaia direction (0ï12 weak dispersion of t:e (br; band ioag tke J'J an.d#J' directions. Tke redacedintexaction of tize sp3 hybrids i'a the back bon.dsZvesrks: Their strongest to states locnqizedat fzrst- aud secoad-layeratozns (3.161. cha,i'adirection, i.e., parallel to tite ,highiuteractioa happens in the (0î1q symmetr.r lines /'J aad J'R. Consequently,the correspondingsurfaceban.ds appeaœia the # and J' pockets, ic particular izl the stomac,hgap arolmd # bnxllcband stmzcture in Fig.3.9b. of the projec'ted
Forces
3.3 Total Energy and Atomic 3.3.1 Bmsic Approximatîozls
Jn Chap. 2 we fokmdthat the smface free e'aergy and tàe thermodpmmic potentiaksof a imlfqpaceare domînated by the total energ.yE of the syste'm, witicll is nearly the interni energy at zero temperatare. A sttrfacesystem coasists of atoznic nuclei and electrons.The electronsare fisFerently bound to the auclei. Therefore,they c,an be divided into tightly bolmd core electrons aud valence electrons.Sincethe chemical bonding and, hence,the smfaceprocesses aze governedby the valenzeelectroits, it is usefutto regard a solid with shprfn.cein terrns of a regular cozectionof ion cores and valence eledrons. For ex-nmple,the free silicon atom with valenceZ = 4 hmsa'n. ion core of càarge 't 6 with the electzonâccoMguration 1s22,,22p6 ) acd a valenceshell of chazge Only in the case of atoms, with -4e with electronic covguration 3.623p2. shaltowtt J, o:r g electronsdoes the picttzre need to be made morè compûcatedby the inclusion of seMcore states that should mostly also be treated as Gence states. One example concerns the Ga3deledrons, in particular in compoundGaN.They are energetically closeto the N2.s the seznicouducting states and, hence, conkibute partially to the chernicalbonds. Apart from of ion cores in thks compication, a hn.lRpaeecaa be regardedas atl ensemble distributed and a valence electron density 'rz@) equilibrpxrnpositions (Rz.). among dsFerent atomic cores. The spatial distribtttkon of the valenceelectrons is usually clmssled i'n.terrnq of metallic, ionic, covalent, molecular, or hydrogen bonds. For the salceof brcvityt here we will refer to ion core.s aa ions and valenceelectrons as eleerons. Anotker basic approximation coazerns the vibrations of the fons with In Chap. 2 we have already learnt resped to their eqpnilibrbxrnsites (.Jk).. that the eiect of the vibratjng lattice on the s'arfacefree energy is s=''n.l!(at for not too high tempeaatures. least, 1tsvadation 9om one phaseto acother) the eqailibrblm geometry of a syste,m Oa thc other hand, i'a detemmlnlmg one lnnmto study many confgtzrawith Nx, NB, atoms of ldnd A, B, tions (Ao.J vith respect to the rnsnl'rnlzationof tEe toti energy B, i.e., the total eneror ELNX, Ns) ...; (.&,.)) ms a hnnction of the atoznic confgaration, aud the accompanying forces(-Vaf(NA, N.B,2..9(A..)))acticg on the ions. ...
...
3.3 Tota) Bnergy and Atomic Forces
103
Tikis couideratiow however,stazts from the assumptionthat the dyaamks of the electrons and the ions can be decoupledso thatj vhatever the tlyzla'a!ics of the ions are, thè eledrons are ia the eluronic ground state of the icstantaaeous geometrs This is the Born-oppenimlrner adiabatic approfmation (3.17,3.1$ It is usnlltlly a good apprnvlrnation becausethe eledroa mass is much smaller t'hn.n that of the ioas. 'I'he electrons respozd idmost instsataneotzsly to chaugesin the positions of the ions. As a consequezme tb.e eledronic an.dionic deree of leedom can be separated.More suctly spensdng,the positions of the ions are parametea's;the total enerr depeencks .
parametrkally
the positions. The Bora-oppenheimerapprov'tmatîonis well jtzstifedfor all stat'ic surface problems. However,it zaay sometim.es breA do= for excitation phe'cal reac tions. For e-xam'ple, irt rnnny clkfprnl'c.al nomena aad clmml reactions aa electronjumpsfrom one enerr surfaceto anothe,r in a lon-adiabatîc way. The approzmation also f>.'11s for non-radiativetrn.ngitionsin solidswherea;n electronfalls from a high-energjrsatrfv.e to a low-enerr surfre not by emitting a photon but by emitting phonons.Electron transpol't (electron curreat, electroa ttlnneling) in surfaceregionsmlty aksogive rise to a Wolation of the Born-oppfanbeimerapproxn'rnation. on
3.3.2 'Potential
Enerr
Surface and Forces l
ta the iirnl't descibed abovetEe total enerr E = ELNX, NB, ..:; (.R./),when is ofïen called.the studied'as a hlndion of the atomic coordmates(Rz.), p0jentïc! energy dur/tzce in the 3 E:=As,.. Jvk-dirnemsioaal. con(PES) (atozaic) fglzrazionspaze, becauseit defmœthe podezztkal enera laad=pe on whic,h that the motion of the the atoms A, B, travel. Since oae hmsassnlrned nuclei is not vezy fast, that the temperatme is very low, aad tlzat the electronic states are at the grotmd state, the termq adiabaticpoterlfùl s'ucface or Bocrt-oppenheimer are atsoused. Jttr/cce Usually a complete PES cxnnot be repr-nted graplzic-allybecatzseof the many coordinates, However, important infovmxtion about the sttdace stmzcture azd energetics can be obtained whena test atom iq.displacedover a surfaze'M:,K a norninn.l geometzy Sucha test atom may be a real adatom or aa atom of the same spedes as i'n the bulk. Then, the coordlmntes of the test atom are âxed in the sudace plane. However, the normal distanceof this atom and tEe coordinatesof the surfaceatozns are allowedto relax. One obtains a spedal PES, that of a suzfacewith a test atom. Axl nvnmple of a resulting total enerr surfre is #venin Fig. 3.12for a Gn,AK(110)1x1 surfu aud a'a Sb test atom. The plm of the energ.yof the tmcoveredstkdaceand the energy of the 1ee (Le., Sb atom is usedas eneror zezo. For that isolated) remson the negative total energy in Fig. 3.12, ...
t)-kaT. Only in this'limd'tdoesthe particle stay long enough i.!l oae dfe, so that equilibration eltn occur. The pzobability of such an elemeuta'cyhoppi'agprocess ks given by the '
behavior Arrhenius (i.e.,activation) r
=
(3.43)
rcexpt-ls/ksTl,
whereihe prefactoris related to the hoppi:agrate'vithout thmrmalactivation. It ksdeterrnlnedby the frequendesof ihe loc.allattice vibrations, thdr interzzal ene'rgyand entropy. Estîmates relate ro to dmaderistic phonon frequencies used are the Einste.inhwuency, t:e Debye of the system. Tjmical valu.e.s vibration (3.26,3.27j. Sucà flwueaqy or ihe fxequencyof the highestsubsirate vith those a proceduregives a n,5nK motion along the (001) direction reqees that adatoms clîmb over tke zig-zag chxinq.In any cwse, the considerable variation between&1: aad EBS.shoid lead to a strong Ysotropy of the sYace dimnsion. Smfacerlllrtlqion also remarlrablylmfluence,sthe homoepiteal p'owtk of a material. This kolds in paztietzlarfor the molecular beam epitaxy of compotmd semiconductorssuc,has Gn.Aqaloag a cubic a'ds. Dependingon the .
lû8
3. Bonding and Eneagetics
growth conditions,whethe,rthey aze Ga-richor As-rich, diferent reconstructions appeaz'.UndermoTe Ga-rjc-hprcpration conditionsone of the most important reconstructions is the GaAs(100)((4x2) Fig. 2.17) smface(sce which has been discoveredrecently (3.29). The potential enerar surfacescomputed for the adsorption of As a'ad Ga atoms (3.304 are plotted irt Fig. 3.14. One fmds the adsorptgonbehnuviorfor ad-cationsto be very dxereut from that of adsorbedanions. The preferredbonding position for a Ga adatom (ACin Fig. 3.14) is located in the tzenches,fourfold coordinated betweenthe doubiy occupieddaagli'agbondsof the Ast-layer As atoms. The calculatedpositiou is supportedby a recent X-ray analysis (3.31), wizich givcseWdencefor a 19% occupation of AC sites by Ga adatozns.However,eve.n for extreme Ga-ric,h surface preparation couclztionsthe adsorption of a Ga atom in the AC posi'
(01k
plïj Fig. 3.14. Potential energy surface for the adsorption of As (>)and Ga (b)on tke Ga-rich GaAs(100))(4x2) surface. The contom spaaing is 0.15 eV. Bright =d, hence, favorable adsorption po(dark)regions indicate m''nlrnn. tmn='mal sitiozzs(trxndtion-state Open tfzlledlcircles represent Ca (As)atonzs. regions). Large'rsmbols indicate topmost atoms. The rn''nn'rnn. AA and AC are describedin the text. From (3.304.
tion increasesthe total energy by 0.1 eV. This type of adsorption does not thereforecoutitate aa eqpll'll'briltmsurfacestrucime. Thc most favorableposition of the As adatom (denoted .&A)correspondsto threefold coordinatioa with three empt.y Ga dangtingbonds.The occttpation of the AA site inueases the total eaerg)rby 0.3 ev tmder Ga-richpreparation conditions.This eaezgy
indicatesa reducedmetaztability of the As adsorption. The PESSin Fig. 3.14 have consequencesfor the dlfhlsion charaderistics of the adatoms.The dlmlsion is rather diFerent for Ga and As atoms. Ga direction,where ehergy atoms preferablymigrate in trenGes along the (011) barriers of only 0.2 ev need to be overcome. The =lnsmpnrn energy barrier for dl'mxsionalong the (01ïjdirection is 0.6 eV. The motion of As adatoms direcdon. is somewhat less anisotropic. It preferably occurs along the L01î) The miniml:rn eneror barzkitrin this direction is O.5eV. lt is smallerthan the
directiom The adatom difh3Ksondepends barrier of about O.7ev in the (011J remarlcablyon the surfacereconstrudion considered.The previouslyaccepted J2(4x2)surface strtzcture (cf.Fig. 2.17)gives rise to a completely dxerent landscapeof the PES for 50th Ga and As adatoms (3.32, 3.334.
3.4
Quantitative Description of Structure aud Stability
3.4.1 Demsity Rlnctional
Theory
Most of modern suzface-strttctttre,stability, ard electronic-structme calculations deali'ag'with the complicated coupled atozaic and electronic problem 'mnllreuse of deztsit.y Gtmctionaltheo'z'y (DFT)(3.34) and a certna'napprovimation for the fhxcitazzgeucozvelation contribution to the total electron-electron approfmatnteraction. Its standarddescziptioais bazedon tize local dezssit.g by adding gradient co=ections However,geuern.lseations tion (LDA)(3.35) or by calculatingthe e-xad exchangeusing the DFT-LDA wave hlnetions aa'e of the electron gas. more hwuently usedto accouat for the im%omogeneity Due to its foz'maland computational simplicitw as well a,s êue to its vezy impressivesuccesses in describi'nggrotmd-statepropertie.s of manpparticle systezns, the DFT-LDA has becomethe domsnn.ntapproackfor calcuhting stmzctttraland electroaicproperties of b'nllcsolids and theîz surfaces.Withia Sect. 3.3.1) for the descdptionof the enthe two basic approvsmations(see of interactiag coz'es and dectrons) the total energy of a sttrfacesystem sembles is divided into .
descibing the repulsion of the bare ions at the positions m cisely, of the
.
or, more
pre-
vith the valence Z.t and the pure electronic contribution Seltrz, (Rz.))dependkg oa the electrondensity. The electronic contribution can be calculated witbln the DFT (3.34,3.35). Tkis theozyis basicatlybuilt on. the Hohenberg-Kobntheorem,whicah proves that the em.ea'gy of a many-electronsystem izzthe Jrotmdstate c.n,m be obtained For a give'n ato'mic confrom lmowledgeof the corred electron density zz@). âgttratiozz(A,.J,the quantity Seltn, is a lznsquefllndional of r.t#. lt (.R,.)) value when obeysa variationi pzinciple in the dtmqity.It acqturesa n75=14731:173 the eledroa densit'yis the corect ttmzel density.The extremum conditjon allows one to map the Mteracting Ne-electron proble,monto the deterrnination of Ne single-pazticleorbitaks.This me--mq eac,helcctron is movi'ag icdependently of the other electroms,but it experience,sa,'n elective potemtial 5r(z)' whic,h emulates att the Mteractions with othe,r electrons. One obtaius the station.ary Kohn-shnm equation (3.354 cores
SkS#J@)= JJMC' (m), :,2
Z= + 1'r@), (3.46) Jm foz' enr'ln independe'atsiugle-peicle state in analog.yto (3-1)but with the Nxs. Here %'(z) and s.f are wave Glnnctionsa'aê Kohn-slmm Hn.rnsltonîn.n eigenvalues,respectively, of norsinteracting âctitious single pazticle,ssuc,h that the correct electron density is obtnlned axs
J:os
=
-
'
zztzl -
J7'zjl#J@)l2,
(3.47)
j
wllere ni denotesthe occupatioa nptmberof the eigenstate that is representedby the one-pazticle wave fhpnction#j(z). I'a the cmse of a 2D trn.nsformxlly replaces the oneiationat system, the e'igenvalueproble,m (3.46) electronScbröasngerequation (3.4)wit: a sjt of one-particlequantdtm nzlmbers J = ?A. The egectiveKohn-shnm potential is deMed as -
(3.48)
+ Vh(œ) + kkctœ). F'(z)= Mo=@)
= The extm'rnalpoteatial Mon@) E( 'tijonlœ R.)is gemeratedat spacepoint due to tEe cores at m with chargeZq.ln z by all Coulombpotentials '?J,Q@) at As' the case that oaly valenceelectronsare studied,thesepotentials 1(oa@l However,sacha stmplefo= of have to be replacedby pseudopotentials(3.36). Fiontz)is only valid for local pseadopotentials.For the non-localpotentials towa'rd non'adedin the majorityof modern applications, a geneaaDzation locat operators is reqaired.The Eartree potential -
coais directly givenby the electrondemsity.The emxc-llaztgetxl-cola'eiationtc) tribution to the total potential follows as the Anmctîonalderivative of the XC contribaiion to the totaz electroaenergy SeT.In.general,thc XC contribution one b.as is ''nsrnown. lu the Famework of the comrnonly used LDA (3.354
and, hence, J dSam@)Exc(n(=))
Wc @)
=
with exc
d. dn
(zz)as
t
hexclzzlg
a=(.)
the axchange-correlationenergy per eledron of
a
tmiform
electzon gas of densit.yrz. The XC eue'rgyoxcln)has been calctzlatedby severalapproaGessuc,has and quactttm Montc Carlo metkods perturbation theoty (3.371 rnnany-body is expressedas an acalytical'htmction For practical calcGtions, excln) (3.381. of the electron density.Tiutsca'a be illustrated by the LDA of the exc,hauge term. Witbsx the X. approirnation (3.392,
9
c
exctzz) '-û'je =
3
(3.51)
-?z K
fozowsvwitiz the parameter tz adjustabiein the intervat 23 S (z S 1, in orde,rto accotmt partially also for tEe electron correlaltiop..Freqientlythe is tzsedfor Exctzzl paramdrization of Perdew and Ztmge,r (3.40) Many mtems such as magnetic transition-metal suzfaces,reconstmzcted semiconductorsurface with remnsnl'ngdaugling boads or the dissociated moleculeson a smface involve unpaired electronsoz molecllln.'rradicals and, th'tzs:require a spin-polarizedmethod.1.nthis context it is the spin demsit.hr hTnctionaltheot'ywitlnp'rlthe local spin densit,gapprovsmation (LSDA) (3.41). of the correlation eae'rgy on the spin polarization is Usually the dependence replacedby the same htearpolationas fotmd for the e-xclzauge energy. For the majorityof surface calculatiorusthe LDA or LSDA gives a sufûdent description of exchangean.dcorrelation in 'ltke gro'tmds'tate. Surface geometriesaad the nattzre of surfacebonds a're reliably described.Eoweverl there are also cases where the original nozslocality of the XC energy lzas to be talcen i'ato account, at least paztially, alread.yfor grolmd-statecalculatioas. E'aa frst step, corrections related to the pactieat of tllc local elHron densityaze addedto the XC energs and the geaeralizedgradientapproximaLs employed. Gradieat-correcteddensity Glmctionalshave been' tion (GGA) Beclce(3.z.t4), and others. For eqltz'librblm suggeted by Perdew (3.42, 3.43), surfacegeometriesthe GGA usually gives results sinnllar to the LDA. Oaly hencel the overestMation the tiny tmderestimationof tlle bond lenzths taad, is lifted an.dthe bondsare weakened.For the descziption of the bond energitws) readions on a stzrface,in particttlr of tmnsition states, where of cl.tarnt'caz bren.ld'ngof old bonds and mn.lrs'ngof new bonds occur, the GGA appears to be supedor to the LDA, i.e., the LDA often gives even (ptnlltatively incorrect .
3.. Bondingan.BEnergetiœ
112
exampleiu this respec't concerns the physisorption of pazticles.Unfortunaiely, vau der Waalsinteractionsnear smfacesare not corzedly describedic b0th LDA a'nd GGA. Moreovea,tEe DFT-LDA and DPT-GGA elmmot cozrectly describeucited electronic states (d. Sect.5.2). In Sec.5.4 anotb.erdeîcit concernicg the imageDke behavior of the singlepazticlepoteutial will be discmssed. resalts
Mothe,r (3.45).
3.4.2 Baud-structttre
ard Interaction
Contributions
The Kohn-en.m theory (3.46) atlows a splitting of the total ene-rg.rof the elHronic sylem ia the groundstate (fora given atomic covguration (.&)) into two physical contribuiions
.E'olt'z, (.R.)) .ELt'n, t.R:)) .àu(0l=
(3-52)
-
T'he band-stmzctureenergy
f'bst'z, (a.)) =
J7zzjv
(3.53)
?
#vesthe total eneror of Ge ron-hteracting system of electronsoccapying the Kohn-shm states j. The sccondcontzibution '
.j:v's(œ) wd:xc (s) z= (pz + EoeLn) ,.:(.) k
j
j
=rs@l
(3.54)
describesthe e:ectron-electronicteraction whic,h acco=ts for double cotmtill (3.48) icg (atletkstin the Hartree contribution (3.49) but also paztially f.n the XC contribution) în the one-electronenergies sj (3.46). The total energy of the suzfacesystem, B, depenas'ngorz the particular atomic conâguradonIJQJ,was de'ned originally as the lrinetic energy of valence eledrons plus the potential emergiesof electronvectron, eledronion, aztdion-iot icteradions. From (3.z14) and (3.52) oae obtxsms + fost'n,: F(.Nk,NB, ..-; fAs'J)= Sbst'n,) (As'.).) (Rz.))
(3.55)
with the elearostatic enerr of the sudacesystem
.E'estn, (.&'))= lion-iontta.l) .feetzzl. -
(3.56)
This splitting of thc total energy is rather convenient for nlpmerîcaltreatment using both a plane-wavebmqisset or a basis set of locldl'medhTnctiozus.The electrostatic energy is partimllarly impohant for the behsvior of surfaces s:ldxces. of compoaads.Examples are the vdzious reconstrtzcvted GGs(100) Their reconstructionsaire remarkablydriven by a temdencyto mimlrnise the loztg-racgeeledrostatic interactions (3.461.
Descriptionof structuzeacd Stabitt 3.4 Quantitative 2D trn.nqlationalsymmet:cy the moment'qmcaa. be apklied.The bazd-stracture ene'rgy space formulation (2.47-3.40) with J = vkt the baad iadex v an.clthe wave keeps its representation (3.53) vector L-i.zzthe smfaee BZ. A vczy powerq:7 approac.his the repeated-slab which allowsone to use a'Et artiîcia,l 3D transapproxqmation (cf.Sec't.3.4.3): is expandedinto pla'n.e Eationalsymmetzy The electrostatic energ.y (3.56) wsves defned by the vectoo of the correspondsngreciprocal lattice, G. One fmds 1 = fzat + VxctGl ëxcllT Jcolyzlztt7l (1 EosLn, (AJJ)
For ordered slTrfnces with
a
V'5,(G)
-
-
-
o
+
akx + 'Yswald
wsth the vobnrneperx atöm, flzt, aud the average valemceIrtrnber,
(3.57) X. The
overcotmting of the electron-electron interaction is govnrned by the Fouzier of the electron densit'y, the Hartree potential, the XC potential trn.nsfoz'ms aud the XC enea'gy'per iectron. The constant a:
=
1
)7 .%% :
d3z
z$e2
+ kqostm) 1œ1
the degreeof repulsivenessof the ionic potentials or pseudopotentials averagedover the atomic basis in the unit cell. The Ewald eaergy measmcbs
v=ld
=
1
2.+:J7 y
.Ekn-ozz((R;)) -
Z?e2 d.3œ*
(3.59)
(œl
is obtained by zemoving the divergenceh'om the ion-ion Coulombrepulsion. tn the emseof a dinmond-stzuctureczystalwith t'wo identical atoms i.nthe 'tmit cell Jswala= -2.6936 Zzeljaoholds with c,: as the lattkc constant g3.50j. The momentlnrn-spaceformn7sqmof the total energy allows a simpMed representation of the Hellmg.nm-Feynman forces(3.39) adiug on 3.21) (3.20, the atoms in the case of non-erlslsbrbtrn atomic coMgtlrations (a.). There are 'two rlilere'at somces contribttting to DmE. One is due to the explicit dependemce of the total energy on I.R.Jand the other to the implicit depenThe dencetbrough the solution %(z) of thc Kohn-slna.mequation (3.46). latter contribation vanishesfor a self-consistentsolution of the Kolm-shn'rn equation, if the bac;isset msedto reprHent (#j@))is complete (3.47, 3.51). Ihis can easily be show'atusingthe LDA, more exactly an XC potential (3.50) The restridioa to tbe 'flrqt contribution corremsponds in an Xo-like form (3.61). to ïhe spplication of the Hellrnn.nn-Feyrtmarttheorem (3.20, 3.21)However, Usthe rcult correspondsakqoto the older force theore,mof Ehreafest(3.521. for the representation of tke eigenvalues; ing the Kohn-sbnamequation (3.46) a rather simple expressionfor the forces one fnds with (3.45) .
3. Bonch'ngand Rnergetics
zszjcz (x; Rt ) Rs% 12IA,' Ri I 3,/ iA,' ,
.% =
-
-
-
-
Re isa,
5-25-2JgJ(G)goo(G)ciGAJ J
with
(1.60)
c
bebzgtke Fom'ie,rtransform of the tonic potential situated at 'J:oa(G)
term ca'a be evatuatedusiug the Ewald plmmatioa method . describesthe force on the ion core at m. d'aeto the bare ions at sîtesRj. The secondte= representsthe forceof the esedive electricEeld
.%. The 'smt 3.53!lt (3.50,
of the density on the (pseudoldemsity indaced by the ionic (pseudolc>n.rge Kence electronsor vice versa. Therefore,the atomic force Tk (3.60) #ves the total Coulombic force excertedoa the ioa at a.. For a given covgaration (.R,.)all quantities apppltrlng in (3.60)are knowa. Only the valenceelectron llas to be calcalated self-consistentlysolviug (3.46). (pseudoldensity 3.4.3 ModelMg of Surfaces
.
Ls lœowp one has to solve the Eve,n if the actual atomic geometzy (.Rs.) for a semi-ln6nstesyste,mself-consisteatly.Since Kohn-sham equation (3.46) the respective uzkitcell is i'n6nitely long in the direction perpendicular to the surface,it contzsnK sn6nitely many atozns. Thus auy standazd bulk bandstractm'e method leads smmediately to (x x c;c matrices that need to be diagorxlszed.Sincethat cauaot be achievedone resorts to either substitate geometriesto simulate a sudace or to alteraative formal approachcswhic.h or do not necessitate the diagonxh'zationof a Harniltonian of the type (3.7) The model of a sernNin6nite soûdis mostly studied together with the (3.46). simple jelliclmn model for the material. In the jelliummodel the positive chaz'ge of the atoznic nuclei is simply represemtedby a llnifoz.m constxnt positive backgmundimsidethe solid a'n.dzero outside an âp'propriately Gosea surface Plane. Maay, but by no men.nm all phenomena iu surface scieace are rclatively shorwrange h ttature normal to the surface. Usually, the surfaceregion cau be restricted to a few atomic layers the nlnrnberof wilicll, howevez,has to be tested carefully depending on the surface pheaomenonstudled. The restriction e-qn be used to model the surface directly in space or by certain pemturbations. lt is possible to choose geometric models wikich aze small enough to be tradable by todayls electTordcstracture methodsbut aa yet still lazge enoughto be physically meaningful.Systemscontnimimgof the order of 10@ l:'nl't cell can be treated a,t a frst-principles level usually atoms per (repeated) basedon DFT 'with today's progrnrnq and compater 'hltrdware.To deal directly
-miczmâucti'ngsuzface.Similar apprnnryes hold for IE-W compoundsor group-rv crystals but vith pseudohydrogenof au appropriate Meace chazge. For group-N atoms tEe dnrgHg bonds 'c-q.n be satmatedwith tnte hydrogen, Z = 1. T'he bottom laye'rsof the pseudohydzogen-covered slab sidesare kept Fozenduriug tite surfaceoptl'mlzzations.They simuhte the bulk regionsof the under cozusidezation. snmsconductors The boading and antibondingstates re.
'
.
(b)
(a) As
As
Ga
Ga
G
0.75 chargedpseudohydrogcn
1.25chargedpseudohydrogen E'ig. 3.18. Lower part of a GaAs(100) slab with satvration by lacdonally charged hydrogen.ls=xlldots). 80th the lowe,rM-temnsnated(a)and Gatemnlnated (b) surfacesare shown.
3,4
Description of Structuzean.dStability Quantitative
119
latedto the cationtanionl-pseudohydrogen bondsshodd beremoved9om the
gap region. This haz to be checlcedin detail. For instance, for cltun diamond slxrfn.cesthis requirement is not valld for the CC-Hantibonding states (3.55). rfhey appear in the Alndxnnentatgap zegion neaz the projected bulk conduction bands. After passivationof one shb side the geometry artd tike electronicstate of the surfaceof i'aterestcan be stuciied.Eowever,two problemsremain. The t'wo slab surfacesare still inequivalent and electric âelcls(although much smn7ler) '
rematn in both the slab and the vacuzlm regions. For thick layers, however, their infuence is small. 1n.the majorityof applicationsthe electronic singlepaztide states àre we,llseparated euergetkallk. Thrtsallows separate studies for the t'wo slab sides.Unfortunately,this is not possiblefor the total sudace energies.According to (2.39) oae calculatesthe s7lm of the absoluteenergies of two diFerent surfaces.In order to divide the sll= i'ato the contributions from the isolateduppe,raad lower surfacesof a non-symmetric shb the eaerr density formalism of Chetty a'ad Maz'tin (3.60) ha,sto be used. Only in cases of lucky coincidencecan absolutesurfaceenergiesbe derivedby combinatioa of results obtaiaed foz diferent shbs (3.551. The iaeqzzivalentsurfaccssalso izdudeprobleznsin tlle calculation of other surfaceproperties: e.g., the opticat propelies of surfaces.Iu this cmse the contribution of the pseudohydrogencoveredslab side to the optical propeztiœ is separated by the Lutroductionof a linear cutoF hïnc'tioa in the optical trnmqition opvator (3.614.
Cluster methods. A sllrfnzte s'yste,mcaa be modeledapproxirn'ately using a seciently large cluster of atoms. In prindplej ezstiug methods of computational qttanttlm chemiqtryor straightfozwardgenerxlizations of sac,hmethods can be employed to deal with .suc,hclusters (3.62-3.64). A cluster-type approac,hh,asthe basic advaatagethat ab initio calculationscan be pedbrmed and total energies zwzm be rnsnsmszedwith regard to atomic covgarations for s'ys-temswith not too rnymyatoms. DMculties, however,arise in studying between genuine localclustersof suocient siz,eto enable the ctiscrlmn'natîon ized sadacestates and states whoseloe.xllzatibnis merely a zesaltof the 'Fnite size of the c'luster. In particalar, clusters whic,h enn be handled in practice d.onot give good bulk referoce energy value.saud the resulttng ene'rg.ygaps are too large. Identifying the enerr position of a given surfacefeatare on a,n absolutescie may be problematic. The enlargedenergy gap in small clusters results Fom the (particle in a tktrcxe-dl'rnemsiolli box' behavior of the eleerons whic,h are reQectedfrom the cluster boundariesgiving rise to standiugwaves insteG of true localizedsmface states. A secondclassof cltuster-typeapproaGes hmsbeen developedto avoid the shortcomingsof 6nite-cluster methods:the smcalledefedive ield methods. The interaction of a Mnsteduster with its real environment is representedby an esectiveîeld to replacethe botmdades(3.651. The clttster-Bethe-lattice method is the most comrnon method used i'a dexlr'ng with this embedding problem (3.65). lt hastmmedout to be a powergtll tecbrllqueespeciallyin the
12O
3. Bondingand.Bnergetics
of amorphous matezisls,for instance Sioa, but has never been directly usedto model slTrflmesor, at least, substTatesbelowsurfaces. periodicity of surfacesor interNote that the remslnî'ng t'wo-dsrnezzsionat thus facesis not exploited $a any of the possiblecluster-type appxoaches, oaly iaformation of an integra: nature (inegectintegrated over the sudace ca'a be obtnlned by thesemethods. Local dcnsitiesof states Bzillollsnzone) in ternns of the Greea's'hmctionof the cluster can be calculated,but not the dispersionof the bands. case
rrrarqfer-matrix method. More iadirect simulations of surfacesare f1-1. quently basedon a representation by layer orbitals azd Green'sRtnctions. The layer orbita'b take advaatageof the 217trn,nnlationn.lsymmetry but varjr with the Ipprnberof atomic layersbeneaththe surface (3.66, 3.6'4.The layerorbital representation therafore combincxsthe advartagesof localizedorbitals solvesthe The Green's'hzncvdonG@) (3.5)with those of Bloch snt'rnq (3.13). = 1 belonging to a homogeneous inhomogeueous equation (S cS)G@) Sfthrödinger-likeequation of type (S cS)#= 0 (3.4,3.6, 3.46)vith the siugle-particle HstrnilioaianS and the overlapmatrh S. Trn.nsfermatrices relate diferent rnxtzix elements of Green's fundions to MC,IIother! itt the layer-orbital representationof surface calcttlations, they couplesuc,hmatrix elementsfor digerent atomiè layers.Using trnrsfer matrices of this type, the Gzeen's Alncvtionof a semi-inGnite crystal can be calculated layer by layer. The trn.nnfer-matzbcmethod avoidssome of the problemsof the slab method and the cllzstirmethod by dealingdirectly with a semi-lr6nlte crystal. Details whereit is of the method hivebeenêescribedby Yndttrnsnand Falicov(3.682: who employed for a modelHpmiltonian, and by Mele an.dJoannopoulos(3.691 surface.Bound states must be found applied it in a study of the GaAs(110) as poles of the Green'sAlnction in the gap amdpocket regions, in contrast to the more direc'tamat-once mspectof the slab method.However,the transfersurfacestates. I'a principle, matrix metlloct is not restricted to wemlocn'.lp'zz,ed. the method also gives an accarate descriptionof resonances, antiresonances, an.dother featuresin the bar.d continua whateve,rtheir localization properties may be. However,identifyin.gsuchfeattzresinvolves subtracting the local densities of states of the semi-snflnite atd sxtfmitesolids wllic,h are large and nlmost equal qlllmtities. The trazmfer-matrix metbod has beea developedto attàck the elecvtronic-stractureproblem of layered systems. lt caa aksobe used to solve the surface geometry problem exadly. Its implementation îs, a, slab method. however, considerablemore cTlrnbersomethaa, for e'xaample, -
-
'
Scattering-theoretionl approach. A.n exad solation of the surfacegeometry problem cac also be obtained by anothe,rmethod which is bmsedon the for localized pertmbations of mystalliae origi'aal Koster-slater idea (3.701 The sohds.This idea was 6r!4t extendedto ideal surfacesby Kouteclc.f (3.711. method starts with an ''nBnite blllk' crystal, whose eigenstatescan be reltraaslational ative'ly emstlydetfarmlnedby exploiting the three-fqsrnension&l symmetzy azd then e-mployingthese solutions to constract the bplk Gzeea's .
3.4
Descdpticn of Strudureand Stability Quantitative
ltmction G0(s).Finally free surfaccsare createdby introducirg an sppropriate short-rangepertlzrbation.The Greem's Atmction G0(s)of the pefect crysta1 entirely contaiu the electronic structure of the unrelaxed surface.Tlms, the method hcorporate,s and retalns all the bllllc propeaties suc,ha,s band continua, ban.dgaps, etc.) and the alterations resulthg from a cleavagecan
be obtainedOectly 9om scattedngtheozybasedon G0@)a'cdthe cleavage potential without subtracting large quantities. Both conceptttally a'aclcomputationalty tMs is a rns.joradvaatagecompared to the othe,rmethods,in wié.c,hone attempts to ft'td the blll'k as well ms the sudace-inducedsolutiors dizectlyj from either a slab, a cluster, or asemi-im6mitesolid.Sizzce the method requires the Green'sRtnction for the perfectin6nite solid.,wllie,hcan be folmd with moderateefoz't by stnrnrningover (partsof)the bTtlk'BzilloTnsm zone, it is computationally Iess clrrnbersome thaa the transfer-matr.ixmethod..The implementatîoa îs not as simple as in the case of the slab metbod.,but once implementedand if thq surface geometr.g is lmown, it is more eëcient and
reliable.
C)
@
@
O
O
*
@
C)
O
O
#.* Nl
O
C)
l
(a)
O
@
@
C)
O
@
@
C)
*
@
C)
èC) '
.
(
u -w cK)
(b) ::
j
Fig. 3-19. SGematicgrapiusshowingthe creation oftwin surfaces:(a)bond-cutting hethod; (b)removal of atomic layers. The resulting uppnvost surface atoms are shaded.The arrows indicate (a)which interatoznic matrjx elementsel'e set to be zero or (b)which intra-atomic matrbc elememtsm'e shifted in eneror by the value u. Mter (3.661.
l22
3. Bonding and Energetjcs
The Mvantagœ of the methodhaverettlted in cation in the theory of relued
a
relativelywide appli-
reconstructed suzface. Ia pazticulaz,Pollmxnn and coworkers(3.6% have employedtlle scattering-theoretiY method in dealing with ideal aud reconstzuctedsmfacesof smmiconductors of (liamond, Gc-blende or wurtzite stzuctttre 'by mtmns of the ETBM ms well as tEe deusity-functioni fozrmlsqm.The basic idea of the method (3.66:3.67) is to divide tàe Hxml'1t,0n1=H = SD + U jnto a bulk ezmtribution S? arzda pertmbatfon U simulatiug the presence of a sarface. Then, one has to solve a Dysonequation G = GQA-GDUG. The b&t reprcentation of the pertmbation U Lsthe layer-orbiial representation. lt allowseazyphysical approaees for U, The e.g., the bond-omingzp,efzWaud the layer-romovdrzzeszoé (3.66,3.72.. tèo approachesare illustrated in Fig.3.19. J.utke 611% method (Fig.3.19a) the ftrstv and second-nearest-neighbor izzteraztioztsia the bdk T-TmiltoGan atomic planesare switGed (3.7)decribhg interadiomsbetweent'wo adjacent o5. Jksa result two equivaleat surfacesare formatly cweatH. In the xcond case (F$g.3.19b) aiomk layers aze forrnnlly (i.e,, with resped to the eaea'gyof the related electronic states) removed fxom the spRe by Klnl'fiingthe atomic eigeaxalues(bya lazgevalue %)far from the eneror region of interest. or
3.5 Bond Breaking: Accompan>g and Atomic Displacements 3.5.1 Characteristic
Charge Transfers
Chnnges ln Total Energy
In order to understand the efect of boneg of orbitals aad brealdngof bonds oa the total enerr (3.55) of slzrfax systmnnn, sevezalmodelshave beendeveloped.l'a the force consta'at model of Chadi (3.734 the cùaage of the electrœtatic emergy (3.56) is daqnribedwithin th.e hxrrnoaic apprnvimation for is exthe geometry changes.The contribution of the geometrical n'hnanges pandedinto powers of Factio:aal bond-lengtàchaqges%j. T:e vadation of the attmber of bonds dkNsondis desm-ibedby an additional erergy contribution with respect to this nl:mber. The fradiom-klbond-leagthchangesof two atoms at m and Hj are deMed as nearest-nefghbor qj
=
1.R; Rj I/Qq -
-
1.
(3.61)
In the presenceof a perturbation, e.g.: a surface,taheralative nearest-neighbor displacementsand the brealdng of bonds nhn.ngethe eledrostatic eaea'gy formally accordlng to (3.56) zâfes =
+ WzNsond, + tfaetl S(rAe< b<J
(3.62)
if orly one coatributioa to the second-orderterm, the raclial one, is hzcluded. Three-bodyterms suchms tkose proportional to 6çJEfa(J# k) a.re ignoredfor
3.5 Bond Brealdqg: Accompanytng ChargeTransfea's
122
to azcoantfor the simplicity.Vaaderbiltaud Jcœmnopottlos suggested (3.744 wiation of the number of bonds in the total eyergyby the tezm W/Nbond. The constant W has to be identiâed with the cohesiveenergy pe,r bond. is calculatedfor a given covgttration The band-structttreenerr Eyn (3.53)
and the dezaitioa of the isotkermazbulk modulus B. From small perturbations of the bulk bond lengïhs one derives The pazameters Uz and Ug foilow 1om the equiiibrium conition (.RJJ.
Uz =
2% with F'
=
-
Dms 0ê
957
ms the
-
,
2=o
:2as 0a?
(3.63)
d.o
volume of the consideredtmit cell and ê as tbe tensor
of (3.61)
the Fadional bond-len#hchauges. Lsbasedon the ehxrge-self-consistent Anothe,rinteresting mod,e,l(3.66,3.754 treatment 'tight-bindingmethod.The basic idea of the charge-self-conskstent can be easily illustrated formal)y asizlg a Hartree-llke (ora Hat-tree-Focklikel treatment of the occupation dependenceof the total e-IIG'gieSand of the slgle-partide eigenvalues.Consideeg a free atom or 1ee ion with a valence coMguration s'Hfp, accmdingto the splitsng in (3.52) one has a total energf Sltom Lslnyn.p )=
swy
(tnj);
t--.a,p
..-
1
ngjnyri.j
-
2
*,i=s,p
with the single-pazticleenergies
sv(.(zz:J) c2+ -
('z.f 'zbôlr-&,f, -
(3.65)
J=ayp
the orbital ener#es i'n the couguratîon s'Zprz; trzoa = ation of the neutral atom. Each c.0n551T> Z), i.e., the grotmd-stateco tion js characterizedby certain occupation nl:mbers na and np. The energies intra-atomic interactions of vaUé; (ï).i= s,p)can be idemtifed with egecwtive lenceelectrons in the same or dsFerelt angulawmomentplm state. They may be calculae as Coulombintevals using the correpondsng wave Amctions. f1llfll1 the Jaaak theorem (3.761, a'ad (3.65) The expressions (3.64) Where &Dsazd ooare
c:
=
+z4
0 stozn nw .B' (.s p,zml &n%. .
(a.66)
The eigenvalueeé equalsthe e'hn.ngeof the total energy q'ith the clkangeof the kccupationnumber zzç of the level ï = s,p. Exw-ions l4smilnr to (3.64) can also be derived for tlke total eaerbes of solids or solids with surface.A system for whiG this can exsily be (3.55) demonstratedis a tetralzedrally coordinatedsemiconductor, e.g., a group-N
124
3. Bondingazd Energeticg
stnzcttzre. The most important contrirnxterial crystnllszing in the din.rnond. butioa to the band-stracture eneror is due to the cherniealbonding gî.1). Thks foEowsapprovimately as the sl'rn of all bonding ener#esof the bonds of two sp3 hybrids pomtim.gto eac,hother along a bond direction. Withim the molecttlemode,l(3.:0)the bonding energymny be written as sbond
=
1 + 3%) V'h+ -4Lâs -
s'Ti
with fs and.G as the orbital enargiesof the hybridizedgroup.N atoras in the covguration skpn. The Xect of the overlapp'mgof the t'wo hybrids) S H Slz, has been taken 'mto accotmt in the lowest llinearlorâer. A representatbn liuear ic the overlap integral has beenGosenintroducing an eective matrix The energy W = -(Ws.- WIV. elemeat5rh= (SzzNlzJ.G)/(1-S%). of t'wo ,sp3hybrids. is the negativeinteractiozt energy(3.28) sfjxpc-svppcl/4 In a itting procedme the te= Skk is usedto model also the repision of the cores. Near the eqllilsbriplrn positioas: Ji rxp 1/J2holds for its vaziation the same Followingthe Hfickeltheory f3.1p with the bond lezrh d (3.6â. 3.32 is valid for the overlap iategra). With the Cottlomb dependenceS rxp 1/(J2 of of sp3 hybrids the total enerr (3.55) iategral U = -Y + Guap+ 9Uvp) 16 (Jss a blhlk'covatentczystalconsistingof N atoms is given by -
-
ba:k(x,(a))-
N zs +
so
-
(deq)2(deql' (a.6s)
43-,:-
d
i4J'J>h
-
su
.
fi and é' The eqttilibrintm bond length doq has been iptroduced. t'a (3.68) denote the corresponkding quNtitiesat the equilibrbnm distances. The parametez'sW and S fcoow 9om the conditions of mechecal equilibrbxm in the bnllk, vauislnn'ngbydrostatic pressure, and the relation of the with respec't to the volzzmeto the btûk modulus secondderivative of (3.68) As a result one fmds S = la The dilerence Scoh = SMOm(s2p2) (3.63). deter.&balktN, (4eq))/Ndeqnesthe cohessveenerr per atom. lt is mainlycsO s0s zninedby the covalent bonding 'w Q and the sp promotion energy -
.
-
Scoh
=
A
2Q @p0s0s)(j.tr7s, 2%p+ Uppt -
-
-
-
.
describes the chaage in the intra-atomic electronThe last term i.zl(3.69) electron interacuon due to electron promotion lozrl an .:2.p2confgaration to the skp3 elec'tron confguration. The remaining fou.r parameters follow h'om the ftrst four atomic ionization poteatials deuecl as ciiFerencesof expxessions (3.64). The resulting tight-bincling paràmcters are lksted in Table 3.1 for diamond,Si, and Ge.
3.5 Bond Breaking: AccompanyingChargeTrausfers
l26
Table 3.1. Tilht-bindtng pammeters es-timatedfrom bulk eqllêibrittzncônditions ard atomic ionzzationpotezttials. M values are in eV. R'om E3.75). Pammeter
system of electronsand cores is esseatiallydetevnined by the caheznieal bondsng.Apart from plomotionazd changesin interactior esécts,the accompanyingbond energy 'U'his usuallytaken equal to tite cob.esive energ.y (3.69) , i.e., the heat of atomization, per nearest-neighborbond. Conscquentlythe sarface energy.of a halfspace (2.39) is therefore given by the excess envergydue to the broken bonds (see atso (3.62))Such a picture hdicates cozrectly tEat the msml'nulrn-enea'gjr surfaceof a covazentdixrnondstructme crystal is the (111) surface tseeFigs.1.6 a'ad 1.15)whch is tite cleavagesurface and.an jmportant growth plane. The simplicity of suc,ha boading picture makes it fmscinatmg.However, lt cxnnot dkectly give details of the smface behavior, such as sudace reconstnlction. Ncvertheless,one may also use it to de-scribe surfacechanges. a
.
The
surface of a (111)
homopolarseraiconductorwith one dangling hybzid pe,r surface atom in the normal diredion allows aq easy Gltzstrationof the eimplifed pictme. Using (2.39), and S = âzone f-nds a surface (3.67), (3.68), enezo-per atom (per1xl tlmit cell)of
A
=
lw -5i 2
Eqfor the 'aaperturbedideal sllrfnne. However,such a
smfacewould be tmstable , Ebecaatuse a single electron is placedin eac,hof the danglinghybrids despite its Ereadivity,Suc,ha surfacerepresents a degenerategrotmdstate. Neglectiagfor : a moment the eledron-electron interaction in neighboeg bybrids, one could alsoequallyput t'wo electronsi'a some hybrids andleaveothersempty without :.
3. Bonding aud Energetics
changingthe total energy.Accordingto the Jnhn-Telle,ztheorem(3.77,3.78) a system having a degenezate Jrotmd state will spont=eously deformto lower its symmetzy unless the degenez'acy is simply a spitt degeaeracy. Iu this simple exmple the surfacerelntion doa not moclifyihe smface Gymmetry.The plane of surface atoms tinds to be displacedslightly inward or outward. TIIJSis accompanied by two efects: a rehybridizationof sllrFnce atoms aad a variatioa of the bonds betweenthe fzst aud secondatomic layecs. The reybridization mnc'nly Mueaces the dangMg-bcndbaad in Fig. 3.9. The sAlrfxce atom displacedoutward (iaward) ends with an a-like @x-likel dangling hybrid, whereasthe three equivatenthybrids in the bnnkbondstend orbitals (cf.Fig.3.20). AII orbitaksare characterip becomeplike tspz-likel with lz for the danglinghybrid and la = hed by the hybrid expression (3.23) la = A4 = A foz the hybrids in the back bonds. Without rehvxtion (v = 0), displacementof tke sudace 1, = .k= 3. In the case of an otztvard linwardl atoms by 'udeq
)/(2z Vp :s)+ Q n$ -0.12. .
-
-
3.5 BondBrealdng:AccompanyingChargeTraafeas
j
(a)
(,)
129
+
(u, Uz
1
(u1 1
U2
3.5.3 Enerr
GG
Fig. 3.22- Efects of buaklîng of a stlrface)e-g-, of Si(111)2x1, on danglingkybl'ids and their occupation. Norrnxl atomic clisplacementstsl aud ua are assttmed.(a) Unbuckled surface with half-fzlled0p3 hybrids; (b)bucldedsurface with dehybridization; =8. (c) buclded surfacewith Qlleds-like ar.d empty pa-like danglirg bonds.
azld Electron Transfer
Ia expression(3.72) a'actFig. 3.22ccomplete electrontrn.nmfer h'om the more plîke danghnghybrid iato the more s-lilte dargling hybrid dllm'ngthe buclding 2x l reconstruction ha-sbeen assnlmed.Tlzis wa.s oaly possible becausethe eEedof the electron-electron iateractioa izl tlle two daugling hybrids, 1 aud 2, has been neglected.In. order to discussthe efect of the întrabybrid electroneledron interaction on the electron transfer, we restrict the considerationsto the two danglin.gorbitals in the 2x 1 tlmst cell of a (111) surface. lnstead of sp3 hybrid energiesthere are the danglingbond energy cz - sds: of the raised atom ar.d the enezgyE'c H aabz of the lowered atom. Asslnrni'ngan electron trlmsfe.rb'n(0K Jn K 1)and, hence;ocmlpation zrlrnbersnz/zàl= 1+Jn and = 1 Sn of the t'wo hybhds, it becomesobvious that tlze danglingn.a'(Jn) bond energies ss a're diFereni fzom the energiœ & bebnging to the slp3 -
confgaration.
130
3. Bonding and Enezgetics
>
X
m
c>x c) = (D c
1 to tite surfaceenea.r of a &4111)2x Fig. 3.23. Correcwtions fzztut)and f2zL% transfer of three quaz'ters of au electron leaving the cation dangling bond empty and 611ingthat of the n'nioa with an electron pair.
An important constrxint whic.hls=l'ts the possiblestoiclkiometziesof compound semiconductorsmfacesgsthe reqttirement that no chargeaccumulates at the surface(4.424. J.nthe limit that sarfacedefectswhich accotmt for the
compensation of charge $nthe spacechargelayer are negligible, tMs holds also on the length scaleof the bond lengths. The relative stability of two surface reconstructions 1511611:1g Principle 1 follows to a large extent from the minimization of the electrostatic interactions i.n such a smface structure (4.43). The tendency to small electrostaticinteractions may somettmesalso drive an interchange of stoms across an interface (4.u). The 'vacancyreconstraction 24.45) snrfaces of IïI-V semicozducof (111) tors (see Fig. 4.6)may be considezedas a consequenceof Principle 2. The removal of one cation generatesarz equal nlpmbezof cations and anions i.c a 2x2 surface pnst cell, a,t least cotmtin.g their dangling bonds. hstead of folzrdanglin.gbondswith a total of three electrons,one obsezves three catiort dangling bonds a'adthree Jmson dangli'ngbonds a,1e.1. formation of a. cation The sidbld rings vacancy. The eledrostatic neutrality is locally g'uaratateed.. of atoms surrotmdl'ngthe corners of the lnnit cell in Fig.4.6 conskstof alternatmg threefold coordt'natedcation and n.nson atoms. The chainsof attezing danglingbonds allow a slmslar electronic s'tzazcttlreas in tie (110)1x 1 =e. ln agreement with Principle 1 the cation tanion.l dangling bonds are empty tflledwith an electron pais.
Pclrzcfple3: A semicondnctormzr/acc tendsto 5eizssulaténg (or.semb-
nonducting).
In principle, all surfacereconstructions f7,1.15111:),g Pricdple 1 and Principle 2 evhibit a tendencyto obey automatically Principle 3. Howevez,there are m=y cas%, in pazticular for quasi-pno-dimeasional surface structtlres, for
and Bondir.g 4.1 Reconstzuction
surfaceafier formatsonof a Fig. 4.6. Top view of a Ga-termlnated GnAR(111)2x2 ae indicated Ga vacancg. A possible2x2 surfacetlnl't cell ksdrawn. Cations taazionsl by open tftlledlecles.
whicb ad.ditional atomic relnvn,tionor recozustructionleadsto a semiconducting instead of a metn.llic eigenvaluespedmpm. For srteras with exterded electronic wave fllndions, metallic grotmê states do not occm in one dt'=en.sion (4.464. One clear examplefor the operation of Principle 3 on tetrahedrally and uctor s'trfaceuris the tilting of the dîmers on Si(100) coordinatedsemicondk Tlb.einteradion of the dirners aloztgthe rows in the (01ï1 Ge(100) (4.47-4.494. direction gives rise to a remarkabledisperstonof the %' and 'ir- bands. If the dimez's are untilted, tke 'zr and g'* bands associatedwith the clsmem overhp i.zïenergy for certxsn 2D Bloch wave vectors. The chnsns of dt'mea'salong the surfaceform a snrplmetal.According to the Peierlsesecvt,semirnetals are unstablein ttrael1D. The l'esulting grouzd state is charadezizedby Peierlsdistoled (i.e.,tilted)dimez's.
4.1.5 Electron Cotmting Rules.
Prindple 3 arzdthe part conceratngthe pusivation of the daagling bonds in Principle 1 aze automatically f7al6lled,if the studied surfacestntd'tzre obeys Suc.ha mfe states that an electron cotmting rule (ECR) 4.50-4,522. (4.42, bonding and non-bonding sttrface states that lie below the Fermi leve,l at the surfacemust be ftlled, whereas the non-bondingaud antibonding states whtc,hlie above the Fermi energy must be empty. This criterion is irt 'clear behaviorof agreement with the condition of the insttlati'ng (semiconductlg) the surface,i.e., Prlnciple 3. Am.ECR cazl be applied directly to detmmnîme
4. RecozzstructionElements
142
m4z
Fig. 4-7. An acioreterminated (100)2xr?z surface of a zhc-blende crys'tal with one are indirnllsing dimer. Aniozzs (cations) cated as full (empty) circles. Hatcheddangling bonds are ftlled.
21)
allowablestkdaeecompositions in agreemèntwith Priaciple 2. Only in order to accotmt to dealwith dopedsemicondudorsmust theserules be genernlszed for defec'tsat the surfacewhich compemsatethe space charge (4.531. surThe electron colmting is well tllustrated for the prototaical (100) The surfaceenergy of thE'se facesof ziac-blemdesemiconductozs(4.51,4.52j. suzfacc is loweredafter d''rnerlzation amdpassivation o.fthe daagling bonds in agreement wîth Prhcipie 1. More precisely,the danglingbondsare ftlled on surfaceaxkionsand are empt'y at surfacecations. Only dsme'rs fo=ed solely aad 2xm (ormx2)reconstrudions by catîorts or aaions (i.e.,homorlirners) are considereêtu the original fo= of such an ECR. The 12'LAa co>equence of the Hsrnerization, while the zrz-foldperiodici'ty is aamnrnedto arise from Lm D) rnsqsi'agdlmeas, leavhzgD fqirnep per qlnit cell where D :; m. Sue,h a reconstruciion is sbown 1. Fig. 4.7. J.norde.rto determsnethe relatioaship tu the aaionbetveen D and m, me cotmt the nlzmbe.rof electrons (4.51). termsnatedcaae of Fig.4.7 ennlntop cllrner requires six electrons, t'wo i'n the c. clinnerbond and two 5.ueach of the remnlnlng dangliugboads forming .n' a'ad z'* statœ. The total number of eledroc on the Hsmezized surfaceshould be equal to that of the blllk-tlnlncated surfacewith a missi'ag aztion pair. This yields -
6.D= 4Ja.D +
4.fcLmD) -
.
He'refradions of electrons
Jc= NA/4' A = NB/4
(4.2)
4.2 ChJ4.lrlA
i'a cation or aaion daaglingbondsllave beeniutroduced (.&+ Ja= 2).XA and NB = 8 Nx are the nxlrnbersof walenceeledrons of the atoms i'a the surfacesone has .A.Bcompound. In the case of cation-term4'nated(100)z?zx2 a'a opposite ftllsngof tke daaglingbonds resulting ia -
2D + Both
8(r?zD) 4#cD+ 4A(rr&D). equations (4.1)and (4.3)give the =
-
(4.3)
-
same
res'alt for the n:nrnberof
aimel.s
D
=
1 Nx
ï Nh
(4.4)
zn..
1
-'
For cliamond-stracturecrystakswith Nx = 4 no iateger solutioncaa be fotmd. This is in agreeaent with the fact that missing dimer reconstructioms are and Ge(l00) sudaces.ln the ease of 111for C(100)j zmt obsemred Si4100) V semiconductors (NA= 3) the rehtion D = 8a/4 is fltlflled for 2x4 and 4x2 rcconstructions with three topmost dsmers. Hdeed, tNe resulting # has beensuggested to explain expnrsrnentalMclings reconstmzction(4.43,4.541 surface.Mfvnwhile, we Hlow that other for the As-ternal'nnnted GaAs(100)2x4 Fig. 2.19) aad that the idea of reconstmction modelsare more favorabletsee Fig. three topmost As Hsrnea.shas surdved in the rough.e,r72 struct'are (see 2.17),where these dlrners are distributed over the flrst a'ad third atomic The 72structure is more stable th= the .precomstraction layers (4.43,4.554. becauseof îts lowe,relectrostatic e'aerr lpdaciple 2). holcls vith NA = 2 and D = m. No top For II-W semicondactom(4.4) dirner should be removedan.dthe smallest possibletlnl't cells shottld be 2x 1 and CdTe(100) surfaces2x 1 a'n.dc(2x 2)reqor 1x2. Hdeed, on ZnSe(100) Larger reconstrudions suchas the constructions were obseznred (4.56-4.591. one are describedby combhed vacaccpdsmer Zn-t-rrnsnated ZnSe(100)4x2 The condition (4.4) of a missing-dsmerteconstruction can structares (4.60j. also not be htlAlled for I-WI compotmds LNx= 1).For instancc CuBr(100) The surface structme is desurfacesshow a c(2x 2) LEED pattera (4.61). scribed by t'wo atozns and vacancies izz tbe top layer of a non-priraitive of electromsiu the dangling bonds is c(2x2)lnnl't cell. The total nApnnbe,r 2 2 Je+ 2 2 A = 8. This nthmberseces to inslzre that fottr anion azld cation dangling bonds will be S'LIHand remain empts respedively. Therefore, a genvalized electron cotmting rule is fctlfllledin the s'pirit of Principle '
.
.
.
.
1.
4.2 Chains 4.2.1 Zig-zag Chains of Cations aud Anlons
EEI-Vazd H-VI compotmd semiconductors,whic,hcry..;Binary W-LV (SiC), planes.1.zlthe bulk taltize ia the cubiczinc-blendelattice, cleavealong (110J
144
4. ReconstrudionElemenis t;l()
k:k$1
E:i'oq 11 gflo
JQ
22 : : : l' :
:
1
:.
1 ; : :w : : 2 :
)1-
: : :. : : :
:' : : 1 ; ; 1
Jc
2
-....
ds
o
1()! E1
.
(joslj
Fig. 4.8- Arrangement of atoms at relxved (110)1xl sutfaces of ziuc-blende semiof bonds in the Srst atomic layer are indicated by conductors. The zig-zag c'knsn!c thiclc lhes. The dotted lines in the top view 5.ve a surface :lnlt cell. The most impozantpairamete,rsof the bord-rotatzon/bond-contration model the rotation angle:zJ.!!tke c'haizz-bunklsng amplittzdezds.kand the bond length ds witht'n the zigzag nhnzns are show'nin the sideview (lower partl.l'ntons: fzlledcircles;cations: opemcircles. Hatching indicatesf1!5ngof dangling bondswith electrons. -
.
-
of these semiconductozs(110)layers consist of plana,r zig-zag clln.7'nq of alanions along a (ï10)diredion (Fig. teznating cations a'n.d. 1.6).Ic a blllk--ltlce covguration eacbv soace atom b.asthree nearest neighborsan.done broken or dangli'ngbond. Two neilboringatoms possessa bond in suc,ha (110) plxne parallel to a E1îî) or (E1R1 direction. The cation tanionl dangliug bonds arc partially Ved vith A = .&4 (A = 'V) eledrons. For (110) surfacesof the zinc-blemdecompolmds the genezalizedelectron cotmtiug rule (Principle 1)can be satisâedwithout chn.ngingthe ) x 1 trauslational symmetry. The Gazacteristic chxngesof tlze atomic geometzyand the bonding behador with respect to the bttl'k termination e-qn be s'Immadzedby a combined sontf-rotcfionvetaantionmodel (4.62,4.63) an.dbond-coniraction
4.2 t7%s.lenq :45
In order to reduce the smfaceenergy tBmsic Prinreïcaation model (4.64J. at tba surfacelayer ciple)accorduto the bond-rotation relaxation model, the n.nson moves away Fom the b'nll- in favor of a p3 bondlng vith the three neighbozingcatiomsrestûtingizl a local pyrnmldalgeometry with bond angles tending to 90o (see Fig.3cz,0a). The cation iu a surface''nit cell, on the othezhand.,moves into the btllk in favor of au 4J2 bonding with the three neigbboringanlons restllti'ag i'a for a loci pln.nltr geometzy with bozd angles dose io 1200 (see a tezzdenc.y Fig. 3.20b). The correspondingbucldiag of the surface zipzag clmlns'is aecompanied by d-like erson dangling bonds and p-like cation dangnngbonds. i.n Sect.3.5.3 the n.nion dangling Therefore,in agreementwith the dkscllrwqkon bond îs completely Xled 'witil eleetrons whtle the cation dangling bond remn.l'ns empty and, hencevPrinciplœ 2 and 3 are also f111611ed. Thjs sftuation Lsindicated in the lower part of ng.4.8. 'nis picture is con6rrnedby t'hemeasurements of fzlledaud empty sttrfn'ce states by mp-emKof STM. The comhination of both state lmngesinclicatesthe formation of zipzag chairts of cations and n.nlons in the FI0)dtredion as sudace (4.654. A czaracteristic demonstratedin Fig.4.9 for the 1nP(110)1xl direction featttreof the zîg-zagchp.5nsis their lateri extent lzrl in tbe Eû01) A the ideal brllkwterrninatect case) Zllj = co/4. also Fig.4.8, top view). Jm. (see further low-ri'ngof the smfaeeenerr (Principle 1)may be a consequenceof a shozleningof the bondsin the zig-zagchains due to the attraswtivehteraction of catiotusand anions.This restfts in bond-contractfonrelaxation. A spatial' represantation of the zipzag cllains is given i.u Fig. 4.10.
Fjg. 4.9. Zig-oag chains of empty' and FIII?.IIdapglingbonds at the rehxed surfaceas measuredby STM for bias wlth opposite signs (4.65q (copp 1'nP(110)1x1 from right (2003), wit: permlnsion Elseviez').
146
4. ReconstructionElements
surface with the resultFig. .4.7.0. Schematic indication of a relaxed GaP(11O)lx1
ing zig-zageahsGoq.
Att key 3tructural paœametersof a relaxed (110)surface are showa in Fig. 4.8. Tïe relative displacememtof the cation and acion dtmnesa vertic.al sllear ïza., the b'anklingamplitade, and a correspondingrotation angleLo ill the (1ï0)plaue. The deviation of the length ds of a cation-anioa bond fzom the bu)k bond length deq= VVo/4characterizesthe Coulombinteraction along a zig-zagchain. The z'ig-zagcharacteris governedby the deviation of In addition, the cations zlllj 9om its blllk 'value co/4(Fig.4.8, top view). azd aaio:as in the subsurfacelayer show a cotmter-relaxation,i.e., cations move away h'om (toward) the bux albeit on a much smaûerscale. taaionsl For coaveationalIII-V compoundsthe vertical separationbetwee,nthe smface interplaaar layer and the subsurfacelayer becomesshorter than the (110) distance ao/(2W)i'n the bulk. relaxValuesfor key parameters of the bond-rotation/bond-contration ation are listed in Table4.1 togethe.rwith the bulk httice constants co and the charge-azymmetrycoeëdeats .ç of the b7llk-bonds. The values are take,a 4.66, 4.67) a'n.dmeasarement of strttdural h'om ab initio caktzlations(4.41, The key parameters indicate that parameters LEED measurements (4.681. the surface rele.xntion is strongly Mueaced by the interplay of covalentand ionic bonding aud, to some degree,the ratio of the atomic radil. The rcu laxation prameters can be arrauged ia t'wo groups. The conventional III-V compotmdsare ard II-W (ZnS) arseaides,aad phosphide-s) (antimozzides, chazacterizedby bond-rotation relaxation with a.n plmost bond-lengthconservation. The layer-tGtanralesul vazy in a relatively small mterval of 25-320 1 surfacesof the while the elnm'otilt 11.u. amouats to 0.6-0.8 A. The (11O)1x compokmdswith anions belongiagto the fzst row of the Periodic Table (zzirela=tion. The trides, SiC)exhibit a mixed bosd-rotatîon/bond-xntrRtion
4.2 Chstl'nq
l47
Table 4.1. Charaderisticpazametersof bulk bonds @o, J) aud geometzyof The stracl surfacestlz-u,az, G/doq 1)for zinc-blendese 'mlconductoz's. (110)1x or LBED 4.66,4.67) t'aral parameters are taken from ab jnitio calculatioms(4.41, The càarge-asymmetry coeëcient .q is taken from (4.26). meaamements (4.68). -
surfaceneazwt-neighborbond lengths are shortenedby 4-8 % with ropec't to the bulk bond lertgth, and muc,hsrnxllez tilt angles tzg = 12-160occlzr. The buckling ll.z of the surfaceclmz'nsis reducedto values of about O.2i. The key parameters are not independentof eachother. Tn fac't
i suw, %/ (ds 8) 2 , ao/ (ds 8)i (1 sias'l.
Zluzzll
-
z
=
v
-
-
(4.s)
-
For that zeason we only discusst'wo pazameters in terms of covalent and ionic bondicg in Fig. 4.11. A.scan be seec tzl Fig.4.11a, there is a fairly good licear dependence betweenthe bunldsngamplitudeïz-u and the lengthof the bonds ds in the sudaceahnsns,except maybe for BN. The gene'raliêea that the bunkls'ngamplitude is Gversely proportional to the bonding strenkthis con6rmed.This relation, howevea.,is somewhatmoeed by the ionicity of the bonds, the atomic sizes and electronic correlation efecwts.One esect of the ionic part of the chemicalbondimgis demonstratedin Fig. 4.11b.The depen.dson the strength of the Cotllombinteraction alongthe zipzag cahnsns of cations alld aztiorts(4.69j. egectiveatomic chargesZc/a = ZVA/B 4(1:F.t.?) Thetr diference Zc Q = 8.: (Nz NAI governs the strength of the bond-lenlh contràdion in the cimsnq.There is an Mlrnost linear dependence. The exceptions BN, AIP, and ZnS are probably a consequenceof the fact -
Fig. 4.18. Band structures in DFT-LDA of tEe (111)2x1 sarfacesdescribt.dv'itb.izz tbe rr-bondedchaiztmodel tchain-left isomez) (4.76j.
4.2 ClmlnK
155
ing the strudklre factor. This caa happenby a conjugation of the zr-bonded c'hainsms for hydrocarbonsof the type =CH-CH=CH-CH=CH-.... This is accompanied by a dlmerization aud, hencetfliferent bond lengths dz and d, Fig. 4.15). The eFectc,a.rt be describedby assuming Alp = j (1+ ,4.'t)irt (see The geometrical strucwture factor becomesIS(i)I2= cpszt%co/zWl + .(4.6). along ?.1 and LS(i)j= gzâà) along <W.The gap open(1A)2sinzfkvaojzsfk iug along J'# is given by 2I1AIfr.TMs egect is clearly demonstratedia ...
Fig.4.16b. For Si ard Ge no indication %r dimezization hmsbeen obsen-ed.In the 1 surfacedirnezization is lmder discussion(4.871. cmse of tlle C(111)2x Performs'nggrazing-in'cidenceX-ray diAactîon studiesHuism= et al. (4.88) favor Shim-sequenc'y tilted chnsmq. experiments (4.8Sj genezationspecwtroscopy e,xpla,in the observedselection rule,sby (I':G buckling. Apaz't from only one
reference(4.87j: convergedAst-principlescalcalations(4.77, fotmd 4.00-4.92) almost undsrnerheda'ad unbuckled -C-C- chains on the diamond surface as a minimlnrn stzrface-free-enerastnzctme. Bucklsng and dimerization of the clmsnsresttlt i'a a deformntion of the tmderyngseven-member azzdfvemembe,rrings and, hencej increase the elastic energy whic.hreducesthe enr erg.g gain due to reconstmzctton.Ths eFect Ls strongest for diamond. The wave înnctions resutting for the ideal x-bonded chai'n structtzre are shownin Fig. 4.19.In qualitative agreementwith the tight-binding estimates (Fjg. 4.16) the accompanyingband structttre in Fig.4.18 resaltiug within DFT-LDA indicatœ a sernsrnetatticbehavior. There is a need for fuzther studie,sin-
1U111
./*
I'ttz:l *
r'i--la:l
1
O
(:r-;c2 (,1::1
1
Fig. 4.19. Plots of the square of the wave llnctîon of the 'C-artibonding and gw bonding surfacestates of C(111)2x1 for a wave vector 0.4 Jt%.The up#erpanels show ihe projectiononto the chain direction and the lower panels a top new of tàe surfacec'hain.Lines of equal densit.gare separatedby 0.075boizr-'. Fmm (4.774.
4. ReconstructionElements
cluding electronic exnhnangeand correlation in a bette,r way (4.40) On the other haad, the data poiats for occupiedsttrfacestates fxomARPES (4.93q a'nd t'wo-photon PES (4.94) have been shown (4.95J to be i.n good agreement vith the b=d dispeoion derivedfrom DFT-LDA resutts for ideal x-bonded chains (4.771. .
4.2.3 Seiwatz Chnlms the (111) smface of diamondstruckt'are semiconductorsmîght be expected if cleavagereslzlts in a triple dangling-bond (TDB)surface.I'n this case the cleavageoccurs between two narrowly spacedlayez's(see Fig. 4.5) , izï coatrast to a single dangling-bond (SDB)smface, which sqpazatesa widely spved bilayer at wlzic,hthe bozlds direction. Ir. the TDB case three boae az'e orîented exactly i:ct the g111)
Qkliteauothe,r2x1
arrrgement
of atoms
on
r rEl E1
:.:':t.l
(b)
12,11 n'.I
4
E11Q
Fig. 4.20. Formationof Seiwatzcllairts of surface atoms resulting in a 2x1 reconstruction of a TDB grolzp-lYtllllsarface:(a)ideal TDB surGne (side view);(b) and (c)Seiwatz chn.lnq(top'eiew). Seiwatzciminq (ssde'=ewl; .
4.: Ckalns
Eig. 4.21. Perspectivcview of the formation of rr-bondedzipzag chains on dangling-bond (114) sudace of a diamond-stmzcurecrystal.
a
triple
brokemper smfaceatom (Fig.4.2Oa). Two of them might saturste when neighbozingrows of the atoms are movedtowardseac,hother tmtil the distance of atoms of one aad the other row approachesthe bullc bond length deq and 4.20c). A perspectiveview of the resalthg stntctme is shown (Figs.4.20b in Fig.4.21. Jt is consistent with the Seiwatzmodel of x-bondedc'haizus (4.964. A 2x1 reconstruction occms. The surfaceatoms form zig-zagczhnt'nq alongthe direction. Two atoms cha,inare paz't of Ae-memberedrmgs. Eac,hc'haiït (ï10q atom is threefold coorrlinxted.The remaining dn.nglingbonds form zc-bonded e'hnsnq similar to that discussedfor the Pandey chains on the SDB sttrface. They aze accompaniedby 'zr-bondikg and x#-rtibondin.g suzface-statebands in the ftmdmental gap of the projectedbulk band strttcture (4.771 Until the discovet'yof the Pandeymodel the eazly Seiwatzcb.ainmodel was usedto explain the Si(111)2x1 surface,thougll it did aot completelydescribe the LEED fndings (4.97). Since 1981a11the properties of tke metastable Si and Ge(111)2x 1 sufaces are explained itt the gamework of the x-bonded ckaiu model of Pandey.Only i'a the case of diamond are the TDB surface azldSeiwatzchnsnsstill tmder discuasion.Hydrogenttarmimatîonmay play an important role, in pazticular fo< C'O-grow.a diamond samples.A tllzoe-step model hmsbeen suggestedfor the chenzisorptjonand desorption of hydroTDB smface,which makesthe creation of a correspondsng gen at the C(111) clean C($11)2x 1 stHaceenergetically possible(4.984. The outstaradingrole oî hydrogenis also obvious from the preparation of diamond61mAsvitlnin a CVD saperstructures have beenobservedafter heats.ag process. (WxW).R3OO ttp surface of polyczystnqll'neBl=s. They are related to TDB s-udaces for (111) spnmetr,y reasons. However,STM ha.salso shownthe coepdstenceof 2x 1 domaizsswith IWXWIJBOO structure.s (4.9% From the thezmodynamic point .of view the SDB C(111)2x ! surface is euergetically much more favorable thaa the TDB C(111)2x1 surface.Their surface energiesdfer by 1.35 ev The TDB smfaceis not likely to occtlr upon per 1x1 sudacelmit cell (4.774. are
.
'
I57
.
158
4. RecomstructionBlemexts
preparedby CVD growth and cleaxingdiamoadbttlk material. For smface,s subsequentAnnen,lingthe situation may be difFeyent,however.
4.3 Dsmers 4.3.1 Sylnlaetric
'
Ilirners
The ideal; bulk-termlnated (100Jsurfaces of diamond- azd zinc-blendeFig. 1.6). stzmcture crystalshave a square ttnit cell, with one sarfaceatom (see Thc seace atozns posseD two dangling.:.p3hybrids. Eowever,suc,han ideal experimentally.Accordângto the fmdingsin structure hasnot beenobsezwed surface a dehybridizationtakes place yieldi:ag a bridgSect. 3.2.2, on a (100) AccordMgto Principlq 1 the a danglingspx orbital (3.36) ing ppjz orbital ain.d. arlmber of dangliag bonds elm be mimimszedby a pairiug mechanism, more s'tdctly by the formation of dimers witk a bond length Q, as iaclicatedia Fig. 4.22. Two atoms that aze secozd-neazest neilbors fxom a bulk poin.t of They form a cr bond emanatiug from view approaG eachothe,r along (011). the two bridog ozbita,lsIbr)in betweenthe dz'merizedsurfaceatoms an.d poHted a weakerzr-like bond emanatiug 9om the t'wo danglin.gozbitals 1d) of the sicgle-electron away from the dimer. In the orbital representation (3.18) Hmn'ltoxf.an the t'wo interatoxnic interactions are approvlmately givea by where the Fjl = (brlslbr)= Fm. and Vk = (dlsid) = lz(Wa.+' V'ppxl, have to be ta,kenat the dsrner bond length Q. matrix elements F'oys(3.21) Neglecti.agdeformations of t:e d=gling-bond orbitakqdnlrsngthe dimemrization and interactionswith tke surrotmdingorbitals) the moleculeHnmiltonian (3.7)e-n.n be written in the form .
0 0 a'd -
s
5Q
W ca
-
e
TV resttltiug aatibonding aald qrlth the intra-atomic matrix elements (3.37). bonding Ievelsare ec-jc = ssr + 1$61 1and s.:r.fr = ca ::: lFki. =
!àooli
we-X'>.
-
x=>
E()r'Il on a Fig. 4.22. Formatîon of 35me,.rboadsalong (011)
surface (schemxtically). (100)
4.3 Dimers
l5S
Ga tr
9
V
G
6
.x.
C.i
%t.3
g: .a
>x
61
P ;
:
o.e
cbr
cur W
c%
-3
:1 r
cd
Fip 4.23. Molecule levels of a climer of eqcvalent surface Ga,,As or C atoms. The sf' hybrid energy' of cazbonis taken as zero energy. The correspondiugmolecule levels of isolated Ga, A.s aud C dimers are = representedin Fig. 4.23. The orbital enertesn (Ga) -11.55 eV) o(Ga)= = -5.67 eV, aa(As) -8.98 eV, ss(C)m -19.38 eV, s= -18.92 eV: sp(As) = -11.07 ev have beentakea from the Solid State Table a'n.dcp(C) (4.1004. The interacticm mnnutrix elements(3.22) have been (nlculated with'dlrner bond lensh Q = 2.49 i for Ga on Gn.AK(1O0)2x4 Q = 2.50 A for As on E4.101),
1.37.1.for C on C(100)2x1 The (4.40,4.95j. energetic ordrktg of the dsme'r levels i.n F1g.4.23 and the nlnrnber NA/s of avazableelectrons determsnethe level occapation. 1n the crase of a carbon 1 surface with Nzvs = 4, eachof the c and 'c' dimer on a din.rnondtloolzx bonding states is occupied with two electroaswith spitsap ='d spia-down. Prindple 1 is also G'lfllled vith respect to the passivation of the dazlgling bonds. I'n the case of the As dsrners, the three lowest levels o't rr, azd x* should be Slled to obey Peciples 1, 2: azd 3. This requires one additional electrop NB + 1 = 61in aveement with the electron cotmting rule (see Sect. The sitaatîon js less cleaœfor Ga dsme'rs becauseof the high-lying 4.1.5). bridge-bondorbital energy sbr, The absolutelevel mlues howeverindicate a tendeacy to donate electroas,at least oae, itt sprfaceAs dimers resulting in Nx 1 = 2 and a'a occupation of the lowest bonding cl5=e.1.level. Thereby, the ECR can be hllfllled. Mtuitively, a 2x 1 reconstruction is obtnsnedfor goup.Nmatezialsby an in Fig. 4.22 (mght arracgeme'ntof dsmers paralle,lto (011j partli'a linearc'hains iong a g0ï1q d/ection. This dimer model was Srst proposedby Srllnllerand Fhrnqwoz'th(4.103) to explnsn2x l reconstructions of (lool-oriented surfaces of silicon and germanblrn. It also chazactfm'mes the clean (100)surfaces of 1, neighboring surface atoms fo= double-bonded dhmozd. For C(100)2x dsrners with a bond length of about 1.37 A (4.40, 4.95j,whiek Lsvery close to the doublmbondlength of 1.34i in, e.g., the Cam molecale.The C=C double.boad is formed by the c- axd 's-like orbita,ls of the C dizner atozns
and Q = GaAs(1O0)2x4 (4.102j,
-
(see Fig.4.22).
Because of the strong bonding of the surface C atoms, the c' and c* states eAnnot modify the eledzonic structme arolmd the projectedAln-
:60
4. ReconstruciionElements
Fig. 4.24. Filled acd empty surface-state bandsof C(l00)2x1 in the fandamentalgap of the projected bulk baudstructure (shaded
area)ku DFT-LDA (4.7$.
dnmental gap. Only surface bands belonging to the %-- and x'*-like states may occuz iu the Atndltrnezztal gap as shown in Fig. 4.24. The dispersion i.e., of these bands is a consequenceof the dimex interaction along (0ï1q, along the dimer rows. Such an interaction V.Sof two neioboringdangli'ag qrbitalsis smallerth= Vkbecauseof the largerdistance.Then approvimately $4$0.37.1.is valid. According to (3.15) the corresponding Vk = 7.1.(QVVc0)2 tight-bindîr.g Hamiltonian is .
.#,r-,r(:)=
cd
+
ccost:aal'/..u
0 -ixtza!z&
aa +
eiEzfavr.k
(4.9)
acost:aalp.u
with @,z= co/W(O, lc 0)and dd = Qtl, 0, 0)and a new Cartesian coordinate and zI1 1ts diagonnlszation gives parallel onesystem zlIp11),:1IE011), (100q. ;:!z Vkl. They show dimeasion.albaads c..y.(i) = cd .k.2 cost/cvco/v/llvk Fig. 1.22) strong ctispersionalozgthe J'X-and J'15lines i.n the sttrfaceBZ (e,f. bbr Q > 0 (the
27.1.one observesquantitative agrecmentwith the enviromment) results of the DFT-LDA calculations iu F'1g.4,24,The indirect gap betxeen the #+ bar.d along #T a'adthe x band along 13.1 is dpBne' d by 2(lFk I 2JQ). The orbital character of the two band states is shown for the X point in Fig.4.25. Their bonflimgor aatibondingnata're is clearly visible. Symmetric dimez'shavenot only been obsezvedon dixmondtlool surfaces, smfaces of zinc-blendematerials. One exampie is thc Sibut also on (100) termiaated surfaceof cubic silicon carbide (SiC), whic,hbesidesthe c(4x2), The 3x2, and 5x2 reconstructions also shows a 2x1 reconstruction (4.105). bonds are partially ionic. TMs is related to a nhnrge tranqfer from dlrner Si atoms to C atorms in the secondatomic layer, which malcu dsmer buckling ltnlskely.J.uaddition, when %surfacedlmers form, axlgular forceson the -
4.3 Dimer:
161
*
ElKI
1
Ct-----â's p
1)0û1
)
:()11
.
q> :
$:1 Eol
Fig. 4.25. Sqlrnad wave fn'nctions for the occupied'r state and the empt,y zrv state at Ge poht K ir. lhp BZ. of the C(1Q0)2x 1 sudace (4.1041 . A pleme contnsm'ngt:e dime.r and the surfacenormal is used. The contour s'pacingis 0.05 bohr-Y.
secondrlayer C atoms
izwolved.They prevent
too strong atomic rearrangement. Moreover, khe lsttsce const=t of SiC is about 20%smaxer than that of silicon. Consmuently, the Si sutface-laye.r atoms move only slightly tovctrds each other (witha dlmer leng'th of yboutda = 2.73 i (4.95, 4.102) = ânsteadof the atozaic distaucq co/'/f 3.06 A on the ideal sudace). The di-staace of the two paired Si atoms is lazger thaa the typical Si-si bond length of abou.t2.35 i. Therefore,one rnnnot really speak about the forrnntion of Si sl:rêlme dMers in this case (4.95,4.1064. The polar (100) surfacesof III-V compoundsalso tmhibit a dlmer reconstzuction to hll6ll Peciple 1. For Hstance, As surfacedimers are impolta'n.t building bloclcsto e'xphln the various reconstractions of the GaAs(10O) smu face (cf.Fig. 2.17a). The surfacedimqrs shouldbe almost tmbuckledto avoid caharging esectsof the surfacein contrmst to Prixciple 2. One important example Lsthe 2 x4 reconstrudiop whic,h aksocontains the basic unit of the are
a
2
nj >
& >
P
Q) c
m0
Fig. 4.26. Surface-statebazlds(solid sarfaceverlinœlfor the GaAs(001)72(2x4) sus the projecwted bulk band stmcture (grayregfons). From DFT-LDA calculations (4.1O2j ttsing pseudopotentio wltîcètsomewhatopen tZe gap.
4. Reconstraction Elements
16t?
that the 72stractme c(2x8)reconstruction. Now, there is ageement (4.55)
2x4 reconstmction appen.nlngkmde,rintermediate As-ric,lzpreparatiou conditions (see Figs. 2.1.9a,and 2.20a). Under more AsZ'iC,h conditionstke c(4x4)structme is stabilized.For lessXs-ric,hconditiotus 2.17a a'ad 2.19a) :Lsstabilized. The top As dlmers the ('(4x2)structme (Figs. mplitude is not of the 72 structttre are @lrnostsymmetric, i.e., the bu The larger thn:n 0.02 .l. acd no twisting in the sudace plane occms (4.1û2j. Hirner bond length of Q = 2.50 i is ver.y closeto the bond dkstanceof 2.51 -t found for bllk- .&swhere tEe atozns are threefold coordinated E4.107q. The /2 structare ia Fig. 2.17a contnsnq two As dinners in each2x4 mkit cell of the top layer. One pair of Ga atoms is removed from the second atomic layer, aad aa additional hs dsrner is formed i'a the third layer. Tlle the electron cotmthg rule (4.4)derived for a missing As p2 structm.e 6111611n dlmer ia the toplayerin a somewEatgenerstlsaed sense. Origiaally,there azc 12 As danglingbonds with A = 24electronsand 4 Ga ctaaglingbon.dswitlz Jc= 24electrozls.The total 18 electrons allow the completeoccupation of the c, '7r, and x* leveltsof the As dimea's,whereasthe Ga dangling bondsremaln empty. ms shovn by the b=d stntctm'e in A semicozducting suzfaceresults (4.102)
exeplnsns the (Fig.2.17a)
-
C1
o
o
@
Q
Ct
e
.
t7
C2
.
o
o V2
V1
too (g;
y,-g.y'g. .
L-
=
'
. '
.Q' V4
73
% So Vè
>.
Q* Uca
(ï*
Fig. 4.277 Cmntoklrplots of the squred wave Glnclions at X for surface states of stmzcttae as indicated in 57.4.26.The ccmtom spacing is the GaAs(100)72(2x4) 10-3 bohr-3. A)l plots are drawn parallel to the surfacenormal. C1is plotted along C2, V2, and V4 are at a tomhyer d''me2. V1 and 5J3 the x4 dizection, i.e., (011). smbols iudicate As (Ga)atoms. are shown at the third-layer d''rnea..Filled (open)
4.3 Dime'rs
163
Fig.4.26. TEe smface-state gap is la'rgerthan the btllk gap hdicating a, gain of batd-structme enera due to recomqtruction. The two lowestempty surface states C1 aud C2 as well p:ksthe four highestS'II?'CI surfacestates V1, V2, V3, and V4 at the # poin.t are plotted irl Fig. 4.27. The highes'toccupiedstates V1 a'ad V2 are related to aatibonding rr' combinationsat the topmost aad third-layer As dl'mers. The states V3 aud V4 represOt bondiugcombinations. The lowestempty state C1 is related to Ga daagliagbonds.Eowever)the nex't one, C29is mnsnly built by antsbondisgc* combinations at top As dsrners. For the GaXs(100)((4x2) s'tzrface the b=d stradttre and the djscvion of the orbital characterof the surfacebacds cau be fotmd in Ref. (4.*54. 4.3.2 Jksylnlnetric Ilitaers
In. contrat to C(1.00), for Si(100) a large nttrnberof reconstrudionshnAbeea fouzd. The most impoztant ones are the c(4x2)a'ndp(2x2) reconstnzctions obsezvedat 1owtemperatmes, and the 2x 1 stzazdlzreat room tempezatme. The revezsiblec(4x2)-F 2x 1 pitasetrn.nKition takes place at a'rolmd 200 K. The Ge(100) surface shows siznilar behavior 24.28, The large,rrecon4.311. structions shoutd be possiblefor crystals vith covalent bonds weaker thau timse iu diamoad.J.nsuchsysorns the surfaceenergy may be further lowered by allowing the d'lmers to buckle o'at of the sT:rf'nce plaae (4.47j or to Mst the dime.razs in the surfaceplane (4.7.081. The c-haractezistlc parameters are zîb a'ad dk (see Fig. 4.28). The accompaayiugenergygain is larger than the correspoaclingloss due to the elastic energy of deformatloa. The driving force for suc,hadditional atomic rearrangements is rehted to Principle 3. For symmetric Si dinners the surfacomormatdangling-bond orbitals form a nearly degeneratepair of rr acd x* bands.A Jn.lnn-rfeller-like distortîon is expected to open a surfacegap between the 'r. and =. bam.ds Totat-energycalciations (4.95, indicate a bunldsngof the (4.47, 4.9,$. 4.1091 du
4àb
T :511
ElDf! -
E01 f
,
:1 f
Fig. 4.28. Top and.ssdeWew of an msymmetric tî''=e.r geometzy Cbnractedstic lengths suc,has the dimer bond lengtà Q: the bu#ingamplitude ïs parallel to and the twisting nmplitude ztk parallel to (011) are indicattd. (100),
164
4. ReconstructionPlements
Fig. 4.29. S'urfaceelectronic states itl the bttlk b=d gàp reglon of the Si and 1 stracture within the asymmetric dime.rmodel. The DFT-LDA bands Ge(100)2x are
taken from
(4.7$.
dirners but no twisting, i.e., lt = 0.'-1naccord=ce with the discuession in Scc't.3.5, one of the asmer atozns is pushed away from the surface,becorning moTe pyrnmz'dal(p3-1ilcel bondedwith a fdled s-lilce dangling bond Dup. The other dimer' atom is pushed inward into a nearly planaz (spz-like bonded) cooguration. The accompanying pa-lilce daagling bon.d Ddonmdonates its electron. The restzltîng dimer ha,sa weake,rbond with a:a iucreaseof the boad to Q = 2.25 k (asymmetric Si length from Q = 2.20i (symmetric Si di'rnea'l dimer).1ts veztical buckqlng is ztlla= 0.6 A, malcing the tilt angle nearly 169. The xsymmetric dimer geometryresults in an energy gsin of approxn'mately 0.12 ev per dlrner (4.10% F1:11re4.29clearly showsthe exmectedband-gap opening betweenthe surfacebandsbui)t mainly by Ddownor Dup orbitals. The eHstence of azymmetric dimezseasily e-xplainslazger reconstrttctioas snd p(2x2)as dlferent covgurations of left- aud right-tilted suc,has c(4x2) dime> (see . They give rise to small additional total-energy gainsof Fig.4.30) 0.05 ev per dimer hom knte-racwtions betweennaighboring dimers in a dsmer rOW an d 0.003ev per dimerâ'om iuteraetions of nearest dimersin Vo aeighIt turns out that the smface eneror is minimal boring dimer rows (4.1101. if tbe Jimer-tilt direction altnrnates along and is perpenclictzlarto the dime,r Tom. The resulting c(4x 2)reconstradion is indeedobservedat 1owtempem .
stures.
Despite ex-tensiveexpmrimentalaad theoretical bwestigatioa, the nature of the Si and Ge(10O) surfacesis still subjectto debate. of the reconstvction This holds in pazticular for the asymmetrjr of the Si dimers studied by me.ans of STM. In early STM expe-rimentsthe dirnezs appearedto be symhetric
4.3 Dimers
165
yX1
l'ïj j $' .y' y' p'
ê'
P(2x2) l 1 1 .1
1
d 1 1 1
.p
I I
I
11
I I I
1
1 1
I
1
I I I
1 1 1
Fig. 4-30- Surfaceunit cells of variotusreco>trudions of (100)sudaces.The formxtîon of dimers is indicated by horlzontal lines. TEe lazge empty and shaded c'irclesindicate the inequivalent tomlayer atoms, e.g., the Cup'and sdown' atoms i'a atorns. an aymmetric dimer. Dots representsecond-layer
Tlzisexperimentalfmdsng,however,is not necessmrllyilz coatradictiou (4.111j.
to msyrnmetric dsrners. Studies of the dynsxnl'csof the sttrface have shown that the dlrnez's may Sip betweeatheir opposite tûtiug diredions (4.112, At room temperabme, thia dimer Nipping happens on a t.l'me scaleof 4.1132. 10-10-10-Ss. Most meaemements do not havethe respective time resolution. At room temperature, most dimersappear symmetric due to tb.ei:rdplnrnlc flipping motion, ar.d only dlme,rs close to defects are plrned in a buckled The nlrrnbe,rof symmetric dz'mez'sdecreazesbelow 129K coMgaration E4.114).
aclddirrlersbuckle alternately witkin eac,hrow, witit the formation of p(2x2) rows inuidezztkalor opposite domains correspon8ing to adjacent âztdc(4x2) ozientations(4.114, 4.115J.
166
4. Reconstruction Elements
Thîs picture has recently been chatlenged. A seriesof low-temperature STM studies b.a'sbeenreported that, while the c(4x2)structme is obsezwed below 120 K, farthe,r cooling below 20 K causc the dimers to again appear tha't.this is a drmmical phenomenon symmetric. It llas been argtted (4.116) causedby a loweeg of the potemtial energy barzie,rbetxreenthe t'wo backled cougurations, which allows the dl'rnez'sto resllme the Eipdop motion czharacteristic of room temperatmz. Other authozz(4.117) claim that the observed symmetz'icdirners are static sincetheir imagesd.oaot exhibit the noise associated svitb.the Sipping motion obsezwed ita the same sample at 11OK. Howevcrl there is also evidencefor a sigocant imAuenceof tlxnnelingprocesses(4.118j. To cla'rpy the low-temperatme bu behaWor1om a theoretical poin.t of view, the subtle apects of electronic conelation near surfaceshave to be taken izzto consideratiom The syznmetric dimer with two g' dangling hybrids, eac,hfllled with one electroa, representsa biradical state. Therefore, static cozbrelation beyozd LDA and includicg gradient corrections, (perhaps, see Sect. 3.4.1) must be properly talcen into accolmt. The clime,r bu is accompeed by electron tzansfer.Thus, the net stabilit.y of the buckledconfguration depen.dson the devee to wilic,htite repulsiveCoulombinteraction (cf.Sect. 3.5)of the two electrons in the apper Si daazglingbonds is dynamically screened.. To accoaat for this dpmmical correlation esect, qttnantllm Monte Carlo studies have been perfo'rmed E4.119J. However, they qualitattveiy gavethe same favorizationof the bttcltled couguration as DFT-LDA or DFT-GGA calculatiou, '
'
4.3.3 Hetetodkmers
On the GaAs(001) surface b0th. Ga and As dsrners f.t geometrically to the underlyiag bulk. The covalen.tradii of Ga, roa = 1.26 i, and As, rxs = 1.20 i, are nearly the same (4.25). The situation is HiFerent for TnP(0û1) with atoms of dl'gerent size, nn = 1.44 1.ard re = 1.06 X An Jn-P mived Jlrner or heterodirne.rshould ft muc,hbette.r to the bulk. Consequently,an In-P e-xchangeshoald be energeticluly favorable. TMs ide.ahas been used by Scbmsdtet al. (4.120, to explain the 2x4 reconstruction occlzrrlng 4.1211 i'a a large range of In-zic,'itprepration conditions tsee Fio. 2.19 and 2.20). Heterodirnezsaksoappear to play a'n. important role i:a Imdeationa'adgrowth sttrfnce.s (4.122) on 1DSb(001) as well as for submonolaye,rgzowtll of Ge on
Si(OO1) :4.1231.
The suggestedmodel for the 1nP(001)2x4 recortraction is shown in Fig. 4.31. The sttdaceis ternninnted by a complete In layer. ln tlle ideal case, eac,h1'n.atom possessest'wo dangling.6p3hybl'ids along (111) and gïï1) with a 6115ng fraswtiom.al Je= QuThe corresponding2D square Bravaislattice is given by a1 = .g-o.a (1,-1, 0)and s,c = e.z (1,1, 0).A mived In-P dirner on a 2x4 cell adds eigb.tclectronsto the available12 electrons in the 16 Iu dazzgli'ag bonds aud forms four back bonds whic.h are occupied by.etht electrons,The top. layer In atoms fo= further four bonds with additional eight electrons. The .
4.3 Dimez's
I
167
(110q -'
'
Eig. 4.31.. Top and side view of t;e rnlxed-dsrne,rmodel of the TnP(001)2x4 suzface. Emqtytslledlcirole'srepresemtln (P)atoxns. Large (srnM1) smbols indicate positionsm tàe &st aad second(thizdand fourth) atomic layers.
dangliug bonds at the I'n atoms being fa'r away from the heterodimer remain empty. Two electrons occupy the c' bonding orbital of J.n and P states of the buckledmsveddimer. The remnsnn'mg two electronsSl thee-lilte dattgling orbital of the P atom in the topmost heterodlrner.One may' concludethat all reconstructionprinciples,Prvciple1, 2, and 3, aqreA116l1ed. Consequently, this is also tzme for the electron counting rtzle in a geueralizectsense. Total-emerg.yoptsmizations witlnl'n a Mt-principles appronnb (4.120,4.121) com6rm the aboveconsiderations.The bond length Q = 2.57 i of the mived fimer is close to, the sum of covalent radii 'ru + 'rp. TH dimer is buclded witiz an angleof j.7b.The ln atom is 0.43 i close,rto the substrate than the P atom. Ss=l'ln.rresults are obtained for the mlxed-clime.rreconstractioa of tïe GaP(OO1)2x4 suzface(4.124) The Ga-P dlmer is characterizedby a bond .
V1
V2
Fîg. 4.32. Contour plots of the squm'edwave functions at R for the two highœt occupied surface states of the mixed-dimer reconstrudion of ïnP(001)2x4. The plots are clrawn in (001)planes 0.8 â below and 0.8 .â.abovethe uppemnost P atom
;.. m be ïmbfeved easily, however, jtzst by adding a threefold coord-lnxted adatom, which co'ald be a,n atom chemlcatlyidentical with a blllk spedes or a subs'titutional atom. Sucll adatoms on (111)aud (0001) surfacesmay occupy two types of sites wlzie.hare illustrated i'a Fig.4.35. These geometries a're distinguished ms hollow (Ha)aud atop sites (T4)dependingoa whether the subs-trateatom bebw the adatom is fotmd i'a the fout'th or secoadatomic layer, respectively.i:a eacZof the geometriezthe adatom is in a threefoldt3latom bdow symmetric site. J.nu the T4 case the adatom a'adthe second-laye,r wemklyinteract what gives neady a foulfoldtzl) coordiuatioa.I'a the geometry Ss the adatom occmpiesa substitutional site in the secondatomic layer beneath a T4 adatom of a bulk constitaent. lt possessesfour neares't neighbors
Fig. 4.35. Adatoznson (111)surfaces of zinc-blendeor diamond czystals i'a Ta, Ha, aud Ss sites. Adatoms are shaded. Top (a)aud side (b)'dews are showa. The coorasnate systems are given in Fig. 4.l3a.
4.4 Adatoms attd Adclttstes's
but is also Huenced by a flfth (forthat reason, index 5)atom :in the T4 position. The illustration of the adsorption sites in F1g.4.35is not only valid 4.11and 611polytres one for (111)surfaœs.ln tke case of (tooly-oriented of the bondsizt the appermost two bilayezsas may expect tie same st shown in the fgure. The Ss geometry Ls only pbservablefor adatoms chnmle-q.llydiserent 9om the substrate atoms. Typical esxamplesfor the renalization of Ss are '
aad ozl recomstruc-tiozlson Si(111) surfaces (4.315 B-induced (WxW).R30O surfaces(4.131). J.nthe pme silicon case the incorSi-termlrnted SiC(111) reaction. The adsorption of poration may be consideredas a'a exclnn.nge other group-lll elements Ak Ga) a'ad Ia lcads also to the formatioa of stractmu. Shce they are tzivalent, eachgroup-/l atom sat(WxW)-R30Q tzrates three dangling boncls of the group.N atoms and, hence,leads to surface passimtiom The T4 site is favored.This pictme of adsozption also remains valid for constituent atoms. Total e'nergy studies of the gcoumW tmnKlational smsurfaceswitbl'n the (Wx W).R3OO adsorption oa 1V(111) show that the T4 adsorbatesite is mue-hfavored metzy (4.130, 4.132-4.136) over the Hz site. Despite usicg diserent calculational approaches,there is agxeementfor Si and Ge that adatom Vqorption in a T4 site is exothermk 1 smface.This is in contrmst to diamond with respect to the relaxed (111)1x smfaceson whic,hadatomsare clearly energetically llnfavorable. (111) T4 adatom geomeOn the other handj the Si and Ge(111) (W x W).R30O tries give zise to ê=gling-bolld-related surface bands itt the Gmdamental gap. They are ha'lf Ved. and pin the Ferms level near midgap. This fact violates Priaciple 3 and the electroncouting rule. Moreover, r.%adatoms on (111)2x2 Sect. 4.4.2). cells shottld be energeticallymore favorablefor Si a'ad.Ge (see However:there is a mystery about the widmband-gapsemiconductorSiC.The same type of,violations occm.s for the Si-termsnated SiC(111)(WxW)&0O
111 E1
(1-:21
E1 l'oq
0* surface (W3xW3)RV F'ig. 4.36. Perspective view o f the si-terrnsnatedskc(1:1) circles. with si T4 adatoms. si (c) atoms are uicated by open (f:1led)
172
> *
>
P *
m
4. Reconstruction Blements
2.5 2.0 j MwX 1.0 0-5 0-0 -0.5 -1.0 -1.6 F -.
**>..
'**
œ v
'
...
wwus
.%
> +
. *
*
.**+
..
l -
M
..-
K
-
F
Fig. 4.37. Ezectronicstructure of the 3G.SiC(11))(WxW)&02 sttrfacewitht'n DFT-LDA (4.139). The projected bulk band structtlre is shownaz the shadedregion. The dotted line representasthe Si danryling-bond-dedved band. surface taad%r the corresponding(0001) ones of the 411and 6H polytypes). Nevelheless,tEe L adatomgeometry shown iu Fig. 4.36 gives a stable reconstructiou for not too Si-ric.h preparation conditions (4.130, TMS 4.1364. result is iu agreement with othe.rtotal-enerq cakulations and e'xpezimental data meastlred for the (WxW).R3OQ recdnstruction of 3&SiC(1l1), 6Han.d4H-SiC(0001) s'urfaces(see also (4.95)). SiC(0001): The stkrfazeb=d structures of the Si(T4) adatom (Wx W)S30O geometries of the Si-tcrminated3C-SiC(111), and 4H-KC(00O1) 61FSiC(00û1) suzfacespossessa dangling-bond-relatedhalf-oed bar.d in the AJndamental One exampleis shown izl gsp, at least withim the DFT-EDA /4.137-4.139). F1g.4.37for the 3C,-SiC(l11) sudace.55='2a,rban.dstmzdares (W x WIA300 are obteed in the cmse of the hexagonalpolytype,s 611and 4H. Only the prolectedfttmdamentalgap is videned by about 1 eV. The theoretical results obtained witlnl'n DFT-LDA are in disaghree-memt with a comblation of ARPES (4.140) and KRIPES $.1414 inve-stigatiomqas well ms scnnml'ngt=nelîng spectroscopy (4.:421. For the hexagonal polytrpesthe expem'ments 'indicate b0th a'n empty aad a ftlled danglinpbond b=d separatedby a surface-state g-apof about 2 eV. The discreparcy is solveé by the mss:nrnm tion of a, Mott-Hubbazdinsulator golmd s'tate of the SiC(Wx surWIS3OQ faces (see Sed. 5.4.2) with a.c on-szte CoulombGteraction paramete,rU rz 2 ev (4.137, 4.139,4.1431. This is in agreementwith Peciple 3.
4.4:2 Adatoms AccompYed
by Rest Atoms
Although e,azhadatomreducesthe dangling-bonddensityVcording to Pri'n.ciple 1, tbe (WxW)J?,30D Tecoastruction of group-lvtllllsufaces csmmot G11611 Priaciple 3 i'a tlze original sense. However,this principle fxltn be obeyed
4.4 Adatomsand Adclcsters
(b)
Fig. 4.38. Adztoms in T4 (a)and Ha (b)sites formsng2x2 lattices on W(1:1) cir dots (second layer, adatom) surfaces.The atoms are kdicated by circles (fzrst cel'ls Adatoms are shaded,rest atoms aze hatched.Rectaurllar (Nmcagonal) layer). lizïes. are iadkated by thizzsolid (dotted)
considering a, 2x2 reconstlmcioruH tMs cmse one adatom occurs for eveayfot:r sarface atoms accomrnodating75%of the broken bonds =d, conseqaently, leaving aa adatom density correspondlng to .24of tha original sarface.The pm'rnl'tivecells have four times the azea of tke ttnrecozlstructedlx1 smface. There are two suc,hcoverings,one with hexagonalsyznmetl'y, the 2x2 stlrface, aud one with rectangular symmetry, the c(2x4)reconstractiom 80th cell types are hdicated iu Fig. 4.38. The hexagoaalcells an.dthe T4 adatom pocitions usuallyTveziseto a lower surfaceenero'. The essentialmechanisp is howeverindepeudeatof the cell shape. 1'atke 2x2 case of gaining enwetrgjr the adatom bonds to three of the surface atoms, leaving one atom with a and Ge(111) the adatom danglingbouduknovn as the rest.atom. For Sî(111) is displaced towaœdtlle btttlc, whereasthe rest atom tends to be pobonded and.,hence, is displacedaway from the bulk. Tlle dangling bond becomes TV rearraugement is acfor the adatom (restatom). more p-like (&1ikel companiedby an electrontr=sfer from the adatom to the rest atom forming Table d.2. Recoastmtction-inducedenera gaizzin eV/lx 1 cell with respect to the clearzrelaxed smface for the 2x1 x-bonded chain modek2x2 Hatom model and.the 717 dimer-adatom(hexagonal cell),the c(2x8)adatommodel (4.38,4.39): from a: izkitiodezusztyGlnctioual calculatioms.' stacking fault model (4.144,4.145) 7x7 c(2x8) -0.18 (4.76)-0.23 (4-76) C(l11) 0.80 (4.76) 0.83 (4.136)-0-10 (4.135) 0.27 (4-76) 0.30 (4-7$ Si(111) 0.24 (4.764 0.28 (4.13410.27 (4.1341 Ge(111)0.22 (4-761 0.20 (4.134J 0.26 (4-761 0.26 (4.76) 0.28 (4.74j 0-27 (4-74j 0.32 (4.742
Surface
2x 1
2x2
;'
4. Reconstraction Elements
lone pair in the daagiing-bondorbîtal of the rest atom. The resultiug sarface geometzy fnllfllls Pzinciples1 aud 3. A surface-state gap appears (4.134). The accompanying lowezingof the sl4rfnne smera comes in the range of values calclzlatedfor the x-bozded c'hn.in reconstrudioa (cf.Tab1e4.2). Only for dlamoadtllll are adatomslessfavorablebecauseof tlle stronger carboa bonds in the bulk. Consequently,the 'r-bonded e'hain reconstruction repreents the grouxd state of the C(111)2x 1 smface. Otaecaa gemerateother coveringsof the adatom density iu? suchas c(2x 8), by decorating larger pm'mz'tivecells. Suc,haa exmmplewitlz T4 adatoms is shownin Fig.4.39. The two pairs of adatomand rest atom pe,r'nn:'t cell oger new degreesof freedom, which lower the total energy. On Ge(111)c(2x8), ab iaitio calculationson the simple adatom model have indicated that the msymmetzy of protrusions fotmd in the STM srnn.gesis mxsnly e'atzsed by the a
.
:1'tlll
ptol
Fig. 4-39. Atoznic structtu'e of a group-IV(111)c(2x8) surface (topview).The simpleadatommodelLsrepresented.The atoms are indicated by cirdes t5.rst layer) or dots (second Adatoms are shaded while rest atoms Erst layer). (ontop) tin Layer) zlnlt are hatched. Possible2x8 and c(2x8) cells m'e demotedby dotted lines.
4.4 Adatoms and Adclustel's
175
Fig. 4.40. Sm'faceb=d structures versus high-symmetrydirec-tionsin the 2D BZ surfaces(simple adatom model) of C, Si, acd Ge. for reconstruded(111)c(2x8) The shadedareas represemt the projectedbullc band structures. From (4.76;. '
The bualdlng bu between the t'wo rest atoms in the llznit cell (4.39j. makes a 2x2 subnlnstmore electron-ric,hthn.n tEe corresponds'ngc(2x4)subltmt't ia the c(2x8)recozlstruction. Thks stabiGcs the c(2x8)trauslational symmetzy ove'r the 2x2 a'ad c(2x 4)suzfacesaccordingto Principle 2. Tlle bunkllng of the rest atoms alsohappensi:a the c(2x8) orderiugon the Si(111) surface (4.76, The chn.racteristic energy gn.s'nsin Table 4.2 clearly in4.146). stabilization of relaxed Si(111) and Ge(111) surfaces dicate an (even bette'r) chalns. Tllis is izl by adatoms in comparison wxiththe formation of zr-bonded dear contrut to dixrnondtl t1) A'n idea about the energy gytlns due to a, c(2x8)adatom reconstnzction surface violates Prhciple 3. In is indicated in Fig. 4.40. The C(111)c(2x8) However, contrast, surface-stategaps al-e openedfor Si a'nd Ge(111)c(2x8) in the Ge càsethe occupied surfacestates are much lower i'a enerr tha,n Therefore,for Gemore baad-stmzctureenergy is the valence-bandmnv'=:lm. gainedthan i.n the Si cmse. More precisely,the adatom-inducedelastic energy '
.
.
(a) J'
(111)
1- (11:
.
f'/e
e
F.
.a o
..
z
(b)
s:'s .
G
.
.
.
qsl .
.
4 CED .
.
:
0.
co .
.
.
Fig. 4-41. Contpmplots of surface s'tates of tke adatom-coveredGe(111)c(2x8) plane contaMng a.u adatom aud a rest atPz-ty sttrfacein tke (110) (a)state of tize lowest empty sarfaceband (Fig. 4.40)at a wave vector on thq CYline; (b)state of in the higllest ocompiedsurface band. at a wave vector on the rF/ lice (Fiz.4.40) the smfaceBZ. The distanceof the Ge atoms to the piane of the adatom and rest atom is mclicated by the varying size of the dots. Fcom (4.78J.
176
4. ReconstrudionElements
is obviouslyovercompensatedmore strongly by lowerirtg the band-stnzcture of the densityof daaglingbords'in the Ge case. eaergydue to tEe redvuctioxz The orbitals beloagingto the lowe-stempt.g smface bacd and Mghest occupied surfaceban.dare plotted iu Fig. 4.41 for the Ge surface. Indeed, as discussedabovefor the 2x2 adatom situation, the wave ftlnction of the high.est occupiedband is mnr'nly localizedat a rest atom (Fig.4.4lb).'The snme holdsfor the localizationat the adatom of the wave Atmctionbelongîngto the lowœt empt.y surfacebaud (Fîg. However, there are also contdbutîons 4.41a). from the neighboring atoms. 4.4.3 Adatomm Combined with Other Reconstmzction Elements
The energy gains in Table4.2 showthat the grotmd state of the Sî(111) surface corresponds to a huge 7x7 reconstmzction. After long and controversial disputesit is nowadaysexplainedby a dsrneoadatom-stacldng fault (DAS) model with corner holes (4.144, as representedizt Fig.4.42. Each 7x 7 4.145) unit mesh contn.ins (i) a stanlrimgfault in one of its triangular subunits, (ii) a corner hole correspondingto one mlrwqiagatom i'a the secondatomic layer and, hence, leaving a dvgling bond at its center atom ill-the third atomic laye'r,(iii)aine dimersforrnsngdomainwalls abng the botmdar.gof one of its t'wo triaugular sabunits: (iv)12 adatoms in T4 sites in a 2x2-1ike environ.ment, and (v)six rest atoms in the ftrst atomic laycr the danglirlg bonds of wlkic,hare not saturated by bondingto the adatoms. Consequently, 42 atozns remah in the flrst atomic layea',among them the rest atoms, 48 atoms are ia the layer beneath,a'ad 12 adatomsi.n the top layer decorate a surfacetmit cell. Tlle DAS model also follows Pzinciple !. The 7x 7 surfacemixtimizesits dangling-bond densit.gby the formation of dimer-row dornnsn.watls whiczh are euergedcallyfavorablebecauseof the relatively 1owenergiesof stackinp fault au.dcorner-holeformation. The decorationwith 12 adatoms passimtes 36 d=gling bonds in the ftrst atomic laye'r. 12 dangling bonds remain at the adatoms,sbc more at the rest atoms, and one at the center atom of the corner hole. However; furthe,r dangling-bond saturation happens via the mechaaismdisccsedin Sect.3.5 in accordancewith bunkling/charge-trxnsfcr Priuciple &. Danglittg bonds locsll-zedat rest atoms acd cqrner-holeatoms corssidera' b1ycontzibute to occupiedsudacebandsjus'tbelow the Ferml level. The contributioa of daugling bon.dssîtuated at adatoms near corner holes (CoF,CoU)or the center of the llnz't mesh (CeF,CeU)are mu' c,lzsmaller, independent of the occurrence in the fautted (F) or nlnfaulted (U)triaugle The empty surfacebands mns'nly adse fzom daugling bonds of (4.78, 4.147). adatoms. Thîs situatioû is indicated in Fig.4.43. In. gece-'ral the loe>lszation of the surfacestates is lesscleaz.Partîal minring of dangling bondsmay occur. Due to the reducedCsv point-groap symmetry the adatoms ar.d zest atoms are no longer equivalent. They HiFer with respec'tto the position, Co or Ce, in a triaugular subllnst ard tlze occurrenvce of the stnrxing failt (4.148) (see '
4.4 Adatomsand Adclusters
*
.
* *
* . '
(a)
@
* *
@ '
*
*
@.
*
-
110
*
.
.
. *
@
.
*
*.
*
.
*
*
..
*
R*
*
*
G
* *'
R
*
e
@'
.
'
.
*
*
é
*
Q
.
R
..
@
.
*
R
. ..
*' @ @
..
*
*
*
* *
q-2y (lq
CeF CeU
CoF
(b)
CoU R
R
1f q1
1'
faulted
>
unfaulted
fault Fig. 4.42. Top view (a)and sideplane vievr (b)of the dimer-nrlntom-stacldng smface. J.nthe top view (a)the shaded circles desip (DAS)model of a Si(ll1)7x7 nate the adatoms.The circ'les'with a letter R designztetite rest atoms. Large open cîrcles designatetriply Bonded atoms izzthe ftrst atomic layer below the adatoms, whereas small open 'mrcles designatefourfold coordfnatedatoms in tke lower part of the same bûayer.The dots designateatoms in the third and fourth atomic layet's beneaththe adatoms. The lower panel (b)correspondsto a plaze view o: the nearest-neighborbonding in a plxne normal to t:e s'arfacecoat 'aningtke long dîagonalof the sttrfaceunit cen. A possiblesmfaceunit cell can be describidby 'lznes connecting the atoms with a dangling bond in the centers of the fottr (notshown) corner
holes. Afte'r (4.14z1J.
Fig. 4.42b) However,there are excellentexpem'mentalstudiH using H''gerent witiclk give information about the methods, e.g., STM (4.1491 or PES (4.:501, dangling-bond-relstedstates. For tlznnelklg out of rnn.ny occupied smface states the STM image (see Fig. 4.44)showstwelve potrustons ia the smface lTmstcell which caxl bc traced bnr,'t-to the adatoms.The corner holes aad the regions betweendimers ixï the domin walls aze also clearly visible in Fig.4.44. but alsoARPES (see, Scanningtl:nneling spectröscopy (4.14% e.g., (4.312) found a baxld whic,h extends up to the Ferrnl' level and pins it at 0.7 ev at abovethe top of the valencebacds. The correspondsr,gstate's are loem7szed adatoms. Naively, the Si(111)7x 7 strttdm'e may not seem to satisfy the re.
.
1'J8
4. Reconstruction Blements '
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surface. T:e Fig- 4-43- Contour plots of the sttrface states Pf the Si(111)7x7 planefor tlzreesurface bands squaresof wave hlnctiomsm'e representedin the (110) near the Fermi level (for detKs see Fig. l in (4.76j). (a)Peially FLIIGIsudace band in the gap, (b)au.d (c)ocyupiedsmfacebandsjust below or above the 'bullc plxne contxl'nq the long diagonal of F1g.4.42but valence-bandmaximl:rn. The (110) is ceuteredon a corne.r bole. Fkom (4.781.
construction pzinciples. The seemsnglymetxllic c'hn.racter indicates residual tmsatisâedvalenciesat the surface,ia contradiction to the spirit of Principles 1 and 3. One possibility to solve the puzzle could be related to the a-s obsenred in the cmAe of occuzzence of strong electron correlation (4.152J A scp.lr'ngof and 3x3 smfaces(4.137,4.139) SiC(111)/(0001)WxW 4.:431. vith the amo'ant of Si iu the surthe Hubbard pamrneter & (cf.Sect.3.5.1) facere#onmay indicate a value of about a tenth of an eV. The accompanyin.g surface-stategap shotzldbe llardly nmmzqtzrable (seealso Sect. 5.4.2). The DAS model described for the 7x7 trnanslationalsymmetzy can be The generalizedto (2n+ 1)x (2n+ 1)reconstractionswith n = 1, 2: 3, 4, corresponaing pnnc'tcezscontns'n (27z+ 1)21x1 surfacellmit cellsand are decorated by zzlzz + 1) adatoms leaving 2a rest atoms. T*e resulting adatom '
....
4.4 Adatoms and Adclusters
179
stzm Fig. 4.o4d;.STM image of a Si(ll1)7x7 nm2) sxrnplebims; face (image ex-tentt 12x12 with -3V). R-om (4.151) (copyrig'ht (2003), permsssionfrom Elsevier).
densit'y is reduced by 14 ,) with respect to the derusit'y.14in a c(2x8) structuxe. The domaia wxllg consisting of dimer rom with 3%Jsmers f:.ll 2/, the sarfaceLsfaalted. A rel1x1 tmit cells,the corner hole 6111one, and %M,16 atively simple attempt to tmdezstandthe variotzsstructures is that suggested by Vanderbilt (4.1531. For Si this author esttmatedthe euergy costs per 4 x 1 nlnst cell with rupec't to thearelaxedsurfaceof the s'tnztlringfault, A.f = 0.06 eV, of the domain wall, zl'tzl= -0.655 eV, and of the corne,r hole lc = 1.40 eV. The enerr of the formation of a'a adatom is a = 0.28 eV (cf.also Table 4.2).Thtus,the total energy gaiu per 1x1 smface region is
zo)
AE :2n+1)x(2a+1)=
1
'j/.f
+
2n,11/2+ mc + ntx+ (2a+ 1)c
1)c .
witiin the simple adatom model fotmd for Ge(111)c(2x8) the energy gain nmounts to
ïfctcxs)
=
1
zc.
allowsthe eonstrttcand (4.11) The comparison of the two energie,s(4.10) tioa of phasediagrams,e.g., 'usingthe relative star'zng-fatzlt formation enerr lj.fj6lj'trLaud the relative corner-holefommyttionenergy Ic/lzl'trlms indepenadatom dent variables.Suc,ha phasediivam is shou in F1g.4.45for a fLXe'd eaezgy.lt exbi'bits a series of DAS stractmes tf I.J is s=n.ll, which have iIs creaskzg(2zz + 1) periodicity as lc increases.At lazge.rvalues of ztu, the stackizlgfault is n'rfn.vorable,and tiere ksa trxnsition to azl orderedadatom stracture, notioaally the c(2x8)onc. .Xn inczeaseof the adatom formation recoastmzctions, e.g., 5x5, more farenergy malcesthe lower (2a+1)x (2zz+1) phmsewith the parameters vorable. The star indicates the stableSi(111)7x7 givea in the text. If the cost of fommingthe stanlrlng fatfts and corner holes ts too high, simple dangting-bond-removing adatom structttres are formed, like the c(2x8)stracture of Ge(1l1) shown in F#g.4.39.lndeed; the formation of stacking fautts cos'tsznore energy in Ge thxn ia Si (4.154). ,A.sa result
18û
4. RecoustrudionElements
0.6
O.5 0.4 ,..SR>
c(a8)
.t1 0.3
Y
.S 0.2 (/)
.%, 3x3 0
0
5x5
1
7x7
9x9
2 3 Cornerhole Ac /IAwl
4
'Xg- 4-45. Phmse(liagram of a (111) surfacewith c(2x8)recorbstzudion(sknple an.d (2zJ adatom model) + 1) x (2r;+ 1) DAS reconstructiozls.The reduced formation energiesof staddng faltlts =d corner holes are taken as variables. The = reducedadatom formation energy is fxed at c/1zl01 0.427. The star indicates n. the Si(11l)7x7 surfacewith the prameters listed in the text.
c(2x8)yeconstruction is more likely tlnn.n for Si, thougk the enera is close to that foy 7x7 (cf.Table4.2) However, the c(2x8)reconstruction can also be obsezared for quemchedSi(111) surfaces(4.146, On the othe,r 4.155). the
.
hand, strnsmedGe(111) overlayersgrowa on Si(111) substrate also show a 7x7 reconstruction (4.156). 80th fac'tsindicatethe outstandimg role of surface stress for the forrnntion of the lonprange reconstructions 7x; and c(2x8)ms disclnrvqed in Sect. 2.2.3. 4.4.4 Trimers
In. order to accotmt for the 12 protrusions
seen
in STM images of the
suface, a few of more complicatedadatom modeks'have been Si(111)7x7 proposed (see the pyraamong the,m tEe milk-stool model (4.1584, (4.157q), midat-cluste,rmodel g4.150), a'c.dthe trimer model (4.160', However, 4.1611. suzfacedismlssedthese models.'T%'me,rs subsequentstudiesof the Sî(111)7x7 on
group-W surfacesagain extered the discussionafter the obselwationof
(WxW)R%Oreconstructions on diamondtllllsurfaces of crystallites in CvD-growa 61mq (4.991. Oneremson was that STM imageshaveshownthat 2x l domnsnqcoexist witE (WxW)A30O strudmes. For (WxW)S30ô reconstructions of the single dangling-boud (SDB) sur'
face, there eldstsno model that allows the half-ftlled dangllngboads to become nearest neighbou. One conceivableway is to reduze the density of the danglingbonds by adsorption of one addition.al carbon atom.pe,rthzeesurface
4.4 Adatomsand Adclustezs atoms ia a Ha or Tz site, still leaving one dangling bond per
(WxW)A3OO
tlnst ce2 tsee Sed. 4.4.1). Suchadatomsare howeverenergeticazytlnfavorable (4.135). Iit contrast to the SDB surfacej the triple dangling-bond (TDB) smface to reconstruct by the suface provides a nattzral way for the (111) l'nst cells without any additional adatom. The formation of (Wx W)S300 reconstmction obset'vedia STM foT dixmoacl curstxl7stesis (WxW3)A30b suggeted to consist of trimer structures that are centered at a hollowll-llsite position on the TDB mtrface. Such a stnzcture is showain 1Rg.4.46. The hollow-site position is characterizedby the absenceof atoms uitderneath the trirner ic the secondsubstratelayer. The atoms fomning the trsmer are bonded with one bond to one earbonatom ia the top substratelayer. The other three dangling bonds per atom pardcipate in a 'very s'trong bonding withp'nthe trjmcr. lt forms an isoscelestriangle witk eqlllllbrip:nnbond lengtk 1.39 .â.of the tvo equally spaced bonds. The elongated bond length equals TEe distortions 1.52i. The angleat the vertex of the triangle is 67O(4.77j. The odd apnmberof dangling bonds on the substrate are ex-treme'lyMmM,11. suggestsa metallic charader of the stuface i.c contrndiction to Priadple 3. Neveztheless) tEe surfaceenera is lowe,rthan that of the Seiwatzclzainstsee Three smfacebands occttr i:a the projectedGlndsrneatalgap. Sed. 4.2.3). The wave llnction of the b=d pinning tbe Ferrn'' level i'a a midgap positiol consists of dangling bondsat the three t='=e'r atoms. Til.ts b=d is very sat, kence,perhapsGdicating the possibitit'y of a Mott-Hubbard metal-insutator transition (4.136,4.138, also Sed. 5.4.2). (see 4.142) Q
'f01 E'I
l .
-
4.46. Top view of a hollow-site trimer reconstruction of the C(l1l) triple dauglirzg-bondsurface. Carbon atoms in the uppermost sur(WxW)A30O facehyer are iuclicatedas large open drclœ, whereasdots and small circles descdbe atoinsin the two layers beneath. Fig.
4. ReconstrudâonEle-ments
182
Other exnmplesfor trimer formation cotkldbe polar suzfacesof compound safaces of 111on (111) semkoaductors. The formation of vacanc'iœ (4,45) V semiconductorstseeFig. 4.6)seenls not to occur at the As-teminated surface,of'tenreferredto as Gau&stlïïl. Whtle the formation of GaAs(111) is exothermic, the formation of As vacaccies Ga vacandes on GaAs(1I1) Unll'ke the situation for Gn.Aq(111), is endothermic(4.1621 on Gn.AR(ïîï) xs WxW3, 2x2, for (>.A.s(1I1) a vaziety of dîferent reconstractions, s'ac-h 3x3 and Ui-fxx/i-f,occuz's depemdingupon the procerssingconditions (see The 2 x2 strucwtm-eis generally believedto be the As trimer strucvture (4.31)). illtustrated in Fig. 4.47. Most of the evidencefor this reconstntction model Iu particlnlltr, the triangular potrusions is obtnsmedby STM (4.163, 4.164). surfacecaa be explainedby seen tu the STM imagesfrom the GauYs(ïîï)2x2 the chernhorptiozz of A.s tz'imezs on the As-terml'nated stvface. Mnn.nwhile, the trimer model has receivedfurthe.t support h'om a trausmisstoneledron micoscopy kwestjgation of Iusblïlïlzxz (4.164. Witbi'n the tzimer mode.lan n.nsoa-terrnl'nated(ïïï) suçfaceis decorated by a'a n.n5oaoverlayercorrespondingto a e = ) coverage.THee anions withâa .
/ J
r
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?
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F/g. 4.48. Perspective view (a) and top <ew (b) of a Si-rich Si-terminated surfacewithin the tetramer-adlayermodel. Dots desigzza'te C atoms. SiC(111)3x3 circ'les designate Si atoms in the subMrate (tom Large open tfzlled, katched) most substrate layer, adlayer). The larger hatcked circies indicate Si atoms izï the tetrnrnez.
hybrids fo=
a
threefold coordination within the Vlayer. There is a tendency
for a reybridization to spl. and p orbitals for one'part of the nrllayer atoms'. Theiz .%2 kybrids form bonds witlnl'n the ftrst nrllny.er? aud the rernnsns'ng p orbital of, a,n adlaye,ratom formq a bond to the substrate. The adlaye.r atoms possessiug oae bond to the trimer basis of the adcluster evhfbit a tendency toward t'wo sp and two p orbitals. The adatom itself appears to be lmhybridizedand bondedby p orbita!s as judgedfrom its bond a'aglesof 90ö. Jn efect, this allows azl energeticallyfavorablef-like drgling bond or orbital. The fm'eence of this half-flled daugliug bond seeaus to contradid b0th Principle 1 and Principle 3. However,it haeeen shownthat the more Si-ric,h 3x7 reconstnlction forms a Mott-Hubbard insulator grou'nd state as the less Si-rich (xCxVi).R30B rcsconstructionof the SiC(111)/(0O01) surfaces(4.13% Scct. .5.4.$. Only the Hubbard Mteradioa parameter U a 1 ev is mue,h (see srnnller becattseof the larger mount of silicon on the surface.The fad that the Si-rich 3x3 reconstrucvtionsatisûesthe Basic Priuciple is iudicated ia accordlng to the Fig, 4.49. This pbn.qediagram b.asbeen constmzcted (4.13$ rules describeê in Sect. 2.5.4. The tetrnmer-adlayer surface is predicted to be the rnsnlmllm gee-energy structme ttnder Si-ric,hpreparatioa conditions. This is con6rmedexperimeatally since this stntct'are is accœsiblein a Si-rich enviromment (4.136,4.167,4.1721.
4.4 Adatoms and Adclusters Nx
.
x
0.0
N
5X1 clean
Nh z'.y x S
x.
X 1./N.
9
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-
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h.x
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.
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x hx a+qp%'h. +lzh-4w ''N. ..>
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A*A. '%#' x. -w
tp x h. x 'h z o x 4w. N %. >g%% N h% Nhv -Nx
=
m
+u> x
-
(i- 2
185
Q'o z
1
I
C-ricb
pbsf P.c '
Sl-rich
Fig. 4.49. Phasediavam of a Si-termlnated mzrfacems a function of the SiC(111) càemicalpotential of the Si or C atolns. Relevantrecozzstrudionmodelshave been seiected.The enerr of the rathe,rdisordered 1x1 struct'are hmsbeenlowered by azz emtropy term assuznirg Sa= 0-15 ks aud T = 1000 K, in orde.r to explain tllc sequenceof pllasetransîtions Wx W -> 1 x 1 -F 3 x 3 with increasir.gSi coverage. From (4.136) .
5. Elementary Excitations 1: Single Electronic Quasiparticles
5.1 Electrons
and Holes
5.1.1 Excitation
and
QllnqiparticleChnracter
Surfacesare many-body syztems consisting of interacting
and electrons 3.3.1).In order to descrîbe rnxny propertiesj in pardctzlar, vound(Sec't. state propez-tie, it is stnlcient to l'eplazethe system of interac'ting electrons by a system of independentparticles (3.2). One exn.mpleis the density Gpnc.tional theor.y (Sect. 3.4.1)witbs'n the local approvsmatioh for the exchange aud correlation contributioa to the total eaergy (3.50). Using the KoMSham equation (3.46), the grotmd state of the electronic subsystem can be particles, more describedby Mdipendent(i.e.,eiectively non-int.eractiag) An stridly eledrons moving in an eobcwtive single-particle poteltiat (3.48). Cindependent'electron i'a a Kohn-shnm state possessesa âxedsingle-particle this eledzonin space. enero- and a defmedprobabilit'y distributioa of 0c111)g Eowever,exdtations of ac electronic system cnmnot correcvtlybe described by the independentKohn-sbam particles in (3.46). A 1ot of experbrnentalstudies are associated with spectroscopies a'ad, A.n elqdron may be added therefore:evitations of the eledronic (sublsystem. to the system or an electronmn.ybe taken away from the s'ystemand.,hence, a h,oleis created.The e'xcitedelectron or the kole strongly Mteracts with the many other electrons of the system. The electronic subestem is polarized and reacts with a redistribution of the e'lecvtroa deasity. Consequently, the enerr of such an electronic exdtation will diler from those for nozsiuteracting particles. lt is renormmlszedwith respect to energy azd to behavior i'a time: distribution. If azt excitation hnuqa, stnlciently long lifei.e., to the specwtral time, it howeverbehaveslike a particle. There-fore, it is calleda cutkdpc,ràïcle, moze strictly a quasielecvtron or quasiholedependingon the occupation of the corresponalng single-particle state before excitation. The properties of the cores
quasiparticleare better describedby a spativy non-localspectralt-weight) hlndion thau by a'a eigenenezgyand a wave hlnction., Suc,h quaaipazticlesare adually observablein severalsurface-sensitive spectroscopies.
The Xst scn.nningt'Tnnelizzgmicroscope was built in 1982 by Binnn'g aad Rohrer (5.1!. The physical phenomenonat the ol'ignof this new inldrralment is tEe brnneting of electrousthrough the vacuplm. J.nsuch a microscope a shar.pmetallie tip is positioned at a distauce d (ofthe order of a few i) h'om the suzfaceof a conduding smple t'Fig.5.1).l'a this way there is au overlap bet'weeathe electronic wave Rmctions of the tip and substrate. A voltage F' is applied to the t'wo electrodesresulting in a tllmne.lcurren.t IT. This can occur 1om the metal tîp to the smface or vice versa, dependiug on the dkedion of the bias.Structaral icformation can be obtainedby sc-nmning, i.e., by moving the tip over the lmrface, e.g., witlu'n conqunvcurreat mode. by vazying the The condition of a constant tllnmel ctzrrent I.c ca'a be A116E.ed dkstanced betwee,nthe tip and the sample. The rœulthlg corrugation hlncoon contnlns information about the sarfacetopography.A topograpidc5Vage of the tke suzfaceis also obtained wit'hT'nconstant-heightmode, by memmnvlng zaartudeof the ttxpneling m3rrentas tke tip is movedacross the st-nce at a microscopy (STM) Lsusaally performed fzxeddistanced. Scxnnlz.gt:tmmfRll'ng ic the constant-current operation mode. More information aboat the eledronic s'tructare of the surface can be obtaiued by studying the dependen- of the STM signat on the siga and magnitudeof tlb.etip-sample voltage. By varying the bims,stonmlng t':nmeo'ag can be done. The sign of the voltagedeterrnineswhether spectroscopy (STS) occupiedor empt,ystates are studied, ms showai:a Fig. 5.2 for a sernsconducvtor snrfnme. For posjtive bias (a),ttmne,ling of electrons rA.n only occur from occupiedtip-metal states inio empty suzfacestates or conduction-bandstates i:a the substrate. lu the opposite case (b) with F < 0, elmsticttnnneling of electrons1om the metal iato the semiconductoris not possible.Only a current with oppYte siga is measurable.The measured'ialnneling curzent I.r originates from occapiedsucfaceor valeace-bandstate,sin the semkonductor. .
1 Tip
V d
Y
Surlœ d
Scan 7
Diredion
Fig- 5-1. PrMcipleof a scltnningtunneling Mcroscope.
5.1 Eleckronsa'adHoles
1%
(a) Energy
Eneay
CBM x
CBM
x
.s-1
sF
eV
-1.y-. .eV
VBM
y/E,,,
a---
----------------------------------;, /,
/
.-,ç
-,''>--u
Ffg- 5.2. Electronic band schemealong a sudacenormal of a semiconductorsttrface an.da meta,ltfp trv,h' valuesof the 'biasvoltagein (a)and (b). (1eft) t) fbr oppos-ste The energles of the conduction-band rnlnlmum (CBM),valezlce-bandmn.vlmum (VBM)a'adFerml' level (ga)ak well as distributions of possiblesurfacestates are indicated for the unbhaaed seMconductor.
By meastlri'ngthe dependenceof the curren.t I'j: on the applied voltage F, one can obtain an image of the energy distribution of the elec-trordc states ia
the surface region. Sizlceone is more interestedi.a the generalspectralbehaviorof the tl:nnel current and not i'a its exact absolutevalue,a simpMed appzoac.h caa be used. At the surface-tip separations of interest in STM or STS, of the order of 4 i or more, the surface-tip interaction is extremely weak. It is natllral theh' to calculate the tlmneling current using time-dependentpezttzrbatioatheozy The result for an elementazytrlnnel processis given by FerrxstsGoldenRttle. Axsplrningnon-interacting electronsJ.nthe tip materii and the sxmple surface o'r of the type chazacterized e.g. ia (3.1) the probabûil per tmit time (3.46), for an Lsoenergetic tptnmelprocess across the barrier betweena surfacestate with energy ss = eu (i) and a tîp state #.r@)' with enerr s,.r #s@)(SL?A) is givezzby
lWs
=
2% ,-
.: 8 @'r+ ekI'J.'Ts I -g-
-
&s))
wherethe trn.nnition matrix element
j
r'as -
d3z4,:t(z)>#s@)
i s intToduccdfor the taaaeling operator t. It may be issentiallyidentised M'ith the current densit'yoperator (5.21. F' denotesthe appliedbias. First we considera 'Fnste negative biks 1//,although tilis is relatively small on the scaleof the tip work fllnction or iomszationenerr of the sample.The tqtal current for tpznnelingfrom the sutface into the tip (7 < 0) is give:a
J@)mq the Fermi dltzibutziomThe factor 2 accolmts for the spin decan be rewzitten a.s generacy.With (5.1)expression (5.3) with
I'2 =
4rre s
+*
L.
a dcltc)%-ulçe eF')J J(c-cv'-e,r)J(s-ss), (5.4) 57IT'z'sl -
T,S
using the properties of Dirac's élhlncwtion. Resukssfrn-llarto (5.3) or (5.4) Eave beenexploited by mrtny groups. However,genernlizationsto obtaia an explicit formllln.for the cl:rrent density have alsobœn usedvand real atoms of the tip
and surfaceinstead of modelpotentials kave beentékea into accout (5.41 A more or lesse-xact way to determl'nethe tltnneling current vithin the singleparticle picture is possible asing a Green'sAlnction formalism aud solvir.g zplmericallythe Lippmaa-seahwiuger equatiou (5.5). However,the zplmerical efort is large and,moreover, a detailedMowledàe of the tip shapeksneeded. The theory of the tnlnnel currem.t (5.4)makesno distinction betweensmfaceand tip states. Howevez,ixl STM or STSthis distiaction is cruciat. Ideally, one is interœted to relate a'a STM image directly to the stlrrlme propezties, wheremqin the abovedescription the current mvolves a convolution of the electronic spectra of surface and tip. nerefore, Tersof aud Hhmn.nn (5.6) proposed= approvsrnn.teway to elirninate the tiphpzoperties. They consideredau ideal tAlnneliagOcroscope with a model for the tip that woald have the highestpossibleresolutiomThis goal is best aœeved assnprnlng that the tip is a rnnthematic,alpoint sotzrce (s-wave 1ts potectial a'n.d approvsrnation). wave Atndions are arbitrnHly locnll''zed.STM experimentsobviously n.1mat atomic resolution. Therdbre, intuition suggeas that the tip must have atomic dimenm'ons as shown in Fig. 5-1.Iu oth.e.rwords,the tip is not point-like on tshe length scateof the axperirnents. Nevelheless,the point-probe appzoxn'rnxtion leadsto a reliabledescriptionof expersmental data, thougk it is not c prscrï cleaz why thks approvirnation holds. Witlnl'rt tlze polt-probe approximatioa for the tip at the positiou œt:p, the trnêndtion strength is given by .
:x: (#s(œt$p)121T$s!2
(5-5)
This approvsmxtion lfo-qrlm directly to the local electrozkicdtansityof state of the surfaceregion of the snmple.Without cotmting the sp% degen(DOS) eracs $t reazitq
p(œ:i)=
1#v/:(z)I2é@ ::.(/2)). X--2
(5.6)
-
vvk
This quautity represents the diagonal elementsp@; c)= A@,tr: sj of a more geuezalquantity, the single-pmicle spectratt-wtAirht) h'ndion A@,zT;e) of the electroic s'yste,m(5.7J. This hlnction also accouuts for the qua-siparticle character of the' exdtations. =* Comsideringthe lirnl't of low temperatures witk J@)(1 .f(s cF)J O(as>c)O(c ey sp)and introduchzgthe giobat electronic DOS of the -
-
tip
-
-
-
5.1 Blectrons and Holes
sy)
=
191
EJ(s cv), -
T
one
falds for the voltage dependemceof the t'lnneling curzem.t
dsots ekrlptœtyp; s). &F+W
J,r =
-
Low temperattzres me-qn that the thermal energy kBT is small comparedto $he position of cp with respect to VBM and CBM. 1.ntlze llmn't of small applied bias voltages or nearly constant DOS of the tip in the region of the Ferznienergy, this expression becomes .
Je
JT =
DLsvj
/ JFVW
dsptœt:p; s).
A ssrnilitr result holds f6r a small positive voltage (7 >
0)
el?+eV'
DLep)dF
I.v (x
'
dcptmup; c).
Eoression (5.9)showsthat for Jzr< O (7 > 0) the tunneling cltrreat îs accompaaiedby the generation of hole (eledroas) ia the rebozlof the spmple smface. Correspondingly,the tl:nneling clTrrint locnlly probes the pazt of of states of the surface that is occupied (unoccapied) the electronic dezusit.g for zero bitzs.Expression (5.9)is derived withsm the single-pacticlepicture, in the tip with the The attractive Coulomb interacbtion of electrons (holes) holes(eledrons) generatedj.n the surfacere#onha.sbeenneglectedfrom the One argument justifyiugthis neglect is the distance of tlb.e vep- be particles, rother the dmmatic redudion of the eledzon-holeiateraction by the metn.llsc screenlng in the tip reson. is proportional to the local eledronic density The tlmneltug current (5.9) of states (DOS) taken at the position of the tip. The local densityof stata is integrated over an energy intezval for F' > 0 (7 < 0)above (below) the Fermi energy, the length of wlzic,his given by the applied voltage. Consequently the restzlting 'STM images depend on the siga artd the magnitude of the applied voltage. Exadly thks dependenceis damonstratedin Fig. 5.3 for Garic,hGaP(001)2x4 surfacereconstructions (5.81. A constrt-height mod.ewith a tip-surface distance of about d = 4 â. is assïlmedfor the simulatiom To accolmt paztially for the nqnideality of the tip, the specvtraare averagedover a small interval Zz = 1.5 A of the normal distances. Two llmstations in modeûng STM images witbin the Tersof-Hn.rnltn'n approach (5.9)have to be mentioned.The tmderlying perturbation theory requires timsn.mpledistances larger thac the decay length of the wave hlnctions into tlle 'vacullm. Larger currents are modifed by multiple scattering of tite eledrons. Mother problem for small distaucesis related to the wavehlnction overlap of tip and sttrface.The resttlti'ag lbondsng'atld t=tibondsng' '
Top Ga Dimer j( 1.t. y: àLt y .tj u:? . . ,r , . (.?s ;.(..'p't
.
)
..
.
!
.
.., k:'';.Jz )..i.)f . :
: '.
. . ..
(tL''.
%%
z ...t. / :(.: .
.
lk'
., ;..(y: ë. .'r' ..!,
.
.'
'
'
1
..,..t. ... j:l ,,. ..' . . .
'
: .. ? s......) .. . .:-.
(
J
'
.
i
.
12
:
i
Ii
'
,
i
I
1 l .
j..v ,.
....'..
t,..,'.
'
G' +-
:
.
EI :1 i!(
.
)
.
!
.
l
f
.
i ,.
. .
1.i'
é.xq
L
1+-d ' '
1.û1 17:.
7.1.53nm
'.
'
i
;
$ 53 nm .
Fig. 5.3. Calctzlated STM imagœ for two dimer geometries of the GaP(OO1)2x4 sklrface.In the cmse of the tomGamdlrner str'ucttlre the P atom in the hetezodsrne.r of the mhed-dimer geometry (shown ita ng. 2.17)Ls replaced by a Ga atom. The negative (yosijive) voltagœ are measured=th respec't to the CBM (RM). The ''brightness indicates the magetude of the 10c,alelectronic DOS. From (5.8).
linear combinatîonsmay #ve rise to a completechangei'n the STM contrast (d. the remarlcsin (5.9q). The memsmement of the spatial variation of the local DOS for a given voltage is a powerfool tool for obtaining local stntctltrat 'Oormation about metal or semiconductor mnrfaces. Moreover, the spectroscopicmode of STM, irt which the tlmneliag current f,r is recordedas a fhlmction of the applied bias 7, gives direct information about the loci electronic strtzctttre of the suzface.Asmtrnsngthat the prefactor of (5.9)is nearly independentof the voltage,diferentiating thks expression with rezpect to F yields the diferential conductance,dfan/dvr (x: Dtsslptœtsp; se + e5rl.As a function of F' this qt,n.ntit'y roughly reproduce.sthe local densitjrof states of the surfacein the neighborhood of the Fermi level. Density-of-states featares in I-V curves appear as various lcin'k-sand b:tmps.They aze obscmedby the fact that the '''
$.1 Electronsand Holes
193
'*'
*
6
4% % 1.
*
r.r . .
zs.
a
..*.
..e .
m
.
.
.
Jea
jj
* @
2
2lb..
%*-z
*>
.
*w .
*
'.
Nà ,. z,
2
.-
hwx lw-k
a
;.ee sv< =
x
*>
A*
**
&
*
0
..4
a
. .
.
*
T:
4
-2.
-1
Tr
0
1
2
3
4
Energy(eV) Fig. 5.4. Relative conductancevezsus electron (hole) eaergyrelative to the Fermi energy s'v measuredby STM on a cleavedSi(lll)2x 1 sttdace.After (5.10).
tunneling cuzrent dependsexponentially on tlb.etip-sample separation and. in a norl'lnea,r mnrl'ne,r on the applied voltage. Most of this dependencecan be removed by computiug the ratio of diferential to total tTnnnelingconductance,
c'1.f 'r dy
rrf = = 'f/'
plztip,Ev + dV')
1
fF
cl'U'qssicv
dz#lttip,&).
Eereonly the restûtfor v' < Ois givem J.nthe opposite case 1/r> 0, tEe energy integratioa hmsto be Ganged accordsng to (5.9b). hdeed, (5.10) gives the local DOS of the surface at a hole (electron) energ.ywhic,h is given by the applied voltage. It :is normalized to the local DOS averagedoveer a,n eneror interval of length elFlTvenby thks voltage. Am examplefor the z'esttltof such a procedttreis shown ia Fig. 5.4 for a Within the region of the G'nndxrnental cleavedSi(111)2x 1 s'arface(5.10J gap .
of bpllk Si two surface-state-relateddoublmpeak stnzctttres appear with peak energies -1.1, -0.3, 0.2, and 1.2 ev with respect to the Fmrrnsenergy. The spedTalfeatttresat higher or lower energiesare due to bulk states. The fotzr central pen.lrKcan be ex-plxinedusing the model of tûted em-bonded citains (see resulting witlnln a simpletight-binding Sect.4.2.$.The band structure (4.7) approxsrnation gives a (normazzed) electronic density of states of zsbonded càains as (é'= 1s1 sz 1, sl + sg = 0)
j
-
'
(5.11) The meastu'edpeakpositions in Fig. 5.4 cac be identïed with the square-root singulHties in thjs DOS.This gives a surface-stategap of about rs)-s2 I = 0.5
194
5. Blementazy Excitatioms 1: SingleEledzonic
Quasiparticlc
ev auda.tè averagebandwidth of a rr* or rr bandof about 0.9eV:i.e., Ikrl= 1.0 W with F' bemg the averagedinteratomk Mteractioa matrix rlement. 5.1.3 Photoemission Speeroscopy and Iuverse Photoeml'ssion' If one bombardsa samplesurfacewith electronsor phètons,electromsand/or photons will be emitted, whick have aa enerr spedmzm. Heuce,Hormation is obtained dkectly or indirec'tly about the jurface electrozzicstates. The most important and widely used experlrnental technlque to gain iaformatioa about occupiedeledronic surface states is photoemlnqionspectroscopy TNe solid surface (PES)(5.112. (or sometimes photoeledzon spectroscopy) is irradiated by monochromatic photons with energy ruo, holE'sare generated izt the sample, and the emitted electrons are analyzedwitit respect to When photons in the tûtraviolet (UV) their eetic enerr ck:u = :2:2/2m. spectral rauge are uset the teelnniqueis cmlledUPS ('UVphoterniAqioaspecBesid%the kiaetic eaezgyone may also use the enzissiondirection troscopy). by the angle9 with the suzface sin/sin8, cosl)de-sczibed kjk = (cos/sin8, normal a'adthe angle / in the surface plaue to charactezizethe geometry of the exp-rime'nt (Fig.5.5a). Vatyiag $ and/or / the method is then kmown as angle-resolved(AR)PES or UPS, ARPES or ARUPS. hwerse photonmsssionspectroscopy' ImESIcan be regarded as a timeIt therefore probes the tmoccupied reversedphotoemissionprocess (5.12J surfacestates. Ia this tenbnsquea be>.= of electronswith ene-rgysktn and wave vector k = àtcos ysinù, ss.a/siap, cos ;) is kcident on a smface (F)g. 5.5b). The electronstrn.nqmittedinside the solid decayto states with lower enerr through tlle Auge,resect or by emitting photons,'which are detected.There are tvo operatbzgmodes:either the energy &zJof the detectedphotons is held .
(b)
(a) /
Ekinl k
ho
/
Z Ekin, k
T'sg. 5.5. schematicrepresenzation of a photoemission sion (b)process.
(a)and inverse photoemis-
5.1 Electrons and Holes
195
collstaat ard the spectrtlrn is obteed by varying cku (isochromat mode), or sk.ln is kept coneant and the spectranrnis taken as a funcwtioa of hu'. 1f, izï additîon,oae takesadvantageof the k-vector raolution, one ca,llsthe method k-resolved(KR.) DES, KR.TPES. A rigordtls theoretical approachof azl elementar,gphotoem7'Fmion (invezse 6111 q'aantllm-mechxnscaltreatment of the process requires a photomml'ssioa) complex coherentHteraction processesstarting with a photon (eledron) and Enishin.gwith an electzon (photon) in the detector ms well as a hole (eledron) ia the smple. Titeoretical approaGes of this k'sndtreat the pkotoemission for instaace a.s a one-step process (5.13-5.151. A more instracvtiveapproachis The optical excitatioa of an electron ts.r. the so-catledthree-step model (5.16). a 6149,1 state fk) and a hole (iLthe izzitiatstate ïi) can simply be desmibed agnin by Fe-rmils Golden Rule. Withln the dipole approvirnxtion aud the iadependent-pazticleapproximatjon the trxmqition probability for the ftrst step is given by 2r
a
i t î&h(&)= -..àk) aytjg)spl (k)( j @.f( 1.,,/.2 '
-
-
.
Withi'n the single-pazticlepicture the pezturbation operator Xint) the lightmatter interactionj is given by the eledron velodty operator 'v = ihlV,a(- tiI1 the cornmutator represeatation) oz' the m'omentamkoperatorp li:athe case of local potentialsi.u the single-particleHxrniltonian Hj and ilzevector potentiaz A of the iuddent electromagneticwave as tin.cgs llnits)Xfnt'= --G.A .p with mc matrix elementé t
J42(:)
e
-
--A mC
.p.s(:),
py:@) d3œ#),@)p#ns@).
(5.13)
=
The gauge of the electromagneticâeld is here chose.nsuch that the scalar potential vanishes. The vector potential .A is nearly spatially constant iu the long-wavelengthl'l=5t (inITPSAhe wavelengthof rmiiation is still à > 10û describesenerr. conservation dlpring a direct A). T:e J-fttndioa in (5.12) optical tmnsition from the Oed initial state sç(i)into the e'mpty ftnal state sJ(i) in the surface bands. For photon energicslazger than the ionization energy of the system, f.nal states with energiesabovetke vacullm enerzysvuc occm. In the caze of blllk states) as a secondstepj the e-xcitedeledroms propagate to a certain extemt to the smface.'This trn.nqpozt probabilsty dependsalso on the eaergy and wave vector. J.naay case, for egLL-) > svac electrons may escape through the stkrfaceinto the vacmtm with a probabiiit,y T/kin,i) as a thhd step. They appear in the vacuum with kinetic enerc slqia a'admomentlnrn hk. The escapeprocessesare elmstic:i.e., dkia = sy(k) emic holds. Together with tlze ettergy corzservation in (5.12), this equation results in the well-lcaown Einstei'a law of the photoeledric eF.ect (5.171 Becauseof the 2D tr:mqlational -
symmetry of the system, the electron transmlAqion through the suz'faceinto form kl g= the vacqrlm requiresmomentl'rn conselwationin the generxliaed k + g for the wave-vector component parallel to tbvesurface,where g 'xs a vedor of the 217reciprocal lattice of the surface.On the other hand, the wave-vector component parazlelto the surfaceoutside the crystal is relatecl sin ù ard )ç:g= to the experimentalpazmmetersclcis, 0,tj by kjj = zmskin/D,z kgk(cos 4,sin/, O).Consequently,not only the energ.yof the initial state (Mth i.a the surface b=d structme but also its wave respect to the vacutkrn level) vector can be determineâfrom the expem'rnentalparameters FlaJ,skin, 3, and
/.
In s , the three s'tepsylelcl an approvsmate expression for the photoelectron current in the vacmlm or for the measurednlTmberN of photopointing in a specfc electronsemitted into a cone (witha small solid a'aglel
of the single-electronspectral-weightfunction of the sl:rfn.ce system are takeu with single-particleeigezzstates '(u@)(withquantzlm nllmbersIzil that have been calculatedin a certnsn approvlrnation of non-interactiug particle-s,e.g., to interactiug withsn DFT-LDA (3.46). Thrusallows a genernlscationof (5.14) electrons ms indicatedby tke replacementé'(c cv(i))--> Xvvti,s)of Dirac's matrix elememts of the spedral-weight by d-iagonaz J-fïtnctionsin (5.14) (5.15) hlnction A(m, a/; s).Jn this my the idemtécation p@; œ;sj of the s) > .A(œ, becomesobvious. local eiectzonicDOS (5.6) with v = #: izl which ortly 1.cthe second spectral factor Jks izz (5.14) occupiediztitial s'tates are considered,it is allowedthat the exc-itedholesinteract with the remairdng electrons. On the way to the Apacutlm the oatgoing photoelectron, v = j', also interacts to a certnsn erent with the electroas in the surface acd barrie'r region and the remnsnlng holr (adiabatic approxHoweverj this ezect shouldbe weakenedwith increasing1dimation (5.18)). so that the replacement of the electron netic energy (suddezz approvlrnation), spectral hlmetion AJJ by a élAlnction is oRen a good approxn'mation.The i'aterpretation of expersmeutsis mostly baseclon the sudden approxn'mation, expressing a photoelectro:a specwtrlprni.zzterms of the one-pvticlespectral hlnctiozls of the iaitial s'tates,Aç(E-, s).However, in genezal,with the excitation interacting hole and electron quasiparticla occur (5.19). -
5.l Eledrousand Hole,s In the simpliîed expression
197
(5.14)'with
tite factoeed representation A:(i, c F?zslWyyt/1, s),the electron-holeinteraction hmsbeenneglectedin agreement with its derivation witqhsmthe origizlal picture of non-interactiug partscsles. Sincethe e-xcitation energiesr?ware muc,hlarger th= the energy of the fl:ndnmemtalgap: the cledron a'nd hole excited i'a a photoemissioc procv are energeticallywell separated.Their coapling in the surfaceshould thereforebe smltll. This holds even more for the interaction of the oatgohg photoalectron with the remainlng hole. Nevertheless,for not too large klnetic intemsit'y vafatiozus of the enertesof the photoelectrons (adiabatie lc'rn-lt) plazmon satellite of the main photoelectron peaakversus energy have been and have beemt'racectbnr!lcto the eseds of the electron-hole observed(5.19) interaeion, at least to vez'te.xcorrectious (5.20, 5.211. stdctly a photoelectron spectznlm in the adiA spectral Glnction (more abatic limjt) for a surface-state band is shovm in Fig. 5.6 for the occuyied The enezgy Hssereacesv Ev(k) is surface(5.221. r-band of the C(111)2x1 intezpreted a,s the binding eaergy of an electron with resped to the Femmi = 50 ev has been performedunde.rsttrfacelevel. The measurement with FlnJ seasitive conditions. The measuredazimutlnxl direction is fxed parallel to (ï10).For 0 = 33.30 66.80the wave vec-tor L thereforevaries along the ?R li'ne from its middle to a,a eqttivalentpoint in the neighboring surface BZ Lg# 0).Despite the broadens'ngof the spectra, the peak positions should be identiîed with surface 'bazd energies, here with those of the cr-bandof the Pandey chnsnmodel.The observeddispersionis in qualitative agreemezzt w'ith that of the calcltlated c-bands in Figs.4.16 and 4.18. Near # the occuied x-band comes closestto the Ferrnl'level whereazin the directionstoward Bz .a strong dispezsionof t:e xr in the sxrne Bz aud.J' ic the adlacent baud towa.d lowerezzergilas is observed.Thc measaredvalue of cx(A)at lewst -
-
-
0= 33.30 K = c =
37.40 41.60 45.60
r=
Q)
49.70 54.00 58.20
c
62.50
w
>h
M
(D
c
T(
5
4
3
2
66.8o
1
0=EF
Binding energy (eV)
-;2
Fig. 5.6- Angle-rœolved photoelectron spectra taken frop a:a Aonealed surface. The photon enC(111)2x1 ergy Ls 50 ev at normal incidence of the incorning light: and the measured =imuthal direction is (ï10q.
Fig. 5-C. Combined photoemission and inverse photoernlqqion spedra for the determsnation of surface band gaRs at the highsymmetry point X? ir the surface BZ of s'zx IH-V(110)1x,1 surfaces. The energy zero is given by the VBM. The photon eneries have been chosento be >= = 21.2 eV (PES)and &zz = 9.9 W IIPESI. From (5.23).
0.5 eV bclow cs hdicates that the C(1l1)2x 1 sarfaceis stamiconductitlg,iu contrast to the DFT-LDA result i'a Fig. 4.18 tle:ît p=el). A.n expression s7'=57arto (5.14) IPES may be derivedfor the tve-reversed spedrnTrn is dorninated by the main peak in tlze process. The corresfonding spectral 'hlndion of the electrons in the empty ân.al state. Neglecting the iniuence of the trnnqition matzix elements and the veatex corrections, the spectral variation of KRIPES is governed by the empty i-vector-resolved density of states Ey AULL, c). 1n contrast: in the cmse of ARPES/ARUPS' the spectra aze governed by the occupiedpart of the Lvector-resolved dettsity
s.1 Blectrons amdHole.s
199
of states of the smfaces'tates,Eç.3ksti, the combinationof $. Conseqaentiy,
ARPES and KRDES allows one to detemnlnethe complete i-vector-resolved smgle-paaâcledemsityof states of a sllrfnme slrstem. An exampleis shown in Fig. 5,7, in wbic,h combiaûdphotoemsAsion aud invezse photoèmlssionspectra (5.23) aœe presOted for the XT point iu the s'arfaceBZ in the enerr region of the ftlled anion-derived dxnglblg-bond band Jls and'the empty catiozl-derivedCa bard (see Sect.4.2.1,Fig.4.12) for slprfnrtes.For tkase surfacesthe expercleavedEFI-Vsemiconductor(J.10)1x1 iment has the aduntage that a cornmon energ)rreferencefor the two applied spedroscopiescoald be es-tablishfd.J.nall cases i'n.Fig.5.7, it is e'ddentthat the anion-derivedsurface s'tate gives rise to a cler peak, whilè the cationderivedfeaturein some casœ is a rathe,rbroad line with a large slope at lower 1 surfaceof these materials energiescThe surface band gap for the (110)1x at the X' poin.t of the surfaceBrilloTlin zone is consideaxblylarger thn.n the fltndamental bttlk gap, bttt the hcrease 1om I'asb to Gap hmsl'oughly the same slope. 5.1.4 Satellites
The genern.lizationof Dirac's élhTnctionsfor electron acd hole excitations in to shgle-quasiparticlespectral flmctions (5.15) allows a moce complex (5.14) view of photoeelectron excitations beyondthe trivial lifetime broadenimgof a J-l:nnction (as,e.g., a porssibledeascriptionof the spectra iu Fig. 5.6).ln the spitit of Sed. 5.1.1 such a geaeralization allows oae to account for egects of the complete electron-elecvtroninteracvtion at the leve,l of single-partide excitations. The accompanyingm=pbody eseds al'e best demonstratedfor the spectroscopy of strongly locltll'zedcore states and a wider enera range. As an exmple, the Sizp and Si2.sphotoeledroa spedra of a Si mystal with (111)7x7 surface are presentedin Fig. 5.8. They are measuredwith a photon energy of Fttzp= 1486.7 ev IAIAQradiation) Tlle large ki(5.24). netic energy mdicates the sudden lx'ml't. The escape depth of the outgomg photoelectronsis varied by cblmrimg the escapedirection from normat emission (#= 00: mav'='lrn depth eqtln.lto the mean 9ee path Arafpof elecvtrons) to grarjing emsxqioa(ù= 80Q,smxll depth 'w àmfpcos 800). Iu. the flmt case, the spectra are dornsnatedby blllk losses.1'athe secondcase, it is clear that the sltrface plays atl esserttialrole for the PES. Beside the miu'n quasipam ticle peak desmibingthe core-holeexcitation witkout losses,the positiou of P wlliclt is usury used to d'R6 ne the core electron bindsng enera, or p QP W ith respect to a referencelevel, .-ggs j e.g., the vacul:rn level avac or the Fprmi level ek- (here: ss),the spedznlrn h.asan incoherentsatellite structme. In dnbonded solids the satellite stractures are mainly due to shake-up of sllrface aud blllk plmsmons.At the side of higher binr?îngenergies, multiple (n= 1, 2, 3, ...)lossesby plasmonsare Hdeed visible. The strongest satellites are due to blllk pMmons at about ssizl .-nop (J= p, .s) with Fopprts 16 eV. In
-T
Si2p
Si2s
Aop O
t= =
pmtop.
71
rM
=
:3 pdhop-è
n be seen from the spectral repzœentationor Lehmn.nnrepresentation of the Gzeen'sftmction,
G>hm/1f)
+co =
z;
' a;), dE?.g.(m a .et ..yjy
with n = cié (J-y +0)for energies above (+, electrons) or below (-, holes) the Fhrmt' enerr, at leas't for zero taperature. 'I'he Green'sR'nction descibes the dylmrnp'cs of the No -> Ne + 1 e-xdtatiozzs, whereNe is the nlxrnbe,r of eledrons in the syste,mgrotmd state (5,72. From its equation of motion with the m=y-body Hstrniltoml'ym i'acludi:agthe f4117 longitudsnaleledron-eledroa inttaraction,the Fourie.r trnnnnformationwith respect to the time wiable leads to the Dysonequatioa 25.31-5.33) e
:2 + ip + zazrk -
-
Mon@)5k@) &@,Z; &) -
=??;s)J@M = d3a//.)J@, 1 a/; &) 6Lm=p) -
with tke external potential of the ions
and the Montœ)
(,5.21) Hartree potential
z/; s) accounts for the exchange (X)and The self-energyELœ, V'H(z)(3.49). correzatioa(C)efec'tsoa the single-particle level It represeats a non-local, complœxtnon-Hermitsxml, and Gergpdependent operator. Equation (5.21)
the correspoadhghomogeneoms oae is e-qlledthe quasipmicle equation. The occarrence of tEe self-enersryJ7 mctieatesthe better treatment of XC efects for excitations th= for iudependentparticles in the electronicg'round or
state (3.46, 3.48). Due to the complicated self-consistentdependenceof E' on G, solviag is a dlec''ult task even for the simplest electronic model systems, e.g., (5.21) jellilxrnwith s'zr-lhœ. Moreover, E Yntnsnn the bare Coulornhinteraction n(m :F) = eajjm a?(to atl orders. The expn.nsionof 2 in the 677d:nonvazsisbl'ngorde,rwith respect to 'o results in the Foc,koperator, the non-local exclnnngeicteraction. The ecpxmqionof E in ttarrnKof r Lsslowly convergent. Anotb.ezapprolmh is more promising. The electronic system kmdezconsideration reacts to the pzesenceof an excitedelectronor hole wsth a redistribution '
-
-
'
5.2 Many-Body Efects
203
of the electrondensity i.e., a polarization of the electroaicsystem. For that reason
it is
more
conveGent to conside,ran epqmnqiozlof X i:a tnrmq of the
hwersedielectric h:ncwitll the spatially non-local and fzequency-dependent œ/;aJ) of the system. The reducvtionof the iuteractioa potential tion c-l (œ, by screening makesit obdous that an expnnRionof E i'a g/mis more rapidly convergent.
The self-consistentdependenceof 11'on G is provided by a closed set of coupledequations, the so-calledsystem of fhlndamentalequatioas (5.% 5.31J which, ia the simplest approximation that iacludesdyzmmicalpolarization processes,are decoupled'by neglectiug the vezte,x corrections. !(nthis case E can be wri'ttem as a convolutionintegral œJ;c) =
E(œ;
i
s
+x -K
.
dec-oo
+
z'; &s), (5.23) G@,m'be &,J)W'(z, -
where the vez'tex corrections have also been negiectedin the calcqzlationof c-1. This correspondsto a descriptionof the screensngwitbs'n the raadom 5.30, 5,31). Tlb.eresulting approvirnation phmseapprovlrnntion IRPAI(5.7, is called the GW approdmation. Unfortunately,> strasghtfoNardtm(5.23) n pf the self-energy provement of the method is Himcult. A furthe,r ex-pn.nqio' i'a powers of the screemedinteraction may yield tmpbysicalresuks such as I1l fact, the expxnsion itse,lfis only condinegative spedral h:nctioxxs (5.33). tionally convergentdue to the long-rauge natme of the Coulomb potential. So f a,r there is no systema'tic way of cboosing wizic.lkdiag'ramsto sum to go beyond the GW approxlmntion.The choiceis usuatly dictated by physical iutuition. Fortunately,already the GW approimation gives extremely good reutts for quasipartide energiesand spedral distributions. 'WhenW' in (5.23) is replacedby the bare Coulombpotential r, the rœult,The energy irg modiâedexpressioaleadsto the Fockoperator for exelns.nge. essentially gives the integral of the single-particle GreeapsAlnctioa in (5.23) exchangedemsi'ty, whicll is related to the of-cliagonalelrents of the spectral A'mction in contrrt to the eledron density (5.19). Consequently,the dfe'rallows one to exctrad emsilythe correlation part emce of the t'wo self-energies of the shgle-particle self-enerr =/; s)= .S'C@,
5.2.2
$ 2x
-swo+gtw dFwJe , g? a.xo,y jw.ta,ttggss;
-
z,tz a/)! (5.24) -
.
QuasiparticleShifts and Spectral Weights
is similar Apaz't from the inhomogene'itythe quasipacticleequation (5.21) to the covesponding homogeneousKohn-sh.am equation of density 'htnc-
Only the XC self-energyoperator X@j:r'; s) has to be (3.46). replacedby the Kolm.-shil.rnXC potential Fkc@ll@m/)(3.50). This sugtionnl theory
-
gests the reprexiation of the swtial dependenceof all q',smtitie in tez'ms of the oabonormallzed aradcomplete set of eigenAtnctioas (W:@)Jof the including aksothose of the empty statas, for a Kohn-shnm equation (3.46), surfacesystem with 2D traaslationn.lsymmetzy.Another reason is the wide availability of the Kolln-slmm (KSIeigenfnndions. The majorityof the opt'rnsqations of the st:Hace structuzes and the cazculationsof total en4rgiesis bmsedon DFT i'a LDA (orGGA), The Kohn-shnm equation is always solved using these Rnnctions. TMS lp.1rl!;to the represeatation ..
Glm,tF;s)
=
t?vzzzli, s)#,:(=)#s!zx(œ'), X'.I7-.7: ukvl
(5-25)
jp
m'; for all other quaatities, e.g., A@,œ';s) and J7(z, s). The og-diagonx!esementsof Gwv'(/7, ej azzdrvrzzlijs)Mth. respect to the b=d index are respousiblefor the fact that, in general, the quasiparticle (QP)wave Ganctionsare dieea'e'atfrom the Kohn-shnanones. However, for of excitations near the h'ndxmentalgap of b:pl'lrsystemsthe near equivalence QP aad KS wave G:nctions has been sbown (:.34j.There is experieace that the equivalemeremains 'valid if the energetic ozdezingof the KS eigenvaluœ sv(i) as well as of the QP energies CQ.,P (:), i.e., the posjtions oî the main peaksin the spectralA'nctions .âs,vti, c),is the snme. However,also in cases o' 1 smface, of violation f tahisordering, as in the case of the GaAs(110)1x the eect remai)ks small due to the small hybriOation of surf-related aud b'ttk-relatedKS states (5.354. In generaljhowever,the QPwave ftlmctionscorrespondingto nearly degeneratestates with diferent degrec of surface/bulk loemlszationmay be h'nfluenced by a stronger hybridization, mnldrtg the QX wave h'nctions signlcazttly dfaent 9om the KS ones. Dxerencesbetweea KS' and QP wave îtmctiozzs can be eoected in any noxsb::l'ksystem, i.e., when there are resonsof space where iEe electrondensit.ygoes to zero. 1:a thœe zetonstlte LDA generaœ an XC potatial with iacorrect asmptotic a rnsmingk/r tail izl the caze X atoms). behavior (e.g., the resultizfgDyson equaDespitethe asmnrnptionof band diàgonnulîaation tion can only be solved approvîmately via an iteration procedure, shce the pertatrbatlon itself Ls a, A:mctional of the Greelpsfl'ndion. The dhgrstmrnxtic representation of the self-energ.g contzibution iu the GWapprov'mation (5.23) Ls showu schematically iu Fig. 5.10. The solid electronline. represen.ts the Gleen'sTnmctionG, while the dnAhedline indicates the scrïvmed Coulombinteraction W. ln orderto begia witù a Green'sG'nction G0 VV (i,c), the spectral fltnction of which is close to the true oneh a shl'ft 1v(k)of the as
.
KS eigenvalueto sQv P(:)=
sv(i) + z;z(t-)
loding to is htroduced (5.36, 5.374,
(5.26)
5.2 ManpBodyEfec'ts
J
#
'
+
(5.25)for the
representation
With +0, for electronstholesl,
and the (3.46)
Green's 'hlnnction and the seklenergy the Dyson
reads (5.21) Gu(:;s) G2s(t-, s)(1+ pz'z,.(:, s) zv(:)jctv(&,s)) with the diagonn) elementst'EvvLk, c) of the rtarnnlnsmgXC self-energy(irL the GW approvsmxtion) equatiol
=
-
zJ; =/; c) ELm, JJ7@, c) J?kctœll@œ') =
-
-
zzot included i'a the KS muation.
Restridicg to the 'Ilrst non-vinishing order 'with rezpec't to the pertmu in the bation /JCv(i, , in whîc: the Green'sfnnction G (5.25) s) z1.(ï,-)q has to be replacedby Gz (5.27), ozle obtnsmq a ptmmaad self-enerr E (5.23) being quadratic i:a GD. ozt the right-bnnd side of the Dyson equation (5.28) 1n.order to avoidata ucphysicaldoublepole of the zesulting Gzeen'sAtnction -
at c
=
aQw P(i) in this terrn, one hmsto set
zv(&)=
Rc è'Euuv @;cQ.P(&)) .
(5.30)
ht this way the nlnnlrnown quasiparticle sbiêk in (5.26)is now defmed by The restriction to the Green's which.llas to be solvedself-consistently. (5.30) fntnction GQ (5.27) 'with the shift (5.30) cau be identfed with a Srst-order pertmbatjozstheozy treatment of the pelurbstion 5.8 (5.29) beyond the exwession (5.30) azd (5.28), givez an exDFT-LDA. Together vrith (5.23) KS eigenplicit de6rition of the QPshkft, whic.hcorrec'ts tlle corrcxspondsng
.
1r.addition.,îzl many practcal Gculations the energy dependenceof the Thjs seY-energy matrkx elementsis asstïrnedto be linea,rarolmd c = cQvP(J&). '= gives a renormalizatioa ,t1v(:) a.(:)ReJ&0.(i,s.(:)) where zv-l(i) = 3,e.,Avitha denominatorof nearlythe spectral t'iRefvovti,
(1 -
g)/&Ie=a.(:)1,
5. ElemeutaryExcitations 1: Single Blectronic Q''nmiparticles
206
0.40
>
..Q 0.35
,
MainQPpeak
c'
X' 0.30 w
Nzvtk)
xs peak ,.1
gsgg
2=
0-20
9 :e$ c ls c .Q -6 0.10
'w (ufetirnel-l
> 0) the chaugeL'E of thû XC selfeaergy wîth respect to the XC esectsalready i'acludedict the XC potential (5.29)is dornsnatedby the conelatioa part EC of the self-enerr (5.24), 62 rks EC. Negsedingthe movcation of .DC due to the dyzmrnicsin the screening, asslnmlmg,e.g., cstqawl - &s(ç, 0),the corrdation xlf-ene'rgy XC is detemninedby the diFerence of tNe scree'nedand baze Coulomb potentîal
(z,z/
>
0), ,
N@, 0) nlœ œ;
-
-
,
m
)
=
-c
=
1 -s
.---.
.P'
-0 1 '
m
ê
X = . c c)
-0 .2
O rs
e.3
/
z
#
J
t d l
:
(.)
>< .0.4
1 ' I
.
J
I I
-o 5 -1.00
'
-0.50
0-00
Dislnce
Piy. 5-14.
0.50 z
1.00
(%y)
Exckange-correlationpotential at
k'
sHple metal surfacecalculatedfor jezrl!'n with am electron gaz paramete.r rs = 3.9318. Tke solid curve represents a local XC potential extracted from the GW self-enera (5.46J1 and the dotted cmwe is the correspohaingLDA potential. The dashedcun'e is the image potential -c7/4(z mm),where zim = 0.07As.is the position of the Gective image p7xne. The distance z fxom the surface is measurediu uztits of the Fermi wavelength Atp= 2a(4zr2/9)T7Z (here:ls = 12.9 aB).After (5.464. a
:
-
.
:ê I ' .
5.3
SurfaceStates Qumsiparticle
correspon.diu.g to the GW self-enerr becomesimagelikeoutdde the surface, wherea.sthe LDA form gives rise to too large potentiat values Lu this reTon.
5.3
Quuipvticle Surface States
5.3.1 Surface Barrîer
'
The escape of a'a electron from > crystal without excitation or ex-ternalperturbation is usually preventedby a surfacebarrier. The formatîon of a stlrface barrie,r ca'a ewsily be ex-plnsnedwithsn the jellizlmmodel of a simple metal (5.471. The positive ion cores are smeare out to a positive background. h the surfacere#onit c-xn be modeledas a stemlike density (Fig.5.15). The electronic wave Glmctionspossess tn.!1Kinto the vamulm. The accompacyiug éxponential decay of the electron density is rœponsiblefor a redttcvtioaof the electron concentration in the region of the positive b ouzd near to its step. lusid.ethe crys'tal the electron density approache,sasymptotianlly the bltlk value tn the form of Fhedel oscitlations(5.484. .Xs a consdquencea net smface dipole layer appears (see Fig. 5.15). The fomnxtionof a smfacedipole layermtu'ns that the electrostaticpoteniial far in the vaclrnrn is higher than the mean electrostatic potential in the bulk. The mn.croscopicdipole and the accompanyin.gelectrostatic potential cu directly be obtxsnedh'om the microsèopic iectrostatitpotential 5Q@): ùr evnmple calculated withsn DFT-LDA (3.48). Jt is defmedby e%
O
ê
XM j.g (U >
V
--
.
...
xxxz
j
X
l
lI t j
:'*-R
g& =
0.5
;H7 Q)
ï d % %
'EP
;'7.(U
.'4n '
&
.-
c.
0
. ..
+
jy
r
a
% X
j
zx
:
zx
X
x N-
-1.0
-0.5
0
gjsjance Z
0.5
1.0
/ Fj
Fig. 5.1.ô. Blectron densit.gproile (dashed liaeland positve backgzound(solid at a jelliumsueace. A metnllsc system with a 1owelectron density, rs = 5ûB) nlikel is considered.The FezmiwavelengthAp = 2rs(4*2/9)1/3 Lsttsed as the rthxracteristsc 'length. After (5.47J .
the local part of the ex-ternal potential Moatœ) and the Hartree potential As one is mnlnly interested in the macroscopic spatial depen5'k@l(3.49). denceof the electrostatic potential in the surfacenormal direction z, it is convenfen.tto perfozm a pln'nar average 1
Ks(z) ; =
A
dzdp7estœl,
where z1.correspondsto the area of the surface tmit cezl. T jte plan.ar averaged fnnction can therz be passed throagh a âlter of lemglh L in orde,r to ex-trad tb.e macroscopic changesof the electrostaticpoteatial (5.492, HTIZ dz/ t = Q Cesl.Zl ). UStt ally' L is chosento be tNe rnsn5rnum thic'ks Jx-zw 5%IZ ness of atomic layers 'glvingclzazgeaeutrality or, as a mncdrmlm, the extent of an irreducible cvystal slab tsee Sect.1.2.2). Examples for the averagedelectrostatic potential are given in Fig. 5.16 for zinc-blendeGaN(111)1x 1 with diFerentGaor N overlayersand SiC(111)Wx W surfaceswith group-lff adsorbates.In the GaN case the surfacestoichiometry is changed.The secondeexample represents the Muence of dsFeront a& sorbates and adsorbategeometries tsee The graphs in Fig.5.16 Fig.4.35). clearly show the ivuezzce of the compound,the surfaceoriemtation, the stoiclliometry oz coverage,a'adthe geometzy on the actual surfacebn.rrser.This holdsixt particula'r for the trNsition rebonitself. 0n the other hand)tlle efect of all these'detailsis weakenedfor the potential step .'A?= Zstxl -'Q(-x) (a)
(b) Q
----
œA
A
ê*
-5 l +4(1
>Q) we
K w'
.
a?
-$5 ,
.
fâ
>
k.
/ #
* -2tl
l @
d
1
tI j
-:ts
J ' 7 't
.
l
t1
*30
$
'-'e.-
.-.
,
? ,
--'----
In (T4) c.o(p n (.q) ; (s$) B tsub.œ
-35 41
:l.,: z':
Distancez
c,i
(atomcc Iayor)
Pig. 5.16. Averagedelectrostatic potentials of semiconâuctorsudases'with difeerent stoickiometry (coverage) sudacesof and/ordiserezztgeometry. (llll-oriented the zinc-blendepolytype of GKN(a)and SiC (b)are chosen.The bulk valeace-band mmvt'vnxrn kstaken as the energy zero in (a),whereasthe vacuum levql desnesthe zero in (b).From DPT-LDA calculations(5.50, 5.514.
5.3
SmfaceStates Quasiparticle
Atomic bilayer Fig. 5.17. Averagedtotal one-electronpotential U(z)(dnmked tineland averaged elecrostatic potential 5Q(z)(solidlinelfor a diamondtlllllx1 surface. The sla'b a'ad'vamrnrr, regbns
are
shown. From
a
DFT-LDA calmzlatîon (5.52).
which ks a dired coasequenceof the macoscopic smface dipole. The detn5lq of the groupqll adsorption(element, atomic geometzy) have pradically no imfluence on the surface barrier. Additionnl overlayezsof Ga (N)on the the surfacedipole by sevâ sttrface)however,reduce(hcrease) GaN(111)1x eral tenths of an ev (several eV).t'a addition, many-body efects, e.g., the modif.gthi absolutevalue of the surface barrier felt image potentii (5.40), Fig. 5.14)but contribute practically nothing to by an escaphg electron (see c'hangesof the surface dipole with geometry and stoickiometzy Exch=ge a'adcorreïa' tîon efects increasethe surfacebarzier, ic pazticu!ar within the DFT-LDA ms indicated in Fig. 5.17. There,besidesthe electrostatic also the total one-electron potential (3.48) averaged over potentiat 5$s(z), the surface plane (5.42) is plotted. 1n.the bltllr region it is much lower in enerr tha'a kk(z) as a consequence of the laœgeattractive contribtttion of e-xchange an.dcorzelatiom I'n the b7llk region b0th potentialsshow oscilhtions normal to 1he sttrface. The wjdths of the oscillations are equal; only their nmpDtudesvary sligEtly. lu this region the dxerence of the (macroscopicilly) taken at the average averagedpotentials is governed by e-xpremion (3.50) eledron derusity.Tllis pammeter is a blllk quantity. It is therefore smportant for tke absoluteenergy position of the energy levels,but it doe not smflueace tlve vadation oî the surfacebarrier wîth orientation and atomic covgttration
(b)near a suzfaace, Fig. 5-7.8.Baud diagrams of a metal (a)and a rveml'coztductor showlng the de6rition of the work hmction *, electron aëzkity xj aud ionization en'era 1. A possibleband bend-tngîzt tite semiconductorcase is not visible on the characteristsclength scale of a surfaceof a few monolayers. 5.3-2 Chntaceteristic Ene-rgîes
.
suc,hEtsthe The surfacebarrier can be characterizedby characteristic ener#es * of a metal or the Lonization ezzergrI au.dthe eïectronclrzfàp 'workfzmction by Narious x in the caze of a norrnetal. Thesequaltiti% can be measured the ln the case of a metal (Fig.5.l8a)z spectroscopiesincluding PES/HVS. energy of the highestoccupiedelectronic level withp'n the crystal is stûl ev, where ev is the Fermi energy calcttlated for tlke ideal l'rfln'lte czystal with the tota,l periodic poteatial V'(z).The lowest enea'o-of an electron outside the curstal mizht be assumedto be zero, siace Ftzl and 1ts(#approe a eledron cazl be constant outsideof the mystal, and the kz'neticenergy of a $1.% madearbitrarily small. We call this level U(x) = Ws(x),the nacmum Jcpreô cvac, au.dusuaxyz'eferthe absolateenerpr positkonof sGgle particle states to 9om emc. Therefore, the rnt'nirnlxrn enerr 1 required to zemove an electron the intedor of the czystalto a point outsid.ethe crystal wéuld be +
=
cvac
sp.
-
(5.z13)
Figure 5.16 icdicates that the position of fvac is ivuenced by the surfaco barzier. Consequently,atso * may be influenced by sttrface relaxand nziqorptiozt of other spedes. In the case of nonation/reconstraction metals, i'n particular for sezniconductozs,it may also depemdon the backon tEe gxolmd doping. Howeverlit is better to avoid a strong dependemce doping level and to defne otb.e.rcharactezisticener#es .
X
=
1=
fvac
cvac-
-
FCBM) svsv.
(5.44)
5.3
SurfaceStates 215 Qumsipartîcle
The clectronnmn7't.yx 'vacuptm
ionization energy J)is the energy of the (photoelectron level referredto the bottom of the conductionbandsrcshc (topof the bands
vateace Iu tMs way, the energiesof the condudion-bazd rn1'n5svBM). mn.r'mthm are generallydefmeêaz quasiparticlevalues mplrn a'ndvalence-baud Startin.gwith the Kohn-sha.mvalueszQPcorrediozss (5.30) have to (5.26). be added. The dferettce of the t'wo band-edgevaluo, Sg = scsM evnu, defmesthe fundamental QP rnerr gap. lt exn be directly comparedv-ith vasuesderived 1om measureddata for x a'ad J. Rmdamental gaps dezived from optical mvuzements are in general somewhats'mxlqer êue to e'xcitonic binding eFects.The b=d line-u.pin Fig. 5.18brepresents tlle situation for a rtxlds.qticseznicönductorfor which defects and fmite temperattziea,re allowed. For ideal systems the introduction of a quantity * (lileren.t from I would not be useful. Witbin the axact DFT of an ideal imsulator at zero temperatllre, the highest occupiedKohn-sham eigenvaluereproents the position of the chemjcalpotential of the electrons (5.53). J.nFig. 5.19 the work hmdions for some typical metals az determlmedby vario'asexperimentalmethods (including PES)are plotted vers!zs the average distance rs of the electrons (5.48). The small variation in the work hmction 6vertlsferent cnrstallographâc faôesLsnot shown.There is a cler itlcreaseof + with i'acrewsiugelectron dtanqî'ty r,). This tread js well pronounced (decremsing for nlkn.li and p.11m.15 eartk metals but 1% signifcant for group-x metals. No clear trend ca,n be observedfor the other elements,e.gk, noble metals. Their -
4-s
F
s 4.o o
g t= ?
.uR
Yo
R . .
:
x
(p o ysrx
zjzxf x
. x
3.5
4.x
i ''v
Nx x
so
N.
.
*
2' s
o .
N
.
x
x
o
2.0
x.
w
N
N
xox. o Nx
N
2
3
4
5
NQ 6
Eledron distance r.j(cs)
.;i' ;E :..
)qFig- 5.19. Memsm'ed work fklndioms * of selectedmetals verstzs averagedistance '(,E:/of the electrozlsin the system. Data are taken from A11m.1't metals: opeu (5.482. .2 tilrcles,nll4n1leartk metals;fllled drcles, noble metalm squares,groupqll elements: siars, group-nr =d. -V dements:open tmnngles,trxnsition metals: 511Mtrianlles ..
.
ind group-lo metals: diamonds.The dashedline iuclicates a quadratic variatïon ihf ô witk ra on average.
and theoretical values f'ig. 5.20. Ex-perimentalionization ener#es (opea syzjbols) or an'dpprovlmaàe calculatedtusinga 61111 GW zpprovsmatlon (5.64) (flledsymbols) verst'ls cubic httice constant,s cm. The data collectzonis taken self-enerr (5.50,5.55) 1 sttrfaces only data for cleaved(11O)1x from (5.564. For zinc-blendeseznicouductoz's valuesfor (100)2x1 are presented.In the ca-se of elemeatalgroup-l'v semiconductot's and (111)2x1 are shown. The dashed lines indicate a lineaz (trianrles) (squarc) variatzon of the experimentalvalues witbln a compoundclass.
+ valuesvctry only izt the range 3,83 4.52 W with elecHzongmspazameters ra gs 2.12-3.0241. The tendency of a:a increaseof * with the electroa density Ls in agreement with the simple pictttre of the fozmationof a surface ctipole izl Fig. 5.15.1ts strength should be proportional to the dezudt'yof particles. Measmxda'n.dcalcalatedionization energiesare pbtted in Fig. 5.20verstts the bulk httice constant. The ionizatkon energies I belonog to a certa.in III-V, or I1-W)decreasewith increasing lattice group of compouadsIIVLIV, of the ppc and constaut. TMs is rnna'nly a consequenceof the l/oo-variation whic,h donninatethe valince-ban.d ppr hteratomic matrix elements(3.22) ms.vp'm:xo Table 4.1) However, the ionicity of the bonds (see sc4(0)(3.34). a.lsoplays a role. For group.W s-sconductors and III-V compoundsthe experiment/ ard quasiparticle valuesfor I agree excezettly. In the caqe of the II-W semiconductorsthe theoretic/ values tmderestimate the measured are not correctly vczlucs.The reason is not clear. Perhaps the QPshifts (5.30) ctescribedby the applied oversimpMqddescription (5.552. Ionization energiesa'adelectron Alnities are plotted in Fig. 5.21for g'roum cation size.rfhe signs of these energiesare chosen HI nitrides versus hlezremsing an' d in sucha my that the plotted levelsrepresenttlze valence-baudmnvirna m5'n5rna. Their eaea'gydistancef 'J' Tvesthe quasiparticle couductîon-band dependweakly on the group-lll atom. This gap Sg. The ioaizatioa eneyrgies is in agreement with the fact that the valence-bandma-'dmum of strongly -
-
,.
5.3 Qumsiparticle SurfaceStates
217
Pig. 5.21. Negative val'aes of the ionizatîon emeroe 1 (valeztce-baud mnvlrnum, and of the electron n.mn''tyx (conduction-band rnlnimtlm, squares) for cubic circles) groumTH nitrides. The level positions wah respect to the vammm leveâhave been determ':ned for (11O)lx1 surfacc in the framework of DFT-LDA (5.57j. They are sllifted by the relevant QPcorrections (5.50) computed within a simpliûed mofel
(5.58) .
ionic compounds is essentiallydetenml'nedby the anion. The electron xmnlty 'varies dramatically with the cation size. Tlb.evariation mn.lnly reûectsthe hugevadation of the fandamenti ettera gap along the seriesBN, A.IN, GaN, IZlN 25.591. The vatuesfor tlze elec'tron aEnity of AIN aad, i'a particular, BN are smalt but not negative. Onemay, however,eomecttbxt a negative eledron n.mnlty can be realizedby the adsoation of suitable atozns. 5.3-3 State LocnliRation
The surfaceof a solid,brexlrsmg the thrce-dimensionalpeziodicit'yof the cvystal, leads to strong modiûcations of the electronic stnzcvture in its viciaity. Beasides the fact that oaly the 2D waye vector k fro:ai the surface Brillotaim sone instead of the 3D vector k, represents good quautlnm nllmbez's(Sect. the actual surfacestnzctme azd chemsqtzymod.if.g the electronic stntc1.3.3), ttzrc as indicated in Fig's.5.16 ar.d 5.17 for the one-electron potential. lt is obviousthat in the neighborhood of the surfacethe eledronic wave Alnctiomq are dferent 9om the Bloch waves of aa i'n6nite czystal. In the presenccof a staface the wave fttmctions of valence electrons are of two maia types: i. The electronic ekgenvalues of the system with surface do aot correspond to Bloch ene,rgiœof the l'n6nnstebulk crystsl. Thei.r enezgieslie iu the forbidden regfon of the. projectedbulk b=d structure. The assodated electron states are accordinglylocn.lszed at the pertmbatioa, the surface,
Fig. 5,22. The tlzree type of eledron state of a trystal witk surface below the vacuum leve) (sckematically). Electrordc bands(solid are skown(a)together lkinesl with the projec-ted butk b=d structure (hatched Re,alparts of the cmrerebons). spondingwave ftmcdozusare also sketchedversus the H'leance z &om the surface
(b)-
and deOy exponemtially1nt,0the bulk tsee These sùates are Fig.5.22). catled ùo'urdstöac.esfxees and result i.n botmd surfacebandswhen var./ing the 2D wave vedor throughout the sudaceBZ. Evnmple for real bozmdstates are #vemia Figs.3.l0, 4.18, 4.24, 4.29, aad 4.37. ii. The eigenvaluescoindde with energiesiu the allowedreson of the projectedbulk band structure. The amplitudesof the associatedwave ftmotion are usually onhaucedat the badacelsee Fig. 5.22) due to rœznaace betveen surface azd bulk state. Mlvlng or Hybridizatioa of thesestatH is axowedfor appropriate symmetries. TKe resulting states describes'urJueeresoncr?,ce (sometimes statœ. They decay into the antirosonanaj bulk. The degreeof the decaydepeadson the actual state and n@n vazy betweentEe two extreme cases also showr in Pig. 5.22. .
The criterion of the euea'gyoverlap of states atlowedin the sezni-sn6nite crystal and balk Blochstates is Tathe,rqttabtative.ne loexlizzktionof dectron stata near a surfaceis essentially êetnrmlnedby the surface bamn'e,r axd it.s mate.idngto the potential enezorof the electrons in thc bulk tsee Fig.'5.17). This e-'m be dronstrated in simple model Yculatjons. For a fzxedk m use a onedimensional repr-ntation to model the total potential F(z)felt by an electron. According t,o Flg. 5.17 tke potentîal deep inside the crystal is describedby a cosine variation 244costzrrz/cl with lattice ,cozustanta azd
5.3 QuMiparticleSurface States
2l9
(b)
V(z) Vco
2Vo 0
-2tz
o
-a
z
-2Vc
-a
0
z
Ylg- 5.23. Model potential enera of an electron with Và> 0 in the presenceof a smface. wko diferent match>g positions z.o = O (a)a'ad zo = -c/2 (b) of the periodicpotentialand the stepike bmier are cahosem.
amplitude 2W.1ts averageval'aemay be usedas the zero of enerr. Onernny
imagine that the attradive ion cores are located at the minirnn. positio:as z = (zz + .1 zlc,oLn= 1, -2. , ) if W > -1 or a = na (n = (),-1, -2, ...) if Fh < 0. The surface around z = 0 îs modelledby matcht'ag aa abrupt potential step V'x = kr(x) (wlziciapproachesthe vacut'rn level at cvacl z = zo to the blllk potential. The resulting total potential is '
-
7(z)
...
V'x
for z > 216costzrrz/c) foz z 0 is asmtmedand the matchsng coaditgoztls chosento be p = 0 (a)or zo
=
-c/2 (b).
The motion of the cledron is describedby a one-dinnensionaz Schrödinger
equation of the type
n,z d2 + 2,,.,dz2
(-
(3.1)or (3.4)
V'/))
#6Z) fV7(I)' =
(5.46)
The eigenvalueproblemis solvedwithin the nearly-lee-electron INFEIap-
proxsmation (5.48j. In the weak bondl'ng l:'ml't the eignnflmctions and eigenMues are oaly Gghtly flifkreat som. those of a 1ee electron îf the wave kector k (along the z-direction with -x(a < k %A'/aif restrided to the BZ) ls not near a Br a gg rvection pln.ne, i.e., near a point k = ::Ex/a,, +3x/c,... If howeverk is neaz suc,ha point, e.g.?'Vc (aswe vdll asmlme i:a the fozowing), the f:r% electron states e iAz and ei(A-2x/c)aare xlmost degenerate.The true state is nearly give'nby their linear combination .
Wave vedor k F'ig. 5-24. The 'two lowest eaea'r bamds(solidlines)of a ID m'ys'talin the NFE The dashedlines indicate the free electronbands. approvlmation (schematically). :' .
2 -4-.7:2
am
-
'
svtk)
Kz k m (
Vi
,4 B
11
z
-
REo2
)
sv
-
(k)
=
(5.4s)
()
with Fo is the matzix elememtof the bulk potemtial with the cosinevariation (5.45)azd the t'wo h'ee-electronstates (5.47)This problam can be easily Near the zone b.otmdar,gk = j+ t%the two bandstake the solvect(5.60-5.621. form .
a
acu
a
('j q ). (j ) +
+
-
nz
2
urp,
( vol
+
+ 4a2
v
x
2
(-c )
(:.49)
.
The degenerayis lifted and the free-electroabards spht. There is an enerpr gap of 2(W ( at k = E (s = 0) betweenthe t'wo bands. The two bands are shownin Fig. 5.24, The correspondiageigenstates are
#cu.I+x(z) =
Aeinz eioz + 1 s:lc -vo
=
(-u,s) +
-
5,2 zm
x
2
(-czc) +
e-i',
(5.50)
.
and The resulting banct-edgestates foTn = Oaud 11 > 0, 4:k.:(z).x, cos (m.z) sbl (Iz),can be describedby trigonometric sanctions. ln 'contrwst to the ba,l'srcmse, in tke presenceof the surface altowed eigenl z states can occ.ar witbi.n the .hlndamentalgap with energiess, -D=a.(I) 1%l < 2 2 s < -#-zw,(I) + L5i L.They can be determiued by applying the method Allowing complex wave vectors 3, 5.64), of the tomnlea àarlé stmtctnre :5.6, ,
-
'
'.
5.3 Quasîparticle SmfaceStates
(0
rr)
22l
Pwq.s) i.e., k = -i(2 < q < L'Zoj inside the ij iq, the solutions (5.50) crystal z < za clescribewaves decayiag L'atothe czystat (z< O).The continul:m of states (5.49) is sometimesalso called the conthmlprnof drtucz gap dfafcd (ViGS) They play an importaut role in the explanation of the (5.561. electronic properties of metal-semiconductor contacts by meial-çndnced .:c7) state.s(YGS)(5.561 WMCJIof thesestates with energies(5.49) actuatty 5.641 exist, dependson the botmdary conditions at the interface. Ia the surface caze represented in Fig. 5.23, the wave Amctions outside the crystal (z> zc)must be exponentially decayingfxlmctioxtstc:1< V'x) tz
=
-
.
.$e.(z)
=
2m,
D exp
-
(1rx sslz -
a
The t'wo general solutions for
z < za
(5.51)
.
(5.50)azd for
z > zo
(5.51)have to
be
matchedat z = zc. OT1chmsto assltrne a particular potential defmedby zc and W. Exn.rnplesare plottedirz Fig. 5.23. The matching conditions require for the wave Rlnctions and for their dezivatives
o;!v/a,-sçlzol'/u (.zn)) =
d'lizzbmla-bq = tf#a.+ -
dz
Z =zo
dz
(5.52)
.
a=z,
The two equations azowtlle detomniuation of two freeparn.meters,the enerr eigenvalue(5.49) or the wave vedor q and the ratio of the coeëcientsA(D. Extemdedstudies of the possible solutioms of the mat problem caa be found in (5.62, 5.65j.For the kwo situations representedin Fig. 5.23 one fmds: '
i. If the surface is positioned smmetrically with respect to tke atoms (Fig.5.231, zo = 0) a botmd surfacestate exists for Fi > 0 but not for Và< 0. T*e bolmd surface state (ex-tsting for 'U8> 0)is deziveâ from the analytical contizmation of the lower baad. ii. The situation is revereedif the surface is not symmetrically located (Fig.5.23b, zo = -c/2).'A boan.dsurfacestate only exists if Fà< 0 and not 7i > 0. This state is also derivedfz'omtlze analytical continuation o'f .
the lower band. Anyway the trivial ex-numple describedby the one-dimensionalSchrödinger csquation(5.46) with the potentiat enerr izzFig. 5.23 makes it obviotls that smface bolmd states, which decay into the blllk- and possess eigenMues withs'n the Apndamentalgap of the bulk, may exîs't, depending on the actual sttrfacebarrie-r. 5.3.4 Quasiparticle Bauds au'd Gaps
The grotmd-state calculationswithin the DFT-LDA (orDFT-GGA) allow for an accuzate deterrnl'nation of many soace properties, in particular stuu facegeometdes. For these geometdes the Kohn-shnameigenvatues(3.46) are
automatically detvmlned. However,there is no rigorous justlcation for the interpretation of the Kohn-sham eigezwaluesas single-particle excitation enas STS, PES and EPES,are related ergies. A)l spectroscopiesdiscussed,'sue,h to the removal or adclitkonof an electron. The cozaupondiugexcitations are the rnn5n quasiparticle peaksof whic-h desczibedby spectzalfnlnc'tions (5.15) Ttather)the peakpositiotasdeaz.c Imt locatedat the KS energies&;z(i) (3.46). whic,hare shifted by .X(i) agnsnqt fm.equasipatrticle ener#es s,zQP(i) (5.26), the KS values.Thesesllifts cnn be computedin a pertmbatîon-theor.gmaze Mcanwhile,there eldst many calculations of such QPshifts for ner (à.30). semiconductor surfaces:including the detemmlnationof complete QP band strudmvs
(5.66-5.77).
with exparlmentalobseo The Kohn-shltrn surfacestate energiesclksagree vations since (i) bacd gaps bet'wee,nempty a'ad occupied surface-state band energiesare too smals(ii) the dispersionof the DFT-LDA smfacebandsis too small i'n some cases, too Iarge in others, acd (iii) the placemen.tof occupied surface-stateenergies is in some cases too high by 0.5 1.0 eV relative Thre'e missing pkysical rnnixn'lrnlrn to the bu.lk vcklence-band I'VBMI(5.66j. efects are crucial for the corzect'energ.gposition of the QPstates and, hence, must be consideredto remove the DFT-LDA failmes. First, the spatial nonis more sensitive to the localization locality of the self-energyoperator (5.23) p roperties of surface states tha'a tie only density-dependentXC potentiat of the DFT-LDA. This requires a proper accoant of the nonlocality of (3.50) This nonlocality leads to a modifed disperthe Green's htnction G (5.20). sion of the quazipeicle enerr bands teoughout the surface BZ. Second, the inclusion of local ûe.l(tsdue to the preseneeof the surface i'a the bwerse and, hence,in the screenedlntezaction W' (5.22) dielectric Annctîons-1 (5.32) is crucial for the QPapproach, siace these local Vids describethe strongly at versus vacullm, see expressioa (5.36) imbomogeneotzsscreeniag (bulk-like Third) a'a adequatetreatment of the dyaamical efects în the tlle surface). to d.owith contriscreenl'ng is more împortant than i'a the bulk cue. This %M.q butionsfzom botk bttlk and surfaceplasmonsand the smaltezenergydistance of botmd s'arface states to the Ferm7'level. A11these efects are importaat. of surface states, ennnot Fbcusingo:a only oae esed, e.g., the loewqlszation #ve a gegerallycorrect nanmwer for the QPshiAs (5.55). b=d s'tractures of intrinsic surfacestates i.nthe prototypicat Qumsiparticle elementalsernlconductorssilicon and diamondare comparedwith DFT-LDA electroaic structttres in Figs. 5.25 and 5.26. The 2xl reconstruded Si(I11) A bucldedzr-bonded sktrfacu are selectedas evnmples(5.72,5.761. aad C(10O) or a symmetric-dsmermode: modelwith a positive bacldizk c'b.a.iu 4.2.2) (Sect. Fig. 5.25als: showsbo'and is applied.In the cmse of Si(111)2x1, 4.3.2) (Sect. surface-statebandsmeasmedby direc't and Yverse photoemission(5.78,5.794. are not izzcluded expevlmentaldata (5.22,5.80,5.81) 1.cthe case of C(100)2xl izk the îg'ttre becauseof the presence of hydrogen axd contradidory Sndiags. Figare 5.25 makes the priucipal situation obvio'as. Just as it happeas -
5.3 Quasipartkle Surface State.s
2 >
-6!2--1 >
p)
>-o r
Lu
.
0 -1
a)
2
b) yy.jg 1IIpgIj
-
. ..
:)down
rJ D up
uq,
Enlil
K
1 0
223
. r:,down
J zn utyu
,
Dup
-1 :
Fig. 5.25. Kokn-sitam (a)and quasiyarticle (b)band strtzcttue of the Si(111)2x1 sudace. Thc hatched areas denote S1bz'llc states. Fkom (5.764. Th: dots denote experimeptal data (5.78, Dup and.Dcjowuare explnsnedjn Sect.4.3.2. 5.795.
h: bltllr semiconductors,DFT-LDA is unable to provide ac accuzate descdption of the ba'nd structure of these sarface êtates.Ozf.y the inclusion of a 6111masy-body treatment of the singvpazticle problam by usicg the GW approvlmation (5.30) allows the reproduction of the expersnneutalelectronic stmzctme. ïn fact, usually QPcorrqdions may be even more important tbn.n in bulk semicondudors (5.821. The igurcxs,Fig. 5.25 and Fig.5.26, iudicate general tendenciesbut also specialities for botmd sudacestates of elementalsemiconductoz's.Ln generak the quasiparticlesblh:sof empt.y (f1led) smfacestates a,rçpositive (negative). However,with resped to the bnl'lr VBM the result depeudson the magnitude of its negative QPshifi:. The empty surface states Ddo'wn and r* are shifted towa'rclshigher enerdes,while the relative shifts rwith respec't to the blll'lr VBMIof the occupiedDap an.â'zr bandsare small. The sign of the net sb7'f'h for Dup (x)is positive (negative). E(nany cwse the indirect surfaceband gaps at J ,-+ 0.57.8azkd0.25,/./1--> X-are openedfrom O.4ev to 0.7 eV (Si(111)2x1)
:
#ig. 5.26. Kolm-sha.m (a)and ql4asiparticle (6)band structtue of the C(100)2x 1 suface. The shadedre#onsindicate the projectedbulk band stracture. From (5.72).
x or even an elementalsemiconductorwith a Amlte diezectrk constant sb. We know 'that the surfaee bltrrser of the system in the grotmd state depeadson the geometric, clmrnîcal and bondlng details TnKid.ethe crys'tal it should of the atoms in the sudace regba (Sect. 5.3.1). smoothly matah with the czystal potentii. Outsidethe czystaâthe potential becomesconstant with a value spon. Studying the localizatioa of crystal states near the sudace (Sec't. 5.3.3)we have approximated the s'arfacebarrie,r by a stemlike potentii. Thks does not rnrnnsn valid for a'a eledron present or exdted outside the crjrstal. The surfacebarrier must havethe image-poteatial the total singleWithim a '?e,rycrude model (5.1021 asmptotic fo= (5.40). particle potential outside the crystaz (z > 0) clm therefore be describedby .
V/)
=
fvac
-
1 ea ss + 1 k-(' cs
.
-
(5.53)
An electron placed at a distaace z tu front of a surface genezatesan electdc âelcl. This ûeld leads to a rearrangement of the chaz'gein the crystal in suc.h a way that the parallel component of the ûe'ldvaaishesa,t the surface, In the vacul:rn the electron an.dthe image charge -@b 1)/@b + 1)placed at -z produce a'a eledric felct whic,his perpendiclzlarto the surface in the wbole between the smfaceplane. This results i:athe attradive interadion in (5.53) electrozzaad Rs image charge.To avoid conhlsion the image plane ksidentiEed with the plnanez = 0, i.e., approvimately Mriththe electroric surface. in (5.53) lt may be consideredas the pl=e at a distancezsm from the last atomic hyer of the cores) in the hallpace. The.n zim is approvlmately given by (position half the atomic spadng in tEe normal directiom The attractive potenti/ (5.53) can give rise to a self-trapping of the elec-' tron with s < e=c by its own image, as long ms zzo allowed states favlqt inside crystal the crystal for the energy of the electron below svac. The projected -
5 4 Strong Electron
Correlation 227
Energy
// Eg
Gap
.%.2 4z
Eu EF
XXX
'xx. Mz
0
Fig. 5.28. The potential energg of an electron in front of a metal surface.The metal the z < 0 imlfkpace::s assumedto possessa gap arozmdthe vacuum level occuyying withm the empty'bands. The gap Sscharacterizedby tYe upper (U)and lower (L) band edgeswith eneriesa-u aztd ss, respedively.
band stmzcture should have a gap or, at least, a pocket for an appropdate electron momentttm paraMe: to the sudace.S'acha situatipnis representedin Fig. 5.28 for a metal. For energies e < svuc (butlazger th= the energy of the highest allowedconduction states as below the vacultm level), tke electron clmmot overcome the vaclrtrn barrier (withappropriate parallel momentum) and becomestrapped in gont of the surface provided that there e-qn be sn a'llowed(image) state. It can only exponentiaEydecayinto the bulk. Neglectiug alsothis penetration, the crystal is taken to be in6nstely zepulsive.Henee, the potential may be approtmated for z < 0 as
7(z)
=
(5.54)
x.
L'
Outsiclethe cryb'tal the motion of the electron parilel to the surface is nearly :Nee.The remaining Gect of the czystal potential on the motion witbl'n the zp-plane e-an be taken into account in the ânmeworkof the Xective-mnss approxsrnation (EMA)(5-48,5.103). The f:-%e'lectronrnnzqs m is replacedby an eective rnxqs m., whic.h is asplmedto be positive am.disotropic for simplicity. The motioa of the elechtronoutside the crystal obeys a Schrödingerequation Then the wave Glnnctionbelongingto a s'tate with enerr j'.' of the fo= (3.1). f',: e is of the form exp(ikq(p)#(z), #(a)being a solution oî the ID Schrödsnger : equation .
0 izl the vidnand (5.54), V'(z),(5.53) hyclrogen-likeenera levels (dotted ity of a surface. The accompan#ng lines)aad mve functions (dazh'ed lines)aa'e also shown. Fig. 5.29- SGematic single-particlepotential
witlz the poteatial enegg.g givea in (5,53) and (5.54). .'I'he energy c =.d. the wave vectof :, kjl = k + g, should lie in a Ipocket or gap of the projected The Annctions#(z) obey the boundary conditîon b1:7kband struâtttre.
,11/) =
(5.56)
O
k' For realistic sarfacedescriptionsthe condition (5.56) can be weakened.J.fthe bpnl'lr band energy and wave vector fall icto tke allowedzegionof the projected structure, the wave 'hlmction of an imagestate slzotzld'fapidly decayhto the 1c. l))a1 is formally eqttivczlemtto that of the determa'nation The problem (5.55) o f the radial part Ii (r) of the a orbitals of the hydrogen atom (5.1041. This becomes obviotts if one sets #(z) = rl(r) ir=z and replaces e by with ,g5 Thus: the bound states of (5.55) 5.106). @b 1)/(sb+ 1)c/2(5.10 energiessL < s < evzc + L2k?jr/(2m*)are rendily obtltimedfrom those of the hydrogen atom. One gets a Rydberg series, the image-potential states. The resulting levels and wave hmctions are iudicated in Fig. 5.29. More precisely, not only levels but two-dimensiomalbands s H sntkjl)('n= 1, 2, ...)result: 52k2 L = cvac + s,z(kj1) . + au, 2r4 2 1 RH sb = fa 16n2 sb + l -
-
'-
with the Rydberg constant Jlu
=
of the hydrogenproblem. meLjLzhlt
5.4 StrongEzectronCorrelation
229
must fall in a gap For a given k;i = L + g the baud energies (5.57) or pocket of the prolected btllk band stnlcture. For vzztishingmomen.tltm k11= O acd cs -F x the Rydberg series of botmd states varies in the rtmge -0.85 ev K c cvac S 0. For Bnste parallel momenta also energies above the 'vacullm level are allowed. The forbidden region in the projectedband stmcture should be arotmd the 'vactstrn levei. Tlze localization of the image from states varies rapidly with the quantllm Iglrnber 1z. The averagedista,nces + the surface, (aw)= possess the Gues (za)= 6n2cs(sb dzzlk,ztzllz, 1)/@b 1).The minzurnm distrce amounts to ab ou t 3 k and inc'reases dramatically with n. lt is obvious that for stzr,hdistance,sthe treatment of or DFT-GGA fails. exchangeand corzelation witlqt'm the DFT-LDA (3.59) Only the msymptoticly cozrec't trestment of XC ia the form of the selfyields a correct description. energ.g(5.40) -
.(*
-
In order to describe the Lmagestates, XC elecwts,i:a particular the electron correlation, have to be treated beyondthe approvsmationschemesof DFTnot included in the LDA or DFT-GGA. OrulyXC self-enera esects (5.40) local XC potential (3.50) #ve rise to a correct description of the eledrons ia plre. Therafore, one may interpret the vacunlrn dose to the smface/image the occurremce of image states a,s a consequence of esects of strong electron
cozavlation. We memtion that the inouenzeof the trae surface barrier on the energies can be talcen into account by replacing of the imag-potentia,l states (5.57) the quantllm number 'n by (zz + :) witlz a quaatttm defec't0 S ; K 0.5 (5.964.
(1h) T
Ni(001)
X P
8
F
6
8
S2
sj
jjj
&
;
ps
4 .e'
,
X
r
Sq xx h
S2 h
S2
R
*
O
w
gu (j ggj
YG
Ekin
x
..-w-==.
;
.--
n
œb
.------x.
1.6
10 nwx
----
A:k...x
w *
----...jyp N
n=3 n=:
% N
1%
Nx
h Qöa
12
Nx
'
x
N
x
1.$ .
.
.4(t.i
n=1
(;.ty
2PPE signal
(a) Schematîcenerr diaram for the excitation steps in two-photon photoernz'mion (2PPE). (b)Enmo-ze-solvedZPPE spectrtzm that recordstke emitPig. 5.31.
ted electrons as a Rxnction of their kinetic eztera. It is obteed after evitation of a Cu(100) surface by photons of emergy ntda = 4.7 ev and F/zzo= 1.57 eV. lnrom (5.1001.
''
5.4 Strong llectron Correlation
Table 5.2. Bindsng,energies,-ers (ineV)an'd efective masses Mage states for various metal surfaces (5.1097 Surface
öhe expeded parabolic depeademce on k;I (5.57). At kjj = 0: it albws the detmrminationof the bindi'ng energy, .-ca, with respect to the vacuum level. Witltsn 2PPE spectroscopyone ttltraviolet photon with e'atargy1.,). exdtes au electron out of an occupiedstate below the Ferm! level cp into the imagepotemtial state with the quant'zm zmmberrz (below cvacfor kII = 0).A second photon with energyhulsixl the ieared (1R)or visible spectral region exdtes the electron to a'n enera above sxac (Fig.5.31a). The eledron leaves the For normazescape,ckis = Flso-l-ca surface,and its l-snetic energyis meastzred.. holds.The escapingelectrons are recorded.As an e-xamplea 2PPB spectrct'm is plotted versus the k-lnetscenerg.gin Fig. 5.31b.It îs measttredfor a Cu(100) surfaceusing tlle photon energiesXaa = 4.7 ev and Tun = 1.57 eV. TV ptu'kpositions allow the determinationof the bindiag energiœ -sp, of the image states ms well as of tite e'Tecdvemasses 'rn' for meazmements with Anite kI1.Correspondingvalu.esare sllnnnnarizedfor several metal surfacesin Table 5.2 E5.109). They indicate that the model calculations (5.57) give the correct trends and correc't order of magzzitude. Howevera quantlam defed 0 %; %0.5 is nececysazy to accotmt for the shapeof the tru.e stMacebarzier. Tùe paraboEc dispersionis con6rrnedfor the electrons in the image-statebands.However, thea'eare only small deviations of their esective masses from the leo-electron kslae.
à.4.cMott-subbara
Bazas
Xb Mtio tefinsques suc,has DH-LDA or DFT-GGA have been employed V-titg'rea,tsuccess to descdbe the electronic stzazctureof weakly correlated .:
materials like semiconductorsor simple metals and their surfaces.There was
necessit.gto take into accotmt additional XC egects in the case of excitations.For more stronglycorrelatedsystems suc,has d- and J-baudsystems, cuprates, etc.: on the other hand, the concepts behind available ab iuitio tecHques are sometimes too lsrnited to correctly descibe the comple.xmanpbody eFects.The question nrsqes whethersurfaceswith electrons i.n rather isolated dangling bonds cazl also show Xects of strong electron correlationbeyondthe esec'tsi'acludedin the standard technlques. D=gling-bond-derived surfacebandswith partial occupation havebeen predfctedfor seve'ralc-lean,reconstructedgroup-l'v surMes
15m.ed at the dangling bond giving rise to the surfaceband of hteerestsFrom th: -
total-enea'gy diFerences E5.112j ozze œtimates a vatue U Q$ 2.1 W D#T-LDA (Vss 1.ûeV)for the Wx W (3x3)suzface.However,the calmklationof total energiu of chargedsupercellssuf'ez'sstrongly from spurious electrêstatic in'
teactions between the supercelks. Hence,the values reprmenta rathe,r cnlde estimate. For the more Si-rielk3x3 surface the value of U appzoaœesmore closely that of pure Si as a consequenceof the rather lazgeSi coverageof tbis strttcture and, hence, incremsed screeaing. The daagliug bon.dsare azranged in hexagonal latticu. For C > 0 the cliagomnlszationof the corres/onding tight-binding l'Ixrnlltozkian Tves a dangling-bozd band with a clispersion a(i)= 2t(1+ 2 where c-ostrc,slqt the parameter s describesthe variation of the V vector iong .f'V, or along the Z'V line (0; s K 1)in the + cos((4r/3)sj) c(k)=2J(2cos((2r/3):) corrupozkdsnghexagona,lsuzfaceBZ. A ît to the dispersionof the dauglingbond b=d calculatedwitbin DFT-LDA yieldsa hoppMg parameter t = 0.014 ev (ï = 0.05 eV)for the 3x3 (WxW) stznzcture. With electron correlation rw C the single-particle problem belonogto the Hubbard Hxmiltotkiaa cn.nnot be solved exactly (5.121) However) in the atomic llmit t the width of the uaoccupiedband of about 0.35 ev g5.116) seems to be slightly laœger 5.xlthe W'kW cwse. On the other haztd, for the 3x3 s'arface,thfeozyfmds a measmabledispersionof 0.13 eV. The ARUPS value is srnnlle,r than the exptm'mental tmcertainties, i.e., smalle.r than 0.1 eV. There is another way to obtain the bazd structure givem in (5.59) from '.. an ab initio meareftelêapproaœ (5.1132. lt Ls necessanr to accmately incorporate tlle long-rangecormlation a'aclscreeningefeds in the electronic self-energy operator (5.21). This can be done in a higltly rehable my by the However, as a basis for the GW calculation one GW appyovl'mation (5.23). hmsSrst to trea,t the sadacesystem withi'n the local spin density approfmation (LSDA) to obtain the fltlly spin-pohrszed covgazra(seeSec't.3.4.1) tion. This already leads to a splitting of the former metatlic DFT-LDA bazd (Fig.5.32, dashedlinelizlto t'wo b=ds separatedby a direc't DFT-LSDA gap of 0.6 ev for Uj'xué tsee Fig.5.32, dot-dashed1i'aesl The reslzlt(5.1131. -
-
-
:'.
:((..:. : Fig. ii'q .
5.32. Quasipazticle band stractme (solîd linœ and hatchedregions) of t:e 6H.: jîc(0001)WxW stkrfve calculated ix a fully spin-polarizedGW approximation. ; For comparison the dangling-bond-relatectbands in DFT-LDA (dmshed 'E' line)or izz :;.bFT-LSDA (dot-dmshed lineslare alsoshovn. From (5.113q. '
QPba'ads(Fig.5.32, solid iines)a're fmther separated. Comparedto the lowerDFFIJSDA band,the ocmpied majority-spin baud.is s'hiffeddownto ing
lower e'aergiesby 0.2 ev while the empty zninority-spin band is s'hlfted up to higher ettergies by 1.15 eV. 'I'his is accompeed by a sEght increase of ihe band widths. 'I'he mpzm direct gap betweu the two bands is iZLCZ'O'XSGI by 1.35eV due to the QPcorrections and amolmts to 1.95 eV. This value is itt good areement with the on-site interaction parameter r) of the Hubbard
model. 'No remxrks are necvary. Fizst, at a real &C(0001) surface the s'pin confguration may not be fltlly polazized.I'n fact, wîthin the DFT-LSDA calculation the totaz energy of the spin-poladzedsmface is neazly the same ms that of the urzpohm'zedone. lt eltn thus be expected tut the spin 1701=hation, if Evorable at all, is easily brokcn by noazero temperatme oz other pertmbations, so it senms likely that the real suzfaceis not spin polrized. Second:suc.ha metal-insulator trlmsqtion.due to strong correlation effecvts and ms dâscussed for SiC(0001)/(1l1) stufazo may alx occur on mxrfnex of ferromagnetic se-mlconductors.One example seems to be the EuO(100) surface(5.126!. Fblally: in ligb.t of the results fotmd for SiC surface.sthere are also some doubts tlla,t tùe Si(111)7x 7 swface tseeSet. 4.4.3) should be the only true metallic surface of a semiconductor. The sîtuation is sirnn'lar to the SiC(0001)Wx W a'ad 3x3 surfaceswith their adatoms or adatom clusters. Only the averagedistanceof the zemnsnsnghalf-fzlleddanglingboncksat adatoms is somewhstlarger than the distaaceof the dangling hybfds in tEe SiC case rd.: probably more important, the surface syreening is mttch lvger. One msy exmect that the drxstic reduction of the U parameter h'om 2 ev to 1 ev (SiC(0091)3x3) is enforcedfor the Si(111)7x7 (SiC(0001)WxW) sttrface,rasultiygizï an extremesy small Mott-Hubbard gap. Moièover, fog smaner ratios J/rtl the gap opeeg îs reduced with respec't to the =lue U relcvant in the atomic limit :5.1224. Probably cx'p'erimeatal tenlnnqquessuch as PFXS elmnot rexlly omtzibute to resolveS'UC.IZ a very mnn.ll gap of the order of 0.1 eV. At room temperatmm the styfaceshould look metallic. The naturazway cotzldbe.to work at very 1owtemperattzres. Unforttmatezy,the 7x7 surface shows a strong suface pbotovoltage shift. whic,hresults iu an tmdeftnH Fermi-levelpositioa in the eedra. TH lowersthe preddon of the low-t'emperatttrePES measurements (5.127, There is aaother compli5.128J catjon for the theoreticaland experimental studio of the Si(11l)7x 7 surface. Many bands appear kl the projectedAlndamentalgap (5.129). Their ene-rr spacing is small resulting i'a problems concerni'agthe peak identfcation in PES. Farthermore, a singlmbandHubbard Hailtonian cannot describe the real situation of strong electron correlation efeds g5.130j. .
'
Excitations II: 6. Elementary Pair and Collective Excitations $.
6.1 Probing
Surfaces by Excitatiolas
6.1.1 Optical Spectroscopies
Optical ypectrdscopies a're emergingms particalarly promising tools to probe surfaces,siace they allow for in sitn, non-destrueive ar.d realrtime monitori'ng tmder challenging conditions as may be encouutered,for instance, dzzring epiteal vowth. For epiteal growth by men.nm of clmrnl'calzeadions, such chemical mpor depositîon IMOCVDI, as, e.g., metal-6rgxnsc optical spectroscopiesprovidethe only possibitity for suchmonitoring. Other advantagesare that the materiaz damagean.dcontxmsnation associatedwith chargedparticle bfoltrnqare avoided. Insulatozs can be studied without the proble,mof chargicg eFects,and buried interfacesare accessible owingto the large penetration depth of the electromagnetic rnrliation. Optical tenbnsques ofer micron lateral spatial resolution a'ad femtosecondtemporal resolution. Howeverj since light peaetration and wavelengt'hare mucbularger than surface titiplcnersses (a few i), sucll tec>nique aze actually poorly snnRitive to surfaces. Some ltrinlm' have to be employedin order to iucreasetheir smface sensitMty. Tite expmrsrnem.tal progress k'athe characterization of s'arfacesusing light hmsbeen splrnmarlzed in a coupleof excellentfeviews ar.d monograpbs(6.1-6.51. rfhcoretical considezationzcan be fouud in review aztides by R. Del Sole (6.6, 6.*4. V The probkg depth of light in a solid, even ia the spectral range of Mghest absorption, is of the order of 10-500 n=. For a charactedstic smface layer of O.5 nm tbânk-ness,the relative surface contribution to the total optical signal only Jtnnotmtsto 10 -10 SeveralapproaGeshave been developed to improve the surface sensitMty. The bazic idea is to mexs'tzre dsFerence signakswhich enhauce tke surface contribution with rcspect to that of the blllk-. Fottr terthniquesare comnnonlytlsed. One is snöace (Jzfererùtïc,l reyectance (SDR)speciroscopy.It is basedon measuring the dsference ia reîectance due to chenkicalmodifeation of the surface,for example,often the Vsorption of oxvgen or hydrogen. The percentagedsFerence is related to the surfacestructare. However,to what exteat it is related to the c'lta-qn or to the c'hetmipsnrbed surfaceand.whether or not it is sensîtive to the spectrnnm of suzfacestates and/orto the atomk structtzre of the smface,is in general dilcult to determîne(6.8). Figure 6.1 shows .
238
6. Elementaz'yExcitatioas 11:Pair and CollectiveBxcstations
6 *
@* o
5
c?-h o
m
4
* * *
.5,
*
W 3 c P 'œ
*
.
X
0
@
@
2
(n œ 1
*
@
*
*
;
*
GWoXOOO 0.4
0.5
0 o
0-6
O
0.7
Photon energy (eV)
6.1- Diferemtial reûectazce of spectra a smgle-domain smface for ligh.t poSi(:.11)2x1 laaqzqdalong tlze tq- 11(711)(open aud v LIgOllq cizcles) (dots)directions. nom (6.91.
Fig.
spectrnlrn that docllrn'ents the breakthot'tgh of SDR spectroscopy because of the lzsc of polnHzed light on Si(111) sampleswith singllz-domai'a2x1 reconstruction (6.9). A:a oxédizedsurface is tusedas a reference.After oxygen chemîsorption,surfacestates are saturated a'ad therefore optical tmnqitions across them occtlr at higher energies,having the DR sigmqlrelated only to the suzfaces'tate.sof the cle-ansurface.Thc measured10O%rmisotropy of the 0.45ev pemkyields s'trongevidencei'a favor of the Pandeycb.ainmodel (see a
4-2.2).
The memsurement of the relative reîectance diference for two orthogm naz ligh.t polarhations (z and p) in the surface pla'ne is cnlled renectance csùsfrop!/ (RA)syedroscoyy (RAS).Since the bulk of cubic materials Jts optically isotropic (atleast, as loag as xni'rvotropies due to the photon wave vedor are negligible), any RA observedfor suck cnrsta.lsmus't be related to the reduced spnmetzy of the surfaceor to another syznmetrsy-inre-q.kl'ng perturbation, for >ample an electric seld. However: RXS is not restricted to surfacœof cubic czystals. For e-x=ple normxl-incideztce spedroscopy phkratle1to the c-azs of a uuia'xial crystal (e.g.: can also be tzsed.The wurtzite) great advantageof the ItA. ten'hnique comparedto the SDR method is that it doesnot involve imdta6nedreferences'arfaces,so that it can indeed be usedto monitor C'O-based epitaxisl growth. The theoreticalinterpretation is also simpler in principle, since impoz'tant atzdpoorly known iuformation) e.g., the atomic structmc of the refereacesurface,is not necessary. The t'wo othe,r methods are ''urlacsphoioabsovption(SPA) whic.hmeasures ambient-inducedchangesiu the npolarized reoectanceat or near the Brewster azzgle,aud ellipsomeiry or spectral :JJ#UIIZ/JeJZ'.!/ whic,h meav (SE)1 slzres the comple.xre:ectanceratio of s- and p-polp.rizeêlight.
6.1 ProbingSurfacesby Excitdtions 11
239
z
.1.
7
#
* '
%
y
X
ds!
c=(f.b), syyltol
Eb(œ) Fig. 6.2. Sckematicconfguration of a rededanceexwriment vith poladzed light. A three-layer system is aumed. Tie propagation clirection of tile light and tâe divisior of zts polnHMationdirectioa with respect to the surfaceare tdicated.
Theoreticaldescriptionsof reâectaaceexperimentsat surfaceswith polmshowni.n Fig. 6.2. ized light often start with the three-layermodel (6.6,6.10) The solid consists of the butk aad of a surface layer with efedive tikicloless ds m'achsmalle,rtibsn thc wavelength,j of the ûght. The bulk is mssalmed The surfaceis describedby a to have all isotzopic dieledric 'hlmctios cbtuJl. gequemzy-dependen.t dielectric tensor with eigenvectorsparallel to z aad p. ?1Yecorrespondingcomple,xdiagonalelementsare a=(aJ) and sw@).The ttpper halfspaceis the vacullm. For normal iccidence L$= 00)the comple,x signal is given by the dslerence betweenrefectance amplitudesat (z = 0Q J1tA. and a = 900, 1-J. azd %,respedkvaly,(6.101 -
.
I
zïiz = zlrridss..@) -
W
y
as
(.;
evv
.y.
(w)
(6 1) .
with ,4f: = f:z Dv and 'F = (irz+ /p)/2.Genernlizationsto a'a nrnbient halfspace,a, non-norma,lligb.t incicteuce,and an nniqotropic substrate cau be i fo'uzd izz g6.111. The standard IRASsetup measkkres the polrm''zation state of the reâected and 1m(z%//). Most eaverimentalists publish spectra ligsht,Le.) Re(.zA///) :éf the real part of the relative variation of the reoectionAmplitudes(6.1) ; = the refectivities J?.z formally related to This quantity is I.i% l .1te(1///). : aud .% = 1/ (2for the two polarizations. This situatiou is schematically ! indicatedin ng. 6.3, though the,irrelative dl.s erence js uot directly measured. Thea -
Re
.4/
-f-
=
1 Z.S
ï
R
(6-2)
6. Elemento Excitatiozts1I: Pxir and CoûectiveExdtations
Rx- Ry
y
9 X
X
Fîg. 6.3. Schemaficreprœentation of the dxerential meamlring tec%nsqtze in a reoectamce-auisotropy spectroscopyexpersmeat.The rlîffbrent light poln.m'zationdirectiomsare indicated.
with M = & % and 2t = (& + G)/2 ms the diFeremceand the mean value of the redectivities for orthogonat polnrszations.Herc the approximate validity of the relation 2Re (,&%*) = has been assqlmed. I/z(2+ IFvI2 Second-orde.r tmrmq rw 1.4/12 are neglected.StacetLe algeaence of two reîectivities is studied in (6.$,RAS is sometimesalso termed rclccfcncz d@àr-
ence
spectroscopy(RDS) Accordsngto (6.7j.
(6.1)the relative Gange of the
polnrlzation-dependent reiectivities Lsgvenas z.a R
=
zlu;ds lm c
s=(u,) swta,l &b(u?)1 -
.
-
(6.a;
The sensîtMt'y o? the reEectanceesotropy to the smface reconstructîon ard stoichiometz'yis demonstratedin Fig. 6.4 for GaAs(100) seaces (6.122In Figs. 2.19and 2.20the reconstructions appenrlng in Fig. 6.4 are relatedto the surfacepreparation conditioas.J.t1 Fig. 2.17 possibleatomic structures are given for the indkated 2D transhtional symmetries. 6.1.2 Light Propagation
in Surfaçes
1n.orclerto detamminereNectanceand transmittaace of a'a electromagnetic wave with Feque'ncwy a7 in a cr.ystal surface,one izwsto solve Maxwell's equations for tEe ezectrk displacementvector DLm, uJ),whic,his related to the electric Eeld ELT, a))by the constitutive relation = a/; w)Jy@/, D0:@, ta)) J') d3a/staptœ, (s).
J7
Heret:s aud J label Cartesiancoordlmatesand eœp(œ, Wb(,J) is the microscopic
dielectric Rnnctionof the vacu4tm-crysta,liutezface. Instead of the longitudinal fh:ncvtionin Sect.5.2,herewe use the generalizationto a tensor in order to accotmt for the transverse character of the light. The dielectric hlrction/tensor
6.1 ProbingSuzfaces by Excitations
Photon energy (eV) 6-4- Refectance antqotropy memsuredfor difrerent reconstnlctions of the sllrfxce. The zr-tT/-la7ds is parallel to E0ï1) The àorizontal linœ ((0111). Ga&s(100) mark the zero level of eacà spectrnlm. From (6.12q.
f'ig.
arbitrazy system accordiagto linear-responsetheActually the temsorcàaracter can alsobe derivedhom the spaceory (6,131 dependent longitudiral response gtndioa in the l5rnl't of small photon wave dielectric of the gequezzey-dependent vectors by relating it to the projection tensor onto the propagation vector. J.nthe case of light propagation ia bulk crysta'ls,thc constitative relation (6.4)greatly simpMes. The electric displacement feld contnsmsionly 10%wavelengtkcomponents.Dgher Fourier components ic the total zaicoscopic spstial homogeadt.yof the system. electzic ûeld cnn be neglectedassttrnn'mg of the dieledric fnlnction. That m/-qanq, Thksresults $nan @-œ/l-dependence one neglectsthe spatial dependence of the electron dersity induced by the atomic struM'are of the crystal, umzallyreferredto as local-feld Gects (6.14, 6.15!Nevezlheless,they play a role i:a m=y crystals, in particalar in the static lirnit (6.16). As an approvlmate resalt tbe dielectric respoase can be œ?;a;). = dccribed by a Fequency-dependeattensor c.4(og) J d3œ/stxygta In cubic crystals, becauseof the symmetrs the dielectric tensoz becomH a, = scaza,rq''p,ntit.g ssta7l witlz s.#(ttJ) sbttbillap. can
be calculated for
a'a
.
.
-
6. ElementaryExcitatioas 11:Pair and Colledive Exdtations
242
In the ceaase of a surface:even when neglectiuglocal-âeldeseds dueto the atomk stractme, the Aniqotropy, the spatial non-loelzi'tya'ad iAomogeneity of the delectdc resporse, i.e., its z- an.d z/-dependeace,have to be takea into accotmt. That makesthe solution of lkfax-well'sequations a non-tri'dal task. 'fhey are eexsilysolvei however,for the rigorously simpved c,ax of an abrupt interface between a snmûln6nite czystal occupying the halfspace takes z < 0 aad the vacuntm foz z > 0. Iû that case, the dielectric Ahmcvtioa the fo=
(6.5)
c@;z)= #(z)+ p(-z)cb@).
.
leadsto the well-knowaFkesThe solution of the light-propagatioa equatsoms 0f comse, in thœe formulasthe microscopic nel formulmsof reâectivity (6.171. feata'resof the surface are lost, since the surface contriimtion i.s completely negleded.Oneway to indude it ksto tzse th.ethree-laye.rmode.ldvribed i.u. ng. 6.2 (6.10,6.11). approvh was taken in 1979by Bagchi, A more general (since microscopic) Barrera and Rajagopal They started 9om the jellblrnmodel for the (6.1S). halfspacez < 0. The tmnncation of the bulk leadsto a moocation of the optical propertiœ in a mtrface region of thinlcness%.BaC.Gon the assllmption. that da 'zs of the ozderof a few bllllc lattice const=ts or less aa.dtkus mtte,h smaller than the light wavelength,% xg. 6-5. Iuelastic scattering process (resectioa) velocity 't1 on a hn.lhmace(z< 0).Possibleenergy lossesare due to the polarhable The electzon trajedoryis quazimedium v'itk a bulk dielectz'icfrnction sb/,(zJ). elastic becauseof the smn7lrless of the energy transfcr.
the langaageof macroscopicelectrodynamjcs,a dieledric displacementse-ld (coride'nngthe most important longitudinaâfeld components and, hence, accordiug to negaleaingretardation egec-ts) 1
c
r(z, tt --V=r@ 't7à) 1= 't:zja(m mt c = e2/Iœ1 with w@) To be surfacesensitivea reoedion scatterlng geometzy and prsrnary energies m172/2 < 50 ev are used (Fig. 6.5) The enerr is =
-
=
-
-
.
.
small that the electrons penetrate only a few Mgstromsinto the solid. Hbwever,these eledroas are accompaniedby a lonprauge Codomb fteld halRpace. The whic,his screenedby the electronsazldions in the poln.m'zable tot/ eledrscEeld E nxn be related to the sceened Coulombpotential W' in so
(5,22)or (5.36)in in (6.12).
a
similar way
as
the D Eeld.to the Cotfomb potential
r
The enerr losse.s of the eledrons pemetrating or approachtngthe solid are to the clynamicsof the screeningprocœses. The total emergjrtransfe,r r#ated Qis give; by the c'haugein the eaergy densityof thc Coalombâeld.ji.c refectfon geometry essentiatlyby that outsid.ethe polazizablehalfspacebolmded The totat (time-htegrated) by the imageplace z = 0 (cf.Sect.5.2.3): energy t'rnnsfe.r1om a'n inelastir-qlly reiected eledron to the solid occupying the halfspaccz < Ocan be written az (6.22, 6.23) ::
Q s -
(6.1a) ae/+-dt/daze(z)s(a,t)o(z,t), -X
î
where b icdicata the time dezivative.The totat etectric seld .E@, t) in near the hxlfxpace z < 0 kclades the efect of the pola6zablemedbnrn.
or
6.1 Probing Smfaccsby Bxc-itations
245
Witbi'n the sxrne approvlrnntions whic,hhavebeenuseêfozthe calculation of the displacementfeld (6.12), tke total electzicûeld ca'a be related to tke screenedCoulombpoteatial W' by (5.22, 5.34)
--lva j j d3a/
X@,:)=
d/ctz,
z?; f
f/). Y)p(z?:
-
(6.14)
Since the dezusit.gof the scattered electron is simply g'iven by Dirac's & R:nction
plm,t) 5Lm.?J1)1 =
one
(6.15)
-
Mds 1 --V.
X(m,1)=
+K
.
, dt , W'(œ,.?J'I t , ). ;t
(6.16)
-
&
-x
=J; 1-Y) = r@-a/)J(t-tJ), Without the polarizable medbxrn,i.e., for Wr@, the eeressions (6.12) ar.d (6.16) becomeidemticàl. The spatiazsymmetry of the scattea.ingproblem and the tgmeintegration in (6.13) sugges'tthe tlse of a Fomier representationof the tvo Nelds.f = D, E sirnt-lnr to that msedi:a Sect. 5.2.3,
.f@,t)= with p as trlmsfer
Q
=
+0*
d2Qciop z
(2.,4
cu e-i-zytq,z, -.
ap)
217vedor i.zl the zr-plazze.One obtes
a
(6.1.7) for the total energy
+x
j.. j dp,g
d2QF=#(Q,(a)
(6.18)
with the scattezing probabitit for the traasfe,rof the eaea'gjr&,dand a vector Q parallel to the surface(image plane)
.ëLQ,u)? -
1
-
s,r,s,
wave
-u,). (6.19) utkj*qjz.eLq,zzu,lnL-q,z, c
The Fomier trsmqformsof the âelds caa be easily calcalatedassllrnsrtgthat the electrontrajectorgm' = 'l:tl is ventially that of an elmsticallyscattered electron. The time t = 0 js takea to be the momem.t of reiection at the plane (z = 0).Tlzeu,becauseof the smallaessof the energy surface/image losses (FâzJ it holds rztf < 0) = Iwl = -rz@> 0) for tbe z 0) tke polnr'zable halfspace. T:e inelmstic scattezing of electrons witic.h pauetrate the m'ystal (z < 0) is ikside the polarizable dominatedby the sceened Coulombpotential (5.36) halfspace(z < 0).T*e correspoadûlgtoti electdc Seld is governed by the In the ll'=l't of weakwave-vector dependence, quantity a(Q,z zûuJ)(5.37). = e-QIz->?I/sb(uJ)j z'b (,g) this te= generates the bulk loss fzlnction z aLQ, -
-
Im (-1/cs@)). The surface scatteakg menhn.nimm consfderedso faz is restricted to the poLzxinieraction of electronswith a homogeneous imnge-potentiaz-mediated ,xp
6.l Probing Surfacesby Excitatious
247
izable hopace. The details of the atomic aud eledroic structuze of the surface are not tnt-nn into account. In a rough approvimation eledron encrgy lossesdue to elementar,gexcitations eo=ected diredly with the suzface itself can be treated xqgnminga three-layersystem as in Fig. 6.2. .&n.additional suzfacelayer with thîcuess ds and anl'qotropicdielectric teasor with disgon.alelementscxtall La= m,y, # is nmnnnedto l)e embeddH bet'weeu The wave-vector dependence of the vacu.tzm and the bulk czystalwith sb(w). t,hesediagoaalelementsshotlld be reglected in the follo
5 o
-)y(;j
3 2
,.j; ,
1 0 A
0.06
.-d
NV m
:
'
@)
o
'& 0.0a
I
o
l,..'.ru .4 . 0'n J
-0.06 0
1
2
3
4
5
6
7
Energy lossLek7 Fig. 6.T- (a) Bnergy loss intemsity measmed for directions of the momectum trxnKfer Q parallel and perpendicularto the dl'rnezeof a MB&g'rOU As-capped sn'vfpce. The inset showsthe LEED patten acquired at 39 eV. GGs(001)c(4x4) with permsssion (b)The relative Jeerence spectmnm.From (6.28) (copyzight (2003), h'om Elsevier).
exceeds100.The mnsn stnlcture in the loss spectrnlm is therefore des2s@) termined by gLQ, the tblnlk' loss fllnction of the aJ) Qdslm(-1/sza@)J, =
thsmxnlsotropic suzfacclayer weightedby the sma)lprefactor Q%.A qualitative explacation of tb.is e/ec't is givea in (6.254. In the ca-se of loss spedra 1 sudace iu Fig. 6.6, ezzLwj Ls real and nearly a constant for the Si(111)2x as a Ahnction of ul (6.274. Consequently:in this case the meastzredanisotropy is dominatedby the diference-sin the flmt te= cw Ilrz/?gttzJl for 7 = y an.cl p = a. One has to mention that formtllas of the type (6.23)are also tlsed to cvaluate the electzon energ.yloss spectra of reconstructedsemiconductor sarfaces(6.28,6.291 The slab approvsrnationis appEedto calculatcthe tensor of the slzrface clieledric R'nction so(aJ) Lp= z, y, z). .
6.1.4 K-qmxn Scattering Nonll'near opticsl methods are also succGetlly apphed to obtain iaformation about sudaces.Oneovnmple'issecond-hnrmonicgenezation(SHG). For czystaks,suG as diamond-stractm'eczystalswith vazkishing bulk coûtdbutioa, the SHG represents a powerhll tool for surfacestudies.Mother prominect lt iuvolvesthe inelasticscatteeg of phoexnmpleksthe Raman efecf(6.302. torus in tke visible or W raage by elemental'yexdtations of tite system: e.g., surf'acephonoas and plnmons. It thereforeallows some additional inso' t into the vibrational azd electronk properties of sudacesif its suzfacesnmq'ltivit'y is mxlnxnced.This tmhancemeat is substaatîal since izt general only a mnn.ll nttmbe,rof photons is inelastially scattered ia a cel'tain solld angle. For snml'conductorsthe typical Qmmn.nelciezlc'y has been estimateêto be In all inehstic light scattczing process about 10-6- 10-? jterad.c-ml-l (6.31J. aa incident photon with emergyLulj aad mMe eaea'gyks trxnsêerredbet'wee,n vector g$ tmd the sample,resukGg 5.&a scatteredphoton of a dseerezlt enea'r es and mme vedor qs. The aotmt of tmneerred energy correspondsto the eigecenerr Mu(Q)of an demeatazy excitation labeled by the izde.xx, e.g., tàe phononbrrch) azd the wave vector Q. Eaergy conservationyields
fàtsli tJsl :l;#=v(Q).
(6.27)
=
-
The t+' sîgn standsfor those Raman processcsin which a'n. elementanr axh citation is generated.Tkese are called Stokesprocesses. The nnnnslnslntion of an elementanrvdtation correspondsto the t-' sign. Thcy are referredto ms anti-stokesproc-es. A correspor.dingR.q.rn= spectmlm is schematically drawn ia Fig. 6.8versus the lequency shlfk (aq -&Js).Generationaad xnnihilation of thealementaryemxcitations dependozt Mmperatlzre i;a a càaracinm'stic the Stokesscatttm''ngis way. For not too high temperatzzresNT L F?z,;v(Q) more tutertse.ID.analog.yto energyconsezvatiop the qumsi-momot!zm conserution iaw gives a corelation of the componerts of the pbotoa wave vectors )'
2% c
.'
S
;
-040)
0
'
(tk(0)
'
q- rz't
Fig. 6.8. Intensit'y of inelaatically scattezedlight versus lequency shl'ff
the sudaceand the 2D wave vedor Q of tha surfaceelementar '
Hdtytion
l 'Q1 '
(6-21
Lq? 4alli &Q=
-
iBecauseof the smaltcessof tie transferredphoton mm vectora practicall p)11 y elementar.yexcitations with waye vedors sear the center r of the stu ,f,1..e BZ are excited. Amqumingtkat vksiblelight is usedto eexcitethe Ramai trans qç>ttexingîn a samplewith rcfractive'index of about 3, the mnr'== terred wave vector IQlis of the order of 10-6 cm-l. This vatueLsabout 1/101 J :6fthe size of the BZ of an Dmeconstmldedsmrface. E2:EA pronnl'ne'nt Rn.rnan scatteeg by optical phonons exampleis (resonance) :.rlthisscattea'c is mediatedby the electroaic system. The photonshterac l . the phoaons are created or xnnilln.ted via the electron. tEvth electrons and ; jj.(. .: Xbnon icteradiota. Six elementaryprocessescontribute to the so-calledone ) Rnmnn scattering. The mœt resonaat prouessis des crsa yc'shoaon (Stokes) 'E'by a Feynma'n diagzam of the type drawn in Fig. 6.9. The correspondia! ,iptàcàttexizzg probabitity ca,a again (cf.seets.5.1.1 aud 5.1.2) be obtained h'on :!' :j s GoldenRlzle (6.32) ,llqFermi '
'
.
.
.'
:
.
..
.
c,,v
a
P@ila7sl -g.Xl lAsol&&l Ysll Jtruai *x(0) &,.zsl,
(6.29
Juotaq, u'sl
(6-30.
=
-
-
'
j' ( '
'
'
(o l/zi-eteq-azill z) (z 1.:-,(0,-(0))1 z-) (z.' 14-(.-,:.,-)1 0). .)....) (&,& (Sz -F70)! EAaps (r.4w .E%)1 -
AJ'
-
-
-
the excitar the izdepemdent-particle picture used in (5.12), tbn of the electzoaicsystem is hezedescribediu terms of electron-holepais i2a tes 1d)with energîes EA. IO)denotes the bzitîal state of tbe scatteriat It is usually identEed with the gzotmdstate of the system, in witicl E>rocess. the electron-photon iater. :/o electron-holepair is e-xcited.Corresptmcllngk, :'' trièfion Hn.mtlton Jytut (c,a7)has beemgenern.4.7ze(j compared with tue siuglo oiitiic)ezepresentation(5.13)The dependenceon the polazization vector e i: àE l'Aplicitlyindicated.The photonwave vector is mssnmedto be negligible.Th
(1nèontrast to
.
, ,
>m''..
.
6. ElementaryBxdtatiozzs11:Pair and CollectlveExcktations
252
ligb.t scatterïg proceedsin three s'teps-In the 6rst step, the incident photou M;L vith polarization ek exdtes the system, e.g.; a s-rn''condttctor surface: into an intermediste state ld) by creati'ag an electron-hole pair. In the secoud step, this eledron-hole pa,tris scattered irto another iatermediate state with vaaisAingwave vector Q ss 0 via the IT) by emittiag a phonon &zx(0) electron-pbononinteraction Hn.rnsltonianX*. Iu the lmststep, the electronradiatively with emission of the scatteredphoton holepair ia JT)recombi'aes after with polarization es. The electrordc subsystemremn.lnq tmc-hanged Fpags the onemhononRmmnxnprocess. Witln'ln the single-particlepfcture three sttrfazeeledrouic bands are bwolved in azl elementary seatt/rimg process. EM% eledron-hole pair izl a:a intermediate state is azsoaiatedwith a conctuction and a valenceband. The electron-phononinteraction cac scatter the electrou band. in anothe,rcocduction (vezlence) t1lolel The most prominent coupling mecltn.nistznsfor bqllk semiconductorsare deformation-potentialscattezing and pola,r Fröblic,h scattering (6.3% 6.321. 80th. zone-cente,r trs.nwerse optîcal(TO)a'acl lon#turlimaloptical (LO) phonozlsshowa deformxtion-potentialeoupliag to the electrons. 1I1systems with pmially ionic boads the longitudinal phononsca'a aksocouple to the electronic syste,m via the accompanying zong-razge electzic feld (Fzlblich dependson the The strength of the Rm.=a,I1Jtmplltude (6.30) m'vtbxnsRm). The Rwtrnazl amplitude depends scsttering geometnr,denoteda.s qitei,eslçs. not only ozt the directiorusof the light polarization ez oz es but also on
the direcdon of the phonon displacemeut.For tEe
(100)surfaceof a, zinc-
()01J
1O41 E0
-
ej
qy
qs es
Fig. 6-10. A possiblepohrs'zation coHg-urationfor backscattering at a face.A.longitudinal phononcnn be obsmwed.
(100)sur-
6.1 Probing Surfacesby Excitations
253
blendeczys'tal lo-symmetryl and a bnnlcscatteeg covguration, tw Ir(ï0()1 '
the created/xommllated b',lk. optîcaalphonozzsare polaœized 11(1OOj, aloz:gthe k010)/(0011 direction (TO)or along the g100J directioa (LO).ln and qs
this coHguration tke T0 phononsare not Rsrnazwactive. The zone-center LO pbonon ca.n be excited via the deformation-potentialmeclmnsqm whec + escsv) (cyvesz # 0 holds for the light polarizations, e.g., whemci 11(010) and es ILjoûlq or ei (1 es IrE011). Suc,ha situation is showain Fig. 6.10. For surfacesthe symmetzy îs reduced..A gene-raldiscussiozz of the symmetqy of the so-calledRxrna'a tezssoz's(6.30) aad the correspondsngselectionrules can be found in articllasby Loudon (6.31, 6.332. The 1bmazt seledion rules for bnztlrscattczingfrom a (110) surface of a zino-blendecrys'tal are completely dlfrereut. ln the blAl1-only TO phonozzs caa contribute 'via the deformn.tion-potentialmeclmnfqmfor polnrssations + figdsvl# 0 Or leiatGz espl + csatc;-,c lEipGz eipl)# 0, for instatlce in a parallel covguration eë 1(es 11 or for perpendicala.rpolarizations (1ï0J aad es Iy(1101. ci 11 For that reason the strongcst peak in the Rnmn.n (001) spectra of a relaxedIA(110)1x1 surfaceizl Fig. 6-11is relatedto blll'lr zoneceater TO phonons (6.344. However,the 'Lwospedra for paralle,land crossed polarizations evhsbit signifca'at dilerenccys. This is due to the.ir surfacesezl= sitivits Thc photon energieshtz;â = 2.96 ev of incident or 3.00ev and Fttz?s scatteredûght are in closere-sonancc qrktitthe electronicsllrface b=d gaps at -
*
R
6s
5
S
S'm *
c
n o
ç-'l
cy o
2
i
f 2 5
: ;
i !
:
*
c: o ..-, =c
(v
E
X ne
.
i ë
!
J -
! ! : i i E :, ë j i : g : i ; : i ; : i
:
E
w
> ...'
i
:
:.
3 ! :,
i
a47 : ;
: :
.
;
: :
'
2,4270
:46
,
; ;
. '
'.
'J 2
-
s
2 :
(j l jj (j .â.
J
-
(110J(1 10)
2 : : : ': : : :
x
i ! i '
30S
100
200
300
Raman shift
2
e I ..j.o s
po-jlg-l lcj 400
(cm-1)
Fig. 6.11. One-phononmamanspectra of the clean 1nP(110)1x1 surfaceobserved în the baclrzcatterlnggeometry for parallel and crossedpoln.rszationcovgurations. The photon energy of the inmdemtlight is huh= 3.00eV. From (6.34J.
6. ElemenàaryExcitationsI1: Pair and Colledive Exeitations
û54
Fig. 5.7 aad Table 5.1).Consequently,the X and X/ in tke sarfaceBZ (see .
generatkon and recombination of electron-kole paiz.s mediating the phonon scattezing happen =n5nly in the locnlization reson of the contributing stu'Fig. 5.22). Rtrthermore,tke surfacesymmetry is faceelectronicstates (see. Table 1.3).Apart Bwered. The poiat group Ls reducedfrom Q to m (see from spectral featmesdaeto multiphonon procves surfacephouortpeAs of and A' symmetry of 69) 146:254, 270, and 347 c'm-1 appear (cf.(6.35-6.371 the discussionin Sec't.6.4).
Pairs: Excitons
6.2 Electron-Hole 6.2.1 Polarization
nAnction
Tlb.eceltzal quaatity ia the calculatioa of a dieledric Ahnction or a polarizabilit'y is the polarizatioa flnnction or irreducible poladzation propagator, P, of the polarhable electroaic system. Accordimgto the dsénstionof the and tite 3D Fourier trxnqformationu dieledric G3nction (5.33) (longitudinal) pohrszability tsusazd d-iezectz'ic the macroscopic dielectric tensor a.#(aJ) ceptibitity) x.p@)e.an be relatedto the correspondinglongitttdl'nal quantity photon wave vectoz in the direction # = g/1çIby depencllngon the vn.niqlnsmg
sçn,u,)
=
)7 tlceco.pta'lt.?p,
(6.31)
o,p
c(o@) %n+ 4rxcg@l =
with
:(4,t(J) =
1
-
55=
q->0
a 'ûLq4d œ
t z d.a= Je-$q(x-=z).p œ (zœ, z ; tz7) (6 aa; .
-
The Fourier tzxnnform 'ïLqj= 4zrc2/(F')ç)2) (F'is the 'volllme of the syshmsbeen i'atroduzedhere. temlof the ba'reCoulombpotential v@)= c2/jm1 I'a contzast to the earlier notation of the polarization fanction in relation the four spatial argametatshdicate that the polsm'zationhtndion (5.33), a/z/; (gJ) 5.s:ia generaz,a four-point response'h'nction i'a P(m,F; uJ)> ?Lxœ, accordrce with its rehtion to the No-particle Green's Al'ncwtioa(6.38-6.401 The horizontal bar oa P indicatesthat, in contrwst to the microscopiccase P, local-feld eseds are now explicitly included in the deterrnimxtion of the polazization Glnctioa of the rnnmoscopic clieledrk h'nction, The combiuaallows the K-draction of the ar.d (6.32) tion of thc two representatiozzs(6.31) complete macroscopic dielectric teanqor.Consequently,the response of the eledronie system to a trrmgveHe pertttrbation suchas light cnn be described density responserepreseuted bz the poby stat'tiug from the (locgitudfnnl) lnn''zatîon fllnction X(a=,a/rt/i aJ). Without Coulomb correlatioa, i.e., witht'n the independentmquasipmicle with approvirnation ttsedin Chap.5,,it holds J'5- Lo (6.22: 6.40)
6.2 Eledron-Hole Px5m:Excitons +x
.!b@lœz,e1 œ'z;Y) -X =
The replacementP
(u
255
,
+ *)J@'z,œz;s). 2rrî t?@l,a?z;f
(6.33)
Ln z:l GG ks often also C'aIIHthe random-pbxqeapHowever,this is somewkatincorred (6.41j Even when provsznatîon(RPA). aad -are negteded,the Koha-sham Tnrniltonia,a (3.46) J quasiparticle eEec'ts llence G contlin efects of both exchaugeand correlation. Ic general, the 'single-partic'leGreen's 'A'nnctioasG for electrons and holesobeythe quasipar'ticle equation (5.21). Thc Coulombcorrelation of aa eledzonand a hole caa öe mcludedonly ic 'an. approximate rnn.nner. 'ne most important long-range mteraction of the two pat't i c les occllrring in P shoald be treated vithl'n the sceenedladdez for This is basedon the GW approvlrnation (5.23) appTovimnAsoa 6.421. (6.40, the setf-energyof atl inclividual particle. The kernelof the most gemerali'nte. gral equation foz P, the so-exlledBethe-salpete,requation (BSE), is given ms a hlnctional dedmtive of the sel-enera with resped to the single-partîcle fllmction. Within the GW approvimation (5.23) the kernelis replaced EGzeen's unde,rthe additional assllmp''ijkthe screenedCoulombinteraction I'F (5.22)1 ïion that the derivativeof the screenedinteradion W' with resped to G ca,n A secondshort-raagecontribationto the k/rnel .%' e'.neglected (6.22,6.40,6.43). or, pf the BSEfor P can be traced to the possibilit,gof particle mccaha'age (6.4% Eihenseamlhsngfor tke mnnrtx%opicdidectaic fnnction (6.3$,to a Coulomb t(' ho restrided to one ''n't ce,n rpjtelatter thteracthon (6.44:6.451. g@)= rtmllsraugrt 'àescribe,s locabûeldegects.For the pobrlqation flmction dependingon only é ($de gequenc'ya closeêBSE follows by negleding the dyunml-csita tlle screenœ?;0)taken at (z2= 0. .: by the static one 'A'@, CQ .; Le., replackv T,F(5.22) Ci'' he BSE then becomu (6,22, 6.42,6.46) =
.
..
.
'.
J'5@zœc, œ/aœ/zif,gl œla(;ts) Jmtœzaa, (6.34) dza œ'za/z:tz?l œ'azaj u')Wr@3,z'a; E.:?:@zœ2, 0)#(auz'3, P @'aa(, 2.:0 @laa,traza; w)'p @t? a?a) zla(;a81 tthe factor 2 incticatesthat the sulet ftmaion is coztsideredazd formany =
-
j Jdz'a -
.
'.
-
.
ry
th'i spin s':rnmation ha,sbeencazriedout. : E ( ;'
'
7d:2 Hnmlltozgau :i.. r 2 'rwo-particle '
*
'
'ètti. e txse o f the C.w approvlmxuou suggets the diagonalapprovlrnxtjon (5.25) for the siagle-particleGreen'sAanction.Thksm/u.r'l that the wave fud (5.27) 'without quaaiparticlee/ects, e.g., l'lttnctions are replved by thorye(#v,@)J E Erttie' solutions of the Kohn-sha.m equation for the surface problem (3.46). r ormoze ' the dîagonruGrœn's 'htnctîon cau 'be replaced by i'ts zvoth'Eltirt,h' ( f....... 'Vdèr (5.27)with respect to the satellite stmzcttzre. Only the eigenvalues ..J ( Ejkiï shrfkedby the quasiparticlecorzections(5.26). A rigorotts proof of this L.IJ. J':' ''
procedtzrecxnrot be Sven. Howevez', it caa be arguedthat tMs approximation js cozzsisten.twith the neglect of the lequency dependenceof T&rLu the BSE Jt has beev shown that the dyzmmicefects of W' in (6.34) laœgely (6.34). comptmqate the stzengthof the satellite stntctures related to the frequeucy of the self-enerr operator (5.31) dependence (6.47) Tke eompleteand orthonormzlszedset of zeroth-orderquasiparticlewave hlndions (%u(z))allows a Bloc,hrepresentation of the polazizabitity ic For the diagonalelementswith respec't to the light polarization direc(6.31). tions one dezivesin the long-waveleugthlimit .
262:2
xpp (=)
-
t
v -
/
)J'b(cw:,'n'c'& ; u') E X''2(Ms-Q(:)pzc,J'v,(R
-
c l z',5c/,v',;'
-ts)1
M(,z,i (k8')23* (ctJ:,'c'c'k'q (:)z4J*v,
with the Bloch matrix elemautsof the velodty operator
'tJ
(6.35) (6.411
tck8l'olvil Mj,(:) sc(k) evqvt -
(6.36)
.
-
The fador 2 h'om spic sïlrnrnationhas beentalceninto account. ln the liznit the matzix elements of the velocity of locaz shgle-particle potentiks (3.48) apa,rt from a operator are identicalto thosecf the momentllm operator (5.13) only hllly factoz given by the Fee-electroamyms. Jn the representation (6.35) occupied valencebands ('n)or empty conductionbands (c) ard interbaud trn.nsitions across tbem are considered.Jntrabandtrn.rqitio:as are discassed in (6.35) we sxl'rn over late,rin the coctext of surfaceplasmons.Conseqpently, pnl'm of electronst'a condudion-baudstates lck)acd holes in valenceband states lwi) that are virtually or physically excited by photozss. The Bloc,hrepresentation of the macoscopic polarization ftmction
These do not on:y resonantly couple eledzon-hole pairs cw azld c''v' but contain also antiresonant Mteradions cw and wfc? as we2 as non-particlecoaserving terms in whidz three conductiomband indices or three valenceresonaat terms are takeminto band izldice,socmm Usually only the 1m.q35mg
6.2 Electroa-HolePairs: Bxcitons '
accotmt
detaileê discussion' see
(fora
257
6.43,6.48,6.40)). The omitted (6.38)
efects are only importaut in special casxsesto appronr'h cxtreme accauvy. M exmmpleis the calculation of plmsmonresoaances where the rnlxn'mgof interband trn.nsitions of both positive ar.d negatkvegequenciesmust be irs cluded (6.50) Becauseof the vaaislasmg photon wave vector in (6.35) the :'mlnomogeneous BSE can be substantially simpved. Since,together with the siugle-particle Green's Annc'tion (5.27), the Bloch representation (6.37) of the polarization hlnctioa for inâepemdentquaziparticles (6.33) leads to azz energy denoMnator, the relev=t BSE c,an be transforrned i'nto a'n. iAomogcneous t'wo-parqcmle Sclnrödsnger equation in this represeatatioa26.4% .
A small dampicg r of the electron-hole pair c''t?iis introduced in (6.39). The solutions of (6.39) deterrns'nethe inter'baud poinrlrzability (6.3$). The He-rznitian quautity Hlcwv, c?rTV) ca'a be interpreted rzs the Bloc,hrepresen(6.40) tation of a t'wo-partscleHamiltooa'a dacribing the intertal interaction of arz electron-holepair (more precisely: quasielectron-vpnsihole pairlar.dits inte'raction with othe: pairs. Aftez real or virtual eaccitationof eledron-hole palrs with photons, the e-xcitudelectrons and holes do not only interact with the stzrrotmris'ngremnsmingmlemceeledrons resulting in the renornmlization to fwasipacticlœ,sv (&)-> gQ,zP(:) there Ls aksoa direct intcraction (cf.(5.26)); IF (6.38) betwee,nthe excited electrons a'ad holes. This contes the bngrange eledzon-hole attraction deterrnimedby the screenedCoulombinteraction x' -W. The additional jmscreeaed) short-rangeiuteraction ,xz 2/ represents a'a electron-holee-xchacge A diagrnmrnaticrepresenyation of (6.42,6.46). the t'wo contributions is given in Fig. 6.12 (6.49;. For a vxnseîng electron-hole
(b) cT
Cj4LL4LL
ck
vf :
, vP
và v'J,
cI'
Fig. 6-12- Seahernn.tic representation of (a) electron-hole attraction a,nd. (b) eleotron-hole mcchange.The screenedjlnqcreened) Coulomb intezaction is indicated by a duhed (dotted) liue.
the polarization Alndioa in'(5.3S) follows to give (6.40) + irlj rsjng tbjs P(i) avQP(:)?à(a) PL=k,v/c/i'; a7) -%c'Xn'%k,( gcoc one obtxlnq the well-kcownformu-la(6.514 qllxntity hzthe stusceptibitity(6.35)
interadion .E H 0 ic =
-
-
.
for the iaterband contribution to the dielectzic tensor i'a the GdependentP quasiparticle approxnemation or, with the replacement sQ,, (:) -> s;ztil,in the independent-particleapproYrnation (6.521. 6.2.3 Excitons
BSE (6.39) can be trn.nRformed kto an e'igenvalueprobThe inlnomogeneo'as lem involving the efective two-particle Hnmlltozzia'a (6.40)
Hl=k, c'w'i')ad*'(#').Eu.4rtil 5--2 )((g
(6.41)
=
YTQ?LV
with eigeuvckluesEA and eigenvectors x4T(i).The iudex .t1represents the set of a2 (plnamttl,'naumbers of a.u i'ateracting electroa-holepair in the b=d states c,i and 'ck. Becauseof the attractivc electron-hole interaction both bouad and scattering states are pos X i'a the regioa of the scattnrlng states (6.6:4. The drastic chamgœ of the optical absorptiond'tze to the electron-hole attradioa are hdicated i.n Figs. 6.15a and b. .&n analytical solution for the absozptionspectmlm can alrsnbe fotmd in the 2D >se (6.64) Fig. 6.15c). I'a the extreme two-dim-nsional llmn't of (see i.a whic,hth: pairs are located i!l a sheet at z = 0 (which may be (6.46), = identifed with a surfacelj the pair e'xcitatioa enertesare (,/1 rzzrl; n = 1, 2, ...; 0 S l'mlS zs 1)(6.65) -
E om
=
f Ssur g
-
JLj
81, .
a sarf
2 rz
1 -
2
(j
where tize gap enerr Egsk're and the dielectric constant ssurf relevaat to tze mlrfncm sheet are introduced. With csurf r? @b + 1)/2 Jk$ sb/2,exprvion predids a sigvezmt increaseof the exdton binding energy by about (6.47) i.n compm'qon' a faztor 16 vith resped to the bulk val'ue .%. Consequeatly, to the bulk situation stronger excitoaic efeds Ootlld appear for suzfaces. 6.2.4 Surface Exciton Botmd States
It shouldbe possibleto s'tudybotmd states of smface e'xcitonsfor the lowest lmrface-state Aansitiozss if these are energetie-qllywell separated from bulk stlrRztc (see s'tates. This ts the cmse for the Si(111)2x 1 and Ge(111)2x1 The sudace bandsDdowntemptyl an.dDup toccuSect. 4.2.2 and F1.5.25). arising from tEe daïzgling-bondstatœ of the buckled ûr-bondedclmsmq pied.) lie essentiallykï the balk ftmdameat/ gap. The surface-iaducedoptical absorption has an onset energy EgsurîLd.Table 5,1)that is smaller th= the
i'adirec't bulk flmdamental gap. As a consequencea well-pronotmcedpeak is observedic the dsFerentialzefedaace (see Fig. 6.1)for the altowedoptical traztsitio:aswith the transition operator paralle,lto the cimsns.A ssmllaz peak J7 l surface(6.66). is observedi'a Fig.6.16 in the N A s:1)4*:*,4-17= of t'he Si(111)2x rne broad peak possves a mnvsmltm at photon energiœ of about &gJ= 0.48 eV. Its Mgll-energytail becomessmall closeto the bulk iudired gap of about 1.1 eV, Sl'mslarvalues of F= = 0.45 W azisefzomch'fvential reoectMty specand phototimmmxldeiection (6.67). On the other hand, 1om a :: troscopy (6.9) combinatioa of dired and iuverse photoern:xqionspectroscopy,a surfacegap E calcttlationsgive :: of about 0.75ev Lsdeterrnl'nedtsee Table5.1). Quuiparticle The dxerence betveen the mn.='mn. :1 a ml 'nimtlm dired gap of 0.69 W (6.68). J in opticaz spectroscopiesazldthe deriyed gaps of about 9.2-0.3 ev can be in.- z.1 tezweted as an iudication of the large bindiûg energy of the surfaceexciton. The dtuation is slm51arfor the Ge(111)2x1 surfaze (6.69). 4,
Exdtons 6.2 Electron-HoleP%1rR:
26:
4.5
4%
4.O
.
@ * @
3.5 *
en b 3.0
.
.
X
@
* @
we
ch 2.5 1'& < w œ
.
% *
2.0
*
.
*
t
**
1.5
* @* **
*
1O
*
*
@
*
@ *
*
().61 ().0 '
().:$ ().:1 0.5
0.6 Q.7 ().19 0-9 1.0
photon energy
:
(eV)
x'ig. 6.16. rlze reqectancexnisotropy spectmlm of a single-domain sitzzklzxl temperatme aud norrnnl inddence. The two poln.rirzationsaa'f :.sllrfmce ibr room : iosen paralle,land perpemdicular to the zr-bondednlnnsnq.The LBBD pattern o: the surface is given in the hsset.n'om (6.66).
.
.:
!'
The physical ptcttu'e has been c-hrsfled (6.43, 6.68jby careftf tpnmeric,as and the t'wo-particlt ttalctûations basedon the solution of the BSE (6.39) :. Hnmsltozf.an(6.40) for the eledronic subsystea of the t'wo staface ban.df the results are presentedin Fig. 6.17. làowzli'a Fig. 5.25. For Si(11l)2x1 E: The forbidden trn.nsitions for ligb.t pohrizatioa perpendicular to the clmlnç. ':Eià: without intezest for the exciton problem. The spedznlm of the alloweé : it Jpt ical transitîons for light polarizatîon paralle,lto the nbxims shows the dra. ir ) tlàtic ''nnuence of the bound exciton states. Thesegeneratethe bz'oadppm.Q:rti''èapee-rentsaz resectMky spectmzmbelowthe single-quwsiparticleabsorptior Egs''rt= 0.69 eV. Above the st:rlhce QPgap, the dlferential re:ectivity tyèike LtL'j ,'..,.:jj muc,h reduced.. Due to the electron-bole interaction, spectri att d osc juw E:'E: strengths iJr are redistributed to smaller energies.Thc =n.6n reason is th ,, .
Photon energy (eV) Fig. 6.17. Deerential refectivity spectmlrn of the Si(11l)2x 1 sudacecalculated for norrnnl inddence. The tclusion (nerslect) of the ezectron-holeinteraction is ldicated by solid (dashed) curves. The two lijht polarizatîons are chosenparallel (a)and perpendicular (b)to the zr-bondedchams..&nartiûcial broadrasngof r?,r= 0.05 ev Lsmcluded. The dots denoteexperimental data by Clniaradiaet a1. (6.g;. From (6.684.
destrudive couplingof tmcorrelatedpair oscillators c'ri Mritit diFerent wave vedors by the eigenvectorsXJ (i) (cf.(6.43)). Bciow the seace QPgap, a n4lrnberof discreteexeiton states are formed. The optical oscillatoz streng'th is, however,nearly completelyconcecttateê in the lowest-energyexciton at 0.43 eV. The dominant sph-siuglet exciton at 0.43ev possessesa,n exciton bintiimg eneror of 0.26 eV. This is more than one order of magnitude larger thaa the value of pbout 15 meV kl bulk Si. About a factor 4 may be due to the reduccd screensngin the surfacereglonwith au esective dielectric constant of csus = The othe,rmairt reason (sb+1)72(cf.the ctiscussionia Sect. 5.2.3an.d(6.47)). for this icczeased bindlng is the spatii conBnement of b0th the eledron and the hole at the stzrf'sce(see also (6.47)). For the direction perpendëcularto thc suface this is already dear hom the 217model (6.47). Becauseof the localization of the surface s'ta'tes derived mns'nly 9om llze Ddownand Dgp orbitals (see Fig.4.19 or 5.25), the b=ds get a partial 2D character with the strongest dispersion parallel to the chains. TMs ID character may further Mcreasethe e-xcitonbizlding.
6.2 Electron-HolePairs: Bxcitors
kqN---hble
'% @
o
2 c -2 W -6
o
:
c,n w
.
X
c
-15
-10
-5
o
0
5
10
so .
15
Distancefrom the hole (# Eig . 6.18. The electror-hole space for a flxed hole position
fht'nction of the lowest-energy exciton in real For detn.17R see text. From (6.685. (irecated). wave
A gencral visualization of the eledron-hole conelation at the sarface2lustrates the electron-holewave hznction dependingon b0th the electron coordinate œe axzdthe hole coordinatezh. It is plotted ia Fig. 6.18for a flxed hole position œjz slightly above one of the ttp atoms in a r-bonded chatu, position whezc the amplitude of the Dup hole state twhicllcontributes strongly to the excitoa) is very high. The contom plot in Fig. 6.18, which shows the distribution of the excited electron relative to the ftxed plane perpendic'ala,rto the Pandey nhn.l'n5k, can hole, m = tre zh, in a (011) i.e., at
a
-
therefore be interpreted pzsa visualization of the wave 'hlmctionof thc internnq motion desczibedby, (6.46). The Arnplitude of the electron is ve.r.glarge on the sxrne Pandey chna'nwhe'rethe hole is located. Ou the neighboring Pandey cAnsnqto the left and to the right, the ampûtudeis mue,hweaker.On the second-aeighbor Pandey clzltlrtK,the xmplitude is already closeto zero. As a eonsequenceof the qllnuqi-lDband structttre with strong dkspersioain clirecthe chain direction the exdton showsFrnnkel-like behaviorizzthe g2îîq tion and, hence, a large binrls'ngenergy. The situation LsdsFerent alongthe dmims (notshown in Fig. 6.18)The probability distribution of fmdm'g the eledron in the (01Iq direction is more extended.The mean square distanvce in the closn direction of 40 J't may be interpreted as a'a indication of the Wannier-Mott characterof thc exdton in this direction. .
6.2.5 Surface-Morln-Ged B111'k'Excitons The interpretatîon of smface optical spedza for photon energiesabovetlb.e butk hlndnrnentatgap is much more complicated. B111lc and surfaceoptical trn.nRitionsi'aterrnsv.Botmd states of one absorption edgeoccur izl the contizmllm of scattering states of other absorption edges.Ic the presence of a resonaace iateraction mediatcdby E this mteraction may give rise $o a Fazo lineshapenear the bolmd states of e-xcitons(6.702. In orde,rto avoid conhtsion
266
6. Elementary Excitations F1:Pair and CollectâveExcitations
Fîg. 6.19. Uppermostatomic layers of t/e hyclrogen-covered Si(110)lx
wé conside,ra, model stuface, such as the hydrogetssaturatedSi(1 surfacerepresentedin Fig. 6.19. Becauseof the bonclingto hydrq the surface states are removed h'om the eaea'or region of the f '
tal bulk gap. The optical reQectancen.niKotropy (6.3)ca'a only trnmqitioas betweensttrfaco-modi6edbulk states in the surfacereg'id cussedin the fozowiug.Accordingto ciculations tkis rezion is ret lessthaa 30 atoznic layers. The conwpoudiug experimental RA sp The spectmlm is rat Fig. 6.20 (6.71) can easily be reproduced(6.72j. sitive to the strtzc'tttral and clmrnicazdetnslqof the smface passiva,t
0.01
'
V2
E1
Experiment rzl
k. The ktraband contzibu.tion to the dielectric glnctions of metals rtn.n also be calculate by ab initio In the limit of pa-rabolic eledzortic-stmctme methods (forcopper see (6.734). describesfor eadzpartially fllled baad v bau.ddispezsion,ex-pression(6,48) the watl-kaow.nLindhard dieledric A:nctgon(6.74J. the intraband In the ll'=l't of small wave vectors (or large h'equencies) contribution cau be approvlmately treated. With the elementsof the ttmqor of the inverse elective mmss
p2sv(k) gpzv-1(k)q., ksok. ,k# 1
=
(6.49)
of the dielectrk flmdion takesthe Drade fo= the intraband pazt (6.48)
sintrat*) (s;
=
1
-
/p21op
(6.75)
(6.5û)
jtx + g (y (uJ iF) o4
with the aaisotropic plmsmaFeqttenc'yof the ealectronicsystem 2
8xv.6 yzlto 1(k)q a J? :7 e (aF' cv(k))qzrz-v P =
-
.
(6.51)
zsir
band u svith isotropic azld parabolic kIn the case of one partially SIIC'CI dispersiot characterizedby the efedive rnnAs zn*, the plazma gequencybecomes
6. ElementaryExcitatsoms1I: Pair and CollectiveExdtations Ya p
4rc2% =
(6.51)'
m.
with the homogeneouselectzoudensit.y n
=
2
p Etayp
-
c.(k))
(6.53)
.
For 1ee electrons ia the jelliammodel we have rn, = m-. Such a'a appcox-: imation allows tEe calckllation of tke intraband clielectric Alnction also foq non-vcmishingwave vectors. One Snds a wave-vedomdependentplcma 1t.4E + 3cp/(5'rrJ*) queucy wp E1 Lq/iupllj (6.76j ,
6.3.2 Pluma
('
Oscillations
For vanishing damping the zesulting isotropic intraband clielectric A'nctio/ TMs men'nq that an eletq sintrwtu,) (6.5c)uas a zero at the frequency so (6.52). troa gash:u an 'm6nttelylazgeresponseto ields appliedwith this frlzqaency. k 1.n.other words, there exist self-sustnsnceng chazgeoscillations of the systeù; E ï' These are jas'ttbe long-wavelemgth plasma osdllations of the eledzordcsuV , the use of the terrn : plasma frequencvy is Jus tjsed 1::) system. Consequentlyj general, zeros in the dielectric flmdion correspondto Gcited states of tV system. They give the eigenenergiesof (longitudsnxl) cozedive excitatiol.' In norrnxl metals the (anergyFztoof thtaseexdtations) the plmsmonsat vanish: i'ng wave vedor, is typically severaèeV. The valuesfor bulk alknlt metals ai/ Fw)p= 8.03 (Li),5.90 (Na),and 4.36 ev (K) (6.76). For dopeâ semicondùèu: tors the plasma gequendesaze muc.h smaller becauseof the srnnller carrier densities involved and the presence of the fmite electronic interba'ad polarizh abili'ty (6.35), mpinly (cb 1)/4x,at such small gequencies,which tscreed'E the plasmafawuency.Moreover,the modïcation of the eecvtiveban.dmais '
.
,
.
'
'
-
has to (6.52)
be takezti'ato consideration. Physically the plasma oscillations correspondto solmdlik-e compzession waves in the electron gas. However, becatlse of the long-range natme of the Coiomb potential, whic,h sustnanKt he osc illations ; their frequencydoeq not approac,hzero at long wavelengthsbut approachesthe 6nlte plmsma::1quency. BG plazma oscillations or bulk phsmozls may be excited icumeta.ts electrons at tlnir fozs. Electrons i'ateract strongly witi by flring Mgh-enea.gy the plasmamodesa'adthe caharacterlstic by stadyhg eaergy lossis obseawable the trsnqmitted elecqzonbtoltrn(e,f. the discussionill Sed. 6.1.3). Another pom sibility Lsthe obsemtion of the satellite stzucvtttresin photoelecxtzonspectfi (seeFig. 5.9) '
,
6.3.3 Surface MoalGcations
The presemzeof a smfacemorlmes the lossesobservedby EELS i'a a reâedits geometl'y (see or the satellite stzudares in PES (see Figs.5.8 and 5:01 (6.21))
6.3 Phsmons
271
ty tnqncation a'ad image-potentialeseds. One type of thesemodl'6cations gives zise to plazma oscillations locxlized in the surface region arotmd z = O (6.771. For the displacement of electrors in sach a sheet during plasma ioscillations the driving force is reducedin compn.rllonwith the 3D cafe due to the restriction of the density fuctuatiors to 2D space. This gives zise to a fedttction of the phsma gequeccy. The eigenlequency of a surfaceplasmpn, als, followsfrom (6.22) using (6.50) a'adneglectMgthe interbandcontribution to the dielectric înnction at this lequency as (6.35) ws
=
wp/v'it
(6.54)
Therefore, the energy of the surface plunfon at vazzis%ingwave vector Q it is diredly related appeal'sto be a pzopert.y of the blllk'. Becauseof (6.52) to the bll'k electzondensity zz. The wave-vecvtor dispersionof the surface ljlasmon enerr is, howevea',completely diserent from that of the b771> (see Sect.6.3.1). Jnsteadof the eue'rgy increasing with the square of the wave ''qedor, a aegative stzrface-plasmon dispersionhas been predicted (6.781. More precisely, a linea.r dispersionrelation a;s (1 Ic1 IQr)occurs for small 2D wave ( -
.
kedozs.
>
= v)
= 1-c)
Qx
.
S
o .1
9.163 A
2.6meV
Q=
' o . c 16 A'@
#'
e
ae
O
1
2
t3
4
Energy Ioss (eV)
fiik. 6.23.
Electron energy lossspedza (measklred in the reèectioa from geometzy) > jiliclc K metal flm grown onto A1(lll) for tsvodiferent Mues of tke momezttzzm transfer and primaz'y electron enezgyof 12 eV. From (6.79J.
Q $1) as a fn'nctîon of IQ1 Eig. 6.24. Memstzredctispersîonrelation of the,surfaceplasmon for K metat The dmshedline correspondsto a tkeoretlcal prediction for the littear From (6.794. te= (6.781.
A soace-sensitiveelectzonenergy loss spedrctm for au alka,limetal is The cozresportdingmeaslzreddispersion relation of shown in Fîg. 6.23 (6.79). betveen the sarfaceplasmonis given in Fig. 6.24.Thereaze two discrepancies theozymentionedabove. The &st is that the expeaimentand the An'rnplifying valu.eof als (6.54) deducedfrom Fig. 6.24is 2.74 ev while the auticipated value for the densityof the 11metalwould have beenabou.t3.0 eV. This djscrepancy may be relatedto the fad that the bulk K metal is not an ideallellium.Bandstructme efects haveto be taken into accout. The seconddiscrepancythe positive dispezsionfor large wave vectors, iudicates tkat Mgher-orderterms ia IQ1becomeimportant. However,also the treatment of the liuear te= has to be improved by the inclusion of exeange and correlatson esects (6.80). We note that the observatîonof surface piasmons oa surfacesof doped A semicondudorsby N'RRELSis more diëcult than for metals (6.81, 6.82j. dead layer of surfaeeplssmonsseems to exist. Moreover,their energiescome and coupled sarface plasmonwitidn the Tange of optical phonon eneargies the sarfaceplmsmonsin sezniconductors phonon lossesappear. Consequently, èrenot sensitiv!to the details of the atomic geometzyatd eledroaic stmzcture of the Srst atomic layers in the smface.An example for sach coapledmodes smface with a bulk eledron is #venin Fig. 6.25for a cleavedInSb(110)1x1 concentration of n = 1.5x 1017cm-S and a'n LO phoron entargjrof 24 mev. The double-ppxkstrudtu'e relatedto coupled suzfacemodesazd its mriation prefactor (6.21) are deazly visîble. due to the wave-vector-dependent
6.4 Pkonons
(a)
273
'
K :3
Eo
.'
.
(b)
L
0
-90 -60 -30
O 30 60 90 120 Energy Ioss (meV)
1 surfacewiG t'wo Fig. 6.25. EELS spectra measuzedozz a deaved HSb(l10)1x and (b) 5 eV. A doped 4:::.yst9with a hiz,h electron p 'nnrnaryener#es(a) 20 ev cm-3 coscentration .p,= 1.5x 1017 is studied. Fkom(6,82i.
6.4 Phonons 6.4.1 I'Iarmonic Lattice Dyzmmics
I'a order to dœcribethe lattice dyzomics n.ear surfaces,the same attempts at modelicg should be used ms i'a the cmse of the electronic-stmzcztttre and are the slab and total-energ.gstudies in Sect. 3.4,3. Promlnem.tRvnmnples scatteriug-theoretical methods. The repeated-slabmethod or the Green's fnrnction method togetb.er with a special mnrface ssnrllntio:a aze commonly The instantaueous position of an atomic core (atom, applied (6.83!. ion)is given by -
m
.
=
m. +
w
(A),
where R.t = R + 'rs (Bravais lattice vector R and atomic basis vector z%) gives the eqlliln'brinrrnposition of the particle and w(A) the time-depemdent displacemeut.Tllese displacementsa're the ceatral quantitîes of the lattice
dynanlics. Withsmthe hltrmonic approvimation the equatioz:sof motion of lattice particles
witll the atoznic rnn.qses i Ci.;) of ià'e' (X,R ?) are dm6 ne d a,s the secondderivatives02Ej@R%.=0.%p) of the consideredsystem iu agreementwith the de6ritotal energy (3.$5) for vauishing forces (3.39) tion of tke cvpilibz-blrnatomic structure (.Rs.) A nlamberof relations between the force constaats foltow from the beha'fu ior of the totat eztea'gya'ad the atomic forces unde,r rigid-body transhtionà J aad rotations (6.84j. Oae of these relations 1(,)a*to aa expression for ï'lli 'self-iateraction' force constants .
'i
-'s R) cutA, .&) C;=çR, =
.
=
l
-
',i Jt ?). C%œ;?IR,
zz4,:
The prlrne indicatesthat ltra-atomic terms are not taken into the s7lmmae tioa.
Becauseof the 2D trnmlational i'avarianceof the s'ystem, the force ccfy!r = JU stants depend only on the aiFeavnce (A .Jr),C'U RJ4 (A X$!) (A, œ;? .# Their tramqlationalpropezty impliesthat the norrnxl mode solutionsto (6.5d) have the form of 2D Bloch waves -
-
u#(Jz,z)
1
=
Jg
)(Te .J(())et(q(z?+.ï)-?.,,(e)tq
where N denotestke n'lmber of xlmit ctallq. J.nthe cmse of bnlllcsystmrnsràhd.. 'a in the repeated-slabapprornnation snrnliltr formu)a,sare valid witll 3D q((: >!l tities. Withi'n the repeated-slabappror'rnationj however,the modesstZ (Vz pend maizslyon the 2D wave vector Q 5.nthe surfaceplane since the dépetïu dence on the third wavewector component is negligible for su/dently laq4 i'ato (6.56) leads to the eigenvaluepToblem slabs. Substitution of (6.58) .
.
.
.
.
JT-IZE/J œp(Q)? s#(Q) dCQ)e'(Q) -
.
''
' .
(6.(59) :
lna
,f.J-3
with tlze Hermitian dynamçcalmcfrâ'r .j
D.,LQ)
=
1
c'it x)e-w(a+r:-r,1 .p( 'X'I u : syj .
(6kt't) 1
x
eigenvalueproblem (6.59): In the generalcase of S atoms pe.'rsupercelkthe pps! 2 ë 35 solutions (as(Q) (/1= 1, 2, .-., 3S)for s' at eachpoint Q in the lst'afà4.dlj: Briliontsn zone, which can be interpreteê as the brauc'he.sof a multiokttd' Eaehnormalmode nQwith frequenc'yw(Q) and dgenviètbz!! 'hlmctiontsztol. ei a excitatioa of the vibratiug lattice of all atomd.: & (Q)desczibes particular is called a pltonon. A quaattlm of the lattice vibrations with energy hu)nLql ..
...
6.4 Phonons
Consequentlyj the relationsexpressed by the equatinnsa7 =
V
Itonorbdfapersïtmrdations.
275
w(Q)aze lcowa
There are many phenomenologicalmodelsto determsnethe force conEétants XJ (A,A/)in (6.56)(6.32j. Contrary to phenomenologicalmodels, ab ktitio caalcalationsrequire an accmate ard parametergeeMowledgeof the microscopic zespoaseto lozen-in lattice vibrations. The basic ide,ato all irst-prindples methods is to detemninetbe inteaatomic force cortstants Nia the total enea'gy(3.55) of the s'ystem with gozen core .coordjnates.ln the list decade,mnany theoretical advrces have beea made toward the appm 'cation of these concepts to lattice dynamlcs.There are two cornmonlyused àpproazhesfor this knd of caLculation:the direct lozen-phonoa method and fhe pemtmbattveapproacz.Thesemethods aze essemtiallybasedon the den.'2 àztrg:nctional theol'y presentedin Sect.3.4.1. Withl''n the hnzen-pltonovt ûpprocch (6.85, one considersthe propa6.86) cation of a phonon wave of a ftxedwave vector whic,his comnnenstlrate with a :r E(,/etikproc,al lattice 'vector. Tbis causes the atomb to vibrate witiz a d.e6nitedis7 ,.ylacement pattfacn. At a given moment the syste,mwill cocespondto a new E' al strtz dure corrcxspondingto the Aozea' vibrationi mode. For suc,ha ',tjtllst jhononmode the total energyand the atomic forcesare calclzlatedas a, fhnrdcof atomic displacemeatsin a supercellappropriate for the new crystal ttickn dtrudare. Using the energy chfkre'ace betweenthe distozled and tmdistorted E E:btructt'trE's or equivalently from the atomic forcesiu the supercellgeometzy ibneconstruc'ts the force constant matrix. Meanwhile,this method hasbeen 'mânedto the placaz force constant method (6.87). Tite atomic planes perto a given wave vector are regardeêa,s rigid bodies, and the lattice E/çndicala.r i'2' D nrnscs can be treated in the spirit of a linear ckta,inmodel. The most Mportant pexurbative approae comsiders the linea.rrezpozxse is?the electronic system to the displacements.Basedon the DFT it is called 'the density Jvrlcfïorlipectncbationtheorl (DFPT) meih'od(6.88,6.891. For a of tNe elecwtrondensit'y jjveu'lattice distortion the resulting chauges.zAn,(z) and 1v'(œ)of the total Kohn-siam potential (3.48) thus needto be (3.47) qvaluated.Thesechangesare Oectly related to the electroGc contributio:ato hn.rmonicforceconstrts. The ionic contribution ca'a be straightforwardly Etffe i uated 9om the Ewald snlnnmationmethod (6.841. :'y: Polar czystals, suc.has semicondudorswith partially ionic bonds and ktjzuc czystals,needaddaionalconsiderations. Irzstzclzsystems,wheredieerent ::' are involved, the long-ranger-haracter of the Coulomb iljrjesof atoms/ions .!. g'ivesrîse to macroscopicelectl'ic îelds for loagitudinaloptical phonons Yrcqs ''trt'the long-wavelengthlz'mx't.As a consequence, the dynamicalmatrbc is aot EValjrtic at q = 0 (ia3D).There is an additional non-analytic contribution the dpormicaLmatzix (6.60) : Etd It %smtke geaezalfo= (6.00) ..
,.
q...l'.
'
.
.
CJ -> D=o Lq
0)
=
z-t ph (n( a-!f k 3 h.=
l 4cr: V' MLM.f -
-
.
z- ,*)
-*7= %)Jç q êx q -
.
-
-
.
(6.61)
6. ElementaryExcitations 1I: Paiz and CoEective Excitatio>
276
The tensors êx and k*ç denotethe ielectzic and the efective dyzomical ion chargetensors, respectively.Thereby,the index x stands for the (Born) purely static eledronic contribution to the screensng.The tensor êco ca'a be calculatedtusing (6.J1)for vanishy'ngfrequency ul = 0. J.'athe cubic llml't acd without the lattice conebution, the scala,rsoo correspondsto the btTlk ,
cozlstant sb discussedabove.A new abbreviation is 'ttsfafltl, since another &.-
electric constrt so hcluding the static contribution of the lattice will late,r be introduced. The qu=tities X* ç an d êx can also be obteed in the framework of the DFPT i'a termn of derivativesof the dielectdcpolnm'zationfeld of the s'ystemwith resped to the dispacementsa'adthe electdc âeld (6.88,6.90) However,at least for cubic systems it has been shown that the screened Born Garges Z;/ sx caa also be derived directly wlthsma supercezcalculation (6-91, 6.92). .
6.4.2 Surface and Bulk Mpdes
The poln.rszaticnvectors eix(Q) of the lattice displacementsiu (6.59) have 35 components, nxmely three cï .(Q) for each pazticle at 'rï izzthe l'nst cell. 5 They indicate how the atom ï vzbratesizl the paztioTlltr mode sQ. Sincethe dpmrnl'calmlttrtx $sHerxnitian,the polarizationvcdozs satisfy the orthonormalit'y condition
et*.(Q)4-.(Q)Jxxz 5-2 'lta
(6-62)
-
and the clomtrerelation
X'le'(Q),f* MJ(Q)
-
nœ
kiè'œp
(6.63)
N
edx(Q)eL.(-Qtand au(Q) *s(-Q)
For ins'tance,the poTv'zation vedoz's of a givec mod.eare normalized to 'nn7't.yove.r the whole thicAmessof a slab. This enablesas to determine, by izzspectîonof the variation of the ei NG (Q)for that mode over the variation of the atomic positions vz parallel to the sttrfacenormal withn'n the thielenessof the slab, the locxll'zation charmode acte,r of the vibrational mode,i..e., whether it is a HJ: mode,a s'urfac.e mode. TMs is quite sre=l'ln.rto what we have lenrmt or a m'ized(rc-s/zwzzcc.l about the zectroaic state of a surfacesystem in Sect. 5.3.3, Consequotly, the identlcation of tkevdsferentmode Garacters r.a.zl also be basedon the of the uu(Q)of the surface system with the b:llk phonon discompazqsozz J.nthe b:711c persioa relations projected onto the suzfaceBZ (see Sckct.1.3.3). case there are three tconstic p/ztmtmbraachesand 3(S 1) opticai Jl?lozltm branches with S ms the Irlrnber of atoms i'a the prlrnstive n'n''t cell of tke crystal. Mong high-symmetzydirectionsin the bphl'lrBZ, suc,haz the (100) directionsin cabic c'nrstals, the phonons can be c-lassifedas transand (111) nerse or longitndino.taccorclingto whethertheir displacements o:r polarization
: Note only theq! that modes. plane. Thesewill be labeledshear-horizontal(SH) polarhation normal to the sagittal plane (siti! SH modeshavea well-defu'aed 17 modesare attti.r.E Fi f- 6 27)) whereasthe so-calledtrn.nsvezse and longiturismnl .( atly coherent rnlx-tm'es of b0th polnvlzations, the name indâcatingwhie,h hl 5: the lazgestrelative amplitude. .
.
'
'
.
polarizafv/ . .. .
'.'
--
,.
.
-
6.4-3 Myleigh
Waves :' :';':':'
T he f o rrnnlism describedin Sect.6.4.1 glves complete iaformation about thètiq: y ,.jg lattice dyzmrnl'csof a surfaces'ystem. However, for ceztna'mfrequeney aù2,), to apply certxirl approsnrnate descrtpttùriïy wave-vector regiorusit ks pozssible that give deepet physicazinsight. TMS holds in partichlln.r for rnnztroscolft' vibrational modes.We lmow from practical exercisesi'a solid state physics(i.èï that izl the lonpwaveleng'tklimt't the equations of motion (6.5à)l e.g. (6.93j) over into the wave equations of an elutic contintplm for a linear chnnneahauge .: ..7i of motiid of elasticii'y the equations of the linea'r theory Witlnsnthe frnmework of an elastic medlum are (6.94) ..
..
..
.
,
'
' ' '' '
' *' '
.
..
o,
pazzralx,t)
=
(-9.., (w,tj )C ozp p
La,p =
z,
v, z),
..:.
(6.6i''
6.4 Phonons
2$9
@ere p is tEe mass
densityof the medium,.u@, of the tj is tite displacmment zzfèililll'n at tbe point m and time 1, aad &#t t) îs the stress tensor. The latte'r
1kgiven by Cœp=
Hooke's law
c.o.,o' -''tG' Dzp'
J'')
ue/,J3/
khere the (tQ,.,,r)
-
eobofqeo, J-.z
(6.65)
,
q.;, according to stbemez:r = $, yy = 2, zz = 3, yz = zy = 4, zz = zz = 5, n = yz = 6, t11'è tkudto iutroduce the elementsof symmetrizedstress an.dstram tensors. The elpstic mectb:rnLsconsideredto occupy the lowe,rhnlRpace z < 0. The surfàce z = 0 ùs azsumed to be stresseee, i.e., cazlzxo = 0 (withcuz in the dkmmetrized form).The electric Neldsf1116l1the botmdaryconditionsof elccaze
.
tvàtatics.
ln order to simplif.g tbe considerationswe study an Lsotvoptcmedbnm 4k'i'h ou t the piezoelearic Kect. There are 'two iudeperdeat ehstic moduli tî:, cza, au.dcu with cu = c11 czz. with Laml's mod.uliit = lz(c11 cza) ttktdà = clc, (6.64) becomes(6.93,6.944 J ,,.
-
:2 = 'tztm, t) &2
njl
-
graddivuta,Lt -
v:2
carl cuzlxztm, tj,
(6.66)
.wrllrethe longituainal and trxnwerse soundvelocitie,s ',(a= 'nt
=
L2g+ à)/p, y/p
(6.67)
1'eltzoduzed.
Equation (6.66) suggets splitthg up the displacamentâeld ''QL# 'tz! +w icto a turbulence-free contribution tcurlw= .7.L, 0)and a source-free '
'.
.
jz't ldivut 0).In the btllk', tEe resulting dzflbrezttialwave eqttatiozlscan be ikved independentlyfor b0th contrsbutioas,giving longituan'nalsotmdwaves l éïi;(jp, aad trxnsvezse (sheaz) sound waves 'tsutz: t) propagatingwith the ( t) .
.
=
'
: ,...
'
sotiid velodties /t& or ,2 t) = 'tzvttm, (9$2
vt.
Eac,hof these soandwaves obeys a
wave
equation
'tf/taltrlt,l/ttm, t).
In the cmse of the elastic halfspace (z < 0) we only seeksolutiozzsof the equations (6.68) 'Lqtpke witic,hrepresent surfacewaves, i.e'., which are characViiked by = ex-ponentialdecay iuto the crystal. With a wave vector Q in i dùrfaceplaue (z= 0)one has
28û
6. ElementazyFzxcitations 11:Pair and CollediveFxcktatioz)s
'tz:/tlz, t)
=
e O2-bJ2/WpJ el/te5(Q=.-a7t)
(6.69)
for Q > Wqyt.The special fo= of the positive decay constant is a consethe conditions quence of the wave'equation. The polarization vectors F7r1611 et
.
e) x
(Q:r1
K%/ï =
Qv,-i
A-t
0, AA= 0, =
Q2 cJz/rtzs -
.
aze called Rayleigh waves (6.952. Ffowevez,its Wavcsof the type (6.69) dividual waves (6.69) cnmmotfttlfll! the botmd.a'cyconditions of a stress-lee stTrfnre. This is only possibleby linear combitmtions('t1: + 'tttl'with a mixed chxracter. Moreover,the bolmdar.yconditionscFmnot longitudlmal-transverse be satisled for shear-hozizontal modes.ln a'a isotropic e-lmsticmedî''m only vector = of thc sagittal-planeE'ayleighwaves are. Vowed. The displacemem.t surfacewave lies iu the plaue spannedby Q and the smface norrnnl zz. = 0 The botmdav conditions,cuzqmc (a= 1, &, # , lead to an eigenvalue equation. Slml'larto the bulk case a linea'r dispezsionrelation
uzawlol rswkol, =
rRw
=
'pt
'
(6.71)
ï
Ls obtained fov the Rayleir,hwaves. The phase velocity luw is #venby the solution ( (0< # < 1)of the polyaomial eqaation
ï
6
-
12
2
16 1 84 + 84 3 2.-.% p 'tl 4
a
-
-
'r
-
-.tz %J :
=
0.
The quactity (' dependson the ratio of the sound velocities. The ratio (14/q)2 VHe.S between 0 and 0.5 for the various mateziaks.Tllis corrœponds to a xariatiozzof #'iu the interval 0.955 0.874. For Gaà.sa value ( = 0.92 is obtaiaed (6.974. J.nany case thc velodty of the Rsyleigh wave zxw is srnnller tha'a the tzansvezsesolmdvelocià rt.. ThksSsalso tmzefor cubk mystals with three hdependent elastic constants (6.96j. Consequently,the Rayleigh mode should be belowthe projeeedacoustic branches,at lemstfor wave vectors in higkssympetry directioms. The rn5vedlongitudical-t:ansversechrader ks seen from the direction of the dîsplacementswhic.h are paztially parallel acd pactially norm/ to the to be parallel to the r-aMs. At the surface, propagatioa direction msslzmed z = 0, the polazization vector is given by -
2
eaw =
1
-
-
($2 2 2
('pt/'t&) ('
-
2 1
-
$2,0, -i$ 2
.
(6.73)
The dispersionrelation of a Rayleighsuzfacephonon Lsshown5.1Fig. 6.28. surface by men.nq TEe dispersionrelation Ls measuredfor a clea,zz Ni(100)
6.4 Phonons
5
.
R
150
R
S .E,
%o
k
k
# 1Oo
O u
ö' @ = g' v:z 2 o
X
% n4 80
->
c:
281
8
3
O c
a u
2
a u L
50
1
ul
n-
0 0.0
D = x
'
0.5
F
1.0
O c C =
œ
()
X
Q1
Q (K )
Fiy. 6.28.
Wave-vectorclispezsiozz of the Rayleighphonon on tke Ni(100) surface. It ls memsured by menx!q of EBLS usitg t*e primav emergiœ180W (squares) and 322 ev tcirclesl. After (6.982,
of M-RMELS26.9% Near the bouad.azyof tile slzrênre BZ êeWatioasfrom the resdt of the continutlm occtzr as in the bullt case. The phnse velocity exhibits a wave-vector dispemion.For wave vedors at the zone botmdary the localization of the Rayleigh phonon near the surfaceiacex'xse,s according to t.he decayconstauts, e.g., Q 1 ê2.J.nsach a Rayleighwave only a few atomic layers beneath the surfaceWbrate.Consequently:Rayleighwaves with short wavelogtkts may also be used to detect efects of bond changes aad reconstruction. One drastic example is the sharp (iip kl the surfacephonon disperskonof W(110) whic,happears 0e,r hydrogensatttration of the stlzface .
-
(6.99!.
6.4.4 Fuchs-Kliewer
Phonons
The surface iHuence onwoptical pbonons can also be studied i'a the longwavelecgth tsud,hence,quui-contiauum) lirnst ) simn'lar to what we have learnt for acortic phonons. We consider the simple case where a bulk Eikactive czystal has tvo atoms in the uait cell with mmsses 3f+ aad M-. Such a crystal could again be a polar smmicondudor or azt ionic c'rystal.The bonds
6. Elementary Exdtations II: Pair and Coiedive Excitations
282
shouldbe partiatly ionic, ard the pair of atoms conwpondsto cation and anlou witlz sm-'ned dynnrnscalion charga e* and -c* with e. = ezijsfc (see force Taking iuto accotmt only the efedive nexet-nearest-neighbor (6.61)). for the disphcements'tz.y and .tzconstant .f, the equationsof motion (6.56)
read ms
M+ M-
:2
z*E,
'tt+
=
-2.f('u+ 'tz-)+
'tz-
=
-2f ('tz- 'tz+) e*B.
d@2 d2
-
-
(6.74)
-
(uz I'a the consideredloug-wavelength l''m'lt the displacementsdo not depend oa the position R of the Atnit cell. 1r. the continutlm limit, however: one may discussthe dependenceon the space coordinatez. Becaaseof the effectîve càarges of the cation aad anlon additional restoring e-lectricforces occuz. Tàe total electric feld E ac'ts at the position of the respective core. It is therefore inBuencedby Bcal-âeld esects wizich wilk however, not be discussedhere (6.90, 6.93).Skce we are interested in lonpwaveleugth opticat phozmnsjthere is only a need for stad>g the relative disphcement 'u = '?z,+ 'u- of the cation-anion pair, whic,hdbrate.swith the reduced mass Mr = M+M-/(M++ M- ) Then -
.
Mr
d.2 dzz
'tz
=
(6.75)
-2.fu + 6*E.
with frequencytzl one obtnl'nq J.11 the ll'ml't of a harmoztictime dependence c*
n,
=
1
V;
u savo
.
ts
-
(6.76)
g.E
with tEe zone-center frequency u'l'o
=
ag'ro
(0)=
2.VMr
of the bulk TO phonons azsumed to be kmdamped. The displaced catioms and anions fo= dipoles with moment henceja dielectric polnrization âeld puaz=
N. F
..e
N
+2
e
xhztfz;) V Mr ww;
ands
(6.78*)
w,.
and tNe relation P lat Becauseof (6.77) susceptibility a,s =
c''tt
1 -
u' a
.
=
x
lat
.s , (6 zr8) cu%es jue (&J) .
latticèE
(6.79)
In additionoatsothe electrons contribute to the total polarization âeld byL .PO: = x e.lE. Fbr frequenciesbelow the absorption edge of the electrdiiv '
6.4 Phonozus
283
Fystemthe electroak polarhability xd = (:x 1)/41ex.n be replacedby the static polarizability with ex ms the higisfreqttenc,yalectcotdcdielectric constant. From the deAmitionof the dielectric dksplacementîeld -
D
E + 4zr,FDM + 4rr.Pd H
=
LbLu)4E
(6.80)
the lcquenca--dependen.t blllk dselectricAlnction Q
sbtapl cx =
1 + Ltoz
u'To
-
-
wz 'ro LJz
(6.81)
k derivedfor lequencie,sbelow the htndamentalgap in the elèctronic band s'tmzctare.He'rethe (sceeaed) ionic plasmalequency 2
2
2
DJLO û7TO = -
4re* N E'xafrF
(6.82)
;i: fntroduced.This dm6nstiong'uaranteesthat the zero of the dielectz'icflcncof the zone-center LO jion,cbt(al= 0, eqz:xlslthe SequencywLo = 1t?z,0(0) With the statie dielectric constant sz = sb(0)j lshonon. that also accotmts fPr thc static lattice pobqrtzabilit'y the Eyddane-sn.nlnK-r-reller , i.e., &a > sx, = tèl>tion tu&o/u&olz ca/cxholds (6.100). Sincemacroscopicallythe crysta,li,seledric-allyneutraz,one 3àkksshw
dîvr with
or
=
cau
0
apply the '
(6-83)
eqttivazently (6.80)
= 0. skhtslltlivf
(6-84)
Nèglectingretardation efects, in addition the Maxwez'sequatioa .' ;' ''';.''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''d
r'E'
curl E
=
O
1i6l&.Equation (6.84) is f1116:1ed when eitheerdâvf
(6.85)
= 0. The 0 oz sstoJl qt case ctivx = 0 implit's that tlze electric âeld E and with (6.76) also i1k4relatkvedisplacemezefeld '?z.kave to be trxnmvezse,div'u = 0..The diJtqqtric A'nction sb(uJ) h.asa resonxnce at ts = tga'o which is assocîatedwith trxiivez'se lattice Wbrations.Thus) az'ro Lscatled the transverse rcsoncp'ce ,y = yknertcy.In the secondcase, div E # 0 (butcuzl.F = 0)and sb(aJ) 0, iiè :eledric Eeld E of the excited waves is longitadinal, Tke chvadetistîc ièquenc'y is EO:E.II at w = fztoo the lonnitudinal rssonance Jrazuerzcr. Shce k''w:r/ss(w),E kscot necessxrlly zero even for D = 0. The displacement iéïd.satisses cuzlvz= 0. Fpr Fequenciestzx-o < ;(J < wuot the lattice polarization bnn simtztane=
,
, l'
:. .
.
.
-
.
ottslfto
An161!
284
6. ElementaryBxdtatîons F1:Pair and Colledive Bxcitations
div Jva,t= 01
cml-rfat
=
(6.86)
û1
zesulting ia slmslar relations for the relative displacementfeld 'tz. These conditions have consequènces for possible displacement felds in a polarizable haltspace(z< 0)with smface(z= û) The felds ia such a system are cilarazterizedby a scalarpotential /@) with E = -grad/. Becauseof (6.83) and the potemtialmust Gal6ll a Laplaceequation (5.85) .
Zm/@)=
(6.87)
0
a'ad.the standardbound.ar.y conditions
/(z)Iz=-0= /@)Iz=+c, cb(Y)s/(=)Iz=-o yr/@)1z-+0
(6.S8)
=
of elec-tros-tatîcs.
.
Waves propagating paraltel to the sudace along a 29 wave vedor Q IL g-azds,whic,hdecayinto the vacuhtrn as well as into the crystal, are described by the fozowing solutions of (6.87)
/(œ)= /(œ) =
yuiozeom BeîQne-Q%
fozz < 0, forz > 0.
(6.89)
The 2D wave vector Q deterrnl'nu the attenuation of the electric îeld and the displacementâeld into the blllk. The boundarjr conditions#ve a1.= B and the eigenvalueequation (6.22) for the dieledric smfaceexc-itatioas.With = -1, leads to the gequency of the (in the eigenvalueequation, sb(uJ) (6.87) the pre-sentapproxlmation dispersioaless) Fizc/ls-fflïelccr phonon (6.1O1q tz'.p'lc= u:sto
+ 1 , soo + l co
whic,h indeed lies ill the intezvezltppc?o llsc phmse41s3.r.g a one-ba'nd Eubbard Hn.rnik tözïia.u i'a one dimension (6.1214. The spedral Annctiozzof the (cf.Sect.5.4.2) iigle-particle Greea'sAlnction (5.20) with spin can be decomposed icto t'wo peskq that changetheir energiescvztk)licearly with the (wwsiktfwsipazticle ùzùmentx'rn Flk. Thesetwo pen,k-qcoincideat the Ffarrn''level ss ozfy, where Efh' e sphon and holon lines intersect. The spinon hmsa laTgergroup ve-locity '8ii (k)/d(Fzk) th= the holon by nlrnost a factor of two. e.
.
*
.
.
--
.
.
-
''
.
-
-
.
.
.
. .
-'
.
'
'
0)oz holesLq< 0)ia the eledronic reservoir.
7.2 Point Defects
da
297
db
dc
Eo01)
s
p'$oj
representation of ar anioravacancy on a III-V sornlcondudor Fig. 7.3. Schnrnn.tic (110)stkrface.Three dacgling bonds (Q, dN?and dcjare created dtlring the formation of a vacancy. Small circles correspond to atoms in the secondhyer. Filled smbols represen.t cations tanionsl. (open)
The Fermi level sv (thes=n.11temperatttre-induzedcliFereace to the chemiis measlzred1om the mlen.ce-band cal potential of the electronsJsnegleded) mnavlrnam evnu. Usuallyan alignmentproceduzeof the energ.gscalesin the Ia exideal system aud the system with a chargedvacancy is applied (7.71. plicit calculations often a shift of the valence-baudmaOmllm due to the cbvging of the dafec'tLqconsidered.lt may be calcuhted by the dlference + &k@)(3.48, of the spatially averagedeledrostatic potentials Montœl 5.41) for the càarged and neutral situations (7.7, 7.81. The charge state of the anion vacaucy i'n the surface layer detnrml'nes the atomic relaxatioa around the dmfec't(in contrast to the bulkl and the energeticsof its formatioa. The three daugljng bonds of an anion 'vacrtzjr VGs(see Fîg. 7.3)are occupiedby (3 qt electrons.Consequentlythe chazged vacancies Vs- azd V+s prefer rebonded geometric whereasthe aeutral N'acar.cy Wsexhibits a weakertemdencyfor rebondtng.The positively charged aaion vaca'aclesa're dominated by an inwazdrelaxation of the neighboring cations (7.8-7.101. However,the symmetrically non-rebondedand the nonsymmetrically rebondedcovgwations are almost deguerate i'n eneror. ln the non-bonded geometry the catiomsa aad b (Fig.7.3)relax inward. In the Sed. 4.3)with the c rebotded geometu, the a or b cation forms a dimer (see cation. In 170th.cases a single empty dangling-bond-derivedstate appears ixt the 'h:ndamentalgap. -
298
7. Delhects
Fcg. 7.4. Formation ener#escalculated for dl'lerently charged P vacalcies öy!. mlrfaces tmder P-rich conditions verstus the Ferrnl'energy. Adapted fibm 1nP4110)
(7.:3-:1.) -
Accozdîngto the formnt,iolk eneror (7.1)tEe most favorableeahazge statq deper.dson the doping level. TMS is klltzstratedin Fig. 7.4 for dlFere/ttk chargedP varnncies on 1nP(110) surfacesprepm'edtmdez plmsphoruscrich bunc conditions, Jzp = pzp On surfaces of pdoped substrates a singlt (7.11J positively cttargedP vacancy JCis stablej wheremsa singlenegatively chargèd P vacancy V- hmsthe lowest formation energg for zsdoped substrates.Fck tmdoped TnP the neutzal P varancy F'eo be the most stable one. Tlzev may Mdsmgsare in agreement with m=y ex-perlmentalobservationsfor Gavj! 1nP, acd GaP(110) surfaces g7.12J. The positions of the Fnrml' level sp at wllich the formatioa enfzrorJ)f 8f a defect iu two cllFe-ren.tchargestates q and q + 1 become.s equal in Fig. z.iï or (7.1) deft'aesthe éonémiion Jewelor charge-transition J6VeJ vitil sLq+ 1jt2) respect to the VBM. According to (7.1) one %n.qg7.7, 7.81 '
.
:(ç + l1ç) ELNX,NU1, qt EINX,NB 1,ç + 1) =
-
-
-
-
Sv'aM
disregardingfor a momest tke alignment procedure. For the acion vacacéîr!q there aœe t'wo gap levels,s(+I0)and s(0E-), as demonstratedin Fig. 7.4 fùt 5'pvon TnP(11O)1x1. The Srst (second) one is related to a more acceptötlg like (donoz-)
state. The euerg.gc(+I0)may be interpreted as the bindiik energy of a hole to the neutral vacancy, while X c(0I-)gives the e'nerj/ o f a'a electron bonded to the neutral vacancjr. Uaforttmately, neither STV'' a'n.dphotovoltage memsurementsnoz ftrst-principlescalculationsgive a uzziqttèE -
pictzlre for the t'wo energy levelsin the gap g7.12). Even for the mode,lsystelE
7.2 Point Defec'ts
(a)
(b)
-):( >
:-01 f
299
y
Q
y j yj
Fig. 7.5. Stmzctuzalmodels (topview)for the C-type ddec.ton Si(10O). Dànglt'ag ibùnds rezatedto rniAsingatoms and located at the furst-hyeraad tbhd-layer atoms ;' E:. o aàeindicated..Bu of the z'mers Ls allowed aa kzdicatedby the dlflkrent shes 6f the circ'les representing Srst-layeratoms. (a)A two half-rl':rner model (7.15, 7.16J, (b)m''qsixgatom in the secondlayer E7.17). '
.
.
v
.
ö?a
sl:rfbnrtethe leve,lpositiozls single .ks vacancy at a cltoltnGaAs(110)1x1 lait'4 the intezwetation of the chaapder of the levels va'cy (7.8-7.10, 7.13). bnx recently remaruble progress ia the calculations been aœeved .k 9wever, t.'LL?Jncerns'ng states (7.112. convergenceand treatmemt of the c-harged Vacaudes are also discussedfor sttrfaces of covalentsmmiconductors, for k 'Li acce the Si(111)2x1 surface(7.14) and Si(100) suzfaces(7.15-7.171 One pvn.nnpleis the so-enlledC-defect on Si(109) wllich is always obsetwed (7.15J i/ room-temperatttre STM lmltges. At positive snmplebias it appears as a Vghprotrusion adjacent to a depressionin the surfacelayer. Two interpreVtionstseeFig. 7.5)have beengiven for these STM images.Izzone case they ake interpreted to inclicate a dimer ddect in. which t'wo adjacent Si atoms kong the dsrner row are rnr'qsiztg,i.e., like t'wo 'hn.lf-fssrners(7.15,7.162. Conscthis defectshould represent a mcaucy pair in thc fkrst atomic layer ktzently? Ciculations (7.171 haveshowatha,t the optirnizedstructtzre start(Fig. .LE 7.5a). ,k#jFomthe two lnnlf-dimezcougttration is llnnble to rèproducethe observed j''FCMimages. Tnqtramzla C-type defect structme izl whic.h only one Si atom i'a ,,
'
.
.
.
3OO
7. Defeds
Fig.. 'J.6. Lltustration
(topview)of a mlqsjng-dimerstractuzej a sty-nn.lledtype.4 defed, on a Si(l0O) smface; (a)idea,ldimer row; (b)cll'mer zow wiih a dimer vacaztc'y.Large (small) atoms. open circles represent Srst-layer (second-layer) the secondlayer is mlqskzg,a,s shownin yaig.7.5b, IhAAbeen proposed.rIn1np's correspondsto an isolated vacancy in the atomic layer bene-aththe dimer layer. Many other defeds disturbing the dirner rows along (OI1j havebeen discussedfor Si(1O0) surfac%.lmportant ones are related to missing dimers, i.e., to patt's of vacancies izz the 6mt atomic layer. A dsrner-mcmmcymodel i'n whic,ht'wo adjacent Si atoms along a (O11j direction are absent is shown in Fig. 1.6. suc,b. a model has beeaproposedby Pandey (7.18) to elurpln.in the type-A defect obsemrecl on Si('1O0) (7.15, 7.16j.Other defect complexes consisting of t'tvo oz more missing dsrnersare observed.) e.g.,the so-calledtype-.s defed (7.15,7.16j, an.dgive rise to confgurations with not too lmgc formation energiesF.19). 7.2.2 Impurities
The abil!ty to kcorporate reprodudbly dopant atoms, in particulaz shallow impurîties as acceptorsand donors,with precisely controllcdconcentrations and.spatial distriimtions is essentialin vadous de (7.3) (7.25J on A,s sites as acceptors. Cozlsequently; compenqation eFectshappen.I'a addition, tlb.eattractive i'ateraztion of Sioa and Sihsshallow impurlties suppoz'ts the formation of Si clustea's.Suc.ha view e-xplnsnm the obsenation of nearly constau.t donor and acceptor concentratior (7.21J itl contrut to theoretici predieionsj whie,husllxlly study uornînally Ssolateddefects and do not také the defect-defectiateractioashto account. Nevertheless, the electrical 5nAuL ence of the other defects in the system is reasonablymodeledby tqb.e chenlical potential of the eledrous. 7.2.3' Antisites
Mtîsite ddbcts
are
not identïed ia Fîg. 7.7. However, the
iccotrpozatipyjEéf
arseaic at low vowth temperatures leadsto the forrnyktionof a hij concen tration of defects.Cross-sectional STM showsdiretztly that the Yfé/' il'' fo=ed by the incorporation of Acess arsenic are mostly isolated Ge2i: antisite (Asoa) defects,aud that tllese antisite defec'tsgive zise to an iatènbytj , bazd of midgap states (7.26,7.271. One of the most stm'kl'ngfeatttresrevéalèà b: the STM imagesare the spatially extendedlocal DOS of the defeds (i.jd)r. u ë Basedon the symmetr.g aud the intensit,y of the local DOS as imagedby.STV) Asca defeds in dllerent sabsurfacelayet'swere distiagaished. The syznmetzyand intezlsft.yof the Iocal DOS also allow the ideutifcatld:( surfaces éf of ardon antlite defects ën the topraost surface layer of (110) excess
,
'
Fig. T.ï. STM ''mxgesof occupied (upper aud empty (lower elecpanels) panels) tronic states of the majordefects on Si-dopedGn.Aq(1l0) surfaces. (a1)a'ad (a2): Ga vacancy; (b1)and (b2): Sioa donor; (c1)and (c2):Sixs acceptor; (d1)and (4i2): Sioa Voa complex;and (e1)and (e2): intersedion line of a plxnn.r Si cluster (perpendictzlarto the surface). Tke tnmnelingvoltagesare -2.4 (al),-2.0 (b1,cl, d1); -2.2 (e1), 1.8 (V), 1.4 (b2jc2, d2),aad 1.5 V (e2).Adapted from (7.21) (copyright lmerican 'by the Physical (2003) Sodety). -
7.3 Line Defects: Steps
a-doped
303
surface. GaP(110)
Four Poa antisite defects by t:e (copyrigît(2Oô3)
GaAs: GaP, azd J.IIPusing STM, in pazticular, by compariqon with results Sf DFT calculations (7.28J. The aaion antisite defect in the surfacelaye,rof smfacesb.asa vezy localized DOS, i'a contrast to the extendedDOS E(110) As an mvnmplethe STM image of a'a zz$f the sabsurfacea'atksites(7.26). cleavczge soace with four Poa antisites is shown i'a Fig. 7.8. Ediped GaP(110) (The constaut-cunent STM image eebits rows of occupieddanglhg boads with E:locnlszed above the P atoms of the lx1 surfacestracture (compare In addition to the regularly spaceddangling bouds, one obsmes k:Fig.4.9). by arrosvs in Pig. 7.8), which i additional mnorima betweenthe rows (indicated may also be identifted with completely Slled P danr;ingbonds =d, Eenze, with Pca aatisite ddects. The lack of any apparen.tEeigkt chnangeor long-rangevoltage-depezden.t contras't around the defect featmes in large-scaleSTM images indëcatethat the kmdealyingdefect type Ls electhcally uachazgedon a11investigaiedztdoped substrates. Consequently,the surfaceaaion antisîte defects aze elect21 ically iaactive a'addo not induce a Fermi-level pinning 'lnlike bullc aatssite ddects. This holds not ozktyfor GaAs but essentiallyalso for Gap and Tnp. .
'
7.3 Line Defects:
Steps
7.3.1 Geometry and Notation
The most impoztant ckmssof liue defectsoa surfacœ are steps i'a whic,h the ledge separates t'wo terraces from eac,hother. Steps'are impoztaut i'a the i.e., surfaceswhic,h are formation of vicinal surfaces(high-inde,x mmhces), with respectto a low-indexsarface.Suc.h orientedat a sma;llangle I (i 100)
3O4
1. Defeds
vicinal surfacesare formedby small Eow-inde,x terraces and a 1nltg. h deusi'ty of regnzla,r steps (see Fig. 7.9). Thetr surfaceenergy shouldbe srnnller thaa that of the high-inde,x surfacewith a certsu'n atomic structure (see Sect. 2.4.1). As an evn.rnple,a 2D kut of a simple cubic stzuctttre with a suzfacenorma,l slightly misorientedwith respec'tto the (901) direction is sbownin Fig. 7.9. The peziodicsuccessjonof terraces and s'teps on a vici'nal surface c.ark be specfed by their correspondingMille.r irdice.s,for i'astance(5,57) for the situation of an fcc crystal (without atomic basîsl in Fig. 7.10 with a tilt angle 0 = 9.40 of the surface aormal with respect to the (111) direction. However, this notation is not ve,zyconvenientsince it doesnot indicate,at Sz'stsight, the tmze geometrical stractme. Such rstaircEtse'stzazcttlresms shoqm i'a Fig. 7.10 are more conveHentlyreferredto tzsi'agthe fmicrofacet'notation (7.2%
p(AkI)x Lhtkllî),
(7.4)
in wizic,hh'kk and Wk'l' are, respectively, the M511e.rindices of tbe terraces and of the ledges, an.dp gives the wzmber of atomic rows in the terrace paralle,lto tlve edge. Hence:an fcctlll) surface may alternatively be labelled 64100) x (111), i.e., a series of six-atom wide (100) terraces separated by (100) x (111) place may be referred to as steps. Sinzla,rly,a'a fcc(331) x (111), i.e., three-atomwide (111) terraces sepazatedby (111)x 3(111) (111) x (001) steps.The suzfacein Fig. 7.10is1thusj a 6(11:) steppedsurfaceof azl fcc czystalwith a lattice cozsstautco. The geometzy of the (111) ar.d (001) planœ of aa fcc crystal are describedin Fig. 1.2. The steps a'e propagating along the (EE21 clirection. The coordinstesof the lattice site,sare given bl Table 1.1. Periodicitrin this dizectionhappens for L = 17cc/(2W) with a distance c = Wcc/W betvee,nt'qro adjacent atomfc rows fa a tevace, the step height is d = (U/W,and the horizontal shifk bet-ween t'wo consecutive terraccxs nmouts
to g
=
co/A/d.
We mention that, contrary to low-index surfaces,the sign a'adthe order of Miller indices in the notatio:a pLhkljx (Wktl')are important ('D30j. For example in a bcc lattice p(110) x (1ï0)is dsFerent 9om p(110) x (011). I'ndeed,it is easy to see that the edge of the Srst step is parallel to the
Eig. 7-9- 2D cat of a simple cubk crystal, showingterrace atd ledge atozns in proftle. The nominal (014) suzfacedecaysinto (001) terrace and (010) steps. The 1. incline angle8 to the low-inde.x (001) plane is gfven by tan8 = 4 .
7.3 Line Defeds:Steps
5Oq E1
E-l ï21
(EI,.f 00:)
periodicltyL ;
-11.1 (5541 E1
(b)
)'h'jl *---
,; a
d
9 L
Fig. 7.10*. Vicinal (557) smfaceof an fcc czystal. (a)Positionsof lattice point.sin a s'teppedsurface. The characteristic lengths and heights are indicated x (001) 6(111) in (b).
diredion., in which the suzcessiveatoms are second-nearestneighbors, (001j whtle for the second step it is pazalle,lto (ï11Jin whic,h the successive ,
atoms are flmt-nearest neighbors. Motker problem occuzs for orientations in the centrat part of the stereographctriaugle (Fig, 2.6).For example,at = x (111) x (100) it sllotf.d be noted that steps and ter2(111) (311)*2(100) pole abng the jo:ïj races becomeindistinguishable.Traveling from the (111) pole, the size of the terraces decreasesand (eveniuatly) direction to tke (100) is referred to a,s terraces becomeslarge'r.Therefore,(311) the size of the (100) the t'lrning pokt of the zonç.The stereographictriaagle may also #veother wsually cor&om the edges) îzlformation. Points witbsn the triangle (away containing regular non-linear respondto k-inkedhigh-inde,xplaues (sïtrênces However,recent studieshave shown that high-inde,xsurfaceson aa steps). edge of the s'te-reographictrsn.ngle,e-g., Si(311) ) or Mrltbs'nthe stereo(7.31j 5 11)(7.3241 can be s'tableagn.inqt decayinto graphic triangle, e.g., GaAs(2 terraces of low-index surfacesseparatedby steps. For realistic cabic czystals, for example for dinmond-stnzcturecrystals, and (331) vichlal surfaceshaa beendiscussedby the stntcture of (211), (311), One hms partimzlarly for steps propagatiag along(îI21and g11l). Chadi (7.33),
7. Defects
306
to mention that for viainxl surfacesof suchcprstals sometimesaksoa deviating nomenclatme is ased (7.34). =d. Si(2(111)(ïï2))suzfaces Sig3(111) (112)q corzis't of regtzlarlyspacedterraces tkat are three/t'wo Si atoms wide iong the or g1E2) directiop separatedby siugle boyer steps. The nomenclatare E11I) is also simplifed aûd s'ac,hstnzdttres aze zeferredto as (11R acd (1ï2j(or more genea'ally(11i)aud (ïï2))steps. -
7.3.2 Steps
-
Si(100)Surfaces
on
We coasidersteps on Si sa'rface.s as prototypical objects. Siliconis availableizi the fo= of the most perfect single crjrstals. lt is used as one of the most common substrates for homo- a'ad heteroepitax'y. There are severi remsons for s'tudying the geometric and electronic structure of steps on atomically cleaziE vicinal Si(1O0) stzrfaces(perhaps after n.nnen.lingat high temperature for sktfa scient time)ms well ms of steps on cleavedSi(l11) surfaces.Fit-st, dedactioy of the azomic s'tructare of the stepsis izztrsmm'rm.lly interestiug. Second,an ux/ derstanding of the step s'tmzct'azeleads naturally to possibleinsights iato the structure of high-indexsmfaces,whic,hare themselvesperiodic arrays of Low ( i'ad e.x steps (7.33). Third , the heteroepitaxial growth of other snmiconductozp suc.haz GGs or Gap ozl silicon appears to dependon the step stnzd'ure 6f the sllrfnne. surfacesthe atomic layerson the terraces arq ln tlle case of vicinal Si(1O0) at equal distances of co/4(d. Sed. 1.2.2 and Fîg. 1.6).At least siugle-layer (S)or mozlatomic aud double-laye,r(D)or biatomic steps shouldbe possiblqt 'lnwodistinc't tgpes (.4,B4may be distinguished accordingto the orientatio/ of the dirners on the adjacen.t (100)terraces. The steps are labeled as Sz, Ss, Dz, and Ds accordi'agto the notation of Kroemer (7.351 The three moàt '
'
.
SA HH-H e
-
-
-
-
-
ezzzmzz
-
-
-
e
-
-ZZTTZTZ ezzzzzT
-
H---
-
**
-
-
-,z,,,,
-
JZ zyz, - ezzzzzz
De
ss
TTTTTTT zzzz zzz
e-
2222222
zzzzzzz
-
-
221r122
ZZTCTTT
-
-
e-
-
-
-
Fig. 7.11- The three most impoz'tant single-laye,r (Sz,Ss)and double-layer(Dsk steps on Wcinal S6(1O0) surfaces (sche-matfc) The step height and the orientation of the dimers oa tlte terrres a'e iadkated..The terraces show 2x 1 ar.d 1x2 rlsrnéf reconstructions. .
7.3 Line Defects;Steps
307
1*
Fîg. 7.12- Top view of tlsmer-vacancy strudares used to model Sx, Ss, azd Ds iarfacesteps of Si(100), The size of the circle.sw.1-1e.s with the depth of the three comsideredatomk layers beneath tke surface. Anom Mdlcate lime,r Sup' atoms: dotted lines the resulting surface uzkit cell, and dashed lines approm'rnxtely the steps. After (7.362. .
*
importaat ones are sczematicallyshownitt Fig. 7.)1. The âguzemakes.evident that the subscripts .4 and B denote ,whethe,r the dimerization d/ection on azf upper tezrace near a step is normal (A)aad pazallelLBtto the step edge. Possibleatomic stmzctures kzsedto model the steps Sx, Sxv and D.s oa 'a sarfacejat least for a high dtmqityof steps, aze shownin Fig. 7.12 Si(100) The vicinal Si(100) surfacesare assllmedto be tûted about g011q and, (7.36!. consequently,the steps appear along the g0î1j diredion. Monatoxnicsteps separate 1x2 an.d2x1 domainsof dimerhation, therefore Sx aa.dSs steps 'bltmz-nate. The Sx s'teps do not require the formation of new or the breaking of existiug bonds. The Ss steps, however, aze more complicated aud nviqt izl three mriatiozts. M titree types az'e observedin snAnning txlnnelLngzaicroscopy,even tllough the rebondedversion has the lowest formntion The comrnonly observedbiatomic steps are of type Ds and anergy (7.37j. rebonded,Tile neighboring terraces show a 2x 1 reconstrudion (forrot too largeterraces and not too low temperatures) '
.
308
7. Defect,s
Experimentatly the occmrence of a ceztnl'n step and hence the step height dependon the temperatttre ar.dtkc misorientation. Surfacesteps have monatomic height below a miscut angle J of 1-20. For larger miscuts.more and more biatomic ste/sare observed.The steps aze zlmost exclusivelybiatomic for 9 e-xceedl'ng6-80 (7.37) J.nthe gn.mework of the tight-bindl'ng approach the formation energy 'yf = qslwith of step) Stidealsttrfacelj/fzs a cerwin step pez llnit step Iength Ls haBbeenciculated (7.384 The values are yf (Sx)= 0.01 ev/fgs, %(Ss)G 0.1: eM(Ls, (Dx)= 0.54 ev/fs, artd 'g/f yf (Ds)= 0.05ev/fs,whereLs = cc/uf = 3.85A Lsthe lx 1 sarfacelattice constart. 111reaâty thesevczluesdepezdon the step-step intezaction, i.e., on .
-
.
the periodicity L of the large surfacelnnit cell usedirt the cazculations(7.394. Thc tendcrc'yis boweve,r obviousfrom the above-mentionedvalues.StepsSx, s:lrfaces to a certna'mextect in Ss, a'ad.Ds sholzldoccur on the tilted Si(100) tlmvranody.cLnml'c eqllilr'brilIrn. However, their occunence is also l'n6uencedby sllrfaces misorientedtoward (O1lJ the smfacepreparation..For (100) oz (011) the eaergy situation is completely chamgedin compadson with the :at sur-' facesprepared by a repeated c'ycleof sputtering a'n:dnannen.lsng h'om a (100) Si wafez.For a tilted surfaceit is fmpossibleto haveonly Sz steps. If such a step doesoccur then a Ss step kstmavoidableacross some botmdar.vbetween terraces. For vazkishing spadng of Sx and Ss steps, howeve'r,a biatomic step of type Ds is created (see Fig. 7.11). Stepss'nflttence many surfaceproperties and, hence, are measurableiu many mzrfacemeaslzrementtenhnsquessuc,has LEED, STM, RAS, etc. The dslezenïcombinations of 2x 1 and 1x2 dimer reconstrudions and the elec)I
!
i
g g 1
l l'
: :
1 1
Tr
1
L' h i ter
l
/
o.co2
12
I
;.n
q
uq Gh-o c
-
S
t :
2
3
:,
t : :r k Tz t
4j :
'
: :,
'1
@)
:1
:
.
: l ; ; : 1 l
(s) (y; j (d)
ll %
w 4
5
6
Photonenergy(eV)
Fig. 7.13. Refectaztce n:n''qotropyspedra measuredfor clean Si(100) sarfacesby a'-merent Dmshedlines mark , and (d) (7-43). (a)(7.402: (b) (7.412, (c)(7.421 poups: the posîtlon of the bulk matical pokn.tenergesEï and F'z. After (7.361.
lated for Sx, Ss, and Ds steps in Fig. 7.12 (K36J; (b)RAS memsuredfor cllokrent miscut angle,s9 (7.41q. Terracecontributions havebeensubtzacted. troaic strttdme of the stepsthcselves sholfd particulazly iHueace the sarface anisotropy wlzich can be deteded using optichl reectance aaisotropy spectzoscopydescribedin Sect. 6.1.2. In. Fig. 7.13 we present the reûectance surfaces of fo'tm dferently prepved Si(10O) aaisotropy (see Sect. 6.1.1) A1l experimentalspedra show comrnon featmes:sach as a max(7.40-7.431. im= at or close to the & critîcal-point eneror ard negative auisotropiœ around the Ez enezgy.The spectra (c)an.d (d)obtainedfor 'vinsnn.lumples havea rninn'mttm around3 ev denotedby S in. Fig. 7.13. The spedra (a)and (b)of tEe higitly orientedsamples,instead,are charadezizedby msnima atc and 8.6 eAC(7*3). 1.6 (:71), 3.1 (S*2), The featmu T1, T%and T3 can be relatedto smfRœslate-relatedoptical surfres with essentiuy c(4x 2)reconstructrausitio:nsof :at, sizlgle-domain tion (d. Sed. 4.3.2). They are Muenced by the interMtion of tbe dirnerz parallel aad perpendictllar to the dimer rom (7.361. T'he RA feature S ks cleady relatedto steps. This is demonstratedin Fig. 7.14. J.nordez to extraswt the pttre step contribution alsofor Sx and Ds steps) for wïc,h the contribueac,hother, the norvnxln'zed tions of tlte upper ard lowerterraces do not omnce,l spedra of singlodomainsurfaceshavebem subtracted.Giventhe lsznitations in particxllnr with respect to the modeledstep density, of the caèettlations(a)s studied experimentally, the compariwhic,his far higberthan at the sample.s son witit the experlmemtal data (b)is gratifging. The surfacesteps, especiuy belov the Ss an.dDs, #vc rise to a broad negative Jtnlqotropy (Sfeature)
310
7. Defects
a minirnam at about 3 eV. This agreeswitk the expprt'men.% energy w1t21
tal ftndiazgs,whic,hshow increasing AnsRotropywith increasing miscut angle 1:1.Experimentatly a positive acisotropy between3.5 aud 4.0 e'V is observed for zniscut anglesj9Ihrger than 4O.A s5m57= featme appears iu the IRA spectrum calculatedfor Ds steps. 7.3.3 Steps on
Si(111)Surfaces
If siticon Rxmplesare ealeaved dn situ along the
or (711j dizections,2xl(2114
reconstruded terraces occur. Theseterraces are botmdedby steps. LEED aûd STM studies obxrve predomlnantly gzlîj-oriente s'tel)s with a step Eeight
s
(c)
-j
4
c@a
y
4 6z '
2 o
()
20 zo 40 scan distancealocg (g1t'z (&
lc
6:
Fig. T-15. STM Lmageof a step on Si(111); aquizedat samplevoltage +1.2 V and a comst=t tttannelingcurren.t of l nA.. (a)perspectiveview; (b)top view of tEe same step! an.d (e)crosmsedionalcut alorg the line indicated 1. (b).The step edgeis identifed by tick maz'lcsat the border of srnsgein (b).After 17.45).
7.3 Liae Defeds: Steps
(a)
..
,
(2-1-(
-
.
.
(b)
.
.
)17*
' Nh .,
f....:; .!
,
'.
,.
.ji?
Fig.. 7.16. Stde<ew of two po%ible con4gurations for
(0l 1)
311
a
m'.
);
,
'.
,.Li:
step lnmning iong (2-1-)1
.
of
az/s/z 3.14 .t
correspoadingto a'a atomic bilayer i'a thd (111q direction A typicat STM hnage of sucha step on a Si(111) surface(7.45) is (7.44,7.45). shown in Fig. 7.15. Two cegions of the step edge are apparent in the uppe,r and lower parts of tke fLgure.80th regions have tlnst pedodicity along the step edge, the (O1Ej direction. The lower region consists of a row of mn.='ma in the image-s,with one maxkzplrn per llnst cell. This row splits into two in the ttpper pazt of the image. The cornzgatsoaalongthese two rows is wealc A cross-sectionthrouglz this uppe,rregion of the image is shown in Fig. 7.1:c. The 'kworows observedalongthe step edgegive rise to Fmxn'rnn. in the STM contotzr, with the rows separatedlateratly by 4.5 + 0.5 A. The STM images indicate that there are two possibleclassesof steps along (01î)and a bilayer height (7.33, Conceivablemodels are shown 7.461. in Fig. 7.16. The outermost stom at the step edge f->tn eithe,rbe tvofoldcoordinatedas sitowain panel (a))or threefold-coordlnated as slmwuin panel =
ng. 7.17- Two possible models of step reconstruction (sideview) a , iztcludz'ng r-bonded cknn'n(ândicated by arrows) reconstrtzctiohof the tercaces. Afte,r a sup ges-tionin
(7.45) .
312
7. Defects
the flrst class,with only six-memberzingsalso occms on high-inde,xsurfacessar,hws (211) and (311) , which may be inter(7.33q preted as s'teppedsoaces with shozt terraces. The daagling bonds of the outerrnos't edgeatom form a drabbît-earttypeazaangement,ideatical to that of the Si(100) surface (Fig.3.11, Sect. 4.3.1). Consequently,a rlsrnerization along the s'tep edgeis expeded. Howeverj it leads to a doubleor quadruple periodicit.galong the edge,in coatradiction to the STM obsezwations. The seconâclass of steps in Fig. 7.16b is Garacterized by a geometry where the closestzizzgbn.q lost onc atom resulting in a sve-memberring a,t the step edge. Suc.hstructures are also discussedfor (211)suzfaces(7.33,
(b).The idealstructtlrej
They 7.464.
l'rnrnediately be combined with a 2x 1 reconstruction withs'n the r-bonded nhnsn model (Sect. terraces. Two clangling 4.2.2)of the (111) bonds remai'a at the two step-edgeatoms ar.d may be completelySlled or empt'y after recoastnzction. A11other danglingbondsare satttrated during the formation of tEe new tezrace topology with Nve-memberand seven-membe'r The dl'lerences T>. Two possiblestractmes are shomt i:rt F'ig.7.17 (7.45J izl the fmrö modelsare due to the topology of the rin.gswith whic,hthe upper and lower terraces approachthe s'tep. Tite step extents in these models of 5.1 at.d 3.6 i come close to the value of the lateral sepazation of rows on rl:'Reren t tezraces of (4' .5 ::E; 0.5)i. However,it sllould be mentionedthat also more abnpt steps havebeen observedby STM (7.454. cltn
Stanlrs'ngfaults are one of the most cornrnon types of planar defectsin crystalline dinmond-typeazd zinc-blende-typesemiconductors,suc,hms Si, Ge, a'ad GaAs,as well as ia many fcc metals. They have vezy low formation energies (ofthe order of 20-70 mJ/m2 and a're created when chacges (7.471) of the atoznic plane stackingsequence in the perfect czystattake place arong the E111) direction, without breltking bonds. For evnmple, dislocations can dissociatef'at?partials and create stacklmg fatzlts. Two types, the intrinsic and ex-trinqic s'tackqng faults, are indicated in Fig. 7.18. They conwpond to one missing or one ex4ra bilayer, respectively,in a'a otherwise pefect crystal with a stanlrs'ngsequence ABC. The correspondinglayer sequencesare .ABCA ICABC. (1SF) or For zinc-blead.eABCAICEBCAB. (ESF). semiconductorsin att attmmaxti've way one cazs view the (dixmond-lstractttre stanking fault a,s consisting of one (>0) layem of tetram intrinsic (exetrinsic) hedra twisted by 180D.The periodic arrangement of such stacldng fatllts gives new htexagonal polytgpes,211(wuztzite) for ISF, 411for ESF, and 611 for triple stacking faots (notshown in Fig. 7.18)as Mdicated in Fig. 7.19. For compouadsemiconductorswith stroager ionic bonds sttchpolytypes re . . .
. .
. .
.
. .
.
,
.
. .
7.4 PlanarDefects:StaddngFattlts
ideal (3C)
ISF
+ + + + + + +
+ + +
+
+
3l2
ESF + + +
+ +
+ + +
+ A BC A B CA B C
A B CA R C A : C
A B C A B CA B C
Fsg. 7.18. Staclcingsequence for an ideal fcc structtzre tleftpanel), fcc withuan htrhsic-stacldng fault (JSF)(middle and fcc 44th an ex-trlndc stacldng panel), fazklt IBSFI(rightq=ell. A, B, C zepresen.tthe three inequivalent positions withc'n a (110) plar.e with'p an irreducible crystal slab (d. Sect. 1.:.2).In the case of semîconductorsa dot repraents a pair of atoms with the connecting bond parallel to
(111).
e'aergeticallymore stable tha'a the 3C (ziac-blende) one, for evnmple 2H for ZnS, GaN, and .A.INas well as 41-1acd 6H for SiC. (wartzite) Stazlrs'ngfaults also occur on mIrfaces. They are not ac-tually a tfault' fault but a,'a importaat reconstrudion element of the d'lrnemadatom-stpcklng model of the Si(111)7x7 surface(see Sect.4.4.3), in contrast to stanMng faults in blll.k Si czystals.In the case of halogen-terrnsnated smfacesthe Si(111)1x1 introduction of a staûlrimgfaalt aloag a (111) step edgeatlowsthose bilayer steps rin the two czystallovaphic directions (ïï2) and (111) to have the sn.rne atomic structure 7.4-2 sî
on
g7.48).
'
si(111)xdx,vY-B
m:rface whic,h Interestingly, Si homoepitaxial vowth on a Si(111)WxVFB includesj monolaye,rof boron showsa tendenc'yfor rotation by 1801of the Si tetrahedra i.'athe interface, i.e., for the generation of the flrst paz't of a stackiug fatllt (see Fig. 7.18), a so-calledtwin boundar,g(7.49, At 1owtempera7.501. ture, the surfacereconstmzctionis partly prcxserved, buried unde,ran epilayer, ar.d the homopitanial Si layer gwws rotated by 1801with rcsped to the suN strate. This situation is indicated in Fig. 7.20b. The rotation is in coatzmst to all other cazes of bomoepitaxy.Usually the epilaye,ris crystatlographically aligned with the substrate, irrespective of the surface reconstrucion, imput'it.g segregation, oz other Gects at the substzatesurface.Tlle origsnnlsmface reconstntctiox is alwaysreorderediuto an thmreeonstructed iaterfacebetweea the substrateand.61m, since epitak'y requires a snTmcientlylligh temperature for surfacedifhlsion to occttr. That means, in prindplc, there is the possibilit.g of growi'n.ghexagonalSi polytypes (7.51) on a Si(111)WxW-B surface,
becausethey are strttdmw incladiag twin botmdariesperioically along the direction tsee Fig. 7.19). (111)
shows that A quatitative analysis of the system Si(I11) (WxW)A3O12B the boron atom pe,r surface lnnit cell occapies a, subsurfacesubsitutioaal S5 adsorption site tsee Sec't.4.4.1) becauseof the smmll covalemtrnrh'uzof boron. would lead to B-Si bondsmuch shorter Adsorption in thé T4 site (Fig. 4.35) than the substrate bonds, giving rise to considerable strnan'n(7.52-7.55). Tkè right panel of Fig. 7.20makesevide'nttkat the mcontn.msnatedsurfaceservej is as a template upon whic.hthe new orientatioa of the twi'a crystal (layer)
(b)
(a)
p ,11;
)
Eif1.lq
substratei Fkg. 7.29. Interfacestracture of a homoepitaxialSi layer and a Si(111) (a)witkout B coverage;(b)after ) monolayerboron coverage. The Wx W reconu struc tion of the interface (dashed lm'e) is introduced by occupying every tlnird site izz a single atomic layer.The bond-stnrllng directions are indicated by thsn solid line. Opentfzlledl mrcles:Si (B)atoms. Mer (7.491.
7.4 Plnnn.tDelectmSkackiugFaults
315
enugetically prefeaa'edj at least during the nudeon stageof the 61m gwwth near 400 QC (7.49, Eighe,rgrowth temperatau'œ of 800 OCor nnnealillg 7.501. st 1000OCmakethe B atoms mobile and #verkqeto a normnl st of the Si bonds and tetrahea- The third-nearestneighbor atmmsof the tetœahedra abovethe contxmînxtedlayer te'adto ocmpy a dte directly abovethe B atonxs j.ûSs sites. TMSis only possible by a twist of the Si tetrahedra by 1800.Two :factsmay stabilizesuch a situation. The smallerB atom reducesthe repulsive forcesbetweenthe third neighboz's. The eledron trnnqfer bemeenboroa and Si atoms may give zise to an attradive Coulombhteraction betweenB and ji atoms abovean S5site. '
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